A general equation of state: Equations for ammonia and steam

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 λ_P,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.     

An equation of state for a system with a single type of transformation

Based on theory of a previous paper, the writer has developed an equation of state for a system with a single type of transformation. This equation is of the form

$\text{h=A+Bv+Cp+Dpv?T(E+Fv+Gp+Hpv)}$

where h = ε + pv is the total heat, p the pressure, v the specific volume, T the temperature, and p, v, T are considered independent variables. A, B, C, etc., are constants for the system. The latent eat at constant (p, T) is given by

$\text{λp,T=(v}{}_2\text{?v}{}_1\text{)(}\text{?h}\text{?v}\text{)P,T= (v}{}_2\text{?v}{}_1\text{)[(B?TF)+p(D?TH)]}$

. These equations are checked with data on saturated and superheated ammonia, and the agreement is good to within a few tenths of a per cent. Also, checks with data on saturated and superheated steam show agreement within several per cent.     

Null controllability of some nonlinear degenerate 1D parabolic equations

The main goal of the present paper is twofold: (i) to establish the well-posedness of a class of nonlinear degenerate parabolic equations and (ii) to investigate the related null controllability and decay rate properties. In a previous step, we consider an appropriate regularized system, where a small parameter α is involved. More precisely, the usual nonlinear term b(x)uu_x is replaced by b(x)zu_x, where $z={(\text{Id}.?{\alpha }^{2}A)}^{?1}u$ and A is a Poisson–Dirichlet operator. We investigate the behavior of the null controls and their associated states as α → 0.     

Periodic solutions to systems of reaction-diffusion equations

In this paper, necessary and sufficient conditions are derived for the existence of temporally periodic “dissipative structure” solutions in cases of weak diffusion with the reaction rate terms dominant in a generic system of reaction-diffusion equations ?c_i/?t = D_i?²c_i+Q_i(c), where the enumerator index i runs 1 to n, c_i = c_i(x, t) denotes the concentration or density of the ith participating molecular or biological species, D_i is the diffusivity constant for the ith species and Q_i(c), an algebraic function of the n-tuple c = (c₁,\3., c_n), expresses the local rate of production of the ith species due to chemical reactions or biological interactions.     

On the two-dimensional steady flow of a compressible viscous fluid far behind a solid symmetrical body

A number of writers have observed that the particle size distributions obtained from some breakage processes appear to be logarithmico-normal. There have been virtually no attempts to explain this phenomenon. In this paper a statistical model is constructed for a breakage mechanism which will generate size distributions which are asymptotically logarithmico-normal. In this model, F_n(x), the distribution function of the particle sizes, after n steps of the breakage process, is the distribution function of the product of (n + 1) independent random variables and hence, by the central limit theorem, is asymptotically logarithmico-normal.     

The inverse lure problem of optimal control

Given the linear system $\text{x}\text{}$ = Ax - bu, y = c^Tx, it is shown that, for a certain non-quadratic cost functional, the optimal control is given by u_opt(x) = h(c^Tx), 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.     

Periodic solutions for a fourth-order p-Laplacian neutral functional differential equation

By means of Mawhin's continuation theorem, we study a kind of fourth-order p-Laplacian neutral functional differential equation with a deviating argument in the form:

(φ_p(x(t)−cx(t−δ))^″)^″=f(x(t))x^′(t)+g(t,x(t−τ(t,|x|_∞)))+e(t).     

Accurate approximate solutions to oscillatory problems with perturbing singular damping

In this paper we attempt to obtain approximate solutions of improved accuracy for a class of differential equations of the form

$\text{d}{}^2\text{y}\text{d}\text{x}{}^2\text{+εμ(x)}\text{d}\text{y}\text{d}\text{x}\text{+ω}{}^2{}_{\mathrm{c}}\text{y = 0}$

, where ε is a real parameter less than unity, ω_c is a positive real constant of order unity and μ(x) is a singular function of x in the region of interest. It does not appear to be possible to find a general analytic expression for the error estimate of the approximate solution. For the case μ(x) = x^?2, however, it is shown that the approximate solution is accurate to 0(ε²), as x → 0^? from negative values, by comparing it with the numerically integrated solution. For the same case, the approximate solution is orders of magnitude more accurate than Poincaré's first-order perturbation solution, which is accurate to $\text{0(}\text{ε}{}^2\text{ln}\text{|x|}\text{|x|}\text{)}$ as x → 0^?. This work arose in search of analytic solutions to a linearized form of the restricted three-body problem.     

On recursive averaging processes and Hilbert space extensions of the contraction mapping principle

If T maps a convex domain D_T into itself, and if {ω_n} is a real sequence with range in (0, 1] then the recursive averaging process,

$\text{X}{}_{\mathrm{n+1}}\text{=(1?omega;}{}_{\mathrm{n}}\text{)}\text{X}{}_{\mathrm{n}}\text{+ω}{}_{\mathrm{n}}\text{+ω}{}_{\mathrm{n}}\text{T}\text{x}{}_{\mathrm{n}}\text{,}\text{x}{}_0\text{=ξ?D}{}_{\mathrm{T}}$

generates a sequence {x?_n}; with range in D_T. Under suitable conditions on D_T, T and {ω_n} the sequence {x?_n} will converge in some sense to a fixed point of T. We prove that if D_T is a closed convex subset of a complex Hilbert space H, if Tω = (1 ? ω) I + ωT is a strict contraction for some ω ? (0, 1], and if {ω_n} satisfies the conditions,

$\text{ω}{}_{\mathrm{n}}\text{→ 0}$

and

$\text{∑}\text{n=0}\text{∞}\text{ω}{}_{\mathrm{n}}\text{=∞}$

then, for arbitrary ξ ? D_T, {x?_n} converges strongly to (the unique) fixed point of T. We also prove that if D_T and {ω_n} satisfy the foregoing conditions, if T has at least one fixed point, and if Tω is non-expansive for some ω ? (0, 1], then for all ξ ? D_T, {x?_n} converges at least weakly to some fixed point of T. Finally, we apply these results to linear equations involving bounded normal operators and obtain an extension of the classical Neumann operator series.     

A matrix generalization of heaviside's expansion theorem

Starting with the energy and dissipation functions of the general n mesh linear bilateral network and using the operational methods of the Laplacian transformation, a solution is obtained for the Lagrangian equations of the system subject to initial boundary conditions. The equations take a particularly simple and general form if matrix notation is used.It is noted that the general case bears a close resemblance to the simple, one mesh, series circuit when the scalar factors which appear in this circuit are generalized to matrix form.     

Shifted-Jacobi Series Analysis of Linear Optimal Control Systems Incorporating Observers

This paper uses the Jacobi series to analyze linear optimal control systems incorporating observers. The method simplifies the system of equations into the successive solution of a set of linear algebraic equations. An illustrative example is included to demonstrate that only a small number (m = 6) of shifted-Jacobi series are needed to obtain an accurate solution.     

Periodic solutions for p-Laplacian neutral Liénard equation with a sign-variable coefficient

Using Lu's continuation theorem, the extension one of Manásevich-Mawhin, we study the existence of periodic solutions for p-Laplacian neutral Liénard equation of the form

(?_p(x(t)-cx(t-σ))^′)^′+f(x(t))x^′(t)+β(t)g(x(t-τ(t))=e(t).     

Periodic solutions for a kind of Rayleigh equation with two deviating arguments

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Rayleigh equation with two deviating arguments of the form

x^″+f(x^′(t))+g₁(t,x(t-τ₁(t)))+g₂(t,x(t-τ₂(t)))=p(t).     

An adaptive online learning algorithm for distributed convex optimization with coupled constraints over unbalanced directed graphs

This paper investigates a distributed optimization problem over multi-agent networks subject to both local and coupled constraints in a non-stationary environment, where a set of agents aim to cooperatively minimize the sum of locally time-varying cost functions when the communication graphs are time-changing connected and unbalanced. Based on dual decomposition, we propose a distributed online dual push-sum learning algorithm by incorporating the push-sum protocol into dual gradient method. We then show that the regret bound has a sublinear growth of O(T^p) and the constraint violation is also sublinear with order of $O\left(T^{1?p/2}\right),$ where T is the time horizon and 0 < p ≤ 1/2. Finally, simulation experiments on a plug-in electric vehicle charging problem are utilized to verify the performance of the proposed algorithm. The proposed algorithm is adaptive without knowing the total number of iterations T in advance. The convergence results are established on more general unbalanced graphs without the boundedness assumption on dual variables. In addition, more privacy concerns are guaranteed since only dual variables related with coupled constraints are exchanged among agents.     

On numerical solutions for hyperbolic–parabolic equations with the multipoint nonlocal boundary condition

A numerical method is proposed for solving multi-dimensional hyperbolic–parabolic differential equations with the nonlocal boundary condition in t and Dirichlet and Neumann conditions in space variables. The first and second order of accuracy difference schemes are presented. The stability estimates for the solution and its first and second orders difference derivatives are established. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of a one-dimensional hyperbolic–parabolic differential equations with variable coefficients in x and two-dimensional hyperbolic–parabolic equation.     

Mathematical models of information system use

This report presents the results from a study of mathematical models relating to the usage of information systems. For each of four models, the papers developed during the study provide three types of analyses: reviews of the literature relevant to the model, analytical studies, and tests of the models with data drawn from specific operational situations. (1) The Cobb-Douglas model: x₀ = ax₁^bx₂^(1?b).This classic production model, normally interpreted as applying to the relationship between production, labor, and capital, is applied to a number of information related contexts. These include specifically the performance of libraries, both public and academic, and the use of information resources by the nation's industry. The results confirm not only the utility of the Cobb-Douglas model in evaluation of the use of information resources, but demonstrate the extent to which those resources currently are being used at significantly less than optimum levels. (2) Mixture of Poissons:

$\text{χ}{}_0\text{=}\text{∑}\text{i=0}\text{n}\text{i}\text{∑}\text{j=0}\text{p}\text{n}{}_{\mathrm{j}}\text{e}{}^{\mathrm{mj}}\text{(m}{}_{\mathrm{j}}\text{)′/i!}$

where x₀ is the usage and (n_j,m_j),j = 0 to p, are the p + 1 components of the distribution. This model of heterogeneity is applied to the usage of library materials and of thesaurus terms. In each case, both the applicability and the analytical value of the model are demonstrated. (3) Inverse effects of distance: x = a e^?md if c(d) = rdx = ad^?m if c(d) = r log(d).These two models reflect different inverse effects of distance, the choice depending upon the cost of transportation. If the cost,c(d), is linear, the usage is inverse exponential; if logarithmic, the usage is inverse power. The literature that discusses the relationship between usage of facilities and the distance from them is reviewed. The models are tested with data from the usage of the Los Angeles Public Library, both Central Library and branches, based on a survey of 3662 users. (4) Weighted entropy:

$\text{S(x}{}_1\text{,x}{}_2\text{,...,x}{}_{\mathrm{n}}\text{)= -}\text{∑}\text{i=1}\text{n}\text{r(x}{}_{\mathrm{i}}\text{P(x}{}_{\mathrm{i}}\text{)}\text{log}\text{(p(x}{}_{\mathrm{i}}\text{)).}$

This generalization of the “entropy measure of information” is designed to accommodate the effects of “relevancy”, as measured by r(x), upon the performance of information retrieval systems. The relevant literature is reviewed and the application to retrieval systems is considered.     

Event-based weighted residual generator design via non-PDC scheme for fault diagnosis in T–S fuzzy systems

This paper is concerned with the event-based weighted residual generator design via non-parallel distribution compensation (PDC) scheme for fault diagnosis in discrete-time T–S fuzzy systems, under consideration of the imperfect premise matching membership functions. An event-triggered mechanism is firstly introduced to save communication resources, which leads to the premise variables of the system and observer to be asynchronous. Then, a fuzzy diagnostic observer with mismatched premise variables is designed to estimate the unmeasurable states of the system. Moreover, by using non-PDC method, a diagnostic observer-based weighted residual generator is established to improve the fault detection (FD) performance by using the information provided by each local system, in which the membership functions structure of the diagnostic observer and residual generator need not to be the same as the systems, and the L_∞/L₂ and L_∞ FD scheme is used to optimize the FD performance. Finally, two simulation results are provided to show the efficiency of the proposed non-PDC method.     

Thermodynamics,part I: The second law from the standpoint of the equation of state

In this paper the concepts “equilibrium” and “reversibility” are examined from the standpoint of the number of independent variables which define a thermodynamic system. The endeavor is made to show that the number of independent variables which are selected is crucial to the basic concepts as well as to the second law of thermodynamics. If this number is less than the minimum required to define the system completely, the equation of state for the internal energy will be incomplete. It is our practice to employ incomplete equations of state, and this results in processes which we consider to be irreversible. The Kelvin-Planck and Clausius principles are shown to be somewhat related to the first law of thermodynamics, and their validity arises from the fact that the equations of state which are used are incomplete. If these equations were complete, that is, if the number of independent variables were not less than the minimum required to describe the systems completely, we should have complete conversion of heat into work, as well as work into heat. The difficulties in realizing complete equations are considered.The entropy concept is considered in detail, and it is shown that there are serious limitations in its application.