scholarly journals Homogeneous orbit closures and applications

2011 ◽  
Vol 32 (2) ◽  
pp. 785-807 ◽  
Author(s):  
ELON LINDENSTRAUSS ◽  
URI SHAPIRA
Keyword(s):  
Real Number ◽  
Unit Volume ◽  
Orbit Closures ◽  
Image Position

AbstractWe give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in ℝd for d≥3 with nice (homogeneous) orbit closures, as well as examples of orbits with explicitly computable but irregular orbit closures. We give Diophantine applications to the former; for instance, we show that, for all γ,δ∈ℝ, where 〈c〉 denotes the distance of a real number c to the integers.

6. On Laplace's Theory of the Internal Pressure in Liquids

1889 ◽  
Vol 15 ◽  
pp. 426-427
Author(s):  
Tait
Keyword(s):  
Internal Pressure ◽  
Unit Volume ◽  
Plane Surface ◽  
Surface Film ◽  
Liquid Particle ◽  
Molecular Force ◽  
Molecular Forces ◽  
Image Position

Laplace, assuming molecular force to be insensible at distances greater than a small quantity a, finds the resultant molecular force on a unit particle at a distance x within the (plane) surface. This being called X, the internal pressure iswhere p is the density of the liquid. But this is evidently the work required to take unit volume of the liquid (particle by particle) from the interior to the surface. And it is easily seen that to carry it from the surface beyond the range of the molecular forces requires just as much more work:—for the density of the surface-film is treated as equal to that of the rest of the liquid.

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION

2020 ◽  
pp. 1-11
Author(s):  
MARTIN BUNDER ◽  
PETER NICKOLAS ◽  
JOSEPH TONIEN
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Exact Formula ◽  
Positive Real ◽  
Image Position

For a positive real number $t$ , define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

1. Note on Ångström's Method for the Conductivity of Bars

1875 ◽  
Vol 8 ◽  
pp. 55-61 ◽  
Author(s):  
Tait
Keyword(s):  
Thermal Conductivity ◽  
Unit Volume ◽  
Transverse Section ◽  
Unit Surface ◽  
Image Position ◽  
Quantity Of Heat

If we assume the excess of temperature above that of the air, v, to be the same throughout a transverse section of the bar, the equation for the flux of heat is—where cρ is the water equivalent of unit volume of the bar, k its thermal conductivity, a its side, and hv the quantity of heat lost by radiation and convection from unit surface of the bar per unit of time, when the excess of temperature is v.

Eigenvalues of Finite Band-Width Hilbert Space Operators and Their Application to Orthogonal Polynomials

1989 ◽  
Vol 41 (1) ◽  
pp. 106-122 ◽  
Author(s):  
Attila Máté ◽  
Paul Nevai
Keyword(s):  
Hilbert Space ◽  
Orthogonal Polynomials ◽  
Real Number ◽  
Main Diagonal ◽  
Band Width ◽  
Positive Part ◽  
Hilbert Space Operators ◽  
Image Position ◽  
Finite Band

The main result of this paper concerns the eigenvalues of an operator in the Hilbert space l2that is represented by a matrix having zeros everywhere except in a neighborhood of the main diagonal. Write (c)+ for the positive part of a real number c, i.e., put (c+ = cif c≧ 0 and (c)+=0 otherwise. Then this result can be formulated as follows. Theorem 1.1. Let k ≧ 1 be an integer, and consider the operator S on l2 such that

An integral which occurs in statistics

1933 ◽  
Vol 29 (2) ◽  
pp. 271-276 ◽  
Author(s):  
A. E. Ingham
Keyword(s):  
Real Number ◽  
Positive Definite ◽  
Direct Evaluation ◽  
Independent Variables ◽  
Symmetrical Matrices ◽  
Image Position

1. In this note we give a direct evaluation of the integralwhose value has been inferred from the theory of statistics. Here A = Ap = (αμν) and C = Cp = (Cμν) are real symmetrical matrices, of which A is positive definite; there are ½ p (p + 1) independent variables of integration tμν (1 ≤ μ ≤ ν ≤ p), and tμν is written also as tνμ for symmetry of notation; in the summation ∑ the variables μ, ν run independently from 1 to p; k is a real number. A word of explanation is necessary with regard to the determination of the power |A − iT|−k. Since A is positive definite and T real and symmetric, the roots of the equation

scholarly journals The dilution assay of viruses

1954 ◽  
Vol 52 (2) ◽  
pp. 189-193 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Maximum Likelihood ◽  
Unit Volume ◽  
Rapid Test ◽  
Data Fitting ◽  
Average Density ◽  
Virus Particles ◽  
Exponential Curve ◽  
Dilution Assay ◽  
Image Position

Suppose that λ is the average density of virus particles per unit volume. If x is a dilution of this and unit volume is applied to an egg (or plate in other problems) the probability that the egg remains sterile isprovided that if a particle is present, it will infect the egg. To make a dilution assay we choose dilutions x1, …, xm (m levels) and apply these to n1, …, nm eggs. If these result in r1, …, rm sterile eggs we can estimate λ by maximum likelihood. The theory has been given by Barkworth & Irwin (1938), and full references to work on this problem will be found in Finney (1952). If we plot the quantities r1/n1, …, rm/nm against x1, …, xm we get a set of point whose fit to the curve (1) can be tested by a χ2 test. In a number of situations, however, it is found that (1) does not give a good fit. The estimation of λ is then completely invalid. In the present paper we consider why this happens, what types of curve may be fitted to the data and what they imply, and we also give a simple rapid test for such data fitting an exponential curve.

The limit theorem for aperiodic discrete renewal processes

1964 ◽  
Vol 4 (1) ◽  
pp. 122-128
Author(s):  
P. D. Finch
Keyword(s):  
Limit Theorem ◽  
Real Number ◽  
Renewal Process ◽  
Positive Real Number ◽  
Random Variables ◽  
Renewal Processes ◽  
Positive Real ◽  
Image Position

A discrete renewal process is a sequence {X4} of independently and inentically distributed random variables which can take on only those values which are positive integral multiples of a positive real number δ. For notational convenience we take δ = 1 and write where .

On the Product of Three Homogeneous Linear Forms. IV

1943 ◽  
Vol 39 (1) ◽  
pp. 1-21 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Linear Transformation ◽  
Linear Forms ◽  
Special Linear ◽  
Absolute Value ◽  
Cubic Equation ◽  
Image Position ◽  
Homogeneous Linear

Let L1, L2, L3 be three homogeneous linear forms in u, v, w with real coefficients and determinant 1. Let M denote the lower bound offor integral values of u, v, w, not all zero. I proved a few years ago (1) thatmore precisely, thatexcept when L1, L2, L3 are of a special type, in which case If we denote by θ, ø, ψ the roots of the cubic equation t3+t2-2t-1 = 0, the special linear forms are equivalent, by an integral unimodular linear transformation, to(in any order), where λ1,λ2,λ3 are real number whose product is In this case, L1L2L3|λ1λ2λ3 is a non-zero integer, and the minimum of its absolute value is 1, giving

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

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