We initiate the study of the relationship between two complexity classes, BQP (BoundedError Quantum Polynomial-Time) and PPAD (Polynomial Parity Argument, Directed). We first give a conjecture that PPAD is contained in BQP, and show a necessary and sufficient condition for the conjecture to hold. Then we prove that the conjecture is not true under the oracle model. In the end, we raise some int...

We show that the universal Gröbner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph.

This paper gives a general survey of ranks of elliptic curves over the field of rational numbers. The rank is a measure of the size of the set of rational points. The paper includes discussions of the Birch and SwinnertonDyer Conjecture, the Parity Conjecture, ranks in families of quadratic twists, and ways to search for elliptic curves of large rank.

For code pairs (A, B); A, B C {0, 1 ..... 7-1}"; with mutually constant parity of the Hamming distances a conjecture of the first author concerning the maximal value of IAIIBI is established.

We discuss the mass spectrum of highly-excited nucleon and ∆ * resonances. The spectrum exhibits parity doublets, pairs of resonances of identical total angular momentum J but of opposite parity. It has been proposed that the parity doublets evidence restoration of chiral symmetry at large baryon excitation energies. We compare this conjecture with the possibility that high-mass states are orga...

We show that the parity conjecture for Selmer groups is invariant under deformation in p-adic families of self-dual pure Galois representations satisfying Pančǐskin’s condition at all primes above p. 2000 Mathematics Subject Classification: 11G40 11R23

In this article, we study the Iwasawa theory for Hilbert modular forms over anticyclotomic extension of a CM field. We prove an one-sided divisibility result toward main conjecture in setting. The proof relies on first and second reciprocity laws relating theta elements to Heegner point Euler systems Shimura curves. As by-product also towards rank 0 case certain Bloch–Kato parity conjecture.

Abstract. We establish the volume conjecture for (m, 2)-cables of the figure 8 knot, when m is odd. For (m, 2)-cables of general knots where m is even, we show that the limit in the volume conjecture depends on the parity of the color (of the Kashaev invariant). There are many cases when the volume conjecture for cables of the figure 8 knot is false if one considers all the colors, but holds tr...

Two players share a connected graph with non-negative weights on the vertices. They alternately take the vertices (one in each turn) and collect their weights. The rule they have to obey is that the remaining part of the graph must be connected after each move. We conjecture that the first player can get at least half of the weight of any tree with an even number of vertices. We provide a strat...