共找到 20 条结果
Morse-Cerf theory considers a one-parameter family of smooth functions defined on a manifold and studies the evolution of their critical points with the parameter. This paper presents an adaptation of Morse-Cerf theory to a family of piecewise-linear (PL) functions. The vertex diagram and Cerf diagram are introduced as representations of the evolution of critical points of the PL function. The characterization of a crossing in the vertex diagram based on the homology of the lower links of vertices leads to the definition of a topological descriptor for time-varying scalar fields. An algorithm for computing the Cerf diagram and a measure for comparing two Cerf diagrams are also described together with experimental results on time-varying scalar fields.
We relate Bruhat numbers in real Morse theory to cluster variables in braid varieties. This provides instances of wall-crossing coordinates in the study of Cerf diagrams.
For a half-unknotted implanted $(i,n-i)$-barbell $β=β_{i,n-i}$ in $M^n$, we construct two specific pseudo-isotopies, which we denote by standard barbell pseudo-isotopies, both resulting in that barbell diffeomorphism, each having a Cerf diagram only containing a single eye and with easily computable Hatcher-Wagoner invariants. We give an explicit formula for $β_{2,n-2}$ and a special class of $β_{3,n-3}$. Using this we show that for $n\geq 6$, every pseudo-isotopy with vanishing first Hatcher-Wagoner invariant can be isotoped to a composition of standard barbell pseudo-isotopies with $i=2$ or $3$. In dimension $n=4$, we further generalize the constructions and computations to half-unknotted immersed barbell diffeomorphisms and prove that for every $s\in \mathbb{Z}_2, σ\in π_2 M,γ\in π_1 M$ with $s=0 \text{ or }w_2^M(σ) eq0$, there exists a standard immersed barbell pseudo-isotopy $f_β$ with the second induced Hatcher-Wagoner invariant $Θ(f_β)=(s,σ)\cdot [γ]$.
We address a recent conjecture stated by Z. Van Herstraeten and N.J. Cerf. They claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant, which can be attained only by Gaussian pure states. We introduce an alternative definition of entropy for all absolutely integrable Wigner functions, which is the Shannon entropy for positive Wigner functions. Moreover, we are able to prove, in arbitrary dimension, that this entropy is indeed bounded below by a positive constant, which is not very distant from the constant suggested by Van Herstraeten and Cerf. We also prove an analogous result for another conjecture stated by the same authors for the Rényi entropy of positive Wigner functions. As a by-product we prove a new inequality for the radar-ambiguity function (and for the Wigner distribution) which is reminiscent of Lieb's inequalities.
Using a rigidity property of the foliations of $S^2 \times [0, 1]$ that are defined by a non-vanishing closed one-form, we give a rather simple proof of a theorem due J. Cerf, going back to 1968, that the group of direct diffeomorphisms of $S^3$ is connected.
We show that the Cerf-Adami inequalities do not necessarily depend on conditional entropies nor any reference to Markov chains. While the latter are not explicit in the original form, they are often implied in certain derivations. We also show that these inequalities are intimately related to at least one interpretation of the second law of thermodynamics. The combination of these results provides added insight into why some quantum systems violate the Cerf-Adami inequalities thereby improving our understanding of the quantum-classical boundary. As a result we suggest that the second law may serve as some type of boundary condition on classical knowledge.
It is shown that the operad maps $E_n\to E_{n+k}$ are formal over the reals for $k\geq 2$ and non-formal for $k=1$. Furthermore we compute the cohomology of the deformation complex of the operad maps $E_{n}\to E_{n+1}$, proving an algebraic version of the Cerf Lemma.
The author previously defined the spectral invariants, denoted by $ρ(H;a)$, of a Hamiltonian function $H$ as the mini-max value of the action functional $Å_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class $a$. The spectrality axiom of the invariant $ρ(H;a)$ states that the mini-max value is a critical value of the action functional $Å_H$. The main purpose of the present paper is to prove this axiom for {\it nondegenerate} Hamiltonian functions in {\it irrational} symplectic manifolds $(M,ω)$. We also prove that the spectral invariant function $ρ_a: H \mapsto ρ(H;a)$ can be pushed down to a {\it continuous} function defined on the universal ({\it étale}) covering space $\widetilde{Ham}(M,ω)$ of the group $Ham(M,ω)$ of Hamiltonian diffeomorphisms on general $(M,ω)$. For a certain generic homotopy, which we call a {\it Cerf homotopy} $\HH = \{H^s\}_{0 \leq s\leq 1}$ of Hamiltonians, the function $ρ_a \circ \HH: s \mapsto ρ(H^s;a)$ is piecewise smooth away from a countable subset of $[0,1]$ for each non-zero quantum cohomology class $a$. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floe
Following a line of reasoning suggested by Eliashberg, we prove Cerf's theorem that any diffeomorphism of the 3-sphere extends over the 4-ball. To this end we develop a moduli-theoretic version of Eliashberg's filling-with-holomorphic-discs method.
We develop an equivariant Cerf theory for Morse functions on finite-dimensional manifolds with group actions, and adapt the technique to the infinite-dimensional setting to study the moduli space of perturbed flat $SU(n)$-connections. As a consequence, we prove the existence of perturbative $SU(n)$ Casson invariants on integer homology spheres for all $n\ge 3$, and write down an explicit formula when $n=4$. This generalizes the previous works of Boden and Herald.
Heegaard splittings and Heegaard diagrams of a closed 3-manifold M are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on M. We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous pseudo-isotopy problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when dim M>2. The main tool that we introduce is an elementary swallow tail lemma which could be useful elsewhere.
The set $\mathcal{Q}$ of quantum correlations is the collection of all possible probability distributions on measurement outcomes achievable by space-like separated parties sharing a quantum state. Since the original work of Tsirelson [Tsirelson, Lett. Math. Phys. 4, 93 (1980)], this set has mainly been studied through the means algebraic and convex geometry techniques. We introduce a perturbative method using Lie-theoretic tools for the unitary group to analyze the response of the evaluations of Bell functionals under infinitesimal unitary perturbations of quantum strategies. Our main result shows that, near classical deterministic points, an $(n, 2, d)$ Bell operator decomposes into a direct sum of $(k, 2, d-1)$ Bell operators which we call \emph{subset games}. We then derive three key insights: (1) in the $(n, 2, 2)$ case, if $p_0$ is classically optimal, it remains locally optimal even among 2-dimensional quantum strategies, implying in turn that the boundary of $\mathcal{Q}$ is flat around classical deterministic points; (2) it suggests a proof strategy for Gisin's open problem on correlations in $\mathcal{Q}(D)$ unattainable by projective strategies of the same dimension; and
We revisit the proof of the de Moivre--Laplace theorem, which is the ancestor of the central limit theorem for the binomial distribution. Our goal is to provide a proof that can be reasonably presented to undergraduate students within a basic course of probability theory. We follow the strategies presented in two classical references, the books of Breiman and Feller, but we replace the arguments involving series expansions of the logarithm or the exponential by the basic inequality $\exp(t)\geq 1+t$. This way we avoid completely the use of uniform convergence and power series. We also avoid using Stirling's formula, instead we use the exact formula for the Wallis integral. As a by product of the proof, we also obtain a non-asymptotic inequality linking the binomial and the Gaussian distributions.
The basic problem of the calculus of variations consists of finding a function that minimizes an energy, like finding the fastest trajectory between two points for a point mass in a gravity field moving without friction under the influence of gravity or finding the best shape of a wing. The existence of a solution may be established in quite abstract spaces, much larger than the space of smooth functions. An important practical problem is that of being able to approach the value of the infimum of the energy. However, numerical methods work with very concrete functions and sometimes they are unable to approximate the infimum: this is the surprising Lavrentiev phenomenon. The papers that ensure the non-occurrence of the phenomenon form a recent saga, and the most general result formulated in the early '90s was actually fully proved just recently, more than 30 years later. Our aim here is to introduce the reader to the calculus of variations, to illustrate the Lavrentiev phenomenon with the simplest known example, and to give an elementary proof of the non-occurrence of the phenomenon.
We define the pivotal set of a Boolean function and we prove a fundamental inequality on its expected size, when the inputs are independent random coins of parameter~$p$. We give two complete proofs of this inequality. Along the way, we obtain the classical Margulis--Russo formula. We give a short proof of the classical Hoeffding inequality for i.i.d. Bernoulli random variables, and we use it to derive more complex deviations inequalities associated to the pivotal set. We follow finally Talagrand's footsteps and we discuss a beautiful inequality that he proved in the uniform case.
Our theoretical understanding of crossover is limited by our ability to analyze how population diversity evolves. In this study, we provide one of the first rigorous analyses of population diversity and optimization time in a setting where large diversity and large population sizes are required to speed up progress. We give a formal and general criterion which amount of diversity is necessary and sufficient to speed up the $(μ+1)$ Genetic Algorithm on LeadingOnes. We show that the naturally evolving diversity falls short of giving a substantial speed-up for any $μ=O(\sqrt{n}/\log^2 n)$. On the other hand, we show that even for $μ=2$, if we simply break ties in favor of diversity then this increases diversity so much that optimization is accelerated by a constant factor.
Viruses present an amazing genetic variability. An ensemble of infecting viruses, also called a viral quasispecies, is a cloud of mutants centered around a specific genotype. The simplest model of evolution, whose equilibrium state is described by the quasispecies equation, is the Moran--Kingman model. For the sharp peak landscape, we perform several exact computations and we derive several exact formulas. We obtain also an exact formula for the quasispecies distribution, involving a series and the mean fitness. A very simple formula for the mean Hamming distance is derived, which is exact and which do not require a specific asymptotic expansion (like sending the length of the macromolecules to $\infty$ or the mutation probability to $0$). We try also to extend these formulas to a general fitness landscape. We obtain an equation involving the covariance of the fitness and the Hamming class number in the quasispecies distribution. With the help of these formulas, we discuss the phenomenon of the error threshold and the notion of quasispecies. We recover the limiting quasipecies distribution in the long chain regime. We go beyond the sharp peak landscape and we consider fitness lands
We show the existence of a majorization ladder in bosonic Gaussian channels, that is, we prove that the channel output resulting from the $n\text{th}$ energy eigenstate (Fock state) majorizes the channel output resulting from the $(n\!+\!1)\text{th}$ energy eigenstate (Fock state). This reflects a remarkable link between the energy at the input of the channel and a disorder relation at its output as captured by majorization theory. This result was previously known in the special cases of a pure-loss channel and quantum-limited amplifier, and we achieve here its nontrivial generalization to any single-mode phase-covariant (or -contravariant) bosonic Gaussian channel. The key to our proof is the explicit construction of a column-stochastic matrix that relates the outputs of the channel for any two subsequent Fock states at its input. This is made possible by exploiting a recently found recurrence relation on multiphoton transition probabilities for Gaussian unitaries [M. G. Jabbour and N. J. Cerf, Phys. Rev. Research 3, 043065 (2021)]. Possible generalizations and implications of these results are then discussed.
We prove a new inequality controlling the large deviations of the empirical measure of a Markov chain. This inequality is based on the martingale used by Donsker and Varadhan and the minimax theorem. It holds for convex sets and it requires to take an infimum over the starting point. In the case of a compact space, this inequality is a partial improvement of the large deviations estimates of Donsker and Varadhan. In the case of a non compact space, we condition on the event that the process visits $n$ times a compact subset of the space and we still obtain a control on the exponential scale.