A new control strategy for a class of piecewise deterministic processes (PDP) is presented. In this class, PDP stochastic processes consist of ordinary differential equations that are subject to random switches corresponding to a discrete Markov process. The proposed strategy aims at controlling the probability density function (PDF) of the PDP. The optimal control formulation is based on the hyperbolic Fokker–Planck system that governs the time evolution of the PDF of the PDP and on tracking objectives of terminal configuration with a target PDF. The corresponding optimization problems are formulated as a sequence of open-loop hyperbolic optimality systems following a model predictive control framework. These systems are discretized by first-order schemes that guarantee positivity and conservativeness of the numerical PDF solution. The effectiveness of the proposed computational control framework is validated considering PDP with dichotomic noise.

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have recently been used to obtain various properties of triangulations that are embedded in a fixed set of points in the plane. We generalize this method to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound of O*(187.53N) (where the O*(⋅) notation hides polynomial factors) for the maximum number of crossing-free straight-edge graphs that can be embedded in any specific set of N points in the plane (improving upon the previous best upper bound 207.85N in Hoffmann, Schulz, Sharir, Sheffer, Tóth and Welzl [14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on the expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded in a specific point set.

We then apply the cross-graph charging-scheme method to graphs that allow certain types of crossings. Specifically, we consider graphs with no set of k pairwise crossing edges (more commonly known as k-quasi-planar graphs). For k=3 and k=4, we prove that, for any set S of N points in the plane, the number of graphs that have a straight-edge k-quasi-planar embedding over S is only exponential in N.

This paper studies ‘pro-excision’ for the $K$ -theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller’s conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller’s conjecture in mixed characteristic, and we establish various finiteness results for the $K$ -groups of singularities, covering both orders in number fields and singular curves over finite fields.

Let G be a finite graph with minimum degree r. Form a random subgraph Gp of G by taking each edge of G into Gp independently and with probability p. We prove that for any constant ε > 0, if $p=\frac{1+\epsilon}{r}$, then Gp is non-planar with probability approaching 1 as r grows. This generalizes classical results on planarity of binomial random graphs.

The mean weight of a cycle in an edge-weighted graph is the sum of the cycle's edge weights divided by the cycle's length. We study the minimum mean-weight cycle on the complete graph on n vertices, with random i.i.d. edge weights drawn from an exponential distribution with mean 1. We show that the probability of the min mean weight being at most c/n tends to a limiting function of c which is analytic for c ≤ 1/e, discontinuous at c = 1/e, and equal to 1 for c > 1/e. We further show that if the min mean weight is ≤ 1/(en), then the length of the relevant cycle is Θp(1) (i.e., it has a limiting probability distribution which does not scale with n), but that if the min mean weight is > 1/(en), then the relevant cycle almost always has mean weight (1 + o(1))/(en) and length at least (2/π2 − o (1)) log2n log log n.

We study the global well-posedness of 3D Navier–Stokes equations for a class of large initial data. This type of data slowly varies in the vertical direction (expressed as a function of $\varepsilon {x}_{3} $ ), and it is ill-prepared in the sense that its norm in ${C}^{- 1} $ will blow up at the rate ${\varepsilon }^{- \alpha } $ for $\frac{1}{2} \lt \alpha \lt 1$ as $\varepsilon $ tends to zero.

Given an abelian variety $A$ of dimension $g$ over a number field  $K$ , and a prime $\ell $ , the ${\ell }^{n} $ -torsion points of $A$ give rise to a representation ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$ . In particular, we get a mod- $\ell $ representation  ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $ -adic representation  ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$ . In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.

Applying our results in dimension $g= 1$ , we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur’s bound on rational isogenies of prime degree to number fields.

We use the heat equation as an illustrative example to show that the unified method introduced by one of the authors can be employed for constructing analytical solutions for linear evolution partial differential equations in one spatial dimension involving non-separable boundary conditions as well as non-local constraints. Furthermore, we show that for the particular case in which the boundary conditions become separable, the unified method provides an easier way for constructing the relevant classical spectral representations avoiding the classical spectral analysis approach. We note that the unified method always yields integral expressions which, in contrast to the series or integral expressions obtained by the standard transform methods, are uniformly convergent at the boundary. Thus, even for the cases that the standard transform methods can be implemented, the unified method provides alternative solution expressions which have advantages for both numerical and asymptotic considerations. The former advantage is illustrated by providing the numerical evaluation of typical boundary value problems.

We define the notion of a trace kernel on a manifold $M$ . Roughly speaking, it is a sheaf on $M\times M$ for which the formalism of Hochschild homology applies. We associate a microlocal Euler class with such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle ${T}^{\ast } M$ over $M$ , and we prove that this class is functorial with respect to the composition of kernels.

This generalizes, unifies and simplifies various results from (relative) index theorems for constructible sheaves, $\mathscr{D}$ -modules and elliptic pairs.

We deal with aspects of direct and inverse problems in parameterized Picard–Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) $G$ is a PPV Galois group over these fields if and only if $G$ contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs $G$, including unipotent groups, $G$ is such a group if and only if it has differential type $0$. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a parameterized linear differential equation if its Galois group has differential type $0$.

In this paper we study the analytical obstructions preventing the germ of a generic analytic family of elliptic ordinary differential equations from being isochronous. The formal obstructions are purely arithmetical and can be set in terms of commutators. The analytical obstructions arise out of the non-convergence of the normalizing coordinates in the Poincaré–Dulac normalization process. The temporal part of the invariant is the obstruction to isochronicity in a particular context.

We study the semiclassical limit of the least energy solutions to the nonlinear Dirac equationfor x ∈ ℝ3. We prove that the equation has least energy solutions for all ħ > 0 small, and, in addition, that the solutions converge in a certain sense to the least energy solution of the associated limit problem as ħ → 0.

For a real-valued function V of the Faddeev–Marchenko class, we prove the norm-resolvent convergence, as ε → 0, of a family Sε of one-dimensional Schrödinger operators on the line of the form

Under certain conditions, the functions ε−2V (x/ε) converge in the sense of distributions as ε → 0 to δ′ (x), and then the limit S0 of Sε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ′.

We prove the following theorem: if w is a quasi-conformal mapping of the unit disc onto itself satisfying elliptic partial differential inequality , then w is Lipschitz continuous. This result extends some recent results where, instead of an elliptic differential operator, only the Laplace operator is considered.

Let n ≥ 3 and let ψλ0 be the radially symmetric solution of Δ log ψ + 2βψ + βx · ∇ψ = 0 in ℝn, ψ(0) = λ(0), for some constants λ0 > 0, β > 0. Suppose u0 ≥ 0 satisfies u0 − ψλ0 ∈ L1 (ℝn) and u0 (x) ≈ (2(n − 2)/β)(log∣x∣/∣x∣2) as ∣x∣ → ∞. We prove that the rescaled solution ũ(x,t) = e2βtu(eβtx, t) of the maximal global solution u of the equation ut = Δ log u in ℝn × (0, ∞), u(x, 0) = u0 (x) in ℝn, converges uniformly on every compact subset of ℝn and in L1 (ℝn) to ψλ0 as t → ∞. Moreover, ∥ũ(·, t) − ψλ0∥L1(ℝn) ≤ e−(n−2)βt∥u0 − ψλ0∥L1(ℝn) for all t ≥ 0.

We study the existence and multiplicity of positive solutions for the following nonlinear Schrödinger–Poisson equations:where 2 < p ≤ 3 or 4 ≤ p < 6, λ > 0 and Q ∈ C(ℝ3). We show that the number of positive solutions is dependent on the profile of Q(x).

We present new results in the theory of the classical theta functions of Jacobi: series expansions and defining ordinary differential equations (ODEs). The proposed dynamical systems turn out to be Hamiltonian and define fundamental differential properties of theta functions; they also yield an exponential quadratic extension of the canonical θ-series. An integrability condition of these ODEs explains the appearance of the modular ϑ-constants and differential properties thereof. General solutions to all the ODEs are given. For completeness, we also solve the Weierstrassian elliptic modular inversion problem and consider its consequences.

On any spacelike surface in a light cone of four-dimensional Lorentz–Minkowski space, a distinguished smooth function is considered. We show how both extrinsic and intrinsic geometry of such a surface are codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must be totally umbilical, deriving a Liebmann-type result. Two remarkable families of examples of spacelike surfaces in a light cone are explicitly constructed. Finally, several results that involve the first eigenvalue of the Laplace operator of a compact spacelike surface in a light cone are obtained.