Related papers: Unit shapes and a wealth of calculus problems

We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. We provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, describe an example…

We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity' maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced.…

Self-similar potentials generalize the concept of shape-invariance which was originally introduced to explore exactly-solvable potentials in quantum mechanics. In this article it is shown that previously introduced algebraic approach to the…

The problem of finding a canonical form of complex matrices up to conjugacy with the set of canonical matrices being a union of affine planes in the matrix space is considered. A solution of the problem is given producing a new canonical…

We consider harmonic maps on simply connected Riemann surfaces into the group $\mathrm{U}(n)$ of unitary matrices of order $n$. It is known that a harmonic map with an associated algebraic extended solution can be deformed into a new…

In this paper we define a numerical shape invariant of a continuous map called shape dimension of a map, which generalizes the shape dimension of a topological space. Some basic properties and applications of this invariant are given. The…

A situation calculus is presented that provides a solution to the frame problem for hierarchical situations, that is, situations that have a modular structure in which parts of the situation behave in a relatively independent manner. This…

Here we look at some situations that are like the unit circle or the real line in some ways, but which can be more complicated or fractal in other ways.

Isomorphism is central to the structure of mathematics and has been formalized in various ways within dependent type theory. All previous treatments have done this by replacing quantification over sets with quantification over groupoids of…

A correspondence between different $Pin$-type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.

Recently, sub-indices and sub-factors of groups with connections to number theory, additive combinatorics, and factorization of groups have been introduced and studied. Since all group subsets are considered in the theory and there are many…

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the…

This is a review of [Michor, Peter W.: The moment mapping for a unitary representation, Ann. Global Anal. Geometry, 8, No 3(1990), 299--313] including a careful description of calculus in infinite dimensions. For any unitary representation…

Here we initiate a program to study relationships between finite groups and arithmetic-geometric invariants in a systematic way. To do this we first introduce a notion of optimal module for a finite group in the setting of holomorphic mock…

We advocate an account of dualities between physical theories: the basic idea is that dual theories are isomorphic representations of a common core. We defend and illustrate this account, which we call a Schema, in relation to symmetries.…

In this note we briefly survey and propose some open problems related to isoparametric theory.

We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex…

For the first time we represent every finite group in the form of a graph in this book. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group.…

Shapes do not define a linear space. This paper explores the linear structure of deformations as a representation of shapes. This transforms shape optimization to a variant of optimal control. The numerical challenges of this point of view…

This paper investigates type isomorphism in a lambda-calculus with intersection and union types. It is known that in lambda-calculus, the isomorphism between two types is realised by a pair of terms inverse one each other. Notably,…