###### Solutions of the time-independent Schrödinger equation by uniformization on the unit circle

## Abstract

The idea presented here of a general quantization rule for bound states is mainly based on the Riccati equation which is a result of the transformed, time-independent, one-dimensional Schrödinger equation. The condition imposed on the logarithmic derivative of the ground state function *W*
_{0} allows to present the Riccati equation as the unit circle equation with winding number equal to one which, by appropriately chosen transformations, can be converted into the unit circle equation with multiple winding number. As a consequence, a completely new quantization condition, which gives exact results for any quantum number, is obtained.

###### Strictly Increasing Additive Generators of the Second Kind of Associative Binary Operations

## Abstract

The class of strictly increasing additive generators of the second kind is defined and analyzed. Necessary and sufficient conditions for a binary operation generated by a strictly increasing additive generator of the second kind to be associative are introduced. The relation between the class of strictly increasing additive generators of the second kind of associative binary operations and the class of discrete upper additive generators of associative binary operations is revealed.

###### A Three Dimensional Modification of the Gaussian Number Field

## Abstract

For vectors in **E**
_{3} we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications.

Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra 𝕋 is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra 𝕋 is isomorphic to direct product ℂ*×*ℝ, and so it contains a subalgebra isomorphic to the Gaussian complex plane.

###### Topological Up-Algebras

## Abstract

In this paper, we introduce the notion of topological UP-algebras and several types of subsets of topological UP-algebras, and prove the generalization of these subsets. We also introduce the notions of quotient topological spaces of topological UP-algebras and topological UP-homomorphisms. Furthermore, we study the relation between topological UP-algebras, Hausdor spaces, discrete spaces, and quotient topological spaces, and prove some properties of topological UP-algebras.

###### Uniqueness Theorem in Complete Residuated Almost Distributive Lattices

## Abstract

Important properties of primary elements in a complete residuated ADL *L* and the uniqueness theorem in a complete complemented residuated ADL *L* are proved.

###### Yet Two Additional Large Numbers of Subuniverses of Finite Lattices

## Abstract

By a subuniverse, we mean a sublattice or the emptyset. We prove that the fourth largest number of subuniverses of an *n*-element lattice is 43 2^{n}^{−6} for *n* ≥ 6, and the fifth largest number of subuniverses of an *n*-element lattice is 85 2^{n}^{−7} for *n* ≥ 7. Also, we describe the *n*-element lattices with exactly 43 2^{n}^{−6} (for *n* ≥ 6) and 85 2^{n}^{−7} (for *n* ≥ 7) subuniverses.

###### The Zariski Topology on the Graded Primary Spectrum Over Graded Commutative Rings

## Abstract

Let *G* be a group with identity *e* and let *R* be a *G*-graded ring. A proper graded ideal *P* of *R* is called *a graded primary ideal* if whenever *r _{g}s_{h}∈P*, we have

*r*or

_{g}∈ P*s*(

_{h}∈ Gr*P*), where

*r*(

_{g},s_{g}∈ h*R*). The

*graded primary spectrum p.Spec*(

_{g}*R*) is defined to be the set of all graded primary ideals of

*R*.In this paper, we define a topology on

*p.Spec*(

_{g}*R*), called Zariski topology, which is analogous to that for

*Spec*(

_{g}*R*), and investigate several properties of the topology.

###### Fully discrete convergence analysis of non-linear hyperbolic equations based on finite element analysis

## Abstract

With the development of modern partial differential equation (PDE) theory, the theory of linear PDE is becoming more and more perfect, . Non-linear PDE has become a research hotspot of many mathematicians. In fact, when describing practical physical problems with PDEs, non-linear problems tend to be more general than linear problems, which are close to real problems and have practical physical significance. Hyperbolic PDEs are a kind of important PDEs describing the phenomena of vibration or wave motion. The solution of hyperbolic PDE can be decomposed into the form of multiplication of vibration and vibration or of exponential function and exponential function. Generally, the energy is infinite. A full discrete convergence analysis method for non-linear hyperbolic equation based on finite element analysis is proposed. Taking second-order and fourth-order non-linear hyperbolic equation as examples, the full discrete convergence of non-linear hyperbolic equation is analysed by finite element method and the super-convergence results are obtained.

###### Investigation of A Fuzzy Problem by the Fuzzy Laplace Transform

## Abstract

This paper is on the solutions of a fuzzy problem with triangular fuzzy number initial values by fuzzy Laplace transform. In this paper, the properties of fuzzy Laplace transform, generalized differentiability and fuzzy arithmetic are used. The example is solved in relation to the studied problem. Conclusions are given.

###### A New Approach For Weighted Hardy’s Operator In VELS

## Abstract

A considerable number of research has been carried out on the generalized Lebesgue spaces *L*
^{p(x)} and boundedness of different integral operators therein. In this study, a new approach for weighted increasing near the origin and decreasing near infinity exponent function that provides a boundedness of the Hardy’s operator in variable exponent space is given.