Fredholm's Minors of Arbitrary Order: Their Representations as a Determinant of Resolvents and in Terms of Free Fermions and an Explicit Formula for Their Functional Derivative

We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the $n$th and $n-1$th minors, whose solution is a representation of the $n$th minor as an $n\times n$ determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit formula for the functional derivative of a Fredholm minor of order $n$ with respect to the kernel. Our formula is a linear combination of the $n$th and the $n\pm 1$th minors.Comment: 17 pages, Latex, no figures connection to supplementary compound matrices mentioned, references added, typos correcte

Beam scanning reflectarray antenna with circular polarization

A novel means of scanning a circularly polarized reflectarray antenna. The reflectarray is an array of metallic elements arranged on a surface designed to compensate for the various path lengths of the optical rays from an illuminating feed to the reflecting surface and then to the antenna aperture. With appropriate design, the phase in the aperture can be made to vary linearly in any desired direction and also to produce a radiated beam normal to the constant phase surface. In the case of circular polarization, this path length compensation can be accomplished by rotation of the individual elements

Singly generated quasivarieties and residuated structures

A quasivariety K of algebras has the joint embedding property (JEP) iff it is generated by a single algebra A. It is structurally complete iff the free countably generated algebra in K can serve as A. A consequence of this demand, called "passive structural completeness" (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if K is relatively semisimple then it is generated by one K-simple algebra. It is a minimal quasivariety if, moreover, it is PSC but fails to unify some finite set of equations. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC iff its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics

Estimating brain volume loss after radiation therapy in children treated for posterior fossa tumors (Corpus callosum and whole brain volume changes following radiotherapy in children).

Background More than half of pediatric tumors of central nervous system (CNS) primarily originate in the posterior fossa and are conventionally treated with radiation therapy (RT).Objectives The objective of this study was to establish whether corpus callosum volumes (CCV) and whole brain volumes (WBV) are correlated and to determine the impact of whole-brain lowvs high-dose RT on brain parenchymal volume loss as assessed using each technique.Material and methods Of the 30 identified children (6-12 years) with newly diagnosed posterior fossa tumors treated with cranial RT, including focal and whole-brain RT, suitable imaging was obtained for 23. Radiotherapy regimens were the following: no whole-brain RT (Group 1, n = 7), low-dose whole-brain RT (30 Gy, Group 3, n = 7) in addition to focal boost. Magnetic resonance images (MRIs) were analyzed at baseline and follow-up (median 14 months). The CCVs were manually segmented on midline sagittal slice (n = 23), while WBVs were segmented semi-automatically using Freesurfer (n = 15). This was done twice (6-month interval) for all baseline CCV measurements and 5 randomly selected WBV measurements to establish measurement reproducibility. Correlations between CCV and WBV were investigated and percentage of children demonstrating reduction in CCV or WBV noted.Results Correlation between baseline CCV and WBV was not significant (p = 0.37). Measurement reproducibility was from 6% to -9% for CCV and from 4.8% to -1.2% for WBV. Among the children studied, 30.4% (7/23) had >9% reduction in CCV at follow-up, while 33.3% (5/15) had >1.2% reduction in WBV. Five of 7 patients with CCV loss were not picked up by WBV measurements. Similarly, 3 of 5 patients with WBV loss were not picked up by CCV measurements.Conclusions The CCV and the WBV are unrelated and may indicate different brain parenchymal losses following RT. Up to a third of posterior fossa tumors treated with RT have measurable CCV or WBV loss; incidence was equivalent in lowvs high-dose whole-brain RT

Multiple Conclusion Rules in Logics with the Disjunction Property

We prove that for the intermediate logics with the disjunction property any basis of admissible rules can be reduced to a basis of admissible m-rules (multiple-conclusion rules), and every basis of admissible m-rules can be reduced to a basis of admissible rules. These results can be generalized to a broad class of logics including positive logic and its extensions, Johansson logic, normal extensions of S4, n-transitive logics and intuitionistic modal logics

∗\ast-SDYM Fields and Heavenly Spaces. I. ∗\ast-SDYM equations as an integrable system

It is shown that the self-dual Yang-Mills (SDYM) equations for the $\ast$-bracket Lie algebra on a heavenly space can be reduced to one equation (the \it master equation\rm). Two hierarchies of conservation laws for this equation are constructed. Then the twistor transform and a solution to the Riemann-Hilbert problem are given.Comment: 25 page