Finitistic dimension and restricted injective dimension

We study the relations between finitistic dimensions and restricted injective dimensions. Let R be a ring and T a left R-module with A = End _R T. If _R T is selforthogonal, then we show that rid(T _A ) ? findim(A _A ) ? findim( _R T) + rid(T _A ). Moreover, if R is a left noetherian ring and T is a finitely generated left R-module with finite injective dimension, then rid(T _A ) ? findim(A _A ) ? fin.inj.dim( _R R)+rid(T _A ). Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.     

Tensor products of tilting modules

We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T ? T′: The main result is that T ? T′ turns out to be an (n + m)-tilting A ? B-module, where T is an m-tilting A-module and T′ is an n-tilting B-module.     

Conjugacy classes in Möbius groups

Let \({\mathbb H^{n+1}}\) denote the n + 1-dimensional (real) hyperbolic space. Let \({\mathbb {S}^{n}}\) denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of \({\mathbb {S}^{n}}\) is denoted by M(n). Let M _o (n) be its identity component which consists of all orientation-preserving elements in M(n). The conjugacy classification of isometries in M _o (n) depends on the conjugacy of T and T ^?1 in M _o (n). For an element T in M(n), T and T ^?1 are conjugate in M(n), but they may not be conjugate in M _o (n). In the literature, T is called real if T is conjugate in M _o (n) to T ^?1. In this paper we classify real elements in M _o (n). Let T be an element in M _o (n). Corresponding to T there is an associated element T _o in SO(n + 1). If the complex conjugate eigenvalues of T _o are given by \({\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}\) , then {θ₁, . . . , θ _k } are called the rotation angles of T. If the rotation angles of T are distinct from each-other, then T is called a regular element. After classifying the real elements in M _o (n) we have parametrized the conjugacy classes of regular elements in M _o (n). In the parametrization, when T is not conjugate to T ^?1 , we have enlarged the group and have considered the conjugacy class of T in M(n). We prove that each such conjugacy class can be induced with a fibration structure.     

Some Structure Theories of Leibniz Triple Systems

In this paper, we investigate the Leibniz triple system T and its universal Leibniz envelope U(T). The involutive automorphism of U(T) determining T is introduced, which gives a characterization of the \(\mathbb {Z}_{2}\)-grading of U(T). We show that the category of Leibniz triple systems is equivalent to a full subcategory of the category of \(\mathbb {Z}_{2}\)-graded Leibniz algebras. We give the relationship between the solvable radical R(T) of T and R a d(U(T)), the solvable radical of U(T). Further, Levi’s theorem for Leibniz triple systems is obtained. Moreover, the relationship between the nilpotent radical of T and that of U(T) is studied. Finally, we introduce the notion of representations of a Leibniz triple system, which can be described by using involutive representations of its universal Leibniz envelope.     

Rates of decay in the classical Katznelson-Tzafriri theorem

The Katznelson-Tzafriri Theorem states that, given a power-bounded operator T, ∥Tⁿ(I ? T)∥ → 0 as n → ∞ if and only if the spectrum σ(T) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T) ∩ T = {1}. The results obtained lead, in particular, to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator R(e^iθ, T) as θ → 0. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case.     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

Holomorphic maps preserving parts of the local spectrum

Let x ₀ be a nonzero vector in \({\mathbb{C}^{n}}\) , and let \({U\subseteq \mathcal{M}_{n}}\) be a domain containing the zero matrix. We prove that if φ is a holomorphic map from U into \({\mathcal{M}_{n}}\) such that the local spectrum of T ∈ U at x ₀ and the local spectrum of φ(T) at x ₀ have always a common value, then T and φ(T) have always the same spectrum, and they have the same local spectrum at x ₀ a.e. with respect to the Lebesgue measure on U. If \({\varphi \colon U\rightarrow \mathcal{M}_{n}}\) is holomorphic with φ(0) = 0 such that the local spectral radius of T at x ₀ equals the local spectral radius of φ(T) at x ₀ for all T ∈ U, there exists \({\xi \in \mathbb{C}}\) of modulus one such that ξT and φ(T) have the same spectrum for all T in U. We also prove that if for all T ∈ U the local spectral radius of φ(T) coincides with the local spectral radius of T at each vector x, there exists \({\xi \in \mathbb{C}}\) of modulus one such that φ(T) = ξT on U.     

Similarity Results for Operators of Class <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>

If T is a multiplicity-free contraction of class C ₀ with minimal function m _T , then it is quasisimilar to the Jordan block S(m _T ). In case m _T is a Blaschke product with simple roots forming a Carleson sequence, we show that the relation between T and S(m _T ) can be strengthened to similarity. Under the additional assumption that u(T) has closed range for every inner divisor \({u\in H^\infty}\) of m _T , the result also holds in the more general setting where the roots have bounded multiplicities.     

Maximum Distance Between Slater Orders and Copeland Orders of Tournaments

Given a tournament T?=?(X, A), we consider two tournament solutions applied to T: Slater’s solution and Copeland’s solution. Slater’s solution consists in determining the linear orders obtained by reversing a minimum number of directed edges of T in order to make T transitive. Copeland’s solution applied to T ranks the vertices of T according to their decreasing out-degrees. The aim of this paper is to compare the results provided by these two methods: to which extent can they lead to different orders? We consider three cases: T is any tournament, T is strongly connected, T has only one Slater order. For each one of these three cases, we specify the maximum of the symmetric difference distance between Slater orders and Copeland orders. More precisely, thanks to a result dealing with arc-disjoint circuits in circular tournaments, we show that this maximum is equal to n(n???1)/2 if T is any tournament on an odd number n of vertices, to (n ²???3n?+?2)/2 if T is any tournament on an even number n of vertices, to n(n???1)/2 if T is strongly connected with an odd number n of vertices, to (n ²???3n???2)/2 if T is strongly connected with an even number n of vertices greater than or equal to 8, to (n ²???5n?+?6)/2 if T has an odd number n of vertices and only one Slater order, to (n ²???5n?+?8)/2 if T has an even number n of vertices and only one Slater order.     

Functional envelope of Cantor spaces

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), F_σ) 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 ent U(FT) of the functional envelope of(Σ, T) is infinity.     

Two classes of <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated with a von Neumann algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P ₁ and P ₂ of τ-measurable operators and investigate their properties. The class P ₂ contains P ₁. If a τ-measurable operator T is hyponormal, then T lies in P ₁; if an operator T lies in P _k , then UTU* belongs to P _k for all isometries U from M and k = 1, 2; if an operator T from P ₁ admits the bounded inverse T ^?1, then T ^?1 lies in P ₁. We establish some new inequalities for rearrangements of operators from P ₁. If a τ-measurable operator T is hyponormal and T ⁿ is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P ₁ coincides with the set of all paranormal operators on H.     

Equicontinuity of maps on a dendrite with finite branch points

Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =y}. In this paper, we show that the following statements are equivalent:(1) f is equicontinuous.(2) ω(x, f) = Ω(x,f) for any x∈T.(3) ∩_(n=1)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T.     

Extremes of Gaussian processes with smooth random expectation and smooth random variance

Let ξ(t), t ∈ [0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup _t∈[0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.     

Coherence relative to a weak torsion class

Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules

$$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$

such that F₀,..., F_n?1 are finitely generated free and K_n?1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext ⁿ _R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor ^R _n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.

     

On the zeros of the Davenport–Heilbronn function

Let N ₀(T) be the number of zeros of the Davenport–Heilbronn function in the interval [1/2, 1/2+ i T]. It is proved that N ₀(T) ? T (ln T)^1/2+1/16?ε, where ε is an arbitrarily small positive number.     

<Emphasis Type="Italic">T</Emphasis>-Coloring of Certain Networks

Given a graph G and a finite set T of non-negative integers containing zero, a T-coloring of G is a non-negative integer function f defined on V(G) such that \(|f(x)-f(y)|\not \in T\) whenever \((x,y)\in E(G)\). The span of T-coloring is the difference between the largest and smallest colors, and the T-span of G is the minimum span over all T-colorings f of G. The edge span of a T-coloring is the maximum value of \(|f(x)-f(y)|\) over all edges \((x,y)\in E(G)\), and the T-edge span of G is the minimum edge span over all T-colorings f of G. In this paper, we compute T-span and T-edge span of crown graph, circular ladder and mobius ladder, generalized theta graph, series-parallel graph and wrapped butterfly network.     

Homological Epimorphisms, Compactly Generated t-Structures and Gorenstein-Projective Modules

The aim of this paper is two-fold. Given a recollement (T′, T, T″, i*, i_*, i^!, j_!, j*, j_*), where T′, T, T″ are triangulated categories with small coproducts and T is compactly generated. First, the authors show that the BBD-induction of compactly generated t-structures is compactly generated when i_* preserves compact objects. As a con-sequence, given a ladder (T′, T, T″, T, T′) of height 2, then the certain BBD-induction of compactly generated t-structures is compactly generated. The authors apply them to the recollements induced by homological ring epimorphisms. This is the first part of their work. Given a recollement (D(B-Mod),D(A-Mod),D(C-Mod), i*, i_*, i^!, j_!, j*, j_*) induced by a homological ring epimorphism, the last aim of this work is to show that if A is Gorenstein, _A B has finite projective dimension and j_! restricts to D ^b (C-mod), then this recollement induces an unbounded ladder (B-Gproj,A-Gproj, C-Gproj) of stable categories of finitely generated Gorenstein-projective modules. Some examples are described.     

Absolute Continuity and Singularity of Two Probability Measures on a Filtered Space

Let μ and ν be fixed probability measures on a filtered space \((\varOmega, \mathcal{F}, \allowbreak(\mathcal{F}_{t} )_{t\in \mathbf{R}^{+} } )\). Denote by μ _T and ν _T (respectively, μ _T? and ν _T?) the restrictions of the measures μ and ν on \(\mathcal{F}_{T} \) (respectively, on \(\mathcal{F}_{T-} \)) for a stopping time T. We find the Hahn decomposition of μ _T and ν _T using the Hahn decomposition of the measures μ, ν and the Hellinger process h _t in the strict sense of order \(\frac{1}{2}\). The norm of the absolutely continuous component of μ _T? with respect to ν _T? is computed in terms of density processes and Hellinger integrals.     

A continuation method for solving symmetric Toeplitz systems

A fast algorithm is proposed for solving symmetric Toeplitz systems. This algorithm continuously transforms the identity matrix into the inverse of a given Toeplitz matrix T. The memory requirements for the algorithm are O(n), and its complexity is O(log κ(T)nlogn), where (T) is the condition number of T. Numerical results are presented that confirm the efficiency of the proposed algorithm.