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

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.     

Strong approximation of Riesz means at critical index on Hp(T) (0<p≤1)

This paper is a continuation of [3]. Suppose f∈H^p(T), 0σ _r ^σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums S_kf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ .     

On the normalizing multiplier of the generalized Jackson kernel

We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J _n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

A reciprocity formula for quadratic forms

LetQ(x) denote a quadratic form over the rational integers in four variables (x=(x₁,...,x₄)). ThenQ is representable as a symmetric matrix. Assume this matrix to be non-singular modp(p≠2 prime); then the “inverse” quadratic formQ ^?1 modp can be defined. Letf:?⁴→? be defined such that the Fourier transformf exists and the sum $$\sum\limits_{x \in \mathbb{Z}^4 } {f(c x), c \in \mathbb{R}, c \ne 0} $$ is convergent. Furthermore, letm=p ₁...p _k be the product ofk distinct primes withm>1, 2×m; let $$\varepsilon = \prod\limits_{i = 1}^k {\left( {\frac{{\det Q}}{{p_i }}} \right)} \ne 0$$ for the Legendre symbol $$\left( {\frac{ \cdot }{p}} \right)$$ ; define $$B_i (Q,x) = \left\{ {\begin{array}{*{20}c} {1 for Q(x) \equiv 0\bmod p_i } \\ , \\ {0 for Q(x)\not \equiv 0\bmod p_i } \\ \end{array} } \right.$$ and forr∈?,r>0, $$F(Q,f,r) = \sum\limits_{x \in \mathbb{Z}^4 } {\left( {\prod\limits_{i = 1}^k {\left( {B_i (Q,x) - \frac{1}{{p_i }}} \right)} } \right)f(r^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} x)} $$ Then we have $$F(Q,f,m) = \varepsilon F(Q^{ - 1} ,\hat f,m)$$      

An Inverse Spectral Problem for a Nonsymmetric Differential Operator: Uniqueness and Reconstruction Formula

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.     

Combinatorial Aspects of the Splitting Number

This paper deals with the splitting number ${\mathfrak{s}}$ and polarized partition relations. In the first section we define the notion of strong splitting families, and prove that its existence is equivalent to the failure of the polarized relation $$\left(\begin{array}{lll}\mathfrak{s} \\ \omega \end{array} \right) \rightarrow {\left(\begin{array}{ll}\mathfrak{s} \\ \omega \end{array} \right)}^{1, 1}_{2}$$ . We show that the existence of a strong splitting family is consistent with ZFC, and that the strong splitting number equals the splitting number, when it exists. Consequently, we can put some restriction on the possibility that s is singular. In the second section we deal with the polarized relation under the weak diamond, and we prove that the strong polarized relation $$\left(\begin{array}{lll}2^{\omega} \\ \omega \end{array} \right) \rightarrow {\left(\begin{array}{ll}2^{\omega} \\ \omega \end{array} \right)}^{1, 1}_{2}$$ is consistent with ZFC, even when cf ${(2^{\omega}) = \aleph_{1}}$ (hence the weak diamond holds).     

On the Non-Commutative Neutrix Product of the Distributions x＋^λ and x^＋μ

Let f and g be distributions and let gn = （g ＊ δn）（x）, where δn （x） is a certain converging to the Dirac delta function. The non-commutative neutrix product fog of f and g to be the limit of the sequence {fgn }, provided its limit h exists in the sense that sequence is defined N-lim n-∞（f（x）g,, （x）, φ（x）〉 = （h（x）, φ（x）},for all functions p in 2. It is proved that （x^λ＋1n^px＋）0（x^μ＋1n^qx＋）=x＋^λμ1n^p＋qx＋,（x^λ-1n^qx-）=x-^λ＋μ1n^p＋qx-,for λ＋μ〈-1; λ,μ, λ＋μ≠-1,-2…and p,q=0,1,2……     

Wave recursive interpolation

In this paper it is studied that the generated theory of wave recursive interpolation of uniform T-subdivi-ston scheme include wave parameter.The paper analyses the convergence of sequences of control polygons produced by wave recursive interpolation T-subdivision scheme of the formj=l,2,…,T-1;m=O,l,…,nTk;k=0,l,2,…,and differentiability of the limit curve.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

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.     

Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation

For the numerical evaluation of Cauchy principal value integrals of the form f(x)(x - λ) -1dx withλ ∈ (-1,1)and f ∈ C1[-1,1], we investigate the quadrature formula Q(n+1)Spl1[·;λ] obtained by replacing the integrand function f by its piecewise linear interpolant at an equidistant set of nodes as proposed by Rabinoivitz (Math. Comp. , 51:741 - 747,1988). We give upper bounds for the Peano - type error constantsfor s∈ {1,2}. These are the best possible constants in inequalities of the typeFurthermore, we prove that our upper bounds are asymptotically sharp.     

Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate

In this paper we consider a p-Laplacian equation with strong Allee effect growth rate and Dirichlet boundary condition $$\left\{\begin{array}{ll} {\rm div} (|\nabla u|^{p-2} \nabla u) + \lambda f(x,u)=0, &\quad x \in \Omega, \\ u=0, &\quad x \in \partial \Omega, \qquad \qquad ^ {(P_\lambda)} \end{array}\right.$$ where Ω is a bounded smooth domain in ${\mathbb{R}^N}$ for ${N \ge 1, p > 1}$ , and λ is a positive parameter. By using variational methods and a suitable truncation technique, we prove that problem (P _λ) has at least two positive solutions for large parameter and it has no positive solutions for small parameter. In addition, a nonexistence result is investigated.     

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 strong approximation by the (C, α)-means of Fourier series. II

стАтьь ьВльЕтсь пРОД ОлжЕНИЕМ пРЕДыДУЩЕИ ОДНОИМЕННОИ РАБОты АВтОРА, гДЕ ИжУ ЧАлсь пОРьДОк ВЕлИЧИН пРИ УслОВИьх, ЧтО α>-1/2, Рα >- 1 И ЧтО МАтРИцАt _nk УДОВлЕтВОРьЕт НЕкОт ОРОМУ УслОВИУ РЕгУльРНОстИ. жДЕсь ДОкАжыВАЕтсь, Ч тО ЕслИf∈H ^Ω, тО ВыпОлНь Етсь ОцЕНкА $$\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left| {\sigma _k^\alpha \left( x \right) - f\left( x \right)} \right|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} = O\left( {\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left( {\frac{1}{k}\mathop \smallint \limits_{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}^{2\pi } \frac{{\omega \left( t \right)}}{{t^2 }}dt} \right)^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} + \left( {\frac{{\lambda _n }}{n}} \right)^\alpha \omega \left( {\frac{1}{n}} \right)} \right)$$ (λ₁=1, λ_n+1-λ_n≦1), А тАкжЕ ЧтО Ёт А ОцЕНкА ОкОНЧАтЕльН А В сВОИх тЕРМИНАх; пОДОБ НыИ РЕжУль-тАт спРАВЕДлИВ тАкжЕ И Дль сОпРьжЕННОИ ФУНкцИИ . ДОкАжыВАЕтсь, ЧтО Усл ОВИьα>?1/2 Иpα>?1, кОтОРыЕ Б ылИ НАлОжЕНы В УпОМьНУтО И ВышЕ ЧАстИ I, сУЩЕстВЕН Ны.     

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 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 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