In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
In mathematics, a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180°. It has only one line of symmetry.
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon. Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones, via the use of mathematical theory.
In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle.
In 2-dimensional geometry, a lens is a convex region bounded by two circular arcs joined to each other at their endpoints. In order for this shape to be convex, both arcs must bow outwards (convex-convex). This shape can be formed as the intersection of two circular disks. It can also be formed as the union of two circular segments, joined along a common chord.
A dynamical billiard is a dynamical system in which a particle alternates between free motion and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed. Billiards are Hamiltonian idealizations of the game of billiards, but where the region contained by the boundary can have shapes other than rectangular and even be multidimensional. Dynamical billiards may also be studied on non-Euclidean geometries; indeed, the first studies of billiards established their ergodic motion on surfaces of constant negative curvature. The study of billiards which are kept out of a region, rather than being kept in a region, is known as outer billiard theory.
Outer billiards is a dynamical system based on a convex shape in the plane. Classically, this system is defined for the Euclidean plane but one can also consider the system in the hyperbolic plane or in other spaces that suitably generalize the plane. Outer billiards differs from a usual dynamical billiard in that it deals with a discrete sequence of moves outside the shape rather than inside of it.
In physics and mathematics, the Hadamard dynamical system is a chaotic dynamical system, a type of dynamical billiards. Introduced by Jacques Hadamard in 1898, and studied by Martin Gutzwiller in the 1980s, it is the first dynamical system to be proven chaotic.
The honeycomb conjecture states that a regular hexagonal grid or honeycomb has the least total perimeter of any subdivision of the plane into regions of equal area. The conjecture was proven in 1999 by mathematician Thomas C. Hales.
The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them. The percolation threshold is the critical value of the occupation probability p, or more generally a critical surface for a group of parameters p1, p2, ..., such that infinite connectivity (percolation) first occurs.
The Fermi–Ulam model (FUM) is a dynamical system that was introduced by Polish mathematician Stanislaw Ulam in 1961.
Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C, and the length or perimeter of C. Since the area A may be small while the length l is large, when C looks elongated, the relationship can only take the form of an inequality. What is more, such an inequality would be an upper bound for A: there is no interesting lower bound just in terms of the length.
In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves include the closed convex curves, the smooth curves that are convex, and the strictly convex curves, which have the additional property that each supporting line passes through a unique point of the curve.
In mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form.
Quantum scarring refers to a phenomenon where the eigenstates of a classically chaotic quantum system have enhanced probability density around the paths of unstable classical periodic orbits. The instability of the periodic orbit is a decisive point that differentiates quantum scars from the more trivial observation that the probability density is enhanced in the neighborhood of stable periodic orbits. The latter can be understood as a purely classical phenomenon, a manifestation of the Bohr correspondence principle, whereas in the former, quantum interference is essential. As such, scarring is both a visual example of quantum-classical correspondence, and simultaneously an example of a (local) quantum suppression of chaos.
A capsule, or stadium of revolution, is a basic three-dimensional geometric shape consisting of a cylinder with hemispherical ends. Another name for this shape is spherocylinder.
In geometry, a circular triangle is a triangle with circular arc edges.
Random sequential adsorption (RSA) refers to a process where particles are randomly introduced in a system, and if they do not overlap any previously adsorbed particle, they adsorb and remain fixed for the rest of the process. RSA can be carried out in computer simulation, in a mathematical analysis, or in experiments. It was first studied by one-dimensional models: the attachment of pendant groups in a polymer chain by Paul Flory, and the car-parking problem by Alfréd Rényi. Other early works include those of Benjamin Widom. In two and higher dimensions many systems have been studied by computer simulation, including in 2d, disks, randomly oriented squares and rectangles, aligned squares and rectangles, various other shapes, etc.
