Bijection | EPFL Graph Search

In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set; there are no unpaired elements between the two sets. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. The term one-to-one correspondence must not be confused with one-to-one function (an injective function; see figures). A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite

Irreducible components of exotic Springer fibres

Neil John Saunders

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson–Schensted bijection, which is a variant of the type C Robinson–Schensted bijection between pairs of same‐shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson–Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces.

2018

Parity-regular Steinhaus graphs

Maxime Augier

Steinhaus graphs on n vertices are certain simple graphs in bijective correspondence with binary {0,1}-sequences of length n-1. A conjecture of Dymacek in 1979 states that the only nontrivial regular Steinhaus graphs are those corresponding to the periodic binary sequences 110 ... 110 of any length n-1 = 3m. By an exhaustive search the conjecture was known to hold up to 25 vertices. We report here that it remains true up to 117 vertices. This is achieved by considering the weaker notion of parity-regular Steinhaus graphs, where all vertex degrees have the same parity. We show that these graphs can be parametrized by an F-2-vector space of dimension approximately n/3 and thus constitute an efficiently describable domain where true regular Steinhaus graphs can be searched by computer.

American Mathematical Society2008

Representation of Radon Shape Diffusions via Hyperspherical Brownian Motion

Victor Panaretos

A framework is introduced for the study of general Radon shape diffusions, that is, shape diffusions induced by projections of randomly rotating shapes. This is done via a convenient representation of unoriented Radon shape diffusions in (unoriented) D.G. Kendall shape space

\tilde{\Sigma}^k_n

through a Brownian motion on the hypersphere. This representation leads to a coordinate system for the generalized version of Radon diffusions since it is shown that shape can be essentially identified with unoriented shape in the projected case. A bijective correspondence between Brownian motion on real projective space and Radon shape diffusions is established. Furthermore, equations are derived for the general (unoriented) Radon diffusion of shape- and-size, and stationary measures are discussed.

2008

Representation of Radon Shape Diffusions via Hyperspherical Brownian Motion

Victor Panaretos

\tilde{\Sigma}^k_n

2008