To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure email@example.com
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
The WAIS (West Antarctic Ice Sheet) Divide deep ice core was recently completed to a total depth of 3405 m, ending 50 m above the bed. Investigation of the visual stratigraphy and grain characteristics indicates that the ice column at the drilling location is undisturbed by any large-scale overturning or discontinuity. The climate record developed from this core is therefore likely to be continuous and robust. Measured grain-growth rates, recrystallization characteristics, and grain-size response at climate transitions fit within current understanding. Significant impurity control on grain size is indicated from correlation analysis between impurity loading and grain size. Bubble-number densities and bubble sizes and shapes are presented through the full extent of the bubbly ice. Where bubble elongation is observed, the direction of elongation is preferentially parallel to the trace of the basal (0001) plane. Preferred crystallographic orientation of grains is present in the shallowest samples measured, and increases with depth, progressing to a vertical-girdle pattern that tightens to a vertical single-maximum fabric. This single-maximum fabric switches into multiple maxima as the grain size increases rapidly in the deepest, warmest ice. A strong dependence of the fabric on the impurity-mediated grain size is apparent in the deepest samples.
Groups of pigs were brought to an abattoir by truck and approximately 25 were killed on each of the next 3 days.
While the pigs were in lairage they were given water but were not fed. After slaughter the caecal contents of all pigs were cultured to detect Salmonella spp. The organism was isolated from 70% of 145 pigs killed after 1 day in lairage, 49% of 143 pigs that had been in lairage for 2 days and 41% of 135 pigs that had been held for 3 days.
Recent advances in ion microprobe instrumentation and techniques have enabled the mapping of C isotope ratios across the whole of a polished plate of a natural diamond from Guaniamo, Venezuela. The resultant map of C isotope variation closely matches the cathodoluminescence image of the growth structure of the diamond and, therefore, indicates an extremely limited scale of diffusion of C atoms sincethetimeof diamond formation. This result is compatible with thelimite d mobility of N atoms shown by theIaAB aggregation stateof thediamond. Inclusions in thediamond aree clogitic, in common with many Guaniamo diamonds with temperatures of formation of around 1200ºC. At such temperature the IaAB aggregation state indicates a mantle residence time on the order of 1 Ga. Such temperatures of formation and mantle residence times are common to many natural diamonds; thus the extremely limited diffusion of C isotopes shown by the mapping indicates that many diamonds will retain the C isotope compositions of their initial formation.
For an optimal control problem with an infinite time horizon, assuming various terminal state conditions (or none), terminal conditions for the costate are obtained when the state and costate tend to limits with a suitable convergence rate. Under similar hypotheses, the sensitivity of the optimum to small perturbations is analysed, and in particular the stability of the optimum when the infinite horizon is truncated to a large finite horizon. An infinite horizon version of Pontryagin's principle is also obtained. The results apply to various economic models.
If a constrained minimization problem, under Lipschitz or uniformly continuous hypotheses on the functions, has a strict local minimum, then a small perturbation of the functions leads to a minimum of the perturbed problem, close to the unperturbed minimum. Conditions are given for the perturbed minimum point to be a Lipschitz function of a perturbation parameter. This is used to study convergence rate for a problem of continuous programming, when the variable is approximated by step-functions. Similar conclusions apply to computation of optimal control problems, approximating the control function by step-functions.
Textured superconducting films of YbBa2Cu3O7-δ supported on single and polycrystalline substrates were prepared by oxidation of a liquid precursor alloy. The substrates were coated by dipping them into a molten alloy (YbBa2Cu3, m.p. ∼870°C), withdrawing them from the melt, then oxidizing the adhering liquid alloy layer to the corresponding oxide phase, i.e., YbBa2Cu3O7-δ. Samples prepared in this way exhibited a superconducting transition at ∼80 K following annealing in pure O2 at 500°C. With SrTiO3 (100) and MgO (100) substrates, evidence was seen for the epitaxial growth of YbBa2Cu3O7-δ crystals having their c-axis parallel to the  direction of the substrate. For polycrystalline MgO, x-ray diffraction and microstructural examination showed that the high-Tc crystallites in the films were also oriented with their c-axis perpendicular to the substrate surface, but the a and b axes directions were randomly oriented rather than epitaxial.
An approximate dual is proposed for a multiobjective optimization problem. The approximate dual has a finite feasible set, and is constructed without using a perturbation. An approximate weak duality theorem and an approximate strong duality theorem are obtained, and also an approximate variational inequality condition for efficient multiobjective solutions.
A simple rigorous approach is given to finding boundary conditions for the adjoint differential equation in an optimal control problem. The boundary conditions for a time-optimal problem are calculated from the simpler conditions for a fixed-time problem.
A ratio game approach to the generalized fractional programming problem is presented and duality relations established. This approach suggests certain solution procedures for solving fractional programs involving several ratios in the objective function.
Lagrangian necessary conditions for optimality, of both Fritz John and Kuhn Tucker types, are obtained for a constrained minimization problem, where the functions are locally Lipschitz and have directional derivatives, but need not have linear Gâteaux derivatives; the variable may be constrained to lie in a nonconvex set. The directional derivatives are assumed to have some convexity properties as functions of direction; this generalizes the concept of quasidifferentiable function. The convexity is not required when directional derivatives are replaced by Clarke generalized derivatives. Sufficient Kuhn Tucker conditions, and a criterion for the locally solvable constraint qualification, are obtained for directionally differentiable functions.
Under suitable hypotheses on the function f, the two constrained minimization problems:
are well known each to be dual to the other. This symmetric duality result is now extended to a class of nonsmooth problems, assuming some convexity hypotheses. The first problem is generalized to:
in which T and S are convex cones, S* is the dual cone of S, and ∂y denotes the subdifferential with respect to y. The usual method of proof uses second derivatives, which are no longer available. Therefore a different method is used, where a nonsmooth problem is approximated by a sequence of smooth problems. This duality result confirms a conjecture by Chandra, which had previously been proved only in special cases.
For both differentiable and nondifferentiable functions defined in abstract spaces we characterize the generalized convex property, here called cone-invexity, in terms of Lagrange multipliers. Several classes of such functions are given. In addition an extended Kuhn-Tucker type optimality condition and a duality result are obtained for quasidifferentiable programming problems.
Optimality conditions and duality results are obtained for a class of cone constrained continuous programming problems having terms with arbitrary norms in the objective and constraint functions. The proofs are based on a Fritz John theorem for constrained optimization in abstract spaces. Duality results for a fractional analogue of such continuous programming problems are indicated and a nondifferentiable mathematical programming duality result, not explicitly reported in the literature, is deduced as a special case.
A simple rigorous approach is given to generalized functions, suitable for applications. Here, a generalized function is defined as a genuine function on a superset of the real line, so that multiplication is unrestricted and associative, and various manipulations retain their classical meanings. The superset is simply constructed, and does not require Robinson's nonstandard real line. The generalized functions go beyond the Schwartz distributions, enabling products and square roots of delta functions to be discussed.
This paper deals with the question of the existence of a solution to the stationary-point problem corresponding to a given nonlinear nondifferentiable program. An existence theorem for the stationary-point problem is presented under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the nonlinear program.
Stampfli and Embry have shown that a point of the numerical range of an operator is extreme if and only if a set of vectors corresponding to it is linear. This is generalized here to show that a point of the closure of the numerical range is extreme if and only if a corresponding set of sequences forms a linear space. A more geometric alternative proof is given for a theorem of Das and Garske concerning weak convergence to zero at the unattained extreme points of the closure of the numerical range.The result is shown to hold also for lone extreme points of the numerical range which lie on line segments on its boundary. Further, a bound is obtained on the norm of the weak limit of the weakly convergent sequences corresponding to points on a line segment on the boundary of numerical range.