Groups with regular automorphisms of order four

Mathematische Zeitschrift -     

The regular points of simple functions

We extend the work of Ioffe and Lewis [A. Ioffe and A. Lewis, Critical points of simple functions, Optimization, 57 (2008), pp. 3–16] and relate it to the previous work of Morse [M. Morse, Topologically non-degenerate functions in a compact n-manifold, J. Anal. Math. 7 (1959), p. 243]. We show that the concept of regularity for piecewise linear functions can be explained in geometric topological terms and this explanation leads to a unified view of several concepts of regularity for such functions.     

Pointwise recurrent homeomorphisms with stable fixed points

We prove that a pointwise recurrent, orientation preserving homeomorphism of the 2-sphere, which is different from the identity and whose fixed points are stable in the sense of Lyapunov must have exactly two fixed points. If moreover there are no periodic points, other than fixed, then every stable minimal set is connected and its complement has exactly two connected components. Finally, we study liftings of the restriction to the complement of the fixed point set to the universal covering space.     

Diffeomorphisms of the plane with stable periodic points

We consider self-diffeomorphisms of the plane with a hyperbolic fixed point and a nontransversal homoclinic point. We show that a neighborhood of the homoclinic point may contain countably many stable periodic sets whose characteristic exponents are bounded away from zero.     

Families of curves with ordinary singular points on regular surfaces

Sunto Fissata una superficie algebrica liscia X sopra un campo algebricamente chiuso e di caratteristica 0, e una curva ridotta C su X, si introduce, per ogni -upla di numeri naturali m₁,..., m, uno schema W=W(C, m₁,..., m), parametrizzante le curve del sistema lineare ¦C¦ con punti multipli ordinari di molteplicità almeno m₁,..., m. Si studia W in relazione con la configurazione delie singolarità e delle componenti irriducibili di C. Si prova in particolare un teorema di «virtuale connessione», che permette di stabilire l'esistenza di curve irriducibili in alcuni di tali schemi W.

---

---

Membro del G.N.S.A.G.A. del C.N.R.     

Stability and regular points of inequality systems

We undertake a general study of regular points of Lipschitz and strictly differentiable mappings with applications to tangent cone analysis, inversion theorems, perturbed optimization problems, and higher-order conditions.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A5116.Thanks are due to J. P. Penot, J. Ch. Pomerol, and a referee for their many helpful comments.     

On the enumeration of chains in regular chain-groups

Let N be a regular chain-group on E (see W. T. Tutte, Canad. J. Math.8 (1956), 13–28); for instance, N may be the group of integer flows or tensions of a directed graph with edge-set E). It is known that the number of proper Z_λ-chains of N (λ ∈ Z, λ ≥ 2) is given by a polynomial in λ, P(N, λ) (when N is the chain-group of integer tensions of the connected graph G, λP(N, λ) is the usual chromatic polynomial of G). We prove the formula: P(N, λ) = Σ_{[E′]∈O(N)⁺/～}Q(R_[E′](N), λ), where O(N)⁺ is the set of orientations of N with a proper positive chain, ～ is a simple equivalence relation on O(N)⁺ (sequence of reversals of positive primitive chains), and Q(R_[E′](N), λ) is the number of chains with values in [1, λ ? 1] in any reorientation of N associated to an element of [E′]. Moreover, each term Q(R_[E′](N), λ) is a polynomial in λ. As applications we obtain: P(N, 0) = (?1)^r(N)∥O(N)⁺/～∥; P(N, ?1) = (?1)^r(N)∥O(N)⁺∥ (a result first proved by Brylawski and Lucas); P(N, λ + 1) ≥ P(N, λ) for λ ≥ 2, λ ∈ Z. Our result can also be considered as a refinement of the following known fact: A regular chain-group N has a proper Z_λ-chain iff it has a proper chain in [?λ + 1, λ ? 1].     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

The moduli space of regular stable maps

We prove that the moduli space of regular stable maps in a complex manifold admits a natural complex orbifold structure. Our proof is based on Hardy decompositions and Fredholm intersection theory. The authors would like to thank the referee for his/her diligent work. We are grateful for the careful attention to detail in the report.     

Modeling viscoelastic response of particulate polymeric composites with high volume fractions of fillers

A generalized self-consistent method is extended to particulate viscoelastic composites with elastomeric matrices and high volume fractions of elastic inclusions. It is shown that the effective bulk modulus of a composite coincides with the bulk modulus of particles. A quadratic operator equation is derived for an analog of the effective shear relaxation kernel. This equation is explicitly solved using the Laplace transform method. The influence of material and geometrical parameters of a composite on its effective viscoelastic moduli is analyzed numerically.     

Fuchs inequalities for systems of linear differential equations with regular singular points

The upper and lower bounds for the sum of all exponents of a system of linear differential equations with regular singular points given on the Riemann sphere are presented. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Characterization of regular Markov chains for asymptotically independent variables

The characterization problem of a homogeneous Markov chain (with either finitely many or a countable number of states) is considered for the class of the variables satisfying the asymptotic independence. An allied result when some state distribution is given beforehand at some step is also presented.     

Determination of regular discrete relaxation spectra from the viscoelastic functions

A procedure for determining regular discrete relaxation spectra from the viscoelastic functions is presented as a generalization of a previously proposed procedure for determining the conditional rheological characteristics in a piecewise-exponential approximation of the experimental creep, relaxation, and – curves. The calculated data are in good agreement with the experimental results.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 801–807, September–October, 1971.     

Plane curves with three or four total inflection points

In this article, we will study plane curves of a certain degreed with three or four total inflection points. In particular,we will study their image in the moduli spaces. Also a resulton curves with five total inflection points is included.