Vector Space Research Papers - Academia.edu (original) (raw)

Topic-based language model has attracted much attention as the propounding of semantic retrieval in recent years. Especially for the ASR text with errors, the topic representation is more reasonable than the exact term representation.... more

Topic-based language model has attracted much attention as the propounding of semantic retrieval in recent years. Especially for the ASR text with errors, the topic representation is more reasonable than the exact term representation. Among these models, Latent Dirichlet Allocation(LDA) has been noted for its ability to discover the latent topic structure, and is broadly applied in many text-related tasks. But up to now its application in information retrieval(IR) is still limited to be a supplement to the standard document models, and furthermore, it has been pointed out that directly employing the basic LDA model will hurt retrieval performance. In this paper, we propose a lexicon-guided two-level LDA retrieval framework. It uses the HowNet to guide the first-level LDA model's parameter estimation, and further construct the second-level LDA models based on the first-level's inference results. We use TRECID 2005 ASR collection to evaluate it, and compare it with the vector space model(VSM) and latent semantic Indexing(LSI). Our experiments show the proposed method is very competitive.

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is... more

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is w-well-covered if all maximal independent sets of G are of the same weight. The set of weight functions w for which a graph is w-well-covered is a vector space. We prove that finding the vector space of weight functions under which an input graph is w-well-covered can be done in polynomial time, if the input graph does not contain cycles of length 4, 5, 6 and 7.

In this paper we introduce COV, a novel information retrieval (IR) algorithm for massive databases based on vector space modeling and spectral analysis of the covariance matrix, for the document vectors, to reduce the scale of the... more

In this paper we introduce COV, a novel information retrieval (IR) algorithm for massive databases based on vector space modeling and spectral analysis of the covariance matrix, for the document vectors, to reduce the scale of the problem. Since the dimension of the covariance matrix depends on the attribute space and is independent of the number of documents, COV can be applied to databases that are too massive for methods based on the singular value decomposition of the document-attribute matrix, such as latent semantic indexing (LSI). In addition to improved scalability, theoretical considerations indicate that results from our algorithm tend to be more accurate than those from LSI, particularly in detecting subtle differences in document vectors. We demonstrate the power and accuracy of COV through an important topic in data mining, known as outlier cluster detection. We propose two new algorithms for detecting major and outlier clusters in databases—the first is based on LSI, and the second on COV. Our implementation studies indicate that our cluster detection algorithms outperform the basic LSI and COV algorithm in detecting outlier clusters.

The scalar triple product of three vectors find a useful way of introducing linear independence/dependence of three vectors which is the central concept in linear algebra. The author successfully used this approach as a primer to... more

The scalar triple product of three vectors find a useful way of introducing linear independence/dependence of three vectors which is the central concept in linear algebra. The author successfully used this approach as a primer to motivate the students before a full course on linear algebra.

To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of... more

To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ r , where r=number of conjugacy classes of symplectic reflections in ΓΓΓ. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ=Sn,theWeylgroupof?=??n.Weconstructa1−parameterdeformationoftheHarish−Chandrahomomorphismfrom?(?)?,thealgebraofinvariantpolynomialdifferentialoperatorson??n,tothealgebraofSn−invariantdifferentialoperatorswithrationalcoefficientsonthespaceCnofdiagonalmatrices.ThesecondorderLaplacianon?goes,underthedeformedhomomorphism,totheCalogero−MoserdifferentialoperatoronCn,withrationalpotential.OurcrucialideaistoreinterpretthedeformedHarish−Chandrahomomorphismasahomomorphism:?(?)?↠sphericalsubalgebrainHκ,whereHκisthesymplecticreflectionalgebraassociatedtothegroupΓ=Sn.Thisway,thedeformedHarish−Chandrahomomorphismbecomesnothingbutadescriptionofthesphericalsubalgebraintermsof‘quantum’Hamiltonianreduction.Inthe‘classical’limitκ→∞,ourconstructiongivesanisomorphismbetweenthesphericalsubalgebrainH∞andthecoordinateringoftheCalogero−Moserspace.WeprovethatallsimpleH∞−moduleshavedimensionn!,andareparametrisedbypointsoftheCalogero−Moserspace.ThefamilyofthesemodulesformsadistinguishedvectorbundleontheCalogero−Moserspace,whosefiberscarrytheregularrepresentationofSn.Moreover,weprovethatthealgebraΓ=S n , the Weyl group of ?=?? n . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ?? n , to the algebra of S n -invariant differential operators with rational coefficients on the space ℂ n of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ n , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=S n . This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction. In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S n . Moreover, we prove that the algebra Γ=Sn,theWeylgroupof?=??n.Weconstructa1−parameterdeformationoftheHarish−Chandrahomomorphismfrom?(?)?,thealgebraofinvariantpolynomialdifferentialoperatorson??n,tothealgebraofSn−invariantdifferentialoperatorswithrationalcoefficientsonthespaceCnofdiagonalmatrices.ThesecondorderLaplacianon?goes,underthedeformedhomomorphism,totheCalogero−MoserdifferentialoperatoronCn,withrationalpotential.OurcrucialideaistoreinterpretthedeformedHarish−Chandrahomomorphismasahomomorphism:?(?)?↠sphericalsubalgebrainHκ,whereHκisthesymplecticreflectionalgebraassociatedtothegroupΓ=Sn.Thisway,thedeformedHarish−Chandrahomomorphismbecomesnothingbutadescriptionofthesphericalsubalgebraintermsof‘quantum’Hamiltonianreduction.Inthe‘classical’limitκ→∞,ourconstructiongivesanisomorphismbetweenthesphericalsubalgebrainH∞andthecoordinateringoftheCalogero−Moserspace.WeprovethatallsimpleH∞−moduleshavedimensionn!,andareparametrisedbypointsoftheCalogero−Moserspace.ThefamilyofthesemodulesformsadistinguishedvectorbundleontheCalogero−Moserspace,whosefiberscarrytheregularrepresentationofSn.Moreover,weprovethatthealgebraH∞ is isomorphic to the endomorphism algebra of that vector bundle.

This paper surveys current technology and research in the area of digital color imaging. In order to establish the background and lay down terminology, fundamental concepts of color perception and measurement are first presented using... more

This paper surveys current technology and research in the area of digital color imaging. In order to establish the background and lay down terminology, fundamental concepts of color perception and measurement are first presented using vector-space notation and terminology. Present-day color recording and reproduction systems are reviewed along with the common mathematical models used for representing these devices. Algorithms for processing color images for display and communication are surveyed, and a forecast of research trends is attempted. An extensive bibliography is provided

8 vector space axioms and examples of how to derive them

Given an n-dimensional Lie algebra gg over a field k⊃Qk⊃Q, together with its vector space basis X10,…,Xn0, we give a formula, depending only on the structure constants, representing the infinitesimal generators, Xi=Xi0t in... more

Given an n-dimensional Lie algebra gg over a field k⊃Qk⊃Q, together with its vector space basis X10,…,Xn0, we give a formula, depending only on the structure constants, representing the infinitesimal generators, Xi=Xi0t in gk⊗k[[t]]g⊗kk[[t]], where t is a formal variable, as a formal power series in t with coefficients in the Weyl algebra AnAn. Actually, the theorem is proved for Lie algebras over arbitrary rings k⊃Qk⊃Q.We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by direct calculations with tensors. This involves a number of interesting combinatorial formulas in structure constants. The final step in calculation is a new formula involving Bernoulli numbers and arbitrary derivatives of coth(x/2)coth(x/2). The dimensions of certain spaces of tensors are also calculated. The second method of proof is geometric and reduces to a calculation of formal right-invariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras.

The second half of the paper is devoted to finding and classifying elementary functions with elementary Fourier transforms when Q is a fixed function with rational Q_*. We consider the simplest case when Q is a monomial, and classify... more

The second half of the paper is devoted to finding and classifying elementary functions with elementary Fourier transforms when Q is a fixed function with rational Q_*. We consider the simplest case when Q is a monomial, and classify combinations of multiplicative characters that can arise. The answer (for real and complex fields) is given in terms of exact covering systems. We also describe examples related to prehomogeneous vector spaces. Finally, we consider examples over p-adic fields, and in particular give a local proof of an integral formula of D.K. that could previously be proved only by a global method.

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided... more

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces , where X is a rack and q is a 2-cocycle on X with values in . Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.

The notion of n-best approximations can be used for error correction in coding theory. The notion of n-eigen values can be used in deterministic modal superposition principle for undamped structures, which can find its applications in... more

The notion of n-best approximations can be used for error correction in coding theory. The notion of n-eigen values can be used in deterministic modal superposition principle for undamped structures, which can find its applications in finite element analysis of mechanical structures with uncertain parameters. Further, it is suggested that the concept of n-matrices can be used in real world problems which adopts fuzzy models like Fuzzy Cognitive Maps, Fuzzy Relational Equations and Bidirectional Associative Memories. The applications of these algebraic structures are given in the third chapter. The fourth chapter suggests problems to further a reader's understanding of the subject.

A system of linear equations over a skew field has properties similar to properties of a system of linear equations over a field. Even noncommutativity of a product creates a new picture the properties of system of linear equations and of... more

A system of linear equations over a skew field has properties similar to properties of a system of linear equations over a field. Even noncommutativity of a product creates a new picture the properties of system of linear equations and of vector space over skew-field have a close relationship.