On the spectrum of a multidimensional periodic magnetic Shrödinger operator with a singular electric potential

We prove absolute continuity of the spectrum of a periodic $n$-dimensional Schrödinger operator for $n\geqslant 4$. Certain conditions on the magnetic potential $A$ and the electric potential $V+\sum f_j\delta_{S_j}$ are supposed to be fulfilled. In particular, we can assume that the following conditions are satisfied. (1) The magnetic potential $A\colon{\mathbb{R}}^n\to{\mathbb{R}}^n$ either has an absolutely convergent Fourier series or belongs to the space $H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-1$, or to the space $C({\mathbb{R}}^n;{\mathbb{R}}^n)\cap H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-2$. (2) The function $V\colon{\mathbb{R}}^n\to\mathbb{R}$ belongs to Morrey space ${\mathfrak{L}}^{2,p}$, $p\in \big(\frac{n-1}{2},\frac{n}{2}\big]$, of periodic functions (with a given period lattice), and $$\lim\limits_{\tau\to+0}\sup\limits_{0<r\leqslant\tau}\sup\limits_{x\in{\mathbb{R}}^n}r^2\bigg(\big(v(B^n_r)\big)^{-1}\int_{B^n_r(x)}|{\mathcal{V}}(y)|^pdy\bigg)^{1/p}\leqslant C,$$ where $B^n_r(x)$ is a closed ball of radius $r>0$ centered at a point $x\in{\mathbb{R}}^n$, $B^n_r=B^n_r(0)$, $v(B^n_r)$ is volume of the ball $B^n_r$, $C=C(n,p;A)>0$. (3) $\delta_{S_j}$ are $\delta$-functions concentrated on (piecewise) $C^1$-smooth periodic hypersurfaces $S_j$, $f_j\in L^p_{\mathrm{loc}}(S_j)$, $j=1,\ldots,m$. Some additional geometric conditions are imposed on the hypersurfaces $S_j$, and these conditions determine the choice of numbers $p\geqslant n-1$. In particular, let hypersurfaces $S_j$ be $C^2$-smooth, the unit vector $e$ be arbitrarily taken from some dense set of the unit sphere $S^{n-1}$ dependent on the magnetic potential $A$, and the normal curvature of the hypersurfaces $S_j$ in the direction of the unit vector $e$ be nonzero at all points of tangency of the hypersurfaces $S_j$ and the lines $\{x_0+te\colon t\in\mathbb{R}\}$, $x_0\in{\mathbb{R}}^n$. Then we can choose the number $p>\frac{3n}{2}-3$, $n\geqslant 4$.

Fixed Point of Rational Contractions and Its Application for Secure Dynamic Routing in Wireless Sensor Networks

Security is one of the major concerns for data communication over wireless sensor networks (WSNs). Dynamic routing algorithms can provide small similarity paths of data delivery between two consecutive transmitted packets, improving data security without adding extra information or control messages. This article illustrates the iteration of the fixed point (FP) of rational contractions and generalized Banach contractions (BC) in the setting of F-metric space (F-MS). It also describes an FP of the said mappings, while restricting the imposition of the contraction only to a subset of the F-MS, the closed ball, rather than executing it on the entire F-MS. The results have been verified and supported by concise examples. Further, the application of the functional equation proved results with randomization is given to find a solution for secure dynamic routing of data transmission in WSNs. The application is a tool to analyze and model a network structure in which sensors can be deployed with high security and low risk in a greater region (sensor field), thus boosting the accuracy.

Crystal structures of a dodecameric multicopper oxidase from Marinithermus hydrothermalis

Multicopper oxidases (MCOs) represent a diverse family of enzymes that catalyze the oxidation of either an organic or a metal substrate with concomitant reduction of dioxygen to water. These enzymes contain variable numbers of cupredoxin domains, two, three or six per subunit, and rely on four copper ions, a single type I copper and three additional copper ions organized in a trinuclear cluster (TNC), with one type II and two type III copper ions, to catalyze the reaction. Here, two crystal structures and the enzymatic characterization of Marinithermus hydrothermalis MCO, a two-domain enzyme, are reported. This enzyme decolorizes Congo Red dye at 70°C in the presence of high halide concentrations and may therefore be useful in the detoxification of industrial waste that contains dyes. In two distinct crystal structures, MhMCO forms the trimers seen in other two-domain MCOs, but differs from these enzymes in that four trimers interact to create a dodecamer. This dodecamer of MhMCO forms a closed ball-like structure and has implications for the sequestration of bound divalent metal ions as well as substrate accessibility. In each subunit of the dodecameric structures, a Trp residue, Trp351, located between the type I and TNC sites exists in two distinct conformations, consistent with a potential role in facilitating electron transfer in the enzyme.

Some Fixed Point Results in Function Weighted Metric Spaces

After the establishment of the Banach contraction principle, the notion of metric space has been expanded to more concise and applicable versions. One of them is the conception of ℱ -metric, presented by Jleli and Samet. Following the work of Jleli and Samet, in this article, we establish common fixed points results of Reich-type contraction in the setting of ℱ -metric spaces. Also, it is proved that a unique common fixed point can be obtained if the contractive condition is restricted only to a subset closed ball of the whole ℱ -metric space. Furthermore, some important corollaries are extracted from the main results that describe fixed point results for a single mapping. The corollaries also discuss the iteration of fixed point for Kannan-type contraction in the closed ball as well as in the whole ℱ -metric space. To show the usability of our results, we present two examples in the paper. At last, we render application of our results.

On Unique and Nonunique Fixed Points and Fixed Circles in M v b -Metric Space and Application to Cantilever Beam Problem

We introduce M v b -metric to generalize and improve M v -metric and unify numerous existing distance notions. Further, we define topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology on M v b -metric space and to create an environment for the survival of a unique fixed point. Also, we introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map and establish fixed circle and fixed disc theorems. Further, we verify all these results by illustrative examples to demonstrate the authenticity of the postulates. Towards the end, we solve a fourth order differential equation arising in the bending of an elastic beam.

About Total Stability of a Class of Nonlinear Dynamic Systems Eventually Subject to Discrete Internal Delays

This paper studies and investigates total stability results of a class of dynamic systems within a prescribed closed ball of the state space around the origin. The class of systems under study includes unstructured nonlinearities subject to multiple higher-order Lipschitz-type conditions which influence the dynamics and which can be eventually interpreted as unstructured perturbations. The results are also extended to the case of presence of multiple internal (i.e., in the state) point discrete delays. Some stability extensions are also discussed for the case when the systems are subject to forcing efforts by using links between the controllability and stabilizability concepts from control theory and the existence of stabilizing linear controls. The results are based on the ad hoc use of Gronwall’s inequality.

Iterating Fixed Point via Generalized Mann’s Iteration in Convex b-Metric Spaces with Application

This manuscript investigates fixed point of single-valued Hardy-Roger’s type F -contraction globally as well as locally in a convex b -metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom (F3) of the F -contraction is removed, and thus the mapping F is relaxed. An important approach used in the article is, though a subset closed ball of a complete convex b -metric space is not necessarily complete, the convergence of the Cauchy sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations.

Fixed point theorems on a closed ball

The aim of the paper is to obtain some fixed point theorems for extended (ϕ, F)-weak type contraction on a closed ball in metric spaces. Our results generalize some recently established results.