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Time reflection and refraction are temporal analogies of the spatial boundary effects derived from Fermat’s principle. They occur when classical waves strike a time boundary where an abrupt change in the properties of the medium is introduced. The main features of time-reflected and time-refracted waves are the shift in frequency and conservation of momentum, which offer a new degree of freedom for steering extreme waves and controlling the phases of matter. The concept was originally proposed for manipulating optical waves more than five decades ago. However, due to the extreme challenges in the ultrafast engineering of optical materials, the experimental realization of the time boundary effects remains elusive. Here we introduce a time boundary into a momentum lattice of ultracold atoms and simultaneously demonstrate time reflection and refraction experimentally. Through launching a Gaussian-superposed state into the Su–Schrieffer–Heeger atomic chain, we observe the time-reflected and time-refracted waves when the input state strikes a time boundary. Furthermore, we detect a transition from time reflection/refraction to localization with increasing strength of disorder and show that the time boundary effects are robust against considerable disorder. Our work opens a new avenue for the future exploration of time boundaries and spatiotemporal lattices, as well as their interplay with non-Hermiticity and many-body interactions. Time reflection and refraction are experimentally observed in ultracold atoms. To this end, the time boundary is formed by imposing an abrupt change in the coupling strength of the atomic chain. Time boundary effects are robust against material disorder.

Reinterpretation of the Fermat principle governing the propagation of light in media within the Ramsey theory is suggested. Complete bi-colored graphs corresponding to light propagation in media are considered. The vertices of the graphs correspond to the points in real physical space in which the light sources or sensors are placed. Red links in the graphs correspond to the actual optical paths, emerging from the Fermat principle. A variety of optical events, such as refraction and reflection, may be involved in light propagation. Green links, in turn, denote the trial/virtual optical paths, which actually do not occur. The Ramsey theorem states that within the graph containing six points, inevitably, the actual or virtual optical cycle will be present. The implementation of the Ramsey theorem with regard to light propagation in metamaterials is discussed. The Fermat principle states that in metamaterials, a light ray, in going from point S to point P, must traverse an optical path length L that is stationary with respect to variations of this path. Thus, bi-colored graphs consisting of links corresponding to maxima or minima of the optical paths become possible. The graphs, comprising six vertices, will inevitably demonstrate optical cycles consisting of the mono-colored links corresponding to the maxima or minima of the optical path. The notion of the “inverse graph” is introduced and discussed. The total number of triangles in the “direct” (source) and “inverse” Ramsey optical graphs is the same. The applications of “Ramsey optics” are discussed, and an optical interpretation of the infinite Ramsey theorem is suggested.

Ellipsoidal and plane‐elliptical surfaces are widely used as reflective, point‐to‐point focusing elements in many optical systems, including X‐ray optics. Here the classical optical path function approach of Fermat is applied to derive a closed‐form expression for these surfaces that are uniquely described by the object and image distances and the angle of incidence at a point on a mirror surface. A compact description facilitates design, modeling, fabrication, and testing to arbitrary accuracy. Congruent surfaces in two useful coordinate systems — a system centered on the ellipsoid's axes of symmetry and a mirror‐centered or `vertex' system with the surface tangent to the xy plane at the mirror's center — are presented. Expressions for the local slope and radii of curvature are derived from the result, and the first several terms of the Maclauren series expansion are provided about the mirror center. The derivation of closed‐form expressions for ellipsoidal mirror surfaces from the Fermat principle, in terms of the object and image distances, and the glancing angle of incidence, in mirror‐centered coordinates is given.

The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if F is a subfield of a local field of characteristic ≠2, then the special upper triangular group ST+(n,F) is minimal precisely when the special linear group SL(n,F) is. We provide criteria for the minimality (and total minimality) of SL(n,F) and ST+(n,F), where F is a subfield of C. Let Fπ and Fc be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for A∈Fπ,Fc: A is finite; ∏Fn∈ASL(Fn−1,Q(i)) is minimal, where Q(i) is the Gaussian rational field; and ∏Fn∈AST+(Fn−1,Q(i)) is minimal. Similarly, denote by Mπ and Mc the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let B∈Mπ,Mc. Then the following conditions are equivalent: B is finite; ∏Mp∈BSL(Mp+1,Q(i)) is minimal; and ∏Mp∈BST+(Mp+1,Q(i)) is minimal.

We explore deeper and analyse in more detail Fermat's and Hamilton's principles. We try to address some questions: Is it possible to have S negative? Is Hamilton's principle always valid for entire path of the system? Is there a relation between Fermat's principle and Hamilton's principle? We assume analogy with Hamilton's principle, is Fermat's principle always valid for entire path of the system? Does a least action take a least time for happening?.

Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat’s Last Theorem to hold over F and also, for the nonexistence of solutions to the unit equation over F. For example, if two totally ramifies and three splits completely in F, then the asymptotic Fermat’s Last Theorem holds over F.

In 1937, the 16-years-old Hungarian mathematician Endre Weiszfeld, in a seminal paper, devised a method for solving the Fermat–Weber location problem—a problem whose origins can be traced back to the seventeenth century. Weiszfeld’s method stirred up an enormous amount of research in the optimization and location communities, and is also being discussed and used till these days. In this paper, we review both the past and the ongoing research on Weiszfed’s method. The existing results are presented in a self-contained and concise manner—some are derived by new and simplified techniques. We also establish two new results using modern tools of optimization. First, we establish a non-asymptotic sublinear rate of convergence of Weiszfeld’s method, and second, using an exact smoothing technique, we present a modification of the method with a proven better rate of convergence.

The systems of nonlinear differential equations of certain types can be simplified to matrix forms. Two types of matrix differential equations will be considered in the paper, one is Fermat type matrix differential equation where n = 2 and n = 3, another is Malmquist type matrix differential equation , where α (≠ 0), β, γ are constants. By solving the systems of nonlinear differential equations, we obtain some properties on the meromorphic matrix solutions of the above matrix differential equations. In addition, we also consider two types of nonlinear differential equations, one of them is called Bi-Fermat differential equation.

In this paper another seven problems from Wacław Sierpiński’s book “250 Problems in Elementary Number Theory” are formalized, using the Mizar formalism, namely: 53, 61, 81, 90, 100, 156, and 167.

Fermat–Weber points with respect to an asymmetric tropical distance function are studied. It turns out that they correspond to the optimal solutions of a transportation problem. The results are applied to obtain a new method for computing consensus trees in phylogenetics. This method has several desirable properties; e.g., it is Pareto and co-Pareto on rooted triplets.