Cylindrical Algebraic Sub-Decompositions

Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic sub-decompositions (sub-CADs), which are subsets of CADs containing all the information needed to specify a solution for a given problem. We define two new types of sub-CAD: variety sub-CADs which are those cells in a CAD lying on a designated variety; and layered sub-CADs which have only those cells of dimension higher than a specified value. We present algorithms to produce these and describe how the two approaches may be combined with each other and the recent theory of truth-table invariant CAD. We give a complexity analysis showing that these techniques can offer substantial theoretical savings, which is supported by experimentation using an implementation in Maple.Comment: 26 page

Diversity in Parametric Families of Number Fields

Let X be a projective curve defined over Q and t a non-constant Q-rational function on X of degree at least 2. For every integer n pick a point P_n on X such that t(P_n)=n. A result of Dvornicich and Zannier implies that, for large N, among the number fields Q(P_1),...,Q(P_N) there are at least cN/\log N distinct, where c>0. We prove that there are at least N/(\log N)^{1-c} distinct fields, where c>0.Comment: Minor inaccuracies detected by the referees are correcte

Advanced water iodinating system

Potable water stores aboard manned spacecraft must remain sterile. Suitable sterilization techniques are needed to prevent microbial growth. The development of an advanced water iodinating system for possible application to the shuttle orbiter and other advanced spacecraft, is considered. The AWIS provides a means of automatically dispensing iodine and controlling iodination levels in potable water stores. In a recirculation mode test, simulating application of the AWIS to a water management system of a long term six man capacity space mission, noniodinated feed water flowing at 32.2 cu cm min was iodinated to 5 + or - ppm concentrations after it was mixed with previously iodinated water recirculating through a potable water storage tank. Also, the AWIS was used to successfully demonstrate its capability to maintain potable water at a desired I2 concentration level while circulating through the water storage tank, but without the addition of noniodinated water

Open circular billiards and the Riemann hypothesis

A comparison of escape rates from one and from two holes in an experimental container (e.g. a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeroes of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.Comment: 5 pages, 4 embedded postscript figures; v2: more explicit on how the Reimann Hypothesis arises from a comparison of one and two hole escape rate

Composite fermion model for entanglement spectrum of fractional quantum Hall states

We show that the entanglement spectrum associated with a certain class of strongly correlated many-body states --- the wave functions proposed by Laughlin and Jain to describe the fractional quantum Hall effect --- can be very well described in terms of a simple model of non-interacting (or weakly interacting) composite fermions.Comment: 6 pages, 2 figure

Evaluation and characterization of the methane-carbon dioxide decomposition reaction

A program was conducted to evaluate and characterize the carbon dioxide-methane (CO2-CH4) decomposition reaction, i.e., CO2 + CH4 = 2C + 2H2O. The primary objective was to determine the feasibility of applying this reaction at low temperatures as a technique for recovering the oxygen (O2) remaining in the CO2 which exits mixed with CH4 from a Sabatier CO2 reduction subsystem (as part of an air revitalization system of a manned spacecraft). A test unit was designed, fabricated, and assembled for characterizing the performance of various catalysts for the reaction and ultraviolet activation of the CH4 and CO2. The reactor included in the test unit was designed to have sufficient capacity to evaluate catalyst charges of up to 76 g (0.17 lb). The test stand contained the necessary instrumentation and controls to obtain the data required to characterize the performance of the catalysts and sensitizers tested: flow control and measurement, temperature control and measurement, product and inlet gas analysis, and pressure measurement. A product assurance program was performed implementing the concepts of quality control and safety into the program effort

A nullstellensatz for sequences over F_p

Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l < p. The critical case l=p is of particular interest. In this context, we prove that whenever l=p and A is nonconstant, the above equation has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The subcritical case l=p-1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.Comment: 23 page

Non‐Homogeneous Cubic Equations

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135443/1/jlms0657.pd

Spinful Composite Fermions in a Negative Effective Field

In this paper we study fractional quantum Hall composite fermion wavefunctions at filling fractions \nu = 2/3, 3/5, and 4/7. At each of these filling fractions, there are several possible wavefunctions with different spin polarizations, depending on how many spin-up or spin-down composite fermion Landau levels are occupied. We calculate the energy of the possible composite fermion wavefunctions and we predict transitions between ground states of different spin polarizations as the ratio of Zeeman energy to Coulomb energy is varied. Previously, several experiments have observed such transitions between states of differing spin polarization and we make direct comparison of our predictions to these experiments. For more detailed comparison between theory and experiment, we also include finite-thickness effects in our calculations. We find reasonable qualitative agreement between the experiments and composite fermion theory. Finally, we consider composite fermion states at filling factors \nu = 2+2/3, 2+3/5, and 2+4/7. The latter two cases we predict to be spin polarized even at zero Zeeman energy.Comment: 17 pages, 5 figures, 4 tables. (revision: incorporated referee suggestions, note added, updated references