Main Theorem Research Articles

A Method for Computing the Edge-Hosoya Polynomial with Application to Phenylenes

The edge-Hosoya polynomial of a graph is the edge version of the famous Hosoya polynomial. Therefore, the edge-Hosoya polynomial counts the number of (unordered) pairs of edges at distance k ≥ 0 in a given graph. It is well known that this polynomial is closely related to the edge-Wiener index and the edge-hyper-Wiener index. As the main result of this paper, we greatly generalize an earlier result by providing a method for calculating the edge-Hosoya polynomial of a graph G which is obtained by identifying two edges of connected bipartite graphs G1 and G2. To show how the main theorem can be used, we apply it to phenylene chains. In particular, we present the recurrence relations and a linear time algorithm for calculating the edge-Hosoya polynomial of any phenylene chain. As a consequence, closed formula for the edge-Hosoya polynomial of linear phenylene chains is derived.

Large deviations for the volume of k-nearest neighbor balls

This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of k-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of M0-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most k in a random geometric graph in the dense regime.

Fixed-point results via $ \alpha_{i}^{j} $-$ \left({\bf D}_{{\mathscr{C}}}\left(\mathfrak{P}_{\hat E}\right)\right) $-contractions in partial $ \flat $-metric spaces

<abstract><p>In this study, we characterize a novel contraction mapping referred to as $ \alpha_{i}^{j} $-$ \left({\bf D}_{{\mathscr{C}}}\left(\mathfrak{P}_{\hat E}\right)\right) $-contraction in light of $ {\bf D}_{\mathscr{C}} $-contraction mappings associated with the Geraghty-type contraction and $ E $-type contraction. Besides, a novel common fixed-point theorem providing such mappings is demonstrated in the context of partial $ \flat $-metric spaces. It is stated that the main theorem is a generalization of the existing literature, and its comparisons with the results are expressed. Additionally, the efficiency of the result of this study is demonstrated through some examples and an application to homotopy theory.</p></abstract>

Polynomial dynamics and local analysis of small and grand orbits

We prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role of the divisible group is taken by the small orbit of a point $\alpha$ where the small orbit by a polynomial $f$ is given by \begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*} Our main theorem is a classification of the algebraic relations that hold between infinitely many pairs of points in $\mathcal {S}_\alpha$ when everything is defined over the algebraic numbers and the degree $d$ of $f$ is at least 2. Our proof relies on a careful study of localisations of the dynamical system and follows an entirely different approach than previous proofs in this area. In particular, we introduce transcendence theory and Mahler functions into this field. Our methods also allow us to classify all algebraic relations that hold for infinitely many pairs of points in the grand orbit \begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*} of $\alpha$ if $|f^{\circ n}(\alpha )|_v \rightarrow \infty$ at a finite place $v$ of good reduction co-prime to $d$.

Generalizations of some $ q $-integral inequalities of Hölder, Ostrowski and Grüss type

<abstract><p>This paper investigates some well-known inequalities for $ q $-$ h $-integrals. These include Hölder, Ostrowski, Grüss and Opial type inequalities. Refinement of the Hadamard inequality for $ q $-$ h $-integrals is also established by applying the definition of strongly convex functions. From main theorems, $ q $-Hölder, $ q $-Ostrowski and $ q $-Grüss inequalities can be obtained in particular cases.</p></abstract>

Kazhdan constants and isomorphic graph pairs

Let G be a finite group, and let Γ be a subset of G. The Kazhdan constant of the pair (G,Γ) is defined to bethe maximum distance we can guarantee that an arbitrary unitvector in an arbitrary nontrivial irreducible unitary representation space of G can be moved by some element of Γ. The Kazhdanconstant relates to the expansion properties of the Cayley graph generated by G and Γ, and has been much studied in this context. Different pairs (G1,Γ1) and (G2,Γ2) may give rise to isomorphic Cayley graphs. In this paper, we investigate the question: To whatextent is the Kazhdan constant a graph invariant? In other words, if the pairs yield isomorphic Cayley graphs, must the corresponding Kazhdan constants be equal? In our main theorem, we constructan infinite family of such pairs where the Kazhdan constants areunequal. Other relevant results are presented as well.

Difference Analogues of Second Main Theorem and Picard Type Theorem for Slowly Moving Periodic Targets

In this paper, we show some Second main theorems for linearly nondegenerate meromorphic mappings over the field ????c1 of c-periodic meromorphic functions having their hyper-orders strictly less than one in ℂm intersecting slowly moving targets in ℙn(ℂ). As an application, we give some Picard type theorems for meromorphic mappings of ℂm into ℙn(ℂ) under the growth condition hyper-order less than one.

Towards coupled coincidence theorems of generalized admissible types of mappings on partial satisfactory cone metric spaces and some applications

<abstract><p>This paper introduces a novel class of generalized $ {\alpha} $-admissible contraction types of mappings in the framework of $ {\theta} $-complete partial satisfactory cone metric spaces and proves the existence and uniqueness of coincidence points for such mappings. In this setting, the topology generated and induced by the partial satisfactory cone metric is associated with semi-interior points rather than interior points of the underlying cone. In addition, some applications of the paper's main coincidence point theorems are given. The results of this paper unify, extend and generalize some previously proved theorems in this generalized setting.</p></abstract>

Equivariant Filters are Equivariant

Observers for systems with Lie group symmetries are an active area of research that is seeing significant impact in a number of practical domains, including aerospace, robotics, and mechatronics. This paper builds on the theory of the recently proposed Equivariant Filter (EqF), which is a general observer design for systems on homogeneous spaces that takes advantage of symmetries to yield significant performance advantages. It is shown that the EqF error dynamics are invariant to transformation of the input signal and equivariant as a parametrised vector field. The main theorem shows that two EqF's with different choices of local coordinates and origins and with equivalent noise modelling yield identical performance. In other words, the EqF is intrinsic to the system equations and symmetry. This is verified in a simulation of a 2D robot localisation problem, which also shows how the ability to choose an origin for the EqF can yield practical performance advantages by mitigating floating point precision errors.

Positivity analysis for mixed order sequential fractional difference operators

<abstract><p>We consider the positivity of the discrete sequential fractional operators $ \left(^{\rm RL}_{a_{0}+1}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ defined on the set $ \mathscr{D}_{1} $ (see (1.1) and <xref ref-type="fig" rid="Figure1">Figure 1</xref>) and $ \left(^{\rm RL}_{a_{0}+2}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ of mixed order defined on the set $ \mathscr{D}_{2} $ (see (1.2) and <xref ref-type="fig" rid="Figure2">Figure 2</xref>) for $ \tau\in\mathbb{N}_{a_{0}} $. By analysing the first sequential operator, we reach that $ \bigl(\nabla {f}\bigr)(\tau)\geqq 0, $ for each $ \tau\in{\mathbb{N}}_{a_{0}+1} $. Besides, we obtain $ \bigl(\nabla {f}\bigr)(3)\geqq 0 $ by analysing the second sequential operator. Furthermore, some conditions to obtain the proposed monotonicity results are summarized. Finally, two practical applications are provided to illustrate the efficiency of the main theorems.</p> </abstract>

Subharmonic solutions for degenerate periodic systems of Lotka-Volterra type with impulsive effects

<abstract><p>In this paper, we are concerned with the existence of subharmonic solutions for the degenerate periodic systems of Lotka-Volterra type with impulsive effects. In our degenerate model, the variation of the predator and prey populations may vanish on a time interval, which imitates the (real) possibility that the predation is seasonally absent. Our proof is based on the Poincaré-Birkhoff theorem. By using phase plane analysis, we can find the large gap in the rotation numbers between the "small" solutions and the "large" solutions, which guarantees a suitable twist property. By applying the Poincaré-Birkhoff theorem, we then obtain the existence of subharmonic solutions. Our main theorem extends the associated results by J. López-Gómez et al.</p></abstract>

The hyperbolic sets with the fitting shadowing property

We explain around C1-stable fitting shadowing property of a diffeomorphism f of C∞ (closed manifold M) denote by SFSP of invariant closed set B in M; we want to prove the main Theorem that f_(|B) satisfies a C^1- stable fitting shadowing property if and only if B is a hyperbolic set with the same index; if partially hyperbolic diffeomorphisms map contains q, r are types of saddle points with different indices, then f does not investigate the fitting shadowing property.

Approximate controllability of Hilfer fractional neutral stochastic systems of the Sobolev type by using almost sectorial operators

<abstract><p>The main aim of this work is to conduct an analysis of the approximate controllability of Hilfer fractional (HF) neutral stochastic differential systems under the condition of an almost sectorial operator with delay. The theoretical ideas linked to stochastic analysis, fractional calculus and semigroup theory, along with the fixed-point technique, are utilized to establish the key results of this article. More precisely, the main theorem of this study is devoted to proving the fact that the relevant linear system is approximately controllable. Finally, to help this research be as comprehensive as possible, we provide a theoretical application and filter system.</p></abstract>

Synchronization analysis of coupled fractional-order neural networks with time-varying delays.

In this paper, the complete synchronization and Mittag-Leffler synchronization problems of a kind of coupled fractional-order neural networks with time-varying delays are introduced and studied. First, the sufficient conditions for a controlled system to reach complete synchronization are established by using the Kronecker product technique and Lyapunov direct method under pinning control. Here the pinning controller only needs to control part of the nodes, which can save more resources. To make the system achieve complete synchronization, only the error system is stable. Next, a new adaptive feedback controller is designed, which combines the Razumikhin-type method and Mittag-Leffler stability theory to make the controlled system realize Mittag-Leffler synchronization. The controller has time delays, and the calculation can be simplified by constructing an appropriate auxiliary function. Finally, two numerical examples are given. The simulation process shows that the conditions of the main theorems are not difficult to obtain, and the simulation results confirm the feasibility of the theorems.

THEORETICAL AND NUMERICAL COMPUTATIONS OF CONVEXITY ANALYSIS FOR FRACTIONAL DIFFERENCES USING LOWER BOUNDEDNESS

This study focuses on the analytical and numerical solutions of the convexity analysis for fractional differences with exponential and Mittag-Leffler kernels involving negative and nonnegative lower bounds. In the analytical part of the paper, we will give a new formula for [Formula: see text] of the discrete fractional differences, which can be useful to obtain the convexity results. The correlation between the nonnegativity and negativity of both of the discrete fractional differences, [Formula: see text] with the convexity of the functions will be examined. In light of the main lemmas, we will define the two decreasing subsets of [Formula: see text], namely [Formula: see text] and [Formula: see text]. The decrease of these sets enables us to obtain the relationship between the negative lower bound of [Formula: see text] and the convexity of the function on a finite time set given by [Formula: see text] for some [Formula: see text] Besides, the numerical part of the paper is dedicated to examine the validity of the sets [Formula: see text] and [Formula: see text] in certain regions of the solutions for different values of [Formula: see text] and [Formula: see text]. For this reason, we will illustrate the domain of the solutions by means of several figures in which the validity of the main theorems are explained.

APPLICATION OF FIXED POINT THEOREM IN THE SOLUTION OF INTEGRO-DIFFERENTIAL EQUATIONS: A COMPLEX VALUED APPROACH

It's remarkable to note that Complex valued Integro-differential and integral type equations are currently intensifying the attention of appreciable researchers due to their comprehensive applications. Thus, this study is fully devoted to the application part of the complex valued controlled, double controlled metric ∂c. We introduce an extended version of the Fisher and Banach type contraction theorem and present some examples to sustain our results. As part of the main theorem's application, we address a common solution with uncertainty in two different folds as follows: [I] Applying the fractional Adams-Bashforth method to the (1.1) FVIdE. [II] Applying it to the integral type equation (1.2) in the setting of the Extended complex valued metric space.

Metric Basis of Four-Dimensional Klein Bottle

The Metric of a graph plays an essential role in the arrangement of different dimensional structures and finding their basis in various terms. The metric dimension of a graph is the selection of the minimum possible number of vertices so that each vertex of the graph is distinctively defined by its vector of distances to the set of selected vertices. This set of selected vertices is known as the metric basis of a graph. In applied mathematics or computer science, the topic of metric basis is considered as locating number or locating set, and it has applications in robot navigation and finding a beacon set of a computer network. Due to the vast applications of this concept in computer science, optimization problems, and also in chemistry enormous research has been conducted. To extend this research to a four-dimensional structure, we studied the metric basis of the Klein bottle and proved that the Klein bottle has a constant metric dimension for the variation of all its parameters. Although the metric basis is variying in 3 and 4 values when the values of its parameter change, it remains constant and unchanged concerning its order or number of vertices. The methodology of determining the metric basis or locating set is based on the distances of a graph. Therefore, we proved the main theorems in distance forms.

Best proximity points for (φ-ψ)-weak contractions and some applications

The principal goal of this paper is to express the existence and uniqueness of the best proximity point for a comprehensive contractive non-self mapping in partially ordered metric spaces. The main result covers a lot of former well-known theorems in related to best proximity point. Moreover, as an interesting application, integral versions of main theorem are obtained.

On the Perov's type $ (\beta, F) $-contraction principle and an application to delay integro-differential problem

<abstract><p>We present Perov's type $ (\beta, F) $-contraction principle and examine the fixed points of the self-operators satisfying Perov's type $ (\beta, F) $-contraction principle in the context of vector-valued $ b $-metrics. A specific instance of the $ (\beta, F) $-contraction principle is the $ F $-contraction principle. We generalize a number of recent findings that are already in the literature and provide an example to illustrate the hypothesis of the main theorem. We apply the obtained fixed point theorem to show the existence of the solution to the delay integro-differential problem.</p></abstract>

The explicit solution of a class of higher order impulsive fractional differential equation involving Hadamard fractional derivative

In this paper, we obtain the explicit solution of a class of singular impulsive fractional differential equation of order [Formula: see text] involving the Hadamard fractional derivatives. In particular, the result is given in the form of necessary and sufficient conditions about the explicit solution. The proof of the main theorem is obtained by converting boundary value problems into integral equations and applying analytical techniques. Finally, some examples explain the application of the main theorem in this paper.