Mathematics

The treated matrix equation (1+a e − ∥X∥ b )X=Y in this short note has its origin in a modelling approach to describe the nonlinear time-dependent mechanical behaviour of rubber. We classify the solvability of (1+a e − ∥X∥ b )X=Y in general normed spaces (E,∥⋅∥) w.r.t. the parameters a,b∈R , b≠0 , and give an algorithm to numerically compute its solutions in E= R m×n , m,n∈N , m,n≥2 , equipped with the Frobenius norm.

This paper shows that, in the critical strip, the Riemann zeta function ζ(s) have the same set of zeros as F(s):= ????0 t s?? ( e t +1 ) ?? dt , and then discusses the behavior of F(s) .

The ordinal sum of t-norms on a bounded lattice has been used to construct other t-norms. However, an ordinal sum of binary operations (not necessarily t-norms) defined on the fixed subintervals of a bounded lattice may not be a t-norm. Some necessary and sufficient conditions are presented in this paper for ensuring that an ordinal sum on a bounded lattice of two binary operations is, in fact, a t-norm. In particular, the results presented here provide an answer to an open problem put forward by Ertuğrul and Yeşilyurt [Ordinal sums of triangular norms on bounded lattices, Inf. Sci., 517 (2020) 198-216].

A double Roman dominating function of a graph G is a function f:V(G)→{0,1,2,3} having the property that for each vertex v with f(v)=0 , there exists u∈N(v) with f(u)=3 , or there are u,w∈N(v) with f(u)=f(w)=2 , and if f(v)=1 , then v is adjacent to a vertex assigned at least 2 under f . The double Roman domination number γ dR (G) is the minimum weight f(V(G))= ∑ v∈V(G) f(v) among all double Roman dominating functions of G . An outer independent double Roman dominating function is a double Roman dominating function f for which the set of vertices assigned 0 under f is independent. The outer independent double Roman domination number γ oidR (G) is the minimum weight taken over all outer independent double Roman dominating functions of G . In this work, we present some contributions to the study of outer independent double Roman domination in graphs. Characterizations of the families of all connected graphs with small outer independent double Roman domination numbers, and tight lower and upper bounds on this parameter are given. We moreover bound this parameter for a tree T from below by two times the vertex cover number of T plus one. We also prove that the decision problem associated with γ oidR (G) is NP-complete even when restricted to planar graphs with maximum degree at most four. Finally, we give an exact formula for this parameter concerning the corona graphs.

This paper examines the use of Lie group and Lie Algebra theory to construct the geometry of pairwise comparisons matrices. The Hadamard product (also known as coordinatewise, coordinate-wise, elementwise, or element-wise product) is analyzed in the context of inconsistency and inaccuracy by the decomposition method. The two designed components are the approximation and orthogonal components. The decomposition constitutes the theoretical foundation for the multiplicative pairwise comparisons. Keywords: approximate reasoning, subjectivity, inconsistency, consistency-driven, pairwise comparison, matrix Lie group, Lie algebra, approximation, orthogonality, decomposition.

We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the connections among the nodes are made based on the divisibility pattern among the numbers. We derive analytical expressions for the centrality measures of this network in terms of the floor function and the divisor functions. We validate these measures with the help of standard methods which make use of the adjacency matrix of the network. Thus how the measures of the network relate to patterns in the behaviour of primes and composite numbers becomes apparent from our study.

In this paper, we propose a mathematical model of subjective experience in terms of classes of hierarchical geometries of representations ("n-awareness"). We first outline a general framework by recalling concepts from higher category theory, homotopy theory, and the theory of (infinity, 1)-topoi. We then state three conjectures that enrich this framework. We first propose that the (infinity, 1)-category of a geometric structure known as perfectoid diamond is an (infinity, 1)-topos. In order to construct a topology on the (infinity, 1)-category of diamonds we then propose that topological localization, in the sense of Grothendieck-Rezk-Lurie (infinity, 1)-topoi, extends to the (infinity, 1)-category of diamonds. We provide a small-scale model using triangulated categories. Finally, our meta-model takes the form of Efimov K-theory of the (infinity, 1)-category of perfectoid diamonds, which illustrates structural equivalences between the category of diamonds and subjective experience (i.e. its privacy, self-containedness, and self-reflexivity). Based on this, we investigate implications of the model. We posit a grammar ("n-declension") for a novel language to express n-awareness, accompanied by a new temporal scheme ("n-time"). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? We also examine the notion of "self" within our framework. A new model of personal identity is introduced which resembles a categorical version of the "bundle theory"; selves are not substances in which properties inhere but (weakly) persistent moduli spaces in the K-theory of perfectoid diamonds.

Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in mathematics is the Periodic Table of Geometric Numbers discussed here. In 1878 William Kingdon Clifford discovered the defining rules for what he called geometric algebras. We show how these algebras, and their coordinate isomorphic geometric matrix algebras, fall into a natural periodic table, sidelining the superfluous definitions based upon tensor algebras and quadratic forms.

We discuss an extension of classical combinatorics theory to the case of spatially distributed objects.

We describe the construction of a new family of developable rollers based on the Platonic solids. In this way kinetic sculptures may be realised, with the Platonic solids quite literally in their heart. We also describe the strong way in which the Platonicons circumscribe the Platonic solids.