Associated primes of graded components of local cohomology modules
Markus
P.
Brodmann;
Mordechai
Katzman;
Rodney
Y.
Sharp
4261-4283
Abstract: The $i$-th local cohomology module of a finitely generated graded module $M$ over a standard positively graded commutative Noetherian ring $R$, with respect to the irrelevant ideal $R_+$, is itself graded; all its graded components are finitely generated modules over $R_0$, the component of $R$ of degree $0$. It is known that the $n$-th component $H^i_{R_+}(M)_n$ of this local cohomology module $H^i_{R_+}(M)$ is zero for all $n>> 0$. This paper is concerned with the asymptotic behaviour of $\operatorname{Ass}_{R_0}(H^i_{R_+}(M)_n)$ as $n \rightarrow -\infty$. The smallest $i$ for which such study is interesting is the finiteness dimension $f$ of $M$ relative to $R_+$, defined as the least integer $j$ for which $H^j_{R_+}(M)$ is not finitely generated. Brodmann and Hellus have shown that $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is constant for all $n < < 0$ (that is, in their terminology, $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is asymptotically stable for $n \rightarrow -\infty$). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that $R$ is a homomorphic image of a regular ring): our answer is precisely the set of contractions to $R_0$ of certain relevant primes of $R$ whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology. Brodmann and Hellus raised various questions about such asymptotic behaviour when $i > f$. They noted that Singh's study of a particular example (in which $f = 2$) shows that $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ need not be asymptotically stable for $n \rightarrow -\infty$. The second main aim of this paper is to determine, for Singh's example, $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ quite precisely for every integer $n$, and, thereby, answer one of the questions raised by Brodmann and Hellus.
Positivity, sums of squares and the multi-dimensional moment problem
S.
Kuhlmann;
M.
Marshall
4285-4301
Abstract: Let $K$ be the basic closed semi-algebraic set in $\mathbb{R}^n$ defined by some finite set of polynomials $S$ and $T$, the preordering generated by $S$. For $K$ compact, $f$ a polynomial in $n$ variables nonnegative on $K$ and real $\epsilon>0$, we have that $f+\epsilon\in T$ [15]. In particular, the $K$-Moment Problem has a positive solution. In the present paper, we study the problem when $K$ is not compact. For $n=1$, we show that the $K$-Moment Problem has a positive solution if and only if $S$ is the natural description of $K$ (see Section 1). For $n\ge 2$, we show that the $K$-Moment Problem fails if $K$ contains a cone of dimension 2. On the other hand, we show that if $K$is a cylinder with compact base, then the following property holds: \begin{displaymath}(\ddagger)\quad\quad\forall f\in \mathbb{R}[X], f\ge 0 \text{... ...hat }\forall \text{ real } \epsilon>0, f+\epsilon q\in T.\quad \end{displaymath} This property is strictly weaker than the one given in [15], but in turn it implies a positive solution to the $K$-Moment Problem. Using results of [9], we provide many (noncompact) examples in hypersurfaces for which ($\ddagger$) holds. Finally, we provide a list of 8 open problems.
Kähler-Einstein metrics for some quasi-smooth log del Pezzo surfaces
Carolina
Araujo
4303-4312
Abstract: Recently Johnson and Kollár determined the complete list of anticanonically embedded quasi-smooth log del Pezzo surfaces in weighted projective $3$-spaces. They also proved that many of those surfaces admit a Kähler-Einstein metric, and that some of them do not have tigers. The aim of this paper is to settle the question of the existence of Kähler-Einstein metrics and tigers for those surfaces for which the question was left open. In order to do so, we will use techniques developed earlier by Nadel, Demailly and Kollár.
Shifted simplicial complexes are Laplacian integral
Art
M.
Duval;
Victor
Reiner
4313-4344
Abstract: We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs.
Tilting theory and the finitistic dimension conjectures
Lidia
Angeleri-Hügel;
Jan
Trlifaj
4345-4358
Abstract: Let $R$ be a right noetherian ring and let $\mathcal{P}^{<\infty}$ be the class of all finitely presented modules of finite projective dimension. We prove that findim $R = n < \infty$ iff there is an (infinitely generated) tilting module $T$ such that pd$T = n$ and $T ^\perp = (\mathcal P^{<\infty})^\perp$. If $R$ is an artin algebra, then $T$ can be taken to be finitely generated iff $\mathcal P^{<\infty}$ is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.
On the Representation Theory of Lie Triple Systems
Terrell
L.
Hodge;
Brian
J.
Parshall
4359-4391
Abstract: In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of [14]. In a final section, we begin a study of the cohomology of Lie triple systems.
An application of the Littlewood restriction formula to the Kostant-Rallis Theorem
Jeb
F.
Willenbring
4393-4419
Abstract: Consider a symmetric pair $(G,K)$ of linear algebraic groups with $\mathfrak{g} \cong \mathfrak{k} \oplus \mathfrak{p}$, where $\mathfrak{k}$ and $\mathfrak{p}$ are defined as the +1 and -1 eigenspaces of the involution defining $K$. We view the ring of polynomial functions on $\mathfrak{p}$ as a representation of $K$. Moreover, set $\mathcal{P}(\mathfrak{p}) = \bigoplus_{d=0}^\infty \mathcal{P}^d(\mathfrak{p})$, where $\mathcal{P}^d(\mathfrak{p})$ is the space of homogeneous polynomial functions on $\mathfrak{p}$ of degree $d$. This decomposition provides a graded $K$-module structure on $\mathcal{P}(\mathfrak{p})$. A decomposition of $\mathcal{P}^d(\mathfrak{p})$is provided for some classical families $(G,K)$ when $d$ is within a certain stable range. The stable range is defined so that the spaces $\mathcal{P}^d(\mathfrak{p})$are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition of $\mathcal{P}^d(\mathfrak{p})$ is interpreted as a $q$-analog of the Kostant-Rallis theorem.
Extensions for finite Chevalley groups II
Christopher
P.
Bendel;
Daniel
K.
Nakano;
Cornelius
Pillen
4421-4454
Abstract: Let $G$ be a semisimple simply connected algebraic group defined and split over the field ${\mathbb{F} }_p$ with $p$ elements, let $G(\mathbb{F} _{q})$ be the finite Chevalley group consisting of the ${\mathbb{F} }_{q}$-rational points of $G$ where $q = p^r$, and let $G_{r}$ be the $r$th Frobenius kernel. The purpose of this paper is to relate extensions between modules in $\text{Mod}(G(\mathbb{F} _{q}))$ and $\text{Mod}(G_{r})$ with extensions between modules in $\text{Mod}(G)$. Among the results obtained are the following: for $r >2$ and $p\geq 3(h-1)$, the $G(\mathbb{F} _{q})$-extensions between two simple $G(\mathbb{F} _{q})$-modules are isomorphic to the $G$-extensions between two simple $p^r$-restricted $G$-modules with suitably ``twisted" highest weights. For $p \geq 3(h-1)$, we provide a complete characterization of $\text{H}^{1}(G(\mathbb{F} _{q}),H^{0}(\lambda))$ where $H^{0}(\lambda)=\text{ind}_{B}^{G} \lambda$ and $\lambda$ is $p^r$-restricted. Furthermore, for $p \geq 3(h-1)$, necessary and sufficient bounds on the size of the highest weight of a $G$-module $V$ are given to insure that the restriction map $\operatorname{H}^{1}(G,V)\rightarrow \operatorname{H}^{1}(G(\mathbb{F} _{q}),V)$ is an isomorphism. Finally, it is shown that the extensions between two simple $p^r$-restricted $G$-modules coincide in all three categories provided the highest weights are ``close" together.
The space of $(\psi,\gamma)$--additive mappings on semigroups
Valerii
A.
Faiziev;
Themistocles
M.
Rassias;
Prasanna
K.
Sahoo
4455-4472
Abstract: In this paper, we introduce the concept of $(\psi,\gamma)$-pseudoadditive mappings from a semigroup into a Banach space, and we provide a generalized solution of Ulam's problem for approximately additive mappings.
Summing inclusion maps between symmetric sequence spaces
Andreas
Defant;
Mieczyslaw
Mastylo;
Carsten
Michels
4473-4492
Abstract: In 1973/74 Bennett and (independently) Carl proved that for $1 \le u \le 2$ the identity map id: $\ell_u \hookrightarrow \ell_2$ is absolutely $(u,1)$-summing, i.e., for every unconditionally summable sequence $(x_n)$in $\ell_u$ the scalar sequence $(\Vert x_n \Vert _{\ell_2})$ is contained in $\ell_u$, which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a $2$-concave symmetric Banach sequence space $E$ the identity map $\text{id}: E \hookrightarrow \ell_2$ is absolutely $(E,1)$-summing, i.e., for every unconditionally summable sequence $(x_n)$ in $E$ the scalar sequence $(\Vert x_n \Vert _{\ell_2})$ is contained in $E$. Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator $T$ on $\ell_2$ with values in a $2$-concave symmetric Banach sequence space $E$ is a multiplier from $\ell_2$ into $E$. Furthermore, we prove an asymptotic formula for the $k$-th approximation number of the identity map $\text{id}: \ell_2^n \hookrightarrow E_n$, where $E_n$ denotes the linear span of the first $n$ standard unit vectors in $E$, and apply it to Lorentz and Orlicz sequence spaces.
Classification of compact complex homogeneous spaces with invariant volumes
Daniel
Guan
4493-4504
Abstract: We solve the problem of the classification of compact complex homogeneous spaces with invariant volumes (see Matsushima, 1961).
A Berger-Green type inequality for compact Lorentzian manifolds
Manuel
Gutiérrez;
Francisco
J.
Palomo;
Alfonso
Romero
4505-4523
Abstract: We give a Lorentzian metric on the null congruence associated with a timelike conformal vector field. A Liouville type theorem is proved and a boundedness for the volume of the null congruence, analogous to a well-known Berger-Green theorem in the Riemannian case, will be derived by studying conjugate points along null geodesics. As a consequence, several classification results on certain compact Lorentzian manifolds without conjugate points on its null geodesics are obtained. Finally, several properties of null geodesics of a natural Lorentzian metric on each odd-dimensional sphere have been found.
Two-weight norm inequalities for the Cesàro means of Hermite expansions
Benjamin
Muckenhoupt;
David
W.
Webb
4525-4537
Abstract: An accurate estimate is obtained of the Cesàro kernel for Hermite expansions. This is used to prove two-weight norm inequalities for Cesàro means of Hermite polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted'' cases. An almost everywhere convergence result is obtained as a corollary. It is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the Cesàro supremum of the means are necessary.
Involutions fixing $\mathbb{RP}^{\text{odd}}\sqcup P(h,i)$, I
Zhi
Lü
4539-4570
Abstract: This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space $\mathbb{RP}^j$ with its normal bundle nonbounding and a Dold manifold $P(h,i)$ with $h>0$ and $i>0$. For odd $h$, the complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on codimension of $P(h,i)$ may not be best possible; for even $h$, the problem may be reduced to the problem for even projective spaces.
Emergence of the Witt group in the cellular lattice of rational spaces
Kathryn
Hess;
Paul-Eugène
Parent
4571-4583
Abstract: We give an embedding of a quotient of the Witt semigroup into the lattice of rational cellular classes represented by formal $2$-cones between $S^{2n}$ and the two-cell complex $X_n=S^{2n}\cup_{[\iota,\iota]}e^{4n}$ ($n\geq1$).
3-connected planar spaces uniquely embed in the sphere
R.
Bruce
Richter;
Carsten
Thomassen
4585-4595
Abstract: We characterize those locally connected subsets of the sphere that have a unique embedding in the sphere -- i.e., those for which every homeomorphism of the subset to itself extends to a homeomorphism of the sphere. This implies that if $\bar G$ is the closure of an embedding of a 3-connected graph in the sphere such that every 1-way infinite path in $G$ has a unique accumulation point in $\bar G$, then $\bar G$ has a unique embedding in the sphere. In particular, the standard (or Freudenthal) compactification of a 3-connected planar graph embeds uniquely in the sphere.
Growth and ergodicity of context-free languages
Tullio
Ceccherini-Silberstein;
Wolfgang
Woess
4597-4625
Abstract: A language $L$ over a finite alphabet $\boldsymbol\Sigma$ is called growth-sensitive if forbidding any set of subwords $F$ yields a sublanguage $L^{F}$ whose exponential growth rate is smaller than that of $L$. It is shown that every ergodic unambiguous, nonlinear context-free language is growth-sensitive. ``Ergodic'' means for a context-free grammar and language that its dependency di-graph is strongly connected. The same result as above holds for the larger class of essentially ergodic context-free languages, and if growth is considered with respect to the ambiguity degrees, then the assumption of unambiguity may be dropped. The methods combine a construction of grammars for $2$-block languages with a generating function technique regarding systems of algebraic equations.
Brownian-time processes: The PDE connection II and the corresponding Feynman-Kac formula
Hassan
Allouba
4627-4637
Abstract: We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth-order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth-order PDEs in which the initial function plays an important role in the PDE itself, not only as initial data. We connect two such types of interesting and new PDEs to BTPs. The first is obtained by running the BTP and then integrating along its path, and the second type of PDEs is related to what we call the Feynman-Kac formula for BTPs. A special case of the second type is a step towards a probabilistic solution to linearized Cahn-Hilliard and Kuramoto-Sivashinsky type PDEs, which we tackle in an upcoming paper.
Gaugeability and conditional gaugeability
Zhen-Qing
Chen
4639-4679
Abstract: New Kato classes are introduced for general transient Borel right processes, for which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Green-tight measures in the classical Brownian motion case. However, the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.
Scaling coupling of reflecting Brownian motions and the hot spots problem
Mihai
N.
Pascu
4681-4702
Abstract: We introduce a new type of coupling of reflecting Brownian motions in smooth planar domains, called scaling coupling. We apply this to obtain monotonicity properties of antisymmetric second Neumann eigenfunctions of convex planar domains with one line of symmetry. In particular, this gives the proof of the hot spots conjecture for some known types of domains and some new ones.