Cube Volume Research Articles

Effect of training on left ventricular structure and function. An echocardiographic study.

To study the cardiac effects of running, 15 healthy non-athletic male volunteers underwent a moderate training schedule for six weeks; thereafter five continued to run at a similar intensity for a further six weeks and 10 stopped. Left ventricular wall thickness and dimension, relaxation, and diastolic function were studied by digitised echocardiography at entry and at two, four, and six weeks, and again at 12 weeks. Significant increases in running ability and maximum oxygen consumption were observed. Maximum oxygen consumption was unchanged in those who continued to run but a reduction was noted in those who stopped. No significant changes in left ventricular dimensions were observed, but the thickness of the posterior wall and the septum increased during six weeks' running. Left ventricular mass (cube volume formula) increased significantly during the first six weeks. Comparison of those who continued to train and those who stopped showed that the former had no change in mass, whereas in the latter mass returned towards pre-exercise values. Systolic function (fractional shortening) was unchanged throughout the study. In addition, no abnormalities of relaxation or diastolic function were detected. Exercise induced left ventricular hypertrophy may develop rapidly during training and the increased myocardial mass resulting from six or 12 weeks' running is not associated with impaired relaxation and diastolic function as found in pathological forms of hypertrophy.

A geometric inequality with applications to linear forms

Let CN be a cube of volume one centered at the origin in RN and let Pκ be a iΓ-dimensional subspace of RN. We prove that CN Π Pκ has iΓ-dimensionai volume greater than or equal to one. As an application of this inequality we obtain a precise version of Minkowski's linear forms theorem. We also state a conjecture which would allow our method to be generalized. I* Introduction* Let CN — [ — 1/2, 1/2]^ be the iV-dimensional cube of volume one centered at the origin in RN and suppose that Pκ is a if-dimensional linear subspace of RN. Dr. Anton Good has conjectured that the iί-dimensional volume of Pκ Π CN is always greater than or equal to one. In case K = N — 1 this has recently been proved by Hensley [6], who also obtained upper bounds for this volume. Our purpose in this paper is to prove the conjecture for arbitrary K and to give some applications to Minkowski's theorem on linear forms. In fact we prove a more general inequality for the product of spheres of various dimensions which contains the conjecture as a special case. We write x for the column vector I I in Rn and

A central limit theorem for exchangeable variates with geometric applications

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN –1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV– 1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.

A central limit theorem for exchangeable variates with geometric applications

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN–1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV–1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.

The van der Waals limit for classical systems. I. A variational principle

We consider the thermodynamic pressurep(μ, γ) of a classical system of particles with the two-body interaction potentialq(r)+γvK(γr), where υ is the number of space dimensions, γ is a positive parameter, and μ is the chemical potential. The temperature is not shown in the notation. We prove rigorously, for hard-core potentialsq(r) and for a very general class of functionsK(s), that the limit γ→0 of the pressurep(μ, γ) exists and is given by where the limit and the supremum can be interchanged. Here ℛ is a certain class of nonnegative, Riemann integrable functions,D is a cube of volume |D|, anda0(ϱ) is the free energy density of a system withK=0 and density ϱ. A similar result is proved for the free energy.

Thermodynamic Properties of Small Systems

We investigate the dependence of the pressure of a homogeneous system, at a given density $\ensuremath{\rho}$ and temperature $T$, on the number of particles $N$. The particles of the system are assumed to interact via forces of finite range $a$ and are confined to a periodic cube of volume ${L}^{3}$, $\ensuremath{\rho}\ensuremath{\equiv}\frac{N}{{L}^{3}}$. We find that there are generally two types of $N$ dependencies in the pressure and other intensive properties of the system. There is a simple dependence which goes essentially as a power series in ($\frac{1}{N}$) and may be computed explicitly in terms of the grand-ensemble averages of these properties where it is absent. The other, more complex, dependence comes from the volume dependence of those cluster integrals which are large enough to wind at least once around the periodic torus. These do not appear in a virial expansion for terms $k\ensuremath{\le}{(\frac{N}{\ensuremath{\rho}{a}^{3}})}^{\frac{1}{3}}$. They play however a dominant role in the $N$ dependence observed by Alder and Wainwright in their machine computations on a hard-sphere gas. While the explicit calculation of these terms is very difficult and has been carried through only in a few special cases, they may be related, approximately at least, to the radial distribution function in an infinite system. We also find an expression for the correlation between the particles of an ideal gas represented by a microcanonical ensemble.

Precise measurement of the density of mercury at 20° C. II. Content method

The determinations described in this paper are complementary to earlier ones (Cook & Stone 1957) in which the density of mercury was found from the mass of mercury displaced by a solid cube of measured volume (displacement method). In the present work (content method) the density was calculated from the mass of mercury filling a hollow cube formed of optically worked blocks of fused silica, the internal dimensions of which could be measured by optical interference. The densities of three samples have been measured by both methods, the mean difference between methods being 0-45 p.p.m. (part per million). The silica cube consists of six optically worked blocks adhering by molecular forces without any cement. The separations of the internal faces are about 7-3 cm, the internal volume is about 390 cm<super>3</super> and the mass of mercury contained in it is about 5-3 kg. The mean separations of the internal faces have been measured with standard deviations of about 2 nm but changes in the separation over periods of 6 months or so are nearly ten times this. The uncertainty of the measured volume of the cube is therefore predominantly due to the changes in dimensions; it corresponds to a standard deviation of 0-18 p.p.m. The cube was filled with mercury under vacuum through a capillary tube in the top face and mercury could be withdrawn through a hypodermic tube inserted through the capillary. The volume of mercury in the cube was found from the height of the meniscus in the capillary when the cube was immersed in a thermostatic bath which had been kept at a temperature constant to 1 to 2 x 10<super>-3</super> deg C for 18 to 24 h. The mass of mercury in the cube was found by weighing the cube empty and full against a similar empty cube as counterpoise. The overall uncertainty of the mass of mercury filling the cube at a specified temperature has a standard deviation of 0-35 p.p.m. The standard deviation of the measured density of any one sample is about 0-3 p.p.m., depending on the number of determinations on the sample. The densities of six samples were measured; three had been prepared for the measurements by the displacement method, one was a sample used to fill primary barometers at the N.P.L., one was supplied by the National Standards Laboratory, Australia, and the sixth by the National Bureau of Standards, Washington, on behalf of the International Union of Pure and Applied Physics. The range of all measurements by both content and displacement methods on all samples is 1.7 p.p.m. The differences between samples are significant; the mean of all measurements is 13.545884 g/cm<super>3</super> at 20 °C and 1 atm pressure. The corresponding value at 0 °C and 1 atm calculated from the expansion formula of Beattie, Blaisdell, Kaye, Gerry & Johnson (1941) is 13.595080 g/cm<super>3</super>. The values assigned to the standard wavelengths used in the interferometry measurements were changed between the displacement and content methods and the results of the former have been recomputed accordingly.

Density Change of K<sub>2</sub>O-Na<sub>2</sub>O-SiO<sub>2</sub> Glasses by Heating

Using purified sand and the materials of reagent grade K2O-Na2O-SiO2 glasses were melted in a platinum crusible.The quenched samples were subjected to reheating by holding at a constant temperature in the range of 300°-525°C, and the change of the density was measured by floating method.The data obtained were analysed in the light of the new theory posturated by the author (cf. J. Ceram. Assoc. Japan, 66, 117, 1958).The density change ΔD may be represented byΔD=_??_pie-t/φi=pe-t/φ4+qe-t/φ3+re-t/φ2+se-t/φ1+…where t is the time, φ relaxation time, P, q, …are the constants whose absolute values increase monotonously with decreasing temperature, having the increments falling off below a certain temperature. φ′s have the respective apparent activation energies and increase with decreasing temperature. The apparent activation energies, however, are possible to become more or less smaller with decreasing temperature.The activation energy of the largest time constant, φ4, have the value close to that of viscosity giving φ4≅480 min at the temperature corresponding to 1014.3 pois. Assuming that the part of the density change represented by this term takes place with the flow of the same structural units as in viscous flow it is possible to calculate the size of the units by applying the Eyring's formula with a result that the volume is about 15 Å3 containing 1-2 O2- ions.Around the strain point it may fairly be presumed that the mobility of separate ions would predominate that of molecular groups.The other terms than the first correspond to the elastic after effect in the range of transformation temperature and below, being probably caused by the expansion, contraction, and rotation of silicate chain anions. There is a possibility of giving a wide distribution of comparatively low activation energies which take part in the density change at low temperatures.In the range of composition covered by the present paper the amount of contraction by heating decreased with increasing alkali content, the trend which reversed at the alkali content of 20-25 mol per cent. This decrease of contraction may be interpreted by the obstructing effect of alkaline ions to the change of network structure, and the point of inflection would correspond to the same trend of the composition-density diagrams.It is highly probable that the density change caused by the migration of alkaline ions would be less than 10-3g/cm3. In mixed alkali-glasses, therefore, the influence of thermal history would be highly reservable owing to the small mobility of mixed alkaline ions, and the larger contribution of the place exchange of alkaline ions to the density change would bring about the larger thermal after effect.The author modified his model representing the energy state of a glass structure by an assembly of small cubes of equal volume each being assigned the minimum free energy of the level, L1, L2, …as follows:(1) Li is composed of the levels l11, l12 having the values of free energy close to each other.(2) lij are composed of lij1*, lij2*.And the transformation L↔L corresponds, for example, to the growth and disappearance of microphases which bring about the change of structure, l↔l to the expansion, contraction, and rotation of the chain structure-units, and l*↔l* to the movement of alkalineions.

On upper bounds for error detecting and error correcting codes of finite length

Upper bounds for error detecting and error correcting codes are obtained in this paper. One upper bound is found by exploiting the geometrical model of coding introduced by Hamming. The volume of an appropriate geometrical body is compared with the volume of the unit cube, in getting the first upper bound. An improvement on this upper bound can be found by introducing a mass density function, and comparing the mass of the body with the mass of the unit cube. A comparison is made with known upper bounds, and with best codes found thus far. The improved upper bound given here is frequently somewhat smaller than previously known upper bounds.

Precise measurements of the density of mercury at 20°C I. Absolute displacement method

This paper gives an account of the absolute measurement of the density of mercury at 20° C in units of length and mass, the result of which is probably correct to one part in a million. The densities of four samples of mercury have been measured by finding the mass of mercury displaced by an accurately formed cube of known volume that just sinks in mercury. The cube is of tungsten carbide sintered with cobalt, its sides are 8.9 cm long, it weighs 9.7 kg in air and 217 g in mercury at 20° C. It was lapped with diamond dust to such an accurate form and good surface finish that the standard deviation of the volume of mercury displaced, as calculated from optical interference measurements of the dimensions of the cube, is 0.15 p.p.m. (part per million). The determined mass of the cube has a standard deviation of 0.1 p.p.m. and the measured weight of the cube in any one sample of mercury at 20° G has a standard deviation of 0.28 p.p.m. The uncertainty in the density of any one sample of mercury is however greater, for two reasons, than the value of 0.3 p.p.m. that these figures indicate. First, all measurements of the temperature of the mercury are subject to a constant error which may reach 0.001 ° C, equivalent to 0.2 p.p.m. of the density of mercury. Secondly, and more important, the effective volume of the cube may be greater than the measured volume through layers of grease adhering to it despite very careful cleaning and the corresponding error probably varied from one group of weighings to another; it is difficult to estimate but is thought to be between 0.2 and 1 p.p.m. The mean density at 20° C and 1 atm pressure of four samples of mercury from different sources is 13.545892 g/cm3, the range of the four values being 1.1 p.p.m. The corresponding value at 0° C, calculated from the thermal expansion formula of Beattie, Blaisdell, Kaye, Gerry & Johnson (1941) is 13.595086 g/cm3. The chemical and isotopic compositions of the samples cannot be specified with an accuracy corresponding to that of the measurements, but evidence is given indicating that our results should represent the density of pure mercury of average isotopic constitution to between 1 and 2 p.p.m. Complementary measurements are now being prepared in which the mass of mercury that fills a box of fused silica is to be found.

Dispersion of Fillers in Rubber

Abstract The activity of a filler in rubber depends on the size and geometrical shape of the particles, and on their chemical constitution. The smaller a particle is, the larger is the total surface area of the filler per unit weight. If, for example, the diameter of a particle of an active carbon black is 30 mµ, the total surface area is about 100 sq. m. per gram. If the particle diverges from spherical shape and has a crystalline form, the surface area is greater, as the relation of the area of a sphere to that of a cube of equal volume clearly shows (surface area of sphere: surface area of cube = 12.5:15.6). Furthermore, corners and edges are present on a particle of crystalline shape, which are particularly active according to the experimental results of interface physics. By means of this phenomenon, the differing activity of various fillers of equal particle diameter can be explained.

On the Volume of a Topological Cube

On the Volume of a Topological Cube