Defect of compactness for non-compact imbeddings of Banach spaces can be expressed in the form of a profile decomposition. Let X be a Banach space continuously imbedded into a Banach space Y, and let D be a group of linear isometric operators on X. A profile decomposition in X, relative to D and Y, for a bounded sequence (x(k))(k is an element of N) subset of X is a sequence (S-k)(k is an element of N), such that (x(k) - S-k)(k is an element of N) is a convergent sequence in Y, and, furthermore, S-k has the particular form S-k = Sigma(n is an element of N)g(k)((n))W((n)) with g(k)((n)) is an element of D and w((n)) is an element of X. This paper extends the profile decomposition proved by Solimini [10] for Sobolev spaces (H) over dot(1,P)(R-N) with 1 < p < N to the non-reflexive case p = 1. Since existence of "concentration profiles" w((n)) relies on weak-star compactness, and the space (H) over dot(1,1) is not a conjugate of a Banach space, we prove a corresponding result for a larger space of functions of bounded variation. The result extends also to spaces of bounded variation on Lie groups.

We show that the Moser functional J(u) = integral Omega(e(4 pi u2) - 1) dx on the set B = {u is an element of H-0(1)(Omega) : parallel to del u parallel to(2) <= 1}, where Omega subset of R-2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Define g(s)w(r) = s(-1/2)w(r(s)) for s > 0. If u(k) -> u in B while lim inf J(u(k)) > J(u), then, with some s(k) -> 0, u(k) = g(sk) [(2 pi)(-1/2) min {1, log1/vertical bar x vertical bar}], up to translations and up to a remainder vanishing in the Sobolev norm. In other words, the weak continuity fails only on translations of concentrating Moser functions. The proof is based on a profile decomposition similar to that of Solimini [16], but with different concentration operators, pertinent to the two-dimensional case.

We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate lambda(1) (lambda(2)) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if lambda(1) = lambda(2), then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable V is an element of (0,1), as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If lambda(1) not equal lambda(2), on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.

We study survival among two competing types in two settings: a planar growth model related to two-neighbor bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncolored sites are given a color at rate 0, 1 or infinity, depending on whether they have zero, one, or at least two neighbors of that color. In the urn scheme, each vertex of a graph G has an associated urn containing some number of either blue or red balls ( but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same color is added to each neighboring urn, and balls in the same urn but of different colors annihilate on a one-for-one basis. We show that, for every connected graph G and every initial configuration, only one color survives almost surely. As a corollary, we deduce that in the two-type growth model on Z(2), one of the colors only infects a finite number of sites with probability one. We also discuss generalizations to higher dimensions and multi-type processes, and list a number of open problems and conjectures.

We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing property for Voronoi percolation at criticality, and an Efron Stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences. As a corollary of the proof, we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive.

Consider a monotone Boolean function f : {0, 1}(n) -> {0, 1} and the canonical monotone coupling {eta(p) : p is an element of [0, 1]} of an element in {0, 1}(n) chosen according to product measure with intensity p is an element of [0, 1]. The random point p is an element of [0, 1] where f (eta(p)) flips from 0 to 1 is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large n, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. We determine this distribution for a number of the Boolean functions which are typically studied and pay particular attention to the functions corresponding to iterated majority and percolation crossings. It turns out that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate probability measure on R arises in this way for some sequence of Boolean functions.

We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter.c. Moreover, For. <.c, the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution. At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets. We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius. For. >.c, the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast. The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.

We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in R-2 with radii determined by an underlying stationary and ergodic random field phi: R-2 -> [0, infinity), independent of the Poisson process. This setting, in which the random field is independent of the point process, is often referred to as geostatistical marking. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of R(2 )does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.

First passage percolation on is a model for describing the spread of an infection on the sites of the square lattice. The infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges. In this paper, the speed of the growth and the shape of the infected set is studied by aid of large-scale computer simulations, with focus on continuous passage time distributions. It is found that the most important quantity for determining the value of the time constant, which indicates the inverse asymptotic speed of the growth, is , where are i.i.d. passage time variables. The relation is linear for a large class of passage time distributions. Furthermore, the directional time constants are seen to be increasing when moving from the axis towards the diagonal, so that the limiting shape is contained in a circle with radius defined by the speed along the axes. The shape comes closer to the circle for distributions with larger variability.

We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automor-phisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno. (C) 2016 Elsevier Inc. All rights reserved.

By methods similar to those used by L. Jeffrey [L. C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys. 147 (1992) 563-604], we compute the quantum SU(N)-invariants for mapping tori of trace 2 homeomorphisms of a genus 1 surface when N = 2, 3 and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the SU(2) case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace -2 homeomorphisms, obtaining - in combination with Jeffrey's results - a proof of the asymptotic expansion conjecture for all torus bundles.

Using the method of spectral decimation and a modified version of Kirchhoff's matrix-tree theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in theorem 3.4. We show how spectral decimation implies the existence of the asymptotic complexity constant and obtain some bounds for it. Examples calculated include the Sierpinski gasket, a non-post critically finite analog of the Sierpinski gasket, the Diamond fractal, and the hexagasket. For each example, the asymptotic complexity constant is found.

We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a uniform random graph with prescribed degrees. Simplicity corresponds to the absence of self-loops and multiple edges. We show that the number of self-loops and multiple edges converges in distribution to two independent Poisson random variables when the second moment of the empirical degree distribution converges. We also provide estimations on the total variation distance between the numbers of self-loops and multiple edges and their limits, as well as between the sum of these values and the Poisson random variable to which this sum converges to. This revisits previous works of Bollobas, of Janson, of Wormald and others. The error estimates also imply sharp asymptotics for the number of simple graphs with prescribed degrees. The error estimates follow from an application of the Stein-Chen method for Poisson convergence, which is a novel method for this problem. The asymptotic independence of self-loops and multiple edges follows from a Poisson version of the Cramer-Wold device using thinning, which is of independent interest. When the degree distribution has infinite second moment, our general results break down. We can, however, prove a central limit theorem for the number of self-loops, and for the multiple edges between vertices of degrees much smaller than the square root of the size of the graph. Our results and proofs easily extend to directed and bipartite configuration models.

We introduce a first order strategy to study boundary value problems of parabolic systems with second order elliptic part in the upper half-space. This involves a parabolic Dirac operator at the boundary. We allow for measurable time dependence and some transversal dependence in the coefficients. We obtain layer potential representations for solutions in some classes and prove new well-posedness and perturbation results. As a byproduct, we prove for the first time a Kato estimate for the square root of parabolic operators with time dependent coefficients. This considerably extends prior results obtained by one of us under time and transversal independence. A major difficulty compared to a similar treatment of elliptic equations is the presence of non-local fractional derivatives in time.

We study parabolic operators $\cH = \partial_t-\div_{\lambda,x} A(x,t)\nabla_{\lambda,x}$ in the parabolic upper half space $\mathbb R^{n+2}_+=\{(\lambda,x,t):\ \lambda>0\}$. We assume that the coefficients are real, bounded, measurable, uniformly elliptic, but not necessarily symmetric. We prove that the associated parabolic measure is  absolutely continuous with respect to the surface measure on $\mathbb R^{n+1}$  in the sense defined by $A_\infty(\mathrm{d} x\d t)$. Our argument also gives a simplified proof of the corresponding result for elliptic measure.

In this paper we propose a definition of "parabolic NTA" for solutions to the degenerate p-parabolic equation. Given this definition we prove the Carleson estimate, originally proved for this equation in Avelin et al. (J Eur Math Soc, 2015) for cylindrical domains. Moreover we study a non-optimal, stronger "outer corkscrew" condition, such that we obtain Holder continuity up to the boundary, for non-negative solutions vanishing on a part of the boundary.

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplace equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part S-T of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.

We show that bounded pseudoconvex domains that are Hölder continuous for all α < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 184 1987) beyond Lipschitz regularity. 

We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.

This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in [26], to prove a generalized Carleson estimate. We also prove boundary Holder continuity and a boundary Harnack type inequality.

We establish boundary estimates for non-negative solutions to the $p$-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA-domains together with sharp boundary decay estimates. If the underlying domain is $C^{1,1}$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting time phenomena present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.

We prove a comparison principle for the porous medium equation in more general open sets in Rn+1 than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.

Weak supersolutions to the porous medium equation are defined by means of smooth test functions under an integral sign. We show that non-negative weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. This shows that weak supersolutions belong to a class of supersolutions defined by a comparison principle.

In this paper we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer share the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context of minimization by stochastic gradient descent, a theoretical foundation for considering Neural ODEs as the deep limit of ResNets. Our proof is based on certain decay estimates for associated Fokker-Planck equations.

This paper deals with different characterizations of sets of nonlinear parabolic capacity zero, with respect to the parabolic p-Laplace equation. Specifically we prove that certain interior polar sets can be characterized by sets of zero nonlinear parabolic capacity. Furthermore we prove that zero capacity sets are removable for bounded supersolutions and that sets of zero capacity have a relation to a certain parabolic Hausdorff measure.

We show that if Ω⊂Rn⁺¹, n≥1, is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of Ω implies the existence of exterior corkscrew points at all scales, so that in fact, Ω is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior corkscrew conditions, and an interior Harnack chain condition. We discuss some implications of this result for theorems of F. and M. Riesz type, and for certain free boundary problems.

Let M be an n-dimensional complex manifold. A holomorphic function f : M -> C is said to be semi-Bloch if for every lambda is an element of C the function g(lambda) = exp(lambda f(z)) is normal on M. We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate M in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.

We consider solutions to nonlinear elliptic equations with measure data and general growth and ellipticity conditions of degenerate type, as considered in Lieberman (Commun Partial Differ Equ 16:311-361, 1991); we prove pointwise gradient bounds for solutions in terms of linear Riesz potentials. As a direct consequence, we get optimal conditions for the continuity of the gradient.

We prove a Harnack inequality for minimisers of a class of non-autonomous functionals with non-standard growth conditions. They are characterised by the fact that their energy density switches between two types of different degenerate phases.

We show that a stochastic (Markov) operator S acting on a Schatten class C-1 satisfies the Noether condition (i.e. S' (A) = A and S' (A(2)) = A(2), where A is an element of C-infinity is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H, <.,.>)), if and only if S(E-A(G)XEA(G)) = E-A (G)S(X)E-A (G) for any state X is an element of C-1 and all Borel sets G subset of R, where E-A (G) denotes the orthogonal projection coming from the spectral resolution A = integral(sigma(A)) zE(A)(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.

We consider the weak convergence of iterates of so-called centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vector-valued traits in populations of inbreeding or hermaphroditic species, whenever the offspring's trait is equal to an additively perturbed arithmetic mean of the parents' traits. It is shown that for the existence of a weak limit, it is sufficient that the distributions of the trait and the perturbation have a finite variance or have tails controlled by a suitable power function. In particular, probability distributions from the domain of attraction of stable distributions have found an application, although in general the limit is not stable.

This paper is devoted to the study of the problem of prevalence in the classof quadratic stochastic operators acting on the L1 space for the uniform topology.We obtain that the set of norm quasi-mixing quadratic stochastic operators is a denseand open set in the topology induced by a very natural metric. This shows the typicallong-term behaviour of iterates of quadratic stochastic operators.

We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the Green's function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a "spinor" observable for random walk started from one of the vertices of the marked edge.

In this paper, we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator. The method is based on an ansatz for the solution which requires nothing but feedforward neural networks and an unconstrained gradient based optimization method such as gradient descent or a quasi-Newton method. We show an example where classical mesh based methods cannot be used and neural networks can be seen as an attractive alternative. Finally, we highlight the benefits of deep compared to shallow neural networks and device some other convergence enhancing techniques.

Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities of interest. A different approach is to measure the quantities of interest and use deep learning to reverse engineer the PDEs which are describing the physical process. In this paper we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. We include examples of data from a known model problem, and real data from weather station measurements. We show how necessary transformations of the input data amounts to coordinate transformations in the discovered PDE, and we elaborate on feature and model selection. It is shown that the dynamics of a non-linear, second order PDE can be accurately described by an ordinary differential equation which is automatically discovered by our deep learning algorithm. Even more interestingly, we show that similar results apply in the context of more complex simulations of the Swedish temperature distribution

In this paper we show how to augment classical methods for inverse problems with artificial neural networks. The neural network acts as a parametric container for the coefficient to be estimated from noisy data. Neural networks are global, smooth function approximators and as such they do not require regularization of the error functional to recover smooth solutions and coefficients. We give detailed examples using the Poisson equation in 1, 2, and 3 space dimensions and show that the neural network augmentation is robust with respect to noisy data, mesh, and geometry.

We consider the Weinstein type equation L(lambda)u = 0 on (0, infinity) X (0, infinity), where L-lambda= delta(2)(t) + delta-lambda(lambda-1)/x(2), In this paper we characterize the solutions of L(lambda)u = = 0 on (0, infinity) x (0, infinity) representable by Bessel-Poisson integrals of BMO-functions as the ones satisfying certain Carleson properties.

In this paper we define square functions (also called Littlewood-Paley-Stein functions) associated with heat semigroups for Schrodinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) L-p-boundedness properties for the square functions to our Banach valued setting by using gamma-radonifying operators. We also prove that these L-p-boundedness properties of the square functions actually characterize the Banach spaces having the UMD property.