Pointwise Relations Research Articles

On the validity of variational inequalities for obstacle problems with non-standard growth


 The aim of the paper is to show that the solutions to variational problems with non-standard growth conditions satisfy a corresponding variational inequality expressed in terms of a duality formula between the constrained minimizers and the corresponding dual maximizers, without any smallness assumptions on the gap between growth and coercitivity exponents. Our results rely on techniques based on Convex Analysis that consist in establishing pointwise relations that are preserved passing to the limit. We point out that we are able to deal with very general obstacle quasi-continuous up to a subset of zero capacity.

A comparative study of atomistic-based stress evaluation

<p style='text-indent:20px;'>This paper presents a comparative study on several issues of the microscopic stress definitions. Firstly, we derived an Irving-Kirkwood formulation for Cauchy stress evaluation in Eulerian coordinates. We showed that quantities, such as density and momentum, should to be defined properly on microscopic level in order to guarantee the conservation relations on macroscopic level. Secondly, the relation between Cauchy and first Piola-Kirchhoff stress was investigated both theoretically and numerically. At zero temperature, classical pointwise relation between these two stress is satisfied both in Virial and Hardy formulation. While at finite temperature, temporal averaging is required to guarantee this relation for Virial formulation. For Hardy formulation, an additional term need to be included in the classical relation between the Cauchy stress and the first Piola-Kirchhoff stress. Meanwhle, the linear relation between the Cauchy stress and the first Piola-Kirchhoff stress with respect to the temperature are obtained in both Virial and Hardy formulations. The thermal expansion coefficients are also studied by using quasi-harmonic approximation. Thirdly, different from that in the Lagrangian coordinates case, where the time averaging procedure can be performed in a post-processing manner when the kernel function is separable, the stress evaluation in Eulerian system must be evaluated spatially and temporally at the same time, even in separable kernel case. This can be seen from the comparison of the two procedures. Numerical examples were provided to illustrate our investigations.

A maximal restriction theorem and Lebesgue points of functions in $\mathcal F(L^p)$

Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the L^q -norm of the restriction of the Fourier transform of a function f in L^p(\mathbb R^n) to a given subvariety S , endowed with a suitable measure. Such estimates allow to define the restriction \mathcal{R} f of the Fourier transform of an L^p -function to S in an operator theoretic sense. In this article, we begin to investigate the question what is the „intrinsic" pointwise relation between \mathcal{R} f and the Fourier transform of f , by looking at curves in the plane, for instance with non-vanishing curvature. To this end, we bound suitable maximal operators, including the Hardy–Littlewood maximal function of the Fourier transform of f restricted to S .

On the dual formulation of obstacle problems for the total variation and the area functional

We investigate the Dirichlet minimization problem for the total variation and the area functional with a one-sided obstacle. Relying on techniques of convex analysis, we identify certain dual maximization problems for bounded divergence-measure fields, and we establish duality formulas and pointwise relations between (generalized) BV minimizers and dual maximizers. As a particular case, these considerations yield a full characterization of BV minimizers in terms of Euler equations with a measure datum. Notably, our results apply to very general obstacles such as BV obstacles, thin obstacles, and boundary obstacles, and they include information on exceptional sets and up to the boundary. As a side benefit, in some cases we also obtain assertions on the limit behavior of p-Laplace type obstacle problems for p↘1.On the technical side, the statements and proofs of our results crucially depend on new versions of Anzellotti type pairings which involve general divergence-measure fields and specific representatives of BV functions. In addition, in the proofs we employ several fine results on (BV) capacities and one-sided approximation.

Conic epipolar constraints from affine correspondences

We derive an explicit relation between local affine approximations resulting from matching of affine invariant regions and the epipolar geometry in the case of a two view geometry. Most methods that employ the affine relations do so indirectly by generating pointwise correspondences from the affine relations. In this paper we derive an explicit relation between the local affine approximations and the epipolar geometry.We show that each affine approximation between images is equivalent to 3 linear constraints on the fundamental matrix and that the linear conditions guarantee the existence of an homography, compatible with the fundamental matrix. We further show that two affine relations constrain the location of the epipole to a conic section. Therefore, the location of the epipole can be extracted from 3 regions by intersecting conics.The result is further employed to derive a procedure for estimating the fundamental matrix, based on the estimated location of the epipole. It is shown to be more accurate and to require less iterations in LO-RANSAC based estimation, than the current point based approaches that employ the affine relation to generate pointwise correspondences and then calculate the fundamental matrix from the pointwise relations.

Pointwise Relations Between Information and Estimation in Gaussian Noise

Many of the classical and recent relations between information and estimation in the presence of Gaussian noise can be viewed as identities between expectations of random quantities. These include the relationship between mutual information and minimum mean square error (I-MMSE) of Guo ; the relative entropy and mismatched estimation relationship of Verdú; the relationship between causal estimation and mutual information of Duncan, and its extension to the presence of feedback by Kadota ; the relationship between causal and non-casual estimation of Guo , and its mismatched version of Weissman. We dispense with the expectations and explore the nature of the pointwise relations between the respective random quantities. The pointwise relations that we find are as succinctly stated as-and give considerable insight into-the original expectation identities. As an illustration of our results, consider Duncan's 1970 discovery that the mutual information is equal to the causal MMSE in the additive white Gaussian noise channel, which can equivalently be expressed saying that the difference between the input-output information density and half the causal estimation error is a zero-mean random variable (regardless of the distribution of the channel input). We characterize this random variable explicitly, rather than merely its expectation. Classical estimation and information theoretic quantities emerge with new and surprising roles. For example, the variance of this random variable turns out to be given by the causal MMSE (which, in turn, is equal to twice the mutual information by Duncan's result).

The use of interactive visualizations to foster the understanding of concepts of calculus: design principles and empirical results

Calculus and functional thinking are closely related. Functional thinking includes thinking in variations and functional dependencies with a strong emphasis on the aspect of change. Calculus is a climax within school mathematics and the education to functional thinking can be seen as propaedeutics to it. Many authors describe that functions at school are mainly treated in a static way, by regarding them as pointwise relations. This often leads to the underrepresentation of the aspect of change at school. Moreover, calculus at school is mainly procedure-oriented and structural understanding is lacking. In this work, two specially designed interactive activities for the teaching and learning of concepts of calculus based on dynamic geometry software are presented. They accentuate the aspect of change and the object aspect of functions using a double stage visualization. Moreover, the activities allow the discovery and exploration of some concepts of calculus in a qualitative-structural way without knowing or using curve-sketching routines. The activities were used in a qualitative study with 10th grade students of age 15–16 in secondary schools in Berlin, Germany. Some pairs of students were videotaped while working with the activities. After transcribing, the interactions of the students were interpreted and analyzed focusing to the use of the computer. The results show how the students mobilize their knowledge about functions working on the given tasks, and using the activities to formulate important concepts of calculus in a qualitative way. Also, some important epistemological obstacles can be detected.

SHARPNESS OF SOME PROPERTIES OF WIENER AMALGAM AND MODULATION SPACES

AbstractWe prove sharp estimates for the dilation operator f(x)⟼f(λx), when acting on Wiener amalgam spaces W(Lp,Lq). Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces Mp,q, as well as the optimality of an estimate for the Schrödinger propagator on modulation spaces.

Variational optimality condition in the problem of control of initial boundary conditions for semilinear hyperbolic systems

Consideration is given to the problem of optimal control of a system of semilinear hyperbolic equations at finite-dimensional (pointwise) relations between initial boundary states of the system and control actions. In this case, the optimality condition in the form of a pointwise maximum principle is invalid. A nonclassical optimality condition of a variational type is proved. The sense of the obtained result is in that an optimal boundary or start control nearly at each point is a solution to a special problem of control of initial conditions for the system of ordinary differential equations constructed on the family of characteristics of the initial hyperbolic system. Constructions of the indicated optimality condition are illustrated by two examples. An iterative method based on the obtained result is stated. The method does not require additional assumptions about the differentiability of parameters of the control problem and convexity of the set of admissible controls necessary in using gradient methods.

Relative rearrangement and Lebesgue spaces [formula omitted] with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L p ( ⋅ ) ( Ω ) with variable exponent. In particular, we show that the maps u ∈ L p ( ⋅ ) ( Ω ) → k ( t ) u ∗ ∈ L p ∗ ( ⋅ ) ( 0 , meas Ω ) and u ∈ L p ( ⋅ ) ( Ω ) → u ∗ ∈ L p ∗ ( ⋅ ) ( 0 , meas Ω ) are locally ϕ-Hölderian ( u ∗ (resp. p ∗ ) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.

Some new applications of the pointwise relations for the relative rearrangement

Let $V=V(\Omega)$ be a set satisfying the Poincaré-Sobolev pointwise inequalities for the relative rearrangement and $\rho$ an arbitrary norm on the set of measurable functions on the interval $\Omega_*=(0,measure(\Omega))$. Then using the associate norm of $\rho$, we give some sufficient conditions to ensure that a function $u$ of $V$ has to be bounded or integrable in an Orlicz space. We show similar results for a solution of quasilinear equation under some conditions between the data and $\rho$. We then give an unified approach for many kinds of estimates. Using a main relation involving some pointwise relations for the relative rearrangement, we prove a regularity result for the time derivative of a parabolic system well posed in a Grand Sobolev space.

General pointwise relations for the relative rearrangement and applications

We prove some new Poincaré—Sobolev pointwise relations for the relative rearrangement. The first consequence is the derivation of Polyà—Szëgo pointwise inequalities which imply in particular inequalities in any normed spaces with a rearrangement invariant Fatou norm. Our method gives an unified approach for many kind of inequalities on bounded or unbounded domains. The second new application for partial differential equation is a simple formalism for having a priori estimates which is based on the pointwise estimate of the relative rearrangement of the gradient with respect to the solution of the P.D.E.

Returns to Scale and Economies of Scale: Further Observations

Despite discussions about the relationship between returns to scale and economies of scale under assumptions of constant and nonconstant input prices, mistakes continue to be made in textbooks on these issues. This article demonstrates the pointwise relation between returns to scale and economies of scale, with an adaption to a calculus-based intermediate microeconomics class.

The Kinematic Formula in Complex Integral Geometry

Given two nonsingular projective algebraic varieties $X,Y \subset {{\mathbf {P}}^n}$, $Y \subset {{\mathbf {P}}^n}$ meeting transversely, it is classical that one may express the Chern classes of their intersection $X \cap Y$ in terms of the Chern classes of $X$ and $Y$ and the Kähler form (hyperplane class) of ${{\mathbf {P}}^n}$. This depends on global considerations. However, by putting a hermitian connection on the tangent bundle of $X$, we may interpret the Chern classes of $X$ as invariant polynomials in the curvature form of the connection. Armed with this local formulation of Chern classes, we now consider two complex submanifolds (not necessarily compact) $X$, $Y \subset {{\mathbf {P}}^n}$, and investigate the geometry of their intersection. The pointwise relation between the Chern forms of $X \cap Y$ and those of $X$ and $Y$ is rather complicated. However, when we average integrals of Chern forms of $X \cap gY$ over all elements $g$ of the group of motions of ${{\mathbf {P}}^n}$, these can be expressed in a universal fashion in terms of integrals of Chern forms of $X$ and $Y$. This is, then, the kinematic formula for the unitary group.

The kinematic formula in complex integral geometry

Given two nonsingular projective algebraic varieties X, Y c P meeting transversely, it is classical that one may express the Chern classes of their intersec- tion X n Y in terms of the Chern classes of X and Y and the Kahler form (hyperplane class) of P. This depends on global considerations. However, by putting a hermitian connection on the tangent bundle of X, we may interpret the Chern classes of X as invariant polynomials in the curvature form of the connec- tion. Armed with this local formulation of Chern classes, we now consider two complex submanifolds (not necessarily compact) X, Y c P, and investigate the geometry of their intersection. The pointwise relation between the Chern forms of X n Y and those of X and Y is rather complicated. However, when we average integrals of Chern forms of X n g Y over all elements g of the group of motions of P, these can be expressed in a universal fashion in terms of integrals of Chem forms of X and Y. This is, then, the kinematic formula for the unitary group.

On the Calculation of Mutual Information

Abstract : Calculating the amount of information about a random function contained in another random function has important uses in communication theory. An expression for the mutual information for continuous time random processes has been given by Gelfand and Yaglom, Chiang, and Perez by generalizing Shannon's result in a natural way. Under a condition of absolute continuity of measures the continuous time expression has the same form as Shannon's result. For two Gaussian processes Gelfand and Yaglom express the mutual information in terms of a mean square estimation error. We generalize this result to diffusion processes and express the solution in a different form which is more naturally related to a corresponding filtering problem. We also use these results to calculate some information rates.