More projective geometry associated with unramified double covers of curves of genus four

Sunto Sia : YY un rivestimento doppio non diramato di una curva di genere quattro definita su C e a moduli generali. Sia il punto di 2-divisione associate a . In questa nota si studia il sistema 2 delle quardriche di contatto al modello canonico di Y, associato al dato rivestimento.e si esplicita una biezione tra l'insieme delle theta-caratteristiche dispari di Y che diffeiscono per e l'insieme dell theta caratteristiche dispari dell curve di genere tre la cui Jacobiana e isomorfa (come v.a.p.p.) alla varieta di Prym P(f Y Y)     

Subgroups of the full linear group over a Dedekind ring

We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net of ideals of R such that, where E() is the subgroup generated by all transvections of the net subgroup G(). and is the normalizer of G(). The subgroup E() is normal in. To study the factor group we introduce an intermediate subgroup F(), E() F() G(). The group is finite and is connected with permutations in the symmetric group. The factor group G()/F() is Abelian — these are the values of a certain determinant. In the calculation of F()/E() appears the SK₁-functor. Results are stated without proof.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 13–20, 1979.     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

Configuration of subgroups in the special linear group over a skew field with infinite center

Let T be a skew field with infinite center, let be the special linear group over T of degree 3, and let be the subgroup of diagonal matrices with unit Dieudonee determinant. It is proved that for each intermediate subgroup H, H , there exists a net of order n such that ( H N().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 5–12, 1989.In conclusion, the author would like to thank his instructor Z. I. Borevich, as well as N. A. Vavilov, for their assistance.     

Subgroups of the general symplectic group containing the group of diagonal matrices

There are described the subgroups of the general symplectic group =GSp(2n, R) over a commutative semilocal ring R, containing the group of symplectic diagonal matrices. For each such subgroup P there is uniquely defined a symplectic D-net a such that ()pN(), where () is the net subgroup in corresponding to (cf. RZhMat, 1977, 5A288), and N() is its normalizer. The quotient group N × ()/() is calculated. There are also considered subgroups in Sp(2n, R). Analogous results for subgroups of the general linear group were obtained earlier in RZhMat, 1978, 9A237.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 31–47, 1980.     

Boundary control of heat conduction

The paper considers control of the heat conduction process u_t — u = g from the initial state u(x, 0) to the final state u(x, t₁) in a fixed (finite) time t₁ via the coefficient (z) in the boundary condition Bu = (u/n) + (x)u. A uniqueness theorem is proved for the problem to find the process—control pair (u, ). The control problem is posed in terms of the coefficient in a boundary condition of the form Bu = (u/n) + (t)u.Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 3, pp. 93–97, 1993.     

Net determinant over a Bezoutian local ring

Let be any D-net of ideals of order n over a commutative local Bezoutian ring R and denote by G() the corresponding net subgroup in the general linear group of degree n over R (RZhMat, 1977, 2A280). We give an explicit computation of the factor group G()/E(), where E() is the subgroup generated by all elementary transvections in G().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 5–13, 1982.     

Die Öffnungsstrecken der Bahnregelflächen geschlossener räumlicher äquiformer Zwangläufe

Let the orientated line of the three-dimensional moving space , trace out a closed ruled surface in the fixed space and let us consider an integral invariant the aperture distance of an orthogonal trajectory of its generators. Then the locus of lines with a given is a cyclic quadratic complex, which reduces to a linear complex in the case =0. Furthermore in this paper some line-geometric Holditch-theorems due toS. Hentschke [6],L. Hering [7] andJ. Hoschek [9], are generalized.     

Extremal problems in the class σ(Τ)

Let L_f(r)={w=f(x), ¦z¦=r}, 1 < r < , be a level line of the function f(z) . Sharp upper bounds are obtained for the diameter of the curve L_f(r) in the class () for functions f(z)=z+₀+₁z^–1+... for which there exists a domain _f that complements the exterior of the unit disk and has conformal radius at the point w=0 satisfying the condition R(_f,0) , 0 < < 1. Also, a set of values is found for the coefficient ₁ in the class ().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 138–153, 1991.     

Complementary permutations for abelian groups

Summary LetG be an additively written abelian group and leth: G G be a given function. M. Hall Jr. (1952) and L. Fuchs (1958) already answered the following question. For what functionsh: G G does the functional equation(x) + (x) = h(x) (x G) have as its solution a pair of permutations and ofG? In this paper, we give explicit constructions of such a pair, in a number of cases, in particular whenh(x) x andG is finite. We further determine the finite groupsG where the latter, can be chosen to be automorphisms.In the case whereG is an infinite topological group, we study in how far and can be chosen as Borel measurable permutations, given thath: G G itself is Borel measurable.     

Characterizations of absolute F σδ-sets

We give several internal characterizations for the metrizable absolute F -spaces. The characterizing conditions involve the existence of compatible bicomplete quasi-metrics, of complete sequences of -discrete closed covers and of compact -discrete closed networks.     

σ-additivity in manuals and orthoalgebras

Notions of -additivity are introduced for orthoalgebras and for manuals. It is shown that the logic of a -additive manual is a -additive orthoalgebra, and that, conversely, every -additive orthoalgebra arises as such a logic. Using this theorem, it is shown that a given orthoalgebra admits at most one reasonable extension of its orthogonal sum operation to countable jointly orthogonal sets. It is also shown that every orthoalgebra can be embedded in a sigma-orthoalgebra and every orthomodular poset, in a -OMP.     

On the Convergence of Stewart's QLP Algorithm for Approximating the SVD

The pivoted QLP decomposition, introduced by Stewart [20], represents the first two steps in an algorithm which approximates the SVD. The matrix A₀ is first factored as A₀=QR, and then the matrix R ^T₁ is factored as R ^T₁=PL ^T, resulting in A=Q₁ LP ^T₀ ^T, with Q and P orthogonal, L lower-triangular, and ₀ and ₁ permutation matrices. Stewart noted that the diagonal elements of L approximate the singular values of A with surprising accuracy. In this paper, we provide mathematical justification for this phenomenon. If there is a gap between _k and _k+1, partition the matrix L into diagonal blocks L ₁₁ and L ₂₂ and off-diagonal block L ₂₁, where L ₁₁ is k-by-k. We show that the convergence of ( _j (L ₁₁)^–1– _j ^–1)/ _j ^–1 for j=1,. . .,k, and of ( _j (L ₂₂)– _k+j )/ _k+j , for j=1,. . .,n–k, are all quadratic in the gap ratio _k+1/ _k . The worst case is therefore at the gap, where the absolute errors L ₁₁ ^–1– _k ^–1 and L ₂₂– _k+1 are thus cubic in _k ^–1 and _k+1, respectively. One order of convergence is due to the rank-revealing pivoting in the first step; then, because of the pivoting in the first step, two more orders are achieved in the second step. Our analysis assumes that ₁=I, that is, that pivoting is done only on the first step. Although our results explain some of the properties of the pivoted QLP decomposition, they hypothesize a gap in the singular values. However, a simple example shows that the decomposition can perform well even in the absence of a gap. Thus there is more to explain, and we hope that our paper encourages others to tackle the problem. The QLP algorithm can be continued beyond the first two steps, and we make some observations concerning the asymptotic convergence. For example, we point out that repeated singular values can accelerate convergence of individual elements. This, in addition to the relative convergence to all of the singular values being quadratic in the gap ratio, further indicates that the QLP decomposition can be powerful even when the ratios between neighboring singular values are close to one.     

Pinning effect in peierls impurity systems with deviation from half-filling of the energy band

A study is made of the effect of deviation from half-filling of the energy band (0) on the Fröhlich collective mode in onedimensional impurity systems. A low impurity concentration is considered, and the infinite series of impurity scattering is taken into account self-consistently in the determination of the collective mode Green's function. The conductivity () is found in terms of this Green's function, and an analytic expression is obtained for () at _T ( _T is the pinning frequency). It is shown that for the ratio Re(()/_max) a universal formula arises. It differs from the results of Kurihara in the expression for _T , which contains an essential dependence on in the incommensurate state of the charge density wave. It is also shown that the width of the peak in the dependence () and its position increase with increasing .Institute of Applied Physics, Moldovan Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 110–122, October, 1994.     

Group automorphisms with finite entropy

A compact metrie abelian group X with the normalized Haar measure is a Lebesgue probability space. A group automorphism ofX is an invertible measure preserving transformation of the probability space. This paper is to show that if the entropy of is finite, then there exist totally disconnected subgroupsH andN, a finite-dimensional subgroupS and a subgroupT satisfying the conditions: (i)H, N, S andT are strictly -invariant, (ii)N=HST, (iii)h(| _N )=0, (iv) ifS/N is non-trivial then it is a finite-dimensional solenoidal group with condition (**) (see the definition in §1), (v) ifT/N is non-trivial then it is connected and locally connected, such thatX/N splits into a direct sumX/N=H/NS/NT/N. This result characterizes the structure of finite entropy automorphisms.     

A characterization of subgroups of the orthogonal group

LetG be a subgroup of the general linear group GL_n(K), where charK 2. Put Kⁿ =V. AssumeG is generated by the setS of all elements inG for which dimV( – 1) = 1, and suppose ²=1_V for each inS. If {V(–1)¦S} contains a simplex, if – 1_V G, and if inG is a product of dim v(–1) elements inS wheneverV(–1) is not contained in the kernel of–1, thenG is a subgroup of an orthogonal group.This research was supported in part by NSERC Canada grant A7251.To Helmut Mäurer on his 60th birthday     

Approximation of functions by partial sums of Fourier series in polynomials orthogonal on an interval

For certain weight functions p(t) and q(t), upper bounds are obtained for the difference between partial sums of Fourier series of a function/ with respect to the systems _p and _q of polynomials orthogonal on [–1, 1] (a comparison theorem is incidentally proved for the systems _p and _q). By using these upper bounds, known asymptotic expressions for the Lebesgue function, and an upper bound (forf W^r H) of the remainder in a Fourier-Chebyshev series, we establish corresponding results for Fourier series with respect to a system _p.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 431–441, October, 1970.     

A skew-product which is Bernoulli

We show that if is the shift on sequences of {0,1} and is the entropy zero transformation used by Ornstein in constructing a counter-example toPinsker's conjecture, then the skew-product transformationT defined byT(x,y)=(x, ^x0 y) is Bernoulli. ThisT is conditionally mixing with respect to the independent generator for , a partition with full entropy.This research was done while the first author was a visitor at Stanford, supported in part by NSF Grant MP-575-08324.     

Local partial regularity theorems for suitable weak solutions of a class of degenerate systems

A partial regularity theorem is established for a particular class of weak solutions to the systemu/t– div(K(u)u)=(u)¦¦², div((u))=0 on a bounded domain inR ^N . Under our assumptions, (u) may exhibit exponential decay, and thus the system may be degenerate. Our proof is based upon a blow-up argument.This work was supported in part by NSF Grant DMS9424448.     

Parabolic subgroups of twisted Chevalley groups over a semilocal ring

It is shown that, under minor additional assumptions, the standard parabolic subgroups of a Chevalley group G (, R) of twisted type =A_l,l odd, D_l, E₆ over a commutative semilocal ring R with involution are in one-to-one correspondence with the -invariant parabolic nets of ideals of R of type , i.e., with the sets, of ideals of R such that: (l) whenever; (2) = for all ; (3) =R for > 0. For Chevalley groups of normal types, analogous results were obtained in Ref. Zh. Mat. 1976, 10A151; 1977, 10A 301; 1978, 6A476.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 21–36, 1979.