Abstract:
Let $\mathbb{F}_{2^m}$ be a finite field of characteristic $2$ and $R=\mathbb{F}_{2^m}[u]/\langle u^k\rangle=\mathbb{F}_{2^m} +u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$ ($u^k=0$) where $k\in \mathbb{Z}^{+}$ satisfies $k\geq 2$. For any odd positive integer $n$, it is known that cyclic codes over $R$ of length $2n$ are identified with ideals of the ring $R[x]/\langle x^{2n}-1\rangle$. In this paper, an explicit representation for each cyclic code over $R$ of length $2n$ is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over $R$ of length $2n$ is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over $R$ of length $2n$ are investigated. (AAECC-1522)

Abstract:
polygons, are useful in survey sampling in terms of balanced sampling plans excluding contiguous units (BSECs) and balanced sampling plans excluding adjacent units (BSAs) when neighboring units in a population provide similar information. In this paper, themethod of cyclic shifts is used and cyclic polygonal designs (CPDs) are constructed with block size k 3 and 1, 2, 3, 4, 6,12 for joint distance 3 and v {21, 22, ,100}treatments.

Abstract:
In 1954 H. S. Shapiro proposed an inequality for a cyclic sum in variables. All the numerical evidence indicates that the inequality is true for even and for odd . We give an analytic proof for the case , which implies the former result. The remaining case remains an open problem.

Abstract:
We prove the freeness conjecture of Broue, Malle and Rouquier for the Hecke algebras associated to the primitive complex 2-reflection groups with a single conjugacy class of reflections.

Abstract:
Let $I\subset \mathbb C[x,y,z]$ be an ideal of height 2 and minimally generated by three homogeneous polynomials of the same degree. If $I$ is a locally complete intersection we give a criterion for $\mathbb C[x,y,z]/I$ to be arithmetically Cohen-Macaulay. Since the setup above is most commonly used when $I=J_F$ is the Jacobian ideal of the defining polynomial of a "quasihomogeneous" reduced curve $Y=V(F)$ in $\mathbb P^2$, our main result becomes a criterion for freeness of such divisors. As an application we give an upper bound for the degree of the reduced Jacobian scheme when $Y$ is a free rank 3 central essential arrangement, as well as we investigate the connections between the first syzygies on $J_F$, and the generators of $\sqrt{J_F}$.

Abstract:
We present a construction method for complete sets of cyclic mutually unbiased bases (MUBs) in Hilbert spaces of even prime power dimensions. In comparison to usual complete sets of MUBs, complete cyclic sets possess the additional property of being generated by a single unitary operator. The construction method is based on the idea of obtaining a partition of multi-qubit Pauli operators into maximal commuting sets of orthogonal operators with the help of a suitable element of the Clifford group. As a consequence, we explicitly obtain complete sets of cyclic MUBs generated by a single element of the Clifford group in dimensions $2^m$ for $m=1,2,...,24$.

Abstract:
We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of "second order freeness" and derive the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.

Abstract:
Let ${\mathcal C}= \bigcup_{i=1}^n C_i \subseteq \mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a large class of arrangements of smooth rational curves, giving an inductive tool for understanding the freeness of the module $\Omega^1({\mathcal C})$ of logarithmic differential forms with pole along ${\mathcal C}$. We also show that the analog of Terao's conjecture (freeness of $\Omega^1({\mathcal C})$ is combinatorially determined if ${\mathcal C}$ is a union of lines) is false in this setting.

Abstract:
For elliptic curves E over the rationals Q, the classification according to their torsion subgroups Etors,(Q) of rational points has been studied. When Etors, (Q) are cyclic groups with even orders, the classification is given with explicit critria, and the generators of the torsion groups are also explicitly presented in each case. These results, together with the recent results of Ono for the non-cyclic torsion groups, have completely solved the problem of the explicit classification withE being a rational point of order 2.