Abstract:
We abstract the definition of the Costas property in the context of a group and study specifically denseCostas sets (named Costas clouds) in groups with the topological property that they are dense in themselves:as a result, we prove the existence of nowhere continuous dense bijections that satisfy the Costas property on ？2, ？2, and ？2, the latter two being based on nonlinear solutions of Cauchy's functional equation, as well as on？, ？, and ？, which are, in effect, generalized Golomb rulers. We generalize the Welch and Golomb constructionmethods for Costas arrays to apply on ？ and ？, and we prove that group isomorphisms on and tensor products of Costas sets result to new Costas sets with respect to an appropriate set of distance vectors. We also give two constructive examples of a nowhere continuous function that satisfies a constrained form of the Costas property (over rational or algebraic displacements only, i.e.), based on the indicator function of a dense subset of ？.

Abstract:
Costas array is a class of radar signal array with perfect correlation property. An open problem about the Costas array proposed by Golomb aud Taylor been solved in this paper.

Abstract:
A Costas latin square of order n is a set of n disjoint Costas arrays of the same order. Costas latin squares are studied here from a construction as well as a classification point of view. A complete classification is carried out up to order 27. In this range, we verify the conjecture that there is no Costas latin square for any odd order n >= 3. Various other related combinatorial structures are also considered, including near Costas latin squares (which are certain packings of near Costas arrays) and Vatican Costas squares.

Abstract:
We extend the definition of the Costas property to functions in the continuum, namely on intervals of the reals or the rationals, and argue that such functions can be used in the same applications as discrete Costas arrays. We construct Costas bijections in the real continuum within the class of piecewise continuously differentiable functions, but our attempts to construct a fractal-like Costas bijection there are successful only under slight but necessary deviations from the usual arithmetic laws. Furthermore, we are able, contingent on the validity of Artin's conjecture, to set up a limiting process according to which sequences of Welch Costas arrays converge to smooth Costas bijections over the reals. The situation over the rationals is different: there, we propose an algorithm of great generality and flexibility for the construction of a Costas fractal bijection. Its success, though, relies heavily on the enumerability of the rationals, and therefore it cannot be generalized over the reals in an obvious way.

Abstract:
We prove that the only Costas arrays that can be constructed by interlacing 2 Costas arrays of smaller orders (either equal or differing by 1) are those of order 2, and that, consequently, no non-trivial Costas arrays result from this method.

Abstract:
Hemos limitado nuestra atención al estudio de las pesquerías tradicionales modestas, realizadas con aparejos de anzuelo en las costas vascas y cantábricas. Las primeras noticias que de ellas nos llegan remontan al siglo XII y XIV, haciendo referencia a las pescas de besugos y cazones, siendo la más antigua una procedente de 1272 que menciona las pesqueras de congrios de Bayona. Recogemos información de los aparejos habitualmente usados para estas pescas, insistiendo en que, en un principio, se utilizaron sólamente aparejos de mano, y que la aparición de los primeros palangres de gran fondo es mucho más reciente de lo que se había pensado. Económicamente, tras la pesca del besugo y la merluza, se explotó la pesca de los entonces denominados peces de cuero (peque os escualos) y, con una intensidad muy superior a la actual, graciasa su fácil conservación, la de los congrios. Se exponen los aparejos utilizados para la captura de estas especies y algunas características de las maniobras que realizan nuestros pescadores, así como algunos modos de comercialización y conservación del pescado. He intentado, a la vez, recoger cuantas voces vascas se refieren al tema, muchas de ellas fonéticamente, ya hace medio siglo, y sus variantes locales. Asimismo las correspondientes voces castellanas.

Abstract:
We investigate the generalization of the Costas property in 3 or more dimensions, and we seek an appropriate definition; the 2 main complications are a) that the number of ``dots'' this multidimensional structure should have is not obvious, and b) that the notion of the multidimensional permutation needs some clarification. After proposing various alternatives for the generalization of the definition of the Costas property, based on the definitions of the Costas property in 1 or 2 dimensions, we also offer some construction methods, the main one of which is based on the idea of reshaping Costas arrays into higher-dimensional entities.

Abstract:
We study several criteria for the (non)linearity of Costas permutations, with or without the imposition of additional algebraic structure in the domain and the range of the permutation, aiming to find one that successfully identifies Costas permutations as more nonlinear than randomly chosen permutations of the same order. 1. Introduction Costas arrays, namely, square arrangements of dots and blanks such that there lies exactly one dot per row and column, and such that no four dots form a parallelogram and no three dots lying on a straight line are equidistant, appeared for the first time in 1965 in the context of SONAR detection [1, 2], when Costas, disappointed by the poor performance of SONAR systems, used them to describe a novel frequency hopping pattern for SONARs with optimal auto-correlation properties. About two decades later, Professor S. Golomb published two generation techniques [3–5] for Costas permutations, both based on the theory of finite fields, known as the Welch and the Golomb method, respectively. These are still the only general construction methods for Costas permutations available today. Despite the intensive mathematical research dedicated to Costas arrays in the last two decades, many key questions about them remain unresolved, and most notably the issue of their existence: do Costas arrays exist for all orders? There is currently no order known for which Costas arrays provably do not exist, while the two smallest orders for which no Costas arrays are known are 32 and 33 [3]. An interesting application of Costas arrays in cryptography was discovered when it was shown that Welch Costas arrays are Almost Perfect Nonlinear (APN) permutations [6]. This prompted further an investigation of the nonlinearity of Welch Costas permutations, in the sense defined in [7, 8], whereby Welch Costas permutations were interpreted as mappings on , the group of integers modulo , and were indeed shown to exhibit high nonlinearity among all such functions/permutations [9]. Costas permutations, however, are not defined over , but rather over , the set of the first nonnegative integers, on which no group structure is imposed. The object of this work is to investigate the correct interpretation and calculation of the (non)linearity of a Costas permutation, and, by extension, of any discrete function, in this context. What does it mean for a discrete function to be linear? How can the concept of linearity be quantified? Can this quantification benefit, in the case of functions on , from the fact that such functions can be extended to functions on ?

Abstract:
Costas arrays are not only useful in radar engineering, but they also present many interesting, and still open, mathematical problems. This work collects in it all important knowledge about them available today: some history of the subjects, density results, construction methods, construction algorithms with full proofs, and open questions. At the same time all the necessary mathematical background is offered in the simplest possible format and terms, so that this work can play the role of a reference for mathematicians and mathematically inclined engineers interested in the field.