Conditions for existence

Acute and Chronic Toxicity. Although chromium displays nine oxidation states, the low oxidation state compounds, -II to I, all require Special conditions for existence and have very short lifetimes in a normal environment. This is also tme for most organ ochromium compounds, ie, compounds containing Cr—C bonds. Chromium compounds that exhibit stabiUty under the usual ambient conditions are limited to oxidation states II, III, IV, V, and VI. Only Cr(III) and Cr(VI) compounds are produced in large quantities and are accessible to most of the population. Therefore, the toxicology of chromium compounds has been historically limited to these two states, and virtually all of the available information is about compounds of Cr(III) and/or Cr(VI) (59,104). However, there is some indication that Cr(V) may play a role in chromium toxicity (59,105—107). Reference 104 provides an overview and summary of the environmental, biological, and medical effects of chromium and chromium compounds as of the late 1980s. 

The classical problem of multiple solutions and undamped oscillations occurring in a continuous stirred-tank reactor, dealt with in the papers by Aris and Amundson (39), involved a single homogeneous exothermic reaction. Their theoretical analysis was extended in a number of subsequent theoretical papers (40, 41, 42). The present paragraph does not intend to report the theoretical work on multiplicity and oscillatory activity developed from analysis of governing equations, for a detailed review the reader is referred to the excellent text by Schmitz (3). To understand the problem of oscillations and multiplicity in a continuous stirred-tank reactor the necessary and sufficient conditions for existence of these phenomena will be presented. For a detailed development of these conditions the papers by Aris and Amundson (39) and others (40) should be consulted. 

Consider the relation dL - R(r) dr If dL is an exact differential prove that dL is the differential of a scalar function F(r) satisfying the relation R(r) - V F(r) and show that the necessary and sufficient condition for existence of a gradient of a scalar function is that it be irrotational V x R - 0. Show that this latter requirement coincides with the necessary and sufficient conditions rendering dL an exact differential. 

Chastened by this example, we state a theorem that provides sufficient conditions for existence and uniqueness of solutions to x = /(x). Existence and Uniqueness Theorem Consider the initial value problem X = fix), X(0) = Xg. ... 

Necessary conditions for the existence of a self-similar solution are that (1) the governing PDE must reduce to an ODE for F as a function ot// alone, and (2) the original boundary and initial conditions must reduce to a number of equivalent conditions for F that are consistent with the order of the ODE. Of course, a proof of sufficient conditions for existence of a selfsimilar solution would require a proof of existence of a solution to the ODE and boundary conditions that are derived for F. In general, however, the problems of interest will be nonlinear, and we shall be content to derive a self-consistent set of equations and boundary conditions and attempt to solve this latter problem numerically rather than seeking a rigorous existence proof. Let us see how the systematic solution scheme based on the general form (3-135) works for the Rayleigh problem. 

The condition for existence of a similarity transformation is that the coefficients of the equation for//must be independent of t. This condition leads to a pair of equations for a and ft,... 

One of the remaining conditions for existence of a similarity solution is thatg(0) = 0. Thus we can see from (10-107) that... 

However, the two coefficients involving g in (10-230) are identical, and the condition for existence of a similarity solution thus reduces to a single, first-order ODE for g(0),... 

Suppose that a is sufficiently small, i.e., We is sufficiently large, that surface tension plays no role in determining the bubble shape, except possibly locally in the vicinity of the rim where the spherical upper surface and the flat lower surface meet. Further, suppose that the Reynolds number is sufficiently large that the motion of the liquid can be approximated to a first approximation, by means of the potential-flow theory. Denote the radius of curvature at the nose of the bubble as R(dX 6 = 0). Show that a self-consistency condition for existence of a spherical shape with radius R in the vicinity of the stagnation point, 0 = 0, is that the velocity of rise of the bubble is... 

Therefore, we see that the boundary conditions at the interface lead to a set of four linear algebraic equations for the constants A, C), B2, D2. The condition for existence of a nontrivial solution of this set of algebraic equations is that the determinant of the coefficient matrix must equal to zero. This condition leads to a complicated algebraic equation relating the dimensionless growth-rate coefficient a to the dimensionless wave number a for specified values of the fluid viscosities, the fluid densities, and the interfacial tension. As usual, stability is determined by the sign of the real part of a. 

Apparently, the condition for existence of non-zero solutions to the system of equations (5.8) is vanishing of the determinant... 

As follows from (7.6), the condition for existence of stationary periodic trains is If (uj) < 0. Tliis means that they are found inside the region of the Arnold tongue, lying between the line B = and the (nearest)... 

Finiteness of the series (4) for p guarantees automatically its convergence, provided that conditions for the expansions are met. These are the conditions for existence of (a) the first n derivatives of. 4 p at dp = 0, plus... 

Conditions for existence of the system Stability to finite perturbations Extremum of function... 

The variability in energy savings and capital costs still depends on the existing evaporator conditions. For existing multi-effect units in which the total amount of first-effect vapor is now used in the second effect, the choice and location of the steam-jet thermocompressor are more complicated because the heat balances and heat-transfer rates are affected for each evaporator reaction. The economics may still be favorable to justify the technical evaluation and modification. Since the limitations of steam-jet thermocompressors are a compression ratio below 1.8 and new heat-transfer rates, the first effect normally becomes the best location. 

Classical Limit for Plasmon Energy and Conditions for Existence of Plasmons... 

Replication of the DNA molecule by the complementary duplication of polynucleotide chains satisfies at least the fundamental condition for existence of living matter in all its forms reproduction and preservation of genetic information, recording nature by means of a nucleotide alphabet or code. The elementary units of this code are known as codons, groups of three nucleotides, each of which codes the position of one amino acid residue in the polypeptide chain. 

The net result of this analysis is quite definite and remarkably simple of all possible shocks that satisfy the jump conditions, eqns (14.9) and (14.10), all and only upwards travelling compression shocks satisfy further necessary conditions for existence. 

A single linear algebraic equation, ax = b, is easily solved, and the condition for existence and uniqueness of the solution x = b/a, a 0, is trivial. For a single nonlinear algebraic equation... 

The conditions for existence of positive real solution in Equation (8) which is quadratic in Np and Ns, are as follows ... 

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