On a Class of Number Fields of Eisenstein Type

In this paper the author determine the factorization of p and the number of p-th subfields of the splitting field K of the irreducible binomial xp ?m(m is p-th powerfree and p | m) and obtain that all p-th subfields of K have power integral bases and the same discriminant.     

Stable K ‐Theory of Finite Fields

It was conjectured in J.Pure Appl.Algebra 96 (1994), pp.709–720, that stable Ktheory of a ring R with finite degree bifunctor coefficients is equal to topological Hochschild homology of R with the same coefficients. The conjecture was proven by Dundas and McCarthy in thecase of bilinear bifunctors. We give a proof of the conjecture in full generality when R equals a finite field.     

A Note on Pseudo Ideals of Fields

Definition A subring A of a ring R is called a pseudo ideal, if for any r∈R and α∈A, r~2α∈A and αr~2∈A. The subring (R)~2={Σ±r_1~2…r_i~2|r_i∈R, t∈N} is called the square closed ring of R. Lemma The square closed ring of a division ring D is the intersection of all nontrivial pseudo ideals of D. Theorem Ⅰ If a simple ring R with a unit has a nontrivial pseudo ideal A containing an invertible element, R has characteristic p=2. In particular, the necessary condition for a division ring containing a nontrivial pseudo ideal is     

Hausdorff-type Measures of the Sample Path of Gaussian Random Fields

Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.     

C ∞-Normal Forms of Local Vector Fields

The problem of a local classification of vector fields is considered. A construction of formal and C -normal forms is described.     

Basic Results of Gabor Frame on Local Fields

Basic facts for Gabor frame {Eu(m)bTu(n)ag}m,n∈p on local field are investigated. Accurately, that the canonical dual of frame {Eu(m)bTu(n)ag}m,n∈p also has the Gabor structure is showed; that the product ab decides whether it is possible for {Eu(m)bTu(n)ag}m,n∈p to be a frame for L2(K) is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L2(K) are established. An example is finally given.     

Distributional Dimension of Fraetal Sets in Local Fields

The distributional dimension of fractal sets in R^n has been systematically studied by Triebel by virtue of the theory of function spaces. In this paper, we first discuss some important properties about the B-type spaces and the F-type spaces on local fields, then we give the definition of the distributional dimension dimD in local fields and study the relations between distributional dimension and Hausdorff dimension. Moreover, the analysis expression of the Hausdorff dimension is given. Lastly, we define the Fourier dimension in local fields, and obtain the relations among all the three dimensions. Keywords local field, B-type space, F-type space, distributional dimension, Hausdorff dimension Fourier dimension     

Explicit Determination of Torsion Subgroups of Elliptic Curves Over Multi-Quadratic Fields

LotKbeanalgebraicnUmberfield,EanellipticcurveoverK.Thewell-knownMordelLWeiltheOemassertsthattheK-rationalpointsE(K)ofEisafinitelygeneratedabeliangroup(MordelgrouP):E(K)2E(K)torsxZ,whereE(K)torsisthetorsionsubgroup,ristherank(anonnegativeinteger),Zistheadditivegroupofrationalintegers.TheaboveisomorphismsometimesisaIsowrittellasequality:E(K)=E(K)torsxZ".TherearemanyliteraturesstudyingthetorsionsubgrouPE(K)tors'EspeciallyL.Merelrecenilyprovedthe"uniformb0undednessconjecture":F0ranyp…     

Volume-preserving Fields and Reeb Fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.     

The Problem on Class Numbers of Quadratic Number Fields

It is a survey of the problem on class numbers of quadratic number fields.     

Group-Theoretic Dimension of Stationary Symmetric α-Stable Random Fields

The growth rate of the partial maximum of a stationary stable process was first studied in the works of Samorodnitsky (Ann. Probab. 32:1438–1468, 2004; Adv. Appl. Probab. 36:805–823, 2004), where it was established, based on the seminal works of Rosiński (Ann. Probab. 23:1163–1187, 1995; 28:1797–1813, 2000), that the growth rate is connected to the ergodic-theoretic properties of the flow that generates the process. The results were generalized to the case of stable random fields indexed by ? ^d in Roy and Samorodnitsky (J. Theor. Probab. 21:212–233, 2008), where properties of the group of nonsingular transformations generating the stable process were studied as an attempt to understand the growth rate of the partial maximum process. This work generalizes this connection between stable random fields and group theory to the continuous parameter case, that is, to fields indexed by ? ^d .     

Inverse-preserving Linear Maps Between Spaces of Matrices over Fields

Suppose F is a field different from F2, the field with two elements. Let Mn（F） and Sn（F） be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn（F）, Mn（F）}, we say that a linear map f from G1 to G2 is inverse-preserving if f（X）^-1 = f（X^-1） for every invertible X ∈ G1. Let L （G1, G2） denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets .L（Sn（F）,Mn（F））, L（Sn（F）,Sn（F））, L （Mn（F）,Mn（F）） and L（Mn （F）, Sn （F）） are characterized.     

Indivisibility of Class Numbers and Iwasawa λ-Invariants of Real Quadratic Fields

Let D>0 be the fundamental discriminant of a real quadratic field, and h(D) its class number. In this paper, by refining Ono's idea, we show that for any prime p>3, {0<D<X|h(D)0(mod p)}>> _p (X)/logX.     

Explicit Formulae for Values of Dedekind Zeta Functions of Two Kinds of Cyclotomic Fields

Let K = Q(m) denote the m-th cyclotomic field, and K+ its maximal real subfield, where m =exp is an m-th primary root of unity. Let K (s) denote the Dedekind zeta function ofK. For prime integers m = p, Fumio Hazama recently in [1] obtained formulae for calculating special values of K(s) and K+(s), i.e., calculating formulae of K+(1 - n) and for positive integers n, which are the newest results of a series of his work in many years (see [1-3]).Here we develop Hazama's work for prime integ…     

Strong Law of Large Numbers for Cosine Weighted Partial Sums of Random Fields

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On the Boundary of the Set of Morse-Smale Vector Fields on Two-dimensional Manifolds

NotatianM a compact,boundaryless,orientable or non—orientable two-dimensional manifold of class C~∞H~r(M) the Banach space of C~r(r≥1)vector fields on M∑~r(M) the set of structurally stable vector fields in H~r(M)MS~r(M) the set of Morse—Smale vector fields in H~r(M)     

Chaotic Dynamics of Homogeneous Yang–Mills Fields with Three Degrees of Freedom

Differential Equations - We consider a scenario of transition to chaotic dynamics in a Hamiltonian system of homogeneous Yang–Mills fields with three degrees of freedom in the presence of the...     

Existence of Some Specific Elements in Finite Fields北大核心CSCD

Let q = pk and Fqn be the extension field of Fq of degree n, where p is an odd prime and n, k are positive integers. The main contribution of this paper is as follows: If n | (q − 1), k ≥ 11, n ≥ 14 or n (q − 1), k ≥ 10, n ≥ 8, then there exists a primitive element α in Fqn such that α + α−1 is a normal element, and 1 + α2 is a square element, and there exists a normal element β, such that β + β−1 is a primitive element, and 1 + β2 is a square element. © 2022 Chinese Academy of Sciences. All rights reserved.     

Lie Symmetries of Quasihomogeneous Polynomial Planar Vector Fields and Certain Perturbations

In this work we study Lie symmetries of planar quasihomogeneous polynomial vector fields from different points of view, showing its integrability. Additionally, we show that certain perturbations of such vector fields which generalize the so-called degenerate infinity vector fields are also integrable.     

Large Deviations for the Boundary Crossing Probabilities of Some Random Fields

For Y_1,Y_2,…i.i.d.with Y_1～N(μ,1) and S_n=sum from 1 to n(Y_i),the large deviations are obtained for the probabilities that conditionally given (i) S_m=0, and (ii) S_(m_i)=Applied these results to the double change-points model with some nuisance parameters, we developed the large deviation for the significance level of the likelihood ratio test.