On C*-algebras from homoclinic equivalences on subshifts of finite type

Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand show that the reduced C^*-algebra C_r^*(■)of■is a unital simple approximately finite(AF)-dimensional C^*-algebra.The shift action G of onΣinduces a canonical automorphism action of G on the C^*-algebra C_r^*(■).We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C^*-algebras,and show that,if G is an amenable group,then the noncommutative topological entropy of the canonical automorphism action of G on C_r^*(■)is equal to the topology entropy of the shift action of G onΣ.We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C^*-dynamical system(C_r^*(■),G).     

Hereditary uniform property Γ

We study the uniform property Γ for separable simple C^*-algebras which have quasitraces and may not be exact. We show that a stably finite separable simple C^*-algebra A with the strict comparison and uniform property Γ has tracial approximate oscillation zero and stable rank one. Moreover in this case,its hereditary C^*-subalgebras also have a version of uniform property Γ. If a separable non-elementary simple amenable C^*-algebra A with strict compariso...     

本刊英文版Vol.39(2023),No.5论文摘要（英文）

<正>Monge-Ampère Type Equations on Almost Hermitian Manifolds Jiao Gen ZHANG Abstract In this paper we consider the Monge-Ampere type equations on compact almost Hermitian manifolds.We derive C^∞ a priori estimates under the existence of an admissible C-subsolution.Finally,we obtain an existence result if there exists an admissible supersolution.     

On the Existence of Positive Solutions ofSingular Second Order BoundaryValue Problems

This paper deals with the existence of positive solutions of the equation u“ f(t,u)=0 with linear boundary conditions. We show the existence of at least onepositive solution if f is neither superlinear nor sublinear on u by a simple application of afixed point Theorem in cones.     

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION

In this paper,for a bounded C² domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C² boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.     

On the classification of certain real rank zero C*-algebras

In this paper,we give a classification of real rank zero C^*-algebras that can be expressed as inductive limits of a sequence of a subclass of Elliott-Thomsen algebras C.     

Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.     

Monge–Ampère Type Equations on Almost Hermitian Manifolds

In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds. We derive C^∞ a priori estimates under the existence of an admissible C-subsolution. Finally,we obtain an existence result if there exists an admissible supersolution.     

对称锥规划的Mehrotra型预估-矫正算法的多项式复杂性（英文）

We establish polynomial complexity bounds of the Mehrotra-type predictorcorrector algorithms for linear programming over symmetric cones. We first slightly modify the maximum step size in the predictor step of the safeguard based Mehrotra-type algorithm for linear programming, that was proposed by Salahi et al[18]. Then, using the machinery of Euclidean Jordan algebras, we extend the modified algorithm to symmetric cones. Based on the Nesterov-Todd direction, we obtain O(r log ε-1) iteration complexity bound of this algorithm, where r is the rank of the Jordan algebras and ε is the required precision. We also present a new variant of Mehrotra-type algorithm using a new adaptive updating scheme of centering parameter and show that this algorithm enjoys the same order of complexity bound as the safeguard algorithm. We illustrate the numerical behaviour of the methods on some small examples.     

Complementarity properties of the Lyapunov transformation over symmetric cones

The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX +XA* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V . Given any element a ∈ V , we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0 -property for La and show that La has the R0 -property if and only if a is invertible. Finally, we provide La with some characterizations of the E0 -property and the nondegeneracy property.     

Born-Infeld理论中的多项式函数模型（英文）

Based on the Lagrangian action density under Born-Infeld type dynamics and motivated by the one-dimensional prescribed mean curvature equation, we investigate the polynomial function model in Born-Infeld theory in this paper with the form of-([1-a(φ)2]φ) = λf(φ(x)),where λ > 0 is a real parameter, f ∈ C²(0, +∞) is a nonlinear function. We are interested in the exact number of positive solutions of the above nonlinear equation. We specifically develop for the problem combined with a...     

STABILITY AND SUPERSTABILITY OF JORDAN HOMOMORPHISMS AND JORDAN DERIVATIONS ON BANACH ALGEBRAS AND C~＊-ALGEBRAS： A FIXED POINT APPROACH

Using fixed point methods, we prove the Hyers-Ulam-Rassias stability and superstability of Jordan homomorphisms (Jordan *-homomorphisms), and Jordan derivations (Jordan *-derivations) on Banach algebras (C*-algebras) for the generalized Jensen-type functional equationwhere r is a fixed positive real number in (1, ∞).     

INVARIANCE AND STABILITY OF THE PROFILE EQUATIONS OF GEOMETRIC OPTICS

The profile equations of geometric optics are described in a form invariant under the natural transformations of first order systems of partial differential equations. This allows us to prove that various strategies for computing profile equations are equivalent. We prove that if L generates an evolution on L 2 the same is true of the profile equations. We prove that the characteristic polynomial of the profile equations is the localization of the characteristic polynomial of the background operator at (y, dφ(y)) where φ is the background phase. We prove that the propagation cones of the profile equations are subsets of the propagation cones of the background operator.     

Linear Maps Preserving Idempotence on Nest Algebras

In this paper, we discuss the rank-l-preserving linear maps on nest algebras of Hilbertspace operators. We obtain several characterizations of such linear maps and apply them to show that a weakly continuous linear hijection on an atomic nest algebra is idempotent preserving if and only if it is a Jordan homomorphism, and in turn, if and only if it is an automorphism or an anti-automorphism.     

非线性分数微分方程边值问题正解的存在性(英文)

In this paper,we study a Dirichlet-type boundary value problem（BVP） of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D^α_o⁺is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel’skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.     

POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM OF FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION

In this paper,we establish sufficient conditions for the existence of positive solutions to a general class of integral boundary value problem(BVP) of nonlinear fractional functional differential equation.A differential operator is taken in the RiemannLiouville sense.Our analysis relies on the Krasnosel'skii fixed-point theorem in cones.We also give examples to illustrate the applicability of our results.     

Functional ergodic limits for occupation time processes of site-dependent branching Brownian motions in R

We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factorσ(·).We focus on the functional ergodic limits for the occupation time processes of the models in R.It is proved that the limiting process has the form ofλξ(·),where A is the Lebesgue measure on IE andξ(·)is a real-valued process which is non-degenerate if and only ifσis integrable.Whenξ(·)is non-degenerate,it is strictly positive for t0.Moreover,ξconverges to O in finite-dimensional distributions if the integral ofσtends to infinity.     

Global multiplicity of solutions to a defocusing quasilinear Schr?dinger equation with the singular term

We consider a class of modified quasilinear Schr?dinger equations ■ with u(x)=0 on ??, where ? ? R^N is a bounded domain with a regular boundary, N≥3, a and b are bounded mensurable functions, 0 < α < 1 < β < 2^*-1 and k, λ≥0 are two parameters. We establish the global existence and multiplicity results of positive solutions in H₀¹ (?) ∩ L^∞ (?) for appropriate classes of parameters k and λ and coefficients a(x) and b(x).     

The minimal measurement number for generalized conjugate phase retrieval

The generalized conjugate phase retrieval problem aims to reconstruct a complex signal x ∈ Cⁿ from quadratic measurements x*A₁x,...,x*A_mx,where A₁,...,A_m∈R^n×n are real symmetric matrices.The equivalent formulation for generalized conjugate phase retrieval along with the minimal measurement number required for accurate retrieval(up to a global phase factor as well as conjugacy) is derived in this paper.We present a set of nine vectors in R⁴ and prove that it is conjugate phase retrievable on C⁴.This result implies the measurement number bound 4n-6 is not optimal for some n,which confirms a conjecture in the article by Evans and Lai(2019).