Let A be a finite-dimensional associative algebra over a field of characteristic 0. Then there exist C ∈ ℚ+ and t ∈ ℤ+ such that gcn(A) ∼ Cntdn as n → ∞, where d = PIexp(A). In particular, Amitsur’s and Regev’s conjectures hold for the codimensions gcn(A) of generalized polynomial identities. 相似文献
In a recent paper by Engel and Schneider, it was asked if, for every n ? 1, A ∈ τ<n> implies (A+D) ∈ τ<n> for every D = diag[d1, d2,… dn] with di ? 0, 1 ? i ? n. We answer this question in the negative. More precisely, we show that for, any n ? 3, the set (τ < n>): = {D ∈Cn,n:(A+D)∈τ < n> for all A∈τ<n>} is exactly given by(Gt<n>) = {γIn:γ ? 0}. 相似文献
In the space A (θ) of all one-valued functions f(z) analytic in an arbitrary region G ? ? (0 ∈ G) with the topology of compact convergence, we establish necessary and sufficient conditions for the equivalence of the operators L1=αnznΔn + ... + α1zΔ+α0E and L2= znan(z)Δn + ... + za1(z)Δ+a0(z)E, where δ: (Δ?)(z)=(f(z)-?(0))/z is the Pommier operator in A(G), n ∈ ?, αn ∈ ?, ak(z) ∈ A(G), 0≤k≤n, and the following condition is satisfied: Σj=sn?1 αj+1 ∈ 0, s=0,1,...,n?1. We also prove that the operators zs+1Δ+β(z)E, β(z) ∈ AR, s ∈ ?, and zs+1 are equivalent in the spaces AR, 0?R?-∞, if and only if β(z) = 0. 相似文献
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). 相似文献
Let z(t) ∈ Rn be a generalized Poisson process with parameter λ and let A: Rn → Rn be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) ∈ Rn which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated. 相似文献
For the lower sigma-exponent of the linear differential system ? = A(t)x, x ∈ Rn, t ≥ 0, defined by the formula Δσ(A) ≡ infλ[Q]≤-σλ1(A + Q), σ > 0, on the basis of the lower characteristic exponents λ1(A+Q) of perturbed linear systems with Lyapunov exponents λ[Q] ≤ ?σ < 0 of perturbations Q, we prove the following general form as a function of the parameter σ > 0. For any nondecreasing bounded function f(σ) of the parameter σ ∈ (0,+∞) that coincides with a constant on some infinite interval (σ0,+∞), σ0 ≥ 0, and satisfies the Lipschitz condition on the complementary interval (0, σ0], we prove the existence of a linear system with coefficient matrix Af (t) bounded on the half-line [0,+∞) whose lower sigma-exponent Δσ(Af) coincides with the function f(σ) on the entire interval (0,+∞). 相似文献
A method developed in Arlinski? (1987) [1] is applied to study the numerical range of quasi-sectorial contractions and to prove three main results. Our first theorem gives characterization of the maximal sectorial generator A in terms of the corresponding contraction semigroup {exp(−tA)}t?0. The second result establishes for these quasi-sectorial contractions a quite accurate localization of their numerical range. We give for this class of semigroups a new proof of the Euler operator-norm approximation: exp(−tA)=limn→∞(I+tA/n)−n, t?0, with the optimal estimate: O(1/n), of the convergence rate, which takes into account the value of the sectorial generator angle (the third result). 相似文献
Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1. 相似文献
Let Xn = {Xn(t): 0 ⩽ t ⩽1}, n ⩾ 0, be a sequence of square-integrable martingales. The main aim of this paper is to give sufficient conditions under which ∫·0fn (An(t), Xn(t)) dXn(t) converges weakly in D[0, 1] to ∫·0f0(A0(t), X0(t)) dX0 (t) as n → ∞, where {An, n ⩾ 0} is some sequence of increasing processes corresponding to the sequence {Xn, n ⩾ 0}. 相似文献
Let Jωx(t) = x + ∝0tbω(s) ds, where bω is planar Brownian motion starting at 0. A Wiener-type criterion is proved for the process Jωx(t): Let K be a compact plane set and let x?K. Then if ∑ 2nM1(An(x)?K) < ∞ (where and M1 denotes one-dimensional Hausdorff content), the process Jωx(t) stays within K for a positive period of time t, a.s. In particular, this applies to almost all x with respect to area in the nowhere dense “Swiss Cheese” sets. The method is based on general potential theory for Markov processes. 相似文献
In the context of the degree/diameter problem, the ‘defect’ of a graph represents the difference between the corresponding Moore bound and its order. Thus, a graph with maximum degree d and diameter two has defect two if its order is n=d2−1. Only four extremal graphs of this type, referred to as (d,2,2)-graphs, are known at present: two of degree d=3 and one of degree d=4 and 5, respectively. In this paper we prove, by using algebraic and spectral techniques, that for all values of the degree d within a certain range, (d,2,2)-graphs do not exist.The enumeration of (d,2,2)-graphs is equivalent to the search of binary symmetric matrices A fulfilling that AJn=dJn and A2+A+(1−d)In=Jn+B, where Jn denotes the all-one matrix and B is the adjacency matrix of a union of graph cycles. In order to get the factorization of the characteristic polynomial of A in Q[x], we consider the polynomials Fi,d(x)=fi(x2+x+1−d), where fi(x) denotes the minimal polynomial of the Gauss period , being ζi a primitive ith root of unity. We formulate a conjecture on the irreducibility of Fi,d(x) in Q[x] and we show that its proof would imply the nonexistence of (d,2,2)-graphs for any degree d>5. 相似文献
Suppose that ?n is the p-dimensional space with Euclidean norm ∥ ? ∥, K (?p) is the set of nonempty compact sets in ?p, ?+ = [0, +∞), D = ?+ × ?m × ?n × [0, a], D0 = ?+ × ?m, F0: D0 → K (?m), and co F0 is the convex cover of the mapping F0. We consider the Cauchy problem for the system of differential inclusions $$\dot x \in \mu F(t,x,y,\mu ),\quad \dot y \in G(t,x,y,\mu ),\quad x(0) = x_0 ,\quad y(0) = y_0$$ with slow x and fast y variables; here F: D → K (?m), G: D → K (?n), and μ ∈ [0, a] is a small parameter. It is assumed that this problem has at least one solution on [0, 1/μ] for all sufficiently small μ ∈ [0, a]. Under certain conditions on F, G, and F0, comprising both the usual conditions for approximation problems and some new ones (which are weaker than the Lipschitz property), it is proved that, for any ε > 0, there is a μ0 > 0 such that for any μ ∈ (0, μ0] and any solution (xμ(t), yμ(t)) of the problem under consideration, there exists a solution uμ(t) of the problem ${\dot u}$ ∈ μ co F0 (t, u), u(0) = x0 for which the inequality ∥xμ(t) ? uμ(t)∥ < ε holds for each t ∈ [0, 1/μ]. 相似文献
We consider the problem of the identification of the time-varying matrix A(t) of a linear m-dimensional differential system y′ = A(t)y. We develop an approximation An,k = ∑nj ? 1cj{Y(tk + τj) Y?1(tk) ? I} to A(tk) for grid points tk = a + kh, k = 0,…, N using specified τj = θjh, 0 < θj < 1, j = 1, …, n, and show that for each tk, the L1 norm of the error matrix is (hn). We demonstrate an efficient scheme for the evaluation of An,k and treat sample problems. 相似文献
Let denote the set of continuous n×n matrices on an interval . We say that is a nontrivial k-involution if where ζ=e-2πi/k, d0+d1++dk-1=n, and with . We say that is R-symmetric if R(t)A(t)R-1(t)=A(t), , and we show that if A is R-symmetric then solving x′=A(t)x or x′=A(t)x+f(t) reduces to solving k independent dℓ×dℓ systems, 0ℓk-1. We consider the asymptotic behavior of the solutions in the case where . Finally, we sketch analogous results for linear systems of difference equations. 相似文献
In this paper we investigate the problem of the equiconvergence on TN = [-π, π)N of the expansions in multiple trigonometric series and Fourier integral of functions f ∈ Lp(TN) and g ∈ Lp(?N), where p > 1, N ≥ 3, g(x) = f(x) on TN, in the case when the “rectangular partial sums” of the indicated expansions, i.e.,–n(x; f) and Jα(x; g), respectively, have indices n ∈ ?N and α ∈ ?N (nj = [αj], j = 1,...,N, [t] is the integer part of t ∈ ?1), in those certain components are the elements of “lacunary sequences”. 相似文献
In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A1(t)u,A2(t)u,?,A?(t)u), (Ai(t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (Ai(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations. 相似文献
Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let Lp(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P ? Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ Lp(M, τ), then A2 ∈ Lp(M, τ). Let n be a positive integer, n > 2, and A = An ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μt(A) belong to the set {0} ∪ [‖An?2‖?1, ‖A‖] for all t > 0; 2) either μt(A) ≥ 1 for all t > 0 or there is a t0 > 0 such that μt(A) = 0 for all t > t0. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z2 = 0, ZP = 0, and PZ = Z. Here, if Q ∈ Lp(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L1(M, τ) and if A = A3 and A ? A2 ∈ M, then τ(A) ∈ R. 相似文献
In this paper, we study nonautonomous Cauchy problems (NCP) {u̇(t) = A(t)u(t)u(s) = x ∈ X for a family of linear operators (A(t))t∈I on some Banach space X by means of evolution semigroups. In particular, we characterize “stability” in the so called “hyperbolic case” on the level of evolution semigroups and derive a product formula for the solutions of (NCP). Moreover, in Section 4 we connect the “hyperbolic” and the “parabolic” case by showing, that integrals ∫tsA(τ) dτ always define generators. This yields another product formula. 相似文献
We prove that if a functionf ∈C(1) (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialPn =Pn(x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ωk (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialPn =Pn (x) of degree ≤n such thatPn (x) ≥ 0,x ∈I, and |f(x) ?Pn(x)| ≤c(k)ωk (f;n?2 +n?1 √1 ?x2),x ∈I. 相似文献
Given an n×n real matrix A with nonnegative off-diagonal entries, the solution to , x0=x(0), t?0 is x(t)=etAx0. The problem of identifying the initial points x0 for which x(t) becomes and remains entrywise nonnegative is considered. It is known that such x0 are exactly those vectors for which the iterates x(k)=(I+hA)kx0 become and remain nonnegative, where h is a positive, not necessarily small parameter that depends on the diagonal entries of A. In this paper, this characterization of initial points is extended to a numerical test when A is irreducible: if x(k) becomes and remains positive, then so does x(t); if x(t) fails to become and remain positive, then either x(k) becomes and remains negative or it always has a negative and a positive entry. Due to round-off errors, the latter case manifests itself numerically by x(k) converging with a relatively small convergence ratio to a positive or a negative vector. An algorithm implementing this test is provided, along with its numerical analysis and examples. The reducible case is also discussed and a similar test is described. 相似文献