Transduction of Automatic Sequences and Applications

We consider the implementation of the transduction of automatic sequences, and their generalizations, in the Walnut software for solving decision problems in combinatorics on words. We provide a number of applications, including (a) representations of n! as a sum of three squares (b) overlap-free Dyck words and (c) sums of Fibonacci representations. We also prove results about iterated running sums of the Thue-Morse sequence

An elementary proof of Hilbert's theorem on ternary quartics

In 1888, Hilbert proved that every non-negative quartic form f=f(x,y,z) with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. Up to now, no elementary proof is known. Here we present a completely new approach. Although our proof is not easy, it uses only elementary techniques. As a by-product, it gives information on the number of representations f=p_1^2+p_2^2+p_3^2 of f up to orthogonal equivalence. We show that this number is 8 for generically chosen f, and that it is 4 when f is chosen generically with a real zero. Although these facts were known, there was no elementary approach to them so far.Comment: 26 page

On Some Conjectures in Additive Number Theory

In the first part of the thesis we prove that every sufficiently large odd integer can be written as a sum of a prime and 2 times a product of at most two distinct odd primes. Together with Chen\u27s theorem and Ross\u27s observation, we know every sufficiently large integer can be written as a sum of a prime and a square-free number with at most three prime divisors, which improves a theorem by Estermann that every sufficiently large integer can be written as a sum of a prime and a square-free number. In the second part of the thesis we prove some results that specialize to confirm some conjectures of Sun, which are related to Fermat\u27s theorem on sums of two squares and other representations of primes in arithmetic progressions that can be represented by quadratic forms. The proof uses the equidistribution of primes in imaginary quadratic fields

Kronecker\u27s Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares

In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker\u27s paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence. In the second chapter we introduce the class number, proper class number and complete class number as well as two refinements, which facilitate the development of a connection with binary quadratic forms. Our third chapter is devoted to deriving several class number formulas in terms of divisors of the determinant. This chapter also contains lower bounds on the class number for bilinear forms and classifies when these bounds are attained. Lastly, we use the class number formulas to rigorously develop Kronecker\u27s connection between binary bilinear forms and binary quadratic forms. We supply purely arithmetic proofs of five results stated but not proven in the original paper. We conclude by giving an application of this material to the number of representations of an integer as a sum of three squares and show the resulting formula is equivalent to the well-known result due to Gauss

Symmetry groups, semidefinite programs, and sums of squares

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete symmetries in semidefinite programs. It is shown that symmetry exploitation allows a significant reduction in both matrix size and number of decision variables. This result is applied to semidefinite programs arising from the computation of sum of squares decompositions for multivariate polynomials. The results, reinterpreted from an invariant-theoretic viewpoint, provide a novel representation of a class of nonnegative symmetric polynomials. The main theorem states that an invariant sum of squares polynomial is a sum of inner products of pairs of matrices, whose entries are invariant polynomials. In these pairs, one of the matrices is computed based on the real irreducible representations of the group, and the other is a sum of squares matrix. The reduction techniques enable the numerical solution of large-scale instances, otherwise computationally infeasible to solve.Comment: 38 pages, submitte

The Small Number System

I argue that the human mind includes an innate domain-specific system for representing precise small numerical quantities. This theory contrasts with object-tracking theories and with domain-general theories that only make use of mental models. I argue that there is a good amount of evidence for innate representations of small numerical quantities and that such a domain-specific system has explanatory advantages when infants’ poor working memory is taken into account. I also show that the mental models approach requires previously unnoticed domain-specific structure and consequently that there is no domain-general alternative to an innate domain-specific small number system