Issue 1, Volume 46, March 2021

2021

1. Singularities and Perfectoid Geometry Yves André PDF

   ▸ Abstract

   These notes give a quick overview of recent work by the speaker and of some subsequent works introducing perfectoid geometry into homological commutative algebra and singularity theory. The emphasis is on big Cohen-Macaulay algebras and applications. The progresses take place primarily in mixed characteristic, but sometimes provide a bridge between characteristic p and characteristic 0.

   1-8
2. Ergodic Theorems for Laminations and Foliations: Recent Results and Perspectives Viêt-Anh Nguyên PDF

   ▸ Abstract

   This report discusses recent results as well as new perspectives in the ergodic theory for Riemann surface laminations, with an emphasis on singular holomorphic foliations by curves. The central notions of these developments are leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles, and Lyapunov exponents. We deal with various ergodic theorems for such laminations: random and operator ergodic theorems, (geometric) Birkhoff ergodic theorems, Oseledec multiplicative ergodic theorem, and unique ergodicity theorems. Applications of these theorems are also given. In particular, we define and study the canonical Lyapunov exponents for a large family of singular holomorphic foliations on compact projective surfaces. Topological and algebro-geometric interpretations of these characteristic numbers are also treated. These results highlight the strong similarity as well as the fundamental differences between the ergodic theory of maps and that of Riemann surface laminations. Most of the results reported here are known. However, sufficient conditions for abstract heat diffusions to coincide with the leafwise heat diffusions (Section 5.2) are new ones.

   9-101
3. Weighted Stepanov-Like Pseudo Almost Automorphic Solutions for Evolution Equations and Applications Nguyen Thieu Huy , Vu Thi Ngoc Ha , Le The Sac , Pham Truong Xuan PDF

   ▸ Abstract

   In this paper, the existence and stability of (weighted) Stepanov-like pseudo almost automorphic solutions are proved for a large class of linear and semilinear evolution equations in interpolation spaces. Our method is based on the one hand on a combination of differential inequalities and interpolations functors for the case of linearized equation, and on the other hand on the fixed point argument for the semi-linear equations to handle the case of (weighted) Stepanov-like pseudo almost automorphic functions. Finally, we apply the abstract results to various problems of incompressible viscous fluid flows.

   103-122
4. Permutation Groups Induced by Derksen Groups in Characteristic Two Keisuke Hakuta PDF

   ▸ Abstract

   We consider the so-called Derksen group which is a subgroup of the polynomial automorphism group of the polynomial ring in n variables over a field. The Derksen group is generated by affine automorphisms and one particular non-linear automorphism. Derksen (1994) proved that if the characteristic of the underlying field is zero and n ≥ 3, then the Derksen group is equal to the entire tame subgroup. The result is called Derksen’s Theorem. It is quite natural to ask whether the same property holds for positive characteristic. In this paper, we point out that the question can be easily answered negatively when the underlying field is of characteristic two. We shall also prove that the permutation group induced by the Derksen group over a finite field of characteristic two is a proper subgroup of the alternating group on the n-dimensional linear space over the finite field. This is a stronger result that Derksen’s Theorem does not hold when the underlying field is a finite field of characteristic two.

   123-132
5. On the Preservation for Quasi-Modularity of Field Extensions El Hassane Fliouet PDF

   ▸ Abstract

   Let $k$ be a field of characteristic $p\not=0$. In 1968, M. E. Sweedler revealed for the first time, the usefulness of the concept of modularity. This notion, which plays an important role especially for Galois theory of purely inseparable extensions, was used to characterize purely inseparable extensions of bounded exponent which were tensor products of simple extensions. A natural extension of the definition of modularity is to say that $K/k$ is $q$-modular (quasi-modular) if $K$ is modular up to some finite extension. In subsequent papers, M. Chellali and the author have studied various property of $q$-modular field extensions, including the questions of $q$-modularity preservation in case $[k:k^p]$ is finite. This paper grew out of an attempt to find analogue results concerning the preservation of $q$-modularity, without the hypothesis on $k$ but with extra assumptions on $K/k$. In particular, we investigate existence conditions of lower (resp. upper) quasi-modular closures for a given $q$-finite extension.

   133-148
6. On Stability of G-Frames in Hilbert Pro-C*-Modules Zahra Ahmadi Moosavi , Akbar Nazari PDF

   ▸ Abstract

   In this paper, we extend the basic stability theorem for frames in Hilbert C*-modules to g-frames in Hilbert pro-C*-modules. We also extend the Casazza-Christensen perturbation theorem to g-frames in Hilbert pro-C*-modules and show that the Casazza-Christensen theorem is not valid for g-Riesz bases in Hilbert pro-C*-modules. Finally, we derive some characterizations of g-frames in Hilbert pro-C*-modules.

   149-161
7. Some Spectral Formulas for Functions Generated by Differential and Integral Operators Ha Huy Bang , Vu Nhat Huy PDF

   ▸ Abstract

   In this paper, we investigate the behavior of the sequence of $L^p$ norm of functions which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function is called its spectrum).

   163-177
8. Some Results for the Windowed Fourier Transform Related to the Spherical Mean Operator Khaled Hleili PDF

   ▸ Abstract

   We define and study the windowed Fourier transform associated with the spherical mean operator. We prove the boundedness and compactness of localization operators of this windowed. Next, we establish new uncertainty principles for the Fourier and the windowed Fourier transforms associated with the spherical mean operator. More precisely, we give a Shapiro-type uncertainty inequality for the Fourier transform that is, for $s>0$ and $\{\alpha_k\}_k$ be an orthonormal sequence in $L^2(dv_{n+1})$ $$\sum\limits_{k=1}^{M}\left( \||(r,x)|^{s}\alpha_{k}\|^{2}_{2,\nu_{n+1}}+\||(\mu,\lambda)|^{s}\mathcal{\tilde{F}}(\alpha_{k})\|^{2}_{2,\nu_{n+1}}\right) \geqslant R_{n,s}M^{1+\frac{s}{2n+1}}.$$ Finally, we prove an analogous inequality for the windowed Fourier transform.

   179-201
9. On the Order of Approximation in Limit Theorems for Negative–Binomial Sums of Strictly Stationary m -Dependent Random Variables Tran Loc Hung , Phan Tri Kien PDF

   ▸ Abstract

   During the last several decades, the results related to geometric random sums of independent identically distributed random variables have become interesting results in probability theory and in insurance, risk theory, queuing theory and stochastic finance, etc. The negative–binomial random sums are extensions of geometric random sums. Up to the present, the negative–binomial random sums of independent identically distributed random variables have attracted much attention since they appear in many fields. However, for the strictly stationary m-dependent summands, such negative–binomial random sums are rarely studied. It seems that the complexity of the dependent structure has limited the use of classical tools based on independent random variables such as characteristic function. Up to now, a number of methods have been used to overcome the dependency such as the truncate method, Stein method, and Heinrich’s method. The paper deals with the order of approximation in weak limit theorems for normalized negative–binomial sums of strictly stationary m-dependent random variables generated by a sequence of independent, identically distributed random variables, using the moving average techniques. The orders of approximation in desired theorems are established in terms of the so-called Zolotarev ζ-metric. The obtained results are extensions and generalizations of several known results related to negative–binomial random sums and geometric random sums of independent, identically distributed random variables.

   203-224