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Let $X$ be a finite or infinite set (of variables) and consider the ring of polynomials $R[X]$ and the ring ...ences|[a2]]], in connection with the combinatorics of plane partitions and descent sets of permutations [[#References|[a3]]]. This time one takes a totally or

...with any given integer, so that the number of solutions of the problem is infinite. Considering the Galois groups of infinite extensions of a given field (cf. [[Galois topological group|Galois topologi

...the one that makes an acute angle with the direction $g(x_p)$ of steepest descent of the objective function (with the vector $g(x_p)=-\left.\frac{d \phi(x)}{ ...thod of feasible directions. Thus, in a gradient-projection method, as the descent direction at the point $x_p$ one takes $s_p=y_p - x_p$, where $y_p$ is the

...ction, including a functional, considered as a function on a corresponding infinite-dimensional functional space, one studies the variation of various topologi gradient descent is commonly employed, i.e. the motion of points in accordance with the [[Gr

coincides with the direction of steepest descent (in gradient) of the surface $ f ( x) = c $ iterations, and it can be regarded as an infinite iterative process for minimizing a functional. Modifications of the calcula

...nt article. Non-linear operator equations connected with the discussion of infinite-dimensional spaces (see, for example [[#References|[4]]]–[[#References|[8 A large number of iteration methods (the so-called descent methods) (see [[#References|[1]]]–[[#References|[3]]], [[#References|[9]]

...the method of steepest descent (cf. [[Steepest descent, method of|Steepest descent, method of]]), etc., as well as the method of variation of the parameter (d ..., let a non-linear operator ordinary differential equation be given on the infinite interval:

are elements of a complete infinite system of linearly independent functions which satisfy the boundary conditi ...w direct methods. The most popular methods are those involving the idea of descent in the space of controls and of successive analysis of variants (of the typ

...agin's method is to construct a family of cohomology classes indexed by an infinite set of square-free integral ideals of the base field $K$. These elements sa ...ditions that the global cohomology classes $c_L$ satisfy. Then Kolyvagin's descent procedure gives good control over corresponding arithmetic objects such as

...onding to a partition of its spectrum. The first results of this sort (for infinite-dimensional spaces) were obtained by F. Riesz (1909), who proposed the foll one can define the concept of a determinant (the infinite product of the eigenvalues). The function $ \mathop{\rm det} ( I - \mu T

...te or finite (by Jensen's theorem the number of irregular prime numbers is infinite [[#References|[4]]]). Kummer [[#References|[5]]] proved the theorem for som

...the interest of scholars. The first proof that the set of prime numbers is infinite is due to Euclid. Only in the middle of the 19th century did P.L. Chebyshev ...ons of it. Fermat (17th century) discovered another method, the "method of descent", and solved by it a number of equations, but the so-called [[Fermat great

...form|Exterior form]]; [[Exterior algebra|Exterior algebra]]) acting on the infinite-dimensional space $ {\mathcal G} $. In this context one now searches for an obstacle to the descent of the determinant $ { \mathop{\rm det} } ( {/\partial } _ \rho ^ {*}

...ame algorithms: coordinate relaxation, sequential overrelaxation, steepest descent, and conjugate gradients (cf. [[#References|[7]]]). ...vectors by no means form an orthogonal basis, rather they are steps in an (infinite) iteration process. Sometimes convergence to an eigen value temporarily hal

...one|brachistochrone]] problem, or the problem of curves of minimum time of descent. This problem is usually considered to be the starting point in the history ...; if the lengths of only two of these geodesics are equal, there exists an infinite number of closed geodesics of equal length. Such problems indicate a close

of rational numbers; then there exists an infinite set of values $ t _{1} ^{0} \dots t _{k} ^{0} \in \mathbf Q $ Another generalization of Hilbert's theorem is Grothendieck's descent theorem; one of its applications in [[Etale topology|étale topology]], whi

realizes the descent of the optimal trajectory $ x ^ {0} ( t) $ ...ty and observability of linear systems have many generalizations to linear infinite-dimensional systems (equations in a Banach space, systems with deviating ar

descent. They were a family of dissenters truths to the relation of the finite to the infinite'.