We study the packing of a large number congruent and non-overlapping circles inside regular polygon. have devised efficient algorithms that allow one to generate configurations N densely packed polygon, we carried out intensive numerical experiments spanning several polygons (the largest sides considered here being 16) up 200 (400 in special cases equilateral triangle hexagon). Some found possibly are not global maxima fraction, particularly for N≫1, due great computational complexity...

We show that it is possible to obtain numerical solutions quantum mechanical problems involving a fractional Laplacian, using collocation approach based on little sinc functions, which discretizes the Schrödinger equation uniform grid. The different boundary conditions are naturally implemented sets of functions with appropriate behavior. Good convergence properties observed. A comparison results Wentzel–Kramers–Brillouin analysis performed.

In this paper we present a straightforward systematic method for the exact and approximate calculation of integrals that appear in formulas period anharmonic oscillators other problems interest classical mechanics.

Using a variational approach, two new series representations for the incomplete Gamma function are derived: first is an asymptotic series, which contains and improves over standard expansion; second uniformly convergent completely analytical, can be used to obtain arbitrarily accurate estimates of $\Gamma(a,x)$ any value $a$ or $x$. Applications these formulas discussed.

In this paper we discuss a new method which can be used to obtain arbitrarily accurate analytical expressions for the deflection angle of light propagating in given metric. Our works by mapping integral into rapidly convergent series and provides extremely approximations already first order. We have derived general order formula generic spherically symmetric static metric tensor tested it four different cases.

I calculate the modes of vibration membranes arbitrary shape using a collocation approach based on little sinc functions. The matrix representation PDE obtained this method is explicit and does not require calculation integrals. To illustrate virtues approach, have considered large number examples, part them are taken from literature, new. When possible, tested accuracy these results by comparing with exact (when available) or literature. In particular, in case L-shaped membrane, first...

In this paper we study the effect of finite size on two-flavor color superconducting state. As well as restricting quarks to a box, project onto states good baryon number and singlets, these being necessary restrictions any observable ``quark nuggets.'' We find that whereas alone has significant for very small boxes, with state often destroyed, projection is restore it again. The infinite-volume limit approximation even quite systems.

We have devised a variational sinc collocation method (VSCM) which can be used to obtain accurate numerical solutions many strong-coupling problems. Sinc functions with an optimal grid spacing are solve the linear and non-linear Schr\"odinger equations lattice $\phi^4$ model in $(1+1)$. Our results indicate that errors decrease exponentially number of points limited effort is needed reach high precision.

We discuss two alternative methods, based on the Lindstedt–Poincaré technique, for removal of secular terms from equations perturbation theory. calculate period an anharmonic oscillator by means both approaches and show that one them is more accurate all values coupling constant. believe present discussion comparison may be a suitable exercise teaching theory in advanced undergraduate courses classical mechanics.

We extend to higher order a recently published method for calculating the deflection angle of light in general static and spherically symmetric metric. have tested our on metric Schwarzschild Reissner-Nordstr\"om black holes, charged hole coupled Born-Infeld electrodynamics Weyl gravity. Since is geometrically convergent, as proved previous work, analytical formulas obtained working fourth are sufficient reach errors few percents even proximity photon sphere.

We analyze two conditionally solvable quantum-mechanical models: a one-dimensional sextic oscillator and perturbed Coulomb problem. Both lead to three-term recurrence relation for the expansion coefficients. show diagrams of distribution their exact eigenvalues with addition accurate ones from variational calculations. discuss symmetry such distributions. also comment on wrong interpretation eigenfunctions by some researchers that has led prediction allowed cyclotron frequencies field intensities.

We develop powerful numerical and analytical techniques for the solution of Helmholtz equation on general domains. prove two theorems: first theorem provides an exact formula ground state arbitrary membrane, while second generalizes this result to any excited membrane. also a systematic perturbative scheme which can be used study small deformations membrane circular or square shapes. discuss several applications, obtaining results.

We analyze the Rayleigh equation for collapse of an empty bubble and provide explanation some recent analytical approximations to model. derive form singularity at second boundary point discuss convergence approximants. also give a rigorous proof asymptotic behavior coefficients power series that are basis approximate expressions.

We have applied the Lindstedt-Poincar\'e method to study limit cycle of van der Pol oscillator, obtaining numerical coefficients series for period and amplitude order $859$. Hermite-Pad\'e approximants been used extract location branch cut with unprecendented accuracy ($100$ digits). Both then resummed using an approach based on Pad\'e approximants, where exact asymptotic behaviors are taken into account. Our results improve drastically all previous obtained this subject.

In this paper we study a new family of sinc--like functions, defined on an interval finite width. These which call ``little sinc'', are orthogonal and share many the properties sinc functions. We show that little functions supplemented with variational approach enable one to obtain accurate results for variety problems. apply them interpolation domain solution Schr\"odinger equation, compare performance present others.

We propose a method for the treatment of two--point boundary value problems given by nonlinear ordinary differential equations. The approach leads to sequences roots Hankel determinants that converge rapidly towards unknown parameter problem. treat several physical interest: field equation determining vortex profile in Ginzburg--Landau effective theory, fixed--point Wilson's exact renormalization group, suitably modified Wegner--Houghton's fixed point local potential approximation, Riccati...

We develop a nonperturbative method that yields analytical expressions for the deflection angle of light in general static and spherically symmetric metric. The works by introducing into problem an artificial parameter, called $\ensuremath{\delta}$, performing expansion this parameter to given order. results obtained are because they do not correspond polynomial expression physical parameters. Already first order $\ensuremath{\delta}$ formulas using our provide at same time accurate...