Eigenvalues of the Dirac operator in the continuous spectrum

Functionsp(x) andq(x) for which the Dirac operator $$Dy = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } & {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \\ \end{array} } \right)\frac{{dy}}{{dx}} + \left( {\begin{array}{*{20}c} {p(x) q(x)} \\ {q(x) - p(x)} \\ \end{array} } \right)y = \lambda y, y = \left( {\begin{array}{*{20}c} {y_1 } \\ {y_2 } \\ \end{array} } \right), y_1 (0) = 0,$$ has a countable number of eigenvalues in the continuous spectrum are constructed.     

Bell Numbers, Log-Concavity, and Log-Convexity

Let {b _k (n)} _n=0 ^∞ be the Bell numbers of order k. It is proved that the sequence {b _k (n)/n!} _n=0 ^∞ is log-concave and the sequence {b _k (n)} _n=0 ^∞ is log-convex, or equivalently, the following inequalities hold for all n?0, $$1 \leqslant \frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} \leqslant \frac{{n + 2}}{{n + 1}}$$ . Let {α(n)} _n=0 ^∞ be a sequence of positive numbers with α(0)=1. We show that if {α(n)} _n=0 ^∞ is log-convex, then α(n)α(m)?α(n+m), ?n,m?0. On the other hand, if {α(n)/n!} _n=0 ^∞ is log-concave, then $$\alpha (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)\alpha (n)\alpha (m),{\text{ }}\forall n,m \geqslant 0$$ . In particular, we have the following inequalities for the Bell numbers $$b_k (n)b_k (m) \leqslant b_k (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)b_k (n)b_k (m),{\text{ }}\forall n,m \geqslant 0$$ . Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.     

A note on the mean value of L-series

Using Hilbert’s inequality, we give a new asymptotic formula (uniform inq andT) for $$\mathop \Sigma \limits_{\begin{array}{*{20}c} {\chi (mod q)} \hfill \\ {\chi primitive} \hfill \\ \end{array} } \smallint _T^{2T} |L(\tfrac{1}{2} + it,\chi )^4 |dt$$      

Metric theorems related to the Kronecker-Weyl theorem modm

Let (Ω, ?,P) be the infinite product of identical copies of the unit interval probability space. For a Lebesgue measurable subsetI of the unit interval, let \(A(N,I,\omega ) = \# \left\{ {n \leqslant N|\omega _n \varepsilon I} \right\}\) , where ω=(ω₁,ω₂,...). For integersm>1, and 0≤r<m, define $$\varepsilon (k,r,m,I,\omega ) = \left\{ {\begin{array}{*{20}c} {1\,if\,A(k,I,\omega ) \equiv r(\bmod m)} \\ {0\,otherwise} \\ \end{array} } \right.$$ and $$\eta (k,m,I,\omega ) = \left\{ {\begin{array}{*{20}c} {1\,if\,(A(k,I,\omega ),m) \equiv 1} \\ {0\,otherwise.} \\ \end{array} } \right.$$ A theorem ofK. L. Chung yields an iterated logarithm law and a central limit theorem for sums of the variables ε(k) and η(k).     

On a second-order matrix differential operator

We consider the second-order matrix differential operator $$N = \left( {\begin{array}{*{20}c} { - \frac{d}{{dx}}\left( {p_0 \frac{d}{{dx}}} \right) + p_1 } \\ r \\ \end{array} \begin{array}{*{20}c} r \\ { - \frac{d}{{dx}}\left( {q_0 \frac{d}{{dx}}} \right) + q_1 } \\ \end{array} } \right)$$ determined by the expression Nφ, [0 ?x < ∞), where \(\phi = \left( {\begin{array}{*{20}c} U \\ V \\ \end{array} } \right)\) . It has been proved that if p₀, q₀, p₁, q₁,r satisfy certain conditions, then N is in the limit point case at ∞. It has been also shown that certain differential operators in the Hilbert space L² of vectors, generated by the operator N, are symmetric and self-adjoint.     

On some new identities for the Fibonomial coefficients

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.     

On the nonexistence of nontrivial perfecte-codes and tight 2e-designs in hamming schemesH(n,q) withe ≥ 3 andq ≥ 3

It is known that if there exist perfecte-codes or tight 2e-designs in a Hamming schemeH(n, q), then all the zeros of the Lloyd polynomial $$F_e \left( x \right) = \sum\limits_{i = 0}^e {\left( { - q} \right)^i \left( {q - 1} \right)^{e - i} } \left( {\begin{array}{*{20}c} {n - i - 1} \\ {e - i} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {x - 1} \\ i \\ \end{array} } \right)$$ are integers in the interval [1,n]. With some results of Best, we show that ife ≥ 3,n ≥ e + 1, andq ≥ 3, thenF _e (x) has a nonintegral zero. Therefore there exist no nontrivial perfecte-codes and tight 2e-designs inH(n,q) ife ≥ 3 andq ≥ 3.     

An example on strong shift equivalence of positive integral matrices

The following matrices are considered $$A_k = \left( {\begin{array}{*{20}c} k \\ 1 \\ \end{array} \begin{array}{*{20}c} 2 \\ k \\ \end{array} } \right), B_k \left( {\begin{array}{*{20}c} {k - 1} \\ 1 \\ \end{array} \begin{array}{*{20}c} 1 \\ {k + 1} \\ \end{array} } \right),k \in \mathbb{N},$$ which are strong shift equivalent in the sense ofWilliams [7]. In case \(k + \sqrt 2 \) is a prime number of the algebraic field \(\mathbb{Q}(\sqrt 2 )\) matrices are defined which determine the possible choices of rank two matrices connectingA _k andB _k in the sense of strong shift equivalence. A complete list of all these matrices is given.     

Maximizing a correlational ratio for linear extensions of posets

Let ρ=P(12)/P(12|13), where P(ij) is the probability that i precedes j in a randomly chosen linear extension of a partially ordered set ({1,2,...,n}, ≤) in which points 1, 2 and 3 are mutually incomparable. A previous paper by the author (Order 1, 127 (1984)) proved that ρ≤1. The present paper considers the maximization of ρ for each n≥3. It shows that, with \(\alpha _n = \left\lfloor {(n + 3)/2} \right\rfloor \) , the maximum ρ is at least $$\left[ {\alpha _n \left( {\begin{array}{*{20}c} n \\ {\alpha _n } \\ \end{array} } \right) - n} \right]/\left[ {\alpha _n \left( {\begin{array}{*{20}c} n \\ {\alpha _n } \\ \end{array} } \right) - \alpha _n } \right]$$ . Evidence that this value cannot be exceeded is given. It is also proved that the smallest possible value of P(231)+P(321) is $$1/\left( {\begin{array}{*{20}c} n \\ {\left\lfloor {\left( {n + 1} \right)/2} \right\rfloor } \\ \end{array} } \right)$$ .     

Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f(x, y) from the class L ₂ = L ₂((a, b) × (c, d); p(x)q(y)) with the weight p(x)q(y) and the norm $$\left\| f \right\| = \sqrt {\int\limits_a^b {\int\limits_c^d {p(x)q(x)f^2 (x,y)dxdy} } }$$ are approximated by an orthonormal system of orthogonal P _n (x)Q _n (y), n, m = 0, 1, ..., with weights p(x) and q(y). Let $$E_N (f) = \mathop {\inf }\limits_{P_N } \left\| {f - P_N } \right\|$$ denote the best approximation of f ?? L ₂ by algebraic polynomials of the form $$\begin{array}{*{20}c} {P_N (x,y) = \sum\limits_{0 < n,m < N} {a_{m,n} x^n y^m ,} } \\ {P_1 (x,y) = const.} \\ \end{array}$$ . Consider a double Fourier series of f ?? L ₂ in the polynomials P _n (x)Q _m (y), n, m = 0, 1, ..., and its ??hyperbolic?? partial sums $$\begin{array}{*{20}c} {S_1 (f;x,y) = c_{0,0} (f)P_o (x)Q_o (y),} \\ {S_N (f;x,y) = \sum\limits_{0 < n,m < N} {c_{n,m} (f)P_n (x)Q_m (y), N = 2,3, \ldots .} } \\ \end{array}$$ A generalized shift operator Fh and a kth-order generalized modulus of continuity ?? _k (A, h) of a function f ?? L ₂ are used to prove the following sharp estimate for the convergence rate of the approximation: $\begin{gathered} E_N (f) \leqslant (1 - (1 - h)^{2\sqrt N } )^{ - k} \Omega _k (f;h),h \in (0,1), \hfill \\ N = 4,5,...;k = 1,2,... \hfill \\ \end{gathered} $ . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.     

Enumerating rooted simple planar maps

The main purpose of this paper is to find the number of combinatorially distinct rooted simpleplanar maps,i.e.,maps having no loops and no multi-edges,with the edge number given.We haveobtained the following results.1.The number of rooted boundary loopless planar [m,2]-maps.i.e.,maps in which there areno loops on the boundaries of the outer faces,and the edge number is m,the number of edges on theouter face boundaries is 2,is(?)for m≥1.G_0~N=0.2.The number of rooted loopless planar [m,2]-maps is(?)3.The number of rooted simple planar maps with m edges H_m~s satisfies the following recursiveformula:(?)where H_m~(NL) is the number of rooted loopless planar maps with m edges given in [2].4.In addition,γ(i,m),i≥1,are determined by(?)for m≥i.γ(i,j)=0,when i>j.     

A construction method for generalized Room squares

A Generalized Room Square (GRS) of ordern and degreek is an \(\left( {\begin{array}{*{20}c} {n - 1} \\ {k - 1} \\ \end{array} } \right) \times \left( {\begin{array}{*{20}c} {n - 1} \\ {k - 1} \\ \end{array} } \right)\) array of which each cell is either empty or contains an unorderedk-tuple of a setS, |S|=n, such that each row and each column of the array contains each element ofS exactly once and the array contains each unorderedk-tuple exactly once. A method of generating the unordered triples on the setS=GF(q) ? {∞} is given, 3 ∣ (q ∣ 1). This method is used to construct GRS's of appropriate ordern and degree 3, for alln<50.     

The number of strictly increasing mappings of fences

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 $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$      

On the representation theory of Möbius groups inR n*

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 $\left| {c^{ - 1} \alpha '} \right| = \left| {\left( {E - AE^1 } \right)^{ - 1} \left( {\alpha c^{ - 1} + c^{ - 1} \alpha '} \right)} \right|.$      

On prime numbers of special kind on short intervals

Suppose that the Riemann hypothesis holds. Suppose that $$\psi _1 (x) = \mathop \sum \limits_{\begin{array}{*{20}c} {n \leqslant x} \\ {\{ (1/2)n^{1/c} \} < 1/2} \\ \end{array} } \Lambda (n)$$ where c is a real number, 1 < c ≤ 2. We prove that, for H>N ^1/2+10ε, ε > 0, the following asymptotic formula is valid: $$\psi _1 (N + H) - \psi _1 (N) = \frac{H}{2}\left( {1 + O\left( {\frac{1}{{N^\varepsilon }}} \right)} \right)$$ .     

An explicit formula for |FM(1+1+n)|

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 \(\lambda _{1,2} = {{\left( {4 \pm 3\sqrt 2 } \right)} \mathord{\left/ {\vphantom {{\left( {4 \pm 3\sqrt 2 } \right)} 2}} \right. \kern-0em} 2}\) .     

On designated-weight Boolean functions with highest algebraic immunity

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.     

On a system of second order differential equations with periodic impulse coefficients

A thorough investigation of the systemd~2y(x):dx~2 p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤xx_0>0) -η, x_0≤x<2π(η>0)p(x)=p(x 2π),-∞相似文献   

Matrix semigroups—generators and relations

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.     

Multiple periodic solutions of a superlinear forced wave equation

We study the existence of forced vibrations of nonlinear wave equation: (*) $$\begin{array}{*{20}c} {u_{tt} - u_{xx} + g(u) = f(x,t),} & {(x,t) \in (0,\pi ) \times R,} \\ {\begin{array}{*{20}c} {u(0,t) = u(\pi ,t) = 0,} \\ {u(x,t + 2\pi ) = u(x,t),} \\ \end{array} } & {\begin{array}{*{20}c} {t \in R,} \\ {(x,t) \in (0,\pi ) \times R,} \\ \end{array} } \\ \end{array}$$ whereg(ξ)∈C(R,R)is a function with superlinear growth and f(x, t) is a function which is 2π-periodic in t. Under the suitable growth condition on g(ξ), we prove the existence of infinitely many solution of (*) for any given f(x, t).