Analytical solution of a class of nonlinear differential equations of the third order in the complex area

В работе приводится доказательство теоремы существования и единственности аналитического решения класса нелинейных дифференциальных уравнений третьего порядка, правая часть которого представлена полиномом шестой степени, в комплексной области. Расширен класс рассматриваемых уравнений за счет новой замены переменных. Получена априорная оценка аналитического приближенного решения. Представлен вариант численного эксперимента оптимизации априорных оценок с помощью апостериорных. The article presents a proof of the theorem of the existence and uniqueness of the analytical solution of the class of nonlinear differential equations of the third order, with a polynomial right-hand side of the sixth degree, in the complex domain. The class of the considered equations has been extended by means of a new change of variables. An a priori estimate of the analytical approximate solution is obtained. A variant of the numerical experiment of optimizing a priori estimates using a posteriori estimates is presented.

The hidden fluxes, that control the fluctuations of scalar fields

The fluctuations of scalar fields, that are invariant under rotations of the worldvolume, in Euclidian signature, can be described by a system of Langevin equations. These equations can be understood as defining a change of variables in the functional integral for the noise, with which the physical degrees of freedom are in equilibrium. The absolute value of the Jacobian of this change of variables therefore repackages the fluctuations. This provides a new way of relating the number and properties of scalar fields with the consistent and complete description of their fluctuations and is another way of understanding the relevance of supersymmetry, which, in this way, determines the minimal number of real scalar fields (e.g. two in two dimensions, four in three dimensions and eight in four dimensions), in order for the system to be consistently closed. The classical action of the scalar fields, obtained in this way, contains a surface term and a remainder, in addition to the canonical kinetic and potential terms. The surface term describes possible flux contributions in the presence of boundaries, while the remainder describes additional interactions, that can’t be absorbed in a redefinition of the canonical terms. It is, however, through its combination with the surface term that the noise fields can be recovered, in all cases. However their identities can be subject to anomalies. What is of particular, practical, interest is the identification of the noise fields, as functions of the scalars, whose correlation functions are Gaussian. This implies new identities, between the scalars, that can be probed in real, or computer, experiments.

Nonexistence of Global Solutions to A Semilinear Wave Equation with Scale Invariant Damping

We obtain a blowup result for solutions to a semilinear wave equation with scale-invariant dissipation. We perform a change of variables that transforms our starting equation into a Generalized Tricomi equation, then apply Kato’s lemma, we can prove a blowup result for solutions to the transformed equation under some assumptions on the initial data. In the critical case, we use the fundamental solutions of the Generalized Tricomi equation to modify Kato’s lemma to deal with it.

Nonlinear Longitudinal Vibrations in Accreting One-dimensional Nanostructures

A governing partial differential equation that describes an accreting nanochain containing an attachment of infinite atoms was considered in this paper. We transformed the space variable u(t,r) → v(τ,x) (for a governing PDE formulated in previous research studies) and introduced a function of linear growth. The boundary conditions were also transformed into the new variables, the left end of the accreting chain was free u(t, r=0)=0 while the right end was fixed. The method of lines was also employed to numerically analyze the governing partial differential equation. We detailed the differential transformation for the change of variables used in obtaining the transformed partial differential equation. We also considered what happens with the introduction of the viscous damping term, (δ). The governing partial differential equation was formulated. Numerical simulations for both cases, was then carried out.

The Distributions of Sums for Shanker, Akash, Ishita, Pranav, Rani, and Ram Awadh

In statistics and probability theory, one of the most important statistics is the sums of random variables. After introducing a probability distribution, determining the sum of n independent and identically distributed random variables is one of the interesting topics for authors. This paper presents the probability density functions for the sum of n independent and identically distributed random variables such as Shanker, Akash, Ishita, Pranav, Rani, and Ram Awadh. In order to determine all aforementioned distributions, the problem-solving methods are applied which is based on the change-of-variables technique.

A generalized Fourier transform by means of change of variables within multilinear approximation

The paper deals with approximations of periodic functions that play a significant role in harmonic analysis. The approach revisits the trigonometric polynomials, seen as combinations of functions, and proposes to extend the class of models of the combined functions to a wider class of functions. The key here is to use structured functions, that have low complexity, with suitable functional representation and adapted parametrizations for the approximation. Such representation enables to approximate multivariate functions with few eventually random samples. The new parametrization is determined automatically with a greedy procedure, and a low rank format is used for the approximation associated with each new parametrization. A supervised learning algorithm is used for the approximation of a function of multiple random variables in tree-based tensor format, here the particular Tensor Train format. Adaptive strategies using statistical error estimates are proposed for the selection of the underlying tensor bases and the ranks for the Tensor-Train format. The method is applied for the estimation of the wall pressure for a flow over a cylinder for a range of low to medium Reynolds numbers for which we observe two flow regimes: a laminar flow with periodic vortex shedding and a laminar boundary layer with a turbulent wake (sub-critic regime). The automatic re-parametrization enables here to take into account the specific periodic feature of the pressure.

Short communication: Inverse isochron regression for Re–Os, K–Ca and other chronometers

Conventional Re–Os isochrons are based on mass spectrometric estimates of 187Re/188Os and 187Os/188Os, which often exhibit strong error correlations that may obscure potentially important geological complexity. Using an approach that is widely accepted in 40Ar/39Ar and U–Pb geochronology, we here show that these error correlations are greatly reduced by applying a simple change of variables, using 187Os as a common denominator. Plotting 188Os/187Os vs. 187Re/187Os produces an “inverse isochron”, defining a binary mixing line between an inherited Os component whose 188Os/187Os ratio is given by the vertical intercept, and the radiogenic 187Re/187Os ratio, which corresponds to the horizontal intercept. Inverse isochrons facilitate the identification of outliers and other sources of data dispersion. They can also be applied to other geochronometers such as the K–Ca method and (with less dramatic results) the Rb–Sr, Sm–Nd and Lu–Hf methods. Conventional and inverse isochron ages are similar for precise datasets but may significantly diverge for imprecise ones. A semi-synthetic data simulation indicates that, in the latter case, the inverse isochron age is more accurate. The generalised inverse isochron method has been added to the IsoplotR toolbox for geochronology, which automatically converts conventional isochron ratios into inverse ratios, and vice versa.

The globalization theorem for the Curvature-Dimension condition

The Lott–Sturm–Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space $$(X,\mathsf {d},{\mathfrak {m}})$$ ( X , d , m ) (so that $$(\text {supp}({\mathfrak {m}}),\mathsf {d})$$ ( supp ( m ) , d ) is a length-space and $${\mathfrak {m}}(X) < \infty $$ m ( X ) < ∞ ) verifying the local Curvature-Dimension condition $${\mathsf {CD}}_{loc}(K,N)$$ CD loc ( K , N ) with parameters $$K \in {\mathbb {R}}$$ K ∈ R and $$N \in (1,\infty )$$ N ∈ ( 1 , ∞ ) , also verifies the global Curvature-Dimension condition $${\mathsf {CD}}(K,N)$$ CD ( K , N ) . In other words, the Curvature-Dimension condition enjoys the globalization (or local-to-global) property, answering a question which had remained open since the beginning of the theory. For the proof, we establish an equivalence between $$L^1$$ L 1 - and $$L^2$$ L 2 -optimal-transport–based interpolation. The challenge is not merely a technical one, and several new conceptual ingredients which are of independent interest are developed: an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order temporal derivative of associated Kantorovich potentials; a surprising third-order theory for the latter Kantorovich potentials, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. As a consequence, numerous variants of the Curvature-Dimension condition proposed by various authors throughout the years are shown to, in fact, all be equivalent in the above setting, thereby unifying the theory.