Some Mathematics for the Method of Pooled PCR Test

At the time of the worldwide COVID-19 disaster, the author learned about the pooled (RT-) PCR test from the news. From the common sense of individual tests, the idea of mixing multiple samples seems taboo, however in fact many samples can be tested with a smaller number of tests by the method. As a retired researcher of mathematical engineering, the author was deeply interested in the idea and absorbed in the mathematical formulation and intensive analysis of the method. Later, he found that the original basic equation was already proposed in the old (1943) treatise [1] and so many related research works have been done and available as materials on the web [2], although many of those seem to be based on qualitative or intuitive analysis. In that sense, some of the analysis here seems to be already known in the field, but some results might be novel, such as boundary conditions, derivation of limit values, estimation of infection rate and adaptive optimization scheme of pool test, strict extension to multi-stage pool test, and explicit derivation of asymptotic approximate solutions of optimal pooling number and achieved efficiency measure, etc. In any case, he decided to put it together here as a material rather than a formal treatise, hoping that the results here would be useful for deeper mathematical insights into and better understanding of the pool inspection, and also in its actual practice.

A Besov algebra calculus for generators of operator semigroups and related norm-estimates

We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille–Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.

Localic remote points revisited

We consider remote points in general extensions of frames, with an emphasis on perfect extensions. For a strict extension ?XL ? L determined by a set X of filters in L, we show that if there is an ultrafilter in X then the extension has a remote point. In particular, if a completely regular frame L has a maximal completely regular filter which is an ultrafilter, then ?L ? L has a remote point, where ?L is the Stone-C?ch compactification of L. We prove that in certain extensions associated with radical ideals and l-ideals of reduced f-rings, remote points induced by algebraic data are exactly non-essential prime ideals or non-essential irreducible l-ideals. Concerning coproducts, we show that if M1 ? L1 and M2 ? L2 are extensions of T1-frames, then each of these extensions has a remote point if the extensionM1?M2?L1?L2 has a remote point.

A completeness theorem for symmetric product phase spaces

In a previous work with Antonio Bucciarelli, we introduced indexed linear logic as a tool for studying and enlarging the denotational semantics of linear logic. In particular, we showed how to define new denotational models of linear logic using symmetric product phase models (truth-value models) of indexed linear logic. We present here a strict extension of indexed linear logic for which symmetric product phase spaces provide a complete semantics. We study the connection between this new system and indexed linear logic.

The kernel relation for a strict extension of certain regular semigroups

Let R be a regular semigroup and denote by (R) its congruence lattice. For , the kernel of pis the set ker . The relation K on (R) defined by λKp if ker λ = ker p is the kernel relation on (R). In general, K is a complete ∩-congruence but it is not a v-congruence. In view of the importance of the kernel-trace approach to the study of congruences on a regular semigroup (the trace of p is its restriction to idempotents of R), it is of considerable interest to determine necessary and sufficient conditions on R in order for K to be a congruence. This being in general a difficult task, one restricts attention to special classes of regular semigroups. For a background on this subject, consult [1].