Hyperreal Delta Functions as a New General Tool for Modeling Systems with Infinitely High Densities

In general, the state of a system in which a physical quantity such as mass is distributed over space can be modeled by a function that represents the density distribution. The purpose of this paper is to introduce special functions that can be applied when in the system to be modeled, where the quantity is distributed over isolated points. For that matter, the expanded real numbers are introduced as an ordered subring of the hyperreal number field that does not contain any infinitesimals, and hyperreal delta functions are defined as special functions from the real numbers to the expanded real numbers satisfying the condition that (i) the support is a singleton, and (ii) the integral over the reals is a nonzero real number. These newly defined hyperreal delta functions, and tensor products thereof, then provide a general tool, applicable for the mathematical modeling of physical systems in which infinitely high densities occur.

ON BINOMIAL SUMS WITH THE TERMS OF SEQUENCES {g_kn} AND {h_kn}

In this paper, we derive sums and alternating sums of products of terms ofthe sequences $\left\{ g_{kn}\right\} $ and $\left\{ h_{kn}\right\} $ withbinomial coefficients. For example,\begin{eqnarray*} &\sum\limits_{i=0}^{n}\binom{n}{i}\left( -1\right) ^{i} \left(c^{2k}\left(-q\right) ^{k}+c^{k}v_{k}+1\right)^{-ai}h_{k\left( ai+b\right) }h_{k\left(ai+e\right) } \\ &=\left\{ \begin{array}{clc} -\Delta ^{\left( n+1\right) /2}g_{k\left( an+b+e\right) }g_{ka}^{n}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is odd,} & \\ \Delta ^{n/2}h_{k\left( an+b+e\right) }g_{ka}^{n}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is even,} & \end{array}% \right.\end{eqnarray*}%and\begin{eqnarray*} &&\sum\limits_{i=0}^{n}\binom{n}{i}i^{\underline{m}}g_{k\left( n-ti\right) }h_{kti} \\ &&=2^{n-m}n^{\underline{m}}g_{kn}-n^{\underline{m}}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{n\left( 1-t\right) }h_{kt}^{n-m}g_{k\left( tm+tn-n\right) },\end{eqnarray*}%where $a, b, e$ is any integer numbers, $c$ is nonzero real number and $m$is nonnegative integer.

On a class of solvable difference equations generalizing an iteration process for calculating reciprocals

The well-known first-order nonlinear difference equation $$ y_{n+1}=2y_{n}-xy_{n}^{2}, \quad n\in {\mathbb {N}}_{0}, $$ y n + 1 = 2 y n − x y n 2 , n ∈ N 0 , naturally appeared in the problem of computing the reciprocal value of a given nonzero real number x. One of the interesting features of the difference equation is that it is solvable in closed form. We show that there is a class of theoretically solvable higher-order nonlinear difference equations that include the equation. We also show that some of these equations are also practically solvable.

On the maximum general sum-connectivity index of trees with a fixed order and maximum degree

The general sum-connectivity index of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text], where [Formula: see text] is the edge connecting the vertices [Formula: see text], [Formula: see text] denotes the degree of the vertex [Formula: see text], and [Formula: see text] is a nonzero real number. This paper is devoted to characterizing the graphs having the maximal [Formula: see text] value among all the trees with order [Formula: see text] and fixed maximum degree.

Log-Like Functions and Uniform Distribution Modulo One

For a function f satisfying f (x) = o((log x) K), K > 0, and a sequence of numbers (qn) n, we prove by assuming several conditions on f that the sequence (αf (qn)) n≥n0 is uniformly distributed modulo one for any nonzero real number α. This generalises some former results due to Too, Goto and Kano where instead of (qn) n the sequence of primes was considered.

Pair Correlations and Random Walks on the Integers

The paper gives conditions for a sequence of fractional parts of real numbers $\left( {\{ a_n x\} } \right)_{n = 1}^\infty $ to satisfy a pair correlation estimate. Here x is a fixed nonzero real number and $\left( {a_n } \right)_{n = 1}^\infty $ is a random walk on the integers.

Intuitionistic fuzzy stability of Jensen-type quadratic functional equations

In this paper, we prove some stability results for Jensen-type quadratic functional equations 2f (x+y/2)+ 2f (x-y/2) = f (x) + f (y), f (ax+ay) + f (ax-ay) = 2a2f(x) + 2a2 f(y) in intuitionistic fuzzy normed spaces for a nonzero real number a with a ? ?1/2

CONTINUOUS AND DISCRETE FLOWS ON OPERATOR ALGEBRAS

Let (N,ℝ,θ) be a centrally ergodic W* dynamical system. When N is not a factor, we show that for each nonzero real number t, the crossed product induced by the time t automorphism θt is not a factor if and only if there exist a rational number r and an eigenvalue s of the restriction of θ to the center of N, such that rst=2π. In the C* setting, minimality seems to be the notion corresponding to central ergodicity. We show that if (A,ℝ,α) is a minimal unital C* dynamical system and A is not simple, then, for each nonzero real number t, the crossed product induced by the time t automorphism αt is not simple if there exist a rational number r and an eigenvalue s of the restriction of α to the center of A, such that rst=2π. The converse is true if, in addition, A is commutative or prime.

Fuzzy Stability of Jensen-Type Quadratic Functional Equations

We prove the generalized Hyers-Ulam stability of the following quadratic functional equations and in fuzzy Banach spaces for a nonzero real number with .

Bifurcations of Symmetric Periodic Orbits Near Equilibrium in Reversible Systems

Consider a family of reversible systems [Formula: see text] with the origin being an equilibrium for each μ. Suppose Dxf(0, 0) has only purely imaginary eigenvalues ±iw1,…,±iwk. We investigate the typical bifurcations of symmetric periodic solutions near the origin. A suitable complex basis is chosen so that Dxf(0, 0) and the involution are in respective simple form. Incorporated with putting f into normal form, a modified version of Lyapunov–Schmidt reduction can be applied to obtain the reduced bifurcation equations. We then focus on the cases in resonance, that is, wj = njw0, where w0 is a nonzero real number and nj is an integer for each j. Some codimension-two bifurcations are illustrated for the system in non-semisimple resonance with nj = 1, 2. A few codimension-one cases are also given for comparison with earlier works by other researchers.