Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description.

In de facto residence definitions this would not be a problem but in de jure definitions individuals risk being recorded on more than one form leading to double counting.

There is a large literature on graphical enumeration: the problem of counting graphs meeting specified conditions.

The primary problem with simple weighted reference counting is that destroying a reference still requires accessing the reference count, and if many references are destroyed this can cause the same bottlenecks we seek to avoid.

A problem is # P-complete if and only if it is in # P, and every problem in # P can be reduced to it by a polynomial-time counting reduction, i. e. a polynomial-time Turing reduction relating the cardinalities of solution sets.

The perfect matching counting problem was the first counting problem corresponding to an easy P problem shown to be # P-complete, in a 1979 paper by Leslie Valiant which also defined the classes # P and # P-complete for the first time.

Cayley's formula is the special case of complete graphs in a more general problem of counting spanning trees in an undirected graph, which is addressed by the matrix tree theorem.

The similar problem of counting all the subtrees regardless of size has been shown to be # P-complete in the general case ().

Andreas Björklund provided an alternative approach using the inclusion – exclusion principle to reduce the problem of counting the number of Hamiltonian cycles to a simpler counting problem, of counting cycle covers, which can be solved by computing certain matrix determinants.

The graph enumeration problem of counting directed acyclic graphs was studied by.

The triangular number T < sub > n </ sub > solves the handshake problem of counting the number of handshakes if each person in a room full of n + 1 total people shakes hands once with each other person.

Weak references ( references which are not counted in reference counting ) may be used to solve the problem of circular references if the reference cycles are avoided by using weak references for some of the references within the group.

For those communities which have financial difficulties in effecting adjustments, there are a number of alternatives any one of which alone, or in combination with others, would minimize if not even eliminate the problem.

Rather, such assignments are made, as they must be, on the basis of certain overall rules and standards, representing to some extent a statistical approach to the problem, taking into account for each situation some of the variables ( e.g., power and station separations ) and averaging out others in order to achieve the balance which must be struck between protection against destructive interference and the assignment of a number of stations large enough to afford optimum radio service to the Nation.

At the same time, every device that can be employed to reduce the number of variables is of the greatest value, and it is one of the attractive features of dynamic programming that room is left for ingenuity in using the special features of the problem to this end.

Another problem was that the gradual identification of more and more chemically similar and indistinguishable lanthanides, which were of an uncertain number, led to inconsistency and uncertainty in the numbering of all elements at least from lutetium ( element 71 ) onwards ( hafnium was not known at this time ).

He studied under a number of teachers, developing his insight into the problem of suffering.

SAT is also easier if the number of literals in a clause is limited to 2, in which case the problem is called 2SAT.

Similarly, if we limit the number of literals per clause to 2 and change the OR operations to XOR operations, the result is exclusive-or 2-satisfiability, a problem complete for SL = L.

The maximum satisfiability problem, an FNP generalization of SAT, asks for the maximum number of clauses which can be satisfied by any assignment.

An example would be the problem of remembering a phone number and recalling it later.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

Because many outstanding problems in number theory, such as Goldbach's conjecture are equivalent to solving the halting problem for special programs ( which would basically search for counter-examples and halt if one is found ), knowing enough bits of Chaitin's constant would also imply knowing the answer to these problems.

In the course of studying the problem, Church and his student Stephen Kleene introduced the notion of λ-definable functions, and they were able to prove that several large classes of functions frequently encountered in number theory were λ-definable.

A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned.

The condition number is a property of the problem.

Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution.

The condition number may also be infinite, in which case the algorithm will not reliably find a solution to the problem, not even a weak approximation of it ( and not even its order of magnitude ) with any reasonable and provable accuracy.

*: The condition number computed with this norm is generally larger than the condition number computed with square-summable sequences, but it can be evaluated more easily ( and this is often the only measurable condition number, when the problem to solve involves a non-linear algebra, for example when approximating irrational and transcendental functions or numbers with numerical methods.

Acquiring a repertoire from the Laboratory library was no problem to one trained to perfect recall.

The discovery of CP violation helped to shed light on this problem by showing that this symmetry, originally thought to be perfect, was only approximate.

Nevertheless, Wells has this very same Time Traveller speak in terms unusual for socialist thought, referring as " perfect " and with no social problem unsolved, to an imagined world of stark class division between the rich assured of their wealth and comfort, and the rest of humanity assigned to lifelong toil: Once, life and property must have reached almost absolute safety.

No doubt in that perfect world there had been no unemployed problem, no social question left unsolved.

Proposing a solution to the theological problem of reconciling the doctrine with that of universal redemption in Christ, he argued that Mary's immaculate conception did not remove her from redemption by Christ ; rather it was the result of a more perfect redemption granted her because of her special role in salvation history.

One solution to avoid this problem is to use a protocol that has perfect forward secrecy.

The same observation can be made for the perfect matching problem.

It was known before that the decision problem " Is there a perfect matching for a given bipartite graph?

Thus the Augustinian theodicist would argue that the problem of evil and suffering is void because God did not create evil ; it was man who chose to deviate from the path of perfect goodness.

The Christofides algorithm follows a similar outline but combines the minimum spanning tree with a solution of another problem, minimum-weight perfect matching.

Due to an increase in many businesses requiring their employees to travel, singles, often young professionals, find online dating websites to be the perfect answer to their " problem ", states Brym and Lenton.

It reconciles the " problem " of the Trinity ( or at least Jesus ) by holding that the Son was not co-eternal with the Father, and that Jesus Christ was essentially granted godhood ( adopted ) for the plans of God and for his own perfect life and works.

Pillai's conjecture concerns a general difference of perfect powers: it is an open problem initially proposed by S. S. Pillai, who conjectured that the gaps in the sequence of perfect powers tend to infinity.

The routing problem is equivalent to the travelling salesman problem, which is NP complete, and therefore not amenable to a perfect solution on a reasonable time scale.

One problem noted with this approach is that there seems to be no fixed basis on deciding what God is not, unless the Divine is understood as an abstract experience of full aliveness unique to each individual consciousness, and universally, the perfect goodness applicable to the whole field of reality.

This seemed like the perfect solution to her problem to reclaim her husband's love from him Iole, the foreign concubine.

Because of simplifications introduced along every step of the way, the corrections are never perfect, but even one cycle of corrections often provides a remarkably better approximate solution to the real problem.

The problem is that critics have been unable to agree on how to reconcile his perfect “ saintliness ” with his obvious adultery with King Arthur ’ s Guinevere.