Difference Quotient Research Articles

The mixed Lipschitz space and its dual for tree metrics

This paper develops a theory of harmonic analysis on spaces with tree metrics, extending previous work in this direction by Gavish, Nadler and Coifman (2010) [30] and Gavish and Coifman (2011, 2012) [28] , [29] . We show how a natural system of martingales and martingale differences induced by a partition tree leads to simple and effective characterizations of the Lipschitz norm and its dual for functions on a single tree metric space. The restrictions we place on the tree metrics are far more general than those considered in previous work. As the dual norm is equal to the Earth Mover's Distance (EMD) between two probability distributions, we recover a simple formula for EMD with respect to tree distances presented by Charikar (2002) [36] . We also consider the situation where an arbitrary metric is approximated by the average of a family of dominating tree metrics. We show that the Lipschitz norm and its dual for the tree metrics can be combined to yield an approximation to the corresponding norms for the underlying metric. The main contributions of this paper, however, are the generalizations of the aforementioned results to the setting of the product of two or more tree metric spaces. For functions on a product space, the notion of regularity we consider is not the Lipschitz condition, but rather the mixed Lipschitz condition that controls the size of a function's mixed difference quotient. This condition is extremely natural for datasets that can be described as a product of metric spaces, such as word-document databases. We develop effective formulas for norms equivalent to the mixed Lipschitz norm and its dual, and extend our results on combining pairs of trees.

Simulation of heat transfer in combustion chamber waterwall tubes of supercritical steam boilers

Abstract The paper presents the results of numerical computations performed for the furnace chamber waterwalls of a supercritical boiler with a steam output of 2400 × 103 kg/h. A model of distributed parameters is proposed for the waterwall operation simulation. It is based on the solution of equations describing the mass, momentum and energy conservation laws. The aim of the calculations was to determine the distribution of enthalpy, mass flow and fluid pressure in tubes. The balance equations can be brought to a form where on the left-hand side space derivatives, and on the right-hand side – time derivatives are obtained. The time derivatives on the right-hand side were replaced with backward difference quotients. This system of ordinary differential equations was solved using the Runge-Kutta method. The calculation also takes account of the variable thermal load of the chamber along its height. This thermal load distribution is known from the calculations of the heat exchange in the combustion chamber. The calculations were carried out with the zone method.

Free analysis and random matrices

We describe the Schwinger–Dyson equation related with the free difference quotient. Such an equation appears in different fields such as combinatorics (via the problem of the enumeration of planar maps), operator algebra (via the definition of a natural integration by parts in free probability), in classical probability (via random matrices or particles in repulsive interaction). In these lecture notes, we shall discuss when this equation uniquely defines the system and in such a case how it leads to deep properties of the solution. This analysis can be extended to systems which approximately satisfy these equations, such as random matrices or Coulomb gas interacting particle systems.

Extending Babuška-Aziz’s theorem to higher-order Lagrange interpolation

We consider the error analysis of Lagrange interpolation on triangles and tetrahedrons. For Lagrange interpolation of order one, Babu\v{s}ka and Aziz showed that squeezing a right isosceles triangle perpendicularly does not deteriorate the optimal approximation order. We extend their technique and result to higher-order Lagrange interpolation on both triangles and tetrahedrons. To this end, we make use of difference quotients of functions with two or three variables. Then, the error estimates on squeezed triangles and tetrahedrons are proved by a method that is a straightforward extension of the original given by Babu\v{s}ka-Aziz.

Some Applications of Fractional Velocities

Abstract Fractional velocity is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. The fractional velocity can be suitable for characterizing singular behavior of derivatives of Hölderian functions and non differentiable functions. Relations to integer-order derivatives and other integral-based definitions are discussed.It is demonstrated that for Hölder functions under certain conditions the product rules deviates from the Leibniz rule. This deviation is expressed by another quantity, fractional co-variation.

Optimal estimation of derivatives in nonparametric regression

We propose a simple framework for estimating derivatives without _tting the regression function in nonparametric regression. Unlike most existing methods that use the symmetric difference quotients, our method is constructed as a linear combination of observations. It is hence very exible and applicable to both interior and boundary points, including most existing methods as special cases of ours. Within this framework, we define the variance-minimizing estimators for any order derivative of the regression function with a fixed bias-reduction level. For the equidistant design, we derive the asymptotic variance and bias of these estimators. We also show that our new method will, for the first time, achieve the asymptotically optimal convergence rate for difference-based estimators. Finally, we provide an effective criterion for selection of tuning parameters and demonstrate the usefulness of the proposed method through extensive simulation studies of the first-and second-order derivative estimators.

Cohesive Dynamics and Brittle Fracture

We formulate a nonlocal cohesive model for calculating the deformation inside a cracking body. In this model a set of physical properties including elastic and softening behavior are assigned to each point in the medium. We work within the small deformation setting and use the peridynamic formulation. Here strains are calculated as difference quotients. The constitutive relation is given by a nonlocal cohesive law relating force to strain. At each instant of the evolution we identify a process zone where strains lie above a threshold value. Perturbation analysis shows that jump discontinuities within the process zone can become unstable and grow. We derive an explicit inequality that shows that the size of the process zone is controlled by the ratio given by the length scale of nonlocal interaction divided by the characteristic dimension of the sample. The process zone is shown to concentrate on a set of zero volume in the limit where the length scale of nonlocal interaction vanishes with respect to the size of the domain. In this limit the dynamic evolution is seen to have bounded linear elastic energy and Griffith surface energy. The limit dynamics corresponds to the simultaneous evolution of linear elastic displacement and the fracture set across which the displacement is discontinuous. We conclude illustrating how aspects of the approach developed here can be applied to limits of dynamics associated with other energies that $\varGamma$ -converge to the Griffith fracture energy.

Multi-step methods for random ODEs driven by Itô diffusions

Linear multi-step methods are derived for random ordinary differential equations (RODEs) driven by the solutions of Itô stochastic differential equations (SODEs) via strong Itô–Taylor schemes for SODEs. Due to the special structure of the RODE–SODE pair it is not necessary to restrict the intensity of the noise. Pathwise convergence is established as well as the B-stability of implicit multi-step methods. Numerical comparisons are provided for explicit schemes applied to a low dimensional RODE and implicit schemes applied to a high dimensional RODE obtained with the method of lines by spatially discretizing a random partial differential equation with finite difference quotients.

Lagrangian flows for vector fields with anisotropic regularity

We prove quantitative estimates for flows of vector fields subject to anisotropic regularity conditions: some derivatives of some components are (singular integrals of) measures, while the remaining derivatives are (singular integrals of) integrable functions. This is motivated by the regularity of the vector field in the Vlasov–Poisson equation with measure density. The proof exploits an anisotropic variant of the argument in [14,20] and suitable estimates for the difference quotients in such anisotropic context. In contrast to regularization methods, this approach gives quantitative estimates in terms of the given regularity bounds. From such estimates it is possible to recover the well posedness for the ordinary differential equation and for Lagrangian solutions to the continuity and transport equations.

Natural convection inside a C-shaped nanofluid-filled enclosure with localized heat sources

Purpose – The purpose of this paper is to investigate natural convection fluid flow and heat transfer inside C-shaped enclosures filled with Cu-Water nanofluid numerically using the finite difference method. Design/methodology/approach – In this investigation, the finite difference method is employed to solve the governing equations with the boundary conditions. Central difference quotients were used to approximate the second derivatives in both the X and Y directions. Then, the obtained discretized equations are solved using a Gauss-Seidel iteration technique. Findings – It was found from the obtained results that the mean Nusselt number increased with increase in Rayleigh number and volume fraction of Cu nanoparticles regardless aspect ratio of the enclosure. Moreover the obtained results showed that the rate of heat transfer increased with decreasing the aspect ratio of the cavity. Also, it was found that the rate of heat transfer increased with increase in nanoparticles volume fraction. Also at low Rayleigh numbers, the effect of Cu nanoparticles on enhancement of heat transfer for narrow enclosures was more than that for wide enclosures. Originality/value – This paper is relatively original for considering C-shaped cavity with nanofluids.

On the first continuous [formula omitted]-cohomology of free group factors

We prove that the first continuous L2-cohomology of free group factors vanishes. This answers a question by Andreas Thom regarding continuity properties of free difference quotients and shows that one cannot distinguish free group factors by means of first continuous L2-Betti number.

A Minty variational principle for set optimization

Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so-called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. Löhne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a “lattice difference quotient.” A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for proofs. The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert's theorem to improper functions.

Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients

The aim of this paper, which deals with a class of singular functionals involving difference quotients, is twofold: deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a kth-order Sobolev space. One of the main theorems is a new characterization of Wk,p(Ω), k∈N and p∈(1,+∞), for arbitrary open sets Ω⊂Rn. In particular, we provide natural generalizations of the results regarding Sobolev spaces summarized in Brézis’ overview article [Brézis (2002)] to the higher-order case, and extend the work [Borghol (2007)] to a more general setting.

Roots of Difference Quotient Forms of Chebyshev Polynomials

AbstractLet ­
Œ Þ
–s be the Chebyshev polynomials of first kind of degree
Π. In this paper, we show that for
ðI , the polynomial with integer coefficients

˜a
­
Œ Þ
˜sa­
Œ Þ
s has all its roots in c
˜c=
 . Key words : Polynomials, Chebyshev Polynomials, Roots 1. Introduction 1 Let
„Þ
–s and
Þ
—s be polynomials in the single independent variables
– and
— with coefficients in the field
g of complex numbers. Cassels et al . [1,2] studied factorizations of
„Þ
–sa
Þ
—s as the polynomial in the pair of variables
– and
— . They also considered a trivial case when
„ and
are the same polynomial since
„Þ
–sa
„Þ
—s is divisible by
–a
— . In this case, obtaining the factors of the polynomial

–a
—
„Þ
–sa
„Þ
—s (1)is in general rather complicated. The polynomial of the form (1) also arises in Bezout matrices that have appeared in the literature for a long time. Given
“Þ
–sa


ˆa×
Œ
“
ˆ
–
ˆ Þ
“
Œ d×si
”Þ
–sa


ˆa×
Œ
”
ˆ
–
ˆ ,let
FÞ
–i
—sa

–a
—
“Þ
–s
”Þ
—sa
“Þ
—s
”Þ
–sa

Retrospective-Prospective Differential Inclusions and Their Control by the Differential Connection Tensors of Their Evolutions: The Trendometer

This study is motivated by two different, yet, connected, motivations. The first one follows the observation that the classical definition of derivatives involves prospective (or forward) difference quotients, not known whenever the time is directed, at least at the macroscopic level. Actually, the available and known derivatives are retrospective (or backward). They co\incide whenever the functions are differentiable in the classical sense, but not in the case of non smooth maps, single-valued or set-valued. The later ones are used in differential inclusions (and thus, in uncertain control systems) governing evolutions in function of time and state. We follow the plea of some physicists for taking also into account the retrospective derivatives to study prospective evolutions in function of time, state and retrospective derivatives, a particular, but specific, example of historical of ''path dependent'' evolutionary systems. This is even more crucial in life sciences, in the absence of experimentation of uncertain evolutionary systems. The second motivation emerged from the study of networks with junctions (cross-roads in traffic networks, synapses in neural networks, banks in financial networks, etc.), an important feature of ''complex systems''. At each junction, the velocities of the incoming (retrospective) and outgoing (prospective) evolutions are confronted. One measure of this confrontation (''jerkiness'') is provided by the product of the retrospective and prospective velocities, negative in ''inhibitory'' junctions, positive for ''excitatory'' ones, for instance. This leads to the introduction of the ''differential connection tensor'' of two evolutions, defined as the tensor product of retrospective and prospective derivatives, which can be used for controlling evolutionary systems governing the evolutions through networks with junctions.

A theory of average growth rate indices

This paper develops an axiomatic theory of an economic variable average growth rate (average rate of change) measurement. The obtained structures generalize the conventional measures for average rate of growth (such as the difference quotient, and the continuously compounded growth rate) to an arbitrary domain of the underlying variable and comprise various models of growth. These structures can be described with the help of intertemporal choice theory by means of parametric families of time preference relations on the “prize-time” space with a parameter representing the subjective discount rate.

Analysis on the Levi-Civita field and computational applications

In this paper, we present an overview of some of our research on the Levi-Civita fields R and C. R (resp. C) is the smallest non-Archimedean field extension of the real (resp. complex) numbers that is Cauchy-complete and real closed (resp. algebraically closed); in fact, R is small enough to allow for the calculus on the field to be implemented on a computer and used in applications such as the fast and accurate computation of the derivatives of real functions as “differential quotients” up to very high orders. We summarize the convergence and analytical properties of power series, showing that they have the same smoothness behavior as real and complex power series; we present a Lebesgue-like measure and integration theory on the Levi-Civita field R; we discuss solutions to one-dimensional and multi-dimensional optimization problems based on continuity and differentiability concepts that are stronger than the topological ones; and we give a brief summary of the results of our ongoing work on developing a non-Archimedean operator theory on a Banach space over C.

Boundary smoothness of analytic functions

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent \(\alpha \), with \(0<\alpha <1\), in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We obtain a result which applies, for example, when the open set admits an interior cone at the special boundary point.

Are the Snapshot Difference Quotients Needed in the Proper Orthogonal Decomposition?

This paper presents a theoretical and numerical investigation of the following practical question: Should the time difference quotients (DQs) of the snapshots be used to generate the proper orthogonal decomposition (POD) basis functions? The answer to this question is important, since some published numerical studies use the time DQs, whereas other numerical studies do not. The criterion used in this paper to answer this question is the optimality of the convergence rate of the error of the reduced order model with respect to the number of POD basis functions. Since to the best of our knowledge a definition of the optimality of the convergence rate is not available, we propose one in Definition 3.1 in this paper. Two cases are considered: the no_DQ case, in which the snapshot DQs are not used, and the DQ case, in which the snapshot DQs are used. For each case, two types of POD bases are used: the $L^2$-POD basis, in which the basis is generated in the $L^2$-norm, and the $H^1$-POD basis, in which the basi...

Solution of dual phase lag equation by means of the boundary element method using discretization in time

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology