# Volume 28 (2020), Issue 1

### 1. Kappa-Slender Modules

For an arbitrary infinite cardinal κ, we define classes of κ-cslender and κ-tslender modules as well as related classes of κ-hmodules and initiate a study of these classes.

### 2. Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type

In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.

### 3. A new parameterized logarithmic kernel function for linear optimization with a double barrier term yielding the best known iteration bound

In this paper, we propose a large-update primal-dual interior point algorithm for linear optimization. The method is based on a new class of kernel functions which differs from the existing kernel functions in which it has a double barrier term. The investigation according to it yields the best known iteration bound O n log ( n ) log ( n ∈ ) \sqrt n \log (n)\log \left( {{n \over \in }} \right) for large-update algorithm with the special choice of its parameter m and thus improves the iteration bound obtained in Bai et al. [2] for large-update algorithm.

### 4. Polynomials and degrees of maps in real normed algebras

Let A be the algebra of quaternions H or octonions O. In thismanuscript an elementary proof is given, based on ideas of Cauchy andD’Alembert, of the fact that an ordinary polynomial f(t) ∈ A[t] has a rootin A. As a consequence, the Jacobian determinant |J(f)| is always nonnegative in A. Moreover, using the idea of the topological degree we showthat a regular polynomial g(t) over A has also a root in A. Finally, utilizingmultiplication (∗) in A, we prove various results on the topological degree ofproducts of maps. In particular, if S is the unit sphere in A and h1, h2 : S →S are smooth maps, it is shown that deg(h1 ∗ h2) = deg(h1) + deg(h2).

### 5. Solutions of the Diophantine Equation7X2 + Y7 = Z2from Recurrence Sequences

Consider the system x 2 − ay 2 = b, P (x, y) = z 2, where P is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation 7X 2 + Y 7 = Z 2 if (X, Y) = (L n , F n ) (or (X, Y) = (F n , L n )) where {F n } and {L n } represent the sequences of Fibonacci numbers and Lucas numbers respectively.

### 6. Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and L1-data

In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic p(·)-Laplacian problem with Dirichlet-type boundary conditions and L 1 data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.