Ambarzumian-type Problems for Discrete Schrödinger Operators

We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.

Sharp bounds for decomposing graphs into edges and triangles

For a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices. The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza (Studia Sci. Math. Hungar.22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput.28 (2019) 465–472) proved via flag algebras that $\pi _3^3(n) \le (1/2 + o(1)){n^2}$ . We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation

This manuscript develops a numerical approach for approximating the solution of the fractional Riccati differential equation (FRDE): $$\begin{align*}D^{\mu}&u(x)+a(x) u^2(x)+b(x) u(x)= g(x),\quad 0\leq \mu \leq 1,\quad 0\leq x \leq t,\\&u(0)=d,\end{align*}$$where u(x) is the unknown function, a(x), b(x) and g(x) are known continuous functions defined in [0,t] and d is a real constant. The proposed method is applied for solving the FRDE with shifted Chebyshev polynomials as basis functions. In addition, the convergence analysis of the suggested approach is investigated. The efficiency of the algorithm is demonstrated by means of several examples and the results compared with those given using other numerical schemes.

On the convergence of solutions to second-order neutral difference equations

A second-order nonlinear neutral difference equation with a quasi-difference is studied. Sufficient conditions are established under which for every real constant there exists a solution of the considered equation convergent to this constant.

Fundamental Solution of Elliptic Equation with Positive Definite Matrix Coefficient

<p>In this paper, we study the fundamental solution of elliptic equations with real constant coefficients </p><p class="Body">where is a positive definite matrix. We obtained by searching the radial solution so that we solved the equation into ordinary differential equations.</p><h1> </h1>

Nonclassical t-Dependent Energy Integral of q″+aq′+b(t)q+c(t)qn=0

In the present note, we address the general nonlinear oscillator of the form q″+aq′+b(t)q+c(t)qn=0, where q as well as the coefficients b(t) and c(t) are complex or real-valued functions of the real variable t; a is a real constant, and n is a real number. By starting with a certain Hamiltonian function, we discuss conditions on the functions b(t) and c(t) for which such differential equation becomes able to admit a nonclassical energy integral – i.e. a t-dependent energy integral. A method to inversely elicit the corresponding conservative solutions of this differential equation is consequently established, and two examples addressing periodic conservative solutions are shown.

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

We consider factorizations of a finite group [Formula: see text] into conjugate subgroups, [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is nilpotent or solvable. We derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group which, for a large class of groups, is linear in the non-solvable length of [Formula: see text]. We also show that every solvable group [Formula: see text] is a product of at most [Formula: see text] conjugates of a Carter subgroup [Formula: see text] of [Formula: see text], where [Formula: see text] is a positive real constant. Finally, using these results we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.

Bounded solutions of nonlinear discrete volterra equations

We give sufficient conditions, under which for every real constant, there exists a solution of the nonlinear discrete Volterra equationconvergent to this constant. We give also conditions under which all solutions are asymptotically constant. Sufficient conditions for the existence of asymptotically periodic solutions of the above equation are also derived.

Existence and multiplicity of solutions for nonlinear elliptic equations of p-Laplace type in RN

In this paper, we discuss the following elliptic equation: -div(?(x,?u)) = ?f (x,u) in RN, where the function ? : RN x RN ? RN is of type |v|p-2 v with a real constant p > 1 and f : RN x R ? R satisfies a Carath?odory condition.