Three-dimensional consistency

Local blow up

Blow up space of initial conditions

Fuzzy relation R

Fuzzy relation R

Fuzzy relation S

Fuzzy relation S

The max-min composition of fuzzy relations R and S

The max-min composition of fuzzy relations R and S

The max-product composition of fuzzy relations R and S

The max-product composition of fuzzy relations R and S

The max-av composition of fuzzy relations R and S

The max-av composition of fuzzy relations R and S

An approximation of a tronquée solution of the fourth Painlevé equation

An approximation of a tronquée solution of the fourth Painlevé equation

A tritronquée solution of the fourth Painlevé equation

A tritronquée solution of the fourth Painlevé equation

5-cube

Orthogonal projection of 5-dimensional hypercube in 2D

Tropic curves on Confocal Quadrics in the Minkowski Space

Double Reflection Theorem

Billiard Trajectories

Tropic curves and their light like tangents

Tesseract

PonceletDarboux Grid

Poncelet Theorem

Lightlike Billiard Trajectories

Interval Exchange Transformation

Fomenko Graph

Ultra-Discrete KdV equation on a two dimensional domain

Propagating from the singularity marked "1" eventually results in a region of singularities trapped by differential points.

Configuration of the bilinear discrete KP equation

This figure shows 3 copies of the bilinear discrete Kadomtsev-Petviashvili system. This is a 3-dimensional lattice equation, which can be considered as relating 6 vertices of a cube.

Staircase reduction

Staircase reduction of the 7-point stencil of the discrete Euler-Lagrange equations of an ABS equation.

Water waves past a submerged source

Free-surface profile for potential flow of infinite depth flowing past a submerged source, containing steady waves generated in the downstream flow profile.

Tritronquée-type solutions to a modified Painlevé equation

Real and imaginary tritronquée-type solutions to a modified Painlevé equation. The tritronquée-type solution is only valid within the indicated region. In the omitted region, elliptic functions are required to describe the solution behaviour.

Time evolution of a 2-compacton initial condition

While collisions of many kinds of solitary waves have been studied, the case of compactons is still an open question.

Exponentially small waves present in the Frenkel Kontorova Model

Exponentially small waves present in the far-field solution of the Frenkel-Kontorova model. The magnitude vanishes on the discrete points for which the solution is defined on

Configuration of the roots of a generalized Okamoto polynomial

Configuration of the roots in the complex plane of one of the generalised Okamoto polynomials. These polynomials are associated with rational solutions of the fourth Painlevé equation. Why these roots are so beautifully organised remains a mystery.

Circle pattern

Circle pattern corresponding to the discrete analogue
of holomorphic functions z^(4/5).

Invariant curve for autonomous q-PIII in complex projective space

These images represent the invariant curve for autonomous q-PIII in complex projective space. They show all 8 base points and potential duplicates.

Second invariant curve for autonomous q-PIII in complex projective space

These images represent the invariant curve for autonomous q-PIII in complex projective space. They show all 8 base points and potential duplicates.

Initial value space of autonomous q-PIII

Initial value space of autonomous q-PIII, wlog, we assume that b>a and d>c. D_i, i=1,2,3,4 are the divisors.

An algebraic curve with 5 ordinary singularities

An algebraic curve with 5 ordinary singularities at (0,1), (0,-1), (1,0), (-1,0) and (0,0).

Algebraic curve with 5 ordinary singularities

f(x,y)=(x^4 + y^4 - x^2 - y^2)^2-9 x^2 y^2, an algebraic curve with 5 ordinary
singularities at (0,1), (0,-1), (1,0), (-1,0) and (0,0).