Classical Fermat Research Articles

Fermat's Lemma on the Hyperbolic Plane

Generalizing classical mathematical problems to other spaces is an important research direction in mathematics. This paper studies Fermat's Lemma on the hyperbolic plane, where the set of hyperbolic numbers is a commutative ring with zero divisors generated by two real numbers, similar to the complex numbers but more complicated. Compared to the classical Fermat’s Lemma, the Fermat’s Lemma in this paper has a higher dimension and is built on a hyperbolic ring whose algebraic structure is more complex than the real number field. On the basis of a generalization of the classical Fermat’s Lemma, the paper overcomes the difficulty that hyperbolic numbers contain zero divisors, and obtains the proof of the first and second Fermat’s Lemma on the hyperbolic plane through the decomposition of hyperbolic numbers, which will give impetus to theoretical research in hyperbolic analysis.

Smooth quotients of generalized Fermat curves

A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $n,p \geq 2$ are integers such that $(p-1)(n-1)>2$, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms with quotient orbifold $S/H$ of genus zero with exactly $n+1$ cone points, each one of order $p$; in this case $H$ is called a generalized Fermat group of type $(p,n)$. In this case, it is known that $S$ is non-hyperelliptic and that $H$ is its unique generalized Fermat group of type $(p,n)$. Also, explicit equations for them, as a fiber product of classical Fermat curves of degree $p$, are known. For $p$ a prime integer, we describe those subgroups $K$ of $H$ acting freely on $S$, together with algebraic equations for $S/K$, and determine those $K$ such that $S/K$ is hyperelliptic.

Holomorphic differentials of generalized Fermat curves

A non-singular complete irreducible algebraic curve Fk,n, defined over an algebraically closed field K, is called a generalized Fermat curve of type (k,n), where n,k≥2 are integers and k is relatively prime to the characteristic p of K, if it admits a group H≅Zkn of automorphisms such that Fk,n/H is isomorphic to PK1 and it has exactly (n+1) cone points, each one of order k. By the Riemann-Hurwitz-Hasse formula, Fk,n has genus at least one if and only if (k−1)(n−1)>1. In such a situation, we construct a basis, called a standard basis, of its space H1,0(Fk,n) of regular forms, containing a subset of cardinality n+1 that provides an embedding of Fk,n into PKn whose image is the fiber product of (n−1) classical Fermat curves of degree k. For p=2, we obtain a lower bound (which is sharp for n=2,3) for the dimension of the space of exact one-forms, that is, the kernel of the Cartier operator. We also do this for (p,k,n)=(3,2,4).

Smooth models of singular $K3$-surfaces

We show that the classical Fermat quartic has exactly three smooth spatial models. As a generalization, we give a classification of smooth spatial (as well as some other) models of singular K3 -surfaces of small discriminant. As a by-product, we observe a correlation (up to a certain limit) between the discriminant of a singular K3 -surface and the number of lines in its models. We also construct a K3 -quartic surface with 52 lines and singular points, as well as a few other examples with many lines or models.

About the Fricke–Macbeath curve

A closed Riemann surface of genus $$g \geqslant 2$$ is called a Hurwitz curve if its group of conformal automorphisms has order $$84(g-1)$$ . In 1895, Wiman noticed that there is no Hurwitz curve of genus $$g=2,4,5,6$$ and, up to isomorphisms, there is a unique Hurwitz curve of genus $$g=3$$ ; this being Klein’s plane quartic curve. Later, in 1965, Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus $$g=7$$ ; this known as the Fricke–Macbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a 4-dimensional family of genus seven closed Riemann surfaces $$S_{\mu }$$ admitting a group of conformal automorphisms so that $$S_{\mu }/G_{\mu }$$ has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Macbeath’s description. We also observe that the jacobian variety of each $$S_{\mu }$$ is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular Fricke–Macbeath curve, we obtain the well-known fact that its jacobian variety is isogenous to $$E^{7}$$ for a suitable elliptic curve E.

Automorphisms of generalized Fermat curves

Let K be an algebraically closed field of characteristic p≥0. A generalized Fermat curve of type (k,n), where k,n≥2 are integers (for p≠0 we also assume that k is relatively prime to p), is a non-singular irreducible projective algebraic curve Fk,n defined over K admitting a group of automorphisms H≅Zkn so that Fk,n/H is the projective line with exactly (n+1) cone points, each one of order k. Such a group H is called a generalized Fermat group of type (k,n). If (n−1)(k−1)>2, then Fk,n has genus gn,k>1 and it is known to be non-hyperelliptic. In this paper, we prove that every generalized Fermat curve of type (k,n) has a unique generalized Fermat group of type (k,n) if (k−1)(n−1)>2 (for p>0 we also assume that k−1 is not a power of p).Generalized Fermat curves of type (k,n) can be described as a suitable fiber product of (n−1) classical Fermat curves of degree k. We prove that, for (k−1)(n−1)>2 (for p>0 we also assume that k−1 is not a power of p), each automorphism of such a fiber product curve can be extended to an automorphism of the ambient projective space. In the case that p>0 and k−1 is a power of p, we use tools from the theory of complete projective intersections in order to prove that, for k and n+1 relatively prime, every automorphism of the fiber product curve can also be extended to an automorphism of the ambient projective space.In this article we also prove that the set of fixed points of the non-trivial elements of the generalized Fermat group coincide with the hyper-osculating points of the fiber product model under the assumption that the characteristic p is either zero or p>kn−1.

Analytic Approach to Optimal Routing for Commercial Formation Flight

This paper explores an analytic, geometric approach to finding optimal routes for commercial formation flight. A weighted extension of the classical Fermat point problem is used to develop a scalable methodology for the formation routing problem, enabling quick calculation of formation costs. This rapid evaluation allows the large-scale fleet assignment problem to be solved via a mixed integer linear program in reasonable time. Weighting schemes for aircraft performance characteristics are first introduced and then extended to allow for differential rates of fuel burn. Finally, a case study for 210 transatlantic flight routes is presented, with results showing possible average fuel-burn savings against solo flight of around 8.7% for formations of two and 13.1% for formations of up to three.

Nonsmooth Algorithms and Nesterov's Smoothing Technique for Generalized Fermat--Torricelli Problems

We present algorithms for solving a number of new models of facility location which generalize the classical Fermat--Torricelli problem. Our first approach involves using Nesterov's smoothing technique and the minimization majorization principle to build smooth approximations that are convenient for applying smooth optimization schemes. Another approach uses subgradient-type algorithms to cope directly with the nondifferentiability of the cost functions. Convergence results of the algorithms are proved and numerical tests are presented to show the effectiveness of the proposed algorithms.

Automorphisms group of generalized Fermat curves of type [formula omitted

The determination of the full group of automorphisms of a closed Riemann surface is in general a very complicated task. For hyperelliptic curves, the uniqueness of the hyperelliptic involution permits one to compute these groups in a very simple manner. Similarly, as classical Fermat curves of degree k admit a unique subgroup of automorphisms isomorphic to Zk2, the determination of the group of automorphisms is not difficult.In this paper we consider a family of non-hyperelliptic Riemann surfaces, obtained as the fibre product of two classical Fermat curves of the same degree k, which exhibit behaviors of both elliptic and hyperelliptic curves. These curves, called generalized Fermat curves of type (k,3), are the highest regular abelian branched covers of orbifolds of genus zero with four cone points, all of the same order k. More precisely, a generalized Fermat curve of type (k,3) is a closed Riemann surface S admitting a group H<Aut(S), called a generalized Fermat group of type (k,3), so that H≅Zk3 and S/H is an orbifold with signature (0,4;k,k,k,k). In this paper we prove the uniqueness of generalized Fermat groups of type (k,3). In particular, this allows the explicit computation of the full group of automorphisms of S.

Derivation of the Evans Wave Equation from the Lagrangian and Action: Origin of the Planck Constant in General Relativity

The Evans wave equation is derived from the appropriate Lagrangian and action, identifying the origin of the Planck constant ħ in general relativity. The classical Fermat principle of least time, and the classical Hamilton principle of least action, are expressed in terms of a tetrad multiplied by a phase factor exp(iS/ħ), where S is the action in general relativity. Wave (or quantum) mechanics emerges from these classical principles of general relativity for all matter and radiation fields, giving a unified theory of quantum mechanics based on differential geometry and general relativity. The phase factor exp(iS/ħ) is an eigenfunction of the Evans wave equation and is the origin in general relativity and geometry of topological phase effects in physics, including the Aharonov-Bohm class of effects, the Berry phase, the Sagnac effect, related interferometric effects, and all physical optical effects through the Evans spin field B(3) and the Stokes theorem in differential geometry. The Planck constant ħ is thus identified as the least amount possible of action or angular momentum or spin in the universe. This is also the origin of the fundamental Evans spin field B(3), which is always observed in any physical optical effect. It originates in torsion, spin and the second (or spin) Casimir invariant of the Einstein group. Mass originates in the first Casimir invariant of the Einstein group. These two invariants define any particle.

On a Fermat principle in general relativity. A Ljusternik-Schnirelmann theory for light rays

In this paper existence and multiplicity results for lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds are proved under intrinsic assumptions. Such results are obtained using an extension to Lorentzian Geometry of the classical Fermat principle in optics. The results are proved using critical point theory on infinite dimensional manifolds. An application to the gravitational lens effect is presented.

Shortening null geodesics in Lorentzian manifolds. Applications to closed light rays

We show the existence of one spatially closed lightlike geodesic on a regular, conformally stationary Lorentzian manifold M , having a non-contractible, light-convex, timelike cylinder C. The result is obtained by using an extension of the classical Fermat's Principle in optics, proven in [2], and a shortening argument similar to that used in [21] for studying the existence of closed geodesics on Riemannian manifolds with boundary.

Fermat–Euler Theorem in Algebraic Number Fields

In this paper a (maximal) generalization of the classical Fermat–Euler theorem for finite commutative rings with identity is proved. Maximal means that we show how to extend the original Fermat–Euler theorem to all of the elements of such rings with the best possible choice of exponents. The proofs are based on an idempotent technique of Schwarz. The results are then applied to Dedekind's ringsRsatisfying the following finiteness condition:[formula]Further specialization of proved results to some special cases of Dedekind's ringsRdepends then upon a good detailed knowledge of the structure of the group of units of the corresponding residue class ringR/I. The most known prototypes of such rings are besides Znthe algebraic number fields. Amongst these the simplest cases represent the quadratic fields, including one of their oldest representatives, the ring of the Gaussian integers.

A Fermat principle for stationary space-times and applications to light rays

We present an extension of the classical Fermat principle in optics to stationary space-times. This principle is applied to study the light rays joining an event with a timelike curve. Existence and multiplicity results of light rays are proved. Moreover, Morse Relations relating the set of rays to the topology of the space-time are obtained, by using the number of conjugate points of the ray. The results hold also for stationary space-times with boundary, in particular the Kerr space-time outside the stationary limit surface.

A general approach to the existence of minimizers of one-dimensional non-coercive integrals of the calculus of variations

We present a general approach to get the existence of minimizers for a class of one-dimensional non-parametric integrals of the calculus of variations with non-coercive integrands. Motivated by the concrete applicative relevance of the problems, we extend the notion of solution to a class of locally absolutely continuous functions with generic boundary values. We extend by lower semi-continuity the functionals and we prove for them representation formulae. Assuming some structure conditions on the partial derivatives of the integrand, we obtain some Wloc1,∞a priori estimates that we use as a main tool to get existence. The results are then applied to get existence theorems for the classical Fermat’s problem and for recent optimal foraging models of behavioural ecology.