XXIV.—On the Equation of Motion of a Free Particle in the Expanding Universe of Kinematical Relativity

1943 ◽  
Vol 61 (3) ◽  
pp. 288-297
Author(s):  
E. A. Milne
Keyword(s):  
Additional Assumption ◽  
A Priori ◽  
Free Particle ◽  
Equation Of Motion ◽  
Expanding Universe ◽  
Image Position ◽  
Special Case

In a recent paper in these Proceedings, Dr G. C. McVittie has published some criticisms of kinematical relativity. These criticisms are to a large extent based on his formula (4.10), namely,It must be stated at the outset that McVittie's interpretation of his derivation of (1) as a derivation of “Milne's formula for the acceleration of a ‘free particle moving in the presence of a substratum,’ for the special case of one spatial co-ordinate only” is wrong. McVittie does not derive the result, as he claims, from what he calls the “axioms of kinematical relativity” alone; he deduces it from these axioms together with an additional assumption, which is equivalent to begging the answer to the whole problem it was my object to solve. Instead of considering a free particle, as I did—that is, a particle whose motion we do not a priori know—he prescribes a priori the motion of his particle as being constrained to obey the rule, in his notation,

scholarly journals On the intersection of free subgroups in free products of groups

2008 ◽  
Vol 144 (3) ◽  
pp. 511-534 ◽  
Author(s):  
WARREN DICKS ◽  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Additional Assumption ◽  
Free Products ◽  
Reduced Rank ◽  
Products Of Groups ◽  
Image Position ◽  
Free Subgroups ◽  
Special Case ◽  
Subgroup Theorem

AbstractLet (Gi | i ∈ I) be a family of groups, let F be a free group, and let $G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$ the free product of F and all the Gi.Let $\mathcal{F}$ denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, $H \cap G_i^{g} = \{1\}.$ By the Kurosh Subgroup Theorem, every element of $\mathcal{F}$ is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as $\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$ To avoid the vacuous case, we make the additional assumption that $\mathcal{F}$ contains a non-cyclic group, and we define We are interested in precise bounds for $\upp$. In the special case where I is empty, Hanna Neumann proved that $\upp$ ∈ [1,2], and conjectured that $\upp$ = 1; fifty years later, this interval has not been reduced.With the understanding that ∞/(∞ − 2) is 1, we define Generalizing Hanna Neumann's theorem we prove that $\upp \in [\fun, 2\fun]$, and, moreover, $\upp = 2\fun$ whenever G has 2-torsion. Since $\upp$ is finite, $\mathcal{F}$ is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that $\upp = \fun$ whenever G does not have 2-torsion.

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

scholarly journals Radially symmetric solutions of a class of singular elliptic equations

1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Open Unit ◽  
Open Unit Ball ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

Existence of Periodic Solution for Equation of Motion of Simple Beams With Harmonically Variable Length

1995 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Function ◽  
Axial Direction ◽  
Equation Of Motion ◽  
The Other ◽  
Sufficient Condition ◽  
Vibrating String ◽  
The Stability ◽  
Special Case

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.

On the estimation of a harmonic component in a time series with stationary dependent residuals

1973 ◽  
Vol 5 (02) ◽  
pp. 217-241 ◽  
Author(s):  
A. M. Walker
Keyword(s):  
Time Series ◽  
Least Squares ◽  
Least Squares Method ◽  
Harmonic Component ◽  
Asymptotic Distributions ◽  
Finite Variance ◽  
Rigorous Derivation ◽  
Image Position ◽  
Vector Valued ◽  
Special Case

Let observations (X 1, X 2, …, Xn ) be obtained from a time series {Xt } such that where the ɛt are independently and identically distributed random variables each having mean zero and finite variance, and the gu (θ) are specified functions of a vector-valued parameter θ. This paper presents a rigorous derivation of the asymptotic distributions of the estimators of A, B, ω and θ obtained by an approximate least-squares method due to Whittle (1952). It is a sequel to a previous paper (Walker (1971)) in which a similar derivation was given for the special case of independent residuals where gu (θ) = 0 for u > 0, the parameter θ thus being absent.

scholarly journals A PRIORI ESTIMATE FOR AN EQUATION WITH FRACTIONAL DERIVATIVES WITH DIFFERENT ORIGINS

2019 ◽  
pp. 41-47
Author(s):  
Л.М. Энеева
Keyword(s):  
Fractional Derivatives ◽  
Model Equation ◽  
A Priori ◽  
A Priori Estimate ◽  
Equation Of Motion ◽  
Point Boundary ◽  
Priori Estimate ◽  
Fractal Media ◽  
Different Origins ◽  
Variable Potential

В работе исследуется обыкновенное дифференциальное уравнение дробного порядка, содержащее композицию дробных производных с различными началами, с переменным потенциалом. Рассматриваемое уравнение выступает модельным уравнением движения во фрактальной среде. Для исследуемого уравнения доказана априорная оценка решения смешанной двухточечной краевой задачи. We consider an ordinary differential equation of fractional order with the composition of leftand right-sided fractional derivatives, and with variable potential. The considered equation is a model equation of motion in fractal media. We prove an a priori estimate for solutions of a mixed two-point boundary value problem for the equation under study.

On Certain Functional Identities in EN

1971 ◽  
Vol 23 (2) ◽  
pp. 315-324 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Euclidean Space ◽  
Taylor Series ◽  
Dimensional Space ◽  
The Real ◽  
Higher Dimensional ◽  
Isolated Points ◽  
Functional Identities ◽  
Image Position ◽  
Special Case ◽  
Real Euclidean Space

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity1.1in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to1.2In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of1.3to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

On a Theorem of Hayman Concerning Quasi-Bounded Functions

1959 ◽  
Vol 11 ◽  
pp. 593-600
Author(s):  
P. B. Kennedy
Keyword(s):  
Meromorphic Functions ◽  
Bounded Function ◽  
Fatou Theorem ◽  
Regular Functions ◽  
Bounded Functions ◽  
Image Position ◽  
Special Case

If f(z) is regular in |z| < 1, the expressionis called the characteristic of f(z). This is the notation of Nevanlinna (4) for the special case of regular functions; in this note it will not be necessary to discuss meromorphic functions. If m(r,f) is bounded for 0 < r < 1, then f(z) is called quasi-bounded in |z| < 1. In particular, every bounded function is quasibounded. The class Q of quasi-bounded functions is important because, for instance, a “Fatou theorem” holds for such functions (4, p. 134).

Note on Best Approximation of |x|

1979 ◽  
Vol 22 (3) ◽  
pp. 363-366
Author(s):  
Colin Bennett ◽  
Karl Rudnick ◽  
Jeffrey D. Vaaler
Keyword(s):  
General Problem ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Best Uniform Approximation ◽  
Linear Fractional Transformations ◽  
Symmetric Complex ◽  
Image Position ◽  
Complex Valued ◽  
Special Case

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximationand the extremal transformations U whenever they exist.

Holomorphic Functions with Positive Real Part

1982 ◽  
Vol 34 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Eric Sawyer
Keyword(s):  
Unit Ball ◽  
Hardy Spaces ◽  
Convex Cone ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Spaces Of Holomorphic Functions ◽  
Extreme Element ◽  
Image Position ◽  
Special Case

The main purpose of this note is to prove a special case of the following conjecture.Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functionsis in Hp(B2) for 0 < p < 3/2.

Sign in / Sign up

Export Citation Format

Share Document