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1.
A new recurrence for the number of subgroups of given index in the modular group is derived. The proof requires the derivation of a recurrence for a sequencea nbn from recurrences for thea n andb n. We show that this is always possible if thea n andb n satisfy polynomial recurrences. We also include a short proof of a result ofW. W. Stothers on the parity of the number of subgroups of given index in the modular group.  相似文献   

2.
We call a set of edgesE of the n-cubeQ n a fundamental set for Q n if for some subgroupG of the automorphism group ofQ n , theG-translates ofE partition the edge set ofQ n .Q n possesses an abundance of fundamental sets. For example, a corollary of one of our main results is that if |E| =n and the subgraph induced byE is connected, then if no three edges ofE are mutually parallel,E is a fundamental set forQ n . The subgroupG is constructed explicitly. A connected graph onn edges can be embedded intoQ n so that the image of its edges forms such a fundamental set if and only if each of its edges belongs to at most one cycle.We also establish a necessary condition forE to be a fundamental set. This involves a number-theoretic condition on the integersa j (E), where for 1 j n, a j (E) is the number of edges ofE in thej th direction (i.e. parallel to thej th coordinate axis).  相似文献   

3.
Assume an additional congruent condition on the coefficients. We prove that the pair 5 of linear equations ∑j=1^5 αλjpj = bλ (λ= 1, 2) has solutions in primes pj satisfying pj 〈〈 (|b1|+|b2|+1) maxλ,j |αλj|^2318+ε. This improves the exponent 79680 without assuming the additional condition of the second author's.  相似文献   

4.
A Steinhaus graph is a graph with n vertices whose adjacency matrix (ai, j) satisfies the condition that ai, j ? aa--1, j--1 + a i--1, j (mod 2) for each 1 < i < jn. It is clear that a Steinhaus graph is determined by its first row. In [3] Bringham and Dutton conjecture that almost all Steinhaus graphs have diameter 2. That is, as n approaches infinity, the ratio of the number of Steinhaus graphs with n vertices having diameter 2 to the total number of Steinhaus graphs approaches 1. Here we prove Bringham and Dutton's conjecture.  相似文献   

5.
LetP be a convexd-polytope without triangular 2-faces. Forj=0,…,d−1 denote byf j(P) the number ofj-dimensional faces ofP. We prove the lower boundf j(P)≥f j(C d) whereC d is thed-cube, which has been conjectured by Y. Kupitz in 1980. We also show that for anyj equality is only attained for cubes. This result is a consequence of the far-reaching observation that such polytopes have pairs of disjoint facets. As a further application we show that there exists only one combinatorial type of such polytopes with exactly 2d+1 facets.  相似文献   

6.
We discuss local solvability of operators of the form where theV j are left-invariant vector fields on the Heisenberg group such that [V j ,V j+n ]=U for 1≤jn andA=(a jk )=A 1+iA 2 is a complex symmetric matrix satisfying the “cone condition” |A 2|≤CA 1. The authors acknowledge the support for this work by the European Commission through the European HCM-program “Fourier Analysis” and the TMR network “Harmonic Analysis”.  相似文献   

7.
Consider a setA of symmetricn×n matricesa=(a i,j) i,jn . Consider an independent sequence (g i) in of standard normal random variables, and letM=Esupa∈Ai,j⪯nai,jgigj|. Denote byN 2(A, α) (resp.N t(A, α)) the smallest number of balls of radiusα for thel 2 norm ofR n 2 (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN 2(A, α))1/4KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN tK(δ)M. Work partially supported by an N.S.F. grant.  相似文献   

8.
Anm×nmatrix =(ai, j), 1≤imand 1≤jn, is called atotally monotonematrix if for alli1, i2, j1, j2, satisfying 1≤i1<i2m, 1≤j1<j2n.[formula]We present an[formula]time algorithm to select thekth smallest item from anm×ntotally monotone matrix for anykmn. This is the first subquadratic algorithm for selecting an item from a totally monotone matrix. Our method also yields an algorithm of the same time complexity for ageneralized row-selection problemin monotone matrices. Given a setS={p1,…, pn} ofnpoints in convex position and a vectork={k1,…, kn}, we also present anO(n4/3logc n) algorithm to compute thekith nearest neighbor ofpifor everyin; herecis an appropriate constant. This algorithm is considerably faster than the one based on a row-selection algorithm for monotone matrices. If the points ofSare arbitrary, then thekith nearest neighbor ofpi, for allin, can be computed in timeO(n7/5 logc n), which also improves upon the previously best-known result.  相似文献   

9.
《Optimization》2012,61(5):729-745
We consider mixed-integer sets of the form X = {(s, y) ∈ ?+ × ? n : s + a j y j b j , ?jN}. A polyhedral characterization for the case where X is defined by two inequalities is provided. From that characterization we give a compact formulation for the particular case where the coefficients of the integer variables can take only two possible integer values a 1, a 2 ∈ ?+ : X n,m = {(s, y) ∈ ?+ × ? n+m : s + a 1 y j b j , ?jN 1, s + a 2 y j b j , jN 2} where N 1 = {1, …, n}, N 2 = {n + 1, …, n + m}. In addition, we discuss families of facet-defining inequalities for the convex hull of X n,m .  相似文献   

10.
We consider aM X/G/1 queueing system withN-policy. The server is turned off as soon as the system empties. When the queue length reaches or exceeds a predetermined valueN (threshold), the server is turned on and begins to serve the customers. We place our emphasis on understanding the operational characteristics of the queueing system. One of our findings is that the system size is the sum of two independent random variables: one has thePGF of the stationary system size of theM X/G/1 queueing system withoutN-policy and the other one has the probability generating function j=0 N=1 j z j/ j=0 N=1 j , in which j is the probability that the system state stays atj before reaching or exceedingN during an idle period. Using this interpretation of the system size distribution, we determine the optimal thresholdN under a linear cost structure.  相似文献   

11.
Let {Ai }and {Bi } be two given families of n-by-n matrices. We give conditions under which there is a unitary U such that every matrix UAiU 1 is upper triangular. We give conditions, weaker than the classical conditions of commutativity of the whole family, under which there is a unitary U such that every matrix UAjU ? is upper triangular. We also give conditions under which there is one single unitary U such that every UAiU 1 and every UBjU ? is upper triangular. We give necessary and sufficient conditions for simultaneous unitary reduction to diagonal form in this way when all the Aj's are complex symmetric and all theBj 's are Hermitian.  相似文献   

12.
Let G=&langle; a1,&ldots; , a n | a i a j a i &ldots; = a_ja_ia_j,&ldots; ,i>j&rangle; be an Artin group and let m ij =m ji be the length of each of the sides of the defining relation involving a i and a j . We show if all m_ij &ges; 7 then G is relatively hyperbolic in the sense of Farb with respect to the collection of its two-generator subgroups a i , a j for which m_ij >&infty;.  相似文献   

13.
It is shown that if, for some 2≦jd-2, all thej-faces of ad-polytopeP are centrally symmetric, then all the faces ofP of every dimension are centrally symmetric.  相似文献   

14.
In this article, we extend the recently developed abstract theory of universal series to include averaged sums of the form \frac1f(n)?j=0n aj xj{\frac{1}{\phi(n)}\sum_{j=0}^{n} a_j x_j} for a given fixed sequence of vectors (x j ) in a topological vector space X over a field \mathbbK{\mathbb{K}} of real or complex scalars, where (f(n)){(\phi(n))} is a sequence of non-zero field scalars. We give necessary and sufficient conditions for the existence of a sequence of coefficients (a j ) which make the above sequence of averaged sums dense in X. When applied, the extended theory gives new analogues to well known classical theorems including those of Seleznev, Fekete and Menchoff.  相似文献   

15.
A matrix ARn×n is called a bisymmetric matrix if its elements ai,j satisfy the properties ai,j=aj,i and ai,j=an-j+1,n-i+1 for 1?i,j?n. This paper considers least squares solutions to the matrix equation AX=B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem.  相似文献   

16.
Let I≥1 be an integer, ω 0=0<ω 1<⋯<ω I π, and for j=0,…,I, a j ∈ℂ, a-j=[`(aj)]a_{-j}={\overline{{a_{j}}}}, ω j =−ω j , and aj 1 0a_{j}\not=0 if j 1 0j\not=0. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form
[(x)\tilde](k) = ?j=-II ajexp(-iwjk) +e(k),     k=-2N,?,2N,\tilde{x}(k)= \sum_{j=-I}^I a_j\exp(-i\omega _jk) +\epsilon (k), \quad k=-2N,\ldots,2N,  相似文献   

17.
A ringR with an involutionaa* which satisfies a polynomial identityp[x 1,…,x d ;x*1, …,x* d ]=0 satisfies- an identity which does not include thex*. This generalizes the result of [1] where the symmetric elements ofR were assumed to satisfy an identity.  相似文献   

18.
Given a finite sequence a{a1, …, aN} in a domain Ω n, and complex scalars v{v1, …, vN}, consider the classical extremal problem of finding the smallest uniform norm of a holomorphic function verifying f(aj)=vj for all j. We show that the modulus of the solutions to this problem must approach its least upper bound along a subset of the boundary of the domain large enough so that its A(Ω)-hull contains a subset of the original a large enough to force the same minimum norm. Furthermore, all the solutions must agree on a variety which contains the hull (in an appropriate, weaker, sense) of a measure supported on the maximum modulus set. An example is given to show that the inclusions can be strict.  相似文献   

19.
Let ξ, ξ1, ξ2, ... be independent identically distributed random variables, and S n :=Σ j=1 n j , $ \bar S $ \bar S := sup n≥0 S n . If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $ \bar S $ \bar S as a → 0 and $ \bar S $ \bar S tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:
1.  The first model assumes that ξ j can be represented as ξ j = ζ j + αη j , where ζ1, ζ 2 , ... and η 1, η 2, ... are two independent sequences of independent random variables, identically distributed in each sequence, such that supn≥0Σ j=1 n ζ j = ∞, sup n≥0Σ j=1 n η j < ∞, and $ \bar S $ \bar S < ∞ almost surely.
2.  In the second model we consider a triangular array scheme with parameter a and assume that the right tail distribution P j t) ∼ V (t) as t→∞ depends weakly on a, while the left tail distribution is P j < −t) = W(t/a), where V and W are regularly varying functions and $ \bar S $ \bar S < ∞ almost surely for every fixed α > 0.
We obtain some results for identically and differently distributed ξ j .  相似文献   

20.
A quasi-ordered setA (i.e. one equipped with a reflexive and transitive relation ) is said to be well-quasi-ordered (wqo) if for every infinite sequencea 1,a 2, ... of elements ofA there are indicesi, j such thati < j anda i a j.Various natural wqo setsQ admit labelling by another wqoA yielding another quasi-ordered setQ(A), which may or may not be wqo. A suitable concept covering this phenomenon is the notion of aQO-category. We have two conjectures aboutQO-categories in the effect that labellingQO-categories by a wqo set can always be reduced to labelling by ordinals. We prove these conjectures for a broad class ofQO-categories and for generalQO-categories we prove weaker forms of these conjectures.  相似文献   

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