Semi-regular Relative Difference Sets with Large Forbidden Subgroups

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that there does not exist a $(2p,p,2p,2)$ relative difference set in any group of order $2p^2$, and an abelian $(4p,p,4p,4)$ relative difference set can only exist in the group $\Bbb{Z}_2^2\times \Bbb{Z}_3^2$. On the other hand, we construct a family of non-abelian relative difference sets with parameters $(4q,q,4q,4)$, where $q$ is an odd prime power greater than 9 and $q\equiv 1$ (mod 4). When $q=p$ is a prime, $p>9$, and $p\equiv$ 1 (mod 4), the $(4p,p,4p,4)$ non-abelian relative difference sets constructed here are genuinely non-abelian in the sense that there does not exist an abelian relative difference set with the same parameters

On vectorial functions mapping strict affine subspaces of their domain into strict affine subspaces of their co-domain, and the strong D-property

Given three positive integers $n<N$ and $M$, we study those vectorial Boolean $(N,M)$-functions $\mathcal{F}$ which map an $n$-dimensional affine space $A$ into an $m$-dimensional affine space where $m<M$ and possibly $m=n$. This provides $(n,m)$-functions $\mathcal{F}_A$ as restrictions of $\mathcal{F}$. We show that the nonlinearity of $\mathcal{F}$ must not be too large for allowing this, and we observe that if it is zero, then it is always possible. In this case, we show that the nonlinearity of the restriction may be large. We then focus on the case $M=N$ and $\mathcal{F}$ of the form $\psi(\mathcal{G}(x))$ where $\mathcal{G}$ is almost perfect nonlinear (APN) and $\psi$ is a linear function with a kernel of dimension $1.$ We observe that the problem of determining the D-property of APN $(N-1,N)$-functions $\mathcal{G}_A$, where $A$ is a hyperplane, is related to the problem of constructing APN $(N-1,N-1)$-functions $\mathcal{F}_A$. For this reason, we introduce the strong D-property defined for $(N,N)$-functions $\mathcal{G}$. We give a characterization of this property for crooked functions and their compositional inverse (if it exists) by means of their ortho-derivatives, and we prove that the Gold APN function in dimension $N$ odd big enough has the strong D-property. We also prove in simpler a way than Taniguchi in 2023 that the strong D-property of the Gold APN function holds for $N$ even big enough. Then we give a partial result on the Dobbertin APN power function, and on the basis of this result, we conjecture that it has the strong D-property as well. We then move our focus to two known infinite families of differentially 4-uniform $(N-1,N-1)$-permutations constructed as the restrictions of $(N,N)$-functions $\mathcal{F}(x)=\psi(\mathcal{G}(x))$ or $\mathcal{F}(x)=\psi(\mathcal{G}(x))+x$ where $\psi$ is linear with a kernel of dimension $1$ and $\mathcal{G}$ is an APN permutation. After a deeper investigation on these classes, we provide proofs (which were missing) that they are not APN in dimension $n=N-1$ even. Then we present our own construction by relaxing some hypothesis on $\psi$ and $\mathcal{G}$

Trapping and Wiggling: Elastohydrodynamics of Driven Microfilaments

We present a general theoretical analysis of semiflexible filaments subject to viscous drag or point forcing. These are the relevant forces in dynamic experiments designed to measure biopolymer bending moduli. By analogy with the ``Stokes problems" in hydrodynamics (fluid motion induced by that of a wall bounding a viscous fluid), we consider the motion of a polymer one end of which is moved in an impulsive or oscillatory way. Analytical solutions for the time-dependent shapes of such moving polymers are obtained within an analysis applicable to small-amplitude deformations. In the case of oscillatory driving, particular attention is paid to a characteristic length determined by the frequency of oscillation, the polymer persistence length, and the viscous drag coefficient. Experiments on actin filaments manipulated with optical traps confirm the scaling law predicted by the analysis and provide a new technique for measuring the elastic bending modulus. A re-analysis of several published experiments on microtubules is also presented.Comment: RevTex, 24 pages, 15 eps figs, uses cite.sty, Biophysical

Signalling Rivalry and Quality Uncertainty in a Duopoly

This paper considers price competition in a duopoly with quality uncertainty. The established firm (the `incumbent') offers a quality that is publicly known; the other firm (the `entrant') offers a new good whose quality is not known by some consumers. The incumbent is fully informed about the entrant's quality. This leads to price signalling rivalry because the incumbent gains and the entrant loses if observed prices make the uninformed consumers more pessimistic about the entrant's quality. When the uninformed consumers' beliefs satisfy the `intuitive criterion' and the `unprejudiced belief refinement', prices signal the entrant's quality only in a two-sided separating equilibrium and are identical to the full information outcome

Double Charge Exchange of Pions on Helium-4

The reaction 4He(π-π+) at an incident π- energy of 165 Mev has been measured at 0° in a search for bound states of the tetraneutron and resonant states in the continuum. No resonant structure was observed, and a cross section of 7 ± 15 nb/sr has been determined for bound tetraneutron production by this reaction. The measured cross section magnitudes are consistent with the related measurements of Falomkin et al and Stetz et al, but are larger than the results of Kaufman et al and the calculations of Gibbs et al by a factor of at least 100. The relevance of these results to identification of the pion double charge exchange reaction mechanism is discussed.</p

Q-Curvature, Spectral Invariants, and Representation Theory

We give an introductory account of functional determinants of elliptic operators on manifolds and Polyakov-type formulas for their infinitesimal and finite conformal variations. We relate this to extremal problems and to the Q-curvature on even-dimensional conformal manifolds. The exposition is self-contained, in the sense of giving references sufficient to allow the reader to work through all details.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA