I'm a 16 year old in the 10th grade and am interested in algorithms. I've looked in multiple places on the web but am unable to find a step by step breakdown of how this equation works. I want to …
Tangents are used to find the slope of a curve at a certain point, while normals are used to find the equation of a curve at a certain point. Understanding the difference between these two concepts is …
5 days ago · Tangents and Normals By nextgenmath.org March 2, 2026 One of the most practical applications of differentiation in Calculus is finding the equations of tangents and normals to a curve.
from Latin modulus, 'a measure'. Models can be divided into physical models (e.g. a ship model) and abstract models (e.g. a set of mathematical equations
Self-similar solution is obtained for propagation of a strong shock, in a flat expanding dusty Friedman universe. Approximate analytic solution was obtained earlier, using relation between self-similar variables, equivalent to the exact energy conser...
A machine-learned (ML) model is developed to enhance the accuracy of turbulence transport equations of Reynolds Averaged Navier Stokes (RANS) solver and applied for periodic hill test case, which involves complex flow regimes, such as attached bounda...
We have compiled list of matchstick puzzles in which you need to complete equation, move matchsticks to solve brain teasers or counting squares rectan
The self-consistency of the assumption of Reggeized form of the production amplitudes in multi-Regge kinematics, which are used in the derivation of the BFKL equation, leads to strong bootstrap conditions. The fulfillment of these conditions opens...
In this paper we present a theory of vessels and its application to the classical inverse scattering of the Sturm-Liouville differential equation. The classical inverse scattering theory, including all its ingredients: Jost solutions, the Gelfand-Lev...
In an earlier letter [Ducharme \textit{et al.} Phys. Rev. Lett. \textbf{126}, 134803 (2021)], a solution to the Dirac equation for a relativistic Gaussian electron beam showed that for a diverging beam the spin of each electron is the sum of fraction...
An infinite family of finite axisymmetric charged dust disks is presented. The disks are obtained by solving the Einstein-Maxwell equations for conformastatic spacetimes by assuming a functional dependency between the time-like component of the ele...
The estimate of coefficients of the Convection-Diffusion Equation (CDE) from experimental measurements belongs in the category of inverse problems, which are known to come with issues of ill-conditioning or singularity. Here we concentrate on a parti...
It is proposed a dimensional Lattice Boltzmann Method (LBM) of wide application for simulating fluid flow and heat transfer problems. The proposed LBM consists in the numerical solution of the discrete lattice Boltzmann equation (LBE) using directly...
Latent stochastic differential equation (SDE) models are important tools for the unsupervised discovery of dynamical systems from data, with applications ranging from engineering to neuroscience. In these complex domains, exact posterior inference of...
We examine the Langevin diffusion confined to a closed, convex domain $D\subset\mathbb{R}^d$, represented as a reflected stochastic differential equation. We introduce a sequence of penalized stochastic differential equations and prove that their inv...
We present a new method of establishing the finite-dimensionality of limit dynamics (in terms of bi-Lipschitz Mane projectors) for semilinear parabolic systems with cross diffusion terms and illustrate it on the model example of 3D complex Ginzburg-L...
We study the global attractors of abstract semilinear parabolic equations and their projections to finite-dimensional planes. It is well-known that the attractor can be embedded into the finite-dimensional inertial manifold if the so-called spectral...
The paper considers a slightly modified one-dimensional infinite mass-in-mass chain. In the case of the long-wave approximation, which corresponds to the transition to a continuous medium, we obtained a system of two equations, which is a generalizat...
This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of Stein's equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximat...
The $Λ$-term in Einstein's equations is a fundamental building block of the `concordance' $Λ$CDM model of cosmology. Even though the model is not free of fundamental problems, they have not been circumvented by any alternative dark energy proposal...
The cosmological constant term, $Λ$, in Einstein's equations has been for three decades a building block of the concordance or standard $Λ$CDM model of cosmology. Although the latter is not free of fundamental problems, it provides a good phenomeno...
The effects of pressure waves-flame collisions and tube aspect ratio on flame evolution and the formation of tulip and distorted tulip flames were investigated using numerical simulations of the fully compressible Navier-Stokes equations coupled with...
We investigate the Jordan-Brans-Dicke action in the cosmological scenario of FLRW spacetime with zero spatially curvature and with an extra scalar field minimally coupled to gravity as matter source. The field equations are studied in two ways. The m...
We investigate Kantowski-Sachs models in Einstein-æther theory with a perfect fluid source using the singularity analysis to prove the integrability of the field equations and dynamical system tools to study the evolution. We find an inflationary so...
For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of which is non-stiff or mildly stiff, and the other part is stiff. Such systems ca...
Foster and Hart proposed an operational measure of riskiness for discrete random variables. We show that their defining equation has no solution for many common continuous distributions including many uniform distributions, e.g. We show how to extend...
In 1908 Thue (1) showed that algebraic numbers of the special form $ξ=\sqrt[n]{\frac{a}{b}}$ can, for every positive $ε$, only be sharply approximated by finitely many rational numbers $\frac{p}{q}$ with the following inequality holding \[ \left|ξ...
Tsunamis are often generated by a moving sea bottom. This paper deals with the case where the tsunami source is an earthquake. The linearized water-wave equations are solved analytically for various sea bottom motions. Numerical results based on th...
We use Maxwell's equations in a sourceless, inhomogeneous medium with continuous permeability $μ(\mathbf{r}) $ and permittivity $% ε(\mathbf{r}) $ to study the wave propagation. The general form of the wave equation is derived and by virtue of some...
Using numerical modeling investigated interaction of solitary waves (solitons) of the regularized long wave equation. For reception the stable model of the nonlinear medium are used methods of the linear prediction and progressive approximation. By...
This erratum addresses a logical mistake in the author's article [Jakob, R. The Willmore flow of Hopf-tori in the $3$-sphere. Journal of Evolution Equations 23, No. 72 (2023)] which resulted in two wrong assertions in parts (II) and (III) of Theorem...
This paper investigates the inelastic response of a yielding structure coupled with a rocking wall which can be vertically restrained. The paper first derives the nonlinear equations of motion of a yielding oscillator coupled with a vertically restra...
Jul 31, 2017 · The argument I am faced with is whether we should use the adverb, similarly, by essentially implicitly encapsulating the entire process to arrive at a new equation as the verb; or …
Among the set of hypersurfaces of degree $d$ and dimension $\ell$ defined by the vanishing of a homogeneous polynomial with coefficients $\pm 1$, we investigate the probability that a hypersurface contains a rational point as $d$ and $\ell$ tend to i...
In the first of two papers, we study the initial boundary-value problem that underlies the theory of the Boltzmann equation for general non-spherical hard particles. In this work, for two congruent ellipses and for a large class of associated boundar...
By studying scattering Lie groups and their associated Lie algebras, we introduce a new method for the characterisation of collision invariants for physical scattering families associated to smooth, convex hard particles in the particular case that t...
This paper investigates the application of piecewise linear approximation for the control of nonlinear systems, particularly focusing on the effective linearization of systems modeled by the differential equation y^(n) + f(y,y',...,y^(n-1)) = u(t). W...
We analyze the dynamics of the forced singularly perturbed differential equation of Duffing's type. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small...
We compute solutions to the Hermitian Yang-Mills equations on holomorphic vector bundles $V$ via an alternating optimisation procedure founded on geometric machine learning. The proposed method is fully general with respect to the rank and structure...
We prove an epsilon-regularity theorem for critical and super-critical systems with a non-local antisymmetric operator on the right-hand side. These systems contain as special cases, Euler-Lagrange equations of conformally invariant variational fun...