Finite vertex-primitive and vertex-biprimitive 2-path-transitive graphs

This paper presents a classification of vertex-primitive and vertex-biprimitive 2-path-transitive graphs which are not 2-arc-transitive. The classification leads to constructions of new examples of half-arc-transitive graphs.     

Locally finite dimensional shift-invariant spaces in

We prove that a locally finite dimensional shift-invariant linear space of distributions must be a linear subspace of some shift-invariant space generated by finitely many compactly supported distributions. If the locally finite dimensional shift-invariant space is a subspace of the Hölder continuous space or the fractional Sobolev space , then the superspace can be chosen to be or , respectively.

     

The algebra

We study a class of endomomorphism algebras of certain -permutation modules over the Hecke algebra of type , whose summands involve both parabolic and quasi-parabolic subgroups, and prove that these algebras are integrally free and quasi-hereditary, and are stable under base change. Some consequences for decomposition numbers are discussed.

     

Character degrees and nilpotence class of finite groups

Let be a finite set of powers of containing 1. It is known that for some choices of , if is a finite -group whose set of character degrees is , then the nilpotence class of is bounded by some integer that depends on , while for some other choices of such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set is class bounding if and only if . In this article we provide a new approach to this problem. Our main result shows the relevance of certain -adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets such that .

     

The maximal orders of finite subgroups in

We give a relatively simple proof that the orthogonal group over the integers is the unique finite subgroup (up to a conjugation) in of the maximal order for .

     

Tarski's finite basis problem via

R. McKenzie has recently associated to each Turing machine a finite algebra having some remarkable properties. We add to the list of properties, by proving that the equational theory of is finitely axiomatizable if halts on the empty input. This completes an alternate (and simpler) proof of McKenzie's negative answer to A. Tarski's finite basis problem. It also removes the possibility, raised by McKenzie, of using to answer an old question of B. Jónsson.

     

The Diophantine equation

Cao has recently proved that, subject to a certain condition on the odd prime , the equation has no solutions in positive integers and , provided also that . It is the object of this note to remove this restriction, and to provide a simple self-contained proof.

     

The action of

We prove that the orbit equivalence relation of the canonical action of on the subsets of is a universal countable Borel equivalence relation.     

and applications

We study an analogue of the classical theory of weights in without assuming that the underlying measure is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderón-Zygmund operators. We also consider commutators of those Calderón- Zygmund operators with bounded mean oscillation functions (), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. Finally, we study self-improving properties of Poincaré-B.M.O. type inequalities within this context; more precisely, we show that if is a locally integrable function satisfying for all cubes , then it is possible to deduce a higher integrability result for , assuming a certain simple geometric condition on the functional .     

Coloring

If and , then define the graph to be the graph whose vertex set is with two vertices being adjacent iff there are distinct such that . For various and and various , typically or , the graph can be properly colored with colors. It is shown that in some cases such a coloring can also have the additional property that if is an isometric embedding, then the restriction of to is a bijection onto .

     

The Chromatic Ext Groups

We compute a certain Ext group related to the chromatic spectral sequence for , the spectrum whose -homology is for each . The answer we get displays a kind of periodicity not seen in the corresponding computation for the sphere spectrum.

     

From

Let be a space of finite type. Set as usual, and define the mod support of by for 0.$"> Call sparse if there is no with

Then we show the relation for any finite type space with being sparse.

As a special case, we have and the main theorem of Ravenel, Wilson and Yagita is also generalized in terms of the mod support.

     

again

It is shown that if, for an entire function,

where , then

In the proof, the zeros of the function are redistributed to minimize the large values of .

     

The

In this paper we analyze the localization of , the fiber of the double suspension map , with respect to . If four cells at the bottom of , the th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum . Let be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map has a lifting to the fiber of and this lifting is shown to be a -periodic equivalence, hence an -equivalence.

     

The number of planar central configurations is finite when mass positions are fixed

In this paper, it is proved that for 2$"> and , if masses are located at fixed points in a plane, then there are only a finite number of -point central configurations that can be generated by positioning a given additional th mass in the same plane. The result is established by proving an equivalent isolation result for planar central configurations of five or more points. Other general properties of central configurations are established in the process. These relate to the amount of centrality lost when a point mass is perturbed and to derivatives associated with central configurations.