Arbitrary Positive Constant Research Articles

Nonlocal combined nonlinear Schrödinger–Gerdjikov–Ivanov model: Integrability, Riemann–Hilbert problem with simple and double poles, Cauchy problem with step-like initial data

In this paper, we propose three new types of the integrable nonlocal combined nonlinear Schrödinger–Gerdjikov–Ivanov (NLS-GI) models. By the Riemann–Hilbert approach, we discuss the Cauchy problem of the reverse-space-time nonlocal combined NLS-GI model with step-like initial data: u(z, 0) = o(1) for z → −∞ and u(z, 0) = A + o(1) for z → +∞, where A is an arbitrary positive constant. First of all, we give an integrable nonlocal combined NLS-GI model and its Lax pair. Then, we consider the analytical and asymptotic behaviors, symmetries, and scattering matrix of the Jost solutions. Finally, we discuss the Cauchy problem for the nonlocal combined NLS-GI model with step-like initial data.

Global Low-Complexity Fault-Tolerant Control for Pure-Feedback Systems with Sensor Faults

A low-complexity global fault-tolerant control method is proposed to solve the tracking problem of uncertain pure-feedback systems in the presence of sensor faults. First, a novel modeling approach is introduced to reconstruct the non-affine term, which removes the restriction that the non-affine function must be differentiable. Second, a novel nonlinear mapping based on inverse-tangent function is utilized in the controller design such that the control parameters are free from initial values of states compared to the traditional prescribed performance control methods, resulting in global fault-tolerant control of pure-feedback systems under sensor and actuator faults. Furthermore, the designed global controller is low-complexity in the sense that no time derivatives of system signals are involved in the controller, and no neural networks or fuzzy logic systems are used, though unknown nonlinearities are present in the considered systems, and the control parameters are allowed to be arbitrary positive constants. Finally, the proposed method is applied to numerical and tailless fly-wing UAV examples, which fully demonstrates the effectiveness of the proposed method.

On fractional sums of the divisor functions

In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by [O. Bordellés, On certain sums of number theory, Int. J. Number Theory 18(9) (2022) 2053–2074; K. Liu, J. Wu and Z. S. Yang, On some sums involving the integral part function, preprint (2021); arXiv:2109.01382v1 [math.NT]]. Precisely, we can show that [Formula: see text] where [Formula: see text] is an arbitrary small positive constant and [Formula: see text] is the number of representations of [Formula: see text] as product of [Formula: see text] natural numbers.

Wave Function for GL(n, ℝ) Hyperbolic Sutherland Model

Abstract We obtain certain Mellin–Barnes integrals that present wave functions for $GL(n,{\mathbb {R}})$ hyperbolic Sutherland model with arbitrary positive coupling constant.

Online k-taxi via Double Coverage and time-reverse primal-dual

We consider the online k-taxi problem, a generalization of the k-server problem, in which k servers are located in a metric space. A sequence of requests is revealed one by one, where each request is a pair of two points, representing the start and destination of a travel request by a passenger. The goal is to serve all requests while minimizing the distance traveled without carrying a passenger. We show that the classic Double Coverage algorithm has competitive ratio 2^k-1 on HSTs, matching a recent lower bound for deterministic algorithms. For bounded depth HSTs, the competitive ratio turns out to be much better and we obtain tight bounds. When the depth is dll k, these bounds are approximately k^d/d!. By standard embedding results, we obtain a randomized algorithm for arbitrary n-point metrics with (polynomial) competitive ratio O(k^cDelta ^{1/c}log _{Delta } n), where Delta is the aspect ratio and cge 1 is an arbitrary positive integer constant. The previous known bound was O(2^klog n). For general (weighted) tree metrics, we prove the competitive ratio of Double Coverage to be Theta (k^d) for any fixed depth d, and in contrast to HSTs it is not bounded by 2^k-1. We obtain our results by a dual fitting analysis where the dual solution is constructed step-by-step backwards in time. Unlike the forward-time approach typical of online primal-dual analyses, this allows us to combine information from both the past and the future when assigning dual variables. We believe this method can also be useful for other problems. Using this technique, we also provide a dual fitting proof of the k-competitiveness of Double Coverage for the k-server problem on trees.

Structural Stability on the Boundary Coefficient of the Thermoelastic Equations of Type III

This paper investigates the spatial behavior of the solutions of thermoelastic equations of type III in a semi-infinite cylinder by using the partial differential inequalities. By setting an arbitrary positive constant in the energy expression, the fast decay rate of the solutions is obtained. Based on the results of decay, the continuous dependence and the convergence results on the boundary coefficient are established by using the differential inequality technique and the energy analysis method. The main work of this paper is to extend the study of continuous dependence to a semi-infinite cylinder, which can be used as a reference for the study of other types of partial differential equations.

Asymptotic confirmation of the Faudree–Lehel conjecture on irregularity strength for all but extreme degrees

AbstractThe irregularity strength of a graph , , is the least admitting a ‐weighting of the edges of assuring distinct weighted degrees of all vertices, or equivalently the least possible maximal edge multiplicity in an irregular multigraph obtained of via multiplying some of its edges. The most well‐known open problem concerning this graph invariant is the conjecture posed in 1987 by Faudree and Lehel that there exists a constant such that for each ‐regular graph with vertices and (while a straightforward counting argument yields ). The best known results towards this imply that for every ‐regular graph with vertices and , while if . We show that the conjecture of Faudree and Lehel holds asymptotically in the cases when is neither very small nor very close to . We in particular prove that for large enough and , , and thereby we show that then. We moreover prove the latter to hold already when , where is an arbitrary positive constant.

Predefined-time synchronization of competitive neural networks

In this paper, the predefined-time synchronization of competitive neural networks (CNNs) is researched based on two different predefined-time stability theorems. In view of the bilayer structure of CNNs, we design two bilayer predefined-time controllers. The first controller utilizes sign function, while the second controller utilizes exponential function and Lyapunov function. In these two controllers, the predefined time is set as a controller parameter, and it can be an arbitrary positive constant. Under these two controllers, the considered CNNs can achieve synchronization within the predefined time regardless of the initial values. A specific example is presented to validate the theoretical results.

A PDE Construction of the Euclidean Phi ^4_3 Quantum Field Theory

We present a new construction of the Euclidean Phi ^4 quantum field theory on {mathbb {R}}^3 based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on {mathbb {R}}^3 defined on a periodic lattice of mesh size varepsilon and side length M. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as varepsilon rightarrow 0, M rightarrow infty . Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder–Schrader axioms for a Euclidean QFT apart from rotation invariance and clustering. Our argument applies to arbitrary positive coupling constant, to multicomponent models with O(N) symmetry and to some long-range variants. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson–Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a distribution on the space of Euclidean fields.

Long-time asymptotics for the nonlocal nonlinear Schrödinger equation with step-like initial data

We study the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equationiqt(x,t)+qxx(x,t)+2q2(x,t)q¯(−x,t)=0 with a step-like initial data: q(x,0)=q0(x), where q0(x)=o(1) as x→−∞ and q0(x)=A+o(1) as x→∞, with an arbitrary positive constant A>0. The main aim is to study the long-time behavior of the solution of this problem. We show that the asymptotics has qualitatively different form in the quarter-planes of the half-plane −∞<x<∞, t>0: (i) for x<0, the solution approaches a slowly decaying, modulated wave of the Zakharov-Manakov type; (ii) for x>0, the solution approaches the “modulated constant”. The main tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem.

Time-Universal Data Compression

Nowadays, a variety of data-compressors (or archivers) is available, each of which has its merits, and it is impossible to single out the best ones. Thus, one faces the problem of choosing the best method to compress a given file, and this problem is more important the larger is the file. It seems natural to try all the compressors and then choose the one that gives the shortest compressed file, then transfer (or store) the index number of the best compressor (it requires log m bits, if m is the number of compressors available) and the compressed file. The only problem is the time, which essentially increases due to the need to compress the file m times (in order to find the best compressor). We suggest a method of data compression whose performance is close to optimal, but for which the extra time needed is relatively small: the ratio of this extra time and the total time of calculation can be limited, in an asymptotic manner, by an arbitrary positive constant. In short, the main idea of the suggested approach is as follows: in order to find the best, try all the data compressors, but, when doing so, use for compression only a small part of the file. Then apply the best data compressors to the whole file. Note that there are many situations where it may be necessary to find the best data compressor out of a given set. In such a case, it is often done by comparing compressors empirically. One of the goals of this work is to turn such a selection process into a part of the data compression method, automating and optimizing it.

Continuum Electromechanical Theory for Nematic Continua with Application to Freedericksz Instability

Of interest in this work are nematic continua that exhibit electromechanical coupling. The first part of this paper presents a novel variational formulation with a potential energy depending on four independent variables (the displacement, director, specific polarization and electric displacement perturbation). Variations of the potential energy with respect to each one of these variables lead to the governing mechanical equilibrium and constitutive relations plus Maxwell’s equations. The proposed variational formulation is next applied to the study of bifurcation of an infinite layer of a nematic liquid crystal confined between two parallel plates and subjected to a uniform electric field perpendicular to these plates under full anchoring boundary conditions. As the electric field exceeds a critical value, the nematic directors which are initially parallel to the plates, rotate and tend to align with the electric field orientation. This phenomenon, termed in the literature as Freedericksz transition, is treated here as a bifurcation problem using a fully 2D formulation. It is shown that the solution corresponding to the lowest applied electric field, also termed the critical load, is uniform in the direction parallel to the plates and that the corresponding bifurcated path is stable near this critical load. This result holds for arbitrary positive constants of the Frank-Oseen energy (and values of electric susceptibility constants that allow bifurcation) and justifies the 1D treatment of the Freedericksz transition in 2D settings that is widely adopted in the liquid crystal literature. An asymptotic analysis of the supercritical, stable bifurcated equilibrium path about the critical load is also presented and compared with the exact bifurcated solution.

Two-dimensional range successor in optimal time and almost linear space

In this article, we revisit the problem of supporting two-dimensional range successor queries. We present a data structure with O(nlg⁡lg⁡n) words of space and O(lg⁡lg⁡n) query time. This improves the work of Nekrich and Navarro (2012) by a factor of lg⁡lg⁡n in query time, or a factor of lgϵ⁡n in space cost, where ϵ is an arbitrary positive constant. Our data structure matches the state of the art for two-dimensional range emptiness queries and achieves the optimal query time. This structure has fruitful applications in computational geometry and text indexing.

Periodically intermittent controlling for finite-time synchronization of complex dynamical networks

In this paper, we consider finite-time synchronization between two complex dynamical networks using periodically intermittent control. Based on finite-time stability theory, some novel and effective finite-time synchronization criteria are derived by applying stability analysis technique. The derivative of the Lyapunov function $$V(t)$$ is smaller than $$\beta V(t)$$ ( $$\beta $$ is an arbitrary positive constant) when no controllers are added into networks. This means that networks can be self-synchronized without control inputs. As a result, the application scope of synchronization is greatly enlarged. Finally, a numerical example is given to verify the effectiveness and correctness of the synchronization criteria.

Approximation scheme for burst scheduling with minimum overhead in time slicing mobile TV

In this paper, we consider the problem of minimizing switching overhead of burst scheduling in time slicing mobile TV broadcast systems. This problem was formulated by Hsu and Hefeeda, and was given an elegant approximation scheme with an approximation ratio of at least two. Our proposed scheme significantly improves the approximation ratio of the previous scheme. More concretely, it generates a burst schedule for mobile TV systems whose switching overhead is at most $$(1/\ln 2)+\epsilon \simeq 1.4427$$ ( 1 / ln 2 ) + ∈ ? 1.4427 times of optimum, where $$\ln $$ ln is the natural logarithm and $$\epsilon $$ ∈ is arbitrary positive constant. Our scheme is a combination of a binary partion of burst cycles and a shifting method which is commonly used for the analysis of approximation algorithms designed for unit disk graphs.

Asymptotic of some integral

Consider an integral I(s) := ∫ b 0 e−s(x 2−icx)x2dx, where c > 0 and b > 0 are arbitrary positive constants. It is proved that I(s) ∼ −2i s3c3 as s→ +∞. Possible applications of this result to the Pompeiu problem are outlined. MSC: 41A60

Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme

In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Euler discretization with $N$ steps is smaller than $O(N^{-2/3+\varepsilon})$ where $\varepsilon$ is an arbitrary positive constant. This rate is intermediate between the strong error estimation in $O(N^{-1/2})$ obtained when coupling the stochastic differential equation and the Euler scheme with the same Brownian motion and the weak error estimation $O(N^{-1})$ obtained when comparing the expectations of the same function of the diffusion and of the Euler scheme at the terminal time $T$. We also check that the supremum over $t\in[0,T]$ of the Wasserstein distance on the space of probability measures on the real line between the laws of the diffusion at time $t$ and the Euler scheme at time $t$ behaves like $O(\sqrt{\log(N)}N^{-1})$.

On Randomized Approximation for Finding a Level Ideal of a Poset and the Generalized Median Stable Matchings

This study is concerned with finding a level ideal (LI) of a partially ordered set (poset). Given a finite poset P, the level of each element p ∈ P is defined as the number of ideals that do not include p, then the problem is to find the ith LI–the ideal consisting of elements whose levels are less than a given integer i. The concept of a level ideal is naturally derived from the generalized median stable matchings, introduced by Teo and Sethuraman [Teo, C. P., J. Sethuraman. 1998. The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4) 874–891] in the context of “fairness” of matchings in a stable marriage problem. Cheng [Cheng, C. T. 2010. Understanding the generalized median stable matchings. Algorithmica 58(1) 34–51] showed that finding the ith LI is #P-hard when i = Θ(N), where N is the total number of ideals of P. This paper shows that finding the ith LI is #P-hard even if i = Θ(N1/c), where c is an arbitrary constant at least one. Meanwhile, we present a polynomial time exact algorithm when i = O((log N)c′), where c′ is an arbitrary positive constant. We also devise two randomized approximation schemes for the ideals of a poset, by using an oracle of an almost-uniform sampler.

Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size

For graph G, let bw(G) denote the branchwidth of G and gm(G) the largest integer g such that G contains a g×g grid as a minor. We show that bw(G)≤3 gm(G) for every planar graph G. This is an improvement over the bound bw(G)≤4 gm(G) due to Robertson, Seymour and Thomas. Our proof is constructive and implies quadratic time constant-factor approximation algorithms for planar graphs for both problems of finding a largest grid minor and of finding an optimal branch-decomposition: (3+ϵ)-approximation for the former and (2+ϵ)-approximation for the latter, where ϵ is an arbitrary positive constant. We also study the tightness of the above bound. We show that for any constant c<2, the bound of ${\mathop {\mathrm {bw}}}(G)\leq c\; {\mathop {\mathrm {gm}}}(G) + o({\mathop {\mathrm {gm}}}(G))$ does not hold in general for a planar graph G.

Interaction Between Tachyon and Hessence (or Hantom) Dark Energies

In this work, we have considered that the universe is filled with tachyon and hessence (or hantom) dark energies. Subsequently we have investigated the interactions between tachyon and hessence (hantom) dark energies and calculated the potentials considering the power law form of the scale factor. It has been revealed that the tachyonic potential always decreases and hessence (or hantom) potential increases with corresponding fields. Furthermore, we have considered a correspondence between the hessence (or hantom) dark energy density and variable modified Chaplygin gas energy density. From this, we have found the expressions of the arbitrary positive constants B 0 and C of variable modified Chaplygin gas.