The Inverse Scattering Method and Systems of Equations qxy = F(q,qy)

Consider equations of the form where q is in general a complex vector and the function F depends nontrivially both on q and on q_y. We show that a family S of such equations can be investigated by the inverse scattering method. If an equation (*) belongs to S, the function F depends linearly on q and algebraically on q_y. We show that the family S contains a subfamily in which each equation can be obtained from the two dimensional Toda lattice equations by a Bäcklund transformation.     

Lehmer pairs of zeros,the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH ₀ (which is closely related to the Riemann -function) to be a Lehmer pair of zeros ofH ₀. With this formulation, we establish that each such pair of zeros gives a lower bound for the de Bruijn-Newman constant (where the Riemann Hypothesis is equivalent to the assertion that 0). We also numerically obtain the following new lower bound for :

Spectroscopy of excited state positronium

Electric dipole transitions in the microwave range have been induced between the fine-structure levels of positronium in the excited staten=2. As an indication of the transitions, we used the increase in Lyman- radiation when the metastable 2³S₁-level is depopulated. The results for the transitions 2³S₁2³P_0,1,2 are ₀=18499.65±1.20±4.00 MHz, ₁=13012.42 ±0.65±1.54 MHz and ₂=8624.38±0.54±1.40 MHz. The first error is statistical and the second systematic. The precision of the present measurement has improved by a factor of 3, compared to previous data. Recent bound state QED-calculations have been extended to the orderR _t8 ⁴ln ^–1. The not yet completely calculated orderR _t8 ⁴ is estimated to contribute less than 1 MHz. Our experimental results are in good agreement with theory. By applying a weak magnetic field, we were able to observe the transition 2³S₁2¹P₁ which is strictly forbidden byC-invariance in zero field. Our result, corrected for Zeeman- and motional Starkeffect, is ₃=11180.0±5.0±4.0 MHz. An upper limit for theC-violating matrix element of MHz could be deduced. Our experiment used moderated slow positrons from the bremsstrahlung and pair production of a pulsed electron linear accelerator (TEPOS facility at the university of Giessen).     

Results and estimates on pseudopowers

Let be a positive integer. We say looks like a power of 2 modulo a prime if there exists an integer such that . First, we provide a simple proof of the fact that a positive integer which looks like a power of modulo all but finitely many primes is in fact a power of . Next, we define an -pseudopower of the base to be a positive integer that is not a power of , but looks like a power of modulo all primes . Let denote the least such . We give an unconditional upper bound on , a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that is about for a certain constant . We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than are also given.

     

Classical irreversibility and mapping of the isosceles triangle billiard

Mapping of the two-dimensional isosceles triangle billiard onto the circular one-dimensional motion of two mass points is described. The singular nature of trajectories directly incident on an acute vertex is discussed in the framework of the present mapping. For an obtuse-angled isosceles triangle, dynamical equations in two-particle space applied to an orbit along a hypotenuse incident on the obtuse vertex suggests irreversible behavior at the critical angle =2/3. Thus it is found that the nonsingular motion of a finite smooth-walled disk on this trajectory exhibits irreversibility. A finite spherical smooth-walled particle moving in a uniform right cylinder whose cross section includes this critical vertex angle likewise exhibits irreversibility. Each such example comprises an irreversible orbit for a single-particle Hamiltonian.     

There are uncountably many universal topological planes

We prove the existence of uncountably many nonisomorphic topological projective planes, each universal in the sense that it contains an isomorphic copy of every pseudoline arrangement.     

A graph efficiency multiproduct model of corn/livestock farming: Accounting for nitrate pollution

This paper estimates a non-parametric production frontier for a population of 117 corn/livestock farms in the Corn Belt region in 1987, employing a hyperbolic graph efficiency approach. There are 7 outputs, 39 variable inputs, 4 fixed inputs, and one bad input (residual nitrogen). Three graph efficiency models are estimated. A profit maximization model is specified to estimate a production frontier constrained only by the fixed factors. Two other models involving tax constraints are also estimated. One involves a tax directly on nitrogen and the other involves a tax directly on residual nitrogen, making the disposal of residual nitrogen costly. The nitrogen tax constraint is more effective in reducing residual nitrogen loadings and causes a larger reduction in income than the residual tax constraint.     

A “Multiplicative Coboundary” theorem for some sequences of random matrices

Schmidt(¹⁰) showed that if the family of distributions of the partial sums of a strictly stationary random sequence is tight, then that random sequence is a coboundary. Here an analog of that result is proved for some sequences of random matrices, with partial sums replaced by matrix products.     

Z 3-Graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d³=0, but d²0. The entities dx ⁱ and d² x ^k generate an associative algebra; we shall suppose that the products dx ⁱ dx ^k are independent of dx ^k dx ⁱ , while theternary products will satisfy the relation: dx ⁱ dx ^k dx ^m =jdx ^k dx ^m dx ⁱ =j ²dx ^m dx ^m dx ⁱ dx ^k , complemented by the relation dx ⁱ d² x ^k =jd² x ^k dx ⁱ , withj:=e^2i/3.We shall attribute grade 1 to the differentials dx ⁱ and grade 2 to the second differentials d² x ^k ; under the associative multiplication law the grades add up modulo 3.We show how the notion ofcovariant derivation can be generalized with a 1-formA so thatD:=d+A, and we give the expression in local coordinates of thecurvature 3-form defined as :=d² A+d(A ²)+AdA+A ³.Finally, the introduction of notions of a scalar product and integration of theZ ₃-graded exterior forms enables us to define the variational principle and to derive the differential equations satisfied by the 3-form . The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensorF _ik and its covariant derivativesD _i F _km .