共查询到20条相似文献,搜索用时 15 毫秒
1.
Wei Cao 《Czechoslovak Mathematical Journal》2007,57(1):253-268
A set S={x
1,...,x
n
} of n distinct positive integers is said to be gcd-closed if (x
i
, x
j
) ∈ S for all 1 ⩽ i, j ⩽ n. Shaofang Hong conjectured in 2002 that for a given positive integer t there is a positive integer k(t) depending only on t, such that if n ⩽ k(t), then the power LCM matrix ([x
i
, x
j
]
t
) defined on any gcd-closed set S={x
1,...,x
n
} is nonsingular, but for n ⩾ k(t) + 1, there exists a gcd-closed set S={x
1,...,x
n
} such that the power LCM matrix ([x
i
, x
j
]
t
) on S is singular. In 1996, Hong proved k(1) = 7 and noted k(t) ⩾ 7 for all t ⩾ 2. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed
set and proves that k(t) ⩾ 8 for all t ⩾ 2. We further prove that k(t) ⩾ 9 iff a special Diophantine equation, which we call the LCM equation, has no t-th power solution and conjecture that k(t) = 8 for all t ⩾ 2, namely, the LCM equation has t-th power solution for all t ⩾ 2. 相似文献
2.
DNA labelled graphs with DNA computing 总被引:2,自引:0,他引:2
Let k≥2, 1≤i≤k andα≥1 be three integers. For any multiset which consists of some k-long oligonucleotides, a DNA labelled graph is defined as follows: each oligonucleotide from the multiset becomes a point; two points are connected by an arc from the first point to the second one if the i rightmost uucleotides of the first point overlap with the i leftmost nucleotides of the second one. We say that a directed graph D can be(k, i;α)-labelled if it is possible to assign a label(l_1(x),..., l_k(x))to each point x of D such that l_j(x)∈{0,...,a-1}for any j∈{1,...,k}and(x,y)∈E(D)if and only if(l_k-i 1(x),..., l_k(x))=(l_1(y),..., l_i(y)). By the biological background, a directed graph is a DNA labelled graph if there exist two integers k, i such that it is(k, i; 4)-labelled. In this paper, a detailed discussion of DNA labelled graphs is given. Firstly, we study the relationship between DNA labelled graphs and some existing directed graph classes. Secondly, it is shown that for any DNA labelled graph, there exists a positive integer i such that it is(2i, i; 4)-labelled. Furthermore, the smallest i is determined, and a polynomial-time algorithm is introduced to give a(2i, i; 4)-labelling for a given DNA labelled graph. Finally, a DNA algorithm is given to find all paths from one given point to another in a(2i, i; 4)-labelled directed graph. 相似文献
3.
Robert S. Strichartz 《Journal of Geometric Analysis》1991,1(3):269-289
Let μ be a measure on ℝn that satisfies the estimate μ(B
r(x))≤cr
α for allx ∈ ℝn and allr ≤ 1 (B
r(x) denotes the ball of radius r centered atx. Let ϕ
j,k
(ɛ)
(x)=2
nj2ϕ(ɛ)(2
j
x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤn, and ∈ ∈E, a finite set, and letP
j
(T)=Σɛ,k
<T,ϕ
j,k
(ɛ)
>ϕ
j,k
(ɛ)
denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls
p(dμ) for 1 ≤p ≤ ∞, we show thatP
j(f dμ) ∈ Lp(dx) and ||P
j
(fdμ)||L
p(dx)≤c2
j((n-α)/p′))||f||L
p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P
j
(fdμ)||L
p(dx) under more restrictive hypotheses.
Communicated by Guido Weiss 相似文献
4.
D. Yu. Grigor'ev 《Journal of Mathematical Sciences》1994,70(4):1873-1880
Suppose given a k1×k2 system of linear equations over the Weyl algebraA
n = F[X1,...X1,D4,...,Dn] or over the algebra of differential operatorsK
n = F[X1,...X1,D4,...,Dn], where the degree of each coefficient of the system is less than d. It is proved that if the system is solvable overA
n, orK
n, respectively, then it has a solution of degree at most (k, d)20(n).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 47–59, 1991. 相似文献
5.
Gy. Maksa 《Aequationes Mathematicae》1999,57(1):50-74
Summary. The solution of the rectangular m ×n m \times n generalized bisymmetry equation¶¶F(G1(x11,...,x1n),..., Gm(xm1,...,xmn)) = G(F1(x11,..., xm1),..., Fn(x1n,...,xmn) ) F\bigl(G_1(x_{11},\dots,x_{1n}),\dots,\ G_m(x_{m1},\dots,x_{mn})\bigr) \quad = \quad G\bigl(F_1(x_{11},\dots, x_{m1}),\dots, \ F_n(x_{1n},\dots,x_{mn}) \bigr) (A)¶is presented assuming that the functions F, Gj, G and Fi (j = 1, ... , m , i = 1, ... , n , m S 2, n S 2) are real valued and defined on the Cartesian product of real intervals, and they are continuous and strictly monotonic in each real variable. Equation (A) is reduced to some special bisymmetry type equations by using induction methods. No surjectivity assumptions are made. 相似文献
6.
L. V. Shabunin 《Algebra and Logic》2001,40(3):194-204
Let m and n be fixed integers, with 1 m < n. A Cantor variety C
m,n is a variety of algebras with m
n-ary and n
m-ary basic operations which is defined in a signature ={g1,...,gm,f1,...,fn} by the identities fig1x1,...,xn),...,gmx1,...,xn) = xi, i=1,...,n, gjf1x1,...,xm),...,fnx1,...,xm)) = xj, j=1,...,m. We prove the following: (a) every partial C
m,n-algebra A is isomorphically embeddable in the algebra G= A; S(A) of C
m,n; (b) for every finitely presented algebra G= A; S in C
m,n, the word problem is decidable; (c) for finitely presented algebras in C
m, the occurrence problem is decidable; (d) C
m,n has a hereditarily undecidable elementary theory. 相似文献
7.
Karin Erdmann 《manuscripta mathematica》1995,88(1):357-386
LetK be an algebraically closed field of characteristic,p>0 and letD
λ be the simple modules of the symmetric groupS
r
overK where λ is a p-regular partition ofr. The dimensions ofD
λ for λ with at mostn parts are the same as the multiplicities of direct summands ofD
⊗r
whereE is the natural module for the groupGL
n
(K). Whenn=2 we determine generating functions for these multiplicities and hence for the dimensions ofD
λ for all partitions λ with two parts. These can be expressed as rational functions of Chebyshev polynomials; and we obtain
explicit formulae for the coefficients. 相似文献
8.
We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that A
k
k[x](x)/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x](x). 相似文献
9.
R. Nair 《Israel Journal of Mathematics》2009,171(1):197-219
We consider a system of “generalised linear forms” defined at a point x = (x
(i, j)) in a subset of R
d
by
for k ≥ 1. Here d = d
1 + ⋯ + d
l
and for each pair of integers (i, j) ∈ D, where D = {(i, j): 1 ≤ i ≤ l, 1 ≤ j ≤ d
i
} the sequence of functions (g
(i, j), k
(x))
k=1∞ are differentiable on an interval X
ij
contained in R. We study the distribution of the sequence on the l-torus defined by the fractional parts X
k
(x) = ({ L
1(x)(k)}, ..., {L
l
(x)(k)}) ∈ T
l
, for typical x in the Cartesian product . More precisely, let R = I
1 × ⋯ × I
l
be a rectangle in T
l
and for each N ≥ 1 define a pair correlation function
and a discrepancy , where the supremum is over all rectangles in T
l
and χ
R
is the characteristic function of the set R. We give conditions on (g
(i, j), k
(x))
k=1∞ to ensure that given ε > 0, for almost every x ∈ T
l
we have Δ
N
(x) = o(N(log N)
l+∈). Under related conditions on(g
(i, j), k
(x))
k =1∞ we calculate for appropriate β ∈ (0, 1) the Hausdorff dimension of the set {x : lim sup
N→∞
N
β Δ
N
(x > 0)}. Our results complement those of Rudnick and Sarnak and Berkes, Philipp, and Tichy in one dimension and M. Pollicott
and the author in higher dimensions. 相似文献
10.
B. Helffer 《Milan Journal of Mathematics》2010,78(2):575-590
Given a bounded open set Ω in
\mathbbRn{\mathbb{R}^n} (or a Riemannian manifold) and a partition of Ω by k open sets D
j
, we can consider the quantity max
j
λ(D
j
) where λ(D
j
) is the groundstate energy of the Dirichlet realization of the Laplacian in D
j
. If we denote by
\mathfrakLk(W){\mathfrak{L}_k(\Omega)} the infimum over all the k-partitions of max
j
λ(D
j
), a minimal (spectral) k-partition is then a partition which realizes the infimum. Although the analysis is rather standard when k = 2 (we find the nodal domains of a second eigenfunction), the analysis of higher k’s becomes non trivial and quite interesting. 相似文献
11.
M. H. Lim 《Linear and Multilinear Algebra》2013,61(4):201-302
Let U be an n-dimensional vector space over an algebraically closed field F. Let U(m) denote the mth symmetric power of U. For each positive integer k≤min{m,n}, let Dk denote the set of all nonzero decomposable elements x1 …xm in U(m) such that dim(x1 …xm ) = k and Ek denote the set of all decomposable elements x1 …xm in U(m) such that dim(x1 …xm ) ≤ k. In this paper we first show that Ek is an algebraic variety with Dk as a dense subset and determine the dimension of Ek . We next use these results to study the structure of linear mappings T on Um such that T(Dk ) ? Dk or T(Ek ) ? Ek for some fixed k. 相似文献
12.
Eric S. Brussel 《Israel Journal of Mathematics》1996,96(1):141-183
LetF be a discretely Henselian field of rank one, with residue fieldk a number field, and letD/F be anF-division algebra. We conduct an exhaustive study of the decomposability of an arbitraryD. Specifically, we prove the following:D has a semiramified (SR)F-division subalgebra if and only ifD has a totally ramified (TR) subfield. However, there may be TR subfields not contained in any SR subalgebra. IfD has prime-power index, thenD is decomposable if and only ifD properly contains a SR division subalgebra. Equivalently,D has a decomposable Sylow factor if and only if ii(D
⊗n
)≠1/n
i(D) for somen dividing the period ofD, that is, if and only if the index fails to mimic the behavior of the period ofD. There exists indecomposableD with prime-power periodp
2 and indexp
3. Every proper division subalgebra ofD is indecomposable. Conversely, every indecomposableF-division algebra ofp-power index embeds properly in someD ofp-power index if and only ifk does not have a certain strengthened form of class field theory’s Special Case. Semiramified division algebras and division
algebras of odd index always properly embed. Finally, these results apply to an extent overk(t), and we prove that there exist indecomposablek(t)-division algebras of periodp
2 and indexp
3, solving an open problem of Saltman.
Dedicated to the memory of Amitsur
Research supported in part by NSF Grant DMS-9100148. 相似文献
13.
A. Taskaraev 《Mathematical Notes》1998,64(5):658-662
The existence and uniqueness of a surface with given geometric characteristics is one of the important topical problems of
global differential geometry. By stating this problem in terms of analysis, we arrive at second-order elliptic and parabolic
partial differential equations. In the present paper we consider generalized solutions of the Monge-Ampère equation ||z
ij
|| = ϕ(x, z, p) in Λ
n
, wherez = z(x
1,...,z
n
) is a convex function,p = (p
1,...,P
n) = (∂z/∂x
1,...,ϖz/ϖx
n), andz
ij =ϖ
2
z/ϖx
i
ϖx
j. We consider the Cayley-Klein model of the space Λ
n
and use a method based on fixed point principle for Banach spaces.
Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 763–768, November, 1998. 相似文献
14.
Abidi Jamel 《Rendiconti del Circolo Matematico di Palermo》2005,54(2):167-194
We prove the following result: If the function Max (log|ω -f
1(z)|, ..., log|ω -f
k(z)|) is plurisubharmonic in the open setD×ℂ (D open of ℂ
n
), thenf
1,...,f
k are analytic functions iff
1,...,f
k are continuous functions onD(k≥2). We prove also some other results. 相似文献
15.
C. Aistleitner I. Berkes R. Tichy 《Proceedings of the Steklov Institute of Mathematics》2012,276(1):3-20
It is known that for any smooth periodic function f the sequence (f(2
k
x))
k≥1 behaves like a sequence of i.i.d. random variables; for example, it satisfies the central limit theorem and the law of the
iterated logarithm. Recently Fukuyama showed that permuting (f(2
k
x))
k≥1 can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal
condition on (n
k
)
k≥1, formulated in terms of the number of solutions of certain Diophantine equations, which ensures the validity of the law of
the iterated logarithm for any permutation of the sequence (f(n
k
x))
k≥1. A similar result is proved for the discrepancy of the sequence ({n
k
x})
k≥1, where {·} denotes the fractional part. 相似文献
16.
Sets of Double and Triple Weights of Trees 总被引:1,自引:0,他引:1
Elena Rubei 《Annals of Combinatorics》2011,15(4):723-734
Let T be a weighted tree with n leaves numbered by the set {1, . . . , n}. Let D
i, j
(T) be the distance between the leaves i and j. Let
Di,j,k(T) = \frac12(Di,j(T)+Dj,k(T)+Di,k(T)){{D_{i,j,k}(T) = \frac{1}{2}(D_{i,j}(T)+D_{j,k}(T)+D_{i,k}(T))}} . We will call such numbers “triple weights” of the tree. In this paper, we give a characterization, different from the previous
ones, for sets indexed by 2-subsets of a n-set to be double weights of a tree. By using the same ideas, we find also necessary and sufficient conditions for a set of
real numbers indexed by 3-subsets of an n-set to be the set of the triple weights of a tree with n leaves. Besides we propose a slight modification of Saitou-Nei’s Neighbour-Joining algorithm to reconstruct trees from the
data D
i, j
. 相似文献
17.
Eduard Boďa 《Mathematica Slovaca》2008,58(2):173-176
Let (R,m) = k[x
1,..., x
n
](x
1,...,x
n
) be a local polynomial ring (k being an algebraically closed field), and Q:= (F
1,..., F
r
)R be a primary ideal in R with respect to a maximal ideal m ⊂ R. In this short note we give a formula for the multiplicity e
0 (QR/(F
1)R, R/(F
1)R).
The author was supported by the grant No. 1/0262/03) of the Slovak Ministry of Education. 相似文献
18.
Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ Nk(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗k k(x) and for some generic modules for Λ.
Received: 24 November 2003; revised: 11 February 2005 相似文献
19.
Michel Talagrand 《Israel Journal of Mathematics》1992,79(2-3):207-224
Consider a setA of symmetricn×n matricesa=(a
i,j)
i,j≤n
. Consider an independent sequence (g
i)
i≤n
of standard normal random variables, and letM=Esupa∈A|Σi,j⪯nai,jgigj|. Denote byN
2(A, α) (resp.N
t(A, α)) the smallest number of balls of radiusα for thel
2 norm ofR
n
2 (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN
2(A, α))1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN
t≤K(δ)M.
Work partially supported by an N.S.F. grant. 相似文献
20.
S. K. Chatterjea 《Annali dell'Universita di Ferrara》1961,10(1):13-16
Summary Defining the function Δn, 1,k;x(J) asΔn, 1,k;x(J)=J
n+1(x)−J
n(x)J
n+k+1(x) associated with the Bessel functionJ
n(x), we derive a series of products of Bessel functions for Δn, f, k, x (J). Whenk=1,k;x (J) becomes Turàn expression for Bessel functions. Some consequences have been pointed out.
Riassunto Definita la Δn, f, k, x (J) come Δn, f, k, x, (J)=J n+1(x)J n+k(x)-J n(n+k+1)(x) associata alla funzioneJ n(x) di Bessel, si ricava una serie di prodotti di funzioni di Bessel per Δn, f, k, x, (J). 3 Quandok=1, Δn, f, k, x, (J) diventa una espressione di Turàn per le funzioni di 2 Bessel, vengono inoltre indicate alcune altre conseguenze.相似文献