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.

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.

On ideal convergence of double sequences in 2-fuzzy n-normed linear space

The purpose of this paper is to define the notions of convergence, Cauchy st–convergence, st–Cauchy, I –convergence and I –Cauchy for double sequences in 2–fuzzy n–normed spaces with respect to α–n–norms and study certain classical and standard properties related to these notions.

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}.

On Some Class of n-Normed Generalized Difference Sequences Related to l_p-Space

In this paper, we introduce the class of n-normed generalized difference sequences related to lp-space. Some properties of this sequence space like solidness, symmetricity, convergence-free etc. are studied. We obtain some inclusion relations involving this sequence space.

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.

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.

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.

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.

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.

非阿基米德n-赋范空间的等距

讨论非阿基米德n-赋范空间中的Mazur-Ulam定理和Aleksandrov问题.证明了非阿基米德n-赋范空间中的满的n-等距映射是反射的,并且保单位距离是等距的充要条件是其保持零-n-距离.

The Discrete Subgroups and Jфrgensen’s Inequality for SL(m, Cp)

In this paper, we give discreteness criteria of subgroups of the special linear group on Qp or Cp in two and higher dimensions. Jorgensen＇s inequality gives a necessary condition for a non- elementary group of M6bius transformations to be discrete. We give a version of JCrgensen＇s inequality for SL（m, Cp）.