Volume 68, Issue 4

The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q -Fibonacci and q -hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a ( p , q )-analogue of the algorithm is introduced and an explicit formula for it in terms of the ( p , q )-binomial coefficient is found.

In this note we characterize Boolean algebras among lattices of type (2, 2, 1) with join, meet and an additional unary operation by means of single two-variable respectively three-variable identities. In particular, any uniquely complemented lattice satisfying any one of these equational constraints is distributive and hence a Boolean algebra.

Weighted uniform densities are a generalization of the uniform density, which is also known as the Banach density. In this paper, we introduce the concept of weighted uniform density ideals and consider the topological complexity of these ideals as well as when they have certain analytical properties related to the ideal convergence of sequences and series. Furthermore, we prove some inequalities between different upper and lower weighted uniform densities and give the answer to the problem concerning the Darboux property of these densities.

In this paper, we consider a polynomial generalization, denoted by uma,b$\begin{array}{} u_m^{a,b} \end{array}$ ( n , k ), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then uma,b$\begin{array}{} u_m^{a,b} \end{array}$ ( n , k ) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by uma,b$\begin{array}{} u_m^{a,b} \end{array}$ ( n , k ) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for uma,b$\begin{array}{} u_m^{a,b} \end{array}$ ( n , k ) as well as for the associated Cauchy numbers.

We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts ζ ( s + i γ k ), k ∈ ℕ, of the Riemann zeta-function, where γ k are imaginary parts of nontrivial zeros of ζ ( s ).

We study distance functions on the set of ordinary (or non-supersingular) elliptic curves in short Weierstrass form (or simplified Weierstrass form) over a finite field of characteristic three. Mishra and Gupta (2008) firstly construct distance functions on the set of elliptic curves in short Weierstrass form over any prime field of characteristic greater than three. Afterward, Vetro (2011) constructs some other distance functions on the set of elliptic curves in short Weierstrass form over any prime field of characteristic greater than three. Recently, Hakuta (2015) has proposed distance functions on the set of ordinary elliptic curves in short Weierstrass form over any finite field of characteristic two. However, to our knowledge, no analogous result is known in the characteristic three case. In this paper, we shall prove that one can construct distance functions on the set of ordinary elliptic curves in short Weierstrass form over any finite field of characteristic three. A cryptographic application of our distance functions is also discussed.

In this paper, we present new explicit formulas for the Drazin inverse of the sum of two matrices, under conditions weaker than those used in some recent results.

In this work, the Green’s function of order two is used together with Fink’s approach in Ostrowski’s inequality to represent the difference between the sides of the Sherman’s inequality. Čebyšev, Grüss and Ostrowski-type inequalities are used to obtain several bounds of the presented Sherman-type inequality. Further, we construct a new family of exponentially convex functions and Cauchy-type means by looking to the linear functionals associated with the obtained inequalities.

Let X be a paracompact topological space and Y be a Banach space. In this paper, we will characterize the Baire-1 functions f : X → Y by their graph: namely, we will show that f is a Baire-1 function if and only if its graph gr( f ) is the intersection of a sequence (Gn)n=1∞ $\begin{array}{} \displaystyle (G_n)_{n=1}^{\infty} \end{array}$ of open sets in X × Y such that for all x ∈ X and n ∈ ℕ the vertical section of G n is a convex set, whose diameter tends to 0 as n → ∞. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces.

The main objective of the present paper, is to obtain some new versions of Young-type inequalities with respect to two weighted arithmetic and geometric means and their reverses, using two inequalities K(ba,2)r≤a∇νba♯νb≤K(ba,2)R,$$\begin{array}{} \displaystyle K\Big(\frac{b}{a},2\Big)^r\leq\frac{a\nabla_{\nu}b}{a\sharp_{\nu}b}\leq K\Big(\frac{b}{a},2\Big)^R, \end{array}$$ where r = min{ ν , 1 – ν }, R = max{ ν ,1 – ν } and K ( t ,2) = (t+1)24t$\begin{array}{} \displaystyle \frac{(t+1)^2}{4t} \end{array}$ is the Kantorovich constant, and e(h−1,ν)≤a∇νba♯νb≤e(h,ν),$$\begin{array}{} \displaystyle e(h^{-1},\nu)\leq\frac{a\nabla_{\nu}b}{a\sharp_{\nu}b}\leq e(h,\nu), \end{array}$$ where h = max {ab,ba}$\begin{array}{} \displaystyle \{\frac{a}{b},\frac{b}{a}\} \end{array}$ and e ( t , ν ) = exp (4 ν (1 – ν )( K ( t ,2)–1) (1−12t)).$\begin{array}{} \displaystyle (1-\frac{1}{2t})\big). \end{array}$ Also some operator versions of these inequalities and some inequalities related to Heinz mean are proved.

In this article, we give the Nevanlinna type hyperbolic characteristics in simply connected domains and angular domains and the Tsuji type hyperbolic characteristics for bounded analytic functions for the first time. The first fundamental theorems are also established concerning hyperbolic derivative for bounded analytic functions in simply connected domains and angular domains. This is a continuous work of Makhmutov [3].

In this paper, we prove a normal criteria for family of meromorphic functions. As an application of that result, we establish a uniqueness theorem for entire function concerning a conjecture of R. Brück. The above uniqueness theorem is an improvement of a problem studied by L. Z. Yang et al. [14]. However, our method differs the method of L. Z. Yang et al. [14]. We mainly use normal family theory and combine it with Nevanlinna theory instead of using only the Nevanlinna theory as in [14].

Our aim is to study the existence of solutions for the following system of nonlocal resonant boundary value problem (φ(x′))′=f(t,x,x′),x′(0)=0,x(1)=∫01x(s)dg(s),$$\begin{array}{} \displaystyle (\varphi (x'))' =f(t,x,x'),\quad x'(0)=0, \quad x(1)=\int\limits_{0 }^{1}x(s){\rm d} g(s), \end{array}$$ where the function ϕ : ℝ n → ℝ n is given by ϕ ( s ) = ( φ p 1 ( s 1 ), …, φ pn ( s n )), s ∈ ℝ n , p i > 1 and φ pi : ℝ → ℝ is the one dimensional p i -Laplacian, i = 1,…, n , f : [0,1] × ℝ n × ℝ n → ℝ n is continuous and g : [0,1] → ℝ n is a function of bounded variation. The proof of the main result is depend upon the coincidence degree theory.

A one-dimensional Keller-Segel system which is defined through uniformly elliptic operators having variable coefficients is considered. In the main theorems, the local existence and uniqueness of the mild solution of the system are proved. The main method to construct the mild solution is an argument of successive approximations by means of strongly continuous semi-groups.

By using variational methods and critical point theory in an appropriate Orlicz-Sobolev setting, the authors establish the existence of infinitely many non-negative weak solutions to a non-homogeneous Neumann problem. They also provide some particular cases and an example to illustrate the main results in this paper.

Our focus is to generalize the definition of the weighted statistical convergence in a wider range of the weighted sequence { t n } n ∈ℕ . We extend the concept of weighted statistical convergence and rough statistical convergence to renovate a new concept namely, rough weighted statistical convergence. On a continuation we also define rough weighted statistical limit set. In the year (2008) Aytar established the following results: The diameter of rough statistical limit set of a real sequence is ≤ 2 r (where r is the degree of roughness) and in general it has no smaller bound. If the rough statistical limit set is non-empty then the sequence is statistically bounded. If x ∗ and c belong to rough statistical limit set and statistical cluster point set respectively, then | x ∗ − c | ≤ r . We investigate whether the above mentioned three results are satisfied for rough weighted statistical limit set or not? Answer is no. So our main objective is to interpret above mentioned different behaviors of the new convergence and characterize the rough weighted statistical limit set. Also we show that this set satisfies some topological properties like boundedness, compactness, path connectedness etc.

In the present paper, we introduce a new sequence of linear positive operators based on ( p , q )-integers. To approximate functions over unbounded intervals, we introduce Baskakov-Durrmeyer type operators using the ( p , q )-Gamma function. We investigate rate of convergence of new operators in terms of modulus of continuities and obtain their approximation behavior for the functions belonging to Lipschitz class. At the end, we present a modification of new operators preserving the test function x .

By an interesting physical perspective and a suitable contraction of the Riemannian curvature tensor in Finsler geometry, Akbar-Zadeh introduced the notion of scalar curvature for the Finsler metrics. A Finsler metric is called of isotropic scalar curvature if the scalar curvature depends on the position only. In this paper, we study the class of generalized 4-th root metrics. These metrics generalize 4-th root metrics which are used in Biology as ecological metrics. We find the necessary and sufficient condition under which a generalized 4-th root metric is of isotropic scalar curvature. Then, we find the necessary and sufficient condition under which the conformal change of a generalized 4-th root metric is of isotropic scalar curvature. Finally, we characterize the Bryant metrics of isotropic scalar curvature.

We show that any compactification of a regular frame can be realised as a frame which is freely generated on a suitable meet semilattice subject to certain relations, and we provide familiar examples of compactifications which can be regarded as such frames.

We consider for integers k ≥ 2 the k –generalized Fibonacci sequences F ( k ) := (Fn(k))n≥2−k,$\begin{array}{} (F_n^{(k)})_{n\geq 2-k}, \end{array} $ whose first k terms are 0, …, 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, we show that there does not exist a quadruple of positive integers a 1 < a 2 < a 3 < a 4 such that a i a j + 1 ( i ≠ j ) are all members of F ( k ) .