Mathematics

For a group G , we denote by G ↔ the coarse space on G endowed with the coarse structure with the base {{(x,y)∈G×G:y∈ x F }:F∈[G ] <ω } , x F ={ z −1 xz:z∈F} . Our goal is to explore interplays between algebraic properties of G and asymptotic properties of G ↔ . In particular, we show that asdim G ↔ =0 if and only if G/ Z G is locally finite, Z G is the center of G . For an infinite group G , the coarse space of subgroups of G is discrete if and only if G is a Dedekind group.

Cographs--defined most simply as complete graphs with colored lines--both dualize and generalize ordinary graphs, and promise a comparably wide range of applications. This article introduces them by examples, catalogues, and elementary properties. Any finite cograph may be realized in several ways, including inner products, polynomials, geometrically, or by "fat intersections." Particular classes then considered include sum cographs (points in Z or Zn; the line C(P,Q) joining points P and Q defined C(P,Q) = P+Q); difference cographs (C(P,Q) = |P-Q|; and intersection cographs (points are sets; C(P,Q) = P intersect Q). Intersection cographs, especially, promise many applications; described here are some to aesthetics. Point-line cographs turn out equivalent to linear spaces. Finally solved here is an interesting group-theoretic problem arising from group cographs (points in a group; C(P,Q) = {PQ,QP}).

A structured approach for the Collatz conjecture is presented using just the odd integers that are, in turn, divided into categories based on the roles they play such as Starter, Intermediary and Terminal. The expression 4x+1 is used as a tool to expose all the hidden and significant characteristics of the conjecture that lead us to its proof. The mixing properties of the iterates are addressed by showing that the Collatz iterates of half of all the odd integers that are of the form 4m+3, on the average, increase by three times the value of the odd integer that was used to start with, while the iterates of those of 4m+1, on the average, decrease by a factor of four. Further, expressions are provided to generate all the sets of odd integers where the Collatz iterate of all the integers in each set is an integer of the of form 6m+1 or 6m+5. The significance of the Collatz net (tree) becomes obvious since it encompasses all the Collatz trajectories.

Halbeisen and Hungerbuhler determined optimal bounds for the length of rational Collatz cycles. Their methods are extended to 3n+c cycles. Another sequence having properties similar to those of Riemann zeta function zeros is introduced.

The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers which inflation propensity remains so unpredictable it could be used to generate reliable proof of work algorithms for the cryptocurrency industry. Here we establish an ad hoc equivalent of modular arithmetic for Collatz sequences to automatically demonstrate the convergence of infinite quivers of numbers, based on five arithmetic rules we prove apply on the entire Collatz dynamic and which we further simulate to gain insight on their graph geometry and computational properties. We then formally demonstrate these rules define an automaton that is playing a Hydra game on the graph of undecided numbers we also prove is embedded in 24N-7, proving that in ZFC the Collatz conjecture is true, before giving a promising direction to also prove it in Peano arithmetic.

We give corrections concerned with the proofs of the theorems from the paper Opial inequality in q-Calculus, where integral inequalities of the q-Opial type were established.

We review a known method of compounding two magic square matrices of order m and n with the all-ones matrix to form two magic square matrices of order mn. We show that these compounded matrices commute. Simple formulas are derived for their Jordan form and singular value decomposition. We verify that regular (associative) and pandiagonal commuting magic squares can be constructed by compounding. In a special case the compounded matrices are similar. Generalization of compounding to a wider class of commuting magic squares is considered. Three numerical examples illustrate our theoretical results.

The \textit{Collatz's conjecture} is an unsolved problem in mathematics. It is named after Lothar Collatz in 1973. The conjecture also known as Syrucuse conjecture or problem. Take any positive integer n . If n is even then divide it by 2 , else do "triple plus one" and get 3n+1 . The conjecture is that for all numbers, this process converges to one. In the modular arithmetic notation, define a function f as follows: f(x)={ n 2 3n+1 if if n??(mod2) n??(mod2). In this paper, we present the proof of the Collatz conjecture for many types of sets defined by the remainder theorem of arithmetic. These sets are defined in mods 6,12,24,36,48,60,72,84,96,108 and we took only odd positive remainders to work with. It is not difficult to prove that the same results are true for any mod 12m, for positive integers m .

The aim of this paper is to discus the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K -Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R.~A. Stoltenberg, Proc. London Math. Soc. \textbf{17} (1967), 226--240, and V.~Gregori and J.~Ferrer, Proc. Lond. Math. Soc., III Ser., \textbf{49} (1984), 36.

We study conformal bi-slant submersions from almost Hermitian manifolds onto Riemannian manifolds as a generalized of conformal anti-invariant, conformal semi-invariant, conformal semi-slant, conformal slant and conformal hemi-slant submersions. We investigated the integrability of distributions and obtain necessary and sufficient conditions for the maps to have totally geodesic fibers. Also we studied the total geodesicity of such maps.