The purpose of this article is to introduce three types of supra compactness and three types of supra Lindelöfness via supra topological spaces based on the supra pre-open sets. With the help of examples, we illustrate the relationships among them and show their relationships with some kinds of supra compactness and supra Lindelöfness given in [3]. We characterize each type of space and investigate the image of them under pre-irresolute mappings. Also, we prove that these spaces are preserved under the finite product spaces, and give a sufficient condition for the equivalence among supra compact, almost supra compact and supra pre-compact spaces. At the end of each section, we provide some examples to demonstrate that the spaces studied and their counterparts, introduced in [9], are independent of each other.

The notion of paracompactness in generalized topology is introduced and studied in [5]. In this paper, we introduce and investigate the notion of $\mu$-paracompact spaces with respect to a hereditary class $\mathcal{H}$, which is a generalization of the notion of $\mu$-paracompact spaces. We study characterizations, subsets, and subspaces of $\mu\mathcal{H}$-paracompact spaces. Also, we investigate the invariants of $\mu\mathcal{H}$-paracompact spaces by functions.

In this short paper, we examine a curious feature of the Collatz function. When the sequences generated by the Collatz function on consecutive integer initial points are arranged into an array, certain arithmetic sequences show up with common differences given by products of twos and threes. The common differences themselves are further related by a formula that depends on even versus odd input. While we do not solve the Collatz Conjecture by this observation, we present our findings as interesting mathematical results in their own rights.

Log-concavity of probability density functions has played an important role in many applications in economics, reliability studies, and many more. Several authors have studied log-concavity of many distribution such as the normal, logistic, exponential, etc., but no attention has been paid to exponential power distributions, a family of elliptical distributions which generalize the normal, double exponential, and Kotz distribution. This distribution has been found to be very useful in applications, especially when flexibility of the tails (heavier or thinner than the normal distribution) are required in modeling of natural phenomenon. This paper was written to fill this void in the literature.

We consider data and boundary conditions and data within the regularity domain of a pseudoparabolic equation with varying time direction. This problem cannot be solved in the sense of J. Hadamard. Based on A. N. Tikhonov's approach, this problem is investigated for conditional correctness, the uniqueness theorem is proved, and the evaluation of conditional stability on the set of correctness is established. This study uses methods of functional analysis and methods of the theory of functions of a complex variable.

In general, finding the sum of an infinite series is not always possible. However, there are infinite series whose sums can be computed. In our earlier paper, we derived a formula for computing the sum of a type of polylogarithm series involving multinomial coefficients. In this paper, we show that the formula leads to an elementary computation for the series $\sum_{k=1}^{\infty}\frac{k^n}{a^k}$ involving numbers obtained by a method similar to Pascal's triangle. We also show that our result is the number of ways of distributing $n$ distinct objects in $n$ or fewer distinct nonempty cells.

Unlike topological groups, hypergroups are not closed under duality. While it has long been known that a hypergroup dual space might be signed, the boundedness of such dual spaces has been an open question. In this paper it is shown that a hypergroup dual space may fail to be bounded. An example will be given of an infinite direct product of finite hypergroups whose dual space is a semi-bounded, but not bounded, generalized hypergroup.

The main objective of this paper is to introduce a new generalized form of topological vector spaces, namely, almost $\alpha$-topological vector spaces by using the concepts of $\alpha$-open sets, regular open sets, and almost $\alpha$-continuous mappings. In the present paper, some important examples of almost $\alpha$-topological vector spaces with basic properties are characterized.

The concept of $\mathbb{N^{\infty}}$-submoduloid has been introduced in [5]. In this paper, the notions of a generating submoduloid, an additive absorption set, a prime $\mathbb{N^{\infty}}$-submoduloid, and an additive $\mathbb{N^{\infty}}$-submoduloid of an $\mathbb{N^{\infty}}$-moduloid are defined and some related results are investigated.

The purpose of this paper is to introduce and investigate strongly fully stable acts relative to an ideal as a concept generalizing strongly fully stable modules relative to an ideal, but is stronger than that of fully stable acts. In this study, we consider some properties and characterizations of the class of strongly fully stable acts relative to an ideal, as well as the relations between this class and other classes. Among these classes are quasi-injective acts, strongly quasi-injective acts, acts which satisfy Baer's criterion, acts satisfying the strongly Baer's criterion, and duo acts. The product of strongly fully stable acts relative to an ideal need not be a strongly fully stable act relative to that ideal. The coproduct of any family of strongly fully stable acts relative to an ideal need not be a strongly fully stable act relative to that ideal. Also, we have that strongly fully stable acts relative to an ideal are equivalent to an $S$-act that satisfies the strongly Baer's criterion relative to an ideal $I$ for cyclic subacts. The strongly fully stable act relative to $I$ is equivalent to the strongly quasi-injective act relative to the ideal $I$ and duo act with a commutative monoid.

An Alternate Cayley-Dickson Algebra, Volume 28, Issue 1 (2016), has been retracted by the author. He has discovered that the doubling product, which is the primary subject of the paper, does not generate a Cayley-Dickson algebra beyond the second doubling. As the paper cannot be repaired, he has asked for the paper's retraction.