Integral Version Research Articles

On some generalizations of nonlinear dynamic inequalities on time scales and their applications

In this article, by using Young's inequality, we prove some new nonlinear dynamic inequalities of Gronwall-Bellman-Pachpatte type on time scales. These inequalities give us the integral and discrete version, and also extend some known dynamic inequalities in certain papers. We can also say, the inequalities proved in this paper can be used as handy tools in the study of qualitative properties of some dynamic equations on time scales. Some examples are introduced to demonstrate the applications of these inequalities.

A CERTAIN DIRICHLET SERIES OF RANKIN–SELBERG TYPE ASSOCIATED WITH THE IKEDA LIFT OF HALF‐INTEGRAL WEIGHT

In this article we obtain an explicit formula for certain Rankin-Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler-Zagier-Ibukiyama correspondence. The integral weight version of the main theorem had been obtained by Katsurada and Kawamura. The result of the integral weight case is a product of $L$-function and Riemann zeta functions, while half-integral weight case is a infinite summation over negative fundamental discriminants with certain infinite products. To calculate explicit formula of such Rankin-Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.

The concept of coordinate strongly convex functions and related inequalities

In this paper, we introduce a new class of functions known as coordinate strongly convex functions. We discuss the relation between strongly convex functions and coordinate strongly convex functions. Also, we present some natural properties of coordinate strongly convex functions. We present Slater’s, Jensen’s and converse of the Jensen inequalities in discrete as well as integral versions for coordinate strongly convex functions. Furthermore, we present Hermite–Hadamard’s type inequalities for coordinate strongly convex functions.

A second-order in time and space particle-based method to solve flow problems on arbitrary meshes

This work presents a novel proposal of a second-order accurate (in time and space) particle-based method for solving transport equations including incompressible flows problems within a mixed Lagrangian–Eulerian formulation. This methodology consists of a symmetrical operator splitting, the use of high-order operators to transfer data between the particles and the background mesh, and an improved version of the eXplicit Integration Following the Streamlines (X–IVS) method. In the case of incompressible flows, a large time-step iterative solver is employed where the momentum equation is split to improve the numerical approximation of the convective term.New interpolation and projection operators are evaluated and quadratically accurate solutions of scalar transport tests are presented. Then, incompressible flow problems are solved where the rate of convergence of the method is assessed using both structured and unstructured background grids. The method is implemented in the open source platform OpenFOAM®allowing employing arbitrary meshes and obtaining reliable computing time comparisons with standardized solvers. The results obtained reveal that the current method is able to obtain a lower level of error than a fast Eulerian alternative, without increasing the total computing time.

Numerical schemes for ordinary delay differential equations with random noise

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) which have a stochastic process in their vector field functions. They have been used in a wide range of applications such as biology, medicine and engineering and play an important role in the theory of random dynamical systems. RODEs can be investigated pathwise as deterministic ODEs, however, the classical numerical methods for ODEs do not attain original order of convergence because the stochastic process has at most Hölder continuous sample paths and the resulting vector is also at most Hölder continuous in time. Recently, Jenzen & Kloeden derived new class of numerical methods for RODEs using integral versions of implicit Taylor-like expansions and developed arbitrary higher order schemes for RODEs. Their idea can be applied to random ordinary delay differential equations (RODDEs) by implementing Taylor-like expansions in the corresponding delay term. In this paper, numerical methods for RODDEs are systematically constructed based on Taylor-like expansions and they are applied to virus dynamics model with random fluctuations and time delay.

On the very accurate evaluation of the Voigt functions

Here we indicate how a particular version of the trapezoidal integration with residue correction for the evaluation of the Voigt functions indicated by us previously can be easily extended to serve the purpose of reference value generation. Specifically by resorting to quadruple precision, we demonstrate a reliable computation with a minimum of 30 digit accuracy for the Voigt functions H(x, a) and G(x, a). Compared to the prescription of Boyer et al., this method reduces the computational cost several fold when very high reference accuracy is the aim. In addition we show that when both the parameters x and a are less than unity, the series of Faddeeva et al. can be profitably used for the required evaluation instead of this quadrature.

A unified treatment of Hilbert–Pachpatte-type inequalities for a class of non-homogeneous kernels

In the present paper we establish a unified treatment of Hilbert–Pachpatte-type inequalities for a class of non-homogeneous kernels. Our results are derived in both discrete and integral versions. A particular emphasis is devoted to constants and weight functions appearing on the right-hand sides of the established inequalities. As an application, we obtain inequalities with constants and weight functions expressed in terms of generalized harmonic numbers, the incomplete Beta and Gamma function, and the logarithmic integral function.

Isogeometric analysis of Mindlin nanoplates based on the integral formulation of nonlocal elasticity

Purpose It has been revealed that application of the differential form of Eringen’s nonlocal elasticity theory to some cases (e.g. cantilevers) leads to paradoxical results, and recourse must be made to the integral version of Eringen’s nonlocal model. The purpose of this paper, within the framework of integral form of Eringen’s nonlocal theory, is to study the bending behavior of nanoscale plates with various boundary conditions using the isogeometric analysis (IGA). Design/methodology/approach The shear deformation effect is taken into account according to the Mindlin plate theory, and the minimum total potential energy principle is utilized in order to derive the governing equations. The relations are obtained in the matrix-vector form which can be easily employed in IGA or finite element analysis. For the comparison purpose, the governing equations are also derived based on the differential nonlocal model and are then solved via IGA. Comparisons are made between the predictions of integral nonlocal model, differential nonlocal model and local (classical) model. Findings The bending analysis of nanoplates under some kinds of edge supports indicates that using the differential model leads to paradoxical results (decreasing the maximum deflection with increasing the nonlocal parameter), whereas the results of integral model are consistent. Originality/value A new nonlocal formulation is developed for the IGA of Mindlin nanoplates. The nonlocal effects are captured based on the integral model of nonlocal elasticity. The formulation is developed in matrix-vector form which can be readily used in finite element method. Comparisons are made between the results of differential and integral models for the bending problem. The proposed integral model is capable of resolving the paradox appeared in the results of differential model.

Maggi-Rubinowicz integral for Fraunhofer diffraction by hexagonal particles: application to cirrus ice crystals.

This paper is devoted to Fraunhofer diffraction by hexagonal ice crystals like those that can be encountered in cirrus clouds. Such crystals are important, for they are the basic building elements of crystals of more complex shape. A new alternative based on the Maggi-Rubinowicz transformation is proposed. This technique, whose basic formula is a line integral version of the Kirchhoff integral, is very easy to implement and is not time consuming. Diffraction patterns of hexagonal ice crystals with (i)fixed arbitrary orientation, (ii)random orientation, and (iii)preferred orientation are given. The topic is of importance since the radiative impact of cirrus clouds, which depends directly on ice crystal optical properties, is still a challenging problem in climate research.

Partial zeta values, Gross's tower of fields conjecture, and Gross–Stark units

We prove a conjecture of Gross regarding the “order of vanishing” of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross–Stark units. This is achieved by introducing an integral version of the Eisenstein cocycle.

Hermite-Hadamard type inequalities pertaining conformable fractional integrals and their applications

In this paper, our main aim is to give results for conformable fractional integral version of Hermite-Hadamard inequality and their applications for mid-point formula and means. First, we prove an identity associated with the Hermite-Hadamard inequality for conformable fractional integrals. By using this identity and convexity of different classes of functions and some well-known inequalities, we obtain several results for the inequality. At the end, we also present applications for some error estimations of the mid-point formula and means. The results obtained in this paper generalize the earlier results. Furthermore, the identity can be used to obtain results for Hermite-Hadamard inequality for some other generalized classes of convex functions.

Trajectory tracking for quadrotor UAV transporting cable-suspended payload in wind presence

In this paper, trajectory tracking for a quadrotor unmanned aerial vehicle (UAV) is considered in the presence of a cable-suspended payload and wind as unknown disturbances. It is assumed that the wind disturbance is slowly time varying and affects quadrotor position and orientation independently. Nonlinear robust strategies, such as backstepping and sliding mode control could be used for trajectory tracking; however, they fail to stabilize the system in the presence of payload or wind, or both together. We have proposed a combined backstepping and super-twisting integral sliding mode strategy to stabilize the system. Conventional sliding mode control suggests discontinuous control signals and suffers from the chattering phenomenon whereas its super-twisting integral version suggests continuous control signals, which makes it implementable and chattering free.

Wokół lwowskiej edycji Pokoju na poddaszu Wandy Wasilewskiej (1940)

The article revolves around the reedition of a youth novel A Room in the Attic (Pokój na poddaszu, 1939) by Wanda Wasilewska, published under the Soviet occupation of Lviv by the Kiev-Lviv National Minorities State Publishing USSR in 1940. It attempts to place it on the background of the entirety of the modest release production of this publishing house in the area of belles-lettres in the years 1940-1941 (among 181 published books only 26 literary positions, including only 7 from Polish literature), pointing most of all to the changes made by the Soviet censorship in regards to the pre-war original (especially the elimination of all mentions of Poland and liberation fights, pejorative descriptions of Russian people in the context of the remembering of 1905 revolution, mentions of religion, the words “pan” and “pani” (lord and lady), and - by adding a few sentences not present in the original - bolstering the “class” context of the text and changing the expression of the ending from moderately optimistic to pessimistic). The post-war reeditions of the novel by Wasilewska in PPR (from 1946-1988) restored the pre-war integral version of the text, rendering the cut-down and changed Lviv version invalid, and making it a documentation of its times.

Integral versions of input-to-state stability for dual-rate nonlinear sampled-data systems

This paper presents versions of integral input-to-state stability and integral input-to-integral-state stability for nonlinear sampled-data systems, under the low measurement rate constraint. In particular, we compensate the lack of measurements using an estimator approximately reconstructing the current state. Interestingly, under certain checkable conditions, we establish that a controller that semiglobally practically integral input-to-(integral-)state stabilizes an approximate discrete-time model of a single-rate nonlinear sampled-data system, also stabilizes the exact discrete-time model of the nonlinear sampled-data system in the same sense implemented in a dual-rate setting. Numerical simulations are given to illustrate the effectiveness of our results.

Quantitative C1-estimates by Bismut formulae

For a C2 function u and an elliptic operator L, we prove a quantitative estimate for the derivative du in terms of local bounds on u and Lu. An integral version of this estimate is then used to derive a condition for the zero-mean value property of Δu. An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings.

Bending of Euler–Bernoulli nanobeams based on the strain-driven and stress-driven nonlocal integral models: a numerical approach

Eringen’s nonlocal elasticity theory is extensively employed for the analysis of nanostructures because it is able to capture nanoscale effects. Previous studies have revealed that using the differential form of the strain-driven version of this theory leads to paradoxical results in some cases, such as bending analysis of cantilevers, and recourse must be made to the integral version. In this article, a novel numerical approach is developed for the bending analysis of Euler–Bernoulli nanobeams in the context of strain- and stress-driven integral nonlocal models. This numerical approach is proposed for the direct solution to bypass the difficulties related to converting the integral governing equation into a differential equation. First, the governing equation is derived based on both strain-driven and stress-driven nonlocal models by means of the minimum total potential energy. Also, in each case, the governing equation is obtained in both strong and weak forms. To solve numerically the derived equations, matrix differential and integral operators are constructed based upon the finite difference technique and trapezoidal integration rule. It is shown that the proposed numerical approach can be efficiently applied to the strain-driven nonlocal model with the aim of resolving the mentioned paradoxes. Also, it is able to solve the problem based on the strain-driven model without inconsistencies of the application of this model that are reported in the literature.

The deformation fields method revisited: Stable simulation of instationary viscoelastic fluid flow using integral models

The implementation of the deformation fields method for integral models within a finite element context [1,2] has been updated with various techniques to have a numerical stability that is comparable to state-of-the-art implementations of differential models. In particular, the time-dependent stability in shear flow, decoupled schemes for zero or small solvent viscosities and the log-conformation representation now have counterparts in the numerical implementation of integral models leading to similar numerical stability. The new techniques have been tested in transient shear flow and the flow around a cylinder confined between two plates for the integral version of upper-convected Maxwell model and for integral models having a non-constant damping function.

Topological polylogarithms and p-adic interpolation of L-values of totally real fields

We develop the topological polylogarithm which provides an integral version of Nori's Eisenstein cohomology classes for $GL_n(\mathbf{Z})$ and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of special values of $L$-functions of totally real fields and a construction of the associated $p$-adic $L$-function. Using a result of Graf, we also apply this to prove some integrality and $p$-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.

The Subset Assignment Problem for Data Placement in Caches

We introduce the subset assignment problem in which items of varying sizes are placed in a set of bins with limited capacity. Items can be replicated and placed in any subset of the bins. Each (item, subset) pair has an associated cost. Not assigning an item to any of the bins is not free in general and can potentially be the most expensive option. The goal is to minimize the total cost of assigning items to subsets without exceeding the bin capacities. The subset assignment problem models the problem of managing a cache composed of banks of memory with varying cost/performance specifications. The ability to replicate a data item in more than one memory bank can benefit the overall performance of the system with a faster recovery time in the event of a memory failure. For this setting, the number n of data objects (items) is very large and the number d of memory banks (bins) is a small constant (on the order of 3 or 4). Therefore, the goal is to determine an optimal assignment in time that minimizes dependence on n. The integral version of this problem is NP-hard since it is a generalization of the knapsack problem. We focus on an efficient solution to the LP relaxation as the number of fractionally assigned items will be at most d. If the data objects are small with respect to the size of the memory banks, the effect of excluding the fractionally assigned data items from the cache will be small. We give an algorithm that solves the LP relaxation and runs in time $$O(\left( {\begin{array}{c}3^d d+1\end{array}}\right) {\text {poly}}(d) n \log (n) \log (nC) \log (Z))$$ , where Z is the maximum item size and C the maximum storage cost.

Unitarily-invariant integrable systems and geometric curve flows in SU (n + 1)/U(n) and SO(2n)/U(n)

Bi-Hamiltonian hierarchies of soliton equations are derived from geometric non-stretching (inelastic) curve flows in the Hermitian symmetric spaces and . The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, new integrable multi-component versions of the Sine–Gordon (SG) equation and the modified Korteveg–de Vries (mKdV) equation, as well as a novel nonlocal multi-component version of the nonlinear Schrödinger (NLS) equation are obtained, along with their bi-Hamiltonian structures and recursion operators. These integrable systems are unitarily invariant and correspond to geometric curve flows given by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map in the case of the SG and mKdV systems, and a generalization of the vortex filament bi-normal equation in the case of the NLS systems.