scholarly journals On the relations between systems of curves which, together, cut their plane into squares

1888 ◽  
Vol 7 ◽  
pp. 2-3
Author(s):  
Tait
Keyword(s):  
Unit Vector ◽  
Common Plane ◽  
The Common ◽  
Image Position

If ρ be the vector of a corner of a square in one system, σ that in a system derived without inversion, we must obviously havek being the unit-vector perpendicular to the common plane.

Optimal selection of stochastic intervals under a sum constraint

1987 ◽  
Vol 19 (2) ◽  
pp. 454-473 ◽  
Author(s):  
E. G. Coffman ◽  
L. Flatto ◽  
R. R. Weber
Keyword(s):  
Decision Rule ◽  
Selection Process ◽  
Random Variables ◽  
Optimal Threshold ◽  
Optimal Selection ◽  
Expected Number ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Selection Of

We model a selection process arising in certain storage problems. A sequence (X1, · ··, Xn) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected.We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En(c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

III.—The Application of Operators to the Solution of the Algebraic Equation

1918 ◽  
Vol 37 ◽  
pp. 18-49
Author(s):  
James Littlejohn
Keyword(s):  
Algebraic Equation ◽  
The Common ◽  
Image Position

The solution of the common quadraticwhich is usually writtenis, when expanded,This expansion will be found to be the work of the operator each term being derived from its predecessor by this operator's agency, so that the whole solution may be written

scholarly journals IX.—Some Identities connected with Alternants, and with Elliptic Functions

1905 ◽  
Vol 40 (1) ◽  
pp. 187-201
Author(s):  
Thomas Muir
Keyword(s):  
Elliptic Functions ◽  
The Common ◽  
Image Position

(l) Cayley in his paper entitled “Note sur l'addition des fonctions elliptiques” obtains among other similar things an expression forin terms ofwhereThe form of the expression is the quotient of two determinants, and as the expression becomes useless for such cases as u = v, u = w, … on account of the simultaneous vanishing of numerator and denominator, he is led to seek a means of throwing out the common evanescent factors.

scholarly journals Improved Bounds for Incidences Between Points and Circles

2014 ◽  
Vol 24 (3) ◽  
pp. 490-520 ◽  
Author(s):  
MICHA SHARIR ◽  
ADAM SHEFFER ◽  
JOSHUA ZAHL
Keyword(s):  
Three Dimensional ◽  
Combinatorial Geometry ◽  
Two Dimensions ◽  
Three Dimensions ◽  
List Type ◽  
Planar Case ◽  
Constant Degree ◽  
Common Plane ◽  
Formula Group ◽  
Image Position

We establish an improved upper bound for the number of incidences betweenmpoints andncircles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, isO*(m2/3n2/3+m6/11n9/11+m+n), where theO*(⋅) notation hides polylogarithmic factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in ℝ3without first improving it in the plane.Nevertheless, we show that if the set of circles is required to be ‘truly three-dimensional’ in the sense that no sphere or plane contains more thanqof the circles, for someq≪n, then for any ϵ > 0 the bound can be improved to\[ O\bigl(m^{3/7+\eps}n^{6/7} + m^{2/3+\eps}n^{1/2}q^{1/6} + m^{6/11+\eps}n^{15/22}q^{3/22} + m + n\bigr). \]For various ranges of parameters (e.g., whenm= Θ(n) andq=o(n7/9)), this bound is smaller than the lower bound Ω*(m2/3n2/3+m+n), which holds in two dimensions.We present several extensions and applications of the new bound.(i)For the special case where all the circles have the same radius, we obtain the improved boundO(m5/11+ϵn9/11+m2/3+ϵn1/2q1/6+m+n).(ii)We present an improved analysis that removes the subpolynomial factors from the bound whenm=O(n3/2−ϵ) for any fixed ϵ < 0.(iii)We use our results to obtain the improved boundO(m15/7) for the number of mutually similar triangles determined by any set ofmpoints in ℝ3.Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.

The Common Fossils of the British Rocks

The Geologist ◽  
1858 ◽  
Vol 1 (2) ◽  
pp. 60-64
Author(s):  
S. J. Mackie
Keyword(s):  
High Interest ◽  
The Past ◽  
Book Of Nature ◽  
The Common ◽  
True Knowledge ◽  
Mechanical Devices ◽  
Image Position ◽  
Holy Book

More things remain to be spoken of fossils; wonders of skill to be presented in their construction and design; marvels of mechanical devices for progression, for strength, for lightness, or for protection to be displayed; and consummate wisdom and benevolent forethought to be exhibited in their adaptation to the various purposes for which they were created—in fact, as much as we find to admire or to consider in the structure of existing animals or plants, in their means of developement or of growth, in the influences of climate and seasons upon them, so much also do we find for equal admiration and reflection in those ancient “medals” of past creations.Even contorted and damaged fossils are not without their evidences.Squeezed on either side or flattened, they do not merely ndicate the pressure to which they have been subjected, but the direction also from which it came. Everything connected with fossils is of high interest; but from first to last the value of fossils is in their teachings; and it is never by pounds, shillings, and pence that we can value them at all. In such a light they are but worthless bits of stone, as fit to mend the roads as to be saved. To minds that esteem them thus, they are no treasures, but merely merchandise. Properly studied, however, they convey their lessons of the past; and when regarded as letters in the vast and holy book of Nature, which must ever be read with solemnity and reverence, they take their places properly in the great sentences and wonderful passages of that mysterious language from which Geology interprets the order, wisdom, goodness, and prescience displayed in the animated worlds that were. It is thus we shall have attained to the true knowledge of the value of fossils, when we shall turn from such readings with adoration to the Great Author of all.

Plastic Deformation of Steel Safety Guardrails

2001 ◽  
Author(s):  
Detlef Neuenhaus ◽  
Christian Kammel ◽  
Urs Joachim Geßler ◽  
Gerhard Sedlacek
Keyword(s):  
Multibody Systems ◽  
Tangential Direction ◽  
Normal Direction ◽  
Virtual Power ◽  
Program Code ◽  
Contact Zones ◽  
Vehicle Impact ◽  
Common Plane ◽  
The Common ◽  
Sophisticated Algorithm

Abstract In case of contact between two colliding bodies various force interactions in the contact area occur. Therefore, the virtual power for contact zones has to comprise impulse or contact in normal direction and friction in tangential direction of the common plane depending on the way of relative movement. Within this paper a sophisticated algorithm for the description of the virtual power in contact zones is presented. Hence, the implementation of this algorithm to the MEPHISTO (Multibody systems with ElasticPlastic HInges and changeable STructure Organization) multibody program code is successfully proved in the particular case of vehicle impact on a safety barrier system.

scholarly journals 5. The axisymmetric case in hydromagnetics

1958 ◽  
Vol 6 ◽  
pp. 46-49
Author(s):  
S. Chandrasekhar ◽  
Kevin H. Prendergast
Keyword(s):  
Magnetic Field ◽  
Recent Work ◽  
Hydrodynamic Equation ◽  
Azimuthal Angle ◽  
Unit Vector ◽  
Velocity Fields ◽  
Laplacian Operator ◽  
Axis Of Symmetry ◽  
Image Position ◽  
The Magnetic Field

Recent work at the Yerkes Observatory has been concerned with the study of configurations in which the magnetic and velocity fields possess a common axis of symmetry. In those cases where the density ρ may be assumed constant, it has proved advantageous to employ a representation suggested by Lüst and Schlüter[1]: in cylindrical co-ordinates (ω, ϕ, z) let and where is a unit vector and T, P, V, and U are independent of the azimuthal angle Φ. The hydrodynamic equation may then be replaced by the pair of equations (cf. Ghandrasekhar [2]) and where Δ5 is the Laplacian operator in 5 dimensions. The equation for the magnetic field, may similarly be replaced by the pair of equations and

An extension of a lemma of Wald

1966 ◽  
Vol 3 (01) ◽  
pp. 272-273 ◽  
Author(s):  
H. Robbins ◽  
E. Samuel
Keyword(s):  
Natural Extension ◽  
Random Variables ◽  
Random Variable ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Independent Identically Distributed

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y 1, y 2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

Pseudo-Measure Energy and Spectral Synthesis

1974 ◽  
Vol 26 (4) ◽  
pp. 985-1001 ◽  
Author(s):  
John J. Benedetto
Keyword(s):  
Harmonic Analysis ◽  
Spectral Synthesis ◽  
Stieltjes Integral ◽  
Natural Notion ◽  
Intimate Relation ◽  
The Common ◽  
Image Position ◽  
Continuous Measures

In this paper we develop a natural notion of continuous pseudo-measure and study the Stieltjes integral with respect to a given pseudo-measure. The common feature to these two topics is the essential appearance in both of integrals having the formSuch integrals come about naturally when one defines the energy of distributions other than measures [6]. The reasons to study continuous pseudo-measures are to find properties analogous with those of continuous measures, and to discover more about the structure of pseudo-measures because of their importance in harmonic analysis, and particularly in spectral synthesis (e.g., [4;15]). The Stieltjes integral with respect to a pseudo-measure is studied because of its intimate relation with spectral synthesis (e.g., §5); the key observations on this matter were initially made by Beurling [6].

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