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Decomposability of Tensors

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ISBN: 9783038975908 / 9783038975915 Year: Pages: 160 DOI: 10.3390/books978-3-03897-591-5 Language: English
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Mathematics --- Physics (General)
Added to DOAB on : 2019-02-15 09:41:46
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Tensor decomposition is a relevant topic, both for theoretical and applied mathematics, due to its interdisciplinary nature, which ranges from multilinear algebra and algebraic geometry to numerical analysis, algebraic statistics, quantum physics, signal processing, artificial intelligence, etc. The starting point behind the study of a decomposition relies on the idea that knowledge of elementary components of a tensor is fundamental to implement procedures that are able to understand and efficiently handle the information that a tensor encodes. Recent advances were obtained with a systematic application of geometric methods: secant varieties, symmetries of special decompositions, and an analysis of the geometry of finite sets. Thanks to new applications of theoretic results, criteria for understanding when a given decomposition is minimal or unique have been introduced or significantly improved. New types of decompositions, whose elementary blocks can be chosen in a range of different possible models (e.g., Chow decompositions or mixed decompositions), are now systematically studied and produce deeper insights into this topic. The aim of this Special Issue is to collect papers that illustrate some directions in which recent researches move, as well as to provide a wide overview of several new approaches to the problem of tensor decomposition.

Alzheimer's Disease and the Fornix

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Book Series: Frontiers Research Topics ISSN: 16648714 ISBN: 9782889199594 Year: Pages: 110 DOI: 10.3389/978-2-88919-959-4 Language: English
Publisher: Frontiers Media SA
Subject: Science (General) --- Neurology
Added to DOAB on : 2016-01-19 14:05:46
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This e-book focuses primarily on the role of the fornix as a functional, prognostic, and diagnostic marker of Alzheimer’s disease (AD), and the application of such a marker in clinical practice. Researchers have long been focused on the cortical pathology of AD, since the most important pathologic features are the senile plaques found in the cortex, and the neurofibrillary tangles and neuronal loss that start from the entorhinal cortex and the hippocampus. In addition to gray matter structures, histopathological studies indicate that the white matter is also altered in AD. The fornix is a white matter bundle that constitutes a core element of the limbic circuits, and is one of the most important anatomical structures related to memory. The fornices originate from the bilateral hippocampi, merge at the midline of the brain, again divide into the left and right side, and then into the precommissural and the postcommissural fibers, and terminate at the septal nuclei, nucleus accumbens (precommissural fornix), and hypothalamus (postcommissural fornix). These functional and anatomical features of the fornix have naturally captured researchers’ attention as possible diagnostic and prognostic markers of AD. Growing evidence indicates that the alterations seen in the fornix are potentially a good marker with which to predict future conversion from mild cognitive impairment to AD, and even from a cognitively normal state to AD. The degree of alteration is correlated with the degree of memory impairment, indicating the potential for the use of the fornix as a functional marker. Moreover, there have been attempts to stimulate the fornix to recover the cognitive function lost with AD. Our goal is to provide information about the status of current research and to facilitate further scientific and clinical advancement in this topic.

Brain Connectivity in Autism

Authors: --- --- --- --- et al.
Book Series: Frontiers Research Topics ISSN: 16648714 ISBN: 9782889192823 Year: Pages: 264 DOI: 10.3389/978-2-88919-282-3 Language: English
Publisher: Frontiers Media SA
Subject: Science (General) --- Neurology
Added to DOAB on : 2015-12-10 11:59:06
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The brain's ability to process information crucially relies on connectivity. Understanding how the brain processes complex information and how such abilities are disrupted in individuals with neuropsychological disorders will require an improved understanding of brain connectivity. Autism is an intriguingly complex neurodevelopmental disorder with multidimensional symptoms and cognitive characteristics. A biological origin for autism spectrum disorders (ASD) had been proposed even in the earliest published accounts (Kanner, 1943; Asperger, 1944). Despite decades of research, a focal neurobiological marker for autism has been elusive. Nevertheless, disruptions in interregional and functional and anatomical connectivity have been a hallmark of neural functioning in ASD. Theoretical accounts of connectivity perceive ASD as a cognitive and neurobiological disorder associated with altered functioning of integrative circuitry. Neuroimaging studies have reported disruptions in functional connectivity (synchronization of activated brain areas) during cognitive tasks and during task-free resting states. While these insights are valuable, they do not address the time-lagged causality and directionality of such correlations. Despite the general promise of the connectivity account of ASD, inconsistencies and methodological differences among studies call for more thorough investigations. A comprehensive neurological account of ASD should incorporate functional, effective, and anatomical connectivity measures and test the diagnostic utility of such measures. In addition, questions pertaining to how cognitive and behavioral intervention can target connection abnormalities in ASD should be addressed. This research topic of the Frontiers in Human Neuroscience addresses “Brain Connectivity in Autism” primarily from cognitive neuroscience and neuroimaging perspectives.

Bridging the gap before and after birth: Methods and technologies to explore the functional neural development in humans

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Book Series: Frontiers Research Topics ISSN: 16648714 ISBN: 9782889196876 Year: Pages: 114 DOI: 10.3389/978-2-88919-687-6 Language: English
Publisher: Frontiers Media SA
Subject: Neurology --- Science (General)
Added to DOAB on : 2016-04-07 11:22:02
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Infant brain damage is a serious condition that affects millions of babies each year. The period from late gestation to the first year of life is the most critical one for the development of central and autonomous nervous systems. Medical conditions such as preterm birth may compromise brain function and the end result usually is that the baby may experience long-term neurological problems related to a wide range of psychological, physical and functional complications, with consequent life-long burdens for the individuals and their families, and a high socio-economic impact for the health care system and the whole of society. During the last years, several techniques have been employed to monitor the brain functional development in utero and after birth. As well, various analytical methods have been used to understand the functional maturation of the brain and the autonomous nervous system. However, in spite of the rapid improvement of diagnostic methods and procedures, there is still a widely recognized, severe shortage of clinically viable means for the high quality monitoring of the brain function in early life with a direct relevance to acute neurological illness and future neurocognitive outcomes. The studies collected in this e-book document the most recent advancements in monitoring systems, analytical methods and clinical diagnostic procedures that contribute to increase our knowledge of the functional development of the human brain and autonomous nervous system during pregnancy and after birth, with the ultimate goal of reducing fetal impairment and improving healthcare in the neonatal and infant period.

Information Geometry

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ISBN: 9783038976325 Year: Pages: 356 DOI: 10.3390/books978-3-03897-633-2 Language: eng
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Mathematics --- Science (General)
Added to DOAB on : 2019-04-05 10:34:31
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This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience.

Geometry of Submanifolds and Homogeneous Spaces

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ISBN: 9783039280001 / 9783039280018 Year: Pages: 128 DOI: 10.3390/books978-3-03928-001-8 Language: eng
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Science (General) --- Mathematics
Added to DOAB on : 2020-01-30 16:39:46
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The present Special Issue of Symmetry is devoted to two important areas of global Riemannian geometry, namely submanifold theory and the geometry of Lie groups and homogeneous spaces. Submanifold theory originated from the classical geometry of curves and surfaces. Homogeneous spaces are manifolds that admit a transitive Lie group action, historically related to F. Klein's Erlangen Program and S. Lie's idea to use continuous symmetries in studying differential equations. In this Special Issue, we provide a collection of papers that not only reflect some of the latest advancements in both areas, but also highlight relations between them and the use of common techniques. Applications to other areas of mathematics are also considered.

Nonparametric Econometric Methods and Application

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ISBN: 9783038979647 / 9783038979654 Year: Pages: 224 DOI: 10.3390/books978-3-03897-965-4 Language: eng
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Science (General) --- Mathematics
Added to DOAB on : 2019-06-26 08:44:06
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The present Special Issue collects a number of new contributions both at the theoretical level and in terms of applications in the areas of nonparametric and semiparametric econometric methods. In particular, this collection of papers that cover areas such as developments in local smoothing techniques, splines, series estimators, and wavelets will add to the existing rich literature on these subjects and enhance our ability to use data to test economic hypotheses in a variety of fields, such as financial economics, microeconomics, macroeconomics, labor economics, and economic growth, to name a few.

Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics

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ISBN: 9783039214099 / 9783039214105 Year: Pages: 254 DOI: 10.3390/books978-3-03921-410-5 Language: eng
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Technology (General) --- General and Civil Engineering
Added to DOAB on : 2019-12-09 11:49:15
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The use of machine learning in mechanics is booming. Algorithms inspired by developments in the field of artificial intelligence today cover increasingly varied fields of application. This book illustrates recent results on coupling machine learning with computational mechanics, particularly for the construction of surrogate models or reduced order models. The articles contained in this compilation were presented at the EUROMECH Colloquium 597, « Reduced Order Modeling in Mechanics of Materials », held in Bad Herrenalb, Germany, from August 28th to August 31th 2018. In this book, Artificial Neural Networks are coupled to physics-based models. The tensor format of simulation data is exploited in surrogate models or for data pruning. Various reduced order models are proposed via machine learning strategies applied to simulation data. Since reduced order models have specific approximation errors, error estimators are also proposed in this book. The proposed numerical examples are very close to engineering problems. The reader would find this book to be a useful reference in identifying progress in machine learning and reduced order modeling for computational mechanics.

Keywords

parameter-dependent model --- surrogate modeling --- tensor-train decomposition --- gappy POD --- heterogeneous data --- elasto-viscoplasticity --- archive --- model reduction --- 3D reconstruction --- inverse problem plasticity --- data science --- model order reduction --- POD --- DEIM --- gappy POD --- GNAT --- ECSW --- empirical cubature --- hyper-reduction --- reduced integration domain --- computational homogenisation --- model order reduction (MOR) --- low-rank approximation --- proper generalised decomposition (PGD) --- PGD compression --- randomised SVD --- nonlinear material behaviour --- machine learning --- artificial neural networks --- computational homogenization --- nonlinear reduced order model --- elastoviscoplastic behavior --- nonlinear structural mechanics --- proper orthogonal decomposition --- empirical cubature method --- error indicator --- symplectic model order reduction --- proper symplectic decomposition (PSD) --- structure preservation of symplecticity --- Hamiltonian system --- reduced order modeling (ROM) --- proper orthogonal decomposition (POD) --- enhanced POD --- a priori enrichment --- modal analysis --- stabilization --- dynamic extrapolation --- computational homogenization --- large strain --- finite deformation --- geometric nonlinearity --- reduced basis --- reduced-order model --- sampling --- Hencky strain --- microstructure property linkage --- unsupervised machine learning --- supervised machine learning --- neural network --- snapshot proper orthogonal decomposition

Cosmological Inflation, Dark Matter and Dark Energy

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ISBN: 9783039217649 / 9783039217656 Year: Pages: 236 DOI: 10.3390/books978-3-03921-765-6 Language: eng
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Science (General) --- Mathematics
Added to DOAB on : 2019-12-09 11:49:16
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Various cosmological observations support not only cosmological inflation in the early universe, which is also known as exponential cosmic expansion, but also that the expansion of the late-time universe is accelerating. To explain this phenomenon, the existence of dark energy is proposed. In addition, according to the rotation curve of galaxies, the existence of dark matter, which does not shine, is also suggested. If primordial gravitational waves are detected in the future, the mechanism for realizing inflation can be revealed. Moreover, there exist two main candidates for dark matter. The first is a new particle, the existence of which is predicted in particle physics. The second is an astrophysical object which is not found by electromagnetic waves. Furthermore, there are two representative approaches to account for the accelerated expansion of the current universe. One is to assume the unknown dark energy in general relativity. The other is to extend the gravity theory to large scales. Investigation of the origins of inflation, dark matter, and dark energy is one of the most fundamental problems in modern physics and cosmology. The purpose of this book is to explore the physics and cosmology of inflation, dark matter, and dark energy.

Differential Geometry

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ISBN: 9783039218004 / 9783039218011 Year: Pages: 166 DOI: 10.3390/books978-3-03921-801-1 Language: eng
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Science (General) --- Mathematics
Added to DOAB on : 2019-12-09 16:10:12
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The present book contains 14 papers published in the Special Issue “Differential Geometry” of the journal Mathematics. They represent a selection of the 30 submissions. This book covers a variety of both classical and modern topics in differential geometry. We mention properties of both rectifying and affine curves, the geometry of hypersurfaces, angles in Minkowski planes, Euclidean submanifolds, differential operators and harmonic forms on Riemannian manifolds, complex manifolds, contact manifolds (in particular, Sasakian and trans-Sasakian manifolds), curvature invariants, and statistical manifolds and their submanifolds (in particular, Hessian manifolds). We wish to mention that among the authors, there are both well-known geometers and young researchers. The authors are from countries with a tradition in differential geometry: Belgium, China, Greece, Japan, Korea, Poland, Romania, Spain, Turkey, and United States of America. Many of these papers were already cited by other researchers in their articles. This book is useful for specialists in differential geometry, operator theory, physics, and information geometry as well as graduate students in mathematics.

Keywords

Euclidean submanifold --- position vector field --- concurrent vector field --- concircular vector field --- rectifying submanifold --- T-submanifolds --- constant ratio submanifolds --- Ricci soliton --- Kähler–Einstein metrics --- compact complex surfaces --- pinching of the curvatures --- statistical manifolds --- Hessian manifolds --- Hessian sectional curvature --- scalar curvature --- Ricci curvature --- Minkowski plane --- Minkowskian length --- Minkowskian angle --- Minkowskian pseudo-angle --- L2-harmonic forms --- Hodge–Laplacian --- manifold with singularity --- L2-Stokes theorem --- capacity --- k-th generalized Tanaka–Webster connection --- non-flat complex space form --- real hypersurface --- lie derivative --- shape operator --- conical surface --- developable surface --- generalized 1-type Gauss map --- cylindrical hypersurface --- inextensible flow --- lightlike surface --- ruled surface --- Darboux frame --- C-Bochner tensor --- generalized normalized ?-Casorati curvature --- Sasakian manifold --- slant --- invariant --- anti-invariant --- trans-Sasakian 3-manifold --- Reeb flow symmetry --- Ricci operator --- Sasakian statistical manifold --- conjugate connection --- Casorati curvature --- framed rectifying curves --- singular points --- framed helices --- centrodes --- circular rectifying curves --- statistical structure --- affine hypersurface --- affine sphere --- conjugate symmetric statistical structure --- sectional ?-curvature --- complete connection --- symplectic curves --- circular helices --- symplectic curvatures --- Frenet frame

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