Nonzero Real Numbers Research Articles

Cacti with maximal general sum-connectivity index

Let V(G) and E(G) be, respectively, the vertex set and edge set of a graph G. The general sum-connectivity index of a graph G is denoted by $$\chi _\alpha (G)$$ and is defined as $$\sum \limits _{uv\in E(G)}(d_u+d_v)^\alpha $$ , where uv is an edge that connect the vertices $$u,v\in V(G)$$ , $$d_u$$ is the degree of a vertex u and $$\alpha $$ is any non-zero real number. A cactus is a graph in which any two cycles have at most one common vertex. Let $$\mathscr {C}_{n,t}$$ denote the class of all cacti with order n and t pendant vertices. In this paper, a maximum general sum-connectivity index ( $$\chi _\alpha (G)$$ , $$\alpha >1$$ ) of a cacti graph with order n and t pendant vertices is considered. We determine the maximum general sum-connectivity index of n-vertex cacti graph. Based on our obtained results, we characterize the cactus with a perfect matching having the maximum general sum-connectivity index.

Approximation by Shifts of Compositions of Dirichlet L-Functions with the Gram Function

In this paper, a joint approximation of analytic functions by shifts of Dirichlet L-functions L ( s + i a 1 t τ , χ 1 ) , … , L ( s + i a r t τ , χ r ) , where a 1 , … , a r are non-zero real algebraic numbers linearly independent over the field Q and t τ is the Gram function, is considered. It is proved that the set of their shifts has a positive lower density.

Various stabilities of reciprocal-septic and reciprocal-octic functional equations

The intention of this paper is to prove various stability results of reciprocal-septic and reciprocal-octic functional equations in non-Archimedean fields and nonzero real numbers relevant to Hyers, Rassias, and Găvruţa stability. Appropriate counter-examples are supplied to invalidate the results in the cases of singularities.

Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences

In this paper, we consider a generalization of Horadam sequence { w_{n} } which is defined by the recurrence relation w_{n}=chi ( n ) w_{n-1}+cw_{n-2}, where chi ( n ) =a if n is even, chi ( n ) =b if n is odd with arbitrary initial conditions w_{0}, w_{1} and nonzero real numbers a, b and c. As a special case, by taking the initial conditions 0, 1 and 2, b we define the sequences { u_{n} } and { v_{n} } , respectively. The main purpose of this study is to derive some basic properties of the sequences { u _{n} } , { v_{n} } and { w_{n} } by using a matrix approach.

Some Bounds on Zeroth-Order General Randić Index

For a graph G without isolated vertices, the inverse degree of a graph G is defined as I D ( G ) = ∑ u ∈ V ( G ) d ( u ) − 1 where d ( u ) is the number of vertices adjacent to the vertex u in G. By replacing − 1 by any non-zero real number we obtain zeroth-order general Randić index, i.e., 0 R γ ( G ) = ∑ u ∈ V ( G ) d ( u ) γ , where γ ∈ R − { 0 } . Xu et al. investigated some lower and upper bounds on I D for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for γ < 0 . The corresponding extremal graphs have also been identified.

Well-defined solutions of the difference equation xn = xn−3kxn−4kxn−5k xn−kxn−2k(±1±xn−3kxn−4kxn−5k)

We investigate the behavior of well-defined solutions of the difference equation [Formula: see text] where the initial conditions [Formula: see text], [Formula: see text] are arbitrary nonzero real numbers. Also, we give some special results and numerical results.

Diophantine Approximation with Mixed Powers of Primes

Let $k$ be an integer with $k \geq 3$. Let $\lambda_1$, $\lambda_2$, $\lambda_3$ be non-zero real numbers, not all negative. Assume that $\lambda_1/\lambda_2$ is irrational and algebraic. Let $\mathcal{V}$ be a well-spaced sequence, and $\delta \gt 0$. In this paper, we prove that, for any $\varepsilon \gt 0$, the number of $\upsilon \in \mathcal{V}$ with $\upsilon \leq X$ such that the inequality \[ |\lambda_1 p_1^2 + \lambda_2 p_2^2 + \lambda_3 p_3^k - \upsilon| \lt \upsilon^{-\delta} \] has no solution in primes $p_1$, $p_2$, $p_3$ does not exceed $O(X^{1-2/(7m_2(k))+2\delta+\varepsilon})$, where $m_2(k)$ relies on $k$. This refines a recent result. Furthermore, we briefly describe how a similar method can refine a previous result on a Diophantine problem with two squares of primes, one cube of primes and one $k$-th power of primes.

Cyclotomic Coincidences

Let denotes the nth cyclotomic polynomial. In this paper, we show that if m and n are distinct positive integers and x is a nonzero real number with , then except when and x = 2. We also observe that 2 appears to be the largest real limit point of the set of values of x for which for some .

On the distribution of zeros of derivatives of the Riemann ξ-function

Abstract For the completed Riemann zeta function ξ ⁢ ( s ) {\xi(s)} , it is known that the Riemann hypothesis for ξ ⁢ ( s ) {\xi(s)} implies the Riemann hypothesis for ξ ( m ) ⁢ ( s ) {\xi^{(m)}(s)} , where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence ( α ⁢ γ m ) {(\alpha\gamma_{m})} , where α is any fixed non-zero real number and γ m {\gamma_{m}} runs over the imaginary parts of the zeros of ξ ( m ) ⁢ ( s ) {\xi^{(m)}(s)} . We also obtain a zero density estimate and an explicit formula for the zeros of ξ ( m ) ⁢ ( s ) {\xi^{(m)}(s)} . In particular, all our results hold uniformly for 0 ≤ m ≤ g ⁢ ( T ) {0\leq m\leq g(T)} , where the function g ⁢ ( T ) {g(T)} tends to infinity with T and g ⁢ ( T ) = o ⁢ ( log ⁡ log ⁡ T ) {g(T)=o(\log\log T)} .

On a k-Fold Beta Integral Formula

In this paper, we investigate some necessary and sufficient conditions which ensure validity of a k-fold Beta integral formula for any $$x^{i}\in \mathbb {R}^n$$ and some nonzero real numbers $$d_i$$ with $$i=1,2,\ldots ,k$$ . We establish that Eq. (0.1) holds if and only if and This yields a complete answer to the question raised by Grafakos and Morpurgo in [4]. In addition, it turns out that formula (0.1) does not hold if $$k\ge 4$$ . For those $$k, d_1, d_2, \ldots , d_k$$ not satisfying (0.2) or (0.3), we prove that the real-valued integral $$\displaystyle {\int _{\mathbb {R}^n}\prod \limits ^k_{i=1}|x^{i}-t|^{-d_i}dt}$$ can be represented as a function of distances of consecutive differences of the sequence $$x^1, x^2, \ldots , x^k$$ .

Proof of a conjecture concerning maximum general sum-connectivity index [formula omitted] of graphs with given cyclomatic number when [formula omitted

The general sum-connectivity index of a graph G is χα(G)=∑uv∈E(G)(d(u)+d(v))α, where d(u) denotes the degree of vertex u∈V(G), and α is a non-zero real number. Ali et al. (2019) proved that the unique graph obtained from the star Sn by adding ν edge(s) between a fixed pendant vertex u and ν other pendant vertices, has the maximum χα value in the collection of all n-vertex connected graphs having cyclomatic number ν with the constraints ν=5,n≥7,α>1 or n≥8,6≤ν≤n−2,α≥2 and conjectured that this result also holds for n≥8,6≤ν≤n−2 and 1<α<2. In this paper, we provide a proof of this conjecture.

On Diophantine approximation with one prime and three squares of primes

Let λ1, λ2, λ3, λ4 be non-zero real numbers, not all of the same sign, ϖ real. Suppose that the ratios λ1/λ2, λ1/λ3 are irrational and algebraic. Then there are infinitely many solutions in primes pj, j = 1, 2, 3, 4, to the inequality |λ1p1 + λ2p22 + λ3p23 + λ4p24 + ϖ| < (max{p1, p22, p23, p24})−5/64. This improves the earlier result.

Higher Order Oscillation and Uniform Distribution

Abstract It is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e 2πiαβn g(β ))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β&gt; 1 (alternatively, for a fixed real number β&gt; 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβ n g(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

On Diophantine approximation by unlike powers of primes

Abstract Suppose that λ1, λ2, λ3, λ4, λ5 are nonzero real numbers, not all of the same sign, λ1/λ2 is irrational, λ2/λ4 and λ3/λ5 are rational. Let η real, and ε &gt; 0. Then there are infinitely many solutions in primes pj to the inequality $\begin{array}{} \displaystyle |\lambda_1p_1+\lambda_2p_2^2+\lambda_3p_3^3+\lambda_4p_4^4+\lambda_5p_5^5+\eta| \lt (\max{p_j^j})^{-1/32+\varepsilon} \end{array}$. This improves an earlier result under extra conditions of λj.

Solutions of Some Difference Equations Systems and Periodicity

In this article, analysis and investigation have been conducted on the periodic nature as well as the type of the solutions of the subsequent schemes of rational difference equations&#x0D; with a nonzero real numbers initial conditions.

On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices

The general sum-connectivity index of a graph $G$ , denoted by $\chi _{_\alpha }(G)$ , is defined as $\sum _{uv\in E(G)}(d(u)+d(v))^{\alpha }$ , where $uv$ is the edge connecting the vertices $u,v\in V(G)$ , $d(w)$ denotes the degree of a vertex $w\in V(G)$ , and $\alpha $ is a non-zero real number. For $\alpha =-1/2$ and $n\geq 11$ , Wang et al. [On the sum-connectivity index, Filomat 25 (2011) 29–42] proved that $K_{2} + \overline {K}_{n-2}$ is the unique graph with minimum $\chi _{_\alpha }$ value among all the $n$ –vertex graphs having minimum degree at least 2, where $K_{2} + \overline {K}_{n-2}$ is the join of the 2-vertex complete graph $K_{2}$ and the edgeless graph $\overline {K}_{n-2}$ on $n-2$ vertices. Tomescu [2-connected graphs with minimum general sum-connectivity index, Discrete Appl. Math. 178 (2014) 135–141] proved that the result of Wang et al. holds also for $n\geq 3$ and $-1\leq \alpha . In this paper, it is shown that the aforementioned result of Wang et al. remains valid if the graphs under consideration are connected, $n\geq 6$ and $-1\leq \alpha , where $\alpha _{0}\approx -0.68119$ is the unique real root of the equation $\chi _{_\alpha }(K_{2} + \overline {K}_{4}) - \chi _{_\alpha }(C_{6})=0$ , and $C_{6}$ is the cycle on 6 vertices.

Symmetric Rogers-Hölder's inequalities on diamond-α calculus

We present symmetric Rogers--Holder's inequalities on time scales when $frac{1}{p}+frac{1}{q}+frac{1}{r}=0$ and $frac{r}{p}+frac{r}{q}$ is not necessarily equal to $1$ where $p,$ $q$ and $r$ are nonzero real numbers.

Log-Like Functions and Uniform Distribution Modulo One

Abstract For a function f satisfying f (x) = o((log x) K ), K &gt; 0, and a sequence of numbers (qn ) n , we prove by assuming several conditions on f that the sequence (αf (qn )) n≥n 0 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.

On the extremal graphs for general sum-connectivity index ([formula omitted]) with given cyclomatic number when [formula omitted

Let V(G) and E(G) be, respectively, the vertex set and edge set of a graph G. The general sum-connectivity index of a graph G is denoted by χα(G) and is defined as ∑uv∈E(G)(du+dv)α, where uv is the edge connecting the vertices u,v∈V(G), du is the degree of the vertex u and α is a non-zero real number. The minimum number of edges of a graph G whose removal makes G as acyclic is known as the cyclomatic number and it is usually denoted by ν. In this paper, it is proved that the unique graph obtained from the star Sn by adding ν edge(s) between a fixed pendant vertex u and ν other pendant vertices, has the maximum χα value in the collection of all n-vertex connected graphs having cyclomatic number ν with the constraints ν=5, n≥6, α>1 or 6≤ν≤n−2, α≥2. It is also proved that only those graphs which consist of (only) vertices of degrees 2, 3, such that no two vertices of degree 3 are adjacent, have the minimum χα value among all n-vertex connected graphs (and also among all n-vertex connected molecular graphs) having cyclomatic number ν with the conditions ν≥3, n≥5(ν−1) and α>1.

General (α,2)-Path Sum-Connectivirty Indices of One Important Class of Polycyclic Aromatic Hydrocarbons

The general ( α , t ) -path sum-connectivity index of a molecular graph originates from many practical problems, such as the three-dimensional quantitative structure–activity relationships (3D QSAR) and molecular chirality. For arbitrary nonzero real number α and arbitrary positive integer t, it is defined as t χ α ( G ) = ∑ P t = v i 1 v i 2 ⋯ v i t + 1 ⊆ G [ d G ( v i 1 ) d G ( v i 2 ) ⋯ d G ( v i t + 1 ) ] α , where we take the sum over all possible paths of length t of G and two paths v i 1 v i 2 ⋯ v i t + 1 and v i t + 1 ⋯ v i 2 v i 1 are considered to be one path. In this work, one important class of polycyclic aromatic hydrocarbons and their structures are firstly considered, which play a role in organic materials and medical sciences. We try to compute the exact general ( α , 2 ) -path sum-connectivity indices of these hydrocarbon systems. Furthermore, we exactly derive the monotonicity and the extremal values of these polycyclic aromatic hydrocarbons for any real number α . These valuable results could produce strong guiding significance to these applied sciences.