The minimal speeds (c) of the KolmogorovPetrovskyPiskunov (KPP) fronts at small diffusion ( 1) in a class of time-periodic cellular flows with chaotic streamlines is investigated in this paper. The variational principle of c reduces the computation to that of a principle eigenvalue problem on a periodic domain of a linear advectiondiffusion operator with spacetime periodic coefficients and small diffusion. To solve the advection dominated time-dependent eigenvalue problem efficiently over large time, a combination of spectral methods and finite element, as well as the associated fast solvers, are utilized to accelerate computation. In contrast to the scaling c= O ( 1/4) in steady cellular flows, a new relation c= O (1) as 1 is revealed in the time-periodic cellular flows due to the presence of chaotic streamlines. Residual propagation speed emerges from the Lagrangian chaos which is quantified as a sub

We establish a compact analogue of the P=W conjecture. For a projective irreducible holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations and the Hodge theory of hyper-Kähler manifolds. We present two applications of our result: one on the cohomology of the base and fibers of a Lagrangian fibration, and the other on the refined Gopakumar–Vafa invariants of a K3 surface. Furthermore, we show that the perverse filtration associated with a Lagrangian fibration is multiplicative under cup product.

We consider the Schrödinger equation with a nondispersive logarithmic nonlinearity and a repulsive harmonic potential. For a suitable range of the coefficients, there exist two positive stationary solutions, each one generating a continuous family of solitary waves. These solutions are Gaussian, and turn out to be orbitally unstable. We also discuss the notion of ground state in this setting: for any natural definition, the set of ground states is empty.

When sample, X = x, is observed from intractable c.d.f. Fθ, or a Black-Box with input θ, an Approximate Bayesian Computation (ABC) method provides approximate posterior, π_ϵ. θ^∗ is included in the support of π_ϵ when the Matching distance ρ(S(x^∗), S(x)) ≤ ϵ; x^∗ is a sample drawn from F_{θ^∗}, θ^∗ is obtained from prior π, S is a summary statistic, ϵ > 0. ABC concerns include: the use of only one sample, x^∗, for each θ^∗; the choices of S, ρ and ϵ; π_ϵ(θ^∗), which is determined by arbitrary kernel, K(x, x^∗; ϵ), creating visual π_ϵ-artifacts. The concerns are accommodated with the introduced Fiducial(F)-ABC for all (θ^∗ drawn): M x^∗ are drawn from F_{θ^∗}; a universal S is used, the empirical measure indexed by sets which activate its sufficiency for exchangeable observations-vectors and have been neglected; a strong, probability distance ρ is used, inherently connected with S and Matching; light is thrown to ϵ’s nature and value; πϵ is obtained from the proportions of x∗ matching x. F-ABC for all posterior is closer to Bayesian philosophy, which does not use θ^∗-exclusions. Under few, mild assumptions, π_ϵ converges to the posterior, π(θ|x), when ϵ ↓ 0, and rates of concentration of T (F_{θ^∗}) to T (F_θ) are obtained when n ↑ ∞; T is a functional. Satisfactory F-ABC for all θ^∗ drawn posteriors are depicted for parametric and data generating models, including Tukey’s (a, b, g, h)-model, a 5-parameter normal mixture and a time series model. Various advantages of the F-ABC for all are presented over coarsened posteriors for observations in R^d, d ≥ 1.

The Matsumoto-Imai (MI) cryptosystem was the first multivariate public key cryptosystem proposed for practical use. Though MI is now considered insecure due to Patarin’s linearization attack, the core idea of MI has been used to construct many variants such as Sflash, which has recently been accepted for use in the New European Schemes for Signatures, Integrity, and Encryption project. Linearization attacks take advantage of the algebraic structure of MI to produce a set of equations that can be used to recover the plaintext from a given ciphertext. In our paper, we present a solution to the problem of finding the dimension of the space of linearization equations, a measure of how much work the attack will require.