We study the large field limit in Schr\"odinger equations with magnetic vector potentials describing translationally invariant $B$-fields with respect to the $z$-axis. Using analytic perturbation theory, we derive an approximate description of the solution, provided the initial data is compactly supported in the Fourier-variable dual to $z\in \mathbb{R}$. The effective dynamics is thereby seen to produce high-frequency oscillations and large magnetic drifts. In a second step we show that this asymptotic description is stable under a fairly general class of singular perturbations by using the theory of almost invariant subspaces. To this end, we also give an elementary proof of the exponential decay of the resolvent and the eigenfunctions of confining Schr\"odinger operators.

In this paper, we study the transverse instability of generalized Zakharov-Kuznetsov equation for the line soliton with critical speed. We derive and justify a normal form reduction for a bifurcation problem of the stationary nonlinear KdV equation on the product space R ? T.

We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of $h$-dependent delta-function potentials on $\mathbb{R}$. In the cases of two or three delta poles, we are able to show that resonances occur along specific lines of the form $\Im z \sim -\gamma h \log(1/h).$ More generally, we use the method of Newton polygons to show that resonances near the real axis may only occur along a finite collection of such lines, and we bound the possible number of values of the parameter $\gamma.$ We present numerical evidence of the existence of more and more possible values of $\gamma$ for larger numbers of delta poles.

In this paper we present modern portfolio theory using basic concepts of linear algebra, differential calculus, statistics and optimization. This theory allows us to measure the return and risk of the investment portfolio, helping to make decisions in the financial market. As an application, we will present a simple investment strategy that aims to minimize the risk of investing in two assets

We study the regularity properties of fermionic equilibrium states at finite positive temperature and show that they satisfy certain semiclassical bounds. As a corollary, we identify explicitly a class of positive temperature states satisfying the regularity assumptions of [J.J. Chong, L. Lafleche, C. Saffirio: arXiv:2103.10946 (2021)].

Thanks to the work of Karin Erdmann, we know a great deal about the representation theory of blocks of finite groups with tame representation type. Our purpose here is to examine the $p$-completed classifying spaces of these blocks and their loop spaces. We pay special attention to the $A_\infty$ algebra structures, and singularity and cosingularity categories.

Let $\omega$ be a Kaehler form on the real $4$-torus $T^4$. Suppose that $\omega$ satisfies an irrationality condition which can be achieved by an arbitrarily small perturbation of $\omega$. This note shows that the smoothly trivial symplectic mapping class group of the one-point symplectic blowup of $(T^4,\omega)$ is infinitely generated.

Public blockchains implement a fee mechanism to allocate scarce computational resources across competing transactions. Most existing fee market designs utilize a joint, fungible unit of account (e.g., gas in Ethereum) to price otherwise non-fungible resources such as bandwidth, computation, and storage, by hardcoding their relative prices. Fixing the relative price of each resource in this way inhibits granular price discovery, limiting scalability and opening up the possibility of denial-of-service attacks. As a result, many prominent networks such as Ethereum and Solana have proposed multi-dimensional fee markets. In this paper, we provide a principled way to design fee markets that efficiently price multiple non-fungible resources. Starting from a loss function specified by the network designer, we show how to compute dynamic prices that align the network's incentives (to minimize the loss) with those of the users and miners (to maximize their welfare), even as demand for these resources changes. Our pricing mechanism follows from a natural decomposition of the network designer's problem into two parts that are related to each other via the resource prices. These results can be used to efficiently set fees in order to improve network performance.

Using the Girard-Newton formulae, I give a simple proof of isotropic square function estimates for extension operators along the moment curve in generic local fields. Using Bezout's Theorem and the Implicit Function Theorem, I give an alternate, sharper proof for real or complex non-degenerate, polynomial curves.

We determine the border rank of each power of any quadratic form in three variables. Since for the case of rank $1$ and rank $2$ quadratic forms the problem is reduced to the determination of the rank of binary forms, we basically focus on the case of non-degenerate quadratic forms. We firstly consider the quadratic form $q_{n}=x_1^{2}+\dots+x_n^{2}$ in an arbitrary number $n$ of variables and we determine the apolar ideal of any power $q_n^s$, proving that it corresponds to the homogeneous ideal generated by the harmonic polynomials of degree $s+1$. Thanks to this result, given any quadratic form in three variables, that without lost of generality can be supposed to be the form $q_3$, we select for each power a specific ideal contained in its apolar ideal and, after verifying some properties, we make use of the recent technique of border apolarity to prove that the border rank of any power $q_3^s$ is equal to the rank of its central catalecticant matrix, that is $(s+1)(s+2)/2$.

We prove several topological and dynamical properties of the boundary of a hierarchically hyperbolic group are independent of the specific hierarchically hyperbolic structure. This is accomplished by proving that the boundary is invariant under a "maximization" procedure introduced by the first two authors and Durham.

This work analyzes the forward and inverse scattering series for scalar waves based on the Helmholtz equation and the diffuse waves from the time-independent diffusion equation, which are important PDEs in various applications. Different from previous works, which study the radius of convergence for the forward and inverse scattering series, the stability, and the approximation error of the series under the $L^p$ norms, we study these quantities under the Sobolev $H^s$ norm, which associates with a general class of $L^2$-based functional spaces. The $H^s$ norm has a natural spectral bias based on its definition in the Fourier domain - the case $s<0$ biases towards the lower frequencies, while the case $s>0$ biases towards the higher frequencies. We compare the stability estimates using different $H^s$ norms for both the parameter and data domains and provide a theoretical justification for the frequency weighting techniques in practical inversion procedures.

We consider the motion of a small rigid object immersed in a viscous compressible fluid in the 3-dimensional Eucleidean space. Assuming the object is a ball of a small radius $\varepsilon$ we show that the behavior of the fluid is not influenced by the object in the asymptotic limit $\varepsilon \to 0$. The result holds for the isentropic pressure law $p(\varrho) = a \varrho^\gamma$ for any $\gamma > \frac{3}{2}$ under mild assumptions concerning the rigid body density. In particular, the latter may be bounded as soon as $\gamma > 3$. The proof uses a new method of construction of the test functions in the weak formulation of the problem, and, in particular, a new form of the so-called Bogovskii operator.

An $R$-module $M$ is called virtually uniserial if for every finitely generated submodule $0 \neq K \subseteq M$, $K/$Rad$(K)$ is virtually simple. In this paper, we generalize virtually uniserial modules by dropping the virtually simple condition and replacing it by the cyclic uniform condition. An $R$-module $M$ is called cyclic-uniform uniserial if $K/$Rad$(K)$ is cyclic and uniform, for every finitely generated submodule $0 \neq K \subseteq M$. Also, $M$ is said to be cyclic-uniform serial if it is a direct sum of cyclic-uniform uniserial modules. Several properties of cyclic-uniform (uni)serial modules and rings are given. Moreover, the structure of Noetherian left cyclic-uniform uniserial rings are characterized. Finally, we study rings $R$ have the property that every finitely generated $R$-module is cyclic-uniform serial.

In 2006, Jeff Smith proposed a theory of ideals for rings in a triangulated symmetric monoidal category such as ring spectra or DGAs. We show that his definition is equivalent to a `central' $R$-$R$-bimodule map $ I \to R$.

A famous result due to I. M. Isaacs states that if a commutative ring $R$ has the property that every prime ideal is principal, then every ideal of $R$ is principal. This motivates ring theorists to study commutative rings for which every ideal is a direct sum of cyclically presented modules. In this paper, we study commutative rings whose ideals are direct sum of cyclically presented modules.

Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $A$ be an abelian surface over $k$. Fix an integer $n \geq 1$ such that $p \nmid n$ and let $K^{[n]}$ be the $n$-th Generalized Kummer Variety associated to $A$. In this article we aim to find the $S$-fundamental group scheme and Nori's fundamental group scheme of $K^{[n]}$.

In this paper we take a look at conditions that make a Riemann soliton trivial, compacity being one of them. We also show that the behaviour at infinity of the gradient field of a non-compact gradient Riemann soliton might cause the soliton to be an Einstein manifold. Finally, we obtain scalar curvature estimates for complete shrinking or steady gradient Riemann solitons whose scalar curvature is bounded from below at infinity.

We show that if $K$ is a compact metrizable space with finitely many accumulation points, then the closed unit ball of $C(K)$ is a plastic metric space, which means that any non-expansive bijection from $B_{C(K)}$ onto itself is in fact an isometry. We also show that if $K$ is a zero-dimensional compact Hausdorff space with a dense set of isolated points, then any non-expansive homeomorphism of $B_{C(K)}$ is an isometry.

We prove the Kudla--Rapoport conjecture for Kr\"amer models of unitary Rapoport--Zink spaces at ramified places. It is a precise identity between arithmetic intersection numbers of special cycles on Kr\"amer models and modified derived local densities of hermitian forms. As an application, we relax the local assumptions at ramified places in the arithmetic Siegel--Weil formula for unitary Shimura varieties, which is in particular applicable to unitary Shimura vartieties associated to unimodular hermitian lattices over imaginary quadratic fields.

In this paper, we propose a modified Kudla-Rapoport conjecture for the Kr\"amer model of unitary Rapoport-Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas of Hermitian local density polynomials, we prove the modified Kudla-Rapoport conjecture when $n=3$. Our conjecture, combining with known results at inert and infinite primes, implies arithmetic Siegel-Weil formula for all non-singular coefficients.

In this paper, we first study the $\alpha-$energy functional, Euler-Lagrange operator and $\alpha$-stress energy tensor. Second, it is shown that the critical points of $\alpha-$ energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an $\alpha-$harmonic map is constructed from any smooth map between Riemannian manifolds under certain assumptions. Next, we determine the conditions under which the fibers of horizontally conformal $\alpha-$ harmonic maps are minimal submanifolds. Then, the stability of any $\alpha-$harmonic map from a Riemannian manifold to a Riemannian manifold with non-positive Riemannian curvature is demonstrated. Finally, the instability of $\alpha-$harmonic maps from a compact manifold to a standard unit sphere is investigated.

In this paper, we study the strong extension groups of Cuntz--Krieger algebras, and present a formula to compute the groups. We also detect the position of the Toeplitz extension of a Cuntz--Krieger algebra in the strong extension group and in the weak extension group to see that the weak extension group with the position of the Toeplitz extension is a complete invariant of the isomorphism class of the Cuntz--Krieger algebra associated with its transposed matrix.

In statistics and probability theory, one the most important statistics is the sums of random variables. After introducing a probability distribution, determining the sum of n independent and identically distributed random variables is one of the interesting topics for authors. This paper presented the probability density function for the sum of n independent and identically distributed random variables such as Shanker, Akash, Ishita, Rani, Pranav and Ram Awadh. In order to determine all aforementioned distributions, the problem-solving methods are applied which is based on the change-of-variables technique. The mth moments for them were also accurately calculated. Besides, the reliability and the mean time to failure of a 1 out of n cold standby spare system has also been evaluated under the Lindley components failure time.

We develop the theory of quasi-$F$-splittings in the context of birational geometry. Amongst other things, we obtain results on liftability of sections and establish a criterion for whether a scheme is quasi-$F$-split employing the higher Cartier operator. As one of the applications of our theory, we prove that three-dimensional klt singularities in large characteristic are quasi-$F$-split, and so, in particular, they lift modulo $p^2$.

We discuss the $qq$-systems, the functional form of the Bethe ansatz equations for the twisted Gaudin model from a new geometric point of view. We use a concept of $G$-Wronskians, which are certain meromorphic sections of principal $G$-bundles on the projective line. In this context, the $qq$-system, similar to its difference analog, is realized as the relation between generalized minors of the $G$-Wronskian. We explain the link between $G$-Wronskians and twisted $G$-oper connections, which are the traditional source for the $qq$-systems.

All energy-momentum type conservation laws are found for a general class of damped nonlinear wave equations in one dimension. The classification shows that conservation laws exist only for linear damping if the wave equation is non-singular. In the simplest cases, the conservation laws generalize ordinary momentum and energy, and also null-energies. An explanation of the existence of these conservation laws is provided by a point transformation that maps the damped nonlinear wave equation to an undamped wave equation with a time-dependent nonlinearity. Specializations to damped nonlinear wave equations that are invariant under time and space translations are summarized. In the general case, the conservation laws describe a further generalization of momentum, energy, and null-energies, as well as a generalized energy-momentum. The latter two types of conservation laws are shown to arise from a mapping to a wave equation that has a null-form damping (namely, a combination of spatial and temporal damping tied to the null lines of the linear wave operator). This represents a certain type of null structure which is closely related to the null-form nonlinearities studied in analysis of nonlinear wave equations.

We study the long time asymptotic behavior for the Cauchy problem of the Camassa-Holm (CH) equation with the appropriate Sobolev initial data. in the solitonic regions. Our main technical tool is the $\bar\partial$ generalization of Deift-Zhou steepest descent method. Through introducing a new scale $(y,t)$ and constructing the RH problem,we derive different long time asymptotic expansion of the solution $u(y,t)$ in different space-time solitonic regions of $\xi=y/t$. We divide the half-plane $\{ (y,t): -\infty <y<\infty, \ t> 0\}$ into two asymptotic regions: 1. phase point absent region: $\xi \in(-\infty,-1/4)\cup(2,+\infty)$,corresponding asymptotic approximations can be characterized with an $N(\Lambda)$-solitons with diverse residual error order $\mathcal{O}(t^{-1+2\rho})$; 2. phase points region: $\xi \in(0,2)$ and $\xi \in(-1/4,0)$.The corresponding asymptotic approximations can be characterized with an $N(\Lambda)$-soliton and an interaction term between soliton solutions and the dispersion term with diverse residual error order $\mathcal{O}(t^{-3/4})$.Our results also confirm the soliton resolution conjecture and asymptotically stability of the N-soliton solutions for the CH equation.

We investigate various spaces of $SL(r+1)$-opers and their deformations. For each type of such opers, we study the quantum/classical duality, which relates quantum integrable spin chains with classical solvable many body systems. In this context, quantum/classical dualities serve as an interplay between two different coordinate systems on the space of opers. We also establish correspondences between the underlying oper spaces, which recently had multiple incarnations in symplectic duality and bispectral duality.

We study the moduli space of flat $GL(1|1)$-connections on a punctured surface from the point of view of graph connections. To each fatgraph, a system of coordinates is assigned, which involves two bosonic and two fermionic variables per edge, subject to certain relations. In the case of trivalent graphs, we provide a closed explicit formula for the Whitehead moves. In addition, we discuss the invariant Poisson bracket.

Power distribution networks are approaching their voltage stability boundaries due to the severe voltage violations and the inadequate reactive power reserves caused by the increasing renewable generations and dynamic loads. In the broad endeavor to resolve this concern, we focus on enhancing voltage stability through stochastic distribution network reconfiguration (SDNR), which optimizes the (radial) topology of a distribution network under uncertain generations and loads. We propose a deep learning method to solve this computationally challenging problem. Specifically, we build a convolutional neural network model to predict the relevant voltage stability index from the SDNR decisions. Then we integrate this prediction model into successive branch reduction algorithms to reconfigure a radial network with optimized performance in terms of power loss reduction and voltage stability enhancement. Numerical results on two IEEE network models verify the significance of enhancing voltage stability through SDNR and the computational efficiency of the proposed method.

The paper studies a reconfigurable intelligent surface (RIS)-assisted multi-user uplink massive multiple-input multiple-output (MIMO) system with imperfect hardware. At the RIS, the paper considers phase noise, while at the base station, the paper takes into consideration the radio frequency impairments and low-resolution analog-to-digital converters. The paper derives approximate expressions for the ergodic achievable rate in closed forms under Rician fading channels. For the cases of infinite numbers of antennas and infinite numbers of reflecting elements, asymptotic data rates are derived to provide new design insights. The derived power scaling laws indicate that while guaranteeing a required system performance, the transmit power of the users can be scaled down at most by the factor 1/M when M goes infinite, or by the factor 1/(MN) when M and N go infinite, where M is the number of antennas and N is the number of the reflecting units. Furthermore, an optimization algorithm is proposed based on the genetic algorithm to solve the phase shift optimization problem with the aim of maximizing the sum rate of the system. Additionally, the optimization problem with discrete phase shifts is considered. Finally, numerical results are provided to validate the correctness of the analytical results.

For $i = 0, 1, 2, \dots, k$, let $\mu_i$ be a Borel probability measure on $[0,1]$ which is equivalent to Lebesgue measure $\lambda$ and let $T_i:[0,1] \rightarrow [0,1]$ be $\mu_i$-preserving ergodic transformations. We say that transformations $T_0, T_1, \dots, T_k$ are uniformly jointly ergodic with respect to $(\lambda; \mu_0, \mu_1, \dots, \mu_k)$ if for any $f_0, f_1, \dots, f_k \in L^{\infty}$, \[ \lim\limits_{N -M \rightarrow \infty} \frac{1}{N-M } \sum\limits_{n=M}^{N-1} f_0 ( T_0^{n} x) \cdot f_1 (T_1^n x) \cdots f_k (T_k^n x) = \prod_{i=0}^k \int f_i \, d \mu_i \quad \text{ in } L^2(\lambda). \] We establish convenient criteria for uniform joint ergodicity and obtain numerous applications, most of which deal with interval maps. Here is a description of one such application. Let $T_G$ denote the Gauss map, $T_G(x) = \frac{1}{x} \, (\bmod \, 1)$, and, for $\beta >1$, let $T_{\beta}$ denote the $\beta$-transformation defined by $T_{\beta} x = \beta x \, (\bmod \,1)$. Let $T_0$ be an ergodic interval exchange transformation. Let $\beta_1 , \cdots , \beta_k$ be distinct real numbers with $\beta_i >1$ and assume that $\log \beta_i \ne \frac{\pi^2}{6 \log 2}$ for all $i = 1, 2, \dots, k$. Then for any $f_{0}, f_1, f_{2}, \dots, f_{k+1} \in L^{\infty} (\lambda)$, \begin{equation*} \begin{split} \lim\limits_{N -M \rightarrow \infty} \frac{1}{N -M } \sum\limits_{n=M}^{N-1} & f_{0} (T_0^n x) \cdot f_{1} (T_{\beta_1}^n x) \cdots f_{k} (T_{\beta_k}^n x) \cdot f_{k+1} (T_G^n x) &= \int f_{0} \, d \lambda \cdot \prod_{i=1}^k \int f_{i} \, d \mu_{\beta_i} \cdot \int f_{k+1} \, d \mu_G \quad \text{in } L^{2}(\lambda). \end{split} \end{equation*} We also study the phenomenon of joint mixing. Among other things we establish joint mixing for skew tent maps and for restrictions of finite Blaschke products to the unit circle.

By introducing height dependency in the surface energy density, we propose a novel regularized variational model to simulate wetting/dewetting problems. The regularized model leads to the appearance of a precursor layer which covers the bare substrate, with the precursor height depending on the regularization parameter $\varepsilon$. The new model enjoys lots of advantages in analysis and imulations. With the help of the precursor layer, the regularized model is naturally extended to a larger domain than that of the classical sharp-interface model, and thus can be solved in a fixed domain. There is no need to explicitly track the contact line motion, and difficulties arising from free boundary problems can be avoided. In addition, topological change events can be automatically captured. Under some mild and physically meaningful conditions, we show the positivity-preserving property of the minimizers of the new model. By using asymptotic analysis and $\Gamma$-convergence, we investigate the convergence relations between the new regularized model and the classical sharp-interface model. Finally, numerical results are provided to validate our theoretical analysis, as well as the accuracy and efficiency of the new regularized model.

We show the invariance of plurigenera for generalized polarized pairs with abundant nef parts and generalized canonical singularities. This is obtained by investigating a type of newly introduced multiplier ideal sheaf which is of bimeromorphic nature.

Let $R$ be a commutative ring with identity. The co-maximal ideal graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertices are proper ideals of $R$ which are not contained in the Jacobson radical of $R$ and two distinct vertices $I, J$ are adjacent if and only if $I+J=R$. In this paper, we use Gallai$^{^,}$s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established.

In this paper, we consider capillary-gravity waves propagating on the interface separating two fluids of finite depth and constant density. The flow in each layer is assumed to be incompressible and of constant vorticity. We prove the existence of small-amplitude solitary wave solutions to this system in the strong surface tension regime via a spatial dynamics approach. We then use a variant of the classical Grillakis--Shatah--Strauss (GSS) method to study the orbital stability/instability of these waves. We find an explicit function of the parameters (Froude number, Bond number, and the depth and density ratios) that characterizes the stability properties. In particular, conditionally orbitally stable and unstable waves are shown to be possible.

In this paper we propose novel optimization-based methods for verifying reach-avoid (or, eventuality) properties of continuous-time systems modelled by ordinary differential equations. Given a system, an initial set, a safe set and a target set of states, we say that the reach-avoid property holds if for all initial conditions in the initial set, any trajectory of the system starting at them will eventually, i.e.\ in unbounded yet finite time, enter the target set while remaining inside the safe set until that first target hit. Based on a discount value function, two sets of quantified constraints are derived for verifying the reach-avoid property via the computation of exponential/asymptotic guidance-barrier functions (they form a barrier escorting the system to the target set safely at an exponential or asymptotic rate). It is interesting to find that one set of constraints whose solution is termed exponential guidance-barrier functions is just a simplified version of the existing one derived from the moment based method, while the other one whose solution is termed asymptotic guidance-barrier functions is completely new. Furthermore, built upon this new set of constraints, we derive a set of more expressive constraints, which includes the aforementioned two sets of constraints as special instances, providing more chances for verifying the reach-avoid properties successfully. When the involved datum are polynomials, i.e., the initial set, safe set and target set are semi-algebraic, and the system has polynomial dynamics, the problem of solving these sets of constraints can be framed as a semi-definite optimization problem using sum-of-squares decomposition techniques and thus can be efficiently solved in polynomial time via interior point methods. Finally, several examples demonstrate the theoretical developments and performance of proposed methods.

We characterize the full classes of M-estimators for semiparametric models of general functionals by formally connecting the theory of consistent loss functions from forecast evaluation with the theory of M-estimation. This novel characterization result opens up the possibility for theoretical research on efficient and equivariant M-estimation and, more generally, it allows to leverage existing results on loss functions known from the literature of forecast evaluation in estimation theory.

The main result of this note is the strengthening of a quite arbitrary a priori Fourier restriction estimate to a multi-parameter maximal estimate of the same type. This allows us to discuss a certain multi-parameter Lebesgue point property of Fourier transforms, which replaces Euclidean balls by ellipsoids. Along the lines of the same proof, we also establish a $d$-parameter Menshov--Paley--Zygmund-type theorem for the Fourier transform on $\mathbb{R}^d$. Such a result is interesting for $d\geq2$ because, in a sharp contrast with the one-dimensional case, the corresponding endpoint $L^2$ estimate (i.e., a Carleson-type theorem) is known to fail since the work of C. Fefferman in 1970. Finally, we show that a Strichartz estimate for a given homogeneous constant-coefficient linear dispersive PDE can sometimes be strengthened to a certain pseudo-differential version.

We describe some representatives of G_2-orbits of pairs of octonions as well as traceless octonions over an algebraically closed field of arbitrary characteristic.

The recent works on a deep learning (DL)-based joint design of preamble set for the transmitters and data-aided active user detection (AUD) in the receiver has demonstrated a significant performance improvement for grant-free sparse code multiple access (GF-SCMA) system. The autoencoder for the joint design can be trained only in a given environment, but in an actual situation where the operating environment is constantly changing, it is difficult to optimize the preamble set for every possible environment. Therefore, a conventional, yet general approach may implement the data-aided AUD while relying on the preamble set that is designed independently rather than the joint design. In this paper, the activity detection error rate (ADER) performance of the data-aided AUD subject to the two preamble designs, i.e., independently designed preamble and jointly designed preamble, were directly compared. Fortunately, it was found that the performance loss in the data-aided AUD induced by the independent preamble design is limited to only 1dB. Furthermore, such performance characteristics of jointly designed preamble set is interpreted through average cross-correlation among the preambles associated with the same codebook (CB) (average intra-CB cross-correlation) and average cross-correlation among preambles associated with the different CBs (average inter-CB cross-correlation).

In this paper, we prove that the trisection genus of the Akbulut cork is $3$ and construct infinitely many corks with trisection genus $3$. These results give the first examples of contractible $4$-manifolds whose trisection genera are determined except for the $4$-ball. We also give a lower bound for the trisection genus of a $4$-manifold with boundary. In addition, we construct low genus relative trisection diagrams of an exotic pair of simply-connected $4$-manifolds with $b_2 = 1$.

A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is exhibited in terms of the Young function defining the Orlicz-Sobolev space. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of non-power type, are also reproduced and, when not sharp, improved.

A class of linear parabolic equations are considered. We give a posteriori error estimates in the maximum norm for a method that comprises extrapolation applied to the backward Euler method in time and finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator.

In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$-\emph{th fractional truncated Laplacian} or the $k$-\emph{th fractional eigenvalue} which are fully nonlinear integral operators whose nonlocality is somehow $k$-dimensional.

We consider second-order ergodic Mean-Field Games systems in the whole space $\mathbb{R}^N$ with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. These MFG systems describe Nash equilibria of games with a large population of indistinguishable rational players attracted toward regions where the population is highly distributed. Equilibria solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for the existence and non existence of classical solutions to the MFG system. By means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential in the Hardy-Littlewood-Sobolev-supercritical regime. On the other hand, using a fixed point argument, we show existence of classical solutions in the Hardy-Littlewood-Sobolev-subcritical regime at least for masses smaller than a given threshold value. In the mass-subcritical regime we show that actually this threshold can be taken to be $+\infty$.

Approximating periodic solutions to the coupled Duffing equations amounts to solving a system of polynomial equations. The number of complex solutions measures the algebraic complexity of this approximation problem. Using the theory of Khovanskii bases, we show that this number is given by the volume of a certain polytope. We also show how to compute all solutions using numerical nonlinear algebra.

We control the behavior of the Poincar{\'e} constant along the Polchinski renormalization flow using a dynamic version of $\Gamma$-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by B. Klartag and E. Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general $\Phi$ 4-measures and discuss the interpretation in terms of transport maps.

In the space of sextic forms in 4 variables with a decomposition of length 18 we determine and describe a closed subvariety which contains all non-identifiable sextics. The description of the subvariety is geometric, but one can derive from that an algorithm which can guarantee that a given form is identifiable.

If $\mathcal{M}$ is a finite abelian category and $\mathbf{T}$ is a linear right exact monad on $\mathcal{M}$, then the category $\mathbf{T}\mbox{-mod}$ of $\mathbf{T}$-modules is a finite abelian category. We give an explicit formula of the Nakayama functor of $\mathbf{T}\mbox{-mod}$ under the assumption that the underlying functor of the monad $\mathbf{T}$ has a double left adjoint and a double right adjoint. As applications, we deduce formulas of the Nakayama functor of the center of a finite bimodule category and the dual of a finite tensor category. Some examples from the Hopf algebra theory are also discussed.

This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is established. Furthermore, this result can be generalized by replacing the class of integral varifolds by some classes of rectifiable varifolds whose density is uniformly bounded from below. However, such decomposition may fail to be unique.

Triality is a classical notion in geometry that arose in the context of the Lie groups of type $D_4$. Another notion of triality, Wilson triality, appears in the context of reflexible maps. We build a bridge between these two notions, showing how to construct an incidence geometry with a triality from a map that admits a Wilson triality. We also extend a result by Jones and Poulton, showing that for every prime power $q$, the group ${\rm L}_2(q^3)$ has maps that admit Wilson trialities but no dualities.

We present convergence theory for corrected quadrature rules on uniform Cartesian grids for functions with a point singularity. We begin by deriving an error estimate for the punctured trapezoidal rule, and then derive error expansions. We define the corrected trapezoidal rules, based on the punctured trapezoidal rule, where the weights for the nodes close to the singularity are judiciously corrected based on these expansions. Then we define the composite corrected trapezoidal rules for a larger family of functions using series expansions around the point singularity and applying corrected trapezoidal rules appropriately. We prove that we can achieve high order accuracy by using a sufficient number of correction nodes around the point singularity and of expansion terms.

This paper encompasses the mathematical derivations of the analytic and generalized formula and recurrence relations to find out the radii of n umber of circles inscribed or packed in the plane region bounded by circular arcs (including sectors, semi and quarter circles) & the straight lines. The values of radii obtained using analytic formula and recurrence relations have been verified by comparing with those obtained using MATLAB codes. The methods used in this paper for packing circles are deterministic unlike heuristic strategies and optimization techniques. The analytic formulae derived for plane packing of tangent circles can be generalized and used for packing of spheres in 3D space and packing of circles on the spherical surface which is analogous to distribution of non-point charges. The packing density of identical circles, externally tangent to each other, the most densely packed on the regular hexagonal and the infinite planes have been formulated and analysed. This study paves the way for mathematically solving the problems of dense packing of circles in 2D containers, the packing of spheres in the voids (tetrahedral and octahedral) and finding the planar density on crystallographic plane.

This paper studies a framework, projected fixed-point method, for graph matching. The framework contains a class of popular graph matching algorithms, including graduated assignment (GA), integer projected fixed-point method (IPFP) and doubly stochastic projected fixed-point method (DSPFP). We propose an adaptive strategy to tune the step size parameter in this framework. Such a strategy improves these algorithms in efficiency and accuracy. Further, it guarantees the convergence of the underlying algorithms. Some preliminary analysis based on distance geometry seems to support that the optimal step size parameter has a high probability of 1 when graphs are fully connected. Secondly, it is observed that a popular projection method, softassign, is sensitive to graphs' cardinality(size). We proposed a dynamical softassign algorithm that is robust to graphs' cardinality. Combining the adaptive step size and the dynamical softassign, we propose a novel graph matching algorithm: the adaptive projected fixed-point method with dynamical softassign. Various experiments demonstrate that the proposed algorithm is significantly faster than several other state-of-art algorithms with no loss of accuracy.

In this paper, a semantic communication framework is proposed for textual data transmission. In the studied model, a base station (BS) extracts the semantic information from textual data, and transmits it to each user. The semantic information is modeled by a knowledge graph (KG) that consists of a set of semantic triples. After receiving the semantic information, each user recovers the original text using a graph-to-text generation model. To measure the performance of the considered semantic communication framework, a metric of semantic similarity (MSS) that jointly captures the semantic accuracy and completeness of the recovered text is proposed. Due to wireless resource limitations, the BS may not be able to transmit the entire semantic information to each user and satisfy the transmission delay constraint. Hence, the BS must select an appropriate resource block for each user as well as determine and transmit part of the semantic information to the users. As such, we formulate an optimization problem whose goal is to maximize the total MSS by jointly optimizing the resource allocation policy and determining the partial semantic information to be transmitted. To solve this problem, a proximal-policy-optimization-based reinforcement learning (RL) algorithm integrated with an attention network is proposed. The proposed algorithm can evaluate the importance of each triple in the semantic information using an attention network and then, build a relationship between the importance distribution of the triples in the semantic information and the total MSS. Compared to traditional RL algorithms, the proposed algorithm can dynamically adjust its learning rate thus ensuring convergence to a locally optimal solution.

We derive a necessary and sufficient condition for stochastic processes to have almost periodic finite dimensional distributions; in particular, we obtain characterizations for infinitely divisible processes to be almost periodic in terms of their characteristic triplets. Furthermore, we derive conditions when the process $(X_t)_{t\in\R}$ defined by the stochastic integral $X_t:= \int_{\R^d} f(t,s) dL(s)$ is almost periodic stationary and also when it is almost periodic in probability, where $f(t,\cdot)\in L^1(\R^d,\R)\cap L^2(\R^d,\R)$ is deterministic and $L$ is a L\'evy basis. Moreover, we discuss almost periodic Ornstein-Uhlenbeck-type processes, and obtain a central limit theorem for $m$-dependent processes with almost periodic finite dimensional distributions.

For immersed surfaces in the four-space, we have a generating set of the Swenton-Hughes-Kim-Miller spatial moves that relate banded singular diagrams of ambient isotopic immersions of those surfaces. We also have Yoshikawa-Kamada-Kawauchi-Kim-Lee planar moves that relate marked graph diagrams of ambient isotopic immersions of those surfaces. One can ask if the former moves form a minimal set and if the latter moves form a generating set. In this paper, we derive a minimal generating set of spatial moves for diagrams of surfaces immersed in the four-space, which translates into a generating set of planar moves. We also show that the complements of two equivalent immersed surfaces can be transformed one another by a Kirby calculus not requiring the 1-1-handle or 2-1-handle slides. This gives a potential room for a stronger immersed surface invariant than the diffeomorphism type of its complement. We also discuss the fundamental group of the immersed surface-link complement in the four-space.

We study the solutions to the Dirac equation for the massive spinor field in the universal covering space of two-dimensional anti-de Sitter space. For certain values of the mass parameter, we impose a suitable set of boundary conditions which make the spatial component of the Dirac operator self-adjoint. Then, we use the transformation properties of the spinor field under the isometry group of the theory, namely, the universal covering group of $\mathrm{SL}(2,\mathbb{R})$, in order to determine which self-adjoint boundary conditions are invariant under this group. We identify the corresponding solution spaces with unitary irreducible representations of this group using the classification given by Pukanzki, and determine which of these correspond to invariant positive- and negative-frequency subspaces and, hence, in a vacuum state invariant under the isometry group. Finally, we examine the cases where the self-adjoint boundary condition leads to an invariant theory with non-invariant vacuum state and determine the unitary representation to which the vacuum state belongs.

In a Hilbert setting, we develop a gradient-based dynamic approach for fast solving convex optimization problems. By applying time scaling, averaging, and perturbation techniques to the continuous steepest descent (SD), we obtain high-resolution ODEs of the Nesterov and Ravine methods. These dynamics involve asymptotically vanishing viscous damping and Hessian driven damping (either in explicit or implicit form). Mathematical analysis does not require developing a Lyapunov analysis for inertial systems. We simply exploit classical convergence results for (SD) and its external perturbation version, then use tools of differential and integral calculus, including Jensen's inequality. By way of illustration of the flexibility of the method we consider the case where the initial dynamics is the regularized Newton method. Then, we show that the technique can be naturally extended to the case of a monotone cocoercive operator. Our approach leads to parallel algorithmic results, which we study in the case of proximal algorithms.

Topological groups whose underlying spaces are basically disconnected, $F$-, or $F'$-spaces but not $P$-spaces are considered. It is proved, in particular, that the existence of a Lindel\"of basically disconnected topological group which is not a $P$-space is equivalent to the existence of a Boolean basically disconnected Lindel\"of group of countable pseudocharacter, that free and free Abelian topological groups of zero-dimensional non-$P$-spaces are never $F'$-spaces, and that the existence of a free Boolean $F'$-group which is not a $P$-space is equivalent to that of selective ultrafilters on $\omega$.

An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.

We present a publicly available software for exponential integrators that computes the $\varphi_l(z)$ functions using polynomial interpolation. The interpolation method at Leja points have recently been shown to be competitive with the traditionally-used Krylov subspace method. The developed framework facilitates easy adaptation into any Python software package for time integration.

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $\alpha$-regular graph for some infinite cardinal $\alpha$ then $D'(G) \le 2$, proving a conjecture of Lehner, Pil\'{s}niak, and Stawiski. We also show that if $G$ is a graph with infinite minimum degree and at most $2^\alpha$ vertices of degree $\alpha$ for every infinite cardinal $\alpha$, then $D'(G) \le 3$. In particular, $D'(G) \le 3$ if $G$ has infinite minimum degree and order at most $2^{\aleph_0}$.

We introduce the concept of braided alternative bialgebra. The theory of unified product for alternative bialgebras is developed. As an application, the extending problem for alternative bialgebra is solved by using some non-abelian cohomology theory.

A famous result due to L. S. Levy provides a classification of all finitely generated indecomposable modules over Dedekind-like rings. This motivates us to outline an approach to the classification of indecomposable pseudo-absorbing primary multiplication modules with finite-dimensional top over certain kinds of pullback rings. In this paper, we give a complete classification, up to isomorphism, of all indecomposable pseudo-absorbing primary multiplication modules with finite-dimensional top over a pullback of two valuation domains with the same residue field. We also find a connection between pseudo-absorbing primary multiplication modules and pure-injective modules over such domains.

We establish existence of infinitely many stationary solutions as well as ergodic stationary solutions to the three dimensional Navier--Stokes and Euler equations in the deterministic as well as stochastic setting, driven by an additive noise. The solutions belong to the regularity class $C(\mathbb{R};H^{\vartheta})\cap C^{\vartheta}(\mathbb{R};L^{2})$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. Moreover, we are able to make conclusions regarding the vanishing viscosity limit and the anomalous dissipation. The result is based on a new stochastic version of the convex integration method which provides uniform moment bounds locally in the aforementioned function spaces.

We study finite-dimensional groups definable in models of the theory of real closed fields with a generic derivation (also known as CODF). We prove that any such group definably embeds in a semialgebraic group. We extend the results to several more general contexts; strongly model complete theories of large geometric fields with a generic derivation, model complete o-minimal expansions of RCF with a generic derivation, open theories of topological fields with a generic derivation. We also give a general theorem on recovering a definable group from generic data in the context of geometric structures.

We consider the problem of designing a feedback controller which robustly regulates an LTI system to an optimal operating point in the presence of unmeasured disturbances. A general design framework based on so-called optimality models was previously put forward for this class of problems, effectively reducing the problem to that of stabilization of an associated nonlinear plant. This paper presents several simple and fully constructive stabilizer designs to accompany the optimality model designs from [1]. The designs are based on a low-gain integral control approach, which enforces time-scale separation between the exponentially stable plant and the controller. We provide explicit formulas for controllers and gains, along with LMI-based methods for the computation of robust/optimal gains. The results are illustrated via an academic example and an application to power system frequency control.

A connected graph $G$ of diameter ${\rm diam}(G) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}|=|W_{yx}|$ for every $x,y\in V(G)$ with $d_{G}(x,y)=\ell$, where $W_{xy}$ is the set of vertices of $G$ that are closer to $x$ than to $y$. We prove that the generalized Petersen graph $GP(n,k)$ is ${\rm diam}(GP(n,k))$-distance-balanced provided that $n$ is large enough relative to $k$. This partially solves a conjecture posed by Miklavi\v{c} and \v{S}parl [20]. Moreover, we determine ${\rm diam}(GP(n,k))$ when $n$ is large enough relative to $k$.

We study the totally asymmetric simple exclusion process with open boundaries in the high density and the low density phase. In the bulk of the two phases, we show that the process on a segment of length $N$ exhibits cutoff at order $N$, while in the intersection of the phases, the coexistence line, the mixing time is of order $N^2$, and no cutoff occurs. In particular, we determine the $\varepsilon$-mixing time in the coexistence line up to constant factors, which do not dependent on $\varepsilon$. Combined with previous results on the maximal current phase, this completes the picture on mixing times for the TASEP with open boundaries.

Functional graphs (FG) allow to model under graph structures the behavior of mapping functions from a discrete set to itself. These functions are used to study real complex phenomena evolving in time. As the systems studied can be large, it can be interesting to decompose and factorise them into several sub-graphs acting together. Polynomial equations over functional graphs can help to define in a formal way this decomposition and factorisation mechanism, and solving them validates or invalidates our hypotheses on their decomposability. The current solution methods breaks done the main equation in a series of basic equations of the form A x X=B, with A, X, and B being FG, but they focus only on the cyclic nodes without taking into account the transient one. In this work, we propose an algorithm which solves these basic equations including also this behavior for FG. We exploit a connection with the cancellation problem over the direct product of digraphs to introduce a first upper bound to the number of solutions for these equations. Then, we introduce a polynomial algorithm able to give some information about the dynamics of all the solutions, and a second exponential version able to concretely find all solutions X for a basic equation. The goal is to make a step forward in the analysis of finite but complex functions such as those used in biological regulatory networks or in systems biology.

In this paper, we prove the non-uniqueness of three-dimensional magneto-hydrodynamic (MHD) system in $C([0,T];L^2(\mathbb{T}^3))$ for any initial data in $H^{\bar{\beta}}(\mathbb{T}^3)$~($\bar{\beta}>0$), by exhibiting that the total energy and the cross helicity can be controlled in a given positive time interval. Our results extend the non-uniqueness results of the ideal MHD system to the viscous and resistive MHD system. Different from the ideal MHD system, the dissipative effect in the viscous and resistive MHD system prevents the nonlinear term from balancing the stress error $(\RR_q,\MM_q)$ as doing in \cite{2Beekie}. We introduce the box type flows and construct the perturbation consisting in six different kinds of flows in convex integral scheme, which ensures that the iteration works and yields the non-uniqueness.

We propose a general framework of proper regularization to solve nonlinear SPDEs with singularities included in both drift and noise coefficients. As applications, the (local and global) existence is presented for a broad class of fluid models driven by pseudo-differential noise, which include the stochastic magnetohydrodynamics (hence Navier-Stokes/Euler) equations, stochastic Camassa-Holm type equations, stochastic aggregation-diffusion equation and stochastic surface quasi-geostrophic equation. Thus, some recent results derived in the literature are considerably extended in a unified way.

In this paper we study grouplike monoids, these are monoids that contain a group to which we add an ordered set of idempotents. We classify finite categories with two objects having grouplike endomorphism monoids, and we give a count of certain categories with grouplike monoids.

We study the ground states of a 2D focusing non-linear Schr\"odinger equation with rotation and harmonic trapping. When the strength of the interaction approaches a critical value from below, the system collapses to a profile obtained from the optimizer of a Gagliardo--Nirenberg interpolation inequality. This was established before in the case of fixed rotation frequency. We extend the result to rotation frequencies approaching, or even equal to, the critical frequency at which the centrifugal force compensates the trap. We prove that the blow-up scenario is to leading order unaffected by such a strong deconfinement mechanism. In particular the blow-up profile remains independent of the rotation frequency.

If $X$ is a smooth projective variety over ${\mathbb R}$, the Hodge ${\mathcal D}$-conjecture of Beilinson asserts the surjectivity of the regulator map to Deligne cohomology with real coefficients. It is known to be false in general but is true in some special cases like Abelian surfaces and $K3$-surfaces - and still expected to be true when the variety is defined over a number field. We prove an analogue of this for Abelian surfaces at a non-Archimedean place where the surface has bad reduction. Here the Deligne cohomology is replaced by a certain Chow group of the special fibre. The case of good reduction is harder and was first studied by Spiess in the case of products of elliptic curve and by me in general.

In this paper we establish a sharp non-uniqueness result for stochastic $d$-dimensional ($d\geq2$) incompressible Navier-Stokes equations. First, for every divergence free initial condition in $L^2$ we show existence of infinite many global in time probabilistically strong and analytically weak solutions in the class $L^\alpha\big(\Omega,L^p_tL^\infty\big)$ for any $1\leq p<2,\alpha\geq1$. Second, we prove the above result is sharp in the sense that pathwise uniqueness holds in the class of $L^p_tL^q$ for some $p\in[2,\infty],q\in(2,\infty]$ such that $\frac2{p}+\frac{d}{q}\leq1$, which is a stochastic version of Ladyzhenskaya-Prodi-Serrin criteria. Moreover, for stochastic $d$-dimensional incompressible Euler equation, existence of infinitely many global in time probabilistically strong and analytically weak solutions is obtained. Compared to the stopping time argument used in \cite{HZZ19, HZZ21a}, we developed a new stochastic version of the convex integration. More precisely, we introduce expectation during convex integration scheme and construct directly solutions on the whole time interval $[0,\infty)$.

We prove existence of twisted K\"ahler-Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when $-K_X$ is big, we obtain a uniform Yau-Tian-Donaldson existence theorem for K\"ahler-Einstein metrics. To achieve this, we build up from scratch the theory of Fujita-Odaka type delta invariants in the transcendental big setting, using pluripotential theory.

We construct indecomposable cycles in the motivic cohomology group $H^3_{{\mathcal M}}(A,{\mathbb Q}(2))$ where $A$ is an Abelian surface over a number field or the function field of a base. When $A$ is the self product of the universal elliptic curve over a modular curve, these cycles can be used to prove algebraicity results for values of higher Green's functions, similar to a conjecture of Gross, Kohnen and Zagier. We formulate a conjecture which relates our work with the recent work of Bruinier-Ehlen-Yang on the conjecture of Gross-Kohnen-Zagier.

The idea behind Poisson approximation to the binomial distribution was used in [J. de la Cal, F. Luquin, J. Approx. Theory, 68(3), 1992, 322-329] and subsequent papers in order to establish the convergence of suitable sequences of positive linear operators. The proofs in these papers are given using probabilistic methods. We use similar methods, but in analytic terms. In this way we recover some known results and establish several new ones. In particular, we enlarge the list of the limit operators and give characterizations of them.

This work discusses parabolic Muckenhoupt weights on spaces of homogeneous type, i.e.\ quasi-metric spaces with both a doubling measure and an additional monotone geodesic property. The main results include a characterization in terms of weighted norm inequalities for parabolic maximal operators, a reverse H\"older inequality, and a Jones-type factorization result for this class of weights. The connection between the space of parabolic bounded mean oscillation and parabolic Muckenhoupt weights is studied by applying a parabolic John--Nirenberg lemma. A Coifman--Rochberg-type characterization of the space of parabolic bounded mean oscillation in terms of parabolic maximal functions is also given. The main challenges in the parabolic theory are related to the time lag in the estimates. The results are motivated by the corresponding Euclidean theory and the regularity theory for parabolic variational problems on metric measure spaces.

Lars Onsager conjectured in 1949 that the Euler equations conserve kinetic energy if the velocity field $\mathbf{u}\in L^3((0,T);C^{0,\alpha}(\mathbb{T}^3))$ with $\alpha>\frac{1}{3}$. In the case that $\alpha<\frac{1}{3}$ energy dissipation can occur. In this work, we pursue an analogue of Onsager's conjecture for the conservation of energy of weak solutions to the hydrostatic Euler equations. Classical spatial analytic solutions of the hydrostatic Euler equations are known to conserve the horizontal kinetic energy $\|u(\cdot,t)\|_{L^2}^2+\|v(\cdot,t)\|^2_{L^2}$, where $(u,v)$ is the horizontal component of the velocity vector field. Unlike the Euler equations, in the case of the hydrostatic Euler equations the vertical velocity $w$ is one degree spatially less regular with respect to the horizontal variables, compared to the horizontal velocity $(u,v)$. Consequently, we introduce the new notions of type II and type III weak solutions (where a type I weak solution refers to the canonical notion of weak solution). As a byproduct, this has its implications for the various formulations of the analogue of the Onsager conjecture for the hydrostatic Euler equations. We first consider the standard notion of weak solution (type I) with the vertical velocity $w\in L^2((0,T);L^2(\mathbb{T}^3))$, and show that if the horizontal velocity $(u,v)\in L^4((0,T);B^\alpha_{4,\infty}(\mathbb{T}^3))$ with $\alpha>\frac{1}{2}$ then the horizontal kinetic energy is conserved. Note that $L^4((0,T);C^{0,\alpha}(\mathbb{T}^3))\subset L^4((0,T);B^\alpha_{4,\infty}(\mathbb{T}^3))$. A plausible explanation for the increase from $\frac{1}{3}$ to $\frac{1}{2}$ in the regularity exponent is due to the aforementioned anisotropic regularity in the velocity field. Finally, we also prove sufficient conditions for energy conservation for type II and type III weak solutions of the hydrostatic Euler equations.

We give a historical perspective on the role of the cyclic category in the development of cyclic theory. This involves a continuous interplay between the extension in characteristic one and in S-algebras, of the traditional development of cyclic theory, and the geometry of the toposes associated with several small categories involved. We clarify the link between various existing presentations of the cyclic and the epicyclic categories and we exemplify the role of the absolute coefficients by presenting the ring of the integers as polynomials in powers of 3, with coefficients in the spherical group ring S[C_2] of the cyclic group of order two. Finally, we introduce the pericyclic category which unifies and refines two conflicting notions of epicyclic space existing in the literature.

Recent progress in deep learning (DL)-based joint source-channel coding (DeepJSCC) has led to a new paradigm of semantic communications. Two salient features of DeepJSCC-based semantic communications are the exploitation of semantic-aware features directly from the source signal, and the discrete-time analog transmission (DTAT) of these features. Compared with traditional digital communications, semantic communications with DeepJSCC provide superior reconstruction performance at the receiver and graceful degradation with diminishing channel quality, but also exhibit a large peak-to-average power ratio (PAPR) in the transmitted signal. An open question has been whether the gains of DeepJSCC come from the additional freedom brought by the high-PAPR continuous-amplitude signal. In this paper, we address this question by exploring three PAPR reduction techniques in the application of image transmission. We confirm that the superior image reconstruction performance of DeepJSCC-based semantic communications can be retained while the transmitted PAPR is suppressed to an acceptable level. This observation is an important step towards the implementation of DeepJSCC in practical semantic communication systems.

In this note, we discuss the following problem: Given a smoothly bounded strongly pseudoconvex domain D in C^n, can we guarantee the existence of geodesics for the Kobayashi--Fuks metric which "spiral around" in the interior of D? We find an affirmative answer to the above question for n=1 when D possesses an infinitely sheeted universal cover.

In this paper we study the contact process on a class of geometric random graphs with scale-free degree distribution, defined on a Poisson point process on $\mathbb{R}^d$. This class includes the age-dependent random connection model and the soft Boolean model. In the ultrasmall regime of these random graphs we provide exact asymptotics for the non-extinction probability when the rate of infection spread is small and show for a finite version of these graphs that the extinction time is of exponential order in the size of the graph.

Periodic integer continued fractions (PICFs) are generalization of the regular periodic continued fractions (RPCFs). It is classical that a RPCF expansion of an irrational number is unique. However, it is no longer unique for a PICF expansion. Hence it is a natural problem to determine all PICF expansions of irrational numbers. In this paper, we determine certain type PICF expansions of square roots of positive square-free integers. To obtain this result, it plays an important role to determine integer points on certain PCF varieties appeared in Brock-Elkies-Jordan. As an application of these results, we obtain fundamental solutions of the Pell equations from PICF expansions of square roots of positive square-free integers as well as the RPCF expansions.

For graphs $H_1,H_2$ by $r^*(H_1,H_2)$ we denote the minimum number of edges in a graph $G$ on $r(H_1,H_2)$ vertices such that $G\to (H_1,H_2)$. We show that for each pair of natural numbers $k,n$, $k\le n$, where $k$ is odd and $n$ is large enough, we have $$r^*(C_n,C_k)=\lceil (n+1)(2n-1)/2\rceil \,.$$

In the setting of the integers, Granville, Harper and Soundararajan showed that the upper bound in Hal\'{a}sz's Theorem can be improved for smoothly supported functions. We derive the analogous result for Hal\'{a}sz's Theorem in $\mathbb{F}_q[t]$, and then consider the converse question of when the general upper bound in this version of Hal\'{a}sz's Theorem is actually attained.

The purpose of this paper is twofold. In the first part, we provide a new proof of the global existence of the solutions to the Vlasov-Maxwell system arising from small data. Our approach relies on vector field methods, together with the Glassey-Strauss decomposition of the electromagnetic field, and does not require any compact support assumption on the initial data. Contrary to previous works on Vlasov systems in dimension 3, we do not modify the linear commutators and avoid then many technical difficulties. In the second part of this paper, we prove a modified scattering result for these solutions. More precisely, we obtain that the electromagnetic field has a radiation field along future null infinity and approaches, for large time, a smooth solution to the vacuum Maxwell equations. As for the Vlasov-Poisson system, in constrast, the distribution function converges to a new smooth density function along modifications of the characteristics of the free relativistic transport equation. In order to define these logarithmic corrections, we identify an effective asymptotic Lorentz force.

Let $k$ be an algebraically closed field of characteristic zero, and $k[[z]]$ the ring of formal power series over $k$. In this paper, we study equations in the semigroup $z^2k[[z]]$ with the semigroup operation being composition. In particular, we answer a question of Horwitz and Rubel about decompositions of "even" formal power series. We also show that every right amenable subsemigroup of $z^2k[[z]]$ is conjugate to a subsemigroup of the semigroup of monomials.

The condition number of computing the invariant left or right singular subspace corresponding to any selected subset of singular values of matrices is determined in the Frobenius norm on the input space of matrices and the chordal, Frobenius, and Procrustes distances on the Grassmannian output space. Up to a small factor, this condition number equals the inverse minimum singular value gap between the singular values selected by the projector and those not selected, including any ghost singular values.

We study the Radial Basis Function (RBF) approximation to differential operators on smooth tensor fields defined on closed Riemannian submanifolds of Euclidean space, identified by randomly sampled point cloud data. We establish the spectral convergence for the classical pointwise RBF discrete non-symmetric approximation of Laplacians. Numerically, we found that this formulation produces a very accurate estimation of leading spectra with large enough data, which leads to a computationally expensive task of solving an eigenvalue problem of not only large but also dense, non-symmetric matrix. However, when the size data is small and/or when the local tangent plane of the unknown manifold is poorly estimated, the accuracy deteriorates. Particularly, this formulation produces irrelevant eigenvalues, in the sense that they are not approximating any of the underlying Laplacian spectra. While these findings suggest that the RBF pointwise formulation may not be reliable to approximate Laplacians for manifold learning, it is still an effective method to approximate general differential operators on smooth manifolds for other applications, including solving PDEs and supervised learning. When the manifolds are unknown, the error bound of the pointwise operator estimation depends on the accuracy of the approximate local tangent spaces. To improve this approximation accuracy, we develop a second-order local SVD technique for estimating local tangent spaces on the manifold that offsets the errors induced by the curvature in the classical first-order local SVD technique. For manifold learning, we introduce a symmetric RBF discrete approximation of the Laplacians induced by a weak formulation on appropriate Hilbert spaces. We establish the convergence of the eigenpairs of both the Laplace-Beltrami operator and Bochner Laplacian in the limit of large data, and provide supporting numerical examples.

Let $d(x,y)$ denote the length of a shortest path between vertices $x$ and $y$ in a graph $G$ with vertex set $V$. For a positive integer $k$, let $d_k(x,y)=\min\{d(x,y), k+1\}$ and $R_k\{x,y\}=\{z\in V: d_k(x,z) \neq d_k(y,z)\}$. A set $S \subseteq V$ is a \emph{distance-$k$ resolving set} of $G$ if $S \cap R_k\{x,y\} \neq\emptyset$ for distinct $x,y\in V$. In this paper, we study the maker-breaker distance-$k$ resolving game (MB$k$RG) played on a graph $G$ by two players, Maker and Breaker, who alternately select a vertex of $G$ not yet chosen. Maker wins by selecting vertices which form a distance-$k$ resolving set of $G$, whereas Breaker wins by preventing Maker from winning. We denote by $O_{R,k}(G)$ the outcome of MB$k$RG. Let $\mathcal{M}$, $\mathcal{B}$ and $\mathcal{N}$, respectively, denote the outcome for which Maker, Breaker, and the first player has a winning strategy in MB$k$RG. Given a graph $G$, the parameter $O_{R,k}(G)$ is a non-decreasing function of $k$ with codomain $\{-1=\mathcal{B}, 0=\mathcal{N}, 1=\mathcal{M}\}$. We exhibit pairs $G$ and $k$ such that the ordered pair $(O_{R,k}(G), O_{R, k+1}(G))$ realizes each member of the set $\{(\mathcal{B}, \mathcal{N}),(\mathcal{B}, \mathcal{M}),(\mathcal{N},\mathcal{M})\}$; we provide graphs $G$ such that $O_{R,1}(G)=\mathcal{B}$, $O_{R,2}(G)=\mathcal{N}$ and $O_{R,k}(G)=\mathcal{M}$ for $k\ge3$. Moreover, we obtain some general results on MB$k$RG and study the MB$k$RG played on some graph classes.

Advances in the development of largely automated microscopy methods such as MERFISH for imaging cellular structures in mouse brains are providing spatial detection of micron resolution gene expression. While there has been tremendous progress made in the field Computational Anatomy (CA) to perform diffeomorphic mapping technologies at the tissue scales for advanced neuroinformatic studies in common coordinates, integration of molecular- and cellular-scale populations through statistical averaging via common coordinates remains yet unattained. This paper describes the first set of algorithms for calculating geodesics in the space of diffeomorphisms, what we term Image-Varifold LDDMM,extending the family of large deformation diffeomorphic metric mapping (LDDMM) algorithms to accommodate the "copy and paste" varifold action of particles which extends consistently to the tissue scales. We represent the brain data as geometric measures, termed as {\em image varifolds} supported by a large number of unstructured points, % (i.e., not aligned on a 2D or 3D grid), each point representing a small volume in space % (which may be incompletely described) and carrying a list of densities of {\em features} elements of a high-dimensional feature space. The shape of image varifold brain spaces is measured by transforming them by diffeomorphisms. The metric between image varifolds is obtained after embedding these objects in a linear space equipped with the norm, yielding a so-called "chordal metric."

We quantify the uniqueness of continuation from Cauchy or interior data. Our approach consists in extending the existing results in the linear case. As by product we obtain a new stability estimate in the linear case. We also show the so-called strong uniqueness of continuation and the uniqueness of continuation from a set of positive measure. These results are derived by using a linearization procedure.

We consider a sequence of Poisson cluster point processes on $\mathbb{R}^d$: at step $n\in\mathbb{N}_0$ of the construction, the cluster centers have intensity $c/(n+1)$ for some $c>0$, and each cluster consists of the particles of a branching random walk up to generation $n$ generated by a point process with mean 1. We show that this 'critical cluster cascade' converges weakly, and that either the limit point process equals the void process (extinction), or it has the same intensity $c$ as the critical cluster cascade (persistence). We obtain persistence, if and only if the Palm version of the outgrown critical branching random walk is locally a.s. finite. This result allows us to give numerous examples for persistent critical cluster cascades.

A broader class of Hardy spaces and Lebesgue spaces have been introduced recently on the unit circle by considering continuous $\|.\|_1$-dominating normalized gauge norms instead of the classical norms on measurable functions a Beurling type result has been proved for the operator of multiplication by the coordinate function. In this paper, we generalize the above Beurling type result to the context of multiplication by a finite Blaschke factor $B(z)$ and also derive the common invariant subspaces of $B^2(z)$ and $B^3(z)$. These results lead to a factorization result for all functions in the Hardy space equipped with a continuous rotationally symmetric norm.

In \cite{SH}, A. L. Shields proved a well-known theorem for the similarity of unilateral weighted shift operators. By using the generalization of this theorem for multivariable weighted shifts and the curvature of holomorphic bundles, we give a necessary and sufficient condition for the similarity of $m$-tuples in Cowen-Douglas class. We also present a necessary condition for commuting $m$-tuples of backward weighted shift operators to be $n$-hypercontractive in terms of the weight sequences.

In the present work we give the construction of the genus field and the extended genus field of an elementary abelian $l$-extension of a field of rational functions, where $l$ is a prime number. In the Kummer case, if $K$ is contained in a cyclotomic funtion field, the construction is given using Leopoldt's ideas by means of Dirichlet characters. Following the definition of Angl\`es and Jaulent of extended Hilbert class field, we obtain the extended genus field of an elementary abelian $l$-extension of a field of rational functions.

For the quiver Hecke algebra $R$ associated with a simple Lie algebra, let $R$-gmod be the category of finite-dimensional graded $R$-modules. It is well-known that it categorifies the unipotent quantum coordinate ring. The localization of $R$-gmod has been defined in [12]. Its Grothendieck ring defines the localized (unipotent) quantum coordinate ring. We shall give a certain crystal structure on the localized quantum coordinate ring by regarding the set of self-dual simple objects in localized $R$-gmod. We also give the isomorphism of crystals to the cellular crystal for an arbitrary reduced word of the longest Weyl group element. This result can be seen as a localized version of the categorification for the crystal of the nilpotent half of quantum algebra by Lauda and Vazirani.

We show that while individual Riesz transforms are two weight norm stable under biLipschitz change of variables on $A_{\infty}$ weights, they are two weight norm unstable under even rotational change of variables on doubling weights. This provides an operator theoretic distinction between $A_{\infty}$ weights and doubling weights. More generally, all iterated Riesz transforms of odd order are rotationally unstable on pairs of doubling weights, thus demonstrating the need for characterizations of iterated Riesz transform inequalities using testing conditions for doubling measures, as opposed to the typically stable 'bump' conditions.

This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic field along the boundary. Based on time-dependent jump conditions of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretising the nonlinear system of boundary integral equations with Runge--Kutta based convolution quadrature in time and Raviart--Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates.

Let $G$ be a connected graph and $g$ be a non-negative integer. The $g$-extra connectivity of $G$ is the minimum cardinality of a set of vertices in $G$, if it exists, whose removal disconnects $G$ and leaves every component with more than $g$ vertices. The strong product $G_1 \boxtimes G_2$ of graphs $G_1=(V_{1}, E_{1})$ and $G_2=(V_{2}, E_{2})$ is the graph with vertex set $V(G_1 \boxtimes G_2)=V_{1} \times V_{2}$, where two distinct vertices $(x_{1}, x_{2}), (y_{1}, y_{2}) \in V_{1} \times V_{2}$ are adjacent in $G_1 \boxtimes G_2$ if and only if $x_{i}=y_{i}$ or $x_{i} y_{i} \in E_{i}$ for $i=1, 2$. In this paper, we obtain the $g$-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles.

The Sturm-Liouville equation represents the linearized form of the first-order Riccati equation. This provides an evidence for the connection between Schwarzian derivative and this first-order nonlinear differential equation. Similar connection is not obvious for higher-order equations in the Riccati chain because the corresponding linear equations are of order greater than two. With special attention to the second- and third-order Riccati equations we demonstrate that Schwarzian derivative has a natural space in higher Riccati equations .

Connection coefficient formulas for special functions describe change of basis matrices under a parameter change, for bases formed by the special functions. Such formulas are related to branching questions in representation theory. The Askey-Wilson polynomials are one of the most general 1-variable special functions. Our main results are connection coefficient formulas for shifting one of the parameters of the nonsymmetric Askey-Wilson polynomials. We also show how one of these results can be used to re-prove an old result of Askey and Wilson in the symmetric case. The method of proof combines establishing a simpler special case of shifting one parameter by a factor of q with using a co-cycle condition property of the transition matrices involved. Supporting computations use the Noumi representation and are based on simple formulas for how some basic Hecke algebra elements act on natural almost symmetric Laurent polynomials.

A partial Latin square of order $n$ can be represented by a $3$-dimensional chess-board of size $n\times n\times n$ with at most $n^2$ non-attacking rooks. Based on this representation, we apply a uniform method to prove the M. Hall's, Ryser's and Cruse's theorems for completion of partial Latin squares. With the help of this proof, we extend the scope of Cruse's theorem to compact bricks, which appear to be independent of their environment. Without losing any completion you can replace a dot by a rook if the dot must become rook, or you can eliminate the dots that are known not to become rooks. Therefore, we introduce primary and secondary extension procedures that are repeated as many times as possible. If the procedures do not decide whether a PLSC can be completed or not, a new necessary condition for completion can be formulated for the dot structure of the resulting PLSC, the BUG condition.

Let G be a simple algebraic group of adjoint type over an algebraically closed field of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs (M,O) where M is the identity component of the centralizer of a semisimple element in G and O is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.

We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by $|\zeta(\frac12+it)|^{2k}$ for $k=1$ and, for test functions with Fourier support in $(-\frac12,\frac12)$, for $k=2$. As a consequence, for $k=1,2$, we deduce under the Riemann hypothesis that $T(\log T)^{1-k^2+o(1)}$ non-trivial zeros of $\zeta$, of imaginary parts up to $T$, are such that $\zeta$ attains a value of size $(\log T)^{k+o(1)}$ at a point which is within $O(1/\log T)$ from the zero.

We provide new proofs based on the Myers-Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics.

In the present paper, one particular attempt is to acquire explicit and analytical solutions of the position-dependent effective mass (PDEM) Schr\"odinger equation for various types of the squared trigonometric potentials. The algebraic procedure entitled as the point canonical transformation (PCT) is implemented in the process. Three different spatially dependent mass configurations are taken into account the establishment of the target system. In the final step, performing the computational tasks lead to the explicit determination of both discrete energy spectra and their corresponding wavefunctions.

When a quantum field theory in $d$-spacetime dimensions possesses a global $(d-1)$-form symmetry, it can decompose into disjoint unions of other theories. This is reflected in the physical quantities of the theory and can be used to study properties of the constituent theories. In this note we highlight the equivalence between the decomposition of orbifold $\sigma$-models and disconnected McKay quivers. Specifically, we show in numerous examples that each component of a McKay quiver can be given definitive geometric meaning through the decomposition formulae. In addition, we give a purely group and representation theoretic derivation of the quivers for the cases where the trivially acting part of the orbifold group is central. As expected, the resulting quivers are compatible with the case of $\sigma$-models on `banded' gerbes.

Sturm's Theorem is a fundamental 19th century result relating the number of real roots of a polynomial $f$ in an interval to the number of sign alternations in a sequence of polynomial division-like calculations. We provide a short direct proof of Sturm's Theorem, including the numerically vexing case (ignored in many published accounts) where an interval endpoint is a root of $f$.

Generalization analyses of deep learning typically assume that the training converges to a fixed point. But, recent results indicate that in practice, the weights of deep neural networks optimized with stochastic gradient descent often oscillate indefinitely. To reduce this discrepancy between theory and practice, this paper focuses on the generalization of neural networks whose training dynamics do not necessarily converge to fixed points. Our main contribution is to propose a notion of statistical algorithmic stability (SAS) that extends classical algorithmic stability to non-convergent algorithms and to study its connection to generalization. This ergodic-theoretic approach leads to new insights when compared to the traditional optimization and learning theory perspectives. We prove that the stability of the time-asymptotic behavior of a learning algorithm relates to its generalization and empirically demonstrate how loss dynamics can provide clues to generalization performance. Our findings provide evidence that networks that "train stably generalize better" even when the training continues indefinitely and the weights do not converge.

Generative design has been growing across the design community as a viable method for design space exploration. Thermal design is more complex than mechanical or aerodynamic design because of the additional convection-diffusion equation and its pertinent boundary interaction. We present a generative thermal design using cooperative multi-agent deep reinforcement learning and continuous geometric representation of the fluid and solid domain. The proposed framework consists of a pre-trained neural network surrogate model as an environment to predict heat transfer and pressure drop of the generated geometries. The design space is parameterized by composite Bezier curve to solve multiple fin shape optimization. We show that our multi-agent framework can learn the policy for design strategy using multi-objective reward without the need for shape derivation or differentiable objective function.

By no fast-forwarding theorem, the simulation time for the Hamiltonian evolution needs to be $O(\|H\| t)$, which essentially states that one can not go across the multiple scales as the simulation time for the Hamiltonian evolution needs to be strictly greater than the physical time. We demonstrated in the context of the semiclassical Schr\"odinger equation that the computational cost for a class of observables can be much lower than the state-of-the-art bounds. In the semiclassical regime (the effective Planck constant $h \ll 1$), the operator norm of the Hamiltonian is $O(h^{-1})$. We show that the number of Trotter steps used for the observable evolution can be $O(1)$, that is, to simulate some observables of the Schr\"odinger equation on a quantum scale only takes the simulation time comparable to the classical scale. In terms of error analysis, we improve the additive observable error bounds [Lasser-Lubich 2020] to uniform-in-$h$ observable error bounds. This is, to our knowledge, the first uniform observable error bound for semiclassical Schr\"odinger equation without sacrificing the convergence order of the numerical method. Based on semiclassical calculus and discrete microlocal analysis, our result showcases the potential improvements taking advantage of multiscale properties, such as the smallness of the effective Planck constant, of the underlying dynamics and sheds light on going across the scale for quantum dynamics simulation.

Pre-shared entanglement can significantly boost communication rates in the high thermal noise and low-brightness transmitter regime. In this regime, for a lossy-bosonic channel with additive thermal noise, the ratio between the entanglement-assisted capacity and the Holevo capacity - the maximum reliable-communications rate permitted by quantum mechanics without any pre-shared entanglement - scales as $\log(1/{\bar N}_{\rm S})$, where the mean transmitted photon number per mode, ${\bar N}_{\rm S} \ll 1$. Thus, pre-shared entanglement, e.g., distributed by the quantum internet or a satellite-assisted quantum link, promises to significantly improve low-power radio-frequency communications. In this paper, we propose a pair of structured quantum transceiver designs that leverage continuous-variable pre-shared entanglement generated, e.g., from a down-conversion source, binary phase modulation, and non-Gaussian joint detection over a code word block, to achieve this scaling law of capacity enhancement. Further, we describe a modification to the aforesaid receiver using a front-end that uses sum-frequency generation sandwiched with dynamically-programmable in-line two-mode squeezers, and a receiver back-end that takes full advantage of the output of the receiver's front-end by employing a non-destructive multimode vacuum-or-not measurement to achieve the entanglement-assisted classical communications capacity.

We prove that the exact closure of SIR pairwise epidemic equations on a configuration model network is possible if and only if the degree distribution is Poisson, Binomial, or Negative Binomial. The proof relies on establishing, for these specific degree distributions, the equivalence of the closed pairwise model and the so-called dynamical survival analysis (DSA) edge-based model which was previously shown to be exact. Indeed, as we show here, the DSA model is equivalent to the well-known edge-based Volz model. We use this result to provide reductions of the closed pairwise and Volz models to the same single equation involving only susceptibles, which has a useful statistical interpretation in terms of the times to infection. We illustrate our findings with some numerical examples.

A general framework with a series of different methods is proposed to improve the estimate of convex function (or functional) values when only noisy observations of the true input are available. Technically, our methods catch the bias introduced by the convexity and remove this bias from a baseline estimate. Theoretical analysis are conducted to show that the proposed methods can strictly reduce the expected estimate error under mild conditions. When applied, the methods require no specific knowledge about the problem except the convexity and the evaluation of the function. Therefore, they can serve as off-the-shelf tools to obtain good estimate for a wide range of problems, including optimization problems with random objective functions or constraints, and functionals of probability distributions such as the entropy and the Wasserstein distance. Numerical experiments on a wide variety of problems show that our methods can significantly improve the quality of the estimate compared with the baseline method.

Quantum Bayesian AI (Q-B) is an emerging field that levers the computational gains available in Quantum computing. The promise is an exponential speed-up in many Bayesian algorithms. Our goal is to apply these methods directly to statistical and machine learning problems. We provide a duality between classical and quantum probability for calculating of posterior quantities of interest. Our framework unifies MCMC, Deep Learning and Quantum Learning calculations from the viewpoint from von Neumann's principle of quantum measurement. Quantum embeddings and neural gates are also an important part of data encoding and feature selection. There is a natural duality with well-known kernel methods in statistical learning. We illustrate the behaviour of quantum algorithms on two simple classification algorithms. Finally, we conclude with directions for future research.

A plethora of modern machine learning tasks require the utilization of large-scale distributed clusters as a critical component of the training pipeline. However, abnormal Byzantine behavior of the worker nodes can derail the training and compromise the quality of the inference. Such behavior can be attributed to unintentional system malfunctions or orchestrated attacks; as a result, some nodes may return arbitrary results to the parameter server (PS) that coordinates the training. Recent work considers a wide range of attack models and has explored robust aggregation and/or computational redundancy to correct the distorted gradients. In this work, we consider attack models ranging from strong ones: $q$ omniscient adversaries with full knowledge of the defense protocol that can change from iteration to iteration to weak ones: $q$ randomly chosen adversaries with limited collusion abilities which only change every few iterations at a time. Our algorithms rely on redundant task assignments coupled with detection of adversarial behavior. For strong attacks, we demonstrate a reduction in the fraction of distorted gradients ranging from 16\%-99\% as compared to the prior state-of-the-art. Our top-1 classification accuracy results on the CIFAR-10 data set demonstrate 25\% advantage in accuracy (averaged over strong and weak scenarios) under the most sophisticated attacks compared to state-of-the-art methods.

Lattice field theory is a very powerful tool to study Feynman's path integral non-perturbatively. However, it usually requires Euclidean background metrics to be well-defined. On the other hand, a recently developed regularization scheme based on Fourier integral operator $\zeta$-functions can treat Feynman's path integral non-pertubatively in Lorentzian background metrics. In this article, we formally $\zeta$-regularize lattice theories with Lorentzian backgrounds and identify conditions for the Fourier integral operator $\zeta$-function regularization to be applicable. Furthermore, we show that the classical limit of the $\zeta$-regularized theory is independent of the regularization. Finally, we consider the harmonic oscillator as an explicit example. We discuss multiple options for the regularization and analytically show that they all reproduce the correct ground state energy on the lattice and in the continuum limit. Additionally, we solve the harmonic oscillator on the lattice in Minkowski background numerically.

We study generic inference on identified linear functionals of nonunique nuisances defined as solutions to underidentified conditional moment restrictions. This problem appears in a variety of applications, including nonparametric instrumental variable models, proximal causal inference under unmeasured confounding, and missing-not-at-random data with shadow variables. Although the linear functionals of interest, such as average treatment effect, are identifiable under suitable conditions, nonuniqueness of nuisances pose serious challenges to statistical inference, since in this setting common nuisance estimators can be unstable and lack fixed limits. In this paper, we propose penalized minimax estimators for the nuisance functions and show they enable valid inference in this challenging setting. The proposed nuisance estimators can accommodate flexible function classes, and importantly, they can converge to fixed limits determined by the penalization, regardless of whether the nuisances are unique or not. We use the penalized nuisance estimators to form a debiased estimator for the linear functional of interest and prove its asymptotic normality under generic high-level conditions, which provide for asymptotically valid confidence intervals.

How complex is an Ising model? Usually, this is measured by the computational complexity of its ground state energy problem. Yet, this complexity measure only distinguishes between planar and non-planar interaction graphs, and thus fails to capture properties such as the average node degree, the number of long range interactions, or the dimensionality of the lattice. Herein, we introduce a new complexity measure for Ising models and thoroughly classify Ising models with respect to it. Specifically, given an Ising model we consider the decision problem corresponding to the function graph of its Hamiltonian, and classify this problem in the Chomsky hierarchy. We prove that the language of this decision problem is (i) regular if and only if the Ising model is finite, (ii) constructive context free if and only if the Ising model is linear and its edge language is regular, (iii) constructive context sensitive if and only if the edge language of the Ising model is context sensitive, and (iv) decidable if and only if the edge language of the Ising model is decidable. We apply this theorem to show that the 1d Ising model and the Ising model on layerwise complete graphs are constructive context free, and the 2d Ising model, the all-to-all Ising model, and the Ising model on perfect binary trees are constructive context sensitive. We also provide a grammar for the 1d and 2d Ising model. This work is a first step in the characterisation of physical interactions in terms of grammars.

A proper coloring of a graph is \emph{proper conflict-free} if every non-isolated vertex $v$ has a neighbor whose color is unique in the neighborhood of $v$. A proper coloring of a graph is \emph{odd} if for every non-isolated vertex $v$, there is a color appearing an odd number of times in the neighborhood of $v$. For an integer $k$, the \textsc{PCF $k$-Coloring} problem asks whether an input graph admits a proper conflict-free $k$-coloring and the \textsc{Odd $k$-Coloring} asks whether an input graph admits an odd $k$-coloring. We show that for every integer $k\geq3$, both problems are NP-complete, even if the input graph is bipartite. Furthermore, we show that the \textsc{PCF $4$-Coloring} problem is NP-complete when the input graph is planar.

Sharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin-Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman's theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar-Parisi-Zhang equation with flat initial profile.