The Experts below are selected from a list of 2607288 Experts worldwide ranked by ideXlab platform
Lionel Vaux - One of the best experts on this subject based on the ideXlab platform.
-
Transport of Finiteness structures and applications
Mathematical Structures in Computer Science, 2018Co-Authors: Christine Tasson, Lionel VauxAbstract:We describe a general construction of Finiteness spaces which subsumes the interpretations of all positive connectors of linear logic. We then show how to apply this construction to prove the existence of least fixpoints for particular functors in the category of Finiteness spaces: these include the functors involved in a relational interpretation of lazy recursive algebraic datatypes along the lines of the coherence semantics of system T.
-
Strong Normalizability as a Finiteness Structure via the Taylor Expansion of λ -terms
2016Co-Authors: Michele Pagani, Christine Tasson, Lionel VauxAbstract:In the folklore of linear logic, a common intuition is that the structure of Finiteness spaces, introduced by Ehrhard, semantically reflects the strong normalization property of cut-elimination. We make this intuition formal in the context of the non-deterministic λ-calculus by introducing a Finiteness structure on resource terms, which is such that a λ-term is strongly normalizing iff the support of its Taylor expansion is finitary. An application of our result is the existence of a normal form for the Taylor expansion of any strongly normalizable non-deterministic λ-term.
-
Strong Normalizability as a Finiteness Structure via the Taylor Expansion of {\lambda}-terms
arXiv: Logic in Computer Science, 2016Co-Authors: Michele Pagani, Christine Tasson, Lionel VauxAbstract:In the folklore of linear logic, a common intuition is that the structure of Finiteness spaces, introduced by Ehrhard, semantically reflects the strong normalization property of cut-elimination. We make this intuition formal in the context of the non-deterministic {\lambda}-calculus by introducing a Finiteness structure on resource terms, which is such that a {\lambda}-term is strongly normalizing iff the support of its Taylor expansion is finitary. An application of our result is the existence of a normal form for the Taylor expansion of any strongly normalizable non-deterministic {\lambda}-term.
-
Primitive recursion in Finiteness spaces
2009Co-Authors: Lionel VauxAbstract:We study iteration and recursion operators in the multiset relational model of linear logic and prove them finitary in the sense of the Finiteness spaces recently introduced by Ehrhard. This provides a denotational semantics of Gödel's system T and paves the way for a systematic study of a large class of algorithms, following the ideas of Girard's quantitative semantics in a standard algebraic setting.
Jacques Theys - One of the best experts on this subject based on the ideXlab platform.
-
an explicit counterexample to the lagarias wang Finiteness conjecture
Advances in Mathematics, 2011Co-Authors: Kevin G Hare, Ian Morris, Nikita Sidorov, Jacques TheysAbstract:Abstract The joint spectral radius of a finite set of real d × d matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the Finiteness property if there exists a periodic product which achieves this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real d × d matrices satisfies the Finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the Finiteness conjecture exist, showing in particular that there exists a family of pairs of 2 × 2 matrices which contains a counterexample. Similar results were subsequently given by V.D. Blondel, J. Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample to the Finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias–Wang Finiteness conjecture. Namely, for the set A α ⁎ : = { ( 1 1 0 1 ) , α ⁎ ( 1 0 1 1 ) } we give an explicit value of α ⁎ ≃ 0.749326546330367557943961948091344672091327370236064317358024 … such that A α ⁎ does not satisfy the Finiteness property.
-
an explicit counterexample to the lagarias wang Finiteness conjecture
arXiv: Optimization and Control, 2010Co-Authors: Kevin G Hare, Ian Morris, Nikita Sidorov, Jacques TheysAbstract:The joint spectral radius of a finite set of real $d \times d$ matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the \emph{Finiteness property} if there exists a periodic product which achieves this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real $d \times d$ matrices satisfies the Finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the Finiteness conjecture exist, showing in particular that there exists a family of pairs of $2 \times 2$ matrices which contains a counterexample. Similar results were subsequently given by V.D. Blondel, J. Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample to the Finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias-Wang Finiteness conjecture. Namely, for the set \[ \mathsf{A}_{\alpha_*}:= \{({cc}1&1\\0&1), \alpha_*({cc}1&0\\1&1)\}\] we give an explicit value of \alpha_* \simeq 0.749326546330367557943961948091344672091327370236064317358024...] such that $\mathsf{A}_{\alpha_*}$ does not satisfy the Finiteness property.
Khalid Bourabee - One of the best experts on this subject based on the ideXlab platform.
-
residual Finiteness growths of virtually special groups
Mathematische Zeitschrift, 2015Co-Authors: Khalid Bourabee, Mark F Hagen, Priyam PatelAbstract:Let \(G\) be a virtually special group. Then the residual Finiteness growth of \(G\) is at most linear. This result cannot be found by embedding \(G\) into a special linear group. Indeed, the special linear group \({{\mathrm{SL}}}_k(\mathbb {Z})\), for \(k > 2\), has residual Finiteness growth \(n^{k-1}\).
-
residual Finiteness growths of virtually special groups
arXiv: Group Theory, 2014Co-Authors: Khalid Bourabee, Mark F Hagen, Priyam PatelAbstract:Let $G$ be a virtually special group. Then the residual Finiteness growth of $G$ is at most linear. This result cannot be found by embedding $G$ into a special linear group. Indeed, the special linear group $\text{SL}_k(\mathbb{Z})$, for $k > 2$, has residual Finiteness growth $n^{k-1}$.
-
quantifying residual Finiteness
Journal of Algebra, 2010Co-Authors: Khalid BourabeeAbstract:Abstract We introduce the notion of quantifying the extent to which a finitely generated group is residually finite. We investigate this behavior for examples that include free groups, the first Grigorchuk group, finitely generated nilpotent groups, and certain arithmetic groups such as SL n ( Z ) . In the context of finite nilpotent quotients, we find a new characterization of finitely generated nilpotent groups.
Pavel Shvartsman - One of the best experts on this subject based on the ideXlab platform.
-
sharp Finiteness principles for lipschitz selections
Geometric and Functional Analysis, 2018Co-Authors: Charles Fefferman, Pavel ShvartsmanAbstract:Let \({(\mathcal{M}, \rho) }\) be a metric space and let Y be a Banach space. Given a positive integer m, let F be a set-valued mapping from \({\mathcal{M}}\) into the family of all compact convex subsets of Y of dimension at most m. In this paper we prove a Finiteness principle for the existence of a Lipschitz selection of F with the sharp value of the Finiteness constant.
-
the whitney extension problem and lipschitz selections of set valued mappings in jet spaces
Transactions of the American Mathematical Society, 2008Co-Authors: Pavel ShvartsmanAbstract:We study a variant of the Whitney extension problem (1934) for the space C k,ω (R n ). We identify C k,ω (R n ) with a space of Lipschitz mappings from R n into the space P k x R" of polynomial fields on R n equipped with a certain metric. This identification allows us to reformulate the Whitney problem for C k,ω (R n ) as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of P k x R". We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for Finiteness numbers in Finiteness theorems for C k,ω (R n ) due to C. Fefferman.
-
the whitney extension problem and lipschitz selections of set valued mappings in jet spaces
arXiv: Functional Analysis, 2006Co-Authors: Pavel ShvartsmanAbstract:We study a variant of the Whitney extension problem for the space $C^{k,\omega}(R^n)$. We identify this space with a space of Lipschitz mappings from $R^n$ into the space $P_k \times R^n$ of polynomial fields on $R^n$ equipped with a certain metric. This identification allows us to reformulate the Whitney problem for $C^{k,\omega}(R^n)$ as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of $P_k \times R^n$. We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for Finiteness numbers in Finiteness theorems for $C^{k,\omega}(R^n)$ due to C. Fefferman.
Kevin G Hare - One of the best experts on this subject based on the ideXlab platform.
-
an explicit counterexample to the lagarias wang Finiteness conjecture
Advances in Mathematics, 2011Co-Authors: Kevin G Hare, Ian Morris, Nikita Sidorov, Jacques TheysAbstract:Abstract The joint spectral radius of a finite set of real d × d matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the Finiteness property if there exists a periodic product which achieves this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real d × d matrices satisfies the Finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the Finiteness conjecture exist, showing in particular that there exists a family of pairs of 2 × 2 matrices which contains a counterexample. Similar results were subsequently given by V.D. Blondel, J. Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample to the Finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias–Wang Finiteness conjecture. Namely, for the set A α ⁎ : = { ( 1 1 0 1 ) , α ⁎ ( 1 0 1 1 ) } we give an explicit value of α ⁎ ≃ 0.749326546330367557943961948091344672091327370236064317358024 … such that A α ⁎ does not satisfy the Finiteness property.
-
an explicit counterexample to the lagarias wang Finiteness conjecture
arXiv: Optimization and Control, 2010Co-Authors: Kevin G Hare, Ian Morris, Nikita Sidorov, Jacques TheysAbstract:The joint spectral radius of a finite set of real $d \times d$ matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the \emph{Finiteness property} if there exists a periodic product which achieves this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real $d \times d$ matrices satisfies the Finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the Finiteness conjecture exist, showing in particular that there exists a family of pairs of $2 \times 2$ matrices which contains a counterexample. Similar results were subsequently given by V.D. Blondel, J. Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample to the Finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias-Wang Finiteness conjecture. Namely, for the set \[ \mathsf{A}_{\alpha_*}:= \{({cc}1&1\\0&1), \alpha_*({cc}1&0\\1&1)\}\] we give an explicit value of \alpha_* \simeq 0.749326546330367557943961948091344672091327370236064317358024...] such that $\mathsf{A}_{\alpha_*}$ does not satisfy the Finiteness property.
