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1.
The paper introduces an algorithm which transforms homogeneous algebraic differential equations into universal differential
equations (in the sense of L. A. Rubel) havingC
n
(ℝ)-solutions. By applications of the algorithm to different initial equations some new universal differential equations
are found, and all the known equations due to R. J. Duffin are rediscovered with this method. Assuming weak conditions one
can find Cn(ℝ)-solutionsy of the differential equation
close to any continuous function such that 1,
with 0 ≤k
1 <k
2 < .... <k
s
≤n are linearly independent over the field of real algebraic numbers at the rational points q1,...,qs. 相似文献
2.
Solutions are obtained for the boundary value problem, y
(n) + f(x,y) = 0, y
(i)(0) = y(1) = 0, 0 i n – 2, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone. 相似文献
3.
A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables 总被引:19,自引:0,他引:19
Qi-Man Shao 《Journal of Theoretical Probability》2000,13(2):343-356
Let {X
i, 1in} be a negatively associated sequence, and let {X*
i
, 1in} be a sequence of independent random variables such that X*
i
and X
i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef(
n
i=1
X
i)Ef(
n
i=1
X*
i
) for any convex function f on R
1 and that Ef(max1kn
n
i=k
X
i)Ef(max1kn
k
i=1
X*
i
) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population. 相似文献
4.
Let X PN be an integral n-dimensional variety and m(X, P, i) (resp. m(X, i)), 1 i N - n + 1, the Hermite invariants of X measuring the osculating behaviour of X at P (resp. at its general point). Here we prove m(X, x) + m(X, y) m(X, x + y) and m(X, P, x) + m(X, y) m(X, P, x + y) for all integers x, y such that x + y N - n + 1, the case n = 1 being known (M. Homma, A. Garcia and E. Esteves).*Partially supported by MIUR and GNSAGA of INdAM (Italy). 相似文献
5.
An estimate for the region of regularity of solutions of certain nonlinear differential inequalities
A. B. Brys'ev 《Journal of Mathematical Sciences》1990,48(1):15-17
In the diskx
2+y
2R
2 of thex, y-plane we consider the differential inequalityz
xxzyy–z
xy
2
–(1+z
x
/2
+z
y
/2
)k, where the constants >0 andk>1. In the case =1 andk=2 this inequality means that the surfacez(x, y) has Gaussian curvatureK1. Efimov has shown that in this case the radius of the disk has an upper bound. In the present article we establish an analogous upper bound for the radiusR of the disk in which the functionz(x, y) satisfies the differential inequality above.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 19–21. 相似文献
6.
A. D. Blinco S. I. El-Zanati G. F. Seelinger P. A. Sissokho L. E. Spence C. Vanden Eynden 《Designs, Codes and Cryptography》2008,48(1):69-77
Let V
n
(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V
n
(q) is a partition of V
n
(q) if every nonzero vector in V
n
(q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the
same size. A partition of V
n
(q) containing a
i
subspaces of dimension n
i
for 1 ≤ i ≤ k induces a uniformly resolvable design on q
n
points with a
i
parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions
that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable
designs on q
n
points where corresponding partitions of V
n
(q) do not exist.
A. D. Blinco—Part of this research was done while the author was visiting Illinois State University. 相似文献
7.
M will be a compact connected n-dimensional Riemannian manifold. If M contains a closed connected k-dimensional, 2 k < n, minimal immersed submanifold of M, we define the kth isoperimetric number of M, Ñ
k
(M), as the infimum of the volumes of all such submanifolds. We obtain a number of interesting estimates for Ñ
k
(M), for both general and special manifolds, which appear to be new.Next we turn to isometric actions and a 1931 theorem of M. H. A. Newman involving the size of orbits of group actions on manifolds. We introduce the higher Newman numbers N
k
(M), 1 k n. Roughly speaking, if M admits isometric actions of compact connected Lie groups with k-dimensional principal orbits, N
k
(M) is defined as the infimum over all such actions of the maximum volume of all maximal dimensional orbits. We observe that N
k
(M) Ñ
k
(M), 2 k < n, provided N
k
(M) is defined; hence our prior estimates for the isoperimetric numbers of M apply directly to the higher Newman numbers.As a best possible candidate we conjecture that N
k
(M) vol S
k
(i(M)/), 1 k n, where i(M) denotes the radius of injectivity of M and S
k
(i(M)/) denotes the standard k-sphere of radius i(M)/. We verify the conjecture for various special cases. We conclude the paper by studying Newman's theorem for compact connected Lie groups with invariant metrics and obtaining a lower bound for the size of small subgroups. 相似文献
8.
Let G be a connected graph. We denote by σ(G,x) and δ(G) respectively the σ-polynomial and the edge-density of G, where
. If σ(G,x) has at least an unreal root, then G is said to be a σ-unreal graph. Let δ(n) be the minimum edgedensity over all n vertices graphs with σ-unreal roots. In this paper, by using the theory of adjoint polynomials, a negative answer to a problem posed by Brenti et
al. is given and the following results are obtained: For any positive integer a and rational number 0≤c≤1, there exists at least a graph sequence {G
i}1≤i≤a
such that G
i is σ-unreal and δ(G
i)→c as n→∞ for all 1 ≤i≤a, and moreover, δ(n)→0 as n→∞.
Supported by the National Natural Science Foundation of China (10061003) and the Science Foundation of the State Education
Ministry of China. 相似文献
9.
Zhaoguo Chen 《应用数学学报(英文版)》1988,4(1):1-12
If we fit a-vector stationary time series using observationsx(1), ...,x(T) with AR models
, then the spectral densityf() of {x(t)} can be estimated byf
k
(T)
()=(2)–
A
k
(T)
(e
–)–1
k
(T)
A
k
(T)
(e
–i)–, where
are estimates of the variance matrix of(t), the residuals of the best linear prediction. By extending some results for the scalar case, this paper treats the asymptotic properties of the estimates in the multichannel case. 相似文献
10.
Séndor J. Kovécs 《Compositio Mathematica》1999,118(2):123-133
Let X be a proper complex variety with Du Bois singularities. Then H(X,
i) H
i(X,
X) is surjective for all i. This property makes this class of singularities behave well with regard to Kodaira type vanishing theorems. Steenbrink conjectured that rational singularities are Du Bois and Kollér conjectured that log canonical singularities are Du Bois. Kollér also conjectured that under some reasonable extra conditions Du Bois singularities are log canonical. In this article Steenbrink's conjecture is proved in its full generality, Kollér's first conjecture is proved under some extra conditions and Kollér's second conjecture is proved under a set of reasonable conditions, and shown that these conditions cannot be relaxed. 相似文献
11.
Let andk be positive integers. A transitively orderedk-tuple (a
1,a
2,...,a
k) is defined to be the set {(a
i, aj) 1i<jk} consisting ofk(k–1)/2 ordered pairs. A directed packing with parameters ,k and index =1, denoted byDP(k, 1; ), is a pair (X, A) whereX is a -set (of points) andA is a collection of transitively orderedk-tuples ofX (called blocks) such that every ordered pair of distinct points ofX occurs in at most one block ofA. The greatest number of blocks required in aDP(k, 1; ) is called packing number and denoted byDD(k, 1; ). It is shown in this paper that
for all even integers , where [x] is the floor ofx. 相似文献
12.
Peter C. Fishburn 《Order》1999,16(4):335-396
Let M
n
(k) denote the family of posets on n points with k ordered pairs that maximize the number of linear extensions among all such posets. Fishburn and Trotter [2] prove that every poset in M
n
(k) is a semiorder and identifies all semiorders in M
n
(k) for k n. The present paper specifies M
n
(k) for all k 2 n – 3. 相似文献
13.
Two finite real sequences (a
1,...,a
k
) and (b
1,...,b
k
) are cross-monotone if each is nondecreasing anda
i+1–a
i
b
i+1–b
i
for alli. A sequence (1,...,
n
) of nondecreasing reals is in class CM(k) if it has disjointk-term subsequences that are cross-monotone. The paper shows thatf(k), the smallestn such that every nondecreasing (1,...,
n
) is in CM(k), is bounded between aboutk
2/4 andk
2/2. It also shows thatg(k), the smallestn for which all (1,...,
n
) are in CM(k)and eithera
k
b
1 orb
k
a
1, equalsk(k–1)+2, and thath(k), the smallestn for which all (1,...,
n
) are in CM(k)and eithera
1b
1...a
k
b
k
orb
1a
1...b
k
a
k
, equals 2(k–1)2+2.The results forf andg rely on new theorems for regular patterns in (0, 1)-matrices that are of interest in their own right. An example is: Every upper-triangulark
2×k
2 (0, 1)-matrix has eitherk 1's in consecutive columns, each below its predecessor, ork 0's in consecutive rows, each to the right of its predecessor, and the same conclusion is false whenk
2 is replaced byk
2–1. 相似文献
14.
Michiel Smid 《Discrete and Computational Geometry》1992,7(1):415-431
A dynamic data structure is given that maintains the minimal distance in a set ofn points ink-dimensional space inO((logn)
k
log logn) amortized time per update. The size of the data structure is bounded byO(n(logn)
k
). Distances are measured in the MinkowskiL
t
-metric, where 1 t . This is the first dynamic data structure that maintains the minimal distance in polylogarithmic time for fully on-line updates.This work was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM). 相似文献
15.
D. W. Müller 《Journal of computational and graphical statistics》2013,22(3):479-492
Abstract Let (x 1, y 1),…, (x n, y n) be a sample of points in , consisting of two subsamples (x 1, y 1),…, (x n1, y n1) and (x n1+1, y n1+1),…, (x n, y n), where n = n 1 + n 2. We consider the problem of separating the two subsamples by a convex contrast curve k…that is, a real-valued convex function x → k(x). For given curve k we consider the relative frequency p 1 of points of the first sample above the graph of k similarly p 2. The goal is to choose k such that the difference in relative frequency p 1 – p 2 becomes maximal. This maximal value can be regarded as a generalized one-sided Kolmogorov-Smirnov test statistic , where p 1(A) = (1/n 1)#{i ≤ n 1 : (xi, y i) ∈ A}, p 2(A) = (1/n 2#{n 1 + 1 ≤ i ≤ n : (x i, y i) ∈ A} and is the class of all epigraphs {(x y) : k(x) ≤ y} of convex functions k. The standard Kolmogorov-Smirnov test statistic corresponds to the class consisting of all sets —that is, epigraphs of constant functions. Test statistics of this form arise in nonparametric analysis of covariance where the regression function is assumed to be convex. In this article we give a recursive relation and an algorithm based on it to compute the value KS and an optimal convex contrast curve in O(n 3) steps. The convexity assumptions on the sets of and on the nonparametric regression function are interrelated; this is explained in the large sample situation where n 1 Λ n 2 → + ∞. 相似文献
16.
Marian Genčev 《Mathematica Slovaca》2009,59(3):365-378
The aim of the paper is the investigation of special infinite series of the form
where (a, b, m
1, m
2, θ, c, P(n)) ∈ ℝ4 × ℂ × {±1} ×
[n] and is a sequence of rational functions. A general summation method for the sum above in the case of the special choice of parameters
a, b and f
n
(n) is included. We find the 2m-tuple of rational numbers α
i
, β
j
(1 ≤ i ≤ m, 1 ≤ j ≤ m) for which
iff
and vice versa.
相似文献
17.
We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x ∈ {0, 1}n, and k parties, who have noisy version of the source bits yi ∈ {0, 1}n, when for all i and j, it holds that P [y = xj] = 1 ? ?, independently for all i and j. That is, each party sees each bit correctly with probability 1 ? ?, and incorrectly (flipped) with probability ?, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on balanced functions fi: {0, 1}n → {0, 1} such that P [f1(y1) = … = fk(yk)] is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The function fi may be thought of as an error correcting procedure for the source x. When k = 2,3, no error correction is possible, as the optimal protocol is given by fi(yi) = y. On the other hand, for large values of k, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to k, n, and ?. Our analysis uses tools from probability, discrete Fourier analysis, convexity, and discrete symmetrization. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005 相似文献
18.
Zi-qing Xie Chuan-miao Chen Institute of Computation Hunan Normal University Changsha Hunan China 《应用数学学报(英文版)》2004,20(2):317-322
The multiple solutions for one-dimensional cubic nonlinear problem u" u~3=0,u(0)=u(π)=0are computed,on the basis of the eigenpairs of-φ"_k=λ_(kφk),k=1,2,3....There exist two nonzero solutions±u_k corresponding to each k,and their Morse index MI(k) for 1(?)k(?)20 is to be exactly determined.It isshown by the numerical results that MI(k)(?)k. 相似文献
19.
V. A. Zalgaller 《Mathematical Notes》1996,59(5):507-510
Let a compact setF
n
contain no less thank points. The functionf
k
:
n
defined by the formulaf
k
(M)=sup
i
=1/k
¦MA
i
¦, whereA
i
are distinct points inF, is convex. Fork=2 its minimum is attained at the center of the smallest ball containingF or on a segment passing through this center. Fork=3 (as well as for any oddk) the minimum point off
k
is unique, whereas for evenk the domain wheref
k
attains its minimum can include a segment.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 703–708, May, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-01044 相似文献