Compressing Mappings on Primitive Sequences over Z/(2) and Its Galois Extension

Let f(x) be a strongly primitive polynomial of degree n over Z/(2^e), η(x₀,x₁,…,x_e−2) a Boolean function of e−1 variables and (x₀,x₁,…,x_e−1)=x_e−1+η(x₀,x₁,…,x_e−2)G (f(x),Z/(2^e)) denotes the set of all sequences over Z/(2^e) generated by f(x), F₂^∞ the set of all sequences over the binary field F₂, then the compressing mapping

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is injective, that is, for , G(f(x),Z/(2^e)), = if and only if Φ( )=Φ( ), i.e., ( ₀,…, _e−1)=( ₀,…, _e−1) mod 2. In the second part of the paper, we generalize the above result over the Galois rings.     

Enumerating Permutation Polynomials I: Permutations with Non-Maximal Degree

Let be a conjugation class of permutations of a finite field _q. We consider the function N (q) defined as the number of permutations in for which the associated permutation polynomial has degree <q−2. In 1969, Wells proved a formula for N_[3](q) where [k] denotes the conjugation class of k-cycles. We will prove formulas for N_[k](q) where k=4,5,6 and for the classes of permutations of type [2 2],[3 2],[4 2],[3 3] and [2 2 2]. Finally in the case q=2ⁿ, we will prove a formula for the classes of permutations which are product of 2-cycles.     

A Class of Polynomials over Finite Fields

Generalizing the norm and trace mappings for _q^r/ _q, we introduce an interesting class of polynomials over finite fields and study their properties. These polynomials are then used to construct curves over finite fields with many rational points.     

Identities on Units of Algebraic Algebras

Let be an algebraic algebra over an infinite field K and let ( ) be its group of units. We prove a stronger version of Hartley's conjecture for , namely, if a Laurent polynomial identity (LPI, for short) f = 0 is satisfied in ( ), then satisfies a polynomial identity (PI). We also show that if is non-commutative, then is a PI-ring, provided f = 0 is satisfied by the non-central units of . In particular, is locally finite and, thus, the Kurosh problem has a positive answer for K-algebras whose unit group is LPI. Moreover, f = 0 holds in ( ) if and only if the same identity is satisfied in . The last fact remains true for generalized Laurent polynomial identities, provided that is locally finite.     

Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

Let Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space (k) shares certain properties with the classical Bergman space L²_α of the unit disk. For example, the weighted Bergman kernels k^β_w(z)=(1−wz)^−β, 1β2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H² (k)L²_a, where the inclusion maps are contractive, and M_ζ, the operator of multiplication with the identity function ζ, defines a contraction operator on (k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace of (k) is a Bergman-type kernel as well. For the weighted Bergman kernels k^β this result even holds for all _ζ-invariant subspace of index 1, i.e., whenever the dimension of /ζ is one. Second, if is any multiplier invariant subspace of (k), and if we set *= z , then M_ζ is unitarily equivalent to M_ζ acting on a space of *-valued analytic functions with an operator-valued reproducing kernel of the type

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where V is a contractive analytic function V :  → ( ,  *), for some auxiliary Hilbert space . Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of (k^β), 1β2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs.     

An upper estimate in Turán's pure power sum problem

Let z₁, z₂, …, z_n be complex numbers, and write for their power sums. Let where the minimum is taken under the condition that . In this paper we prove that .     

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.     

Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes

We obtain some effective lower and upper bounds for the number of (n,k)-MDS linear codes over _q. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over _q of the uniform matroid or, alternately, the number of _q-rational points of certain open strata of Grassmannians. The techniques used in the determination of bounds for the number of MDS codes are applied to deduce several geometric properties of certain sections of Grassmannians by coordinate hyperplanes.     

Dilations of C*-Correspondences and the Simplicity of Cuntz–Pimsner Algebras

We develop a dilation theory for C*-correspondences, showing that every C*-correspondence E over a C*-algebra A can be universally embedded into a Hilbert C*-bimodule X_E over a C*-algebra A_E such that the crossed product A_E  is naturally isomorphic to A_EXE  . The Cuntz–Pimsner algebra _E is isomorphic to _{E E}  where _E and _E are quotients of A_E, resp. X_E.  If E is full and the left action is by generalized compact operators, then _E is an equivalence bimodule or, equivalently, an invertible C*-correspondence. In general, _E is merely an essential Hilbert C*-bimodule. Slightly extending previous results on crossed products by equivalence bimodules, we apply our dilation theory to show that for full C*-correspondences over unital C*-algebras, _E is simple if and only if E is minimal and nonperiodic, extending and simplifying results of Muhly and Solel and Kajiwara, Pinzari, and Watatani.     

Young measures generated by a class of integrands: a narrow epicontinuity and applications to Homogenization

Given a subset E of convex functions from into which satisfy growth conditions of order p>1 and an open bounded subset of , we establish the continuity of a map μΦ_μ from the set of all Young measures on equipped with the narrow topology into a set of suitable functionals defined in and equipped with the topology of Γ-convergence. Some applications are given in the setting of periodic and stochastic homogenization.     

On Identity Bases of Epigroup Varieties

Let C _n and C _n be the varieties of all completely regular and of all completely simple semigroups, respectively, whose idempotent generated subsemigroups are periodic with period n. We use Ol'shanski 's theory of geometric group presentations to show that for large odd n these varieties (and similarly defined varieties of epigroups) do not have finitely axiomatizable equational theories.     

Analytic regularity for an operator with Treves curves

We study a partial differential operator with analytic coefficients, which is of the form “sum of squares”. is hypoelliptic on any open subset of , yet possesses the following properties: (1) is not analytic hypoelliptic on any open subset of that contains 0. (2) If u is any distribution defined near with the property that is analytic near 0, then u must be analytic near 0. (3) The point 0 lies on the projection of an infinite number of Treves curves (bicharacteristics).These results are consistent with the Treves conjectures. However, it follows that the analog of Treves conjecture, in the sense of germs, is false.As far as we know, is the first example of a “sum of squares” operator which is not analytic hypoelliptic in the usual sense, yet is analytic hypoelliptic in the sense of germs.     

Levels of multi-continued fraction expansion of multi-formal Laurent series

The multi-continued fraction expansion of a multi-formal Laurent series is a sequence pair consisting of an index sequence and a multi-polynomial sequence . We denote the set of the different indices appearing infinitely many times in by H_∞, the set of the different indices appearing in by H₊, and call |H_∞| and |H₊| the first and second levels of , respectively. In this paper, it is shown how the dimension and basis of the linear space over F(z) (F) spanned by the components of are determined by H_∞ (H₊), and how the components are linearly dependent on the mentioned basis.     

Generalized Green Classes

Generalized Green classes are introduced; some basic properties of members in a generalized Green class are studied. Finally, we apply the results to (Λ), the Ringel–Hall algebra of a finite-dimensional hereditary algebra Λ over a finite field. In particular, it is proved that (Λ) belongs to a suitable generalized Green class, and that there is direct decomposition of spaces (Λ) =  (Λ)  J, where (Λ) is the composition algebra of Λ and J is a twisted Hopf ideal of (Λ), which is exactly the orthogonal complement of (Λ).     

-Modules on Smooth Toric Varieties

Let X be a smooth toric variety. Cox introduced the homogeneous coordinate ring S of X and its irrelevant ideal . Let A denote the ring of differential operators on Spec(S). We show that the category of -modules on X is equivalent to a subcategory of graded A-modules modulo -torsion. Additionally, we prove that the characteristic variety of a -module is a geometric quotient of an open subset of the characteristic variety of the associated A-module and that holonomic -modules correspond to holonomic A-modules.     

The Number of Rational Points of a Class of Artin–Schreier Curves

We determine the number of _q-rational points of a class of Artin–Schreier curves by using recent results concerning evaluations of some exponential sums. In particular, we determine infinitely many new examples of maximal and minimal plane curves in the context of the Hasse–Weil bound.     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Abelian fourfold of Mumford-type and Kuga-Satake varieties

Let X be a complex abelian fourfold of Mumford-type and let V = H¹(X, ). The complex Mumford-Tate group of X is isogenous to SL(2)³. We recover information about the Hodge structure of X using representations of the Lie algebras (2)³ and (8) acting on V . Using these techniques we show that there is a Kuga-Satake variety A associated to X in such a way that A is isogenous to X³².     

A mixed hook-length formula for affine Hecke algebras

Let be the affine Hecke algebra corresponding to the group GL_l over a p-adic field with residue field of cardinality q. We will regard as an associative algebra over the field . Consider the -module W induced from the tensor product of the evaluation modules over the algebras and . The module W depends on two partitions λ of l and μ of m, and on two non-zero elements of the field . There is a canonical operator J acting on W; it corresponds to the trigonometric R-matrix. The algebra contains the finite dimensional Hecke algebra H_l+m as a subalgebra, and the operator J commutes with the action of this subalgebra on W. Under this action, W decomposes into irreducible subspaces according to the Littlewood–Richardson rule. We compute the eigenvalues of J, corresponding to certain multiplicity-free irreducible components of W. In particular, we give a formula for the ratio of two eigenvalues of J, corresponding to the “highest” and the “lowest” components. As an application, we derive the well known q-analogue of the hook-length formula for the number of standard tableaux of shape λ.     

Bounds for the Quality Parameter of Digital Shift Nets over 2

Until now, the concept of digital (t,m,s)-nets is the most powerful concept for the construction of low-discrepancy point sets in the s-dimensional unit cube. In this paper we consider a special class of digital nets over ₂, the so-called shift nets introduced by W. Ch. Schmid, and give bounds for the quality parameter t of such nets.