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1.
E. A. Ramos 《Discrete and Computational Geometry》1996,15(2):147-167
We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR
d
, there arek hyperplanes so that each orthant contains a fraction 1/2
k
of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2
k
−1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2
k−1. We are able to prove that Δ(j, k)=⌈j(2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2
n
withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove
a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5).
This research was supported by the National Science Foundation under Grant CCR-9118874. 相似文献
2.
Let ξ, ξ1, ξ2, ... be independent identically distributed random variables, and S
n
:=Σ
j=1
n
,ξ
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:
We obtain some results for identically and differently distributed ξ
j
. 相似文献
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(ξ j ≥ t) ∼ V (t) as t→∞ depends weakly on a, while the left tail distribution is P(ξ 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. |
3.
Let π = (d
1, d
2, ..., d
n
) and π′ = (d′
1, d′
2, ..., d′
n
) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π ⊲ π′, if and only if π ≠ π′, Σ
i=1
n
d
i
= Σ
i=1
n
d′
i
, and Σ
i=1
j
d
i
≤ Σ
i=1
j
d′
i
for all j = 1, 2, ..., n. Weuse C
π
to denote the class of connected graphs with degree sequence π. Let ρ(G) be the spectral radius, i.e., the largest eigenvalue of the adjacent matrix of G. In this paper, we extend the main results of [Liu, M. H., Liu, B. L., You, Z. F.: The majorization theorem of connected
graphs. Linear Algebra Appl., 431(1), 553–557 (2009)] and [Bıyıkoğlu, T., Leydold, J.: Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(1), R119 (2008)]. Moreover, we prove that if π and π′ are two different non-increasing degree sequences of unicyclic graphs with π ⊲ π′, G and G′ are the unicyclic graphs with the greatest spectral radii in C
π
and C′
π
, respectively, then ρ(G) < ρ(G′). 相似文献
4.
An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed
convex proper cone inR
n and −Γ′ be the antipodes of the dual cone of Γ. Let
be a partial differential operator with constant coefficients inR
n, whereQ(ζ)≠0 onR
n−iΓ′ andP
i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R
n−iΓ′;P
j(ζ)=0, gradP
j(ζ)≠0} contains some real point on which gradP
j≠0 and gradP
j∉Γ∪(−Γ). LetC be an open cone inR
n−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in
{ξ∈R
n;P(ξ)=0}. Ifu∈ℒ′∩L
loc
2
(R
n−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition
implies that the support ofu is contained in Γ. 相似文献
5.
Huang Cheng-gui 《数学学报(英文版)》1992,8(3):225-235
SupposeX and the coefficientsA
1, …,A
m aren×n matrices. LetB be anmn×mn matrix as in (7). LetJ be the Jordan canonical matrix ofB andB=PJP
−. LetE
i denote thei×i unit matrix.V is defined bydV/dt=JV andV(t=0)=E
mn. ThenZ=PV satisfiesdZ/dt=BZ.P
* is a matrix which consists of the firstn rows ofP. The author proves: There is a solution of (1) ↔ there are anmn×n matrixC, ann×n matrixQ and ann×n function matrixN such thatP
*VC=QN, where detQ≠0 andN is defined byN(t=0)=E
n anddN/dt=RN, in whichR is ann×n Jordan canonical matrix. There are three cases regarding the solutions of (1): No solution, finitek solutions, 1<k<C
n
m
, and infinite solutions which containj parameters, 1<-j<-mn
2. 相似文献
6.
For p∈{3,4} and all p′>p, with p′ coprime to p, we obtain fermionic expressions for the combination χ
1,s
p,p′+q
Δ
χ
p−1,s
p,p′ of Virasoro (W
2) characters for various values of s, and particular choices of Δ. Equating these expressions with known product expressions, we obtain q-series identities which are akin to the Andrews–Gordon identities. For p=3, these identities were conjectured by Bytsko. For p=4, we obtain identities whose form is a variation on that of the p=3 cases. These identities appear to be new.
The case (p,p′)=(3,14) is particularly interesting because it relates not only to W
2, but also to W
3 characters, and offers W
3 analogues of the original Andrews–Gordon identities. Our fermionic expressions for these characters differ from those of
Andrews et al. which involve Gaussian polynomials.
BF is partially supported by grant number RFBR 05-01-01007, and OF by the Australian Research Council (ARC). 相似文献
7.
S. M. Ertel 《Mathematical Notes》1995,58(1):762-769
Forn pointsA
i
,i=1, 2, ...,n, in Euclidean space ℝ
m
, the distance matrix is defined as a matrix of the form D=(D
i
,j)
i
,j=1,...,n, where theD
i
,j are the distances between the pointsA
i
andA
j
. Two configurations of pointsA
i
,i=1, 2,...,n, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices
of anm-dimensional cube. In the first case, the inverse matrix is obtained in explicit form. In the second case, it is shown that
the complete set of eigenvectors is composed of the columns of the Hadamard matrix of appropriate order. Using the fact that
distance matrices in Euclidean space are nondegenerate, several inequalities are derived for solving the system of linear
equations whose matrix is a given distance matrix.
Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 127–138, July, 1995. 相似文献
8.
Assume thatf is an integer transcendental solution of the differential equationP
n
(z, f, f′)=P
n−1(z, f, f′, ... f
(p)), whereP
n
andP
n−1 are polynomials in all variables, the degree ofP
n
with respect tof andf′ is equal ton, and the degree ofP
n−1 with respect tof, f′, ... f
(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E
*, whereE
* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z
1/2 (β+o(1)), β∈C, holds forz=re
iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re
iν)‖=o(r
1/2),r→+∞,r>0,
, where Δ is a certain set on the semiaxisr>0 with mes Δ<∞.
“L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77,
January, 1999. 相似文献
9.
Yu. A. Kilizhekov 《Mathematical Notes》1996,60(4):378-382
LetW
n
2
M be the class of functionsf: Δ
n
→ ℝ (when Δ
n
is ann-simplex) with bounded second derivative (whose absolute value does not exceedM>0) along any direction at an arbitrary point of the simplex Δ
n
. LetP
1,n
(f;x) be the linear polynomial interpolatingf at the vertices of the simplex. We prove that there exists a functiong ∈ W
n
2
M such that for anyf ∈W
n
2
M and anyx ∈ Δ
n
one has |f
(x)−P
1,
n
(f;x)|≤g(x).
Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 504–510, October, 1996.
I thank Yu. N. Subbotin for posing the problem and for his attention to my work. 相似文献
10.
11.
We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with
and
and let A = ‖a
ij
‖
n×n
, where a
ij
∈ P for i, j = 1,..., n. Let A* = ‖a
ij
′
‖
n×n
and
for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ P in (P, ≤). A matrix A has a right inverse over (P, ≤) if and only if A · A* = E over (P, ≤). If A has a right inverse over (P, ≤), then A* is the greatest right inverse of A over (P, ≤). The matrix A has a right inverse over (P, ≤) if and only if A is a column orthogonal over (P, ≤). The matrix D = A · A* is the greatest diagonal such that A is a left divisor of D over (P, ≤).
Invertible matrices over a distributive lattice (P, ≤) form the general linear group GL
n
(P, ≤) under multiplication. Let (P, ≤) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P,≤) −
, ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL
a
(P, ≤) ≅ = S
n
k
.
We give some further results concerning inversion of matrices over a pseudocomplemented lattice.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005. 相似文献
12.
W. G. Bridges 《Israel Journal of Mathematics》1972,12(4):369-372
Bounds on the number of row sums in ann×n, non-singular (0,1)-matrixA sarisfyingA
tA=diag (k
1-λ1,…,k
n-λn),k
j>λj>0,λ1=…=λe,λe+1=…=λn are obtained which extend previous results for such matrices. 相似文献
13.
Qi Yu Sun 《数学学报(英文版)》2001,17(1):1-14
Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z
0 be a subset of Z such than n∈Z
0 implies n + 1 ∈Z
0. Denote the space of all compactly supported distributions by D′, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G
n
and H
n
, n∈Z
0, in D′, define the corresponding nonstationary nonhomogeneous refinement equation
Φ
n
=H
n
*Φ
n+1
(A·)+G
n
for all n∈Z
0
where Φ
n
, n∈Z
0, is in a bounded set of D′. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets,
and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional
solutions Φ
n
, n∈Z
0, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution of the linear equations
for all n∈Z
0
where the matrices S
n
and the vectors , n∈Z
0, can be constructed explicitly from H
n
and G
n
respectively. The results above are still new even for stationary nonhomogeneous refinement equations.
Received December 30, 1999, Accepted June 15, 2000 相似文献
14.
A. Račkauskas 《Lithuanian Mathematical Journal》1997,37(4):402-415
Let (ξ
k
,F
k
) be a martingale difference sequence. The paper concerns the tail behavior of the quadratic formS
n
= ∑
k=1
n
∑
j=1
k−1
β
n
k−j
χ
k
χ
j
, where β
n
asn→∞. The main conclusions aboutP}n
−1
S
n
>x
n
}, wherex
n
→∞, asn→∞, are obtained using the tail behavior of a martingale with values in a certain Hilbert space.
Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published
in Lietuvos Matematikos Rinkinys, Vol. 37, No. 4, pp. 532–549, October–December, 1997. 相似文献
15.
We give an example of two distinct stationary processes {X
n} and {X′
n} on {0, 1} for whichP[X0=1|X−1=a−1,X−2=a−2, …]=P[X′0=1|X′−1=a−1,X′−2=a−2, …] for all {a
i},i=−1, −2, …, even though these probabilities are bounded away from 0 and 1, and are continuous in {a
i}.
Supported in part by NSF Grant DMS 89-01545.
Supported in part by the US Army Research Office. 相似文献
16.
Simeon M. Berman 《Annals of the Institute of Statistical Mathematics》1984,36(1):301-321
Summary Let {X
n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate
mixing conditions, it is shown thatS
n=Xn,1+…+X
n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f
n(x)∼ defined on a stationary sequence {X
j∼, whereX
n.f=fn(Xj). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of
a general type.
This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship
of the National Sciences Foundation, Grant MCS 82-01119. 相似文献
17.
John C. Clements 《Numerische Mathematik》1992,63(1):165-171
Summary AC
2 parametric rational cubic interpolantr(t)=x(t)
i+y(t)
j,t[t
1,t
n] to data S={(xj, yj)|j=1,...,n} is defined in terms of non-negative tension parameters
j
,j=1,...,n–1. LetP be the polygonal line defined by the directed line segments joining the points (x
j
,y
j
),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always exist
j
,j=1,...,n–1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.This research was supported in part by the natural Sciences and Engineering Research Council of Canada. 相似文献
18.
Shichao Chen 《The Ramanujan Journal》2009,18(1):103-112
Let Λ={λ
1≥⋅⋅⋅≥λ
s
≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ
i
nodes in the ith row. Let λ
j
′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ
i
+λ
j
′−i−j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2
t
-core partitions of n are obtained. A simple convolution identity for t-cores is also given.
相似文献
19.
Let M be a smooth compact oriented Riemannian manifold of dimension n without boundary, and let Δ be the Laplace–Beltrami operator on M. Say , and that f (0) = 0. For t > 0, let K
t
(x, y) denote the kernel of f (t
2 Δ). Suppose f satisfies Daubechies’ criterion, and b > 0. For each j, write M as a disjoint union of measurable sets E
j,k
with diameter at most ba
j
, and measure comparable to if ba
j
is sufficiently small. Take x
j,k
∈ E
j,k
. We then show that the functions form a frame for (I − P)L
2(M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly
quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ (I − P)L
2 is in space and in frequency, we can describe which terms in the summation are so small that they can be neglected. If n = 2 and M is the torus or the sphere, and f (s) = se
−s
(the “Mexican hat” situation), we obtain two explicit approximate formulas for the φ
j,k
, one to be used when t is large, and one to be used when t is small.
A. Mayeli was partially supported by the Marie Curie Excellence Team Grant MEXT-CT-2004-013477, Acronym MAMEBIA. 相似文献
20.
Raphael Yuster 《Graphs and Combinatorics》2001,17(3):579-587
We prove that for every ε>0 and positive integer r, there exists Δ0=Δ0(ε) such that if Δ>Δ0 and n>n(Δ,ε,r) then there exists a packing of K
n
with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn
2 edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let αΔ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥αΔ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K
n
).
Received: March 15, 1999?Final version received: October 22, 1999 相似文献