Completely solvable imbedding problems for local fields

Conditions for imbedding a p-extension of local fields in a field with a given p-group are investigated. Some reduction theorems are proved, and sufficient conditions are found which guarantee such solvability.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 121–126, 1978.     

A case of imbedding problem with a solvable kernel

Journal of Mathematical Sciences - It is shown for the solution of the imbedding problem of algebraic number fields with a finite solvable kernel one can take a field if the orders of the kernel...     

On the imbedding problem for local fields

The imbedding problem of local fields is considered for the case where the whole of the group is a p-group having as many generators as the Galois group of the extension and the extension consists of a primitive root of 1 of degree equal to the period of the kernel. It is proved that it is necessary and sufficient for the solvability of this problem that a concordance condition (and even a weaker condition) be satisfied (see [4]).Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 91–94, July, 1972.The author is thankful to A. V. Yakovlev for his valuable advice.When this article was in press, the author proved that Theorem 1 is valid even without Condition IV. He has also found an example showing that Condition III cannot be discarded.     

The embedding problem with non-Abelian kernel for local fields

The embedding problem of local fields with p-groups is equivalent to its associated Abelian problem if the inequality d ≥ r + 2 is valid; here d and r are the numbers of generators of the Demushkin group and of the Galois group of an embedded field. Bibliography: 6 titles.     

Semidirect imbedding problem with a metabelian kernel

In the paper one proves that in the case of a number field, for the solution of the semidirect imbedding problem with a metabelian kernel one can take a field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 81–85, 1980.     

Completely Decomposable Abelian Groups with Clean Endomorphism Rings

Conditions for cleanness of endomorphism rings of completely decomposable Abelian groups are established.     

An interpretation of the additional condition in the theory of embedding of fields with Abelian kernel

We give an interpretation of A. V. Yakovlev's additional embeddability condition, which, together with a compatibility condition, is necessary and sufficient for the solvability of the embedding problem in Galois theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 116–118, 1979.     

Completely Decomposable Quotient Divisible Abelian Groups with Isomorphic Endomorphism Semigroups

Mathematical Notes - Let $$\Lambda$$ be a class of Abelian groups. A group $$A\in\Lambda$$ is said to be determined by its endomorphism semigroup $$E^\star(A)$$ in the class $$\Lambda$$ if every...     

On the universally solvable embedding problem with cyclic kernel

It is proved that the universally solvable embedding problem with cyclic kernel is semidirect. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 173–179.     

Abelian integrals for cubic vector fields

It is proved in this paper that the lowest upper bound of the number of the isolated zeros of the Abelian integral

Interval length imbedding for nonlinear two-point boundary-value problems

In problems of optimal sequential estimation, in the study of fluid and electrolyte systems, in nonlinear mechanics, and throughout applied mathematics we are confronted with solving nonlinear two-point boundary-value problems. A new approach is provided which seems especially useful when solutions are desired for a variety of interval lengths.     

Tauberian and Abelian Theorems for random fields with strong dependence

We prove Tauberian and Abelian theorems for Hankel-type integral transformations.     

Discrete invariant imbedding for singular boundary value problems with a regular singularity

We describe a discrete invariant imbedding method for solving a two point boundary value problem in the interval [0,b] for a linear second order ordinary differential equation with a singularity of the first kind at x = 0. By employing the series expansion on (0, δ], where δ is near the singularity, we first replace it by a regular problem on some interval [δ, b]. The discrete invariant imbedding method is then described to solve the problem over the reduced interval. The stability analysis of the method is discussed. Some model problems are solved, and the numerical results compared with those of other methods.     

Factorization and imbedding for general linear boundary value problems

A new direction in the theory of general linear boundary value problems is explored. The starting point is an explicit Volterra factorization of the Green's matrix (and related kernels) associated with the problem. This result leads to (1) imbedding of the boundary value problems, (2) initial value algorithms for their solution, and (3) comparison theorems relating two different boundary value problems with a common boundary condition. Extensions and connections with earlier work in this area are presented.