An analogue of the Descartes-Euler formula for infinite graphs and Higuchi's conjecture

Let be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let be the set of vertices, and for every , let denote the (Gaussian) curvature of : minus the sum of incident polygon angles. Descartes showed that whenever may be realized as the surface of a convex polytope in . More generally, if is made of finitely many polygons, Euler's formula is equivalent to the equation where is the Euler characteristic of . Our main theorem shows that whenever converges and there is a positive lower bound on the distance between any pair of vertices in , there exists a compact closed 2-manifold and an integer so that is homeomorphic to minus points, and further .

In the special case when every polygon is regular of side length one and for every vertex , we apply our main theorem to deduce that is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless is a prism, antiprism, or the projective planar analogue of one of these that . This resolves a recent conjecture of Higuchi.

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

Differentiability of spectral functions for symmetric -stable processes

Let be a signed Radon measure in the Kato class and define a Schrödinger type operator on . We show that its spectral bound is differentiable if and is Green-tight.

     

The flat model structure on complexes of sheaves

Let be the category of chain complexes of -modules on a topological space (where is a sheaf of rings on ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on . As a corollary, we have a general framework for doing homological algebra in the category of -modules. I.e., we have a natural way to define the functors and in .

     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

Effective cones of quotients of moduli spaces of stable -pointed curves of genus zero

Let , the moduli space of -pointed stable genus zero curves, and let be the quotient of by the action of on the last marked points. The cones of effective divisors , , are calculated. Using this, upper bounds for the cones generated by divisors with moving linear systems are calculated, , along with the induced bounds on the cones of ample divisors of and . As an application, the cone is analyzed in detail.

     

The McMullen domain: Rings around the boundary

In this paper we show that there are infinitely many rings , around the McMullen domain in the parameter plane for the family of complex rational maps of the form where and . These rings converge to the boundary of the McMullen domain as . The rings contain parameter values that lie at the center of Sierpinski holes. That is, these parameters lie at the center of an open set in the parameter plane in which all of the corresponding maps have Julia sets that are Sierpinski curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic of period either or .

     

Bases for some reciprocity algebras I

For a complex vector space , let be the algebra of polynomial functions on . In this paper, we construct bases for the algebra of all highest weight vectors in , where and for all , and the algebra of highest weight vectors in .

     

Complex symmetric operators and applications II

A bounded linear operator on a complex Hilbert space is called complex symmetric if , where is a conjugation (an isometric, antilinear involution of ). We prove that , where is an auxiliary conjugation commuting with . We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition also extends to the class of unbounded -selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.

     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.

     

Quadratic maps and Bockstein closed group extensions

Let be a central extension of the form where and are elementary abelian -groups. Associated to there is a quadratic map , given by the -power map, which uniquely determines the extension. This quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of . We say that is Bockstein closed if is an ideal closed under the Bockstein operator.

We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map given by yield Bockstein closed extensions.

On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension for some -lattice . In this situation, one may write for a ``binding matrix' with entries in . We find a direct way to calculate the module structure of in terms of . Using this, we study extensions where the lattice is diagonalizable/triangulable and find interesting equivalent conditions to these properties.

     

On the unique representation of families of sets

Let and be uncountable Polish spaces. represents a family of sets provided each set in occurs as an -section of . We say that uniquely represents provided each set in occurs exactly once as an -section of . is universal for if every -section of is in . is uniquely universal for if it is universal and uniquely represents . We show that there is a Borel set in which uniquely represents the translates of if and only if there is a Vitali set. Assuming there is a Borel set with all sections sets and all non-empty sets are uniquely represented by . Assuming there is a Borel set with all sections which uniquely represents the countable subsets of . There is an analytic set in with all sections which represents all the subsets of , but no Borel set can uniquely represent the sets. This last theorem is generalized to higher Borel classes.

     

Minimal polynomials and radii of elements in finite-dimensional power-associative algebras

In the first section of this paper we revisit the definition and some of the properties of the minimal polynomial of an element of a finite-dimensional power-associative algebra over an arbitrary field . Our main observation is that , the minimal polynomial of , may depend not only on , but also on the underlying algebra. More precisely, if is a subalgebra of , and if is the minimal polynomial of in , then may differ from , in which case we have .

In the second section we restrict attention to the case where is either the real or the complex numbers, and define , the radius of an element in , to be the largest root in absolute value of the minimal polynomial of . We show that possesses some of the familiar properties of the classical spectral radius. In particular, we prove that is a continuous function on .

In the third and last section, we deal with stability of subnorms acting on subsets of finite-dimensional power-associative algebras. Following a brief survey, we enhance our understanding of the subject with the help of our findings of the previous section. Our main new result states that if , a subset of an algebra , satisfies certain assumptions, and is a continuous subnorm on , then is stable on if and only if majorizes the radius defined above.

     

Deformed preprojective algebras of generalized Dynkin type

We introduce the class of deformed preprojective algebras of generalized Dynkin graphs (), (), , , and () and prove that it coincides with the class of all basic connected finite-dimensional self-injective algebras for which the inverse Nakayama shift of every non-projective simple module is isomorphic to its third syzygy .

     

Torsion on elliptic curves in isogeny classes

Let be an elliptic curve over a number field and its -isogeny class. We are interested in determining the orders and the types of torsion groups in . For a prime , we give the range of possible types of -primary parts of when runs over . One of our results immediately gives a simple proof of a theorem of Katz on the order of maximal -primary torsion in .

     

Structural interactions of conjugacy closed loops

We study conjugacy closed loops by means of their multiplication groups. Let be a conjugacy closed loop, its nucleus, the associator subloop, and and the left and right multiplication groups, respectively. Put . We prove that the cosets of agree with orbits of , that and that one can define an abelian group on . We also explain why the study of finite conjugacy closed loops can be restricted to the case of nilpotent. Group is shown to be a subgroup of a power of (which is abelian), and we prove that can be embedded into . Finally, we describe all conjugacy closed loops of order .

     

Invariance in and

We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.