Strong CHIP, normality, and linear regularity of convex sets
Andrew
Bakan;
Frank
Deutsch;
Wu
Li
3831-3863
Abstract: We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at $x$ is bounded away from 0 uniformly over all points in the intersection of these convex sets.
Ramsey families of subtrees of the dyadic tree
Vassilis
Kanellopoulos
3865-3886
Abstract: We show that for every rooted, finitely branching, pruned tree $T$of height $\omega$ there exists a family $\mathcal{F}$ which consists of order isomorphic to $T$ subtrees of the dyadic tree $C=\{0,1\}^{<\mathbb{N} }$ with the following properties: (i) the family $\mathcal{F}$ is a $G_\delta$ subset of $2^C$; (ii) every perfect subtree of $C$ contains a member of $\mathcal{F}$; (iii) if $K$ is an analytic subset of $\mathcal{F}$, then for every perfect subtree $S$ of $C$ there exists a perfect subtree $S'$ of $S$ such that the set
$L^p$ improving estimates for some classes of Radon transforms
Chan
Woo
Yang
3887-3903
Abstract: In this paper, we give $L^p-L^q$ estimates and the $L^p$ regularizing estimate of Radon transforms associated to real analytic functions, and we also give estimates of the decay rate of the $L^p$ operator norm of corresponding oscillatory integral operators. For $L^p-L^q$estimates and estimates of the decay rate of the $L^p$ operator norm we obtain sharp results except for extreme points; however, for $L^p$regularity we allow some restrictions on the phase function.
The Poincaré metric and isoperimetric inequalities for hyperbolic polygons
Roger
W.
Barnard;
Petros
Hadjicostas;
Alexander
Yu.
Solynin
3905-3932
Abstract: We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincaré density) of polygons on the hyperbolic plane. Our results include, as limit cases, the isoperimetric inequality for the conformal radius of Euclidean $n$-gons conjectured by G. Pólya and G. Szegö in 1951 and a similar inequality for the hyperbolic $n$-gons of the maximal hyperbolic area conjectured by J. Hersch. Both conjectures have been proved in previous papers by the third author. Our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by A. Yu. Solynin. We also employ the dissymmetrization transformation of V. N. Dubinin. As an important part of our proofs, we obtain monotonicity and convexity results for special combinations of the Euler gamma functions, which appear to have a significant interest in their own right.
The uniform companion for large differential fields of characteristic 0
Marcus
Tressl
3933-3951
Abstract: We show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic $0$, which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields. As an application, we introduce differentially closed ordered fields, differentially closed p-adic fields and differentially closed pseudo-finite fields.
Turing patterns in the Lengyel-Epstein system for the CIMA reaction
Wei-Ming
Ni;
Moxun
Tang
3953-3969
Abstract: The first experimental evidence of the Turing pattern was observed by De Kepper and her associates (1990) on the CIMA reaction in an open unstirred gel reactor, almost 40 years after Turing's prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. In this paper we report some fundamental analytic properties of the Lengyel-Epstein system. Our result also indicates that if either of the initial concentrations of the reactants, the size of the reactor, or the effective diffusion rate, are not large enough, then the system does not admit nonconstant steady states. A priori estimates are fundamental to our approach for this nonexistence result. The degree theory was combined with the a priori estimates to derive existence of nonconstant steady states.
Double forms, curvature structures and the $(p,q)$-curvatures
M.-L.
Labbi
3971-3992
Abstract: We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the $(p,q)$-curvatures. They are a generalization of the $p$-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for $p=0$, the $(0,q)$-curvatures coincide with the H. Weyl curvature invariants, for $p=1$ the $(1,q)$-curvatures are the curvatures of generalized Einstein tensors, and for $q=1$ the $(p,1)$-curvatures coincide with the $p$-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension $n\geq 4$, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.
Coloring-flow duality of embedded graphs
Matt
DeVos;
Luis
Goddyn;
Bojan
Mohar;
Dirk
Vertigan;
Xuding
Zhu
3993-4016
Abstract: Let $G$ be a directed graph embedded in a surface. A map $\phi : E(G) \rightarrow \mathbb{R}$ is a tension if for every circuit $C \subseteq G$, the sum of $\phi$ on the forward edges of $C$ is equal to the sum of $\phi$ on the backward edges of $C$. If this condition is satisfied for every circuit of $G$ which is a contractible curve in the surface, then $\phi$ is a local tension. If $1 \le \vert\phi(e)\vert \le \alpha-1$ holds for every $e \in E(G)$, we say that $\phi$ is a (local) $\alpha$-tension. We define the circular chromatic number and the local circular chromatic number of $G$ by $\chi_{\rm c}(G) =\inf \{\alpha \in \mathbb{R}\mid \mbox{$G$has an$\alpha$-tension} \}$and $\chi_{\rm loc}(G) = \inf \{ \alpha \in \mathbb{R}\mid \mbox{$G$\space has a local$\alpha$-tension} \}$, respectively. The invariant $\chi_{\rm c}$ is a refinement of the usual chromatic number, whereas $\chi_{\rm loc}$ is closely related to Tutte's flow index and Bouchet's biflow index of the surface dual $G^*$. From the definitions we have $\chi_{\rm loc}(G) \le \chi_{\rm c}(G)$. The main result of this paper is a far-reaching generalization of Tutte's coloring-flow duality in planar graphs. It is proved that for every surface $\mathbb{X}$ and every $\varepsilon > 0$, there exists an integer $M$ so that $\chi_{\rm c}(G) \le \chi_{\rm loc}(G)+\varepsilon$ holds for every graph embedded in $\mathbb{X}$ with edge-width at least $M$, where the edge-width is the length of a shortest noncontractible circuit in $G$. In 1996, Youngs discovered that every quadrangulation of the projective plane has chromatic number 2 or 4, but never 3. As an application of the main result we show that such `bimodal' behavior can be observed in $\chi_{\rm loc}$, and thus in $\chi_{\rm c}$for two generic classes of embedded graphs: those that are triangulations and those whose face boundaries all have even length. In particular, if $G$ is embedded in some surface with large edge-width and all its faces have even length $\le 2r$, then $\chi_{\rm c}(G)\in [2,2+\varepsilon] \cup [\frac{2r}{r-1},4]$. Similarly, if $G$ is a triangulation with large edge-width, then $\chi_{\rm c}(G)\in [3,3+\varepsilon] \cup [4,5]$. It is also shown that there exist Eulerian triangulations of arbitrarily large edge-width on nonorientable surfaces whose circular chromatic number is equal to 5.
Inequalities for finite group permutation modules
Daniel
Goldstein;
Robert
M.
Guralnick;
I.
M.
Isaacs
4017-4042
Abstract: If $f$ is a nonzero complex-valued function defined on a finite abelian group $A$ and $\hat f$ is its Fourier transform, then $\vert\operatorname{supp}(f)\vert\vert\operatorname{supp}({\hat f})\vert \ge \vert A\vert$, where $\operatorname{supp}(f)$ and $\operatorname{supp}({\hat f})$ are the supports of $f$ and $\hat f$. In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group $A$ is replaced by a transitive right $G$-set, where $G$ is an arbitrary finite group. We obtain stronger inequalities when the $G$-set is primitive, and we determine the primitive groups for which equality holds. We also explore connections between inequalities of this type and a result of Chebotarëv on complex roots of unity, and we thereby obtain a new proof of Chebotarëv's theorem.
Inverse spectral problem for normal matrices and the Gauss-Lucas theorem
S.
M.
Malamud
4043-4064
Abstract: We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss-Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle's theorem, conjectured by Schoenberg.
The cohomology of the Steenrod algebra and representations of the general linear groups
Nguyên
H. V.
Hung
4065 - 4089
Abstract: Let $Tr_k$ be the algebraic transfer that maps from the coinvariants of certain $GL_k$-representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\mathbb{V} _k)_+) \to \pi_*^S(S^0)$. It has been shown that the algebraic transfer is highly nontrivial, more precisely, that $Tr_k$ is an isomorphism for $k=1, 2, 3$ and that $Tr= \bigoplus_k Tr_k$ is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree $d$ and apply $Sq^0$ repeatedly at most $(k-2)$ times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the $GL_k$-representations. As a consequence, every finite $Sq^0$-family in the coinvariants has at most $(k-2)$ nonzero elements. Two applications are exploited. The first main theorem is that $Tr_k$ is not an isomorphism for $k\geq 5$. Furthermore, for every $k>5$, there are infinitely many degrees in which $Tr_k$ is not an isomorphism. We also show that if $Tr_{\ell}$ detects a nonzero element in certain degrees of $\text{Ker}(Sq^0)$, then it is not a monomorphism and further, for each $k>\ell$, $Tr_k$ is not a monomorphism in infinitely many degrees. The second main theorem is that the elements of any $Sq^0$-family in the cohomology of the Steenrod algebra, except at most its first $(k-2)$ elements, are either all detected or all not detected by $Tr_k$, for every $k$. Applications of this study to the cases $k=4$ and $5$ show that $Tr_4$ does not detect the three families $g$, $D_3$ and $p'$, and that $Tr_5$ does not detect the family $\{h_{n+1}g_n \vert\; n\geq 1\}$.
Poincaré-Hopf inequalities
M.
A.
Bertolim;
M.
P.
Mello;
K.
A.
de Rezende
4091-4129
Abstract: In this article the main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse inequalities to hold. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.
Nonuniform hyperbolicity for singular hyperbolic attractors
Wilmer
J.
Colmenárez Rodriguez
4131-4140
Abstract: In this paper we show nonuniform hyperbolicity for a class of attractors of $C^2$ flows in dimension three. These attractors are partially hyperbolic with central direction being volume expanding, contain dense periodic orbits and hyperbolic singularities of the associated vector field. Classical expanding Lorenz attractors are the main examples in this class.
On the finite embeddability property for residuated ordered groupoids
W.
J.
Blok;
C.
J.
van Alten
4141-4157
Abstract: The finite embeddability property (FEP) for integral, commutative residuated ordered monoids was established by W. J. Blok and C. J. van Alten in 2002. Using Higman's finite basis theorem for divisibility orders we prove that the assumptions of commutativity and associativity are not required: the classes of integral residuated ordered monoids and integral residuated ordered groupoids have the FEP as well. The same holds for their respective subclasses of (bounded) (semi-)lattice ordered structures. The assumption of integrality cannot be dropped in general--the class of commutative, residuated, lattice ordered monoids does not have the FEP--but the class of $n$-potent commutative residuated lattice ordered monoids does have the FEP, for any $n < \omega$.
Prescribing analytic singularities for solutions of a class of vector fields on the torus
Adalberto
P.
Bergamasco;
Sérgio
Luís
Zani
4159-4174
Abstract: We consider the operator $L=\partial_t+(a(t)+ib(t))\partial_x$ acting on distributions on the two-torus $\mathbb T^2,$ where $a$ and $b$ are real-valued, real analytic functions defined on the unit circle $\mathbb T^1.$We prove, among other things, that when $b$ changes sign, given any subset $\Sigma$ of the set of the local extrema of the local primitives of $b,$ there exists a singular solution of $L$ such that the $t-$projection of its analytic singular support is $\Sigma;$ furthermore, for any $\tau\in\Sigma$ and any closed subset $F$ of $\mathbb T^1_x$ there exists $Lu\in C^\omega(\mathbb T^2)$ and $\operatorname{sing\, supp_A}(u)=\{\tau\}\times F.$ We also provide a microlocal result concerning the trace of $u$ at $t=\tau.$
Classification of regular maps of negative prime Euler characteristic
Antonio
Breda
d'Azevedo;
Roman
Nedela;
Jozef
Sirán
4175-4190
Abstract: We give a classification of all regular maps on nonorientable surfaces with a negative odd prime Euler characteristic (equivalently, on nonorientable surfaces of genus $p+2$ where $p$is an odd prime). A consequence of our classification is that there are no regular maps on nonorientable surfaces of genus $p+2$where $p$ is a prime such that $p\equiv 1$ (mod $12$) and $p\ne 13$.
Minkowski valuations
Monika
Ludwig
4191-4213
Abstract: Centroid and difference bodies define $\operatorname{SL}(n)$ equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of $\operatorname{SL}(n)$equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of $\operatorname{SL}(n)$contravariant Minkowski valuations and of $L_p$-Minkowski valuations.
Saari's conjecture for the collinear $n$-body problem
Florin
Diacu;
Ernesto
Pérez-Chavela;
Manuele
Santoprete
4215-4223
Abstract: In 1970 Don Saari conjectured that the only solutions of the Newtonian $n$-body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.
Elliptic planar vector fields with degeneracies
Abdelhamid
Meziani
4225-4248
Abstract: This paper deals with the normalization of elliptic vector fields in the plane that degenerate along a simple and closed curve. The associated homogeneous equation $Lu=0$ is studied and an application to a degenerate Beltrami equation is given.