Speeding up Cylindrical Algebraic Decomposition by Gr\"obner Bases

Gr\"obner Bases and Cylindrical Algebraic Decomposition are generally thought of as two, rather different, methods of looking at systems of equations and, in the case of Cylindrical Algebraic Decomposition, inequalities. However, even for a mixed system of equalities and inequalities, it is possible to apply Gr\"obner bases to the (conjoined) equalities before invoking CAD. We see that this is, quite often but not always, a beneficial preconditioning of the CAD problem. It is also possible to precondition the (conjoined) inequalities with respect to the equalities, and this can also be useful in many cases.Comment: To appear in Proc. CICM 2012, LNCS 736

Reliability of Harvested Rainfall as an Auxiliary Source of Non-potable Water

AbstractRainwater harvesting is an ancient practice that involves collecting, storing, and using precipitationto meeton-site water needs. This paper develops and demonstrates guidelines for sizing the capacity of storage tanks to provide areliable continuous supply of harvested rainwater for residential households. Operation of the rainwater harvesting system is simulated with a stochastic mass balance performed on an Excel spreadsheet. The daily volume of rainwater in an unbounded tank is tracked to determine the maximum accumulated deficit on a monthly basis. Results are summarized in dimensionless charts showing the minimum size of a rainwater storage tankneeded to meet water demands at specified levels of reliability

Constructing Mutually Unbiased Bases in Dimension Six

The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex 6x6 Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.Comment: As published version. Added discussion of the impact of numerical approximations and corrected the number of triples existing for non-affine families (cf Table 3

Hipster: Integrating Theory Exploration in a Proof Assistant

This paper describes Hipster, a system integrating theory exploration with the proof assistant Isabelle/HOL. Theory exploration is a technique for automatically discovering new interesting lemmas in a given theory development. Hipster can be used in two main modes. The first is exploratory mode, used for automatically generating basic lemmas about a given set of datatypes and functions in a new theory development. The second is proof mode, used in a particular proof attempt, trying to discover the missing lemmas which would allow the current goal to be proved. Hipster's proof mode complements and boosts existing proof automation techniques that rely on automatically selecting existing lemmas, by inventing new lemmas that need induction to be proved. We show example uses of both modes

On elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation

The Conte-Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a priory restriction is to simplify calculations by means of the use of a few Laurent series solutions instead of one and the use of the residue theorem. The application of our approach to the quintic complex one-dimensional Ginzburg-Landau equation (CGLE5) allows to find elliptic solutions in the wave form. We also find restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions for the CGLE5. Using the investigation of the CGLE5 as an example, we demonstrate that to find elliptic solutions the analysis of a system of differential equations is more preferable than the analysis of the equivalent single differential equation.Comment: LaTeX, 21 page

A CDCL-style calculus for solving non-linear constraints

In this paper we propose a novel approach for checking satisfiability of non-linear constraints over the reals, called ksmt. The procedure is based on conflict resolution in CDCL style calculus, using a composition of symbolical and numerical methods. To deal with the non-linear components in case of conflicts we use numerically constructed restricted linearisations. This approach covers a large number of computable non-linear real functions such as polynomials, rational or trigonometrical functions and beyond. A prototypical implementation has been evaluated on several non-linear SMT-LIB examples and the results have been compared with state-of-the-art SMT solvers.Comment: 17 pages, 3 figures; accepted at FroCoS 2019; software available at <http://informatik.uni-trier.de/~brausse/ksmt/

Complotype affects the extent of down-regulation by Factor I of the C3b feedback cycle in vitro.

Sera from a large panel of normal subjects were typed for three common polymorphisms, one in C3 (R102G) and two in Factor H (V62I and Y402H), that influence predisposition to age-related macular degeneration and to some forms of kidney disease. Three groups of sera were tested; those that were homozygous for the three risk alleles; those that were heterozygous for all three; and those homozygous for the low-risk alleles. These groups vary in their response to the addition of exogenous Factor I when the alternative complement pathway is activated by zymosan. Both the reduction in the maximum amount of iC3b formed and the rate at which the iC3b is converted to C3dg are affected. For both reactions the at-risk complotype requires higher doses of Factor I to produce similar down-regulation. Because iC3b reacting with the complement receptor CR3 is a major mechanism by which complement activation gives rise to inflammation, the breakdown of iC3b to C3dg can be seen to have major significance for reducing complement-induced inflammation. These findings demonstrate for the first time that sera from subjects with different complement alleles behave as predicted in an in-vitro assay of the down-regulation of the alternative complement pathway by increasing the concentration of Factor I. These results support the hypothesis that exogenous Factor I may be a valuable therapeutic aid for down-regulating hyperactivity of the C3b feedback cycle, thereby providing a treatment for age-related macular degeneration and other inflammatory diseases of later life.This is the author's accepted manuscript and will be under embargo until the 13th of August 2015. This final version is available from Wiley in Clinical & Experimental Immunology at http://onlinelibrary.wiley.com/doi/10.1111/cei.12437/abstract

Exotic solutions in string theory

Solutions of classical string theory, correspondent to the world sheets, mapped in Minkowsky space with a fold, are considered. Typical processes for them are creation of strings from vacuum, their recombination and annihilation. These solutions violate positiveness of square of mass and Regge condition. In quantum string theory these solutions correspond to physical states |DDF>+|sp> with non-zero spurious component.Comment: accepted in Il Nuovo Cimento A for publication in 199