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1.
V. A. Testov 《Mathematical Notes》1997,62(6):762-766
In the note we consider ordered groupoids with the Riesz interpolation property, that is, ifa
i
≤b
j
(i, j=1,2), then there exists ac such thata
i
≤c≤b
j
(i, j=1,2). For such groupoids possessing the descending chain condition for the positive cone and the property
a theorem analogous to the fundamental theorem of arithmetic is proved. The result is a generalization of known results for
lattice-ordered monoids, loops, and quasigroups.
Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 910–915, December, 1997.
Translated by A. I. Shtern 相似文献
2.
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
3.
Strongly Closed Subgraphs in a Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1 总被引:1,自引:1,他引:0
Akira Hiraki 《Graphs and Combinatorics》2008,24(6):537-550
Let Γ be a distance-regular graph of diameter d ≥ 3 with c
2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:
Suppose that the condition (SC)
m
holds. Then it has been known that the condition (BB)
i
holds for all i with 1 ≤ i ≤ m. Similarly we can show that the condition (CA)
i
holds for all i with 1 ≤ i ≤ m. In this paper we prove that if the conditions (BB)
i
and (CA)
i
hold for all i with 1 ≤ i ≤ m, then the condition (SC)
m
holds. Applying this result we give a sufficient condition for the existence of a dual polar graph as a strongly closed subgraph
in Γ. 相似文献
(SC) m : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them. | |
(BB) m : Let (x, y, z) be a triple of vertices with ∂ Γ (x, y) = 1 and ∂ Γ (x, z) = ∂ Γ (y, z) = m. Then B(x, z) = B(y, z). | |
(CA) m : Let (x, y, z) be a triple of vertices with ∂ Γ (x, y) = 2, ∂ Γ (x, z) = ∂ Γ (y, z) = m and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z). |
4.
Let X
1, X
2, ... be i.i.d. random variables. The sample range is R
n
= max {X
i
, 1 ≤ i ≤ n} − min {X
i
, 1 ≤ i ≤ n}. If for a non-degenerate distribution G and some sequences (α
k
), (β
k
) then we have
and
almost surely for any continuity point x of G and for any bounded Lipschitz function f: R → R.
相似文献
5.
Let K be a field of characteristic 0 and let p, q, G
0
, G
1
, P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G
n
(x))
n=0
∞ be defined by the second order linear recurring sequence
In this paper we give conditions under which the diophantine equation G
n
(x) = G
m
(P(x)) has at most exp(1018) many solutions (n, m) ε ℤ2, n, m ⩾ 0. The proof uses a very recent result on S-unit equations over fields of characteristic 0 due to Evertse, Schlickewei and Schmidt [14]. Under the same conditions we
present also bounds for the cardinality of the set
In the last part we specialize our results to certain families of orthogonal polynomials.
This work was supported by the Austrian Science Foundation FWF, grant S8307-MAT.
The second author was supported by the Hungarian National Foundation for Scientific Research Grants No 16741 and 38225.
Received June 5, 2001; in revised form February 26, 2002
RID="a"
ID="a" Dedicated to Edmund Hlawka on the occasion of his 85th birthday 相似文献
6.
Hideyo Sasaki 《The Ramanujan Journal》2009,18(1):73-80
Let
be a real quadratic field over Q with m a square-free positive rational integer and
be the integer ring in F. A totally positive definite integral n-ary quadratic form f=f(x
1,…,x
n
)=∑1≤i,j≤n
α
ij
x
i
x
j
(
) is called universal if f represents all totally positive integers in
. Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m=2,3,5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants
are greater than 12.
In this paper we prove that there are only two quaternary universal forms (up to equivalence) over
. For the proof of universality we apply the theory of quadratic lattices.
相似文献
7.
Let us consider the class of hypoelliptic operators
where z=(x,t) ∈ℝ
N
+1, 0 > m
0≤N the coefficients a
i,j
belong to the space of vanishing mean oscillation functions (VMO
L
) and B=(b
i,j
) is a constant real matrix. In this paper we prove that a strong solution to the differential equation Lu=f, with the known term f in the Morrey space L
p
, λ, belongs to a suitable Sobolev–Morrey space S
p
, λ. Then we prove some Morrey-type imbedding results that give a local H?lder continuity of the solution u.
Received: 14 July 1997 / Revised version: 30 January 1998 相似文献
8.
I. Assani 《Israel Journal of Mathematics》1998,103(1):111-124
We prove the following: Let (X, β, μ,T) be a weakly mixing dynamical system such that the restriction ofT to its Pinsker algebra has singular spectrum, then for all positive integersH, for allf
i ∈L
∞, 1≤i≤H, the averages
.
Research supported in part by NSF Grant #DMS 9305754 相似文献
9.
10.
Akira Hiraki 《Graphs and Combinatorics》2009,25(1):65-79
Let Γ be a distance-regular graph of diameter d ≥ 3 with c
2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:
In [12] we have shown that the condition (SC)
m
holds if and only if both of the conditions (BB)
i
and (CA)
i
hold for i = 1,...,m. In this paper we show that if a
1 = 0 < a
2 and the condition (BB)
i
holds for i = 1,...,m, then the condition (CA)
i
holds for i = 1,...,m. In particular, the condition (SC)
m
holds. Applying this result we prove that a distance-regular graph with classical parameters (d, b, α, β) such that c
2 > 1 and a
1 = 0 < a
2 satisfies the condition (SC)
i
for i = 1,...,d − 1. In particular, either (b, α, β) = (− 2, −3, −1 − (−2)
d
) or holds. 相似文献
(SC) m : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them. | |
(BB) m : Let (x, y, z) be a triple of vertices with ∂Γ(x, y) = 1 and ∂Γ(x, z) = ∂Γ(y, z) = m. Then B(x, z) = B(y, z). | |
(CA) m : Let (x, y, z) be a triple of vertices with and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z). |
11.
Shaofang Hong 《Linear algebra and its applications》2006,416(1):124-134
Let S = {x1, … , xn} be a set of n distinct positive integers and f be an arithmetical function. Let [f(xi, xj)] denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its i, j-entry and (f[xi, xj]) denote the n × n matrix having f evaluated at the least common multiple [xi, xj] of xi and xj as its i, j-entry. The set S is said to be lcm-closed if [xi, xj] ∈ S for all 1 ? i, j ? n. For an integer x > 1, let ω(x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = {x1, … , xn} is an lcm-closed set satisfying , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying (resp. f(p) ? p) for any prime p, then the matrix [f(xi, xj)] (resp. (f[xi, xj])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1-14], we also obtain reduced formulas for det(f(xi, xj)) and det(f[xi, xj]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices. 相似文献
12.
Let ℤ2N={0, ..., 2N-1} denote the group of integers modulo 2N, and let L be the space of all real functions of ℤ2N which are supported on {0,...N−1}. The spectral phase of a function f:ℤ2N→ℝ is given by φf(k)=arg
for k ∈ ℤ2N, where
denotes the discrete Fourier transforms of f.
For a fixed s∈L let Ks denote the cone of all f:ℤ2N→ℝ which satisfy φf ≡ φs and let Ms be its linear span. The angle αs between Ms and L determines the convergence rate of the signal restoration from phase algorithm of Levi and Stark [3]. Here we prove
the following conjectures of Urieli et al. [7] who verified them for the N≤3 case:
Acknowledgments and Notes. Nir Cohen-Supported by CNPq grant 300019/96-3. Roy Meshulam-Research supported by the Fund for the Promotion of Research
at the Technion. 相似文献
1. | α (Ms, L)≤π/4 for a generic s∈L. |
2. | If s∈L is geometric, i.e., s(j)=qj for 0≤j≤N−1 where ±1≠q∈ℝ, then α(Ms, L)=π/4. |
13.
M. Eie Y. L. Ong 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》1997,67(1):149-157
Suppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersB
n(n = 1,2,3,...) assert that (1-p
m-1)B
m/m isp-integral and
ifm ≡ n (mod (p-1)p
n). In this paper, we shall prove that for any positive integerk relatively prime top and non-negative integers α, β such that α +jk =pβ for some integerj with 0 ≤j ≤p-l.Then for any non-negative integerN,
ifp-1 is not a divisor ofm andm ≡ n (mod (p-1)p
n). HereB
n(x) (n = 0,1,2,...) are Bernoulli polynomials. This of course contains the Kummer’s congruences. Furthermore, it contains new congruences
for Bernoulli polynomials of odd indices. 相似文献
((1)) |
((2)) |
14.
Consider the probability spaceW={−1, 1}
n
with the uniform (=product) measure. Letf: W →R be a function. Letf=Σf
IXI be its unique expression as a multilinear polynomial whereX
I=Π
i∈I
x
i. For 1≤m≤n let
=Σ|I|=m
f
IXI. LetT
ɛ
(f)=Σf
Iɛ|I|
X
I where 0<ɛ<1 is a constant. A hypercontractive inequality, proven by Bonami and independently by Beckner, states that
This inequality has been used in several papers dealing with combinatorial and probabilistic problems. It is equivalent to
the following inequality via duality: For anyq≥2
In this paper we prove a special case with a slightly weaker constant, which is sufficient for most applications. We show
where
. Our proof uses probabilistic arguments, and a generalization of Shearer’s Entropy Lemma, which is of interest in its own
right.
Supported partially by NSF Award Abstract #0071261. 相似文献
15.
16.
A. V. Ustinov 《Journal of Mathematical Sciences》2006,137(2):4722-4738
Statistical properties of continued fractions for numbers a/b, where a and b lie in the sector a, b ≥ 1, a2 + b2 ≤ R2, are studied. The main result is an asymptotic formula with two meaning terms for the quantity
where sx(a/b) = |{j ε {1, …, s}: [0; tj, …, ts] ≤ x}| is the Gaussian statistic for the fraction a/b = [t0; t1, …, ts]. Bibliography: 12 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 186–211. 相似文献
17.
We establish necessary and sufficient conditions for numerical functions αj(x), j ∈ N, x ∈ X, under which the conditions K(f
j
⊂ K(f
1) ∀j≥2 and yield The functions fj(x) are uniformly bounded on the set X and take values in a boundedly compact space L, and K(fj) is the kernel of the function fj. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where X = N, α
j
(x) ≡ αj, and the space L is the m-dimensional Euclidean space. 相似文献
18.
Let B
0,B
1, ⋯ ,B
n
be independent standard Brownian motions, starting at 0. We investigate the tail of the capture time
where 0<b
i
≤ 1, 1 ≤i≤n. In particular, we have ?τ3=∞ and ?τ5<∞. Various generalizations are also studied.
Received: 10 January 2000 / Revised version: 12 January 2001 /?Published online: 14 June 2001 相似文献
19.
Copositive approximation of periodic functions 总被引:1,自引:0,他引:1
Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y
i
∈ Y:= {y
i
}
i∈ℤ such that for x ∈ [y
i
, y
i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P
n
of order ≦ n, changing its sign at the same points y
i
∈ Y as f, and
where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω
3(f, t) is the third modulus of smoothness of f and ∥ · ∥ is the max-norm.
This work was done while the first author was visiting CPT-CNRS, Luminy, France, in June 2006. 相似文献