New invariant dominates Jones, Kuperberg, and arrow polynomials.
problem Dominating known knot invariants.
method Picture formalism leading to a new invariant.
result New invariant outperforms Jones, Kuperberg, and arrow polynomials.
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48 results for “picture formalism”
Deform moment map on symplectic connections using star product algebras.
problem Understanding symplectic connections and their deformations.
method Study vector bundle of Fedosov star product algebras, formal connection, curvature, and star product trace.
result Showed star product trace as a formal symplectic form and moment map.
Deform quantization recovers scalar curvature in complex structures.
problem Recovering scalar curvature in complex structures.
method Formal moment map construction on almost complex structures.
result Formal moment map deforms scalar curvature moment map in integrable cases.
Principles of financial product synthesis from a few basic financial products constitute an interesting research topic inspired by Islamic finance. We make an effort to answer general questions that should be answered before starting to investigate the main issues concerning this topic with the formalization of financi…
The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of th…
The paper identifies five extreme learning regimes for large linear autoencoders.
problem Understanding the learning dynamics of large weight-tied linear autoencoders.
method Formal loss-expansion hierarchy and analysis of gradient flow.
result Five extreme regimes associated with faces of a triangular prism.
These notes were inspired by the course ''Quantum Field Theory from a Functional Integral Point of View'' given at the University of Zurich in Spring 2017 by Santosh Kandel. We describe Feynman's path integral approach to quantum mechanics and quantum field theory from a functional integral point of view, where the mai…
The geometrical structure known as the Tulczyjew triple has proved to be very useful in describing mechanical systems, even those with singular Lagrangians or subject to constraints. Starting from basic concepts of variational calculus, we construct the Tulczyjew triple for first-order Field Theory. The important featu…
Clarifies mathematical aspects of Picture Changing Operators.
problem Understanding the geometric properties of PCOs.
method Showed PCOs are chain maps between differential and integral forms on supermanifolds.
result PCOs are chain maps, providing a new perspective on their structure.
The aim of this paper is to provide a gentle introduction to Chabauty topology, while very little background knowledge is assumed. As an example, we provide pictures for the Chabauty space of C*. Note that the description of this space is not new; however the pictures are novel.
On odd Laplace operators. IImath.DG
We analyze geometry of the second order differential operators, having in mind applications to Batalin--Vilkovisky formalism in quantum field theory. As we show, an exhaustive picture can be obtained by considering pencils of differential operators acting on densities of all weights simultaneously. The algebra of densi…
With the aid of a computer, we provide a motion picture of the twist-spun trefoil which exhibits the periodicity well.
We prove that constant scalar curvature Kähler metric "adjacent" to a fixed Kähler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T picture described by Donaldson. We prove that the Calabi flow near a cscK metric exi…
We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it …
We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a fami…
The paper develops a method to map knots in a cylinder to virtual-flat knots.
problem How to map knots in a cylinder to virtual knots.
method Construct a diagram on a cylinder with invisible crossings, then pull back invariants.
result Developed a method to map knots in a cylinder to virtual-flat knots.
Surveying Hitchin representations of Fuchsian groups.
problem Understanding representations of Fuchsian groups.
method Survey and conjectural geometric description.
result Conjectural geometric picture of an augmented Hitchin component.
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is gi…
Pfaffian groupoidsmath.DG
This thesis is about the study of Lie groupoids endowed with a compatible (multiplicative) differential 1-form. The motivation and scope of the present work is to study the geometry of PDEs using the formalism of Lie groupoids and multiplicative forms; as such, ideas from the two theories have to be introduced and expl…
We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean 3-space. Since humans cannot visualise Euclidean spaces of higher dimensions, it appears to be impossible to give pictorial representations of higher-dimensional manifolds. However, one can in fact encode the topology of a surface in a 1-dime…
Bitcoin price prediction models fail to outperform a simple 'today's price' baseline, especially at longer horizons.
problem Lack of robust models that consistently outperform a naive price predictor at various horizons.
method Surveyed peer-reviewed papers, categorized by evaluation methodology, contrasted with social media discourse, and proposed methodological standards.
result No peer-reviewed study has shown robust superiority over the naive baseline across multiple market regimes at short-to-medium horizons.
Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of…
Graph Weighted Models (GWMs) have recently been proposed as a natural generalization of weighted automata over strings and trees to arbitrary families of labeled graphs (and hypergraphs). A GWM generically associates a labeled graph with a tensor network and computes a value by successive contractions directed by its e…
Unified geometric formulation of Maxwell-Vlasov system using presymplectic and symmetry reduction.
problem Unified geometric formulation of Maxwell-Vlasov system.
method Skinner-Rusk formalism, presymplectic geometry, reduction by diffeomorphism group, affine Hamiltonian controls.
result Unified geometric structure unifying Lagrangian, Hamiltonian, gauge, reduction, and control-theoretic aspects.
Non-Reidemeister Knot Theory and Its Applications in Dynamical Systems, Geometry, and Topologymath.GT
Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to diagrams of the initial object. The main tool is the notion of {\em free -braid …
A pictorial introduction to differential geometry, leading to Maxwell's equations as three picturesmath.DG
In this article we present pictorially the foundation of differential geometry which is a crucial tool for multiple areas of physics, notably general and special relativity, but also mechanics, thermodynamics and solving differential equations. As all the concepts are presented as pictures, there are no equations in th…
In [14], the second named author constructed the bracket invariant [.] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams K, the following formula holds: [K]=K', where K' is the underlying graph of the diagram, i.e., the…
Extends Lie bialgebroids for string and M theories with new calculus framework.
problem Formalize calculus on algebroids for string and M theories.
method Reinterpret matched pairs of Leibniz algebroids, examine algebroid axioms, construct double on direct sum.
result Construct Drinfel'd double of Lie bialgebroids for general algebroids.
New braid representations using virtual knot theory.
problem No classical features in virtual knot theory.
method Construct new braid representations using virtual knot theory.
result New representations of classical braids.
The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the …
We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A_n, B_n=C_n and D_n and the affine diagrams tilde{A}_n, tilde{B}_n, tilde{C}_n and tilde{D}_n as subgroups of the braid gr…
The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is TM, is based on the existence of canonical symplectic isomorphisms of d…
Quantizes the relationship between Koszul and Schouten brackets in Poisson geometry.
problem Quantizing the relationship between Koszul and Schouten brackets in Poisson geometry.
method Employing Voronov's thick morphism technique and quantum Mackenzie-Xu transformations in the framework of -algebroids.
result Quantizes the -morphism into a single linear operator, a formal Fourier integral operator.
We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold of dimension at most seven, can degenerate. Loosely speaking, our results show …
Extends Satoh's map to virtual m-links and constructs geometric pictures.
problem Virtual m-links and their geometric representations.
method Generalization of Satoh's map to virtual m-links and construction of geometric pictures.
result New geometric pictures of virtual m-links.
The paper clarifies the approximation of SGD with Ito SDEs for finite learning rates.
problem Theoretical justification and experimental verification of the Ito SDE approximation for finite learning rates in SGD.
method An efficient simulation algorithm SVAG and a necessary condition test for the SDE approximation.
result The Ito SDE approximation can meaningfully capture training and generalization properties of deep nets with finite learning rates.
A Group-Theoretic Approach to Computational Abstraction: Symmetry-Driven Hierarchical Clusteringcs.LG
Abstraction plays a key role in concept learning and knowledge discovery; this paper is concerned with computational abstraction. In particular, we study the nature of abstraction through a group-theoretic approach, formalizing it as symmetry-driven---as opposed to data-driven---hierarchical clustering. Thus, the resul…
The paper describes new types of picture-valued invariants and their applications.
problem Understanding and categorizing picture-valued invariants on diagrams.
method General description and classification of derivations and functorial maps.
result Two new examples of functorial maps are introduced, including the order and lifting maps.
Introduces D-branes in para-Hermitian geometries using T-duality.
problem Defines D-branes in a new geometric framework.
method Uses para-Hermitian geometry and metric algebroids to define D-branes as conformal boundary conditions for open strings.
result Reveals D-branes as para-complex versions of topological A/B-branes.
The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of `the categorification of the Jones polynomial'. For the same low cost we also provide some computations, including one that shows that Khovanov's…
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. A…
We study the effect of a relevant double-trace deformation on the partition function (and conformal anomaly) of a CFT at large N and its dual picture in AdS. Three complementary previous results are brought into full agreement with each other: bulk and boundary computations, as well as their formal identity. We show th…
New conditions for stabilizing systems are shown to be independent but stronger under certain conditions.
problem Necessary conditions for stabilizing feedback control systems.
method Extensions of Brockett's and Coron's conditions in fiber and vector bundle pictures.
result The conditions are independent in the fiber bundle picture but stronger in the vector bundle picture under specific conditions.
Unified framework connects physical laws and machine learning.
problem Combining physical laws and machine learning for scientific applications.
method Universal Differential Equations (UDEs) as a unifying framework.
result Wide variety of applications can be efficiently handled through UDE formalism.
Financial market risk is modeled using thermodynamics, linking economic quantities to thermodynamic variables.
problem Model risk in financial markets caused by external information sources.
method Established a thermodynamic analogy between financial market variables and economic quantities, linking financial risk to entropy changes.
result Derivation of worst-case risk characterization rules using thermodynamics, leading to equilibrium and non-equilibrium thermodynamic evaluations.
We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.
Boosted DT classifiers become DP with new calibrated loss.
problem Making boosted DT classifiers differentially private.
method Crafted M-loss and objective calibration.
result Significantly outperforms random forests in DP settings.
We explore the perspective of a bug living on the two-dimensional surface of a polyhedron. Images of various kinds of effects like lensing and cloaking are shown via color pictures of three viewpoints: the first person perspective of the bug, a map of the bug's viewpoint, and a look at the bug on the embedded polyhedro…