arXiv research

Study Swan modules and homotopy types, resolving Wall and Dyer questions.

Extends Serre-Swan theorem to all finitely generated modules over smooth functions.

The paper proves a Serre-Swan Theorem for coisotropic algebras.

Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an …

The Serre-Swan theorem provides the link between projective modules of finite rank and vector bundles over compact manifolds, and plays a prominent role in non-commutative geometry. Its extension to non-compact manifolds is discussed.

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…

The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac operators, exterior algebra bundles and Connes' differential algebras in the commutative case…

Defines super projective modules and explores their properties.

The aim of this note is to analyse the structure of the $L^0$-normed $L^0$-modules over a metric measure space. These are a tool that has been introduced by N. Gigli to develop a differential calculus on spaces verifying the Riemannian Curvature Dimension condition. More precisely, we discuss under which conditions an …

The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an et…

Reduces Lie (bi-)algebroids and Dirac manifolds using constraint vector bundles.

Study adds memory effect to Solow-Swan model for more accurate economic growth modeling.

New risk models use chaotic attractors to predict extreme events.

We develop a simple test for deviations from power law tails, which is based on the asymptotic properties of the empirical distribution function. We use this test to answer the question whether great natural disasters, financial crashes or electricity price spikes should be classified as dragon kings or 'only' as black…

Develops differential K-theory for noncommutative algebras.

Society's drive toward ever faster socio-technical systems, means that there is an urgent need to understand the threat from 'black swan' extreme events that might emerge. On 6 May 2010, it took just five minutes for a spontaneous mix of human and machine interactions in the global trading cyberspace to generate an unp…

Out-of-control information technology (IT) projects have ended the careers of top managers, such as EADS CEO Noel Forgeard and Levi Strauss' CIO David Bergen. Moreover, IT projects have brought down whole companies, like Kmart in the US and Auto Windscreen in the UK. Software and other IT is now such an integral part o…

In this paper, we present Neural Phrase-based Machine Translation (NPMT). Our method explicitly models the phrase structures in output sequences using Sleep-WAke Networks (SWAN), a recently proposed segmentation-based sequence modeling method. To mitigate the monotonic alignment requirement of SWAN, we introduce a new …

The paper shows how to construct non-Gaussian Martingales using hyperbolic diffusion.

The cost-benefit analysis formulates the holy trinity of objectives of project management - cost, schedule, and benefits. As our previous research has shown, ICT projects deviate from their initial cost estimate by more than 10% in 8 out of 10 cases. Academic research has argued that Optimism Bias and Black Swan Blindn…

In the presence of a layer of metaprobabilities (from uncertainty concerning the parameters), the asymptotic tail exponent corresponds to the lowest possible tail exponent regardless of its probability. The problem explains "Black Swan" effects, i.e., why measurements tend to chronically underestimate tail contribution…

QTMRL uses RL with multi-indicators to improve trading adaptability.

We show that, under mild assumptions, some unimaginable events - which we refer to as Black Swan events - must necessarily occur. It follows as a corollary of our theorem that any computational model of decision-making under uncertainty is incomplete in the sense that not all events that occur can be taken into account…

Applying the classical Serre-Swan theorem, as this is extended to topological (non-normed) algebras, one attains a classification of elementary particles via their spin-structure. In this context, our argument is virtually based on a ``correspondence principle'' of S. A. Selesnick, formulated herewith in a sheaf-theore…

We study the interplay among Wall's $D(2)$ problem, normal generation conjecture (the Wiegold Conjecture) of perfect groups and Swan's problem on partial Euler characteristic and deficiency of groups. In particular, for a 3-dimensional complex $X$ of cohomological dimension 2 with a finite fundamental group, assuming t…

Commissioned by MIT's in-house artist Jane Philbrick, we evolve an abstract 2D surface (resembling Marta Pan's 1961 "Sculpture Flottante I") under mean curvature, all the while calculating the eigenmodes and eigenvalues of the Laplace-Beltrami operator on the resulting shapes. These are then synthesized into a sound-wa…

Study finds 'Dragon Kings' in stock market volatility during major economic crises.

Study reveals jumps in crypto markets predict future prices.

The adaptation of numerical wind wave models to the local time-spatial conditions is a problem that can be solved by using various calibration techniques. However, the obtained sets of physical parameters become over-tuned to specific events if there is a lack of observations. In this paper, we propose a robust evoluti…

Model for assembly map of bordism-invariant functors.

We suggest an original physical approach to describe the mechanism of market pricing. The core of our approach is to consider pricing at different time scales separately, using independent equations of motion. Such an approach leads to a pricing model that not only allows estimating the volatility of future market pric…

Classifies modules of surface-knots in terms of their properties.

Proves finiteness and holonomicity of skein modules for 3-manifolds.

We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.

A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is …

A general theory of innovation and progress in human society is outlined, based on the combat between two opposite forces (conservatism/inertia and speculative herding "bubble" behavior). We contend that human affairs are characterized by ubiquitous ``bubbles'', which involve huge risks which would not otherwise be tak…

Neural Module Networks, originally proposed for the task of visual question answering, are a class of neural network architectures that involve human-specified neural modules, each designed for a specific form of reasoning. In current formulations of such networks only the parameters of the neural modules and/or the or…

Paper compares skein modules to Kauffman bracket modules.

Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills of homologies and homotopies, distant mountains - the Kauffman bracket skein mo…

Generalized Steinberg module presentation for Gaussian and Eisenstein integers.

Enhanced Alexander module detects linking numbers in links.

We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. A 2-crossed module bundle 2-gerbe over a manifold is defined in terms of a so called 2-crossed module bundle gerbe, which is a crossed module bundle gerbe equipped with an extra sructure. It is shown that s…

Combinatorial approach to compute satellite knot invariants using graph theory.

New sl(2) action defined on a mathematical module.

Introduces admissible skein modules for non-semisimple categories.

Studies modules over a category of Jacobi diagrams in handlebodies.

This paper generalizes L2 cohomology theory for complex manifolds.

Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad …