We’re a big step closer to a Canon EOS R5 release announcement now, as Nokishita Tweets that it has passed its Bluetooth certification. The belief is that the EOS R5 was originally scheduled to ship in July, and Canon Rumors reports that they’ve been told that’ll still happen. With lockdowns still in effect in much […]

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Are there still folks among you who, like me, prefer handwriting to typing? If you’re in this group, you’ll love this new feature on Google Lens. The app now lets you scan your handwritten notes, copy them, and paste them straight to your computer. I gave it a spin, and I bring you my impressions […]

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It feels like forever since Nikon announced their newest flagship DSLR; the Nikon D6. It’s actually only been three months, but that hasn’t stopped some people getting anxious. Recently, customers were being told that the D6 would start shipping right about now, but now Nikon has officially come out to announce that the Nikon D6 […]

The post Nikon has confirmed that their flagship D6 DSLR will start shipping on May 21st appeared first on DIY Photography.

With the coronavirus pandemic, many folks switched to working online. Things like teaching, business meetings and other face-to-face activities have been replaced with video calls. Home has become both home and workplace, and admit it: your wardrobe totally reflects this. Creative duo The Workmans shows this “fashion crossover” in their latest photo series #COVIDwear. The […]

The post #COVIDwear: a hilarious photo series showing quarantine fashion of remote workers appeared first on DIY Photography.

How we read pie charts is still an open question: is it angle? Is it area? Is it arc length? In a study I'm presenting as a short paper at the IEEE VIS conference in Vancouver next week, I tried to tease the visual cues apart – using modeling and 3D pie charts. The big […]

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We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.

We prove a Marstrand type slicing theorem for the subsets of the integer square lattice. This problem is the dual of the corresponding projection theorem, which was considered by Glasscock, and Lima and Moreira, with the mass and counting dimensions applied to subsets of $mathbb{Z}^{d}$. In this paper, more generally we deal with a subset of the plane that is $1$ separated, and the result for subsets of the integer lattice follow as a special case. We show that the natural slicing question in this setting is true with the mass dimension.

One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker--Planck equation (FPE) egin{align*} &u_t-Delta (eta(u))+{mathrm{ div}}(D(x)b(u)u)=0, quad tgeq0, xinmathbb{R}^d, d e2, \ &u(0,cdot)=u_0,mbox{in }mathbb{R}^d, end{align*} where $u_0in L^1(mathbb{R}^d)$, $etain C^2(mathbb{R})$ is a nondecreasing function, $bin C^1$, bounded, $bgeq 0$, $Din(L^2cap L^infty)(mathbb{R}^d;mathbb{R}^d)$ with ${ m div}, Din L^infty(mathbb{R}^d)$, and ${ m div},Dgeq0$, $eta$ strictly increasing, if $b$ is not constant. Moreover, $t o u(t,u_0)$ is a semigroup of contractions in $L^1(mathbb{R}^d)$, which leaves invariant the set of probability density functions in $mathbb{R}^d$. If ${ m div},Dgeq0$, $eta'(r)geq a|r|^{alpha-1}$, and $|eta(r)|leq C r^alpha$, $alphageq1,$ $alpha>frac{d-2}d$, $dgeq3$, then $|u(t)|_{L^infty}le Ct^{-frac d{d+(alpha-1)d}} |u_0|^{frac2{2+(m-1)d}},$ $t>0$, and the existence extends to initial data $u_0$ in the space $mathcal{M}_b$ of bounded measures in $mathbb{R}^d$. The solution map $mumapsto S(t)mu$, $tgeq0$, is a Lipschitz contractions on $mathcal{M}_b$ and weakly continuous in $tin[0,infty)$. As a consequence for arbitrary initial laws, we obtain weak solutions to a class of McKean-Vlasov SDEs with coefficients which have singular dependence on the time marginal laws.

We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to the case of exponentially decaying potentials. That means that the eigenvalues of $-Delta + V - iepsilon x^2$, $|V(x)|leq e^{-2gamma |x|}$ converge, as $ epsilon o 0+ $, to the poles of the meromorphic continuation of $ ( -Delta + V -lambda^2 )^{-1} $ uniformly on compact subsets of $ extrm{Re},lambda>0$, $ extrm{Im},lambda>-gamma$, $arglambda > pi/8$.

In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.

The $(q, mathbf{Q})$-current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic $q$-Schur algebras. In this paper, we study the $(q, mathbf{Q})$-current algebra $U_q(mathfrak{sl}_n^{langle mathbf{Q} angle}[x])$ associated with the special linear Lie algebra $mathfrak{sl}_n$. In particular, we classify finite dimensional simple $U_q(mathfrak{sl}_n^{langle mathbf{Q} angle}[x])$-modules.

Let $1 le r < n$ be integers. We give a proof that the group $mathop{mathrm{Aut}}({X_{n}^{mathbb{N}}, sigma_{n}})$ of automorphisms of the one-sided shift on $n$ letters embeds naturally as a subgroup $mathcal{h}_{n}$ of the outer automorphism group $mathop{mathrm{Out}}(G_{n,r})$ of the Higman-Thompson group $G_{n,r}$. From this, we can represent the elements of $mathop{mathrm{Aut}}({X_{n}^{mathbb{N}}, sigma_{n}})$ by finite state non-initial transducers admitting a very strong synchronizing condition.

Let $H in mathcal{H}_{n}$ and write $|H|$ for the number of states of the minimal transducer representing $H$. We show that $H$ can be written as a product of at most $|H|$ torsion elements. This result strengthens a similar result of Boyle, Franks and Kitchens, where the decomposition involves more complex torsion elements and also does not support practical extit{a priori} estimates of the length of the resulting product.

We also give new proofs of some known results about $mathop{mathrm{Aut}}({X_{n}^{mathbb{N}}, sigma_{n}})$.

In this article, we prove that the completeness of the Hamilton flow and essential self-dajointness are equivalent for real principal type operators on the circle. Moreover, we study spectral properties of these operators.

Consider that $u_0$ nodes are aware of some piece of data $d_0$. This note derives the expected time required for the data $d_0$ to be disseminated through-out a network of $n$ nodes, when communication between nodes evolves according to a graphical Markov model $overline{ mathcal{G}}_{n,hat{p}}$ with probability parameter $hat{p}$. In this model, an edge between two nodes exists at discrete time $k in mathbb{N}^+$ with probability $hat{p}$ if this edge existed at $k-1$, and with probability $(1-hat{p})$ if this edge did not exist at $k-1$. Each edge is interpreted as a bidirectional communication link over which data between neighbors is shared. The initial communication graph is assumed to be an Erdos-Renyi random graph with parameters $(n,hat{p})$, hence we consider a emph{stationary} Markov model $overline{mathcal{G}}_{n,hat{p}}$. The asymptotic "$u_0$-expected flooding time" of $overline{mathcal{G}}_{n,hat{p}}$ is defined as the expected number of iterations required to transmit the data $d_0$ from $u_0$ nodes to $n$ nodes, in the limit as $n$ approaches infinity. Although most previous results on the asymptotic flooding time in graphical Markov models are either emph{almost sure} or emph{with high probability}, the bounds obtained here are emph{in expectation}. However, our bounds are tighter and can be more complete than previous results.

The paper considers constrained linear systems with stochastic additive disturbances and noisy measurements transmitted over a lossy communication channel. We propose a model predictive control (MPC) law that minimizes a discounted cost subject to a discounted expectation constraint. Sensor data is assumed to be lost with known probability, and data losses are accounted for by expressing the predicted control policy as an affine function of future observations, which results in a convex optimal control problem. An online constraint-tightening technique ensures recursive feasibility of the online optimization and satisfaction of the expectation constraint without bounds on the distributions of the noise and disturbance inputs. The cost evaluated along trajectories of the closed loop system is shown to be bounded by the optimal predicted cost. A numerical example is given to illustrate these results.

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse structures on $omega$, about the interplay between properties of a coarse space and its Higson corona, about some special ultrafilters ($T$-points and cellular $T$-points) related to finitary coarse structures on $omega$, about partitions of coarse spaces into thin pieces, and also about coarse groups having some extremal properties.

We show weak-strong uniqueness and stability results for the motion of a two or three dimensional fluid governed by the Navier-Stokes equation interacting with a flexible, elastic plate of Koiter type. The plate is situated at the top of the fluid and as such determines the variable part of a time changing domain (that is hence a part of the solution) containing the fluid. The uniqueness result is a consequence of a stability estimate where the difference of two solutions is estimated by the distance of the initial values and outer forces. For that we introduce a methodology that overcomes the problem that the two (variable in time) domains of the fluid velocities and pressures are not the same. The estimate holds under the assumption that one of the two weak solutions possesses some additional higher regularity. The additional regularity is exclusively requested for the velocity of one of the solutions resembling the celebrated Ladyzhenskaya-Prodi-Serrin conditions in the framework of variable domains.

We derive the non-relativistic $c oinfty$ and ultra-relativistic $c o 0$ limits of the $kappa$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $kappa$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $kappa$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincar'e, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding $kappa$-Newtonian and $kappa$-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the $kappa$-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter $kappa$, the curvature parameter $eta$ and the speed of light parameter $c$.

Let $k ,=, mathbb{Q}(sqrt[5]{n},zeta_5)$, where $n$ is a positive integer, $5^{th}$ power-free, whose $5-$class group is isomorphic to $mathbb{Z}/5mathbb{Z} imesmathbb{Z}/5mathbb{Z}$. Let $k_0,=,mathbb{Q}(zeta_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $zeta_5$. Let $C_{k,5}^{(sigma)}$ the group of the ambiguous classes under the action of $Gal(k/k_0)$ = $<sigma>$. The aim of this paper is to determine all integers $n$ such that the group of ambiguous classes $C_{k,5}^{(sigma)}$ has rank $1$ or $2$.

This paper deals with an optimal control problem associated to the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation.

Willems et al.'s fundamental lemma asserts that all trajectories of a linear system can be obtained from a single given one, assuming that a persistency of excitation condition holds. This result has profound implications for system identification and data-driven control, and has seen a revival over the last few years. The purpose of this paper is to extend Willems' lemma to the situation where multiple (possibly short) system trajectories are given instead of a single long one. To this end, we introduce a notion of collective persistency of excitation. We will then show that all trajectories of a linear system can be obtained from a given finite number of trajectories, as long as these are collectively persistently exciting. We will demonstrate that this result enables the identification of linear systems from data sets with missing data samples. Additionally, we show that the result is of practical significance in data-driven control of unstable systems.

Miller and Sheffield introduced a notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $mathbb{C}$ and Gaussian free fields (GFFs) on them under conformal equivalence. They considered the situation in which the conformal transformations are given by a chordal Schramm--Loewner evolution (SLE). In the present paper, we construct processes of GFF on $mathbb{H}$ (the upper half-plane) and $mathbb{O}$ (the first orthant of $mathbb{C}$) by coupling zero-boundary GFFs on these domains with stochastic log-gases defined on parts of boundaries of the domains, $mathbb{R}$ and $mathbb{R}_+$, called the Dyson model and the Bru--Wishart process, respectively, using multiple SLEs evolving in time. We prove that the obtained processes of GFF are stationary. The stationarity defines an equivalence relation between GFFs, and the pairs of time-evolutionary domains and stationary processes of GFF will be regarded as generalizations of the imaginary surfaces studied by Miller and Sheffield.

We investigate quasistatic evolution in finite plasticity under the assumption that the plastic strain is compatible. This assumption is well-suited to describe the special case of dislocation-free plasticity and entails that the plastic strain is the gradient of a plastic deformation map. The total deformation can be then seen as the composition of a plastic and an elastic deformation. This opens the way to an existence theory for the quasistatic evolution problem featuring both Lagrangian and Eulerian variables. A remarkable trait of the result is that it does not require second-order gradients.

We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space $mathbb C^n$ to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in $mathbb C^3$ that is Kobayashi hyperbolic and has a nonempty core. In particular, this domain is not biholomorphic to a bounded domain in $mathbb C^3$ and the mentioned above sufficient condition for Kobayashi hyperbolicity is not necessary.

In this paper we consider the "quasidensity" of a subset of the product of a Banach space and its dual, and give a connection between quasidense sets and sets of "type (NI)". We discuss "coincidence sets" of certain convex functions and prove two sum theorems for coincidence sets. We obtain new results on the Fitzpatrick extension of a closed quasidense monotone multifunction. The analysis in this paper is self-contained, and independent of previous work on "Banach SN spaces". This version differs from the previous version because it is shown that the (well known) equivalence of quasidensity and "type (NI)" for maximally monotone sets is not true without the monotonicity assumption and that the appendix has been moved to the end of Section 10, where it rightfully belongs.

We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node in the class of methods with optimal number of communication steps takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique we show that all proposed methods with stochastic oracle can be additionally parallelized at each node.

The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel-Roiter measure $mu$ in an abelian length category $mathcal{A}$, there exists an object $X'$ which depends on $X$ and $mu$, such that $Gamma = operatorname{End}_{mathcal{A}}(X oplus X')$ has finite global dimension. Analogously to Iyama's original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

We consider the stochastic 2-dimensional Cahn-Hilliard equation which is driven by the derivative in space of a space-time white noise. We use two different approaches to study this equation. First we prove that there exists a unique solution $Y$ to the shifted equation (see (1.4) below), then $X:=Y+{Z}$ is the unique solution to stochastic Cahn-Hilliard equaiton, where ${Z}$ is the corresponding O-U process. Moreover, we use Dirichlet form approach in cite{Albeverio:1991hk} to construct the probabilistically weak solution the the original equation (1.1) below. By clarifying the precise relation between the solutions obtained by the Dirichlet forms aprroach and $X$, we can also get the restricted Markov uniquness of the generator and the uniqueness of martingale solutions to the equation (1.1).

The article is devoted to the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(kinmathbb{N})$ based on the generalized iterated Fourier series. The case of Fourier-Legendre series as well as the case of trigonotemric Fourier series are considered in details. The obtained expansion provides a possibility to represent the iterated Stratonovich stochastic integral in the form of iterated series of products of standard Gaussian random variables. Convergence in the mean of degree $2n$ $(nin mathbb{N})$ of the expansion is proved. Some modifications of the mentioned expansion were derived for the case $k=2$. One of them is based of multiple trigonomentric Fourier series converging almost everywhere in the square $[t, T]^2$. The results of the article can be applied to the numerical solution of Ito stochastic differential equations.

We study flows on C*-algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C*-algebras satisfying certain technical properties, which hold for many C*-algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto's conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, strongly purely infinite C*-algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable $KK$-contractible C*-algebras: Two Rokhlin flows on such a C*-algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate.

We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $ int_Omega u^{2gamma-alpha-eta}Delta u^alphaDelta u^eta dx geq cint_Omega|Delta u^gamma |^2dx $, which seem to be of interest on their own right.

We prove small data modified scattering for the Vlasov-Poisson system in dimension $d=3$ using a method inspired from dispersive analysis. In particular, we identify a simple asymptotic dynamic related to the scattering mass.

In this paper we introduce various types of harmonic Finsler manifolds and study the relation between them. We give several characterizations of such spaces in terms of the mean curvature and Laplacian. In addition, we prove that some harmonic Finsler manifolds are of Einstein type and a technique to construct harmonic Finsler manifolds of Rander type is given. Moreover, we provide many examples of non-Riemmanian Finsler harmonic manifolds of constant flag curvature and constant $S$-curvature. Finally, we analyze Busemann functions in a general Finsler setting and in certain kind of Finsler harmonic manifolds, namely asymptotically harmonic Finsler manifolds along with studying some applications. In particular, we show the Busemann function is smooth in asymptotically harmonic Finsler manifolds and the total Busemann function is continuous in $C^{infty}$ topology.

We present a method for optimal path planning of human walking paths in mountainous terrain, using a control theoretic formulation and a Hamilton-Jacobi-Bellman equation. Previous models for human navigation were entirely deterministic, assuming perfect knowledge of the ambient elevation data and human walking velocity as a function of local slope of the terrain. Our model includes a stochastic component which can account for uncertainty in the problem, and thus includes a Hamilton-Jacobi-Bellman equation with viscosity. We discuss the model in the presence and absence of stochastic effects, and suggest numerical methods for simulating the model. We discuss two different notions of an optimal path when there is uncertainty in the problem. Finally, we compare the optimal paths suggested by the model at different levels of uncertainty, and observe that as the size of the uncertainty tends to zero (and thus the viscosity in the equation tends to zero), the optimal path tends toward the deterministic optimal path.

We present a systematic derivation of the abelianity conditions for the $q$-deformed $W$-algebras constructed from the elliptic quantum algebra $mathcal{A}_{q,p}(widehat{gl}(N)_{c})$. We identify two sets of conditions on a given critical surface yielding abelianity lines in the moduli space ($p, q, c$). Each line is identified as an intersection of a countable number of critical surfaces obeying diophantine consistency conditions. The corresponding Poisson brackets structures are then computed for which some universal features are described.

Axial algebras are non-associative algebras generated by semisimple idempotents whose adjoint actions obey a fusion law. Axial algebras that are generated by two such idempotents play a crucial role in the theory. We classify all primitive 2-generated axial algebras whose fusion laws have two eigenvalues and all graded primitive 2-generated axial algebras whose fusion laws have three eigenvalues. This represents a significant broadening in our understanding of axial algebras.

Given an one-dimensional Lorenz-like expanding map we prove that the conditionlinebreak $P_{top}(phi,partial mathcal{P},ell)<P_{top}(phi,ell)$ (see, subsection 2.4 for definition), introduced by Buzzi and Sarig in [1] is satisfied for all continuous potentials $phi:[0,1]longrightarrow mathbb{R}$. We apply this to prove that quasi-H"older-continuous potentials (see, subsection 2.2 for definition) have at most one equilibrium measure and we construct a family of continuous but not H"older and neither weak H"older continuous potentials for which we observe phase transitions. Indeed, this class includes all H"older and weak-H"older continuous potentials and form an open and [2].

In this paper we consider Wronskian polynomials labeled by partitions that can be factorized via the combinatorial concepts of $p$-cores and $p$-quotients. We obtain the asymptotic behavior for these polynomials when the $p$-quotient is fixed while the size of the $p$-core grows to infinity. For this purpose, we associate the $p$-core with its characteristic vector and let all entries of this vector simultaneously tend to infinity. This result generalizes the Wronskian Hermite setting which is recovered when $p=2$.

We show that any toric K"ahler cone with smooth compact cross-section admits a family of Calabi-Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi-Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki-Einstein metrics.

Classical studies on aspiration-based dynamics suggest that a dissatisfied individual changes strategy without taking into account the success of others. This promotes defection spreading. The imitation-based dynamics allow individuals to imitate successful strategies without taking into account their own-satisfactions. In this article, we propose to study a dynamic based on aspiration which takes into account imitation of successful strategies for dissatisfied individuals. This helps cooperative members to resist. Individuals compare their success to their desired satisfaction level before making a decision to update their strategies. This mechanism helps individuals with a minimum of self-satisfaction to maintain their strategies. If an individual is dissatisfied, it will learn from others by choosing successful strategies. We derive an exact expression of the fixation probability in well-mixed populations as in structured populations in networks. As a result, we show that selection may favor cooperation more than defection in well-mixed populations as in populations ranged over a regular graph. We show that the best scenario is a graph with small connectivity.

We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for $E/F$ a quadratic extension of p-adic fields the associated unitary group $G=mathrm{U}(n,n)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $mathrm{GL}_n(E)$. Let $pi$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $iota_P^G pi$ is reducible.

The characterization of phase dynamics in coupled oscillators offers insights into fundamental phenomena in complex systems. To describe the collective dynamics in the oscillatory system, order parameters are often used but are insufficient for identifying more specific behaviors. We therefore propose a topological approach that constructs quantitative features describing the phase evolution of oscillators. Here, the phase data are mapped into a high-dimensional space at each time point, and topological features describing the shape of the data are subsequently extracted from the mapped points. We extend these features to time-variant topological features by considering the evolution time, which serves as an additional dimension in the topological-feature space. The resulting time-variant features provide crucial insights into the time evolution of phase dynamics. We combine these features with the machine learning kernel method to characterize the multicluster synchronized dynamics at a very early stage of the evolution. Furthermore, we demonstrate the usefulness of our method for qualitatively explaining chimera states, which are states of stably coexisting coherent and incoherent groups in systems of identical phase oscillators. The experimental results show that our method is generally better than those using order parameters, especially if only data on the early-stage dynamics are available.

We introduce a Gaussian measure formally preserved by the 2-dimensional Primitive Equations driven by additive Gaussian noise. Under such measure the stochastic equations under consideration are singular: we propose a solution theory based on the techniques developed by Gubinelli and Jara in cite{GuJa13} for a hyperviscous version of the equations.

We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle system in random environment on one dimensional discrete lattice. We establish the convergence result and show a local in time well-posedness of the limit stochastic PDE with spatial white noise. It turns out that our limit stochastic PDE does not require any renormalization. We also show a comparison theorem for the limit equation.

In this work, the generalized $N$-component Fokas-Lenells(FL) equations, which have been studied by Guo and Ling (2012 J. Math. Phys. 53 (7) 073506) for $N=2$, are first investigated via Riemann-Hilbert(RH) approach. The main purpose of this is to study the soliton solutions of the coupled Fokas-Lenells(FL) equations for any positive integer $N$, which have more complex linear relationship than the analogues reported before. We first analyze the spectral analysis of the Lax pair associated with a $(N+1) imes (N+1)$ matrix spectral problem for the $N$-component FL equations. Then, a kind of RH problem is successfully formulated. By introducing the special conditions of irregularity and reflectionless case, the $N$-soliton solution formula of the equations are derived through solving the corresponding RH problem. Furthermore, take $N=2,3$ and $4$ for examples, the localized structures and dynamic propagation behavior of their soliton solutions and their interactions are discussed by some graphical analysis.

In this paper, we give an alternate definition of the well-known Parry measure on an aperiodic subshift of finite type using correlation between the forbidden words. We use the concept of the local escape rate to obtain this definition. We also compute Perron eigenvectors corresponding to the Perron root of the associated adjacency matrix.

Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to $p$-Laplace double obstacle problems involving the Schr"odinger term: $-Delta_p u + mathbb{V}|u|^{p-2}u$ with bound constraints $psi_1 le u le psi_2$ in non-smooth domains. This problem has its own interest in mathematics, engineering, physics and other branches of science. Our approach makes a novel connection between the study of Calder'on-Zygmund theory for nonlinear Schr"odinger type equations and variational inequalities for double obstacle problems.

Let $Nge 2$ and $ hoin(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system

[

left{f_i(x)= ho x+frac{i(1- ho)}{N-1}: i=0,1,ldots, N-1 ight}.

]

Let $s=dim_H E$ be the Hausdorff dimension of $E$, and let $mu=mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $xin E$, the pointwise lower $s$-density $Theta_*^s(mu,x)$ and upper $s$-density $Theta^{*s}(mu, x)$ of $mu$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ for the sets

[

E_*(a)=left{xin E: Theta_*^s(mu, x)ge a ight}quad extrm{and}quad E^*(b)=left{xin E: Theta^{*s}(mu, x)le b ight}

] respectively, such that $dim_H E_*(a)>0$ if and only if $a<a_c$, and that $dim_H E^*(b)>0$ if and only if $b>b_c$. We emphasize that both values $a_c$ and $b_c$ are related to the Thue-Morse type sequences, and our strategy to find them relies on ideas from open dynamics and techniques from combinatorics on words.