Stochastic quantization and regularization

In this paper the method of stochastic quantization introduced by Parisi and Wu is extended to field theories that include fermions and are supersymmetric. A new non-perturbative regulator based on stochastic quantization is introduced. This regulator preserves all the symmetries of the lagrangian, including gauge, chiral, and supersymmetries, at the expense of introducing non-locality.     

On the correlation dimension of the spectral measure for the thue-morse sequence

Using the relationship between the decay rate of autocorrelation and the characteristics of singular Fourier spectra, we show that the correlation dimension of the spectral measure for the infinite Thue-Morse symbolic sequence equals 3 − log(1 + √17)/log 2=0.64298….     

Pairs of Hamiltonian circuits in 5-connected planar graphs

Settling a problem raised by B. Grünbaum, J. Malkevitch, and the author, we present 5-valent 5-connected planar graphs that admit no pairs of edgedisjoint Hamiltonian circuits; our smallest example has 176 vertices. This is used to construct an infinite family of 5-valent 5-connected planar graphs, in which every member has the property that any pair of Hamiltonian circuits in it share at least about $\text{1}\text{168}$ of their edges. We construct 4- and 5-valent, 3-connected non-Hamiltonian planar graphs.     

On the 1-factors of n-connected graphs

L. W. Beineke and M. D. Plummer have recently proved [1] that every n-connected graph with a 1-factor has at least n different 1-factors. The main purpose of this paper is to prove that every n-connected graph with a 1-factor has at least as many as n(n − 2)(n − 4) … 4 · 2, (or: n(n − 2)(n − 4) … 5 · 3) 1-factors. The main lemma used is: if a 2-connected graph G has a 1-factor, then G contains a vertex V (and even two such vertices), such that each edge of G, incident to V, belongs to some 1-factor of G.     

A note on the variational formulation for quasi-linear initial-value problems

A variational formulation, equivalent to an initial-value problem for nonlinear systems, not necessarily self-adjoint, is derived with applications to linear and nonlinear wave problems and transonic aerodynamics.

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Résumé Nous présentons une formulation variationnelle pour les systèmes quasi-linéaires et leur applications aux ondes non-linéaires et l'écoulement transonique stationnaire.

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Work partially supported by Air-Force Grant No. AFOSR-73-2561.     

6-Valent analogues of Eberhard’s theorem

It is shown that for every sequence of non-negative integers (p _n|1≦n≠3) satisfying the equation {ie19-1} (respectively, =0) there exists a 6-valent, planar (toroidal, respectively) multi-graph that has preciselyp _n n gonal faces for alln, 1≦n≠3. This extends Eberhard’s theorem that deals, in a similar fashion, with 3-valent, 3-connected planar graphs; the equation involved follows from the famous Euler’s equation.     

Restricted left principal ideal rings

A ring is an LD-ring ifR is left bounded, ifR/J is a left Artinian left principal ideal ring for every proper idealJ inR, and ifR has finite left Goldie dimension. IfR is non-Artinian thenR is an order in a simple Artinian ringS. The ideal theory of LD-rings is investigated, and we discuss some conditions under which an LD-ring is an hereditary ring, and some under which an LD-ring is a Noetherian, bounded, maximal Asano order. A central localization of an LD-ring is an LD-ring, and the center of some LD-rings is a Krull-domain. This research was supported in part by the National Science Foundation Grant GP 23861.     

Variational multiple-tensor fitting of fiber-ambiguous diffusion-weighted magnetic resonance imaging voxels

Partial volume effects are often experienced in diffusion-weighted MRI of biologic tissue. This is when the signal attenuation reflects a mixture of diffusion processes, originating from different tissue compartments, residing in the same voxel. Decomposing the mixture requires elaborated models that account for multiple compartments, yet the fitting problem for those models is usually ill posed. We suggest a novel approach for stabilizing the fitting problem of the multiple-tensors model by a variational framework that adds biologically oriented assumption of neighborhood alignments. The framework is designed to address fiber ambiguity caused by a number of neuronal fiber compartments residing in the same voxel. The method requires diffusion data acquired by common, clinically feasible MRI sequences, and is able to derive familiar tensor quantities for each compartment. Neighborhood alignment is performed by adding piece-wise smooth regularization constraints to an energy function. Minimization with the gradient descent method produces a set of diffusion-reaction partial differential equations that describe a tensor-preserving flow towards a best approximation of the data while maintaining the constraints. We analyze fiber compartment separation capabilities on a synthetic model of crossing fibers and on brain areas known to have crossing fibers. We compare the results with diffusion tensor imaging analysis and discuss applications for the framework.     

Some Functions of Points in Projective Spaces

Extending earlier results in the plane, we prove that every n-polygon in sufficiently general position in d-dimensional projective space, n d + 2, gives rise to a derived n-polygon, which preserves a few functions; these functions are the cyclial product of (actually affine) ratios of various points, obtained by proper projections on suitable lines.