Equation of state and singularities in FLRW cosmological models

We consider FLRW cosmological models with standard Friedmann equations, but leaving free the equation of state. We assume that the dark energy content of the universe is encoded in an equation of state p = f(ρ), which is expressed with most generality in the form of a power expansion. The inclusion of this expansion in Friedmann equations allows us to construct a perturbative solution and to relate the coefficients of the equation of state with the formation of singularities of different types

Cosmological singularities and modified theories of gravity

We consider perturbative modifications of the Friedmann equations in terms of density corresponding to modified theories of gravity proposed as an alternative route to comply with the observed accelerated expansion of the universe. Assuming that the present matter content of the universe is a pressureless fluid, the possible singularities that may arise as the final state of the universe are surveyed. It is shown that, at most, two coefficients of the perturbative expansion of the Friedman equations are relevant for the analysis. Some examples of application of the perturbative scheme are included

Non-singular developable triangular Bézier patches

We show a characterisation of developable surfaces in the form of B´ezier triangular patches. • Constructions used for rectangular patches are not useful, since they produce degenerate triangular patches. • Explicit constructions of non-degenerate developable triangular patches are provided. • Interpolation of a developable triangle between a curve c(u), the last ruling and initial velocity of the other bounding curve d(u)

Past Singularities in Phantom Theories

FLRW models filled with just dark energy are shown to have a finite past, since causal geodesics cannot be extended beyond a certain proper time. It is shown that curvature measured along causal geodesics becomes infinity on travelling to the past, though curvature scalars tend to zero. Furthermore the time measured by free-falling observers from coincidence time to Big Rip is shown to be as short as wished by increasing their linear momentum

Docencia no presencial como alternativa a la clase magistral en los primeros cursos de ingeniería

La implantación de los nuevos grados en la Universidad Politécnica de Madrid ha supuesto la reducción del número de créditos destinados a las asignaturas básicas. En las titulaciones anteriores se dedicaban numerosas horas prácticas de matemáticas a la resolución de problemas en el aula por parte de los alumnos, con el apoyo de profesores y materiales. Estas tutorías colectivas tienen difícil encaje en nuestras nuevas titulaciones, dado que las asignaturas de matemáticas no sólo se han reducido en número de créditos, sino que también se ha reducido el número de horas presenciales de trabajo de los alumnos para facilitar su aprendizaje autónomo. Nuestro objetivo es, pues, conciliar una disminución de las horas presenciales con el aprendizaje autónomo, manteniendo las clases participativas de titulaciones anteriores. Para ello, hemos recurrido a nuestra experiencia previa de la enseñanza no presencial: los Cursos Masivos Online en Abierto (MOOC). En ellos, la clase magistral se desmenuza en vídeos y presentaciones cortas, de unos pocos minutos de duración, las llamadas píldoras educativas, al final de las cuales el alumno tiene que responder a unas pocas preguntas que muestren que ha captado las ideas que se desea transmitir

Clase magistral y docencia no presencial en matemáticas en los primeros cursos de los grados de ingeniería

En esta comunicación se analizan los problemas detectados en la docencia de asignaturas de matemática aplicada en primeros cursos de los nuevos grados en ingeniería y se propone una iniciativa piloto en curso tendente a solventarlos., basada en el aprendizaje mixto y el uso de píldoras educativas, al estilo de los MOOC

Triangular Bézier Developable Patches

Developable surfaces are defined as zero gaussian curvature surfaces (intrinsically flat). That is, plane patches that are curved by just folding, rolling or cutting, but without stretching or combing. Useful for depicting steel plates in naval industry, cloth in textile industry. . . But they are difficult to include in the NURBS formulation for the zero curvature requirement

Superficies desarrollables racionales a trozos

Con esta charla se pretende revisar avances recientes en el estudio de las superficies desarrollables NURBS, en particular, la extensión a superficies racionales del formalismo desarrollado para superficies desarrollables por el autor. Aparte de la inclusión de pesos en la parametrización de las superficies desarrollables, este formalismo permite tratar con facilidad problemas como la elevación del grado de la superficie desarrollable

w-cosmological singularities

In this paper we characterize barotropic index singularities of homogeneous isotropic cosmological models [M. P. Da̧browski and T. Denkiewicz, Phys. Rev. D 79, 063521 (2009).]. They are shown to appear in cosmologies for which the scale factor is analytical with a Taylor series in which the linear and quadratic terms are absent. Though the barotropic index of the perfect fluid is singular, the singularities are weak, as it happens for other models for which the density and the pressure are regular

Singularities around w=-1

Astronomical observations of luminosity distances derived from Type Ia supernovae, CMB spectrum and global matter distribution provide evidence of cosmic speed up of the Universe. Alternatively, cosmic acceleration might be due to an exotic fluid filling the Universe, known as dark energy. These have given rise to a collection of new cosmological evolutions, future singularites being the most perplexing ones (“big rip”, “sudden singularities”. . .)