The hybrid real number system

The origin of the concepts that we employ such as natural numbers, rational and irrational numbers, zero, infinity and the place value system among others originate in the Vedas, or Veda Samhitas to be correct [Veda Samhitas]. However, due to our second-hand reception from the Arabs and the belief that everything originated in Greece, this appears to have lost the original source and meaning [V. Lakshmikantham, S. Leela, J. Vasundhara Devi, The Origin and History of Mathematics, Cambridge Scientific Publishers, Cambridge, UK, 2005. [5]]. As a consequence, the so-called rational thinking or logic that applies to concrete situations when employed for concepts that are beyond our sensory perceptions has created some paradoxes [E.E. Escultura, FTG XVII: The new mathematics and physics, J. Appl. Math. Comput. 138 (2003) 127–149; E.E. Escultura, FTG XXXIV. Foundations of analysis and the new arithmetic, Nonlinear Anal. Phenom. III (2006) (in press)]. It is therefore necessary to return to the origin to clearly understand the basis.In this paper, we plan to propose a hybrid real number system utilizing the original ideas so as to shed some light on the existing system and strengthen it.     

Some presentations of the real number field

It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) .     

Revisiting the stabilization of a three-dimensional hybrid magnetoelastic structure

In this article, we are concerned with the stabilization of a three-dimensional–two-dimensional hybrid magnetoelastic structure for a reference configuration of rectangular type. We use energy methods to establish the polynomial stability of the energy associated with the interactive system of partial differential equations that govern the motions of the structure. The coupled nature of the model presents some intricacies in the construction of a Lyapunov functional.     

A note about the average number of real roots of a Bernstein polynomial system

We prove that the real roots of normal random homogeneous polynomial systems with n+1$n+1$ variables and given degrees are, in some sense, equidistributed in the projective space P(Rⁿ⁺¹)$\mathbb{P}\left({\mathbb{R}}^{n+1}\right)$. From this fact we compute the average number of real roots of normal random polynomial systems given in the Bernstein basis.     

Revisiting the average number of divisors of a quadratic polynomial

Monatshefte für Mathematik - In this work, we employ a well known result due to Zagier to derive an asymptotic formula for the average number of divisors of a quadratic polynomial of the form...     

Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).     

On the number of real quadratic fields with class number divisible by 3

We find a lower bound for the number of real quadratic fields whose class groups have an element of order . More precisely, we establish that the number of real quadratic fields whose absolute discriminant is and whose class group has an element of order is improving the existing best known bound of R. Murty.

     

On the Iwasawa invariants of totally real number fields

We shall show two sufficient conditions under which the Iwasawa invariants λ _k and μ _k of a totally real fieldk vanish for an odd primel, based on the results obtained in [1], [3] and [4]. LetK _n be the composite ofk and thel ⁿ-th cyclotomic extension of the fieldQ of rational numbers. LetC _n be the factor group of thel-class group ofK _n by a subgroup generated by ideals whose prime factors divide the principal ideal (l). Let ϕ₁ be an idempotent of the group ringZ _l[Gal(K ₁/k)] defined in the below. We shall prove λ _k = μ _k =0 if there is a natural numbern such that ε₁ C _n vanishes, under additional conditions concerning ramifications inK _n/k.     

The intersection number of real polynomial mappings

Let X=V(f₁,…,fn−m)⊂Rn be a compact real algebraic set and g:X→R2m be a continuous function. If the diagonal in X×X is isolated in the set of self-intersection points of g, we define the intersection number of g. In the case where X is a manifold and g is an immersion it is the intersection number defined by Whitney. In the case where g is a polynomial mapping, we present an effective formula for this number.     

Jacobi forms over totally real number fields

We define Jacobi forms over a totally real algebraic number field K and construct examples by first embedding the group and the space into the symplectic group and the symplectic upper half space respectively. Then symplectic modular forms are created and Jacobi forms arise by taking the appropriate Fourier coefficients. Also some known relations of Jacobi forms to vector valued modular forms over rational numbers are extended to totally real fields.     

The conjugacy problem of the modular group and the class number of real quadratic number fields

We shall discuss the conjugacy problem of the modular group, and show how its solution, in conjunction with a theorem of Olga Taussky can be used to compute the class number of certain real quadratic number fields.     

Computing the rank of elliptic curves over real quadratic number fields of class number 1

In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over . Several examples are included.

     

Classification of three-dimensional zeropotent algebras over the real number field

A nonassociative algebra is defined to be zeropotent if the square of any element is zero. In this paper, we give a complete classification of three-dimensional zeropotent algebras over the real number field up to isomorphism. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional real Lie algebras, which is in accordance with the Bianchi classification. Moreover, three-dimensional zeropotent algebras over a real closed field are classified in the same manner as those over the real number field.     

On the Pythagoras number of a real irreducible algebroid curve

This paper is based on the author's doctoral dissertation, written under the supervision of Prof. J. Ruiz     

On sign determinations in real algebraic number fields

Summary Given an ordered fieldF and a finite ordered extensionE. If it is possible to perform constructively the four rational operations inF and to determine the sign for an arbitrary element ofF, a rule is given for the sign determination of an arbitrary element ofE.     

Dedekind zeta motives for totally real number fields

Let k be a totally real number field. For every odd n≥3, we construct an element in the category MT(k) of mixed Tate motives over k out of the quotient of a product of hyperbolic spaces by an arithmetic group. By a volume calculation, we prove that its period is a rational multiple of $\pi^{n[k:\mathbb{Q}]}\zeta^{*}_{k}(1-n)$ , where $\zeta^{*}_{k}(1-n)$ denotes the special value of the Dedekind zeta function of k. We deduce that the group $\mathrm {Ext}^{1}_{\mathrm {MT}(k)} (\mathbb{Q}(0),\mathbb{Q}(n))$ is generated by the cohomology of a quadric relative to hyperplanes, and that $\zeta^{*}_{k}(1-n)$ is a determinant of volumes of geodesic hyperbolic simplices defined over k.     

Theta series of a real algebraic number field

We investigate transformation formulas for theta series with spherical functions on a Hilbert-Siegel space. As an application we show that some of Hilbert-Siegel modular varieties are of general type.