We construct affine varieties over \mathbb{Q} and imaginary quadratic number fields \mathbb{K} with a finite of \alpha-lattice points for fixed \alpha\in \mathcal{O}_\mathbb{K}, where \mathcal{O}_\mathbb{K} denotes the ring algebraic integers \mathbb{K}. These arise from equations form F(y) = F(g(x_1,x_2,\ldots, x_k))+r(x_1,x_2\ldots, x_k), F is rational function, g r are polynomials \mathbb{K}...

A quantum theoretic representation of real and complex numbers is described here as equivalence classes of Cauchy sequences of quantum states of finite strings of qubits. There are 4 types of qubits each with associated single qubit annihilation creation (a-c) operators that give the state and location of each qubit type on a 2 dimensional integer lattice. The string states, defined as finite p...

Are human perception and decision biases grounded in a form of rationality? You return to your camp after hunting or gathering. You see the grass moving. You do not know the probability that a snake is in the grass. Should you cross the grass — at the risk of being bitten by a snake — or make a long, hence costly, detour? Based on this storyline, we consider a rational decision maker maximizing...

Let E/Q be an elliptic curve over the field of rational numbers, with EndQ̄(E) = Z. Let K be a fixed imaginary quadratic field over Q, and x a positive real number. For each prime p of good reduction for E, let πp(E) be a root of the characteristic polynomial of the Frobenius endomorphism of E over the finite field Fp. Let ΠE(K;x) be the number of primes p ≤ x such that the field extension Q(πp(...

They are sums of zeta functions for prehomogeneous vector spaces and generalizations of Epstein zeta functions. For the rational numbers and imaginary quadratic fields one can define these functions also for SL(2, ^-equivalence, which for convenience we call 1equivalence. The second function arises in the calculation of the Selberg trace formula for integral operators on L(PSL(2, (!?)\H) where ...

Rational exponential integrators (REXI) are a class of numerical methods that well suited for the time integration linear partial differential equations with imaginary eigenvalues. Since these can be parallelized in (in addition to spatial parallelization is commonly performed) they exploit modern high performance computing systems. In this paper, we propose novel REXI scheme drastically improv...

Let $K$ be a quadratic field which is not an imaginary of class number one. We describe algorithm to compute the primes $p$ for there exists elliptic curve over admitting $K$-rational $p$-isogeny. This builds on work David, Larson-Vaintrob, and Momose. Combining this with Bruin-Najman, \"{O}zman-Siksek, most recently Box, we determine above set three fields $\mathbb{Q}(\sqrt{-10})$, $\mathbb{Q}...

Let E/Q be an elliptic curve over the field of rational numbers, with EndQ̄ (E) = Z. Let K be a fixed imaginary quadratic field over Q, and x a positive real number. For each prime p of good reduction for E, let πp(E) be a root of the characteristic polynomial of the Frobenius endomorphism of E over the finite field Fp. Let ΠE(K; x) be the number of primes p ≤ x such that the field extension Q(π...