Communications in Mathematics |

In the present paper, seventh order Caudrey-Dodd-Gibbon (CDG) equation is solved by Lie symmetry analysis. All the geometry vector fields of seventh order KdV equations are presented. Using Lie transformation seventh order CDG equation is reduced into ordinarydifferential equations. These ODEs are solved by power series method to obtain exact solution. The convergence of the power series is also discussed.

For a fixed integer $k$, we define the multiplicative function\[D_{k,\omega}(n) := \frac{d(n)}{k^{\omega(n)}}, \]where $d(n)$ is the divisorfunction and $\omega (n)$ is the number of distinct prime divisors of $n$. Themain purpose of this paper is the study of the mean value of the function$D_{k,\omega}(n)$ by using elementary methods.

In this paper we obtain asymptotic expansion for the geometric mean of thevalues of positive strongly multiplicative function $f$ satisfying$f(p)=\alpha(d)\,p^d+O(p^{d-\delta})$ for any prime $p$ with $d$ real and$\alpha(d),\delta>0$.

In the present paper we consider a family of non-Volterra quadraticstochastic operators depending on a parameter $\alpha$ and study theirtrajectory behaviors. We find all fixed points for a non-Volterra quadraticstochastic operator on a finite-dimensional simplex. We construct some Lyapunovfunctions. A complete description of the set of limit points is given, and weshow that such operators have the ergodic property.

The main objective of the present paper is to introduce and study thefunction $_pR_q(A, B; z)$ with matrix parameters and investigate theconvergence of this matrix function. The contiguous matrix function relations,differential formulas and the integral representation for the matrix function$_pR_q(A, B; z)$ are derived. Certain properties of the matrix function$_pR_q(A, B; z)$ have also been studied from fractional calculus point of view.Finally, we emphasize on the special cases namely the generalized matrix$M$-series, the Mittag-Leffler matrix function and its generalizations and somematrix polynomials.

We introduce maximal and average coherence on lattices by analogy with thesenotions on frames in Euclidean spaces. Lattices with low coherence can be ofinterest in signal processing, whereas lattices with high orthogonality defectare of interest in sphere packing problems. As such, coherence andorthogonality defect are different measures of the extent to which a latticefails to be orthogonal, and maximizing their quotient (normalized for thenumber of minimal vectors with respect to dimension) gives lattices withparticularly good optimization properties. While orthogonality defect is afairly classical and well-studied notion on various families of lattices,coherence is not. We investigate coherence properties of a nice family ofalgebraic lattices coming from rings of integers in cyclotomic number fields,proving a simple formula for their average coherence. We look at some examplesof such lattices and compare their coherence properties to those of thestandard root lattices.

In this paper, we study regularity of weak solutions to the incompressibleNavier-Stokes equations in $\mathbb{R}^{3}\times (0,T)$. The main goal is toestablish the regularity criterion via the gradient of one velocity componentin multiplier spaces.

Recall that a ring R is called strongly pi-regular if, for every a in R,there is a positive integer n, depending on a, such that a^n belongs to theintersection of a^{n+1}R and Ra^{n+1}. In this paper we give a further study ofthe notion of a strongly pi-star-regular ring, which is the star-version ofstrongly pi-regular rings and which was originally introduced by Cui-Wang in J.Korean Math. Soc. (2015). We also establish various properties of these ringsand give several new characterizations in terms of (strong) pi-regularity andinvolution. Our results also considerably extend recent ones in the subject dueto Cui-Yin in Algebra Colloq. (2018) proved for pi-star-regular rings and dueto Cui-Danchev in J. Algebra Appl. (2020) proved for star-periodic rings.

We consider Complex Ginzburg-Landau equations with a polynomial nonlinearityin the real line. We use splitting-methods to prove well-posedness for a subsetof almost periodic spaces. Specifically, we prove that if the initial conditionhas multiples of an irrational phase, then the solution of the equationmaintains those same phases.

In this paper we use Faa di Bruno's formula to associate Bell polynomialvalues to differential equations of the form $y^{\prime}=f(y)$. That is, we usepartial Bell polynomials to represent the solution of such an equation and usethe solution to compute special values of partial Bell polynomials.

In this paper, we consider p-biharmonic submanifolds of aspace form. We givethe necessary and sufficient conditions for a submanifold to be p-biharmonic ina space form. We present some new properties for the stress p-bienergy tensor.

Without imposing any conditions on dependence structure, we give a seemingly overlooked simple sufficient condition for $L^{p}$ random variables $X_{1}, X_{2}, \dots$ with given $1 \leq p \leq +\infty$ to satisfy \[\frac{1}{a_{n}}\sum_{i=1}^{b_{n}}(X_{i} - \mathbb{E} X_{i}) \overset{L^{p}}\to 0 \,\,\, \mathrm{as}\, n \to \infty,\]where $(a_{n})_{n \in \mathbb{N}}, (b_{n})_{n \in \mathbb{N}}$ are prespecified unbounded sequences of positive integers.Some unexpected convergences of sample means follow.

This paper, we define the Mus-Gradient metric on tangent bundle $TM$ by adeformation non-conform of Sasaki metric over an n-dimensional Riemannianmanifold $(M, g)$. First we investigate the geometry of the Mus-Gradient metricand we characterize a new class of proper biharmonic maps. Examples of properbiharmonic maps are constructed when all of the factors are Euclidean spaces.

Let $\mathscr{R}$ be a prime ring of Char$(\mathscr{R}) \neq 2$ and $m\neq 1$be a positive integer. If $S$ is a nonzero skew derivation with an associatedautomorphism $\mathscr{T}$ of $\mathscr{R}$ such that $([S([a, b]), [a,b]])^{m} = [S([a, b]), [a, b]]$ for all $a, b \in \mathscr{R}$, then$\mathscr{R}$ is commutative.

In this work, we determined the general terms of all almost balancing numbersof first and second type in terms of balancing numbers and conversely wedetermined the general terms of all balancing numbers in terms of all almostbalancing numbers of first and second type. We also set a correspondencebetween all almost balancing numbers of first and second type and Pell numbers.

We introduce poly-Bernoulli polynomials in two variables by using ageneralization of Stirling numbers of the second kind that we studied in aprevious work. We prove the bi-variate poly-Bernoulli polynomial version ofsome known results on standard Bernoulli polynomials, as the addition formulaand the binomial formula. We also prove a result that allows us to obtainpoly-Bernoulli polynomial identities from polynomial identities, and we usethis result to obtain several identities involving products of poly-Bernoulliand/or standard Bernoulli polynomials. We prove two generalized recurrences forbi-variate poly-Bernoulli polynomials, and obtain some corollaries from them.

In this work, approximate analytic solutions for different types of KdV equations are obtained using the homotopy analysis method (HAM). The convergence control parameter h helps us to adjust the convergence region of the approximate analytic solutions. The solutions are obtained in the form of power series. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. We have compared the approximate analytical results which are determined by HAM, with the exact solutions and shown graphically with their absolute errors. By choosing an appropriate value of the convergence control parameter, we can obtain the solution in few iterations. All the computations have been performed using the software package MATHEMATICA.

In this work, we determine the general solution of the quinqueviginticfunctional equation and also investigate its stability of this equation in thesetting of matrix normed spaces and the framework of matrix non-Archimedeanfuzzy normed spaces by using the fixed point method.

We consider the following Schrödinger-Bopp-Podolsky system in $\mathbbR^{3}$ $$\left\{ \begin{array}{c} -\varepsilon^{2} \Delta u + V(x)u + \phi u =f(u)\\ -\varepsilon^{2} \Delta \phi + \varepsilon^{4} \Delta^{2}\phi =4\pi\varepsilon u^{2}\\ \end{array} \right.$$ where $\varepsilon > 0$ with $ V:\mathbb{R}^{3} \rightarrow \mathbb{R},f:\mathbb{R} \rightarrow \mathbb{R}$ satisfy suitable assumptions. By usingvariational methods, we prove that the number of positive solutions isestimated below by the Ljusternick-Schnirelmann category of $M$, the set ofminima of the potential $V$.

Let $n$ be a positive integer and let $C_n$ be the cycle indicator of thesymmetric group $S_n$. Carlitz proved that if $p$ is a prime, and if $r$ is anon negative integer, then we have the congruence $C_{r+np}\equiv(X_1^p-X_p)^nC_r \mod{pZ_p[X_1,\cdots,X_{r+np}]},$ where $Z_p$ is the ring of $p$-adicintegers. We prove that for $p\neq 2$, the preceding congruence holds modulo$npZ_p[X_1,\cdots,X_{r+np}]$. This allows us to prove a Junod's conjecture forMeixner polynomials.

In this paper we recall some results and some criteria on the convergence ofmatrix continued fractions. The aim of this paper is to give some propertiesand results of continued fractions with matrix arguments. Then we givecontinued fraction expansions of the error function erf(A) where A is a matrix.At the end, some numerical examples illustrating the theoretical results arediscussed.

This paper studies the stationary analysis of a Markovian queuing system with Bernoulli feedback, interruption vacation, linear impatient customers, strong and weak disaster with the server's repair during the server's operational vacation period. Each customer has its own impatience time and abandons the system as soon as that time ends. When the queue is not empty, the server's operational vacation can be interrupted if the service is completed and the server starts a busy period with a probability q or continues the operational vacation with a probability q. A strong disaster forces simultaneously all present customers (waiting and served) to abandon the system permanently with a probability p but a weak disaster is that all customers decide to be patient by staying in the system, and wait during the repair time with a probability p, where arrival of a new customer can occur. As soon as the repair process of the server is completed, the server remains providing service in the operational vacation period. We analyze this proposed model and derive the probabilities generating functions of the number of customers present in the system together with explicit expressions of some performance measures such as the mean and the variance of the number of customers in the different states, together with the mean sojourn time. Finally, numerical results are presented to show the influence of the system parameters on some studied performance measures.

Legendre curves play a very important and special role in geometry andtopology of almost contact manifolds.There are certain results known forLegendre curves in 3-dimensional normal almost contact manifolds. The aim ofthis paper is to study Legendre curves of three-dimensional $C_{12}$-manifoldswhich are non-normal almost contact manifolds and classifying all biharmonicLegendre curves in these manifolds.

It is well-known that the antipode $S$ of a commutative or cocommutative Hopfalgebra satisfies $S^{2}=\operatorname*{id}$ (where $S^{2}=S\circ S$).Recently, similar results have been obtained by Aguiar, Lauve and Mahajan forconnected graded Hopf algebras: Namely, if $H$ is a connected graded Hopfalgebra with grading $H=\bigoplus_{n\geq0}H_n$, then each positive integer $n$satisfies $\left( \operatorname*{id}-S^2\right)^n \left( H_n\right) =0$ and(even stronger) \[ \left( \left( \operatorname{id}+S\right) \circ\left(\operatorname{id}-S^2\right)^{n-1}\right) \left( H_n\right) = 0. \] For somespecific $H$'s such as the Malvenuto--Reutenauer Hopf algebra$\operatorname{FQSym}$, the exponents can be lowered. In this note, we generalize these results in several directions: We replacethe base field by a commutative ring, replace the Hopf algebra by a coalgebra(actually, a slightly more general object, with no coassociativity required),and replace both $\operatorname{id}$ and $S^2$ by "coalgebra homomorphisms" (ofsorts). Specializing back to connected graded Hopf algebras, we show that theexponent $n$ in the identity $\left( \operatorname{id}-S^2\right) ^n \left(H_n\right) =0$ can be lowered to $n-1$ (for $n>1$) if and only if $\left(\operatorname{id} - S^2\right) \left( H_2\right) =0$. (A sufficient conditionfor this is that every pair of elements of $H_1$ commutes; this is satisfied,e.g., for $\operatorname{FQSym}$.)

In this paper we introduce and study the theories of pseudo links andsingular links in the Solid Torus, ST. Pseudo links are links with some missingcrossing information that naturally generalize the notion of knot diagrams, andthat have potential use in molecular biology, while singular links are linksthat contain a finite number of self-intersections. We consider pseudo linksand singular links in ST and we set up the appropriate topological theory inorder to construct invariants for these types of links in ST. In particular, weformulate and prove the analogue of the Alexander theorem for pseudo links andfor singular links in ST. We then introduce the mixed pseudo braid monoid andthe mixed singular braid monoid, with the use of which, we formulate and provethe analogue of the Markov theorem for pseudo links and for singular links inST. \smallbreak Moreover, we introduce the pseudo Hecke algebra of type A,$P\mathcal{H}_n$, the cyclotomic and generalized pseudo Hecke algebras of typeB, $P\mathcal{H}_{1, n}$, and discuss how the pseudo braid monoid (cor. themixed pseudo braid monoid) can be represented by $P\mathcal{H}_{n}$ (cor. by$P\mathcal{H}_{1, n}$). This is the first step toward the construction ofHOMFLYPT-type invariants for pseudo links in $S^3$ and in ST. We also introducethe cyclotomic and generalized singular Hecke algebras of type B,$S\mathcal{H}_{1, n}$, and we present two sets that we conjecture that theyform linear bases for $S\mathcal{H}_{1, n}$. Finally, we […]

In this article we study the homology of nilpotent groups. In particular acertain vanishing result for the homology and cohomology of nilpotent groups isproved.

In this work, we consider the existence of global solution and theexponential decay of a nonlinear porous elastic system with time delay. Thenonlinear term as well as the delay acting in the equation of the volumefraction. In order to obtain the existence and uniqueness of a global solution,we will use the semigroup theory of linear operators and under a certainrelation involving the coefficients of the system together with a Lyapunovfunctional, we will establish the exponential decay of the energy associated tothe system.

Let $d(n)$ and $d^{\ast}(n)$ be the numbers of divisors and the numbers ofunitary divisors of the integer $n\geq1$. In this paper, we prove that \[\underset{n\in\mathcal{B}}{\underset{n\leq x}{\sum}}\frac{d(n)}{d^{\ast}%(n)}=\frac{16\pi% %TCIMACRO{\U{b2}}% %BeginExpansion {{}^2}% %EndExpansion}{123}\underset{p}{\prod}(1-\frac{1}{2p% %TCIMACRO{\U{b2}}% %BeginExpansion{{}^2}% %EndExpansion }+\frac{1}{2p^{3}})x+\mathcal{O}\left(x^{\frac{\ln8}{\ln10}+\varepsilon }\right) ,~\left(x\geqslant1,~\varepsilon>0\right) , \] where $\mathcal{B}$ is the set whichcontains any integer that is not a multiple of $5,$ but some permutations ofits digits is a multiple of $5.$

In this study we find all solutions of the Diophantine equation$B_{n_{1}}+B_{n_{2}}=2^{a_{1}}+2^{a_{2}}+2^{a_{3}}$ in positive integervariables $(n_{1},n_{2},a_{1},a_{2},a_{3}),$ where $B_{n}$ denotes the $n$-thbalancing number.

This survey is devoted to necessary and suffcient conditions for a rationalnumber to be representable by a Cantor series. Necessary and suffcientconditions are formulated for the case of an arbitrary sequence $(q_k)$.