Showing papers in "Transactions of the American Mathematical Society in 2012"
TL;DR: In this paper, it was shown that there is a constant L(d,r) such that the density at which percolation becomes likely in any (fixed) number of dimensions.
Abstract: In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the d-dimensional grid $[n]^d$. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine $p_c([n]^d,r)$, the density at which percolation (infection of the entire vertex set) becomes likely. In this paper we prove, for every pair $d ge r ge 2$, that there is a constant L(d,r) such that $p_c([n]^d,r) = [(L(d,r) + o(1)) / log_(r-1) (n)]^{d-r+1}$ as $n o infty$, where $log_r$ denotes an r-times iterated logarithm. We thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. Moreover, we determine L(d,r) for every pair (d,r).
192 citations
TL;DR: In this paper, the number and size of conjugacy classes in finite Chevalley groups and their variations were shown to be polynomial in the number of simple groups.
Abstract: We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third).
152 citations
TL;DR: In this article, it was shown that the irregular set for any β-shift or β-transformation is either empty or has full topological entropy and Hausdorff dimension.
Abstract: Let (X, d) be a compact metric space, f : X → X be a continuous map satisfying a property we call almost specification (which is slightly weaker than the g-almost product property of Pfister and Sullivan), and φ : X → R be a continuous function. We show that the set of points for which the Birkhoff average of φ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every β-shift satisfies almost specification and we show that the irregular set for any β-shift or β-transformation is either empty or has full topological entropy and Hausdorff dimension.
145 citations
TL;DR: In this paper, it was shown that the complete non-compact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.
Abstract: In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.
139 citations
TL;DR: In this article, a complete classification of all continuous GL(n) contravariant Minkowski valuations is established and a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality is presented.
Abstract: A complete classification of all continuous GL(n) contravariant Minkowski valuations is established. As an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality.
130 citations
TL;DR: In this article, the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system were analyzed, and it was shown that generic solutions blow up as t→∞ in the sense that the energy and the L-norms of the velocity field grow to infinity.
Abstract: In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as t→∞ in the sense that the energy and the L-norms of the velocity field grow to infinity for large time for 1 ≤ p < 3. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from (u0, θ0) with zero-mean for the initial temperature θ0 have a special behavior as |x| or t tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.
109 citations
105 citations
TL;DR: In this article, it was shown that if Δu 2 is replaced by a suitable norm, namely Δu + u 2, then the supremum of ∫ Ω(e 32πu − 1) dx over all functions u ∈ W 2, 2 0 (Ω) with Δu 2 ≤ 1 is bounded by a constant independent of the domain Ω.
Abstract: Adams’ inequality for bounded domains Ω ⊂ R4 states that the supremum of ∫ Ω e 32πu dx over all functions u ∈ W 2, 2 0 (Ω) with ‖Δu‖2 ≤ 1 is bounded by a constant depending on Ω only This bound becomes infinite for unbounded domains and in particular for R4 We prove that if ‖Δu‖2 is replaced by a suitable norm, namely ‖u‖ := ‖ − Δu + u‖2, then the supremum of ∫ Ω(e 32πu − 1) dx over all functions u ∈ W 2, 2 0 (Ω) with ‖u‖ ≤ 1 is bounded by a constant independent of the domain Ω Furthermore, we generalize this result to any W m, n m 0 (Ω) with Ω ⊆ Rn and m an even integer less than n
103 citations
TL;DR: In this paper, a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to �+b� �/2 was established.
Abstract: For d � 1 and � 2 (0,2), consider the family of pseudo differential operators f �+b� �/2 ;b 2 (0,1)g on R d that evolves continuously fromto � + � �/2 . In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to �+b� �/2 (or equivalently, the sum of a Brownian motion and an independent symmetric�-stable process with constant multipleb 1/� ) inC 1,1 open sets. Here a "uniform" BHP means that the comparing constant in the BHP is independent of b 2 (0,1). Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to � +b� �/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques.
85 citations
TL;DR: In this paper, it was shown that if the linear commutator of a Calderon-Zygmund operator does not map continuously into the subspace of the Calderon Zygmund operators, then it is possible to construct a bounded subbilinear operator that does not depend on the operator.
Abstract: Let $b$ be a $BMO$-function. It is well-known that the linear commutator $[b, T]$ of a Calderon-Zygmund operator $T$ does not, in general, map continuously $H^1(\mathbb R^n)$ into $L^1(\mathbb R^n)$. However, Perez showed that if $H^1(\mathbb R^n)$ is replaced by a suitable atomic subspace $\mathcal H^1_b(\mathbb R^n)$ then the commutator is continuous from $\mathcal H^1_b(\mathbb R^n)$ into $L^1(\mathbb R^n)$. In this paper, we find the largest subspace $H^1_b(\mathbb R^n)$ such that all commutators of Calderon-Zygmund operators are continuous from $H^1_b(\mathbb R^n)$ into $L^1(\mathbb R^n)$. Some equivalent characterizations of $H^1_b(\mathbb R^n)$ are also given. We also study the commutators $[b,T]$ for $T$ in a class $\mathcal K$ of sublinear operators containing almost all important operators in harmonic analysis. When $T$ is linear, we prove that there exists a bilinear operators $\mathfrak R= \mathfrak R_T$ mapping continuously $H^1(\mathbb R^n)\times BMO(\mathbb R^n)$ into $L^1(\mathbb R^n)$ such that for all $(f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n)$, we have \begin{equation}\label{abstract 1} [b,T](f)= \mathfrak R(f,b) + T(\mathfrak S(f,b)), \end{equation} where $\mathfrak S$ is a bounded bilinear operator from $H^1(\mathbb R^n)\times BMO(\mathbb R^n)$ into $L^1(\mathbb R^n)$ which does not depend on $T$. In the particular case of $T$ a Calderon-Zygmund operator satisfying $T1=T^*1=0$ and $b$ in $BMO^{\rm log}(\mathbb R^n)$-- the generalized $\BMO$ type space that has been introduced by Nakai and Yabuta to characterize multipliers of $\BMO(\bR^n)$ --we prove that the commutator $[b,T]$ maps continuously $H^1_b(\mathbb R^n)$ into $h^1(\mathbb R^n)$. Also, if $b$ is in $BMO(\mathbb R^n)$ and $T^*1 = T^*b = 0$, then the commutator $[b, T]$ maps continuously $H^1_b (\mathbb R^n)$ into $H^1(\mathbb R^n)$. When $T$ is sublinear, we prove that there exists a bounded subbilinear operator $\mathfrak R= \mathfrak R_T: H^1(\mathbb R^n)\times BMO(\mathbb R^n)\to L^1(\mathbb R^n)$ such that for all $(f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n)$, we have \begin{equation}\label{abstract 2} |T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|. \end{equation} The bilinear decomposition (\ref{abstract 1}) and the subbilinear decomposition (\ref{abstract 2}) allow us to give a general overview of all known weak and strong $L^1$-estimates.
82 citations
TL;DR: In this paper, a multiplicity Tutte polynomial M(x,y) was introduced for zonotopes and toric arrangements and proved to satisfy a deletion-restriction recurrence and has positive coefficients.
Abstract: We introduce a multiplicity Tutte polynomial M(x,y), with applications to zonotopes and toric arrangements. We prove that M(x,y) satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincare' polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x,y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x,1) computes the volume and the number of integral points of the associated zonotope.
TL;DR: In this article, the authors explore the connection between Gentzen systems and residuated frames and illustrate how frames provide a uniform treatment for semantic proofs of cut-elimination, the finite model property and the finite embeddability property.
Abstract: Residuated frames provide relational semantics for substructural logics and are a natural generalization of Kripke frames in intuitionistic and modal logic, and of phase spaces in linear logic. We explore the connection between Gentzen systems and residuated frames and illustrate how frames provide a uniform treatment for semantic proofs of cut-elimination, the finite model property and the finite embeddability property. We use our results to prove the decidability of the equational and/or universal theory of several varieties of residuated lattice-ordered groupoids, including the variety of involutive
TL;DR: In this paper, the authors considered weighted norm inequalities for multilinear Fourier multipliers and showed that the result can be interpreted as a multinear version of the result by Kurtz and Wheeden.
Abstract: In this paper, we consider weighted norm inequalities for multilinear Fourier multipliers. Our result can be understood as a multilinear version of the result by Kurtz and Wheeden.
TL;DR: In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1 vertical bar 2) in the theory of Dunkl operators is obtained, which leads to a Dirac operator depending on three parameters.
Abstract: In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1 vertical bar 2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator.
TL;DR: In this article, the authors consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces.
Abstract: We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or “regular” infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called “consecutive” ordering, the Vershik map is not strongly mixing on all finite rank diagrams. MSC: 37B05, 37A25, 37A20.
TL;DR: In this article, the stability properties of the Haagerup property and of coarse embeddability in a Hilbert space under certain semidirect products were studied. And they were shown to be stable under taking standard wreath products.
Abstract: We study stability properties of the Haagerup property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking standard wreath products. Our construction also allows for a characterization of subsets with relative Property T in a standard wreath product.
TL;DR: In this article, a modified version of the two-player "tug-of-war" game introduced by Peres, Schramm, Sheffield, and Wilson is considered.
Abstract: We present a modified version of the two-player "tug-of-war" game introduced by Peres, Schramm, Sheffield, and Wilson. This new tug-of-war game is identical to the original except near the boundary of the domain $\partial \Omega$, but its associated value functions are more regular. The dynamic programming principle implies that the value functions satisfy a certain finite difference equation. By studying this difference equation directly and adapting techniques from viscosity solution theory, we prove a number of new results. We show that the finite difference equation has unique maximal and minimal solutions, which are identified as the value functions for the two tug-of-war players. We demonstrate uniqueness, and hence the existence of a value for the game, in the case that the running payoff function is nonnegative. We also show that uniqueness holds in certain cases for sign-changing running payoff functions which are sufficiently small. In the limit $\epsilon \to 0$, we obtain the convergence of the value functions to a viscosity solution of the normalized infinity Laplace equation. We also obtain several new results for the normalized infinity Laplace equation $-\Delta_\infty u = f$. In particular, we demonstrate the existence of solutions to the Dirichlet problem for any bounded continuous $f$, and continuous boundary data, as well as the uniqueness of solutions to this problem in the generic case. We present a new elementary proof of uniqueness in the case that $f>0$, $f< 0$, or $f\equiv 0$. The stability of the solutions with respect to $f$ is also studied, and an explicit continuous dependence estimate from $f\equiv 0$ is obtained.
TL;DR: In this paper, it was shown that if an operator T is bounded on weighted Lebesgue space L 2 (w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator T k b = (b;T k 1 b ) with a function b in BMO will obey a quadratic bound.
Abstract: We show that if an operator T is bounded on weighted Lebesgue space L 2 (w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator (b;T ) with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that the kth-order commutator T k b = (b;T k 1 b ) will obey a bound that is a power (k + 1) of the A2 constant of the weight. Sharp extrapolation provides corresponding L p (w) estimates. The results are sharp in terms of the growth of the operator norm with respect to the Ap constant of the weight for all 1 < p <1, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.
TL;DR: In this paper, a Hopf monad ZT on an autonomous category C, the centralizer of T, and a canonical distributive law Ω: TZT → ZTT is given.
Abstract: The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of a Hopf monad naturally arises if one tries to reconstruct the structure of Z(C) in terms of its monad Z: we show that Z is a quasitriangular Hopf monad on C and Z(C) is isomorphic to the braided category Z − C of Z-modules. More generally, let T be a Hopf monad on an autonomous category C. We construct a Hopf monad ZT on C, the centralizer of T , and a canonical distributive law Ω: TZT → ZTT . By Beck’s theory, this has two consequences. On one hand, DT = ZT ◦Ω T is a quasitriangular Hopf monad on C, called the double of T , and Z(T − C) DT − C as braided categories. As an illustration, we define the double D(A) of a Hopf algebra A in a braided autonomous category in such a way that the center of the category of A-modules is the braided category of D(A)-modules (generalizing the Drinfeld double). On the other hand, the canonical distributive law Ω also lifts ZT to a Hopf monad Z Ω T on T−C, and ZΩ T ( , T0) is the coend of T−C. For T = Z, this gives an explicit description of the Hopf algebra structure of the coend of Z(C) in terms of the structural morphisms of C. Such a description is useful in quantum topology, especially when C is a spherical fusion category, as Z(C) is then modular.
TL;DR: In this paper, it was shown that a diffeomorphism f has a C 1 -robust homoclinic tangency if there is a C 2 -neighbourhood U of f such that every diffeomorphic in U has a hyperbolic setg, depending contin- uously on g, such that the stable and unstable manifolds ofg have some non-transverse intersection.
Abstract: A diffeomorphism f has a C 1 -robust homoclinic tangency if there is a C 1 -neighbourhood U of f such that every diffeomorphism in g ∈ U has a hyperbolic setg, depending contin- uously on g, such that the stable and unstable manifolds ofg have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a lo- cal mechanism (blender-horseshoes) generating diffeomorphisms with C 1 -robust homoclinic tangencies. Using blender-horseshoes, we prove that homoclinic classes of C 1 -generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display C 1 -robust homoclinic tangencies.
TL;DR: In this paper, the authors investigated the regularity of weighted Bergman projections on the unit disc and in higher dimensions, and they showed that the latter is the same as the latter.
Abstract: We investigate $L^p$ regularity of weighted Bergman projections on the unit disc and $L^p$ regularity of ordinary Bergman projections in higher dimensions.
TL;DR: In this article, the authors established extreme value statistics for functions with multiple maxima and some degree of regularity on certain non-uniformly expanding dynamical systems via a general lifting theorem.
Abstract: We establish extreme value statistics for functions with multiple maxima and some degree of regularity on certain non-uniformly expanding dynamical systems. We also establish extreme value statistics for time-series of observations on discrete and continuous suspensions of certain non-uniformly expanding dynamical systems via a general lifting theorem. The main result is that a broad class of observations on these systems exhibit the same extreme value statistics as i.i.d processes with the same distribution function.
TL;DR: In this paper, it was shown that the expected value of C is at most a constant multiple of the largest hitting time of an element in the state space, where C is the first time at which all walkers have coalesced into a single cluster.
Abstract: Consider a system of coalescing random walks where each individual performs random walk over a finite graph G, or (more generally) evolves according to some reversible Markov chain generator Q. Let C be the first time at which all walkers have coalesced into a single cluster. C is closely related to the consensus time of the voter model for this G or Q. We prove that the expected value of C is at most a constant multiple of the largest hitting time of an element in the state space. This solves a problem posed by Aldous and Fill and gives sharp bounds in many examples, including all vertex-transitive graphs. We also obtain results on the expected time until only k � 2 clusters remain. Our proof tools include a new exponential inequality for the meeting time of a reversible Markov chain and a deterministic trajectory, which we believe to be of independent interest.
TL;DR: In particular, this paper showed that all such representations are tensor products of evaluation representations and one-dimensional representations, and established conditions ensuring that they are all evaluation representations for equivariant map algebras.
Abstract: Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The corresponding equivariant map algebra is the Lie algebra M of equivariant regular maps from X to g. We classify the irreducible finite-dimensional representations of these algebras. In particu- lar, we show that all such representations are tensor products of evaluation representations and one-dimensional representations, and we establish conditions ensuring that they are all evaluation representations. For example, this is always the case if M is perfect. Our results can be applied to multiloop algebras, current algebras, the Onsager algebra, and the tetrahedron algebra. Doing so, we easily recover the known classifications of irreducible finite- dimensional representations of these algebras. Moreover, we obtain previously unknown classifica- tions of irreducible finite-dimensional representations of other types of equivariant map algebras, such as the generalized Onsager algebra.
TL;DR: In this paper, the authors considered both the defocusing and focusing cubic nonlinear Klein-Gordon equations in two dimensions for real-valued initial data and showed that in the focusing case, solutions are global and have finite global spacetime bounds.
Abstract: We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the defocusing case, solutions are global and have finite global $L^4_{t,x}$ spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state.
These results rely on analogous statements for the two-dimensional cubic nonlinear Schrodinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional.
It was previously shown by Nakanishi that spacetime bounds for Klein--Gordon equations imply the same for nonlinear Schrodinger equations.
TL;DR: In this paper, the authors studied the long-term behavior of the mild solution to the stochastic heat equation with the assumption that the initial data u0 is a bounded and measurable function and σ is nonconstant and Lipschitz continuous.
Abstract: We consider the stochastic heat equation of the following form ∂ ∂t ut(x) = (Lut)(x) + b(ut(x)) + σ(ut(x))Ḟt(x) for t > 0, x ∈ R, where L is the generator of a Levy process and Ḟ is a spatially-colored, temporally white, gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE. For the most part, we work under the assumptions that the initial data u0 is a bounded and measurable function and σ is nonconstant and Lipschitz continuous. In this case, we find conditions under which the preceding stochastic PDE admits a unique solution which is also weakly intermittent. In addition, we study the same equation in the case that Lu is replaced by its massive/dispersive analogue Lu − λu where λ ∈ R. And we describe accurately the effect of the parameter λ on the intermittence of the solution in the case that σ(u) is proportional to u [the “parabolic Anderson model”]. We also look at the linearized version of our stochastic PDE, that is the case when σ is identically equal to one [any other constant works also]. In this case, we study not only the existence and uniqueness of a solution, but also the regularity of the solution when it exists and is unique.
TL;DR: In this paper, a new equivalent relation between two dust-like self-similar sets called matchable condition is introduced, which is a necessary condition for Lipschitz equivalence.
Abstract: In this paper we investigate the Lipschitz equivalence of dust-like self-similar sets in Rd. One of the fundamental results by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223– 233] establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. In this paper we extend the study by examining deeper such connections. A key ingredient of our study is the introduction of a new equivalent relation between two dust-like self-similar sets called matchable condition. Thanks to a certain measure-preserving property of bi-Lipschitz maps between dust-like self-similar sets, we show that the matchable condition is a necessary condition for Lipschitz equivalence. Using the matchable condition we prove several conditions on the Lipschitz equivalence of dust-like self-similar sets based on the algebraic properties of the contraction ratios, which include a complete characterization of Lipschitz equivalence when the multiplication groups generated by the contraction ratios have full rank. We also completely characterize the Lipschitz equivalence of dust-like self-similar sets with two branches (i.e. they are generated by IFS with two contractive similarities). Some other results are also presented, including a complete characterization of Lipschitz equivalence when one of the self-similar sets has uniform contraction ratio.
TL;DR: This paper establishes Fernique-type inequalities and utilizes them to study the global and local moduli of continuity for anisotropic Gaussian random fields and applications to fractional Brownian sheets and to the solutions of stochastic partial differential equations are investigated.
Abstract: This paper is concerned with sample path properties of anisotropic Gaussian random fields. We establish Fernique-type inequalities and utilize them to study the global and local moduli of continuity for anisotropic Gaussian random fields. Applications to fractional Brownian sheets and to the solutions of stochastic partial differential equations are investigated.
TL;DR: In this article, a method to prove that a generalized Newtonian system admits a unique, local, strong solution in the Lp-setting is presented in the case of generalized Newtonians.
Abstract: Consider the system of equations describing the motion of a rigid body immersed in a viscous, incompressible fluid of Newtonian or generalized Newtonian type. The class of fluids considered includes in particular shearthinning or shear-thickening fluids of power-law type of exponent d ≥ 1. We develop a method to prove that this system admits a unique, local, strong solution in the Lp-setting. The approach presented in the case of generalized Newtonian fluids is based on the theory of quasi-linear evolution equations and requires that the exponent p satisfies the condition p > 5.
TL;DR: In this article, the authors studied the isomorphism classes of Artinian Gorenstein local rings with socle degree three by means of Macaulay's inverse system and proved that their classification is equivalent to the projective classification of the hypersurfaces of P n of degree three.
Abstract: In this paper we study the isomorphism classes of Artinian Gorenstein local rings with socle degree three by means of Macaulay’s inverse system. We prove that their classification is equivalent to the projective classification of the hypersurfaces of P n of degree three. This is an unexpected result because it reduces the study of this class of local rings to the homogeneous case. The result has applications in problems concerning the punctual Hilbert scheme Hilbd(P n ) and