Brouwer's Theory Research Articles

Positive solution for a class of Choquard equations with Hardy–Littlewood–Sobolev critical exponent in exterior domains

In this paper we show existence of positive solutions for a class of problems involving the Choquard term in exterior domain and the nonlinearity with critical growth. We prove the main results using variational method combined with Brouwer theory of degree and Deformation lemma.

Existence of positive solutions for a class of fractional Choquard equation in exterior domain

<p style='text-indent:20px;'>In this paper we show existence of positive solutions for a class of problems involving the fractional Laplacian in exterior domain and Choquard type nonlinearity. We prove the main results using variational method combined with Brouwer theory of degree and Deformation Lemma..</p>

Handel’s Fixed Point Theorem: A Morse Theoretical Point of View

Michael Handel has proved in [10] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that turned out to be an efficient tool in the study of the dynamics of surface homeomorphisms. The present article fits into a series of articles by the author [13] and by Juliana Xavier [21, 22], where proofs were given, related to the classical Brouwer Theory, instead of the Homotopical Brouwer Theory used in the original article. Like in [13, 21] and [22], we will use “free brick decompositions” but will present a more conceptual Morse theoretical argument. It is based on a new preliminary lemma, that gives a nice “condition at infinity” for our problem.

Existence of a positive solution for a class of fractional elliptic problems in exterior domains involving critical growth

In this paper we show existence of positive solutions for a class of problems involving the fractional Laplacian in exterior domain and the nonlinearity with critical growth. We prove the main results using variational method combined with Brouwer theory of degree and Deformation lemma.

Existence of positive solution for a fractional elliptic equation in exterior domain

In this paper, we show the existence of positive solutions for nonlinear Schrodinger equation with fractional Laplacian(−Δ)su+λu=|u|p−2uinΩ, where Ω⊂RN is an unbounded domain, ∂Ω≠∅ is bounded, λ∈R+, s∈(0,1), N>2s and 2<p<2s⁎. Precisely, we show that the problem has at least one positive solutions. We achieved our results by using variational method together with Brouwer theory of degree. Moreover, we also prove a version to the Fractional operator in unbounded domain of the Global Compactness result due to Struwe (see [24]).

Forcing theory for transverse trajectories of surface homeomorphisms

This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations. This allows us to derive new proofs for some known results as well as some new applications, among which we note the following: we extend Franks and Handel’s classification of zero entropy maps of $$\mathbb {S}^2$$ for non-wandering homeomorphisms; we show that if f is a Hamiltonian homeomorphism of the annulus, then the rotation set of f is either a singleton or it contains zero in the interior, proving a conjecture posed by Boyland; we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms, proving a first case of the Franks–Misiurewicz conjecture; we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms.

Some Proposition that Links Ferromagnetic Models with Cantorian Set Theory

&lt;p class="1Body"&gt;In this paper, we established a link between ferromagnetic models and set theory. Three dimension spins "S&lt;sub&gt;X&lt;/sub&gt;, S&lt;sub&gt;y&lt;/sub&gt;, S&lt;sub&gt;z&lt;/sub&gt;" are considered in Heisenberg model but it is restricted to z_ axis&lt;strong&gt;&lt;em&gt; &lt;/em&gt;&lt;/strong&gt;" S&lt;sub&gt;z&lt;/sub&gt;" in Ising mode. By using Brouwer theory, fractal motion of magnetic domains is predicted theoretically in the materials that their magnetism are explained by Ising model. In addition, we achieved that the experimental data of the fractal motion of magnetic domains in the thin films are agreement with the theoretical assumptions that are proposed.&lt;/p&gt;

Conjugacy invariants for Brouwer mapping classes

We give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set called the set ofwalls, which splits the plane intomaximal translation areasandirreducible areas. We then focus on Brouwer mapping classes relative to four orbits and describe them explicitly by adding atangleto Handel’s diagram and to the set of walls. This is essentially an isotopy class of simple closed curves in the cylinder minus two points.

An index for Brouwer homeomorphisms and homotopy Brouwer theory

We use the homotopy Brouwer theory of Handel to define a Poincaré index between pairs of orbits for an orientation-preserving fixed-point-free homeomorphism of the plane. Furthermore, we prove that this index is almost additive.

Handel’s fixed point theorem revisited

AbstractMichael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.

Free planar actions of the Klein bottle group

We describe the structure of the free actions of the Klein bottle group by orientation preserving homeomorphisms of the plane. This group is generated by two elements $a,b$, where the conjugate of $b$ by $a$ equals the inverse of $b$. The main result is that $a$ must act properly discontinuously, while $b$ cannot act properly discontinuously. As a corollary, we describe some torsion free groups that cannot act freely on the plane. We also find some properties which are reminiscent of Brouwer theory for the infinite cyclic group $Z$, in particular that every free action is virtually wandering.

Une nouvelle preuve du théorème de point fixe de Handel

M Handel has proved in [Topology 38 (1999) 235–264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel’s fixed point theorem, based on Brouwer theory and some plane topology arguments. We will slightly improve the theorem by proving the existence of a simple closed curve of index 1. This index result was known to be true under an additional hypothesis and has been used by different authors (J Franks [NYJM 2 (1996) 1–19, Trans.AMS 348 (1996) 2637–2662] S Matsumoto [Topol. Appl. 104 (2000) 191–214]) to study homeomorphisms of surfaces.

A New Method of determining Initial Conditions for Satellite Formation Flying

Satellite formation flying is the placing micro-satellites with the same mission into nearby orbits to form a cluster. Clohessy-Wiltshire equations are used to describe the relative motion and control strategies between satellites within a cluster, which are known as Hill's equations. Even though Hill's equations are powerful in determining initial conditions for the satellite formation flying, they can not accurately express the relative motion under J2 perturbation. Some methods have been developed for the determination of initial conditions to avoid limits of Hill's equation. This paper gives a new method of determining initial conditions using mean elements. For this research mean elements were transformed to osculating elements using Brouwer's theory and the orbit was propaeated with the consideration of J2-J8 to get a relative position. The results show that satellites within a cluster are maintained in the desired boundary for long period and the method is effective on a fuel saving for satellite formation flying.

Defining concurrent processes constructively

This paper proposes a constructive logic in which a concurrent system can be defined as a proof of a specification. The logic is defined by adding stream types and several rules for them to a simple constructive logic based on intuitionism. The unique feature of the obtained system is in the ( MPST) rule, which is a kind of structural induction on streams. The ( MPST) rule is based on the idea of Brouwer's theory of choice sequences in intuitionism and it allows to define a concurrent process, which is actually a Burge's mapstream function, as a proof of a specification with a good intuition on computation. Several techniques for defining stream-based concurrent programs are also presented through various exmaples.

The theory of empirical sequences

In this paper I want to accomplish three things: First, I want to present a 'formal system' for L. E. J. Brouwer's conception of elementary analysis. Secondly, I want to prove two meta-theorems about the proposed system; and finally, I want to use those results to speculate about a historical question in Brouwer scholarship that was left open at the end of Posy (1976). By elementary analysis I mean the theory of the arithmetic of real numbers which treats them as infinite sequences of natural numbers. Thus the paper revolves around a portrayal of Brouwer's conception of numerical sequences, but before discussing the special features of the formal system that follows let me mention two devices that figure centrally in my portrayal of Brouwer's thought about that sort of choice sequence. My primary device is a version of Brouwer's theory of the creative subject. I shall develop that theory here and also generalize its applications to analysis. In (1976) I1 distinguished two versions of the theory of choice sequences: the theory of general sequences (Brouwer's explicit theory throughout his early and middle carreer) and the theory of epistemic sequences (which occured explicitly only after 1948). The latter study concerns sequences which are themselves generated by empirical observations about the activity of the creative subject. The theory of empirical sequences, which I formalize here, is a natural generalization of the latter epistemic approach, and allows any series of events at all or more precisely any series of trials with a determinate number of possible outcomes to generate a choice sequence. The theory of empirical sequences implicitly alluded to by Brouwer and Heyting was first explicitly proposed by Myhill (in 1968a and 1970) and mentioned in Troelstra (1969), but the present paper explores it systematically for the first time. Of course in the more general empirical approach of this paper the basic idea of the creative subject has not vanished. He is still around both as an observer and occasionally (though now merely a special case) as a generator

A note on ?The main problem of satellite theory for small eccentricities, by A. Deprit and A. Rom, 1970?

Criticism of Deprit and Rom's (1970) extension of Brouwer's (1959) first-order satellite theory to the third order in J sub 2. It is noted that Deprit and Rom's theory suffers from the drawback that the perturbations of all orders are expressed as infinite power series in the eccentricity e. It is demonstrated that Deprit and Rom's reported failure to extend Brouwer's theory to the second order in a closed form by means of Lie transforms is due to an oversight and is not caused by difficulties inherent in the problem or in the particular perturbation method used.

Some possible modifications of Brouwer's theory of the general perturbations in rectangular coordinates

The method of general perturbations in rectangular coordinates is the most direct of all methods of expansion of the perturbations into series because it is intimately associated with the computation of ephemerides. In addition, unlike the method of variation of elliptic elements, the method of coordinates does not have the zero eccentricity as a singularity. In Brouwer's theory of the general perturbations in rectangular coordinates the variation of elements in the canonical form is used. However, if the perturbations are being developed into trigonometric series with purely numerical coefficients, the use of canonical elements is not of any advantage. This fact was recognized by Davis, who rewrote Brouwer's formulas in terms of the standard elliptic elements. Davis's formula contains two terms of order −1 in the eccentricity. The presence of these terms causes considerable numerical inconvenience in the study of nearly circular orbits. We suggest here the use of the Eckert-Brouwer formula for the orbit correction as a foundation of a planetary theory. The application of this formula leads directly to a vectorial expression for perturbations which is free from the disadvantages mentioned above and is also convenient for the numerical computations. The method of iteration is suggested for computing the effects of higher orders. The inclusion of the higher-order terms is important not only in the planetary case but also in the case of artificial celestial bodies moving in orbits in cislunar space, far away from the earth. Such bodies in their motions resemble planets or comets more than satellites.

Small eccentricities or inclinations in the Brouwer theory of the artificial satellite

view Abstract Citations (87) References (2) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Small eccentricities or inclinations in the Brouwer theory of the artificial satellite Lyddane, R. H. Abstract Brouwer's theory of the artificial satellite is re-examined for small eccentricities and inclinations and im- proved formulas are obtained. The validity of the new formulas is confirmed by comparison with the results of numerical integration. Publication: The Astronomical Journal Pub Date: October 1963 DOI: 10.1086/109179 Bibcode: 1963AJ.....68..555L full text sources ADS |

Numerical comparison between Brouwer's theory and solution by Cowell's method for the orbit of an artificial satellite

Numerical comparison between Brouwer's theory and solution by Cowell's method for the orbit of an artificial satellite

Prediction of theoretical radiants from very short periodic comets.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Prediction of theoretical radiants from very short periodic comets. Hamid, Salah Abstract Jupiter's secular perturbations on certain comets cause a rotation of their lines of nodes in the plane of the ecliptic and a periodic motion in their inclinations while the aphelion and perihelion distances of the comets are very slightly affected. Therefore, it is possible for such comets occasionally to cross the ecliptic at the intersections with the earth's orbit and give rise to showers of meteors, provided that the comets' perihelia lie inside the earth's orbit and provided their aphelia lie inside Jupiter's orbit. An application of Brouwer's theory of secular perturbation of Jupiter on comets, which is valid for comets of inclination less than 360, has been made by the author. Lucke's comet and Comet Grigg-Skjellerup are the only two periodic comets that satisfy the necessary criteria. Encke's comet gives rise to known night showers, the Taurids, as shown by Whipple, while calculation indicates that it can give rise also to two daylight showers: June 12, ot = 65?7, 6 = 26?I and May 19, a = 43?5, 6 = 13?2. The first appears to be detected with radar by Lovell. Comet Grigg-Skjellerup gives rise to two night showers: January 21, a = 1070, 6 = February 9, a = 1310, 6 = both of which seemed to be observed visually by Opik and the American Meteoric Society. Also it produces two other daylight radiants, June 21, a = 1050, 6 = 150; May 31, a = 83?5, 6 = 70; both of which appear to be detected with radar by Clegg, Huges, and Lovell. Harvard College Observatory, Cambridge, Mass. Publication: The Astronomical Journal Pub Date: 1948 DOI: 10.1086/106162 Bibcode: 1948AJ.....54R..39H full text sources ADS |