$$ H^{p}_{w} - L^{p}_{w} $$ Boundedness of Marcinkiewicz Integral

The Marcinkiewicz integral is essentially a Littlewood-Paley g-function, which plays a very important role in harmonic analysis. In this paper we give weaker smoothness conditions assumed on Ω to imply the boundedness of the Marcinkiewicz integral operator μ_Ω, where w belongs to the Muckenhoupt weight class.     

Scaling of conformal blocks and generalized theta functions over $$\overline{\mathcal {M}}_{g,n}$$

By way of intersection theory on \(\overline{\mathcal {M}}_{g,n}\), we show that geometric interpretations for conformal blocks, as sections of ample line bundles over projective varieties, do not have to hold at points on the boundary. We show such a translation would imply certain recursion relations for first Chern classes of these bundles. While recursions can fail, geometric interpretations are shown to hold under certain conditions.     

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$

Maximum distance separable (MDS) convolutional codes are characterized through the property that the free distance meets the generalized Singleton bound. The existence of free MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) was recently discovered in Oued and Sole (IEEE Trans Inf Theory 59(11):7305–7313, 2013) via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over \({\mathbb {Z}}_{p^{r}}\) from a new perspective. We introduce the notions of p-standard form and r-optimal parameters to derive a novel upper bound of Singleton type on the free distance. Moreover, we present a constructive method for building general (non necessarily free) MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) for any given set of parameters.     

On Linear Codes over $${\mathbb{Z}}_{2^{s}}$$

In an earlier paper the authors studied simplex codes of type α and β over and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over The above codes are also shown to satisfy the chain condition.A part of this paper is contained in his Ph.D. Thesis from IIT Kanpur, India     

Elliptic near-MDS codes over $${\mathbb{F}}_5$$

Let Γ₆ be the elliptic curve of degree 6 in PG(5, q) arising from a non-singular cubic curve of PG(2, q) via the canonical Veronese embedding

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

On free $${\mathbb{Z}_{p}}$$ actions on products of spheres

In this paper we consider free actions of large prime order cyclic groups on the product of any number of spheres of the same odd dimension and on products of two spheres of differing odd dimensions. We require only that the action be free on the product as a whole and not each sphere separately. In particular we determine equivariant homotopy type, and for both linear actions and for even numbers of spheres the simple homotopy type and simple structure sets. The results are compared to the analysis and classification done for lens spaces. Similar to lens spaces, the first k-invariant generally determines the homotopy type of many of the quotient spaces, however, the Reidemeister torsion frequently vanishes and many of the homotopy equivalent spaces are also simple homotopy equivalent. Unlike lens spaces, which are determined by their ρ-invariant and Reidemeister torsion, the ρ-invariant here vanishes for even numbers of spheres and linear actions and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and sets of normal invariants are computed in the process.     

On some permutation binomials and trinomials over $$\mathbb {F}_{2^n}$$

In this work, we completely characterize (1) permutation binomials of the form \(x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb {F}_{2^n}[x], n = 2^st, a \in \mathbb {F}_{2^{2t}}^{*}\), and (2) permutation trinomials of the form \(x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb {F}_{2^t}[x]\), where s, t are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.     

Rank two aCM bundles on general determinantal quartic surfaces in $${\mathbb {P}^{3}}$$

Let \(F\subseteq {\mathbb {P}^{3}}\) be a smooth determinantal quartic surface which is general in the Nöther–Lefschetz sense. In the present paper we give a complete classification of locally free sheaves \({\mathcal E}\) of rank 2 on F such that \(h^1(F,{\mathcal E}(th))=0\) for \(t\in \mathbb {Z}\).     

Measures and submeasures on $$\hbox {d}_{0}$$-algebras

The structure of \(\hbox {d}_{0}\)-algebra is a generalization of a D-lattice. We extend to this structure the definitions of measure and submeasure, and investigate their properties. We also investigate the relationships between uniformities and measures (or submeasures).     

Regular complete permutation polynomials over $${\mathbb {F}}_{2^{n}}$$ F 2 n

Designs, Codes and Cryptography - Applications of permutation polynomials in cryptography are closely related to their cycle structures. For example, many block ciphers use permutation polynomials...     

Classification of multiple unilateral weighted shifts by $$\mathbb{A}_{\aleph _0 }$$

In this paper, it is characterized when a multiple unilateral weighted shift belongs to the classes \mathbbA_n ( 1 \leqslant n \leqslant à₀ )\mathbb{A}_n \left( {1 \leqslant n \leqslant \aleph _0 } \right). As a result, we perfect and generalize the previous conclusions given by H. Bercovici, C. Foias, and C. Pearcy. Moreover, we remark that Question 21 posed by Shields has been negatively answered.     

On some classes of linear codes over $${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$$ and their covering radii

In this paper, we define the simplex and MacDonald codes of types \(\alpha \) and \(\beta \) over \({\mathbb {Z}}_{2}{\mathbb {Z}}_{4}\). We also examine the covering radii of these codes. Further, we study the binary images of these codes and prove that the binary image of the simplex codes of type \(\alpha \) meets the Gilbert bound.     

$${\mathcal{C}}_{0}$$ ( X)-algebras,stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

We study permanence properties of the classes of stable and so-called -stable -algebras, respectively. More precisely, we show that a (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for -algebras absorbing the Jiang–Su algebra tensorially). Furthermore, we prove that if is a K ₁-injective strongly self-absorbing -algebra, then A absorbs tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a -algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of -absorbing -algebras. Research supported by: Deutsche Forschungsgemeinschaft (through the SFB 478), by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280), and by the Center for Advanced Studies in Mathematics at Ben-Gurion University     

NONVANISHING OF CONFORMAL BLOCKS DIVISORS ON $$ {\overline{\mathrm{M}}}_{0,n} $$

We introduce and study the problem of finding necessary and sufficient conditions under which a conformal blocks divisor on \( {\overline{\mathrm{M}}}_{0,n} \) is nonzero, solving the problem completely for \( \mathfrak{s}{\mathfrak{l}}_2 \). We give necessary nonvanishing conditions in type A, which are sufficient when theta and critical levels coincide. We also show divisors are subject to additive identities, reflecting a decomposition of the weights and level.     

Weight Function for the Quantum Affine Algebra Uq( $$\widehat{\mathfrak{s}\mathfrak{l}}_3$$ )

We give a precise expression for the universal weight function of the quantum affine algebra U _q ( ). The calculations use the technique of projecting products of Drinfeld currents on the intersections of Borel subalgebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 3–34, October, 2005.     

Determination of simple CM abelian surfaces defined over $${\mathbb{Q}}$$

It is known that in the moduli space of elliptic curves, there exist precisely nine -rational points represented by an elliptic curve with complex multiplication by the maximal order of an imaginary quadratic field. In Murabayashi and Umegaki (J Algebra 235:267–274, 2001) and Umegaki [Determination of all -rational CM-points in the moduli spaces of polarized abelian surfaces, Analytic number theory (Beijng/Kyoto, 1999). Dev. Math., vol 6. Kluwer, Dordrecht, pp 349–357, 2002] we determined all -rational points in (the moduli space of d-polarized abelian surfaces) represented by a d-polarized abelian surface whose endomorphism ring is isomorphic to the maximal order of a quartic CM-field by using the result in Murabayashi (J Reine Angew Math 470:1–26, 1996). In this paper, we prove that polarized abelian surfaces corresponding to these -rational CM points have a -rational model by constructing certain Hecke characters.     

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (\mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂)/(xⁿ-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general.