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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]

Making meaning with math in physics requires blending physical conceptual knowledge with mathematical symbology. Students in introductory physics classes often struggle with this, but it is an essential component of learning how to think with math. Teaching the dimensionality of measured quantities and dimensional analysis (DA) is a valuable first step in helping them learn to appreciate this difference. In this paper I discuss some of the issues associated with learning dimensional analysis and show some ways we can modify our instruction to help. This paper is one of a series on how to help students develop the scientific thinking skills required for learning to use math in science. [ABSTRACT FROM AUTHOR]

The key difference between math as math and math in science is that in science we blend our physical knowledge with our knowledge of math. This blending changes the way we put meaning to math and even the way we interpret mathematical equations. Learning to think about physics with math instead of just calculating involves a number of general scientific thinking skills that are often taken for granted (and rarely taught) in physics classes. In this paper, I give an overview of my analysis of these additional skills. I propose specific tools for helping students develop these skills in subsequent papers. [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]

Music is an important part of education and most students are ardent fans. However, the students are not necessarily aware of the artistic and technical details it takes to create what they hear. These details are what necessitate music and technology coming together. Part of science, technology, engineering, and math (STEM) education should be to help students appreciate the practical applications that come from each of the entities in STEM, namely, science, technology, engineering, and math. Music provides the connection of these elements with a topic that interests the students and motivates their learning. This paper outlines ways for educators to integrate music with STEM topics and put the "A" in "STEAM" (science, technology, engineering, arts, and math). [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]

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]

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]

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]

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]

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]

Learning to create, use, and evaluate models is a central element of becoming a scientist. In physics, we often begin an analysis of a complicated system with highly simplified or toy models. In introductory physics classes, we tend to use them without comment or motivation. Some students infer that physics is irrelevant to their understanding of the real world and are discouraged from making the cognitive blend of physics concepts with math symbology essential for making sense of physics. In this paper, I discuss the often-hidden barriers that make it difficult for our students to accept and understand the value of toy models and suggest instructional approaches that can help. [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]

An important step in learning to use math in science is learning to see symbolic equations not just as calculational tools, but as ways of expressing fundamental relationships among physical quantities, of coding conceptual information, and of organizing physics knowledge structures. In this paper, I propose "anchor equations" as a construct to support teaching and learning in introductory physics. I define anchor equation, provide examples, and suggest ways anchor equations can be used in instruction to support the development of students' mathematical sense-making. [ABSTRACT FROM AUTHOR]

The present paper studies two various models with two different types: the nonlinear Schrödinger equation with power-law nonlinearity and the (3 + 1)-dimensional nonlinear Schrödinger equation. We perform the modified ( G ′ G ) -expansion method to find some exact solutions and to construct various types of solitary wave phenomena for each equation. The received aspects contribute to the firm mathematical foundation and might be essential to the soliton waves. As a result, we obtain all the solutions from Wazwaz [Math. Comput. Modell. 43, 178–184 (2016)] and also obtain some new soliton solutions. [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]

In this paper, we present a study on a network-based susceptible-infected-recovered (SIR) epidemic model with a saturated treatment function. It is well known that treatment can have a specific effect on the spread of epidemics, and due to the limited resources of treatment, the number of patients during severe disease outbreaks who need to be treated may exceed the treatment capacity. Consequently, the number of patients who receive treatment will reach a saturation level. Thus, we incorporated a saturated treatment function into the model to characterize such a phenomenon. The dynamics of the present model is discussed in this paper. We first obtained a threshold value R 0 , which determines the stability of a disease-free equilibrium. Furthermore, we investigated the bifurcation behavior at R 0 = 1. More specifically, we derived a condition that determines the direction of bifurcation at R 0 = 1. If the direction is backward, then a stable disease-free equilibrium concurrently exists with a stable endemic equilibrium even though R 0 < 1. Therefore, in this case, R 0 < 1 is not sufficient to eradicate the disease from the population. However, if the direction is forward, we find that for a range of parameters, multiple equilibria could exist to the left and right of R 0 = 1. In this case, the initial infectious invasion must be controlled to a lower level so that the disease dies out or approaches a lower endemic steady state. [ABSTRACT FROM AUTHOR]

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]

We respond to the comment on 'Classification of Lie point symmetries for quadratic Liénard type equation ẍ + f(x)ẋ2 + g(x) = 0' [J. Math. Phys. 61, 044101 (2013)] by Iacono regarding linearizability and isochronicity. We assert here that the condition for linearization of the equation ẍ + f(x)ẋ2 + g(x) = 0 given by us in our paper is correct with the condition g 1 = ω 0 2 > 0. We present the explicit form of local and nonlocal transformations that transform the quadratic Liénard equation x ̈ + F + 1 1 − x x ̇ 2 + x (1 − x) (1 + D x) = 0 into the harmonic oscillator equation for the four cases mentioned in the comment and confirm the statements given in our paper are all valid. [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]

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]

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]

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]

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]

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]

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 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]

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]