Galois Reconstruction of Finite Quantum Groups

Let be a (small) category and let F:  →  alg_f be a functor, where alg_f is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra A_aut(F) such that F factorizes through a functor :  →  coalg_f(A_aut(F)), where coalg_f(A_aut(F)) is the category of finite-dimensional measured A_aut(F)-comodule algebras. This general reconstruction result allows us to recapture a finite-dimensional Hopf algebra A from the category coalg_f(A) and the forgetful functor ω: coalg_f(A) →  alg_f: we have A  A_aut(ω). Our universal construction is also done in a C*-algebra framework, and we get compact quantum groups in the sense of Woronowicz.     

Density Modulo 1 of Sublacunary Sequences

We prove the existence of real numbers badly approximated by rational fractions whose denominators form a sublacunar sequence. For example, for the ascending sequence s _n , n = 1, 2, 3, ..., generated by the ordered numbers of the form 2ⁱ3^j, i, j = 1, 2, 3, ..., we prove that the set of real numbers α such that inf _n∈ℕ n‖s _n α‖ > 0 is a set of Hausdorff dimension 1. The divergence of the series implies that the Lebesgue measure of those numbers is zero.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 803–813.Original Russian Text Copyright ©2005 by R. K. Akhunzhanov, N. G. Moshchevitin.     

Operators on <Emphasis Type="Italic">C</Emphasis>[0,1] preserving copies of asymptotic ℓ<Subscript>1</Subscript> spaces

Given separable Banach spaces X, Y, Z and a bounded linear operator T:X→Y, then T is said to preserve a copy of Z provided that there exists a closed linear subspace E of X isomorphic to Z and such that the restriction of T to E is an into isomorphism. It is proved that every operator on C([0,1]) which preserves a copy of an asymptotic ℓ₁ space also preserves a copy of C([0,1]).     

On a family of cubics over imaginary quadratic fields</o:p>

Summary We consider the relative Thue equations \[X^3 - t X^2 Y - (t+1) X Y^2 -Y^3=\mu,\] where the parameter <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>t$, the root of unity $\mu$ and the solutions $X$ and $Y$ are integers in the same imaginary quadratic number field. We use Baker's method to find all solutions for $|t|> 2.88 \cdot 10^{33}$.     

A New Method for Constructing Williamson Matrices

For every prime power q 1 (mod 4) we prove the existence of (q; x, y)-partitions of GF(q) with q=x²+4y² for some x, y, which are very useful for constructing SDS, DS and Hadamard matrices. We discuss the transformations of (q; x,y)-partitions and, by using the partitions, construct generalized cyclotomic classes which have properties similar to those of classical cyclotomic classes. Thus we provide a new construction for Williamson matrices of order q².The research supported by NSF of China (No. 10071029).     

3-Wise Exactly 1-Intersecting Families of Sets

Let f(l, t, n) be the maximal size of a family such that any l2 sets of have an exactly t1-element intersection. If l3, it trivially comes from [8] that the optimal families are trivially intersecting (there is a t-element core contained by all the members of the family). Hence it is easy to determine Let g(l,t,n) be the maximal size of an l-wise exaclty t-intersecting family that is not trivially t-intersecting. We give upper and lower bounds which only meet in the following case: g(3, 1, n) = n^2/3(1 + o(1)).     

Coverings by convex bodies and inscribed balls

Let be a Hilbert space. For a closed convex body denote by the supremum of the radiuses of balls contained in . We prove that for every covering of a convex closed body by a sequence of convex closed bodies , . It looks like this fact is new even for triangles in a 2-dimensional space.

     

On singularity of energy measures on self-similar sets

We provide general criteria for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type measures. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on 2-dimensional Sierpinski carpets.Partially supported by Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Encouragement of Young Scientists, 15740089.Mathematics Subject Classification (2000): 28A80 (60G30, 31C25, 60J60)     

On decompositions of continuity and some weaker forms of continuity via idealization

Summary We give decompositions of continuity and some weaker forms of continuity via idealization using the concepts of <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathcal{A_{I}}$-sets, $\alpha \mathcal{A_{I}}$-sets, $B_{1I}$-sets, $B_{2I}$-sets, $B_{3I}$-sets, $\alpha C_{I}$-sets and $WLC_{I}$-sets.     

Second order parabolic systems, optimal regularity, and singular sets of solutions

We present a new, complete approach to the partial regularity of solutions to non-linear, second order parabolic systems of the form

u_t−divA(x,t,u,Du)=0.