Périodes locales et propagation de périodes dans un mot

On square-free arithmetic progressions in infinite words

The palindromic length $\text{PL}(v)$ of a finite word $v$ is the minimal number of palindromes whose concatenation is equal to $v$. In 2013, Frid, Puzynina, and Zamboni conjectured that: If $w$ is an infinite word and $k$ is an integer such that $\text{PL}(u)\leq k$ for every factor $u$ of $w$ then $w$ is ultimately periodic. Suppose that $w$ is an infinite word and $k$ is an integer such $\text{PL}(u)\leq k$ for every factor $u$ of $w$. Let $\Omega(w,k)$ be the set of all factors $u$ of $w$ that have more than $\sqrt[k]{k^{-1}\vert u\vert}$ palindromic prefixes. We show that $\Omega(w,k)$ is an infinite set and we show that for each positive integer $j$ there are palindromes $a,b$ and a word $u\in \Omega(w,k)$ such that $(ab)^j$ is a factor of $u$ and $b$ is nonempty. Note that $(ab)^j$ is a periodic word and $(ab)^ia$ is a palindrome for each $i\leq j$. These results justify the following question: What is the palindromic length of a concatenation of a suffix of $b$ and a periodic word $(ab)^j$ with "many" periodic palindromes? It is known that $\lvert\text{PL}(uv)-\text{PL}(u)\rvert\leq \text{PL}(v)$, where $u$ and $v$ are nonempty words. The main result of our article shows that if $a,b$ are palindromes, $b$ is nonempty, $u$ is a nonempty suffix of $b$, $\vert ab\vert$ is the minimal period of $aba$, and $j$ is a positive integer with $j\geq3\text{PL}(u)$ then $\text{PL}(u(ab)^j)-\text{PL}(u)\geq 0$.

The treatment of ventricular arrhythmia often requires detailed information about the location of ectopic beats. A noninvasive procedure is adopted to achieve this purpose. The aim of this work is the reconstruction of ectopic foci by using the critical point theory introduced by Greensite and Huiskamp [1]. The reliability and adaptability of the obtained simulation results are evaluated with regard to the reconstruction error.The reconstruction of bioelectrical sources from measured body surface potentials is an ill-posed problem and requires regularization. The advantage of the presented method is to deal with a well-posed formulation of the problem. Locations of ectopic beats can be detected by the critical point theorem. Four simulated ectopic centers have been localized to evaluate the method. The influence of Gaussian noise is considered.The reconstruction depends on the effective rank of a singular value decomposition (SVD) of the multi-channel ECG matrix. Regarding lower ranks, many critical points presenting ectopic foci can be observed. For higher ranks, the detection leads more and more to a stable estimation of ectopic locations. The detected critical points are shown to be reliable approximations of the simulated ectopic foci.Keywordsectopic beatslocalizing critical pointsinverse problem of electrocardiography

Autocorrelation Function of a Family of Complex Sequences

Multiplicity of solutions for nonlinear coercive problems

The aim of the contemporary variational theory is to transform the problem of searching the solution of equation or equation system into the problem of investigating the critical point of the corresponding energy function in a suitable space. This paper concerns the existence of solution for a singular quasilinear elliptic system involving two critical Sobolev Hardy exponents. Using Sobolev Hardy inequality, Ekeland’s variation principle and the critical point theorem, the existence of solution was proved under the certain conditions that the coefficients and nonlinear term of the equations meet.KeywordsNonlinear termCritical Sobolev-Hardy exponentElliptic equation systemsEkeland’s variational principle

A critical points approach for the existence of multiple solutions of a Dirichlet quasilinear system

The article surveys some decidability results for DPDAs on infinite words\n(omega-DPDA). We summarize some recent results on the decidability of the\nregularity and the equivalence problem for the class of weak omega-DPDAs.\nFurthermore, we present some new results on the parity index problem for\nomega-DPDAs. For the specification of a parity condition, the states of the\nomega-DPDA are assigned priorities (natural numbers), and a run is accepting if\nthe highest priority that appears infinitely often during a run is even. The\nbasic simplification question asks whether one can determine the minimal number\nof priorities that are needed to accept the language of a given omega-DPDA. We\nprovide some decidability results on variations of this question for some\nclasses of omega-DPDAs.\n

Integral rigid sets and periods of nonexpansive maps

The existence of at least three bounded weak solutions is established for a nonlinear elliptic equation with unbounded coefficients. The approach is based on variational methods and critical point theorems.

Three solutions for a perturbed Dirichlet boundary value problem involving the [formula omitted]-Laplacian

In this paper, we consider a class of fractional advection–dispersion equations involving instantaneous and non-instantaneous impulses. The existence of nontrivial solutions is established via Bonanno and D’Aguì’s critical point theorem. Under suitable conditions, we further prove the nonexistence of nontrivial solutions, which is the new result. Additionally, the application of our main results is demonstrated through two examples.

formula omitted]-Laplacian equations satisfying Cerami condition

Let L be the differential operator, Lu = <TEX>$u_{tt}+u_{xxxx}$</TEX>. We consider nonlinear beam equations, Lu + <TEX>$bu^+$</TEX> = j, in H, where H is the Hilbert space spanned by eigenfunctions of L. We reveal the existence of multiple solutions of the nonlinear beam problems by critical point theorems.

This paper is concerned with the existence of solutions for Sturm‐Liouville boundary value problem of a class of second‐order impulsive differential equations, under different assumptions on the nonlinearity and impulsive functions, existence criteria of single and multiple solutions are established. The main tools are variational method and critical point theorems. Some examples are also given to illustrate the main results.