Abstract:
An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we study 4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of skew-symmetric amorphous association schemes, submitted for publication], we discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables and intersection numbers for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S. Y. Song [Commutative association schemes whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996]. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that no 2-class Johnson scheme can admit a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.

Abstract:
We establish the nonexistence of extremizers for a local Fourier restriction inequality on a certain class of planar convex curves whose curvature satisfies a natural assumption. We accomplish this by studying the local behavior of the triple convolution of the arclength measure on the curve with itself, and show in particular that every extremizing sequence concentrates at a point on the curve.

Abstract:
Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated symmetric design is trivial. We present several non-amorphous examples, which are either cyclotomic association schemes, or their fusion schemes. Special properties of symmetric designs guarantee the existence of further fusions, and the two known non-amorphous association schemes of class 4 discovered by van Dam and by the authors, are recovered in this way. We also give another pseudocyclic non-amorphous association scheme of class 7 on GF(2^{21}), and a new pseudocyclic amorphous association scheme of class 5 on GF(2^{12}).

Abstract:
Suzuki (1998) showed that an imprimitive Q-polynomial association scheme with first multiplicity at least three is either Q-bipartite, Q-antipodal, or with four or six classes. The exceptional case with four classes has recently been ruled out by Cerzo and Suzuki (2009). In this paper, we show the nonexistence of the last case with six classes. Hence Suzuki's theorem now exactly mirrors its well-known counterpart for imprimitive distance-regular graphs.

Abstract:
It is shown that linked systems of symmetric designs with $a_1^*=0$ and mutually unbiased bases (MUB) are triply regular association schemes. In this paper, we characterize triple regularity of linked systems of symmetric designs by its Krein number. And we prove that maximal MUB carries a quadruply regular association scheme and characterize the quadruple regularity of MUB by its parameter.

Abstract:
E. R. van Dam gave an example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the cyclotomic scheme of class $45$ on $￥GF(2^{12})$. The aim of this paper is to present a new example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the cyclotomic scheme of class $75$ on $￥GF(2^{20})$. We also propose an infinite family of parameters of association schemes containing both of these two examples.

Abstract:
We present a fully conservative, skew-symmetric finite difference scheme on transformed grids. The skew-symmetry preserves the kinetic energy by first principles, simultaneously avoiding a central instability mechanism and numerical damping. In contrast to other skew-symmetric schemes no special averaging procedures are needed. Instead, the scheme builds purely on point-wise operations and derivatives. Any explicit and central derivative can be used, permitting high order and great freedom to optimize the scheme otherwise. This also allows the simple adaption of existing finite difference schemes to improve their stability and damping properties.

Abstract:
Here, we utilize facts about the big Chern classes discovered by M. Kapranov and independently by M. V. Nori to prove certain results about the nonexistence of certain morphisms from Grassmannian to Grassmannian in characteristic 0. In particular, we show that the filtration F_. on CH^l(X) where F_rCH^l(X) is the linear span of ch_l(V) for all vector bundles V of rank <=r is nontrivial as a theory by showing that for a fixed l>=2, for infinitely many r, for a Grassmannian of r dimensional quotient spaces in an n dimensional vector space (n sufficiently large) over a field of charactersitic 0, ch_l(Q) is in F_rCH^l(G(r,n)) but not in F^(r-1)CH^l(G(r,n)) where Q is the universal quotient bundle of the Grassmannian G(r,n). Separately but using similar methods, we show that for r>=2, if Q denotes the universal quotient bundle of the Grassmannian of r dimensional quotients G(r,n), n>=2r+1, then the class [\psi^pQ] can never be equal to the class of a genuine vector bundle in K(G). In parallel, we obtain a formula for the bid Chern classes in terms of the ch_l. This leads to the identification of proper subfunctors of the Hodge functors H^q(X, \Omega^p) in characteristic 0.

Abstract:
in this paper mainly important properties of skew lattices and symmetric lattices is obtained. a necessary and sufficient condition for skew lattice to be symmetric is obtained. maximal element of a skew lattice is also obtained.

Abstract:
In this paper mainly important properties of skew lattices and symmetric Lattices is obtained. A necessary and sufficient condition for skew lattice to be symmetric is obtained. Maximal element of a skew lattice is also obtained.