A Diophantine m-tuple is a set {a₁, a₂, …, a_m} of positive integers such that a_ia_j+1 is a perfect square for all 1 ≤ i < j ≤ m. Let E_k be the elliptic curve induced by Diophantine triple {F_2k, 5F_2k+2, 3F_2k + 7F_2k+2}. In this paper, we find the structure of a torsion group of E_k, and find all integer points on E_k under assumption that rank(E_k(ℚ)) = 1 and some further conditions.

The Study determinant is known as one of replacements for the determinant of matrices with entries in a noncommutative ring. In this paper, we give a generalization of the Study determinant and show its relationship with the Cayley-Dickson process. We also give some properties of a non-associative ring obtained by the Cayley-Dickson process with a not necessarily commutative, but associative ring as the initial ring.

In this paper, a new form of poly-Genocchi polynomials is defined by means of polylogarithm, namely, the Apostol-type poly-Genocchi polynomials of higher order with parameters a, b and c. Several properties of these polynomials are established including some recurrence relations and explicit formulas, which are used to express these higher order Apostol-type poly-Genocchi polynomials in terms of Stirling numbers of the second kind, Apostol-type Bernoulli and Frobenius polynomials of higher order. Moreover, certain differential identity is obtained that leads this new form of poly-Genocchi polynomials to be classified as Appell polynomials and, consequently, draw more properties using some theorems on Appell polynomials. Furthermore, a symmetrized generalization of this new form of poly-Genocchi polynomials that possesses a double generating function is introduced. Finally, the type 2 Apostolpoly-Genocchi polynomials with parameters a, b and c are defined using the concept of polyexponential function and several identities are derived, two of which show the connections of these polynomials with Stirling numbers of the first kind and the type 2 Apostol-type poly-Bernoulli polynomials.

In this paper we study the existence and long-time behavior of weak solutions to a class of quasilinear degenerate parabolic equations involving weighted p-Laplacian operators with a new class of nonlinearities. First, we prove the existence and uniqueness of weak solutions by combining the compactness and monotone methods and the weak convergence techniques in Orlicz spaces. Then, we prove the existence of global attractors by using the asymptotic a priori estimates method.

Let 𝓗(𝔻) be the space of analytic functions on the unit disc 𝔻. Let 𝜓 = (𝜓_j)ⁿ_j=0 and 𝚽 = (𝚽_j)ⁿ_j=0 be such that 𝜓_j, 𝚽_j ∈ 𝓗(𝔻). The linear differential operator is defined by T_𝜓(f) = ∑ⁿ_j=0 𝜓_jf^(j), f ∈ 𝓗(𝔻). We characterize the boundedness and compactness of the difference operator (T_𝜓 - T_𝚽)(f) = ∑ⁿ_j=0 (𝜓_j - 𝚽_j) f^(j) between weighted-type spaces of analytic functions. As applications, we obtained boundedness and compactness of the difference of multiplication operators between weighted-type and Bloch-type spaces. Also, we give examples of unbounded (non compact) differential operators such that their difference is bounded (compact).

In this article, we prove the existence of a solution to some classes of integral equations of generalized convolution type generated by the Kontorovich-Lebedev (K) transform, the Laplace (𝓛) transform and the Fourier (F) transform in some appropriate function spaces and represent it in a closed form.

The hypergeometric series of four variables are introduced and studied by Bin-Saad and Younis recently. In this line, we derive several fractional derivative formulas, integral representations and operational formulas for new quadruple hypergeometric series.

The paper has been devoted to study the uniqueness problem of meromorphic function and its linear differential polynomial sharing two values. We have pointed out gaps in one of the theorem due to [1]. We have further extended the corrected form of Chen-Li-Li's result which in turn extend the an earlier result of [8] in a large extent. In fact, we have subtly use the notion of weighted sharing of values in this particular section of literature which was unexplored till now. A handful number of examples have been provided by us pertinent to different discussions. Specially we have given an example to show that one condition in a theorem can not be dropped.

In this paper we prove the existence of global weak solutions, the exponential stability of a stationary solution and the existence of a global attractor for the three-dimensional convective Brinkman-Forchheimer equations with finite delay and fast growing nonlinearity in bounded domains with homogeneous Dirichlet boundary conditions.

We show the solvability of the ordinary differential equations describing curves on sphere or hyperbolic space. Making use of the geometric properties of the equations, we derive explicit solution formulae.

In this present paper, we consider four integrals and differentials containing the Gauss' hypergeometric ₂F₁(x) function in the kernels, which extend the classical Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential operators. Formulas (images) for compositions of such generalized fractional integrals and differential constructions with the n-times product of the generalized modified Bessel function of the second type are established. The results are obtained in terms of the generalized Lauricella function or Srivastava-Daoust hypergeometric function. Equivalent assertions for the Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential are also deduced.

In this paper, we introduce the Berger type deformed Sasaki metric on the cotangent bundle T^*M over an anti-paraKähler manifold (M, 𝜑, g) as a new natural metric with respect to g non-rigid on T^*M. Firstly, we investigate the Levi-Civita connection of this metric. Secondly, we study the curvature tensor and also we characterize the scalar curvature.

The aim of the present paper is to establish an intrinsic generalization of Cartan connection in Finsler geometry. This connection is called the vertical recurrent Finsler connection. An intrinsic proof of the existence and uniqueness theorem for such connection is investigated. Moreover, it is shown that for such connection, the associated semi-spray coincides with the canonical spray and the associated nonlinear connection coincides with the Barthel connection. Explicit intrinsic expression relating this connection and Cartan connection is deduced. We also investigate some applications concerning the fundamental geometric objects associated with this connection. Finally, three important results concerning the curvature tensors associated to a special vertical recurrent Finsler connection are studied.

In the Galilean space G₃, we study a special kind of tube surfaces, called tube-like surfaces. They are defined by sweeping a space curve along another central space curve. In this setting, we investigate some equations in terms of Gaussian and mean curvatures, showing some relevant theorems. Our theoretical results are illustrated with some plotted examples.

We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic structures to define invariants of Legendrian knots with explicit computational examples. We classify Legendrian structures on racks with 3 and 4 elements. We use Legendrian racks to distinguish certain Legendrian knots which are equivalent as smooth knots.