Related papers: Unit shapes and a wealth of calculus problems
We define the bounded coarse structure attached to a family of pseudometrics and give some counterexamples to conjectures that arise naturally.
An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the…
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…
The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…
We introduce an original approach to geometric calculus in which we define derivatives and integrals on functions which depend on extended bodies in space--that is, paths, surfaces, and volumes etc. Though this theory remains to be fully…
In this paper, on envelopes created by circle families in the plane, all four basic problems (existence problem, representation problem, problem on the number of envelopes, problem on relationships of definitions) are solved.
Assembling parts into an object is a combinatorial problem that arises in a variety of contexts in the real world and involves numerous applications in science and engineering. Previous related work tackles limited cases with identical unit…
In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct interesting actions…
We survey a variety of results about partially isometric matrices. We focus primarily on results that are distinctly finite-dimensional. For example, we cover a recent solution to the similarity problem for partial isometries. We also…
We present a generalization of the notion of an algebra norm relevant to real finite-dimensional unital associative algebras. Among other things, this leads to a novel set of algebra isomorphism invariants, some of which are computationally…
We give a conceptual explanation of universal deformation formulas for unital associative algebras and prove some results on the structure of their moduli spaces. We then generalize universal deformation formulas to other types of algebras…
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties…
This paper provides an extensive study of the homotopy theory of types of algebras with units, like unital associative algebras or unital commutative algebras for instance. To this purpose, we endow the Koszul dual category of curved…
A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally…
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we…
We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential…
Family of replica matrices, related to general ultrametric spaces, is introduced. These matrices generalize the known Parisi matrices. Some functionals of replica approach are computed.
We give a general account of family algebras over a finitely presented linear operad, this operad together with its presentation naturally defining an algebraic structure on the set of parameters.
We study the collection of points on the modular surface obtained from the logarithm embeddings of the groups of units in totally real cubic number fields. We conjecture that this set is dense and show that its closure contains countably…
This is a largely expository article based on our previous work on arithmetic diagonal cycles on unitary Shimura varieties. We define a class of Shimura varieties closely related to unitary groups which represent a moduli problem of abelian…