Factorization formulae on counting zeros of diagonal equations over finite fields

Let be the number of solutions of the equation over the finite field , and let be the number of solutions of the equation . If , let be the least integer represented by . and play important roles in estimating . Based on a partition of , we obtain the factorizations of and , respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for in some special cases.

     

Root closed function algebras on compacta of large dimension

Let be a Hausdorff compact space and let be the algebra of all continuous complex-valued functions on , endowed with the supremum norm. We say that is (approximately) -th root closed if any function from is (approximately) equal to the -th power of another function. We characterize the approximate -th root closedness of in terms of -divisibility of the first Cech cohomology groups of closed subsets of . Next, for each positive integer we construct an -dimensional metrizable compactum such that is approximately -th root closed for any . Also, for each positive integer we construct an -dimensional compact Hausdorff space such that is -th root closed for any .

     

Varieties with a reducible hyperplane section whose two components are hypersurfaces

We classify smooth complex projective varieties of dimension admitting a divisor of the form among their hyperplane sections, both and of codimension in their respective linear spans. In this setting, one of the following holds: 1) is either the Veronese surface in or its general projection to , 2) and is contained in a quadric cone of rank or , 3) and .

     

On the finiteness properties of extension and torsion functors of local cohomology modules

Let be a commutative Noetherian ring with non-zero identity, and ideals of with , and a finitely generated -module. In this paper, for fixed integers and , we study the finiteness of and in several cases.

     

Spreading of quasimodes in the Bunimovich stadium

We consider Dirichlet eigenfunctions of the Bunimovich stadium , satisfying . Write where is the central rectangle and denotes the ``wings,' i.e., the two semicircular regions. It is a topic of current interest in quantum theory to know whether eigenfunctions can concentrate in as . We obtain a lower bound on the mass of in , assuming that itself is -normalized; in other words, the norm of is controlled by times the norm in . Moreover, if is an quasimode, the same result holds, while for an quasimode we prove that the norm of is controlled by times the norm in . We also show that the norm of may be controlled by the integral of along , where is a smooth factor on vanishing at . These results complement recent work of Burq-Zworski which shows that the norm of is controlled by the norm in any pair of strips contained in , but adjacent to .

     

Minimal rank and reflexivity of operator spaces

Let be an -dimensional space of linear operators between the linear spaces and over an algebraically closed field . Improving results of Larson, Ding, and Li and Pan we show the following.

Theorem. Let be a basis of . Assume that every nonzero operator in has rank larger than . Then a linear operator belongs to if and only if for every , is a linear combination of .

     

Henselian valuations and orderings of a commutative ring

The purpose of this paper is to investigate the interplay between henselian valuations and orderings (or semiorderings) of a ring. As a main result, it is proved that for a henselian valuation on a ring , the following statements are equivalent: (1) is compatible with every semiordering of ; (2) is compatible with every ordering of ; (3) Every real prime ideal of is contained in the core of .

     

Isomorphism of Borel full groups

Suppose that and are Polish groups which act in a Borel fashion on Polish spaces and . Let and denote the corresponding orbit equivalence relations, and and the corresponding Borel full groups. Modulo the obvious counterexamples, we show that .

     

Algebraic reflexivity of linear transformations

Let be the set of all linear transformations from to , where and are vector spaces over a field . We show that every -dimensional subspace of is algebraically -reflexive, where denotes the largest integer not exceeding , provided is less than the cardinality of .

     

A structure theorem for quasi-Hopf comodule algebras

If is a quasi-Hopf algebra and is a right -comodule algebra such that there exists a morphism of right -comodule algebras, we prove that there exists a left -module algebra such that . The main difference when comparing to the Hopf case is that, from the multiplication of , which is associative, we have to obtain the multiplication of , which in general is not; for this we use a canonical projection arising from the fact that becomes a quasi-Hopf -bimodule.

     

A condition under which follows from

We will give some sufficient conditions for a -hyponormal operator, , to be normal, and a sufficient condition for a triplet of operators , , with , self-adjoint and unitary such that necessarily satisfies .

     

Boundary spectra of uniform Frostman Blaschke products

A closed set in the unit circle is the boundary spectrum of a uniform Frostman Blaschke product if and only if is nowhere dense in .

     

Densely algebraic bounds for the exponential function

An upper bound for that implies the inequality between the arithmetic and geometric means is generalized with the introduction of a new parameter . The new upper bound is smoothly and densely algebraic in , and valid for for arbitrarily large positive provided that () is sufficiently close to . The range of its validity for negative is investigated through the study of a certain family of quadrinomials.

     

Intersections of conjugacy classes and subgroups of algebraic groups

We show that if is a reductive group, then th roots of conjugacy classes are a finite union of conjugacy classes, and that if is an algebraic overgroup of , then the intersection of with a conjugacy class of is a finite union of -conjugacy classes. These results follow from results on finiteness of unipotent classes in an almost simple algebraic group.

     

Holes in the spectrum of functions generating affine systems

Given a expansive dilation matrix , a measurable set is called a -dilation generator of if is tiled (modulo null sets) by the collection . Our main goal in this paper is to prove certain results relating the support of the Fourier transform of functions generating a wavelet or orthonormal affine system associated with the dilation to an arbitrary set which is a -dilation generator of .

     

Asymptotically harmonic spaces in dimension 3

Let be a Hadamard manifold of dimension whose sectional curvature satisfies and whose curvature tensor satisfies for suitable constants and . We show that is of constant sectional curvature provided is asymptotically harmonic. This was previously only known if admits a compact quotient.

     

A Schottky-type theorem for starlike domains in Banach spaces

We show that if is a starlike domain in a Banach space and is a family of holomorphic functions on that omit two distinct values and is bounded at the origin, then is uniformly bounded on each -bounded set.

     

Steinhaus tiling problem and integral quadratic forms

A lattice in is said to be equivalent to an integral lattice if there exists a real number such that the dot product of any pair of vectors in is an integer. We show that if and is equivalent to an integral lattice, then there is no measurable Steinhaus set for , a set which no matter how translated and rotated contains exactly one vector in .

     

Lipschitz representations of subsets of the cube

We show that for any class of uniformly bounded functions with a reasonable combinatorial dimension, the vast majority of small subsets of the -dimensional combinatorial cube cannot be represented as a Lipschitz image of a subset of , unless the Lipschitz constant is very large. We apply this result to the case when consists of linear functionals of norm at most one on a Hilbert space.

     

Rigidity of surfaces whose geodesic flows preserve smooth foliations of codimension 1

Let be a closed orientable surface. Assume that there exists a codimension one foliation of class in the unit tangent bundle of , whose leaves are invariant under the geodesic flow of . Then, the curvature of is a nonpositive constant.