In this paper, we derived the parametric solution of Euler and Elkies, xyz(x+y+z) = a, in an elementary manner. In addition we proved there are infinitely many parametric solutions of Euler's and Elkies's family of solutions....
What is electron affinity. Learn its equation, examples, values, and change across a period & down a group in the periodic table. Also, learn its exceptions.
The wide applications of higher dimensional gravity and gauge/gravity duality have fuelled the search for new stationary solutions of the Einstein equation (possibly coupled to matter). In this topical review, we explain the mathematical foundations...
In this paper, we propose a mean score equation-based approach to estimate the the volume under the receiving operating characteristic (ROC) surface (VUS) of a diagnostic test, under nonignorable (NI) verification bias. The proposed approach involves...
We analyze combined Quasi-Monte Carlo quadrature and Finite Element approximations in Bayesian estimation of solutions to countably-parametric operator equations with holomorphic dependence on the parameters as considered in [Cl.~Schillings and Ch.~S...
This paper is a sequel to the author's earlier work and investigates the homogeneous complex Monge--Ampere equation (HCMA) on the product space $X \times D$, where $X$ is an asymptotically locally Euclidean (ALE) Kahler manifold and $D subset C$ is t...
Few-nucleon correlations in nuclear matter at finite densities and temperatures are explored. Using the Dyson equation approach leads to effective few-body equations that include self energy corrections and Pauli blocking factors in a systematic wa...
Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (Archilla, 1998). However, recently I have shown one way that centre manifold theory may be...
Factorial Schur functions are generalizations of Schur functions that have, in addition to the usual variables, a second family of "shift" parameters. We show that a factorial Schur function times a deformation of the Weyl denominator may be expresse...
Neuronal dynamics is driven by externally imposed or internally generated random excitations/noise, and is often described by systems of random or stochastic ordinary differential equations. Such systems admit a distribution of solutions, which is (p...
Deep learning has been proposed as an efficient alternative for the numerical approximation of PDE solutions, offering fast, iterative simulation of PDEs through the approximation of solution operators. However, deep learning solutions have struggle...
For an electron, a spin-1/2 particle, the spin charge $\mathbf{s}$, a real pseudovector with constant length, could determine the spin polarization properties in quantum mechanics. Since spin density $ρ_{\mathbf{s}}$ could be expressed as the prod...
I compare three methods for transmitting signals over interstellar distances: radio transmitters, lasers and artificial transits. The quantitative comparison is based on physical quantities depending on energy cost and transmitting time L, the last p...
The Hardy-Weinberg equation relies on the absence of sexual selection, the absence of genetic flow or mutation, and a large population such that the probabilities are equal to the frequencies.
Hardy-Weinberg law, an algebraic equation that describes the genetic equilibrium within a population. It was discovered independently in 1908 by Wilhelm Weinberg, a German physician, and Godfrey …
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Online scientific calculator from MathDA: perform calculations with basic math to equation solver, integral, derivative, matrix, vector, statistics...
Ascertaining the core-collapse supernova mechanism is a complex, and yet unsolved, problem dependent on the interaction of general relativity, hydrodynamics, neutrino transport, neutrino-matter interactions, and nuclear equations of state and reactio...
Fokker-Planck equations (forward Kolmogorov equations) evolve probability densities in time from an initial condition. For distributions over the real line, these evolution equations can sometimes be transformed into dynamics over the incomplete zero...
In earlier work, we derived a nonlinear, nonlocal Fokker-Planck equation for the Yard-Sale Model of asset exchange. In the absence of redistribution, we showed that the Gini coefficient is a Lyapunov functional for this model, tending to one in the t...
In recent work, Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of th...
We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservati...
We point out incorrect equations derived in a paper published in this journal Ref. [1] (E. O. Silva, Eur. Phys. J. Plus (2018) 133 : 530) for the Klein-Gordon equation with the Aharonov-Bohm and Coulomb potentials in a Gödel-type space-time. We deri...
We introduce a generalization of team semantics which provides a framework for manipulationist theories of causation based on structural equation models, such as Woodward's and Pearl's; our causal teams incorporate (partial or total) information abou...
A class of optimal control problems governed by semilinear parabolic equations with mixed constraints and a box constraint for control variable is considered. We show that if the separation condition is satisfied, then both optimality conditions of K...
A class of parametric optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is investigated. The perturbations appear in the objective functional, the state equation and in mixed pointwise constraints. B...
Recent observations of neutron stars (NS) with radio, X-rays and gravitational waves have begun to constrain the equation of state (EoS) for nuclear matter beyond the nuclear saturation density. To one's surprise, there exist approximate universal re...
In this paper, we investigate conformal Killing's vectors (CKVs) admitted by some plane symmetric spacetimes. Ten conformal Killing's equations and their general forms of CKVs are derived along with their conformal factor. The existence of conformal...
This article reviews the use of differential forms and Lie derivatives to find symmetries of differential equations, as originally presented by Harrison and Estabrook, J. Math. Phys., 12 (1971), 653. An outline of the method is given, followed by e...
In this paper, we introduce a new extended version of the shallow water equations with surface tension which is skew-symmetric with respect to the L2 scalar product and allows for large gradients of fluid height. This result is a generalization of th...
Symplectic integrators offer vastly superior performance over traditional numerical techniques for conservative dynamical systems, but their application to \emph{dissipative} systems is inherently difficult due to dissipative systems' lack of symplec...
We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of usin...
Following on our previous work [S. Delong and B. E. Griffith and E. Vanden-Eijnden and A. Donev, Phys. Rev. E, 87(3):033302, 2013], we develop temporal integrators for solving Langevin stochastic differential equations that arise in fluctuating hydro...
A generalization of the kinetic equation is proposed for explaining observed shapes of wind wave spectra. The approach allows to fix a critical uncertainty in modeling wind wave spectra using a condition of equilibrium of nonlinear transfer and wave...
A skew meadow is a non-commutative ring with an inverse operator satisfying two special equations and in which the inverse of zero is zero. All skew fields and products of skew fields can be viewed as skew meadows. Conversely, we give an embedding...
A meadow is a zero totalised field (0^{-1}=0), and a cancellation meadow is a meadow without proper zero divisors. In this paper we consider differential meadows, i.e., meadows equipped with differentiation operators. We give an equational axiomati...
We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $Π=\{x=(x_{h}, x_3)\ |\ |x_{h} |>1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymm...
The properties of linear instabilities in the Large Plasma Device [W. Gekelman et al., Rev. Sci. Inst., 62, 2875 (1991)] are studied both through analytic calculations and solving numerically a system of linearized collisional plasma fluid equations...
I've been working on replacing standard O(n²) self-attention with something based on wave equation dynamics. Wanted to share the journey because the debugging process might be interesting to peop...
Wanted to share a PyTorch implementation of an alternative attention mechanism based on wave physics. How it works: 1. QKV projection (standard) 2. Bilinear scatter — deposit values onto a co...