Related papers: The Pseudo-hyperresolution and Applications
Autoencoders have been extensively used in the development of recent anomaly detection techniques. The premise of their application is based on the notion that after training the autoencoder on normal training data, anomalous inputs will…
We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed…
Real algebraic geometry is the study of semi-algebraic sets, subsets of $\R^k$ defined by Boolean combinations of polynomial equalities and inequalities. The focus of this thesis is to study quantitative results in real algebraic geometry,…
Majority of the current dimensionality reduction or retrieval techniques rely on embedding the learned feature representations onto a computable metric space. Once the learned features are mapped, a distance metric aids the bridging of gaps…
We define a basic class of algebras which we call homotopy path algebras. We find that such algebras always admit a cellular resolution and detail the intimate relationship between these algebras, stratifications of topological spaces, and…
Modular functors, i.e. consistent systems of projective representations of mapping class groups of surfaces, have been constructed for non-semisimple modular categories already decades ago. Concepts from homological algebra have not been…
We extend the BMS(4) group by adding logarithmic supertranslations. This is done by relaxing the boundary conditions on the metric and its conjugate momentum at spatial infinity in order to allow logarithmic terms of carefully designed form…
In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of…
Motivated by the classical correspondence between short exact sequences and splitting properties in module theory, this paper examines the projective and injective analogues within the category of Lie algebras. We first show that no Lie…
The main result of this paper is an application of the topology of the space $Q(X)$ to obtain results for the cohomology of the symmetric group on $d$ letters, $\Sigma_d$, with `twisted' coefficients in various choices of Young modules and…
A method for dimension reduction with clustering, classification, or discriminant analysis is introduced. This mixture model-based approach is based on fitting generalized hyperbolic mixtures on a reduced subspace within the paradigm of…
We formally introduce a systematic (de/re)-composition approach, based on the algebraic formalism of "Multi-Dimensional Homomorphisms (MDHs)". Our approach is designed as general enough to be applicable to a wide range of data-parallel…
We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method…
This paper describes an approach to computer aided calculations in the cohomology of arithmetic groups. It complements existing literature on the topic by emphasizing homotopies and perturbation techniques, rather than cellular subdivision,…
We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the…
We introduce a cohomology theory of grading-restricted vertex algebras. To construct the {\it correct} cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the…
In this work, we develop systematically the ``Dirichlet Hopf algebra of arithmetics'' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra…
The holomorph of a discrete group $G$ is the universal semi-direct product of $G$. In chapter 1 we describe why it is an interesting object and state main results. In chapter 2 we recall the classical definition of the holomorph as well as…
The geometric models for the module category and derived category of any gentle algebra were introduced to realize the objects in module category and derived category by permissible curves and admissible curves respectively. The present…
Following our previous work, we suggest here a large class of algebras of scalars in which simultaneous and correlated computations can be performed owing to the existence of surjective algebra homomorphisms. This may replace the currently…