An abstract characterization of Thompson’s group F

We show that Thompson’s group F is the symmetry group of the ‘generic idempotent’. That is, take the monoidal category freely generated by an object A and an isomorphism A ? A→A; then F is the group of automorphisms of A.     

Distortion of Wreath Products in Thompson’s Group F

The authors use geometric techniques to prove that the restricted wreath product F■Z is a quasi-isometrically embedded subgroup of Thompson's group F.     

On the conjugacy problem in the group F/N 1 ∩ N 2

Let N ₁ (N ₂) be the normal closure of a finite symmetrized set R ₁ (R ₂, respectively) in a finitely generated free group F = F(A). As is known, if R _i satisfies condition C(6), then the conjugacy problem is decidable in F/N _i . In the paper, it is proved that, if one adds to condition C(6) on the set R ₁ ∪ R ₂ the atoricity condition for the presentation 〈A | R ₁, R ₂〉, then the conjugacy problem is decidable in the group F/N ₁ ∩ N ₂ as well. In particular, for the decidability of the conjugacy problem in F/N ₁ ∩ N ₂, it is sufficient to assume that the set R ₁ ∪ R ₂ satisfies condition C(7).     

Thompson’s conjecture for alternating group of degree 22

For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial center, and N(G) = N(L), then L and G are isomorphic. In this paper, it is proved that Thompson’s conjecture is true for the alternating group A ₂₂ with connected prime graph.     

Generators and normal forms of Richard Thompson’s group F and the four-color theorem

We use a result of Kauffman to explore the connection between Richard Thompson’s Group F and the four-color theorem.     

Reducing conjugacy in the full diffeomorphism group of ℝ to conjugacy in the subgroup of orientation-preserving maps

Let Diffeo = Diffeo(ℝ) denote the group of infinitely differentiable diffeomorphisms of the real line ℝ, under the operation of composition, and let Diffeo⁺ be the subgroup of diffeomorphisms of degree +1, i.e., orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements f, g ∈ Diffeo are conjugate in Diffeo to associated conjugacy problems in the subgroup Diffeo⁺. The main result concerns the case when f and g have degree −1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in Diffeo⁺, in order to ensure that f is conjugated to g by an element of Diffeo⁺. The methods involve formal power series and results of Kopell on centralisers in the diffeomorphism group of a half-open interval. Bibliography: 4 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 231–237.     

The Dehn function of Baumslag’s metabelian group

Baumslag??s group is a finitely presented metabelian group with a ${\mathbb Z \wr \mathbb Z}$ subgroup. There is an analogue with an additional torsion relation in which this subgroup becomes ${C_m \wr \mathbb Z}$ . We prove that Baumslag??s group has an exponential Dehn function. This contrasts with the torsion analogues which have quadratic Dehn functions.     

The linear symmetries of Hill’s lunar problem

Archiv der Mathematik - A symmetry of a Hamiltonian system is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. For the planar and spatial Hill lunar problem, four...     

The numerical solution of Newton’s problem of least resistance

In this paper we consider Newton’s problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in \({\mathbb {R}}^{2}\) . We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized problem could be solved by standard optimization software efficiently. Second, we conjecture that the optimal body has a certain structure. We exploit this structure and obtain a variational problem in \({\mathbb {R}}^{1}\) . Deriving its Euler–Lagrange equation yields a program with two unknowns, which can be solved quickly.     

Irreducibility and extensions of Ostrowski’s Theorem

In this paper an extension of Ostrowski’s Theorem for complex square irreducible matrices is presented. Also extensions of similar statements for square complex matrices are analyzed and completed. Most of the statements in this work cover also the case of reducible matrices.     

The solution of graham’s greatest common divisor problem

The following conjecture of R. L. Graham is verified: Ifn≧n ₀, wheren ₀ is an explicitly computable constant, then for anyn distinct positive integersa ₁,a ₂, ...,a _n we have a _i /(a _i ,a _j ) ≧ ≧n, and equality holds only in two trivial cases. Here (a _i ,a _j ) stands for the greatest cnmmon divisor ofa _i anda _j .     

Some extensions of Brioschi’s double alternant

We evaluate some determinants related to Brioschi’s extension of the classical double alternant of Cauchy, and Scott’s extension of Brioschi’s.     

Conjugacy and dynamics in Thompson’s groups

We give a unified solution to the conjugacy problem for Thompson’s groups \(F, \,T\), and \(V\). The solution uses “strand diagrams”, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson’s groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson’s groups, and we use this correspondence to investigate the dynamics of elements of \(F\). Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.     

On Hilbert’s solution of Waring’s problem

In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x ₁ ^k + x ₂ ^k + … + x _g ^k , where the x _i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit bounds on the least permissible value of g. We show how to modify Rieger’s argument, using ideas of F. Dress, to obtain a better explicit bound. While far stronger bounds are available from the powerful Hardy-Littlewood circle method, it seems of some methodological interest to examine how far elementary techniques of this nature can be pushed.     

A brief account of Klein’s icosahedral extensions

We present an alternative relatively easy way to understand and determine the zeros of a quintic polynomial whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in his famous Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade are here shortened via Heymann’s theory of resolvents. Also, we give a complete explanation of the so-called icosahedral equation and its solution in terms of Gaussian hypergeometric functions. As an innovative element, we construct this solution by using algebraic transformations of hypergeometric series.     

Waring’s problem in prime numbers

The existence of a constant V(n) such that any sufficiently large natural number can be represented as a sum of nth degrees of primes in total quantity not exceeding this constant is proved.     

Parametric stability in Robe’s problem

In this work we have given a Hamiltonian formulation to Robe’s problem, obtaining again the classic results. We have computed the resonances existing in the circular case and obtained some information with regard to the linear stability of the central equilibrium of Robe’s problem in the elliptic case. In some critical cases we have constructed, in the parameter plane, the boundary curves that separate the regions of stability and instability.     

On the negativity of higher order derivatives of Dirichlet’s energy in plateau’s problem

We calculate higher order derivatives of Dirichlet’s Energy at a branched minimal surface in the direction of Forced Jacobi Fields discovered by the author and R. Böhme. We show that, under certain conditions these derivatives can be made negative, while all lower order derivatives vanish. This is the first time that derivatives of order greater than three have been calculated.