Four-manifold systoles and surjectivity of period map

P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic invariant of a 4-manifold X with indefinite intersection form. The estimate turns out to be polynomial, rather than logarithmic, in \chi(X), if the conjectured surjectivity of the period map is correct. Such surjectivity is targeted by the current work in gauge theory. The surjectivity allows one to insert suitable lattices with metric properties prescribed in advance, into the second de Rham cohomology group of X, as its integer lattice. The idea is to adapt the well-known Lorentzian construction of the Leech lattice, by replacing the Leech lattice by the Conway-Thompson unimodular lattices which define asymptotically dense packings. The final step can be described, in terms of the successive minima \lambda_i, as deforming a \lambda_2-bound into a \lambda_1-bound.Comment: 16 page

Superluminal motion and Lorentzian symmetry breaking and repairing in two-metric theories

The new results by OPERA collaboration claim the discovery of superluminal neutrinos. Superluminal particles have to break Lorentzian symmetry or causality principle. The method discussed gives us the possibility to reintroduce Lorentzian symmetry without breaking of causality.Comment: 5 pages. Extra references are added in v

A quantitative obstruction to collapsing surfaces

We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound. Keywords: curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distanceComment: 4 pages. Published in Open Mathematic

On uniqueness of quantum measurement theory

The paper discuss the structure of quantum mechanics and uniqueness of its postulates. The Born rule for quantum probabilities is fixed by requirement of nonexistence of quantum telepathy. Von Neumann projection postulate describes the transformation of quantum state under the condition of no-interaction measurement. Projection postulate could be considered as transition to conditional probability under the condition of a certain result of quantum measurement.Comment: 8 page

Torus cannot collapse to a segment

In earlier work, we analyzed the impossibility of codimension-one collapse for surfaces of negative Euler characteristic under the condition of a lower bound for the Gaussian curvature. Here we show that, under similar conditions, the torus cannot collapse to a segment. Unlike the torus, the Klein bottle can collapse to a segment; we show that in such a situation, the loops in a short basis for homology must stay a uniform distance apart.Comment: 8 pages, Journal of Geometry 111, Article number: 13 (2020

Systolic inequalities and Massey products in simply-connected manifolds

We show that the existence of a nontrivial Massey product in the cohomology ring H^*(X) imposes global constraints upon the Riemannian geometry of a manifold X. Namely, we exhibit a suitable systolic inequality, associated to such a product. This generalizes an inequality proved in collaboration with Y. Rudyak, in the case when X has unit Betti numbers, and realizes the next step in M. Gromov's program for obtaining geometric inequalities associated with nontrivial Massey products. The inequality is a volume lower bound, and depends on the metric via a suitable isoperimetric quotient. The proof relies upon W. Banaszczyk's upper bound for the successive minima of a pair of dual lattices. Such an upper bound is applied to the integral lattices in homology and cohomology of X. The possibility of applying such upper bounds to obtain volume lower bounds was first exploited in joint work with V. Bangert. The latter work deduced systolic inequalities from nontrivial cup-product relations, whose role here is played by Massey products.Comment: 14 pages, to appear in Israel J. Mat

Physics and technology system of units for electrodynamics

The contemporary practice is to favor the use of the SI units for electric circuits and the Gaussian CGS system for electromagnetic field. A modification of the Gaussian system of units (the Physics and Technology System) is suggested. In the Physics and Technology System the units of measurement for electrical circuits coincide with SI units, and the equations for the electromagnetic field are almost the same form as in the Gaussian system. The XXIV CGMP (2011) Resolution "On the possible future revision of the International System of Units, the SI" provides a chance to initiate gradual introduction of the Physics and Technology System as a new modification of the SI.Comment: 12 pages. Misprints in table at page 10 are correcte

Search for photon bubble oscillations in V0332+53

We report results of our search for fast oscillations in lightcurve of one of the brightest accretion powered pulsars on the sky V0332+53 with the help of data of the PCA spectrometer of the RXTE observatory. In course of this search we have carefully explored complications appearing if one uses only sub-bands of the total bandpass of the PCA spectrometer. We show that lightcurves collected in the soft sub-band of the PCA spectrometer contains an additional instrumental noise, lightcurves of harder sub-bands lack some fraction of the anticipated Poisson noise. We show that this noise is caused by a cross-talk of energy bands, which lasts up to ~200usec. One hypothesis is that these effects are caused by temporarily drop of the PCA detector gain after any occurred event due to slowly moving ions in the detector volume. In order to avoid this effect we searched for fast oscillations in flux of V0332+53 only in the total bandpass of the PCA spectrometer 2-60 keV. We have not detected any quasi-periodic oscillations in lightcurve of the source with an upper limit at the level of 0.5% in the Fourier frequency range 200-1500 Hz.Comment: 7 pages, 11 figures, accepted for publication in MNRA

Lattices of homomorphisms and pro-Lie groups

Early this century K. H. Hofmann and S. A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, all locally compact abelian groups, and all connected locally compact groups and is closed under the formation of products and closed subgroups. They defined a topological group $G$ to be almost connected if the quotient group of $G$ by the connected component of its identity is compact. We show here that all almost connected pro-Lie groups as well as their continuous homomorphic images are $R$-factorizable and \textit{$\omega$-cellular}, i.e.~every family of $G_\delta$-sets contains a countable subfamily whose union is dense in the union of the whole family. We also prove a general result which implies as a special case that if a topological group $G$ contains a compact invariant subgroup $K$ such that the quotient group $G/K$ is an almost connected pro-Lie group, then $G$ is $R$-factorizable and $\omega$-cellular. Applying the aforementioned result we show that the sequential closure and the closure of an arbitrary $G_{\delta,\Sigma}$-set in an almost connected pro-Lie group $H$ coincide.Comment: 22 page

Infinitesimals, Imaginaries, Ideals, and Fictions

Leibniz entertained various conceptions of infinitesimals, considering them sometimes as ideal things and other times as fictions. But in both cases, he compares infinitesimals favorably to imaginary roots. We agree with the majority of commentators that Leibniz's infinitesimals are fictions rather than ideal things. However, we dispute their opinion that Leibniz's infinitesimals are best understood as logical fictions, eliminable by paraphrase. This so-called syncategorematic conception of infinitesimals is present in Leibniz's texts, but there is an alternative, formalist account of infinitesimals there too. We argue that the formalist account makes better sense of the analogy with imaginary roots and fits better with Leibniz's deepest philosophical convictions. The formalist conception supports the claim of Robinson and others that the philosophical foundations of nonstandard analysis and Leibniz's calculus are cut from the same cloth.Comment: 34 page