On the sphericity of the subgroup PSL2(ℝ) in the group of diffeomorphisms of the circle

It is shown that the group PSL₂(?) is a spherical subgroup in the group of C³-diffeomorphisms of the circle.     

On the constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem

We study the rate of convergence in von Neumann’s ergodic theorem. We obtain constants connecting the power rate of convergence of ergodic means and the power singularity at zero of the spectral measure of the corresponding dynamical system (these concepts are equivalent to each other). All the results of the paper have obvious exact analogs for wide-sense stationary stochastic processes.     

Global indeterminacy of the equilibrium in the Chamley model of endogenous growth in the vicinity of a Bogdanov–Takens bifurcation

This paper studies the dynamics implied by the Chamley (1993) model, a variant of the two-sector model with an implicit characterization of the learning function. We first show that under some “regularity” conditions regarding the learning function, the model has (a) one steady state, (b) no steady states or (c) two steady states (one saddle and one non-saddle). Moreover, via the Bogdanov–Takens theorem, we prove that for critical regions of the parameters space, the dynamics undergoes a particular global phenomenon, namely the homoclinic bifurcation. Because these findings imply the existence of a continuum of equilibrium trajectories, all departing from the same initial value of the predetermined variable, the model exhibits global indeterminacy.     

One-loop calculation of the β-function in the modified formulation of the Yang-Mills theory

In the one-loop approximation for the modified Yang-Mills theory, we calculate contributions to the two- and three-point Green’s functions of the gauge field A _μ , to the Green’s function of the anticommuting fields e and b, and to the vertex function Γ _Aeb. We find the renormalization constants Z ₁ and Z ₂ and the corresponding constants \(\bar Z_1\) and \(\bar Z_2\) for the anticommuting fields and show that they satisfy the same equation as in the standard Yang-Mills theory. We demonstrate that the β-functions in the standard and modified theories coincide.     

Existence of the Self-Similar Solutions in the Heat Flow of Harmonic Maps

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A General Theorem of the Alternative in the Context of Set-Valued Function

1.IntroductionTheclassicalworkOiltheoremsofthealternativeforsingle-valuedfunctionbyG..d..II]andMotzkin[2]wereextendbymanyresearchersinvariousways(suchasJeyakumar['l'],Lily],andTamminen[6]),whereconditionsofconvexityaremoreorlessrelaxed.Met's..rk[:]givesemphasistotheoremsofthealternativeinthesettingofset-valuedfunctionwhereconditionofconvexityissettobecone-convex.Thispaperaimsatprovidinganunifiedextensionoftheabove-mentionedtheorems,andgivesoutageneraltheoremofthealternativeinthecontextofset…     

On the Dimension of the Space of Dark States in the Tavis–Cummings Model

Mathematical Notes - The space of minimal energy of a qubit system is the dark subspace of quantum states of a system of two-level atoms in the finite-dimensional Tavis–Cummings (TC) model of...     

Reducing conjugacy in the full diffeomorphism group of ℝ to conjugacy in the subgroup of orientation-preserving maps

Let Diffeo = Diffeo(ℝ) denote the group of infinitely differentiable diffeomorphisms of the real line ℝ, under the operation of composition, and let Diffeo⁺ be the subgroup of diffeomorphisms of degree +1, i.e., orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements f, g ∈ Diffeo are conjugate in Diffeo to associated conjugacy problems in the subgroup Diffeo⁺. The main result concerns the case when f and g have degree −1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in Diffeo⁺, in order to ensure that f is conjugated to g by an element of Diffeo⁺. The methods involve formal power series and results of Kopell on centralisers in the diffeomorphism group of a half-open interval. Bibliography: 4 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 231–237.     

Separability of the subgroups of residually nilpotent groups in the class of finite π-groups

Siberian Mathematical Journal - Given a nonempty set π of primes, call a nilpotent group π-bounded whenever it has a central series whose every factor F is such that: In every quotient...     

Some Generalizations of Semimartingales in the Plane

51.IntroductionTwo(parametersemi-martingaleswereinvestigated,forexample,inthepaperbyBrennan[21,Cairoli[3]andZakai[81.Inthispaperwegeneralizetheconceptofatwo-parametersemimartingaleandinvestigatesomepropertiesofthegeneralizedsemi-martingaoes.Wealsodefineaclassofpositivemeasuresassociatedwiththegeneralizedsemi-martingales.InSectionII,followingtheterminologygivenin[21,wedefinestrongandweakA--processesoforderp,alsofor15p相似文献   

On the order of the operators in the Douglas–Rachford algorithm

The Douglas–Rachford algorithm is a popular method for finding zeros of sums of monotone operators. By its definition, the Douglas–Rachford operator is not symmetric with respect to the order of the two operators. In this paper we provide a systematic study of the two possible Douglas–Rachford operators. We show that the reflectors of the underlying operators act as bijections between the fixed points sets of the two Douglas–Rachford operators. Some elegant formulae arise under additional assumptions. Various examples illustrate our results.     

Revisit of the Theorem of Image Trajectories in the Earth-Moon Space

On the occasion of the 50th anniversary of the theorem of the image trajectories in the Earth-Moon space, the author revisits the theorem and clarifies the relation between the class of image trajectories and the class of symmetric free-return trajectories, which were employed in the Apollo program. In a nutshell, the symmetric free-return trajectories are those image trajectories that intersect the Earth-Moon axis orthogonally at some point above the far side of the Moon. Optimization implications are pointed out.     

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Čech compactification β S is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient contains a linear isometric copy of ℓ _∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.     

On the Post-Darwin Approximation of the Maxwell–Lorentz Equations of Motion of Point Charges in the Absence of Neutrality

Ukrainian Mathematical Journal - The existence of holomorphic (in time) solutions of the nonrelativistic equations of motion of nonneutral systems of point charges that do not contain inverse...     

Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative

The purpose of this study was to gain insight into 30, first year calculus students’ understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up group interviews. Students’ personal meanings of the vertex, including misconceptions, were explored, along with students’ understanding to solve problems pertaining to the derivative of a quadratic function. Results give evidence of students’ weak schema of the vertex, lack of connection between different problem types and the importance of linguistics in relation to levels of APOS theory. A preliminary genetic decomposition was developed based on the results. Future research is suggested as a continuation to improve student understanding of the relationship between the vertex of quadratic functions and the derivative.     

On the influence of the potential on the convergence rate of expansions in root functions of the Shrödinger operator

We consider the one-dimensional Schrödinger operator with integrable potential. We analyze the rate of the uniform equiconvergence of the biorthogonal expansion of an absolutely continuous function in the root functions of this operator with its Fourier trigonometric series on a compact set. For this convergence rate, we obtain an estimate depending on the modulus of continuity of the potential. We extract subclasses of absolutely continuous functions on which these estimates can be improved.     

Conditions for the uniqueness of statistic's distribution in the class of spherical distridutions

This paper gives the necessary and sufficient conditions, which are very simple for checking the uniqueness of the distribution of a statistic in those classes of the matrix spherical distributions and improve the results of Kariya [1].