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In [Ammari et al., SIAM J Math Anal., 52 (2020), pp. 5441--5466], the first author with collaborators proved the existence of Dirac dispersion cones at subwavelength scales in bubbly honeycomb phononic crystals. In this paper, we study the time-evolution of wave packets that are spectrally concentrated near such conical points. We prove that the wave packets dynamics is governed by a time-dependent effective Dirac system, which still depends, but in a simple way, on the subwavelength scale., Comment: Comments are welcome

We investigate the boundary-value problem of atmospheric Ekman flows with piecewise-uniform eddy viscosity. In addition we present a method for finding more general solutions by considering eddy viscosity as an arbitrary step-function. We discuss the existence and uniqueness of the solutions obtained through this method, providing detailed proofs for cases with one and two "jumps" in eddy viscosity. For scenarios with more "jumps," we establish results inductively. Furthermore, we examine the angle between the bottom surface of the Ekman layer and geostrophic winds by extremizing variables such as the eddy viscosity and its point of change. These calculations reveal how the angle can differ from \(45^\circ\), demonstrating that the extreme values of \(0^\circ\) and \(90^\circ\) are achievable, indicating the potential range of the deflection angle.

We study via microscopic considerations and symmetry principles the optical response of the alternating twisted trilayer graphene. The layer-resolved optical conductivities are expressed in terms of contributions from effective twisted bilayer and single-layer systems along with their coupling. We show the emergence of constitutive laws for: (i) an in-plane magnetic response proportional to the above coupling; and (ii) an effective twisted-bilayer electro-magnetic response that involves a chirality parameter. We estimate the in-plane magnetic response to be negligible, but finite for twists close to the magic angle near charge neutrality. Our approach makes use of a unitary transformation for the trilayer Hamiltonian, and the Kubo formulation of linear response theory. We discuss other implications of our analytical results such as features of bulk plasmonic modes., Comment: 19 pages, 1 figure

We use an exact Moreau-Yosida regularized formulation to obtain the exchange-correlation potential for periodic systems. We reveal a profound connection between rigorous mathematical principles and efficient numerical implementation, which marks the first computation of a Moreau-Yosida-based inversion for physical systems. We develop a mathematically rigorous inversion algorithm including error bounds that are verified numerically in bulk silicon. This unlocks a new pathway to analyze Kohn-Sham inversion methods, which we expect in turn to foster mathematical approaches for developing approximate functionals., Comment: 6 pages, 4 figures, supporting information see https://github.com/mfherbst/supporting-my-inversion

A standard task in solid state physics and quantum chemistry is the computation of localized molecular orbits known as Wannier functions. In this manuscript, we propose a new procedure for computing Wannier functions in one-dimensional crystalline materials. Our approach proceeds by first performing parallel transport of the Bloch functions using numerical integration. Then, using novel analysis, we show that a simple correction can be analytically computed that yields the optimally localized Wannier function. The resulting scheme is robust and capable of achieving high-order accuracy. We illustrate this in a number of numerical experiments.

Estimating the fidelity between a desired target quantum state and an actual prepared state is essential for assessing the success of experiments. For pure target states, we use functional representations that can be measured directly and determine the number of copies of the prepared state needed for fidelity estimation. In continuous variable (CV) systems, we utilise the Wigner function, which can be measured via displaced parity measurements. We provide upper and lower bounds on the sample complexity required for fidelity estimation, considering the worst-case scenario across all possible prepared states. For target states of particular interest, such as Fock and Gaussian states, we find that this sample complexity is characterised by the $L^1$-norm of the Wigner function, a measure of Wigner negativity widely studied in the literature, in particular in resource theories of quantum computation. For discrete variable systems consisting of $n$ qubits, we explore fidelity estimation protocols using Pauli string measurements. Similarly to the CV approach, the sample complexity is shown to be characterised by the $L^1$-norm of the characteristic function of the target state for both Haar random states and stabiliser states. Furthermore, in a general black box model, we prove that, for any target state, the optimal sample complexity for fidelity estimation is characterised by the smoothed $L^1$-norm of the target state. To the best of our knowledge, this is the first time the $L^1$-norm of the Wigner function provides a lower bound on the cost of some information processing task., Comment: 40 pages main text, 4 pages of appendices

This letter presents new nontrivial stochastic identities for random isotropic second rank tensor fields. They can be considered as criteria of statistical isotropy in turbulent flows of any nature. The case of axial symmetry is also considered. We confirm the validity of the identities using different direct numerical simulations of turbulent flows., Comment: 7 pages, 4 figures

The paper investigates a generalization of the classical Sitnikov problem, concentrating on the movement of a satellite along the Z-axis as it interacts with $n$ primary bodies in periodic motion. It establishes the existence of an infinite number of even and anti-periodic solutions with increasing periods. The proof employs the Leray-Schauder degree theory to trace the critical points of action functionals, using a homotopy from solutions when the primary bodies are transformed into circular orbits., Comment: 21 pages, 2 figures

Quantum channel capacities give the fundamental performance limits for information flow over a communication channel. However, the prevalence of superadditivity is a major obstacle to understanding capacities, both quantitatively and conceptually. Examples of additivity, while rare, provide key insight into the origins of nonadditivity and enable our best upper bounds on capacities. Degradable channels, which have additive coherent information, are some of the only channels for which we can calculate the quantum capacity. In this paper we construct non-degradable quantum channels that nevertheless have additive coherent information and therefore easily calculated quantum capacity. The first class of examples is constructed by generalizing the Platypus channel introduced by Leditzky et al. The second class of examples, whose additivity follows from a conjectured stability property, is based on probabilistic mixture of degradable and anti-degradable channels.

In this work, we obtain the Demkov-Fradkin tensor of symmetries for the quantum curved harmonic oscillator in a space with constant curvature given by a parameter $\kappa$. In order to construct this tensor we have firstly found a set of basic operators which satisfy the following conditions: i) their products give symmetries of the problem; in fact the Hamiltonian is a combination of such products; ii) they generate the space of eigenfunctions as well as the eigenvalues in an algebraic way; iii) in the limit of zero curvature, they come into the well known creation/annihilation operators of the flat oscillator. The appropriate products of such basic operators will produce the curved Demkov-Fradkin tensor. However, these basic operators do not satisfy Heisenberg commutators but close another Lie algebra. As a by-product, the classical Demkov-Fradkin tensor for the classical curved harmonic oscillator has been obtained by the same method. The case of two dimensions has been worked out in detail: the operators close a $so_\kappa(4)$ Lie algebra; the spectrum and eigenfunctions are explicitly solved in an algebraic way and in the classical case the trajectories have been computed., Comment: 25 pages, 10 figures

Multi-color solitons that are parametrically created in dual-pumped microresonators generate interleaved frequency combs that can be used to obtain combs at new frequencies and when synchronized can be used for low-noise microwave generation and potentially as an element in a chip-scale clockwork. Here, we first derive three-wave equations that describe multi-color solitons that appear in microresonators with a nearly quartic dispersion profile. These solitons are characterized by a single angular group velocity and multiple angular phase velocities. We then use these equations to explain the interleaved frequency combs that are observed at the output of the microresonator. Finally, we used these equations to describe the experimentally-observed soliton-OPO effect. In this effect, the pump frequency comb interacts nonlinearly with a signal frequency comb to create an idler frequency comb in a new frequency range, analogous to an optical parametric oscillation (OPO) process. We determine the conditions under which we expect this effect to occur. We anticipate that the three-wave equations and their extensions will be of use in designing new frequency comb systems and determining their stability and noise performance.

We discuss local magnetic field quenches using perturbative methods of finite time path field theory in the following spin chains: Ising and XY in a transverse magnetic field. Their common characteristics are: i) they are integrable via mapping to second quantized noninteracting fermion problem; ii) when the ground state is nondegenerate (true for finite chains except in special cases) it can be represented as a vacuum of Bogoliubov fermions. By switching on a local magnetic field perturbation at finite time, the problem becomes nonintegrable and must be approached via numeric or perturbative methods. Using the formalism of finite time path field theory based on Wigner transforms of projected functions, we show how to: i) calculate the basic "bubble" diagram in the Loschmidt echo of a quenched chain to any order in the perturbation; ii) resum the generalized Schwinger-Dyson equation for the fermion two point retarded functions in the "bubble" diagram, hence achieving the resummation of perturbative expansion of Loschmidt echo for a wide range of perturbation strengths under certain analiticity assumptions. Limitations of the assumptions and possible generalizations beyond it and also for other spin chains are further discussed., Comment: 28 pages, 2 figures

We define a path integral over Dirac operators that averages over noncommutative geometries on a fixed graph, as the title reveals, partially using quiver representations. We prove algebraic relations that are satisfied by the expectation value of the respective observables, computed in terms of integrals over unitary groups, with weights given by the spectral action. These equations generalise the Makeenko-Migdal equations -- the constraints of lattice gauge theory -- from lattices to arbitrary graphs. As a perspective, these constraints are combined with positivity conditions (on a matrix of parametrised by composition of Wilson loops). A simple example of this combination known as `bootstrap' is fully worked out., Comment: 11 pp, comments welcome

The goal of this paper is to settle the study of non-commutative optimal transport problems with convex regularization, in their static and finite-dimensional formulations. We consider both the balanced and unbalanced problem and show in both cases a duality result, characterizations of minimizers (for the primal) and maximizers (for the dual). An important tool we define is a non-commutative version of the classical $(c,\psi)$-transforms associated with a general convex regularization, which we employ to prove the convergence of Sinkhorn iterations in the balanced case. Finally, we show the convergence of the unbalanced transport problems towards the balanced one, as well as the convergence of transforms, as the marginal penalization parameters go to $+\infty$.

We consider the derivative of the characteristic polynomial of $N \times N$ Haar distributed unitary matrices. In the limit $N \to \infty$, we give a formula for general non-integer moments of the derivative for values of the spectral variable inside the unit disc. The formula we obtain is given in terms of a confluent hypergeometric function. We give an extension of this result to joint moments taken at different points, again valid for non-integer moment orders. For integer moments, our results simplify and are expressed as a single Laguerre polynomial. Using these moment formulae, we derive the mean counting function of zeros of the derivative in the limit $N \to \infty$. This motivated us to consider related questions for the derivative of the Riemann zeta function. Assuming the Lindel\"of hypothesis, we show that integer moments away from but approaching the critical line agree with our random matrix results. Within random matrix theory, we obtain an explicit expression for the integer moments, for finite matrix size, as a sum over partitions. We use this to obtain asymptotic formulae in a regime where the spectral variable approaches or is on the unit circle., Comment: 45 pages

We begin with an exact expression for the entropy of a system of hard spheres within the Hamming space. This entropy relies on probability marginals, which are determined by an extended set of Belief Propagation (BP) equations. The BP probability marginals are functions of auxiliary variables which are introduced to model the effects of loopy interactions on a tree-structured interaction graph. We explore various reasonable and approximate probability distributions, ensuring they align with the exact solutions of the BP equations. Our approach is based on an ansatz of (in)homogeneous cavity marginals respecting the permutation symmetry of the problem. Through thorough analysis, we aim to minimize errors in the BP equations. Our findings support the conjecture that the maximum packing density asymptotically conforms to the lower bound proposed by Gilbert and Varshamov, further validated by the solution of the loopy BP equations., Comment: 31 pages, 4 figures

In this project, we provide a deep-learning neural network (DNN) based biophysics model to predict protein properties. The model uses multi-scale and uniform topological and electrostatic features generated with protein structural information and force field, which governs the molecular mechanics. The topological features are generated using the element specified persistent homology (ESPH) while the electrostatic features are fast computed using a Cartesian treecode. These features are uniform in number for proteins with various sizes thus the broadly available protein structure database can be used in training the network. These features are also multi-scale thus the resolution and computational cost can be balanced by the users. The machine learning simulation on over 4000 protein structures shows the efficiency and fidelity of these features in representing the protein structure and force field for the predication of their biophysical properties such as electrostatic solvation energy. Tests on topological or electrostatic features alone and the combination of both showed the optimal performance when both features are used. This model shows its potential as a general tool in assisting biophysical properties and function prediction for the broad biomolecules using data from both theoretical computing and experiments.

We analyze the spectrum of the hexagonal lattice graph with a vertex coupling which manifestly violates the time reversal invariance and at high energies it asymptotically decouples edges at even degree vertices; a comparison is made to the case when such a decoupling occurs at odd degree vertices. We also show that the spectral character does not change if the equilateral elementary cell of the lattice is dilated to have three different edge lengths, except that flat bands are absent if those are incommensurate., Comment: 12 pages, 4 figures

For a given region, and specified boundary flux and density rate of an extensive property, the optimal flux field that satisfies the balance conditions is considered. The optimization criteria are the $L^{p}$-norm and a Sobolev-like norm of the flux field. Finally, the capacity of the region to accommodate various boundary fluxes and density rates is defined and analyzed.

In this paper, we develop a mathematical framework for generating strong customized field concentration locally around the inhomogeneous medium inclusion via surface transmission resonance. The purpose of this paper is twofold. Firstly, we show that for a given inclusion embedded in an otherwise uniformly homogeneous background space, we can design an incident field to generate strong localized field concentration at any specified places around the inclusion. The aforementioned customized field concentration is crucially reliant on the peculiar spectral and geometric patterns of certain transmission eigenfunctions. Secondly, we prove the existence of a sequence of transmission eigenfunctions for a specific wavenumber and they exhibit distinct surface resonant behaviors, accompanying strong surface-localization and surface-oscillation properties. These eigenfunctions as the surface transmission resonant modes fulfill the requirement for generating the field concentration.

One of the fundamental problems in quantum mechanics is finding the correct quantum image of a classical observable that would correspond to experimental measurements. We investigate for the appropriate quantization rule that would yield a Hamiltonian that obeys the quantum analogue of Hamilton's equations of motion, which includes differentiation of operators with respect to another operator. To give meaning to this type of differentiation, Born and Jordan established two definitions called the differential quotients of first type and second type. In this paper we modify the definition for the differential quotient of first type and establish its consistency with the differential quotient of second type for different basis operators corresponding to different quantizations. Theorems and differentiation rules including differentiation of operators with negative powers and multiple differentiation were also investigated. We show that the Hamiltonian obtained from Weyl, simplest symmetric, and Born-Jordan quantization all satisfy the required algebra of the quantum equations of motion.

We discuss two distinct operator-theoretic settings useful for describing (or defining) propagators associated with a scalar Klein-Gordon field on a Lorentzian manifold $M$. Typically, we assume that $M$ is globally hyperbolic, but we will also consider examples where it is not. Here, the term propagator refers to any Green function or bisolution of the Klein-Gordon equation pertinent to Classical or Quantum Field Theory. These include the forward, backward, Feynman and anti-Feynman propagtors, the Pauli-Jordan function and 2-point functions of Fock states. The first operator-theoretic setting is based on the Hilbert space $L^2(M)$. This setting leads to the definition of the operator-theoretic Feynman and anti-Feynman propagators, which often (but not always) coincide with the so-called out-in Feynman and in-out anti-Feynman propagator. On some special spacetimes, the sum of the operator-theoretic Feynman and anti-Feynman propagator equals the sum of the forward and backward propagator. This is always true on static stable spacetimes and, curiously, in some other cases as well. The second setting is the Krein space $\mathcal{W}_{\rm KG}$ of solutions of the Klein-Gordon equation. Each linear operator on $\mathcal{W}_{\rm KG}$ corresponds to a bisolution of the Klein-Gordon equation, which we call its Klein-Gordon kernel. In particular, the Klein-Gordon kernels of projectors onto maximal uniformly definite subspaces are 2-point functions of Fock states, and the Klein-Gordon kernel of the identity is the Pauli-Jordan function. After a general discussion, we review a number of examples: static and asymptotically static spacetimes, FLRW spacetimes (reducible by a mode decomposition to 1-dimensional Schr\"odinger operators), deSitter space and anti-deSitter space, both proper and its universal cover., Comment: 82 pages, 2 figures

Through the rotational invariance of the 2-d critical bond percolation exploration path on the square lattice we express Smirnov's edge parafermionic observable as a sum of two new edge observables. With the help of these two new edge observables we can apply the discrete harmonic analysis and conformal mapping theory to prove the convergence of the 2-d critical bond percolation exploration path on the square lattice to the trace of $SLE_6$ as the mesh size of the lattice tends to zero., Comment: 112 pages, 10 figures

These are expanded notes of a course on basics of quantum field theory for mathematicians given by the author at MIT., Comment: 206 pages, latex

It is shown by analyzing the $1D$ Schr\"odinger equation the discontinuities in the coupling constant can occur in both the energies and the eigenfunctions. Surprisingly, those discontinuities, which are present in the energies versus the coupling constant, are of three types only: (i) discontinuous energies (similar to the 1st order phase transitions), (ii) discontinuous first derivative in the energy while the energy is continuous (similar to the 2nd order phase transitions), (ii) the energy and all its derivatives are continuous but the functions are different below and above the point of discontinuity (similar to the infinite order phase transitions). Supersymmetric (SUSY) Quantum Mechanics provides a convenient framework to study this phenomenon., Comment: 15 pages, 13 figures

In this study, we conduct parameter estimation analysis on a data assimilation algorithm for two turbulence models: the simplified Bardina model and the Navier-Stokes-{\alpha} model. Our approach involves creating an approximate solution for the turbulence models by employing an interpolant operator based on the observational data of the systems. The estimation depends on the parameter alpha in the models. Additionally, numerical simulations are presented to validate our theoretical results, Comment: 24 pictures

Motivated by the spectral theory of relativistic atoms, we prove matching upper and lower bounds for the transition density of Hardy perturbations of subordinated Bessel heat kernels. The analysis is based on suitable supermedian functions, in particular invariant functions., Comment: 59 pages

We introduce a method that ensures efficient computation of one-dimensional quantum systems with long-range interactions across all temperatures. Our algorithm operates within a quasi-polynomial runtime for inverse temperatures up to $\beta={\rm poly}(\ln(n))$. At the core of our approach is the Density Matrix Renormalization Group algorithm, which typically does not guarantee efficiency. We have created a new truncation scheme for the matrix product operator of the quantum Gibbs states, which allows us to control the error analytically. Additionally, our method is applied to simulate the time evolution of systems with long-range interactions, achieving significantly better precision than that offered by the Lieb-Robinson bound., Comment: 7 pages + 9 pages Supplementary materials, 1 figure

Why humans fight has no easy answer. However, understanding better how humans fight could inform future interventions, hidden shifts and casualty risk. Fusion-fission describes the well-known grouping behavior of fish etc. fighting for survival in the face of strong opponents: they form clusters ('fusion') which provide collective benefits and a cluster scatters when it senses danger ('fission'). Here we show how similar clustering (fusion-fission) of human fighters provides a unified quantitative explanation for complex casualty patterns across decades of Israel-Palestine region violence, as well as the October 7 surprise attack -- and uncovers a hidden post-October 7 shift. State-of-the-art data shows this fighter fusion-fission in action. It also predicts future 'super-shock' attacks that will be more lethal than October 7 and will arrive earlier. It offers a multi-adversary solution. Our results -- which include testable formulae and a plug-and-play simulation -- enable concrete risk assessments of future casualties and policy-making grounded by fighter behavior., Comment: Comments welcome. Working paper

In recent years it has been shown for hard sphere gas that, by retaining the correlation information, dynamical fluctuation and large deviation of empirical measure around Boltzmann equation could be proved, in addition to the classical kinetic limit result by Lanford. After taking low-density limit, the correlation information can be encoded into a functional Hamilton-Jacobi equation. The results above are restricted to short time. This paper establishes global-in-time construction of a solution of the Hamilton-Jacobi equation, by analyzing a system of coupled Boltzmann equations. The global solution converges to a non-trivial stationary solution of the Hamilton-Jacobi equation in the long-time limit under proper assumptions.

We address justification and solitary wave solutions of the cylindrical KdV equation which is formally derived as a long wave approximation of radially symmetric waves in a two-dimensional nonlinear dispersive system. For a regularized Boussinesq equation, we prove error estimates between true solutions of this equation and the associated cylindrical KdV approximation in the L2-based spaces. The justification result holds in the spatial dynamics formulation of the regularized Boussinesq equation. We also prove that the class of solitary wave solutions considered previously in the literature does not contain solutions in the L2-based spaces. This presents a serious obstacle in the applicability of the cylindrical KdV equation for modeling of radially symmetric solitary waves since the long wave approximation has to be performed separately in different space-time regions., Comment: 27 pages; 2 figures

We analyze the spectral stability of the standing periodic waves in the massive Thirring model in laboratory coordinates. Since solutions of the linearized MTM equation are related to the squared eigenfunctions of the linear Lax system, the spectral stability of the standing periodic waves can be studied by using their Lax spectrum. Standing periodic waves are classified based on eight eigenvalues which coincide with the endpoints of the spectral bands of the Lax spectrum. Combining analytical and numerical methods, we show that the standing periodic waves are spectrally stable if and only if the eight eigenvalues are located either on the imaginary axis or along the diagonals of the complex plane., Comment: 42 pages; 19 figures

Cutting mechanics in soft solids present a complex mechanical challenge due to the intricate behavior of soft ductile materials as they undergo crack nucleation and propagation. Recent research has explored the relationship between the cutting force needed to continuously cut a soft material and the radius of the wire (blade). A typical simplifying assumption is that of material incompressibility, albeit no material in nature is really incompressible. In this study, we relax this assumption and examine how material (in)compressibility influences the correlation between cutting forces and material properties like toughness and modulus. The ratio {\mu}/\k{appa}, where {\mu} and \k{appa} are the shear and bulk moduli, indicates the material's degree of compressibility, where incompressible materials have {\mu}/\k{appa}=0, and larger {\mu}/\k{appa} provide higher volumetric compressibility. Following previous observations, we obtain two cutting regimes: (i) high toughness or small wire radius, and (ii) low toughness or large wire radius. Regime (i) is dominated by frictional dissipation, while regime (ii) is dominated by adhesive debonding and/or the wear resistance of the material. These regimes are controlled by the ratio between the wire radius and the elasto-cohesive length of the material: the critical crack opening displacement at fast fracture under uniaxial tension. In the large radius, regime (ii), our theoretical findings reveal that incompressible materials require larger forces. Notably, however, the elasto-cohesive length of the material, defining the transition wire radius between regimes (i) and (ii), is larger for compressible materials, which are therefore more likely to be cut in regime (i), and thus requiring larger cutting forces.

We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $\alpha,\beta$ or $\alpha\pm\beta$ assume integer values. Exceptional Jacobi operators are in one-to-one correspondence with spectral diagrams, a combinatorial object that describes the number and asymptotic behaviour at the endpoints of $(-1,1)$ of all quasi-rational eigenfunctions of the operator. With a convenient indexing scheme for spectral diagrams, explicit Wronskian and integral construction formulas are given to build the exceptional operators and polynomials from the information encoded in the spectral diagram. In the fully degenerate class $\alpha,\beta\in\mathbb N_0$ there exist exceptional Jacobi operators with an arbitrary number of continuous parameters. The classification result is achieved by a careful description of all possible rational Darboux transformations that can be performed on exceptional Jacobi operators.

We solve inverse problems from the D-N map for the quantum graph on a finite domain in a square lattice and that on a hexagonal lattice, as well as inverse scattering problems from the S-matrix for a locally perturbed square lattice and a hexagonal lattice.

Necking localization under quasi-static uniaxial tension is experimentally observed in ductile thin-walled cylindrical tubes, made of soft polypropylene. Necking nucleates at multiple locations along the tube and spreads throughout, involving higher-order modes, evidencing trefoil and fourthfoiled (but rarely even fifth-foiled) shaped cross-sections. No evidence of such a complicated necking occurrence and growth was found in other ductile materials for thin-walled cylinders under quasi-static loading. With the aim of modelling this phenomenon, as well as all other possible bifurcations, a twodimensional formulation is introduced, in which only the mean surface of the tube is considered, paralleling the celebrated Fl\"ugge treatment of axially-compressed cylindrical shells. This treatment is extended to include tension and a broad class of nonlinear-hyperelastic constitutive law for the material, which is also assumed to be incompressible. The theoretical framework leads to a number of new results, not only for tensile axial force (where necking is modelled and, as a particular case, the classic Consid\`ere formula is shown to represent the limit of very thin tubes), but also for compressive force, providing closed-form formulae for wrinkling (showing that a direct application of the Fl\"ugge equation can be incorrect) and for Euler buckling. It is shown that the J2-deformation theory of plasticity (the simplest constitutive assumption to mimic through nonlinear elasticity the plastic branch of a material) captures multiple necking and occurrence of higher-order modes, so that experiments are explained. The presented results are important for several applications, ranging from aerospace and automotive engineering to the vascular mechanobiology, where a thin-walled tube (for instance an artery, or a catheter, or a stent) may become unstable not only in compression, but also in tension.

This paper presents a new model for the generation of axisymmetric concentrated vortices. The solution of a nonlinear equation for internal gravity waves in an unstable stratified atmosphere is obtained and analyzed within the framework of ideal hydrodynamics. The corresponding expressions describing the dependences on the radius for the radial and vertical velocity components in the inner and outer regions of the vortex include combinations of Bessel functions and modified Bessel functions. The proposed new nonlinear analytical model makes it possible to study the structure and nonlinear dynamics of vortices in the radial and vertical regions. The vortex is limited in height. The maximum vertical velocity component is reached at a certain height. Below this height, radial flows converge towards the axis, and above it, an outflow occurs. The resulting instability in the stratified atmosphere leads to an increase in the radial and vertical velocity components according to the hyperbolic sine law, which turns into exponential growth. The characteristic growth time is determined by the inverse growth rate of the instability. The formation of vortices with finite velocity components, which increase with time, is analyzed. The radial structure of the azimuthal velocity is determined by the structure of the initial perturbation and can change with height. The maximum rotation is reached at a certain height. The growth of the azimuth velocity occurs according to a super-exponential law., Comment: 17 pages, 4 figures. arXiv admin note: text overlap with arXiv:2408.14210

The study of symmetries in the realm of manifolds can be approached in two different ways. On one hand, Killing vector fields on a (pseudo-)Riemannian manifold correspond to the directions of local isometries within it. On the other hand, from an algebraic perspective, global symmetries of such manifolds are associated with group elements. Although the connection between these two concepts is well established in the literature, this work aims to build an unexplored bridge between the Killing vector fields of n-dimensional maximally symmetric spaces and their corresponding isometry Lie groups, anchoring primarily on the definition of induced vector fields. As an application of our main result, we explore two specific examples: the three-dimensional Euclidean space and Minkowski spacetime.

The GKSL master equation for N-level systems provides a necessary and sufficient form for the generator of a quantum dynamical semigroup in the Schrodinger picture where the underlying Hilbert space is $\mathbb{C}^N$. In this paper we provide a detailed, self-contained, and elementary construction of the GKSL master equation for an N-level system. We also provide necessary and sufficient conditions for forms of generators of semigroups which have some, but not all, of the defining properties of quantum dynamical semigroups. We do this in such a way to illuminate how each defining property of a quantum dynamical semigroup contributes to the form of the generators., Comment: 18 pages

The representation theory of the Nappi-Witten VOA was initiated in arXiv:1104.3921 and arXiv:2011.14453. In this paper we use the technique of inverse quantum hamiltonian reduction to investigate the representation theory of the Nappi-Witten VOA $ V^1(\mathfrak h_4)$. We first prove that the quantum hamiltonian reduction of $ V^1(\mathfrak h_4)$ is the Heisenberg-Virasoro VOA $L^{HVir}$ of level zero investigated in arXiv:math/0201314 and arXiv:1405.1707. We invert the quantum hamiltonian reduction in this case and prove that $ V^1(\mathfrak h_4)$ is realized as a vertex subalgebra of $L^{HVir} \otimes \Pi$, where $\Pi$ is a certain lattice-like vertex algebra. Using such an approach we shall realize all relaxed highest weight modules which were classified in arXiv:2011.14453. We show that every relaxed highest weight module, whose top components is neither highest nor lowest weight $\mathfrak h_4$-module, has the form $M_1 \otimes \Pi_{1} (\lambda)$ where $M_1$ is an irreducible, highest weight $L^{HVir}$-module and $\Pi_{1} (\lambda)$ is an irreducible weight $\Pi$-module. Using the fusion rules for $L^{HVir}$-modules and the previously developed methods of constructing logarithmic modules we are able to construct a family of logarithmic $V^1(\mathfrak h_4)$-modules. The Loewy diagrams of these logarithmic modules are completely analogous to the Loewy diagrams of projective modules of weight $L_k(\mathfrak{sl}(2))$-modules, so we expect that our logarithmic modules are also projective in a certain category of weight $ V^1(\mathfrak h_4)$-modules., Comment: 22 pages

Using the ratios theorems, we calculate the leading order terms in $N$ for the following averages of the characteristic polynomial and its derivative: $\left< \left|\Lambda_A(1 )\right| ^{r} \frac{ \Lambda_A'(\mathrm{e}^{\mathrm{i} \phi}) }{ \Lambda_A(\mathrm{e}^{\mathrm{i} \phi})} \right>_{SO(2N)}$ and $\left< \left|\Lambda_A(1 )\right| ^{r} \frac{ \Lambda_A'(\mathrm{e}^{\mathrm{i} \phi}) }{ \Lambda_A(\mathrm{e}^{\mathrm{i} \phi})} \right>_{USp(2N)}$. Our expression, derived for integer $r$, permits analytic continuation in $r$ and we conjecture that this agrees with the above averages for non-integer exponents. We use this result to obtain an expression for the one level density of the `excised ensemble', a subensemble of $SO(2N)$, to next-to-leading order in $N$. We then present the analogous calculation for the one level density of quadratic twists of elliptic curve $L$-functions, taking into account a number theoretical bound on the central values of the $L$-functions. The method we use to calculate the above random matrix averages uses the contour integral form of the ratios theorems, which are a key tool in the growing literature on averages of characteristic polynomials and their derivatives, and as we evaluate the next-to-leading term for large matrix size $N$, this leads to some multi-dimensional contour integrals that are slightly asymmetric in the integration variables, which might be useful in other work.

Typical dualities in arbitrary dimensions are understood through a Hilbert-space extension method. By these results, we rigorously dualize the quantum ingappabilities to discrete height model in one dimension which is inaccessible by earlier work such as flux-insertion arguments. It turns out that the ingappabilities of quantum discrete height model is protected by an exotic "modulating" translation symmetry, which is a combination of modulating internal symmetry transformation and the conventional lattice translation. It can be also generalize to higher-form gauge fields in arbitrary dimensions, e.g., $\mathbb{Z}$-gauge theory in two dimensions with $\mathbb{Z}$ one-form symmetry and a modulating translation symmetry.

We consider quantum transport on a tight-binding model on the Bethe lattice of a finite generation, or the Cayley tree, which may model the energy transport in a light-harvesting molecule. As a new feature to analyze the quantum transport, we add complex potentials for sources on the peripheral sites and for a drain on the central site. We find that the eigenstates that can penetrate from the peripheral sites to the central site are quite limited to the number of generation. All the other eigenstates are localized around the peripheral sites and cannot reach the central site. The former eigenstates can carry the current, which reduces the problem to the quantum transport on a parity-time ($PT$)-symmetric tight-binding chain. When the number of links is common to all generations, the current takes the maximum value at the exceptional point for the zero-energy states, which emerges because of the non-Hermiticity due to the $PT$-symmetric complex potentials. As we introduce randomness in the number of links in each generation of the tree, the resulting linear chain is a random-hopping tight-binding model. We find that the current reaches its maximum not exactly but approximately for a zero-energy state, although it is no longer located at an exceptional point in general., Comment: 41 pages, 17 figures

This paper is devoted to the study of the nonlinear Schr\"odinger-Poisson system with a doping profile. We are interested in the existence of stable standing waves by considering the associated $L^2$-minimization problem. The presence of a doping profile causes a difficulty in the proof of the strict sub-additivity. A key ingredient is to establish the strict sub-additivity by adapting a scaling and an iteration argument, which is inspired by \cite{ZZou}. When the doping profile is a characteristic function supported on a bounded smooth domain, smallness of some geometric quantity related to the domain ensures the existence of stable standing waves.

In this paper, we consider the rectangular random matrix $X=(x_{ij})\in \mathbb{R}^{N\times n}$ whose entries are iid with tail $\mathbb{P}(|x_{ij}|>t)\sim t^{-\alpha}$ for some $\alpha>0$. We consider the regime $N(n)/n\to \mathsf{a}>1$ as $n$ tends to infinity. Our main interest lies in the right singular vector corresponding to the smallest singular value, which we will refer to as the "bottom singular vector", denoted by $\mathfrak{u}$. In this paper, we prove the following phase transition regarding the localization length of $\mathfrak{u}$: when $\alpha<2$ the localization length is $O(n/\log n)$; when $\alpha>2$ the localization length is of order $n$. Similar results hold for all right singular vectors around the smallest singular value. The variational definition of the bottom singular vector suggests that the mechanism for this localization-delocalization transition when $\alpha$ goes across $2$ is intrinsically different from the one for the top singular vector when $\alpha$ goes across $4$.

Following Hill and Leblond, the aim of our work is to show, for isotropic nonlinear elasticity, a relation between the corotational Zaremba-Jaumann objective derivative of the Cauchy stress $\sigma$, i.e. \begin{equation} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[\sigma] = \frac{{\rm d}}{{\rm d}{t}}[\sigma] - W \, \sigma + \sigma \, W, \qquad W = {\rm skew}(\dot F \, F^{-1}) \end{equation} and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor $F ={\rm D} \varphi$, the left Cauchy-Green tensor $B = F \, F^T$, and the strain-rate tensor $D = {\rm sym}(\dot F \, F^{-1})$, we show that \begin{equation} \label{eqCPSdef} \begin{alignedat}{2} \forall \,D\in{\rm Sym}(3) \! \setminus \! \{0\}: ~ \langle{\frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[\sigma]},{D}\rangle > 0 \quad &\iff \quad \log B \longmapsto \widehat\sigma(\log B) \;\textrm{is strongly Hilbert-monotone} &\iff \quad {\rm sym} {\rm D}_{\log B} \widehat \sigma(\log B) \in{\rm Sym}^{++}_4(6) \quad \text{(TSTS-M$^{++}$)}, \end{alignedat} \tag{1} \end{equation} where ${\rm Sym}^{++}_4(6)$ denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality ``corotational stability postulate'' (CSP), a novel concept, which implies the \textbf{T}rue-\textbf{S}tress \textbf{T}rue-\textbf{S}train strict Hilbert-\textbf{M}onotonicity (TSTS-M$^+$) for $B \mapsto \sigma(B) = \widehat \sigma(\log B)$, i.e. \begin{equation} \langle \widehat\sigma(\log B_1)-\widehat\sigma(\log B_2),{\log B_1-\log B_2} \rangle> 0 \qquad \forall \, B_1\neq B_2\in{\rm Sym}^{++}(3) \, . \end{equation} In this paper we expand on the ideas of Hill and Leblond, extending Leblonds calculus to the Cauchy elastic case.

Projective structures on topological surfaces support the structure of 2d CFTs with a degree of technical simplification. Arguments for their existence are reviewed along with their essential properties. We propose a complex analytic manifold $\mathcal{P}_g$ biholomorphic to $T^*_{(1,0)} \mathcal{M}_g$ as a pseudo moduli space of the projective structures of the genus $g$ topological surface. Explicit computations at $g=1$ including the analysis of transformations under the modular group support this proposal, and show that $\mathcal{P}_{g=1}$ naturally resolves the orbifold locus of the affine structure moduli space. For $g \geq 2$, whether $\mathcal{P}_g$ contains redundancy at each value of the complex structure moduli remains open. Physically, the space $\mathcal{P}_g$ represents the bundle of universal, stationary, chiral hydrodynamic flows spatially confined to compact genus-$g$ Riemann surfaces., Comment: 44 pages, no figure

We present a classical algorithm for estimating expectation values of arbitrary observables on most quantum circuits across all circuit architectures and depths, including those with all-to-all connectivity. We prove that for any architecture where each circuit layer is equipped with a measure invariant under single-qubit rotations, our algorithm achieves a small error $\varepsilon$ on all circuits except for a small fraction $\delta$. The computational time is polynomial in qubit count and circuit depth for any small constant $\varepsilon, \delta$, and quasi-polynomial for inverse-polynomially small $\varepsilon, \delta$. For non-classically-simulable input states or observables, the expectation values can be estimated by augmenting our algorithm with classical shadows of the relevant state or observable. Our approach leverages a Pauli-path method under Heisenberg evolution. While prior works are limited to noisy quantum circuits, we establish classical simulability in noiseless regimes. Given that most quantum circuits in an architecture exhibit chaotic and locally scrambling behavior, our work demonstrates that estimating observables of such quantum dynamics is classically tractable across all geometries., Comment: Main text: 8 pages, 3 figures. Appendices: 25 pages, 1 figure

In 1983, Hartle and Hawking introduced a conceptually intriguing idea involving signature-type change, which led to the no-boundary proposal for the initial conditions of the universe. According to this proposal, the universe has no beginning because there is no singularity or boundary in spacetime; however, there is an origin of time. Mathematically, this entails signature-type changing manifolds where a Riemannian region smoothly transitions to a Lorentzian region at the surface where time begins. We present a coherent framework for signature-type changing manifolds characterized by a degenerate yet smooth metric. We then adapt firmly established Lorentzian tools and results to the signature-type changing scenario, introducing new definitions that carry unforeseen causal implications. A noteworthy consequence is the presence of locally closed time-reversing loops through each point on the hypersurface. By imposing the constraint of global hyperbolicity on the Lorentzian region, we demonstrate that for every point $p\in M$, there exists a pseudo-timelike loop with point of self-intersection $p$. Or put another way, there always exists a closed pseudo-timelike path in $M$ around which the direction of time reverses, and a consistent designation of future-directed and past-directed vectors cannot be defined., Comment: 33 pages, 8 figures

In this paper, we develop a systematic deep learning approach to solve two-dimensional (2D) stationary quantum droplets (QDs) and investigate their wave propagation in the 2D amended Gross-Pitaevskii equation with Lee-Huang-Yang correction and two kinds of potentials. Firstly, we use the initial-value iterative neural network (IINN) algorithm for 2D stationary quantum droplets of stationary equations. Then the learned stationary QDs are used as the initial value conditions for physics-informed neural networks (PINNs) to explore their evolutions in the some space-time region. Especially, we consider two types of potentials, one is the 2D quadruple-well Gaussian potential and the other is the PT-symmetric HO-Gaussian potential, which lead to spontaneous symmetry breaking and the generation of multi-component QDs. The used deep learning method can also be applied to study wave propagations of other nonlinear physical models., Comment: 17 pages, 12 figures (Proc. R. Soc. A, accepted for publication). arXiv admin note: text overlap with arXiv:2409.01124