We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.

In this paper we study the following set[A={p(n)+2^nd mod 1: ngeq 1}subset [0.1],] where $p$ is a polynomial with at least one irrational coefficient on non constant terms, $d$ is any real number and for $ain [0,infty)$, $a mod 1$ is the fractional part of $a$. By a Bernoulli decomposition method, we show that the closure of $A$ must have full Hausdorff dimension.

This work presents a novel data-driven framework for constructing eigenfunctions of the Koopman operator geared toward prediction and control. The method leverages the richness of the spectrum of the Koopman operator away from attractors to construct a rich set of eigenfunctions such that the state (or any other observable quantity of interest) is in the span of these eigenfunctions and hence predictable in a linear fashion. The eigenfunction construction is optimization-based with no dictionary selection required. Once a predictor for the uncontrolled part of the system is obtained in this way, the incorporation of control is done through a multi-step prediction error minimization, carried out by a simple linear least-squares regression. The predictor so obtained is in the form of a linear controlled dynamical system and can be readily applied within the Koopman model predictive control framework of [12] to control nonlinear dynamical systems using linear model predictive control tools. The method is entirely data-driven and based purely on convex optimization, with no reliance on neural networks or other non-convex machine learning tools. The novel eigenfunction construction method is also analyzed theoretically, proving rigorously that the family of eigenfunctions obtained is rich enough to span the space of all continuous functions. In addition, the method is extended to construct generalized eigenfunctions that also give rise Koopman invariant subspaces and hence can be used for linear prediction. Detailed numerical examples with code available online demonstrate the approach, both for prediction and feedback control.

Let $hat G$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $hat G$ over its field of definition. We explore the overgroup structure of $hat G$ in $mathrm{GL}(V)$, and the submodule structure of the exterior square (and sometimes the third Lie power) of $V$. When $hat G$ is defined over a field of odd prime order $p$, this allows us to construct the smallest (with respect to certain properties) $p$-groups $P$ such that the group induced by $mathrm{Aut}(P)$ on $P/Phi(P)$ is either $hat G$ or its normaliser in $mathrm{GL}(V)$.

We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato's original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the one-boundary Temperley-Lieb algebra (also known as the blob algebra). This provides a first step in generalizing the geometric versions of Khovanov's arc algebra to the exotic setting.

We derive finite rational formulas for the traces of cycle integrals of certain meromorphic modular forms. Moreover, we prove the modularity of a completion of the generating function of such traces. The theoretical framework for these results is an extension of the Shintani theta lift to meromorphic modular forms of positive even weight.

Gabber and Joseph introduced a ladder diagram between two natural sequences of extensions. Their diagram is used to produce a 'twisted' sequence that is applied to old and new results on extension groups in category $mathcal{O}$.

In this work, we propose a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators. Our method converges under the same assumptions as Tseng's forward-backward-forward method, namely, it does not require cocoercivity of the single-valued operator. Moreover, each iteration only requires one forward evaluation rather than two as is the case for Tseng's method. Variants of the method incorporating a linesearch, relaxation and inertia, or a structured three operator inclusion are also discussed.

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).

Tree balance plays an important role in different research areas like theoretical computer science and mathematical phylogenetics. For example, it has long been known that under the Yule model, a pure birth process, imbalanced trees are more likely than balanced ones. Therefore, different methods to measure the balance of trees were introduced. The Sackin index is one of the most frequently used measures for this purpose. In many contexts, statements about the minimal and maximal values of this index have been discussed, but formal proofs have never been provided. Moreover, while the number of trees with maximal Sackin index as well as the number of trees with minimal Sackin index when the number of leaves is a power of 2 are relatively easy to understand, the number of trees with minimal Sackin index for all other numbers of leaves was completely unknown. In this manuscript, we fully characterize trees with minimal and maximal Sackin index and also provide formulas to explicitly calculate the number of such trees.

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.

High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are abundant.

In this work, we construct new families of bounded degree high dimensional expanders obeying the local spectral expansion property. This property has a number of important consequences, including geometric overlapping, fast mixing of high dimensional random walks, agreement testing and agreement expansion. Our construction also yields new families of expander graphs which are close to the Ramanujan bound, i.e., their spectral gap is close to optimal.

The construction is quite elementary and it is presented in a self contained manner; This is in contrary to the highly involved previously known construction of the Ramanujan complexes. The construction is also very symmetric (such symmetry properties are not known for Ramanujan complexes) ; The symmetry of the construction could be used, for example, in order to obtain good symmetric LDPC codes that were previously based on Ramanujan graphs.

The main tool that we use for is the theory of coset geometries. Coset geometries arose as a tool for studying finite simple groups. Here, we show that coset geometries arise in a very natural manner for groups of elementary matrices over any finitely generated algebra over a commutative unital ring. In other words, we show that such groups act simply transitively on the top dimensional face of a pure, partite, clique complex.

In this paper, we are concerned with the numerical solution of one type integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. As an application, we will try to simulate the estimation of ruin probability with an unknown parameter driven not by the classical L'evy process but by the mixed fractional Brownian motion.

We use Ricci flow to obtain a local bi-Holder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson-Cheeger-Colding-Tian, describing how Ricci limit spaces in three dimensions must be homeomorphic to manifolds, and we obtain this in the most general, locally non-collapsed case. The proofs build on results and ideas from recent papers of Hochard and the current authors.

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.

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $mathfrak{sl}_2$, where they use singular blocks of category $mathcal{O}$ for $mathfrak{sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $mathfrak{sl}_n$ over a field $ extbf{k}$ of characteristic $p$ with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig's conjectures for representations of Lie algebras in positive characteristic.

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.

The paper proves the Riemann Hypothesis.

We address the problem of optimal path planning for a simple nonholonomic vehicle in the presence of obstacles. Most current approaches are either split hierarchically into global path planning and local collision avoidance, or neglect some of the ambient geometry by assuming the car is a point mass. We present a Hamilton-Jacobi formulation of the problem that resolves time-optimal paths and considers the geometry of the vehicle.

In this paper, we show that the difference between the number of parts in the odd partitions of $n$ and the number of parts in the distinct partitions of $n$ satisfies Euler's recurrence relation for the partition function $p(n)$ when $n$ is odd. A decomposition of this difference in terms of the total number of parts in all the partitions of $n$ is also derived. In this context, we conjecture that for $k>0$, the series

$$

(q^2;q^2)_infty sum_{n=k}^infty frac{q^{{kchoose 2}+(k+1)n}}{(q;q)_n}

egin{bmatrix}

n-1\k-1

end{bmatrix}

$$ has non-negative coefficients.

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.

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 use the classical Fourier analysis to introduce analytic families of weighted differential operators on the unit sphere. These operators are polynomial functions of the usual Beltrami-Laplace operator. New inversion formulas are obtained for totally geodesic Funk transforms on the sphere and the correpsonding lambda-cosine transforms.

We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to c-groups and to a c-crossed module, which is a connected, special and strict c-crossed module in the sense defined by us. The results obtained here will be applied in the proof of an equivalence of the categories of categorical groups and connected, special and strict c-crossed modules.

Let $X$ (resp. $Y$) be a curve of genus 1 (resp. 2) over a base field $k$ whose characteristic does not equal 2. We give criteria for the existence of a curve $Z$ over $k$ whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of $X$ and $Y$. Moreover, we give algorithms to construct the curve $Z$ once equations for $X$ and $Y$ are given. The first of these involves the use of hyperplane sections of the Kummer variety of $Y$ whose desingularization is isomorphic to $X$, whereas the second is based on interpolation methods involving numerical results over $mathbb{C}$ that are proved to be correct over general fields a posteriori. As an application, we find a twist of a Jacobian over $mathbb{Q}$ that admits a rational 70-torsion point.

In cite{CC} the authors, investigating a model of population dynamics, find the following result. Let $Omegasubset mathbb{R}^N$, $Ngeq 1$, be a bounded smooth domain. The weighted eigenvalue problem $-Delta u =lambda m u $ in $Omega$ under homogeneous Dirichlet boundary conditions, where $lambda in mathbb{R}$ and $min L^infty(Omega)$, is considered. The authors prove the existence of minimizers $check m$ of the principal positive eigenvalue $lambda_1(m)$ when $m$ varies in a class $mathcal{M}$ of functions where average, maximum, and minimum values are given. A similar result is obtained in cite{CCP} when $m$ is in the class $mathcal{G}(m_0)$ of rearrangements of a fixed $m_0in L^infty(Omega)$. In our work we establish that, if $Omega$ is Steiner symmetric, then every minimizer in cite{CC,CCP} inherits the same kind of symmetry.

Among all dissipative solutions of the multi-dimensional isentropic Euler equations there exists at least one that minimizes the acceleration, which implies that the solution is as close to being a weak solution as possible. The argument is based on a suitable selection procedure.

Let $Isubseteq S=K[x_1,ldots,x_n]$ be a homogeneous ideal equipped with a monomial order $<$. We show that if $operatorname{in}_<(I)$ is a square-free monomial ideal, then $S/I$ and $S/operatorname{in}_<(I)$ have the same connectedness dimension. We also show that graphs related to connectedness of these quotient rings have the same number of components. We also provide consequences regarding Lyubeznik numbers. We obtain these results by furthering the study of connectedness modulo a parameter in a local ring.

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 $p=2$.

A reaction-diffusion model was developed describing the spread of the COVID-19 virus considering the mean daily movement of susceptible, exposed and asymptomatic individuals. The model was calibrated using data on the confirmed infection and death from France as well as their initial spatial distribution. First, the system of partial differential equations is studied, then the basic reproduction number, R0 is derived. Second, numerical simulations, based on a combination of level-set and finite differences, shown the spatial spread of COVID-19 from March 16 to June 16. Finally, scenarios of unlockdown are compared according to variation of distancing, or partially spatial lockdown.

We consider the most general class of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of an arbitrary order whose solutions and right-hand sides belong to appropriate Sobolev spaces. For parameter-dependent problems from this class, we prove a constructive criterion for their solutions to be continuous in the Sobolev space with respect to the parameter. We also prove a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem.

In this article, we prove that if $R$ is a finitely generated ring over $mathbb{Z}$ of dimension $d, dgeq2, frac{1}{d!}in R$, then any unimodular row over $R[X]$ of length $d+1$ can be mapped to a factorial row by elementary transformations.

We offer an alternative proof of a result of Conlon, Fox, Sudakov and Zhao on solving translation-invariant linear equations in dense Sidon sets. Our proof generalises to equations in more than five variables and yields effective bounds.

We consider two special cases of the connection problem for the second Painlev'e equation (PII) using the method of uniform asymptotics proposed by Bassom et al.. We give a classification of the real solutions of PII on the negative (positive) real axis with respect to their initial data. By product, a rigorous proof of a property associate with the nonlinear eigenvalue problem of PII on the real axis, recently revealed by Bender and Komijani, is given by deriving the asymptotic behavior of the Stokes multipliers.

Given spherically symmetric characteristic initial data for the Einstein-scalar field system with a positive cosmological constant, we provide a criterion, in terms of the dimensionless size and dimensionless renormalized mass content of an annular region of the data, for the formation of a future trapped surface. This corresponds to an extension of Christodoulou's classical criterion by the inclusion of the cosmological term.

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.

In this paper we study removable singularities for regular $(1,1/2)$-Lipschitz solutions of the heat equation in time varying domains. We introduce an associated Lipschitz caloric capacity and we study its metric and geometric properties and the connection with the $L^2$ boundedness of the singular integral whose kernel is given by the gradient of the fundamental solution of the heat equation.

A natural problem in the context of the coupon collector's problem is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provide explicit asymptotic expressions for the moments of that maximum, as well as of the maximum of exponential random variables with corresponding parameters. We also deal with the probability of each of the variables being the maximal one.

The calculations lead to expressions involving Hurwitz's zeta function at certain special points. We find here explicitly the values of the function at these points. Also, the distribution function of the maximum we deal with is closely related to the generating function of the partition function. Thus, our results (and proofs) rely on classical results pertaining to the partition function.

We show that every 4-uniform hypergraph with $n$ vertices and minimum pair degree at least $(5/9+o(1))n^2/2$ contains a tight Hamiltonian cycle. This degree condition is asymptotically optimal.

This is the second of two papers in which we construct formal power series solutions in external parameters to the vacuum Einstein equations, implementing one bounce for the Belinskii-Khalatnikov-Lifshitz (BKL) proposal for spatially inhomogeneous spacetimes. Here we show that spatially inhomogeneous perturbations of spatially homogeneous elements are unobstructed. A spectral sequence for a filtered complex, and a homological contraction based on gauge-fixing, are used to do this.

We construct a smooth moduli stack of tuples consisting of genus two nodal curves, line bundles, and twisted fields. It leads to a desingularization of the moduli of genus two stable maps to projective spaces. The construction of this new moduli is based on systematical application of the theory of stacks with twisted fields (STF), which has its prototype appeared in arXiv:1906.10527 and arXiv:1201.2427 and is fully developed in this article. The results of this article are the second step of a series of works toward the resolutions of the moduli of stable maps of higher genera.

In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution $(u,b)$ becomes regular provided that one velocity and one current density component of the solution satisfy% egin{equation} u_{3}in L^{frac{30alpha }{7alpha -45}}left( 0,T;L^{alpha ,infty }left( mathbb{R}^{3} ight) ight) ext{ with }frac{45}{7}% leq alpha leq infty , label{eq01} end{equation}% and egin{equation} j_{3}in L^{frac{2eta }{2eta -3}}left( 0,T;L^{eta ,infty }left( mathbb{R}^{3} ight) ight) ext{ with }frac{3}{2}leq eta leq infty , label{eq02} end{equation}% which generalize some known results.

We construct a 2-equivalence $mathfrak{CohTheory}^ ext{op} simeq mathfrak{TypeSpaceFunc}$. Here $mathfrak{CohTheory}$ is the 2-category of positive theories and $mathfrak{TypeSpaceFunc}$ is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in $mathfrak{CohTheory}$. The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is `the same' as the collection of its type spaces (i.e. its type space functor).

In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory.

The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories.

A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this paper we introduce the q-analogue of a PMD and its properties. In order to do that, we first establish new cryptomorphic definitions for q-matroids. We show that q-Steiner systems are examples of q-PMD's and we use this matroid structure to construct subspace designs from q-Steiner systems. We apply this construction to S(2, 13, 3; q) Steiner systems and hence establish the existence of subspace designs with previously unknown parameters.

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

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 give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form $$ F(x,u,Du,D^2u) = 0 $$ under suitable structure conditions on the equation allowing for non-Lipschitz growth in the gradient terms. In case of smooth boundaries, we also prove the Hopf lemma, the boundary Harnack inequality and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply to weak solutions of an eigenvalue problem for the variable exponent $p$-Laplacian.

A subset of vertices in a graph $G$ is called a maximum dissociation set if it induces a subgraph with vertex degree at most 1 and the subset has maximum cardinality. The dissociation number of $G$, denoted by $psi(G)$, is the cardinality of a maximum dissociation set. A subcubic tree is a tree of maximum degree at most 3. In this paper, we give the lower and upper bounds on the dissociation number in a subcubic tree of order $n$ and show that the number of maximum dissociation sets of a subcubic tree of order $n$ and dissociation number $psi$ is at most $1.466^{4n-5psi+2}$.

The main aim of this paper is to detect dynamical properties of the Gy"orgyi-Field model of the Belousov-Zhabotinsky chemical reaction. The corresponding three-variable model given as a set of nonlinear ordinary differential equations depends on one parameter, the flow rate. As certain values of this parameter can give rise to chaos, the analysis was performed in order to identify different dynamics regimes. Dynamical properties were qualified and quantified using classical and also new techniques. Namely, phase portraits, bifurcation diagrams, the Fourier spectra analysis, the 0-1 test for chaos, and approximate entropy. The correlation between approximate entropy and the 0-1 test for chaos was observed and described in detail. Moreover, the three-stage system of nested subintervals of flow rates, for which in every level the 0-1 test for chaos and approximate entropy was computed, is showing the same pattern. The study leads to an open problem whether the set of flow rate parameters has Cantor like structure.