Ideal Class Groups and Subgroups of Real Quadratic Function Fields

In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n, and obtained eight series of such fields. The ideal class numbers h(O K) of K in the series all have a factor n.[

Halton-type sequences from global function fields

For any prime power q and any dimension s≥1, a new construction of (t, s)-sequences in base q using global function fields is presented. The construction yields an analog of Halton sequences for global function fields. It is the first general construction of (t, s)-sequences that is not directly based on the digital method. The construction can also be put into the framework of the theory of (u, e, s)-sequences that was recently introduced by Tezuka and leads in this way to better discrepancy bounds for the constructed sequences.

Lower Bound for Ideal Class Numbers of Real Quadratic Function Fields

In this paper, the theory of continued fractions of algebraic functions will be used to give a general theorem on lower bounds for class numbers of real quadratic function fields K=k(D). The bounds are given more explicitly for six types of real quadratic function fields. As a consequence, six classes of real quadratic function fields with ideal class number greater than one are given.[

Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.

SUBGROUPS OF CLASS GROUPS OF ALGEBRAIC QUADRATIC FUNCTION FIELDS

Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both real and imaginary fields K. Then several series of functionfields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, aregiven, for which the class groups H(K) are proved to contain cyclic subgroups of order n.

Parametrization of the Quadratic Function Fields Whose Divisor Class Numbers are Divisible by Three

A parametrization of quadratic function fields whose divisor class numbers are divisible by 3 is obtained by using free parameters when the characteristics of the fields are not 3.

Molecular properties and potential energy function model of BH under external electric field

Using the density functional B3P86/cc-PV5Z method, the geometric structure of BH molecule under different external electric fields is optimized, and the bond lengths, dipole moments, vibration frequencies, and other physical properties parameters are obtained. On the basis of setting appropriate parameters, scanning single point energies are obtained by the same method and the potential energy curves under different external fields are also obtained. These results show that the physical property parameters and potential energy curves may change with external electric field, especially in the case of reverse direction electric field. The potential energy function without external electric field is fitted by Morse potential, and the fitting parameters are obtained which are in good agreement with experimental values. In order to obtain the critical dissociation electric parameter, the dipole approximation is adopted to construct a potential model fitting the corresponding potential energy curve of the external electric field. It is found that the fitted critical dissociation electric parameter is consistent with numerical calculation, so that the constructed model is reliable and accurate. These results will provide important theoretical and experimental reference for further studying the molecular spectrum, dynamics, and molecular cooling with Stark effect.

Subgroups of ideal class groups of real quadratic algebraic function fields

Necessary and sufficient condition on real quadratic algebraic function fields K is given for theirideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic functionfields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal classnumbers of these function fields are divisible by n.

ON THE MINKOWSKI UNIT IN FUNCTION FIELDS

Suppose that K is a cyclic number field with odd prime degree. The unit group U_K of the integer ring O_K is U_K={+1}×V_K (direct product), where V_K is the group of units with norm 1. By Dirichet Unit Theorem, V_K is free Abelian

Construction Theory of Function on Local Fields

We establish the construction theory of function based upon a local field Kp as underlying space. By virture of the concept of pseudo-differential operator, we introduce ＂fractal calculus＂ （or, p-type calculus, or, Gibbs-Butzer calculus）. Then, show the Jackson direct approximation theorems, Bermstein inverse approximation theorems and the equivalent approximation theorems for compact group D（C Kp） and locally compact group Kp^＋-（= Kp）, so that the foundation of construction theory of function on local fields is established. Moreover, the Jackson type, Bernstein type, and equivalent approximation theorems on the HOlder-type space C^σ（Kp）, σ 〉0, are proved; then the equivalent approximation theorem on Sobolev-type space Wr（Kp）, σ≥0, 1≤r 〈∞, is shown.

The Analytical Potential Energy Function of NH Radical Molecule in External Electric Field

The geometric structures of an Nit radical in different external electric fields are optimized by using the density functional B3P86/cc-PVSZ method, and the bond lengths, dipole moments, vibration frequencies and IR spectrum are obtained. The potential energy curves are gained by the CCSD （T） method with the same basis set. These results indicate that the physical property parameters and potential energy curves may change with the external electric field, especially in the reverse direction electric field. The potential energy function of zero field is fitted by the Morse potential, and the fitting parameters are in good accordance with the experimental data. The potential energy functions of different external electric fields are fitted adopting the constructed potential model. The fitted critical dissociation electric parameters are shown to be consistent with the numerical calculation, and the relative errors are only 0.27% and 6.61%, hence the constructed model is reliable and accurate. The present results provide an important reference for further study of the molecular spectrum, dynamics and molecular cooling with Stark effect.

On Decomposition Law in Carlitz-Kummer Function Fields

Let Fq be a finite field of q elements with characteristic p. Let k=Fq（T） be a ra-tional function field, and kac, a fixed algebraic closure of k. Let M be a monic polynomialin R=Fq[T]. The Carlitz action of M on a∈kac is defined

QUADRATIC FUNCTION FIELDS AND CODES

For the hyperelliptic curve y^2=f(x) over F_q with deg(f)≥5, there is a singular point (0, 1, 0), such that a curve is not smooth. But the codes which are constructed by using hyperelliptic curves can be discussed basing on the theory of arithmetic of quadratic function fields.

MAXIMAL INDEPENDENT SYSTEM OF UNITS IN CYCLOTOMIC FUNCTION FIELDS

A series of maximal independent systems of cyclotomic units for cyclotomic functionfields and their subfields are constructed as an analogue of Ramachandra’s and Levesque’scyclotomic unit systems in cyclotomic number fields. The indexes of subgroups generated bythese unit systems in the whole unit groups are calculated and compared.

ON MAXIMAL INDEPENDENT SYSTEM OF UNITS IN CYCLOTOMIC FUNCTION FIELDS

This note is concerned in constructing a series of maximal independent systems of cyclotomic units in cyclotomic function fields and their subfields. Let us introduce basic facts on cyclotomic function fields briefly (see Refs. [1—3] for detail).

Class number relation between type (l,l/,…, l) function fields over F_q(T) and their subfields

Let L/Fq(T) be a tame abelian extension of type (l, l,...l). The ratio of the degree zero divisor dass number (as well as the ideal class number) of L to the product of corresponding class numbers of all cydic subfields of L is clearly determined.

A NOTE ON THE S-UNIT EQUATION OVER FUNCTION FIELD

In this note we show that the Mason’s upper bound of deg(u) for S-units u andv satisfying u + v = 1 over function field K = k(C) can be attained for any algebraicallyclosed coefficient field k and for some hyperelliptic or Artin-Schreier curve C over k withany given genus g≥0.

BERNOULLI-GOSS POLYNOMIAL AND CLASS NUMBER OF CYCLOTOMIC FUNCTION FIELDS

Let k = Fq(T),q=pn, and let K=k((?)p)be the cyclotomic function field with conduc-tor P = P(T), and suppose K+ is the maximal real subfield of K, hp(h+p) is the class number of divisor group (of degree zero) of K(K+), and h-p=hp/h+p(∈ Ⅱ). This paper proves that for any fixed q≥3, there exist infinite many irreducible manic polynomial P∈Fq[T] such that p\h+p and pq-2\h-p. In addition, all regular quadratic irreducible polynomials in Fq[T] for 2≤p≤269 are determined.

Numerical modeling calculation for the spatial distribution characteristics of horizontal field transfer functions

Applying 3-dimension finite difference method, the distribution characteristics of horizontal field transfer functions for rectangular conductor have been computed, and the law of distribution for Re-part and Im-part has been given. The influences of source field period, the conductivity, the buried depth and the length of the conductor on the transfer functions were studied. The extrema of transfer functions appear at the center, the four corners and around the edges of conductor, and move with the edges. This feature demonstrates that around the edges are best places for transfer functions' observation.

A FUNCTION MINIMIZATION CALCULATION OF CYCLONE 10 MAGNETIC FIELDS

By app(?)ing function minimization calculation method, two function expressions are used to simulate the magnetic field measured for cyclotron Cyclone 10 in azimuth and radius The numerical fitting curves are consistent with magnetic field measured. In most pl(?)es, the accuracies are several thousandth, except those errors to be pointed out in paper.