The Experts below are selected from a list of 52227 Experts worldwide ranked by ideXlab platform
Clément De Seguins Pazzis - One of the best experts on this subject based on the ideXlab platform.
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Products of Involutions of an Infinite-Dimensional Vector Space
Canadian Journal of Mathematics, 2019Co-Authors: Clément De Seguins PazzisAbstract:AbstractWe prove that every automorphism of an infinite-Dimensional Vector Space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.
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Sums of Quadratic Endomorphisms of an Infinite-Dimensional Vector Space
Proceedings of the Edinburgh Mathematical Society, 2018Co-Authors: Clément De Seguins PazzisAbstract:We prove that every endomorphism of an infinite-Dimensional Vector Space over a field splits into the sum of four idempotents and into the sum of four square-zero endomorphisms, a result that is optimal in general.
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Sums of quadratic endomorphisms of an infinite-Dimensional Vector Space
arXiv: Rings and Algebras, 2016Co-Authors: Clément De Seguins PazzisAbstract:We prove that every endomorphism of an infinite-Dimensional Vector Space splits as the sum of four idempotents and as the sum of four square-zero endomorphisms, a result that is optimal in general.
Jesse Alama - One of the best experts on this subject based on the ideXlab platform.
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the rank nullity theorem
Formalized Mathematics, 2007Co-Authors: Jesse AlamaAbstract:Summary. The rank+nullity theorem states that, if T is a linear transformation from a nite-Dimensional Vector Space V to a nite-Dimensional Vector Space W , then dim(V ) = rank(T ) + nullity(T ), where rank(T ) = dim(im(T )) and nullity(T ) = dim(ker(T )). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T ) and extend it to a basis B of V , and then show that dim(im(T )) is equal tojB Aj, and that T is one-to-one on B A.
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The Rank+Nullity Theorem
Formalized Mathematics, 2007Co-Authors: Jesse AlamaAbstract:Summary. The rank+nullity theorem states that, if T is a linear transformation from a nite-Dimensional Vector Space V to a nite-Dimensional Vector Space W , then dim(V ) = rank(T ) + nullity(T ), where rank(T ) = dim(im(T )) and nullity(T ) = dim(ker(T )). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T ) and extend it to a basis B of V , and then show that dim(im(T )) is equal tojB Aj, and that T is one-to-one on B A.
Edoardo Ballico - One of the best experts on this subject based on the ideXlab platform.
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Tensor ranks and symmetric tensor ranks are the same for points with low symmetric tensor rank
Archiv der Mathematik, 2011Co-Authors: Edoardo BallicoAbstract:Fix integers n ≥ 1, d ≥ 2. Let V be an (n + 1)-Dimensional Vector Space over a field with characteristic zero. Fix a symmetric tensor \({T\in S^d(V)\subset V^{\otimes d}}\). Here we prove that the tensor rank of T is equal to its symmetric tensor rank if the latter is at most (d + 1)/2.
Chris Eckl - One of the best experts on this subject based on the ideXlab platform.
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sentic medoids organizing affective common sense knowledge in a multi Dimensional Vector Space
International Symposium on Neural Networks, 2011Co-Authors: Erik Cambria, Amir Hussain, Thomas Mazzocco, Chris EcklAbstract:Existing approaches to opinion mining and sentiment analysis mainly rely on parts of text in which opinions and sentiments are explicitly expressed such as polarity terms and affect words. However, opinions and sentiments are often conveyed implicitly through context and domain dependent concepts, which make purely syntactical approaches ineffective. To overcome this problem, we have recently proposed Sentic Computing, a multi-disciplinary approach to opinion mining and sentiment analysis that exploits both computer and social sciences to better recognize and process opinions and sentiments over the Web. Among other tools, Sentic Computing includes AffectiveSpace, a language visualization system that transforms natural language from a linguistic form into a multi-Dimensional Space. In this work, we present a new technique to better cluster this Vector Space and, hence, better organize and reason on the affective common sense knowledge in it contained.
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senticSpace visualizing opinions and sentiments in a multi Dimensional Vector Space
International Conference on Knowledge-Based and Intelligent Information and Engineering Systems, 2010Co-Authors: Erik Cambria, Amir Hussain, Catherine Havasi, Chris EcklAbstract:In a world in which millions of people express their feelings and opinions about any issue in blogs, wikis, fora, chats and social networks, the distillation of knowledge from this huge amount of unstructured information is a challenging task. In this work we build a knowledge base which merges common sense and affective knowledge and visualize it in a multi-Dimensional Vector Space, which we call SenticSpace. In particular we blend ConceptNet and WordNet-Affect and use Dimensionality reduction on the resulting knowledge base to build a 24-Dimensional Vector Space in which different Vectors represent different ways of making binary distinctions among concepts and sentiments.
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KES (4) - SenticSpace: visualizing opinions and sentiments in a multi-Dimensional Vector Space
Knowledge-Based and Intelligent Information and Engineering Systems, 2010Co-Authors: Erik Cambria, Amir Hussain, Catherine Havasi, Chris EcklAbstract:In a world in which millions of people express their feelings and opinions about any issue in blogs, wikis, fora, chats and social networks, the distillation of knowledge from this huge amount of unstructured information is a challenging task. In this work we build a knowledge base which merges common sense and affective knowledge and visualize it in a multi-Dimensional Vector Space, which we call SenticSpace. In particular we blend ConceptNet and WordNet-Affect and use Dimensionality reduction on the resulting knowledge base to build a 24-Dimensional Vector Space in which different Vectors represent different ways of making binary distinctions among concepts and sentiments.
Erik Cambria - One of the best experts on this subject based on the ideXlab platform.
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sentic medoids organizing affective common sense knowledge in a multi Dimensional Vector Space
International Symposium on Neural Networks, 2011Co-Authors: Erik Cambria, Amir Hussain, Thomas Mazzocco, Chris EcklAbstract:Existing approaches to opinion mining and sentiment analysis mainly rely on parts of text in which opinions and sentiments are explicitly expressed such as polarity terms and affect words. However, opinions and sentiments are often conveyed implicitly through context and domain dependent concepts, which make purely syntactical approaches ineffective. To overcome this problem, we have recently proposed Sentic Computing, a multi-disciplinary approach to opinion mining and sentiment analysis that exploits both computer and social sciences to better recognize and process opinions and sentiments over the Web. Among other tools, Sentic Computing includes AffectiveSpace, a language visualization system that transforms natural language from a linguistic form into a multi-Dimensional Space. In this work, we present a new technique to better cluster this Vector Space and, hence, better organize and reason on the affective common sense knowledge in it contained.
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senticSpace visualizing opinions and sentiments in a multi Dimensional Vector Space
International Conference on Knowledge-Based and Intelligent Information and Engineering Systems, 2010Co-Authors: Erik Cambria, Amir Hussain, Catherine Havasi, Chris EcklAbstract:In a world in which millions of people express their feelings and opinions about any issue in blogs, wikis, fora, chats and social networks, the distillation of knowledge from this huge amount of unstructured information is a challenging task. In this work we build a knowledge base which merges common sense and affective knowledge and visualize it in a multi-Dimensional Vector Space, which we call SenticSpace. In particular we blend ConceptNet and WordNet-Affect and use Dimensionality reduction on the resulting knowledge base to build a 24-Dimensional Vector Space in which different Vectors represent different ways of making binary distinctions among concepts and sentiments.
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KES (4) - SenticSpace: visualizing opinions and sentiments in a multi-Dimensional Vector Space
Knowledge-Based and Intelligent Information and Engineering Systems, 2010Co-Authors: Erik Cambria, Amir Hussain, Catherine Havasi, Chris EcklAbstract:In a world in which millions of people express their feelings and opinions about any issue in blogs, wikis, fora, chats and social networks, the distillation of knowledge from this huge amount of unstructured information is a challenging task. In this work we build a knowledge base which merges common sense and affective knowledge and visualize it in a multi-Dimensional Vector Space, which we call SenticSpace. In particular we blend ConceptNet and WordNet-Affect and use Dimensionality reduction on the resulting knowledge base to build a 24-Dimensional Vector Space in which different Vectors represent different ways of making binary distinctions among concepts and sentiments.