The energy of a vertex in a graph plays a very important role in terms of its contribution to the total energy of a graph, a significant graph invariant in the field of chemical graph theory. Particularly, the computation of vertex energies of various types of graphs in general and integral graphs has been carried out in recent studies. Working in this direction, we determine the vertex energy of all connected non-regular non-bipartite integral graphs with maximum vertex degree four in this article.

Three-valued conditional logic (CL), defined by Guzm ́an and Squier (1990) and based on McCarthy’s noncommutative connectives, axiomatises a short-circuit logic (SCL), that is, a logic that prescribes shortcircuit evaluation of conjunction and disjunction. CL defines more identities than three-valued MSCL (Memorising SCL, which also has a two-valued variant). This follows from the fact that the definable connective that prescribes full left-sequential conjunction is commutative in CL. We observe that CL also has a two-valued variant of which the full left-sequential connectives and negation define a commutative logic that is weaker than propositional logic because the absorption laws do not hold. Next, we show that the original, equational axiomatisation of CL is not independent and give several alternative, independent axiomatisations. Finally, we show that in CL, the full left-sequential connectives and negation define Bochvar’s three-valued logic. The paper ends with an appendix on the use of Prover9 and Mace4.

An entirely novel virus that causes severe acute respiratory syndrome began to spread around the world at the end of 2019. Since there are no curative treatments to treat the viral infection, efforts are being made globally to stop the coronavirus from spreading. The coronavirus strain is also known as 2019-nCoV and officially identified as SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2). Several distinctive characteristics of graphs have been used to differentiate the construction of entropy-based measurements from the structure of chemical graphs. Graph entropies have emerged as the information-theoretic quantities for quantifying the fundamental information of chemical graphs. The prospective applicability of the graph entropy measure in discrete mathematics, biology and chemistry has drawn attention from the scientific community. In this work, we study the structural properties of one of the known antiviral drugs, Molnupiravir, and introduce the concept of entropy measure for this drug. We evaluate the degree-based and neighborhood degree sum-based entropy by considering the values of topological indices of the chemical compound. Furthermore, we present the graphical representations of these entropy measures. A comparative analysis of degree-based and neighborhood degree sum-based topological indices, along with their associated entropy measures, is presented for Molnupiravir through graphical illustrations using numerical data.

This paper concerns an expansion of first-order Belnap-Dunn logic, called BD^{\supset,F}_{\bot}, and an application of this logic in the area of relational database theory. The notion of a relational database, the notion of a query applicable to a relational database, and several notions of an answer to a query with respect to a relational database are considered from the perspective of this logic, taking into account that a database may be an inconsistent database and/or a database with null values. The chosen perspective enables among other things the definition of a notion of a consistent answer to a query with respect to a possibly inconsistent database without resort to database repairs. For each of the notions of an answer considered, being an answer to a query with respect to a database of the kind considered is decidable.

Shrinkage estimators modify classical statistical estimators by scaling them towards zero in order to decrease their prediction error. We propose shrinkage estimators for linear regression models which explicitly take into account the presence of the intercept term, shrinking it independently from other coefficients. This is different from current shrinkage estimators, which treat the intercept just as an ordinary regression coefficient. We demonstrate that the proposed approach brings systematic improvements in prediction accuracy if the true intercept term differs in magnitude from other coefficients, which is often the case in practice. We then generalize the approach to uplift regression which aims to predict the causal effect of a specific action on an individual with given characteristics. In this case the proposed estimators improve prediction accuracy over previously proposed shrinkage estimators and achieve impressive performance gains over original models.

The reticulation L(R) of a commutative ring R was introduced by Joyal in 1975, then the theory was developed by Simmons in a remarkable paper published in 1980. L(R) is a bounded distributive algebra whose main property is that the Zariski prime spectrum Spec(R) of R and the Stone prime spectrum SpecId (L(R)) of L(R) are homeomorphic. The construction of the lattice L(R) was generalized by Belluce for each unital ring R and the reticulation was defined by axioms. In a recent paper we generalized the Belluce construction for algebras in a semidegenerate congruence-modular variety V. For any algebra A ∈ V we defined a bounded distributive lattice L(A), but in general the prime spectrum Spec(A) of A is not homeomorphic with the prime spectrum SpecId (L(A)). We introduced the quasi-commutative algebras in the variety V (as a generalization of Belluce’s quasi-commutative rings) and proved that for any algebra A ∈ V, the spectra Spec(A) and SpecId (L(A)) are homeomorphic. In this paper we define the reticulation A ∈ V by four axioms and prove that any two reticulations of A are isomorphic lattices. By using the uniqueness of reticulation and other results from the mentioned paper, we obtain a characterization theorem for the algebras A ∈ V that admit a reticulation: A is quasi-commutative if and only if A admits a reticulation. This result is a universal algebra generalization of the following Belluce theorem: a ring R is quasi-commutative if and only if R admits a reticulation. Another subject treated in this paper is the spectral closure of the prime spectrum Spec(A) of an algebra A ∈ V, a notion that generalizes the Belluce spectral closure of the prime spectrum of a ring.

A topological index is a numerical quantity that defines a chemical descriptor to report several physical, biological and chemical properties of a chemical structure.In recent literature, various degree-based topological indices of a molecular structure are easily calculated by deriving a M-polynomial of that structure.In this paper, we first determine the expression of a M-polynomial of the triangular Hexderived network of type three of dimension n and then obtain the corresponding degree-based topological indices from the closed form of M-polynomial.In addition, we use Maple software to represent the M-polynomial and the concerned degree-based topological indices pictorially for different dimensions.

Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) a role which can be played by a number. Sumterm splitting operators are introduced and it is argued that without further precautions the presence of these operators gives rise to an instance of the so-called sum splitting paradox. A survey of solutions to the sum splitting paradox is given.

<p>The notion of a most general algebraic specification of an arithmetical datatype of characteristic zero is introduced.Three examples of such specifications are given. A preference is formulated for a specification by means of infinitely many equations which can be presented via a finite number of so-called schematic equations phrased in terms of an infinite signature. On the basis of the latter specification three topics are discussed: (i) fracterm decomposition operators and the numerator paradox,  (ii) foundational specifications of arithmetical datatypes, and  (iii) poly-infix operations.</p>

A stay point of a moving entity is a region in which it spends a significant amount of time. In this paper, we identify all stay points of an entity in a certain time interval, where the entity is allowed to leave the region but it should return within a given time limit. This definition of stay points seems more natural in many applications of trajectory analysis than those that do not limit the time of entity's absence from the region. We present an O(n log n) algorithm for trajectories in R 1 with n vertices and a (1 + )-approximation algorithm for trajectories in R 2 to identify all such stay points. Our algorithm runs in O(kn 2 ), where k depends on and the ratio of the duration of the trajectory to the allowed gap time. We also present an algorithm to answer stay point queries in logarithmic time, after an O(kn log n) time preprocessing.