1 2 Contents 0. One-variable calculus 1. The derivative 2. Inverse function and implicit function theorem 3. Fundamental local existence theorem for ODE 4. The Riemann integral in n variables 5. Integration on surfaces 6. Differential forms 7. Products and exterior derivatives of forms 8. The general Stokes formula 9. The classical Gauss, Green, and Stokes formulas 10. Holomorphic functions and...

We consider the problem of enumerating polynomials over Fq, that have certain coefficients prescribed to given values and permute certain substructures of Fq. In particular, we are interested in the group of N -th roots of unity and in the submodules of Fq. We employ the techniques of Konyagin and Pappalardi to obtain results that are similar to their results in [Finite Fields and their Applica...

In this paper a 2-dimensional mapping is investigated in the complex field C for the existence of analytic invariant curves. Employing the method of majorant series, we need to discuss the eigenvalue α of the mapping at a fixed point. Besides the hyperbolic case |α| = 1, we focus on those α on the unit circle S1, i.e., |α| = 1. We discuss not only those α at resonance, i.e., at a root of the un...

The idea to derive topological invariants of smooth manifolds from partitions functions of certain action functionals was suggested by A. Schwarz (1978) and highlighted by E. Witten (1988). Witten interpreted the Jones polynomial of links in the 3-sphere S as a partition function of the Chern-Simons field theory. Witten conjectured the existence of mathematically defined topological invariants ...

We apologize the readers for the mistake made in equation (53) on page 531, which was found by us in paper [1] after its publication in the Robotica Journal. Namely, values of function s defined in equation (53) are always greater or equal to unity: s(t) ≥ 1 for all t ≥ 0. On the other hand, the inverse 1/s(t), which was used in the left-hand side of (53), belongs to the set (0, 1] (function s ...

This paper introduces a new method in the study of representations of groups into affine algebraic groups. In this work we focus on the study of SLnrepresentations and postpone the consideration of the theory in full to future papers. Let G be a group and let R be a commutative ring with unity. There is a commutative R-algebra Repn(G), called the universal representation algebra, and the univer...

social capital that exists in relations among social actors and the existence of fortified civil society, with a modern, powerful and developmental state can advance the development process in developing countries with minimum damage and in shortest time. the elements are the three sides of one single triangle. if unity and balance exist and an organic relationship is made between those element...

In an earlier work, the authors have determined all possible weights n for which there exists a vanishing sum ζ1 + · · · + ζn = 0 of m th roots of unity ζi in characteristic 0. In this paper, the same problem is studied in finite fields of characteristic p. For given m and p, results are obtained on integers n0 such that all integers n ≥ n0 are in the “weight set” Wp(m). The main result (1.3) i...

The existence is shown of a set of (p~ -1) generalized Hadamard matrices H(p, p~'~) of order p2'~, each of which is symmetric and regular. When normalized to become unitary matrices, they form a multiplicative group of order p'~, simply isomorphic to the additive group of GF(pm). The rows of these (p~ 1) matrices are shown to be the image, under the well-known isomorphic mapping relating the pt...

In this paper we propose different notions of Fζ-geometry, for ζ a root of unity, generalizing notions of F1-geometry (geometry over the “field with one element”) based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate Fζ-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as ...