Tabulation of cubic function fields via polynomial binary cubic forms

We present a method for tabulating all cubic function fields over $\mathbb{F}_q(t)$ whose discriminant $D$ has either odd degree or even degree and the leading coefficient of $-3D$ is a non-square in $\mathbb{F}_{q}^*$, up to a given bound $B$ on the degree of $D$. Our method is based on a generalization of Belabas' method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires $O(B^4 q^B)$ field operations as $B \rightarrow \infty$. The algorithm, examples and numerical data for $q=5,7,11,13$ are included.Comment: 30 pages, minor typos corrected, extra table entries added, revamped complexity analysis of the algorithm. To appear in Mathematics of Computatio

Universal Factorization of 3n−j(j>2)3n-j (j > 2) Symbols of the First and Second Kinds for SU(2) Group and Their Direct and Exact Calculation and Tabulation

We show that general $3n-j (n>2)$ symbols of the first kind and the second kind for the group SU(2) can be reformulated in terms of binomial coefficients. The proof is based on the graphical technique established by Yutsis, et al. and through a definition of a reduced $6-j$ symbol. The resulting $3n-j$ symbols thereby take a combinatorial form which is simply the product of two factors. The one is an integer or polynomial which is the single sum over the products of reduced $6-j$ symbols. They are in the form of summing over the products of binomial coefficients. The other is a multiplication of all the triangle relations appearing in the symbols, which can also be rewritten using binomial coefficients. The new formulation indicates that the intrinsic structure for the general recoupling coefficients is much nicer and simpler, which might serves as a bridge for the study with other fields. Along with our newly developed algorithms, this also provides a basis for a direct, exact and efficient calculation or tabulation of all the $3n-j$ symbols of the SU(2) group for all range of quantum angular momentum arguments. As an illustration, we present teh results for the $12-j$ symbols of the first kind.Comment: Add tables and reference

Polynomials with prescribed bad primes

We tabulate polynomials in Z[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and infinity. We explain how these sets of polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification. Finally we discuss the similar but technically more complicated tabulation problem corresponding to removing the auxiliary conditions.Comment: 26 pages, 3 figure

N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces

We establish a correspondence between generalized quiver gauge theories in four dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups as exemplified by those which arise in Moonshine. We analyze in detail the case of index 24, where modular elliptic K3 surfaces emerge: here, the elliptic j-invariants can be recast as dessins d'enfant which dictate the Seiberg-Witten curves.Comment: 42+1 pages, 5 figures; various helpful comments incorporate

Hardward and algorithm architectures for real-time additive synthesis

Additive synthesis is a fundamental computer music synthesis paradigm tracing its origins to the work of Fourier and Helmholtz. Rudimentary implementation linearly combines harmonic sinusoids (or partials) to generate tones whose perceived timbral characteristics are a strong function of the partial amplitude spectrum. Having evolved over time, additive synthesis describes a collection of algorithms each characterised by the time-varying linear combination of basis components to generate temporal evolution of timbre. Basis components include exactly harmonic partials, inharmonic partials with time-varying frequency or non-sinusoidal waveforms each with distinct spectral characteristics. Additive synthesis of polyphonic musical instrument tones requires a large number of independently controlled partials incurring a large computational overhead whose investigation and reduction is a key motivator for this work. The thesis begins with a review of prevalent synthesis techniques setting additive synthesis in context and introducing the spectrum modelling paradigm which provides baseline spectral data to the additive synthesis process obtained from the analysis of natural sounds. We proceed to investigate recursive and phase accumulating digital sinusoidal oscillator algorithms, defining specific metrics to quantify relative performance. The concepts of phase accumulation, table lookup phase-amplitude mapping and interpolated fractional addressing are introduced and developed and shown to underpin an additive synthesis subclass - wavetable lookup synthesis (WLS). WLS performance is simulated against specific metrics and parameter conditions peculiar to computer music requirements. We conclude by presenting processing architectures which accelerate computational throughput of specific WLS operations and the sinusoidal additive synthesis model. In particular, we introduce and investigate the concept of phase domain processing and present several “pipeline friendly” arithmetic architectures using this technique which implement the additive synthesis of sinusoidal partials