Control subspaces of minimal dimension. Elementary introduction. Discotheca

In this paper there is introduced and studied the following characteristic of a linear operator A acting on a Banach space Χ: , where Cyc A=R∶R is a subspace of Χ, dim R<+∞. Spqn (AⁿR∶n?0)=χ. Always disc A ?μ_A=(the multiplicity of the spectrum of the operator (dim R∶R∈Cyc A), where (by definition) in each A-cyclic subspace there is contained a cyclic subspace of dimension ? disc A. For a linear dynamical system x(t)=Ax(t)+Bu,(t) which is controllable, the characteristic disc A of the evolution operator A shows how much the control space can be diminished without losing controllability. In this paper there are established some general properties ofdisc (for example, conditions are given under which disc(A⊕B))=max(discA, disc B); disc is computed for the following operators: S (S is the shift in the Hardy space H²); disc S=2, (but μ_S=i); disc S _n ^* =n (butμ=1), where S_n=S⊕. ⊕S; disc S=2, (but μ_S=1), where S is the bilateral shift. It is proved that for a normal operator N with simple spectrum, disc N=μ_N=1 ? (the operator N is reductive). There are other results also, and also a list of unsolved problems.     

On Real Forms of Belyi Surfaces With Symmetric Groups of Automorphisms

In virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces. Here we study the actions of the symmetric groups S _n on Belyi Riemann surfaces. We show that such surfaces are symmetric and we calculate the number of connected components of the corresponding real forms.     

Approximation of solutions of the Monge-Ampère equations by surfaces reduced to developable surfaces

We consider an approximate construction of the surface S that is the graph of a C ²-smooth solution z = z(x, y) of the parabolic Monge-Ampère equation

Multidimensional Renewal Theory in the Non-Centered Case. Application to Strongly Ergodic Markov Chains

Let (S _n ) _n?≥?0 be a ? ^d -valued random walk (d?≥?2). Using Babillot’s method (Babillot, Ann Inst Henri Poincaré, B, Tome 24(4):507–569, 1988), we give general conditions on the characteristic function of S _n under which (S _n ) _n?≥?0 satisfies the same renewal theorem as in the independent case (i.e. the same conclusion as in the case when the increments of (S _n ) _n?≥?0 are assumed to be independent and identically distributed). This statement is applied to additive functionals of strongly ergodic Markov chains under the non-lattice condition and (almost) optimal moment conditions.     

Conjugacy in Permutation Representations of the Symmetric Group

Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e., that conjugacy classes of S _n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S _n  × S _n . We also consider the permutation representations of S _n which arise from the action of S _n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.     

Rigidity of minimal surfaces inS 3

Isometric deformations of compact minimal surfaces in the standard three-sphere are studied. It is shown that a given surface admits only finitely many noncongruent minimal immersions intoS ³ with the same first fundamental form.     

Willmore Surfaces in S n

A surface x> : M S ⁿ is called a Willmore surface if it is a critical surface of the Willmore functional. It is well known that any minimal surface is a Willmore surface and that many nonminimal Willmore surfaces exists. In this paper, we establish an integral inequality for compact Willmore surfaces in S ⁿ and obtain a new characterization of the Veronese surface in S ⁴ as a Willmore surface. Our result reduces to a well-known result in the case of minimal surfaces.     

(B n * GnSn) — Geometrien

A consequence of BENZ [3], [4] and ARTZY [2] is that the class of all (B ₂ ^* S₂)-geometries coinzides with the class of the chain geometries (K,K×K), K a commutative field. Therefore the tangency theorem holds in every (B ₂ ^* S₂)-geometry.In this paper we present a corresponding tangency theorem for (B ₃ ^* G₃S₃)-geometries and prove that up to isomorphisms the (B _n ^* G_nS_n)-geometries, n>1, are exactly the chain geometries (K,Kⁿ).     

The Semi-Arc Automorphism Group of a Graph with Application to Map Enumeration

A map is a connected topological graph cellularly embedded in a surface. For a given graph Γ, its genus distribution of rooted maps and embeddings on orientable and non-orientable surfaces are separately investigated by many researchers. By introducing the concept of a semi-arc automorphism group of a graph and classifying all its embeddings under the action of its semi-arc automorphism group, we find the relations between its genus distribution of rooted maps and genus distribution of embeddings on orientable and non-orientable surfaces, and give some new formulas for the number of rooted maps on a given orientable surface with underlying graph a bouquet of cycles B_n, a closed-end ladder L_n or a Ringel ladder R_n. A general scheme for enumerating unrooted maps on surfaces(orientable or non-orientable) with a given underlying graph is established. Using this scheme, we obtained the closed formulas for the numbers of non-isomorphic maps on orientable or non-orientable surfaces with an underlying bouquet B_n in this paper.     

Embeddings of bipartite graphs

If G is a bipartite graph with bipartition A, B then let G_m,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a₁, …, a_m, each vertex b of B by an independent set b₁,…, b_n, and each edge ab of G by the complete bipartite graph with edges a_ib_j (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of G_m,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess-current graphs.” They include new genus embeddings for a large class of bipartite graphs.     

Quasi-direct sums of rings and their radicals

Let A, B be rings and P a radical property. Call B an A-Algebra if B is an A-bimodule such that (ba)b₁ = b(ab₁), (bb₁)a = b(b₁a), a(bb₁) = (ab)b₁ for any a ∈ A and any b,b₁ ∈ B. A ring R, written as R = A ? B, is called a quasi-direct sum of (A, B) if A is a subring of R, B is an ideal of R and R is a direct sum of A and B as additive groups. The following results are obtained: 1. A quasi-direct sum of (A, B) is uniquely determined by an A-Algebra B (up to isomorphism); 2. The P-radical of the Algebra B is the same as the P-radical of the ring B; 3. P(A ? B) = P(A) +(B) if and only if P(A)B + BP(A) ? P(B); 4. If B has an identity e then P(A ? B) = P(A)(1?e) + P(B); 5. If P(Z) = 0 for the integer ring Z, then P(M_n(R)) = M_n(P(R)) holds for all rings R if and only if the above equality holds for all unitary rings R. In addition, some relationships of radicals between rings (or algebras over a field, semigroup algebras, etc.) and their corresponding identity extensions are discussed.     

On the reconstruction of the variation of the metric tensor of a surface on the basis of a given variation of christoffel symbols of the second kind under infinitesimal deformations of surfaces in the euclidean space <Emphasis Type="Italic">E</Emphasis><Subscript>3</Subscript>

We investigate the problem of reconstruction of the variation of the metric tensor of a surface on the basis of a given variation of Christoffel symbols of the second kind under infinitesimal deformations of surfaces in the Euclidean space E ₃.     

Nonequilibrium statistical mechanics of a solid immersed in a continuum

We study actions of the symmetric group S₄ on K3 surfaces X that satisfy the following condition: there exists an equivariant birational contraction \(\bar r:X \to \bar X\) to a K3 surface \(\bar X\) with ADE singularities such that the quotient space \(\bar X\)/S₄ is isomorphic to P². We prove that up to smooth equivariant deformations there exist exactly 15 such actions of the group S₄ on K3 surfaces, and that these actions are realized as actions of the Galois groups on the Galoisations \(\bar X\) of the dualizing coverings of the plane which are associated with plane rational quartics without A₄, A₆, and E₆ singularities as their singular points.     

A property of conformal infinitesimal deformations of multidimensional surfaces in Riemannian space

It is proved that conformal infinitesimal deformations of a surfaceF ^kin Riemannian space, and they only, are areally recurrent infinitesimal deformations. All areally recurrent deformations of the hypersphereS ⁿ–1 inE ⁿ are described.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 284–290, February, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 95-01-00228a.     

Fragmented deformations of primitive multiple curves

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that Y _red is smooth. We study the deformations of Y to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity n of Y). We are particularly interested in deformations to n disjoint smooth irreducible components, which are called fragmented deformations. We describe them completely. We give also a characterization of primitive multiple curves having a fragmented deformation.     

Moment inequalities and weak convergence for negatively associated sequences

A probability inequality for S_n and somepth moment (p⩾2) inequalities for |S_n| and max 1⩽k⩽n | S_k| are established. Here S_n is the partial sum of a negatively associated sequence. Based on these inequalities, a weak invariance principle for strictly stationary negatively associated sequences is proved under some general conditions. Project supported by the National Natural Science Foundation of China, the Doctoral Program Foundation of the State Education Commission of China and the High Eductional Natural Science Foundation of Guangdong Province.     

Dimension de l’anneau des polynomes a valeurs entieres

Let A be a commutative domain with quotient field K and A_S the ring of integer-valued polynomials thus A_S={f∈K[X]; f(A)⊂A}; we show that the Krull dimension of A_S is such that dim A_S≥dim A[X]-1 and give examples where dim A_S=dim A[X]-1. Considering chains of primes of A_S above a maximal idealm of finite residue field, we give also examples where the length of such a chain can arbitrarily be prescribed (whereas in A[X] the length of such chains is always 1). To provide such examples we consider a pair of domains A⊂B sharing an ideal I such that A/I is finite; we give sufficient conditients to have A_S⊂B[X] and show that in this case dim A_S=dim B[X]. At last, as a generalisation of Noetherian rings of dimension 1, we consider domains with an ideal I such that A/I is finite and a power Iⁿ of I is contained in a proper principal ideal of A; for such domains we show that every prime of A_S above a primem containing I is maximal.      

同态模Hom(A,B)的同调维数

本文从研究函子(?)与Hom的联系入手,来考虑求Hom(A,B)的弱维数与投射维数。当K为域时,且条件(a)[R:K]∞,A是有限生成右R模;(b)·[R:K]<∞,S是右凝聚代数:(c)[S:K]∞,R是右Noether代数,有一成立得到1.wd_R(?)SHom(A,B)r.id_RA+1.wd_sB。     

Über Deformationen von analytischen Abbildungskeimen

The notion of deformations of germs of k-analytic mappings generalizes the one of deformations of germs of k-analytic spaces. Using algebraic terms, we prove:

1. The morphism f: A→B of analytic algebras is rigid, iff it is infinitesimally rigid. Moreover, this is equivalent to Ex_A (B,B)=0. This theorem generalizes a result of SCHUSTER [11].
2. Let A be a regular analytic algebra. Then f is rigid iff there exists a rigid analytic algebra B_o such that f is equivalent to the canonic injection A→A?B_o.
3. If f is “almost everywhere” rigid or smooth, then the injection Ext _B ^l (Ω_B|A, Bⁿ)→Ex_A(B, Bⁿ) is an isomorphism.

     

Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

A Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $ \mathbb{E}_2^4 $ \mathbb{E}_2^4 and in neutral pseudo 4-sphere S ₂⁴ (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H ₂⁴ (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H ₂⁴ (−1). Conversely, every parallel Lorentz surface in H ₂⁴ (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.