The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

Given the requirements of weight, and fragment density required this dictated using approximately 700 fragments, and being able to aim the mine with an accuracy of around two feet ( 0. 6 m ) at the center of the target zone.

Given the practical weight of suitable power sources, the accelerations given by ion thrusters are frequently less than one thousandth of standard gravity.

Given their large size, Great Danes continue to grow ( mostly gaining weight ) longer than most dogs.

: Given two sets, A and T, of equal size, together with a weight function C: A × T → R. Find a bijection f: A → T such that the cost function:

Given the explosion of medical knowledge in the 20th century, it is easily appreciated that this led to a vast expansion of the book, which threatened to collapse under its own weight in a metaphorical and physical sense.

Given the amount of evidence suggesting a link between antipsychotic medications and metabolic syndrome, several agencies have recommended regular clinical monitoring of weight, symptoms of hyperglycemia, glucose, hypertension and signs and symptoms of cardiovascular disease in chronically medicated patients with schizophrenia.

Given that the tank was reaching its viable limit, to avoid a corresponding weight increase, the appliqué steel plates were removed from its side armor, which instead had its base thickness increased to.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Given a modular form f ( z ) of weight k, the mth Hecke operator acts by the formula

: Given a weighted directed graph and an integer K, is there a Hamiltonian path ( or Hamiltonian cycle if the salesman must return home ) with total weight less than or equal to K?

Given an integral weight, one of two cases occur: ( 1 ) There is no such that is dominant, equivalently, there exists a nonidentity such that ; or ( 2 ) There is a unique such that is dominant.

Given in weight %

* Given a graph where all its edges have distinct weights, is a given edge in the minimum weight spanning forest?

Given the very light weight of the " pin top ," the. 22 is quite capable of removing the target from the table with a single well-placed shot.

: Given two sets, P (" facilities ") and L (" locations "), of equal size, together with a weight function w: P × P → R and a distance function d: L × L → R. Find the bijection f: P → L (" assignment ") such that the cost function:

Given the size and weight of the memorial, even in its early design stages, a solid foundation for the structure was critical.

Given their lighter weight, Pony models were known for their ability to deliver excellent fuel economy — mileage upwards of in highway driving was not uncommon.

In a confusion when it was time for Richard Parker to catch a train ride to find Thirsty a home, the woman at the ticket counter confused the tigers name to be Richard Parker, and the hunters name to be Thirsty with his last name being " None Given.

Given both stores, pilots can train rocket and bomb fire on a single sortie.

Given the fortuitous timing ( Kaplan's parting with his Las Vegas team ), he agreed to go in the hopes of putting together a new local team that he could train and manage.

Given that the train was a demonstrator, it was likely that production designs or future infrastructure plans would have dealt with this.

Given the choices, Governor Scott Bone authorized the transport of 300, 000 units of serum in Anchorage to Nenana by train, where it was picked up by the first of twenty mushers and more than one hundred dogs who relayed the serum the remaining 674 miles ( 1, 085 km ) to Nome.

Given the tonal design of the train whistle, the sound level, how often trains pass through a given community, the number of grade crossings in proximity, and the time of day ( night ) of occurrence, community residents residing near crossing sometimes feel that train whistles have a serious detrimental effect on the quality of life despite the gain in safety that sounding the horn provides to motorists.

Given the imminent liberation of Paris by the Allies, they need only delay the train for a few days — still, it is an extremely dangerous operation and it must be done in such a way that does not risk damaging the priceless cargo.

Given wartime exigencies, officials would not transport him to prison or even pay his train fare, so he hitchhiked to the Arizona prison where he was sentenced to reside.

Given estimates of the image gradients, a search is then carried out to determine if the gradient magnitude assumes a local maximum in the gradient direction.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

Given the rugged terrain of Ngazidja and Nzwani, and the dedication of extensive tracts to agriculture on all three islands, population pressures on Comoros are becoming increasingly critical.

Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

* Frederick Douglass lecture on Haiti – Given at the World's Fair in Chicago, January 1893.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given these, a new term Anthropocene, is specifically proposed and used informally for the latest part of this epoch since approximately synchronous lithospheric evidence, or more recently atmospheric evidence, of human impacts have been found on the Earth and its ecosystems ; these impacts may be considered of global significance for future evolution of living species.

Given that this etymology does not appear in the usual scholarly works on etymology, this claim is usually dismissed as a false etymology.

Given the dual status of science as objective knowledge and as a human construct, good historiography of science draws on the historical methods of both intellectual history and social history.

Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if

Given its heavy dependence on imported energy, Japan has aimed to diversify its sources.

Along with a hand-picked escort led by Given Campbell, Davis and his wife were captured on May 10, 1865, at Irwinville in Irwin County, Georgia.

Given the repeated failures and frustrations of workers ' revolutions and movements, Marx also sought to understand capitalism, and spent a great deal of time in the reading room of the British Museum studying and reflecting on the works of political economists and on economic data.

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given that spinning a Dyson Sphere would result in the atmosphere pooling around the equator, the Ringworld removes all the extraneous parts of the structure, leaving a spinning band landscaped on the sun-facing side, with the atmosphere and inhabitants kept in place through centrifugal force and 1000 mile high perimeter walls ( rim walls ).