Function Ofx Research Articles

Inclusive production of lambda in the proton fragmentation region from K−p→ΛX at 4.2 GeV/c

A study of Λ production has been made for the process\(p\underrightarrow {K^ - }\Lambda \) from K−p interactions at 4.2 GeV/c. The total Λ production cross-section is (4.5±0.2) mb and the cross-section in which Λ is associated with\(K\bar K\) in the final state is found to be (0.44±0.03) mb. A study of the polarization is made for the off-mass-shell strangeness annihilations\( \ll K\bar K \gg \to pions\) and strangeness nonannihilations\( \ll K\bar K \gg \to K\bar K + pions\); the polarization, when plotted as a function ofx,pT and (recoiling mass)2, exhibits differences between the two processes. The polarization observed in the strangeness nonannihilation process shows identical behaviour to that in\(p\bar p\) reactions. An attempt has been made to understand the behaviour of the polarized and the unpolarized cross-sections as a function of (recoiling mass)2 in the triple-Regge framework.

An integral criterion for oscillation of linear differential equations of second order

It is proved that if for some n>2 the function x1−nAn(x), where An(x) is the n-th primitive ofa(x), is not bounded above, then the equation y″ +a(x)y = 0 oscillates.

A simple model for studying the magnetism of an actinide impurity in a disordered transition host

We present a theoretical model able to describe the occurrence of localized magnetism in disordered systems such as A x B1−x Ac, where A and B are two transition elements and Ac is an actinide impurity. The disordered host is treated within the coherent potential approximation (CPA); thed level of the actinide impurity creates a Slater-Koster-liked scattering potential and thef states of the actinide impurity are broadened by thed–f hybridization. The occurrence of magnetic-nonmagnetic transitions as a function ofx is discussed within the Hartree-Fock approximation. In the limiting casesx=0 andx=1, this model improves the previous theories for the magnetism of an actinide impurity in a pure transition host and in the general case it permits some predictions, especially for the system Pd x Th1−x Pu.

Atomic volume dependence of the153Eu isomer shift in ScxY1?x

Mossbauer studies of dilute153Eu in Sm0.05ScxY0.95−x (0≤x≤0.95) reveal large changes in the isomer shift as a function ofx. The strong dependence of the isomer shift on alloy composition, is interpreted in terms of changes in the charge density at the Eu nucleus (ρ(0)), which result from changes in atomic volume. The isomer shift was found to depend linearly on ‡V/V (‡V/V changes between 0 and −0.3). A value of (−17.5±2.0)a0−3 was obtained for dρ(0)/d lnV.

A study of some exact solutions to Einstein's field equations which yield separable Hamilton-Jacobi equations

Exact solutions to Einstein's field equations, which give rise to a Stackel-separable Hamilton-Jacobi equation of the form $$,y,z)\left[ {X(x)\left( {\frac{{\partial S}}{{\partial x}}} \right)^2 - 2\left( {\frac{{\partial S}}{{\partial x}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) - 2\left( {\frac{{\partial S}}{{\partial y}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) + Z(z)\left( {\frac{{\partial S}}{{\partial z}}} \right)^2 - 2\left( {\frac{{\partial S}}{{\partial z}}} \right)\left( {\frac{{\partial S}}{{\partial t}}} \right) - F(x,y,z)\left( {\frac{{\partial S}}{{\partial t}}} \right)^2 } \right] = \lambda $$ are considered. It is shown that there are no solutions for whichD is a function ofx orz, orx andz. The exact solutions are of Petrov typeN and are plane polarized waves without rotation. Some of the solutions are given explicitly, up to two arbitary functions. For these solutions the Hamilton-Jacobi equation is reduced to an uncoupled set of first-order ordinary differential equations.

Studies of the dielectric constant in solid H2,4He, and Ne

Measurements of the static dielectric constant e and of the resulting polarizability αCM defined by the Clausius-Mosotti relation are reported for several molecular solids under conditions of quasiconstant density. The range of measurements was between 0.4 and about 15 K, and pressures up to about 1 kbar were used. In solid H2, the changes of α as a function of ortho concentration are reported at 4.2 and 12 K, from which it is concluded that [α(ortho)−α(para)]/α≃0.005, which is approximately 2 1/2 times larger than in the liquid phase. Furthermore the temperature change of αCM is studied for various densities at comparable ortho concentrationsX. Particular attention is paid to the orientational order-disorder transition in solid H2 withX≃0.65, where αCM shows an abrupt change that appears to be a function ofX, density, and thermal history. In general αCM is lower in the ordered (cubic) phase than in the disordered (hexagonal) phase. The temperature change of αCM in almost pure para-H2, measured between 4 and 0.4 K, is presented and is compared with other recently reported experiments. It is concluded that the mechanism that produces the change of αCM withT and at the order-disorder phase transition is not understood. In solid Ne, measurements of αCM are reported for one sample assumed to be at zero pressure and one where the density is quasiconstant. There is a strong temperature dependence in αCM for the former above 5 K (in agreement with previous findings), while for the latter αCM appears to be constant within the uncertainties introduced by strain-induced hysteresis effects. The sample of solid4He, investigated between its melting point at 4.3 and 0.8 K at constant density, shows a very weak temperature dependence of αCM, which is also consistent with previously reported measurements.

Inclusive production of lambda and neutral kaon in K−p interactions at 3.93 GeV/c and 14.3 GeV/c

Lambda and neutral-kaon production is studied in K−p interactions at 3.93 and 14.3 GeV/c. Topological inclusive cross-sections are given at both energies. Invariant cross-sections for lambdas and neutral kaons are presented as a function of the Feynmanx-variable and are found to decrease in magnitude and vary their shapes with energy. The\(AK\overline K \) production, which represents (9±2)% and (29±3)% of the lambda inclusive cross-section at 3.93 GeV/c and 14.3 GeV/c respectively, yields mainly backward lambdas. The shapes of the transverse-momentum invariant distributions vary withx, but are roughly independent of energy. The lambda polarization is studied as a function ofx at both incident momenta: for fast lambdas it is negative, as observed in reactions where nucleon exchange is dominant.

Solution of a game with a right to the first move and inexact information on the partner's aim

The present paper examines a modification of the game with unopposed interests first stated and solved in [ll. Our statement of the game differs from that in [II in just one respect: it is assumed that the first player does not know precisely the second player’s pay-off function, he only knows its range of variation, i.e. he has information on the functions fi’(x,, x2), fz+(Xl, x2), where f,+ (x1, x,) >/f2(xI, x2) >/f2-(x,, x,); the functions f,, fi’ and f2+ are assumed continuous, while x1 and x, belong to compacta X, and X,. AS in [II, therefore, the first player, counting on his information on the concrete choice of the second player 1c, s X,, chooses his strategy x,(x,> and communicates it to the second player. The final choice of strategy x,(x2) is fully determined by the first player’s attempt to ensure himself under these conditions the best guaranteed result, starting from the pay-off function fi(x,, n,). The second player’s behaviour is fully described by his attempt to increase his pay-off function f2(xl, x,) (not exactly known to the first player), by selecting x, E X,, knowing the function x1(x,).

On the Decomposition of Moments by Conditional Moments

M\'any practical problems in applied statistics involve the determination of the mean and variance of a certain statistic. The usual procedure is to attempt to find the distribution, i.e. probability density function or characteristic function, of the statistic, and then, if successful, to carry out the appropriate integration or differentiation. In many practical situations, information about the distribution may not be complete, or the sample may not consist of independent and identically distributed random variables, but of random variables whose distributions differ and which are correlated. The method of decomposition by conditional moments which requires only the knowledge of certain conditional moments may in some cases be helpful. Let X1, . . . , Xk be random variables of dimensions n, ... ., n, respectively. Assume that the means and covariances of the conditional random variables Xi I (Xi+1, .. ., Xk) for i =1,...,k-1 as well as the unconditional random variable Xk are known. If Z, and Z2 are any txvo random variables which are functions of X1, . . , Xk then for j = 1, 2,

Velocity of sound in solid hexagonal close-packed H2 and D2

Sound velocity measurements are reported on samples of hexagonal close-packed H2 and D2, believed to be single crystals. In the first part of the experiments, both the longitudinal and transverse velocities (v 1 and v t ) of crystals grown at a given melting pressureP M were measured at 4.2 K. For each mode, the velocities plotted vs.P M were found to lie between approximately parallel lines that represented the maximum and minimum observable velocities under the given experimental conditions set by the acoustic cell dimensions. From these extreme velocities v 1max, v 1min, v tmax, and v tmin an estimation of the elastic constants was made. Knowledge of these constants in turn permitted the calculation of the velocity surface, the bulk modulus, and the Debye temperature. In spite of the uncertainties in the analysis of the present data, a meaningful comparison could be made with results from other experiments. For both and D 2 , the elastic constants are on the average ∼20% below those derived from neutron scattering data. The transverse velocities, both for H 2 and D 2 , are consistent with those of Bezuglyi and Minyafaev for polycrystalline samples, but the longitudinal velocities are about 10% smaller than those of these authors. The bulk modulus and the Debye temperature from our experiments are compared with those from equation-of-state and calorimetric experiments. In the second part of this research, we have investigated the temperature dependence of the sound velocities at constant volume for both H 2 and D 2 with low concentrationsX of molecules with rotational angular momentumJ=1. From the measurements in H 2 , we are able to calculate the temperature change in the bulk modulus, which is then compared with that calculated from calorimetric data assuming the validity of the Gruneisen equation of state. Good agreement is obtained. In solid D 2 , were were able to deduce the temperature dependence of the adiabatic and isothermal modulusc 33 and make a rough estimation for that ofc 13 . In the third part we present measurements of v 1 and v t at 42 K and at constant volume as a function ofX in solid H 2 . It is found that as the orthopara conversion proceeds,v 1 increases andv t decreases. From these measurements the bulk modulus change withX is calculated. Comparison with the bulk modulus change calculated from static pressure measurements, assuming electric quadrupole-quadrupole interaction between the (J=1) molecules, shows very good agreement. The calculated dependence of the Debye temperature onX, as obtained from a combination of sound velocity and static measurements, is smaller than that reported from recent neutron scattering results. A short section is then devoted to absorption measurements in solid H 2 , which gave only qualitative results. Finally, in Section 4, we present measurements of the sound velocity in liquid D 2 at saturated vapor pressure as a function of temperature and for both liquid H 2 and D 2 along the melting curve.

Substitution of Fe3+ by Al3+ in the trigonal sites of M-type hexagonal ferrites

Mössbauer measurements in Ba1−ySryFe12−xAlxO19 (0<x<11, 0 <y < 1) compounds nave been made above their Curie temperatures. The number of Fe3+ ions in the trigonal sites of the structure, as a function both ofx andy, has been deduced. These data are discussed together with similar ones relative to gallium-substituted strontium ferrites. For y = 0, the Fe3+ ions in trigonal sites are not substituted by Al3+ ions up tox = 4, while above the miscibility gap (x > 7.2) they rapidly decrease and go down to zero forx - 10. On the contrary, for a givenx, the number of Fe3+ ions in trigonal sites increases with increasingy.

Gauge invariance of the Dirac equation

The Dirac equation in its most usual form can be derived by the minimum action principle from the Lagrangian density where γk (k =1,2,3,4) are the Dirac 4 × 4 matrices satisfying relations; ψ(x) is a column matrix with 4 complex elements being functions of x1, x2, x3, x4, coordinates of the pseudo-Euclidean space with metric gik.(

On the convergence of optimum stratifications for empiric distribution function in univariate case

Let {X n },n=1,2,..., be a sequence of independent random variables distributed according to a distribution functionF(x) with finite variance,F n (x) be the empiric distribution function ofX 1,...,X n for eachn, andφ () * andφ * be optimum stratifications corresponding toF n (x) andF(x) respectively. It is shown in this paper thatφ () * tends almost surely toφ * under a suitable criterion.

High density shot noise and gaussianity

The distance from Gaussianity of the shot noise processis considered, wheretiare the random times of a Poisson process with average densityλ(t).WithF(x) the distribution function ofx(t) andG(x) that of a normal process with the same mean and variance asx(t) it is shown thatwhereIf the processx(t) is stationary with λ(t) =λandh(t, τ) =h(t – τ) and the functionh(t) is bandlimited by ωc, then the above yields

On asymptotic independence of the partial sums of positive and negative parts of independent random variables

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {Xn} be a sequence of independent identically distributed random variables with common distribution function F, assume F is in the domain of attraction of a stable distribution with characteristic exponent α, 0 &lt; α ≦ 2, and let {Bn} be normalizing coefficients for F. Let us denote Xn+ = XnI[Xn &gt; 0] and Xn− = − XnI[Xn&lt;0], so that Xn = Xn+ - Xn−. Let F+ and F− denote the distribution functions of X1+ and X1− respectively, and let Sn(+) = X1+ + · · · + Xn+, Sn(-) = X1− + · · · + Xn−. The problem considered here is to find under what conditions there exist sequences of real numbers {an} and {bn} such that the joint distribution of (Bn-1Sn(+) + an, Bn-1Sn(-) + bn) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α &lt; 2 and when α = 2. When α &lt; 2, there always exist such sequences so that the above is true, and in this case both U and V are stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 &lt; ∫ x2dF(x) &lt; ∞, then there always exist such sequences such that the above convergence takes place; both U and V are normal (possibly one is a constant), and if neither is a constant, then U and V are not independent. If α = 2 and ∫ x2dF(x) = ∞, then at least one of F+, F− is in the domain of partial attraction of the normal distribution, and a modified statement on the independence of U and V holds. Various specialized results are obtained for α = 2.

Enclosure Theorems for Eigenvalues of Elliptic Operators

LetLbe the linear, elliptic, self-adjoint partial differential operator given bywhereDjdenotes partial differentiation with respect toxj, 1 ≤j≤n,bis a positive, continuous real-valued function ofx= (x1,…,xn) inn-dimensional Euclidean spaceEn, theaijare real-valued functions possessing uniformly continuous first partial derivatives inEnand the matrix {aij} is everywhere positive definite. A solutionuofLu= 0 is assumed to be of class C1.

On the first boundary problem for a hyperbolic equation in an arbitrary cylinder

A study is made of the solutions of a second-order hyperbolic equation which vanish on the boundary of an arbitrary domain in the space of the variables x1..., xn The degree of smoothness in the initial conditions, necessary and sufficient to guarantee the same degree of smoothness in the solution (considered as a function of x1..., xn for all t, is ascertained.

Solution of the equations of a regular electrostatic beam in the presence of emission from an arbitrary surface

An analytic solution of the equations of a regular electrostatic beam in the presence of emission from an arbitrary surface under total space charge conditions is given. It is assumed that the emitter is a coordinate surface x1= const in an orthogonal system xi (i=1, 2, 3), and the emission-current density J is a given function J(x2, x3). The solution is represented in the form of series in x with coefficients that are functions of x2, x3 and determined from recurrence relations. In expansion along the length of an arc of the curvilinear axis x1, which is orthogonal to the emitter, the first correction of the Child-Langmuir 3/2 law is determined only by the total curvature (the sum of the principal curvatures) of the emitting surface. Solution of the problem in the formulation in question permits determination of the collector shape that ensures the given distribution of the emissioncurrent density over the given surface.

Scattering of two plane electromagnetic waves in the non-linear Born-Infeld electrodynamics

The problem of scattering of two plane waves is exactly solved in the Born-Infeld electrodynamics. The shape and the direction of the plane waves after scattering turn out to be unchanged. There is only change in the wave phase by the value which is the product of its frequency and the momentum of the incident wave. It is shown that in the interaction region the solution becomes a multiple-valued function ofx, t when the electromagnetic field exceeds a certain value which plays the role of an absolute field scale.

On a type of Riemannian space

In connection with Relativity, Kottler [2] introduced the space V4 whose metric tensor is given by xi being space coordinates and Ф being a function of x1 only, and showed that if Where a and b are arbitrary constants, then the V4 is an Einstein space.