Möbius’s functional equation and Schur’s lemma with applications to the complex unit disk

Möbius addition is defined on the complex open unit disk by

$$\begin{aligned} a\oplus _M b = \dfrac{a+b}{1+\bar{a}b} \end{aligned}$$

and Möbius’s exponential equation takes the form \(L(a\oplus _M b) = L(a)L(b)\), where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Möbius’s exponential equation is connected to Cauchy’s exponential equation. Möbius’s exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Möbius addition, considered as a nonassociative group-like structure. This suggests studying Schur’s lemma in a more general setting.

     

Leray’s problem on the stationary Navier–Stokes equations with inhomogeneous boundary data

Consider the stationary Navier–Stokes equations in a bounded domain whose boundary consists of multi-connected components. We investigate the solvability under the general flux condition which implies that the total sum of the flux of the given data on each component of the boundary is equal to zero. Based on our Helmholtz–Weyl decomposition, we prove existence of solutions if the harmonic part of the solenoidal extension of the given boundary data is sufficiently small in L ³ compared with the viscosity constant. Dedicated to Professor Giovanni P. Galdi on the occasion of his 60th birthday.     

Generalized and Variational Statements of the Riemann Problem with Applications to the Development of Godunov’s Method

Computational Mathematics and Mathematical Physics - This paper is devoted to the numerical method for solving the fluid dynamics equations proposed by Godunov more than 60 years ago. This method...     

A Categorical Interpretation of the Intuitionistic,Typed, First Order Logic with Hilbert’s $${\varepsilon}$$-Terms

We introduce a typed version of the intuitionistic epsilon calculus. We give a categorical semantics of it introducing a class of categories which we call \({\varepsilon}\)-categories. We compare our results with earlier ones of Bell (Math Logic Quart 39(3):323–337, 1993).     

A general estimator for the right endpoint with an application to supercentenarian women’s records

We extend the setting of the right endpoint estimator introduced in Fraga Alves and Neves (Statist. Sinica 24, 1811–1835, 2014) to the broader class of light-tailed distributions with finite endpoint, belonging to some domain of attraction induced by the extreme value theorem. This stretch enables a general estimator for the finite endpoint, which does not require estimation of the (supposedly non-positive) extreme value index. A new testing procedure for selecting max-domains of attraction also arises in connection with the asymptotic properties of the general endpoint estimator. The simulation study conveys that the general endpoint estimator is a valuable complement to the most usual endpoint estimators, particularly when the true extreme value index stays above ?1/2, embracing the most common cases in practical applications. An illustration is provided via an extreme value analysis of supercentenarian women data.     

On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities

We show how Noether conservation laws can be obtained from the particle relabelling symmetries in the Euler–Poincaré theory of ideal fluids with advected quantities. All calculations can be performed without Lagrangian variables, by using the Eulerian vector fields that generate the symmetries, and we identify the time-evolution equation that these vector fields satisfy. When advected quantities (such as advected scalars or densities) are present, there is an additional constraint that the vector fields must leave the advected quantities invariant. We show that if this constraint is satisfied initially then it will be satisfied for all times. We then show how to solve these constraint equations in various examples to obtain evolution equations from the conservation laws. We also discuss some fluid conservation laws in the Euler–Poincaré theory that do not arise from Noether symmetries, and explain the relationship between the conservation laws obtained here, and the Kelvin–Noether theorem given in Sect. 4 of Holm et al. (Adv. Math. 137:1–81, 1998).     

New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

In this paper, by using the modified mapping method and the extended mapping method, we derive some new exact solutions of the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity, which are the linear combination of two different Jacobi elliptic functions and we also consider the solutions in the limit cases.     

Application of Avery–Peterson fixed point theorem to nonlinear boundary value problem of fractional differential equation with the Caputo’s derivative

In this paper, by means of the Avery–Peterson fixed point theorem, we establish the existence result of at least triple positive solutions of four-point boundary value problem of nonlinear differential equation with Caputo’s fractional order derivative. An example illustrating our main result is given. Our results complements previous work in the area of boundary value problems of nonlinear fractional differential equations.     

It’s about how the task is set: the inclusion–exclusion effect and accountability in preprocessing management information

Central European Journal of Operations Research - Managerial decision-making often rests on information that is preprocessed and therefore considerably reduced compared to the raw data. In our...     

A generalized Duffing equation with the Coulomb’s friction law and Signorini–type contact conditions

This work provides mathematical and numerical analyses for a spring–mass system, in which Signorini–type contact conditions and Coulomb’s friction law with thermal effects are taken into consideration. The motion of a mass attached to a viscoelastic (Kelvin–Voigt type) nonlinear spring is described by a generalized Duffing equation. Signorini contact conditions are understood as extended complementarity conditions (CCs), where convolution is incorporated, allowing to consider thermal aspects of an obstacle. We prove the existence of global weak solutions for the highly nonlinear differential equation system with all the conditions, based on the regularized differential equation and the normal compliance condition with the standard mollifier. In addition, we investigate what side effects produce higher singularities of contact forces in dynamic contact problems, which is also supported by numerical evidences. Numerical schemes are proposed and then several groups of data are selected for the display of our numerical simulations.     

Further Remarks on the Luo–Hou’s Ansatz for a Self-similar Solution to the 3D Euler Equations

It is shown that the self-similar ansatz proposed by T. Hou and G. Luo to describe a singular solution of the 3D axisymmetric Euler equations leads, without assuming any asymptotic condition on the self-similar profiles, to an overdetermined system of partial differential equations that produces two families of solutions: a class of trivial solutions in which the vorticity field is identically zero, and a family of solutions that blow-up immediately, where the vorticity field is governed by a stationary regime. In any case, the analytical properties of these solutions are not consistent with the numerical observations reported by T. Hou and G. Luo. Therefore, this result is a refinement of the previous work published by D. Chae and T.-P. Tsai on this matter, where the authors found the trivial class of solutions under a certain decay condition of the blow-up profiles.     

Computing the topological entropy of continuous maps with at most three different kneading sequences with applications to Parrondo’s paradox

We introduce an algorithm to compute the topological entropy of piecewise monotone maps with at most three different kneading sequences, with prescribed accuracy. As an application, we compute the topological entropy of 3-periodic sequences of logistic maps, disproving a commutativity formula for topological entropy with three maps, and analyzing the dynamics Parrondo’s paradox in this setting.     

Hölder–Lipschitz Norms and Their Duals on Spaces with Semigroups,with Applications to Earth Mover’s Distance

We introduce a family of bounded, multiscale distances on any space equipped with an operator semigroup. In many examples, these distances are equivalent to a snowflake of the natural distance on the space. Under weak regularity assumptions on the kernels defining the semigroup, we derive simple characterizations of the Hölder–Lipschitz norm and its dual with respect to these distances. As the dual norm of the difference of two probability measures is the Earth Mover’s Distance (EMD) between these measures, our characterizations give simple formulas for a metric equivalent to EMD. We extend these results to the mixed Hölder–Lipschitz norm and its dual on the product of spaces, each of which is equipped with its own semigroup. Additionally, we derive an approximation theorem for mixed Lipschitz functions in this setting.     

Uniqueness of a Probability Solution to the Kolmogorov Equation with a Diffusion Matrix Satisfying Dini’s Condition

Doklady Mathematics - Abstract—In this note we study the stationary Kolmogorov equation and prove that, in the case where the diffusion matrix satisfies Dini’s condition and the drift...