共查询到20条相似文献,搜索用时 578 毫秒
1.
Leonid G. Fel 《Functional Analysis and Other Mathematics》2006,1(2):119-157
We find the matrix representation of the set Δ(d
3), where d
3=(d
1,d
2,d
3), of integers that are unrepresentable by d
1,d
2,d
3 and develop a diagrammatic procedure for calculating the generating function Φ(d
3;z) for the set Δ(d
3). We find the Frobenius number F(d
3), the genus G(d
3), and the Hilbert series H(d
3;z) of a graded subring for nonsymmetric and symmetric semigroups
and enhance the lower bounds of F(d
3) for symmetric and nonsymmetric semigroups.
相似文献
2.
Kyriakos Keremedis 《Mathematical Logic Quarterly》2012,58(3):130-138
Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:
- (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
- (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
- (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
- (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.
3.
A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication,
and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial
mapping u: Z
d
→ R
l
has a representation u(n) = f(ϕ(n)x), n ∈ Z
d
, where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Z
d
-action by translations on X. This fact is used to show that the sequence u(n), n ∈ Z
d
, is well distributed on a piecewise polynomial surface (with respect to the Borel measure on that is the image of the Lebesgue measure under the piecewise polynomial function defining ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine
approximations extending the work of van der Corput and of Furstenberg–Weiss. 相似文献
4.
Saugata Basu 《Discrete and Computational Geometry》2008,40(4):481-503
Let
be an o-minimal structure over ℝ,
a closed definable set, and
the projection maps as depicted below:
For any collection
of subsets of
, and
, let
denote the collection of subsets of
where
. We prove that there exists a constant C=C(T)>0 such that for any family
of definable sets, where each A
i
=π
1(T∩π
2−1(y
i
)), for some y
i
∈ℝ
ℓ
, the number of distinct stable homotopy types amongst the arrangements
is bounded by
while the number of distinct homotopy types is bounded by
This generalizes to the o-minimal setting, bounds of the same type proved in Basu and Vorobjov (J. Lond. Math. Soc. (2) 76(3):757–776,
2007) for semi-algebraic and semi-Pfaffian families. One technical tool used in the proof of the above results is a pair of topological
comparison theorems reminiscent of Helly’s theorem in convexity theory. These theorems might be of independent interest in
the quantitative study of arrangements.
The author was supported in part by NSF grant CCF-0634907. 相似文献
5.
J. B. Lasserre 《TOP》2012,20(1):119-129
We consider the semi-infinite optimization problem:
f*:=minx ? X {f(x):g(x,y) £ 0, "y ? Yx},f^*:=\min_{\mathbf{x}\in\mathbf{X}} \bigl\{f(\mathbf{x}):g(\mathbf{x},\mathbf{y}) \leq 0, \forall\mathbf{y}\in\mathbf {Y}_\mathbf{x}\bigr\}, 相似文献
6.
Hiroaki Minami 《Archive for Mathematical Logic》2010,49(4):501-518
We investigate splitting number and reaping number for the structure (ω)
ω
of infinite partitions of ω. We prove that
\mathfrakrd £ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and
\mathfraksd 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency
\mathfrakrd < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and
\mathfraksd < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants
\mathfrakrpair{\mathfrak{r}_{pair}} and
\mathfrakspair{\mathfrak{s}_{pair}} . We also study the relation between
\mathfrakrpair, \mathfrakspair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that
cov(M),cov(N) £ \mathfrakrpair £ \mathfraksd,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and
\mathfraks £ \mathfrakspair £ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} . 相似文献
7.
The toric ideals of 3×3 transportation polytopes
Trc\mathsf{T}_{\mathbf{rc}}
are quadratically generated. The only exception is the Birkhoff polytope B
3.
If
Trc\mathsf{T}_{\mathbf{rc}}
is not a multiple of B
3, these ideals even have square-free quadratic initial ideals. This class contains all smooth 3×3 transportation polytopes. 相似文献
8.
Richard Nickl 《Journal of Theoretical Probability》2009,22(1):38-56
Let μ
n
be a sequence of random finite signed measures on the locally compact group G equal to either
or ℝ
d
. We give weak conditions on the sequence μ
n
and on functions K such that the convolution product μ
n
*K, and its derivatives, converge in law, in probability, or almost surely in the Banach spaces
or L
p
(G). Examples for sequences μ
n
covered are the empirical process (possibly arising from dependent data) and also random signed measures
where
is some (nonparametric) estimator for the measure ℙ, including the usual kernel and wavelet based density estimators with
MISE-optimal bandwidths. As a statistical application, we apply the results to study convolutions of density estimators.
相似文献
9.
Witold Bednorz 《Journal of Theoretical Probability》2007,20(4):917-934
For a Young function φ and a Borel probability measure m on a compact metric space (T,d) the minorizing metric is defined by
10.
We prove a dichotomy between absolute continuity and singularity of the Ginibre point process \(\mathsf {G}\) and its reduced Palm measures \(\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}\), namely, reduced Palm measures \(\mathsf {G}_{\mathbf {x}}\) and \(\mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x} \in \mathbb {C}^{\ell }\) and \(\mathbf {y} \in \mathbb {C}^{n}\) are mutually absolutely continuous if and only if \(\ell = n\); they are singular each other if and only if \(\ell \not = n\). Furthermore, we give an explicit expression of the Radon–Nikodym density \(d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\). 相似文献
11.
Sang-Eon Han 《Acta Appl Math》2008,104(2):177-190
In order to study digital topological properties of a k-surface in Z
n
, we generalize the topological number in Bertrand (Pattern Recogn. Lett. 15:1003–1011, 1994). Furthermore, we show that a local (k
0,k
1)-isomorphism preserves some digital-topological properties, such as a generalized topological number and a simple k
0-point, and prove that a local (k
0,k
1)-isomorphism takes a simple k
0-surface in
into a simple k
1-surface in
.
相似文献
12.
A. V. Bogomol'naya 《Journal of Mathematical Sciences》1995,73(6):633-637
We consider
,mE > 0,G(E) is a certain subspace of L
1
(E) consisting of functions concentrated on E and integrable, and {dk}, (k ∈ ℤ) in a summable sequence of positive numbers. It is proved that if G(E)=Lp(E), p≥2, then there exists f∈G(E) such that |f(n)|≥dn,
(one of the questions involved in the majorization problem). Sufficient conditions are obtained for certain other function
classes G(E). We study the question of partial majorization. Bibliography: 2 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 42–48. 相似文献
13.
Svante Linusson John Shareshian Volkmar Welker 《Journal of Algebraic Combinatorics》2008,27(3):331-349
For positive integers k,n, we investigate the simplicial complex
of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The
number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially
defined complexes are described. In addition we study for positive integers r,s and k the simplicial complex
of all bipartite graphs G on bipartition
such that there is no matching of size k in G, and obtain results similar to those obtained for
.
S. Linusson and V. Welker supported by EC’s IHRP program through grant HPRN-CT-2001-00272. J. Shareshian partially supported
by National Science Foundation grants DMS-0070757 and DMS-0030483. 相似文献
14.
Sascha Orlik 《Inventiones Mathematicae》2008,172(3):585-656
Let be Drinfeld’s upper half space over a finite extension K of ℚ
p
. We construct for every GL
d+1-equivariant vector bundle on ℙ
d
K
, a GL
d+1(K)-equivariant filtration by closed subspaces on the K-Fréchet . This gives rise by duality to a filtration by locally analytic GL
d+1(K)-representations on the strong dual . The graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect
to maximal parabolic subgroups. This paper generalizes the cases of the canonical bundle due to Schneider and Teitelbaum [ST1]
and that of the structure sheaf by Pohlkamp [P]. 相似文献
15.
Sergey Bereg Prosenjit Bose Adrian Dumitrescu Ferran Hurtado Pavel Valtr 《Discrete and Computational Geometry》2009,41(4):513-532
Given a finite set of points S in ℝ
d
, consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the minimum number of segments (links).
We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is
that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S, and let G
n
d
be an n×…×n grid in ℤ
d
. Kranakis et al. (Ars Comb. 38:177–192, 1994) showed that L(G
n
2)=2n−1 and
and conjectured that, for all d≥3,
We prove the conjecture for d=3 by showing the lower bound for L(G
n
3). For d=4, we prove that
For general d, we give new estimates on L(G
n
d
) that are very close to the conjectured value. The new lower bound of
improves previous result by Collins and Moret (Inf. Process. Lett. 68:317–319, 1998), while the new upper bound of
differs from the conjectured value only in the lower order terms.
For arbitrary point sets, we include an exact bound on the minimum number of links needed in an axis-aligned path traversing
any planar n-point set. We obtain similar tight estimates (within 1) in any number of dimensions d. For the general problem of traversing an arbitrary set of points in ℝ
d
with an axis-aligned spanning path having a minimum number of links, we present a constant ratio (depending on the dimension d) approximation algorithm.
Work by A. Dumitrescu was partially supported by NSF CAREER grant CCF-0444188.
Work by F. Hurtado was partially supported by projects MECMTM2006-01267 and Gen. Cat. 2005SGR00692.
Work by P. Valtr was partially supported by the project 1M0545 of the Ministry of Education of the Czech Republic. 相似文献
16.
Evgenii E. Mukhin Vitaly O. Tarasov Alexander N. Varchenko 《Functional Analysis and Other Mathematics》2006,1(1):47-69
Let
be a space of quasipolynomials of dimension N=N
1+⋅⋅⋅+N
n
. We define the regularized fundamental operator of V as the polynomial differential operator D=∑
i=0
N
A
N−i
(x)∂
x
i
annihilating V and such that its leading coefficient A
0 is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials
whose regularized fundamental operator is the differential operator ∑
i=0
N
u
i
A
N−i
(∂
u
). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity)
of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical
points of the two master functions associated with the
-dual Gaudin models and also between the corresponding Bethe vectors.
The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460).
The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579). 相似文献
17.
Preda Mihailescu. 《Mathematics of Computation》2008,77(262):1199-1221
Arithmetic in large ring and field extensions is an important problem of symbolic computation, and it consists essentially of the combination of one multiplication and one division in the underlying ring. Methods are known for replacing one division by two short multiplications in the underlying ring, which can be performed essentially by using convolutions.
However, while using school-book multiplication, modular multiplication may be grouped into operations (where denotes the number of operations of one multiplication in the underlying ring), the short multiplication problem is an important obstruction to convolution. It raises the costs in that case to . In this paper we give a method for understanding and bypassing this problem, thus reducing the costs of ring arithmetic to roughly when also using fast convolutions. The algorithms have been implemented with results which fit well the theoretical prediction and which shall be presented in a separate paper. 18.
R. Kh. Sadikova 《Mathematical Notes》1997,61(6):724-730
Suppose (T, Σ, μ) is a space with positive measure,f: ? → ? is a strictly monotone continuous function, and &;(T) is the set of real μ-measurable functions onT. Letx(·) ∈ &;(T) andf ○x)(·) ∈L 1(T,μ). Comparison theorems are proved for the means $\mathfrak{M}_{(T,{\mathbf{ }}\mu ,{\mathbf{ }}f)} (x( \cdot ))$ and the mixed means $\mathfrak{M}_{(T_1 ,{\mathbf{ }}\mu _1 ,{\mathbf{ }}f_1 )} (\mathfrak{M}_{(T_2 ,{\mathbf{ }}\mu _2 ,{\mathbf{ }}f_2 )} (x( \cdot )))$ these inequalities imply analogs and generalizations of some classical inequalities, namely those of Hölder, Minkowski, Bellman, Pearson, Godunova and Levin, Steffensen, Marshall and Olkin, and others. These results are a continuation of the author's studies. 相似文献
19.
George Voutsadakis 《Order》2007,24(1):15-29
A completion of an n-ordered set is defined, by analogy with the case of posets (2-ordered sets), as a pair , where Q is a complete n-lattice and is an n-order embedding. The Basic Theorem of Polyadic Concept Analysis is exploited to construct a completion of an arbitrary n-ordered set. The completion reduces to the Dedekind–MacNeille completion in the dyadic case, the case of posets. A characterization
theorem is provided, analogous to the well-known dyadic one, for the case of joined n-ordered sets. The condition of joinedness is trivial in the dyadic case and, therefore, this characterization theorem generalizes
the uniqueness theorem for the Dedekind–MacNeille completion of an arbitrary poset.
相似文献
20.
Shan-tao Liao 《Frontiers of Mathematics in China》2006,1(1):1-52
Let M
n
be an n-dimensional compact C
∞-differentiable manifold, n ≥ 2, and let S be a C
1-differential system on M
n
. The system induces a one-parameter C
1 transformation group φ
t
(−∞ < t < ∞) over M
n
and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M
n
. The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group.
Among various results established in the paper, we mention here only the following, which might describe quite well the nature
of our study.
(A) Let M be the set of regular points in M
n
of the differential system S. With respect to a given C
∞ Riemannian metric of M
n
, we consider the bundle
of all (n−2) spheres Q
x
n−2, x∈M, where Q
x
n−2 for each x consists of all unit tangent vectors of M
n
orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ
t
#
(−∞<t<∞) of
. For an l-frame α = (u
1, u
2,⋯, u
l
) of M
n
at a point x in M, 1 ≥ l ≥ n−1, each u
i
being in
, we shall denote the volume of the parallelotope in the tangent space of M
n
at x with edges u
1, u
2,⋯, u
l
by υ(α), and let
. This is a continuous real function of t. Let
|