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1

Sleep EEG analysis utilizing inter-channel covariance matrices

Gopan K. Gopika, Prabhu Sathvik S., Sinha Neelam

Biocybernetics and Biomedical Engineering

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2020

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Vol. 40, no. 1

527--545

EN

Background: Sleep is vital for normal body functions as sleep disorders can adversely affect a person. Electroencephalographic (EEG) signals indicate brain functions and have characteristic signatures for various sleep stages. These enable the use of EEG as an effective tool for in-depth studies about sleep. Sleep stages are broadly divided as rapid eye movement (REM) and non-rapid eye movement (NREM). NREM is further divided into 3 stages. The objective of the work is to distinguish the given EEG epoch as wake, NREM1, NREM2, NREM3 and REM. DREAMS Subject Database containing 5 EEG channels is used here. This work focuses on utilizing EEG by exploiting variations in inter-dependencies of different brain regions during sleep. New method: Covariance matrices of the wavelet-decomposed channels are used to obtain the variations in inter-dependencies. The feature sets are: (1) simple matrix properties(MF) like trace, determinant and norm, (2) eigen-values (E1), (3) eigen-vector corresponding to the largest eigen-value (E2) and (4) tangent vectors obtained using Riemann geometry (RG-TS). The features are input to ensemble classifier with bagging. Subject-specific, All-subjects-combined and Leave-one-subject-out methods of analysis are carried out. Results: In all methods of analysis, RG-TS features give maximum accuracy (80.05%, 83.05% and 61.79%), closely followed by E1 (79.49%, 77.14% and 58.34%). Comparison with existing method: The proposed method obtains higher and/or comparable accuracy. This work also ensures no biasing of classifier due to unequal class distribution. Conclusion: The performances of RG-TS and E1 features reveal that the changes in interdependencies of pre-frontal and occipital lobe along with the central lobe can be used to distinguish the different sleep stages.

2

Kernel-based Fisher discriminant analysis on the Riemannian manifold for nuclear atypia scoring of breast cancer

Das Asha, Nair Madhu S., Peter David

Biocybernetics and Biomedical Engineering

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2019

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Vol. 39, no. 3

728--741

EN

Breast carcinoma is the most prevalent type of malignancy among women worldwide. Breast cancer grading often termed as Nuclear Atypia Scoring (NAS) forms a significant factor in determining individualized treatment plans and in the prognosis of the disease. For addressing the problem of breast cancer grading, we attempt to model the variations in features between histopathological images of different cancer grades and thereby explore the discriminative information concealed in these variations. In this regard, we aggregate multiple correlated features from the images using the geodesic geometric mean of the region covariances, to obtain the gmRC descriptors. As these gmRC descriptors are symmetric positive definite (SPD) matrices lying on the non-Euclidean Riemannian manifold, the discriminant analysis techniques developed for the Euclidean framework may not be appropriate. Hence, we propose a kernel-based Fisher discriminant analysis on the Riemannian manifold (KFDAR), that exploits the kernel trick for embedding the non-linear Riemannian manifold M into a higher dimensional linear Hilbert space H, which are then reduced to a low-dimensional and more discriminative subspace, where the samples become linearly separable. The kernel approach formulated for the Hilbert space embedding and for the kernel discriminant analysis is based on three Riemannian distance metrics: the log-Euclidean metric and the two symmetrized Bregman divergences – Stein and Jeffrey divergences. The experimental results show that this mapping to a highly discriminative space has succeeded in well-separating the histopathological images belonging to different cancer grades and hence it qualitatively and quantitatively outperforms the existing algorithms for cancer grading.

3

Selected geometrical and physical aspects of conformal transformations

Dudek M. K.

Problemy Nauk Stosowanych

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2015

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T. 3

91--106

EN

Introduction and aims: The conformal transformations play the crucial role in the analysis of the global structure of the space-time. The main goal of this article is to present the consequences of the conformal transformation of the metric like the creation of the energy and momentum for the gravitational field or the creation of the matter. Material and methods: In the paper have been proved the transformation formulas for some geometrical and physical objects. The analytical methods have been used in the paper. Results: In the paper there were proved the transformation formulas of the affine connection, the covariant derivative, the geodesics, the curvature tensor, the Ricci tensor, the curvature scalar and the Weyl tensor. The Einstein’s equations were proved not to be invariant under the conformal rescaling of the metric, as well as the Landau-Lifshitz pseudotensor. Conclusion: The article shows that the conformal rescaling of the metric creates a new matter and an additional energy and momentum of the gravity. There is also possibility of creation of the Friedman universes from the vacuum.

PL

Wstęp i cele: Transformacje konforemne odgrywają istotną rolę w analizie globalnej struktury czasoprzestrzeni. Głównym celem tego artykułu było przedstawienie konsekwencji konforemnego przeskalowania metryki takich jak kreacja energii i pędu dla pola grawitacyjnego oraz kreacja materii. Materiał i metody: W pracy zostały udowodnione wzory transformacyjne dla niektórych obiektów geometrycznych i fizycznych. Metody analityczne zostały wykorzystane w tym artykule. Wyniki: W pracy zostały udowodnione prawa transformacyjne dla koneksji afinicznej, pochodnej kowariantnej, linii geodezyjnych, tensora krzywizny, tensora Ricciego, skalara krzywizny i tensora Weyla. Udowodniono, że równania Einsteina jak również tensor Landaua-Lifszyca nie są konforemnie niezmiennicze w wyniku konforemnego przeskalowania metryki. Wniosek: Artykuł pokazuje, że konforemne przeskalowanie metryki kreuje nową materię i dodatkową energię oraz pęd grawitacji. Istnieje również możliwość tworzenia wszechświatów Friedmana z próżni.

4

On weakly symmetric generalized trans-sasakian manifold

Das L. S., Nivas R., Agnihotri R.

Commentationes Mathematicae

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2014

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Vol 54, No. 1

141--145

EN

In this paper, we have defined the weakly symmetric generalized Trans-Sasakian manifold G(WS)n and it has been shown that on such manifold if any two of the vector fields λ,γ,τ, defined by equation (0.3) are orthogonal to ξ, then the third will also be orthogonal to ξ. We have also proved that the scalar curvature r of weakly symmetric generalized Trans-Sasakian manifold G(WS)n, (n>2) satisfies the equation r=2n(α2−β2), where α and β are smooth function and γ≠τ.

5

On almost pseudo conformally symmetric manifolds

De U., Gazi A.

Demonstratio Mathematica

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2009

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Vol. 42, nr 4

869-886

EN

The object of the present paper is to study a type of non-conformally flat semi-Riemannian manifolds called almost pseudo conform ally symmetric manifold. The existence of an almost pseudo conformally symmetric manifold is also shown by a non-trivial example.

6

Theory of residual stresses with application to an arterial geometry

Klarbring A., Olsson T., Stalhand J.

Archives of Mechanics

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2007

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Vol. 59, nr 4-5

341-364

EN

This paper presents a theory of residual stresses, with applications to biomechanics, especially to arteries. For a hyperelastic material, we use an initial local deformation tensor K as a descriptor of residual strain. This tensor, in general, is not the gradient of a global deformation, and a stress-free reference configuration, denoted ..., therefore, becomes incompatible. Any compatible reference configuration ... will, in general, be residually stressed. However, when a certain curvature tensor vanishes, there actually exists a compatible and stress-free configuration, and we show that the traditional treatment of residual stresses in arteries, using the opening-angle method, relates to such a situation. Boundary value problems of nonlinear elasticity are preferably formulated on a fixed integration domain. For residually stressed bodies, three such formulations naturally appear: (i) a formulation relating to ... with a non-Euclidean metric structure; (ii) a formulation relating to ... with a Euclidean metric structure; and (iii) a formulation relating to the incompatible configuration ... . We state these formulations, show that (i) and (ii) coincide in the incompressible case, and that an extra term appears in a formulation on ... , due to the incompatibility.

7

Almost locally conformal Kaehler product manifolds

Sahin B.

Demonstratio Mathematica

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2006

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Vol. 39, nr 1

211-218

EN

It is known that the product of two locally conformal Kaehler manifolds is not a locally conformal Kaehler manifold, ([1], P:46 ). In this paper we introduce an almost locally conformal Kaehler product manifold and show that the product of two locally conformal Kaehler manifolds is an almost locally conformal Kaehler manifold. Moreover, we investigate properties of curvature tensors of an almost locally conformal Kaehler Product manifold.

8

A modulus and an extremal form of a foliation

Blachowska D.

Demonstratio Mathematica

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2004

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Vol. 37, nr 4

939--954

EN

We prove that the p-modulus of the foliation is conformal invariant. We study the problem of existing of the extremal form for a foliaton on Riemannian manifold. We also compute the value of p-modulus and the extremal form for k-dimensional foliation given by a submersion.

9

Some gradient estimates on covering manifolds

Dungey N.

Bulletin of the Polish Academy of Sciences. Mathematics

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2004

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Vol. 52, nr 4

437-443

EN

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.

10

Solving Parabolic Equations by Using the Method of Fast Convergent Iterations

Bondarenko V., Bidyuk P., Bernatska J.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 2

333-344

EN

The paper describes an approach to solving parabolic partial differential equations that generalizes the well-known parametrix method. The iteration technique proposed exhibits faster convergence than the classical parametrix approach. A solution is constructed on a manifold with the application of the Laplace-Beltrami operator. A theorem is formulated and proved to provide a basis for finding a unique solution. Simulation results illustrate the superiority of the proposed approach in comparison with the classical parametrix method.

11

Inverse/observability estimates for Schroedinger equations with variable coefficients

Triggiani R., Yao P.

Control and Cybernetics

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1999

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Vol. 28, no 3

627-664

EN

We consider a general Schroedinger equation defined on an open bounded domain [Omega is a subset of R^n] with variable coefficients in both the elliptic principal part and in the first-order terms as well. At first, no boundary conditions (B.C.) are imposed. Our main result (Theorem 3.5) is a reconstruction, or inverse, estimate for solutions w: under checkable conditions on the coefficients of the principal part, the H[sup l](Omega)-energy at time t = T, or at time t = 0, is dominated by the L[sub2](Sigma)-norms of the boundary traces [...] and w[sub t] modulo an interior lower-order term. Once homogeneous B.C. are imposed, our results yield - under a uniqueness theorem, needed to absorb the lower order term - continuous observability estimates for both the Dirichlet and Neumann case, with an arbitrarily short observability time ; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditions are imposed on the controlled part of the boundary in the Neuman case. In contrast to existing literature, the first step of our method employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in [Omega is a subset of R^n] to a problem on an appropriate Riemannian manifold (determined by the coefficients of the principal part), where the principal part is the Laplacian. In our second step, we employ explicit Carleman estimates at the differential level to take care of the va.riable first-order (energy level) terms. In our third step, we employ micro-local analysis yielding a sharp trace estimate to remove artificial geometrical conditions on the controlled part of the boundary in the Neumann case.