In this note, we define material-uniform hyperelastic bodies (in the sense of Noll) containing discrete disclinations and dislocations, and study their properties. We show in a rigorous way that the size of a disclination is limited by the symmetries of the constitutive relation; in particular, if the symmetry group of the body is discrete, it cannot admit arbitrarily small, yet non-zero, disclinations. We then discuss the application of these observations to the derivations of models of bodies with continuously-distributed defects.

We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing condition $\mathbf{\mathbf{\nabla }}\left( h\mathbf{B} \right) \neq 0$ for the one-dimensional problem, which has been used to overcome unphysical behaviors. The hyperbolic system of partial differential equations depends on two parameters: the constant gravitational potential $g$ and the Coriolis term $f_{0}$, related to the constant rotation of the reference frame. For four different cases, namely $g=0,~f_{0}=0$; $g\neq 0\,,~f_{0}=0$; $g=0$, $f_{0}\neq 0$; and $g\neq 0$, $f_{0}\neq 0$ the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the $L^{10}=\left\{ A_{3,3}\rtimes A_{2,1}\right\} \otimes _{s}A_{5,34}^{a}$; $% L^{8}=A_{2,1}\rtimes A_{6,22}$; $L^{7}=A_{3,5}\rtimes\left\{ A_{2,1}\rtimes A_{2,1}\right\} $; and $L^{6}=A_{3,5}\rtimes A_{3,3}~$respectively, where we use the Morozov-Mubarakzyanov-Patera classification scheme. For the general case where $f_{0}g\neq 0$, we derive all the invariants for the Adjoint action of the Lie algebra $L^{6}$ and its subalgebras, and we calculate all the elements of the one-dimensional optimal system. These elements are then considered to define similarity transformations and construct analytic solutions for the hyperbolic system.

The modern algebra concepts are used to construct tables of algebraic spinors related to Clifford algebra multivectors with real and complex coefficients. The following data computed by Mathematica are presented in form of tables for individual Clifford geometric algebras: 1. Initial idempotent; 2. Two-sided ideal; 3. Left ideal basis (otherwise projector, or spinor basis); 4. Matrix representations (reps) for basis vectors in Clifford algebras in spinor basis; 5. General spinor; 6. Spinor in matrix form; 7. Squared hermitian norm of the spinor. Earlier in 1998, only the first four items computed by Maple were published by R. Ablamowicz.

Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a finite-dimensional integrable hierarchy on an equal footing. This formulation allows a streamlined one-step derivation of both the multi-time Euler-Lagrange equations and the closure relation (encoding integrability). We argue that any Lagrangian one-form for a finite-dimensional system can be recast in our new framework. We also show that our new framework easily extends to non-commuting flows corresponding to nonabelian Lie groups. Thus Hamiltonian Lie group actions can be derived from a variational principle.

The term material with memory is generally used to indicate materials whose mechanical and/or thermodynamical behaviour depends not only on the process at the present time but also on the history of the process itself. Crucial in heat conductors with memory is the heat relaxation function which models the thermal response of the material. The present study is concerned about a thermodynamical problem with memory "aging"; that is, we analyze the temperature evolution within a rigid heat conductor with memory whose relaxation function takes into account the aging of the material. In particular, we account for variations of the relaxation function due to a possible deterioration of the thermal response of the material related to its age.

In this paper, we solve unbalanced optimal transport (UOT) problem on surfaces represented by point clouds. Based on alternating direction method of multipliers algorithm, the original UOT problem can be solved by an iteration consists of three steps. The key ingredient is to solve a Poisson equation on point cloud which is solved by tangent radial basis function (TRBF) method. The proposed TRBF method requires only the point cloud and normal vectors to discretize the Poisson equation which simplify the computation significantly. Numerical experiments conducted on point clouds with varying geometry and topology demonstrate the effectiveness of the proposed method.

We consider the cyclic representations $\Omega_{rs}$ of $ U_q(\widehat{\mathfrak{sl}}_2)$ at $q^N=1$ that depend upon two points $r,s$ in the chiral Potts algebraic curve. We show how $\Omega_{rs}$ is related to the tensor product $\rho_r\otimes \bar{\rho}_s$ of two representations of the upper Borel subalgebra of $U_q(\widehat{\mathfrak{sl}}_2)$. This result is analogous to the factorization property of the Verma module of $U_q(\widehat{\mathfrak{sl}}_2)$ at generic-$q$ in terms of two q-oscillator representation of the Borel subalgebra - a key step in the construction of the Q-operator. We construct short exact sequences of the different representations and use the results to construct Q operators that satisfy TQ relations for $q^N=1$ for both the 6-vertex and $\tau_2$ models.

Topological zero modes in topological insulators or superconductors are exponentially localized at the phase transition between a topologically trivial and nontrivial phase. These modes are solutions of a Jackiw-Rebbi equation modified with an additional term which is quadratic in the momentum. Moreover, localized fermionic modes can also be induced by harmonic potentials in superfluids and superconductors or in atomic nuclei. Here, by using inverse methods, we consider in the same framework exponentially-localized zero modes, as well as gaussian modes induced by harmonic potentials (with superexponential decay) and polynomially decaying modes (with subexponential decay), and derive the explicit and analytical form of the modified Jackiw-Rebbi equation (and of the Schr\"odinger equation) which admits these modes as solutions. We find that the asymptotic behavior of the mass term is crucial in determining the decay properties of the modes. Furthermore, these considerations naturally extend to the nonhermitian regime. These findings allow us to classify and understand topological and nontopological boundary modes in topological insulators and superconductors.

Three-dimensional supergravity in the Batalin--Vilkovisky formalism is constructed by showing that the theory including the Rarita--Schwinger term is equivalent to an AKSZ theory.

There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. However, the theory of fracton orders remains in its early stages, and the complex landscape of examples is far from being mapped out. Here we introduce the class of planon-modular (p-modular) fracton orders, a relatively simple yet still rich class of quantum orders that encompasses several well-known examples of type I fracton order. The defining property is that any non-trivial point-like excitation can be detected by braiding with planons. From this definition, we uncover a significant amount of general structure, including the assignment of a natural number (dubbed the weight) to each excitation of a p-modular fracton order. We identify simple new phase invariants, some of which are based on weight, which can easily be used to compare and distinguish different fracton orders. We also study entanglement renormalization group (RG) flows of p-modular fracton orders, establishing a close connection with foliated RG. We illustrate our general results with an analysis of several exactly solvable fracton models that we show to realize p-modular fracton orders, including Z_n versions of the X-cube, anisotropic, checkerboard, 4-planar X-cube and four color cube (FCC) models. We show that each of these models is p-modular and compute its phase invariants. We also show that each example admits a foliated RG at the level of its non-trivial excitations, which is a new result for the 4-planar X-cube and FCC models. We show that the Z_2 FCC model is not a stack of other better-studied models, but predict that the Z_n FCC model with n odd is a stack of 10 4-planar X-cubes, possibly plus decoupled layers of 2d toric code. We also show that the Z_n checkerboard model for n odd is a stack of three anisotropic models.

We develop a class of functions Omega_N(x; mu, nu) in N-dimensional space concentrated around a spherical shell of the radius mu and such that, being convoluted with an isotropic Gaussian function, these functions do not change their expression but only a value of its 'width' parameter, nu. Isotropic Gaussian functions are a particular case of Omega_N(x; mu, nu) corresponding to mu = 0. Due to their features, these functions are an efficient tool to build approximations to smooth and continuous spherically-symmetric functions including oscillating ones. Atomic images in limited-resolution maps of the electron density, electrostatic scattering potential and other scalar fields studied in physics, chemistry, biology, and other natural sciences are examples of such functions. We give simple analytic expressions of Omega_N(x; mu, nu) for N = 1, 2, 3 and analyze properties of these functions. Representation of oscillating functions by a sum of Omega_N(x; mu, nu) allows calculating distorted maps for the same cost as the respective theoretical fields. We give practical examples of such representation for the interference functions of the uniform unit spheres for N = 1, 2, 3 that define the resolution of the respective images. Using the chain rule and analytic expressions of the Omega_N(x; mu, nu) derivatives makes simple refinement of parameters of the models which describe these fields.

We define a fractional Ito stochastic integral with respect to a randomly scaled fractional Brownian motion via an $S$-transform approach. We investigate the properties of this stochastic integral, prove the Ito formula for functions of such stochastic integrals and apply this Ito formula for investigation of related generalized time-fractional evolution equations.

We discuss a large class of non-relativistic continuum field theories where the Euclidean spatial symmetry of the classical theory is violated in the quantum theory by an Adler-Bell-Jackiw-like anomaly. In particular, the continuous translation symmetry of the classical theory is broken in the quantum theory to a discrete symmetry. Furthermore, that discrete symmetry is extended by an internal symmetry, making it non-Abelian. This presentation streamlines and extends the discussion in [1]. In an Appendix, we present an elementary introduction to 't Hooft and Adler-Bell-Jackiw anomalies using a well-known system.

We formulate a positivity conjecture relating the Verlinde ring associated with an untwisted affine Lie algebra at a positive integer level and a subcategory of finite-dimensional representations over the corresponding quantum affine algebra with a cluster algebra structure. Specifically, we consider a ring homomorphism from the Grothendieck ring of this representation category to the Verlinde ring and conjecture that every object in the category has a positive image under this map. We prove this conjecture in certain cases where the underlying simple Lie algebra is simply-laced with level 2 or of type $A_1$ at an arbitrary level. The proof employs the close connection between this category and cluster algebras of finite cluster type. As further evidence for the conjecture, we show that for any level, all objects have positive quantum dimensions under the assumption that some Kirillov-Reshetikhin modules have positive quantum dimensions.

We study stationary solutions of McKean-Vlasov equation on a high-dimensional sphere and other compact Riemannian manifolds. We extend the equivalence of the energetic problem formulation to the manifold setting and characterize critical points of the corresponding free energy functional. On a sphere, we employ the properties of spherical convolution to study the bifurcation branches around the uniform state. We also give a sufficient condition for an existence of a discontinuous transition point in terms of the interaction kernel and compare it to the Euclidean setting. We illustrate our results on a range of system, including the particle system arising from the transformer models and the Onsager model of liquid crystals.

We study the evolution of a system of many point particles initially concentrated in a small region in $d$ dimensions. Particles undergo overdamped motion caused by pairwise interactions through the long-ranged repulsive $r^{-s}$ potential; each particle is also subject to Brownian noise. When $s

We propose a novel and systematic recurrence method for the energy spectra of non-Hermitian systems under open boundary conditions based on the recurrence relations of their characteristic polynomials. Our formalism exhibits better accuracy and performance on multi-band non-Hermitian systems than numerical diagonalization or the non-Bloch band theory. It also provides a targeted and efficient formulation for the non-Hermitian edge spectra. As demonstrations, we derive general expressions for both the bulk and edge spectra of multi-band non-Hermitian models with nearest-neighbor hopping and under open boundary conditions, such as the non-Hermitian Su-Schrieffer-Heeger and Rice-Mele models and the non-Hermitian Hofstadter butterfly - 2D lattice models in the presence of non-reciprocity and perpendicular magnetic fields, which is only made possible by the significantly lower complexity of the recurrence method. In addition, we use the recurrence method to study non-Hermitian edge physics, including the size-parity effect and the stability of the topological edge modes against boundary perturbations. Our recurrence method offers a novel and favorable formalism to the intriguing physics of non-Hermitian systems under open boundary conditions.

We explore the relation between a classical periodic Hamiltonian system and an associated discrete quantum system on a torus in phase space. The model is a sinusoidally perturbed Harper model and is similar to the sinusoidally perturbed pendulum. Separatrices connecting hyperbolic fixed points in the unperturbed classical system become chaotic under sinusoidal perturbation. We numerically compute eigenstates of the Floquet propagator for the associated quantum system. For each Floquet eigenstate we compute a Husimi distribution in phase space and an energy and energy dispersion from the expectation value of the unperturbed Hamiltonian operator. The Husimi distribution of each Floquet eigenstate resembles a classical orbit with a similar energy and similar energy dispersion. Chaotic orbits in the classical system are related to Floquet eigenstates that appear ergodic. For a hybrid regular and chaotic system, we use the energy dispersion to separate the Floquet eigenstates into ergodic and integrable subspaces. The distribution of quasi-energies in the ergodic subspace resembles that of a random matrix model. The width of a chaotic region in the classical system is estimated by integrating the perturbation along a separatrix orbit. We derive a related expression for the associated quantum system from the averaged perturbation in the interaction representation evaluated at states with energy close to the separatrix.

$A$-type Little String Theories (LSTs) are engineered from parallel M5-branes on a circle $\mathbb{S}_\perp^1$, probing a transverse $\mathbb{R}^4/\mathbb{Z}_M$ background. Below the scale of the radius of $\mathbb{S}_\perp^1$, these theories resemble a circular quiver gauge theory with $M$ nodes of gauge group $U(N)$ and matter in the bifundamental representation (or adjoint in the case of $M=1$). In this paper, we study these LSTs in the presence of a surface defect, which is introduced through the action of a $\mathbb{Z}_N$ orbifold that breaks the gauge groups into $[U(1)]^N$. We provide a combinatoric expression for the non-perturbative BPS partition function for this system. This form allows us to argue that a number of non-perturbative symmetries, that have previously been established for the LSTs, are preserved in the presence of the defect. Furthermore, we discuss the Nekrasov-Shatashvili (NS) limit of the defect partition function: focusing in detail on the case $(M,N)=(1,2)$, we analyse two distinct proposals made in the literature. We unravel an algebraic structure that is responsible for the cancellation of singular terms in the NS limit, which we generalise to generic $(M,N)$. In view of the dualities of higher dimensional gauge theories to quantum many-body systems, we provide indications that our combinatoric expression for the defect partition are useful in constructing and analysing quantum integrable systems in the future.

Based on the Newtonian mechanics, in this article, we present a heuristic derivation of the Friedmann equations, providing an intuitive foundation for these fundamental relations in cosmology. Additionally, using the first law of thermodynamics and Euler's equation, we derive a set of equations that, at linear order, coincide with those obtained from the conservation of the stress-energy tensor in General Relativity. This approach not only highlights the consistency between Newtonian and relativistic frameworks in certain limits but also serves as a pedagogical bridge, offering insights into the physical principles underlying the dynamics of the universe.

We study a class of Gaussian random band matrices of dimension $N \times N$ and band-width $W$. We show that delocalization holds for bulk eigenvectors and that quantum diffusion holds for the resolvent, all under the assumption that $W \gg N^{8/11}$. Our analysis is based on a flow method, and a refinement of it may lead to an improvement on the condition $W \gg N^{8/11}$.