A Sharp Version of Price’s Law for Wave Decay on Asymptotically Flat Spacetimes

We prove Price’s law with an explicit leading order term for solutions $$\phi (t,x)$$ ϕ ( t , x ) of the scalar wave equation on a class of stationary asymptotically flat $$(3+1)$$ ( 3 + 1 ) -dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that $$\phi (t,x)=c t^{-3}+{\mathcal {O}}(t^{-4+})$$ ϕ ( t , x ) = c t - 3 + O ( t - 4 + ) for bounded |x|, where $$c\in {\mathbb {C}}$$ c ∈ C is an explicit constant. This decay also holds along the event horizon on Kerr spacetimes and thus renders a result by Luk–Sbierski on the linear scalar instability of the Cauchy horizon unconditional. We moreover prove inverse quadratic decay of the radiation field, with explicit leading order term. We establish analogous results for scattering by stationary potentials with inverse cubic spatial decay. On the Schwarzschild spacetime, we prove pointwise $$t^{-2 l-3}$$ t - 2 l - 3 decay for waves with angular frequency at least l, and $$t^{-2 l-4}$$ t - 2 l - 4 decay for waves which are in addition initially static. This definitively settles Price’s law for linear scalar waves in full generality. The heart of the proof is the analysis of the resolvent at low energies. Rather than constructing its Schwartz kernel explicitly, we proceed more directly using the geometric microlocal approach to the limiting absorption principle pioneered by Melrose and recently extended to the zero energy limit by Vasy.

Lower Bounds on OBDD Proofs with Several Orders

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

Counting multiplicative groups with prescribed subgroups

We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime [Formula: see text] and a finite abelian [Formula: see text]-group [Formula: see text], we consider the set of integers [Formula: see text] such that the Sylow [Formula: see text]-subgroup of the multiplicative group [Formula: see text] is isomorphic to [Formula: see text]. We show that the counting function of this set of integers is asymptotic to [Formula: see text] for explicit constants [Formula: see text] and [Formula: see text] depending on [Formula: see text] and [Formula: see text]. Second, we consider the set of integers [Formula: see text] such that the multiplicative group [Formula: see text] is “maximally non-cyclic”, that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to [Formula: see text] for an explicit constant [Formula: see text], where [Formula: see text] is Artin’s constant. As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory.

An improvement of convergence rate in the local limit theorem for integral-valued random variables

Let $X_{1}, X_{2}, \ldots , X_{n}$ X 1 , X 2 , … , X n be independent integral-valued random variables, and let $S_{n}=\sum_{j=1}^{n}X_{j}$ S n = ∑ j = 1 n X j . One of the interesting probabilities is the probability at a particular point, i.e., the density of $S_{n}$ S n . The theorem that gives the estimation of this probability is called the local limit theorem. This theorem can be useful in finance, biology, etc. Petrov (Sums of Independent Random Variables, 1975) gave the rate $O (\frac{1}{n} )$ O ( 1 n ) of the local limit theorem with finite third moment condition. Most of the bounds of convergence are usually defined with the symbol O. Giuliano Antonini and Weber (Bernoulli 23(4B):3268–3310, 2017) were the first who gave the explicit constant C of error bound $\frac{C}{\sqrt{n}}$ C n . In this paper, we improve the convergence rate and constants of error bounds in local limit theorem for $S_{n}$ S n . Our constants are less complicated than before, and thus easy to use.

Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

In this paper we prove that for every integer [Formula: see text], there exists an explicit constant [Formula: see text] such that the following holds. Let [Formula: see text] be a number field of degree [Formula: see text], let [Formula: see text] be any rational prime that is totally inert in [Formula: see text] and [Formula: see text] any elliptic curve defined over [Formula: see text] such that [Formula: see text] has potentially multiplicative reduction at the prime [Formula: see text] above [Formula: see text]. Then for every rational prime [Formula: see text], [Formula: see text] has an irreducible mod [Formula: see text] Galois representation. This result has Diophantine applications within the “modular method”. We present one such application in the form of an Asymptotic version of Fermat’s Last Theorem that has not been covered in the existing literature.

Sharp error estimates for spline approximation: Explicit constants, n-widths, and eigenfunction convergence

In this paper, we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under [Formula: see text]-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.

Random Planar Maps and Graphs with Minimum Degree Two and Three

We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to the core of a random planar graph is of order $c \log(n)$ for an explicit constant $c$. These results provide new information on the structure of random planar graphs.

Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-Interpolation

This paper provides a discrete Poincaré inequality innspace dimensions on a simplexKwith explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero onKand all integrals of jumps zero along all interior sides by its Lebesgue norm times{C(n)\operatorname{diam}(K)}. The explicit constant{C(n)}depends only on the dimension{n=2,3}in case of an adaptive triangulation with the newest vertex bisection. The second part of this paper proves the stability of an enrichment operator, which leads to the stability and approximation of a (discrete) quasi-interpolator applied in the proofs of the discrete Friedrichs inequality and discrete reliability estimate with explicit bounds on the constants in terms of the minimal angle{\omega_{0}}in the triangulation. The analysis allows the bound of two constants{\Lambda_{1}}and{\Lambda_{3}}in the axioms of adaptivity for the practical choice of the bulk parameter with guaranteed optimal convergence rates.

Upper bounds for some Brill–Noether loci over a finite field

Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.