In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z2n−1, and we described the binary weights of the inverses of the Gold and Kasami exponents. We studied the function Invd (n), which for a fixed positive integer d maps integers n ≥ 1 to the least positive residue of the inverse of d modulo 2n − 1, if it exists. In particular, we showe...

We construct dual descriptions of (0, 2) gauged linear sigma models. In some cases, the dual is a (0, 2) Landau-Ginzburg theory, while in other cases, it is a non-linear sigma model. The duality map defines an analogue of mirror symmetry for (0, 2) theories. Using the dual description, we determine the instanton corrected chiral ring for some illustrative examples. This ring defines a (0, 2) ge...

We study the map which sends vectors of polynomials into their Wronski determinants. This defines a projection map of a Grassmann variety which we call a Wronski map. Our main result is computation of degrees of the real Wronski maps. Connections with real algebraic geometry and control theory are described.

We construct Adams operations on higher algebraic K-groups induced by operations such as symmetric powers on any suitable exact category, by constructing an explicit map of spaces, combinatorially deened. The map uses the S-construction of Waldhausen, and deloops (once) earlier constructions of the map.

This paper treats a set of equations of the form X + AF(X)A = Q, where F maps positive definite matrices either into positive definite matrices or into negative definite matrices, and satisfies some monotonicity property. Here A is arbitrary and Q is a positive definite matrix. It is shown that under some conditions an iteration method converges to a positive definite solution. An estimate for ...

Some important examples and explications of Grothendieck duality which should have been included in Chapter 5 of [C1] were overlooked. Several such topics are treated here. One question which can be asked about the trace map H(X,ΩX/k) → k for a proper smooth scheme X of pure dimension n over a field k is how it relates to the trace map for a smooth divisor. Generalizing a little, one can ask wh...

For cryptographic applications, the behaviour of the difference map is of importance and since there are no perfect nonlinear (PN) permutations, almost perfect nonlinear (APN) permutations have attracted a lot of attention in the last decades. Costas permutations, which are also extreme with respect to the behaviour of their difference maps, were originally introduced in connection with applica...

There are many methods for solving a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several third and fourth order methods to find simple zeros. The relationship between the basins of attraction and the c...

Keywords: Basin of attraction Simple roots Nonlinear equations Halley method Euler–Chebyshev method a b s t r a c t There are many methods for solving a nonlinear algebraic equation. Here we introduce a family of Halley-like methods and show that Euler–Chebyshev and BSC are just members of the family. We discuss the conjugacy maps and the effect of the extraneous roots on the basins of attracti...

‎let $r$ be a $*$-prime ring with center‎ ‎$z(r)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $r$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $r$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎suppose that $u$ is an ideal of $r$ such that $u^*=u$‎, ‎and $c_{sigma,tau}={cin‎ ‎r~|~csigma(x)=tau(x)c~mbox{for~all}~xin r}.$ in the present paper‎, ‎it is shown that...