Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's 'effective dimension'. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component lsing (and more generally Griffiths-Simon class) spin variables. In particular, it is established that even in dimensions d < 4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as Jx = 1/Ixl a+~ with 0 < a~< d/2, then the exponents ~, 6, ? and A 4 exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with try> 2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.

& Frisch (1980), by a shift of origin. It offers the advantage of the symmetric Fourier representation (1 2 ) . Early-time behaviour of the inviscid flowThe symmetries of the TG vortex are listed in appendix A. Here we emphasize only those that help to visualize the qualitative features of the flow and those that may be important in making this flow atypical of general three-dimensional flow. First, for all times, no fluid crosses any of the boundaries x , y or z = nn, where n is an integer.Therefore the flow can be visualized as flow in the box 0 < x , y , z < n with impermeable stress-free faces. I n the following discussion, the region 0 < x, y , z < n is termed the impermeable box, as it confines the flow, while the region 0 < x , y , z < 2n is termed the periodicity box, as it reflects the periodicity of the Fourier series (1.2). Also, because of the symmetries listed in appendix A, the flow at any point in space can be inferred from its values in the fundamental box 0 < x, y, z < in.Secondly, if near each face we write the velocity field in terms of components parallel or perpendicular to the face, i.e. u = u,, + vl, then ul and au,,/an vanish on the face. This implies that the vorticity on each face is normal to that face so it may be written w = @, where fi is the unit normal. Note that g must vanish on all edges of the box where faces meet. I n contrast, a general incompressible flow will have only isolated points of vanishing vorticity. (Both velocity and vorticity also vanish for all time a t the centre x = y = z = in.)Thirdly, the vanishing of ul and au,,/an on each face also implies that the tensor V v is partly diagonal. One principal axis of the strain rate or symmetric part of this tensor is then perpendicular to the face. Furthermore, the magnitude of the strain rate along this axis determines the fractional growth rate of the normal vorticity on the face, i.e. d % = --~r * v --lnlcl. an ' Idt (2.1) 14-2

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Abstract. We calculate very long low-and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the five-particle contribution χ (5) and six-particle contribution χ (6) . These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. The series for χ (low-and high-temperature regime), χ (5) and χ (6) are now extended to 2000 terms. In addition, for χ (5) , 10000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by χ (5) modulo a prime.A diff-Padé analysis of the 2000 terms series for χ (5) and χ (6) confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n-particle components of the susceptibility, up to a small set of "additional" singularities. The exponents at all the singularities of the Fuchsian linear ODE of χ (5) and the (as yet unknown) ODE of χ (6) are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of χ (5) , and w 2 = 1/8 for the ODE of χ (6) , which are not singularities of the "physical" χ (5) and χ (6) , that is to say the series-solutions of the ODE's which are analytic at w = 0.Furthermore, analysis of the long series for χ (5) (and χ (6) ) combined with the corresponding long series for the full susceptibility χ yields previously conjectured singularities in some χ (n) , n ≥ 7. The exponents at all these singularities are also seen to be rational numbers.We also present a mechanism of resummation of the logarithmic singularities of the χ (n) leading to the known power-law critical behaviour occurring in the full χ, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility χ.

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