A few years ago, I defined a squarefree module over a polynomial ring S = k[x1, . . . , xn] generalizing the Stanley-Reisner ring k[∆] = S/I∆ of a simplicial complex ∆ ⊂ 2. This notion is very useful in the StanleyReisner ring theory. In this paper, from a squarefree S-module M , we construct the k-sheaf M on an (n − 1) simplex B which is the geometric realization of 2. For example, k[∆] is (th...

A few years ago, I defined a squarefree module over a polynomial ring S = k[x1, . . . , xn] generalizing the Stanley-Reisner ring k[∆] = S/I∆ of a simplicial complex ∆ ⊂ 2. This notion is very useful in the StanleyReisner ring theory. In this paper, from a squarefree S-module M , we construct the k-sheaf M on an (n − 1) simplex B which is the geometric realization of 2. For example, k[∆] is (th...

‎A $mu$-way $(v,k,t)$ $trade$ of volume $m$ consists of $mu$‎ ‎disjoint collections $T_1$‎, ‎$T_2‎, ‎dots T_{mu}$‎, ‎each of $m$‎ ‎blocks‎, ‎such that for every $t$-subset of $v$-set $V$ the number of‎ ‎blocks containing this t-subset is the same in each $T_i (1leq‎ ‎i leq mu)$‎. ‎In other words any pair of collections ${T_i,T_j}$‎, ‎$1leq i< j leq mu‎$ is a $(v,k,t)$ trade of volume $m$. In th...

In [24] the first named author developed the theory of singular fibers of generic differentiable maps between manifolds of negative codimension. Here, the codimension of a map f W M !N between manifolds is defined to be k D dim N dim M . For k 0, the fiber over a point in N is a discrete set of points, as long as the map is generic enough, and we can study the topology of such maps by using the...

We examine the conjecture that an 11d E8 bundle, appearing in the calculation of phases in the M-Theory partition function, plays a physical role in M-Theory, focusing on consequences for the classification of string theory solitons. This leads for example to a classification of IIA solitons in terms of that of LE8 bundles in 10d. Since K(Z , 2) approximates LE8 up to π14, this reproduces the K...

The random matrix uniformly distributed over the set of all mby-n matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called k-good random matrices, is studied. It is shown that a k-good random m-by-n matrix with a distribution of minimum support size is uniformly distributed over a maximum-rank-dista...

We study the decays K → ππ in one-loop two-flavour Chiral Perturbation Theory. We provide arguments why the calculation of the coefficient of the pionic chiral logarithm ℓ M = M 2 log M 2 is unique and then perform the calculation. As a check we perform the reduction of the known three-flavour result. Our result can be used to perform the extrapolation to the physical pion mass of direct lattic...