Related papers: Primal and Dual Combinatorial Dimensions
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…
We continue the study of model-independent constraints on the unitary Conformal Field Theories in 4-Dimensions, initiated in arXiv:0807.0004. Our main result is an improved upper bound on the dimension \Delta of the leading scalar operator…
Extremely long-lived, time-dependent, spatially-bound scalar field configurations are shown to exist in $d$ spatial dimensions for a wide class of polynomial interactions parameterized as $V(\phi) = \sum_{n=1}^h\frac{g_n}{n!}\phi^n$.…
A planar boundary introduced \`a la Symanzik in the 5D topological BF theory, with the only requirement of locality and power counting, allows to uniquely determine a gauge invariant, non topological 4D Lagrangian. The boundary condition on…
We establish an other upper bounding for packing dimension in the framework of the vectorial multifractal formalism that is in some cases finer than that established by J. Peyriere.
The VC-dimension of a family of sets is a measure of its combinatorial complexity used in machine learning theory, computational geometry, and even model theory. Computing the VC-dimension of the $k$-fold union of geometric set systems has…
In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…
We adapt concepts, methodology, and theory originally developed in the areas of multidimensional scaling and dimensionality reduction for multivariate data to the functional setting. We focus on classical scaling and Isomap -- prototypical…
This paper deals with some basic constructions of linear and multilinear algebra on finite-dimensional diffeological vector spaces. We consider the diffeological dual formally checking that the assignment to each space of its dual defines a…
We provide a self-contained exposition of the well-known multifractal formalism for self-similar measures satisfying the strong separation condition. At the heart of our method lies a pair of quasiconvex optimization problems which encode…
The categorified theories known as "doctrines" specify a category equipped with extra structure, analogous to how ordinary theories specify a set with extra structure. We introduce a new framework for doctrines based on double category…
Two-dimensional conformal field theory (CFT) has several sources: the search for simple examples of quantum field theory, the description of surface critical phenomena, the study of (super)string vacua. In the present overview of the…
For monomial special multiserial algebras, which in general are of wild representation type, we construct radical embeddings into algebras of finite representation type. As a consequence, we show that the representation dimension of…
We study adaptive data-dependent dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are…
We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic…
Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…
We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not…
Multi-valued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS…
We show that abelian bosonization of 1+1 dimensional fermion systems can be interpreted as duality transformation and, as a conseguence, it can be generalized to arbitrary dimensions in terms of gauge forms of rank $d-1$, where $d$ is the…
The correlations in classical multi-component ionic mixtures with spatial dimension $d\geq 2$ are studied by using a restricted grand-canonical ensemble and the associated hierarchy equations for the correlation functions. Sum rules for the…