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1.
P. D. Seymour 《Combinatorica》1982,2(1):91-97
De Bruijn and Erdős proved that ifA
1, ...,A
k
are distinct subsets of a set of cardinalityn, and |A
i
∩A
j
|≦1 for 1≦i<j ≦k, andk>n, then some two ofA
1, ...,A
k
have empty intersection. We prove a strengthening, that at leastk /n ofA
1, ...,A
k
are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary.
Partially supported by N. S. F. grant No. MCS—8103440 相似文献
2.
Amiram Braun 《Israel Journal of Mathematics》1986,55(3):345-349
An example is given of a ringR (with 1) satisfying the standard identityS
6[x
1, ...,x
6] butM
2(R), the 2 × 2 matrix ring overR, does not satisfyS
12[x
1, ...,x
12]. This is in contrast to the caseR=M
n (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS
2n [x
1, ...,x
2n] andM
2(R) satisfiesS
4n
[x
1, ...,x
n].
Part of this work was done while the author enjoyed the hospitality of the University of California at San Diego, the University
of Texas at Austin and the University of Washington at Seattle. 相似文献
3.
Kh. D. Ikramov 《Computational Mathematics and Mathematical Physics》2009,49(5):743-747
Let s 1, ..., s n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s i is an eigenvalue of the leading principal submatrix A i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag(s 1, ..., s n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A i and A i + 1. 相似文献
4.
In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g)
S
= V(f) A
V(g)
T
+ <u, f
(N)
g
(N)V(f) =(f(c
0), f "(c
0), ..., f
(n – 1)
0(c
0), ..., f(c
p
), f "(c
p
), ..., f
(n – 1)
p(c
p
))
u is a regular linear functional on the linear space P of real polynomials, c
0, c
1, ..., c
p
are distinct real numbers, n
0, n
1, ..., n
p
are positive integer numbers, N=n
0+n
1+...+n
p
, and A is a N × N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation. 相似文献
5.
Marc Levine 《K-Theory》1992,6(2):113-175
LetR be a commutative, semi-local ring,I
1, ...,I
s
ideals. In this paper, we define therelative Milnor K-groups of (R;I
1, ...,I
s
),K
p
M
(R;I
1, ...,I
s
), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK
p
M
(R;I
1, ...,I
s
) is isomorphic to the weightp portion of the relative QuillenK-groupK
p
(R;I
1, ...,I
s
), after inverting (p–1)!. We also define the relative group homology of GL
n
(R;I
1, ...,I
s
), and show thatK
p
M
(R;I
1, ...,I
s
) is isomorphic toH
p
(GLp(R;I
1, ...,I
s
))/Im(H
p
(GL
p–1 (R;I
1, ...,I
s
))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK
p
M
(F)/l
v
to
, and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:
相似文献
6.
Phil Hanlon 《Inventiones Mathematicae》1986,86(1):131-159
Summary LetA+(k) denote the ring [t]/t
k+1
and letG be a reductive complex Lie algebra with exponentsm
1, ...,m n. This paper concerns the Lie algebra cohomology ofGA
+(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA
+(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m
s
+1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m
s
+1, one has weight 0 and the others have weights (k+1)m
s
+t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of A
+(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA
n
,B
n
,C
n
, orD
n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.Partially supported by NSF grant number MCS-8401718 and a Bantrell Fellowship 相似文献
7.
E. Kissin D. S. Potapov V. S. Shulman F. A. Sukochev 《Functional Analysis and Its Applications》2011,45(2):157-159
It is proved that, for any Lipschitz function f(t
1, ..., t
n
) of n variables, the corresponding map f
op: (A
1, ...,A
n
) → f(A
1, ..., A
n
) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal S
p
, p ∈ (1,∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz
functions of one variable preserve domains of closed derivations with values in S
p
. It is also proved that the map f
op is Fréchet differentiable in the norm of S
p
if f is continuously differentiable. 相似文献
8.
Jianqiang Zhao 《数学学报(英文版)》2008,24(10):1593-1616
In this paper, we shall define the renormalization of the multiple q-zeta values (MqZV) which are special values of multiple q-zeta functions ζq (s1,..., Sd) when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang (Renormalization of Multiple Zeta Values, arxiv: math/0606076v3). We show that our renormalization process produces the same values if the MqZVs are well-defined originally and that these renormalizations of MqZV satisfy the q-stuffle relations if we use shifted-renormalizations for all divergent ζq(S1,..., sd) (i.e., s1 ≤ 1). Moreover, when q ↑ 1 our renormalizations agree with those of Guo and Zhang. 相似文献
9.
Cesarina Marchionna Tibiletti 《Annali di Matematica Pura ed Applicata》1978,44(1):233-244
Sunto Si precisa anzitutto cosa si intenda per gruppo G=A
i
... As, prodotto ordinato di s gruppi A
i
, ..., As, e si definiscono certi particolari prodotti ordinati (con fattori a 2 a 2 permutabili, a catena normale, a catena principale).
Dati s gruppi A
i
*, ..., As* si determinano tutti i gruppi G=A
i
... As che siano prodotti ordinati di s gruppi Ai rispettivamente isomorfi ai gruppi Ai. Precisamente tale determinazione consiste nel riconoscere che i suddetti gruppi G sono i sottogruppi di un prodotto completoΓ (noto a partire dai dati) aventi determinati caratteri. Lo stesso problema viene poi trattato nel caso in cui i gruppi G
siano prodotti ordinati di tipo particolare (con fattori a 2 a 2 permutabili, a catena normale, a catena principale); anche
in queste ipotesi i gruppi G vengono riconosciuti essere i sottogruppi del precedente prodotto completoΓ aventi certi caratteri (che si precisano). 相似文献
10.
We show that for every n > 0 there is a planar topological disk A0 and n translates A1, A2, ..., An of A0 such that the interiors of A0, ..., An are pairwise disjoint, but with each Ai touching A0 for 1 ≤ i ≤ n. 相似文献
11.
We consider a linear time-invariant finite-dimensional system x=Ax+Bu with multi-inputu, in which the matricesA andB are in canonical controller form. We assume that the system is controllable andB has rankm. We study the Lyapunov equationPA+A
T
P+Q=0, withQ>0, and investigate the properties thatP must satisfy in order that the canonical controller matrixA be Hurwitz. We show that, for the matrixA being Hurwitz, it is necessary and sufficient thatB
T
PB>0 and that the determinant ofB
T
PW be Hurwitz, whereW=block diag[w
1,...,w
m
], with elementw
i
=[s
k
i
–1,s
k
i
–2,...,s, 1]
T
; here, the symbolsk
i
,i=1, 2, ...,m, denote the Kronecker invariants with respect to the pair {A, B}. This result has application in designing robust controllers for linear uncertain systems. 相似文献
12.
Paul Deheuvels 《Probability Theory and Related Fields》1981,58(1):1-6
Summary If X
1, X
2, ..., are i.i.d. random variables and Y
n
=Max(X
1, ..., X
n
); if for some sequences A
n
, Bn, n=1, 2, ..., E
n
(t)=AnY[nt]+Bn is such that E
n
(1) weakly converges to a non degenerate limit distribution, then we prove that it is possible to construct a sequence of replicates of extremal processes E
(n)(t) on the same probability space, such that d(E
n
(.), E
(n)(.))0 a.s., with the Levy metric. We give the rates of consistency of the approximations. 相似文献
13.
A quasigroup (Q,) satisfying the identityx(yx) =y (or the equivalent identity (xy)x =y) is called semisymmetric. Ann-quasigroup (Q, A) satisfying the identityA(A(x
1, ...,x
n
),x
1, ...,x
n–1) =x
n
is called cyclic. So, cyclicn-quasigroups are a generalization of semisymmetric quasigroups. In this paper, self-orthogonal cyclicn-quasigroups (SOCnQs) are considered. Some constructions ofSOCnQs are described and the spectrum of suchn-quasigroups investigated. 相似文献
14.
We prove the existence of real numbers badly approximated by rational fractions whose denominators form a sublacunar sequence. For example, for the ascending sequence s
n
, n = 1, 2, 3, ..., generated by the ordered numbers of the form 2i3j, i, j = 1, 2, 3, ..., we prove that the set of real numbers α such that inf
n∈ℕ
n‖s
n
α‖ > 0 is a set of Hausdorff dimension 1. The divergence of the series
implies that the Lebesgue measure of those numbers is zero.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 803–813.Original Russian Text Copyright ©2005 by R. K. Akhunzhanov, N. G. Moshchevitin. 相似文献
15.
H. Y. Huang 《Journal of Optimization Theory and Applications》1975,16(5-6):429-445
A computationally stable method for the general solution of a system of linear equations is given. The system isA
Tx–B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA
T and the augmented matrix [A
T,B] are of the same rankm, wheremn, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx
m and a symmetricn ×n matrixH
m of rankn–m, so thatx=x
m+H
my satisfies the system for ally, ann-vector. Whenm=n, the matrixH
m reduces to zero andx
m becomes the unique solution of the system.The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.This research was supported by the National Science Foundation, Grant No. GP-41158. 相似文献
16.
17.
ZHU Bin Department of Mathematical Sciences Tsinghua University Beijing China 《中国科学A辑(英文版)》2006,49(12):1839-1854
Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types. 相似文献
18.
Generalized perfect arrays and menon difference sets 总被引:1,自引:0,他引:1
Jonathan Jedwab 《Designs, Codes and Cryptography》1992,2(1):19-68
Given an s
1 × ... × s
rinteger-valued array A and a (0, 1) vector z = (z
1, ..., z
r), form the array A from A by recursively adjoining a negative copy of the current array for each dimension i where z
i = 1. A is a generalized perfect array type z if all periodic autocorrelation coefficients of A are zero, except for shifts (u
1, ..., u
r) where u
i, - 0 (mod s
i) for all i. The array is perfect if z = (0, ..., 0) and binary if the array elements are all ±1. A nontrivial perfect binary array (PBA) is equivalent to a Menon difference set in an abelian group.Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on their existence. We show that a generalized PBA whose type is not (0, ..., 0) is equivalent to a relative difference set in an abelian factor group. We recursively construct several infinite families of generalized PBAs, and deduce nonexistence results for generalized PBAs whose type is not (0, ..., 0) from well-known nonexistence results for PBAs. A central result is that a PBA with 22y
32u
elements and no dimension divisible by 9 exists if and only if no dimension is divisible by 2
y+2. The results presented here include and enlarge the set of sizes of all previously known generalized PBAs. 相似文献
19.
In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT−1, TBT−1). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set
0n,2,r,π Mn × Mn (Mn = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø1,…,øs in the entries of A and B whose values are constant on all pairs similar in
n,2,r,π to (A, B). The values of the functions øi(A, B), I = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in
n,2,r,π. Let Sn be the subspace of complex symmetric matrices in Mn. For (A, B) ε Sn × Sn we consider the similarity class (TATt, TBTt), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ε Sn × Sn. 相似文献
20.
C. Coker 《Graphs and Combinatorics》1993,9(2-4):117-134
In this paper we study a class of symmetric matricesT indexed by positive integers m≥ n≥2 and defined as follows: for any positive integersp andq let ?p,q be the set of partitions ofU = {1,2,3, ...,pq} into p blocks each of sizeq. Letm ≥n ≥ 2 be positive integers. By atransversal of α = A1/A2/.../An ∈ ?n,m we mean a partitionß = B1/B2/.../Bm ∈ ? m,n such that ‖A i ∩B j = 1 for every i= 1,2, ...,n and everyj = 1,2, ...,m. LetM be the zero-one matrix with rows indexed by the elements of ?n,m and columns indexed by the elements of ?m,n such that Mαß = 1 iffß is a transversal of α. We are interested in finding the eigenvalues and eigenspaces of the symmetric matrixT = MMt. The nonsingularity ofT implies Foulkes’s Conjecture (for these values of m andn). In the casen = 2 we completely determine the eigenvalues and eigenspaces of T and in so doing demonstrate the non-singularity ofT. Forn = 3 we develop a fast algorithm for computing the eigenvalues ofT, and give numerical results in the cases m = 3,4, 5, 6. 相似文献
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