Linear forms and quadratic uniformity for functions on ℤ<Subscript><Emphasis Type="Italic">N</Emphasis></Subscript>

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW10a] and [GW10b], [GW10c], we gave a complete answer to this question for groups of the form G = F _p ⁿ , provided p is not too small. In ℤ _N , substantial extra difficulties arise, of which the most important is that an “inverse theorem” even for the uniformity norm || ·||_U³{\left\| \cdot \right\|_{{U^3}}} requires a more sophisticated “local” formulation. When N is prime, ℤ _N is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW10a]. Moreover, we obtain a doubly exponential bound.     

Complete classification of the positive solutions of −Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We study the equation −Δu + u ^q = 0, q > 1, in a bounded C ² domain Ω ⊂ ℝ ^N . A positive solution of the equation is moderate if it is dominated by a harmonic function and σ-moderate if it is the limit of an increasing sequence of moderate solutions. It is known that in the subcritical case, 1 < q <, q _c = (N + 1)/(N − 1), every positive solution is σ-moderate [32]. More recently, Dynkin proved, by probabilistic methods, that this remains valid in the supercritical case for q ≤ 2, [15]. The question remained open for q > 2. In this paper, we prove that for all q ≥ q _c , every positive solution is σ-moderate. We use purely analytic techniques, which apply to the full supercritical range. The main tools come from linear and non-linear potential theory. Combined with previous results, our result establishes a one-to-one correspondence between positive solutions and their boundary traces in the sense of [36].     

Uniform geometric estimates of sublevel sets

This paper reconsiders the uniform sublevel set estimates of Carbery, Christ, and Wright [7], Phong, Stein, and Sturm [23], and Carbery and Wright [8] from a geometric perspective. This perspective leads one to consider a natural collection of homogeneous, nonlinear differential operators, which generalize mixed derivatives in ℝ _d . As a consequence, it is shown that, in comparison to these previous works, improved uniform estimates are possible in all but certain explicitly “flat” situations.     

Deducing the multidimensional Szemerédi theorem from an infinitary removal lemma

We offer a new proof of the Furstenberg-Katznelson multiple recurrence theorem for several commuting probability-preserving transformations T ₁, T ₂, …, T _d : ℤ ↷ (X, ∑, μ) ([6]), and so, via the Furstenberg correspondence principle introduced in [5], a new proof of the multi-dimensional Szemerédi Theorem. We bypass the careful manipulation of certain towers of factors of a probability-preserving system that underlies the Furstenberg-Katznelson analysis, instead modifying an approach recently developed in [1] to pass to a large extension of our original system in which this analysis greatly simplifies. The proof is then completed using an adaptation of arguments developed by Tao in [13] for his study of an infinitary analog of the hypergraph removal lemma. In a sense, this addresses the difficulty, highlighted by Tao, of establishing a direct connection between his infinitary, probabilistic approach to the hypergraph removal lemma and the infinitary, ergodic-theoretic approach to Szemerédi’s Theorem set in motion by Furstenberg [5].     

On Weyl-Heisenberg orbits of equiangular lines

An element is called fiducial if {gz:g∈G} is a set of lines with only one angle between each pair, where G ≅ℤ _d ×ℤ _d is the one-dimensional finite Weyl-Heisenberg group modulo its centre. We give a new characterization of fiducial vectors. Using this characterization, we show that the existence of almost flat fiducial vectors implies the existence of certain cyclic difference sets. We also prove that the construction of fiducial vectors in prime dimensions 7 and 19 due to Appleby (J. Math. Phys. 46(5):052107, 2005) does not generalize to other prime dimensions (except for possibly a set with density zero). Finally, we use our new characterization to construct fiducial vectors in dimension 7 and 19 whose coordinates are real. Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).     

On the Classification of <Emphasis Type="Italic">N</Emphasis>-Points Concentrating Solutions for Mean Field Equations and the Symmetry Properties of the <Emphasis Type="Italic">N</Emphasis>-Vortex Singular Hamiltonian on the Unit Disk

Motivated by the analysis of the multiple bubbling phenomenon (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7–8):1241–1265, 2004) for a singular mean field equation on the unit disk (Bartolucci and Montefusco in Nonlinearity 19:611–631, 2006), for any N≥3 we characterize a subset of the 2π/N-symmetric part of the critical set of the N-vortex singular Hamiltonian. In particular we prove that this critical subset is of saddle type. As a consequence of our result, and motivated by a recently posed open problem (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7–8):1241–1265, 2004), we can prove the existence of a multiple bubbling sequence of solutions for the singular mean field equation.     

Diagrammes de Diamond et (<Emphasis Type="Italic">φ</Emphasis>, Γ)-modules

Let ρ be a 2-dimensional continuous semi-simple generic representation of Gal(̅ℚ _p /ℚ _p ) over ̅F _p . The modulo p Langlands correspondence for GL₂(ℚ _p ) defined in [5], as realized in [9], can be reformulated as a quite simple recipee giving back the (φ, Γ)-module of the dual of ρ starting from the “Diamond diagram” associated to ρ. Let F be a finite unramified extension of ℚ _p and ρ a 2-dimensional continuous semi-simple generic representation of Gal(̅ℚ _p /F) over ̅F _p . When one formally extends this recipee to the Diamond diagrams associated to ρ in [6], we show that one essentially finds the (φ, Γ)-module of the tensor induction from F to ℚ _p of the dual of ρ.     

On planarity of compact, locally connected, metric spaces

Independently, Claytor [Ann. Math. 35 (1934), 809–835] and Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2-connected, compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K ₅ or K _3;3. The “thumbtack space” consisting of a disc plus an arc attaching just at the centre of the disc shows the assumption of 2-connectedness cannot be dropped. In this work, we introduce “generalized thumbtacks” and show that a compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K ₅, K _3;3, or any generalized thumbtack, or the disjoint union of a sphere and a point.     

The <Emphasis Type="Italic">f</Emphasis>-Vector of the Descent Polytope

For a positive integer n and a subset S⊆[n−1], the descent polytope DP  _S is the set of points (x ₁,…,x _n ) in the n-dimensional unit cube [0,1] ⁿ such that x _i ≥x _i+1 if i∈S and x _i ≤x _i+1 otherwise. First, we express the f-vector as a sum over all subsets of [n−1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,…}∩[n−1]. We derive a generating function for F _S (t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.     

Moment-angle complexes from simplicial posets

We extend the construction of moment-angle complexes to simplicial posets by associating a certain T ^m -space Z _S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z _S has many important topological properties of the original moment-angle complex Z _K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z _S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z _S from below by proving the toral rank conjecture for the moment-angle complexes Z _S .     

Newton–Hensel Interpolation Lifting

The main result of this paper is a new version of Newton-Hensel lifting that relates to interpolation questions. It allows one to lift polynomials in ℤ[x] from information modulo a prime number p ≠ 2 to a power p^k for any k, and its originality is that it is a mixed version that not only lifts the coefficients of the polynomial but also its exponents. We show that this result corresponds exactly to a Newton--Hensel lifting of a system of 2t generalized equations in 2t unknowns in the ring of p-adic integers ℤ_p. Finally, we apply our results to sparse polynomial interpolation in ℤ[x].     

Stable Subconstructs: A Correction and New Results

In Lowen and Wuyts (Appl Categ Struct 8:235–245, 2000) the authors studied the simultaneously concretely reflective and concretely coreflective subconstructs of the category Ap of approach spaces. For the sake of shortness we call such subconstructs stable. Using a technique introduced in Herrlich and Lowen (1999) it was possible to explicitly describe such stable subconstructs by a condition on the objects which used certain subsets of [0, ∞ ]. Thus each stable subconstruct Ap _m described in [9] corresponds to the subset {0} ∪ [m, ∞ ] ⊂ [0, ∞ ] for m ∈ [0, ∞ ]. Although this characterization is correct, Theorem 4.7 in [9] stating that the subconstructs Ap _m were the only stable subconstructs of Ap is not. The main results, which together prove that the only stable subconstructs are those where a restriction is put on the range of the distances of the objects, are upheld, but it turns out that not only the sets {0} ∪ [m, ∞ ], but actually each closed subsemigroup of [0, ∞ ] determines a stable subconstruct (albeit again in exactly the same way as characterized in [9]). In the first part of our paper, Sections 1 and 2, we develop the general technique, which is totally different to the one from [3], and in Theorem 2.13 we prove the main result for the case of approach spaces. The technique which we develop is also applicable to other cases. Thus, in Section 3, more precisely in Theorems 3.9 and 3.11, we give the complete solution to the corresponding characterization problem for the constructs pq Met ^∞ of pseudo-quasi-metric spaces and p Met ^∞ of pseudometric spaces and in Section 4 we briefly sketch how the technique can be adapted and used to also completely solve the problem in the case of more general types of approach spaces and metric spaces. At the same time, in all cases, we are able to give necessary and sufficient conditions under which two stable subconstructs of one of these topological constructs are concretely isomorphic. It turns out that in all cases there are 2^à₀2^{\aleph_0} non-concretely isomorphic stable subconstructs.     

Selection from poisson processes

Summary There are givenk Poisson processes with mean arrival times 1/λ₁,...1/λ _k . Let λ_[1]≦λ_[2]≦...≦λ_[k] denote the ordered set of values λ₁...,λ_[k]. We consider three procedures for selecting the process corresponding to λ_[k]. The processes are observed until there areN arrivals from any of the given processes, when the processes are observed continuously, or until there are at leastN arrivals, when the processes are observed at successive intervals of time whereN is a pre-determined positive integer. In the continuous case, the process for which theNth arrival time is shortest, is selected. In the discrete case, the selection involves certain randomization. Given (λ_[k]/λ_[k-1])≧0>1, it is shown that the probability of a correct selection (Pcs) is minimized whenθλ_[1]=θλ_[2]=...=θλ_[k-1]=θλ_[k]=θλ, say. The Pcs for this configuration is independent of λ for two of the given procedures, and monotone increasing in λ for the third. The value ofN is determined by a lower bound placed on the value of the Pcs. The problem of selecting from given Poisson processes for the discrete case is related to the problem of selecting from given Poisson populations. An application of the given procedures to a problem of selecting the “most probable event” from a multinomial population, is considered.     

Globalwell-posedness and I method for the fifth order Korteweg-de Vries equation

The Kawahara equation has fewer symmetries than the KdV equation; in particular, it has no invariant scaling transform and is not completely integrable. Thus its analysis requires different methods. We prove that the Kawahara equation is locally well posed in H ^−7/4, using the ideas of an [`(F)] ^s{\overline F ^s}-type space [8]. Then we show that the equation is globally well posed in H ^s for s ≥ −7/4, using the ideas of the “I-method” [7].     

Large deviations and concentration properties for ∇ϕ interface models

We consider the massless field with zero boundary conditions outside D _N ≡D∩ (ℤ ^d /N) (N∈ℤ⁺), D a suitable subset of ℝ ^d , i.e. the continuous spin Gibbs measure ℙ _N on ℝ ^ℤd/N with Hamiltonian given by H(ϕ) = ∑ _{x,y:|x−y|=1} V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈D _N ^C . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D _N . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ _N = ϕ/N. We study various concentration and relaxation properties of the family of random surfaces {ξ _N } and of the induced family of gradient fields ∇ ^N ξ _N as the discretization step 1/N tends to zero (N→∞). In particular, we prove a large deviation principle for {ξ _N } and show that the corresponding rate function is given by ∫ _D σ(∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ℙ _N under the volume constraint, i.e. the constraint that (1/N ^d ) ∑ _x∈DN ξ _N (x) stays in a neighborhood of a fixed volume v > 0, and the hard–wall constraint, i.e. ξ _N (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ _N of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇ ^N ξ _N (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ? _p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer–Sj?strand [31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields. Received: 3 March 1999 / Revised version: 9 August 1999 / Published online: 30 March 2000     

Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing

In compressed sensing, we seek to gain information about a vector x∈ℝ ^N from d ≪ N nonadaptive linear measurements. Candes, Donoho, Tao et al. (see, e.g., Candes, Proc. Intl. Congress Math., Madrid, 2006; Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006; Donoho, IEEE Trans. Inf. Theory 52:1289–1306, 2006) proposed to seek a good approximation to x via ℓ ₁ minimization. In this paper, we show that in the case of Gaussian measurements, ℓ ₁ minimization recovers the signal well from inaccurate measurements, thus improving the result from Candes et al. (Commun. Pure Appl. Math. 59:1207–1223, 2006). We also show that this numerically friendly algorithm (see Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006) with overwhelming probability recovers the signal with accuracy, comparable to the accuracy of the best k-term approximation in the Euclidean norm when k∼d/ln N.     

Analysis of μ R,D -Orthogonality in Affine Iterated Function Systems

Let μ _R,D be a self-affine measure associated with an expanding integer matrix R∈M _n (ℤ) and a finite subset D⊆ℤ ⁿ . In the present paper we study the μ _R,D -orthogonality and compatible pair conditions. We also show that any set of μ _R,D -orthogonal exponentials contains at most 3 elements on the generalized plane Sierpinski gasket and the number 3 is the best.      

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

On Borell-Brascamp-Lieb Inequalities on Metric Measure Spaces

In this paper we introduce the notion of a Borell-Brascamp-Lieb inequality for metric measure spaces (M,d,m) denoted by BBL(K,N) for two numbers K,N ∈ ℝ with N ≥ 1. In the first part we prove that BBL(K,N) holds true on metric measure spaces satisfying a curvature-dimension condition CD(K,N) developed and studied by Lott and Villani in (Ann Math 169:903–991, 2007) as well as by Sturm in (Acta Math 196(1):133–177, 2006). The aim of the second part is to show that BBL(K,N) is stable under convergence of metric measure spaces with respect to the L ₂-transportation distance.