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Let $X$ be a Noetherian separated and finite dimensional scheme over a field $\mathbb{K}$ of characteristic zero. The goal of this paper is to study deformations of $X$ over a differential graded local Artin $\mathbb{K}$-algebra by using local Tate-Quillen resolutions, i.e., the algebraic analog of the Palamodov's resolvent of a complex space. The above goal is achieved by describing the DG-Lie algebra controlling deformation theory of a diagram of differential graded commutative algebras, indexed by a direct Reedy category., Final version. To appear in Indagationes Mathematicae

Following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure, Isbell, in his pioneering work on injective metric spaces, attempted a characterization of cellular complexes admitting the structure of an injective metric space. A bit later, Mai and Tang confirmed Isbell's conjecture that a simplicial complex is injectively metrizable if and only if it is collapsible. Considerable advances in the understanding, classification and applications of injective envelopes have since been made by Dress, Huber, Sturmfels and collaborators, and most recently by Lang. Unfortunately a combination theory for injective polyhedra is still unavailable. Here we expose a connection to the duality theory of cubings -- simply connected non-positively curved cubical complexes -- which provides a more principled and accessible approach to Mai and Tang's result, providing one with a powerful tool for systematic construction of locally-compact injective metric spaces: Main Theorem. Any complete pointed Gromov--Hausdorff limit of locally-finite piecewise-$\ell_\infty$ cubings is injective. This result may be construed as a combination theorem for the simplest injective polytopes, $\ell_\infty$-parallelopipeds, where the condition for retaining injectivity is the combinatorial non-positive curvature condition on the complex. Thus it represents a first step towards a more comprehensive combination theory for injective spaces. In addition to setting the earlier work on injectively metrizable complexes within its proper context of non-positively curved geometry, this paper is meant to provide the reader with a systematic review of the results~ ---~ otherwise scattered throughout the geometric group theory literature~ ---~ on the duality theory and the geometry of cubings, which make this connection possible., Significantly expanded and revised. 40 pages, 6 figures, to appear in Expositiones Mathematicae (electronic https://doi.org/10.1016/j.exmath.2018.06.001)

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.

In this paper, the second in a series, we continue to study the generalized Lame equation with the Treibich-Verdier potential y ″ ( z ) = [ ∑ k = 0 3 n k ( n k + 1 ) ℘ ( z + ω k 2 | τ ) + B ] y ( z ) , n k ∈ Z ≥ 0 from the monodromy aspect. We prove the existence of a pre-modular form Z r , s n ( τ ) of weight 1 2 ∑ n k ( n k + 1 ) such that the monodromy data ( r , s ) is characterized by Z r , s n ( τ ) = 0 . This generalizes the result in [17] , where the Lame case (i.e. n 1 = n 2 = n 3 = 0 ) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δ u + e u = 16 π δ 0 and Δ u + e u = 8 π ∑ k = 1 3 δ ω k 2 on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $${\rm P.V.} \int_{\mathbb R^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy = 0.$$ Solutions are defined via integration by parts with test functions, as viscosity solutions or via comparison. Our main result states that for bounded solutions, the three different notions coincide., Comment: 21 pages, to appear in Journal de Math\'ematiques Pures et Appliqu\'ees

This paper establishes the mapping properties of integral operators on space of bounded mean oscillation and Campanato spaces. In particular, we have the Hardy’s inequality and the boundedness of the Hadamard fractional integrals on space of bounded mean oscillation and Campanato spaces.

In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, R -graph, central graph. Also, it generalizes the construction of overlay graph (Somodi et al., 2017), and consequently, Q -graph, total graph, and quasitotal graph. We denote this new graph by [ S ( G ) ] H 2 H 1 , where G is a graph and, H 1 and H 2 are suitable graphs corresponding to G . Further, we define several new unary graph operations which becomes particular cases of this construction. We determine the Adjacency and Laplacian spectra of [ S ( G ) ] H 2 H 1 for some classes of graphs G , H 1 and H 2 . From these results, we derive the L -spectrum of the graphs obtained by the unary graph operations mentioned above. As applications, these results enable us to compute the number of spanning trees and Kirchhoff index of these graphs.

This paper introduced the calculus of mappings on finite product of spaces, allowing different degrees of differentiability in the different factors. This enables us to prove an important feature in the infinite-dimensional Lie theory, the exponential law in generalized setting for locally convex spaces and for manifolds modelled on locally convex spaces.

On the real line, one of the interesting problems in wavelet analysis is to obtain a characterization for the orthonormality of a wavelet system. In this paper, we wish to characterize the orthonormality of the wavelet system on the Heisenberg group. We also attempt to study the completeness of this system. Further, we consider the wavelet system associated with twisted translations and dilations on L 2 ( C ) and study its orthonormality.

The Laguerre–Sobolev polynomials form an orthogonal polynomial system with respect to a Sobolev-type inner product associated with the Laguerre measure on the positive half-axis and two point masses M , N > 0 at the origin involving functions and derivatives. These polynomials have attracted much interest over the last two decades, since they became known to satisfy, for any value of the Laguerre parameter α ∈ N 0 , a spectral differential equation of finite order 4 α + 10 . In this paper we establish a new explicit representation of the corresponding differential operator which consists of a number of elementary components depending on α , M , N . Their interaction reveals a rich structure both being useful for applications and as a model for further investigations in the field. In particular, the Laguerre–Sobolev differential operator is shown to be symmetric with respect to the Sobolev-type inner product.

Let n be a nonzero integer. A set of nonzero integers { a 1 , … , a m } such that a i a j + n is a perfect square for all 1 ≤ i j ≤ m is called a D ( n ) - m -tuple. In this paper, we consider the question, for a given integer n which is not a perfect square, how large and how small can be the largest element in a D ( n ) -quadruple. We construct families of D ( n ) -quadruples in which the largest element is of order of magnitude | n | 3 , resp. | n | 2 ∕ 5 .

In this paper, we prove the hot spots conjecture for long rotationally symmetric domains in R n by the continuity method. More precisely, we show that the odd Neumann eigenfunction in x n associated with lowest nonzero eigenvalue is a Morse function on the boundary, which has exactly two critical points and is monotone in the direction from its minimum point to its maximum point. As a consequence, we prove that the Jerison and Nadirashvili's Conjecture 8.3 holds true for rotationally symmetric domains and are also able to obtain a sharp lower bound for the Neumann eigenvalue.

Bessel processes $(X_{t,k})_{t\ge0}$ in $N$ dimensions are classified via associated root systems and multiplicity constants $k\ge0$. They describe interacting Calogero-Moser-Suther\-land particle systems with $N$ particles and are related to $\beta$-Hermite and $\beta$-Laguerre ensembles. Recently, several central limit theorems were derived for fixed $t>0$, fixed starting points, and $k\to\infty$. In this paper we extend the CLT in the A-case from start in 0 to arbitrary starting distributions by using a limit result for the corresponding Bessel functions. We also determine the eigenvalues and eigenvectors of the covariance matrices of the Gaussian limits and study applications to CLTs for the intermediate particles for $k\to\infty$ and then $N\to\infty$., Comment: 20 pages

In this paper we consider weighted polynomial approximation on unbounded multidimensional domains in the spirit of the weighted version of the Weierstrass trigonometric theorem according to which every continuous function on the real line with equal finite limits at ± ∞ is a uniform limit on R of weighted algebraic polynomials of degree 2 n with varying weights ( 1 + t 2 ) − n . We will verify a similar statement in the multivariate setting for a general class of convex weights. We also consider the similar problem of multivariate polynomial approximation with varying weights for some typical non convex weights. In case of non convex weights of the form w α ( x ) ≔ ( 1 + | x 1 | α + . . . + | x d | α ) 1 α , 0 α 1 in order for weighted polynomial approximation to hold for a given continuous function it is necessary that the function vanishes on a certain exceptional set consisting of all coordinate hyperplanes and ∞ . Moreover, in case of rational α this condition is also sufficient for weighted polynomial approximation to hold.

One of the most influential versions of the classical Schwarz–Pick Lemma is probably that of Ahlfors. Pulling back a conformal semimetric on a Riemann surface under any holomorphic map from the open unit disk equipped with a Poincare metric, the curvature of which is assumed to bound from above the curvature of the Riemann surface, he successfully showed that a conformal semimetric to be compared with the Poincare metric is obtained. In the present paper, we give a comparison theorem between two conformal semimetrics of variable curvature in the same spirit. Our main theorem is a local one by its nature, but global results can be derived therefrom.

In this paper, we first study generalizations of Fujita vanishing theorems for q-ample divisors. We then apply them to study positivity of subvarieties with nef normal bundle in the sense of intersection theory. After Ottem's work on ample subschemes, we introduce the notion of a nef subscheme, which generalizes the notion of a subvariety with nef normal bundle. We show that restriction of a pseudoeffective (resp. big) divisor to a nef subvariety is pseudoeffective (resp. big). We also show that ampleness and nefness are transitive properties. We define the weakly movable cone as the cone generated by the pushforward of cycle classes of nef subvarieties via proper surjective maps. This cone contains the movable cone and shares similar intersection-theoretic properties with it, thanks to the aforementioned properties of nef subvarieties. On the other hand, we prove that if Y ⊂ X is an ample subscheme of codimension r and D | Y is q-ample, then D is ( q + r ) -ample. This is analogous to a result proved by Demailly-Peternell-Schneider and Kuronya.

The exact boundary controllability for hyperbolic systems can not be realized generally on a network with loops (see [16] ). In this paper we consider the exact boundary controllability of nodal profile on a network with loops. Precisely speaking, on a network with a triangle-like loop, when nodal profiles are given at various kinds of nodes, different constructive methods can be used to get the corresponding exact boundary controllability of nodal profile for Saint-Venant system by means of boundary controls acting on suitable nodes, respectively. This reveals that the exact boundary controllability of nodal profile is quite different from the usual exact boundary controllability, and has relatively distinctive behaviors and characters.

We make systematic developments on Lawson-Osserman constructions relating to the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977 Acta paper. In particular, we show the existence of boundary functions for which infinitely many analytic solutions and at least one nonsmooth Lipschitz solution exist simultaneously. This newly-discovered amusing phenomenon enriches the understanding on the Lawson-Osserman philosophy., Supercedes arXiv:1610.08162. To appear in the Journal de Math\'ematiques Pures et Appliqu\'ees

We prove a partial fraction decomposition of a quotient of two functions E α ( i t x ) and I α ( i t ) which are defined in terms of the Bessel functions J α and J α + 1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure | x | 2 α + 1 d x 2 α + 1 Γ ( α + 1 ) in [ − 1 , 1 ] , which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials E n , α ( x ) of degree n are defined by considering E α ( t x ) ∕ I α ( t ) as a generating function. It is shown that the sum ∑ m = 1 ∞ 1 ∕ j m , α 2 k , where j m , α are the positive zeros of J α , is equal (up to an explicit factor) to E 2 k − 1 , α ( 1 ) . For α = 1 ∕ 2 this leads to classical results of Euler since the function E 1 ∕ 2 ( x ) is the exponential function and E n , 1 ∕ 2 ( x ) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel’nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [ − 1 , 1 ] .

In this paper we investigate the extent to which the Lovasz Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lovasz Local Lemma is used to produce a function f : X → Y with certain properties, where X is some underlying combinatorial structure and Y is a (typically finite) set. Can this function f be chosen to be Borel or μ -measurable for some probability Borel measure μ on X (assuming that X is a standard Borel space)? In the positive direction, we prove that if the set of constraints put on f is, in a certain sense, “locally finite,” then there is always a Borel choice for f that is “ e -close” to satisfying these constraints, for any e > 0 . Moreover, if the combinatorial structure on X is “induced” by the [ 0 ; 1 ] -shift action of a countable group Γ, then, even without any local finiteness assumptions, there is a Borel choice for f which satisfies the constraints on an invariant conull set (i.e., with e = 0 ). A direct corollary of our results is an upper bound on the measurable chromatic number of the graph G n generated by the shift action of the free group F n that is asymptotically tight up to a factor of at most 2 (which answers a question of Lyons and Nazarov). On the other hand, our result for structures induced by measure-preserving group actions is, at least for amenable groups, sharp in the following sense: a probability measure-preserving action of a countably infinite amenable group satisfies the measurable version of the Lovasz Local Lemma if and only if it admits a factor map to the [ 0 ; 1 ] -shift action. To prove this, we combine the tools of the Ornstein–Weiss theory of entropy for actions of amenable groups with concepts from computability theory, specifically, Kolmogorov complexity.

Corresponding to the Legendre ellipsoid and the LYZ ellipsoid, two new sine ellipsoids are introduced in this paper. These four ellipsoids are closely related in the Pythagorean relation and duality. Several volume inequalities and the valuation properties are obtained for two new ellipsoids.

Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{\bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${\bf H}(W)$ have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of $H_{\bf q}(W)$-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the "Demazure" part of ${\bf H}(W)$ is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra., Comment: AMSLaTex 67 pages, to appear in Advances in Mathematics

In this paper, we study the topological structure of the Hahn field F whose elements are functions from the additive abelian group of rational numbers Q to the real numbers field R , with well-ordered support. After reviewing the algebraic and order structures of the field F , we introduce different vector topologies on F that are induced by families of semi-norms, all of which are weaker than the order or valuation topology. We compare those vector topologies and we identify the weakest one which we denote by τ w and whose properties are similar to those of the weak topology on the Levi-Civita field (Shamseddine, 2010). In particular, we state and prove a convergence criterion for power series in F , τ w that is similar to that for power series on the Levi-Civita field in its weak topology (Shamseddine, 2013).

Let f = h + g ‾ be a normalized and sense-preserving convex harmonic mapping in the unit disk D . In a recent paper, Ponnusamy and Sairam Kaliraj conjectured that there is a θ ∈ [ 0 , 2 π ) such that the function h + e i θ g is convex in D . In this article, we first disprove a more flexible conjecture: “Let f = h + g ‾ be a convex harmonic mapping in the disk D . Then there is a θ ∈ [ 0 , 2 π ) such that the function h + e i θ g is starlike in D ”. In addition, we present an example to show that there exists a harmonic automorphism f = h + g ‾ of a disk such that the function h + e i θ g is convex in only one direction for θ ≠ 0 , and that the analytic function h + g is not starlike therein. The article concludes with a new conjecture.

We study the class P of positive linear functionals φ on L ∞ ( [ 1 , ∞ ) ) for which φ ( f ) = α if lim x → ∞ 1 x ∫ 1 x f ( t ) d t = α . The semigroup of translations f ( x ) ↦ f ( r x ) on L ∞ ( [ 1 , ∞ ) ) , where r ∈ [ 1 , ∞ ) , plays a crucial role in the study of P . In particular, we give three different expressions of their extremal values, which can be considered main results of this paper. We also study linear functionals on l ∞ , the set of all real-valued bounded functions on natural numbers N , which extend Cesaro mean and give similar results about their extremal values, including a functional analytic proof of the classical result of Polya.

The study of the restricted isometry property (RIP) of corrupted random matrices is particularly important in the field of compressed sensing (CS) with corruptions. If a matrix still satisfies the RIP after that a certain portion of rows are erased, then we say that this matrix has the strong restricted isometry property (SRIP). In the field of compressed sensing, random matrices which satisfy certain moment conditions are of particular interest. Among these matrices, those with entries generated from i.i.d. Gaussian or symmetric Bernoulli random variables are often typically considered. Recent studies have shown that matrices with entries generated from i.i.d. Gaussian random variables satisfy the SRIP under arbitrary erasure of rows with high probability. In this paper, we study the erasure robustness property of Bernoulli random matrices. Our main result shows that with overwhelming probability, the SRIP holds for Bernoulli random matrices. Moreover, our analysis leads to a robust version of the famous Johnson–Lindenstrauss lemma for Bernoulli random matrices.

Let F Θ = G ∕ P Θ be a generalized flag manifold, where G is a real non-compact semi-simple Lie group and P Θ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow F Θ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in R n and use them to compute the boundary operator ∂ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of ∂ are 0 or ± 2 (so that Z 2 -homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse–Witten complex is solved.

$\newcommand{\poly}{_{\operatorname{poly}}^{\bullet}}\newcommand{\td}{(\operatorname{td}_{L/A}^{\nabla})^{\frac{1}{2}}}\newcommand{\cx}[1]{\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee)\otimes_R\mathcal{#1}\poly\big)}\newcommand{\cy}[1]{\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{#1}\poly)}$Kontsevich's formality theorem states that there exists an $L_\infty$ quasi-isomorphism from the dgla $T\poly(M)$ of polyvector fields on a smooth manifold $M$ to the dgla $D\poly(M)$ of polydifferential operators on $M$, which extends the classical Hochschild--Kostant--Rosenberg map. In this paper, we extend Kontsevich's formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and $\mathfrak{g}$-manifolds. The spaces $\cx{T}$ and $\cx{D}$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to $L_\infty$ isomorphism. These two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators, respectively. Their corresponding cohomology groups $\cy{T}$ and $\cy{D}$ admit canonical Gerstenhaber algebra structures. We establish the following formality theorem for Lie pairs: there exists an $L_\infty$ quasi isomorphism from $\cx{T}$ to $\cx{D}$ whose first Taylor coefficient is equal to $\operatorname{hkr}\circ\td$. Here $\td$ acts on $\cx{T}$ by contraction. Furthermore, we prove a Kontsevich--Duflo type theorem for Lie pairs: the Hochschild--Kostant--Rosenberg map twisted by the square root of the Todd class of the Lie pair $(L,A)$ is an isomorphism of Gerstenhaber algebras from $\cy{T}$ to $\cy{D}$. As applications, we establish formality theorems and Kontsevich--Duflo type theorems for complex manifolds, foliations, and $\mathfrak{g}$-manifolds. In the case of complex manifolds, we recover the Kontsevich--Duflo theorem of complex geometry., Comment: 55 pages, several typos corrected, some references added, some minor cosmetic changes in the presentation

In this paper we characterize the surjective linear variation norm isometries on JB-algebras. Variation norm isometries are precisely the maps that preserve the maximal deviation, the quantum analogue of the standard deviation, which plays an important role in quantum statistics. Consequently, we characterize the Hilbert's metric isometries on cones in JB-algebras., Comment: 21 pages

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/function field analogy to predict coincidences for limiting homological densities of various sequences Z n ( d 1 , … , d m ) ( X ) of spaces of 0-cycles on manifolds X. The main theorem in this paper is that these topological predictions, which seem strange from a purely topological viewpoint, are indeed true. One obstacle to proving such a theorem is the combinatorial complexity of all possible “collisions” of points. This problem does not arise in the simplest (and classical) case ( m , n ) = ( 1 , 2 ) of configuration spaces. To overcome this obstacle we apply the Bjorner–Wachs theory of lexicographic shellability from algebraic combinatorics.

In this paper, we study smooth complex projective varieties X such that some exterior power ⋀ r T X of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If T X is strictly nef, then X isomorphic to the projective space P n . If ⋀ 2 T X is strictly nef and if X has dimension at least 3, then X is either isomorphic to P n or a quadric Q n .

Given an ample action of an inverse semigroup on a locally compact and Hausdorff topological space, we study the ideal structure of the crossed product algebra associated with it. By developing a theory of induced ideals, we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras. This can be interpreted as an algebraic version of the Effros-Hahn conjecture. Finally, as an application of our result, we study the ideal structure of a Steinberg algebra associated with an ample groupoid by interpreting it as an inverse semigroup crossed product algebra., Comment: This paper has resulted from the author's work as a PhD student in Florianopolis under the advice of professor Ruy Exel. arXiv admin note: substantial text overlap with arXiv:1605.07540 by other authors

In this paper, we construct a class of weighted Bergman spaces such that they can generate H ∞ in the sense of Mobius invariance. H ∞ is also equal to the intersection of these spaces. Applying the relation between H ∞ and these spaces, we show that the characterization via higher order derivatives does not hold for some of these weighted Bergman spaces.

In this paper, we prove that the spectral curve Γ n of the generalized Lame equation with the Treibich–Verdier potential y ″ ( z ) = [ ∑ k = 0 3 n k ( n k + 1 ) ℘ ( z + ω k 2 | τ ) + B ] y ( z ) , n k ∈ Z ≥ 0 can be embedded into the symmetric space Sym N E τ of the N-th copy of the torus E τ , where N = ∑ n k . This embedding induces an addition map σ n ( ⋅ | τ ) from Γ n onto E τ . The main result is to prove that the degree of σ n ( ⋅ | τ ) is equal to ∑ k = 0 3 n k ( n k + 1 ) / 2 . This is the first step toward constructing the pre-modular form associated with this generalized Lame equation.

It seems that the index theory for non-compact spaces has found its ultimate formulation in the realm of coarse spaces and K-theory of related operator algebras. Relative and partitioned index theorems may be mentioned as two important and interesting examples of this program. In this paper we formulate a combination of these two theorems and establish a partitioned-relative index theorem.

Let X be a variety with an action by an algebraic group G. In this paper we discuss various properties of G-equivariant D -modules on X, such as the decompositions of their global sections as representations of G (when G is reductive), and descriptions of the categories that they form. When G acts on X with finitely many orbits, the category of equivariant D -modules is equivalent to the category of finite-dimensional representations of a finite quiver with relations. We describe explicitly these categories for irreducible G-modules X that are spherical varieties, and show that in such cases the quivers are almost always representation-finite (i.e. with finitely many indecomposable representations).

The Boussinesq a b c d system is a 4-parameter set of equations posed in R t × R x , originally derived by Bona, Chen and Saut [11] , [12] as first order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation [17] . Among many particular regimes, depending each of them in terms of the value of the parameters ( a , b , c , d ) present in the equations, the generic regime is characterized by the setting b , d > 0 and a , c 0 . If additionally b = d , the a b c d system is Hamiltonian. The equations in this regime are globally well-posed in the energy space H 1 × H 1 , provided one works with small solutions [12] . In this paper, we investigate decay and the scattering problem in this regime, which is characterized as having (quadratic) long-range nonlinearities, very weak linear decay O ( t − 1 / 3 ) because of the one dimensional setting, and existence of non scattering solutions (solitary waves). We prove, among other results, that for a sufficiently dispersive a b c d systems (characterized only in terms of parameters a , b and c), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone | x | ≤ | t | . We prove this result by constructing three suitable virial functionals in the spirit of works [27] , [28] , and more precisely [42] (valid for the simpler scalar “good Boussinesq” model), leading to global in time decay and control of all local H 1 × H 1 terms. No parity nor extra decay assumptions are needed to prove decay, only small solutions in the energy space.

The Bialynicki-Birula decomposition of the space of λ-connections restricts to the Morse stratification on the moduli space of Higgs bundles and to the partial oper stratification on the de Rham moduli space of holomorphic connections. For both the Morse and partial oper stratifications, every stratum is a holomorphic Lagrangian fibration over a component of the space of complex variations of Hodge structure. In this paper, we generalize known results for the Hitchin section and the space of opers to arbitrary strata. These include the following: a biholomorphic identification of the fibers of the two strata over a stable variation of Hodge structure via the “ħ-conformal limit” of Gaiotto, a proof that the fibers of the Morse and partial oper stratifications are transverse at the base point, and an explicit parametrization of the fibers as half-dimensional affine spaces.

In arxiv:1602.04254, we have defined polynomial Witt vectors functor from vector spaces over a perfect field $k$ of positive characteristic $p$ to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild-Witt complex $WCH_*(A)$ for any associative unital $k$-algebra $A$, with homology groups $WHH_*(A)$. We prove that the group $WHH_0(A)$ coincides with the group of non-commutative Witt vectors defined by Hesselholt, while if $A$ is commutative, finitely generated, and smooth, the groups $WHH_i(A)$ are naturally identified with the terms $W\Omega^i_A$ of the de Rham-Witt complex of the spectrum of $A$., Comment: 64 pages, LaTeX2e

The Eichler-Shimura isomorphism relates holomorphic modular cusps forms of positive even integral weight to cohomology classes. The Haberland formula uses the cup product to give a cohomological formulation of the Petersson scalar product. In this paper we extend Haberland's formula to modular cusp forms of positive real weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle. We may also consider the cup product of two Eichler cocycles. In the classical situation this cup product is almost always zero. However we show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by duality with a space of entire modular forms. The cup product yields a bilinear map over C from pairs of holomorphic modular forms (not necessarily of the same weight, one of them may have large growth at the cusps) to coinvariants in infinite-dimensional modules. To investigate whether this bilinear map is non-trivial we test the result against entire modular forms of a suitable weight. Under some conditions on the weights, this leads to an explicit triple integral, which can be investigated numerically, thus providing evidence that the cup product is non-trivial at least in some situations.

Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and H\"older's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct a frame decomposition for the functions belonging to the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ associated to $L$, and for functions in the Lebesgue spaces $L^p$, $1, Comment: 37 pages, to appear in Journal of Approximation Theory

In this paper, we will study the maximality of functions and a new Cegrell class N m ( Ω ) which is between the classes F m ( Ω ) and E m ( Ω ) . We study the Dirichlet problem, the subsolution theorem and the stability theorem for this class.

In this paper, we show a stochastic expression for the radial processes of Brownian motions on RCD ⁎ ( K , N ) -spaces. It corresponds to the one on Riemannian manifolds. The expression holds for all starting points without exceptional sets. It can be extended beyond the hitting time to the reference point provided the reference point is sufficiently regular. We further prove that the regularity condition is satisfied for almost every reference point. Our results extend the comparison theorems over Alexandrov spaces proved in [36] , [29] .

For a prime p, we call a positive integer n a Frobenius p-number if there exists a finite group with exactly n subgroups of order p^a for some $a\ge 0$. Extending previous results on Sylow's theorem, we prove in this paper that every Frobenius p-number $n\equiv 1\pmod{p^2}$ is a Sylow p-number, i.e., the number of Sylow p-subgroups of some finite group. As a consequence, we verify that 46 is a pseudo Frobenius 3-number, that is, no finite group has exactly 46 subgroups of order 3^a for any $a\ge 0$., Comment: 6 pages, expository, to appear in Expo. Math

In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliation in dimension two and their invariant curves. It appears a characterization of second type foliations and generalized curve foliations as well as a description of the G S V -index in terms of polar curves. We also interpret the proofs concerning the Poincare problem with polar invariants.

Recently, a new partition function associated with Ramanujan's third order mock theta function ω ( q ) was discovered, and subsequently its overpartition analogue was introduced. In this paper, we prove an intriguing identity arising from the study of that analogue, which involves a double series of Hecke-Rogers type. In addition, two further identities will be given. Our proofs heavily rely on formulas from the work of Liu [12] .

This paper studies connections among observable sets, the observability inequality, the Holder-type interpolation inequality and the spectral inequality for the heat equation in R n . We present the characteristic of observable sets for the heat equation. In more detail, we show that a measurable set in R n satisfies the observability inequality if and only if it is γ-thick at scale L for some γ > 0 and L > 0 . We also build up the equivalence among the above-mentioned three inequalities. More precisely, we obtain that if a measurable set in R n satisfies one of these inequalities, then it satisfies others. Finally, we get some weak observability inequalities and weak interpolation inequalities where observations are made over a ball.

Abelian categories provide a self-dual axiomatic context for establishing homomorphism theorems (such as the isomorphism theorems and homological diagram lemmas) for abelian groups, and more generally, modules. In this paper we describe a self-dual context which allows one to establish the same theorems in the case of non-abelian group-like structures; the question of whether such a context can be found has been left open for seventy years. We also formulate and prove in our context a universal isomorphism theorem from which all other isomorphism theorems can be deduced.

The metric d ( A , B ) = tr A + tr B − 2 tr ( A 1 ∕ 2 B A 1 ∕ 2 ) 1 ∕ 2 1 ∕ 2 on the manifold of n × n positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to matrix analysis.

This paper is devoted to the study of the deviation of the (random) average L 2 − error associated to the least-squares regressor over a family of functions F n (with controlled complexity) obtained from n independent, but not necessarily identically distributed, samples of explanatory and response variables, from the minimal (deterministic) average L 2 − error associated to this family of functions, and to some of the corresponding consequences for the problem of consistency. In the i.i.d. case, this specializes as classical questions on least-squares regression problems, but in more general cases, this setting permits a precise investigation in the direction of the study of nonasymptotic errors for least-squares regression schemes in nonstationary settings, which we motivate providing background and examples. More precisely, we prove first two nonasymptotic deviation inequalities that generalize and refine corresponding known results in the i.i.d. case. We then explore some consequences for nonasymptotic bounds of the error both in the weak and the strong senses. Finally, we exploit these estimates to shed new light into questions of consistency for least-squares regression schemes in the distribution-free, nonparametric setting. As an application to the classical theory, we provide in particular a result that generalizes the link between the problem of consistency and the Glivenko-Cantelli property, which applied to regression in the i.i.d. setting over non-decreasing families ( F n ) n of functions permits to create a scheme which is strongly consistent in L 2 under the sole (necessary) assumption of the existence of functions in ∪ n F n which are arbitrarily close in L 2 to the corresponding regressor.