The geometry of Whitney’s critical sets

In this paper, the complete geometric characterizations, including decomposition and compression theorems, are obtained for a connected and compact set to be a critical set in Whitney’s sense, i.e., a set such that there exists a differentiable function critical but not constant on it. The problem to characterize these sets geometrically was posed by H. Whitney [21] in 1935. We also provide a complete geometrical characterization for monotone Whitney arc, i.e., there exists a differentiable function critical but also increasing along the arc. All examples appearing in the literature are monotone Whitney arcs, for example, the examples by Whitney [21] and Besicovitch [2], Norton’s t-quasi-arcs with Hausdorff dimension > t [14], and self-similar arcs [19]. Furthermore, after introducing the notion of homogeneous Moran arc, we can completely characterize all the monotone Whitney arcs of criticality > 1, which include t-quasi arcs and self-conformal arcs. Some applications to arcs which are attractors of Iterated Function Systems are discussed, including self-conformal arcs, self-similar arcs and self-affine arcs. Finally, we give an example of critical arc such that any of its subarcs fails to be a t-quasi-arc for any t, providing an affirmative answer to an open question by Norton.     

The asymptotic behavior of the variance of the “surface area” of a random field

We study the asymptotics of the second moment of a geometric functional of surface area type of a realization of a homogeneous isotropic Gaussian random field over a disk of radius r as the latter is increased.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 106–110.     

The mystery of ten wooden blocks: Hjelmslev’s geometry of reality

Mathematische Semesterberichte - I shall argue that a collection of wood models at the University of Copenhagen was made by Johannes Hjelmslev as an aid in his elementary...     

On the geometry of O’N

We classify the rank two BCDL ₂₀₀₃-geometries of O’Nan and show that the maximal rank of a BCDL ₂₀₀₃-geometry for O’Nan is 4. This bound is sharp since it is satisfied by the rank four geometry given by Buekenhout (Contemp Math 45:1–32, 1985).     

A proof of Parisi’s conjecture on the random assignment problem

An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1+1/4+1/9+...+1/k ² conjectured by G. Parisi for the case k=m=n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n. AcknowledgementWe thank Mireille Bousquet-Mélou and Gilles Schaeffer for introducing the problem to us. We also thank an anonymous referee for suggesting some improvements of the exposition.     

Chung’s law of the iterated logarithm for anisotropic Gaussian random fields

In this paper we establish Chung’s law of the iterated logarithm for a class of anisotropic Gaussian random fields with stationary increments. This result is applicable to space–time Gaussian random fields and solution to the stochastic fractional heat equation.     

‘There are great alterations in the geometry of late’. The rise of Isaac Newton’s early Scottish circle†

This paper traces the rise of three Scottish mathematicians – Colin Campbell, John Craig, and David Gregory – to become key figures in the dissemination and promotion of Newton’s mathematical ideas and natural philosophy in the 1680s. Two medical men – Archibald Pitcairne and his former student George Cheyne – both likewise captivated by the Principia, played minor roles in the story of Newton’s mathematics, while at the same time promoting the concept of mathematical medicine derived from his philosophical thought. Drawing on contemporary correspondence and previously unpublished papers, it considers how these men contributed to the scholarly perception of Newton and how, conversely, Newton used his increasing influence in order to encourage their work, most notably obtaining for Gregory the vacant chair in astronomy at Oxford in 1691.     

On the Poisson geometry of the Adler-Gel’fand-Dikii brackets

In this paper, we consider a symplectic leaf that goes through a singular point of the Adler-Gel’fand-Dikii Poisson bracket associated to SL(n,R). We find a finite-dimensional transverse section2 at the singular point and we prove that one can induce a Poisson structure on2 (the transverse structure) that is linearizable and equivalent to the Lie-Poisson structure on sl(n,R)*. This problem is closely related to finding normal forms for nth order scalar differential operators with periodic coefficient. We partially generalize a well-known result for Hill’s operators to the higher order case.     

Convergence field of Abel’s summation method

Let F(A) denote the set of all bounded sequences summable by Abel’s method. It is known, that F(A) is a linear subspace of the linear metric space (S, ρ) of all bounded sequences endowed with the sup metric. It is shown in [KOSTYRKO, P.: Convergence fields of regular matrix transformations 2, Tatra Mt. Math. Publ. 40 (2008), 143–147] that the convergence field of a regular matrix transformation is a σ-porous set. We show that F(A) is very porous in S.     

The geometry of the Gauss–Picard modular group

We give a construction of a fundamental domain for PU(2,1,mathbbZ [i]){{rm PU}(2,1,mathbb{Z} [i])}, that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers mathbbZ [i]{mathbb{Z} [i]}. We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.     

The Gaussian free field and Hadamard’s variational formula

We relate the Gaussian free field on a planar domain to the variational formula of Hadamard which explains the change of the Green function under a perturbation of the domain. This is accomplished by means of a natural integral operator—called the Hadamard operator—associated with a given flow of growing domains.     

Birational geometry of O’Grady’s six dimensional example over the Donaldson–Uhlenbeck compactification

We determine the birational geometry of O’Grady’s six dimensional example over the Donaldson–Uhlenbeck compactification, by looking at the locus of non-locally-free sheaves on the relevant moduli space.     

Time-periodic solutions of the Einstein’s field equations II: geometric singularities

In this paper, we construct several kinds of new time-periodic solutions of the vacuum Einstein’s field equations whose Riemann curvature tensors vanish, keep finite or take the infinity at some points in these space-times, respectively. The singularities of these new time-periodic solutions are investigated and some new physical phenomena are discovered.     

Time-periodic solutions of the Einstein’s field equations Ⅲ:physical singularities

In this paper we construct a new time-periodic solution of the vacuum Einstein’s field equations, this solution possesses physical singularities, i.e., the norm of the solution’s Riemann curvature tensor takes the infinity at some points. We show that this solution is intrinsically time-periodic and describes a time-periodic universe with the time-periodic physical singularity. By calculating the Weyl scalars of this solution, we investigate new physical phenomena and analyze new singularities for this univ...     

On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations

In this paper we improve the regularity in time of the gradient of the pressure field arising in Brenier’s variational weak solutions (Comm Pure Appl Math 52:411–452, 1999) to incompressible Euler equations. This improvement is necessary to obtain that the pressure field is not only a measure, but a function in . In turn, this is a fundamental ingredient in the analysis made by Ambrosio and Figalli (2007, preprint) of the necessary and sufficient optimality conditions for the variational problem by Brenier (J Am Mat Soc 2:225–255, 1989; Comm Pure Appl Math 52:411–452, 1999).     

How small are the increments of partial sums of a discrete random field?

In this paper we establish inferior limit results and path properties for the increments of partial sums of a strictly stationary and linearly positive quadrant dependent (LPQD) or linearly negative quadrant dependent (LNQD) discrete random field with multidimensional indices.