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1.
The solution of the differential equation y″ + 2Ry′ + n2y = E cos pt is written in a new form which clearly exhibits many important facts thus far overlooked by theoretical and experimental investigators. Writing s = n ? p, and Δn = n ? √n2 ? R2, it is found: (a) When s ≠ Δn, there are “beats,” and the first “beat” maximum is greater than any later maximum while the first “beat” minimum is less than any later “beat” minimum. The “beat” frequency is (s ? Δn)2π. (b) When n2 ? p2 = R2, there are no “beats,” and the resultant amplitude grows monotonically from zero to the amplitude of the forced vibration, (c) At resonance, when n = p, we still have maxima which occur with a frequency Δn2π in a damped system. (d) The absence of “beats” is neither a sufficient nor a necessary condition for resonance in a damped system.In the experimental investigation the upper extremity of a simple pendulum was moved in simple harmonic motion and photographic records obtained of the motion of the pendulum bob. Different degrees of damping were used, ranging from very small to critical.The experimental results are in excellent agreement with theory.  相似文献   

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3.
Numerous relatively simple physical systems give rise under appropriate circumstances to oscillations which obey the equation y″ + ?(1 + k cos t)y = 0 (Mathieu's equation). These oscillations may be either stable, periodic, or unstable, depending upon parameters of the physical system as expressed by the parameters ? and k in the basic equation. It has been customary to distinguish between the stable and unstable states by diagrams of the type of Fig. 1, from which it is possible to tell whether a given set of values of the parameters ?, k will yield a stable or unstable solution. In this paper are given curves which not only present this information, but in addition give for an important part of the stable state the values of the characteristic exponent μ. The solution of the equation y″ + ?(1 + k cos t)y = 0 depends to a large extent on this exponent, and the availability of values of μ should greatly facilitate the practical application of the equation.  相似文献   

4.
The rate of evaporation of electrons, νe, from a thoriated tungsten filament depends on the temperature T and on θ, the fraction of the surface covered by thorium atoms. The relation of νe to θ and T has been given by Brattain and Becker. From νe the change in contact potential V of the filament surface produced by the adsorbed thorium can be calculated by the Boltzmann equation. Knowing σ the number of thorium atoms per cm.3, the dipole moment M of each is given by V = 2τσM. By an equation already used for Cs films on tungsten, the 2-dimensional equation of state of the adsorbed film can be calculated from M. Then by Gibbs' adsorption equation the relation of the atom evaporation rate νa to θ and T can be determined. It is thus calculated that in the range from θ = 0.2 to θ = 0.6, νa varies in proportion to ?Hθ where H = 8.1 in good agreement with the value H = 7.8 given by Brattain and Becker's measurements. A recalculation of the data of numerous experiments in 1921–1923, using the new relations of νe to θ1 gives data on νa as a function of T and θ in good agreement with the values of νa calculated from νe.The diffusion coefficients of Th through tungsten crystals, along grain boundaries and over the free filament surface, are calculated. A theory is given for the cause of the variation in the surface diffusion coefficient with σ. The probable mechanism of the production of the metallic thorium within the filament is discussed. At 2400° the thorium which arrives at the surface along grain boundaries, for some unknown reason, does not spread out over the surface as it does at lower temperatures (1900–2100°).  相似文献   

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6.
This paper is a continuation of a previous paper published in this JOURNAL. The basic idea in the two papers is to enlarge the assemblage of thermodynamic states by including the so-called “metastable” states. Considering a system, in one or two phases, which has a single type of transformation, the writer develops an equation of state of the form η = a + by + cp + dpv + (e + fv + gp + hpv) In T, where p, v, T are three independent variables, and a, b, c, etc. are constants.The latent heat at p, T = constant is λP,T = I (v2 -v1) [b+db+(f+hp) InT), which is derived from the equation of state.The available thermodynamic data on ammonia and steam are used to check these equations. It is found that within the saturated region the agreement is quite satis factory, whereas for the superheated region the agreement is not so good.  相似文献   

7.
The general mth order difference equation X(n+m)+a1X(n+m?1)+…+amX(n) = F[n,X(n),…,X(n+m?1)] is considered. The stability properties of its solutions are studied using the discrete form of Liapunov's direct method. A quadratic form is selected as a possible Liapunov function V(n,X) and a scheme is developed for determining appropriate conditions on this function to insure that its total difference ΔV(n,X) is negative semi-definite or negative definite with respect to the difference equation. The approach is applied to the fourth-order difference equation in full detail to illustrate the method for determining the conditions which imply either uniform stability or uniform asymptotic stability and specific results are obtained. Several comments on, and extensions of, the work done by Puri and Drake for the cases m = 2 and m = 3 are presented.The results of the present approach in the homogeneous case where F[n,X(n),…,X(n+m?1)] = 0 are compared with the usual Schur-Cohn criteria and are shown to be at least as good.  相似文献   

8.
Instead of the commonly accepted inverse square law, Lotka's original formulation was based on a more general inverse power law: xn·y = c. The exponent and the constant must be estimated from the given set of author productivity data. A step-by-step outline is presented for testing the applicability of Lotka's law. Steps include the computation of the values of the exponent and the constant based on Lotka's method, and the test for significance of the observed frequency distribution against the estimated theoretical distribution derived from Lotka's formula.  相似文献   

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10.
The plasmonic response of gold clusters with atom number (N) = 100–70 000 was investigated using scanning transmission electron microscopy-electron energy loss spectroscopy. For decreasing N, the bulk plasmon remains unchanged above N = 887 but then disappears, while the surface plasmon firstly redshifts from 2.4 to 2.3 eV above N = 887 before blueshifting towards 2.6 eV down to N = 300, and finally splitting into three fine features. The surface plasmon''s excitation ratio is found to follow N0.669, which is essentially R2. An atomically precise evolution picture of plasmon physics is thus demonstrated according to three regimes: classical plasmon (N = 887–70 000), quantum confinement corrected plasmon (N = 300–887) and molecule related plasmon (N < 300).  相似文献   

11.
The periodic differential equation (1+ε cos t)y&#x030B; + py = 0, hereby termed the Carson–Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided |ε|<1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p–ε plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to ε3. Two-timing theory is employed to obtain solutions for values of k near the critical points k = ±12, ±32, ±52. Three-timing is employed to extend the solution near k = ±12. The solutions of the Carson–Cambi equation are compared with the solutions of the corresponding Mathieu equation.  相似文献   

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13.
The usual model for (Poissonian) linear birth-death processes is extended to multiple birth-death processes with fractional birth probabilities in the form λi(Δt)α+o((Δt)α, 0<α<1. The probability generating function for the time dependent population size is provided by a fractional partial differential equation. The solution of the latter is obtained and comparison with the usual model is made. The probability of ultimate extinction is obtained. One considers the special case of fractional Poissonian processes with individual arrivals only, and then one outlines basic results for continuous processes defined by fractional Poissonian noises. The key is the Taylor’s series of fractional order f(x+h)=Eα(hαDxα)f(x), where Eα(·) is the Mittag-Leffler function, and Dxα is the modified Riemann-Liouville fractional derivative, as previously introduced by the author.  相似文献   

14.
BackgroundC4ST-1 catalyzes the transfer of sulfate groups in the sulfonation of chondroitin during chondroitin sulfate synthesis. Chondroitin sulfate consists of numerous copies of negatively charged sulfonic acid groups that participate in the nucleation process of biomineralization. In the present study, we obtained two CHST11 genes (PmCHST11a and PmCHST11b) which encoded the C4ST-1 and explored the functions of these genes in the synthesis of chondroitin sulfate and in the formation of the nacreous layer of shells.ResultsBoth PmCHST11a and PmCHST11b had a sulfotransferase-2 domain, a signal peptide and a transmembrane domain. These properties indicated that these genes localize in the Golgi apparatus. Real-time PCR revealed that both PmCHST11a and PmCHST11b were highly expressed in the central zone of the mantle tissue. Inhibiting PmCHST11a and PmCHST11b via RNA interference significantly decreased the expression levels of these genes in the central zone of the mantle tissue and the concentration of chondroitin sulfate in extrapallial fluid. Moreover, shell nacre crystallized irregularly with a rough surface after RNA interference.ConclusionsThis study indicated that PmCHST11a and PmCHST11b are involved in the nacre formation of Pinctada fucata martensii through participating in the synthesis of chondroitin sulfate.  相似文献   

15.
Montgomery multiplication over rings   总被引：1，自引：0，他引：1  
Montgomery multiplication of two elements a and b of a finite field Fq is defined as abr-1 where r is a fixed field element in . In this paper we define Montgomery multiplication of elements a(x) and b(x) in a polynomial ring modulo the ideal generated by a reducible polynomial f(x). We then show that Montgomery multiplication over a field represented by a polynomial ring modulo an irreducible pentanomial can be performed more efficiently in terms of time delay by embedding the field in a quotient of a polynomial ring modulo a reducible trinomial. The trinomial has a degree that is slightly higher than that of the pentanomial, thereby increasing the number of gates in the multiplier by a small amount.  相似文献   

16.
Given the linear system x = Ax - bu, y = cTx, it is shown that, for a certain non-quadratic cost functional, the optimal control is given by uopt(x) = h(cTx), where the function h(y) must satisfy the conditions ky2?h(y)y>0 for y≠0, h(0) = 0 and existence of h-1 everywhere. The linear system considered must satisfy the Popov condition 1/k + (1 +?ωβ) G(?ω)>0 for all ω, G(s) being the y(s)/u(s) transfer function.  相似文献   

17.
The deformation of a strong elastic lithosphere supported by an underlying weak magma is calculated for a long mountainous load. The deformations, produce vertical superposed stresses in the lithosphere that contribute to the support of the mountain and distribute its weight over a large area.Employing the principle of isobaric equilibrium, the distribution of the vertical supporting stress and the associated gravity anomalies may be determined by analysis.The results of the analysis are applied to the Hawaiian chain and it is shown that the calculated physical properties and gravity anomalies are remarkably like those observed. It is inferred from the study that: (a) a strong lithosphere underlies the Hawaiians and this section is practically identical with those underlying other regions; (b) the lithosphere will support one-sided stresses approximating 109 dynes/cm.2 for long periods of time; (c) the distribution of gravity anomaly may be described quantitatively in terms of the calculated deformations; (d) the principle of isobaric equilibrium is obeyed throughout the region whereas the isostatic principle is generally violated; and (e) the distribution of compensating masses is not at all like that demanded by the principle of isostasy.It is concluded that isostasy leads to results that are clearly wrong when applied to mountains or other deformed regions. The principle of isobaric equilibrium appears, however, to be well established by the investigation.  相似文献   

18.
In Part I properties of the scale coördinate, of the form: B(n + θ) are discussed. n is shown to be associated with the operation of counting scale marks, θ with the operation of estimating between them, and B, with the operational and configurational aspects of that part of apparatus which lies adjunct to the scale system.In Part II three types of measurement codification are discussed: (a) the differential interval; (b) the finite amorphous interval; (c) the scale interval; a relationship among them is postulated.In Part III the finite differences in scale coördinates are defined and simple theorems are used to illustrate these definitions. Simple difference equations in scale coördinates are solved to illustrate macroscopic “selection principles” arising partly out of the methodology of codifying a coincidence in scale coördinates.In Part IV an example of causally related dimensional systems is described by use of the scale coördinate. This example is taken from the perfect gas law and Van der Waals' gas law.  相似文献   

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20.
Control problems in Hilbert spaces are treated in a measure-theoretical framework; instead of dealing with a set of admissible trajectory-control pairs, a set of measures defined by the boundary conditions and the differential equations of the problem are considered. The concept of weak controllability is introduced; a system has this property if every pair of initial and final points, (ta,xa) and (tb,xb) can be weakly joined; this is possible if a set of linear equalities involving measures has a solution. In turn, this is shown to be equivalent to the possibility of extending a linear functional in a positive manner. Necessary and sufficient conditions for controllability are derived, and applied to the study of a finite-dimensional system with the control appearing linearly.  相似文献   

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