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Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Cited by 46 (19 self)
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
Partition Algebras
, 2005
"... The partition algebra CAk(n) is the centralizer algebra of Sn acting on the kfold tensor product V ⊗k of its ndimensional permutation representation V.The partition algebra CA 1 (n) k+ 2 is the centralizer algebra of the restriction of V ⊗k to Sn−1 ⊆ Sn. Weapply the theory of the basic constructio ..."
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Cited by 34 (10 self)
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The partition algebra CAk(n) is the centralizer algebra of Sn acting on the kfold tensor product V ⊗k of its ndimensional permutation representation V.The partition algebra CA 1 (n) k+ 2 is the centralizer algebra of the restriction of V ⊗k to Sn−1 ⊆ Sn. Weapply the theory of the basic construction (generalized matrix algebras) to the tower of partition algebras CA0(n) ⊆ CA 1 (n) CA1(n) ⊆ CA 1 (n) ⊆···.Ourmainresults are: 2 1 2 (a) a presentation on generators and relations for CAk(n); (b) aderivation of “Specht modules ” from the basic construction; (c) a proof that CAk(n) is semisimple if and only if k ≤ (n + 1)/2 (except for a few special cases); (d) Murphy elements for CAk(n);and (e) an exposition on the theory of the basic construction and semisimple algebras.
The BoyerMoore Theorem Prover and Its Interactive Enhancement
, 1995
"... . The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. First we introduce the logic in ..."
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Cited by 31 (0 self)
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. The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. First we introduce the logic in which theorems are proved. Then we briefly describe the two mechanized theorem proving systems. Next, we present a simple but illustrative example in some detail in order to give an impression of how these systems may be used successfully. Finally, we give extremely short descriptions of a large number of applications of these systems, in order to give an idea of the breadth of their uses. This paper is intended as an informal introduction to systems that have been described in detail and similarly summarized in many other books and papers; no new results are reported here. Our intention here is merely to present Nqthm to a new audience. This research was supported in part by ONR Contract N...
A Theorem Prover for a Computational Logic
, 1990
"... We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of line ..."
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Cited by 24 (0 self)
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We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of linear resolution, rewriting, and arithmetic decision procedures. We describe some applications of the prover, including a proof of the correct implementation of a higher level language on a microprocessor defined at the gate level. We also describe the ongoing project of recoding the entire prover as an applicative function within its own logic.
Proof checking the RSA public key encryption algorithm
 American Mathematical Monthly
, 1984
"... The authors describe the use of a mechanical theoremprover to check the published proof of the invertibility of the public key encryption algorithm of Rivest, Shamir and Adleman: (M mod n) mod N=M, provided n is the product of two distinct primes p and q, M
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Cited by 22 (9 self)
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The authors describe the use of a mechanical theoremprover to check the published proof of the invertibility of the public key encryption algorithm of Rivest, Shamir and Adleman: (M mod n) mod N=M, provided n is the product of two distinct primes p and q, M<n, and e and d are multiplicative inverses in the ring of integers modulo (p1)*(q1). Among the lemmas proved mechanically and used in the main proof are many familiar theorems of number theory, including Fermat’s theorem: M mod p=1, when p M. The axioms underlying the proofs are those of Peano arithmetic and ordered pairs. The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules. Godel [11] But formalized mathematics cannot in practice be written down in full, and therefore we must have confidence in what might be called the common sense of the mathematician... We shall therefore very quickly abandon formalized mathematics... Bourbaki [1] 1.
Derived Categories of Quadric Fibrations and Intersections of Quadrics
, 2005
"... We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algeb ..."
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Cited by 13 (4 self)
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We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration generalizing the Kapranov’s description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety (P(S 2 W ∗), B0), where B0 is the universal sheaf of even parts of Clifford algebras, is Homologically Projectively Dual to the projective space P(W) in the double Veronese embedding P(W) → P(S 2 W). Using the properties of the Homological Projective Duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics.
Constructing the real numbers in HOL
, 1992
"... This paper describes a construction of the real numbers in the HOL theoremprover by strictly definitional means using a version of Dedekind's method. It also outlines the theory of mathematical analysis that has been built on top of it and discusses current and potential applications in verificatio ..."
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Cited by 7 (1 self)
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This paper describes a construction of the real numbers in the HOL theoremprover by strictly definitional means using a version of Dedekind's method. It also outlines the theory of mathematical analysis that has been built on top of it and discusses current and potential applications in verification and computer algebra. Keywords: Mathematical Logic; Deduction and Theorem Proving 1 The real numbers For some mathematical tasks, the natural numbers N = f0; 1; 2; : : :g are sufficient. However for many purposes it is convenient to use a more extensive system, such as the integers (Z) or the rational (Q ), real (R) or complex (C ) numbers. In particular the real numbers are normally used for the measurement of physical quantities which (at least in abstract models) are continuously variable, and are therefore ubiquitous in scientific applications. 1.1 Properties of the real numbers We can characterize the reals as the unique `complete ordered field'. More precisely, the reals are a set ...
Universal differential calculus on ternary algebras
"... General concept of ternary algebras is introduced in this article, along with several examples of its realization. Universal envelope of such algebras is defined, as well as the concept of trimodules over ternary algebras. The universal differential calculus on these structures is then defined and ..."
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Cited by 7 (0 self)
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General concept of ternary algebras is introduced in this article, along with several examples of its realization. Universal envelope of such algebras is defined, as well as the concept of trimodules over ternary algebras. The universal differential calculus on these structures is then defined and its basic properties investigated. MSC2000: 46L87, 58B34, 20N10. Key words: ternary algebras, trimodules, universal ternary differential.
Covering Data and Higher Dimensional Global Class Field Theory. arXive: math/0804.3419
"... Abstract: For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism ρX: CX → πab 1 (X), which is surjective and whose kernel is the connected component of the identity. The (topological) group CX is explicitly given and built solely out of data a ..."
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Cited by 6 (2 self)
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Abstract: For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism ρX: CX → πab 1 (X), which is surjective and whose kernel is the connected component of the identity. The (topological) group CX is explicitly given and built solely out of data attached to points and curves on X. A similar but weaker statement holds for smooth varieties over finite fields. Our results are based on earlier work of G. Wiesend. To the memory of Götz Wiesend 1 The aim of global class field theory is the description of abelian extensions of arithmetic schemes (i.e. regular schemes X of finite type over Spec(Z)) in terms of arithmetic invariants attached to X. The solution of this problem in the case dim X = 1 was one of the major achievements of number theory in the first part of the previous century. In the 1980s, mainly due to K. Kato and S. Saito [8], a generalization to higher dimensional schemes has been found. The description of
Inductive Theory of Vision
, 1996
"... In spite of the fact that some of the outstanding physiologists and neurophysiologists (e.g. Hermann von Helmholtz and Horace Barlow) insisted on the central role of inductive learning processes in vision as well as in other sensory processes, there are absolutely no (computational) theories of visi ..."
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Cited by 3 (0 self)
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In spite of the fact that some of the outstanding physiologists and neurophysiologists (e.g. Hermann von Helmholtz and Horace Barlow) insisted on the central role of inductive learning processes in vision as well as in other sensory processes, there are absolutely no (computational) theories of vision that are guided by these processes. It appears that this is mainly due to the lack of understanding of what inductive learning processes are. We strongly believe in the central role of inductive learning processes, around which, we think, all other (intelligent) biological processes have evolved. In this paper we outline a (computational) theory of vision completely built around the inductive learning processes for all levels in vision. The development of such a theory became possible with the advent of the formal model of inductive learningevolving transformation system (ETS). The proposed theory is based on the concept of structured measurement device, which is motivated by the formal model of inductive learning and is a farreaching generalization of the concept of classical measurement device