We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to Ad whenever X* has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized.
Given a topological dynamical system (X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of (X, T) is the system (S(X), FT), where FT is defined by FT(?)=T?? for any ?∈S(X). We show that (1) If (Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ; (2) If (S(Σ), Fσ) is transitive then it is Devaney chaos, where (Σ, σ) is a subshift of finite type; (3) If (Σ, T) has shadowing property, then (SU(Σ), FT) has shadowing property, where Σ is any closed subset of a Cantor set and T a selfmap of Σ; (4) If (X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T:X→X is continuous, then (SU(X), FT) is sensitive; (5) If Σ is a closed subset of a Cantor set with infinite points and T:Σ→Σ is positively expansive then the entropy entU (FT) of the functional envelope of (Σ, T) is infinity.
The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.
The Casimir element of a fusion ring (R, B) gives rise to the so called Casimir matrix C of (R, B). This enables us to construct a generalized Cartan matrix D-C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D-C of finite (resp. affine) type. It turns out that there exists a fusion ring with D-C being of finite (resp. affine) type if and only if D-C has only the form A2 (resp. A1(1). We also realize all fusion rings with D-C being a particular generalized Cartan matrix of indefinite type.
For any integer n≥2, let P(n) be the largest prime factor of n. In this paper, we prove that the number of primes p≤x with P(p-1)≥pc is more than (1-c+o(1))π(x) for 0 < c < (1)/(2). This extends a recent result of Luca, Menares and Madariaga for (1)/(4)≤c≤(1)/(2). We also pose two conjectures for further research.
In this paper, we study some algebraic and spectral properties of dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space of the unit disk.
The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, we study the permanence properties of strong embeddability for metric spaces. We show that strong embeddability is coarsely invariant and it is closed under taking subspaces, direct products, direct limits and finite unions. Furthermore, we show that a metric space is strongly embeddable if and only if it has weak finite decomposition complexity with respect to strong embeddability.
For the strongly coupled system of M≥3 competing species:-Δ[di+βijuj)ui=(ai-bi)ui-kuiuj, i=1,..., M, we prove the uniqueness of the limiting configuration as k→∞ under suitable conditions. Moreover, we prove that the limiting configuration minimizes a variational problem associated to the strongly coupled system among the segregated states with the same boundary conditions.
In this paper, by the Aubry-Mather theory, it is proved that there are many periodic solutions and usual or generalized quasiperiodic solutions for relativistic oscillator with anharmonic potentials models , where p(t)∈C0(R1) is 1-periodic and α>0.
Let (X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f) and Ω(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold:(1) If x∈Ω(f)-Ω(fn) for some n≥2, then x∈EP(f). (2) Ω(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D))=ℵ0 (the cardinal number of the set of positive integers).