Universally concordant Galois extensions

A Galois extension is called universally concordant of period q, if for any imbedding problem of this extension whose kernel is an Abelian group of period q the concordance condition is satisifed. A necessary and sufficient condition is given for the imbeddability of one universally concordant extension into another. For a universally concordant extension of period of an algebraic number field containing no roots of 1 of degrees p₁, ..., p_m the solvability of any imbedding problem with solvable kernel of period q is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 133–152, 1977.     

Completely solvable imbedding problems with Abelian kernel for local fields

The imbedding problem for p-groups is considered in which the fields are assumed to be local and the kernel commutative. Additional conditions are investigated under which a solvable imbedding problem has a field as solution. Sufficient conditions are found for such solvability in the form of inequalities imposed on the number of generators of certain groups.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 67–73, 1978.     

Imbedding problem with given localizations and ramification restrictions

For a solvable imbedding problem (L/K,G,A) of a number field with Abelian kernel A and a finite set of points S of the field K, a finite set T of points of the field K, is found which has the following property: for any solvable imbedding problem with given localizations, corresponding to the problem (L/K,G,A), there exists a solution which is unramified outside of SuT.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 85–99, 1976.     

Compactness properties of Sobolev imbeddings for rearrangement invariant norms

Compactness properties of Sobolev imbeddings are studied within the context of rearrangement invariant norms. Attention is focused on the extremal situation, namely, when the imbedding is considered as defined on its optimal Sobolev domain (with the range space fixed). The techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator (of just one variable).

     

The imbedding index of a graph

The natural extension of MacLane's combinatorial approach to planar imbeddings is seen to yield a combinatorial formulation of imbedding of a graph in a pseudosurface. This leads to a combinatorially defined parameter for all graphs, called the imbedding index. A generalization of the Heaword inequality is then proved for this parameter.     

Local extrema in genus-stratified graphs

Beyond the obvious organization of all the orientable imbeddings of a graph according to the genus of the imbedding surface, there are several seemingly natural ways to ascribe proximity of imbeddings. One of these is to stipulate that two imbeddings are adjacent if one imbedding can be obtained from the other by moving one end of an edge in its rotation. If one associates “altitude” with genus, then one might hope to construct algorithms for minimum genus and maximum genus by descent and ascent, respectively. This investigation of the structure of the system of orientable graph imbeddings reveals that although there may occur arbitrarily deep traps among the local minima, there cannot exist any strict local maxima. These new discoveries seem consistent with a known contrast in the computational complexity of the maximum genus and minimum genus problems. That is, whereas Furst, Gross, and McGeoch have devised a polynomial-time algorithm to find the maximum genus of an arbitrary graph, Thomassen has proved that the problem of finding the minimum genus is NP-complete.     

Imbedding problem with a noncommutative kernel of order p4. II

The problem of imbedding number fields is investigated for p-groups, where the kernel is a non-Abelian group of order p⁴ with two generators , and relationsIt is shown that the solvability of this problem is equivalent to the simultaneous solvability of all the collateral local problems and the collateral Abelian problem obtained by the factorization of the kernel by the derived group.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 101–113, 1991.     

Components of branched coverings of current graphs

Desired graph imbeddings are obtained as branched coverings of simpler imbeddings, continuing earlier work of the authors. This paper inaugurates a systematic approach to the problem of devising appropriate current assignments. Chief results include the establishment of a bijective correspondence between the components of the cover and the cosets of the isotropy group of an arbitrary component, leading to a proof that the components are mutually isomorphic regular branched coverings of the dual of the current graph imbedding. Also given is a characterization of the isotropy group via a face labeling technique that aids in the construction of imbeddings.     

Hierarchy for imbedding-distribution invariants of a graph

Most existing papers about graph imbeddings are concerned with the determination of minimum genus, and various others have been devoted to maximum genus or to highly symmetric imbeddings of special graphs. An entirely different viewpoint is now presented in which one seeks distributional information about the huge family of all cellular imbeddings of a graph into all closed surfaces, instead of focusing on just one imbedding or on the existence of imbeddings into just one surface. The distribution of imbeddings admits a hierarchically ordered class of computable invariants, each of which partitions the set of all graphs into much finer subcategories than the subcategories corresponding to minimum genus or to any other single imbedding surface. Quite low in this hierarchy are invariants such as the average genus, taken over all cellular imbeddings, and the average region size, where “region size” means the number of edge traversals required to complete a tour of a region boundary. Further up in the hierarchy is the multiset of duals of a graph. At an intermediate level are the “imbedding polynomials.” The hierarchy is explored, and several specific calculations of the values of some of the invariants are provided. The main results are concerned with the amount of work needed to derive one invariant from another, when possible, and with principles for computing the algebraic effect of adding an edge or of otherwise combining two graphs.     

Hermitian Metric Rigidity for Foliated Vector Bundles

We show that the canonical isometric imbedding of the symplectic group Sp(n) into R⁴n² gives the least-dimensional isometric imbedding into the Euclidean space, even in the local standpoint. We prove this result by calculating the quantity p_G determined by the curvature of Sp(n), which serves as an obstruction to the existence of local isometric imbeddings. We also exhibit the estimates on the value p_G for the remaining compact classical simple Lie groups, and improve the previous results on the codimension of local isometric imbeddings of these groups.     

Local Isometric Imbeddings of Symplectic Groups

We show that the canonical isometric imbedding of the symplectic group Sp(n) into R ^{4n 2} gives the least-dimensional isometric imbedding into the Euclidean space, even in the local standpoint. We prove this result by calculating the quantity p_G determined by the curvature of Sp(n), which serves as an obstruction to the existence of local isometric imbeddings. We also exhibit the estimates on the value p_G for the remaining compact classical simple Lie groups, and improve the previous results on the codimension of local isometric imbeddings of these groups.     

On Pierce Problem for Reduced <Emphasis Type="Italic">p</Emphasis>-Groups

One of the problems in the study of Abelian groups and their endomorphism rings is the problem of constructing an appropriate structural theory. The structural theorems obtained for Abelian groups and their endomorphism rings allowto reduce the study of complex objects to simpler ones. A radical is one of the tools possessing this property. In this paper, we describe the Jacobson radical of the endomorphism rings of reduced p-groups from some class, which is a solution to the Pierce problem for this class of groups.     

The concordance condition in the field imbedding problem

Under certain additional assumptions (which are always satisfied for local fields and number fields) it is proved that the concordance condition for the field imbedding problem is equivalent to the concordance condition for the imbedding problem obtained from the original by factorization by the commutant of the kernel.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 155–162, 1977.     

S-TORSION FREE COVERS OF MODULES

Over Pruüfer domains every module has a torsion free cover with a flat kernel. In this paper we show that the commutative Noetherian rings that have this property are the Gorenstein rings with Krull dimension at most one. Here we use the torsion theory defined by the nonzero-divisors of the commutative ring R.     

Dual imbeddings and wrapped quasi-coverings of graphs

This paper gives a new method for constructing imbeddings of graphs which are “nearly” coverings of given imbedded graphs. The method is based on the dual theories of current graphs and voltage graphs. Some applications are given, in particular the following theorem: Let G be any graph which has a triangular imbedding in the sphere. Then there are infinitely many integers n for which the composition G[nK₁] has a triangular (orientable surface) imbedding.     

复合形在欧氏空间中的同痕问题(Ⅰ)

<正> 作者曾经指出,一个空间的约化积这一概念对于非同伦性的拓扑问题颇为有用,并曾应用之以研究空间在一欧氏空间中的实现问题,局部实现问题,同痕与同位问题,以及其他一些有关问题(参阅例如[1],与该处文献).近来作者又曾证明对于一个有限复合形在一欧氏空间中的任两线性实现,从复合形的约化积可导出一些不变量来,它们之为0给出了两个实现线性同痕的必要条件,而在临界情形,即欧氏空间的维数等于复合形维数的两倍加1的情形,(复合形维数假定>1)这些必要条件又同时是充分的.本文及以下一     

Imbedding posets in the integers

An imbedding of a poset P in the integers is a one-to-one order presevring map from P into the integers. Such a map always exists when P is finite and, moreover, certain imbeddings of subsets of finite P can be extended to imbeddings of the whole of P. D. E. Daykin has asked when an imbedding in the integers of a finite subset of a countable poset can be extended to the whole poset. This paper answers Daykin's question and some related questions.     

交换群层的正合序列与同态定理

在交换群层（简称层）中给出了层的单（满）同态核与上核的泛性定理及正合交换图的一系列结论,进而证明了交换群层的同态（同构）定理。