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We establish the general form of a geometric comparison principle for n-fold convolutions of certain singular measures in ℝd which holds for arbitrary n and d. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [3].
In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.
We present the second-order necessary and sufficient conditions for quasiconvex and pseudoconvex functions in terms of their second-order regular subdifferentials.
By an influential theorem of Boman, a function
$f$
on an open set
$U$
in
$\mathbb{R}^{d}$
is smooth (
${\mathcal{C}}^{\infty }$
) if and only if it is arc-smooth, that is,
$f\,\circ \,c$
is smooth for every smooth curve
$c:\mathbb{R}\rightarrow U$
. In this paper we investigate the validity of this result on closed sets. Our main focus is on sets which are the closure of their interior, so-called fat sets. We obtain an analogue of Boman’s theorem on fat closed sets with Hölder boundary and on fat closed subanalytic sets with the property that every boundary point has a basis of neighborhoods each of which intersects the interior in a connected set. If
$X\subseteq \mathbb{R}^{d}$
is any such set and
$f:X\rightarrow \mathbb{R}$
is arc-smooth, then
$f$
extends to a smooth function defined on
$\mathbb{R}^{d}$
. We also get a version of the Bochnak–Siciak theorem on all closed fat subanalytic sets and all closed sets with Hölder boundary: if
$f:X\rightarrow \mathbb{R}$
is the restriction of a smooth function on
$\mathbb{R}^{d}$
which is real analytic along all real analytic curves in
$X$
, then
$f$
extends to a holomorphic function on a neighborhood of
$X$
in
$\mathbb{C}^{d}$
. Similar results hold for non-quasianalytic Denjoy–Carleman classes (of Roumieu type). We will also discuss sharpness and applications of these results.
This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous
$f$
-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the
$f$
-divergence is proposed, and related functional affine isoperimetric inequalities are established.
Let Q be the open unit square in ℝ2. We prove that in a natural complete metric space of BV homeomorphisms f : Q → Q with f|∂Q = Id, residually many homeomorphisms (in the sense of Baire categories) map a null set onto a set of full measure, and vice versa. Moreover, we observe that for 1 ⩽ p < 2, the family of W1,p homemomorphisms satisfying the above property is of the first category.
In this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilation A⊕tQ as t converges to 0. Here A and Q are subsets of n-dimensional Euclidean space, A has finite perimeter, and Q is finite. If Q consists of two points only, n and n+u, say, this derivative coincides up to a sign with the directional derivative of the covariogram of A in direction u. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of A. We extend this result to finite sets Q and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at 0. The proofs are based on an approximation of the indicator function of A by smooth functions of bounded variation.
Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions that are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that SBDp functions are approximately continuous -almost everywhere away from the jump set. On the negative side, we construct a function that is BD but not in BV and has distributional strain consisting only of a jump part, and one that has a distributional strain consisting of only a Cantor part.
We show that, in the two-dimensional case, every objective, isotropic and isochoric energy function that is rank-one convex on GL+(2) is already polyconvex on GL+(2). Thus, we answer in the negative Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasi-convexity. Our methods are based on different representation formulae for objective and isotropic functions in general, as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.
Let be an open set in ℝn and suppose that is a Sobolev homeomorphism. We study the regularity of f–1 under the Lp-integrability assumption on the distortion function Kf. First, if is the unit ball and p > n – 1, then the optimal local modulus of continuity of f–1 is attained by a radially symmetric mapping. We show that this is not the case when p ⩽ n – 1 and n ⩾ 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for ∣Df–1∣ in terms of the Lp-integrability assumptions of Kf.
We give an L2 x L2 → L2 convolution estimate for singular measures supported on transversal hypersurfaces in ℝn, which improves earlier results of Bejenaru et al. as well as Bejenaru and Herr. The quantities arising are relevant to the study of the validity of bilinear estimates for dispersive partial differential equations. We also prove a class of global, nonlinear Brascamp–Lieb inequalities with explicit constants in the same spirit.
We prove that, for an interval X ⊆ ℝ and a normed space Z, diagonals of separately absolutely continuous mappings f : X2 → Z are exactly mappings g : X → Z, which are the sums of absolutely convergent series of continuous functions.
In the present paper, a coupled algorithm refining recursively the Hermite–Hadamard inequality on a simplex is investigated. Our approach allows us to express the integral mean value
$M_{f}$
of a convex function
$f$
on a simplex as both the limit of sequences and sum of series involving iterative lower and upper bounds of
$M_{f}$
. Two examples of interest are discussed.
In this paper we establish concavity properties of two extensions of the classical notion of the outer parallel volume. On the one hand, we replace the Lebesgue measure by more general measures. On the other hand, we consider a functional version of the outer parallel sets.
We study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.
Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function
${K}_{ir} (x)$
of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of
${K}_{ir} (x)$
and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of
$r$
. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of
${K}_{ir} (x)$
.
We define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.
The aim of this paper is to analyze a class of random processes which models the motion of a particle on the real line with random velocity and subject to the action of friction. The speed randomly changes when a Poissonian event occurs. We study the characteristic and moment generating functions of the position reached by the particle at time t > 0. We are able to derive the explicit probability distributions in a few cases. The moments are also widely analyzed. For the random motions having an explicit density law, further interesting probabilistic interpretations emerge if we consider randomly varying time. Essentially, we consider two different types of random time, namely Bessel and gamma times, which contain, as particular cases, some important probability distributions (e.g. Gaussian, exponential). For the random processes built by means of these compositions, we derive the probability distributions for a fixed number of Poisson events. Some remarks on possible extensions to random motions in higher spaces are proposed. We focus our attention on the persistent planar random motion.
Blackwell (1951), in his seminal work on comparison of experiments, ordered two experiments using a dilation ordering: one experiment, Y, is ‘more spread out’ in the sense of dilation than another one, X, if E(c(Y))≥E(c(X)) for all convex functions c. He showed that this ordering is equivalent to two other orderings, namely (i) a total time on test ordering and (ii) a martingale relationship E(Yʹ | Xʹ)=Xʹ, where (Xʹ,Yʹ) has a joint distribution with the same marginals as X and Y. These comparisons are generalized to balayage orderings that are defined in terms of generalized convex functions. These balayage orderings are equivalent to (i) iterated total integral of survival orderings and (ii) martingale-type orderings which we refer to as k-mart orderings. These comparisons can arise naturally in model fitting and data confidentiality contexts.