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1.
Let Γn, n ≥ 2, denote the symmetrized polydisc in ?n, and Γ1 be the closed unit disc in ?. We provide some characterizations of elements in Γn. In particular, an element (s1,..., sn?1, p) ∈ ?n is in Γn if and only if \({s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p\), j = 1,..., n ? 1, for some (β1,..., βn?1) ∈ Γn?1, and |p| ≤ 1. 相似文献
2.
Mordechay B. Levin 《Israel Journal of Mathematics》2010,178(1):61-106
In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ? ? s , we construct Kronecker-like and van der Corput-like ergodic transformations T 1,Γ and T 2,Γ of [0, 1) s . We prove that for admissible lattices Γ, (T ν,Γ n (x))n≥0 is a low discrepancy sequence for all x ∈ [0, 1) s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ? [0, 1) s , for almost all lattices Γ ∈ L s = SL(s,?)/SL(s, ?) (in the sense of the invariant measure on L s ), the following asymptotic formula holds with arbitrary small ? > 0, for all x ∈ [0, 1) s , and ν ∈ {1, 2}.
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$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$
3.
Z. I. Bezhaeva V. L. Kulikov E. F. Olekhova V. I. Oseledets 《Proceedings of the Steklov Institute of Mathematics》2017,297(1):28-34
Let A ∈ M n (?) be a matrix with eigenvalues greater than 1 in absolute value. The ? n -valued random variables ξ t , t ∈ ?, are i.i.d., and P(ξ t = j) = p j , j ∈ ? n , 0 < p 0 < 1, ∑ j p j = 1. We study the properties of the distributions of the ? n -valued random variable ζ 1 = ∑ t=1 ∞ A ?t ξ t and of the random variable ζ = ∑ t=0 ∞ A t ξ ?t taking integer A-adic values. We obtain a necessary and sufficient condition for the absolute continuity of these distributions. We define an invariant Erd?s measure on the compact abelian group of A-adic integers. We also define an A-invariant Erd?s measure on the n-dimensional torus. We show the connection between these invariant measures and functions of countable stationary Markov chains. In the case when |{j: p j ≠ 0}| < ∞, we establish the relation between these invariant measures and finite stationary Markov chains. 相似文献
4.
In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators \(\mathcal{M}^+\) and \(\mathcal{M}^-\). More precisely, we prove that \(\mathcal{M}^+\) and \(\mathcal{M}^-\) map W 1,p (?) → W 1,p (?) with 1 < p < 1, boundedly and continuously. In addition, we show that the discrete versions M + and M ? map BV(?) → BV(?) boundedly and map l 1(?) → BV(?) continuously. Specially, we obtain the sharp variation inequalities of M + and M ?, that is if f ∈ BV(?), where Var(f) is the total variation of f on ? and BV(?) is the set of all functions f: ? → ? satisfying Var(f) < 1.
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$$Var\left( {{M^ + }\left( f \right)} \right) \leqslant Var\left( f \right)andVar\left( {{M^ - }\left( f \right)} \right) \leqslant Var\left( f \right)$$
5.
A. A. Illarionov 《Proceedings of the Steklov Institute of Mathematics》2017,299(1):96-108
Functional equations of the form f(x + y)g(x ? y) = Σ j=1 n α j (x)β j (y) as well as of the form f1(x + z)f2(y + z)f3(x + y ? z) = Σ j=1 m φ j (x, y)ψ j (z) are solved for unknown entire functions f, g,α j , β j : ? → ? and f1, f2, f3, ψ j : ? → ?, φ j : ?2 → ? in the cases of n = 3 and m = 4. 相似文献
6.
On <Emphasis Type="Italic">s</Emphasis>-Elementary Super Frame Wavelets and Their Path-Connectedness
A super wavelet of length n is an n-tuple (ψ 1,ψ 2,…,ψ n ) in the product space \(\prod_{j=1}^{n} L^{2}(\mathbb{R})\), such that the coordinated dilates of all its coordinated translates form an orthonormal basis for \(\prod_{j=1}^{n} L^{2} (\mathbb{R})\). This concept is generalized to the so-called super frame wavelets, super tight frame wavelets and super normalized tight frame wavelets (or super Parseval frame wavelets), namely an n-tuple (η 1,η 2,…,η n ) in \(\prod_{j=1}^{n}L^{2} (\mathbb{R})\) such that the coordinated dilates of all its coordinated translates form a frame, a tight frame, or a normalized tight frame for \(\prod_{j=1}^{n} L^{2}(\mathbb{R})\). In this paper, we study the super frame wavelets and the super tight frame wavelets whose Fourier transforms are defined by set theoretical functions (called s-elementary frame wavelets). An m-tuple of sets (E 1,E 2,…,E m ) is said to be τ-disjoint if the E j ’s are pair-wise disjoint under the 2π-translations. We prove that a τ-disjoint m-tuple (E 1,E 2,…,E m ) of frame sets (i.e., η j defined by \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\) is a frame wavelet for L 2(?) for each j) lead to a super frame wavelet (η 1,η 2,…,η m ) for \(\prod_{j=1}^{m} L^{2} (\mathbb{R})\) where \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\). In the case of super tight frame wavelets, we prove that (η 1,η 2,…,η m ), defined by \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\), is a super tight frame wavelet for ∏1≤j≤m L 2(?) with frame bound k 0 if and only if each η j is a tight frame wavelet for L 2(?) with frame bound k 0 and that (E 1,E 2,…,E m ) is τ-disjoint. Denote the set of all τ-disjoint s-elementary super frame wavelets for ∏1≤j≤m L 2(?) by \(\mathfrak{S}(m)\) and the set of all s-elementary super tight frame wavelets (with the same frame bound k 0) for ∏1≤j≤m L 2(?) by \(\mathfrak{S}^{k_{0}}(m)\). We further prove that \(\mathfrak{S}(m)\) and \(\mathfrak{S}^{k_{0}}(m)\) are both path-connected under the ∏1≤j≤m L 2(?) norm, for any given positive integers m and k 0. 相似文献
7.
The generalized k-connectivity κ k (G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λ k (G) = min{λ(S): S ? V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T 1, T 2, …, T l in G such that S ? V (T i ) for 1 ? i ? l. In this paper we prove that for any two connected graphs G and H we have λ 3(G □ H) ? λ 3(G) + λ 3(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes. 相似文献
8.
Let G be a simple algebraic group of type G2 over an algebraically closed field of characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ2 and an infinite family of pairwise non-conjugate homomorphisms ρ: Γ → G whose restrictions to Γ2 are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of Γ on a connected unipotent group V such that the map of 1-cohomologies H1(Γ, V) → H1(Γp, V) induced by restriction of 1-cocycles has an infinite fibre. 相似文献
9.
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. 相似文献
10.
The system ? i = ? i (?) + x i+2, \(i \in \overline {1,n - 2} \), ? n?1 = ? n?1(?) + u 1, ? n = ? n (?) + u 2,where ? i (·) are nonanticipating functionals of an arbitrary nature with the following properties—\(\left| {{\varphi _i}\left( \cdot \right)} \right| \leqslant c\sum\nolimits_{k = 1}^i {\left| {{x_k}\left( t \right)} \right|} \), \(i \in \overline {1,n} \), c = const—and u 1 and u 2 are the controls is considered. It is assumed that only the outputs x 1 and x 2 are measurable. The problem of synthesis of both continuous and impulsive controls u1 and u2, which make the system globally asymptotically stable, is solved. The solution of the problem is based on the construction of the observer-based equations, the quadratic Lyapunov function, and the averaging method. 相似文献
11.
Set \(A\subset {\mathbb N}\) is less than \(B\subset {\mathbb N}\) in the colex ordering if m a x(A△B)∈B. In 1980’s, Frankl and Füredi conjectured that the r-uniform graph with m edges consisting of the first m sets of \({\mathbb N}^{(r)}\) in the colex ordering has the largest Lagrangian among all r-uniform graphs with m edges. A result of Motzkin and Straus implies that this conjecture is true for r=2. This conjecture seems to be challenging even for r=3. For a hypergraph H=(V,E), the set T(H)={|e|:e∈E} is called the edge type of H. In this paper, we study non-uniform hypergraphs and define L(H) a generalized Lagrangian of a non-uniform hypergraph H in which edges of different types have different weights. We study the following two questions: 1. Let H be a hypergraph with m edges and edge type T. Let C m,T denote the hypergraph with edge type T and m edges formed by taking the first m sets with cardinality in T in the colex ordering. Does L(H)≤L(C m,T ) hold? If T={r}, then this question is the question by Frankl and Füredi. 2. Given a hypergraph H, find a minimum subhypergraph G of H such that L(G) = L(H). A result of Motzkin and Straus gave a complete answer to both questions if H is a graph. In this paper, we give a complete answer to both questions for {1,2}-hypergraphs. Regarding the first question, we give a result for {1,r 1,r 2,…,r l }-hypergraph. We also show the connection between the generalized Lagrangian of {1,r 1,r 2,? ,r l }-hypergraphs and {r 1,r 2,? ,r l }-hypergraphs concerning the second question. 相似文献
12.
Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) = r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C p (3). Also if Γ(G) = Γ(B 3(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained. 相似文献
13.
Aladár Heppes 《Discrete and Computational Geometry》2011,45(2):321-328
Let K denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of K is said to have property T(k) if for every subset of at most k translates there exists a common line transversal intersecting all of them. Property T means that a transversal exists for all members of the family. Two translates, K i and K j are said to be φ-disjoint, φ>0, if the concentric φ-enlarged copies of K i and K j are disjoint. It is well known that in a φ-disjoint family of congruent discs T(3)?T if \(\varphi>\sqrt{2}\), and \(T(3)\not\Rightarrow T\) if \(\varphi<\sqrt{2}\). In this note finite φ-disjoint T(3)-families of translates of an oval will be investigated. 相似文献
14.
We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so–called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self–adjoint integral operator in a suitable L 2(μ)–space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X H with Hurst parameter H∈(0,1), indexed by a self–similar set T?? N of Hausdorff dimension D. This rate turns out to be of order n ?H/D (log?n)1/2. In the case T=[0,1] N we present a concrete wavelet representation of X H leading to an approximation of X H with the optimal rate n ?H/N (log?n)1/2. 相似文献
15.
We investigate the equiconvergence on TN = [?π, π)N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ Lp(TN) and g ∈ Lp(RN), p > 1, N ≥ 3, g(x) = f(x) on TN, in the case where the “partial sums” of these expansions, i.e., Sn(x; f) and Jα(x; g), respectively, have “numbers” n ∈ ZN and α ∈ RN (nj = [αj], j = 1,..., N, [t] is the integral part of t ∈ R1) containing N ? 1 components which are elements of “lacunary sequences.” 相似文献
16.
Let L=?Δ+V be a Schrödinger operator on ? d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L p (? d ), for some p>d /2. Let K t be the semigroup generated by ?L. We say that an L 1(? d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?t>0|K t f| belongs to L 1(? d ). We prove that \(f\in H^{1}_{L}\) if and only if R j f∈L 1(? d ) for j=1,…,d, where R j =(?/? x j )L ?1/2 are the Riesz transforms associated with L. 相似文献
17.
Considering the linear system of elasticity equations describing the wave propagation in the half-space ? + 3 = {x ∈ ?3 | x 3 > 0} we address the problem of determining the density and elastic parameters which are piecewise constant functions of x 3. The shape is unknown of a point-like impulse source that excites elastic oscillations in the half-space. We show that under certain assumptions on the source shape and the parameters of the elastic medium the displacements of the boundary points of the half-space for some finite time interval (0, T) uniquely determine the normalized density (with respect to the first layer) and the elastic Lamé parameters for x 3 ∈ [0, H], where H = H(T). We give an algorithmic procedure for constructing the required parameters. 相似文献
18.
A proper edge coloring of a graph G is said to be acyclic if there is no bicolored cycle in G.The acyclic edge chromatic number of G,denoted byχ′a(G),is the smallest number of colors in an acyclic edge coloring of G.Let G be a planar graph with maximum degree.In this paper,we show thatχ′a(G)+2,if G has no adjacent i-and j-cycles for any i,j∈{3,4,5},which implies a result of Hou,Liu and Wu(2012);andχ′a(G)+3,if G has no adjacent i-and j-cycles for any i,j∈{3,4,6}. 相似文献
19.
Let (j1,..., jn) be a permutation of the n-tuple (1, ..., n). A system of differential equations \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) in which each function fi is continuous on ? is considered. This system is said to have the property of generation of solutions with a small period if, for any number M > 0, there exists a number ω0 = ω0(M) > 0 such that if 0 < ω ≤ ω0 and hi(t, x1, ..., xn) are continuous functions on ? × ?n ω-periodic in t that satisfy the inequalities |hi| ≤ M the system \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) has an ω-periodic solution. It is shown that a system has the property of generation of solutions with a small period if and only if fi(?) = ? for i = 1,..., n. It is also shown that the smallness condition on the period is essential. 相似文献
20.
A (v, β o , μ)-design over regular graph G = (V, E) of degree d is an ordered pair D = (V, B), where |V| = v and B is the set of maximum independent sets of G called blocks such that if i, j ∈ V, i ≠ j and if i and j are not adjacent in G then there are exactly μ blocks containing i and j. In this paper, we study (v, β o , μ)-designs over the graphs K n × K n , T(n)-triangular graphs, L 2(n)-square lattice graphs, Petersen graph, Shrikhande graph, Clebsch graph and the Schläfli graph and non-existence of (v, β o , μ)-designs over the three Chang graphs T 1(8), T 2(8) and T 3(8). 相似文献