Haken 3-Manifolds in Small Covers*

The authors prove that a 3-dimensional small cover M is a Haken manifold if and only if M is aspherical or equivalently the underlying simple polytope is a flag polytope. In addition, they find that M being Haken is also equivalent to the existence of a Riemannian metric with non-positive sectional curvature on M.     

Reduced Crossed Products of Pro-Banach Algebras*

This paper gives the concept of the reduced pro-Banach algebra crossed product associated with inversely pro-Banach algebra dynamical system, and shows that the reduced crossed product is an inverse limit of an inverse system of Banach algebra crossed products. Also, the authors show that if the locally compact group is amenable, then the crossed product and the reduced crossed product are isometrically isomorphic.     

On permutation polytopes

In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices.We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify ?4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online.We conclude with several questions suggested by a general finiteness result.     

On the Kernel of Restriction of Characters*

Let G be a finite group, H be a proper subgroup of G, and S be a unitary subring of C. The kernel of the restriction map S[Irr(G)] → S[Irr(H)] as a ring homomorphism is studied. As a corollary, the main result in [Isaacs, I. M. and Navarro, G., Injective restriction of characters, Arch. Math., 108, 2017, 437–439] is reproved.     

On Fridman Invariants and Generalized Squeezing Functions*

In this paper, the authors introduce the notion of generalized squeezing function and study the basic properties of generalized squeezing functions and Fridman invariants.They also study the comparison of these two invariants, in terms of the so-called quotient invariant.     

On Geometric Realization of the General Manakov System*

It is well-known that the general Manakov system is a 2-components nonlinear Schr¨odinger equation with 4 nonzero real parameters. The analytic property of the general Manakov system has been well-understood though it looks complicated. This paper devotes to exploring geometric properties of this system via the prescribed curvature representation in the category of Yang-Mills’ theory. Three models of moving curves evolving in the symmetric Lie algebras u(2, 1) = k_α ⊕ m_α(α =...     

C*-Isomorphisms Associated with Two Projections on a Hilbert C*-Module*

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C^*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C^*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of L(H), where L(H) denotes the set of all adjointable operators on H. The C^*-subalgebras of L(H) generated by elements in {P - P ∧ Q, Q - P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation (π, X e) of i(P, Q, H) such that e π(P - P ∧ Q) = π(P) - π(P) ∧ π(Q),eπ(Q - P ∧ Q) = π(Q) - π(P) ∧ π(Q).When (P, Q) is semi-harmonious, that is, R(P + Q) and R(2I - P - Q) are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I - Q, I - P, H) are unitarily equivalent via a unitary operator in L(H). A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I - Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C^*-modules are also provided.     

Survey on Path-Dependent PDEs*

In this paper,the authors provide a brief introduction of the path-dependent partial differential equations(PDEs for short) in the space of continuous paths,where the path derivatives are in the Dupire(rather than Fréchet) sense.They present the connections between Wiener expectation,backward stochastic differential equations(BSDEs for short) and path-dependent PDEs.They also consider the well-posedness of path-dependent PDEs,including classical solutions,Sobolev solutions and viscosity solution...     

Range-Renewal Processes: SLLNs and Power Laws*

Given n samples(viewed as an n-tuple) of a γ-regular discrete distribution π,in this article the authors concern with the weighted and unweighted graphs induced by the n samples. They first prove a series of SLLN results(of Dvoretzky-Erd¨os’ type). Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1 + γ. They also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distribu...     

A Theory of Orbit Braids*

In this paper, the authors systematically discuss orbit braids in M × I with regards to orbit configuration space FG(M, n), where M is a connected topological manifold of dimension at least 2 with an effective action of a finite group G. These orbit braids form a group, named orbit braid group, which enriches the theory of ordinary braids.The authors analyze the substantial relations among various braid groups associated to those configuration spaces FG(M, n), F(M/G, n) and F(M, n). They also co...     

On the Graph Connectivity of Skeleta of Convex Polytopes

Given a d-dimensional convex polytope P and nonnegative integer k not exceeding d−1, let denote the simple graph on the node set of k-dimensional faces of P in which two such faces are adjacent if there exists a (k+1)-dimensional face of P which contains them both. The graph is isomorphic to the dual graph of the (d−k)-dimensional skeleton of the normal fan of P. For fixed values of k and d, the largest integer m such that is m-vertex-connected for all d-dimensional polytopes P is determined. This result generalizes Balinski’s theorem on the one-dimensional skeleton of a d-dimensional convex polytope. Supported by the 70/4/8755 ELKE Research Fund of the University of Athens.     

Bochner-Martinelli Formula for Higher Spin Operators of Several R6 Variables*

The higher spin operator of several R⁶ variables is an analogue of the ■-operator in theory of several complex variables.The higher spin representation of so₆(C)is ⊙^kC⁴ and the higher spin operator D_k acts on ⊙^kC⁴-valued functions.In this paper,the authors establish the Bochner-Martinelli formula for higher spin operator D_k of several R⁶variables.The embedding of R⁶ⁿ into the space of co...     

Lattice-Based Cryptography: A Survey*

Most of current public key cryptosystems would be vulnerable to the attacks of the future quantum computers.Post-quantum cryptography offers mathematical methods to secure information and communications against such attacks,and therefore has been receiving a significant amount of attention in recent years.Lattice-based cryptography,built on the mathematical hard problems in(high-dimensional) lattice theory,is a promising post-quantum cryptography family due to its excellent efficiency,moderate s...     

Homomorphisms into N *

We study the properties of continuous homomorphisms from β S into N ^* and from N ^* into N ^* . We show that the image C of N ^* under a continuous homomorphism which does not arise from the continuous extension of a homomorphism mapping N to itself, has the property that C+C is a singleton.     

On the Range of Certain ASH Algebras of Real Rank Zero*

In this paper, the authors consider the range of a certain class of ASH algebras in [An, Q., Elliott, G. A., Li, Z. and Liu, Z., The classification of certain ASH C*-algebras of real rank zero, J. Topol. Anal., 14(1), 2022, 183–202], which is under the scheme of the Elliott program in the setting of real rank zero C*-algebras. As a reduction theorem, they prove that all these ASH algebras are still the AD algebras studied in [Dadarlat, M. and Loring, T. A., Classifying C*-algebras via ordered, m...     

Equicontinuity and Sensitivity of Group Actions*

Let(X, G) be a dynamical system(G-system for short), that is, X is a topological space and G is an infinite topological group continuously acting on X. In the paper,the authors introduce the concepts of Hausdorff sensitivity, Hausdorff equicontinuity and topological equicontinuity for G-systems and prove that a minimal G-system(X, G) is either topologically equicontinuous or Hausdorff sensitive under the assumption that X is a T₃-space and they provide a classification of transitive d...     

Dual-holomorphic Functions and Problems of Lifts*

The main purpose of this paper is to study the differential geometrical objects on tangent bundle corresponding to dual-holomorphic objects of dual-holomorphic manifold.As a result of this approach, the authors find a new class of lifts(deformed complete lifts)in the tangent bundle.     

On Minimal Triangulations of Products of Convex Polygons

We give new lower bounds for the minimal number of simplices needed in a triangulation of the product of two convex polygons, improving the lower bounds in Bowen et al. (Topology 44:321–339, 2005).     

Diophantine方程y~2=px(x~2+2)

设p是大于3的奇素数.本文证明了:当p≡5或7(mod 8)时,方程y~2=px(x~2+2)无正整数解(x,y);当p≡1(mod 8)时,该方程至多有1组解;当p≡3(mod 8)时,该方程至多有2组解.