scholarly journals The determinant of the Gram matrix for a Specht module

1979 ◽  
Vol 59 (1) ◽  
pp. 222-235 ◽  
Author(s):  
G.D James ◽  
G.E Murphy
Keyword(s):  
1990 ◽  
Vol 33 (3) ◽  
pp. 337-366 ◽  
Author(s):  
Harry Dym ◽  
Nicholas Young

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.


Author(s):  
Dongwei Li ◽  
Jinsong Leng ◽  
Tingzhu Huang

In this paper, we give some new characterizations of g-frames, g-Bessel sequences and g-Riesz bases from their topological properties. By using the Gram matrix associated with the g-Bessel sequence, we present a sufficient and necessary condition under which the sequence is a g-Bessel sequence (or g-Riesz basis). Finally, we consider the excess of a g-frame and obtain some new results.


1946 ◽  
Vol 53 (1) ◽  
pp. 21 ◽  
Author(s):  
C. J. Everett ◽  
H. J. Ryser
Keyword(s):  

Author(s):  
Xuanlu Xiang ◽  
Zhipeng Wang ◽  
Zhicheng Zhao ◽  
Fei Su

In this paper, aiming at two key problems of instance-level image retrieval, i.e., the distinctiveness of image representation and the generalization ability of the model, we propose a novel deep architecture - Multiple Saliency and Channel Sensitivity Network(MSCNet). Specifically, to obtain distinctive global descriptors, an attention-based multiple saliency learning is first presented to highlight important details of the image, and then a simple but effective channel sensitivity module based on Gram matrix is designed to boost the channel discrimination and suppress redundant information. Additionally, in contrast to most existing feature aggregation methods, employing pre-trained deep networks, MSCNet can be trained in two modes: the first one is an unsupervised manner with an instance loss, and another is a supervised manner, which combines classification and ranking loss and only relies on very limited training data. Experimental results on several public benchmark datasets, i.e., Oxford buildings, Paris buildings and Holidays, indicate that the proposed MSCNet outperforms the state-of-the-art unsupervised and supervised methods.


2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Shan Liu ◽  
Yun Bo ◽  
Lingling Huang

With the further development of the social economy, people pay more attention to spiritual and cultural needs. As the main place of people’s daily life, the family is very important to the creation of its cultural atmosphere. In fact, China has fully entered the era of interior decoration, and people are paying more and more attention to decorative effects and the comfort and individual characteristics of decoration. Therefore, it is of practical significance to develop the application of decorative art in interior space design. However, the transfer effect of current interior decoration art design tends to be artistic, which leads to image distortion, and image content transfer errors are easy to occur in the process of transfer. The application of image style transfer in interior decoration art can effectively solve such problems. This paper analyzes the basic theory of image style transfer through image style transfer technology, Gram matrix, and Poisson image editing technology and designs images from several aspects such as image segmentation, content loss, enhanced style loss, and Poisson image editing constrained image spatial gradient. The application process of style transfer in interior decoration art realizes the application of image style transfer in interior decoration art. The experimental results show that the application of image style transmission in interior decoration art design can effectively avoid the contents of the interior decoration errors and distortions and has a good style transfer effect.


10.37236/3709 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Jacob G. Martin ◽  
E. Rodney Canfield

A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The random Gram matrix of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.


2016 ◽  
Vol Volume 23 - 2016 - Special... ◽  
Author(s):  
Abdelwaheb Ifa ◽  
Michel Rouleux

International audience We revisit in this Note the well known Bohr-Sommerfeld quantization rule (BS) for a 1-D Pseudo-differential self-adjoint Hamiltonian within the algebraic and microlocal framework of Helffer and Sjöstrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the " flux norm " is not invertible. Dans le cadre algébrique et microlocal élaboré par Helffer et Sjöstrand, on propose une ré-écriture de la règle de quantification de Bohr-Sommerfeld pour un opérateur auto-adjoint h-Pseudo-différentiel 1-D; elle s'exprime par la non-inversibilité de la matrice de Gram d'un couple de solutions WKB dans une base convenable, pour le produit scalaire associé à la " norme de flux " .


2018 ◽  
Vol 10 (01) ◽  
pp. 1850002 ◽  
Author(s):  
Kenji Kume ◽  
Naoko Nose-Togawa

Singular spectrum analysis (SSA) is a nonparametric spectral decomposition of a time series into arbitrary number of interpretable components. It involves a single parameter, window length [Formula: see text], which can be adjusted for the specific purpose of the analysis. After the decomposition of a time series, similar series are grouped to obtain the interpretable components by consulting with the [Formula: see text]-correlation matrix. To accomplish better resolution of the frequency spectrum, a larger window length [Formula: see text] is preferable and, in this case, the proper grouping is crucial for making the SSA decomposition. When the [Formula: see text]-correlation matrix does not have block-diagonal form, however, it is hard to adequately carry out the grouping. To avoid this, we propose a novel algorithm for the adaptive orthogonal decomposition of the time series based on the SSA scheme. The SSA decomposition sequences of the time series are recombined and the linear coefficients are determined so as to maximizing its squared norm. This results in an eigenvalue problem of the Gram matrix and we can obtain the orthonormal basis vectors for the [Formula: see text]-dimensional subspace. By the orthogonal projection of the original time series on these basis vectors, we can obtain adaptive orthogonal decomposition of the time series without the redundancy of the original SSA decomposition.


2019 ◽  
Vol 30 (10) ◽  
pp. 1950048 ◽  
Author(s):  
Michael Penn ◽  
Christopher Sadowski ◽  
Gautam Webb

This is the third in a series of papers studying the vertex-algebraic structure of principal subspaces of twisted modules for lattice vertex operator algebras. We focus primarily on lattices [Formula: see text] whose Gram matrix contains only non-negative entries. We develop further ideas originally presented by Calinescu, Lepowsky, and Milas to find presentations (generators and relations) of the principal subspace of a certain natural twisted module for the vertex operator algebra [Formula: see text]. We then use these presentations to construct exact sequences involving this principal subspace, which give a set of recursions satisfied by the multigraded dimension of the principal subspace and allow us to find the multigraded dimension of the principal subspace.


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