The real tau‐conjecture is true on average

Koiran's real τ‐conjecture claims that the number of real zeros of a structured polynomial given as a sum of m products of k real sparse polynomials, each with at most t monomials, is bounded by a polynomial in mkt. This conjecture has a major consequence in complexity theory since it would lead to superpolynomial lower bounds for the arithmetic circuit size of the permanent. We confirm the conjecture in a probabilistic sense by proving that if the coefficients involved in the description of f are independent standard Gaussian random variables, then the expected number of real zeros of f is (mk2t).EC/H2020/787840/EU/Complexity and Condition in Algebra and Numerics/COCANTU Berlin, Open-Access-Mittel – 202

Sqrt{shat}_{min} resurrected

We discuss the use of the variable sqrt{shat}_{min}, which has been proposed in order to measure the hard scale of a multi parton final state event using inclusive quantities only, on a SUSY data sample for a 14 TeV LHC. In its original version, where this variable was proposed on calorimeter level, the direct correlation to the hard scattering scale does not survive when effects from soft physics are taken into account. We here show that when using reconstructed objects instead of calorimeter energy and momenta as input, we manage to actually recover this correlation for the parameter point considered here. We furthermore discuss the effect of including W + jets and t tbar+jets background in our analysis and the use of sqrt{shat}_{min} for the suppression of SM induced background in new physics searches.Comment: 23 pages, 9 figures; v2: 1 figure, several subsections and references as well as new author affiliation added. Corresponds to published versio

Non-integrability of the mixmaster universe

We comment on an analysis by Contopoulos et al. which demonstrates that the governing six-dimensional Einstein equations for the mixmaster space-time metric pass the ARS or reduced Painlev\'{e} test. We note that this is the case irrespective of the value, $I$, of the generating Hamiltonian which is a constant of motion. For $I < 0$ we find numerous closed orbits with two unstable eigenvalues strongly indicating that there cannot exist two additional first integrals apart from the Hamiltonian and thus that the system, at least for this case, is very likely not integrable. In addition, we present numerical evidence that the average Lyapunov exponent nevertheless vanishes. The model is thus a very interesting example of a Hamiltonian dynamical system, which is likely non-integrable yet passes the reduced Painlev\'{e} test.Comment: 11 pages LaTeX in J.Phys.A style (ioplppt.sty) + 6 PostScript figures compressed and uuencoded with uufiles. Revised version to appear in J Phys.

A network centrality method for the rating problem

We propose a new method for aggregating the information of multiple reviewers rating multiple products. Our approach is based on the network relations induced between products by the rating activity of the reviewers. We show that our method is algorithmically implementable even for large numbers of both products and consumers, as is the case for many online sites. Moreover, comparing it with the simple average, which is mostly used in practice, and with other methods previously proposed in the literature, it performs very well under various dimension, proving itself to be an optimal trade--off between computational efficiency, accordance with the reviewers original orderings, and robustness with respect to the inclusion of systematically biased reports.Comment: 25 pages, 8 figure

On the Physics of the Riemann Zeros

We discuss a formal derivation of an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicates that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and discuss a physical interpretation of this result within the Hilbert-Polya approach. In this context we also outline a relation between string theory and the Riemann Hypothesis.Comment: 8 pages, LaTeX, Quantum Theory and Symmetries 6 conference proceeding

Computational Complexity and Phase Transitions

Phase transitions in combinatorial problems have recently been shown to be useful in locating "hard" instances of combinatorial problems. The connection between computational complexity and the existence of phase transitions has been addressed in Statistical Mechanics and Artificial Intelligence, but not studied rigorously. We take a step in this direction by investigating the existence of sharp thresholds for the class of generalized satisfiability problems defined by Schaefer. In the case when all constraints are clauses we give a complete characterization of such problems that have a sharp threshold. While NP-completeness does not imply (even in this restricted case) the existence of a sharp threshold, it "almost implies" this, since clausal generalized satisfiability problems that lack a sharp threshold are either 1. polynomial time solvable, or 2. predicted, with success probability lower bounded by some positive constant by across all the probability range, by a single, trivial procedure.Comment: A (slightly) revised version of the paper submitted to the 15th IEEE Conference on Computational Complexit

A Multi-Agent Model of Tax Evasion with Public Expenditure

We develop a model where heterogeneous agents maximize their individual utility based on (after tax) income and on the level of public expenditure (as in Cowell, Gordon, 1988). Agents are different in risk aversion and in the relative preference for public expenditure with respect to personal income. In each period, an agent can optimally conceal some income based on conjectures on the perceived probability of being subject to audits, the perceived level of public expenditure and the perceived amount of tax paid by other individuals. As far as the agent-based model is concerned, we assume that the Government sets the tax rate and the penalties, uses all the revenue to finance public expenditure (with no inefficiency) and fights evasion by controlling a (random) fraction of agents. We show that, through computational experiments based on micro-simulations, stable configurations of tax rates and public expenditure endogenously form in this case as well. In such equilibrium-like situations we find: • a positive relationship between the tax rate and evasion still arises. • tax compliance mainly depends on the distribution of personal features like risk-aversion and the degree of preference for public expenditure. • an endogenous level of tax evasion that is almost not affected by reasonable rates of control. A proper choice of the tax rate results instead in voluntary partial compliance. • the enforcement of higher compliance rates requires unrealistic and costly large-scale audits.Tax evasion, public expenditure, agent-based models