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In this paper, we consider a random one-dimensional Schrödinger operator H ω on the half line which has, as its potential term, Gaussian white noise multiplied by a decaying factor. Although the potential term is not an ordinary function, but a distribution, it is possible to realize H ω as a symmetric operator in L2([0, ∞); dt) as was pointed out by the present author [Minami, Lect. Notes Math. 1299, 298 (1986)], and it will be shown that H ω is actually self-adjoint with probability one. When the white noise in H ω is replaced by random functions of a specific type, [Kotani and Ushiroya, Commun. Math. Phys. 115, 247 (1988)] made a precise analysis of the positive part of the spectrum. According to them, if the decaying factor is not square integrable, the positive part of the spectrum typically consists of singular continuous and dense pure-point parts, which are separated by a threshold number. On the other hand, the positive part of the spectrum is purely absolutely continuous when the decaying factor is square integrable. In this work, we shall focus on the case of square integrable decaying factor, and prove the absolute continuity of the positive part of the spectrum of H ω . We shall further prove that the negative part of the spectrum of H ω is discrete, with no accumulation points other than 0. [ABSTRACT FROM AUTHOR]

In an earlier work [K. Castillo et al., J. Math. Anal. Appl. 514, 126358 (2022)], we give positive answer to the first, and apparently more easy, part of a conjecture of M. Ismail concerning the characterization of the continuous q-Jacobi polynomials, Al-Salam-Chihara polynomials or special or limiting cases of them. In this note we present an example that disproves the second part of such a conjecture, and so this issue is definitively closed. [ABSTRACT FROM AUTHOR]

We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type over the unitary group U(N) with respect to Haar measure, for finite N and including fixed external matrices. When depending only on the eigenvalues of the unitary matrix, such integrals can be related to a Toeplitz determinant with jump singularities. After introducing fixed deterministic matrices as external sources, the integrals can no longer be solved using Andréiéf's integration formula. Resorting to the character expansion as put forward by Balantekin, we derive explicit determinantal formulae containing Kummer's confluent and Gauß' hypergeometric function. They depend only on the eigenvalues of the deterministic matrices and are analytic in the parameters of the jump singularities. Furthermore, unitary two-matrix integrals of the same type are proposed and solved in the same manner. When making part of the deterministic matrices random and integrating over them, we obtain similar formulae in terms of Pfaffian determinants. This is reminiscent to a unitary group integral found recently by Kanazawa and Kieburg [J. Phys. A: Math. Theor. 51(34), 345202 (2018)]. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of small-amplitude unique solutions near the Maxwellian for the Fokker–Planck–Boltzmann equation in a finite channel with specular reflection boundary conditions. The solution space we consider is denoted as L k ̄ 1 L T ∞ L x 1 , v 2 , introduced in Duan et al. [Commun. Pure Appl. Math. 74(5), 932–1020 (2021)]. In addition, we investigate the long-time behavior of solutions for both hard and soft potentials. [ABSTRACT FROM AUTHOR]

Sub-quadratic repulsive potentials accelerate quantum particles and can relax the decay rate in the x of the external potentials V that guarantee the existence of the quantum wave operators. In the case where the sub-quadratic potential is −|x|α with 0 < α < 2 and the external potential satisfies |V(x)| ≤ C(1 + |x|)−(1−α/2)−ɛ with ɛ > 0, Bony et al. [J. Math. Pures Appl. 84, 509–579 (2005)] determined the existence and completeness of the wave operators, and Itakura [J. Math. Phys. 62, 061504 (2021)] then obtained their results using stationary scattering theory for more generalized external potentials. Based on their results, we naturally expect the following. If the decay power of the external potential V is less than −(1 − α/2), V is included in the short-range class. If the decay power is greater than or equal to −(1 − α/2), V is included in the long-range class. In this study, we first prove the new propagation estimates for the time propagator that can be applied to scattering theory. Second, we prove that the wave operators do not exist if the power is greater than or equal to −(1 − α/2) and that the threshold expectation of −(1 − α/2) is true using the new propagation estimates. [ABSTRACT FROM AUTHOR]

Localization results for a class of random Schrödinger operators within the Hartree–Fock approximation are proved in two regimes: Large disorder and weak disorder/extreme energies. A large disorder threshold λHF analogous to the threshold λAnd obtained in Schenker [Lett. Math. Phys. 105(1), 1–9 (2015)] is provided. We also show certain stability results for this large disorder threshold by giving examples of distributions for which λHF converges to λAnd, or to a number arbitrarily close to it, as the interaction strength tends to zero. [ABSTRACT FROM AUTHOR]

Motivated by recent Physical Review Letters of Wang and Liu [Phys. Rev. Lett. 128(17), 176403 (2022)] and Ledwith, Vishwanath, and Khalaf [Phys. Rev. Lett. 128(17), 176404 (2022)], we study [G. Tarnopolsky, A. Kruchkov, and A. Vishwanath, Phys. Rev. Lett. 122(10), 106405 (2019)] chiral model of two sheets of n-layer Bernal stacked graphene twisted by a small angle using the framework developed by Becker et al. [Probab. Math. Phys. 3(1), 69 (2022)]. We show that magic angles of this model are exactly the same as magic angles of chiral twisted bilayer graphene with multiplicity. For small inter-layer tunneling potentials, we compute the band separation at Dirac points as we turning on the tunneling parameter. Flat band eigenfunctions are also constructed using a new theta function argument and this yields a complex line bundle with the Chern number −n. [ABSTRACT FROM AUTHOR]

In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In case of matrix algebras, we characterize all variants of Choi matrices which retain the usual correspondences between k-superpositivity and Schmidt number ≤ k as well as k-positivity and k-block-positivity. We also compare de Pillis' definition [Pac. J. Math. 23, 129–137 (1967)] and Choi's definition [Linear Algebra Appl. 10, 285–290 (1975)], which arise from different bilinear forms. [ABSTRACT FROM AUTHOR]

Cubic blocks are studied assembled from linear operators R acting in the tensor product of d linear "spin" spaces. Such operator is associated with a linear transformation A in a vector space over a field F of a finite characteristic p, like "permutation-type" operators studied by Hietarinta [J. Phys. A: Math. Gen. 30, 4757–4771 (1997)]. One small difference is that we do not require A and, consequently, R to be invertible; more importantly, no relations on R are required of the type of Yang–Baxter or its higher analogues. It is shown that, in d = 3 dimensions, a pn × pn × pn block decomposes into the tensor product of operators similar to the initial R. One generalization of this involves commutative algebras over F and allows to obtain, in particular, results about spin configurations determined by a four-dimensional R. Another generalization deals with introducing Boltzmann weights for spin configurations; it turns out that there exists a non-trivial self-similarity involving Boltzmann weights as well. [ABSTRACT FROM AUTHOR]

We are concerned with the existence as well as the exponential stability in H1-level for the damped defocusing Schrödinger equation posed in a two-dimensional exterior domain Ω with smooth boundary ∂Ω. The proofs of the existence are based on the properties of pseudo-differential operators introduced in Dehman et al. [Math. Z. 254, 729–749 (2006)] and a Strichartz estimate proved by Anton [Bull. Soc. Math. Fr. 136, 27–65 (2008)], while the exponential stability is achieved by combining arguments firstly considered by Zuazua [J. Math. Pures Appl. 9, 513–529 (1991)] for the wave equation adapted to the present context and a global uniqueness theorem. In addition, we proved propagation results for the linear Schrödinger equation for any dimensional and for any (reasonable) boundary conditions employing microlocal analysis. [ABSTRACT FROM AUTHOR]

We prove a continuation criterion for the free boundary problem of three-dimensional incompressible ideal magnetohydrodynamic (MHD) equations in a bounded domain, which is analogous to the theorem given in Beale, Kato, and Majda [Commun. Math. Phys. 94, 61–66 (1984)]. We combine the energy estimates of our previous works [C. Hao and T. Luo, Arch. Ration. Mech. Anal. 212(3), 805–847 (2014)] on incompressible ideal MHD and some analogous estimates in Ginsberg [SIAM J. Math. Anal. 53, 3366–3384 (2021); arXiv:1811.06154] to show that the solution can be continued as long as the curls of the magnetic field and velocity, the second fundamental form and injectivity radius of the free boundary and some norms of the pressure remain bounded, provided that the Taylor-type sign condition holds. [ABSTRACT FROM AUTHOR]

Stimulated by improved oscillation estimates of the potential function and the scalar curvature on compact gradient Ricci solitons introduced in a recent study by Cheng, Ribeiro, and Zhou [Proc. Am. Math. Soc. Ser. B 10, 33–45 (2023)], we give several new sufficient conditions for compact four-dimensional normalized shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Our new conditions refine the validity of the Hitchin–Thorpe inequality obtained by Tadano [J. Math. Phys. 58, 023503 (2017)], Tadano [J. Math. Phys. 59, 043507 (2018)], and Tadano [Differ. Geom. Appl. 66, 231–241 (2019)]. [ABSTRACT FROM AUTHOR]

The solutions to degenerate dynamical systems of the form A (x) x ̇ = f (x) are studied by considering the equation as a differential inclusion. The set Z = { det (A (x)) = 0 } , called the singular set, is assumed to have an empty interior. The reasons leading us to the definition of the sets used for differential inclusion are exposed in detail. This definition is then applied on the one hand to generic cases and on the other hand to the particular cases resulting from physics, which can be found in Saavedra, Troncoso, and Zanelli [J. Math. Phys. 42, 4383 (2001)]. It is shown that generalized solutions may enter, leave, or remain in the singular locus. [ABSTRACT FROM AUTHOR]

A derivation error in the article cited in the title of this Comment is pointed out and corrected. In addition, the Maxwell–Cattaneo based model assumed therein is extended to include expected Joule heating effects; an alternative theory of second-sound that allows the same modeling to be performed, but with fewer assumptions, is noted and applied; and the difference between ordinary solitary waves and solitons is recalled. [ABSTRACT FROM AUTHOR]

In this paper, we will show that the Dubrovin–Frobenius prepotentials on the orbit space of the Coxeter group B n constructed by Arsie et al. [Sel. Math. New Ser. 29, 1 (2023)] coincide with the solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations associated with the constrained KP hierarchy. [ABSTRACT FROM AUTHOR]

In this work, we investigate the Cauchy problem for the pressureless Euler–Navier–Stokes system in R 3 . We first establish the global small solutions of this system with critical regularity and then obtain the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). The proof crucially depends on non-standard product estimates and interpolations. In comparison with previous studies about time-decay by Choi and Jung [J. Math. Fluid Mech. 23, 99 (2021); arXiv:2112.14449], the smallness requirement of the low frequencies of initial data could be removed. [ABSTRACT FROM AUTHOR]

Since K(h, )-1 (X) <= (X ) for m <= X <= M and h = M/m by Ref. 1, Lemma 4.3, it follows thatTr[ ( )]=Tr[ ( )1/2 (( -1/2 -1/2) ) ( )1/2]>=K(h2, )-1Tr[ ( )1/2 ( -1/2 -1/2) ( )1/2]=K(h2, )-1Tr[ ( ) ( )],where (X) = ( )-1/2 ( 1/2X 1/2) ( )-1/2, and hence, we have the corrected Theorem 7.4.Furthermore, since Ref. 3, Lemma 8.1 is derived from Ref. 3, Lemma 5.2, it is not valid in general. By the corrected Lemma 5.1, since I t i SP I i sp is operator convex for I i [-1, 0) [1, 2], the condition for inequality (5.2) in Ref. [3], Lemma 5.2 to hold is modified to I i [-1, 0] [1, 2]. (5.3) Therefore, it follows from the corrected Lemma 5.2 that Ref. 3, Theorem 5.3 (data processing inequality for -Rényi divergence) holds for the following restricted conditions. [Extracted from the article]

In this paper, we consider the long-time behavior of the nonclassical diffusion equation with perturbed parameter and memory on a bounded domain Ω ⊂ R n (n ≥ 3). The main feature of this model is that the equation contains a dissipative term with perturbation parameters −νΔu and the nonlinearity f satisfies the polynomial growth of arbitrary order. By using the nonclassical operator method and a new analytical method (or technique) (Lemma 2.7), the existence and regularity of uniform attractors generated for this equation are proved. Furthermore, we also get the upper semicontinuity of the uniform attractors when the perturbed parameter ν → 0. It is remarkable that if ν = 0, we can get the same conclusion as in the works of Toan et al. [Acta Appl. Math. 170, 789–822 (2020)] and Conti et al. [Commun. Pure Appl. Anal. 19, 2035–2050 (2020)], but the nonlinearity is critical. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem of compressible Navier–Stokes–Vlasov equations under a local alignment regime in a one-dimensional bounded domain. Based on the relative entropy method and compactness argument, we prove that a weak solution of the initial-boundary value problem converges to a strong solution of the limiting two-phase fluid system. This work extends in some sense the previous work of Choi and Jung [Math. Models Methods Appl. Sci. 31(11), 2213–2295 (2021)], which considered the diffusive term ∂ξξfɛ in the kinetic equation. Note that the diffusion term was not considered in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of tri-dimensional compressible Navier–Stokes equations and construct global smooth solutions by choosing a class of new special initial velocity and density whose B ̇ 2 , ∞ − s -norm can be arbitrarily large and improve the previous result in Li et al. [J. Math. Fluid Mech. 24, 22 (2022)]. Our main idea is splitting the linearized equations from the compressible Navier–Stokes equations and exploring the damping effect of the linearized system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to get an estimation of decay rates to first-order and second-order derivatives of space for large-solutions to 3D compressible magnetohydrodynamic system. While the condition (σ0 − 1, u0, Q0) ∈ L1 ∩ H2 is satisfied via a classical energy method and Fourier splitting method, first-order and second-order derivatives of space for large-solutions tending to 0 by L2-rate (1 + t) − 5 4 are shown. It is a necessary supplement to the result of Gao, Wei, and Yao [Appl. Math. Lett. 102, 106100 (2020)] in which they only obtained an estimation of decay rates to magnetic fields. Meanwhile, compared with the work of Gao, Wei, and Yao [Physica D 406, 132506 (2020)], we find that the appearance of magnetic fields does not have any bad effect on the estimation of decay rates to both the velocity field and density. [ABSTRACT FROM AUTHOR]

Given a random subspace Hn chosen uniformly in a tensor product of Hilbert spaces Vn ⊗ W, we consider the collection Kn of all singular values of all norm one elements of Hn with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of Hn, Vn tending to infinity at the same speed by Belinschi, Collins, and Nechita [Commun. Math. Phys. 341(3), 885–909 (2016)]. In this paper, we provide measure concentration estimates in this context. The probabilistic study of Kn was motivated by important questions in quantum information theory and allowed us to provide the smallest known dimension for the dimension of an ancilla space, allowing for Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where the violation of MOE occurs. [ABSTRACT FROM AUTHOR]

We consider a problem of separation of variables for Lax-integrable Hamiltonian systems governed by gl(n) ⨂ gl(n)-valued classical r-matrices r(u, v). We find a new class of classical non-skew-symmetric non-dynamical gl(n) ⨂ gl(n)-valued r-matrices rJ(u, v) for which the "magic recipe" of Sklyanin [Prog. Theor. Phys. Suppl. 118, 35 (1995)] in the theory of variable separation is applicable, i.e., for which standard separating functions A(u) and B(u) of Gekhtman [Commun. Math. Phys. 167, 593 (1995)] and Scott ["Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain," arXiv:hep-th 940303] produce a complete set of canonical coordinates satisfying the equations of separation. We illustrate the corresponding separation of variable theory by the example of the anisotropically deformed An Toda models proposed in the work of Skrypnyk [J. Phys. A: Math. Theor. 38, 9665–9680 (2005)] and governed by the r-matrices rJ(u, v) and by the generalized Gaudin models [T. Skrypnyk, Phys. Lett. A 334(5–6), 390 (2005)] governed by the same classical r-matrices. The n = 2 and n = 3 cases are considered in detail. [ABSTRACT FROM AUTHOR]

We provide a transparent, simple, and unified treatment of recent results on the equality conditions for the data-processing inequality of the sandwiched quantum Rényi divergence, including the statement that the equality in the data-processing implies recoverability via the Petz recovery map for the full range of the Rényi parameter α recently proven by Jenčová [J. Phys. A: Math. Theor. 50, 085303 (2017)]. We also obtain a new set of equality conditions, generalizing a previous result by Leditzky et al. [Lett. Math. Phys. 107, 61 (2017)]. [ABSTRACT FROM AUTHOR]

The object under consideration in this article is the total volume Vg,n(x1, ..., xn) of the moduli space of hyperbolic surfaces of genus g with n boundary components of lengths x1, ..., xn, for the Weil–Petersson volume form. We prove the existence of an asymptotic expansion of the quantity Vg,n(x1, ..., xn) in terms of negative powers of the genus g, true for fixed n and any x1, ..., xn ≥ 0. The first term of this expansion appears in the work of Mirzakhani and Petri [Comment. Math. Helvetici 94, 869–889 (2019)], and we compute the second term explicitly. The main tool used in the proof is Mirzakhani's topological recursion formula, for which we provide a comprehensive introduction. [ABSTRACT FROM AUTHOR]

The 1971 Fortuin–Kasteleyn–Ginibre inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008, one of us (Sahi) conjectured an extended version of this inequality for all n > 2 monotone functions on a distributive lattice. Here, we prove the conjecture for two special cases: for monotone functions on the unit square in R k whose upper level sets are k-dimensional rectangles and, more significantly, for arbitrary monotone functions on the unit square in R 2 . The general case for R k , k > 2 , remains open. [ABSTRACT FROM AUTHOR]

A large family of Flett potentials is investigated. Formally, these potentials are negative powers of the operators id + |x|1−1/mΔk, where Δk is the Dunkl Laplace (differential and difference) operator on R. Here, k ≥ 0 and m ∈ N \ { 0 }. In the (k = 0, m = 1) case, our family of potentials reduces to the classical one studied by Flett [Proc. London Math. Soc. s3-22, 385–451 (1971)]. An explicit inversion formula of the Flett potentials is obtained for functions belonging to C 0 (R) and weighted Lp spaces, 1 ≤ p < ∞. As a tool, we use a wavelet-like transforms generated by a Poisson type semigroup and signed Borel measures. In this context, a fundamental theorem proving an almost everywhere convergence of a convolution operator for an approximate identity was given. The k = 0 case is already new. [ABSTRACT FROM AUTHOR]

In particular, the main problem of the Bogoliubov transformation comes when, with the parameters HT ht and HT ht , one tries to verify if the transformation is unitary. In fact, if one computes the commutators between HT ht and HT ht , an expression similar to Eq. (3) is obtained. [Extracted from the article]

In this paper, we have illustrated the construction of a real structure on a fuzzy sphere S * 2 in its spin-1/2 representation. Considering the SU(2) covariant Dirac and chirality operator on S * 2 given by U. C. Watamura and Watamura [Commun. Math. Phys. 183, 365–382 (1997) and Commun. Math. Phys. 212, 395–413 (2000)], we have shown that the real structure is consistent with other spectral data for KO dimension-4 fulfilling the zero order condition, where we find it necessary to enlarge the symmetry group from SO(3) to the full orthogonal group O(3). However, the first order condition is violated, thus paving the way to construct a toy model for an SU(2) gauge theory to capture some features of physics beyond the standard model following Chamseddine et al. (J. High Energy Phys. 2013, 132). [ABSTRACT FROM AUTHOR]

In this paper, we prove the uniqueness of an energy conservative Hölder continuous weak solution to a general quasilinear wave equation by the analysis of characteristics. This result has no restriction on the size of solutions, i.e., it is a large data result. This paper is a companion paper of Cai, Chen, and Shen [J. Math. Pures Appl. (submitted); arXiv:2007.15201] addressing the Lipschitz continuous dependence of solution. [ABSTRACT FROM AUTHOR]

This paper aims at studying the inelastic Boltzmann equation without Grad's angular cutoff assumption, where the well-posedness theory of the Cauchy problem is established for the Maxwellian molecules in a space of probability measure defined by Cannone and Karch [Commun. Pure Appl. Math. 63, 747–778 (2010)] via Fourier transform and the infinite energy solutions are not a priori excluded. The key strategy is to construct a brand new geometric relation of the inelastic collision mechanism to extend the result of Cannone and Karch from moderate singularity of the non-cutoff collision kernels to strong singularity and simultaneously handle more general restitution coefficients. Moreover, we extend the self-similar solution to the Boltzmann equation with infinite energy shown by Bobylev and Cercignani [J. Stat. Phys. 106, 1039–1071 (2002)] to the inelastic case by using a constructive approach, which is also proved to be the large-time asymptotic steady solution with the help of an asymptotic stability result in a certain sense. [ABSTRACT FROM AUTHOR]

A linear map between matrix algebras corresponds to the Choi matrix in the tensor product of two matrix algebras whose definition depends on matrix units. Paulsen and Shultz [J. Math. Phys. 54, 072201 (2013)] considered the question if one can replace matrix units by another basis of matrix algebras in the definition of the Choi matrix to retain the correspondence between the complete positivity of maps and the positivity of Choi matrices and gave a sufficient condition on basis under which this is true. In this note, we provide necessary and sufficient conditions to see that the Paulsen–Shultz condition is also necessary. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of critical fractional Kirchhoff-type equations with singular potential. With a range of parameters, we propose several existence results. Our work extends the results of Li and Su [Z. Angew. Math. Phys. 66, 3147 (2015)]. [ABSTRACT FROM AUTHOR]

Whether the three dimensional isentropic compressible Navier–Stokes equations admit weak solutions for arbitrary initial data with adiabatic exponent γ > 1 remains a challenging problem. The only available results under γ > 1 were achieved by either assuming the initial data with small energy due to Hoff [J. Differ. Equations 120(1), 215–254 (1995)] or under the spherically symmetric condition by Jiang and Zhang [Commun. Math. Phys. 215, 559–581 (2001)] and Huang [J. Differ. Equations 262, 1341–1358 (2017)]. In this paper, we establish the existence of weak solutions with higher regularity of the three-dimensional periodic compressible isentropic Navier–Stokes equations in small time for the adiabatic exponent γ > 1 in the presence of vacuum. It can be viewed as a local version of Hoff's work and also extends the result of Desjardins [Commun. Partial Differ. Equations 22(5–6), 977–1008 (1997)] by removing the assumption of γ > 3. [ABSTRACT FROM AUTHOR]

This work studies an inhomogeneous Schrödinger coupled system in the mass-super-critical and energy-sub-critical regimes. In the focusing sign, a sharp dichotomy of global existence and scattering vs finite time blow-up of solutions is obtained using some variational methods, a sharp Gagliardo–Nirenberg-type inequality, and a new approach of Dodson and Murphy [Proc. Am. Math. Soc. 145(11), 4859–4867 (2017)]. In the defocusing sign, using a classical Morawetz estimate, the scattering of global solutions in the energy space is proved. [ABSTRACT FROM AUTHOR]

Kruskal [J. Math. Phys. 3, 806 (1962)] showed that each nearly periodic dynamical system admits a formal U(1) symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly invariant manifolds of each order, near which rapid oscillations are suppressed. We study the nonlinear normal stability of these slow manifolds for nearly periodic Hamiltonian systems on barely symplectic manifolds—manifolds equipped with closed, non-degenerate 2-forms that may be degenerate to leading order. In particular, we establish a sufficient condition for long-term normal stability based on second derivatives of the well-known adiabatic invariant. We use these results to investigate the problem of embedding guiding center dynamics of a magnetized charged particle as a slow manifold in a nearly periodic system. We prove that one previous embedding and two new embeddings enjoy long-term normal stability and thereby strengthen the theoretical justification for these models. [ABSTRACT FROM AUTHOR]

We study emergent behaviors of Cucker–Smale (CS) flocks on the hyperboloid H d in any dimensions. In a recent work [Ha et al., J. Math. Phys. 61(4), 042701 (2020)], a first-order aggregation model on the hyperboloid was proposed and sufficient conditions for emergent dynamics were proposed in terms of initial configuration and system parameters. In this paper, we are interested in the second-order modeling of CS flocks on the hyperboloid. For this, we derive our second-order model from the abstract CS model on complete and smooth Riemannian manifolds via explicit identifications of geodesic and parallel transport. Velocity alignment has been shown by combining general velocity alignment estimates for the abstract CS model on manifolds and verifications of the a priori estimate of the second derivative of the energy functional. For the two-dimensional case H 2 , similar to the recent result by Ahn, Ha, and Shim [Kinet. Relat. Models 14(2), 323–351 (2021)], asymptotic flocking admits only two types of asymptotic scenarios, either convergence to a rest state or a state lying on the same plane (coplanar state). We also provide several numerical simulations to illustrate an aforementioned dichotomy on the asymptotic dynamics of the hyperboloid CS model on H 2 . [ABSTRACT FROM AUTHOR]

Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [Ballesteros et al., Classical Quantum Gravity 25, 165005 (2008)], the Taub–NUT system [Ballesteros et al., SIGMA 7, 048 (2011)], and all the 17 superintegrable systems for the four types of Darboux spaces as determined by Kalnins et al. [J. Math. Phys. 44, 5811–5848 (2003)]. [ABSTRACT FROM AUTHOR]

We study the probability that all the eigenvalues of n × n Hermitian matrices, from the Laguerre unitary ensemble with the weight x γ e − 4 n x , x ∈ 0 , ∞ , γ > − 1 , lie in the interval [0, α]. By using previous results for finite n obtained by the ladder operator approach of orthogonal polynomials, we derive the large n asymptotics of the largest eigenvalue distribution function with α ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159, 151–174 (1994)] and later proved by Deift, Its, and Krasovsky [Commun. Math. Phys. 278, 643–678 (2008)]. Our conclusions are reduced to those of Deift et al. when γ = 0. It should be pointed out that our derivation is straightforward but not rigorous, and hence, the above results are stated as conjectures. [ABSTRACT FROM AUTHOR]

We study symmetry properties and the possibility of exact integration of Klein–Gordon equations in external electromagnetic fields on 3D de Sitter background dS3. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra extensions. Based on the well-known classification of the subalgebras of the algebra s o (1 , 3) , we classify all electromagnetic fields on dS3 for which the corresponding Klein–Gordon equations admit first-order symmetry algebras. Then, we select the integrable cases, and for each of them, we construct exact solutions using the noncommutative integration method developed by Shapovalov and Shirokov [Theor. Math. Phys. 104, 921–934 (1995)]. We also propose an original algebraic method for constructing the special local coordinates on de Sitter space dS3, in which basis vector fields for subalgebras of the Lie algebra s o (1 , 3) have the simplest form. [ABSTRACT FROM AUTHOR]

We present a fresh look at the methods introduced by Boccato, Brennecke, Cenatiempo, and Schlein [Commun. Math. Phys. 359(3), 975–1026 (2018); Acta Math. 222(2), 219–335 (2019); Commun. Math. Phys. 376, 1311 (2020)] concerning the trapped Bose gas and give a conceptually very simple and concise proof of Bose–Einstein condensation in the Gross–Pitaevskii limit for small interaction potentials. [ABSTRACT FROM AUTHOR]

We revisit the free field construction of the deformed W-superalgebras W q , t (s l (2 | 1)) by Ding and Feigin, Contemp. Math. 248, 83–108 (1998), where the basic W-current and screening currents have been found. In this paper, we introduce higher W-currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin diagrams for the superalgebra s l (2 | 1) , although the screening currents are not. This allows us to define W q , t (s l (2 | 1)) by generators and relations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Schrödinger–Poisson system −Δu + ϕu = λu + μ|u|2u and −Δϕ = u2 setting on a bounded domain Ω ⊂ R 3 with smooth boundary and λ , μ ∈ R being parameters. By using variational techniques in combination with the nodal Nehari manifold method, we show the existence of μ ̄ > 0 such that for all (λ , μ) ∈ (− ∞ , λ 1 ) × ( μ ̄ , + ∞) , the above system has one least energy sign-changing solution, where λ1 > 0 is the first eigenvalue of − Δ , H 0 1 (Ω) . The results of this paper are complementary to those in Alves and Souto [Z. Angew. Math. Phys. 65, 1153–1166 (2014)]. [ABSTRACT FROM AUTHOR]

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 the total nodal length of this ensemble relative to the rotation invariant model of random spherical harmonics. Jean Bourgain's research, and his enthusiastic approach to the nodal geometry of Laplace eigenfunctions, has made a crucial impact in the field and the current trends within. His works on the spectral correlations {Theorem 2.2 in the work of Krishnapur et al. [Ann. Math. 177(2), 699–737 (2013)]} and Bombieri and Bourgain [Int. Math. Res. Not. (IMRN) 11, 3343–3407 (2015)] have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavor, such as the torus or the square. Furthermore, Bourgain's work [J. Bourgain, Isr. J. Math. 201(2), 611–630 (2014)] on toral Laplace eigenfunctions, also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts. [ABSTRACT FROM AUTHOR]

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 continuity is shown to be a special case of q-MLR. Quantum Schnorr randomness is introduced. A quantum analog of the law of large numbers is shown to hold for quantum Schnorr random states. [ABSTRACT FROM AUTHOR]

The present work is dedicated to the global solutions to the incompressible Oldroyd-B model without damping on the stress tensor in R n (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 covers the recent result of Chen and Hao [J. Math. Fluid Mech. 21, 42 (2019)]. [ABSTRACT FROM AUTHOR]

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 that even minimally superintegrable systems in three-dimensional flat space have hidden symmetries that make them linearizable. [ABSTRACT FROM AUTHOR]

Reference [1] defined spin-weighted spherical functions as functions on the spin group Spin(3) SU(2) or the rotation group SO(3), and described an operator I K i analogous to the usual angular-momentum operator I L i . The equation defining the raising and lowering operators of I K i SB I z i sb contained a typo. Erratum: "How should spin-weighted spherical functions be defined?". [Extracted from the article]