Euclidean Setting Research Articles

Asymptotical stability of Muckenhoupt weights through Gurov-Reshetnyak classes

We show that weights in the Gurov-Reshetnyak class GR( ) satisfy a weak reverse Holder inequality with an explicit and asymptotically sharp bound for the exponent, thus extending classical results from the Euclidean setting to doubling metric mea- sure spaces. As an application, we study asymptotical behaviour of embeddings between Muckenhoupt classes and reverse Holder classes.

2d affine XY-spin model/4d gauge theory duality and deconfinement

Abstract We introduce a duality between two-dimensional XY-spin models with symmetry-breaking perturbations and certain four-dimensional SU(2) and SU(2)/ $ {\mathbb{Z}_2} $ gauge theories, compactified on a small spatial circle $ {\mathbb{R}^{{^{{{1},{2}}}}}} $ × $ {\mathbb{S}^{{^{{1}}}}} $ , and considered at temperatures near the deconfinement transition. In a Euclidean set up, the theory is defined on $ {\mathbb{R}^{{^{{2}}}}} $ × $ {\mathbb{T}^{{^{{2}}}}} $ . Similarly, thermal gauge theories of higher rank are dual to new families of “affine” XY-spin models with perturbations. For rank two, these are related to models used to describe the melting of a 2d crystal with a triangular lattice. The connection is made through a multi-component electric-magnetic Coulomb gas representation for both systems. Perturbations in the spin system map to topological defects in the gauge theory, such as monopole-instantons or magnetic bions, and the vortices in the spin system map to the electrically charged W-bosons in field theory (or vice versa, depending on the duality frame). The duality permits one to use the two-dimensional technology of spin systems to study the thermal deconfinement and discrete chiral transitions in four-dimensional SU(N c ) gauge theories with n f ≥1 adjoint Weyl fermions.

Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space

We study the Emden–Fowler equation −Δu = |u| p−1 u on the hyperbolic space $${{\mathbb H}^n}$$ . We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is p = (n + 2)/(n − 2) as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers (Bhakta and Sandeep, Poincaré Sobolev equations in the hyperbolic space, 2011; Mancini and Sandeep, Ann Sci Norm Sup Pisa Cl Sci 7(5):635–671, 2008) consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.

Newtonian Lorentz metric spaces

This paper studies Newtonian Sobolev-Lorentz spaces. We prove that these spaces are Banach. We also study the global p, q-capacity and the p, q-modulus of families of rectifiable curves. Under some additional assumptions (that is, X carries a doubling measure and a weak Poincare inequality), we show that when 1 ≤ q 1. We provide a counterexample for the density result in the Euclidean setting when 1 < p ≤ n and q =∞.

Ultraviolet–Infrared Mixing on the Noncommutative Minkowski Space in the Yang–Feldman Formalism

We study infrared divergences due to ultraviolet-infrared mixing in quantum field theory on Moyal space with Lorentzian signature in the Yang-Feldman formalism. Concretely, we are considering the phi^4 and the phi^3 model in arbitrary even dimension. It turns out that the situation is worse than in the Euclidean setting, in the sense that we find infrared divergences in graphs that are finite there. We briefly discuss the problems one faces when trying to adapt the nonlocal counterterms that render the Euclidean model renormalizable.

On fully nonlinear CR invariant equations on the Heisenberg group

In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author in Li and Li (2003) [21]. We also prove a comparison principle for solutions of second order fully nonlinear CR invariant equations defined on bounded domains of the Heisenberg group and a comparison principle for solutions of a family of second order fully nonlinear equations on a punctured ball.

A Spline Framework for Estimating the EEG Surface Laplacian Using the Euclidean Metric

This letter develops a framework for EEG analysis and similar applications based on polyharmonic splines. This development overcomes a basic problem with the method of splines in the Euclidean setting: that it does not work on low-degree algebraic surfaces such as spherical and ellipsoidal scalp models. The method's capability is illustrated through simulations on the three-sphere model and using empirical data.

SEMILOCAL CONVERGENCE OF NEWTON'S METHOD FOR SINGULAR SYSTEMS WITH CONSTANT RANK DERIVATIVES

We provide a semilocal convergence result for approximating a solution of a singular system with constant rank derivatives, using Newton's method in an Euclidean space setting. Our approach uses more precise estimates and a combination of two Lipschitz-type conditions leading to the following advantages over earlier works [13], [16], [17], [29]: tighter bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples are also provided in this study.

Bayesian Fusion on Lie Groups

An increasing number of real-world problems involve the measurement of data, andthe computation of estimates, on Lie groups. Moreover, establishing confidence in the resultingestimates is important. This paper therefore seeks to contribute to a larger theoretical frameworkthat generalizes classical multivariate statistical analysis from Euclidean space to the setting of Liegroups. The particular focus here is on extending Bayesian fusion, based on exponential familiesof probability densities, from the Euclidean setting to Lie groups. The definition and properties ofa new kind of Gaussian distribution for connected unimodular Lie groups are articulated, and ex-plicit formulas and algorithms are given for finding the mean and covariance of the fusion modelbased on the means and covariances of the constituent probability densities. The Lie groups thatfind the most applications in engineering are rotation groups and groups of rigid-body motions.Orientational (rotation-group) data and associated algorithms for estimation arise in problemsincluding satellite attitude, molecular spectroscopy, and global geological studies. In robotics andmanufacturing, quantifying errors in the position and orientation of tools and parts are impor-tant for task performance and quality control. Developing a general way to handle problemson Lie groups can be applied to all of these problems. In particular, we study the issue of howto ‘fuse’ two such Gaussians and how to obtain a new Gaussian of the same form that is ‘closeto’ the fused density.This is done at two levels of approximation that result from truncating theBaker-Campbell-Hausdorff formula with different numbers of terms. Algorithms are developedand numerical results are presented that are shown to generate the equivalent fused density withgood accuracy

Semidefinite complementarity reformulation for robust Nash equilibrium problems with Euclidean uncertainty sets

Consider the N-person non-cooperative game in which each player's cost function and the opponents' strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player's decision-making based on the robust optimization policy. In this paper, we focus on the robust Nash equilibrium problem in which each player's cost function is quadratic, and the uncertainty sets for the opponents' strategies and the cost matrices are represented by means of Euclidean and Frobenius norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give some numerical example to illustrate the behavior of robust Nash equilibria.

Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α > 0 such that \({|\Delta(E)| \gtrsim q}\) whenever \({|E| \gtrsim q^{\alpha}}\), where \({E \subset {\mathbb {F}}_q^d}\), the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here \({\Delta(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x,y \in E\}}\). Iosevich and Rudnev (Trans Am Math Soc 359(12):6127–6142, 2007) established the threshold \({\frac{d+1}{2}}\), and in Hart et al. (Trans Am Math Soc 363:3255–3275, 2011) proved that this exponent is sharp in odd dimensions. In two dimensions we improve the exponent to \({\tfrac{4}{3}}\), consistent with the corresponding exponent in Euclidean space obtained by Wolff (Int Math Res Not 10:547–567, 1999). The pinned distance set \({\Delta_y(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x\in E\}}\) for a pin \({y\in E}\) has been studied in the Euclidean setting. Peres and Schlag (Duke Math J 102:193–251, 2000) showed that if the Hausdorff dimension of a set E is greater than \({\tfrac{d+1}{2}}\), then the Lebesgue measure of Δy(E) is positive for almost every pin y. In this paper, we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set \({\Pi_y(E)=\{x\cdot y: x\in E\}}\). Under the additional assumption that the set E has Cartesian product structure we improve the pinned threshold for both distances and dot products to \({\frac{d^2}{2d-1}}\). The pinned dot product result for Cartesian products implies the following sum-product result. Let \({A\subset \mathbb F_q}\) and \({z\in \mathbb F^*_q}\). If \({|A|\geq q^{\frac{d}{2d-1}}}\) then there exists a subset \({E'\subset A\times \dots \times A=A^{d-1}}\) with \({|E'|\gtrsim |A|^{d-1}}\) such that for any \({(a_1,\dots, a_{d-1}) \in E'}\), $$ |a_1A+a_2A+\dots +a_{d-1}A+zA| > \frac{q}{2}$$ where \({a_j A=\{a_ja:a \in A\},j=1,\dots,d-1}\). A generalization of the Falconer distance problem is to determine the minimal α > 0 such that E contains a congruent copy of a positive proportion of k-simplices whenever \({|E| \gtrsim q^{\alpha}}\). Here the authors improve on known results (for k > 3) using Fourier analytic methods, showing that α may be taken to be \({\frac{d+k}{2}}\).

A counterexample for the geometric traveling salesman problem in the Heisenberg group

We are interested in characterizing the compact sets of the Heisenberg group that are contained in a curve of finite length. Ferrari, Franchi and Pajot recently gave a sufficient condition for those sets, adapting a necessary and sufficient condition due to P. Jones in the Euclidean setting. We prove that this condition is not necessary.

On excitable β-skeletons

A β-skeleton, β ≥ 1, is a planar proximity undirected graph of an Euclidean point set where nodes are connected by an edge if their lune-based neighborhood contains no other points of the given set. Parameter β determines size and shape of the nodes’ neighborhoods. In an excitable β-skeleton every node takes three states—resting, excited and refractory, and updates its state in discrete time depending on states of its neighbors. We design families of β-skeletons with absolute and relative thresholds of excitability and demonstrate that several distinct classes of space–time excitation dynamics can be selected using β. The classes include spiral and target waves of excitation, branching domains of excitation and oscillating localizations.

Routes towards emergent gravity

We discuss a toy model for an emergent non-relativistic gravitational theory. Within a certain class of Bose–Einstein condensates, it is possible to show that, in a suitable regime, a modified version of non-relativistic Newtonian gravity does effectively describes the low energy dynamics of the coupled system condensate/quasi-particles. Furthermore, we study the role of Lorentz and diffeomorphism invariance in emergent gravity scenarios by developing a toy model showing an emergent Lorentzian signature from a Euclidean setting and simultaneously and emergent gravitational dynamics of the Nördstrom type (scalar gravity). Some lessons about the crucial challenges awaiting in the future the emergent gravity proposal are finally drawn.

Gravitational fixed points and asymptotic safety from perturbation theory

The fixed point structure of the renormalization flow in Einstein gravity and higher derivative gravity is investigated in terms of the background effective action. A refined perturbative framework is developed consisting of: use of a covariant operator regularization that keeps track of powerlike divergences, a non-minimal subtraction ansatz for the originally dimensionful couplings in combination with a ‘Wilsonian’ matching condition, and the construction of a one-loop effective action exactly gauge-independent on-shell in regularized form. Using this framework strictly positive fixed points for the dimensionless Newton constant gN and the cosmological constant λ can be identified already in one-loop perturbation theory. The renormalization flow is asymptotically safe with respect to the nontrivial fixed points in both cases. In Einstein gravity a residual gauge dependence of the fixed points is unavoidable while in higher derivative gravity both the fixed point and the flow equations are universal. Along this flow spectral positivity of the Hessians can be satisfied, thereby meeting an essential condition for a well-defined Euclidean field theory setting. Dependence on O(10) initial data is erased to accuracy 0.5% after O(100) units of the renormalization mass scale and the flow settles on a λ(gN) orbit.

Information-theoretic inequalities on unimodular Lie groups

Classical inequalities used in information theory such as those of de Bruijn, Fisher, Cramér, Rao, and Kullback carry over in a natural way from Euclidean space to unimodular Lie groups. These are groups that possess an integration measure that is simultaneously invariant under left and right shifts. All commutative groups are unimodular. And even in noncommutative cases unimodular Lie groups share many of the useful features of Euclidean space. The rotation and Euclidean motion groups, which are perhaps the most relevant Lie groups to problems in geometric mechanics, are unimodular, as are the unitary groups that play important roles in quantum computing. The extension of core information theoretic inequalities defined in the setting of Euclidean space to this broad class of Lie groups is potentially relevant to a number of problems relating to information gathering in mobile robotics, satellite attitude control, tomographic image reconstruction, biomolecular structure determination, and quantum information theory. In this paper, several definitions are extended from the Euclidean setting to that of Lie groups (including entropy and the Fisher information matrix), and inequalities analogous to those in classical information theory are derived and stated in the form of fifteen small theorems. In all such inequalities, addition of random variables is replaced with the group product, and the appropriate generalization of convolution of probability densities is employed. An example from the field of robotics demonstrates how several of these results can be applied to quantify the amount of information gained by pooling different sensory inputs.

Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology

In this paper, we propose a new framework for quantum field theory in terms of consistency conditions. The consistency conditions that we consider are associativity or factorization conditions on the operator product expansion (OPE) of the theory, and are proposed to be the defining property of any quantum field theory. Our framework is presented in the Euclidean setting, and is applicable in principle to any quantum field theory, including non-conformal ones. In our framework, we obtain a characterization of perturbations of a given quantum field theory in terms of a certain cohomology ring of Hochschild-type. We illustrate our framework by the free field, but our constructions are general and apply also to interacting quantum field theories. For such theories, we propose a new scheme to construct the OPE which is based on the use of non-linear quantized field equations.

Schauder estimates for sub-elliptic equations

We establish Holder estimates of second derivatives for a class of sub-elliptic partial differential operators in \({\mathbb{R}^{N}}\) of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the Xj’s are smooth vector fields in \({\mathbb{R}^{N}}\), and aij is a uniformly elliptic matrix. It is assumed that the Xj’s satisfy homogeneity conditions with respect to a group of dilations δr which yield the existence of a composition law \({\circ}\) in \({\mathbb{R}^{N}}\) making the triplet \({\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}\) an homogeneous Lie group on which the Xj’s are left translation invariant. The Holder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on \({\mathbb{G}}\), some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderon and Zygmund in the Euclidean setting.

Generalized Mean Curvature Flow in Carnot Groups

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [4] and [12]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.

Principal points and elliptical distributions from the multivariate setting to the functional case

In this paper, we present an extension of the properties of principal points, self-consistent points and elliptical distribution results obtained in the euclidean setting to the functional elliptical distribution case, i.e., when dealing with random elements over a separable Hilbert space H .