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The (scalar) Ruelle operator theory is well-known both in fractal geometry and dynamical systems. In this paper we consider vector-valued Ruelle operators for weakly contractive iterative function systems associated with Dini matrix potentials. We generalize the result in the paper (J. Math. Anal. Appl. 299 341–56) to weakly contractive iterated function system. More exactly, our main theorem gives a sufficient condition for the vector-valued Ruelle operator with the Perron–Frobenius property.

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map.

We consider affine iterated function systems in a locally compact non-Archimedean field F . We establish the theory of singular value decomposition in F and compute the box and Hausdorff dimensions of self-affine sets in F n , in generic sense, which is an analogy of Falconer’s result for the real case. In R n , the box and Hausdorff dimensions of self-affine sets can be obtained only when the norms of linear parts of affine transformations are strictly less than 1 2 . However, in a locally compact non-Archimedean field, the same result can be obtained without the restriction of the norms.

Oceanic internal waves often have curvilinear fronts and propagate over vertically sheared currents. We present the first study of long weakly-nonlinear internal ring waves in a three-layer fluid in the presence of a horizontally uniform background current with a constant vertical shear. The leading order of this theory leads to the angular adjustment equation—a nonlinear first-order ordinary differential equation describing the dependence of the linear long-wave speed on its angle to the direction of the current. The compact ring waves, well studied in the absence of a current, correspond to the singular solution (envelope of the general solution) of this equation, and they can exist only under certain conditions. The constructed solutions reveal qualitative differences in the shapes of the wavefronts of the two baroclinic modes: the wavefront of the faster mode is elongated in the direction of the current, while the wavefront of the slower mode is squeezed. Moreover, depending on the vorticity strength, several different regimes have been identified. When the vertical shear is weak, part of the wavefront is able to propagate upstream, while when the shear is strong enough, the whole wavefront propagates downstream. A richer pattern of behaviour is observed for the slower mode. As the shear increases, singularities of the swallowtail-type may arise and, eventually, solutions with compact wavefronts crossing the downstream axis cease to exist. We show that the latter is related to the long-wave instability of the base flow. We obtain the cKdV-type amplitude equation and examine analytical expressions for its coefficients. Using this cKdV-type equation we numerically model the evolution of the waves for both modes. The initial stage of the evolution is in agreement with the leading-order predictions for the deformations of the wavefronts. Then, as the wavefronts expand, strong dispersive effects occur in the upstream direction. Moreover, when nonlinearity is enhanced, fission of waves is observed in the upstream part of the ring waves.

In this paper we first prove a nonexistence result on iterative roots, which presents several sufficient conditions for identifying self-maps on arbitrary sets that have no iterative roots of any order. Then, using this result, we prove that when X is [ 0 , 1 ] m , R m or S 1 every non-empty open set of the space of continuous self-maps on X endowed with the compact-open topology contains a map that does not have even discontinuous iterative roots of order n ⩾ 2 . This, in particular, proves that the complement of , the set of non-iterates, is dense in for these X.

We affirmatively settle the question on existence of a real-valued higher order spectral shift function for a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $V(H-iI)^{-1}$ belongs to a Schatten-von Neumann ideal $\mathcal{S}^n$ of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace formula as in the known case of $V\in\mathcal{S}^n$ and that it is unique up to a polynomial summand of order $n-1$. Our result significantly advances earlier partial results where counterparts of the spectral shift function for noncompact perturbations lacked real-valuedness and aforementioned uniqueness as well as appeared in more complicated trace formulas for much more restrictive sets of functions. Our result applies to models arising in noncommutative geometry and mathematical physics., Comment: added proof details, corrected errors

We describe a method, based on "hard" contact topology, of showing the existence of semi-infinite trajectories of contact Hamiltonian flows which start on one Legendrian submanifold and asymptotically converge to another Legendrian submanifold. We discuss a mathematical model of non-equilibrium thermodynamics where such trajectories play a role of relaxation processes, and illustrate our results in the case of the Glauber dynamics for the mean field Ising model., Comment: 40 pages, 5 figures, improved exposition, mild revision of the section on Glauber dynamics

Motivated by the question of what spectral properties of dynamically defined Schrodinger operators may be preserved under periodic perturbations, we study ergodic Schrodinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. The scenario given by a periodic background potential corresponds to a separable structure in which the sampling function is the sum of two pieces, each of which depends only on a single factor of the product system. However, in each case that we study, our methods apply more generally to sampling functions that allow non-trivial dependencies between the product factors.In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for the Boshernitzan condition under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. This is new, even in the case in which the periodic background vanishes entirely. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e. frequency and phase) when the coupling is large.

The study on the large-time behaviour of solutions to the 3D incompressible anisotropic Navier–Stokes (ANS) equations is very recent. Powerful tools designed for the Navier–Stokes equations with full Laplacian dissipation such as the Fourier splitting method no longer apply to the case when there is only horizontal dissipation. For the whole space R 3 , as t → ∞ , solutions of the ANS equations converge to the trivial solution and the convergence rate is algebraic. This paper is devoted to the case when the spatial domain Ω is T 2 × R . Our results reveal that the large-time behaviour for T 2 × R is quite different from that for R 3 . We show that any small initial velocity field u 0 ∈ H 2 ( Ω ) leads to a unique global solution u that remains small in H 2 ( Ω ) . More importantly, as t → ∞ , the velocity field u converges to a nontrivial steady state. The first two components of the steady state are given by the horizontal average of the first two components of u 0 while the third component vanishes. In addition, this convergence is exponentially fast.

Rate-induced tipping (R-tipping) occurs when time-variation of input parameters of a dynamical system interacts with system timescales to give genuine nonautonomous instabilities. Such instabilities appear as the input varies at some critical rates and cannot, in general, be understood in terms of autonomous bifurcations in the frozen system with a fixed-in-time input. This paper develops an accessible mathematical framework for R-tipping in multidimensional nonautonomous dynamical systems with an autonomous future limit. We focus on R-tipping via loss of tracking of base attractors that are equilibria in the frozen system, due to crossing what we call regular R-tipping thresholds. These thresholds are anchored at infinity by regular R-tipping edge states: compact normally hyperbolic invariant sets of the autonomous future limit system that have one unstable direction, orientable stable manifold, and lie on a basin boundary. We define R-tipping and critical rates for the nonautonomous system in terms of special solutions that limit to a compact invariant set of the autonomous future limit system that is not an attractor. We focus on the case when the limit set is a regular edge state, introduce the concept of edge tails, and rigorously classify R-tipping into reversible, irreversible, and degenerate cases. The central idea is to use the autonomous dynamics of the future limit system to analyse R-tipping in the nonautonomous system. We compactify the original nonautonomous system to include the limiting autonomous dynamics. Considering regular R-tipping edge states that are equilibria allows us to prove two results. First, we give sufficient conditions for the occurrence of R-tipping in terms of easily testable properties of the frozen system and input variation. Second, we give necessary and sufficient conditions for the occurrence of reversible and irreversible R-tipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping edge states in the autonomous compactified system.

COVID-19 virus disease is an infectious disease caused by SARS-Cov-2 virus, as the diagnostic imaging department play a remarkable role in detecting and diagnosing of COVID-19 in early stages, therefore, medical imaging workers should be aware of infection control guidelines to prevent the transmission of the disease. A cross-sectional descriptive study survey design was conducted from July 15, 2020 to September 15, 2020, in diagnostic imaging departments in Palestine, aimed to evaluate the impact of COVID-19 pandemic on medical imaging workers; in terms of infection sources, awareness and commitment to relevant safety guidelines. From a 205 valid responses, there was 54.5% governmental, 27.2% private, and 18.3% from NGOs sector. Female were 26.8% and 81.9% holding BA degree, about 41.2% are less than five years of experience. 36.1% had a previous positive COVID-19 result, of them there was 61.6% due to nosocomial infection. Only 63.2% were confidence of their hospital preparedness for COVID-19, 65.1% felt that their department is capable of dealing with COVID-19 patients, 43.3% revealed that they had training on handling COVID-19 patients, 58.7% reported that selfreading was their source of information about COVID-19, inadequate training or knowledge about hand hygiene and proper use of PPE was noted. Our findings emphasize the vital need of expanded COVID-19 pandemic training for medical imaging workers, as well as related protective measures

The main aim of this study was to develop indirect enzyme linked immune sorbent assay based on the use of bacterial cloned SARS CoV-2 spike protein as a cheap and continuously available source of antigen. This test was proved to be useful for a quantitative measurement and evaluation of antibody immune response among SARS-CoV-2 infected individuals. Standard cloning procedures had been used in cloning two different segments of the spike gene and its expression. The size of the two cloned segments were of 700 base pair and of 500 base pair, which were named clone 8 and clone 105, respectively. These DNA segments were cloned in pET28a plasmid and then expressed in Bl21 E. coli. Different serum samples were tested from: current, previous infection, vaccinated and non-vaccinated patients using the amplified expressed proteins in enzyme linked immune sorbent assay. The expressed proteins from each clone were responded with varying degrees of sensitivity against COVID-19 positive human sera, and we attempted to validate which of the two recombinant proteins is the best to be used in Corona IgG and IgM antibody detection. Based on the results of indirect enzyme linked immunoassay, most of the tested samples had greater antibody titers with clone 8, which was found to have a higher similarity (99% resemblance) to the severe acute respiratory syndrome coronavirus 2 surface protein using BLAST search. We recommend clone 8 with high potential to be used for large-scale screening for COVID-19 outbreak; nevertheless, it requires greater sensitivity and specificity validation

COVID-19 virus disease is an infectious disease caused by SARS-Cov-2 virus, as the diagnostic imaging department play a remarkable role in detecting and diagnosing of COVID-19 in early stages, therefore, medical imaging workers should be aware of infection control guidelines to prevent the transmission of the disease. A cross-sectional descriptive study survey design was conducted from July 15, 2020 to September 15, 2020, in diagnostic imaging departments in Palestine, aimed to evaluate the impact of COVID-19 pandemic on medical imaging workers; in terms of infection sources, awareness and commitment to relevant safety guidelines. From a 205 valid responses, there was 54.5% governmental, 27.2% private, and 18.3% from NGOs sector. Female were 26.8% and 81.9% holding BA degree, about 41.2% are less than five years of experience. 36.1% had a previous positive COVID-19 result, of them there was 61.6% due to nosocomial infection. Only 63.2% were confidence of their hospital preparedness for COVID-19, 65.1% felt that their department is capable of dealing with COVID-19 patients, 43.3% revealed that they had training on handling COVID-19 patients, 58.7% reported that selfreading was their source of information about COVID-19, inadequate training or knowledge about hand hygiene and proper use of PPE was noted. Our findings emphasize the vital need of expanded COVID-19 pandemic training for medical imaging workers, as well as related protective measures

In this paper, we study the following Hamiltonian elliptic system − ε 2 Δ u + u = H v ( x , u , v ) in R N , − ε 2 Δ v + v = H u ( x , u , v ) in R N , u ( x ) → 0 , v ( x ) → 0 a s | x | → ∞ , where ɛ > 0 is a small parameter, H is a super-quadratic sub-critical Hamiltonian. Our investigation focuses on the cases that H ( x , u , v ) = 1 2 K ( x ) ( u 2 + v 2 ) + F ( u , v ) and H ( x , u , v ) = K ( x ) F ( u , v ) , where K(x) possesses saddle points. By introducing a new penalization associated with the barycenters of functions and applying the Leray–Schauder degree theory, we adopt the local variational arguments in Byeon and Jeanjean (2007 Arch. Ration. Mech. Anal. 185 185–200); Byeon and Tanaka (2013 J. Eur. Math. Soc. 15 1859–99; 2014 Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations (Providence, RI: American Mathematical Society)) and obtain the existence of semi-classical states for the Hamiltonian elliptic system, which concentrate around the saddle points of V as ɛ → 0.

We consider Lagrangians that are defined only on the horizontal distribution of a sub-riemannian manifold. The associated Hamiltonian is neither strictly convex nor coercive. We prove a result on homogenization of the Hamilton–Jacobi equation following the approach in Contreras et al (2015 Calc. Var. Partial Differ. Equ. 52 237–52). To establish the result, we need to extend weak KAM and Aubry–Mather theories to the sub-riemannian setting. We obtain a Tonelli theorem to dispend of the Lagrangian dynamics tools used in standard weak KAM theory. In the way towards our main result we observe that, the long time convergence of the Lax-Oleinik semigroup, and the vanishing discount convergence of the discounted value function hold.

The paper extends the notion of braided set and its close relative - the Yang-Baxter set - to the category of vector spaces and explore structure aspects of such a notion as morphisms and extensions. In this way we describe a family of solutions for the Yang-Baxter equation on the product of B and C if given B and C correspond to two linear (set-theoretic) solutions of the Yang-Baxter equation. One of the key observation is the relation of this question with the virtual pure braid group., Comment: 29 pages, 3 figures

The packing of hard-core particles in contact with their neighbors offers the statically determinate problem which allows analytical investigation of the stress tensor distribution. We construct the stress probability functional and derive the complete set of equations for the tensor components in two-dimensions., 7 pages

In this paper, we study the role of periodic measures with large support in the sense of density in invariant measures and the exponential growth for systems with periodic shadowing property or periodic approximate product property. These results are more refined versions of the Sigmund’s density result of periodic measures and Bowen’s entropy formula between the growth of periodic measures and entropy. Furthermore, we provide an abstract framework such that the results also hold for general homoclinic classes and shifts with non-uniform structure.

Operators with zero dimensional spectral measures appear naturally in the theory of ergodic Schr\"odinger operators. We develop the concept of a complete family of Hausdorff measure functions in order to analyze and distinguish between these measures with any desired precision. We prove that the dimension of spectral measures of half-line operators with positive upper Lyapunov exponent are at most logarithmic for every possible boundary phase. We show that this is sharp by constructing an explicit operator whose spectral measure obtains this dimension. We also extend and improve some basic results from the theory of rank one perturbations and quantum dynamics to encompass generalized Hausdorff dimensions., Comment: 35 pages

In this paper, we study a Landis-type conjecture for the general fractional Schrödinger equation ((−P)s+q)u=0. As a byproduct, we also prove the additivity and boundedness of the linear operator (−P)s for non-smooth coefficents. For differentiable potentials q, if a solution decays at a rate exp (−∣x∣1+), then the solution vanishes identically. For non-differentiable potentials q, if a solution decays at a rate exp (−∣x∣4s−14s+), then the solution must again be trivial. The proof relies on delicate Carleman estimates. This study is an extension of the work by Rüland and Wang (2019). peerReviewed

We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the Discrete Ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory. Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations.

In this paper, we study the Dirichlet-to-Neumann operators on infinite subgraphs of graphs. For an infinite graph, we prove Cheeger-type estimates for the bottom spectrum of the Dirichlet-to-Neumann operator, and the higher order Cheeger estimates for higher order eigenvalues of the Dirichlet-to-Neumann operator., 30 pages

The purpose of this article is to develop a Hadamard-Riesz type parametrix for the wave propagator in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in \cite{MaMe82} and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works of \cite{MaMe82} and \cite{Popov1994}, is also corrected. In those papers, the principal symbol was never actually computed and to our knowledge, this paper contains the first explicit formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard-Riesz type parametrix with explicit symbol and phase calculations in the interior appears to be new in the literature, with the exception of the author's previous work \cite{Vig18} in the special case of elliptical domains. This allows us to circumvent the symbol calculus in \cite{DuGu75} and \cite{HeZe12} when computing trace formulas, which are instead derived from integrating our explicit parametrix over the diagonal., Comment: 63 pages, 4 figures. The Hadamard variational formula approach has been replaced by a shorter infinitesimal version. The demonstration of 8 orbits near the diagonal of the boundary has been updated to include an exposition in the Friedlander model. References are updated and other minor errors corrected. The new version will appear in the Journal of Spectral Theory

We prove that if a Riemannian covering preserves the bottom of the spectrum of a Schr\"{o}dinger operator, which belongs to the discrete spectrum of the operator on the base manifold, then the covering is amenable., Comment: 26 pages

Measure-theoretic slow entropy is a more refined invariant than the classical measure-theoretic entropy to characterize the complexity of dynamical systems with subexponential growth rates of distinguishable orbit types. In this paper we prove flexibility results for the values of upper and lower polynomial slow entropy of rigid transformations as well as maps admitting a good cyclic approximation. Moreover, we show that there cannot exist a general upper bound on the lower measure-theoretic slow entropy for systems of finite rank., 57 pages, revisions, adding references

In this paper we study various entropies for factor maps of amenable group actions. We prove firstly theorem 3.1 inequalities linking relative topological entropy and conditional topological entropy (for factor maps of amenable group actions) without any additional assumption, which strengthens conditional variational principles (Zhu 2021 Nonlinearity 34 5163–85, theorems 2.12 and 3.9) proved by Zhu under additional assumptions. Then along the line of Misiurewicz (1976 Studia Math. 55 175–200), we introduce a new invariant called relative topological tail entropy and prove a Ledrappier’s type variational principle concerning it (for factor maps of amenable group actions); consequently, any factor map with zero relative topological tail entropy admits invariant measures with maximal relative entropy, which provides a nontrivial sufficient condition for the existence of invariant measures with maximal relative entropy in the setting of factor maps of amenable group actions.

We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial $p_{\,\mathrm{Feig}}(z)=z^2+c_{\,\mathrm{Feig}}$. In addition to that, we construct a piecewise constant function on $[-2,2]$ that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials $p_c(z) = z^2+c$ with $c \in [-2,2]$. Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial $p_c(z)=z^2+c$, is a $C^1$-smooth function of the real parameter $c$ on the interval $c\in(c_{\,\mathrm{Feig}},-3/4)$., Comment: 23 pages, 7 figures

In this paper we prove Morse index theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Our formulas compute the difference of Morse indices of two Hessians related to two different graphs or two different sets of boundary conditions. Several applications such as the iteration formulas or lower bounds for the index are proved., 30 pages, 4 figures

We consider the 3D cubic nonlinear Schr\"odinger equation (NLS) with a strong toroidal trap. In the first part, we show that as the confinement is strengthened, a large class of global solutions to the time-dependent model can be described by 1D flows solving the 1D periodic NLS (Theorem 1.4). In the second part, we construct a steady state as a constrained energy minimizer, and prove its dimension reduction to the well-known 1D periodic ground state (Theorem 1.6 and 1.7). Then, employing the dimension reduction limit, we establish the local uniqueness and the orbital stability of the 3D ring soliton (Theorem 1.8). These results justify the emergence of stable quasi-1D periodic dynamics for Bose-Einstein condensates on a ring in physics experiments., Comment: 52 pages

We adapt methodology of Tosio Kato to establish necessary and sufficient conditions for the solutions to the Navier–Stokes equations with Dirichlet boundary conditions to converge in a strong sense to a solution to the Euler equations in the presence of a boundary as the viscosity is taken to zero. We extend existing conditions for no-slip boundary conditions to allow for nonhomogeneous Dirichlet boundary conditions and curved boundaries, establishing several new conditions as well. We give a brief comparison of various correctors appearing in the literature used for similar purposes.