Certain decompositions of matrices over Abelian rings

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ?. We prove that M ⁿ (R) is nil clean if and only if R/J(R) is Boolean and M _n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ?₃, B or ?₃ ⊕ B where B is a Boolean ring, and that M _n (R) is weakly nil clean if and only if M _n (R) is nil clean for all n ≥ 2.     

On the stability and regularity of the multiplier ideals of monomial ideals

Let a ? ?[x ₁, . . . , x _d ] be a monomial ideal and J (a) its multiplier ideal which is also a monomial ideal. It is proved that if a is strongly stable or squarefree strongly stable then so is J (a). Denote the maximal degree of minimal generators of a by d(a). When a is strongly stable or squarefree strongly stable, it is shown that the Castelnuovo-Mumford regularity of J (a) is less than or equal to d(a). As a corollary, one gets a vanishing result on the ideal sheaf]\(\widetilde {\mathcal{J}\left( a \right)}\) on ? ^d–1 associated to J (a) that H ⁱ(? ^d–1;\(\widetilde {\mathcal{J}\left( a \right)}\)(s–i)) = 0, for all i > 0 and s ≥ d(a).     

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group, π _e (G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(π _e (G)) = k < ∞, otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M ₁₂, M ₂₂, J ₂, He, Suz, M ^c L and O′N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M ₁₂, M ₂₂, He, Suz or O′N, then h(π _e (Aut(M))) ∈¸ {1,∞}.     

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L₁(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.     

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (?L^J,D).     

Semigroup graded algebras and graded PI-exponent

Let S be a semigroup. We study the structure of graded-simple S-graded algebras A and the exponential rate PIexp ^S-gr(A):= lim_n→∞ \(\sqrt[n]{{c_n^{S - gr}\left( A \right)}}\) of growth of codimensions c _n ^S-gr (A) of their graded polynomial identities. This is of great interest since such algebras can have non-integer PIexp ^S-gr(A) despite being finite dimensional and associative. In addition, such algebras can have a non-trivial Jacobson radical J(A). All this is in strong contrast with the case when S is a group since in the group case J(A) is trivial, PIexp ^S-gr(A) is always integer and, if the base field is algebraically closed, then PIexp ^S-gr(A) equals dimA. Without any restrictions on the base field F, we classify graded-simple S-graded algebras A for a class of semigroups S which is complementary to the class of groups. We explicitly describe the structure of J(A) showing that J(A) is built up of pieces of a maximal S-graded semisimple subalgebra of A which turns out to be simple. When F is algebraically closed, we get an upper bound for \({\overline {\lim } _{n \to \infty }}\sqrt[n]{{c_n^{S - gr}\left( A \right)}}\). If A/J(A) ≈ M ₂(F) and S is a right zero band, we show that this upper bound is sharp and PIexp ^S-gr(A) indeed exists. In particular, we present an infinite family of graded-simple algebras A with arbitrarily large non-integer PIexp ^S-gr(A).     

Classification of finite commutative rings with planar,toroidal, and projective line graphs associated with Jacobson graphs

Let R be a commutative ring with nonzero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by J _R , is a graph with vertex-set R J(R), such that two distinct vertices a and b in R J(R) are adjacent if and only if 1 ? ab is not a unit of R. Also, the line graph of the Jacobson graph is denoted by L(J _R ). In this paper, we characterize all finite commutative rings R such that the graphs L(J _R ) are planar, toroidal or projective.     

Structure of Abelian rings

Let R be a ring with identity. We use J(R); G(R); and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R; respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ? J(R) of R; that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2; 3; 4; and 5 orbits under the left regular action on X(R) by G(R): For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R); then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.     

Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic

It was proved that the complexity of square root computation in the Galois field GF(3^s), s = 2^kr, is equal to O(M(2^k)M(r)k + M(r) log₂r) + 2^kkr^1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3^s) is equal to O(M(2^k)M(r)) and O(M(2^k)M(r)) + r^1+o(1), respectively. If the basis in the field GF(3^r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2^kkr^1+o(1) and r^1+o(1) can be omitted. For M(n) one may take n log₂nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form O_r (M (s) log ₂s) = s (log ₂s)²ψ(s).     

Kazhdan-Lusztig cells in some weighted Coxeter groups

Let(W,S) be a Coxeter group with S = I■J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W_I of W with w_0 the longest element of W_I. We describe all the left cells and two-sided cells of the weighted Coxeter group(W,S,L) that have non-empty intersection with W_J,where the weight function L of(W, S) is in one of the following cases:(i) max{L(s) | s ∈J} min{L(t)|t∈I};(ii) min{L(s)|s ∈J} ≥L(w_0);(iii) there exists some t ∈ I satisfying L(t) L(s) for any s ∈I-{t} and L takes a constant value L_J on J with L_J in some subintervals of [1, L(w_0)-1]. The results in the case(iii) are obtained under a certain assumption on(W, W_I).     

Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures

Let (M, g, J) be a compact Hermitian manifold and \(\Omega\) the fundamental 2-form of (g, J). A Hermitian manifold (M, g, J) is called a locally conformal Kähler manifold if there exists a closed 1-form α such that \(d\Omega=\alpha \wedge \Omega\) . The purpose of this paper is to give a completely classification of locally conformal Kähler nilmanifolds with left-invariant complex structures.     

Isometries with dense windings of the torus in <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">M</Emphasis>)

Let C(M) be the space of all continuous functions on M? ?. We consider the multiplication operator T: C(M) → C(M) defined by Tf(z) = zf(z) and the torus

$$O(M) = \left\{ {f:M \to \mathbb{C} \ntrianglelefteq \left\| f \right\| = \left\| {\frac{1}{f}} \right\| = 1} \right\}$$

. If M is a Kronecker set, then the T-orbits of the points of the torus ½O(M) are dense in ½O(M) and are ½-dense in the unit ball of C(M).

     

On the geometry of spaces of oriented geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kähler structure on L(M).     

The general rigidity result for bundles of <Emphasis Type="Italic">A</Emphasis>-covelocities and <Emphasis Type="Italic">A</Emphasis>-jets

Let M be an m-dimensional manifold and A = D _k ^r /I = R⊕N _A a Weil algebra of height r. We prove that any A-covelocity T _x ^A f ∈ T _x ^A *M, x ∈ M is determined by its values over arbitrary max{width A,m} regular and under the first jet projection linearly independent elements of T _x ^A M. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result T ^A *M ? T ^r *M without coordinate computations, which improves and generalizes the partial result obtained in Tomá? (2009) from m ≥ k to all cases of m.We also introduce the space J ^A (M,N) of A-jets and prove its rigidity in the sense of its coincidence with the classical jet space J ^r (M,N).     

Estimates of reachable sets of control systems with nonlinearity and parametric perturbations

In the Banach space L₁(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L₂(M, τ) is introduced, and its main properties are established. A convergence criterion in L₂(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L₁(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖₁ ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L₁(M, τ) is complemented. The convergence in L₂(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained.     

An Extension of a Simply Inequality Between von Neumann–Jordan and James Constants in Banach Spaces

For a non-trivial Banach space X, let J(X), CNJ(X), C_(NJ)~(p)(X) respectively stand for the James constant, the von Neumann–Jordan constant and the generalized von Neumann–Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann–Jordan constants, and establish an inequality between them: C_(NJ)~(p)(X) ≤J(X) with p ≥ 2, which covers the well-known inequality CNJ(X) ≤ J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.     

Arithmetical rank of binomial ideals

In this paper, we investigate the arithmetical rank of a binomial ideal J. We provide lower bounds for the binomial arithmetical rank and the J-complete arithmetical rank of J. Special attention is paid to the case where J is the binomial edge ideal of a graph. We compute the arithmetical rank of such an ideal in various cases.     

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.     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.