Zariski's formulation was shown to be equivalent to the original problem, for X normal.

Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests.

( see below ) But, by Zariski's connectedness theorem, the last part in the above says that the fiber is connected for any.

The book was published in 1935 and reissued 36 years later, with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed.

Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry.

According to Pierre Cartier, the word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski's ideas ; and it was André Martineau who suggested to Serre the move to the current spectrum of a ring in general.

* Zariski's theorem on holomorphic functions

After completing a doctorate under the supervision of Alexander Grothendieck, he worked with him at the Institut des Hautes Études Scientifiques ( IHÉS ) near Paris, initially on the generalization within scheme theory of Zariski's main theorem.

( This part is sometimes referred to as Zariski's connectedness theorem.

It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry.

At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin and Steven Kleiman — thus spanning the main areas of advance in singularity theory, moduli theory and cohomology in the next generation.

Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics.

After Hilbert's initial formulation, the later development of abstract Hilbert space and the spectral theory of a single normal operator on it did very much go in parallel with the requirements of physics ; particularly at the hands of von Neumann.

It is considered by some to be a better formulation of Hilbert's fifth problem, than the characterisation in the category of topological groups of the Lie groups often cited as a solution.

Here Hume tackles the problem of how liberty may be reconciled with metaphysical necessity ( otherwise known as a compatibilist formulation of free will ).

The first proof was given by (), where the formulation of the problem was attributed to Ulam.

In its initial formulation, it made universal statements meaningless, and this was seen as a problem for science.

As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science, engineering, business, and other areas of mathematical practice.

While it is possible to define some arbitrary, ad hoc cost function, frequently a particular cost will be used, either because it has desirable properties ( such as convexity ) or because it arises naturally from a particular formulation of the problem ( e. g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost ).

Statistical models describe the sources of data and can have different types of formulation corresponding to these sources and to the problem being studied.

David Hume's formulation of the problem of evil in Dialogues Concerning Natural Religion:

Also important are Plemelj's contributions to the theory of analytic functions in solving the problem of uniformization of algebraic functions, contributions on formulation of the theorem of analytic extension of designs and treatises in algebra and in number theory.

The knife-edge effect is an outgrowth of the half-plane problem, originally solved by Arnold Sommerfeld using a plane wave spectrum formulation.

" Presupposition and Anaphora: Remarks on the formulation of the projection problem ," Linguistic Inquiry 40 ( 3 ): 367-386.

Such a formulation is called an optimization problem or a mathematical programming problem ( a term not directly related to computer programming, but still in use for example in linear programming-see History below ).

Also like Newton's Second law, the Schrödinger equation describes time in a way that is inconvenient for relativistic theories, a problem that is not as severe in matrix mechanics and completely absent in the path integral formulation.

Determining optimality for rational behavior requires a quantifiable formulation of the problem, and the making of several key assumptions.

* Top-down approach: This is the direct fall-out of the recursive formulation of any problem.

* Monge-Kantorovich problem, a more general formulation

Since the precise notion of " manifold " was not available to Hilbert, there is room for some debate about the formulation of the problem in contemporary mathematical language.

For this reason, one must either modify the formulation to ensure that it's a minimization problem ( for example, by extremizing the square of the gradient of the Lagrangian as below ), or else use an optimization technique that finds stationary points ( such as Newton's method without an extremum seeking line search ) and not necessarily extrema.

In the above formulation, a problem emerged, which is rendering something obligatory on the Divine being — something that seems to directly conflict with Divine omnipotence.

The replica exchange MD ( REMD ) formulation tries to overcome the multiple-minima problem by exchanging the temperature of non-interacting replicas of the system running at several temperatures.

Because, the charge continuity issue with Ampère's original formulation is no longer a problem.

Dynamic CSPs ( DCSPs ) are useful when the original formulation of a problem is altered in some way, typically because the set of constraints to consider evolves because of the environment.

For technical reasons the Itō integral is the most useful for general classes of processes but the related Stratonovich integral is frequently useful in problem formulation ( particularly in engineering disciplines.

Edmund Gettier's formulation of the problem was important as it coincided with the rise of the sort of philosophical naturalism promoted by W. V. O.

As a finite connected vertex-transitive graph that does not have a Hamiltonian cycle, the Petersen graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation of the conjecture asks for a Hamiltonian path and is verified by the Petersen graph.

An equivalent formulation in extremal graph theory asks for the maximum number of edges in a bipartite graph with no complete bipartite subgraph K < sub > a, b </ sub >.

Unlike an opinion poll, which asks whom the voter plans to vote for or some similar formulation, an exit poll asks whom the voter actually voted for.

This raised numerous questions about how various formulations relate to one another, how adequate the formulation is to various philosophical tasks ( in particular, the task of formulating physicalism ), and whether it avoids or entails reductionism.

Using a different formulation of Gödel's theorems, namely, that of Raymond Smullyan and Emil Post, Webb shows one can derive convincing arguments for oneself of both the truth and falsity of p. He furthermore argues that all arguments about the philosophical implications of Gödel's theorems are really arguments about whether the Church-Turing thesis is true.

The Knudsen number is useful for determining whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used: If the Knudsen number is near or greater than one, the mean free path of a molecule is comparable to a length scale of the problem, and the continuum assumption of fluid mechanics is no longer a good approximation.

They also transform the nonogram into a digital tomography problem by replacing the sequence of block lengths in each row and column by its sum, and use a maximum flow formulation to determine whether this digital tomography problem combining all of the rows and columns has any squares whose state can be determined or pairs of squares that can be connected by an implication relation.

This led to the most important part of Kant's ethics, the formulation of the Categorical Imperative, which is the criterion for whether a maxim is good or bad.

He thinks that the current formulation, based on Justice Holmes ' conception of free speech as a marketplace “ disserves the aspirations of those who wrote America ’ s founding document .” The purpose of this reformulation would be to “ reinvigorate processes of democratic deliberation, by ensuring greater attention to public issues and greater diversity of views .” He is concerned by the present “ situation in which like-minded people speak or listen mostly to one another ,” and thinks that in “ light of astonishing economic and technological changes, we must doubt whether, as interpreted, the constitutional guarantee of free speech is adequately serving democratic goals .” He proposes a “ New Deal for speech would draw on Justice Brandeis ' insistence on the role of free speech in promoting political deliberation and citizenship .”

Nevertheless, it remains likely that formulation errors will result in nutritional deficiencies in a wide range of commercially available diets from time to time, whether meat-based, vegetarian or vegan.

This code uses an " Arbitrary Lagrangian-Eulerian " ( ALE ) formulation on an unstructured mesh, allowing one to designate whether material should flow through a stationary mesh ( pure Eulerian ), whether the mesh should move with the material ( pure Lagrangian ), or whether the mesh should be allowed to move independently of the material motion ( arbitrary ).

If, however, Justice Marshall's formulation is guiding, then it is unclear whether methods derived from law and economics or the Moral Constitution might be implicated.