Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA fast modulo primes algorithm for searching perfect cuboids and its implementation
78
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
A perfect cuboid is a rectangular parallelepiped whose all linear extents are given by integer numbers, i. e. its edges, its face diagonals, and its space diagonal are of integer lengths. None of perfect cuboids is known thus far. Their non-existence is also not proved. This is an old unsolved mathematical problem. Three mathematical propositions have been recently associated with the cuboid problem. They are known as three cuboid conjectures. These three conjectures specify three special subcases in the search for perfect cuboids. The case of the second conjecture is associated with solutions of a tenth degree Diophantine equation. In the present paper a fast algorithm for searching solutions of this Diophantine equation using modulo primes seive is suggested and its implementation on 32-bit Windows platform with Intel-compatible processors is presented.
rate research
Read More
Let $p>3$ be a prime, and let $(frac{cdot}p)$ be the Legendre symbol. Let $binmathbb Z$ and $varepsilonin{pm 1}$. We mainly prove that $$left|left{N_p(a,b): 1<a<p text{and} left(frac apright)=varepsilonright}right|=frac{3-(frac{-1}p)}2,$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with ${x^2+b}_p>{ax^2+b}_p$, and ${m}_p$ with $minmathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.
Recently, Bruinier and Ono classified cusp forms $f(z) := sum_{n=0}^{infty} a_f(n)q ^n in S_{lambda+1/2}(Gamma_0(N),chi)cap mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-C
ohen Bracket, we extend this result to modular forms of half integral weight for primes $p geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $pgeq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newmans conjecture for overpartitions.
We propose an adiabatic quantum algorithm capable of factorizing numbers, using fewer qubits than Shors algorithm. We implement the algorithm in an NMR quantum information processor and experimentally factorize the number 21. Numerical simulations indicate that the running time grows only quadratically with the number of qubits.
Let $K$ be a number field, let $phi in K(t)$ be a rational map of degree at least 2, and let $alpha, beta in K$. We show that if $alpha$ is not in the forward orbit of $beta$, then there is a positive proportion of primes ${mathfrak p}$ of $K$ such that $alpha mod {mathfrak p}$ is not in the forward orbit of $beta mod {mathfrak p}$. Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $alpha$ by a hypersurface, such as the ramification locus of a morphism $phi : {mathbb P}^{n} to {mathbb P}^{n}$.
Collision between rigid three-dimensional objects is a very common modelling problem in a wide spectrum of scientific disciplines, including Computer Science and Physics. It spans from realistic animation of polyhedral shapes for computer vision to the description of thermodynamic and dynamic properties in simple and complex fluids. For instance, colloidal particles of especially exotic shapes are commonly modelled as hard-core objects, whose collision test is key to correctly determine their phase and aggregation behaviour. In this work, we propose the OpenMP Cuboid Sphere Intersection (OCSI) algorithm to detect collisions between prolate or oblate cuboids and spheres. We investigate OCSIs performance by bench-marking it against a number of algorithms commonly employed in computer graphics and colloidal science: Quick Rejection First (QRI), Quick Rejection Intertwined (QRF) and SIMD Streaming Extensions (SSE). We observed that QRI and QRF significantly depend on the specific cuboid anisotropy and sphere radius, while SSE and OCSI maintain their speed independently of the objects geometry. While OCSI and SSE, both based on SIMD parallelization, show excellent and very similar performance, the former provides a more accessible coding and user-friendly implementation as it exploits OpenMP directives for automatic vectorization.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
