ISOMETRY AND PHASE-ISOMETRY OF NON-ARCHIMEDEAN NORMED SPACES

In this paper,we study isometries and phase-isometries of non-Archimedean normed spaces.We show that every isometry f:Sr(X)→Sr(X),where X is a finite-dimensional non-Archimedean normed space and Sr(X)is a sphere with radius r∈||X||,is surjective if and only if is spherically complete and k is finite.Moreover,we prove that if X and Y are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with|2|=1,any phase-isometry f:X→Y is phase equivalent to an isometric operator.

Non-Archimedean空间中三次方程的稳定性

研究了三次方程Ef(x,y):=f(mx+y)+f(mx-y)-mf(x+y)-mf(x-y)-2m(m2-1)f(x)=0在non-Archimedean空间中的稳定性问题,其中m为任一实数且m≠1且m≠0.证明了如果G为一赋范空间,X为一完备的non-Archimedean空间.f:G→X是一个映射,δ>0,使得当x,y∈G时,有‖Ef(x,y)‖≤δ(ρ(‖x‖3)+ρ(‖y‖3)),其中ρ:[0,+∞)→[0,+∞)满足ρ(|m|t)≤ρ(|m|)ρ(t)及ρ(|m|)<|m|,那么存在惟一三次映射Q:G→X,使得‖f(x)-Q(x)‖≤δ2|m|3ρ(‖x‖3)(x∈G).

APPROXIMATION OF A CAUCHY-JENSEN ADDITIVE FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN NORMED SPACES

Using the fixed point and direct methods, we prove the Hyers-Ulam stability of the following Cauchy-Jensen additive functional equation 2f（p∑i=1xi＋q∑j=1yj＋2d∑k=1zk/2）=p∑i=1f（xi）＋q∑j=1f（yj）＋2d∑k=1f（zk）,where p, q, d are integers greater than 1, in non-Archimedean normed spaces.

A non-Archimedean number system to characterize the structurally inhomogeneous rock behavior nearby a tunnel

The development of mathematical models of structurally inhomogeneous media leads to the necessity to consider structure of space itself where deformation occurs, i.e. change of mathematical apparatus itself. The space, whose coordinate axes are non-Archimedean straight lines, has been considered. Refusing the fulfillment of Archimedes’s law allows to describe multi-scaling of the space, and so to consider deformation processes on different scale levels. The construction of two-scale mathematical model of rock masses has been considered as an example. The constitutive equations have been formulated on micro-and macro-levels and interaction condition between different levels as well. On micro-level, the elastic behavior of grains and plastic sliding between grains with possible softening are taken into account. On macro-level, the model represents a nonlinear system of equations describing the anisotropic rock mass behavior. On the basis of model, the numerical algorithm and code have been carried out to solve the plane boundary value problems. Examples of numerical simulations of stress-strain state of structural rock masses nearby a tunnel opening are presented. The deformation contours and isolines of stresses are plotted.

A PICARD-TYPE THEOREM AND A UNIQUENESS THEOREM OF NON-ARCHIMEDEAN ANALYTIC CURVES IN PROJECTIVE SPACE

In this article, we prove a Picard-type Theorem and a uniqueness theorem for non-Archimedean analytic curves in the projective space Pn(F), where the characteristic of F is 0 or positive. In the main results of this article, we ignore the zeros with large multiplicities.

APPROXIMATE SOLUTION OF A p-th ROOT FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN (2,β)-BANACH SPACES

In this paper, using the Brzdek's fixed point theorem [9,Theorem 1] in non-Archimedean(2,β)-Banach spaces, we prove some stability and hyperstability results for an p-th root functional equation ■where p∈{1, …, 5}, a_1,…, a_k are fixed nonzero reals when p ∈ {1,3,5} and are fixed positive reals when p ∈{2,4}.

Non-Archimedean Analysis on the Extended Hyperreal Line *R_dand the Solution of Some Very Old Transcendence Conjectures over the Field Q

In 1980 F. Wattenberg constructed the Dedekind completion *Rd of the Robinson non-archimedean field *R and established basic algebraic properties of *Rd. In 1985 H. Gonshor established further fundamental properties of *Rd. In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion *Rd in transcendental number theory were considered. Given any analytic function of one complex variable , we investigate the arithmetic nature of the values of at transcendental points . Main results are: 1) the both numbers and are irrational;2) number ee is transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtained.

The Aleksandrov Problem in Non-Archimedean 2-Fuzzy 2-Normed Spaces

We introduce the definition of non-Archimedean 2-fuzzy 2-normed spaces and the concept of isometry which is appropriate to represent the notion of area preserving mapping in the spaces above. And then we can get isometry when a mapping satisfies AOPP and (*) (in article) by applying the Benz’s theorem about the Aleksandrov problem in non-Archimedean 2-fuzzy 2-normed spaces.

二次-三次函数方程在Non-Archimedean赋范空间的Ulam稳定性

讨论了混合二次-三次函数方程6f(x+y)-6f(x-y)+4f(3y)=3f(x+2y)-3f(x-2y)+9f(2y)

Non-Archimedean赋范空间中一些映射的不动点

本文证明了在球性完备的Non－Archimedean赋范空间中，B.E.Rhoades的第（6），（10），（13），（17），（20），（22）。

Functional Inequalities in Non-Archimedean Normed Spaces

In this paper, we introduce an additive functional inequality and a quadratic functional inequality in normed spaces, and prove the Hyers-Ulam stability of the functional inequalities in Banach spaces. Furthermore, we introduce an additive functional inequality and a quadratic functional inequality in non-Archimedean normed spaces, and prove the Hyers-Ulam stability of the functional inequalities in non-Archimedean Banach spaces.

Range inclusion of operators on non-archimedean Banach space

In this paper, we establish some range inclusion theorems for non-archimedean Banach spaces over general valued fields. These theorems provide close relationship among range inclusion, majorization and factorization for bounded linear operators. It is found that these results depend strongly on a continuous extension property, which is always true in the classical archimedean case, but may fail to hold for the non-archimedean setting. Several counterexamples are given to show that our results are sharp in some sense.

On Tame Operators between Non-Archimedean Power Series Spaces

Let p ∈ {1,∞}. We show that any continuous linear operator T from Al（a） to Ap（b） is tame, i.e., there exists a positive integer c such that supx ｜｜Tx｜｜k/｜x｜ck 〈 ∞ for every k ∈ N. Next we prove that a similar result holds for operators from Am（a） to Ap（b） if and only if the set Mb,a of all finite limit points of the double sequence （bi/aj）i,j∈N is bounded. Finally we show that the range of every tame operator from A∞ （a） to A∞ （b） has a Schauder basis.

The Finite-dimensional Decomposition Property in Non-Archimedean Banach Spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property（OFDDP） if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck＇s approximation theory,where an open problem is the following： Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E.Does D have the OFDDP？ In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP.Next we prove that,however,for certain classes of Banach spaces of countable type,the OFDDP is preserved by taking finite-codimensional subspaces.

The Moduli Space of Stable Coherent Sheaves via Non-archimedean Geometry

We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry,where we use the notion of Berkovich non-archimedean analytic spaces.The motivation for our construction is Tony Yue Yu’s non-archimedean enumerative geometry in Gromov-Witten theory.The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson-Thomas invariants.In this paper we give the moduli construction over a non-archimedean field K.We use the machinery of formal schemes,that is,we define and construct the formal moduli stack of(semi)-stable coherent sheaves over a discrete valuation ring R,and taking generic fiber we get the non-archimedean analytic moduli space of semistable coherent sheaves over the fractional non-archimedean field K.We generalize Joyce’s dcritical scheme structure in[37]or Kiem-Li’s virtual critical manifolds in[38]to the world of formal schemes,and Berkovich non-archimedean analytic spaces.As an application,we provide a proof for the motivic localization formula for a d-critical non-archimedean K-analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes.This generalizes Maulik’s motivic localization formula for the motivic Donaldson-Thomas invariants.

含非阿基米德无穷小量DEA模型的研究综述

1978年,Charnes等提出了第一个DEA模型,该模型不含有非阿基米德无穷小量;1981年,Charnes等将非阿基米德无穷小量引入DEA模型,构建了含非阿基米德无穷小量DEA模型.但关于含非阿基米德无穷小量DEA模型的争论一直持续至今,这些争论主要包括两方面,一方面是关于模型本身的存在性,另一方面是关于模型的求解方法.本文对含非阿基米德无穷小量DEA模型的研究进行了综述.

概率收缩偶与非阿基米德Menger概率赋范空间中非线性方程组的解

本文在非阿基米德Menger概率赋范空间中引入了概率收缩偶的概念,研究了非阿基米德Menger概率赋范空间中具概率收缩偶的非线性方程组的解的存在性与唯一性.发展和改进了引文[1～5]的相应结果.

基于理想方案排序法的黑启动方案相对有效性评价

为了简便有效地筛选出合适的电力系统黑启动方案,在改进的数据包络分析模型(DEA)基础之上提出了理想方案排序法(TRIS)。首先构建新的改进DEA模型,分别计算出各个黑启动方案在改进DEA模型所得到公共权重下的效率值,再构造出排队指标值。通过对排队指标值进行排序,即可得出各个黑启动方案的相对有效性排序。该方法保持了原方法计算简便的优点,克服了由于所选非阿基米德无穷小量不同使黑启动方案优劣排序顺序不同的缺点。最后通过实例验证了该方法的可行性和有效性。

线性n-范空间上的Mazur-Ulam定理和Riesz引理(英文)

给出了严格凸的non-Archimedean域上n-范空间和p严格凸的non-Archimedean上的(n,p)范空间上的Mazur-Ulam定理,同时证明了Riesz引理在实线性n-范空间上也是成立的.

A MODEL FOR WHITE NOISE ANALYSIS IN P-ADIC NUMBER FIELDS

We propose a simple model for white noise analysis in the p-adic case based on the p-adic analogue of the space L(2)(R(infinity), d mu).