Three coupled non-linear differential equations are given, which exhibit what appears to be chaotic behavior. A simple coordinate transformation reduces the number of coupled variables to two; however, the ''chaos'' persists, characterized by a bounded, irregular motion displaying a positive Lyapunov exponent. This behavior, seemingly a violation of the Poincare-Bendixson theorem, is seen to be a result of non-differentiability (with respect to the dependent variables) at a singular point. Any physically realizable trajectory necessarily passes through this point, about which the dynamics shows an extreme sensitivity to any perturbation, regardless of the magnitude. It is concluded that any physical system governed by these (or similar) equations will undergo a stochastic change in its dynamical trajectory whenever it passes near the singularity, rendering even relatively short-term predictability impossible.

The success of nonrelativistic, Schrödinger equation (with relativistic corrections) fits of the charmonium spectra suggests the value of a relativistic extension to allow a treatment of the low-mass meson trajectories. We propose a covariant generalization of the two-particle Dirac equation, including a covariant generalization of the famed «Coulomb-plus-linear» potential. After separating this equation and relating the eigenvalues of the separation to the observable mass of a bound state, we found that the combination of relativistic kinematics, a four-vector tensor form for the interaction, and a potential linear in the proper separations(=(r2−t2)1/2) of the two particles for larges implies asymptotically linear Regge trajectories. By solving the equation we were able to reproduce the π, σ, K and K* trajectories very accurately; our potential parameters were similar to those found in heavy-quark studies. A consideration of the nonrelativistic limit of our equation provides hints as to the source of the long-range weakening of the spin-spin part of the four-vector interaction which has been observed in connection with the hyperfine structure of the heavy-quark spectra.

The recently proposed α-expansion method for calculation of the hadronic spectrum in QCD is tested by the simple example of the harmonic oscillator. It appears that three terms of the expansion of the generalized S-matrix at large momenta are sufficient to calculate the whole spectrum to very good accuracy.

The eclosion rhythm of Drosophila pseudoobscura under 24 h photo-periods is investigated. Pupae emerge within a few hours of dawn for all day to night (L/D) ratios. Well established properties of the endogenous, self-sustained master clock (A-oscillator) coupled to a hour-glass lead to predictions which would fail badly. On the other hand, it is shown that a coupled oscillator model, with a resonant slave oscillator (B-oscillator) predicts emergence near dawn. The adaptive value of B-oscillators is argued and the relevance for photoperiodism is conjectured.

Using an effective-range-type expansion in the $j$ plane for the $N$ and $D$ functions, the $t$-channel partial-wave amplitude $A(j,t) (=\frac{N}{D})$ is expressed as $\frac{[{F}_{1}+{F}_{2}(j\ensuremath{-}{\ensuremath{\alpha}}_{c})+{G}_{1}{(j\ensuremath{-}{\ensuremath{\alpha}}_{c})}^{\frac{1}{2}}]}{[j\ensuremath{-}{\ensuremath{\alpha}}_{0}+\ensuremath{\epsilon}{(j\ensuremath{-}{\ensuremath{\alpha}}_{c})}^{\frac{1}{2}}]}$, where a square-root singularity is assumed for simplicity. For ${\ensuremath{\alpha}}_{0}$ linear in $t$ the following results are obtained: (1) The Regge poles (= ${\ensuremath{\alpha}}_{\ifmmode\pm\else\textpm\fi{}}$) are complex below a certain $t$ value. (ii) The amplitude $A(s,t)$ in the scattering region is of the form ${a}_{+}(s,t){s}^{{\ensuremath{\alpha}}_{+}}+{a}_{\ensuremath{-}}(s,t){s}^{{\ensuremath{\alpha}}_{\ensuremath{-}}}$, with ${a}_{\ifmmode\pm\else\textpm\fi{}}(s,t)$ expressible as a sum of two terms. The first term is one-half the residue of the complex pole, whether the pole be on the physical or unphysical sheet, the second term is a series involving the product $({\ensuremath{\alpha}}_{\ifmmode\pm\else\textpm\fi{}}\ensuremath{-}{\ensuremath{\alpha}}_{c})\mathrm{ln}s$. It is found by explicit calculation that if $\ensuremath{\epsilon}$ is small ($\ensuremath{\approx}0.1$) then only one or two terms of the series are important up to quite high $s(\ensuremath{\approx}200 {\mathrm{BeV}}^{2})$. Only at asymptotic $s$ will the series sum up to give the tip of the cut contribution, $\frac{{s}^{{\ensuremath{\alpha}}_{c}}}{{(\mathrm{ln}s)}^{\frac{3}{2}}}$. At presently available energies, therefore, the $s$ dependence is largely given by ${s}^{{\ensuremath{\alpha}}_{\ifmmode\pm\else\textpm\fi{}}}$. The results remain unchanged if the pole is on the real axis ($\ensuremath{\epsilon}=0$). (iii) At the $t$ value where the poles collide, the $s$ dependence is of a typical double pole from ${s}^{\ensuremath{\alpha}}\mathrm{ln}s$. (iv) It is observed that the strength of the cut is manifested through $|\frac{{F}_{2}}{{F}_{1}}|$ and $|\frac{{G}_{1}}{{F}_{1}}|$ as well as through $\ensuremath{\epsilon}$. (v) For small, fixed $\ensuremath{\epsilon}$ the ${F}_{2}$ term plays a crucial role in shifting the zeros in $t$ of $A(s,t)$ from their simple pole values. (vi) The sign of the cut is intimately connected with the phase of the complex residues for $t\ensuremath{\le}0$, the width of the $t$ channel resonances, and with the question of determining the sheet on which the poles are located for $t\ensuremath{\le}0$. (vii) Finally, $A(s,t)$ is in general not factorizable but can be written as a sum of (complex conjugate) factorized quantities.

Assuming that the trajectories on which $\ensuremath{\rho}$ and ${f}^{0}$ mesons lie are degenerate [i.e., assuming exchange degeneracy and $\mathrm{SU}(3)$ symmetry for the $\ensuremath{\rho}$ and ${P}^{\ensuremath{'}}$ trajectories], the zeroth-moment finite-energy sum rule is used for $I=1$ in the crossed channel of the $\ensuremath{\pi}\ensuremath{\pi}$ system. If we ignore the $I=2$ contribution and assume resonance saturation in the direct channel, then a single trajectory, assumed to be of the form $\ensuremath{\alpha}(x)=\overline{a}+bx$ (where $x=s or t$), dominates the direct channel ($s$ channel) at low energies. We also assume that a single trajectory dominates the crossed channel ($t$ channel). It is found that for any given $t$ in the sum rule, there is no unique (bootstrap) solution but rather a continuous range of solutions for $\overline{a}$ and $b$. The solutions depend on the choice of $t$ and are sensitive to the functional form of the residue $\ensuremath{\beta}$. Even with the most complete form of $\ensuremath{\beta}$, a unique bootstrap solution is not possible within the single-trajectory approximation. For a given $t$, a unique solution is obtained only when $b$ (and ${m}_{\ensuremath{\pi}}$) are given a priori. For $t'\mathrm{s}$ such that $\ensuremath{\alpha}(t)=1$, where the results are independent of the functional form of $\ensuremath{\beta}$, if $b$ is assumed to have the experimental value of 1 Be${\mathrm{V}}^{\ensuremath{-}2}$, then $\overline{a}$ turns out to be 0.64, which is close to the experimental value of \ensuremath{\simeq} 0.5. For $\ensuremath{\alpha}(t)$ in the neighborhood of 1, if $b$ varied from 0 to 5 Be${\mathrm{V}}^{\ensuremath{-}2}$, $\overline{a}$ varies only between 0.5 and 0.7.

A bootstrap calculation for the width of the $\ensuremath{\rho}$ resonance is performed using a simplified version of a Reggeized bootstrap theory proposed recently by the authors. A phenomenological Pomeranchuk input trajectory has been assumed. The $I=2$ channel is eliminated from the appropriate crossing relation and no statements about this channel are necessary. The $2\ensuremath{-}\ensuremath{\pi}$ continuum states are assumed to be dominated by the $\ensuremath{\rho}$ for the $I=1$ and by the ${f}^{0}$ for the $I=0$. The input $\ensuremath{\rho}$ trajectory is parametrized to produce the $\ensuremath{\rho}$ resonance at the observed mass. The $\ensuremath{\rho}$ width is then determined by the maximum satisfaction of the crossing relations. The calculation yields a "best" width of 125 MeV for the $\ensuremath{\rho}$. The problem concerning the simultaneous bootstrap of the $\ensuremath{\rho}$ mass and width is briefly discussed, and a systematic procedure for obtaining the "generalized potential" of the modified Cheng representation is given.

The infinitely many threshold poles that approachλ=0 asν→0 are shown to produce, using a product representation of theS-matrix obtained recently by Cheng, the correct threshold behaviour of theS-matrix; 1 for Reλ>0 and exp[2πiλ] for Reλ<0. Even though the contribution of a single threshold pole to the zero-energy scattering amplitude is vanishingly small, their total contribution is found to be, in general, quite significant. The contribution of these poles to the scattering length is calculated for some soluble cases. For parametrization of low-energy scattering, according to this analysis, a formalism which automatically includes the threshold poles is more reliable than one based on a few Regge poles alone.

A method described in a previous paper is tested in the light of potential theory. The method is based on dispersion relations for Regge pole parameters. The approximations consist in coupling the $\ensuremath{\beta}'\mathrm{s}$ to the $\ensuremath{\alpha}'\mathrm{s}$ by applying unitarity at $l=\ensuremath{\alpha}$ and considering only a few poles. When the generalized potential is replaced by a nonrelativistic potential, the coupling equations and solutions to the integral equations can be compared to exact results. Various representations are tested, and it is found that the "modified Khuri" representation for $A(l, s)$ gives good results for ${\ensuremath{\alpha}}_{1}(s)$ in the one-trajectory approximation for Yukawa potentials strong enough to cause bound $S$ states. The results for ${\ensuremath{\beta}}_{1}(s)$ are less satisfactory. The effect of coupling in the second trajectory is considered.

The connection between continuous Regge trajectories and the vanishing of renormalization constants is explored. It is found that if, in a field theory ${Z}_{1}\ensuremath{\rightarrow}0$ and ${Z}_{3}\ensuremath{\rightarrow}0$ in such a way that $\frac{{Z}_{1}}{{Z}_{3}\ensuremath{\rightarrow}0}$, then a Regge trajectory moves smoothly under an elementary particle pole so that the particle becomes dynamical in the Regge sense. Thus a bootstrapped world may perhaps equally well be defined by its satisfying a field theory with all renormalization constants set equal to zero, as by saying that all particles lie on Regge trajectories.

A study of some problems in nonrelativisitic multichannel scattering is made in the context of complex angular momentum. A generalization of the Mandelstam symmetry is obtained. It is shown that the well-known "cusp" behavior in the elastic cross section when an inelastic channel opens up is not only present for $S$ waves, but is a persistent feature as the (real) angular momentum is varied continuously from $\ensuremath{-}\frac{3}{2}$ to +\textonehalf{}. The existence of the "indeterminacy points" is indicated. Using the factorizability condition on the residues of the $S$ matrix, it is explicitly exhibited how a part of the $S$ matrix decouples when a resonance pole drifts to the real axis.

A recent representation by Cheng of the partial-wave scattering amplitude has been modified to display the large-energy behavior explicitly in order to improve convergence in terms of Regge poles. The resulting representation has the following properties regardless of the number of trajectories included: (a) It is unitary for all $l$ and $s>0$. (b) It gives the correct threshold behavior as $s\ensuremath{\rightarrow}0$ for the real and imaginary parts of the amplitude. (c) It reproduces the appropriate analytic properties for the total amplitude in the $cos\ensuremath{\theta}$ plane. (d) It converges rapidly.

The behavior of Regge trajectories is studied for certain classes of potentials. The asymptotic properties are found to depend on the behavior of the potential at the origin as expected. In particular weak potentials with attractive and repulsive cores are studied. In all cases there are indeterminacy points (crossings of αn(k) and αm(-k) at positive and negative energies and branch points (crossings of αn(k) and αm(-k). Some trajectories are seen to «lie down» at negative integer and half-integerl-values.

Some properties of Jost functions are reviewed for real and complex angular momentum and compared to the determinantal expansion. The exact phase shifts and Jost functions in the complexk-plane are calculated forS- andP-waves and compared to the first and second-order determinantal expansion.