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Kak qislo v ume, na peske ostavl sled, okean gromozdits vo t me, milliony let mertvo i zyb ba ka wepku. I esli rezko xagnut s debarkadera vbok, vovne, budex dolgo padat , ruki po xvam; no ne vosposleduet vspleska. J. Brodsky AbstractWe construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K 2 -delic and K 1 K 1 -delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.Key Words: topological Milnor K-groups, explicit two-dimensional class field theory, adeles for arithmetic schemes, translation invariant integration and harmonic analysis, two-dimensional adelic analysis, zeta functions, zeta integral, analytic duality on arithmetic surfaces, boundary term, elliptic curves over global fields.

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In this work we extend the first part of the previous paper [F4] to higher dimensional local fields. We introduce a nontrivial translation invariant measure on the additive group of higher dimensional local fields, and then develop elements of integration and harmonic analysis. We also discuss its relation with several other measure theories.For an n-dimensional local field F a translation invariant measure µ is defined on a certain ring A of measurable sets and takes values in R ((X 1 )) . . . ((X n−1 )) (which itself is an n-dimensional local field whose last residue field is the archimedean field R). The algebra A in the case of higher dimensional fields with finite last residue field is the algebra generated by characteristic functions of shifts of fractional ideals, i.e. a + bO with a, b ∈ F and O the ring of integers of F with respect to the n-dimensional structure. The measure is countably additive in a refined sense (see sections 7 and 8). Elements of integration theory are introduced in sections 9-13.The additive group of a higher dimensional field has a certain topology on it with respect to which it is not locally compact for n > 1. This topology plays a key role in higher class field theory [F1-F3]. The additive group of F is self dual, which together with the measure and integration leads to analogs of many classical results in Fourier analysis (section 14). In particular, for functions in certain space we introduce their transform F(f )(α) = ψ(αβ) f (β) dµ(β) and show that F 2 (f )(α) = f (−α).In section 16 we discuss the theory in the case where the last residue field is R or C.In sections 17-21 we extend the previous theory to the case of generalized algebraic loop and path spaces (including in particular the complexified space of smooth loops), and indicate how the measure of this work can be used to extend p-adic zeta integrals associated to schemes over complete discrete valuation fields, and how it is related to nonarchimedean measures on spaces of arcs, which have recently found applications in algebraic geometry.Theory of this paper can be viewed as a part of yet unknown general theory of harmonic analysis on certain classes of non locally compact groups. It is expected to find further applications, in particular towards integration on path spaces, which is important for mathematical explanation of quantum physics (see section 18).

Coronavirus COVID-19 spreads through the population mostly based on social contact. To gauge the potential for widespread contagion, to cope with associated uncertainty and to inform its mitigation, more accurate and robust modelling is centrally important for policy making.We provide a flexible modelling approach that increases the accuracy with which insights can be made. We use this to analyse different scenarios relevant to the COVID-19 situation in the UK. We present a stochastic model that captures the inherently probabilistic nature of contagion between population members. The computational nature of our model means that spatial constraints (e.g., communities and regions), the susceptibility of different age groups and other factors such as medical pre-histories can be incorporated with ease. We analyse different possible scenarios of the COVID-19 situation in the UK. Our model is robust to small changes in the parameters and is flexible in being able to deal with different scenarios.This approach goes beyond the convention of representing the spread of an epidemic through a fixed cycle of susceptibility, infection and recovery (SIR). It is important to emphasise that standard SIR-type models, unlike our model, are not flexible enough and are also not stochastic and hence should be used with extreme caution. Our model allows both heterogeneity and inherent uncertainty to be incorporated. Due to the scarcity of verified data, we draw insights by calibrating our model using parameters from other relevant sources, including agreement on average (mean field) with parameters in SIR-based models.We use the model to assess parameter sensitivity for a number of key variables that characterise the COVID-19 epidemic. We also test several control parameters with respect to their influence on the severity of the outbreak. Our analysis shows that due to inclusion of spatial heterogeneity in the population and the asynchronous timing of the epidemic across different areas, the severity of the epidemic might be lower than expected from other models.We find that one of the most crucial control parameters that may significantly reduce the severity of the epidemic is the degree of separation of vulnerable people and people aged 70 years and over, but note also that isolation of other groups has an effect on the severity of the epidemic. It is important to remember that models are there to advise and not to replace reality, and that any action should be coordinated and approved by public health experts with experience in dealing with epidemics.The computational approach makes it possible for further extensive scenario-based analysis to be undertaken. This and a comprehensive study of sensitivity of the model to different parameters defining COVID-19 and its development will be the subject of our forthcoming paper. In that paper, we shall also extend the model where we will consider different probabilistic scenarios for infected people with mild and severe cases.