Collapsed cone convolution and analytical anisotropic algorithm dose calculations compared to VMC++ Monte Carlo simulations in clinical cases

The purpose of this work was to study and quantify the differences in dose distributions computed with some of the newest dose calculation algorithms available in commercial planning systems. The study was done for clinical cases originally calculated with pencil beam convolution (PBC) where large density inhomogeneities were present. Three other dose algorithms were used: a pencil beam like algorithm, the anisotropic analytic algorithm (AAA), a convolution superposition algorithm, collapsed cone convolution (CCC), and a Monte Carlo program, voxel Monte Carlo (VMC++). The dose calculation algorithms were compared under static field irradiations at 6 MV and 15 MV using multileaf collimators and hard wedges where necessary. Five clinical cases were studied: three lung and two breast cases. We found that, in terms of accuracy, the CCC algorithm performed better overall than AAA compared to VMC++, but AAA remains an attractive option for routine use in the clinic due to its short computation times. Dose differences between the different algorithms and VMC++ for the median value of the planning target volume (PTV) were typically 0.4% (range: 0.0 to 1.4%) in the lung and -1.3% (range: -2.1 to -0.6%) in the breast for the few cases we analysed. As expected, PTV coverage and dose homogeneity turned out to be more critical in the lung than in the breast cases with respect to the accuracy of the dose calculation. This was observed in the dose volume histograms obtained from the Monte Carlo simulations.

Computer algebra for x-ray spectral reconstruction between 6 and 25 MV

A previously presented algorithm for the reconstruction of bremsstrahlung spectra from transmission data has been implemented into MATHEMATICA. Spectra vectorial algebra has been used to solve the matrix system A * F = T. The new implementation has been tested by reconstructing photon spectra from transmission data acquired in narrow beam conditions, for nominal energies of 6, 15, and 25 MV. The results were in excellent agreement with the original calculations. Our implementation has the advantage to be based on a well-tested mathematical kernel. Furthermore it offers a comfortable user interface.

Convolution roots and differentiability of isotropic positive definite functions on spheres

We prove that any isotropic positive definite function on the sphere can be written as the spherical self-convolution of an isotropic real-valued function. It is known that isotropic positive definite functions on d-dimensional Euclidean space admit a continuous derivative of order [(d − 1)/2]. We show that the same holds true for isotropic positive definite functions on spheres and prove that this result is optimal for all odd dimensions.

Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥0 is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.

A feasible theory of truth over combinatory algebra

We define an applicative theory of truth TPT which proves totality exactly for the polynomial time computable functions. TPT has natural and simple axioms since nearly all its truth axioms are standard for truth theories over an applicative framework. The only exception is the axiom dealing with the word predicate. The truth predicate can only reflect elementhood in the words for terms that have smaller length than a given word. This makes it possible to achieve the very low proof-theoretic strength. Truth induction can be allowed without any constraints. For these reasons the system TPT has the high expressive power one expects from truth theories. It allows embeddings of feasible systems of explicit mathematics and bounded arithmetic. The proof that the theory TPT is feasible is not easy. It is not possible to apply a standard realisation approach. For this reason we develop a new realisation approach whose realisation functions work on directed acyclic graphs. In this way, we can express and manipulate realisation information more efficiently.

Algebra of Classical Theoretical Term Reductions in the Sciences

An elementary algebra identifies conceptual and corresponding applicational limitations in John Kemeny and Paul Oppenheim’s (K-O) 1956 model of theoretical reduction in the sciences. The K-O model was once widely accepted, at least in spirit, but seems afterward to have been discredited, or in any event superceeded. Today, the K-O reduction model is seldom mentioned, except to clarify when a reduction in the Kemeny-Oppenheim sense is not intended. The present essay takes a fresh look at the basic mathematics of K-O comparative vocabulary theoretical term reductions, from historical and philosophical standpoints, as a contribution to the history of the philosophy of science. The K-O theoretical reduction model qualifies a theory replacement as a successful reduction when preconditions of explanatory adequacy and comparable systematicization are met, and there occur fewer numbers of theoretical terms identified as replicable syntax types in the most economical statement of a theory’s putative propositional truths, as compared with the theoretical term count for the theory it replaces. The challenge to the historical model developed here, to help explain its scope and limitations, involves the potential for equivocal theoretical meanings of multiple theoretical term tokens of the same syntactical type.