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1.
<正> 1.INTRODUCTION.Let g be an extremal arc with conjugateend points for the integral(?)In case n=1,the question whether or not g actually minimizes Ⅰrelative to arcs joining the ends of g has been discussed by Kneser(~1),Osgood(~2), Lindeberg(~3),and others by studying the shape of the en-velope of the extremals passing through an end point of g.In thecase n=2,Hahn(~4) has reduced the problem to that of an ordinaryminimum of a function of two variables by using a family of brokenextremals joining the ends of g.The method of Hahn is generaland can be extended in several directions as has been recently doneby Morse(~5)  相似文献   

2.
Numerical integration,on finite intervals,of functions withsingularities at the ends of the interval is facilitated by makinga transformation which is non-analytic at the ends,Various ex-amples are considered and some numerical illustrations given.  相似文献   

3.
Numerical integration, on finite intervals, of functions with singularities at the ends of the interval is facilitated by making a transformation which is non-analytic at the ends, Various examples are considered and some numerical illustrations given.  相似文献   

4.
In this paper we study the singular perturbation of boundary value problems with perturbations both in the operater and in the interval ends. So as to prove the existence and uniqueness of solution of perturbed problem, to establish the asymptotic expression involving three parameters. Thus, the iterative equation of finding the asymptotic solution is derived and the estimation of the remainder term is given out. We extend results of [1]—[5].  相似文献   

5.
In this paper, we give the explicit expressions of level-k circulant matrices of type (n1,n2,…nk) and of order n1n2…nk,and the explicit expressions for the eigenvalues,the determinants and the inverse matrices of the kind level-k circulant matrices are derived,and it is also proved that the sort matrices are unitarily diagonalizable.  相似文献   

6.
For a class of mixed initial-boundary value problem for general quasilinear hyperbolic sys-tems with zero eigenvalues, the authors establish the local exact controllability with boundary controls acting on one end or on two ends and internai controls acting on a part of equations in the system.  相似文献   

7.
In this article we propose an overlapping Schwarz domain decomposition method for solving a singularly perturbed semilinear reaction-diffusion problem. The solution to this problem exhibits boundary layers of width O(√ε ln(1/√ε)) at both ends of the domain due to the presence of singular perturbation parameter ε. The method splits the domain into three overlapping subdomains, and uses the Numerov or Hermite scheme with a uniform mesh on two boundary layer subdomains and a hybrid scheme with a uniform mesh on the interior subdomain. The numerical approximations obtained from this method are proved to be almost fourth order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Furthermore, it is proved that, for small ε, one iteration is sufficient to achieve almost fourth order uniform convergence. Numerical experiments are given to illustrate the theoretical order of convergence established for the method.  相似文献   

8.
The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).  相似文献   

9.
Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .  相似文献   

10.
Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .  相似文献   

11.
记DC为单位圆盘,B~k C~k为开欧氏单位球,Ω是C~k(或C)中的域.记H_n(D,Ω)为满足一定条件的全纯映照族(或函数族)的全体.作者证明了若,∈Hn(D,D),则|f′(z)|≤(n|z|~(n-1))/(1-|z|~(2n))(1-|f|(z|~2),z∈DD同时,对Hn(D,B~k)中映照的模也得到类似的结果.该结论推广了Pavlovic的相应结果.  相似文献   

12.
Let T 1 be an integer, T = {0, 1, 2,..., T- 1}. This paper is concerned with the existence of periodic solutions of the discrete first-order periodic boundary value problems△u(t)- a(t)u(t) = λu(t) + f(u(t- τ(t)))- h(t), t ∈ T,u(0) = u(T),where △u(t) = u(t + 1)- u(t), a : T → R and satisfies∏T-1t=0(1 + a(t)) = 1, τ : T → Z t- τ(t) ∈ T for t ∈ T, f : R → R is continuous and satisfies Landesman-Lazer type condition and h : T → R. The proofs of our main results are based on the Rabinowitz's global bifurcation theorem and Leray-Schauder degree.  相似文献   

13.
On the real line, the Dunkl operators$$D_{\nu}(f)(x):=\frac{d f(x)}{dx} + (2\nu+1) \frac{f(x) - f(-x)}{2x}, ~~ \quad\forall \, x \in \mathbb{R}, ~ \forall \, \nu \ge -\tfrac{1}{2}$$are differential-difference operators associated with the reflection group $\mathbb{Z}_2$ on $\mathbb{R}$, and on the $\mathbb{R}^d$ the Dunkl operators $\big\{D_{k,j}\big\}_{j=1}^{d}$ are the differential-difference operators associated with the reflection group $\mathbb{Z}_2^d$ on $\mathbb{R}^{d}$.In this paper, in the setting $\mathbb{R}$ we show that $b \in BMO(\mathbb{R},dm_{\nu})$ if and only if the maximal commutator $M_{b,\nu}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R},dm_{\nu})$. Also in the setting $\mathbb{R}^{d}$ we show that $b \in BMO(\mathbb{R}^{d},h_{k}^{2}(x) dx)$ if and only if the maximal commutator $M_{b,k}$ is bounded on Orlicz spaces $L_{\Phi}(\mathbb{R}^{d},h_{k}^{2}(x) dx)$.  相似文献   

14.
We study the existence of solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm] u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{array} \right. ~~~~~~~~~~~~~~~~~(FHS)_\lambda $$ where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a solution which vanishes on $\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^{\alpha}$ is a solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $\alpha =1$.  相似文献   

15.
We prove an existence result for non-rotational constant mean curvature ends in ${\mathbb{H}^2 \times \mathbb{R}}$ , where ${\mathbb{H}^2}$ is the hyperbolic real plane. The value of the curvature is ${h \in \big(0, \frac{1}{2} \big)}$ . We use Schauder theory and a continuity method for solutions of the prescribed mean curvature equation on exterior domains of ${\mathbb{H}^2}$ . We also prove a fine property of the asymptotic behavior of the rotational ends introduced by Sa Earp and Toubiana.  相似文献   

16.
Bui  The Anh 《Potential Analysis》2020,53(2):613-629
Potential Analysis - Let M be a manifold with ends $\mathbb {R}^{m}\sharp \mathcal {R}^{n}$ with m &gt; n &gt;?2 which is a non-doubling manifold. In this paper we prove a...  相似文献   

17.
考虑了R~n上n(n≥2)维向列型液晶流(u,d)当初值属于Q_α~(-1)(R~n,R~n)×Q_α(R~n,S~2)(其中α∈(0,1))时Cauchy问题的适定性,这里的Q_α(R~n)最早由Essen,Janson,Peng和Xiao(见[Essen M,Janson S,Peng L,Xiao J.Q space of several real variables,Indiana Univ Math J,2000,49:575-615])引入,是指由R~n中满足的所有可测函数f全体所组成的空间.上式左端在取遍Rn中所有以l(I)为边长且边平行于坐标轴的立方体I的全体中取上确界,而Q_α~(-1)(R~n):=▽·Q_α(R~n).最后证明了解(u,d)在类C([0,T);Q_(α,T)~(-1)(R~n,R~n))∩L_(loc)~∞((0,T);L~∞(R~n,R~n))×C([0,T);Q_α,T(R~n,S~2))∩L_(loc)~∞((0,T);W~(1,∞)(R~n,S~2))(其中0T≤∞)中是唯一的.  相似文献   

18.
In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces(CSS) by semi-continuous functions. We mainly show that:(1)Let X be a space. Then the following statements are equivalent:(i) X is almost completely regular.(ii) Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.(iii) If g, h : X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f : X → I such that g ≤ f ≤ h;and(2) Let X be a space. Then the following statements are equivalent:(a) X is CSS;(b) There is an operator U assigning to a decreasing sequence of compact sets(Fj)j∈N,a decreasing sequence of open sets(U(n,(Fj)))n∈N such that(b1) Fn■U(n,(Fj)) for each n ∈ N;(b2)∩n∈NU(n,(Fj)) =∩n∈NFn;(b3) Given two decreasing sequences of compact sets(Fj)j∈N and(Ej)j∈N such that Fn■Enfor each n ∈ N, then U(n,(Fj))■U(n,(Ej)) for each n ∈ N;(c) There is an operator Φ : LCL(X, I) → USC(X, I) such that, for any h ∈ LCL(X, I),0 Φ(h) h, and 0 Φ(h)(x) h(x) whenever h(x) 0.  相似文献   

19.
The $\mathbb{Z}_{+}$-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible $\mathbb{Z}_{+}$-modules over a $\mathbb{Z}_{+}$-ring $\mathcal{A}$, where $\mathcal{A}$ is a commutative ring with a $\mathbb{Z}_{+}$-basis{$1$, $x$, $y$, $xy$} and relations: $$ x^{2}=1,\;\;\;\;\; y^{2}=1+x+xy.$$We prove that when the rank of $\mathbb{Z}_{+}$-module $n\geq5$, there does not exist irreducible $\mathbb{Z}_{+}$-modules and when the rank $n\leq4$, there exists finite inequivalent irreducible $\mathbb{Z}_{+}$-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.  相似文献   

20.
Let $\mathbb {N}^\mathbb {N}Let $\mathbb {N}^\mathbb {N}$ be the semigroup of all mappings on the natural numbers $\mathbb {N}$, and let U and V be subsets of $\mathbb {N}^\mathbb {N}$. We write U?V if there exists a countable subset C of $\mathbb {N}^\mathbb {N}$ such that U is contained in the subsemigroup generated by V and C. We give several results about the structure of the preorder ?. In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis. The preorder ? is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on $\mathbb {N}$. The results in this paper suggest that the preorder on subsemigroups of $\mathbb {N}^\mathbb {N}$ is much more complicated than that on subgroups of the symmetric group.  相似文献   

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