The term “particle–hole symmetry” is beset with conflicting meanings in contemporary physics. Conceived and written from a condensed-matter standpoint, the present paper aims to clarify and sharpen the terminology. In that vein, we propose to define the operation of “particle–hole conjugation” as the tautological algebra automorphism that simply swaps single-fermion creation and annihilation operators

In order to study stochastic viscosity solutions for a class of semilinear stochastic integral-partial differential equations (SIPDEs), a new class of generalized backward doubly stochastic differential equations with general jumps is investigated. The definition of stochastic viscosity solutions of SIPDEs is introduced. A probabilistic representation for stochastic viscosity solutions of semilinear

In this paper, we investigate existence of global-in-time strong solutions to the kinetic Cucker–Smale model coupled with the Stokes equations in the whole space. By introducing a weighted Sobolev space and using space–time estimates for the linear non-stationary Stokes equations, we present a complete analysis on the existence of global-in-time strong solutions to the coupled model without any smallness

The present work is dedicated to the global solutions to the incompressible Oldroyd-B model without damping on the stress tensor in Rn(n=2,3). This result allows us to construct global solutions for a class of highly oscillating initial velocities. The proof uses the special structure of the system. Moreover, our theorem extends the previous result of Zhu [J. Funct. Anal. 274, 2039–2060 (2018)] and

Global stability of the spherically symmetric nonisentropic compressible Euler equations with positive density around global-in-time background affine solutions is shown in the presence of free vacuum boundaries. Vacuum is achieved despite a non-vanishing density by considering a negatively unbounded entropy, and we use a novel weighted energy method, whereby the exponential of the entropy will act

In this paper, we investigate the quantum many-body dynamics with a linear combination of many-body interactions. We derive rigorously the nonlinear Schrödinger equation with a general nonlinearity as the mean-field limit of this model. Due to the complex interaction structure, we establish a new energy estimate for 0<β<1(m−1)d, which is efficient to handle the case of many-body interactions and allows

This paper surveys recent progress in the analysis of nonlinear partial differential equations using Anderson localization and semi-algebraic sets method. We discuss the application of these tools from linear analysis to nonlinear equations such as the nonlinear Schrödinger equations, the nonlinear Klein–Gordon equations (nonlinear wave equations), and the nonlinear random Schrödinger equations on

In this article, we consider the swelling problem in porous elastic soils with fluid saturation. We study the well-posedness of the problem based on the semigroup theory, show that the energy associated with the system is dissipative, and establish the stability of the system in the exponential way. To guarantee the stability of the systems, we consider both viscous damping and the time delay term

Inspired by the optical phenomenon of conical refraction, discovered by Hamilton in 1832, we study the existence of singular optical phenomena associated with linear differential operators acting on vector fields on a surface. We do this by studying the singularities of the Fresnel hyper-surface associated with the differential operator and show that the existence of these singularities can be accounted

In this paper, we prove the existence of full-dimensional invariant tori for a non-autonomous, almost-periodically forced nonlinear beam equation with a periodic boundary condition via Kolmogorov–Arnold–MoserAM theory.

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of viscous incompressible magnetohydrodynamic flow with the no-slip boundary condition of velocity and perfectly conducting walls for magnetic fields. The convergence is shown under various Sobolev norms, including the physically important space–time uniform norm

In one-dimensional unbounded domains, we prove the global existence of strong solutions to the compressible Navier–Stokes system for a viscous radiative gas, when the viscosity μ is a constant and the heat conductivity κ is a power function of the temperature θ according to κ(θ)=κ̃θβ, with β ≥ 0 and κ̃>0. Our result generalizes Zhao and Liao’s result [Y. K. Liao and H. J. Zhao, J. Differ. Equations

This work is motivated by the relation between the KP and BKP integrable hierarchies, whose τ-functions may be viewed as flows of sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate the Cartan map, which, for a vector space V of dimension N, embeds the Grassmannian GrV0(V+V*) of maximal isotropic subspaces of

We show that any polynomial tau-function of the s-component KP and the BKP hierarchies can be interpreted as a zero-mode of an appropriate combinatorial generating function. As an application, we obtain explicit formulas for all polynomial tau-functions of these hierarchies in terms of Schur polynomials and Q-Schur polynomials, respectively. We also obtain formulas for polynomial tau-functions of the

A recent series of works characterized the Standard Model (SM) gauge group GSM as the subgroup of SO(9) that, in the octonionic model of the latter, preserves the split O=C⊕C3 of the space of octonions into a copy of the complex plane plus the rest. This description, however, proceeded via the exceptional Jordan algebras J3(O),J2(O) and, in this sense, remained indirect. One of the goals of this paper

We show that the complex absorbing potential method for computing scattering resonances applies to the case of exponentially decaying potentials. This means that the eigenvalues of −Δ + V − iɛx2 and |V(x)| ≤ Ce−2γ|x| converge, as ɛ → 0+, to the poles of the meromorphic continuation of (−Δ+V−λ2)−1 uniformly on compact subsets of Re λ > 0, Im λ > − γ, and arg λ > −π/8.

Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every Gaussian symmetry admits a Klauder–Bargmann integral representation in terms of coherent states. Furthermore, Gaussian states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup E2(H) of bounded operators in Γ(H). This yields a common parametrization

We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Netočný and Redig and the cluster expansion approach to designing algorithms due to Helmuth, Perkins, and Regts. Similar results have previously been obtained by related methods, and our main contribution is a simple and

Quantum Martin-Löf randomness (q-MLR) for infinite qubit sequences was introduced by Nies and Scholz [J. Math. Phys. 60(9), 092201 (2019)]. We define a notion of quantum Solovay randomness, which is equivalent to q-MLR. The proof of this goes through a purely linear algebraic result about approximating density matrices by subspaces. We then show that random states form a convex set. Martin-Löf absolute

With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials Xα1,…,Xαr and their traces tr(Xα1,…,Xαr). Our method rests on translating the action of the symmetric group on tensor product spaces

Rational solutions of the Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations of associativity are given in terms of configurations of vectors, which satisfy certain algebraic conditions known as ⋁-conditions [A. P. Veselov, Phys. Lett. A 261, 297–302 (1999)]. The simplest examples of such configurations are the root systems of finite Coxeter groups. In this paper, conditions are derived that ensure

We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere, satisfying the Dirichlet boundary conditions along the equator. For this model, we find a precise asymptotic law for the corresponding zero density functions, in both short range (around the boundary) and long range (far away from the boundary) regimes. As a corollary, we were able to find a logarithmic negative bias for

In this paper, we consider the p-Laplacian equations with hereditary effects and the nonlinear term f satisfying the polynomial growth of arbitrary order q − 1 (q ≥ 2). We analyze the well-posedness of solutions and establish the existence of the (CL2(Ω),CL2(Ω)) and (CL2(Ω),CLq(Ω))-pullback attractors by applying the idea of the bi-spaces and the asymptotic a priori estimate method.

In this paper, we use the Cartan estimate for meromorphic functions to prove Anderson localization for a class of long-range operators with singular potentials.

This paper mainly investigates the asymptotic behavior of non-autonomous stochastic complex Ginzburg–Landau equations on unbounded thin domains. We first prove the existence and uniqueness of random attractors for the considered equation and its limit equation. Due to the non-compactness of Sobolev embeddings on unbounded domains, the pullback asymptotic compactness of such a stochastic equation is

This paper deals with the well-posedness and existence of attractors of a class of stochastic diffusion equations with fractional damping and time-varying delay on unbounded domains. We first prove the well-posedness and the existence of a continuous non-autonomous cocycle for the equations and the uniform estimates of solutions and the derivative of the solution operators with respect to the time-varying

We present a rigorous solution of the Boltzmann equation for the electron–phonon scattering problem in three spatial dimensions in the limit of low temperatures. The different temperature scaling of the various scattering rates turns the temperature into a control parameter that is not available in classical kinetic theory and allows for a rigorous proof of Bloch’s T5 law. The relation between the

A generalized version of Groeneveld’s convergence criterion for the virial expansion and generating functionals for weighted two-connected graphs is proven. This criterion works for inhomogeneous systems and yields bounds for the density expansions of the correlation functions ρs (a.k.a. distribution functions or factorial moment measures) of grand-canonical Gibbs measures with pairwise interactions

In this work, we present the analytical approach for the evaluation of the conditional measure Wiener path integral. We consider the time-dependent model parameters. We find the differential equation for the variable, determining the behavior of the harmonic as well as the an-harmonic parts of the oscillator. We present the an-harmonic part of the result in the form of the operator function.

We consider possible quantum effects for infinite systems implied by variations of the multiplication law in the algebra of observables. Using the algebraic formulation of quantum theory, we study the behavior of states ω under changes in the defining relations of the canonical commutation relations (CCR-algebra). These defining relations of the multiplication law depend explicitly on the symplectic

In this paper, we study the following critical Dirac equation −iε∑k=13αk∂ku+aβu+V(x)u=P(x)f(|u|)u+Q(x)|u|u,x∈R3, where ɛ > 0 is a small parameter; a > 0 is a constant; α1, α2, α3, and β are 4 × 4 Pauli–Dirac matrices; and V, P, Q, and f are continuous but are not necessarily of class C1. We prove the existence and concentration of semiclassical solutions under suitable assumptions on the potentials

In this article, we survey recent results (with Duyckaerts and Merle) on the long-time behavior of radial solutions of the energy critical nonlinear wave equation in odd dimensions.

Large-scale electrical and thermal currents in ordinary metals are well approximated by effective medium theory: global transport properties are governed by the solution to homogenized coupled diffusion equations. In some metals, including the Dirac fluid of nearly charge neutral graphene, microscopic transport is not governed by diffusion, but by a more complicated set of linearized hydrodynamic equations

The symmetry classifications of two fractional higher dimensional nonlinear systems, namely, (3 + 1)-dimensional incompressible non-hydrostatic Boussinesq equations and (3 + 1)-dimensional Boussinesq equations with viscosity, are discussed. Both the fractional Boussinesq equations are considered to have derivatives with respect to all variables of fractional type, and some exact solutions are reported

This paper concerns the existence and uniqueness of a two-dimensional compressible subsonic cavity flow behind a given obstacle with a free surface in an infinitely long nozzle. More precisely, for any given atmospheric pressure pe > 0, we first show that there exists a critical value mcr for the incoming mass flux m0 such that if m0 < mcr, then there exists a unique smooth subsonic cavity flow behind

We consider the Kirchhoff type equation with steep potential well and critical growth. By developing some techniques in variational methods, we obtain existence, multiplicity, and concentration behavior of positive solutions under suitable conditions.

In this paper, we study a class of nonlinear wave equations with variable exponent sources. By introducing a family of potential wells, we first prove the global existence of solutions with initial data in the potential wells and the finite time blow-up for solutions starting in the unstable sets. The boundedness and asymptotic behavior of global solutions are also concerned. Finally, we obtain the

We study the equations of a planar magnetohydrodynamic (MHD) compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity being proportional to a positive power of the temperature. In particular, we obtain the global existence of the unique strong solutions to the Cauchy problem or the initial-boundary-value one under natural conditions on the initial data

We consider a simple model describing the interaction of a quantum particle with a vibrational environment, which eventually acts as a friction on the particle. This equation admits soliton-like solutions, and we numerically investigate their stability when subjected to a small initial impulsion. Our findings illustrate the analogies with the behavior of classical particles and the relevance of asymptotic

For every simple Lie algebra g, we consider the associated Takiff algebra gℓ defined as the truncated polynomial current Lie algebra with coefficients in g. We use a matrix presentation of gℓ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra U(gℓ). A similar matrix presentation for the affine Kac–Moody algebra ĝℓ is then used to

We consider charge transport for interacting many-body systems with a gapped ground state subspace that is finitely degenerate and topologically ordered. To any locality-preserving, charge-conserving unitary that preserves the ground state space, we associate an index that is an integer multiple of 1/p, where p is the ground state degeneracy. We prove that the index is additive under composition of

We present first a set of commutator relationships involving the joint quantum, semi-quantum, and number operators generated by a finite family of random variables, having finite moments of all orders, and show how these commutators can be used to recover the joint quantum operators from the semi-quantum operators. We show that any linear operator defined on an algebra of polynomials or the polynomial

We analyze simultaneous quantum estimations of multiple parameters with postselection measurements in terms of a trade-off relation. The system, or a sensor, is characterized by a set of parameters, interacts with a measurement apparatus (MA), and then is postselected onto a set of orthonormal final states. Measurements of the MA yield an estimation of the parameters. We first derive classical and

We use a discrete derivative to introduce a time operator for non-relativistic quantum systems with point spectrum. The symmetry requirement on the time operator leads to well-defined time values related to the dynamics of discrete quantum systems. As an illustration, we find travel times between hits with the walls for the quantum particle in a box model. These times suggest a classical analog of

We examine a completely positive and trace preserving evolution of a finite dimensional open quantum system coupled to a large environment via the periodically modulated interaction Hamiltonian. We derive a corresponding Markovian master equation under the usual assumption of weak coupling using the projection operator techniques in two opposite regimes of very small and very large modulation frequency

We obtain new Faber–Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber–Krahn inequality for the Schrödinger operator with point interaction: the optimizer is the ball with the point interaction supported at its center. Next, we establish three-dimensional Faber–Krahn inequalities for a one- and two-body

We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with the code distance being the linear system size is decomposed by a local Clifford circuit of constant depth into a finite number of copies of the toric code stabilizer group (Abelian discrete gauge theory). This means that under local Clifford

We derive a U(1)B−L-extension of the standard model from a generalized Connes–Lott model with algebra C⊕C⊕H⊕M3(C). This generalization includes the Lorentzian signature, the presence of a real structure, and the weakening of the order 1 condition. In addition to the SM fields, it contains a ZB−L′ boson and a complex scalar field σ that spontaneously breaks the new symmetry. This model is the smallest

The solution xnt, n = 1, 2, of the initial-value problem is reported for the autonomous system of two coupled first-order ordinary differential equations with homogeneous cubic polynomial right-hand sides, ẋn=cn1x13+cn2x12x2+cn3x1x22+cn4x23,n=1,2, when the eight (time-independent) coefficients cnℓ are appropriately defined in terms of seven arbitrary parameters, which then also identify the solution

All maximally superintegrable Hamiltonian systems in three-dimensional flat space derived in the work of Evans [Phys. Rev. A 41, 5666–5676 (1990)] are shown to possess hidden symmetries leading to their linearization, likewise the maximally superintegrable Hamiltonian systems in two-dimensional flat space as shown in the work of Gubbiotti and Nucci [J. Math. Phys. 58, 012902 (2017)]. We conjecture

We study five-body central configurations. The three-body equilateral triangle Lagrange’s central configuration and four-body equilateral tetrahedron Lehmann-Filhès’s central configuration were generalized to an equilateral five-body central configuration in four dimensions. Our working tool is a coordinate system that had many useful properties in considering central configurations of three, four

In this work, we are concerned with the asymptotic properties of solutions for an impulsive neutral stochastic functional integro-differential equation. By applying the theory of resolvent operators, the Banach fixed point principle theorem, and results on stochastic analysis, we study respectively the existence, uniqueness, and global attracting and quasi-invariant sets of mild solutions for the considered

We consider the non-equilibrium steady state induced by two infinitely extended quantum thermal reservoirs at different inverse temperatures β + Δβ, β − Δβ and derive a work relation. We consider global cyclic operations and derive an upper bound of the work density in one-dimensional quantum lattice systems. This relation reproduces the second law of thermodynamics in the equilibrium limit Δβ → 0

Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form T0 + iγsn on a Hilbert space, where sn is strongly convergent to the identity operator and γ > 0. We work in both an abstract setting and a more concrete Sturm–Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps.

For some metric spaces of self-adjoint operators, it is shown that the set of operators whose spectral measures have simultaneous zero upper-Hausdorff and one lower-packing dimension contains a dense Gδ subset. Applications include sets of limit-periodic operators.

In this work, the density operator diagonal representation in the coherent states basis, known as the Glauber–Sudarshan-P representation, is used to study harmonic oscillator quantum systems and models of spinless electrons moving in a two-dimensional noncommutative space, subject to a magnetic field background coupled with a harmonic oscillator. Relevant statistical properties such as the Q-Husimi

The complex Monge–Ampère (CMA) equation in a two-component form is treated as a bi-Hamiltonian system. I present explicitly the first nonlocal symmetry flow in each of the two hierarchies of this system. An invariant solution of the CMA equation with respect to these nonlocal symmetries is constructed, which, being a noninvariant solution in the usual sense, does not undergo symmetry reduction in the

S-Heun operators on linear and q-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The continuous Hahn and big q-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Miller’s structure operators. These four

We establish new Lipschitz-type bounds for functions of operators with noncompact perturbations that produce Schatten class resolvent differences. The results apply to suitable perturbations of Dirac and Schrödinger operators, including some long-range and random potentials, and to important classes of test functions. The key feature of these bounds is an explicit dependence on the Lipschitz seminorm

q-deformed fuzzy Dirac and chirality operators on quantum fuzzy four-sphere SqF4 are studied in this article. Using the q-deformed fuzzy Ginsparg–Wilson algebra, the q-deformed fuzzy Dirac and chirality operators in an instanton and no-instanton sector are studied. In addition, gauged Dirac and chirality operators in both cases have also been constructed. It has been shown that in each step, our results