Bilinear Space Research Articles

High‐precision real‐time urine crystallization recognition based on dilated bilinear space pyramid ConvNext

AbstractUrinary sediment image detection, as one of the three major routine clinical tests in medical practice, is an important method for physical examination and diagnosis of urinary system diseases. Crystalluria detection is a subtask of urinary sediment image detection, focusing on detecting and identifying crystalline components in urine. To address the issues of low accuracy and inefficiency caused by small crystal granularity in crystalluria detection, we propose the Dilated Bilinear Space Pyramid ConvNext Network (DBSPC‐Net), which achieves high‐precision real‐time crystalluria detection. DBSPC‐Net ingeniously combines dilated convolution pooling with bilinear space pyramid, introducing Dilated Bilinear Space Pyramid Pooling (DBSPP) to enlarge the receptive field and capture information at multiple scales. Additionally, we utilize the Normalized Gaussian Wasserstein Distance Loss (NWDLoss) instead of Intersection over Union (IoU) to enhance the recognition of small targets. Finally, the ConvNext module is employed to fuse local and global features, enhancing urine crystal recognition accuracy and speed. The crystalluria dataset is sourced from 400 actual patients in a hospital. It comprises five main types of urine crystals, namely calcium oxalate dihydrate, calcium oxalate monohydrate, uric acid, ammonium magnesium phosphate, and cystine. Experimental results demonstrate that the proposed improved model achieves an average precision of 87.34% and a detection time of 7.9 ms per urine crystal image. DBSPC‐Net can accurately and rapidly identify crystalluria objects in scenarios involving microscope mica compensation, meeting the requirements of algorithmic detection accuracy and real‐time performance in crystalluria detection.

On unit signatures and narrow class groups of odd degree abelian number fields

For an abelian number field of odd degree, we study the structure of its 2 2 -Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed.

On the problem of zero-divisors in Artinian rings

Let R be a commutative Artinian ring and let be the compressed zero-divisor graph associated to R. The question of when the clique number was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff. They proved that if (where is the largest length of any of its chains of ideals), then When they gave an example of a local ring R where is possible by using the trivial extension of an Artinian local ring by its dualizing module. The question of what happens when was stated as an open question. We show that if then We first reduce the problem to the case of a local ring We then enumerate all possible Hilbert functions of R and show that the k-vector space admits a symmetric bilinear form in some cases of the Hilbert function. This allows us to relate the orthogonality in the bilinear space with the structure of zero-divisors in R. For instance, in the case when is principal and we show that R is Gorenstein if and only if the symmetric bilinear form on is non-degenerate. Moreover, in the case when our techniques also yield a simpler and shorter proof of the finiteness of avoiding, for instance, the Cohen structure theorem.

ON THE SPLITTING PRINCIPLE FOR COHOMOLOGICAL INVARIANTS OF REFLECTION GROUPS

Let k0 be a field of characteristic not two, (V, b) a finite-dimensional regular bilinear space over k0, and W a subgroup of the orthogonal group of (V, b) with the property that the subring of W-invariants of the symmetric algebra of V is a polynomial algebra over k0. We prove that Serre’s splitting principle holds for cohomological invariants of W with values in Rost’s cycle modules.

The development of inner product spaces and its generalization: a survey

An inner product space is a vector space with an additional structure called the inner product. This additional structure associates each vector pair in space with a scalar quantity known as the product. This paper will discuss a survey related to the development of the inner product space and its generalization. These generalization include semi-inner product space, sesquillinear space, indefinite inner product space, and bilinear space.

Continuous-Like Linear Operators on Bilinear Spaces

This paper introduces continuous-like linear operators on bilinear spaces as a generalization of continuous linear operators on Hilbert spaces. It is well known that the existence of the adjoint of a linear operator on a Hilbert space is equivalent to the operator being continuous. In this paper, this result is extended to the class of linear operators on bilinear spaces. It is shown that the existence of the adjoint of a linear operator on a bilinear space is guaranteed if and only if the operator is continuous-like.

Discriminating between CP and family transformations in the bilinear space of the N -Higgs-doublet model

The scalar potential of the N-Higgs-doublet model (NHDM) is best analyzed not in the space of N complex doublets $\phi_a$ but in the $N^2$-dimensional space of real-valued bilinears constructed of $\phi_a^\dagger \phi_b$. In particular, many insights have been gained into CP violation in the 2HDM and 3HDM by studying how generalized CP transformations (GCPs) act in this bilinear space. These insights relied on the fact that GCPs, which involved an odd number of mirror reflection, could be clearly distinguished from Higgs family transformations by the sign of the determinant of the transformation matrix. It was recently pointed out that this criterion fails starting from 4HDM, where the reflection/rotation dichotomy does not exist anymore. In this paper, we restore intuition by finding a different quantity which faithfully discriminates between GCPs and Higgs family transformations in the bilinear space for any number of Higgs doublets. We also establish the necessary and sufficient conditions for an orthogonal transformation in the bilinear space to represent a viable transformation back in the space of N doublets, which is helpful if one prefers to build an NHDM directly in the bilinear space.

On the motivic commutative ring spectrum $\mathbf {BO}$

We construct an algebraic commutative ring T -spectrum BO which is stably fibrant and (8, 4)-periodic and such that on SmOp/S the cohomology theory (X, U) 7→ BO(X+/U+) and Schlichting’s hermitian K-theory functor (X, U) 7→ KO [q] 2q−p(X, U) are canonically isomorphic. We use the motivic weak equivalence Z×HGr ∼ −→ KSp relating the infinite quaternionic Grassmannian to symplectic K-theory to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is SpecZ[ 1 2 ], this monoid structure and the induced ring structure on the cohomology theory BO are the unique structures compatible with the products KO [2m] 0 (X)× KO [2n] 0 (Y ) → KO [2m+2n] 0 (X × Y ). on Grothendieck-Witt groups induced by the tensor product of symmetric chain complexes. The cohomology theory is bigraded commutative with the switch map acting on BO(T∧T ) in the same way as multiplication by the Grothendieck-Witt class of the symmetric bilinear space 〈−1〉.

Riesz Representation Theorem on Bilinear Spaces of Truncated Laurent Series

In this study a generalization of the Riesz representation theorem on non-degenerate bilinear spaces, particularly on spaces of truncated Laurent series, was developed. It was shown that any linear functional on a non-degenerate bilinear space is representable by a unique element of the space if and only if its kernel is closed. Moreover an explicit equivalent condition can be identified for the closedness property of the kernel when the bilinear space is a space of truncated Laurent series.

201. Understanding Altered Eye Gaze Perception in Schizophrenia Using Dynamic Causal Modeling

Abstract Background: Previous studies have shown abnormal eye gaze perception in schizophrenia, associated with compromised social functioning. However, the neural mechanisms of altered gaze perception in schizophrenia remain unclear, limiting our ability to design effective interventions. This study aims to use dynamic causal modeling (DCM) of functional MRI time-series data to reveal aberrant dynamics between brain regions associated with eye gaze perception in schizophrenia. Methods: Twenty-five patients with schizophrenia or schizoaffective disorder (SZ) and 20 healthy controls (HC) completed a gaze perception task during BOLD fMRI. Participants viewed faces with different gaze angles and decided either the gaze direction (Gaze: “Looking at you?” yes/no) or gender of the face (Gender: male/female). Four nodes were identified as regions of interest for DCM as they showed differential activation between Gaze and Gender: secondary visual cortex (Vis), temporo-parietal junction (TPJ), anterior insula (aIns), and medial prefrontal cortex (mPFC). A bilinear model space was generated with these 4 nodes, as well as visual input of face, self-connections, bidirectional connections between nodes, and modulation of gaze on the connections. Results: Bayesian model selection revealed a winning model, in which Vis showed bottom-up influence on mPFC, which in turn had direct top-down influences on aIns and TPJ and indirect top-down influence on Vis through TPJ; attending to gaze modulated the Vis-mPFC and mPFC-TPJ connections. Group difference was observed in the Gaze modulation on the Vis-mPFC connection, such that attending to gaze inhibited the bottom-up connection from Vis to mPFC but this inhibition was weakened in SZ relative to HC. Conclusion: These findings suggest that when determining the self-referential nature of social information, it is critical to suppress stimulus-driven processing and rely more on conceptually-driven cognition formulated from past experience. A reduced ability to do so may underlie the abnormalities observed in eye gaze perception in schizophrenia.

The Witt ring of a curve with good reduction over a non-dyadic local field

In this work, we present a generalization to varieties and sheaves of the fundamental ideal of the Witt ring of a field by defining a sheaf of fundamental ideals I˜ and a sheaf of Witt rings W˜ in the obvious way. The Milnor conjecture then relates the associated graded of W˜ to Milnor K-theory and so allows the classical invariants of a bilinear space over a field to be extended to our setting using étale cohomology. As an application of these results, we calculate the Witt ring of a smooth curve with good reduction over a non-dyadic local field.

Adaptive Bilinear Element Finite Volume Methods for Second-Order Elliptic Problems on Nonmatching Grids

In this article, we propose and analyze two kinds of adaptive bilinear element finite volume methods for second-order elliptic problems on nonmatching grids. One of them chooses the piecewise bilinear finite element space as the trial function space, which is continuous on the matching part of a grid and is discontinuous on the nonmatching part of it. The other directly uses discontinuous piecewise bilinear element space for the trial function space. A priori estimations ensure the convergence and a posteriori estimations pave the way for adaptive methods. Several numerical experiments are presented to conform our theoretical results.

Perbandingan Koefisien NMF dan Proyeksi Bilinear Space Sebagai Fitur pada Pengenalan Ekspresi Wajah Manusia

NMF is one new developed method to make the part-based representation of non-negative data, such as human face image. NMF can reduce the dimension of high dimensional data such as multimedia data. In many researches,NMF can also used as a classification technique done by utilizing the extracted feature through NMF process. This article discusses about the classification technique of human face expression using NMF. This is done using NMF coeffisient and bilinear projection of face image. Some researches show the use of NMF coefficient in classification and some others use bilinear space projection. This research is conducted by simulating face espression recognition to the two available approaches and then comparing the accuracy and time efficiency aspect of the two methods. Through this research, it can be concluded that the use of NMF coefficient results in better accuracy compared to bilinear space projection, but bilinear space projection obtains better time efficiency.

NMF Coefficient and Bilinear Space Projection as Features For Human Facial Expression Recognition

NMF Coefficient and Bilinear Space Projection as Features For Human Facial Expression Recognition

GLOBAL WELL-POSEDNESS OF THE 1D DIRAC–KLEIN–GORDON SYSTEM IN SOBOLEV SPACES OF NEGATIVE INDEX

We prove that the Cauchy problem for the Dirac–Klein–Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of "almost conservation law" and "I-method" introduced by Colliander, Keel, Staffilani, Takaoka, and Tao. Our proof also relies on the null structure in the system, and bilinear space–time estimates of Klainerman–Machedon type.

Bilinear space–time estimates for linearised KP-type equations on the three-dimensional torus with applications

A bilinear estimate in terms of Bourgain spaces associated with a linearised Kadomtsev–Petviashvili-type equation on the three-dimensional torus is shown. As a consequence, time localized linear and bilinear space–time estimates for this equation are obtained. Applications to the local and global well-posedness of dispersion generalised KP-II equations are discussed. Especially it is proved that the periodic boundary value problem for the original KP-II equation is locally well-posed for data in the anisotropic Sobolev spaces H x s H y ε ( T 3 ) , if s ⩾ 1 2 and ε > 0 .

SMALL ZEROS OF QUADRATIC FORMS OVER $\overline{\mathbb{Q}}$

Let N ≥ 2 be an integer, F a quadratic form in N variables over [Formula: see text], and [Formula: see text] an L-dimensional subspace, 1 ≤ L ≤ N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z,F). This provides an analogue over [Formula: see text] of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over [Formula: see text]. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over [Formula: see text]. This extends previous results of the author over number fields. All bounds on height are explicit.

On Effective Witt Decomposition and the Cartan–Dieudonné Theorem

AbstractLetKbe a number field, and letFbe a symmetric bilinear form in 2Nvariables overK. LetZbe a subspace ofKN. A classical theorem of Witt states that the bilinear space (Z,F) can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights ofFandZ. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan–Dieudonné theorem. Namely, we show that every isometry σ of a regular bilinear space (Z,F) can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights ofF,Z, and σ.

Structure of isometry group of bilinear spaces

We describe the structure of the isometry group G of a finite-dimensional bilinear space over an algebraically closed field of characteristic not two. If the space has no indecomposable degenerate orthogonal summands of odd dimension, it admits a canonical orthogonal decomposition into primary components and G is isomorphic to the direct product of the isometry groups of the primary components. Each of the latter groups is shown to be isomorphic to the centralizer in some classical group of a nilpotent element in the Lie algebra of that group. In the general case, the description of G is more complicated. We show that G is a semidirect product of a normal unipotent subgroup K with another subgroup which, in its turn, is a direct product of a group of the type described in the previous paragraph and another group H which we can describe explicitly. The group H has a Levi decomposition whose Levi factor is a direct product of several general linear groups of various degrees. We obtain simple formulae for the dimensions of H and K.

An elementary proof of Gabriel's theorem on degenerate bilinear forms and its generalization

A theorem due to P. Gabriel shows that every bilinear space decomposes in a unique manner, up to equivalence, as the orthogonal direct sum of indecomposable degenerate spaces and a nondegenerate space. This has been generalized, by C. Riehm and M.A. Shrader-Frechette, to sesquilinear modules over a semisimple ring with an anti-automorphism. We give an elementary proof of these facts which avoids the use of Kronecker modules.