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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]

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]

The Gauss–Bonnet–Chern mass was defined and studied by Ge, Wang, and Wu [Adv. Math. 266, 84–119 (2014)]. In this paper, we consider the evolution of Gauss–Bonnet–Chern mass along the Ricci flow and the Yamabe flow. [ABSTRACT FROM AUTHOR]

Continuing the work of Monnier [Israel J. Math. 129, 189–207 (2002)], we determine the Gerstenhaber algebra structure over the Poisson cohomology groups for a large class of Poisson structures with isolated singularities over the plane. It reveals that there exists a GAGA-type phenomenon. We give an explicit description via generators and relations for the case of simple singularities. [ABSTRACT FROM AUTHOR]

We study moments of the logarithmic derivative of characteristic polynomials of orthogonal and symplectic random matrices. In particular, we compute the asymptotics for large matrix size, N, of these moments evaluated at points that are approaching 1. This follows the work of Bailey et al. [J. Math. Phys. 60(8), 083509 (2019)] where they computed these asymptotics in the case of unitary random matrices. [ABSTRACT FROM AUTHOR]

The α–z Rényi relative entropies are a two-parameter family of Rényi relative entropies that are quantum generalizations of the classical α-Rényi relative entropies. In the work [Adv. Math. 365, 107053 (2020)], we decided the full range of (α, z) for which the data processing inequality (DPI) is valid. In this paper, we give algebraic conditions for the equality in DPI. For the full range of parameters (α, z), we give necessary conditions and sufficient conditions. For most parameters, we give equivalent conditions. This generalizes and strengthens the results of Leditzky et al. [Lett. Math. Phys. 107, 61–80 (2017)]. [ABSTRACT FROM AUTHOR]

We analyze some polynomial solutions to the Schrödinger equation with a central-field potential derived recently by means of the so-called extended Nikiforov–Uvarov method. We show that this approach does not produce the spectrum of the non-relativistic quantum-mechanical model but merely some eigenvalues for particular values of the model parameters. In addition, the approach just mentioned appears to overlook the restrictions that reveal the problem as conditionally solvable. [ABSTRACT FROM AUTHOR]

Quasi-symmetry of a steady magnetic field means integrability of first-order guiding-center motion by a spatial symmetry. Here, we derive many restrictions on the possibilities for a quasi-symmetry. We also derive an analog of the Grad–Shafranov equation for the flux function in a quasi-symmetric magnetohydrostatic field. [ABSTRACT FROM AUTHOR]

We prove sharp lower bounds on the spectral gap of 1-dimensional Schrödinger operators with Robin boundary conditions for each value of the Robin parameter. In particular, our lower bounds apply to single-well potentials with a centered transition point. This result extends the work of Cheng et al. [Comput. Math. Appl. 60(9), 2556–2563 (2010)] and Horváth [Proc. Am. Math. Soc. 131(4), 1215–1224 (2002)] in the Neumann and Dirichlet endpoint cases to the interpolating regime. We also build on the recent work by Andrews, Clutterbuck, and Hauer (arXiv:2002.06900) in the case of convex and symmetric single-well potentials. In particular, we show that the spectral gap is an increasing function of the Robin parameter for symmetric potentials. [ABSTRACT FROM AUTHOR]

We consider the problem of variable separation for the classical integrable Hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Poisson brackets of Freidel and Maillet [Phys. Lett. B 262(2-3), 278 (1991)] with spectral-parameter dependent a-b-c-d tensors. We formulate, in terms of the corresponding a-b-c-d tensors, a sufficient condition that guarantees that the separating functions A(u), B(u) of Sklyanin [Commun. Math. Phys. 150(1), 181 (1992)], Scott [e-print arXiv:hep-th 940303 (1994)], and Gekhtman [Commun. Math. Phys. 167, 593 (1995)] produce canonical coordinates. We consider an important subcase of a-b-c-d algebras, namely, the case of classical reflection equation algebras [E. Sklyanin, J. Phys. A: Math. Gen. 21(10), 2357 (1988)], and formulate, in terms of the corresponding r-s-matrices, the analogous sufficient condition that guarantees the canonicity of the constructed coordinates. For the case s = 0, we recover our previous results on the variable separation for quadratic Sklyanin brackets [B. Dubrovin and T. Skrypnyk, J. Math. Phys. 60, 093506 (2019)]. We consider an example of gl(n) ⊗ gl(n)-valued trigonometric a-b-c-d tensors that satisfy the considered condition and find a class of Lax matrices for them for which the obtained set of canonical coordinates is complete. [ABSTRACT FROM AUTHOR]

Newtonian and Schrödinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrödinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrödinger dynamics into a unified geometric framework is rigid in the sense that the Schrödinger dynamics is a unique extension of the Newtonian one. Under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. The mathematics of the discovered relationship between the classical and the quantum is reviewed and investigated here in detail and applied to the process of measurement of spin and position observables. [ABSTRACT FROM AUTHOR]

In this paper, a doubly nonlinear parabolic problem involving nonstandard growth conditions under homogeneous Dirichlet boundary conditions is studied. To be a little precise, first, some energy estimates and inequalities are exploited to obtain energy decay estimates of the solution. Then, it is proved that the rest field u(x, t) = 0 is asymptotically stable in terms of natural energy associated with the solution. Second, we give two new blow-up criteria that one of them is obtained to remove the exponent restriction by using an extended form of the concavity method and the other one is a blow-up result of weak solutions with arbitrarily high initial energy. Moreover, an extinction result is also showed by establishing a new auxiliary lemma. The present results of this paper complement and improve a recent paper investigated by Liu et al. [J. Math. Phys. 59, 121504 (2018)]. [ABSTRACT FROM AUTHOR]

The initial-boundary value problem (ibvp) for the mth order Korteweg–de Vries (KdVm) equation on the half-line is studied by extending a novel approach recently developed for the well-posedness of KdV on the half-line, which is based on the solution formula produced via the Fokas unified transform method for the associated forced linear ibvp. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space–time regularity of the Cauchy problem of the linear KdVm equation give an iteration map for the ibvp, which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV equation on the line by Kenig, Ponce, and Vega [J. Am. Math. Soc. 4, 323–347 (1991)]. [ABSTRACT FROM AUTHOR]

We prove decomposition rules for quantum Rényi mutual information, generalizing the relation I(A : B) = H(A) − H(A|B) to inequalities between Rényi mutual information and Rényi entropy of different orders. The proof uses Beigi's generalization of Reisz–Thorin interpolation to operator norms [J. Math. Phys. 54(12), 122202 (2013)] and a variation of the argument developed by Dupuis [J. Math. Phys. 56, 022203 (2015)], which was used to show chain rules for conditional Rényi entropies. The resulting decomposition rule is then applied to establish an information exclusion relation for Rényi mutual information, generalizing the original relation by Hall [Phys. Rev. Lett. 74, 3307 (1995)]. [ABSTRACT FROM AUTHOR]

We investigate the existence of nontrivial solutions for a class of Schrödinger–Poisson systems with steep potential well V and the nonlinearity a(x)|u|p−2u(2 < p < 4) in R 3 . Such a problem cannot be studied by applying variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold, because Palais–Smale sequences may not be bounded. In this paper, by developing a novel constraint method, we find one positive solution under suitable assumptions when 2 < p < 4. In particular, one positive ground state solution is found when 1 + 73 3 ≤ p < 4. An interesting point is that unlike some methods that only work for the case of either 2 < p ≤ 3 or 3 < p < 4 {see the work by Jiang and Zhou [J. Differ. Equations 251, 582–608 (2011)]; Sun et al. [Z. Angew. Math. Phys. 68, 73 (2017)]; and Zhao et al. [J. Differ. Equations 255, 1–23 (2013)]}, our method is applicable to all 2 < p < 4. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and instability of normalized standing waves for the fractional Schrödinger equation i∂tψ = (−Δ)sψ − f(ψ), where 0 < s < 1, f(ψ) = |ψ|pψ with 4 s N < p < 4 s N − 2 s or f(ψ) = (|x|−γ*|ψ|2)ψ with 2s < γ < min{N, 4s}. To do this, we consider normalized solutions of the associated stationary equation (−Δ)su + ωu − f(u) = 0. By constructing a suitable submanifold of a L2-sphere and considering an equivalent minimizing problem, we prove the existence of normalized solutions. In particular, based on this equivalent minimizing problem, we can easily obtain the sharp threshold of global existence and blow-up for the time-dependent equation. Moreover, we can show that all normalized ground state standing waves are strongly unstable by blow-up. Our results are a complementary to the results of Peng and Shi [J. Math. Phys. 59, 011508 (2018)] and Zhang and Zhu [J. Dyn. Differ. Equations 29, 1017–1030 (2017)], where the existence and stability of normalized standing waves have been studied in the L2-subcritical case. [ABSTRACT FROM AUTHOR]

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. We propose a universal test function method that works for many nonlinear hyperbolic systems arising from physical applications. We first present a simpler proof of the main result in the work of Sideris [Commun. Math. Phys. 101(4), 475–485 (1985)]: the global classical solution is non-existent for compressible Euler equations even for some small initial data. Then, we apply this approach to nonlinear magnetohydrodynamics in two space dimensions. Finally, we consider second order quasilinear hyperbolic systems with quadratic nonlinearity arising from elastodynamics of isotropic hyperelastic materials by ignoring the cubic and higher order terms. Under some restriction on the coefficients of the nonlinear terms that imply genuine nonlinearity, we show that the classical solutions to these equations can still blow up in finite time even if the initial data are small enough. [ABSTRACT FROM AUTHOR]

Stochastic Schrödinger equations are a special type of stochastic evolution equations in complex Hilbert spaces, which arise in the study of open quantum systems. Quantum Bernoulli noises refer to annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. In this paper, we investigate a linear stochastic Schrödinger equation of exclusion type in terms of quantum Bernoulli noises. Among others, we prove the well-posedness of the equation, illustrate the results with examples, and discuss the consequences. Our main work extends that of Chen and Wang [J. Math. Phys. 58(5), 053510 (2017)]. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with non-isentropic Navier–Stokes equations governing compressible viscous and heat-conductive gases. We formulate an additional regularity assumption of low frequencies in R d (d ≥ 3) and then establish the sharp time-weighted inequality, which allows us to get the time-decay estimates of global strong solutions in the Lp critical Besov spaces. Precisely, it is shown that if the initial data belong to some Besov space B ̇ 2 , ∞ − s 1 with s 1 ∈ (1 − d 2 , s 0 ] ( s 0 ≜ 2 d p − d 2 ) , then the Lp norm of the critical global solutions admits the time decay t − s 1 2 − d 2 ( 1 2 − 1 p ) (in particular, t − d 2 p if s1 = s0), which coincides with that of the heat kernel in the Lp framework. In comparison with the work of Danchin and Xu [J. Math. Fluid Mech. 20, 1641–1665 (2018)], the low-frequency assumption is improved such that the regularity s1 belongs to the whole range (1 − d 2 , s 0 ]. [ABSTRACT FROM AUTHOR]

We study the vanishing viscosity limit of a non-strictly hyperbolic system of balance laws known as the 1D Saint-Venant model whose solution admits δ-waves. The balance term depends on both the variables x and t and is unbounded in the space variable x unlike the work of Olenik [Usp Math. Nauk. 12(3), 3–73 (1957) (in Russian)]. We also studied the large time behavior when the viscosity parameter is positive. When the balance term depends only on the time variable, we obtained an explicit formula for the solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and asymptotic properties of solutions to the fractional Schrödinger equation (− Δ) σ u = λ u + | u | q − 2 u + μ I α * | u | p | u | p − 2 u under the normalized constraint ∫ R N u 2 = a 2 , where N ≥ 2, σ ∈ (0, 1), α ∈ (0, N), q ∈ (2 + 4 σ N , 2 N N − 2 σ ] , p ∈ [ 1 + 2 σ + α N , N + α N − 2 σ ) , a, μ > 0, Iα(x) = |x|α−N, and λ ∈ R appears as a Lagrange multiplier. By using a refined version of the min-max principle, we show that the above problem admits a mountain pass type solution ûμ for some λ ̂ < 0 under suitable assumptions on the related parameters. In particular, we can prove that ûμ is a ground state if p ≤ q 2 + α N . Furthermore, we give some asymptotic properties of the solutions. We mainly extend the results in the work of Bhattarai [J. Differ. Equations 263, 3197–3229 (2017)] and Feng et al. [J. Math. Phys. 60, 1–12(2019)] concerning the above problem from the L2-subcritical setting to L2-critical and L2-supercritical settings with respect to p, involving the Sobolev critical case q = 2 N N − 2 σ especially. [ABSTRACT FROM AUTHOR]

Reciprocal transformations are introduced for two super Camassa–Holm (CH) equations. Under these transformations and appropriate changes of dependent variables, the super CH equation, proposed by Geng et al. [Stud. Appl. Math. 130, 1 (2013)], is converted to a negative member of the super Korteweg–de Vries (KdV) hierarchy studied by Geng and Wu in 2010 [Appl. Math. Lett. 23, 716 (2010)], while the other super CH equation, due to Zhang and Zuo [J. Math. Phys. 52, 073503 (2011)], is related to a new super KdV hierarchy. In the latter case, algebraic properties of this new super KdV hierarchy are established, including Hamiltonian operators, a recursion operator, and conserved quantities. [ABSTRACT FROM AUTHOR]

In this article, we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive Kirkwood–Salsburg-type equations for the PTCF and write the solutions of these equations as a sum of contributions labeled by certain forest graphs, the connected components of which are tree graphs. We generalize the method developed by Minlos and Poghosyan [Theor. Math. Phys. 31, 408 (1977)] in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF, which were obtained earlier for lattice spin systems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the compressible primitive equations (CPEs) with density-dependent viscosity for large initial data. The CPE model can be derived from the 3D compressible and anisotropic Navier–Stokes equations by hydrostatic approximation. Motivated by the work of Vasseur and Yu [SIAM J. Math. Anal. 48, 1489–1511 (2016); Invent. Math. 206, 935–974 (2016)], in which the global existence of weak solutions to the compressible Navier–Stokes equations with degenerate viscosity was obtained, we construct approximate solutions and prove the global existence of weak solutions to the CPE in this paper. In our proof, we first present the vertical velocity as a function of density and horizontal velocity, which plays a role in using the Faedo–Galerkin method to obtain the global existence of the approximate solutions. Then, we obtain the key estimates of lower bound of the density, the Bresch–Desjardins entropy on the approximate solutions. Finally, we apply compactness arguments to obtain global existence of weak solutions by vanishing the parameters in our approximate system step-by-step. [ABSTRACT FROM AUTHOR]

The main goal is to understand the quantitative spatial decay of the solution to the linearized Boltzmann equation for hard potentials with cutoff. We obtain the quantitative space-time behavior of the linearized Boltzmann equation under some slow velocity decay assumption but without any regularity assumption on the initial data. This result extends the classical results of Liu and Yu [Commun. Pure Appl. Math. 57, 1543–1608 (2004); Bull. Inst. Math. Acad. Sin. 1, 1–78 (2006); and ibid.6, 151–243 (2011)] from hard sphere to hard potential under the same assumption on the initial condition [compactly support in the space variable and polynomial decay in the velocity variable with power β = (3/2)+]. [ABSTRACT FROM AUTHOR]

We derive two equivalent analytical expressions for an lth partial-wave scattering length al for central potentials with long-range tails of the form V (r) = − ℏ 2 2 m B r n − 4 (r n − 2 + R n − 2) 2 − ℏ 2 2 m C r 2 (r n − 2 + R n − 2) , (r ⩾ rs, R > 0). For C = 0, this family of potentials reduces to the Lenz potentials discussed in a similar context in our earlier works [R. Szmytkowski, Acta Phys. Pol. A 79, 613 (1991); J. Phys. A: Math. Gen. 28, 7333 (1995)]. The formulas for al that we provide in this paper depend on the parameters B, C, and R characterizing the tail of the potential, on the core radius rs, as well as on the short-range scattering length als, the latter being due to the core part of the potential. The procedure, which may be viewed as an analytical extrapolation from als to al, is relied on the fact that the general solution to the zero-energy radial Schrödinger equation with the potential given above may be expressed analytically in terms of the generalized associated Legendre functions. [ABSTRACT FROM AUTHOR]

We prove that in finite dimensions, a Parity-Time (PT)-symmetric Hamiltonian is necessarily pseudo-Hermitian regardless of whether it is diagonalizable or not. This result is different from Mostafazadeh's result [J. Math. Phys. 43, 205−214 (2002)], which requires the Hamiltonian to be diagonalizable. PT-symmetry breaking often occurs at exceptional points where the Hamiltonian is not diagonalizable. Our result implies that PT-symmetry breaking is equivalent to the onset of instabilities of pseudo-Hermitian systems, which was systematically studied by Krein et al. [Dokl. Akad. Nauk SSSR N.S. 73, 445 (1950)]. In particular, we show that the mechanism of PT-symmetry breaking is the resonance between two eigenmodes with opposite signs of actions. [ABSTRACT FROM AUTHOR]

We show how to establish a correspondence between the recently introduced formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters (without reference to a potential function) and the convention formulation. In this work, we obtain the potential function for the newly introduced quantum system associated with the Wilson and Racah polynomials [A. D. Alhaidari and T. J. Taiwo, J. Math. Phys. 58, 022101 (2017)] using any of the proposed procedures in the work of Alhaidari [Commun. Theor. Phys. 68, 711 (2017)]. To achieve that, we use the matrix elements of the potential function and the basis in which these were calculated. [ABSTRACT FROM AUTHOR]

We defined the notion of the quantum double inclusion [S. De, J. Math. Phys. 60, 071701 (2019)] associated with a finite-index and finite-depth subfactor, which is closely related to that of Ocneanu's asymptotic inclusion, and studied the quantum double inclusion associated with the Kac algebra subfactor RH ⊂ R, where H is a finite-dimensional Kac algebra acting outerly on the hyperfinite II1 factor R and RH denotes the fixed-point subalgebra. In this article, we analyze quantum double inclusions associated with the family of Kac algebra subfactors given by { R H ⊂ R ⋊ H ⋊ H * ⋊ ⋯ ︸ m times : m ≥ 1 }. For each m > 2, we construct a model N m ⊂ M for the quantum double inclusion of R H ⊂ R ⋊ H ⋊ H * ⋊ ⋯ ︸ m − 2 times , where N m = ((⋯ ⋊ H − 2 ⋊ H − 1 ) ⊗ ( H m ⋊ H m + 1 ⋊ ⋯ )) ′ ′ , M = (⋯ ⋊ H − 1 ⋊ H 0 ⋊ H 1 ⋊ ⋯ ) ′ ′ , and for any integer i, the notation Hi stands for H or H* according as i is odd or even. In this article, we give an explicit description of the subfactor planar algebra associated with N m ⊂ M (m > 2) which turns out to be a planar subalgebra of P * (m) ( H m ) (the adjoint of the m-cabling of the planar algebra of Hm). We then show that for each m > 2, the depth of N m ⊂ M is always two. Observing that N m ⊂ M is reducible for all m > 2, we study in great detail the weak Kac algebra structure of the relative commutant ( N m ) ′ ∩ M 2 . [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the relativistic string equations in de Sitter spacetime. Since the background spacetime is nonflat, the governing system of equations is semilinear under the usual isothermal coordinates, contrary to the linear wave equations in the Minkowski spacetime. Based on the null condition enjoyed by the system and the method of [Luli et al., Adv. Math 329, 174–188 (2018)], we succeeded in proving the stability of the traveling wave solution to the relativistic string equations. We note that the traveling wave solution should be small in the sense of weighted energy. [ABSTRACT FROM AUTHOR]

In this paper, we consider a 5D quadratic nonlinear Schrödinger system: iut + Δu = −2vū, ivt + kΔv = −u2, and (t, x) ∈ R × R5, k > 0. If k = 1 2 , we say the system satisfies the mass-resonance condition. Using the argument of Dodson and Murphy [Math. Res. Lett. 25, 1805 (2018)], we prove that the solution (u·v) scatters if initial data (u0, v0) ∈ H1(R5) × H1(R5) satisfy E(u0, v0)M(u0, v0) < E(Q1, Q2)M(Q1, Q2) and K(u0, v0)M(u0, v0) < K(Q1, Q2)M(Q1, Q2) and k lies in some small neighborhood of 1 2 . Under the condition of radial initial data, the range of k can be extended to k > 0 based on the idea of Dodson and Murphy [Proc. Am. Math. Soc. 145, 4859 (2017)]. [ABSTRACT FROM AUTHOR]

In this article, we extend Eardley and Moncrief's L∞ estimates [Commun. Math. Phys. 83(2), 193–212 (1982)] for the conformally invariant Yang–Mills–Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localize their estimates and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global L∞ estimates on the cylinder and extend Choquet-Bruhat and Christodoulou's [Ann. Sci. Éc. Norm. Supér. 14(4), 481–506 (1981)] small data well-posedness result to large data. Finally, by employing an inverse conformal transformation, we deduce exponential decay rates for Yang–Mills–Higgs fields on de Sitter space and inverse polynomial decay rates on Minkowski space. [ABSTRACT FROM AUTHOR]

We study the compressible quantum Navier-Stokes (QNS) equations with degenerate viscosity in the three dimensional periodic domains. On the one hand, we consider QNS with additional damping terms. Motivated by the recent works of Li and Xin [e-print arXiv:1504.06826] and P. Antonelli and S. Spirito [Arch. Ration. Mech. Anal. 225, 1161–1199 (2017)], we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the BD entropy estimate. Using this system, we obtain the global existence of weak solutions to the compressible QNS equations with damping terms for large initial data. Moreover, we obtain some new a priori estimates, which can avoid using the assumption that the gradient of the velocity is a well-defined function, which was indeed used directly in the work of Vasseur and Yu [SIAM J. Math. Anal. 48, 1489–1511 (2016); Invent. Math. 206, 935–974 (2016)]. On the other hand, in the absence of damping terms, we also prove the global existence of weak solutions to the compressible QNS equations without the lower bound assumption on the dispersive coefficient, which improves the previous result of P. Antonelli and S. Spirito [Arch. Ration. Mech. Anal. 225, 1161–1199 (2017)]. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a blow-up criterion for classical solutions to the two-dimensional Prandtl equations under a monotonicity assumption in weighted Sobolev spaces. Our proof is based on nonlinear energy estimates inspired by Masmoudi and Wong [Commun. Pure Appl. Math. 68, 1683–1741 (2015)] and maximum principles for parabolic equations. [ABSTRACT FROM AUTHOR]

Abstract interior-boundary conditions allow for the direct description of the domain and the action of Hamiltonians for a certain class of ultraviolet-divergent models in quantum field theory. This method, which comes without the need for renormalization, was recently applied to the nonrelativistic Nelson model. Here, we demonstrate the flexibility of the novel approach by applying it to two relativistic variants of Nelson's model. The existence of Hamiltonians for these models was shown using renormalization techniques by Eckmann [Commun. Math. Phys. 18, 247 (1970); J. Math. Phys. 15, 190 (1974)]. However, many important properties, such as the domain and the action of the Hamiltonian, were not known. We construct the Hamiltonians for such models via abstract boundary conditions, describing their action explicitly for the first time. In addition, we obtain a detailed characterization of their domain. [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]

We analyze the application of the "tridiagonal representation approach" (TRA) to the Schrödinger equation for some simple, exactly solvable, quantum-mechanical models. In the case of the Kratzer–Fues potential, the expression for the energy does not appear to be correct. In addition, the well known Frobenius method, which resembles the TRA, is far simpler, clearer, and more elegant, in addition to giving the correct result. [ABSTRACT FROM AUTHOR]

We comment on the validity of Noether's theorem and on the conclusions of Anerot et al. [J. Math. Phys. 61(11), 113502 (2020)]. [ABSTRACT FROM AUTHOR]

In this paper, we study the following quasilinear Schrödinger equations − Δ u + V (x) u + κ 2 Δ ( u 2 ) u = f (u) + μ | u | 2 * − 2 u , x ∈ R N , where N ≥ 3, κ > 0, μ ≥ 0, and V : R N → R satisfy suitable assumptions. First, by using a change of variable and some new skills, we obtain the ground states for this problem with subcritical growth via the Pohozaev manifold. Second, we establish the existence of ground state solutions with critical growth via L∞estimates, which use the method developed by Brezis and Nirenberg [Commun. Pure Appl. Math. 36, 437–477 (1983)] and Jeanjean [Proc. R. Soc. Edinburgh, Sect A. 129, 787–809 (1999)]. Moreover, we give the nonexistence of positive solutions for this problem, where the nonlinear term allow general asymptotically linear growth. Our results extend and supplement the results obtained by Severo et al. [J. Differ. Equations 263, 3550–3580 (2017)], Xu and Chen [J. Differ. Equations 265, 4417–4441 (2018)], and Lehrer and Maia [J. Funct. Anal. 266, 213–246 (2014)] and some other related literature. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global well-posedness of modified 2D nonhomogeneous Navier-Stokes equations. Our results improve the results of Zhang [Int. Math. Res. Not. 2008(1), rnn098] to some extent. [ABSTRACT FROM AUTHOR]

The work of Kolmogorov, Arnold, and Moser appeared just before the renormalization group approach to statistical mechanics was proposed by Wilson [Phys. Rev. B 4, 3174–3183 (1971)]: it can be classified as a multiscale approach which also appeared in works on the convergence of Fourier's series [L. Carleson, Acta Math. 116, 135–157 (1966) and C. Fefferman, Ann. Math. 98, 551–571 (1973)] or construction of Euclidean quantum fields [E. Nelson, J. Math. Phys. 5, 1190 (1964)] or the scaling analysis of the short scale behavior of Navier-Stokes fluids [L. Caffarelli, R. Kohn, and L. Nirenberg, Commun. Pure Appl. Math. 35, 771–831 (1982)] to name a few, which resulted in numerous problems. In this review, the KAM theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a "trivial" fixed point. [ABSTRACT FROM AUTHOR]

In this paper, we study the orbital stability of standing waves for the fractional Schrödinger equations with combined power-type and Choquard-type nonlinearities. By using variational methods, when one nonlinearity is focusing and L2-critical, the other is defocusing and L2-supercritical, we prove that there exist the orbitally stable standing waves. We extend the results of Bhattarai [J. Differ. Equations 263, 3197–3229 (2017)] and Feng-Zhang [Comput. Math. Appl. 75, 2499–2507 (2018)] to the L2-critical and L2-supercritical nonlinearities. [ABSTRACT FROM AUTHOR]

We point out a misleading treatment in a recent paper published in this journal [M. Eshgi and H. Mehraban, J. Math. Phys. 57, 082105 (2016)] concerning solutions for the two-dimensional Dirac-Weyl equation with a q-deformed pseudoscalar magnetic barrier. The authors misunderstood the full meaning of the potential and made erroneous calculations, and this fact jeopardizes the main results in this system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mth order Cauchy-Euler's differential equation involving a para-Grassmann variable of order p. In the Grassmann case (i.e., p = 1), we determine the solution for arbitrary order m. In the case of arbitrary order p, we give a solution for the cases m = 1, 2. [ABSTRACT FROM AUTHOR]