arXiv subjects

Sandor Kristyan

Publications and source records attributed to Sandor Kristyan.

11 recordsLinked to original sources

arXiv2024

On the Simple Divisibility Restrictions by Polynomial Equation a n+bn=cn Itself in Fermat Last Theorem for Integer/Complex/Quaternion Triples

The divisibility restrictions in the famous equation a n+bn=cn in Fermat Last Theorem (FLT, 1637) is analyzed how it selects out many triples to be Fermat triple (i.e. solutions) if n greater than 2, decreasing the cardinality of Fermat triples. In our analysis, the restriction on positive integer (PI) solutions ((a,b,c,n) up to the point when there is no more) is not along with restriction on power n in PI as decreasing sets {PI } containing {odd} containing {primes} containing {regular primes}, etc. as in the literature, but with respect to exclusion of more and more c in PI as increasing sets {primes p} in {p k} in {PI}. The divisibility and co-prime property in Fermat equation is analyzed in relation to exclusion of solutions, and the effect of simultaneous values of gcd(a,b,c), gcd(a+b,cn), gcd(c-a,bn) and gcd(c-b,an) on the decrease of cardinality of solutions is exhibited. Again, our derivation focuses mainly on the variable c rather than on variable n, oppositely to the literature in which the FLT is historically separated via the values of power n. Among the most famous are the known, about 2500 years old, existing Pythagorean triples (a,b,c,n=2) and the first milestones as the proved cases (of non-existence as n=3 by Gauss and later by Euler (1753) and n=4 by Fermat) less than 400 years ago. As it is known, Wiles has proved the FLT in 1995 in an abstract roundabout way. The n<0, n:=1/m, as well as complex and quaternion (a,b,c) cases focusing on Pythagoreans are commented. Odd powers FLT over quaternions breaks.

math.GM↗

arXiv2024

Generalization of Cantor Pairing Polynomials (Bijective Mapping Among Natural Numbers) from N02 to N0 to Z2 to N0 and N03 to N

The Cantor pairing polynomials are extended to larger 2D sub-domains and more complex mapping, of which the most important property is the bijectivity. If corners are involved inside (but not the borders of) domain, more than one connected polynomials are necessary. More complex patterns need more complex subsequent application of math series to obtain the mapping polynomials which are more and more inconvenient, although elementary. A tricky polynomial fit is introduced (six coefficients are involved like in the original Cantor polynomials with rigorous but simple restrictions on points chosen) to buy out the regular treatment of math series to find the pairing polynomials instantly. The original bijective Cantor polynomial C1(x,y)= (x2+2xy+y2+3x+y)/2: N02 to N0 (=positive integers) which is 2-fold and runs in zig - zag way along lines x+y=N is extended e.g. to the bijective P(x,y)= 2x2+4sgn(x)sgn(y)xy+2y2-2H(x)sgn(y)x-y+1: Z2 to N0 (with sign and Heaviside functions, Z is integers) running in spiral way along concentric rhombuses, or to the bijective P3D(x,y,z)= [x3+y3+z3 +3(xz2+yz2 +zx2+2xyz +zy2+yx2+xy2) +3(2x2+2y2 +z2+2xz +2yz+4xy) +5x+11y+2z]/6: N03 to N0 which is 6-fold and runs along plains x+y+z=N. Storage device for triangle matrices is also commented as cutting the original Cantor domain to half along with related Diophantine equations.

math.GM↗

arXiv2019

Solving the non-relativistic electronic Schrodinger equation with manipulating the coupling strength parameter over the electron-electron Coulomb integrals

The non-relativistic electronic Hamiltonian, H(a)= Hkin + Hne + aHee, extended with coupling strength parameter (a), allows to switch the electron-electron repulsion energy off and on. First, the easier a=0 case is solved and the solution of real (physical) a=1 case is generated thereafter from it to calculate the total electronic energy (Etotal electr,K) mainly for ground state (K=0). This strategy is worked out with utilizing generalized Moller-Plesset (MP), square of Hamiltonian (H2) and Configuration interactions (CI) devices. Applying standard eigensolver for Hamiltonian matrices (one or two times) buys off the needs of self-consistent field (SCF) convergence in this algorithm, along with providing the correction for basis set error and correlation effect. (SCF convergence is typically performed in the standard HF-SCF/basis/a=1 routine in today practice.)

physics.chem-ph↗

arXiv2019

Quasi-linear buildup of Coulomb integrals via the coupling strength parameter in the non-relativistic electronic Schrodinger equation

The non-relativistic electronic Hamiltonian, Hkin + Hne + aHee, is linear in coupling strength parameter (a), but its eigenvalues (electronic energies) have only quasi-linear dependence on it. Detailed analysis is given on the participation of electron-electron repulsion energy (Vee) in total electronic energy (Etotal electr,k) in addition to the well-known virial theorem and standard algorithm for vee(a=1)=Vee calculated during the standard- and post HF-SCF routines. Using a particular modification in the SCF part of the Gaussian package, we have analyzed the ground state solutions via the parameter a. Technically, with a single line in the SCF algorithm, operator was changed as 1/rij-> a/rij with input a. The most important findings are, 1, vee(a) is quasi-linear function of a, 2, the extension of 1st Hohenberg-Kohn theorem (PSI0(a=1)<=>Hne<=>Y0(a=0)) and its consequences in relation to a. The latter allows an algebraic transfer from the simpler solution of case a=0 (where the single Slater determinant Y0 is the accurate form) to the physical case a=1. Moreover, we have generalized the emblematic Hund rule, virial-, Hohenberg-Kohn- and Koopmans theorems in relation to the coupling strength parameter.

physics.chem-ph↗

arXiv2019

Semi-analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian expansion of Distance Operators W= R$_{C1}^{-n}$R$_{D1}^{-m}$, R$_{C1}^{-n}$r$_{12}^{-m}$, r$_{12}^{-n}$r$_{13}^{-m}$

The equations derived help to evaluate semi-analytically (mostly for k=1,2 or 3) the important Coulomb integrals Int rho(r1)...rho(rk) W(r1,...,rk) dr1...drk, where the one-electron density, rho(r1), is a linear combination (LC) of Gaussian functions of position vector variable r1. It is capable to describe the electron clouds in molecules, solids or any media/ensemble of materials, weight W is the distance operator indicated in the title. R stands for nucleus-electron and r for electron-electron distances. The n=m=0 case is trivial, the (n,m)=(1,0) and (0,1) cases, for which analytical expressions are well known, are widely used in the practice of computation chemistry (CC) or physics, and analytical expressions are also known for the cases n,m=0,1,2. The rest of the cases - mainly with any real (integer, non-integer, positive or negative) n and m - needs evaluation. We base this on the Gaussian expansion of |r|^-u, of which only the u=1 is the physical Coulomb potential, but the u.ne.1 cases are useful for (certain series based) correction for (the different) approximate solutions of Schrodinger equation, for example, in its wave-function corrections or correlation calculations. Solving the related linear equation system (LES), the expansion |r|^-u about equal to SUM(k=0toL)SUM(i=1toM) Cik r^2k exp(-Aik r^2) is analyzed for |r| = r12 or RC1 with least square fit (LSF) and modified Taylor expansion. These evaluated analytic expressions for Coulomb integrals (up to Gaussian function integrand and the Gaussian expansion of |r|^-u) are useful for the manipulation with higher moments of inter-electronic distances via W, even for approximating Hamiltonian.

physics.chem-ph↗

arXiv2018

Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, $R_{C1}^{-n}R_{D1}^{-m}$, $R_{C1}^{-n}r_{12}^{-m}$ and $r_{12}^{-n}r_{13}^{-m}$ with real $(n,m$) and the Descartes product of 3 dim. common density functional numerical integration scheme

Analytical solutions to integrals are far more useful than numeric, however, the former is not available in many cases. We evaluate integrals indicated in the title numerically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the n, m= 0 case is trivial, the (n, m)= (1,0) or (0,1) cases are well known, a fundamental milestone in the integration and widely used in computational quantum chemistry, as well as analytical integration is possible if Gaussian functions are used. For the rest of the cases the analytical solutions are restricted, but worked out for some, e.g. for n, m= 0,1,2 with Gaussians. In this work we generalize the Becke-Lebedev-Voronoi 3 dimensions numerical integration scheme (commonly used in density functional theory) to 6 and 9 dimensions via Descartes product to evaluate integrals indicated in the title, and test it. This numerical recipe (up to Gaussian integrands with seed exp(-|r1|2), as well as positive and negative real n and m values) is useful for manipulation with higher moments of inter-electronic distances, for example, in correlation calculations; more, our numerical scheme works for Slaterian type functions with seed exp(-|r1|) as well.

physics.chem-ph↗

arXiv2018

Optimizing single Slater determinant for Electronic Hamiltonian with Lagrange multipliers and Newton-Raphson methods as an alternative to ground state calculations via Hartree-Fock self consistent field

Considering the emblematic Hartree-Fock (HF) energy expression with single Slater determinant and the ortho-normal molecular orbits (MO) in it, expressed as a linear combination (LC) of atomic orbits (LCAO) basis set functions, the HF energy expression is in fact a 4th order polynomial of the LCAO coefficients, which is relatively easy to handle. The energy optimization via the Variation Principle can be made with a Lagrange multiplier method to keep the ortho-normal property and the Newton-Raphson (NR) method to find the function minimum. It is an alternative to the widely applied HF self consistent field (HF-SCF) method which is based on unitary transformations and eigensolver during the SCF, and seems to have more convenient convergence property. This method is demonstrated for closed shell (even number of electrons and all MO are occupied with both, alpha and beta spin electrons) and restricted (all MOs have single individual spatial orbital), but the extension of the method to open shell and/or unrestricted cases is straightforward.

physics.chem-ph↗

arXiv2017

Eigenvalue equations from the field of theoretical chemistry and correlation calculations

Many equations have been introduced and derived by the author indicated in the title in relation to multi-electron densities between the Hohenberg-Kohn theorems and variational principle, conversion of the non-relativistic electronic Schrodinger equation to scaling correct moment functional of ground state one-electron density to estimate ground state electronic energy, participation of electron-electron repulsion energy operator in the non-relativistic electronic Schrodinger equation via the coupling strength parameter along with generalizing the Hund rule, the emblematic theorems virial-, Moller-Plesset-, Hohenberg-Kohn-, Koopmans-, Brillouin theorem and configuration interactions formalism, and with analytic evaluation of Coulomb integrals for one, two and three-electron operators, as well as focusing on ground state electronic energy, correlation energy and zero point energy of stationer molecular systems.

physics.chem-ph↗

arXiv2017

Analytic evaluation of Coulomb integrals for one, two and three-electron distance operators, $R_{C1}^{-n}R_{D1}^{-m}$, $R_{C1}^{-n}r_{12}^{-m}$ and $r_{12}^{-n}r_{13}^{-m}$ with $n, m=0,1,2$

The state of the art for integral evaluation is that analytical solutions to integrals are far more useful than numerical solutions. We evaluate certain integrals analytically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the $(n,m)=(0,0)$ case is trivial, the $(n,m)=(1,0)$ and (0,1) cases are well known, fundamental milestone in integration and widely used in computation chemistry, as well as based on Laplace transformation with integrand exp(-$a^2t^2$). The rest of the cases are new and need the other Laplace transformation with integrand exp(-$a^2t$) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations.

physics.chem-ph↗

arXiv2017

Generalization of Brillouin theorem for the non-relativistic electronic Schrödinger equation in relation to coupling strength parameter, and its consequences in single determinant basis sets for configuration interactions

The Brillouin theorem has been generalized for the extended non-relativistic electronic Hamiltonian (Hkin+ Hne+ aHee) in relation to coupling strength parameter (a), as well as for the configuration interactions (CI) formalism in this respect. For a computation support, we have made a particular modification of the SCF part in the Gaussian package: essentially a single line was changed in an SCF algorithm, wherein the operator rij-1 was overwritten as 1/rij to a/rij, and a was used as input. The case a=0 generates an orto-normalized set of Slater determinants which can be used as a basis set for CI calculations for the interesting physical case a=1, removing the known restriction by Brillouin theorem with this trick. The latter opens a door from the theoretically interesting subject of this work toward practice.

physics.chem-ph↗

arXiv2017

On the statistical distribution of prime numbers, a view from where the distribution of prime numbers is not erratic

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples.

math.GM↗