Existence of a solution of a difference scheme for one variational problem

The problem of minimizing the functional (A) $${}_a\smallint ^b \varphi (x,y,y',y'')dx$$ under the conditions (B) $$y(a) = a_0 ,y'(a) = a_1 ,y(b) = b_0 ,y'(b) = b_1$$ is replaced by the problem of finding the vector (y₁,y₂,...,y_n?1) on which the sum (C) $$\sum\limits_{\kappa = 0}^n {C_\kappa \varphi (x_\kappa ,y_\kappa ,\left. {\frac{{y_{\kappa + 1} - y_\kappa }}{h},\frac{{y_{\kappa + 1} - 2y_\kappa + y_{\kappa + 1} )}}{{h^2 }}} \right)}$$ takes a minimal value. Under certain conditions on ? andC _k it is proved that a solution exists for the difference scheme constructed. The method of differentiation with respect to a parameter is used for the proof.     

A functional equation related to the product in a quadratic number field

The functional equation $$f(x_{1},y_{1})f(x_{2},y_{2})=f(x_{1}x_{2}+\alpha y_{1}y_{2},x_{1}y_{2}+x_{2}y_{1}),\ (x_{1},y_{1}),\,(x_{2},y_{2})\in \mathbb{ R}^{2}$$ arises from the formula for the product of two numbers in the quadratic field ${\mathbb{Q}(\sqrt{\alpha})}$ . The general solution ${f:\mathbb{R}\rightarrow \mathbb{R}}$ to this equation is determined. Moreover, it is shown that no more general equations arise from a change of basis in the field.     

A new modification of the Hermite-Fejér interpolation

пУсть жАДАНы Ужлы $$ - \infty< x_1< x_2< ...< x_k< x_{k + 1}< ...< x_n< + \infty ,$$ , И пУстьx ₁ ^* <x ₂ ^* <...<x _n-1 ^* — кОРНИ МНОгО ЧлЕНА Ω′(х). гДЕ $$\omega (x) = \prod\limits_{k = 1}^n {(x - x_k ).} $$ В РАБОтЕ ИсслЕДУЕтсь жАДАЧА: кАк ОпРЕДЕлИт ь МНОгОЧлЕНР(х) МИНИМАльНОИ стЕп ЕНИ, Дль кОтОРОгО ВыпОлНь Утсь слЕДУУЩИЕ ИНтЕР пОльцИОННыЕ УслОВИь гДЕ {y _k И {y _k′}-жАДАННы Е сИстЕМы жНАЧЕНИИ.     

Nearly Quartic Mappings in β-Homogeneous F-Spaces

In this paper, we investigate the Hyers–Ulam stability of the following quartic equation $$\begin{array}{ll} {\sum\limits^{n}_{k=2}}\left({\sum\limits^{k}_{i_{1}=2}}{\sum\limits^{k+1}_{i_{2}=i_{1}+1}} \ldots {\sum\limits^{n}_{i_{n-k+1}=i_{n-k}+1}}\right)\\ \quad\times f \left({\sum\limits^{n}_{i=1,i \neq i_{1},\ldots,i_{n-k+1}}} x_{i}-{\sum\limits^{n-k+1}_{r=1}}x_{i_{r}}\right) + f \left({\sum\limits^{n}_{i=1}}x_{i}\right)\\ \quad-2^{n-2}{\sum\limits^{}_{1 \leq{i} \leq{j} \leq{n}}}(f(x_{i} + x_{j}){+f(x_{i} - x_{j})){+2^{n-5}(n - 2){\sum\limits^{n}_{i=1}}f(2x_{i})}} = \theta \end{array} $$ $({n \in \mathbb{N}, n \geq 3})$ in β-homogeneous F-spaces.     

An asymmetric inequality for non-homogeneous ternary quadratic forms

LetQ(x,y,z) be an indefinite ternary quadratic form of type (2,1) and determinantD(<0). Let 0≤t≤1/3 and \(f(t) = \frac{4}{{(1 + t)^2 (1 + 5t)}}\) . Then given any real numbersx ₀,y ₀,z ₀ there exist integersx,y,z satisfying $$ - t(f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}< Q (x + x_0 ,y + y_0 ,z + z_0 ) \leqslant (f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} $$ All the cases when equality holds are also obtained.     

On integral graphs which belong to the class\overline {\alpha K_a \cup \beta K_b }

LetG be a simple graph and let $\bar G$ denotes its complement. We say thatG is integral if its spectrum consists entirely of integers. If $\overline {\alpha K_a \cup \beta K_b } $ is integral we show that it belongs to the class of integral graphs $$\overline {[\frac{{kt}}{\tau }x_o + \frac{{mt}}{\tau }z]K(t + \ell n)k + \ell m \cup [\frac{{kt}}{\tau }y_o + \frac{{(t + \ell n)k + \ell m}}{\tau }z]nK\ell m,} $$ where (i) t, k, l, m, n ∈ ? such that (m, n) =1, (n, t) =1 and (l, t)=1; (ii) τ=((t+ln)k+lm, mt) such that τ| kt; (iii) (x₀, y₀) is aparticular solution of the linear Diophantine equation ((t+ln)k+lm)x-(mt)y=τ and (iv) z≥z₀ where z₀ is the least integer such that $(\frac{{kt}}{\tau }x_0 + \frac{{mt}}{\tau }z_0 ) \geqslant 1$ and $(\frac{{kt}}{\tau }y_0 + \frac{{(t + \ell n)k + \ell m}}{\tau }z_0 ) \geqslant 1$ .     

A strict minimax inequality criterion and some of its consequences

In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let Y be a finite-dimensional real Hilbert space, J : Y ?? R a C ¹ function with locally Lipschitzian derivative, and ${\varphi : Y \to [0, + \infty[}$ a C ¹ convex function with locally Lipschitzian derivative at 0 and ${\varphi^{-1}(0) = \{0\}}$ . Then, for each ${x_0 \in Y}$ for which J??(x ₀)??? 0, there exists ???> 0 such that, for each ${r \in ]0, \delta[}$ , the restriction of J to B(x ₀, r) has a unique global minimum u _r which satisfies $$J(u_r)\leq J(x)-\varphi(x-u_r)$$ for all ${x \in B(x_0, r)}$ , where ${B(x_0, r) = \{x \in Y : \|x-x_0\|\leq{r}\}.}$      

Identities of symmetric and skew-symmetric matrices in characteristicp

It is well known how the Kostant-Rowen Theorem extends the validity of the famous Amitsur-Levitzki identity to skew-symmetric matrices. Here we give a general method, based on a graph theoretic approach, for deriving extensions of known permanental-type identities to skew-symmetric and symmetric matrices over a commutative ring of prime characteristic. Our main result has a typical Kostant-Rowen flavour: IfM≥p[n+1/2] then $C_M (X,Y) = \sum\limits_{\alpha ,\beta \in Sym(M)} {x_{\alpha (1)} y_{\beta (1)} x_{\alpha (2)} y_{\beta (2)} } ...x_{\alpha (M)} y_{\beta (M)} = 0$ is an identity onM _n ^? (Ω), the set ofnxn skew-symmetric matrices over a commutative ring Ω withp1_Ω=0 (provided that $P > \sqrt {[n + 1/2)} $ ). Otherwise, the stronger conditionM≥pn implies thatC _M(X,Y)=0 is an identity on the full matrix ringM _n(Ω).     

A note on approximation to the identity with iterated kernels

Пустьl ₁ иl ₂ — неотрицательные убывающие функции на (0, ∞). Допустим, что $$\int\limits_0^\infty {S^{n_i - 1} l_i (S)\left( {1 + \log + \frac{1}{{S^{n_i } l_i (S)}}} \right)dS}< \infty ,$$ , гдеn ₁ иn ₂ — натуральные числа. Тогда для каждой функции \(f \in L^1 (R^{n_1 + n_2 } )\) при почти всех (x₀, у₀) мы имеем $$\mathop {\lim }\limits_{\lambda \to \infty } \lambda ^{n_1 + n_2 } \int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_1 } } (\lambda |x|)l_2 (\lambda |y|)f(x_0 - x,y_0 - y)dx dy = f(x_0 ,y_0 )\int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_i (|x|)l_2 } } (|y|)dx dy.$$      

Determination of the infinite Jacobi matrix with respect to two-spectra

The inverse problem about two-spectra for the equation (1) $$\begin{gathered} b_0 y_0 + a_0 y_1 = \lambda y_0 , \hfill \\ a_{n - 1} y_{n - 1} + b_n y_n + a_n y_{n + 1} = \lambda y_n \left( {n = 1, 2, 3, ...} \right), \hfill \\ \end{gathered} $$ where {y_n} ₀ ^∞ is the desired solution, λ is a complex parameter and $$a_n > 0, \operatorname{Im} b_n = 0 \left( {n = 0, 1 ,2, ...} \right)$$ is studied. Necessary and sufficient conditions for the solvability of the inverse problem about two-spectrafor Eq. (1) are established and also the procedure of reconstruction of the equation from its two-spectra is indicated.     

The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let ${\mathcal{S} = \{T(s): 0 \leq s < \infty\}}$ be a nonexpansive semigroup on E such that ${Fix(\mathcal{S}) := \cap_{t\geq 0}Fix( T(t) ) \not= \emptyset}$ , and f is a contraction on E with coefficient 0 <  α <  1. Let F be δ-strongly accretive and λ-strictly pseudo-contractive with δ + λ >  1 and ${0 < \gamma < \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$ . When the sequences of real numbers {α _n } and {t _n } satisfy some appropriate conditions, the three iterative processes given as follows : $${\left.\begin{array}{ll}{x_{n+1} = \alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1} = \alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$ and $$ z_{n+1} = T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$ converge strongly to ${\tilde{x}}$ , where ${\tilde{x}}$ is the unique solution in ${Fix(\mathcal{S})}$ of the variational inequality $${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$ Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065–3071, 2009) and Chen and He (Appl Math Lett 20:751–757, 2007) and many others.     

Global asymptotic stability of a second-order system of difference equations

In this paper a sufficient condition is obtained for the global asymptotic stability of the following system of difference equations $$x_{n + 1} = \frac{{x_n y_{n - 1}^b + 1}} {{x_n + y_{n - 1}^b }}, y_{n + 1} = \frac{{y_n x_{n - 1}^b + 1}} {{y_n + x_{n - 1}^b }}n = 0,1,2 \ldots$$ where the parameter b ∈ [0, ∞) and the initial values (x _k , y _k ) ∈ (0, ∞) (for k = ?1, 0).     

A problem of Carlitz and its generalizations

Let ${\mathbb{F}_q}$ be the finite field of characteristic p > 2 with q elements. Carlitz proposed the problem of finding an explicit formula for the number of solutions to the equation $$(x_1+ x_2+\cdots+x_n)^2=a\, x_1x_2\cdots x_n,$$ where ${a\in \mathbb{F}_q^*}$ and n ≥ 3. By using the augmented degree matrix and Gauss sums, we consider the generalizations of the above equation and partially solve Carlitz’s problem. Moreover, the technique developed in this paper may be applied to other equations of the form ${h_1^\lambda=h_2}$ with ${h_1, h_2 \in \mathbb{F}_q[x_1,\ldots,x_n]}$ and ${\lambda \in \mathbb{N}}$ .     

Two Linear Transformations each Tridiagonal with Respect to an Eigenbasis of the other; Comments on the Parameter Array

Let ${\mathbb K} $ denote a field. Let it d denote a nonnegative integer and consider a sequence p=( $\theta_i, \theta^*_i,i=0...d; \varphi_j, \phi_j,j=1...{\it d})$ consisting of scalars taken from ${\mathbb K} $ . We call p a parameter array whenever: (PA1) $\theta_i \not=\theta_j, \; \theta^*_i\not=\theta^*_j$ if $$i\not=j$, $(0 \leq i, j\leq d)$; (PA2) $ \varphi_i\not=0$, $\phi_i\not=0$ $(1 \leq i \leq d)$; (PA3) $\varphi_i = \phi_1 \sum_{h=0}^{i-1} ({\theta_h-\theta_{d-h}})/({\theta_0-\theta_d}) + (\theta^*_i-\theta^*_0)(\theta_{i-1}-\theta_d)$ $(1 \leq i \leq d)$; (PA4) $\phi_i = \varphi_1 \sum_{h=0}^{i-1} ({\theta_h-\theta_{d-h}})/({\theta_0-\theta_d}) + (\theta^*_i-\theta^*_0)(\theta_{d-i+1}-\theta_0)$ $(1 \leq i \leq d)$; (PA5) $(\theta_{i-2}-\theta_{i+1})(\theta_{i-1}-\theta_i)^{-1}$, $(\theta^*_{i-2}-\theta^*_{i+1})(\theta^*_{i-1}-\theta^*_i)^{-1}$$ are equal and independent of i for $2 \leq i \leq d-1$ . In Terwilliger, J. Terwilliger, Linear Algebra Appl., Vol. 330(2001) p. 155 we showed the parameter arrays are in bijection with the isomorphism classes of Leonard systems. Using this bijection we obtain the following two characterizations of parameter arrays. Assume p satisfies PA1 and PA2. Let A, B,A^*, B^* denote the matrices in ${Mat}_{{\it d}+1}$ ( ${\mathbb K} $ ) which have entries A _ii =θ _i , B _ii =θ _d-i , A ^* _ii =θ^* _i , B ^* _ii =θ^* _i (0 ≤ i ≤ d), A _i,i-1=1, B _i,i-1=1, A ^* _i-1,i =φ _i , B ^* _i-1,i =? _i (1 ≤ i ≤ d), and all other entries 0. We show the following are equivalent: (i) p satisfies PA3–PA5; (ii) there exists an invertible G ∈Mat _d+1( ${\mathbb K} $ ) such that G ^?1 AG=B and G ^?1 A ^* G=B ^*; (iii) for 0 ≤ i ≤ d the polynomial $$ \sum_{n=0}^i \frac{ (\lambda-\theta_0) (\lambda-\theta_1) \cdots (\lambda-\theta_{n-1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\varphi_1\varphi_2\cdots \varphi_n}$$ is a scalar multiple of the polynomial $$\sum_{n=0}^i \frac{ (\lambda-\theta_d) (\lambda-\theta_{d-1}) \cdots (\lambda-\theta_{d-n+1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\phi_1\phi_2\cdots \phi_n}.$$ We display all the parameter arrays in parametric form. For each array we compute the above polynomials. The resulting polynomials form a class consisting of the q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, Bannai/Ito, and Orphan polynomials. The Bannai/Ito polynomials can be obtained from the q-Racah polynomials by letting q tend to ?1. The Orphan polynomials have maximal degree 3 and exist for ( ${\mathbb K} $ )=2 only. For each of the polynomials listed above we give the orthogonality, 3-term recurrence, and difference equation in terms of the parameter array.     

On a class of smooth Frechet subalgebras of C * -algebras

The paper contributes to understanding the differential structure in a C ^*-algebra. Refining the Banach $(D_p^*)$ -algebras investigated by Kissin and Shulman as noncommutative analogues of the algebra C ^p [a,b] of p-times continuously differentiable functions, we investigate a Frechet $(D_\infty^*)$ -subalgebra $\ensuremath{{\mathcal B}}$ of a C ^*-algebra as a noncommutative analogue of the algebra C ^?∞?[a,b] of smooth functions. Regularity properties like spectral invariance, closure under functional calculi and domain invariance of homomorphisms are derived expressing $\ensuremath{{\mathcal B}}$ as an inverse limit over n of Banach $(D^*_n)$ -algebras. Several examples of such smooth algebras are exhibited.     

On Waring's Problem for Six Cubes and Higher Degrees

It is proved that the equation $$n = x_{\,1}^3 + x_{\,2}^3 + x_{\,3}^3 + x_{\,4}^3 + x_{\,5}^3 + x_{\,6}^3 + u^4 + v^9$$ has nonnegative integral solutions if $n \equiv 1\left( {\bmod 5} \right)$ is even and sufficiently large. Bibliography: 8 titles.     

Eigenfunctions and Fundamental Solutions of the Fractional Laplace and Dirac Operators: The Riemann-Liouville Case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator \(\Delta _+^{(\alpha , \beta , \gamma )}:= D_{x_0^+}^{1+\alpha } +D_{y_0^+}^{1+\beta } +D_{z_0^+}^{1+\gamma },\) where \((\alpha , \beta , \gamma ) \in \,]0,1]^3\), and the fractional derivatives \(D_{x_0^+}^{1+\alpha }, D_{y_0^+}^{1+\beta }, D_{z_0^+}^{1+\gamma }\) are in the Riemann–Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator \(\Delta _+^{(\alpha ,\beta ,\gamma )}\) in classes of functions admitting a summable fractional derivative. Making use of the Mittag–Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.     

Interior W 1,p Regularity and Hölder Continuity of Weak Solutions to a Class of Divergence Kolmogorov Equations with Discontinuous Coefficients

We consider a class of Kolmogorov equation $$Lu={\sum^{p_0}_{i,j=1}{\partial_{x_i}}(a_{ij}(z){\partial_{x_j}}u)}+{\sum^{N}_{i,j=1}b_{ij}x_{i}{\partial_{x_j}}u-{\partial_t}u}={\sum^{p_0}_{j=1}{\partial_{x_j}}F_{j}(z)}$$ in a bounded open domain ${\Omega \subset \mathbb{R}^{N+1}}$ , where the coefficients matrix (a _ij (z)) is symmetric uniformly positive definite on ${\mathbb{R}^{p_0} (1 \leq p_0 < N)}$ . We obtain interior W ^1,p (1 < p < ∞) regularity and Hölder continuity of weak solutions to the equation under the assumption that coefficients a _ij (z) belong to the ${VMO_L\cap L^\infty}$ and ${({b_{ij}})_{N \times N}}$ is a constant matrix such that the frozen operator ${L_{z_0}}$ is hypoelliptic.     

Cones and error bounds for linear iterations

Consider an ordered Banach space with a cone of positive elementsK and a norm ∥·∥. Let [a,b] denote an order-interval; under mild conditions, ifx*∈[a,b] then $$||x * - \tfrac{1}{2}(a + b)|| \leqslant \tfrac{1}{2}||b - a||.$$ This inequality is used to generate error bounds in norm, which provide on-line exit criteria, for iterations of the type $$x_r = Ax_{r - 1} + a,A = A^ + + A^ - ,$$ whereA ⁺ andA ^? are bounded linear operators, withA ⁺ K ?K andA ^? K ? ?K. Under certain conditions, the error bounds have the form $$\begin{gathered} ||x * - x_r || \leqslant ||y_r ||,y_r = (A^ + - A^ - )y_{r - 1} , \hfill \\ ||x * - x_r || \leqslant \alpha ||\nabla x_r ||, \hfill \\ ||x * - \tfrac{1}{2}(x_r + x_{r - 1} )|| \leqslant \tfrac{1}{2}||\nabla x_r ||. \hfill \\ \end{gathered} $$ These bounds can be used on iterative methods which result from proper splittings of rectangular matrices. Specific applications with respect to certain polyhedral cones are given to the classical Jacobi and Gauss-Seidel splittings.     

New embedded boundary-type quadrature formulas for the simplex

In this paper we consider the problem of the approximation of the integral of a smooth enough function f(x,y) on the standard simplex $\Delta _{2} \subset \mathbb{R}^{2}$ by cubature formulas of the following kind: $${\int\limits_{\Delta _{2} } {f{\left( {x,y} \right)}dxdy} } = {\sum\limits_{\alpha = 1}^3 {{\sum\limits_{i,j} {A_{{\alpha ij}} \frac{{\alpha ^{{i + j}} }}{{\alpha x^{i} \alpha y^{j} }}f{\left( {x_{\alpha } ,y_{\alpha } } \right)} + E{\left( f \right)}} }} }$$ where the nodes $\left( x_{\alpha},y_{\alpha}\right) ,\alpha=1,2,3$ are the vertices of the simplex. Such kind of quadratures belong to a more general class of formulas for numerical integration, which are called boundary-type quadrature formulas. We discuss three classes of such formulas that are exact for algebraic polynomials and generate embedded pairs. We give bounds for the truncation errors and conditions for convergence. Finally, we show how to organize an algorithm for the automatic computation of the quadratures with estimate of the errors and provide some numerical examples.