The supercritical phase of percolation is well behaved

1990 ◽  
Vol 430 (1879) ◽  
pp. 439-457 ◽  
Keyword(s):  
General Result ◽  
Critical Probability ◽  
Percolation Process ◽  
The Subject ◽  
Supercritical Phase

We prove a general result concerning the critical probabilities of subsets of a lattice L . It is a consequence of this result that the critical probability of a percolation process on L equals the limit of the critical probability of a slice of L as the thickness of the slice tends to infinity. This verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation.

scholarly journals IV. Comparison of some recently determined refractive indices with theory

1860 ◽  
Vol 10 ◽  
pp. 199-204
Keyword(s):  
General Result ◽  
Dispersive Media ◽  
Refractive Indices ◽  
Numerical Comparison ◽  
Great Problem ◽  
Extensive Range ◽  
Complete Form ◽  
The Subject ◽  
Dispersive Power

In a series of papers inserted in the Philosophical Transactions (1835,1836,1837), and afterwards, in a more correct and complete form, in my Treatise ‘On the Undulatory Theory applied to the Di­spersion of Light’ (1841), I endeavoured to investigate the great problem of the explanation of the unequal refrangibility of light on the principles of the undulatory theory, as proposed by M. Cauchy about 1830, by numerical comparison with the indices observed, more especially in cases of the most highly dispersive media then examined. The general result then arrived at was, that while the theory applied perfectly through an extensive range of media of low and moderate dispersive power, it did not apply well to those of higher; and to the highest in the scale (which of course formed the true test of the theory) it did not apply within any allowable limits of accuracy. Since that time little has been done towards prosecuting the subject.

XIX.—On the Composition of Milk as affected by Increase of the Amount of Calcium Phosphate in the Rations of Cows

1915 ◽  
Vol 35 ◽  
pp. 195-202 ◽  
Author(s):  
A. Lauder ◽  
T. W. Fagan
Keyword(s):  
Calcium Phosphate ◽  
Phosphoric Acid ◽  
General Result ◽  
Food Supply ◽  
General Consensus ◽  
The Subject ◽  
Mineral Constituents

The various factors which are supposed to influence the composition of milk have already been the subject of numerous investigations. The general result of these has been to show that within very wide limits the composition of milk is very little affected by the nature of the food supplied. As regards the mineral constituents the results of some of the investigations are rather conflicting, and while there is a general consensus of opinion that the composition and amount of the mineral constituents are independent of the food-supply, certain investigators claim to have been able to increase both the calcium and phosphoric acid in the milk by slight amounts.

scholarly journals IV.—On the Amount of Sand brought up by Lobworms to the Surface

1891 ◽  
Vol 8 (11) ◽  
pp. 489-493 ◽  
Author(s):  
Charles Davison
Keyword(s):  
General Result ◽  
Charles Darwin ◽  
Short Stay ◽  
The Subject

Ten years ago Charles Darwin published his last work, the result of more than forty years' observations on the habits of earthworms and the rate at which they bring up soil to the surface. To ascertain this rate, he made use of two methods. In the first place he measured the rate at which layers of lime or cinders were covered over by the spread-out castings of worms; and, later, he collected and weighed all the castings thrown up over a definite area within a given time. As is well known, the general result of his investigation was greatly to exalt our ideas of the importance of earthworms and of the work they do upon the surface of the land.The work performed by lobworms on the surface of tidal sands seeming not less worthy of study, I made some observations on the subject during a short stay in Holy Island last August.

Critical sponge dimensions in percolation theory

1981 ◽  
Vol 13 (02) ◽  
pp. 314-324 ◽  
Author(s):  
G. R. Grimmett
Keyword(s):  
Positive Constant ◽  
Percolation Theory ◽  
Square Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Percolation Process ◽  
Open Path ◽  
Image Position

In the bond percolation process on the square lattice, with let S(k) be the probability that some open path joins the longer sides of a sponge with dimensions k by a log k. There exists a positive constant α = αp such that Consequently, the subset of the square lattice {(x, y):0 ≦ y ≦ f(x)} which lies between the curve y = f(x) and the x-axis has the same critical probability as the square lattice itself if and only if f(x)/log x → ∞ as x → ∞.

scholarly journals XIII. Hydraulic investigations, subservient to an intended Croonian Lecture on the motion of the blood

1808 ◽  
Vol 98 ◽  
pp. 164-186 ◽  
Keyword(s):  
General Result ◽  
Royal Society ◽  
Analytical Formula ◽  
Elastic Tube ◽  
Future Time ◽  
Late Professor ◽  
Separate Paper ◽  
The Subject ◽  
Circulation Of The Blood ◽  
Different Parts

Having lately fixed on the discussion of the nature of in­flammation, for the subject of an academical exercise, I found it necessary to examine attentively the mechanical principles of the circulation of the blood, and to investigate minutely and comprehensively the motion of fluids in pipes, as affected by friction, the resistance occasioned by flexure, the laws of the propagation of an impulse through the fluid contained in an elastic tube, the magnitude of a pulsation in different parts of a conical vessel, and the effect of a contraction advancing progressively through the length of a given canal. The phy­siological application of the results of these inquiries I shall have the honour of laying before the Royal Society at a future time; but I have thought it advisable to communicate, in a separate paper, such conclusions, as may be interesting to some persons, who do not concern themselves with disquisitions of a physiological nature; and I imagine it may be as agreeable to the Society that they should be submitted at pre­sent to their consideration, as that they should be withheld until the time appointed for the delivery of the Croonian Lecture. It has been observed by the late Professor Robison, that the comparison of the Chevalier Dubuat's calculations with his experiments is in all respects extremely satisfactory; that it exhibits a beautiful specimen of the means of expressing the general result of an extensive series of observations in an analytical formula, and that it does honour to the penetra­tion, skill, and address of Mr. Dubuat, and of Mr. de St. Honore', who assisted him in the construction of his expres­sions: I am by no means disposed to dissent from this enco­mium; and I agree with Professor Robison, and with all other late authors on hydraulics, in applauding the unusually accurate coincidence between these theorems and the experi­ments from which they were deduced.

Random fractals and their intersection with winning sets

2021 ◽  
pp. 1-30
Author(s):  
YIFTACH DAYAN
Keyword(s):  
Hausdorff Dimension ◽  
General Result ◽  
Countable Collection ◽  
Percolation Process ◽  
Fractal Percolation ◽  
Random Fractals ◽  
Affine Hyperplane ◽  
Badly Approximable

Abstract We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$ ), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$ , $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$ , where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$ . We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$ . We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.

Critical sponge dimensions in percolation theory

1981 ◽  
Vol 13 (2) ◽  
pp. 314-324 ◽  
Author(s):  
G. R. Grimmett
Keyword(s):  
Positive Constant ◽  
Percolation Theory ◽  
Square Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Percolation Process ◽  
Open Path ◽  
Image Position

In the bond percolation process on the square lattice, with let S(k) be the probability that some open path joins the longer sides of a sponge with dimensions k by a log k. There exists a positive constant α = αp such that Consequently, the subset of the square lattice {(x, y):0 ≦ y ≦ f(x)} which lies between the curve y = f(x) and the x-axis has the same critical probability as the square lattice itself if and only if f(x)/log x → ∞ as x → ∞.

scholarly journals Observations on certain cases of elliptic polarization of light by reflection

1843 ◽  
Vol 4 ◽  
pp. 436-437
Keyword(s):  
General Result ◽  
Analytical Investigation ◽  
Experimental Examination ◽  
Intermediate Point ◽  
Reflected Light ◽  
Elliptic Polarization ◽  
The Subject ◽  
Reflection Of Light ◽  
Polarization Of Light ◽  
Curious Property

The author, by way of introduction, passes in review the labours of various inquirers on the subject of the elliptic polarization of light, and notices more particularly those of Sir David Brewster, who first discovered this curious property, as recorded in the Philosophical Transactions for 1830; of Mr. Airy, in the Cambridge Transactions for 1831 and 1832; and of Professor Lloyd, in the Philosophical Transactions for 1840, and in the Reports of the British Association for 1841. He then proceeds to give an account of his own experimental examination of the phenomena of elliptic polarization in the reflection of light from various surfaces, by observing the modifications of the polarized rings under different conditions, both of surface and of incidence, and by endeavouring to ascertain both the existence and amount of ellipticity, as shown by the dislocation of those rings, and to determine its peculiar chataracter, as indicated by the direction in which the dislocation takes place; the protrusion of the alternate quadrants appearing it certain cases, in one direction, and in others in the opposite. These observations are reducible to two classes; first, those designed to contribute to the inquiry, what substances possess the property of elliptic polarization, by examining the light reflected from bodies; and second, those made on certain cases on of films of several kinds, including those formed on metals by oxidation or other action upon the metal itself, as well as by extraneous deposition. The author found the general result, in all these cases, to be, that from any one tint to another, through each entire order of tints, the form of the rings in the reflected light undergoes certain regular changes, passing from a dislocation in one direction to that in the opposite, through an intermediate point of no dislocation, or of plane polarization; and thus exhibiting a dark and a bright centred system alternately, as long as the order of tints are preserved pure. These changes in the form of the rings, he observes, are precisely those expressed by successive modifications of Mr. Airy's formula, corresponding to the increments in the retardation which belong to the periodical colours of the films. The remaining portion of the paper is occupied by a description of the apparatus and mode of conducting the experiments; and of the observations made on mica, on decomposed glass, plumbago, daguerreotype, and other metallic plates, and on the coloured films produced on steel and on copper by the action of heat, and of voltaic electricity. The author gives, in conclusion, an analytical investigation of Mr. Airy’s general formula.

Experiments on the chemical constitution of several bodies which undergo the vinous fermentation, and on certain results of the chemical action

1843 ◽  
Vol 4 ◽  
pp. 149-151
Keyword(s):  
General Result ◽  
Carbonic Acid ◽  
Grape Juice ◽  
Chemical Changes ◽  
Chemical Action ◽  
Chemical Constitution ◽  
The Subject ◽  
The Common ◽  
The Difference

The special object of this paper is to show, first, that sugar is not constituted of carbon and water only; secondly, that during the vinous fermentation water is decomposed; thirdly, that neither pure carbonic acid nor alcohol is, in the common acceptation of the term, the product of this chemical action; and fourthly, that fermented liquors owe some of their valuable qualities to peculiar products formed during fermentation. In order to trace the various chemical changes which occur in this part of his research, the author has had recourse to numerous experiments, the details of which are recorded in tabular forms. The first table exhibits the analysis of different kinds of sugar, honey, treacle, grape-juice and extract of malt and hops, the general result of which is that all these compounds contain oxygen in excess above the proportion in which it exists in water, and that they also contain a small quantity of nitrogen. He shows, by two independent modes of experimenting, that these bodies, when in solution, cannot be the only compounds undergoing decomposition during that fermentation, which has for its product spirit and carbonic acid; and in proof of this proposition he recapitulates the different elements in the compounds at the commencement and at the conclusion of the experiments. He finds that when the proximate elements are made the subject of calculation, the weight of the alcohol (constituted of two equivalents of carbon, three of hydrogen and one of oxygen) added to that of the carbonic acid and undecomposed sugar, exceeds the weight of the sugar employed by about 7 per cent. On recapitulating the ultimate elements, he finds that the hydrogen and the oxygen in the compounds after the fermentation exceed their quantity in the sugar experimented upon, by 15 per cent, of the former, and nearly 14 per cent, of the latter; and as a proof that no material error is occasioned by the mode of experimenting, it is found that the difference between the quantity of carbon at the first and at the last is very small.

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