Modular Decomposition of the Orlik-Terao Algebra

Let ${\mathcal{A}}$ be a collection of n linear hyperplanes in ${\mathbb{k}^\ell}$ , where ${\mathbb{k}}$ is an algebraically closed field. The Orlik-Terao algebra of ${\mathcal{A}}$ is the subalgebra ${{\rm R}(\mathcal{A})}$ of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of ${\mathcal{A}}$ . It determines an irreducible subvariety ${Y (\mathcal{A})}$ of ${\mathbb{P}^{n-1}}$ . We show that a flat X of ${\mathcal{A}}$ is modular if and only if ${{\rm R}(\mathcal{A})}$ is a split extension of the Orlik-Terao algebra of the subarrangement ${\mathcal{A}_X}$ . This provides another refinement of Stanley’s modular factorization theorem [34] and a new characterization of modularity, similar in spirit to the fibration theorem of [27]. We deduce that if ${\mathcal{A}}$ is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.     

On The Structure of A Commutative Banach Algebra Generated By Toeplitz Operators With Quasi-Radial Quasi-Homogeneous Symbols

Let ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ denote the standard weighted Bergman space over the unit ball ${\mathbb{B}^n}$ in ${\mathbb{C}^n}$ . New classes of commutative Banach algebras ${\mathcal{T}(\lambda)}$ which are generated by Toeplitz operators on ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ have been recently discovered in Vasilevski (Integr Equ Oper Theory 66(1):141?C152, 2010). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of ${\mathbb{B}^n}$ . In the present paper we analyze in detail the internal structure of such an algebra in the lowest dimensional case n?=?2. We explicitly describe the maximal ideal space and the Gelfand map of ${\mathcal{T}(\lambda)}$ . Since ${\mathcal{T}(\lambda)}$ is not invariant under the *-operation of ${\mathcal{L}(\mathcal{A}_{\lambda}^2(\mathbb{B}^n))}$ its inverse closedness is not obvious and is proved. We remark that the algebra ${\mathcal{T}(\lambda)}$ is not semi-simple and we derive its radical. Several applications of our results are given and, in particular, we conclude that the essential spectrum of elements in ${\mathcal{T}(\lambda)}$ is always connected.     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type

Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .     

Forbidden Induced Subgraphs for Perfect Matchings

Let ${\mathcal{F}}$ be a family of connected graphs. A graph G is said to be ${\mathcal{F}}$ -free if G is H-free for every graph H in ${\mathcal{F}}$ . We study the problem of characterizing the families of graphs ${\mathcal{F}}$ such that every large enough connected ${\mathcal{F}}$ -free graph of even order has a perfect matching. This problems was previously studied in Plummer and Saito (J Graph Theory 50(1):1–12, 2005), Fujita et al. (J Combin Theory Ser B 96(3):315–324, 2006) and Ota et al. (J Graph Theory, 67(3):250–259, 2011), where the authors were able to characterize such graph families ${\mathcal{F}}$ restricted to the cases ${|\mathcal{F}|\leq 1, |\mathcal{F}| \leq 2}$ and ${|\mathcal{F}| \leq 3}$ , respectively. In this paper, we complete the characterization of all the families that satisfy the above mentioned property. Additionally, we show the families that one gets when adding the condition ${|\mathcal{F}| \leq k}$ for some k ≥ 4.     

An example related to Gregory’s Theorem

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π ₁ ¹ set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model.     

On the functional codes defined by quadrics and Hermitian varieties

In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219–233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729–1739, 2010; Hallez and Storme, Finite Fields Appl 16:27–35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes ${C_2(\mathcal{H})}$ , with ${\mathcal{H}}$ a non-singular Hermitian variety in PG(N, q ²). The codewords of this code are defined by evaluating the points of ${\mathcal{H}}$ in the quadratic polynomials defined over ${\mathbb{F}_{q^2}}$ . We now present the similar results for the functional code ${C_{Herm}(\mathcal{Q})}$ . The codewords of this code are defined by evaluating the points of a non-singular quadric ${\mathcal{Q}}$ in PG(N, q ²) in the polynomials defining the Hermitian varieties of PG(N, q ²).     

Vector partition functions and generalized dahmen and micchelli spaces

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory. Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space $ \mathcal{F}(X) $ contains the partition function $ {\mathcal{P}_{(X)}} $ . We prove a “localization formula” for any f in $ \mathcal{F}(X) $ , inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function $ {\mathcal{P}_{(X)}} $ is a quasi-polynomial on the Minkowski differences $ \mathfrak{c} - B(X) $ , where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Topological Aspects of Differential Chains

In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus??the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or? $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U?M, but not generally in the strong dual topology of? $\mathcal{B}'(U)$ .     

Inscribing nonmeasurable sets

Our main inspiration is the work in paper (Gitik and Shelah in Isr J Math 124(1):221?C242, 2001). We will prove that for a partition ${\mathcal{A}}$ of the real line into meager sets and for any sequence ${\mathcal{A}_n}$ of subsets of ${\mathcal{A}}$ one can find a sequence ${\mathcal{B}_n}$ such that ${\mathcal{B}_{n}}$ ??s are pairwise disjoint and have the same ??outer measure with respect to the ideal of meager sets??. We get?also generalization of this result to a class of ??-ideals posessing Suslin property. However, in that case we use additional set-theoretical assumption about non-existing of quasi-measurable cardinal below continuum.     

Weighted estimates for the multisublinear maximal function

A formulation of the Carleson embedding theorem in the multilinear setting is proved which allows obtaining a multilinear analogue of Sawyer’s two weight theorem for the multisublinear maximal function $\mathcal{M }$ introduced by Lerner et al. (Adv Math 220:1222–1264, 2009). A multilinear version of the $B_p$ theorem from Hytönen and Pérez (Anal PDE, 2013) is also obtained and a mixed $A_{\overrightarrow{ P}}-W_{\overrightarrow{ P}}^{\infty }$ bound for $\mathcal{M }$ is proved as well.     

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$ . If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$ , then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$ , then ${I(\mathcal Y)}$ , the vanishing ideal of ${\mathcal Y}$ , is generated by ${I_2(\mathcal Y)}$ , the homogeneous elements of degree two in ${I(\mathcal Y)}$ . If ${n >2 \deg(E)}$ , then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$ , where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$ . These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$ , an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.     

Necessary and Sufficient Conditions for Associated Differential Operators in Quaternionic Analysis and Applications to Initial Value Problems

This paper deals with the initial value problem of type $$\begin{array}{ll} \qquad \frac{\partial u}{\partial t} = \mathcal{L} u := \sum \limits^3_{i=0} A^{(i)} (t, x) \frac{\partial u}{\partial x_{i}} + B(t, x)u + C(t, x)\\ u (0, x) = u_{0}(x)\end{array}$$ in the space of generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation $$\mathcal{D}_{\lambda}u := \mathcal{D} u + \lambda u = 0,$$ where ${t \in [0, T]}$ is the time variable, x runs in a bounded and simply connected domain in ${\mathbb{R}^{4}, \lambda}$ is a real number, and ${\mathcal{D}}$ is the Cauchy-Fueter operator. We prove necessary and sufficient conditions on the coefficients of the operator ${\mathcal{L}}$ under which ${\mathcal{L}}$ is associated with the operator ${\mathcal{D}_{\lambda}}$ , i.e. ${\mathcal{L}}$ transforms the set of all solutions of the differential equation ${\mathcal{D}_{\lambda}u = 0}$ into solutions of the same equation for fixedly chosen t. This criterion makes it possible to construct operators ${\mathcal{L}}$ for which the initial value problem is uniquely soluble for an arbitrary initial generalized regular function u ₀ by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also generalized regular for each t.     

Quantitative Stratification and the Regularity of Mean Curvature Flow

Let ${\mathcal{M}}$ be a Brakke flow of n-dimensional surfaces in ${\mathbb{R}^N}$ . The singular set ${\mathcal{S} \subset \mathcal{M}}$ has a stratification ${\mathcal{S}^0 \subset \mathcal{S}^1 \subset \cdots \mathcal{S}}$ , where ${X \in \mathcal{S}^j}$ if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata ${\mathcal{S}^j_{\eta, r}}$ satisfying ${\cup_{\eta>0} \cap_{0<r} \mathcal{S}^j_{\eta, r} = \mathcal{S}^j}$ . Sharpening the known parabolic Hausdorff dimension bound ${{\rm dim} \mathcal{S}^j \leq j}$ , we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of ${\mathcal{S}^j_{\eta, r}}$ satisfies ${{\rm Vol} (T_r(\mathcal{S}^j_{\eta, r}) \cap B_1) \leq Cr^{N + 2 - j-\varepsilon}}$ . Our primary application of this is to higher regularity of Brakke flows starting at k-convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of k-convex hypersurfaces, any backwards selfsimilar limit flow with at least k symmetries is in fact a static multiplicity one plane. Then, denoting by ${\mathcal{B}_r \subset \mathcal{M}}$ the set of points with regularity scale less than r, we prove that ${{\rm Vol}(T_r(\mathcal{B}_r)) \leq C r^{n+4-k-\varepsilon}}$ . This gives L ^p -estimates for the second fundamental form for any p < n + 1 ? k. In fact, the estimates are much stronger and give L ^p -estimates for the reciprocal of the regularity scale. These estimates are sharp. The key technique that we develop and apply is a parabolic version of the quantitative stratification method introduced in Cheeger and Naber (Invent. Math., (2)191 2013), 321–339) and Cheeger and Naber (Comm. Pure. Appl. Math, arXiv:1107.3097v1, 2013).     

Geometric flows and differential Harnack estimates for heat equations with potentials

Let $M$ be a closed Riemannian manifold with a Riemannian metric $g_{ij}(t)$ evolving by a geometric flow $\partial _{t}g_{ij} = -2{S}_{ij}$ , where $S_{ij}(t)$ is a symmetric two-tensor on $(M, g(t))$ . Suppose that $S_{ij}$ satisfies the tensor inequality $2{\mathcal H}(S, X)+{\mathcal E}(S,X) \ge 0$ for all vector fields $X$ on $M$ , where ${\mathcal H}(S, X)$ and ${\mathcal E}(S,X)$ are introduced in Definition 1 below. Then, we shall prove differential Harnack estimates for positive solutions to time-dependent forward heat equations with potentials. In the case where $S_{ij} = R_{ij}$ , the Ricci tensor of $M$ , our results correspond to the results proved by Cao and Hamilton (Geom Funct Anal 19:983–989, 2009). Moreover, in the case where the Ricci flow coupled with harmonic map heat flow introduced by Müller (Ann Sci Ec Norm Super 45(4):101–142, 2012), our results derive new differential Harnack estimates. We shall also find new entropies which are monotone under the above geometric flow.     

On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility

What is an adequate extension of an operator ideal $\mathcal{I }$ to the polynomial and multilinear settings? This question motivated the appearance of the interesting concepts of coherent sequences of polynomial ideals and compatibility of a polynomial ideal with an operator ideal, introduced by D. Carando et al. We propose a different approach by considering pairs $(\mathcal{U }_{k},\mathcal{M }_{k})_{k=1}^{\infty }$ , where $(\mathcal{U }_{k})_{k=1}^{\infty }$ is a polynomial ideal and $(\mathcal{M }_{k})_{k=1}^{\infty }$ is a multi-ideal, instead of considering just polynomial ideals. It is our belief that our approach ends a discomfort caused by the previous theory: for real scalars the canonical sequence $(\mathcal{P }_{k})_{k=1}^{\infty }$ of continuous $k$ -homogeneous polynomials is not coherent according to the definition of Carando et al. We apply these new notions to test the pairs of ideals of nuclear and integral polynomials and multilinear operators, the factorisation method and different classes that generalise the concept of absolutely summing operator.     

Harmonic maps of foliated Riemannian manifolds

We study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps between foliated Riemannian manifolds ${(M, {\mathcal{F}}, g)}$ and ${(N, {\mathcal{G}}, h)}$ i.e. smooth critical points ? : M → N of the functional ${E_T (\phi ) = \frac{1}{2} \int_M \| d_T \phi \|^2 \,d \, v_g}$ with respect to variations through foliated maps. In particular we study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic morphisms i.e. smooth foliated maps preserving the basic Laplace equation Δ _B u =  0. We show that CR maps of compact Sasakian manifolds preserving the Reeb flows are weakly stable ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps. We study ${({\mathcal{F}}, {\mathcal{G}}_0 )}$ -harmonic maps into spheres and give foliated analogs to Solomon’s (cf., J Differ Geom 21:151–162, 1985) results.     

On Transversally Elliptic Operators and the Quantization of Manifolds with f-Structure

Given an f-structure ${\varphi}$ on a manifold M, together with a compatible metric g and connection ${\nabla}$ on M, we construct an odd firstorder differential operator D whose principal symbol is of the type considered in [13]. In the special case of a CR-integrable almost ${\mathcal {S}}$ -structure, we show that when ${\nabla}$ is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator D is given by D = ${{\sqrt {2} (\overline {\partial}_b + \overline{\partial}_{b}^{\ast})}}$ , where ${\overline {\partial}_b}$ is the tangential Cauchy-Riemann operator. We then describe two types of “quantization” of manifolds with f-structure that reduce to familiar methods in symplectic geometry in the case that ${\varphi}$ is a compatible almost complex structure, and to the contact quantizations defined in [16] when ${\varphi}$ comes from a contact metric structure.