This paper is a summary of the research on the characterizations of central function spaces by the author and his collaborators in the past ten years. More precisely, the author gives some characterizations of central Campanato spaces via the boundedness and compactness of commutators of Hardy operator.

We first prove that if a is both left (b; c)-invertible and left (c; b)- invertible, then a is both (b; c)-invertible and (c; b)-invertible in a *-monoid, which generalizes the recent result about the inverse along an element by L. Wang and D. Mosić [Linear Multilinear Algebra, Doi.org/10.1080/03081087. 2019.1679073], under the conditions (ab)* = ab and (ac)* = ac: In addition, we consider that ba is (c; b)-invertible, and at the same time ca is (b; c)-invertible under the same conditions, which extend the related results about Moore- Penrose inverses studied by J. Chen, H. Zou, H. Zhu, and P. Patrício [Mediterr J. Math., 2017, 14: 208] to (b; c)-inverses. As applications, we obtain that under condition (a²)* = a²; a is an EP element if and only if a is one-sided core invertible, if and only if a is group invertible.

With the aid of P-index iteration theory, we consider the minimal period estimates on P-symmetric periodic solutions of nonlinear P-symmetric Hamiltonian systems with mild superquadratic growth.

Biquadratic tensors play a central role in many areas of science. Examples include elastic tensor and Eshelby tensor in solid mechanics, and Riemannian curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor, respectively. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition, and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.

The regularity of random attractors is considered for the nonautonomous fractional stochastic FitzHugh-Nagumo system. We prove that the system has a pullback random attractor that is compact in H^s({\mathrm{ℝ}}^n)\times L^{\mathrm{2}}({\mathrm{ℝ}}^n) and attracts all tempered random sets of L^s({\mathrm{ℝ}}^n)\times L^{\mathrm{2}}({\mathrm{ℝ}}^n) in the topology of H^s({\mathrm{ℝ}}^n)\times L^{\mathrm{2}}({\mathrm{ℝ}}^n) with s\in (\mathrm{0},\mathrm{1}). By the idea of positive and negative truncations, spectral decomposition in bounded domains, and tail estimates, we achieved the desired results.

We prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by S. Cuccagna, V. Georgiev, and N. Visciglia [Comm. Pure Appl. Math., 2013, 6: 957–980] to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J., 1997, 86(1): 109–142] to control the high moments of the solutions emerging from the vector fields method.

Let \{Z_n,\mathrm{}n\ge \mathrm{0}\}be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for log(Z_{n+n_{\mathrm{0}}}/Z_{n_{\mathrm{0}}}) uniformly in n_{\mathrm{0}}\in \mathbb{Z},which extend the corresponding results by I. Grama, Q. Liu, and M. Miqueu [Stochastic Process. Appl., 2017, 127: 1255–1281] established for n_{\mathrm{0}}=\mathrm{}\mathrm{0}. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of log(Z_{n+n_{\mathrm{0}}}/Z_{n_{\mathrm{0}}}) and n.

The first aim of the paper is to study the Hermitizability of secondorder differential operators, and then the corresponding isospectral operators. The explicit criteria for the Hermitizable or isospectral properties are presented. The second aim of the paper is to study a non-Hermitian model, which is now well known. In a regular sense, the model does not belong to the class of Hermitizable operators studied in this paper, but we will use the theory developed in the past years, to present an alternative and illustrated proof of the discreteness of its spectrum. The harmonic function plays a critical role in the study of spectrum. Two constructions of the function are presented. The required conclusion for the discrete spectrum is proved by some comparison technique.

Abundant exact interaction solutions, including lump-soliton, lumpkink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2+1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions.

We introduce a class of singular integral operators on product domains along twisted surfaces. We prove that the operators are bounded on L^p provided that the kernels satisfy weak conditions.

Let k>2 be an integer and P be a 2n×2n symplectic orthogonal matrix satisfying P^k = I_2n and ker(P^j - I_2n) = 0; 1≤j <k: For any compact convex hypersurface \sum \subset {ℝ}^{2n} with n≥2 which is P-cyclic symmetric, i.e., x\mathrm{\in }\sum implies P_x\mathrm{\in }\sum ; we prove that if \sumis (r;R)-pinched with ,then there exist at least n geometrically distinct P-cyclic symmetric closed characteristics on \sum for a broad class of matrices P:

We give a recursive algorithm to compute the multivariable Zassenhaus formula {\mathrm{e}}^{{{\displaystyle X}}_{\mathrm{1}}+{{\displaystyle X}}_{\mathrm{2}}+\mathrm{...}+{{\displaystyle X}}_n}={\mathrm{e}}^{{{\displaystyle X}}_{\mathrm{1}}}{\mathrm{e}}^{{{\displaystyle X}}_{\mathrm{2}}}\mathrm{...}{\mathrm{e}}^{{{\displaystyle X}}_n}{{\displaystyle \mathrm{\Pi }}}_{k=\mathrm{2}}^{\infty }{\hspace{0.17em}\mathrm{e}}^{{{\displaystyle W}}_k} and derive ane effective recursion formula of {{\displaystyle W}}_k.

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of almost equal fourth prime powers.

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton–Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvaturedimension condition RCD(0;N) with N 2 R and N>1: In fact, we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K;N); where K;N 2 R and N>1: Along the way to the proofs, we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Carathéodory spaces which may have independent interests.

We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest (signless) Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges (2d≥k≥3), we show that its largest (signless) Laplacian Z-eigenvalue is d.

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel ?(t, s) = (t - s)^-μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233: 938-950], the error analysis for this approach is carried out for 0<μ<1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-type but also establish the error estimates under a more general regularity assumption on the exact solution.

Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computing the sub-maximal eigenpair is also introduced. The key ideas for each critical improvement are explained. To illustrate the present algorithm and compare it with the related algorithms, more than 10 examples are included.

The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order ?-tensors. In this paper, we establish important properties of diagonally dominant tensors and ?-tensors. Distributions of eigenvalues of nonsingular symmetric ?-tensors are given. An ?₊-tensor is semi-positive, which enlarges the area of semi-positive tensor from ?-tensor to ?₊-tensor. The spectral radius of Jacobi tensor of a nonsingular (resp. singular) ?-tensor is less than (resp. equal to) one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular ?-tensor if and only if all of its principal sub-tensors are nonsingular ?-tensors. An irreducible tensor Ais an ?-tensor if and only if it is quasi-diagonally dominant.

The nonnegative tensor (matrix) factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.

We establish a new characterization of the Musielak–Orlicz–Sobolev space on {ℝ}^n; which includes the classical Orlicz–Sobolev space, the weighted Sobolev space, and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hästö and A. M. Ribeiro [Commun. Contemp. Math., 2017, 19: 1650022] via weakening the assumption f\in L^{\mathrm{1}}({ℝ}^n) into f\in L_{}^{\mathrm{1}}({ℝ}^n), which was conjectured to be true by Hästö and Ribeiro in the aforementioned same article.

In this survey, we give a neat summary of the applications of the multi-resolution analysis to the studies of Besov-Q type spaces {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n) and Triebel-Lizorkin-Q type spaces {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n). We will state briefly the recent progress on the wavelet characterizations, the boundedness of Calderón-Zygmund operators, the boundary value problem of {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n) and {\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n). We also present the recent developments on the well-posedness of fluid equations with small data in {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n) and {\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n).

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six, and ten, respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.

This survey article on Dr. Ky Fan summarizes his versatile achievements and fundamental contributions in the fields of topological groups, nonlinear and convex analysis, operator theory, linear algebra and matrix theory, mathematical programming, and approximation theory, etc., and as well reveals Fan’s exemplary mathematical formation opening up the beauty of pure mathematics, with natural conditions, concise statements and elegant proofs. This article contains a brief biography of Dr. Fan and epitomizes his life. He was not only a great mathematician, but also a very serious teacher known to be extremely strict to his students. He loved his motherland and made generous donations for promoting mathematical development in China. He devoted his life to mathematics, continued his research and published papers till 85 years old.

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case (one-dimensional diffusions) and ends at the discrete one (birth-death processes). Possible generalization of the results studied here is discussed at the end of the paper.

In this paper, the author introduces some late results and puts forward a few problems on commutators of many important operators in harmonic analysis, included the Bochner-Riesz operator below the critical index, the strongly singular integral operator, the pseudo-differential operator, a class of convolution operators with oscillatory kernel, the Marcinkiewicz integral operator, and the fractional integral operator with rough kernel.

For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers (ADMM) has exhibited its efficiency and its convergence is well understood. When either the involved number of separable functions is more than two, or there is a nonconvex function, ADMM or its direct extended version may not converge. In this paper, we consider the multi-block separable optimization problems with linear constraints and absence of convexity of the involved component functions. Under the assumption that the associated function satisfies the Kurdyka- Lojasiewicz inequality, we prove that any cluster point of the iterative sequence generated by ADMM is a critical point, under the mild condition that the penalty parameter is sufficiently large. We also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.

We study the real and complex geometric simplicity of nonnegative irreducible tensors. First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an evenorder nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.