Improved algorithms for the continuous tree edge-partition problems and a note on ratio and sorted matrices searches

On graphs that are not PCGs

Suppose {X(t), t ∈ G d } is a random field in d dimensions, with d ∈ ℤ+; that is, {X(t), t ∈ G d } is a collection of random variables X(t) taking values in a state space S, defined on a probability space (Ω, A, P), and indexed by the variable t ∈ G d . Throughout this chapter, G will stand for either the set of real numbers ℝ, or the set of integers ℤ; thus, the random field {X(t)} is allowed to “run” in either continuous or discrete “time.” Similarly, G+ will denote ℝ+ or ℤ+ (the sets of positive real numbers and integers, respectively) according to whether G = ℝ or G = ℤ.

The celebrated Ackermann encoding of hereditarily finite sets is generalized to a parametric formula designed to map not only these sets but also hereditarily finite multisets and hypersets into the non-negative real numbers. This extension suggests a novel approach to the graph canonization problem by reducing it to a simple comparison of real values. By suitably varying the sole parameter of this formula, both the original Ackermann encoding and another previously studied map emerge as special cases. When the parameter is chosen from the natural numbers, the function yields a bijective encoding of a subuniverse of hereditarily finite multisets into the natural numbers. If, instead, the parameter is chosen to be transcendental and lies within a specific interval on the positive real line, the function is conjectured to provide an injective encoding of both multisets and hypersets.

The article is devoted to the problem of the real numbers' decomposition to a linear combination of natural numbers' squares. First of all the theorem giving indirectly the conditions of existing natural numbers' two equal squares is formulated and proved. Basing on this theorem two algorithms designed for finding couples of integer numbers generating eąual sums of sąuares and determining the multiplicity of that couples are presented. Finally the calculation of the algorithms' classes and estimation of the calculation errors is made. Also an example output of presented algorithms was shown.

Real or natural number interpretation and their effect on complexity

This is a continuation part of our investigation in which the second order nonlinear rational difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1), n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0, is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that gives the number of of the space. Recently, Leinster and Shulman introduced a homology theory for metric spaces, called magnitude homology, which categorifies the magnitude of a space. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and it is completely encoded in the Euler characteristic of magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.

In this paper, we investigate the global attractivity of the difference equation x n + 1 = A − B x n − 2 C + D x n − 1 , n = 0 , 1 , … , where A, B are nonnegative real numbers, C, D are positive real numbers and C + D x n − 1 ≠ 0 for all n ≥ 0.

We consider the higher‐order nonlinear difference equation xn+1 = (α + xn)/(A + Bxn + xn−k), n = 0, 1, …, where parameters are positive real numbers and initial conditions x−k, …, x0 are nonnegative real numbers, k ≥ 2. We investigate the periodic character, the invariant intervals, and the global asymptotic stability of all positive solutions of the abovementioned equation. We show that the unique equilibrium of the equation is globally asymptotically stable under certain conditions.

In this paper, we deal with the global behavior of the positive solutions of the system of k -difference equations u n + 1 1 = α 1 u n − 1 1 / β 1 + α 1 u n − 2 2 r 1 , u n + 1 2 = α 2 u n − 1 2 / β 2 + α 2 u n − 2 3 r 2 , … , u n + 1 k = α k u n − 1 k / β k + α k u n − 2 1 r k , n ∈ ℕ 0 , where the initial conditions u − l i l = 0,1,2 are nonnegative real numbers and the parameters α i , β i , γ i , and r i are positive real numbers for i = 1,2 , … , k , by extending some results in the literature. By the end of the paper, we give three numerical examples to support our theoretical results related to the system with some restrictions on the parameters.

This paper is a study of the structure of a group $G$ equipped with a ’length’ function from $G$ to the nonnegative real numbers. The properties that we require this function to satisfy are derived from Lyndon’s work on groups with integer-valued functions. A real length function is a function which assigns to each $g \in G$ a nonnegative real number $|g|$ such that the following axioms are satisfied: $|x| < |xx|$ if $x \ne 1$. $|x| = 0$ if and only if $x = 1$. $|{x^{ - 1}}| = |x|$. $c(x,y) \geq 0$ where $c(x,y) = 1/2(|x| + |y| - |x y^{-1}|)$. $c(x,y) \geq m$ and $c(y,z) \geq m$ imply $c(x,z) \geq m$. In this paper structure theorems are obtained for the cases when $G$ is abelian and when $G$ can be generated by two elements. We first prove that if $G$ is abelian, then $G$ is isomorphic to a subgroup of the additive group of the real numbers. Then we introduce a reduction process based on a generalized notion of Nielsen transformation. We apply this reduction process to finite sets of elements of $G$. We prove that if $G$ can be generated by two elements, then $G$ is either free or abelian.

Necessary and sufficient conditions for oscillations of partial difference equations with continuous variables

In this article, we study the fundamental solution of the operator $$\left((\diamond+m^2)\left(\frac{\triangle^2+\boxdot^2}{2}\right)\right)^{k}$$, iterated $k$-times and is defined by (\ref{odot}),where $m$ is a nonnegative real number, and $k$ is a nonnegative integer. After that, we study the Fourier transform of the operator $ \left((\diamond+m^2)\left(\frac{\triangle^2+\boxdot^2}{2}\right)\right)^{k}\delta$.

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The aim of this paper is to investigate the global stability and periodic nature of the positive solutions of the difference equation $$x_{n + 1} = \frac{{A + Bx_{n - 2k - 1} }} {{C + D\prod\limits_{i = 1}^k {x_{n - 2i} } }}, n = 0,1,2, \ldots ,$$ where A, B are nonnegative real numbers, C,D &gt; 0 and l, k are nonnegative integers such that l ≤ k.