Abstract:
The paper summarizes the contributions of the three philosophies of mathematics—logicism, intuitionism-constructivism (constructivism for short) and formalism and their rectification—which constitute the new foundations of mathematics. The critique of the traditional foundations of mathematics reveals a number of errors including inconsistency (contradiction or paradox) and undefined and vacuous concepts which fall under ambiguity. Critique of the real and complex number systems reveals similar defects all of which are responsible not only for the unsolved long standing problems of foundations but also of traditional mathematics such as the 379-year-old Fermat’s last theorem (FLT) and 274-year-old Goldbach’s conjecture. These two problems require rectification of these defects before they can be resolved. One of the major defects is the inconsistency of the field axioms of the real number system with the construction of a counterexample to the trichotomy axiom that proved it and the real number system false and at the same time not linearly ordered. Indeed, the rectification yields the new foundations of mathematics, constructivist real number system and complex vector plane the last mathematical space being the rectification of the complex real number system. FLT is resolved by a counterexample that proves it false and the Goldbach’s conjecture has been proved both in the constructivist real number system and the new real number system. The latter gives to two mathematical structures or tools—generalized integral and generalized physical fractal. The rectification of foundations yields the resolution of problem 1 and the solution of problem 6 of Hilbert’s 23 problems.

Abstract:
Theorem 1 is a formula expressing the mean number of real roots of a random multihomogeneous system of polynomial equations as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form expressions for the mean in special cases that include earlier results of Shub and Smale (for the general homogeneous system) and Rojas (for ``unmixed'' multihomogeneous systems). Theorem 3 gives upper and lower bounds for the mean number of roots, where the lower bound is the square root of the generic number of complex roots, as determined by Bernstein's theorem. These bounds are derived by induction from recursive inequalities given in Theorem 4.

Abstract:
Extensions of real numbers in more than two dimensions, in particular quaternions and octonions are finding applications in physics due to the fact that they naturally capture certain symmetries of physical systems. Here it is shown that the property of closure of coset groups can be used to generate the basis and general multiplication rules for extensions of real numbers in a systematic way. The coset approach has the advantage that multiplication rules follow directly from group closure instead of being postulated. In this approach, constraints on multiplication parameters can be formulated in ways that capture the symmetry features of the coset group. A complete classification of numbers systems is therefore obtained based on possible group structures of a given order. General matrix representations are also obtained through the coset procedure and by construction, the form of these matrices is invariant under matrix addition and multiplication. Since group symmetries are captured naturally into each number system, the coset group approach can add insight into the utility of multidimensional number systems in describing symmetries in nature.

Abstract:
Hand gesture recognition is one of the key techniques in developing user-friendly interfaces for human-computer interaction. Static hand gestures are the most essential facets of gesture recognition. User independence is among the important requirements for realizing a real time gesture recognition system in human-computer interaction. Applications of hand gesture recognition range from teleported control to hand diagnostic and rehabilitation or to speaking aids for the deaf. In today’s era of communication, sign language is one of the major tools of communication for physically challenged people. This paper proposes a system of hand gesture recognition for Devanagari Sign Language (DSL) Number System with comparison of feature extraction techniques, Discrete Cosine Transform (DCT) & Edge Oriented Histogram (EOH).

Abstract:
We present an algorithm for constructing infinite series expansion for real numbers, which yields generalized versions of three famous series expansions, namely, Sylvester series, Engel series, and Lüroth series expansions. Using series of rationals, a generalized model for the real number system is also constructed. 1. Introduction According to [1, 2], it is well known that each is uniquely representable as an infinite series expansion called Sylvester series expansion, which is of the form where Moreover, if and only if for all sufficiently large . An analogous representation (see [1–3]) also states that every real number has a unique representation as an infinite series expansion called Engel series expansion, which is of the form where Moreover, if and only if for all sufficiently large . For the last representation (see [1, 2]), it is also known that each is uniquely representable as an infinite series expansion called Lüroth series expansion, which is of the form where Moreover, if and only if is periodic. In 1988, A. Knopfmacher and J. Knopfmacher [4] further derived some elementary properties of the Engel series expansion and Sylvester series expansion and then developed two new methods for constructing new models for the real number system from the ordered field of rational numbers. These methods are partly similar to the one introduced by Rieger [5] for constructing the real numbers via continued fractions. In the present work, we will first introduce an algorithm for constructing an infinite series expansion for real numbers called Sylvester-Engel-Lüroth series expansion or SEL series expansion for short which yields generalized versions of three series expansions, namely, Sylvester series expansion, Engel series expansion, and Lüroth series expansion. Then we will establish some elementary properties of the SEL series expansion and develop a method for constructing a generalized model for the real number system using series of rationals, which yields generalized versions of Knopfmachers' models. 2. SEL Series Expansion Given any real number , write it as , where and . Then recursively define where is a positive rational number, which may depend on , for all . Using this algorithm and the same proof as in [1, 2], we have the following. Theorem 1. Let and assume that for all . Then is uniquely representable as an infinite series expansion called SEL series expansion, which is of the form where and for all . Lemma 2. Any series where converges to a real number such that . Furthermore, . By setting , and , for all in Theorem 1, and by setting ,

Abstract:
We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of $\mathbb Q$. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown to hold for {\it all} extensions of degree 2 of totally real number fields. The definability and undecidability results for integral closures of ``small'' and ``big'' subrings of number fields in the infinite algebraic extensions of $\mathbb Q$, previously shown by the author to hold for totally real fields, are extended to a large class of extensions of degree 2 of totally real fields. This class includes infinite cyclotomics and abelian extensions with finitely many ramified rational primes.

Abstract:
The main result of this paper is that if E is a field extension of finite odd degree over a real field Q, and if E is a repeated radical extension of Q, then every intermediate field is also a repeated radical extension of Q. This paper also contains a number of other results about repeated radical extensions.

Abstract:
With the wide application of real-time operating systems and the rapid development of Linux, more and more efforts are put on issues to develop real-time Linux. In this paper, we discuss the key problems, such as scheduling strategy, kernel reentry, interrupt handling, memory management etc, which are closely related to the extension of Linux to a real-time operating system. Then we analyze the main implementations of two promising realtime Linux, RT-Linux and KURT-Linux, in detail. We also present their own characteristics as well as the fundamental differences between them. The future research work is proposed at last.

Abstract:
Algebraic methods are used to construct families of unramified abelian extensions of some families of number fields with specified Galois groups.