Isoperimetric inequalities in crystallography

Given a cubic space group (viewed as a finite group of isometries of the torus ), we obtain sharp isoperimetric inequalities for -invariant regions. These inequalities depend on the minimum number of points in an orbit of and on the largest Euler characteristic among nonspherical -symmetric surfaces minimizing the area under volume constraint (we also give explicit estimates of this second invariant for the various crystallographic cubic groups ). As an example, we prove that any surface dividing into two equal volumes with the same (orientation-preserving) symmetries as the A. Schoen minimal Gyroid has area at least (the conjectured minimizing surface in this case is the Gyroid itself whose area is ).

     

Ideals in a perfect closure, linear growth of primary decompositions, and tight closure

This paper is concerned with tight closure in a commutative Noetherian ring of prime characteristic , and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal of has linear growth of primary decompositions, then tight closure (of ) `commutes with localization at the powers of a single element'. It is shown in this paper that, provided has a weak test element, linear growth of primary decompositions for other sequences of ideals of that approximate, in a certain sense, the sequence of Frobenius powers of would not only be just as good in this context, but, in the presence of a certain additional finiteness property, would actually imply that tight closure (of ) commutes with localization at an arbitrary multiplicatively closed subset of .

Work of M. Katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the Frobenius powers of has only finitely many maximal members. This paper develops, through a careful analysis of the ideal theory of the perfect closure of , strategies for showing that tight closure (of a specified ideal of ) commutes with localization at an arbitrary multiplicatively closed subset of and for showing that the union of the associated primes of the tight closures of the Frobenius powers of is actually a finite set. Several applications of the strategies are presented; in most of them it was already known that tight closure commutes with localization, but the resulting affirmative answers to Katzman's question in the various situations considered are believed to be new.

     

Flow Polynomials and Their Asymptotic Limits for Lattice Strip Graphs

We present exact calculations of flow polynomials F(G,q) for lattice strips of various fixed widths L _y 4 and arbitrarily great lengths L _x , with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face fl in the infinite-length limit. We study the zeros of F(G,q) in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite-length limit for the various families of strips. The function fl is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case L _y =1, the maximal point, q _cf , where crosses the real axis, is universal on cyclic and Möbius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q _cf =3 for the infinite square lattice.     

A Complete Solution to a Conjecture on the β-Polynomials of Graphs

In 1990, Gutman and Mizoguchi conjectured that all roots of the -polynomial (G,C,x) of a graph G are real. Since then, there has been some literature intending to solve this conjecture. However, in all existing literature, only classes of graphs were found to show that the conjecture is true; for example, monocyclic graphs, bicyclic graphs, graphs such that no two circuits share a common edge, graphs without 3-matchings, etc, supporting the conjecture in some sense. Yet, no complete solution has been given. In this paper, we show that the conjecture is true for all graphs, and therefore completely solve this conjecture.     

On a Characterization of Spaces of Differentiable Functions

In this paper, we generalize Bernstein's theorem characterizing the space by means of local approximations. The closed interval is partitioned into disjoint half-intervals on which best approximation polynomials of degree divided by the lengths of these half-intervals taken to the power are considered. The existence of the limits of these ratios as the lengths of the half-intervals tend to zero is a criterion for the existence of the th derivative of a function. We prove the theorem in a stronger form and extend it to the spaces .     

Weighted exponential polynomial approximation

A necessary and sufficient condition for completeness of systems of exponentials with a weight in Lp is established and a quantitative relation between the weight and the system of exponential in Lp is obtained by using a generalization of Malliavin's uniqueness theorem about Watson's problem.     

The Paley--Wiener Theorem for the Generalized Radon Transform on the Plane

We consider the problem of reconstructing a function on the disk from its integrals over curves close to straight lines, i.e., the inversion problem for the generalized Radon transform. Necessary and sufficient conditions on the range of the generalized Radon transform are obtained for functions supported in a smaller disk under the additional condition that the curves that do not meet coincide with the corresponding straight lines.     

Testing Lack-of-fit for a Polynomial Errors-in-variables Model

When a regression model is applied as an approximation of underlying model of data, the model checking is important and relevant. In this paper, we investigate the lack-of-fit test for a polynomial error-in-variables model. As the ordinary residuals are biased when there exist measurement errors in covariables,we correct them and then construct a residual-based test of score type. The constructed test is asymptotically chi-squared under null hypotheses. Simulation study shows that the test can maintain the significance level well.The choice of weight functions involved in the test statistic and the related power study are also investigated.The application to two examples is illustrated. The approach can be readily extended to handle more general models.     

Divergent Cesàro and Riesz means of Jacobi and Laguerre expansions

We show that for below certain critical indices there are functions whose Jacobi or Laguerre expansions have almost everywhere divergent Cesàro and Riesz means of order .

     

On bifurcation points of a complex polynomial

Let be a polynomial of degree . Assume that the set there is a sequence s.t. and is finite. We prove that the set of generalized critical values of (hence in particular the set of bifurcation points of ) has at most points. Moreover, We also compute the set effectively.