On groups with conjugacy classes of distinct sizes

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following:

1. , for 1a5, a≠2.

2. A₈.

3. PSL(3,4)^e, for 1e10.

4. A₅×PSL(3,4)^e, for 1e10.

Based on this result, we virtually show that if G is an ah-group with π(G) 2,3,5,7 , then F(G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S₃, then the non-abelian socle of G is either trivial or isomorphic to one of the following:

1. , for 3a5.

2. PSL(3,4)^e, for 1e10.

Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z₂S_n. These results are of independent interest and are located in appendices for greater autonomy.     

Anomalous dispersion in the Belousov-Zhabotinsky reaction: Experiments and modeling

We report results on dispersion relations and instabilities of traveling waves in excitable systems. Experiments employ solutions of the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction confined to thin capillary tubes which create a pseudo-one-dimensional system. Theoretical analyses focus on a three-variable reaction-diffusion model that is known to reproduce qualitatively many of the experimentally observed dynamics. Using continuation methods, we show that the transition from normal, monotonic to anomalous, single-overshoot dispersion curves is due to an orbit flip bifurcation of the solitary pulse homoclinics. In the case of “wave stacking”, this anomaly induces attractive pulse interaction, slow solitary pulses, and faster wave trains. For “wave merging”, wave trains break up in the wake of the slow solitary pulse due to an instability of wave trains at small wavelength. A third case, “wave tracking” is characterized by the non-existence of solitary waves but existence of periodic wave trains. The corresponding dispersion curve is a closed curve covering a finite band of wavelengths.     

Cortical disinhibition occurs in chronic neuropathic,but not in chronic nociceptive pain

Background  

The aim of this study was to examine the relationship between chronic neuropathic pain after incomplete peripheral nerve lesion, chronic nociceptive pain due to osteoarthritis, and the excitability of the motor cortex assessed by transcranial magnetic stimulation (TMS). Hence in 26 patients with neuropathic pain resulting from an isolated incomplete lesion of the median or ulnar nerve (neuralgia), 20 patients with painful osteoarthritis of the hand, and 14 healthy control subjects, the excitability of the motor cortex was tested using paired-pulse TMS to assess intracortical inhibition and facilitation. These excitability parameters were compared between groups, and the relationship between excitability parameters and clinical parameters was examined.     

Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

We consider Dirichlet series z_g,a(s)=?_n=1^￥ g(na) e^{-l_n s}{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ _n  = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ?_n=1^￥ g(na) zⁿ{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series z_g,a(s) = ?_n=1^￥ g(n a)/n^s{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ ₀ = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ ₀ satisfies σ ₀ ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ _g,α (s) has an analytic continuation to the entire complex plane.     

On the imaginary simple roots of the Borcherds algebra II9,1

In a recent paper [O. Bärwald, R.W. Gebert, M. Günaydin and H. Nicolai, preprint KCL-MTH-97-22, IASSNS-HEP-97/20, PSU-TH-178, AEI-029, hep-th/9703084, to appear in Commun. Math. Phys.] it was conjectured that the imaginary simple roots of the Borcherds algebra _II9,1 at level 1 are its only ones. We here propose an independent test of this conjecture, establishing its validity for all roots of norm − 8. However, the conjecture fails for roots of norm −10 and beyond, as we show by computing the simple multiplicities down to norm −24, which turn out to be remarkably small in comparison with the corresponding E₁₀ multiplicities. Our derivation is based on a modified denominator formula combining the denominator formulas for E₁₀ and _II9,1, and provides an efficient method for determining the imaginary simple roots. In addition, we compute the E₁₀ multiplicities of all roots up to height 231, including levels up to l = 6 and norms −42.     

Variational Quantum Chemistry Programs in JaqalPaq

We present example quantum chemistry programs written with JaqalPaq, a python meta-programming language used to code in Jaqal (Just Another Quantum Assembly Language). These JaqalPaq algorithms are intended to be run on the Quantum Scientific Computing Open User Testbed (QSCOUT) platform at Sandia National Laboratories. Our exemplars use the variational quantum eigensolver (VQE) quantum algorithm to compute the ground state energies of the H2, HeH+, and LiH molecules. Since the exemplars focus on how to program in JaqalPaq, the calculations of the second-quantized Hamiltonians are performed with the PySCF python package, and the mappings of the fermions to qubits are obtained from the OpenFermion python package. Using the emulator functionality of JaqalPaq, we emulate how these exemplars would be executed on an error-free QSCOUT platform and compare the emulated computation of the bond-dissociation curves for these molecules with their exact forms within the relevant basis.     

A-stable Runge–Kutta methods for semilinear evolution equations

We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale. Under the same assumptions, we prove that a class of implicit, A-stable Runge–Kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale. Under the additional assumption that the linear part of the evolution equation is normal or sectorial, we prove full order convergence of the semidiscretization in time for initial data on open sets. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrödinger equation.     

On Transfer Operators for Continued Fractions with Restricted Digits

For any non-empty subset I of the natural numbers, let _I denotethose numbers in the unit interval whose continued fractiondigits all lie in I. Define the corresponding transfer operator for , where Re (rß) = _I is the abscissa of convergence of the series . When acting on a certain Hilbert space H_I, rß, weshow that the operator L_I, rß is conjugate to an integraloperator K_I, rß. If furthermore rß is real,then K_I, rß is selfadjoint, so that L_I, rß: H_I, rß H_I, rß has purely real spectrum.It is proved that L_I, rß also has purely real spectrumwhen acting on various Hilbert or Banach spaces of holomorphicfunctions, on the nuclear space C [0, 1], and on the Fréchetspace C [0, 1]. The analytic properties of the map rß L_I, rßare investigated. For certain alphabets I of an arithmetic nature(for example, I = primes, I = squares, I an arithmetic progression,I the set of sums of two squares it is shown that rß L_I, rß admits an analytic continuation beyond thehalf-plane Re rß > _I. 2000 Mathematics SubjectClassification 37D35, 37D20, 30B70.     

A conjectural extension of Hecke’s converse theorem

By using specific subsequences of two different types of generalized Stern polynomials, we obtain several related classes of finite and infinite continued fractions involving a single term \(z^{t^j}\) in their partial numerators, where z is a complex variable and t is a positive integer. This approach is extended to other, sparser, subsequences of Stern polynomials, based on certain Lucas functions; this then leads to further infinite classes of continued fractions.