Differential Manifolds

In this non-academic study, we will cover the manifold concept with some non-trivial examples and introduce calculus on manifolds, namely, exterior derivative, integration with the Stokes' theorem. Then we will cover the notion of curvature with the Gauss - Bonnet theorem.

Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic R 4 , and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called smooth quantum gravity by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard R 4 to the exotic R 4 is a quantization of geometry.

The aim of this study which consist of four sections is to examine some
characterizations for surfaces and submanifolds with simple geodesics in
n -dimensional and four dimensional Euclidean spaces. Theorems given in both
of the papers such as “Surfaces of Euclidean 4-space whose geodesics are Wcurves”
by Kim and Lee and “Submanifolds in Euclidean space with simple
geodesics” by Ferus and Schirrmacher have been studied in detail in this work.
In the introduction, the historical development and applications of subject have
been explained. Basic concepts and theorems required in this work have been
given in the following section.
The characterization of submanifolds with simple geodesics in
n -dimensional and 4-dimensional Euclidean spaces has been emphasized in the
third chapter. Also, having obtained parametric equation of W-curve with even
rank, rank of some W-curves has been examined at this chapter.
In the last chapter, surfaces with simple geodesisc in 4-dimensional
Euclidean space have been examined. In this section characterization of
geodesics of helical surfaces and Blaschke manifold have been expressed and
the correspondings of complete, compact and connected surfaces in
4-dimensional Euclidean space are presented.

M/F-theory manifolds admit (O,gamma)=(G, g) X (M, g_E) product orbifolds (G a Calabi-Yau 4-orbifold with special holonomy (8-D) producted with a 2T Einstein
manifold M with signature (3,2) for F-Theory and G a 7-D G-2 an eliiptically fibered orbifold with M a 1T Einstein manifold with signature (3,1) for M-Theory). They serve as models for compacted 13-D 2T F-Theory or 11-D 1T M-Theory.  Einstein metric tensors g_E are proportional to their Einstein Ricci tensors R_E. Invoking stationarity on these manifolds with respect to the metric tensor curvature will imply that the variational acceleration is time-invariant and flat. The corresponding Euler-Lagrange operator expressed as a chained partial derivative with respect to both time dimensions and spatial coordinates implies that the metric tensor variational is 0. Discretizing the metric tensor variations modulo integer multiplies of q_e and approximating them utilizing partial sums of variations, can serve as combinatorial representations for their respective Dp-brane fields. These modulo sum combinations are then posited to create physical fields in their variational quantization.

In this paper, we will describe a topological model for elementary particles based on 1 3-manifolds. Here, we will use Thurstons geometrization theorem to get a simple picture: fermion as 2 hyperbolic knot complements (a complement C(K) = S 3 \ (K × D 2) of a knot K carrying a hyperbolic 3 geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection 4 to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used 5 in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. 6 Every 3-manifold can be described by a 3-fold branched cover of S 3 branched along a knot. In case 7 of knot complements, one will obtain a 3-fold branched cover of the 3-disk D 3 branched along a 8 3-braid or 3-braids describe fermions. The whole approach will uncover new symmetries as induced 9 by quantum and discrete groups. Using the Drinfeld-Turaev quantization, we will also construct a 10 quantization so that quantum states correspond to knots. Particle properties like the electric charge 11 must be expressed by topology and we will obtain the right spectrum of possible values. Finally, we 12 will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic 13 algebras with relations to 3-braids (Bilson-Thompson model).

In this paper we introduce B k α,β-manifolds as generalizations of the notion of smooth man-ifolds with G-structure or with k-bounded geometry. These are C k-manifolds whose transition functions ϕ ji = ϕ j • ϕ −1 i are such that ∂ µ ϕ ji ∈ B α(r) ∩ C k−β(r) for every |µ| = r, where B = (B r) r∈Γ is some sequence of presheaves of Fréchet spaces endowed with further structures, Γ ⊂ Z ≥0 is some parameter set and α, β are functions. We present embedding theorems for the presheaf category of those structural presheaves B. The existence problem of B k α,β-structures on C k-manifolds is studied and it is proved that under certain conditions on B, α and β, the forgetful functor from C k-manifolds to B k α,β-manifolds has adjoints.

A novel methodology for accelerating the solution of PDE-constrained optimization is introduced. It is based on an offline construction of database of local ROMs and an online interpolation within the database. The online flexibility of the ROM database approach makes it amenable to speedingup optimization-intensive applications such as robust optimization, multi-objectives optimization, and multi-start strategies for locating global optima. The accuracy of the ROM database model can be tuned in the offline phase where the database of local ROMs is constructed through a greedy procedure. In this work, a novel greedy algorithm based on saturation assumption is introduced to speed-up the ROM database construction procedure. The ROM database approach is applied to a
realistic wing design problems and leads to a large online speed-up.

In this paper we consider the existence problem of affine connections
on $C^{k}$-manifolds $M$ whose coefficients are as regular as one
needs. We show that if $M$ admits a suitable subatlas, meaning a
$\mathcal{B}_{\alpha,\beta}^{k}$-structure for a certain presheaf
of Fr\'echet spaces $B$ and for certain functions $\alpha$ and
$\beta$, then the existence of such regular connections can be established.
It is also proved that if the $\mathcal{B}_{\alpha,\beta}^{k}$-structure
is actually nice (in the sense of \citep{Bk_manifolds}), then a multiplicity
result can also be obtained by means of Thom's transversality arguments.

We construct an explicit example of dimensional reduction of the free massless Dirac operator with an internal SU(3) symmetry, defined on a 12-dimensional manifold that is the total space of a principal SU(3)-bundle over a four-dimensional (nonflat) pseudo-Riemannian manifold. Upon dimensional reduction the free twelve-dimensional Dirac equation is transformed into a rather nontrivial four-dimensional one: a pair of massive Lorentz spinor SU(3)-octets interacting with an SU(3)-gauge field with a source term depending on the curvature tensor of the gauge field. The SU(3) group is complicated enough to illustrate features of the general case. It should not be confused with the color SU(3) of quantum chromodynamics where the fundamental spinors, the quark fields, are SU(3) triplets rather than octets.

The emergence of low-cost sensor architectures for diverse modalities has made it possible to deploy sensor arrays that capture a single event from a large number of vantage points and using multiple modalities. In many scenarios, these sensors acquire very high-dimensional data such as audio signals, images, and video. To cope with such high-dimensional data, we typically rely on low-dimensional models. Manifold models provide a particularly powerful model that captures the structure of high-dimensional data when it is governed by a low-dimensional set of parameters. However, these models do not typically take into account dependencies among multiple sensors. We thus propose a new joint manifold framework for data ensembles that exploits such dependencies. We show that simple algorithms can exploit the joint manifold structure to improve their performance on standard signal processing applications. Additionally, recent results concerning dimensionality reduction for manifolds enable us to formulate a network-scalable data compression scheme that uses random projections of the sensed data. This scheme efficiently fuses the data from all sensors through the addition of such projections, regardless of the data modalities and dimensions.

In this paper, we focus on some aspects of smooth manifolds, which appear of fundamental importance for the developments of differential geometry and its applications to Theoretical Physics, Special and General Relativity, Economics and Finance.

This paper was originally written around 2011, and it is now being adapted to the system of equations of difference between two arc lengths (a liberated system with without constraints) instead of difference between the circumferences of two circles (a hinged system with paradox) .

In this paper we determine the cosmological constant as a topo-logical invariant by applying certain techniques from low dimensional differential topology. We work with a small exotic R 4 which is embedded into the standard R 4. Any exotic R 4 is a Riemannian smooth manifold with necessary non-vanishing curvature tensor. To determine the invariant part of such curvature we deal with a canonical construction of R 4 where it appears as a part of the complex surface K3#CP (2). Such R 4 's admit hyperbolic geometry. This fact simplifies significantly the calculations and enforces the rigidity of the expressions. In particular, we explain the smallness of the cosmolog-ical constant with a value consisting of a combination of (natural) topological invariant. Finally, the cosmological constant appears to be a topologically supported quantity.

It is well-known that the Harish-Chandra transform, f → Hf, is a topological isomorphism of the spherical (Schwartz) convolution algebra C p (G//K) (where K is a maximal compact subgroup of any arbitrarily chosen group G in the Harish-Chandra class and 0 < p ≤ 2) onto the (Schwartz) multiplication algebra ¯ Z(F ϵ) (of w−invariant members of Z(F ϵ), with ϵ = (2/p) − 1). The same cannot however be said of the full Schwartz convolution algebra C p (G), except for few specific examples of groups (notably G = SL(2, R)) and for some notable values of p (with restrictions on G and/or on C p (G)). Nevertheless the full Harish-Chandra Plancherel formula on G is known for all of C 2 (G) =: C(G). In order to then understand the structure of Harish-Chandra transform more clearly and to compute the image of C p (G) under it (without any restriction) we derive an absolutely convergent series expansion (in terms of known functions) for the Harish-Chandra transform by an application of the full Plancherel formula on G. This leads to a computation of the image of C(G) under the Harish-Chandra transform which may be seen as a concrete realization of Arthur's result and be easily extended to all of C p (G) in much the same way as it is known in the work of Trombi and Varadarajan. 2010 Mathematics Subject Classification: 43A85, 22E30, 22E46

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement C ( K ) = S 3 \ ( K × D 2 ) of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of S 3 branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk D 3 branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so ...

The emergence of low-cost sensing architectures for diverse modalities has made it possible to deploy sensor networks that capture a single event from a large number of vantage points and using multiple modalities. In many scenarios, these networks acquire large amounts of very high-dimensional data. For example, even a relatively small network of cameras can generate massive amounts of high-dimensional image and video data. One way to cope with this data deluge is to exploit low-dimensional data models. Manifold models provide a particularly powerful theoretical and algorithmic framework for capturing the structure of data governed by a small number of parameters, as is often the case in a sensor network. However, these models do not typically take into account dependencies among multiple sensors. We thus propose a new joint manifold framework for data ensembles that exploits such dependencies. We show that joint manifold structure can lead to improved performance for a variety of signal processing algorithms for applications including classification and manifold learning. Additionally, recent results concerning random projections of manifolds enable us to formulate a scalable and universal dimensionality reduction scheme that efficiently fuses the data from all sensors.

We propose a framework for exploiting dimension-reducing random projections in detection and classification problems. Our approach is based on the generalized likelihood ratio test; in the case of image classification, it exploits the fact that a set of images of a fixed scene under varying articulation parameters forms a low-dimensional, nonlinear manifold. Exploiting recent results showing that random projections stably embed a smooth manifold in a lower-dimensional space, we develop the multiscale smashed filter as a compressive analog of the familiar matched filter classifier. In a practical target classification problem using a single-pixel camera that directly acquires compressive image projections, we achieve high classification rates using many fewer measurements than the dimensionality of the images.

Presentación en formato PDF y PPT editable. En ella intento resumir las alucinantes confusiones en que se extravía Gilles Deleuze en las obras escritas con Félix Guattari, en particular en Mil Mesetas. Una de las más desopilantes se presenta cuando Deleuze se ve empujado por su propia retórica a proporcionar dos ejemplos de naciones inclinadas a la organización rizomática y opuestas por ello al despotismo propio del régimen arbóreo. Los mejores ejemplos que se le ocurren son China y la India: la primera con su Palacio Prohibido y más tarde con el régimen maoísta que engendra la revolución cultural, la segunda con un sistema de castas que ni siquiera la retórica rizomática puede justificar.

Otro caso bochornoso envuelve al concepto deleuziano de multiplicidad, originado en un error de traducción de la palabra "Mannigfaltigkeit" (o sea "manifold"="variedad"),la cual carece de los atributos que Deleuze y los deleuzianos imaginan. Un manifold es un concepto que en el contexto de la geometría diferencial tiene que ver con la curvatura (lo que llevaría a la teoría de la relatividad y a la geometría no euclideana): en la filosofía deleuziana, en cambio, se lo toma como una idea referida a números colectivos.

Para ensalzar los rizomas y denigrar a sus contrapartidas (los árboles binarios) Deleuze se ve en el brete de cuestionar la gramática generativa de Chomsky, a la que dedica todo un capítulo en el que resulta evidente que no ha leído (ni citado) un solo libro escrito por el lingüista. No es la primera ni la única vez que Deleuze brinda cátedra sobre temas de los que no tiene la menor idea. Mi pieza favorita en este sentido es su fallida conferencia sobre la armonía musical dictada en Vincennes en 1987 en la que olvida tratar el único concepto musical que él cree que conoce (y que es, por supuesto, el ritornello). Asegurar que el barroco (y con ello el pliegue de Leibniz) se origina en torno a la idea armónica es ignorar que el período, musicalmente hablando, sólo es posible a partir de la monodia, la cual sentó las bases tanto para la música concertante como para la ópera..

En la presentación se hacen referencia a muchas confusiones de esa índole, que se tratan con más detalle en el libro que comparto en el siguiente vínculo:  https://www.academia.edu/20169560/Relectura_cr%C3%ADtica_de_la_antropolog%C3%ADa_perspectivista_y_de_los_giros_ontol%C3%B3gicos_en_la_ciencia_pos_social_Viveiros_de_Castro_Philippe_Descola_Bruno_Latour_2a_edici%C3%B3n_2018

En cuanto a su curso sobre la armonía, finalmente intervenido y rescatado de la catástrofe por cursantes que verdadera sabían de lo estaban hablando, puede verse el video subido a https://www.youtube.com/watch?v=_JBMX6uECxc.

Sobre otras imposturas intelectuales, en fin, puede verse el contenido de mi conferencia.

In this paper a general methodology is introduced for the determination of potential prototype curves used for the drawing of prehistoric wall-paintings. The approach includes a) preprocessing of the wall-paintings contours to properly partition them, according to their curvature, b) choice of prototype curves families, c) analysis and optimization in 4-manifold for a first estimation of the form of these prototypes, d) clustering of the contour parts and the prototypes, to determine a minimal number of potential guides, e) further optimization in 4-manifold, applied to each cluster separately, in order to determine the exact functional form of the potential guides, together with the corresponding drawn contour parts. The introduced methodology simultaneously deals with two problems: a) the arbitrariness in data-points orientation and b) the determination of one proper form for a prototype curve that optimally fits the corresponding contour data. Arbitrariness in orientation has been dealt with a novel curvature based error, while the proper forms of curve prototypes have been exhaustively determined by embedding curvature deformations of the prototypes into 4-manifolds. Application of this methodology to celebrated wall-paintings excavated at Tyrins, Greece and the Greek island of Thera, manifests it is highly probable that these wall-paintings had been drawn by means of geometric guides that correspond to linear spirals and hyperbolae. These geometric forms fit the drawings' lines with an exceptionally low average error, less than 0.39mm. Hence, the approach suggests the existence of accurate realizations of complicated geometric entities, more than 1000 years before their axiomatic formulation in Classical Ages.

We propose a framework for exploiting dimension-reducing random projections in detection and classification problems. Our approach is based on the generalized likelihood ratio test; in the case of image classification, it exploits the fact that a set of images of a fixed scene under varying articulation parameters forms a low-dimensional, nonlinear manifold. Exploiting recent results showing that random projections stably embed a smooth manifold in a lower-dimensional space, we develop the multiscale smashed filter as a compressive analog of the familiar matched filter classifier. In a practical target classification problem using a single-pixel camera that directly acquires compressive image projections, we achieve high classification rates using many fewer measurements than the dimensionality of the images.

The emergence of low-cost sensing architectures for diverse modalities has made it possible to deploy sensor networks that acquire large amounts of very high-dimensional data. To cope with such a data deluge, manifold models are often developed that provide a powerful theoretical and algorithmic framework for capturing the intrinsic structure of data governed by a low-dimensional set of parameters. However, these models do not typically take into account dependencies among multiple sensors. We thus propose a new joint manifold framework for data ensembles that exploits such dependencies. We show that joint manifold structure can lead to improved performance for manifold learning. Additionally, we leverage recent results concerning random projections of manifolds to formulate a universal, network-scalable dimensionality reduction scheme that efficiently fuses the data from all sensors.

In this non-rigorous, possibly inconsistent paper where I talk about quite a few things I do not understand properly, I build a set to describe a very exciting notion: "Infinite-dimensional manifolds are compact surfaces without interiors (and exteriors). Motivated by the implications (that I write about), I am pretty fascinated by such a new perspective.

Let M be a topological space.A covering of M is a collection of open subsets of M whose union is M.A covering is said to be locally finite if each p has a nhood which intersects only finitely many of the sets. A Hausdorff space M is called paracompact if for each covering there exists a locally finite covering which is a refinement.It is known that a locally compact Hausdorff space which has a countable base is paracompact. A locally Euclidean space is a topological space M such that each point has a nhood homeomorphic to an open subset of the Euclidean n-space.In particular such a space is localy compact and paracompact. If ϕ is a homeomorphism of a connected open subset U onto an open subset of n-space then U is called a coordinate nhood;ϕ is called a coordinate map. A smooth atlas on a locally Euclidean space M is a collection of coordinate systems satisfying i. and ii. a topological manifold of dimension n is a Hausdorff,second countable, locally

We give the exact contributions of Harish-Chandra transform, (Hf)(λ), of Schwartz functions f to the harmonic analysis of spherical convolutions and the corresponding L p − Schwartz algebras on a connected semisimple Lie group G (with finite center). One of our major results gives the proof of how the Trombi-Varadarajan Theorem enters into the spherical convolution transform of L p − Schwartz functions and the generalization of this Theorem under the full spherical convolution map.

In this paper, we propose novel gossip algorithms for the low-rank decentralized matrix completion problem. The proposed approach is on the Riemannian Grassmann manifold that allows local matrix completion by different agents while achieving asymptotic consensus on the global low-rank factors. The resulting approach is scalable and parallelizable. Our numerical experiments show the good performance of the proposed algorithms on various benchmarks.

Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite, number of loss functions. In this paper, we propose a novel Riemannian extension of the Euclidean stochastic variance reduced gradient algorithm (R-SVRG) to a compact manifold search space. To this end, we show the developments on the Grassmann manifold. The key challenges of averaging, addition, and subtraction of multiple gradients are addressed with notions like logarithm mapping and parallel translation of vectors on the Grassmann manifold. We present a global convergence analysis of the proposed algorithm with decaying step-sizes and a local convergence rate analysis under fixed step-size with some natural assumptions. The proposed algorithm is applied on a number of problems on the Grassmann manifold like principal components analysis, low-rank matrix completion, and the Karcher mean computation. In all these cases, the proposed algorithm outperforms the standard Riemannian stochastic gradient descent algorithm.

In this paper, a general methodology is introduced for the determination of potential prototype curves used for the drawing of prehistoric wall paintings. The approach includes 1) preprocessing of the wall-paintings contours to properly partition them, according to their curvature, 2) choice of prototype curves families, 3) analysis and optimization in 4-manifold for a first estimation of the form of these prototypes, 4) clustering of the contour parts and the prototypes to determine a minimal number of potential guides, and 5) further optimization in 4-manifold, applied to each cluster separately, in order to determine the exact functional form of the potential guides, together with the corresponding drawn contour parts. The methodology introduced simultaneously deals with two problems: 1) the arbitrariness in data-points orientation and 2) the determination of one proper form for a prototype curve that optimally fits the corresponding contour data. Arbitrariness in orientation has been dealt with a novel curvature based error, while the proper forms of curve prototypes have been exhaustively determined by embedding curvature deformations of the prototypes into 4--manifolds. Application of this methodology to celebrated wall paintings excavated at Tyrins, Greece, and the Greek island of Thera manifests that it is highly probable that these wall paintings were drawn by means of geometric guides that correspond to linear spirals and hyperbolae. These geometric forms fit the drawings' lines with an exceptionally low average error, less than 0.39 mm. Hence, the approach suggests the existence of accurate realizations of complicated geometric entities more than 1,000 years before their axiomatic formulation in the Classical Ages.

Topological information is a promising resource to research in shape-understanding as it provides a high-level description of the characteristics of a shape, and such high-level description often has strong association with the semantics of an object. The Shape Acquisition and Processing (SAP) ontology has been designed to maintain useful information of a model. We propose here an extension to the SAP ontology, that addresses the non-manifold properties of a model. Useful information about the connectivity of an object can be obtained based on an analysis of its non-manifold properties, because the structure of a non-manifold object can be considered as a graph of manifold parts connected together at non-manifold joints. The manifold parts are often pieces that have strong semantic associations. In this work, we describe the type of non-manifold properties, the various types of connected components in a non-manifold object and their semantical significance. We address how the Euler' characteristics of a non-manifold object can be found based on such information. All such information are extractable from a model using TopMesh, a tool that we have developed. In this work, we also describe the features of TopMesh, namely, all the topological properties it extracts.

While manifold structure is often exploited for dimensionality reduction or feature extraction, this structure is rarely used by classification algorithms. We present a class of algorithms that utilize the low-dimensional manifold nature of signal ensembles and result in improved classification performance. The algorithms are built within theoretical frameworks that take into consideration prior knowledge of geometric structure in both labeled and unlabeled data points. Additionally, these frameworks can exploit recent results on random projections of smooth manifolds to ensure computational feasibility on extremely high-dimensional problems.