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Abstract: In our original paper the incorrect labelling of the number of vertices of the line graph L(G) in the proof of Theorem 2 has led to the inequality PubDate: 2022-12-01

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Abstract: Abstract We present a class of multiplicative functions \(f:\mathbb {N}\rightarrow \mathbb {C}\) with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by 1. The key feature is that they pretend to be the constant function 1 and that for some prime q, \(\sum _{k=0}^\infty \frac{f(q^k)}{q^k}=0\) . These combined with other conditions guarantee that these functions are periodic and have sum equal to zero inside each period. Further, we study the class of multiplicative functions \(f=f_1*f_2\) , where each \(f_j\) is multiplicative and periodic with bounded partial sums. We show an omega bound for the partial sums \(\sum _{n\le x}f(n)\) and an upper bound that is related with the error term in the classical Dirichlet divisor problem. PubDate: 2022-12-01

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Abstract: Abstract In this paper we study differentiable equivalences of germs of singular holomorphic foliations in dimension two. We prove that the Camacho–Sad indices are invariant by such equivalences. We also prove that the Baum–Bott index is a differentiable invariant for some classes of foliations. As a corollary we show that generic degree two holomorphic foliations of \({\mathbb {P}}^2\) are differentiably rigid. PubDate: 2022-12-01

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Abstract: Abstract Let M be a connected compact oriented surface with one boundary component. A \(C^\infty \) map \(f:M \rightarrow {\mathbb {R}}^2\) is weakly admissible if it behaves locally like a stable map on a neighbourhood of the boundary \(\partial M\) of M. The notions of the half boundary winding number of a weakly admissible \(C^\infty \) map \(M \rightarrow {\mathbb {R}}^2\) and weakly-admissibly homotopic among weakly admissible \(C^\infty \) maps \(M \rightarrow {\mathbb {R}}^2\) are introduced. Then, we show that weakly admissible \(C^\infty \) maps \(f_0\) and \(f_1:M \rightarrow {\mathbb {R}}^2\) are weakly-admissibly homotopic if and only if the half boundary winding number of \(f_0\) coincides with that of \(f_1\) . Furthermore, we introduce a formula concerning the genus g of M for a stable map \(M \rightarrow {\mathbb {R}}^2\) . PubDate: 2022-12-01

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Abstract: Abstract We study the point pair function in subdomains G of \({\mathbb {R}}^n\) . We prove that, for every domain \(G\subsetneq {\mathbb {R}}^n\) , this function is a quasi-metric with the constant less than or equal to \(\sqrt{5}/2\) . Moreover, we show that it is a metric in the domain \(G={\mathbb {R}}^n{\setminus }\{0\}\) with \(n\ge 1\) . We also consider generalized versions of the point pair function, depending on a parameter \(\alpha >0\) , and show that in some domains these generalizations are metrics if and only if \(\alpha \le 12\) . PubDate: 2022-12-01

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Abstract: Abstract In this present paper, we investigate some essential results, in particular, involving the Nehari manifold and functional coercivity. In this sense, we attack our main result, that is, the existence of at least two positive bounded solutions for weighted singular fractional p-Laplace operator via Nehari manifold in the space \(\psi \) -fractional \({\mathbb {H}}^{\alpha ,\beta ;\psi }_{p}([0,T],{\mathbb {R}})\) . PubDate: 2022-12-01

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Abstract: Abstract In this paper we study a 3D non-stationary Boussinesq system with Navier-slip boundary conditions for the velocity field and Neumann boundary conditions for the temperature. Considering the viscosity and diffusion coefficient as explicit functions depending on the temperature, we prove the existence of weak solutions and present a regularity result that allow us obtain global-in-time strong solutions. PubDate: 2022-12-01

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Abstract: Abstract We study germs of holomorphic distributions with “separated variables”. In codimension one, a well know example of this kind of distribution is given by $$\begin{aligned} dz=(y_1dx_1-x_1dy_1)+\dots +(y_mdx_m-x_mdy_m), \end{aligned}$$ which defines the canonical contact structure on \({\mathbb {C}}{\mathbb {P}}^{2m+1}\) . Another example is the Darboux distribution $$\begin{aligned} dz=x_1dy_1+\dots +x_mdy_m, \end{aligned}$$ which gives the normal local form of any contact structure. Given a germ \({\mathcal {D}}\) of holomorphic distribution with separated variables in \(({\mathbb {C}}^n,0)\) , we show that there exists , for some \(\kappa \in {\mathbb {Z}}_{\ge 0}\) related to the Taylor coefficients of \({\mathcal {D}}\) , a holomorphic submersion $$\begin{aligned} H_{{\mathcal {D}}}:({\mathbb {C}}^n,0)\rightarrow ({\mathbb {C}}^{\kappa },0) \end{aligned}$$ such that \({\mathcal {D}}\) is completely non-integrable on each level of \(H_{{\mathcal {D}}}\) . Furthermore, we show that there exists a holomorphic vector field Z tangent to \({\mathcal {D}}\) , such that each level of \(H_{{\mathcal {D}}}\) contains a leaf of Z that is somewhere dense in the level. In particular, the field of meromorphic first integrals of Z and that of \({\mathcal {D}}\) are the same. Between several other results, we show that the canonical contact structure on \({\mathbb {C}}{\mathbb {P}}^{2m+1}\) supports a Legendrian holomorphic foliation whose generic leaves are dense in \({\mathbb {C}}{\mathbb {P}}^{2m+1}\) . So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense. PubDate: 2022-12-01

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Abstract: Abstract In his work on birational classification of foliations on projective surfaces, Brunella showed that every regular foliation on a rational surface is algebraically integrable with rational leaves. This led Touzet to conjecture that every regular foliation on a rationally connected manifold is algebraically integrable with rationally connected leaves. Druel proved this conjecture for the case of weak Fano manifolds. In this paper, we extend this result showing that Touzet’s conjecture is true for codimension one foliations on threefolds with nef anti-canonical bundle. PubDate: 2022-12-01

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Abstract: Abstract In this note we concern with the Lyapunov exponent for critical circle maps. We prove that for any \(C^3\) critical circle map with irrational rotation number, the Lyapunov exponent at \(x \in S^1\) equals 0 if and only if the orbit of x never hits the critical points. This implies that the Lyapunov exponent at the critical value for unicritical circle map with irrational rotation number is 0, which answers a question of de Faria and Guarino. The idea is to verify Tsujii’s slow recurrence condition. PubDate: 2022-12-01

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Abstract: Abstract In this article, using techniques of Lie groups and dynamical systems, it is shown that lattices of a family of split solvable subgroups of PSL(N+1,C) are complex Kleinian. Also, it is shown that there exists a minimal limit set for the action of these lattices on the complex projective space and that there are exactly two maximal discontinuity regions. PubDate: 2022-12-01

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Abstract: Abstract This paper deals with the existence of insensitizing controls for a 1D free-boundary problem of the Stefan kind for a semilinear PDE. The insensitizing problem consists in finding a control function such that some energy functional of the system is locally insensitive to a perturbation of the initial data. As usual, this problem can be reduced to a nonstandard null controllability problem of some nonlinear coupled system governed by a semilinear parabolic equation with a free-boundary and a linear parabolic equation. Nevertheless, in order to solve the later Stefan problem by the fixed point technique, we need to establish the null controllability of the linear coupled system in a non-cylindrical domain. An observability estimate for the corresponding coupled system in a non-cylindrical domain is established, whose proof relies on a new global Carleman estimate. PubDate: 2022-12-01

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Abstract: Abstract Given an one-dimensional Lorenz-like expanding map we describe a class \({\mathcal {A}}\) of potentials \(\phi :[0,1]\longrightarrow {\mathbb {R}}\) admitting at most one equilibrium measure and we construct a family of continuous but not weak-Hölder continuous potentials for which we observe phase transitions. This give a certain generalization of the results proved in Pesin and Zhang (J Stat Phys 122(6):1095–1110, 2006), where the authors have proved this for a smaller class of potentials, that is, for uniformly expanding maps and weak-Hölder continuous potentials. Indeed, the class \({\mathcal {A}}\) form an open and dense subset of \(C([0,1],{\mathbb {R}})\) , with the usual \({\mathcal {C}}^0\) topology. PubDate: 2022-12-01

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Abstract: Abstract We prove that, for \(g\ge 19\) the mapping class group of a nonorientable surface of genus g, \(\mathrm{Mod}(N_g)\) , can be generated by two elements, one of which is of order g. We also prove that for \(g\ge 26\) , \(\mathrm{Mod}(N_g)\) can be generated by three involutions. PubDate: 2022-12-01

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Abstract: Abstract Let p, q, k and \(\ell \) be positive integers. The \((p,q,k,\ell )\) -Fibonacci sequence \((F_{k,\ell ,p,q})_{n\ge 0}\) is the four-parameter sequence defined by the following recurrence $$\begin{aligned} F_{k,\ell ,p,q}(n)=kF_{k,\ell ,p,q}(n-p)+\ell F_{k,\ell ,p,q}(n-q), \end{aligned}$$ with appropriate initial conditions. In this paper, we study the geometric, algebraic, and analytic aspects of the roots of the characteristic polynomial of this sequence, namely, \(f(x)=x^q-kx^{q-p}-\ell \) . PubDate: 2022-12-01

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Abstract: Abstract The integer sequence defined by \(P_{n+3}=P_{n+1}+P_n\) with initial conditions \(P_{0}=1\) and \(P_{1}=P_{2}=0\) is known as the Padovan sequence \(\left\{ P_{n}\right\} _{n\in \mathbb {Z}}\) . A recurrence sequence \(\{u_{n}\}_{n\in {\mathbb Z}}\) is said to be of Padovan-type if it satisfies the same recurrence relation as the Padovan sequence but with arbitrary initial values \(u_{0},u_{1},u_{2}\) not all zero. The most famous Padovan-type sequence is the Perrin sequence given by \(u_{0}=3\) , \(u_{1}=0\) and \(u_{2}=2\) . We show that every Padovan-type sequence has at most 2 zeros, except for nonzero multiples of shifts of the Padovan sequence which has 0-multiplicity 5. We also show that \(\left\{ P_{n}\right\} _{n\in {\mathbb Z}}\) has total multiplicity 62, i.e., there are 62 pairs \((m,n)\in \mathbb {Z}^{2}\) with \(m<n\) for which \(P_{m}=P_{n}\) . As a consequence, we found that \(\{P_{n}\}_{n\in \mathbb {Z}}\) has multiplicity 9, being 1 the most repeated Padovan number. Finally, we prove that the Perrin sequence has exactly 1 zero, total multiplicity 23 and multiplicity 4. PubDate: 2022-10-18

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Abstract: Abstract We introduce the concept of mixed random-quasiperiodic linear cocycles. We characterize the ergodicity of the base dynamics and establish a large deviations type estimate for certain types of observables. For the fiber dynamics we prove the uniform upper semicontinuity of the maximal Lyapunov exponent. This paper is meant to introduce a model to be studied in depth in further projects. PubDate: 2022-09-30

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Abstract: Abstract Let r, l, m be positive integers such that \(2\not \mid r\) , \(3\mid l\) and \(r>l\) , and let \(e(r,l,m)=\min \{rlm^{2}-1,r(r-l)m^{2}+1\}\) . In 2020, Kizildere, Le and Soydan proved if \(3\not \mid rm\) and \(e(r,l,m)>30\) , then the exponential Diophantine equation \((rlm^{2}-1)^{x}+(r(r-l)m^{2}+1)^{y}=(rm)^{z}\) \((*)\) has only the positive integer solution \((x,y,z)=(1,1,2)\) . In this paper, we improve the above result by removing the condition \(e(r,l,m)>30\) . Namely, we show that if \(3\not \mid rm\) , then the exponential Diophantine equation \((*)\) has only the positive integer solution \((x,y,z)=(1,1,2)\) . Moreover, using elementary methods and some classical results of generalized Ramanujan–Nagell equations, we prove similar result is true for analogous equation \((a(a-2r)m^{2}-1)^{x}+(2arm^{2}+1)^{y}=(am)^{z}\) under the conditions \(3\not \mid am\) and \(ar\equiv -1\pmod 3\) , where a, r, m be positive integers such that \(2\not \mid a\) and \(a\equiv 2r\pmod 3\) . We point out that our result is different from the result of Kizildere-Le-Soydan and there are infinitely many pairs (a, r) satisfying the conditions of Theorem. Some examples are also given in this paper to demonstrate the existence of pairs (a, r). In particular, when r is an odd prime and \(l=3\) , we deduce that if \(3\not \mid m\) or \(m\ge 247r\) , then the exponential Diophantine equation \((*)\) has only the positive integer solution \((x,y,z)=(1,1,2)\) . As a corollary, we derive that the exponential Diophantine equation \((21m^{2}-1)^{x}+(28m^{2}+1)^{y}=(7m)^{z}\) has only the positive integer solution \((x,y,z)=(1,1,2)\) . These extend and improve earlier results. PubDate: 2022-09-30

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Abstract: Abstract The distance Randić matrix of a connected graph G, denoted by \( D^{R}(G) \) , was defined in Díaz and Rojo (Bull Braz Math Soc New Ser, 2021) as follows $$\begin{aligned} D^{R}(G)=Tr(G)^{-1/2}D(G)Tr(G)^{-1/2}, \end{aligned}$$ where D(G) is the distance matrix and Tr(G) is the diagonal matrix of the transmission degrees of G. The matrix \( D^{R}(G) \) is real symmetric and the set of its eigenvalues including multiplicities is the distance Randić spectrum (or \( D^{R}\) -spectrum) of G. In the present article, we find several interesting properties of the eigenvalues of the distance Randić matrix of G. We characterize the graphs with two distance Randić eigenvalues and partially characterize the graphs with three distinct distance Randić eigenvalues. We find some new upper and lower bounds for the distance Randić energy of G and characterize the graphs attaining these bounds. One of our results improve a known upper bound of Díaz and Rojo (Bull Braz Math Soc New Ser, 2021) . PubDate: 2022-09-30

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Abstract: Abstract The purpose of this manuscript is to present a general framework of asymptotic behavior for topological cocycles by introducing the notion of directional attractor. The set-up consists of a topological cocycle conducted by a semigroup action. A directional attractor depends on a direction on the conductor semigroup, which generalizes the notions of past and future attractors. The main result assures the existence and uniqueness of a directional attractor by the preexistence of an absorbing set. Illustrating examples are presented. PubDate: 2022-09-24 DOI: 10.1007/s00574-022-00311-x