We provide a hierarchy of tree languages recognised by nondeterministic parity tree automata with priorities in {0, 1, 2}, whose length exceeds the first fixed point of the ε operation (that itself enumerates the fixed points of x 7→ ω). We conjecture that, up to Wadge equivalence, it exhibits all regular tree languages of index [0, 2].

We give infinite families of elliptic curves over Q such that each curve has infinitely many non-isomorphic quadratic twists of rank at least 4. Assuming the Parity Conjecture, we also give elliptic curves over Q with infinitely many non-isomorphic quadratic twists of odd rank at least 5.

Let f be a binary word and let Fd(f) be the set of words of length d which do not contain f as a factor (alias words that avoid the pattern f). A word is called even/odd if it contains an even/odd number of 1s. The parity index of f (of dimension d) is introduced as the difference between the number of even words and the number of odd words in Fd(f). A word f is called prime if every nontrivial...

On Threshold Circuits for Parity Ramamohan Paturi and Michael E. Saks* Department of Computer Science and Engineering University of California, San Diego; La Jolla, Ca 92093 Motivated by the problem of understanding the limitations of neural networks for representing Boolean functions, we consider sizedepth trade-offs for threshold circuits that compute the parity function. We give an almost op...

We consider the question of giving an asymptotic estimate for the number of odd parity quadratic twists of a fixed elliptic curve that have (analytic) rank greater than 1. The use of Waldspurger’s formula and modular forms of weight 3/2 has allowed the accrual of a large amount of data for the even parity analogue of this, and we use the method of Heegner points to fortify the data in the odd p...

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

Recently, Hirschhorn and the first author considered parity of function [Formula: see text] which counts number integer partitions wherein each part appears with odd multiplicity. They derived an effective characterization based solely on properties In this paper, we quickly reprove their result, then extend it to explicit for all We also exhibit some infinite families congruences modulo 2 foll...

Let p be a prime and F totally real field in which is unramified. We consider mod Hilbert modular forms for F, defined as sections of automorphic line bundles on varieties level to characteristic p. For Hecke eigenform arbitrary weight (without parity hypotheses), we associate two-dimensional representation the absolute Galois group give conjectural description set weights all eigenforms from i...

Let K / F be a finite Galois extension of number fields and let σ an absolutely irreducible, self-dual, complex valued representation Gal ( ) . p odd prime consider two elliptic curves E 1 , 2 defined over Q with good, ordinary reduction at primes above equivalent mod- representations. In this article, we study the variation parity multiplicities in space associated to ∞ -Selmer groups We also ...