共查询到20条相似文献,搜索用时 62 毫秒
1.
Yair Glasner 《Israel Journal of Mathematics》2017,219(1):215-270
Let Γ < GL n (F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty topology. An invariant random subgroup (IRS) of Γ is a conjugation invariant Borel probability measure on Sub(Γ). An IRS is called non-trivial if it does not have an atom in the trivial group, i.e. if it is non-trivial almost surely. We denote by IRS0(Γ) the collection of all non-trivial IRS on Γ.
Theorem 0.1: With the above notation, there exists a free subgroup F < Γ and a non-discrete group topology on Γ such that for every μ ∈ IRS0(Γ) the following properties hold:
Φ: (Sub(Γ), μ) → (Sub(F),Φ*μ) Δ → Δ ∩ F is an F-invariant isomorphism of probability spaces.A more technical version of this theorem is valid for general countable linear groups. We say that an action of Γ on a probability space, by measure preserving transformations, is almost surely non-free (ASNF) if almost all point stabilizers are non-trivial.Corollary 0.2: Let Γ be as in the Theorem above. Then the product of finitely many ASNF Γ-spaces, with the diagonal Γ action, is ASNF.Corollary 0.3: Let Γ < GLn(F) be a countable linear group, A Δ Γ the maximal normal amenable subgroup of Γ — its amenable radical. If μ ∈ IRS(Γ) is supported on amenable subgroups of Γ, then in fact it is supported on Sub(A). In particular, if A(Γ) = <e> then Δ = <e>, μ almost surely. 相似文献
μ-almost every subgroup of Γ is open
- F ·Δ = Γ for μ-almost every Δ ∈ Sub(Γ).
- F ∩ Δ is infinitely generated, for every open subgroup. In particular, this holds for μ-almost every Δ ∈ Sub(Γ).
- The map
2.
Given a connected edge-regular graph Γ with parameters (v, k, λ) and b 1 = k ? λ ? 1, we prove that in the case k ≥ 3b 1 ?2 either |Γ2(u)|(k?2b 1 + 2) < kb 1 for every vertex u or Γ is a polygon, the edge graph of a trivalent graph without triangles that has diameter greater than 2, the icosahedral graph, the complete multipartite graph K r×2, the 3 × 3-grid, the triangular graph T(m) with m ≤ 7, the Clebsch graph, or the Schläfli graph. 相似文献
3.
V. A. Bykovskii 《Doklady Mathematics》2016,94(2):527-528
Given any nonzero entire function g: ? → ?, the complex linear space F(g) consists of all entire functions f decomposable as f(z + w)g(z - w)=φ1(z)ψ1(w)+???+ φn(z)ψn(w) for some φ1, ψ1, …, φn, ψn: ? → ?. The rank of f with respect to g is defined as the minimum integer n for which such a decomposition is possible. It is proved that if g is an odd function, then the rank any function in F(g) is even. 相似文献
4.
Hiroshi Suzuki 《Journal of Algebraic Combinatorics》2009,30(3):401-413
Let Γ=(X,R) be a distance-regular graph of diameter d. A parallelogram of length i is a 4-tuple xyzw consisting of vertices of Γ such that ?(x,y)=?(z,w)=1, ?(x,z)=i, and ?(x,w)=?(y,w)=?(y,z)=i?1. A subset Y of X is said to be a completely regular code if the numbers depend only on i=?(x,Y) and j. A subset Y of X is said to be strongly closed if Hamming graphs and dual polar graphs have strongly closed completely regular codes. In this paper, we study parallelogram-free distance-regular graphs having strongly closed completely regular codes. Let Γ be a parallelogram-free distance-regular graph of diameter d≥4 such that every strongly closed subgraph of diameter two is completely regular. We show that Γ has a strongly closed subgraph of diameter d?1 isomorphic to a Hamming graph or a dual polar graph. Moreover if the covering radius of the strongly closed subgraph of diameter two is d?2, Γ itself is isomorphic to a Hamming graph or a dual polar graph. We also give an algebraic characterization of the case when the covering radius is d?2.
相似文献
$\pi_{i,j}=|\Gamma_{j}(x)\cap Y|\quad (i,j\in \{0,1,\ldots,d\})$
$\{x\mid \partial(u,x)\leq \partial(u,v),\partial(v,x)=1\}\subset Y,\mbox{ whenever }u,v\in Y.$
5.
In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and \({\mathcal{F}}\) a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair \({f(z),g(z)\in \mathcal{F}}\) and one of the following conditions holds: (1) P(z) ? α(z 0) has at least three distinct zeros for any \({z_{0}\in D}\); (2) There exists \({z_{0}\in D}\) such that P(z) ? α(z 0) has at most two distinct zeros and α(z) is nonconstant. Assume that β 0 is a zero of P(z) ? α(z 0) with multiplicity p and that the multiplicities l and k of zeros of f(z) ? β 0 and α(z) ? α(z 0) at z 0, respectively, satisfy k ≠ lp, for all \({f(z)\in\mathcal{F}}\). Then \({\mathcal{F}}\) is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion. 相似文献
6.
We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Kar?owicz (in [?]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected iff for some a ∈ X ∩ Y the order-type of a ∩ X is different from that of a ∩ Y.In addition to the chromatic number x(G) of these graphs we study χ κ (G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph. 相似文献
7.
V. M. Buchstaber 《Proceedings of the Steklov Institute of Mathematics》2016,294(1):176-200
We explicitly construct polynomial vector fields Lk, k = 0, 1, 2, 3, 4, 6, on the complex linear space C6 with coordinates X = (x2, x3, x4) and Z = (z4, z5, z6). The fields Lk are linearly independent outside their discriminant variety Δ ? C6 and are tangent to this variety. We describe a polynomial Lie algebra of the fields Lk and the structure of the polynomial ring C[X,Z] as a graded module with two generators x2 and z4 over this algebra. The fields L1 and L3 commute. Any polynomial P(X,Z) ∈ C[X,Z] determines a hyperelliptic function P(X,Z)(u1, u3) of genus 2, where u1 and u3 are the coordinates of trajectories of the fields L1 and L3. The function 2x2(u1, u3) is a two-zone solution of the Korteweg–de Vries hierarchy, and ?z4(u1, u3)/?u1 = ?x2(u1, u3)/?u3. 相似文献
8.
On the total multiplicity of zeros of generalized polynomials related to nonoscillating trajectories
For a continuous curve L = {x: x = Z(t), t ∈ [a, b]} in R n , we study the number of zeros of the function l h (t) = 〈h, Z(t)〉, where h ∈ R n . We introduce the notion of multiple zeros for such functions and study the possibility of estimating the total multiplicity of such zeros under the assumption that the system {z 1(t), z 2(t), …, z n (t)} of coordinates of the function Z(t) is a Chebyshev system on [a, b]. 相似文献
9.
B. Simon 《Functional Analysis and Its Applications》2007,41(2):143-153
We show that the parameters a n , b n of a Jacobi matrix have a complete asymptotic expansion , where 1 < |µj| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ?1) is an entire meromorphic function. We relate the poles of u to the µj’s.
相似文献
$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $
10.
Yohei Sato 《Calculus of Variations and Partial Differential Equations》2007,29(3):365-395
We study the nonlinear Schrödinger equations: \(-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).\) where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has “critical frequency” \(\inf_{x\in{\bf R}^{N}} V(x)=0\). We also assume that V(x) satisfies \(0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty\) and V(x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V(x). Their limiting profiles—which depend on the local behavior of the potential V(x)—are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u ε(x) as follows: rescaled function \(w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)\) converges to a least energy solution of ?Δw + V 0(y) w = w p , w > 0 in Ω0, \(w\in H^{1}_0(\Omega_0)\). Here g(ε), V 0(x) and Ω0 depend on the local behaviors of V(x). 相似文献
11.
An arithmetic function f is called a sieve function of range Q if its Eratosthenes transform g = f * μ is supported in [1,Q] ∩ N, where g(q) ? ε q ε (?ε > 0). We continue our study of the distribution of f(n) over short arithmetic bands, n ≡ ar + b (mod q), with n ∈ (N,2N] ∩ N, 1 ≤ a ≤ H = o(N) and r,b ∈ Z such that g:c:d:(r,q) = 1. In particular, the optimality of some results is discussed. 相似文献
12.
In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u0(x) = u(x, 0) and the boundary data g0(t) = u(0, t), g1(t) = ux(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.
相似文献
$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$
13.
Let L be a lattice of finite length, ξ = (x 1,…, x k )∈L k , and y ∈ L. The remoteness r(y, ξ) of y from ξ is d(y, x 1)+?+d(y, x k ), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x 1∨?∨x k . In other words, L satisfies the so-called c 1 -median property. 相似文献
14.
It is proved that if an entire function f: ? → ? satisfies an equation of the form α 1(x)β 1(y) + α 2(x)β 2(y) + α 3(x)β 3(y), x,y ∈ C, for some α j , β j : ? → ? and there exist no \({\widetilde \alpha _j}\) and ?\({\widetilde \beta _j}\) for which \(f\left( {x + y} \right)f\left( {x - y} \right) = {\overline \alpha _1}\left( x \right){\widetilde \beta _1}\left( y \right) + {\overline \alpha _2}\left( x \right){\widetilde \beta _2}\left( y \right)\), then f(z) = exp(Az 2 + Bz + C) ? σ Γ(z - z 1) ? σ Γ(z - z 2), where Γ is a lattice in ?; σ Γ is the Weierstrass sigma-function associated with Γ; A,B,C, z 1, z 2 ∈ ?; and \({z_1} - {z_2} \notin \left( {\frac{1}{2}\Gamma } \right)\backslash \Gamma \). 相似文献
15.
R. Nair 《Periodica Mathematica Hungarica》2012,64(1):39-51
Let S be a countable semigroup acting in a measure-preserving fashion (g ? T g ) on a measure space (Ω, A, µ). For a finite subset A of S, let |A| denote its cardinality. Let (A k ) k=1 ∞ be a sequence of subsets of S satisfying conditions related to those in the ergodic theorem for semi-group actions of A. A. Tempelman. For A-measureable functions f on the measure space (Ω, A, μ) we form for k ≥ 1 the Templeman averages \(\pi _k (f)(x) = \left| {A_k } \right|^{ - 1} \sum\nolimits_{g \in A_k } {T_g f(x)}\) and set V q f(x) = (Σ k≥1|π k+1(f)(x) ? π k (f)(x)|q)1/q when q ∈ (1, 2]. We show that there exists C > 0 such that for all f in L 1(Ω, A, µ) we have µ({x ∈ Ω: V q f(x) > λ}) ≤ C(∫Ω | f | dµ/λ). Finally, some concrete examples are constructed. 相似文献
16.
In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u~3+ β(x)v~2u, x ∈R~N,-?v + V_2(x)v = μ_2(x)v~3+ β(x)u~2v, x ∈R~N,u, v ∈H~1(R~N),where N = 1, 2, 3; V_1(x) and V_2(x) are positive and continuous, but may not be well-shaped; and μ_1(x), μ_2(x)and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), μ_i(x)(i = 1, 2) and β(x) satisfy some additional conditions. 相似文献
17.
A. S. Blagoveschenskiĭ 《Siberian Mathematical Journal》2009,50(4):596-602
We pose and solve an inverse problem of finding a coefficient in the wave equation in the inhomogeneous semispace on the scattering data of a plane wave incident from the homogeneous semispace. The unknown coefficient is a sum of a deterministic summand of one variable (the “depth” z) and a small random summand α(x, z). We look for the deterministic summand, the expectation E(α(x, z)) =: m(z), and the second moment r(x 1 t - x 2, z 1, z 2):= E(α(x 1, z 1)α(x 2, z 2)). Here the symbol E(·) stands for expectation. The stratification property of a medium means that (i) the deterministic summand depends only on z, (ii) m(z) depends only on z, and (iii) the second moment for fixed z 1 and z 2 depends only on x 1 ? x 2. 相似文献
18.
ZhangJie Wang 《中国科学 数学(英文版)》2017,60(4):593-612
Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values. 相似文献
19.
Let R be a right coherent ring and D~b(R-Mod) the bounded derived category of left R-modules. Denote by D~b(R-Mod)_([G F,C]) the subcategory of D~b(R-Mod) consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K~b(F ∩ C) the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category D~b(R-Mod)_([G F,C])/K~b(F ∩ C) is triangle-equivalent to the stable category GF ∩ C of the Frobenius category of all Gorenstein flat and cotorsion left R-modules. 相似文献
20.
Let d ≥ 1 and Z be a subordinate Brownian motion on R~d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L~b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p~b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L~b, C_c~∞(R~d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution. 相似文献