Embedding of a universally consistent 2-extension in a universally consistent extension

A Galois extension is considered universally consistent with the period q if for any problem of embedding of this extension with an abelian kernel of the period q the consistency condition holds. Let K be a universally consistent extension of the period 2n + 1 of an algebraic number field k, such that 2 completely splits in K, and let (K/k, φ) be an embedding problem with the cyclic kernel of order 2. It is proven that (under some group-theoretical restrictions) there exists a solution of this embedding problem that is universally consistent with the period 2n.     

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.     

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.     

The Brauer imbedding problem for commutative rings

The imbedding problem with Abelian kernel in the Galois theory of commutative rings is considered. A necessary and sufficient condition for the solvability of the Brauer imbedding problem for commutative rings is found. The group of equivalence classes of imbeddings for given Abelian kernel and the subgroup of semidirect imbeddings are studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 31–50, 1976.     

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.     

Weighted Estimates for Singular Integral Operators Satisfying Hörmander’s Conditions of Young Type

The following open question was implicit in the literature: Are there singular integrals whose kernels satisfy the L^r-Hörmander condition for any r > 1 but not the L^∞-Hörmander condition? We prove that the one-sided discrete square function, studied in ergodic theory, is an example of a vector-valued singular integral whose kernel satisfies the L^r-Hörmander condition for any r > 1 but not the L^∞-Hörmander condition. For a Young function A we introduce the notion of L^A-Hörmander. We prove that if an operator satisfies this condition, then one can dominate the L^p(w) norm of the operator by the L^p(w) norm of a maximal function associated to the complementary function of A, for any weight w in the A_∞ class and 0 < p < ∞. We use this result to prove that, for the one-sided discrete square function, one can dominate the L^p(w) norm of the operator by the L^p(w) norm of an iterate of the one-sided Hardy-Littlewood Maximal Operator, for any w in the A⁺ _∞ class.     

齐型空间上带非光滑核的奇异积分算子的双权不等式

该文在齐型空间( X, d,μ)上建立带非光滑核的奇异积分算子的双权、弱型不等式, 即对于1< p≤ q <∞, 此算子是L^p( X, v)到 L^q,∞( X, u)有界的, 只要权函数对(u, v)满足在权 u 上增加一个"Orlicz-bump" '的 A_p 型条件.     

Tian’s invariant of the Grassmann manifold

We prove that Tian’s invariant on the complex Grassmann manifold G _p,q(?)is equal to 1/(p+ q).The method introduced here uses a Lie group of holomorphic isometries which operates transitively on the considered manifolds and a natural imbedding of (?¹ (?))^p in G _p,q (?).     

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.     

Nielsensche Realisierungssätze für Seifertfaserungen

Let M be a closed aspherical manifold and A a finite subgroup of the outer automorphism group Out ₁M of ₁M. A necessary (and in many cases also sufficient) condition for realising A by the induced action of an isomorphic group of homeomorphisms of M is the existence of an extension 1₁MEA1 to the abstract kernel (A,₁M, AOut ₁M). If the center of ₁M is nontrivial, this condition need not be fulfilled ([14]). We showed in [25] however that one can always find a surjection BA of a finite group B with abelian kernel such that there exists an extension to the abstract kernel (B,₁M,BAOut₁M), and one can try to realize B instead of A. The main result of the present paper is a characterisation of all such groups B (for a given A) which can be realized by a group of homeomorphisms. The class of manifolds considered here consists of certain Seifert fiber spaces in arbitrary dimensions but the main result is purely algebraic and can be applied to other classes of manifolds, for example to flat Riemannian manifolds.     

Optimal query error of quantum approximation on some Sobolev classes

We study the approximation of the imbedding of functions from anisotropic and general-ized Sobolev classes into Lq([0,1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from LpN to LNq , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wpr ([0,1]d)) to Lq([0,1]d) space for all 1 q,p ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.     

Toeplitz–Schur Multipliers of the Class Sp(L2G) for p< 1

We study Toeplitz–Schur multipliers of the Schatten–von Neumann class S _p for 0 < p < 1. We describe all functions F on an arbitrary commutative locally compact group G satisfying the following condition: for any integral operator in S _p with kernel function k(x,y), the kernel function F(x-y)k(x)k(y) defines also an integral operator in S _p. Bibliography: 4 titles.     

Optimal Control of Nonlinear Fredholm Integral Equations

Optimal control problems with nonlinear equations usually do not possess optimal solutions, so that their natural (i.e., continuous) extension (relaxation) must be done. The relaxed problem may also serve to derive first-order necessary optimality condition in the form of the Pontryagin maximum principle. This is done here for nonlinear Fredholm integral equations and problems coercive in an L ^p-space of controls with p<+. Results about a continuous extension of the Uryson operator play a key role.     

On Galois groups of unramified pro-p extensions

Let p be an odd prime satisfying Vandiver’s conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z _p -extensions of Q(μ _p ) and the Galois group of the maximal unramified pro-p extension of Q . We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for to be abelian. The bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p-units. We obtain in particular that, for p  <  1,000, Greenberg’s conjecture that X is pseudo-null holds and is in fact abelian.     

Labelling of some planar graphs with a condition at distance two

The problem of vertex labeling with a condition at distance two in a graph, is a variation of Hale’s channel assignment problem, which was first explored by Griggs and Yeh. For positive integerp ≥q, the λ _p,q -number of graph G, denoted λ(G;p, q), is the smallest span among all integer labellings ofV(G) such that vertices at distance two receive labels which differ by at leastq and adjacent vertices receive labels which differ by at leastp. Van den Heuvel and McGuinness have proved that λ(G;p, q) ≤ (4q-2) Δ+10p+38q-24 for any planar graphG with maximum degree Δ. In this paper, we studied the upper bound of λ _p ,q-number of some planar graphs. It is proved that λ(G;p, q) ≤ (2q?1)Δ + 2(2p?1) ifG is an outerplanar graph and λ(G;p,q) ≤ (2q?1) Δ + 6p - 4q - 1 if G is a Halin graph.     

The conjugation problem on an infinite set of intervals

It is known that if path of integration consists of a finite number of intervals, then: (1) in the case of a Fredholm-type kernel, the index of the Fredholm operator is zero; (2) in the case of a Cauchy-type kernel, the index of the singular integration operator is a finite number (possible zero). Study of the conjugate boundary-value problem on an infinite set of intervals brings out new facts. The following may be noted: (1) A homogeneous boundary-value problem is always solvable in the classK, which is a natural generalization of that of piecewise analytic functions [1]. (2) Associated (conjugated) homogeneous boundary-value problems have any number of linearly independent solutions in the associated (conjugated) classes, so that the notion of class index is no longer relevant. (3) Associated (conjugated) homogeneous singular integral equations have any number of linearly independent solutions in the associated (conjugated) spacesL _p, L_q, p^?1+q^?1=1, so that the notion of operator index is no longer relevant The general theory of the problems under consideration is satisfactorily illustrated by the simplest case—a set of intervals on the real axis. For this reason the line of discontinuities (integration path) in the present paper is part of the real axis. The paper generalizes the results of [2–4]. Relevant work includes [5].     

Continuous spectrum for a class of nonhomogeneous differential operators

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p ₁, p ₂, and q satisfy 1 < p ₂(x) < q(x) < p ₁(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any is an eigenvalue, while any is not an eigenvalue of the above problem.     

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.     

Two remarks concerning the equation Πp(Χ,·)=Ip(Χ,·)

It is proved that the analog of Grothendieck's theorem is valid for a diskalgebra “up to a logarithmic factor.” Namely, if Tε? (C_A, L¹) and then π₂(T)?C (1+logn) ¦T¦. The question of whether the logarithmic factor is actually necessary remains open. It is also established that C _A ^* is a space of cotype q for any q, q > 2. The proofs are based on a theorem of Mityagin-Pelchinskii: π_p(T)?C·p·i_p(T), p?2 for any operator T acting from a disk-algebra to an arbitrary Banach space.     

Tame Kernels of Quaternion Number Fields

Let E/F be a Galois extension of number fields with the quaternion Galois group Q ₈. In this paper, we prove some relations connecting orders of the odd part of the kernel of the transfer map of the tame kernel of E with the same orders of some of its subfields. Let E/? be a Galois extension of number fields with the Galois group Q ₈ and p an odd prime such that p ≡ 3 (mod 4). We prove that if there is at most one quadratic subfield such that the p-Sylow subgroup of the tame kernel is nontrivial, then p ^r -rank(K ₂(E/K)) is even, i.e., 2|p ^r -rank(K ₂(𝒪 _E )) ? p ^r -rank(K ₂(𝒪 _K )), where K is the quartic subfield of E.