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Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number p we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo p . Therefore, Paley graphs are naturally associated with the Legendre symbol at p which is a quadratic Dirichlet character of conductor p . In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of L -functions, we provide an effective upper bound for their Cheeger number. As a by-product of our approach, we settle a question raised in [Cramer et al.: The isoperimetric and Kazhdan constants associated to a Paley graph , Involve 9 (2016), 293–306] about the size of this upper bound.

In this paper, we define and study the variety of tense modal pseudocomplemented De Morgan algebras. This variety is a proper subvariety of the variety of tense tetravalent modal algebras. A tense modal pseudocomplemented De Morgan algebra is a modal pseudocomplemented De Morgan algebra endowed with two tense operators G and H satisfying additional conditions. Also, the variety of tense modal pseudocomplemented De Morgan algebras is intimately connected with some well-known varieties of De Morgan algebras with tense operators.

Let ( F n ) n ≥0 and ( L n ) n ≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b , we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations F n = 10 d F m + L k and F n = 10 d L m + F k in nonnegative integers ( n , m , k ), where d denotes the number of digits of L k and F k , respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.

We prove general formulae in composition theory. We study the number of restricted compositions of a positive integer n in k parts, where the parts are in very general integer sequences. Note that in compositions the order of the parts is considered.

Filbert and Lilbert matrices are defined by terms of linear Fibonacci-like sequences. By considering terms of such linear recurrences and additional free parameters, we define a nonlinear variant of these matrices via ratios of q -forms of terms of Fibonacci-like sequences whose indices are in nonlinear forms. We derive explicit formulae for the matrices L and U come from LU -decomposition, their inverses, inverse of the main matrices and their determinants.

In this paper, we consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions.

We give asymptotic analysis of power series associated with lacunary partition functions. New partition theoretic interpretations of some basic hypergeometric series are offered as examples.

In this paper, we investigate the parallel surfaces of the nonlightlike solution of vortex filament equations, namely, the parallel surfaces to Hasimoto surfaces in E13 $\begin{array}{} \displaystyle \mathbb{E}_{1}^{3} \end{array}$. We examine the relation between the geometric structures of Hasimoto surfaces and parallel surfaces to them. Moreover, we give concluding remarks about the parameter curves of parallel surface to Hasimoto surfaces.

In terms of a very-well-poised 6 ϕ 5 summation formula, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we establish four new q -supercongruences for truncated basic hypergeometric series. One of these results is a new q -analogue of the (F.2) supercongruence of Van Hamme.

In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients.

In this paper, we obtain some Korovkin type approximation theorems for double sequences of positive linear operators on two-dimensional weighted spaces via statistical type convergence method with respect to power series method. Additionally, we calculate the rate of convergence. As an application, we provide an approximation using the generalization of Gadjiev-Ibragimov operators for P p -statistical convergence. Our results are meaningful and stronger than those previously given for two-dimensional weighted spaces.

We consider an eigenvalue problem driven by the anisotropic ( p , q )-Laplacian and with a Carathéodory reaction which is ( p ( z ) − 1)-sublinear as x → + ∞. We look for positive solutions. We prove an existence, nonexistence and multiplicity theorem which is global in the parameter λ > 0, that is, we prove a bifurcation-type theorem which describes in an exact way the changes in the set of positive solutions as the parameter λ varies on ℝ̊ + = (0, + ∞).

We propose some theoretical explanations pertaining to the representation for the solution of the system of the higher-order difference equations xn+1=1A− yn− k,yn+1=1A− xn− k,n,k∈ N0, $$\begin{array}{} \displaystyle x_{n+1}=\frac{1}{A-y_{n-k}},\quad y_{n+1}=\frac{1}{A-x_{n-k}},\qquad n, k\in \mathbb{N}_0, \end{array}$$ where ℕ 0 = ℕ ∪ {0}, A ∈ ℕ 1 and the initial conditions x – k , x – k +1 , …, x 0 , y – k , y – k +1 , …, y 0 are non zero real numbers such that their solution is related to a generalized Balancing numbers. We also study the stability character and asymptotic behavior of this system.

In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.

Assume that k ≤ d is a positive integer and 𝓒 is a finite collection of convex bodies in ℝ d . We prove a Helly-type theorem: If for every subfamily 𝓒 * ⊂ 𝓒 of size at most max{ d + 1, 2( d – k + 1)} the set ⋂ 𝓒 * contains a k -dimensional cone, then so does ⋂ 𝓒. One ingredient in the proof is another Helly-type theorem about the dimension of lineality spaces of convex cones.

We employ closure operators associated with n -ary relations, n > 1 an integer, to provide the digital space ℤ 3 with connectedness structures. We show that each of the six inscribed tetrahedra obtained by canonical tessellation of a digital cube in ℤ 3 with edges consisting of 2 n − 1 points is connected. This result is used to prove that certain bounding surfaces of the polyhedra in ℤ 3 that may be face-to-face tiled with such tetrahedra are digital Jordan surfaces (i.e., separate ℤ 3 into exactly two connected components). An advantage of these Jordan surfaces over those with respect to the Khalimsky topology is that they may possess acute dihedral angles π 4 $\begin{array}{} \displaystyle \pi\over 4 \end{array}$ while, in the case of the Khalimsky topology, the dihedral angles may never be less than π 2 $\begin{array}{} \displaystyle \pi\over 2 \end{array}$.

One point compactification is studied in the light of ideal of subsets of ℕ. 𝓘-proper map is introduced and showed that a continuous map can be extended continuously to the one point 𝓘-compactification if and only if the map is 𝓘-proper. Nowhere tallness, introduced by P. Matet and J. Pawlikowski in [J. Symb. Log. 63(3) (1998), 1040–1054], plays an important role in this article to study various properties of 𝓘-proper maps. It is seen that one point 𝓘-compactification of a topological space may fail to be Hausdorff even if the underlying topological space is Hausdorff but a class {𝓘} of ideals has been identified for which one point 𝓘-compactification coincides with the one point compactification if the underlying topological space is metrizable. Let’s speak our minds that the results in this article will look elegant if one looks at it from a topological angle.

Given a compactum X , a map f : X → X and x ∈ X , it is defined ω ( x , f ) = { y ∈ X : there exists an increasing sequence ( n i ) i ∈ℕ ⊆ ℕ such that limi→ ∞ $\begin{array}{} \displaystyle \lim_{i\to\infty} \end{array}$ f n i ( x ) = y }; and Ω( x , f ) = { y ∈ X : there exist an increasing sequence ( n i ) i ∈ℕ ⊆ ℕ and a sequence ( x i ) i ∈ℕ in X such that limi→ ∞ $\begin{array}{} \displaystyle \lim_{i\to\infty} \end{array}$ x i = x and limi→ ∞ $\begin{array}{} \displaystyle \lim_{i\to\infty} \end{array}$ f n i ( x i ) = y }. It is well known that both ω ( x , f ) and Ω( x , f ) are nonempty compact subsets of X . 2 X denotes the collection of all the nonempty compact subsets of X endowed with the Hausdorff metric. In this paper, we consider ω f : X → 2 X and Ω f : X → 2 X defined for each x ∈ X by ω f ( x ) = ω ( x , f ) and Ω f ( x ) = Ω( x , f ), respectively. We study the continuity of these functions, when f is defined on some kind of continua.

Let 𝓟 n := H * ((ℝ P ∞ ) n ) ≅ ℤ 2 [ x 1 , x 2 , …, x n ] be the graded polynomial algebra over ℤ 2 , where ℤ 2 denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra 𝓟 n , viewed as a graded left module over the mod-2 Steenrod algebra, 𝓐. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this article, we study the hit problem for the case n = 6 in the generic degree d r = 6(2 r − 1) + 4.2 r with r an arbitrary non-negative integer. By considering ℤ 2 as a trivial 𝓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ 2 -vector space ℤ 2 ⊗ 𝓐 𝓟 n . The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ 2 vector space ℤ 2 ⊗ 𝓐 𝓟 6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2 r − 1) + 4.2 r is also discussed at the end of this paper.

In this work, we construct a new bivariate statistical distribution by the conditionally specified model approach. The conditional distributions follow the well-known generalized exponential distribution which includes the ordinary exponential distribution and is more flexible than gamma and Weibull in some ways. The newly defined distribution has four parameters that increase the flexibility of the model in data fitting. By equating the dependence parameter to zero, the marginal distributions become independent generalized exponential distributions. The new bivariate distribution depends on the classical exponential integral function which is not difficult to evaluate numerically. The basic properties of the distribution such as distribution functions, moments and stress-strength reliability are derived. The parameters are estimated by the method of maximum likelihood. Two real data fitting applications prove its usefulness in case of negatively correlated bivariate data modelling.

In this paper, we are concerned with the problem of stabilization of a degenerate parabolic equation with a Dirichlet control, evolving in bounded domain 𝓞 ⊂ ℝ d , d ≥ 2. We apply the proportional control design technique based on the spectrum of the linear operator which governs the evolution equation. The stabilizing feedback control, we design here, is linear, of finite-dimensional structure, easily manageable from the computational point of view.