We study the global regularity of classical solutions to the Cauchy problem of planar magnetohydrodynamics equations when the viscosity coefficients λ, μ, magnetic diffusion coefficient η and the heat conductivity coefficient κ depend on the specific volume v and the temperature θ which are proportional to h(v)θα for certain non-degenerate smooth function h. Under the condition of regularity criterion ∫0 +∞||b||L∞2ds < +∞, when α is small, we prove the existence of global classical solution with large initial data.

We consider the prescribing problem for symmetric function of Ricci tensor. Suppose a closed Einstein manifold (M, g) is not σ₂(Ric) singular. Let f ∈ C^∞(M) and it changes sign. We prove that there exists a metric g^* such that σ₂(Ric_g^*)=f. Then, as a corollary, we have an existence result for the prescribing problem for Einstein manifold with negative scalar curvature.

Let f:M^m → R₁^m+1 be an umbilic-free spacelike hypersurface. Four basic conformal invariants of M^m are the conformal metric g, the conformal 1-form C, the conformal second fundamental form B, and the conformal Blaschke tensor A. M^m is called the para-isotropic spacelike hypersurface, if A + μB=λg for some constant μ and smooth function λ. We not only constructed examples which are para-isotropic spacelike hypersurfaces but also classified completely all para-isotropic spacelike hypersurfaces under the conformal transformal group of R₁^m+1 in this paper.

We proved that the Lusternik-Schnirelmann category of any n-dimensional small cover is n and that the Lusternik-Schnirelmann category of any 2n-dimensional toric manifold is also n. Our result depends on the two well-known facts, one is the inequality cup(M) ≤ cat(M) ≤ dim(M)/r which leads us to compute cup(M). The other is explicit expression on the cohomology ring of small cover which makes the calculation of cup(M) easier. Furthermore, we generalized the relevant conclusions.

Let a_K(n) be the number of non-zero integral ideals in Z[i] with norm n, l ∈ Z⁺. In this paper, we establish an Erdös-Kac type theorem with weight a_K(n)^l in short intervals, and we get an asymptotic formula for the average behavior of a_K(n)^l in short intervals.

Let f(z) be a meromorphic function in the complex plane, let c be a nonzero finite complex number, and let a(z) be a small function with respect to f(z). It is studied that the relationship between the exponent of convergence of zeros of f(z)-a(z), f(z + c)-a(z), and Δ_cⁿ f(z)-a(z) (n ∈ N⁺) and the order of f(z). This improves some results in value distribution of meromorphic functions concerning derivatives and differences.

We investigate the canonical bundle and Kodaira dimension of nearly Kähler 6-manifolds. We prove that the canonical bundle of a strictly nearly Kähler 6-manifold is pseudoholomorphically trivial. Therefore, the Kodaira dimension is zero. As a corollary, we show the existence of non-integrable almost complex structure on CP³ whose Kodaira dimension is not -∞. We also construct explicit generating sections of the canonical bundle of homogeneous strictly nearly Kähler 6-manifolds and prove that the Hodge numbers h^1,0, h^2,0, h^2,3, h^1,3 of the homogeneous strictly nearly Kähler F³ and CP³ are all zeros.

In this paper, the numbers of nontrivial solutions to superlinear fourthorder impulsive elastic beam equations are obtained. We get two theorems via variational methods and corresponding two-critical-points theorems. Combining with the Newton-iterative method, two examples are presented to illustrate the value of the obtained theorems.

We consider the initial boundary value problem of the two-dimensional incompressible magnetohydrodynamic (MHD) equations with partial dissipation and magnetic diffusion. The global and unique strong solution of the model in a bounded domain is justified when the dissipation and magnetic diffusion coefficient in all directions are nonnegative. In addition, the global well-posedness of the system can be extended into the periodic boundary.

We give a novel characterization of BMO functions via weighted norm. As an application, we obtain the boundedness of commutators generated by bilinear Hardy operator and BMO functions on products of weighted Herz-Morrey spaces with variable exponents.

For i=1, 2, …, N, let λ_i be nonzero number, and D the unit open disk of complex plane C, φ_i is analytic self-maps of D. In this paper, the compactness of linear combinations of composition operators ∑_i=1^N λ_iC_φi on the weighted Bloch space is discussed.

In this paper, we construct a large family of pseudorandom binary lattices by using the quadratic character in finite fields, and study the cryptography properties:pseudorandom measure, collision and avalanche effect.

For any positive integer n, let φ(n) be the Euler function and S(n) be the Smarandache function. Bai and Liao[On the solutions for several classes of equations related to the smarandache function, Acta Math. Sin., Chin. Series, 2019, 62(2):247-254] proved that the equation φ(n)= ∑_d|n S(d) has only two solutions:n=2⁵ and n=3×2⁵. In this paper, we point out that both numbers are not solutions of the equation, and point out that this mistake was caused by that the authors misunderstood the Möbius inversion formula.

Metric and linear properties are significant properties of normed spaces, so the study of the relationship between linear operators and isometric operators has become an important research topic in the field of functional analysis. In this paper, we will study a special F-space, b_p⁽²⁾ space, and give the representation theorem for the onto isometric mapping on the unit sphere of the b_p⁽²⁾ space, then give a result about the isometric linear extension from unit sphere in the b_p⁽²⁾.

In this paper, the definitions and properties of torus and toral rank are generalized from modular Lie algebras to super-case. Taking advantage of the restricted envelopes, we obtain certain important results concerning toral rank of modular Lie superalgebras. As an application, the absolute toral ranks of the classical linear Lie superalgebra sl_m|n, the finite-dimensional restricted Lie superalgebras W(m,n,1) and S(m,n,1) of Cartan type and the toral rank of S(m,n,1) in W(m,n,1) are computed.

In this paper, some best proximity point theorems for generalized weakly contractive mappings which satisfy certain conditions by using three control functions in partially ordered Menger PM-spaces are obtained, and sufficient conditions to guarantee the uniqueness of the best proximity points are also given. Moreover, some corollaries are derived as consequences of the main results.

We first introduce the concept and the existence criterion of a random uniform exponential attractor for a non-autonomous random dynamical system. Then we prove the existence of a random uniform exponential attractor for FitzHugh-Nagumo system with additive noise and quasi-periodic external forces defined in Rⁿ.

In this paper, the notion of a generalized central α-Armendariz ring is introduced, and this paper also obtains some basic properties of such rings. At the same time, the relations between generalized central α-Armendariz rings and other rings are studied.

In this paper, the chaotic criteria of discrete dynamical systems is studied in complete metric spaces. It is showed that if f is continuous map from a complete metric space X into itself and has regular nondegenrate snap-back repellers or heteroclinic repellers, then there exists an invariant subset Λ of f such that (Λ, f) is topologically conjugate to the one-side symbolic system (Σ⁺₂, σ). Therefore, f is Devaney's chaos, distributional chaos, ω-chaos and has positive entropy. These results improve the related results.

We obtain a sharp second order subcritical Adams inequality in Lorentz Sobolev space W ²L^2,q(R⁴). Moreover, the lower and upper bounds asymptotically for the subcritical Adams functional is obtained. Our approach is based on the rearrangement free argument developed by Lam and Lu[A new approach to sharp MoserTrudinger and Adams type inequalities:a rearrangement-free argument, J. Diff. Equ., 2013, 255(3):298-325].

We mainly discuss K-g-frames and its duality. First, we explore the relationship between K-g-frames and g-frames. Then, we give some sufficient conditions under which the sum of K-g-frames and g-Bessel sequences with bounded linear operator or nonzero complex bounded sequence is still K-g-frames. In addition, we also give two special forms about the sum of K-g-frames. Finally, we research the duality of K-g-frames in closed subspace R(K), and the ways of constructing the K-g-frames by using the approximate duality.

There are two open problems on the global existence results of non-isentropic gas dynamics. One is whether the weak solutions exist globally with small initial data containing vacuum, the other is whether the global existence results hold with arbitrary large initial data. By introducing a scaling framework, we give the equivalence of the two problems above. For vanishing viscosity solutions, the positive answer to the first question naturally implies the positive answer to the second one. And this scaling framework can be applied to most systems of conservation laws with physical background.

We consider a new form of the localized Baskakov operators, and obtain some convergence properties of the new operators. We also obtain a new estimate for the kernel of the Baskakov operators by making use of one of the central limit theorems in probability theory.

We study the following fractional Schrödinger-Poisson system

where s ∈ (4/3, 1), t ∈ (0, 1), the continuous function f is superlinear at zero and subcritical at infinity and the exponent q ≥ 2_s^*=(3-2s)/6. We obtain a positive solution of the above problem for small λ > 0 via the variational method. Our main contribution is that we can deal with the supercritical case.

Some geometric conditions in terms of the characteristic of convexity, the normal structure coefficient, the James type constant and the García-Falset coefficient were considered in the paper, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some well known results and give affirmative answers to some open questions.

We consider the following Hausdorff operator H_φf(x)=∫_Rⁿφ(u₁,..., u_n) · f(u₁x₁,..., u_nx_n)du₁ · · · du_n, where φ can be considered as a distribution on Rⁿ. When n ≥ 2 and φ is a Schwartz function, we show that H_φ is bounded on H^p(Rⁿ) for some p ∈ (0, 1) if and only if φ ≡ 0. Furthermore, when n ≥ 2, if φ is just a continuous function and H_φ can be defined suitable, then we can also prove that H_φ is bounded on H^p(Rⁿ) for some p ∈ ((n+1)/n, 1) if and only if φ equals to a constant. These facts mean that H_φ is very complicated on H^p(Rⁿ) (n ≥ 2). Moreover, we establish a result of the boundedness of H_φ on L^p(Rⁿ), p > 1. The key idea used here is to reformulate H_φ as a convolution operator.

We describe a close relation between the p-fusion frames and the p-frames on Banach spaces. Using the analysis operators and the synthesis operators, we provide the equivalent descriptions of the p-fusion Bessel sequences, p-fusion frames and q-fusion Riesz bases.

Let R be a commutative ring. Then the small finitistic projective dimension of R is defined as fPD(R)=sup{pd_RM|M ∈ FPR}. In this paper, it is shown that if R is a connected strong Prüfer ring, then fPD(R) ≤ 1. It is also shown that if R is a strong Prüfer ring, and if M is a Q-torsion module with M ∈ FPR, then pd_RM ≤ 1.

In this paper, we study the following fractional Kirchhoff-type equation with critical growth ε^2sM(ε^2s-3 ∫∫_R³×_R³(|u(x)?u(y)|²)/(|x-y|^3+2s)dxdy)(-Δ)^su + V (x)u=λW(x)·f(u) + K(x)|u|^2_s*-2u, x ∈ R³, where M is a continuous and positive Kirchhoff function, λ > 0 is a parameter, (-Δ)^s is the fractional Laplace operator with 3/4 < s < 1, V (x), W(x) and K(x) are all positive potentials. Under some assumptions on potentials, we obtain the existence of a positive ground state solution for ε > 0 small and λ large. Moreover, we show that these ground state solutions concentrate at a special set characterized by potentials. Finally, we study the decay estimate of ground state solutions.

Let T be the singular integral operator whose kernel satisfies a variant of Lipschitz regularity introduced by Grubb and Moore, and T^* be the associated maximal singular integral operator. In this paper, by establishing the weak type endpoint estimates for the grand maximal operators corresponding to T and T^*, the author establishes some quantitative weighted bounds and weighted weak type endpoint bounds in terms of the A_p constants for the operators T and T^*.

This paper is devoted to studing Bergman spaces induced by regular-weight A_ω1,2^p(M) (1 < p < ∞) on annular and positive Toeplitz operators on these spaces. The dual spaces of Bergman spaces induced by regular-weight are characterized. We also obtain equivalent conditions for boundedness and compactness of positive Toeplitz operators between these regular-weighted Bergman spaces.

In this paper, the numerical radius of bounded block operator matrices on Hilbert space is studied. First, the generalized form of numerical radius inequalities of off-diagonal block operator matrix is studied, and taking advantage of the unitary similarity invariance of numerical radius and the generalized mixed Schwarz inequality, the inequalities of the numerical radius of sum of two bounded linear operators are considered. Then, numerical radius inequalities for 2×2 bounded block operator matrices are given. Finally, the conclusion is applied in the bounded infinite dimensional Hamiltonian operator and the inequalities of its numerical radius are obtained.

Image registration is fundamental to image processing. The vector field regularization model performs relatively well among a large number of registration methods. However, it still can't correspond to all interested regions across images correctly. Therefore, we hope to study the theory of the vector field regularization model to see whether there are some problems with the design of the model. Moreover, as there are two unknowns which are related by an initial value problem in the regularization model, it is novel in mathematics. The vector field regularization model takes the form min_v {α||v||_H² + ρ(T (y^v(τ)), S)}, where T is a template image, S is a reference image, y^v(τ):x ? y^v(τ;0, x) is a transformation determined by the solution y^v(s;0, x) of the initial value problem dy/ds=v(s, y), y(0)=x, ρ is a similarity functional, α> 0 is a regularization parameter and H is a Hilbert space. In this paper, we firstly show the vector field regularization model has stable solutions and then demonstrate its convergence. The above results can be obtained by the standard arguments of regularized problems together with the convergence relation of y^v(τ) and v. However, the requirements for ρ, S and T are relatively strong under the existing regularization theory. We give relatively weak conditions for ρ, S and T by taking full advantage of the good properties of y^v(τ). In addition, we verify that three commonly used similarity functionals in image registration satisfy the given conditions.

For a Banach sequence lattice λ and a Banach lattice X, let λ⊗_|π| X (resp. λ⊗_|ε| X) denote the positive projective (resp. injective) tensor product of λ and X. In the paper we prove that if λ is a σ-Levi space then λ⊗_|π| X (resp. λ⊗_|ε| X) is order or σ-order continuous if and only if both λ and X are order or σ-order continuous. We also prove that if λ is σ-order continuous then λ⊗_|π| X is a Levi or σ-Levi space if and only if both λ and X are Levi or σ-Levi spaces.

Let ?_n ∈ C (D), ?_j ∈ C (T) and K ≥ 1, where n ≥ 2 is an integer, j ∈ {1,..., n -1}. In this paper, we establish a Schwarz-Pick type inequality for the K-quasiconformal self-mapping f of the unit disk D satisfying the inhomogeneous polyharmonic equation Δⁿf=?_n with the associated Dirichlet boundary value condition:Δ^n-1f|_T=?_n-1,..., Δ¹f|_T=?₁ and f(0)=0. Furthermore, we prove that this result is asymptotically sharp in the sense that||?_j||_∞ → 0 (j=1,..., n) and K → 1⁺, where||?_n||_∞:=sup_z∈D|?_n(z)|and||?_j||_∞:=sup_z∈T|?_j(z)|(j=1, 2,..., n -1).

This paper investigates the initial value problem for a class of set differential equations in Fréchet space F. Based on that the set K_c(F) of all compact convex subsets of a Fréchet space F is considered as a projective limit of semilinear metric spaces K_c(Eⁱ), and the properties of projective limit, we introduce the notions of the Fréchet partial derivative, hyperconcave and hyperconvex of set-valued functions. By using the method of quasilinearization and comparison principle, we construct two monotone iterative sequences in K_c(F), and obtain the sequences of approximate solutions which converge uniformly and rapidly to the unique solution of the problem. The obtained results enrich and develop the theory of set-valued differential equations in Fréchet space F.

A non-increasing sequence π=(d₁,...,d_n of nonnegative integers is said to be graphic if it is realizable by a simple graph G on n vertices. A graphic sequence π=(d₁,...,d_n is said to be potentially ₃C_l-graphic if there is a realization of π containing cycles of every length r, 3 ≤ r ≤ l. It is well-known that if the nonincreasing degree sequence (d₁,..., d_l) of a graph G on l vertices satisfies the Pósa condition that d_{l +1-i} ≥ i + 1 for every i with 1 ≤ i < l/2, then G is either pancyclic or bipartite. In this paper, we obtain a Pósa-type condition of potentially ₃C_l-graphic sequences, that is, we prove that if l ≥ 5 is an integer, n ≥ l and π=(d₁,...,d_n is a graphic sequence with d_{l +1-i} ≥ i + 1 for every i with 1 ≤ i < l/2, then π is potentially ₃C_l-graphic. We show that this result is an asymptotic solution to a problem due to Li et al.[Adv. Math. (China), 2004, 33(3):273-283]. As an application, we also show that this result completely implies the value σ(C_l, n) for l ≥ 5 and n ≥ l due to Lai[J. Combin. Math. Combin. Comput., 2004, 49:57-64].

Let P be an isolated singularity of multiplicity 4 of a complex surface Y. It is well-known that there is a locally irreducible finite covering π:(Y, P) → (X, p) with π^-1(p)=P, and a Jung's resolution f:? → Y. Let W_p be the exceptional divisor of (π?f)^-1(p). We will prove that W_p has a unique decomposition into fundamental cycles W_p=2Z₁ or W_p=Σ_α=1^l Z_α satisfying some conditions. We will define a local index w_p for π at p and compute it by the above decomposition of W_p. In particular, we will show that (Y, P) is singular iff w_p ≥ 1. As another application of the decomposition of W_p, we also compute the number of blown-downs needed to get the minimal resolution from ?.

An optimal planar drawing of a graph is an embedding in the plane so that the number of crossings is as small as possible. The number of crossings in an optimal planar drawing of a graph G is the crossing number cr(G) of G. A graph is k-planar if it can be embedded in the planar so that each edge is crossed at most k times. Zhang et al. (2012) proved that the crossing number of any 1-planar graph on n vertices is at most n -2, and this upper bound is best possible. Czap, Harant and Hudák (2014) proved that the crossing number of any 2-planar graph on n vertices is at most 5(n-2). In this paper, we give a better upper bound for the crossing number of 2-planar graphs and show from the point of view of combinatorics that K_n is 2-planar if and only if n ≤ 7 (surprisedly, this was an open problem until 2019, in when Angelini solved it with computer assistance).

We investigate the nonlinear differential-difference equations of form f(z)ⁿ+ L(z, f)=q(z)e^p(z), where n ≥ 2, L(z, f)(≢ 0) is a linear differential-difference polynomial in f(z), with small functions as its coefficients, p(z) and q(z) are non-vanishing polynomials. We first obtain that n=2 and f(z) satisfies λ(f)=σ(f)=deg p(z) if the equation possesses a transcendental entire solution of hyper order σ₂(f)< 1. Furthermore, the exact form of the entire solutions of the equation is also obtained.