A new Löwenheim-Skolem theorem
Matthew
Foreman;
Stevo
Todorcevic
1693-1715
Abstract: This paper establishes a refinement of the classical Löwenheim-Skolem theorem. The main result shows that any first order structure has a countable elementary substructure with strong second order properties. Several consequences for Singular Cardinals Combinatorics are deduced from this.
Compactness of isospectral potentials
Harold
Donnelly
1717-1730
Abstract: The Schrödinger operator $-\Delta+V$, of a compact Riemannian manifold $M$, has pure point spectrum. Suppose that $V_0$ is a smooth reference potential. Various criteria are given which guarantee the compactness of all $V$satisfying $\operatorname{spec}(-\Delta+V)=\operatorname{spec}(-\Delta+V_0)$. In particular, compactness is proved assuming an a priori bound on the $W_{s,2}(M)$ norm of $V$, where $s>n/2-2$ and $n=\dim M$. This improves earlier work of Brüning. An example involving singular potentials suggests that the condition $s>n/2-2$ is appropriate. Compactness is also proved for non-negative isospectral potentials in dimensions $n\le 9$.
Coisotropic and polar actions on complex Grassmannians
Leonardo
Biliotti;
Anna
Gori
1731-1751
Abstract: The main result of the paper is the complete classification of the compact connected Lie groups acting coisotropically on complex Grassmannians. This is used to determine the polar actions on the same manifolds.
Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions
Wen-Xiu
Ma;
Yuncheng
You
1753-1778
Abstract: A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.
From \boldmath$\Gamma$-spaces to algebraic theories
Bernard
Badzioch
1779-1799
Abstract: The paper examines semi-theories, that is, formalisms of the type of the $\Gamma$-spaces of Segal which describe homotopy structures on topological spaces. It is shown that for any semi-theory one can find an algebraic theory describing the same structure on spaces as the original semi-theory. As a consequence one obtains a criterion for establishing when two semi-theories describe equivalent homotopy structures.
Existence and asymptotic behavior for a singular parabolic equation
Juan
Dávila;
Marcelo
Montenegro
1801-1828
Abstract: We prove global existence of nonnegative solutions to the singular parabolic equation $u_t -\Delta u + \raise 1.5pt\hbox{$\chi$}_{ \{ u>0 \} } ( -u^{-\beta} + \lambda f(u) )=0$ in a smooth bounded domain $\Omega\subset\mathbb{R} ^N$ with zero Dirichlet boundary condition and initial condition $u_0 \in C(\Omega)$, $u_0 \geq 0$. In some cases we are also able to treat $u_0 \in L^\infty(\Omega)$. Then we show that if the stationary problem admits no solution which is positive a.e., then the solutions of the parabolic problem must vanish in finite time, a phenomenon called ``quenching''. We also establish a converse of this fact and study the solutions with a positive initial condition that leads to uniqueness on an appropriate class of functions.
The baseleaf preserving mapping class group of the universal hyperbolic solenoid
Chris
Odden
1829-1858
Abstract: Given a closed surface $X$, the covering solenoid $\mathbf{X}_\infty$ is by definition the inverse limit of all finite covering surfaces over $X$. If the genus of $X$ is greater than one, then there is only one homeomorphism type of covering solenoid; it is called the universal hyperbolic solenoid. In this paper we describe the topology of $\Gamma(\mathbf{X}_\infty)$, the mapping class group of the universal hyperbolic solenoid. Central to this description is the notion of a virtual automorphism group. The main result is that there is a natural isomorphism of the baseleaf preserving mapping class group of $\mathbf{X}_\infty$ onto the virtual automorphism group of $\pi_1(X,*)$. This may be regarded as a genus independent generalization of the theorem of Dehn, Nielsen, Baer, and Epstein that the pointed mapping class group $\Gamma(X,*)$ is isomorphic to the automorphism group of $\pi_1(X,*)$.
Graphs of zeros of analytic families
Alexander
Brudnyi
1859-1875
Abstract: Let $\mathcal{F}:=\{f_{\lambda}\}$ be a family of holomorphic functions in a domain $D\subset\mathbb{C}$ depending holomorphically on $\lambda\in U\subset\mathbb{C}^{n}$. We study the distribution of zeros of $\{f_{\lambda}\}$ in a subdomain $R\subset\subset D$ whose boundary is a closed non-singular analytic curve. As an application, we obtain several results about distributions of zeros of families of generalized exponential polynomials and displacement maps related to certain ODE's.
Poset fiber theorems
Anders
Björner;
Michelle
L.
Wachs;
Volkmar
Welker
1877-1899
Abstract: Suppose that $f:P \to Q$ is a poset map whose fibers $f^{-1}(Q_{\le q})$ are sufficiently well connected. Our main result is a formula expressing the homotopy type of $P$ in terms of $Q$ and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, Cohen-Macaulay, and equivariant versions are given, and some applications are discussed.
Plane Cremona maps, exceptional curves and roots
Maria
Alberich-Carramiñana
1901-1914
Abstract: We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection $-1$ and $-2$, respectively) on a smooth complex projective surface $S$ which admits a birational morphism $\pi$ to $\mathbb{P} ^{2}$. The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of $C$ is in the $W$-orbit of the class of the total transform of some point blown up by $\pi$ if $C$ is exceptional, or in the $W$-orbit of a simple root if $C$ is root, where $W$ is the Weyl group acting on $\operatorname{Pic}S$; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor $C$ is a necessary and sufficient condition in order that $\pi_{\ast} (C)$ could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.
Blow-up examples for second order elliptic PDEs of critical Sobolev growth
Olivier
Druet;
Emmanuel
Hebey
1915-1929
Abstract: Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$, and $\Delta_g = -div_g\nabla$ be the Laplace-Beltrami operator. Let also $2^\star$ be the critical Sobolev exponent for the embedding of the Sobolev space $H_1^2(M)$ into Lebesgue's spaces, and $h$ be a smooth function on $M$. Elliptic equations of critical Sobolev growth such as \begin{displaymath}(E)\qquad\qquad\qquad\qquad\qquad\qquad\Delta_gu + hu = u^{2^\star-1} \qquad\qquad\qquad\qquad\qquad\qquad\end{displaymath} have been the target of investigation for decades. A very nice $H_1^2$-theory for the asymptotic behaviour of solutions of such equations has been available since the 1980's. The $C^0$-theory was recently developed by Druet-Hebey-Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of $(E)$. It was used as a key point by Druet to prove compactness results for equations such as $(E)$. An important issue in the field of blow-up analysis, in particular with respect to previous work by Druet and Druet-Hebey-Robert, is to get explicit nontrivial examples of blowing-up sequences of solutions of $(E)$. We present such examples in this article.
Associativity of crossed products by partial actions, enveloping actions and partial representations
M.
Dokuchaev;
R.
Exel
1931-1952
Abstract: Given a partial action $\alpha$ of a group $G$ on an associative algebra $\mathcal{A}$, we consider the crossed product $\mathcal{A}\rtimes _\alpha G$. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of $\mathcal{A}\rtimes_\alpha G$ obtained in the context of $C^*$-algebras. In particular, we prove that $\mathcal{A} \rtimes_{\alpha} G$ is associative, provided that $\mathcal{A}$ is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.
Extension-orthogonal components of preprojective varieties
Christof
Geiß;
Jan
Schröer
1953-1962
Abstract: Let $Q$ be a Dynkin quiver, and let $\Lambda$ be the corresponding preprojective algebra. Let ${\mathcal C} = \{ C_i \mid i \in I \}$ be a set of pairwise different indecomposable irreducible components of varieties of $\Lambda$-modules such that generically there are no extensions between $C_i$ and $C_j$ for all $i,j$. We show that the number of elements in ${\mathcal C}$ is at most the number of positive roots of $Q$. Furthermore, we give a module-theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.
Toric residue and combinatorial degree
Ivan
Soprounov
1963-1975
Abstract: Consider an $n$-dimensional projective toric variety $X$defined by a convex lattice polytope $P$. David Cox introduced the toric residue map given by a collection of $n+1$ divisors $(Z_0,\dots,Z_n)$ on $X$. In the case when the $Z_i$ are $\mathbb{T}$-invariant divisors whose sum is $X\setminus\mathbb{T}$, the toric residue map is the multiplication by an integer number. We show that this number is the degree of a certain map from the boundary of the polytope $P$ to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals $I$ of the homogeneous coordinate ring of $X$. We give a necessary and sufficient condition for a homogeneous polynomial of semiample degree to belong to $I$in terms of geometry of toric varieties and combinatorics of fans. Both results have applications to the problem of constructing an element of residue one for semiample degrees.
Threefolds with vanishing Hodge cohomology
Jing
Zhang
1977-1994
Abstract: We consider algebraic manifolds $Y$ of dimension 3 over $\mathbb{C}$ with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the $D$-dimension of $X$ is not zero, then $Y$ is a fibre space over a smooth affine curve $C$ (i.e., we have a surjective morphism from $Y$to $C$ such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of $X$ is $-\infty$ and the $D$-dimension of $X$ is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of $Y$.
Finite quotients of rings and applications to subgroup separability of linear groups
Emily
Hamilton
1995-2006
Abstract: In this paper we apply results from algebraic number theory to subgroup separability of linear groups. We then state applications to subgroup separability of free products with amalgamation of hyperbolic $3$-manifold groups.
Cut numbers of $3$-manifolds
Adam
S.
Sikora
2007-2020
Abstract: We investigate the relations between the cut number, $c(M),$ and the first Betti number, $b_1(M),$ of $3$-manifolds $M.$ We prove that the cut number of a ``generic'' $3$-manifold $M$ is at most $2.$ This is a rather unexpected result since specific examples of $3$-manifolds with large $b_1(M)$ and $c(M)\leq 2$ are hard to construct. We also prove that for any complex semisimple Lie algebra $\mathfrak g$ there exists a $3$-manifold $M$ with $b_1(M)=dim\, \mathfrak g$ and $c(M)\leq rank\, \mathfrak g.$ Such manifolds can be explicitly constructed.
Mansfield's imprimitivity theorem for full crossed products
S.
Kaliszewski;
John
Quigg
2021-2042
Abstract: For any maximal coaction $(A,G,\delta)$ and any closed normal subgroup $N$ of $G$, there exists an imprimitivity bimodule $Y_{G/N}^G(A)$ between the full crossed product $A\times_\delta G\times_{\widehat\delta\vert}N$ and $A\times_{\delta\vert}G/N$, together with $\operatorname{Inf}\widehat{\widehat\delta\vert}-\delta^{\text{dec}}$ compatible coaction $\delta_Y$ of $G$. The assignment $(A,\delta)\mapsto (Y_{G/N}^G(A),\delta_Y)$implements a natural equivalence between the crossed-product functors `` ${}\times G\times N$'' and `` ${}\times G/N$'', in the category whose objects are maximal coactions of $G$ and whose morphisms are isomorphism classes of right-Hilbert bimodule coactions of $G$.
On the theory of elliptic functions based on ${}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{1}{2};z)$
Li-Chien
Shen
2043-2058
Abstract: Based on properties of the hypergeometric series ${}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{1}{2};z)$, we develop a theory of elliptic functions that shares many striking similarities with the classical Jacobian elliptic functions.
Small deviations of weighted fractional processes and average non--linear approximation
Mikhail
A.
Lifshits;
Werner
Linde
2059-2079
Abstract: We investigate the small deviation problem for weighted fractional Brownian motions in $L_q$-norm, $1\le q\le\infty$. Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$. If $1/r:=H+1/q$, then our main result asserts \begin{displaymath}\lim_{\varepsilon\to 0} \varepsilon^{1/H}\log \mathbb{P}\left... ...c(H,q)\cdot\left\Vert{\rho}\right\Vert _{L_r(0,\infty)}^{1/H}, \end{displaymath} provided the weight function $\rho$satisfies a condition slightly stronger than the $r$-integrability. Thus we extend earlier results for Brownian motion, i.e. $H=1/2$, to the fractional case. Our basic tools are entropy estimates for fractional integration operators, a non-linear approximation technique for Gaussian processes as well as sharp entropy estimates for $l_q$-sums of linear operators defined on a Hilbert space.
Generalized spherical functions on reductive $p$-adic groups
Jing-Song
Huang;
Marko
Tadic
2081-2117
Abstract: Let $G$ be the group of rational points of a connected reductive $p$-adic group and let $K$ be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on $G$ are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of $K$. In this paper we show that spaces of generalized spherical functions are finite dimensional. We compute dimensions of spaces of generalized spherical functions on a Zariski open dense set of infinitesimal characters. As a consequence, we get that on that Zariski open dense set of infinitesimal characters, the dimension of the space of generalized spherical functions is constant on each connected component of infinitesimal characters. We also obtain the formula for the generalized spherical functions by integrals of Eisenstein type. On the Zariski open dense set of infinitesimal characters that we have mentioned above, these integrals then give the formula for all the generalized spherical functions. At the end, let as mention that among others we prove that there exists a Zariski open dense subset of infinitesimal characters such that the category of smooth representations of $G$ with fixed infinitesimal character belonging to this subset is semi-simple.