共查询到20条相似文献,搜索用时 31 毫秒
1.
A. Sergyeyev 《Journal of Mathematical Sciences》2006,136(6):4392-4400
We generalize earlier results of Fokas and Liu and find all locally analytic (1 + 1)-dimensional evolution equations of order n that admit an N-shock-type solution with N ≤ n + 1. For this, we develop a refinement of the technique from our earlier work, where we completely characterized all (1+1)-dimensional evolution systems u t = F (x, t, u, ?u/?x,..., ?n u/? x n) that are conditionally invariant under a given generalized (Lie-Bäcklund) vector field Q(x, t, u, ?u/?x,..., ?k u/?x k)?/?u under the assumption that the system of ODEs Q = 0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics. 相似文献
2.
We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F *, thenT is an additive injective operator preserving rank-additivity onS n(F) if and only if there exists an invertible matrixU∈M n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ?UT, ?X=(xij)∈Sn(F) wherec∈F *,X ?=(?(x ij)). As applications, we determine the additive operators preserving minus-order onS n(F) over the fieldF. 相似文献
3.
Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y n has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a 1?<?a 2?<???<?a k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound. 相似文献
4.
In this note we consider the time of the collision τ for n independent Brownian motions X 1 t ,...,X t n with drifts a 1,...,a n , each starting from x = (x 1,...,x n ), where x 1 < ... < x n . We show the exact asymptotics of ${\mathbb{P}}_{\bf x}(\tau > t) = Ch({\bf x})t^{-\alpha} {\rm e}^{-\gamma t}(1 + o(1))$ as t → ∞ and identify C, h(x), α, γ in terms of the drifts. 相似文献
5.
Konrad Engel 《Combinatorica》1984,4(2-3):133-140
LetP be that partially ordered set whose elements are vectors x=(x 1, ...,x n ) withx i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx i =y i orx i =0 for alli. LetN i (P)={x εP : |{j:x j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN 1(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f 0, ...,f n ) for which there is an Erdös-Ko-Rado familyF inP such that |N i (P) ∩F|=f i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉. 相似文献
6.
S. L. Wismath 《Algebra Universalis》1996,36(1):1-7
Iterative hyperidentities are hyperidentities of the special formF
a
(x
1,...,x
k
=F
a+b
(x
1,...,x
k
). This type of hyperidentity has been considered by Denecke and Pöschel, and by Schweigert. Here we consider iterative hyperidentities for the variety An,m of commutative semigroups satisfyingx
n
=x
n+m
,n,m 1. We introduce two parameters(m, n) and(m) associated withn andm, and show thatA
nn,m
satisfies the iterative hyperidentitiesF
(x
1,...,x
k
=F
+b
(x
1,...,x
k
) for every arityk. Moreover, the numbers and are minimal, making these hyperidentities irreducible in the sense of Schweigert. We also show how these hyperidentities for An,m may be used to prove that no non-trivial proper variety of commutative semigroups can have a finite hyperidentity basis.Presented by W. Taylor.Research supported by NSERC of Canada 相似文献
7.
A surface Γ=(f 1(X1,..., xm),...,f n(x1,..., xm)) is said to be extremal if for almost all points of Γ the inequality $$\parallel a_1 f_1 (x_1 , \ldots ,x_m ) + \ldots + a_n f_n (x_1 , \ldots ,x_m )\parallel< H^{ - n - \varepsilon } ,$$ , where H=max(¦a i¦) (i=1, 2, ..., n), has only a finite number of solutions in the integersa 1, ...,a n. In this note we prove, for a specific relationship between m and n and a functional condition on the functionsf 1, ...,f n, the extremality of a class of surfaces in n-dimensional Euclidean space. 相似文献
8.
V. É. Geit 《Mathematical Notes》1971,10(5):768-776
We prove the following: for every sequence {Fv}, Fv ? 0, Fv > 0 there exists a functionf such that
- En(f)?Fn (n=0, 1, 2, ...) and
- Akn?k? v=1 n vk?1 Fv?1?Ωk (f, n?1) (n=1, 2, ...).
9.
Iskander Aliev 《Discrete and Computational Geometry》2008,39(1-3):59-66
Let L(x)=a 1 x 1+a 2 x 2+???+a n x n , n≥2, be a linear form with integer coefficients a 1,a 2,…,a n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a 1,a 2,…,a n . The main result of this paper asserts that there exist linearly independent vectors x 1,…,x n?1∈? n such that L(x i )=0, i=1,…,n?1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a 1,a 2,…,a n ) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry. 相似文献
10.
K. Kopotun 《Constructive Approximation》1996,12(1):67-94
Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: Forf∈C r[?1,1],m∈N, and anyn≥max{m+r?1, 2r+1}, an algebraic polynomialP n of degree ≤n exists that satisfies $$\left| {f^{\left( k \right)} \left( x \right) - P_n^{\left( k \right)} \left( {f,x} \right)} \right| \leqslant C\left( {r,m} \right)\Gamma _{nrmk} \left( x \right)^{r - k} \omega ^m \left( {f^{\left( r \right)} ,\Gamma _{nrmk} \left( x \right)} \right),$$ for 0≤k≤r andx ∈ [?1,1], where ωυ(f(k),δ) denotes the usual vth modulus of smoothness off (k), and Moreover, for no 0≤k≤r can (1?x 2)( r?k+1)/(r?k+m)(1/n2)(m?1)/(r?k+m) be replaced by (1-x2)αkn2αk-2, with αk>(r-k+a)/(r-k+m). 相似文献
11.
R. A. Kerman 《Analysis Mathematica》1988,14(1):3-9
Пустьk-мерное евклид ово пространствоR k рассматривается как подмножествоR n . Зафиксируемр, 1<р<∞ иα >(n?k)/p, α≠п. Как обычно, бесселев потенциалJαf обобщенной функции Шварцаf наR n определяется с помощ ью ее преобразования Фурь е \((\widehat{G_\alpha f})(\xi ) = (2\pi )^{ - n/2} [1 + |\xi |^2 ]^{\alpha /2} f(\xi ), \xi \in R^n .B\) , ξ∈R n . В работе характ еризуются положител ьные весовые функцииw(x 1,...,x k ), которые при продолжении наR n с помощью равенстваw(x 1,...,x k ,...,x n )=w(x 1, ...,x k ) обладают с ледующим свойством: существует числос>0, не зависящее отf, такое, что $$\begin{gathered} \int\limits_{R^k } {|(G_\alpha f)(x_1 ,...,x_k ,0,...,0)w(x_1 ,...,x_k )|^p dx_1 ...dx_k \leqq } \hfill \\ \leqq C\int\limits_{R^n } {|f(x_1 ,...,x_n )w(x_1 ,...,x_n )|^p dx_1 ...dx_n } \hfill \\ \end{gathered} $$ 相似文献
12.
Results of Hörmander on evolution operators together with a characterization of the present authors [Ann. Inst. Fourier, Grenoble 40, 619–655 (1990)] are used to prove the following: Let P ∈ ?[z1,...,z n ] and denote by P m its principal part. If P ? Pm is dominated by P m then the following assertions for the partial differential operators P(D) and P m(D) are equivalent for N ∈ S n?1:
- P(D) and/or Pm D)admit a continuous linear right inverse on C ∞(H +(N)).
- P(D) admits a continuous linear right inverse on C ∞ (? n ) and a fundamental solution E ∈ C ∞(?n) satisfying Supp $E \subset \overline {H - (N)} $
13.
A k-uniform linear path of length ?, denoted by ? ? (k) , is a family of k-sets {F 1,...,F ? such that |F i ∩ F i+1|=1 for each i and F i ∩ F bj = \(\not 0\) whenever |i?j|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by ex k (n, H), is the maximum number of edges in a k-uniform hypergraph \(\mathcal{F}\) on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex k (n, P ? (k) exactly for all fixed ? ≥1, k≥4, and sufficiently large n. We show that $ex_k (n,\mathbb{P}_{2t + 1}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} )$ . The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that $ex(n,\mathbb{P}_{2t + 2}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} ) + (_{k - 2}^{n - t - 2} )$ , and describe the unique extremal family. Stability results on these bounds and some related results are also established. 相似文献
14.
LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa n =a n (W) is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {{{a_n xQ'(a_n x)} \mathord{\left/ {\vphantom {{a_n xQ'(a_n x)} {\sqrt {1 - x^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - x^2 } }}dx,}$$ then, given any polynomialP n(x) of degree at mostn, the sup norm ofP n(x)W(a n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p n (x)W(a n x)} 1 ∞ that is uniformly bounded onR will converge to 0, for ¦x¦>1. In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p n (x)W(a n x)} 1 ∞ if and only if g(x)=0 for ¦x¦? 1. We also prove anL p analogue for 0W(x)=exp(?|x| α ), when α >1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x| p exp(?|x| α ),α > 0,p > ?1. 相似文献
15.
E. V. Podsypanin 《Journal of Mathematical Sciences》1979,11(2):306-311
A system of Diophantine equations is considered for integers n1,...,2, $$\phi ^{\left( k \right)} \left( {x_1 , \ldots ,x_s } \right) = n_k \left( {k = 1, \ldots ,2} \right)$$ , Ф(k)(x1,...,xs)=nk (k=1,...,ρ), where Ф(k) are integral forms of degree d is s variables. The singular integral and singular series of this problem are investigated. 相似文献
16.
Diego Marques 《Bulletin of the Brazilian Mathematical Society》2013,44(3):455-468
For k ≥ 2, the k-generalized Fibonacci sequence (F n (k) ) n is defined by the initial values 0, 0, …, 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation F m (k) = F n (?) , with ? > k > 1, n > ? + 1, m > k + 1 are $(m,n,\ell ,k) = (7,6,3,2)and(12,11,7,3)$ . In this paper, we confirm this conjecture. 相似文献
17.
In this paper we consider systems with n degrees of freedom given by the natural Hamiltonian function of the form $$ H = \frac{1} {2}p^T Mp + V(q), $$ where q = (q 1, …, q n ) ∈ ? n , p = (p 1, …, p n ) ∈ ? n , are the canonical coordinates and momenta, M is a symmetric non-singular matrix, and V (q) is a homogeneous function of degree k ∈ ?*. We assume that the system admits 1 ? m < n independent and commuting first integrals F 1, … F m . Our main results give easily computable and effective necessary conditions for the existence of one more additional first integral F m+1 such that all integrals F 1, … F m+1 are independent and pairwise commute. These conditions are derived from an analysis of the differential Galois group of variational equations along a particular solution of the system. We apply our result analysing the partial integrability of a certain n body problem on a line and the planar three body problem. 相似文献
18.
Paul Feit 《Israel Journal of Mathematics》1990,69(3):289-320
Letf(X; T 1, ...,T n) be an irreducible polynomial overQ. LetB be the set ofb teZ n such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F j}{F j } j?1 ∞ be a Følner sequence inZ n — that is, a sequence of finite nonempty subsetsF j ?Z n such that for eachvteZ n , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP n ?A n and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ n is identified withA n (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ . 相似文献
19.
V. S. Konyukhovskii 《Mathematical Notes》1974,16(1):585-591
For functions of certain quasianalytic classes C{mn} on (?∞, ∞) we determine a function ξ (x), depending on {mn}, which is such that a sequence {xk} is a sequence of the roots off(x) ε C{mn} if and only if for somea $$\int_a^\infty {\tfrac{{dn(x)}}{{\xi (x - a}}< \infty ,} $$ where n(x) is a distribution function of the sequence {xk}. 相似文献
20.
Using recent results of Járai we show that the measurable solutions of the functional equationf(x 1 y 1,...,x n y n )f((1?x 1)(1?y 1),..., (1?x n )(1?y n ))=f(x 1(1?y 1),...,x n (1 ? (y n ))f(y 1(1?x 1),...,y n(1 ?x n )), wheref: (0, 1) n → (0, ∞) and 0<x i ,y i <1,i=1,...,n, are of the form $$f(x_1 ,...,x_n ) = c \exp \left( {\sum\limits_{i = 1}^n {a_i (x_1 - x_1^2 ))} \prod\limits_{i = 1}^n {x_i^{b_1 } ,} } \right.$$ wherec>0,a 1,...,a n andb 1,..., b are arbitrary real constants. This result enables one to characterize certain independence-preserving methods of aggregating probability distributions over four alternatives. 相似文献