{"title":"On the $K$-theory of $C^*$-algebras associated to substitution tilings","authors":"D. Gonçalves, Maria Ramirez-Solano","doi":"10.4064/dm800-4-2020","DOIUrl":"https://doi.org/10.4064/dm800-4-2020","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"551 1","pages":"1-133"},"PeriodicalIF":1.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized $T^p_u$ spaces: On the trail of Calderón and Zygmund","authors":"L. Loosveldt, S. Nicolay","doi":"10.4064/dm798-4-2020","DOIUrl":"https://doi.org/10.4064/dm798-4-2020","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"554 1","pages":"1-64"},"PeriodicalIF":1.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70158182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The algebra of differential invariants under $SA_3(mathbb{R})$ of generic parabolic surfaces $S^2 subset mathbb{R}^3$ with nonvanishing Pocchiola $4^{text{th}}$ invariant $W$ is shown to be generated, through invariant differentiations, by only one other invariant, $M$, of order $5$, having $57$ differential monomials. The proof is based on Fels-Olver's recurrence formulas, pulled back to the parabolic jet bundles.
{"title":"On differential invariants of parabolic surfaces","authors":"Zhangchi Chen, J. Merker","doi":"10.4064/DM816-8-2020","DOIUrl":"https://doi.org/10.4064/DM816-8-2020","url":null,"abstract":"The algebra of differential invariants under $SA_3(mathbb{R})$ of generic parabolic surfaces $S^2 subset mathbb{R}^3$ with nonvanishing Pocchiola $4^{text{th}}$ invariant $W$ is shown to be generated, through invariant differentiations, by only one other invariant, $M$, of order $5$, having $57$ differential monomials. The proof is based on Fels-Olver's recurrence formulas, pulled back to the parabolic jet bundles.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43600764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Kadets, Miguel Martín, Javier Merí, Antonio Pérez, Alicia Quero
Given Banach spaces $X$ and $Y$, and a norm-one operator $Gin mathcal{L}(X,Y)$, the numerical index with respect to $G$, $n_G(X,Y)$, is the greatest constant $kgeq 0$ such that $$max_{|w|=1}|G+wT|geq 1 + k |T|$$ for all $Tin mathcal{L}(X,Y)$. We present some results on the set $mathcal{N}(mathcal{L}(X,Y))$ of the values of the numerical indices with respect to all norm-one operators on $mathcal{L}(X,Y)$. We show that $mathcal{N}(mathcal{L}(X,Y))={0}$ when $X$ or $Y$ is a real Hilbert space of dimension greater than one and also when $X$ or $Y$ is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. For complex Hilbert spaces $H_1$, $H_2$ of dimension greater than one, we show that $mathcal{N}(mathcal{L}(H_1,H_2))subseteq {0,1/2}$ and the value $1/2$ is taken if and only if $H_1$ and $H_2$ are isometrically isomorphic. Besides, $mathcal{N}(mathcal{L}(X,H))subseteq [0,1/2]$ and $mathcal{N}(mathcal{L}(H,Y))subseteq [0,1/2]$ when $H$ is a complex infinite-dimensional Hilbert space and $X$ and $Y$ are arbitrary complex Banach spaces. We also show that $mathcal{N}(mathcal{L}(L_1(mu_1),L_1(mu_2)))subseteq {0,1}$ and $mathcal{N}(mathcal{L}(L_infty(mu_1),L_infty(mu_2)))subseteq {0,1}$ for arbitrary $sigma$-finite measures $mu_1$ and $mu_2$, in both the real and the complex cases. Also, we show that the Lipschitz numerical range of Lipschitz maps can be viewed as the numerical range of convenient bounded linear operators with respect to a bounded linear operator. Further, we provide some results which show the behaviour of the value of the numerical index when we apply some Banach space operations, as constructing diagonal operators between $c_0$-, $ell_1$-, or $ell_infty$-sums of Banach spaces, composition operators on some vector-valued function spaces, and taking the adjoint to an operator.
{"title":"On the numerical index with respect to an operator","authors":"V. Kadets, Miguel Martín, Javier Merí, Antonio Pérez, Alicia Quero","doi":"10.4064/dm805-9-2019","DOIUrl":"https://doi.org/10.4064/dm805-9-2019","url":null,"abstract":"Given Banach spaces $X$ and $Y$, and a norm-one operator $Gin mathcal{L}(X,Y)$, the numerical index with respect to $G$, $n_G(X,Y)$, is the greatest constant $kgeq 0$ such that $$max_{|w|=1}|G+wT|geq 1 + k |T|$$ for all $Tin mathcal{L}(X,Y)$. We present some results on the set $mathcal{N}(mathcal{L}(X,Y))$ of the values of the numerical indices with respect to all norm-one operators on $mathcal{L}(X,Y)$. We show that $mathcal{N}(mathcal{L}(X,Y))={0}$ when $X$ or $Y$ is a real Hilbert space of dimension greater than one and also when $X$ or $Y$ is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. For complex Hilbert spaces $H_1$, $H_2$ of dimension greater than one, we show that $mathcal{N}(mathcal{L}(H_1,H_2))subseteq {0,1/2}$ and the value $1/2$ is taken if and only if $H_1$ and $H_2$ are isometrically isomorphic. Besides, $mathcal{N}(mathcal{L}(X,H))subseteq [0,1/2]$ and $mathcal{N}(mathcal{L}(H,Y))subseteq [0,1/2]$ when $H$ is a complex infinite-dimensional Hilbert space and $X$ and $Y$ are arbitrary complex Banach spaces. We also show that $mathcal{N}(mathcal{L}(L_1(mu_1),L_1(mu_2)))subseteq {0,1}$ and $mathcal{N}(mathcal{L}(L_infty(mu_1),L_infty(mu_2)))subseteq {0,1}$ for arbitrary $sigma$-finite measures $mu_1$ and $mu_2$, in both the real and the complex cases. Also, we show that the Lipschitz numerical range of Lipschitz maps can be viewed as the numerical range of convenient bounded linear operators with respect to a bounded linear operator. Further, we provide some results which show the behaviour of the value of the numerical index when we apply some Banach space operations, as constructing diagonal operators between $c_0$-, $ell_1$-, or $ell_infty$-sums of Banach spaces, composition operators on some vector-valued function spaces, and taking the adjoint to an operator.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41688177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations. In Chapter 2 we collect all the definitions and results regarding derivations that are essential while studying this area. In Chapter 3 we intend to show that derivations can be characterized by one single functional equation. More exactly, we study here the following problem. Let $Q$ be a commutative ring and let $P$ be a subring of $Q$. Let $lambda, muin Qsetminusleft{0right}$ be arbitrary, $fcolon Prightarrow Q$ be a function and consider the equation [ lambdaleft[f(x+y)-f(x)-f(y)right]+ muleft[f(xy)-xf(y)-yf(x)right]=0 quad left(x, yin Pright). ] In this chapter it will be proved that under some assumptions on the rings $P$ and $Q$, derivations can be characterized via the above equation. Chapter 4 is devoted to the additive solvability of a system of functional equations. Moreover, the linear dependence and independence of the additive solutions $d_{0},d_{1},dots,d_{n} colonmathbb{R}tomathbb{R}$ of the above system of equations is characterized. Finally, the closing chapter deals with the following problem. Assume that $xicolon mathbb{R}to mathbb{R}$ is a given differentiable function and for the additive function $fcolon mathbb{R}to mathbb{R}$, the mapping [ varphi(x)=fleft(xi(x)right)-xi'(x)f(x) ] fulfills some regularity condition on its domain. Is it true that in such a case $f$ is a sum of a derivation and a linear function?
这项工作的主要目的是通过函数方程来表征导数。这部作品由五章组成。在第一章中，我们总结了函数方程理论中最重要的概念和结果。在第2章中，我们收集了在研究这一领域时必不可少的关于导数的所有定义和结果。在第三章中，我们打算证明导数可以用一个函数方程来表征。更确切地说，我们在这里研究以下问题。设$Q$是交换环，设$P$是$Q$的子环。设$lambda，muinQsetminusleft｛0right｝$是任意的，$fcolon Prightarrow Q$是一个函数，并考虑方程[lambdaleft[f（x+y）-f（x）-f（y）right]+muleft[f（xy）-xf（y）-yf（x）rigight]=0quadleft（x，yinPright）在本章中，将证明在环$P$和$Q$的一些假设下，导数可以通过上述方程来表征。第四章研究函数方程组的加性可解性。此外，刻画了上述方程组的加性解$d_{0}，d_{1}，dots，d_{n}colonmathbb{R}tomathbb{R}$的线性依赖性和独立性。最后，最后一章讨论了以下问题。假设$neneneba xi colonmathbb｛R｝tomathbb{R｝$是一个给定的可微函数，并且对于加法函数$fcolonmath bb｛R｝to mathbb｝$，映射[varphi（x）=fleft（nenenebb xi（x）right）-nenenebc xi’（x）f（x）]在其域上满足一些正则性条件。在这种情况下，$f$是一个导数和一个线性函数的和，这是真的吗？
{"title":"Characterizations of derivations","authors":"E. Gselmann","doi":"10.4064/DM775-9-2018","DOIUrl":"https://doi.org/10.4064/DM775-9-2018","url":null,"abstract":"The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations. In Chapter 2 we collect all the definitions and results regarding derivations that are essential while studying this area. \u0000In Chapter 3 we intend to show that derivations can be characterized by one single functional equation. More exactly, we study here the following problem. Let $Q$ be a commutative ring and let $P$ be a subring of $Q$. Let $lambda, muin Qsetminusleft{0right}$ be arbitrary, $fcolon Prightarrow Q$ be a function and consider the equation [ lambdaleft[f(x+y)-f(x)-f(y)right]+ muleft[f(xy)-xf(y)-yf(x)right]=0 quad left(x, yin Pright). ] In this chapter it will be proved that under some assumptions on the rings $P$ and $Q$, derivations can be characterized via the above equation. \u0000Chapter 4 is devoted to the additive solvability of a system of functional equations. Moreover, the linear dependence and independence of the additive solutions $d_{0},d_{1},dots,d_{n} colonmathbb{R}tomathbb{R}$ of the above system of equations is characterized. \u0000Finally, the closing chapter deals with the following problem. Assume that $xicolon mathbb{R}to mathbb{R}$ is a given differentiable function and for the additive function $fcolon mathbb{R}to mathbb{R}$, the mapping [ \u0000varphi(x)=fleft(xi(x)right)-xi'(x)f(x) ] fulfills some regularity condition on its domain. Is it true that in such a case $f$ is a sum of a derivation and a linear function?","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43525701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Albiac, J. L. Ansorena, P. M. Berná, P. Wojtaszczyk
The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems (also known as Markushevich bases) in quasi-Banach spaces from a functional-analytic point of view. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (nontrivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations amongst them are carefully discussed.
{"title":"Greedy approximation for biorthogonal systems in quasi-Banach spaces","authors":"F. Albiac, J. L. Ansorena, P. M. Berná, P. Wojtaszczyk","doi":"10.4064/DM817-11-2020","DOIUrl":"https://doi.org/10.4064/DM817-11-2020","url":null,"abstract":"The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems (also known as Markushevich bases) in quasi-Banach spaces from a functional-analytic point of view. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (nontrivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations amongst them are carefully discussed.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42698439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the existence and regularity of the invariant graphs for bundle maps (or bundle correspondences with generating bundle maps motivated by ill-posed differential equations) having some relatively partial hyperbolicity in non-trivial bundles without local compactness. The regularity includes (uniformly) $ C^0 $ continuity, Holder continuity and smoothness. A number of applications to both well-posed and ill-posed semi-linear differential equations and the abstract infinite-dimensional dynamical systems are given to illustrate its power, such as the existence and regularity of different types of invariant foliations (laminations) including strong stable laminations and fake invariant foliations, the existence and regularity of holonomies for cocycles, $ C^{k,alpha} $ section theorem and decoupling theorem, etc, in more general settings.
{"title":"Existence and regularity of invariant graphs for cocycles in bundles: partial hyperbolicity case","authors":"Deliang Chen","doi":"10.4064/DM799-4-2020","DOIUrl":"https://doi.org/10.4064/DM799-4-2020","url":null,"abstract":"We study the existence and regularity of the invariant graphs for bundle maps (or bundle correspondences with generating bundle maps motivated by ill-posed differential equations) having some relatively partial hyperbolicity in non-trivial bundles without local compactness. The regularity includes (uniformly) $ C^0 $ continuity, Holder continuity and smoothness. A number of applications to both well-posed and ill-posed semi-linear differential equations and the abstract infinite-dimensional dynamical systems are given to illustrate its power, such as the existence and regularity of different types of invariant foliations (laminations) including strong stable laminations and fake invariant foliations, the existence and regularity of holonomies for cocycles, $ C^{k,alpha} $ section theorem and decoupling theorem, etc, in more general settings.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45619343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The formulations that Peano gave to many mathematical notions at the end of the 19th century were so perfect and modern that they have become standard today. A formal language of logic that he created, enabled him to perceive mathematics with great precision and depth. He described mathematics axiomatically basing the reasoning exclusively on logical and set-theoretical primitive terms and properties, which was revolutionary at that time. Yet, numerous Peano’s contributions remain either unremembered or underestimated.
{"title":"The forgotten mathematical legacy of Peano","authors":"S. Dolecki, G. H. Greco","doi":"10.4064/DM769-4-2018","DOIUrl":"https://doi.org/10.4064/DM769-4-2018","url":null,"abstract":"The formulations that Peano gave to many mathematical notions at the end of the 19th century were so perfect and modern that they have become standard today. A formal language of logic that he created, enabled him to perceive mathematics with great precision and depth. He described mathematics axiomatically basing the reasoning exclusively on logical and set-theoretical primitive terms and properties, which was revolutionary at that time. Yet, numerous Peano’s contributions remain either unremembered or underestimated.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48466508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study $N$-ary non-commutative notions of independence, which are given by trees and which generalize free, Boolean, and monotone independence. For every rooted subtree $mathcal{T}$ of the $N$-regular tree, we define the $mathcal{T}$-free product of $N$ non-commutative probability spaces and we define the $mathcal{T}$-free additive convolution of $N$ non-commutative laws. These $N$-ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities (such as the relation between free, monotone, and subordination convolutions studied by Lenczewski) can be reduced to combinatorial manipulations of trees. We also develop a theory of $mathcal{T}$-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor.
{"title":"An operad of non-commutative independences defined by trees","authors":"David Jekel, Weihua Liu","doi":"10.4064/dm797-6-2020","DOIUrl":"https://doi.org/10.4064/dm797-6-2020","url":null,"abstract":"We study $N$-ary non-commutative notions of independence, which are given by trees and which generalize free, Boolean, and monotone independence. For every rooted subtree $mathcal{T}$ of the $N$-regular tree, we define the $mathcal{T}$-free product of $N$ non-commutative probability spaces and we define the $mathcal{T}$-free additive convolution of $N$ non-commutative laws. These $N$-ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities (such as the relation between free, monotone, and subordination convolutions studied by Lenczewski) can be reduced to combinatorial manipulations of trees. We also develop a theory of $mathcal{T}$-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47824797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif dans le cas rationnel","authors":"François Ballaÿ","doi":"10.4064/dm781-5-2019","DOIUrl":"https://doi.org/10.4064/dm781-5-2019","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":"7 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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