On the van der Waerden numbers w(2;3,t)

We present results and conjectures on the van der Waerden numbers w(2;3,t) and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. Motivated by such reglarities, we introduce *palindromic van der Waerden numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good partitions), defined as reading the same from both ends. Different from the situation for ordinary van der Waerden numbers, these "numbers" need actually to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide lower bounds, which we conjecture to be exact, for t <= 35. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory. Especially we introduce a novel (open-source) SAT solver, the tawSolver, which performs best on the SAT instances studied here, and which is actually the original DLL-solver, but with an efficient implementation and a modern heuristic typical for look-ahead solvers (applying the theory developed in the SAT handbook article of the second author).Comment: Second version 25 pages, updates of numerical data, improved formulations, and extended discussions on SAT. Third version 42 pages, with SAT solver data (especially for new SAT solver) and improved representation. Fourth version 47 pages, with updates and added explanation

Modified Whittle Estimation of Multilateral Models on a Lattice

In the estimation of parametric models for stationary spatial or spatio-temporal data on a d-dimensional lattice, for d >= 2, the achievement of asymptotic efficiency under Gaussianity, and asymptotic normality more generally, with standard convergence rate, faces two obstacles. One is the "edge effect", which worsens with increasing d. The other is the possible difficulty of computing a continuous-frequency form of Whittle estimate or a time domain Gaussian maximum likelihood estimate, due mainly to the Jacobian term. This is especially a problem in "multilateral" models, which are naturally expressed in terms of lagged values in both directions for one or more of the d dimensions. An extension of the discrete-frequency Whittle estimate from the time series literature deals conveniently with the computational problem, but when subjected to a standard device for avoiding the edge effect has disastrous asymptotic performance, along with finite sample numerical drawbacks, the objective function lacking a minimum-distance interpretation and losing any global convexity properties. We overcome these problems by first optimizing a standard, guaranteed non-negative, discrete-frequency, Whittle function, without edge-effect correction, providing an estimate with a slow convergence rate, then improving this by a sequence of computationally convenient approximate Newton iterations using a modified, almost-unbiased periodogram, the desired asymptotic properties being achieved after finitely many steps. The asymptotic regime allows increase in both directions of all d dimensions, with the central limit theorem established after re-ordering as a triangular array. However our work offers something new for "unilateral" models also. When the data are non-Gaussian, asymptotic variances of all parameter estimates may be affected, and we propose consistent, non-negative definite estimates of the asymptotic variance matrix.spatial data, multilateral modelling, Whittle estimation, edge effect, consistent variance estimation

Two-Photon Quantum Interference Polarization Spectroscopy: Measurements of Transition Matrix Elements in Atomic Rubidium

The estimation of the adequacy of theoretical calculations on the atomic structure requires availability of the precise experimental data on radiative properties of the atoms. Such data is also required in astronomy and some important areas of technology. The lack of precision of traditional spectroscopic studies of atom presents a fundamental obstacle for progress in these areas. For example, in atomic rubidium, the best precision of the traditional spectroscopic results is on the order of about 1 - 5%, which does not allow for clear assessment of the latest sophisticated theoretical calculations on atomic rubidium structure, with emphasis on different, in nature, effects. This situation is typical for atomic physics in general. The purpose of present study is obtaining the experimental data on the radiative properties of atomic rubidium with precision considerably higher than that of the traditional spectroscopic methods. This is accomplished by means of the two-photon quantum interference polarization spectroscopy. A two-photon polarization spectrum of the rubidium atom is obtained in the range of detunings -417 cm-1 to +99 cm-1 from atomic 5s2S1/2-5p 2P3/2-*8s2S1/2resonance. From analysis of the spectra the relativistic and many body effects on the wavefiinctions are revealed in the form of a uniquely defined parameter q = 2 x 10-6 (5) cm and an exact relation between parameters R and p which quantitatively describes the process: R = 1.01756 (57) + 81.466 (15) p where R is dimensionless and p is in cm. The obtained results can be thought of as specific experimentally established two-photon sum rules and can be used for testing the accuracy of the theoretical wavefiinctions. The experimental technique has important advantages comparing to some traditional spectroscopic methods and is essentially free of systematic effects

ON EQUIVALENCY REASONING FOR CONFLICT DRIVEN CLAUSE LEARNING SATISFIABILITY SOLVERS

Satisfiability problem or SAT is the problem of deciding whether a Boolean function evaluates to true for at least one of the assignments in its domain. The satisfiability problem is the first problem to be proved NP-complete. Therefore, the problems in NP can be encoded into SAT instances. Many hard real world problems can be solved when encoded efficiently into SAT instances. These facts give SAT an important place in both theoretical and practical computer science. In this thesis we address the problem of integrating a special class of equivalency reasoning techniques, the strongly connected components or SCC based reasoning, into the class of conflict driven clause learning or CDCL SAT solvers. Because of the complications that arise from integrating the equivalency reasoning in CDCL SAT solvers, to our knowledge, there has been no CDCL solver which has applied SCC based equivalency reasoning dynamically during the search. We propose a method to overcome these complications. The method is integrated into a prominent satisfiability solver: MiniSat. The equivalency enhanced MiniSat, Eq-MiniSat, is used to explore the advantages and disadvantages of the equivalency reasoning in conflict clause learning satisfiability solvers. Different implementation approaches for Eq-MiniSat are discussed. The experimental results on 16 families of instances shows that equivalency reasoning does not have noticeable effects for the instances in one family. The equivalency reasoning enables Eq-MiniSat to outperform MiniSat on eight classes of instances. For the remaining seven families, MiniSat outperforms Eq- MiniSat. The experimental results for random instances demonstrate that almost in all cases the number of branchings for Eq-Minisat is smaller than Minisat

Hilbert schemes: construction and pathologies

In questo elaborato viene presentata la costruzione degli schemi di Hilbert, attraverso le nozioni di polinomio di Hilbert, piattezza, cambiamento di schema di base e regolarità secondo Castelnuovo-Mumford. Vengono poi presentate alcune proprietà fondamentali di tali schemi, come il Teorema di Connessione di Hartshorne, alcuni esempi standard per concludere con una breve discussione di alcuni comportamenti inaspettati e la Legge di Murphy per schemi di Hilbert

Scattering Amplitudes and Form Factors in Effective Field Theories

The central theme of the thesis is the application of modern on-shell techniques to compute Scattering Amplitudes and Form Factors in various Effective Field Theories. In particular, we apply such techniques in the context of the Standard Model Effective Field Theories, focusing on the renormalisation group evolution of irrelevant operators, and the study of the classical binary problem in gravitational theories, beyond General Relativity including higher derivative interactions. We first show how to find a basis of EFT interactions from a purely on-shell point of view. From these EFT building blocks, any tree-level amplitude can be computed using a recursive algorithm which requires only the knowledge of lower-point amplitudes. Starting from these results, modern (generalised) unitarity techniques allow for the computations of higher loop amplitudes which can be used to characterise precision observables both for gravitational waves and for collider experiments. We will focus on the computation of form factors in the context of Standard Model Effective Field Theory which allowed us to compute for the first time the one-loop mixing matrix for all the dimension-eight operators in the theory. Then, we will show how to compute the deflection angle and the time delays induced by higher-derivative corrections to the Einstein-Hilbert action from the eikonal form of gravitational scattering amplitudes

Cache-Friendly, Modular and Parallel Schemes For Computing Subresultant Chains

The RegularChains library in Maple offers a collection of commands for solving polynomial systems symbolically with taking advantage of the theory of regular chains. The primary goal of this thesis is algorithmic contributions, in particular, to high-performance computational schemes for subresultant chains and underlying routines to extend that of RegularChains in a C/C++ open-source library. Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client procedure, not all polynomials of the chain, or not all coefficients of a given subresultant, may be needed. Based on that observation, we design so-called speculative and caching strategies which yield great performance improvements within our polynomial system solver. Our implementation of these techniques has been highly optimized. We have implemented optimized core arithmetic routines and multithreaded subresultant algorithms for univariate, bivariate and multivariate polynomials. We further examine memory access patterns and data locality for computing subresultants of multivariate polynomials, and study different optimization techniques for the fraction-free LU decomposition algorithm to compute subresultants based on determinant of Bezout matrices. Our code is publicly available at www.bpaslib.org as part of the Basic Polynomial Algebra Subprograms (BPAS) library that is mainly written in C, with concurrency support and user interfaces written in C++

Toward the nonmetal-to-metal phase transition of helium

Properties of helium under high pressure are of great importance for a variety of astrophysical objects. This work presents results of extensive ab initio calculations for high-pressure helium regarding the equation of state, the melting line, the electrical conductivity, the reflectivity, and the band gap. The calculations determined a higher-order phase transition from an insulating to a conducting phase which has been discussed for decades. This work finally proposes an update of the phase diagram of helium and discusses its implications on dwarf stars and giant planets.Die Eigenschaften von Helium unter hohem Druck sind wichtig für eine Vielzahl von astrophysikalischen Objekten. Diese Arbeit präsentiert die Ergebnisse umfangreicher Ab-initio-Berechnungen für Helium. Insbesondere wurde die Zustandsgleichung, die elektrische Leitfähigkeit, das Reflexionsvermögen und die Bandlücke berechnet. Die Ergebnisse weisen einen kontinuierlichen Phasenübergang zwischen isolierendem und elektrisch leitfähigem Helium nach. Schließlich zeigt diese Arbeit ein aktualisiertes Phasendiagramm von Helium und diskutiert die Konsequenzen für Zwergsterne und Riesenplaneten