Bouncing Branes

Two classical scalar fields are minimally coupled to gravity in the Kachru-Shulz-Silverstein scenario with a rolling fifth radius. A Tolman wormhole solution is found for a R x S^3 brane with Lorentz metric and for a R x AdS_3 brane with positive definite metric.Comment: 6 pages, LaTe

Cyclic Universe with an Inflationary Phase from a Cosmological Model with Real Gas Quintessence

Phase-plane stability analysis of a dynamical system describing the Universe as a two-fraction fluid containing baryonic dust and real virial gas quintessence is presented. Existence of a stable periodic solution experiencing inflationary periods is shown. A van der Waals quintessence model is revisited and cyclic Universe solution again found.Comment: 21 pages, 8 figures, to appear in Physical Review

Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes

Presented is a two-tier analysis of the location of the real roots of the general quartic equation $x^4 + ax^3 + bx^2 + cx + d = 0$ with real coefficients and the classification of the roots in terms of $a$, $b$, $c$, and $d$, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and solving these, together with some of the resolvent quadratic equations, allows the full classification of the roots of the general quartic and also the determination of the isolation interval of each root. These isolation intervals involve the stationary points of the quartic (among others) and, by solving some of the resolvent quadratic equations, the isolation intervals of the stationary points of the quartic are also determined. Each possible case has been carefully studied and illustrated with a detailed figure containing a description of its specific characteristics, analysis based on solving cubic equations and analysis based on solving quadratic equations only. As the analysis of the roots of the quartic equation is done by studying the intersection points of the "sub-quartic" $x^4 + ax^3 + bx^2$ with a set of suitable parallel lines, a beautiful Pythagorean analogy can be found between these intersection points and the set of parallel lines on one hand and the musical notes and the staves representing different musical pitches on the other: each particular case of the quartic equation has its own short tune

New Bounds on the Real Polynomial Roots

The presented analysis determines several new bounds on the roots of the equation $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0$ (with $a_n > 0$). All proposed new bounds are lower than the Cauchy bound max$\{1, \sum_{j=0}^{n-1} |a_j/a_n| \}$. Firstly, the Cauchy bound formula is derived by presenting it in a new light -- through a recursion. It is shown that this recursion could be exited at earlier stages and, the earlier the recursion is terminated, the lower the resulting root bound will be. Following a separate analysis, it is further demonstrated that a significantly lower root bound can be found if the summation in the Cauchy bound formula is made not over each one of the coefficients $a_0, a_1, \ldots, a_{n-1}$, but only over the negative ones. The sharpest root bound in this line of analysis is shown to be the larger of 1 and the sum of the absolute values of all negative coefficients of the equation divided by the largest positive coefficient. The following bounds are also found in this paper: max$\{ 1, ( \sum_{j = 1}^{q} B_j/A_l )^{1/(l-k)}\}$, where $B_1, B_2, \ldots B_q$ are the absolute values of all of the negative coefficients in the equation, $k$ is the highest degree of a monomial with a negative coefficient, $A_l$ is the positive coefficient of the term $A_l x^l$ for which $k< l \le n$

New Bounds on the Real Polynomial Roots

The presented analysis determines several new bounds on the roots of the equation $a_n x^n + a_{n−1} x^{n−1} + · · · + a_0 = 0$ (with $a_n \u3e 0$). All proposed new bounds are lower than the Cauchy bound max $\{ 1, sum_{j=0}^{n-1} | a_j / a_n | \}$. Firstly, the Cauchy bound formula is derived by presenting it in a new light — through a recursion. It is shown that this recursion could be exited at earlier stages and, the earlier the recursion is terminated, the lower the resulting root bound will be. Following a separate analysis, it is further demonstrated that a significantly lower root bound can be found if the summation in the Cauchy bound formula is made not over each one of the coefficients $a_0, a_1, . . . , a_{n−1}$, but only over the negative ones. The sharpest root bound in this line of analysis is shown to be the larger of 1 and the sum of the absolute values of all negative coefficients of the equation divided by the largest positive coefficient. The following bounds are also found in this paper: max $\{ 1, (sum_{j=1}^{q} | B_j / A_l |)^{1/(l-k)} \}$ where $B_1, B_2, . . . B_q$ are all of the negative coefficients in the equation, $k$ is the highest degree of a monomial with a negative coefficient, $A_l$ is the positive coefficient of the term $A_l x^l$ for which $k \u3c l \le n$

On the Cubic Equation with its Siebeck--Marden--Northshield Triangle and the Quartic Equation with its Tetrahedron

The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck--Marden--Northshield equilateral triangle and regular tetrahedron. The Vi\`ete trigonometric formulae for the roots of the cubic are established through the rotation of the triangle by variation of the free term of the cubic. A very detailed complete root classification for the quartic $x^4 + ax^3 + bx^2 + cx + d$ is proposed for which the conditions are imposed on the individual coefficients $a$, $b$, $c$, and $d$. The maximum and minimum lengths of the interval containing the four real roots of the quartic are determined in terms of $a$ and $b$. The upper and lower root bounds for a quartic with four real roots are also found: no root can lie farther than $(\sqrt{3}/4)\sqrt{3a^2 - 8b}\,$ from $-a/4$. The real roots of the quartic are localized by finding intervals containing at most two roots. The end-points of these intervals depend on $a$ and $b$ and are roots of quadratic equations -- which makes this localization helpful for quartic equations with complicated parametric coefficients.Comment: 29 pages, 4 figure

On the Cosmological Models with Matter Creation

The matter creation model of Prigogine--Géhéniau--Gunzig--Nardone is revisited in terms of a redefined creation pressure which does not lead to irreversible adiabatic evolution at constant specific entropy. With the resulting freedom to choose a particular gas process, a flat FRWL cosmological model is proposed based on three input characteristics: (i) a perfect fluid comprising of an ideal gas, (ii) a quasi-adiabatic polytropic process, and (iii) a particular rate of particle creation. Such model leads to the description of the late-time acceleration of the expanding Universe with a natural transition from decelerating to accelerating regime. Only the Friedmann equations and the laws of thermodynamics are used and no assumptions of dark energy component is made. The model also allows the explicit determination as functions of time of all variables, including the entropy, the non-conserved specific entropy and the time the accelerating phase begins. A form of correspondence with the dark energy models (quintessence, in particular) is established via the Om diagnostics. Parallels with the concordance cosmological ΛCDM model for the matter-dominated epoch and the present epoch of accelerated expansion are also established via slight modifications of both models

A remark on Schwarz's topological field theory

The standard evaluation of the partition function $Z$ of Schwarz's topological field theory results in the Ray--Singer analytic torsion. Here we present an alternative evaluation which results in Z=1. Mathematically, this amounts to a novel perspective on analytic torsion: it can be formally written as a ratio of volumes of spaces of differential forms which is formally equal to 1 by Hodge duality. An analogous result for Reidemeister combinatorial torsion is also obtained.Comment: 9 pages, latex, to appear in Lett. Math. Phy