Real Length Functions in Groups

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

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.

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

For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures, including as special cases Choquet and Sugeno integrals. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the greatest universal integrals are given. Finally, another construction method for obtaining universal integrals is introduced.

Let T be an edge weighted tree, let dT(u, v) be the sum of the weights of the edges on the path from u to v in T, and let d min and d max be two non-negative real numbers such that d min ≤ d max . Then a pairwise compatibility graph of T for d min and d max is a graph G = (V, E), where each vertex u' ∈ V corresponds to a leaf u of T and there is an edge (u', v') ∈ E if and only if d min ≤ dT(u, v) ≤ d max . A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers d min and d max such that G is a pairwise compatibility graph of T for d min and d max . Kearney et al. conjectured that every graph is a PCG [3]. In this paper, we refute the conjecture by showing that not all graphs are PCG s . Moreover, we recognize several classes of graphs as pairwise compatibility graphs. We identify two restricted classes of bipartite graphs as PCG. We also show that the well known tree power graphs and some of their extensions are PCGs.

A set S equipped with two algebraic operations: addition and multiplication is called a semiring if the following conditions are satisfied: both operations are associative; the addition is commutative; the multiplication is distributive with respect to the addition a.(b+ c) = (a.b) + (a.c) and (a+ b).c = (a.c) + (b.c) ∀a, b, c ∈ S . A semiring S is called an idempotent semiring if a+ a = a for all a ∈ S. Employing additively idempotent semirings semirings as a platform for a cryptographic scheme arose several years ago. In the present work we show how to apply different dual pairs of idempotent semirings for constructing new cryptographic protocol. We are interested in four idempotent semirings: Max-plus semiring: Consider Rmax = R ∪ {−∞}. Given a, b ∈ Rmax, define: a ⊕ b = max{a, b}; a⊙ b = a+ b. If we add the top element ⊤ = +∞ to this set, the resulting semiring is complete and denoted by Rmax. Min-plus semiring: Consider Rmin = R ∪ {+∞}. Given a, b ∈ R+∞, define: a ⊕ b = min{a, b}; a⊙ b = a+ b. If we add the top element ⊥ = −∞ to this set, the resulting semiring is complete and denoted by Rmin. Max-time semiring: Consider Rmax,× = R ∪ {+∞} (non-negative real numbers). Given a, b ∈ Rmax,×, define: a⊕ b = max{a, b}; a⊗ b = a.b. Min-time semiring: Consider Rmin,× = R∪{+∞} (non-negative real numbers). Given a, b ∈ Rmin,×, define: a⊕ b = min{a, b}; a⊗ b = a.b. We give a generalization of the Diffie-Hellman key exchange protocol, to the context of semigroup actions. Our protocol in its most general form consists of the following: two commutative semirings S1, S2 act on a set X i.e. ((S1 × S2)×X) → X. We propose two practical realizations of this scheme based on different dual pairs of idempotent semirings. For the semirings S1 and S2 we suggest commutative semirings generated by two given matrices M and N. These semirings are semirings of polynomials in M and N with coefficient of two dual pairs of idempotent semirings. Set X should be selected carefully with respect to the chosen semirings. In the first protocol: S1 = Mn(Rmax), S2 = Mn(Rmin), M ∈ Mn(Rmax), N ∈ Mn(Rmin),X ∈ Mn(R). In the second protocol: S1 = Mn(Rmax,×), S2 = Mn(Rmin,×), M ∈ Mn(Rmax,×), N ∈ Mn(Rmin,×), X ∈ Mn(R).

In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.

CHAPTER 5 - FUNCTIONS OF A REAL VARIABLE

CHAPTER II - Functions

Evolution is a highly complex multilevel process and mathematical modeling of evolutionary phenomenon requires proper abstraction and radical reduction to essential features. Examples are natural selection, Mendel's laws of inheritance, optimization by mutation and selection, and neutral evolution. An attempt is made to describe the roots of evolutionary theory in mathematical terms. Evolution can be studied in vitro outside cells with polynucleotide molecules. Replication and mutation are visualized as chemical reactions that can be resolved, analyzed, and modeled at the molecular level, and straightforward extension eventually results in a theory of evolution based upon biochemical kinetics. Error propagation in replication commonly results in an error threshold that provides an upper bound for mutation rates. Appearance and sharpness of the error threshold depend on the fitness landscape, being the distribution of fitness values in genotype or sequence space. In molecular terms, fitness landscapes are the results of two consecutive mappings from sequences into structures and from structures into the (nonnegative) real numbers. Some properties of genotype-phenotype maps are illustrated well by means of sequence-structure relations of RNA molecules. Neutrality in the sense that many RNA sequences form the same (coarse grained) structure is one of these properties, and characteristic for such mappings. Evolution cannot be fully understood without considering fluctuations--each mutant originates form a single copy, after all. The existence of neutral sets of genotypes called neutral networks, in particular, necessitates stochastic modeling, which is introduced here by simulation of molecular evolution in a kind of flowreactor.

Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1 , then the number # ( | y | , N ) of 1-bits in the expansion of | y | through bit position N satisfies # ( | y | , N ) > C N 1 / D for a positive number C (depending on y ) and sufficiently large N . This in itself establishes the transcendency of a class of reals ∑ n ≥ 0 1 / 2 f ( n ) where the integer-valued function f grows sufficiently fast; say, faster than any fixed power of n . By these methods we re-establish the transcendency of the Kempner–Mahler number ∑ n ≥ 0 1 / 2 2 n , yet we can also handle numbers with a substantially denser occurrence of 1’s. Though the number z = ∑ n ≥ 0 1 / 2 n 2 has too high a 1’s density for application of our central result, we are able to invoke some rather intricate number-theoretical analysis and extended computations to reveal aspects of the binary structure of z 2 .