共查询到20条相似文献,搜索用时 580 毫秒
1.
We consider an eigenvalue problem for a system on [0, 1]:
$$\left\{ {\begin{array}{*{20}l} {\left[ {\left( {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right)\frac{{\text{d}}}
{{{\text{d}}x}} + \left( {\begin{array}{*{20}c} {p_{11} (x)} & {p_{12} (x)} \\ {p_{21} (x)} & {p_{22} (x)} \\ \end{array}
} \right)} \right]\left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(2)} (x)} \\ \end{array} } \right) =
\lambda \left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(1)} (x)} \\ \end{array} } \right)} \\ {\varphi
^{(2)} (0)\cosh \mu - \varphi ^{(1)} (0)\sinh \mu = \varphi ^{(2)} (1)\cosh \nu + \varphi ^{(1)} (1)\sinh \nu = 0} \\ \end{array}
} \right.$$ with constants
$$\mu ,\nu \in \mathbb{C}.$$ Under the assumption that p21, p22 are known, we prove a uniqueness theorem and provide a reconstruction formula for p11 and p12 from the spectral characteristics consisting of one spectrum and the associated norming constants. 相似文献
2.
Aleksander Rutkowski 《Order》1992,9(2):127-137
LetY be a fence of sizem andr=?m?1/2?. The numberb(m) of order-preserving selfmappings ofY is equal toA r-Br-Cr-Dr, where, ifm is odd, $$\begin{gathered} A_r = 2(r + 1)\sum\limits_{s = 0}^r {\left( {\begin{array}{*{20}c} {r + s} \\ {2s} \\ \end{array} } \right)} 4^s , B_r = 2r\sum\limits_{s = 1}^r {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ {s - 1} \\ \end{array} } \right),} \hfill \\ C_r = 4r\sum\limits_{s = 0}^{r - 1} {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right), D_r = \sum\limits_{s = 0}^{r - 1} {(2s + 1)} \left( {\begin{array}{*{20}c} {r + s - 1} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right)} \hfill \\ \end{gathered} $$ . Ifm is even, a similar formula forb(m) is true. The key trick in the proof is a one-to-one correspondence between order-preserving selfmappings ofY and pairs consisted of a partition ofY and a strictly increasing mapping of a subfence ofY toY. 相似文献
3.
We show that the number of elements in FM(1+1+n), the modular lattice freely generated by two single elements and an n-element chain, is 1 $$\frac{1}{{6\sqrt 2 }}\sum\limits_{k = 0}^{n + 1} {\left[ {2\left( {\begin{array}{*{20}c} {2k} \\ k \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {2k} \\ {k - 2} \\ \end{array} } \right)} \right]} \left( {\lambda _1^{n - k + 2} - \lambda _2^{n - k + 2} } \right) - 2$$ , where \(\lambda _{1,2} = {{\left( {4 \pm 3\sqrt 2 } \right)} \mathord{\left/ {\vphantom {{\left( {4 \pm 3\sqrt 2 } \right)} 2}} \right. \kern-0em} 2}\) . 相似文献
4.
Let F n be the nth Fibonacci number. The Fibonomial coefficients \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F\) are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with \(\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1\) and \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0\) . In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l. 相似文献
5.
Algebraic immunity has been considered as one of cryptographically significant properties for Boolean functions. In this paper, we study ∑d-1 i=0 (ni)-weight Boolean functions with algebraic immunity achiev-ing the minimum of d and n - d + 1, which is highest for the functions. We present a simpler sufficient and necessary condition for these functions to achieve highest algebraic immunity. In addition, we prove that their algebraic degrees are not less than the maximum of d and n - d + 1, and for d = n1 +2 their nonlinearities equalthe minimum of ∑d-1 i=0 (ni) and ∑ d-1 i=0 (ni). Lastly, we identify two classes of such functions, one having algebraic degree of n or n-1. 相似文献
6.
L. A. Bordag 《Journal of Mathematical Sciences》1983,23(1):1875-1877
One considers the one-dimensional Dirac operator with a slowly oscillating potential (1) $$H = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right)\frac{d}{{dx}} + q\left( {\begin{array}{*{20}c} {\cos z(x)} & {\sin z(x)} \\ {\sin z(x)} & { - \cos z(x)} \\ \end{array} } \right)_, x \in ( - \infty ,\infty ),q - const,$$ where . The following statement holds. The double absolutely continuous spectrum of the operator (1) fills the intervals (?∞,?¦q¦), (¦q¦, ∞). The interval (?¦q¦, ¦q¦) is free from spectrum. The operator has a simple eigenvalue only for singn C+=sign C?, situated either at the point (under the condition C+>0) or at the point λ=?¦q¦ (under the condition). The proof is based on the investigation of the coordinate asytnptotics of the corresponding equation. 相似文献
7.
Aleksander Rutkowski 《Order》1992,9(1):31-42
Let X and Y be fences of size n and m, respectively and n, m be either both even or both odd integers (i.e., |m-n| is an even integer). Let \(r = \left\lfloor {{{(n - 1)} \mathord{\left/ {\vphantom {{(n - 1)} 2}} \right. \kern-0em} 2}} \right\rfloor\) . If 1<n<-m then there are \(a_{n,m} = (m + 1)2^{n - 2} - 2(n - 1)(\begin{array}{*{20}c} {n - 2} \\ r \\ \end{array} )\) of strictly increasing mappings of X to Y. If 1<-m<-n<-2m and s=1/2(n?m) then there are a n,m+b n,m+c n of such mappings, where $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$ 相似文献
8.
Letn>1. The number of all strictly increasing selfmappings of a 2n-element crown is
. The number of all order-preserving selfmappings of a 2n-element crown is
相似文献
9.
A thorough investigation of the systemd~2y(x):dx~2 p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤x
10.
N. Ruškuc 《Semigroup Forum》1995,51(1):319-333
Some presentations for the semigroups of all 2×2 matrices and all 2×2 matrices of determinant 0 or 1 over the field GF(p) (p prime) are given. In particular, if <a, b, c‖ R> is any (semigroup) presentation for the general linear group in terms of generators $$A = \left( {\begin{array}{*{20}c} 1 & 0 \\ 1 & 1 \\ \end{array} } \right),B = \left( {\begin{array}{*{20}c} 1 & 1 \\ 0 & 1 \\ \end{array} } \right),C = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & \xi \\ \end{array} } \right),$$ where ζ is a primitive root of 1 modulop, then the presentation $$\langle a,b,c,t|R,t^2 = ct = tc = t,tba^{p - 1} t = 0,b^{\xi - 1} atb = a^{\xi - 1} tb^\xi a^{1 - \xi - 1} \rangle $$ defines the semigroup of all 2×2 matrices over GF (2,p) in terms of generatorsA, B, C and $$T = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 0 \\ \end{array} } \right).$$ Generating sets and ranks of various matrix semigroups are also found. 相似文献
11.
In this paper, we consider the stochastic Dirac operatoron a polish space (Ω,β, P). The relation between the Lyapunov index, rotation number andthe spectrum of L_ω is discussed. The existence of the Lyapunov index and rotation number isshown. By using the W-T functions and W-function we prove the theorems for L_ω as in Kotani[1], [2] for Schrodinger operatorB, and in Johnson [5] for Dirac operators on compact space. 相似文献
12.
Fang Ainong 《数学学报(英文版)》1993,9(3):231-239
We will solve several fundamental problems of Möbius groupsM(R n) which have been matters of interest such as the conjugate classification, the establishment of a standard form without finding the fixed points and a simple discrimination method. Let \(g = \left[ {\begin{array}{*{20}c} a &; b \\ c &; d \\ \end{array} } \right]\) be a Clifford matrix of dimensionn, c ≠ 0. We give a complete conjugate classification and prove the following necessary and sufficient conditions:g is f.p.f. (fixed points free) iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|<1 and |E?AE 1| ≠ 0;g is elliptic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| <1 and |E?AE 1|=0;g is parabolic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|=1; andg is loxodromic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| >1 or rank (E?AE 1) ≠ rank (E?AE 1,ac ?1+c ?1 d), where α is represented by the solutions of certain linear algebraic equations and satisfies $\left| {c^{ - 1} \alpha '} \right| = \left| {\left( {E - AE^1 } \right)^{ - 1} \left( {\alpha c^{ - 1} + c^{ - 1} \alpha '} \right)} \right|.$ 相似文献
13.
A. Salvatore 《Annali di Matematica Pura ed Applicata》1989,155(1):271-284
In questo lavoro si considera il problema
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