Results in Mathematics最新文献

We examine the problems associated with the Collatz map T from the point of view of functional analysis. We associate with T a certain linear operator (mathcal {T}) and show that cycles and (hypothetical) divergent trajectories (generated by T) correspond to certain classes of fixed points of the operator (mathcal {T}). We also show the relationship between the dynamic properties of the operator (mathcal {T}) and the map T. We prove that the absence of non-trivial cycles of T leads to hypercyclicity of the operator (mathcal {T}). In the second part, we show that the index of the operator (Id-mathcal {T}in mathcal {L}(H^2(D))) provides an upper estimate for the number of cycles of T. For the proof, we consider the adjoint operator (mathcal {F}=mathcal {T}^*)

which was first introduced by Berg, Meinardus in [3], and show that it has no non-trivial fixed points in (H^2(D)). Furthermore, we calculate the resolvent of the operator (mathcal {F}) and derive the equation for the characteristic function of the total stopping time (sigma _{infty }) as an application. In addition, we construct an invariant measure for (mathcal {T}) in a slightly different setup, and investigate how the operator (mathcal {T}) acts on generalized arithmetic progressions.

The k–generalized Fibonacci sequence ({F_n^{(k)}}_{nge 2-k}) is the linear recurrent sequence of order k whose first k terms are (0, ldots , 0, 1) and each term afterwards is the sum of the preceding k terms. The case (k=2) corresponds to the well known Fibonacci sequence ({F_n}_{nge 0}). In this paper we extend the study of the exponential Diophantine equation (left( F_{n+1}right) ^x+left( F_{n}right) ^x-left( F_{n-1}right) ^x=F_{m}) with terms (F_r^{(k)}) instead of (F_r), where (rin {n+1,n,n-1,m}).

In this manuscript, we study the operator moment problem for a block tridiagonal non-symmetric matrix. To study this problem, we used the relation between the resolvent function and a sequence of left-matrix orthogonal polynomials with respect to the sesquilinear form defined from the operator.

A compact set (Esubset mathbb {C}^N) satisfies the Markov inequality if the supremum norm on E of the gradient of a polynomial p can be estimated from above by the norm of p multiplied by a constant polynomially depending on the degree of p. This inequality is strictly related to the Bernstein approximation theorem, Schur-type estimates and the extension property of smooth functions. Additionally, the Markov inequality can be applied to the construction of polynomial grids (norming sets or admissible meshes) useful in numerical analysis. We expect such an inequality with similar consequences not only on polynomially determining compacts but also on some nowhere dense sets. The primary goal of the paper is to extend the above definition of Markov inequality to the case of compact subsets of algebraic varieties in (mathbb {C}^N). Moreover, we characterize compact sets satisfying such a Markov inequality on algebraic hypersurfaces as well as on certain varieties defined by several algebraic equations. We also prove a division inequality (a Schur-type inequality) on these sets. This opens up the possibility of establishing polynomial grids on algebraic sets. We also provide examples that complete and ilustrate the results.

For any Wulff shape W, its dual Wulff shape and spherical Wulff shape (widetilde{W}) can be defined naturally. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, we prove that a polytope is self-dual if and only if its spherical Wulff shape is a spherical convex body of constant width. We also prove that a smooth Wulff shape is self-dual if and only if for any interior points P of (widetilde{W}) and for any point Q of the intersection of the boundary of (widetilde{W}) and the graph of its spherical support function (with respect to P), the image of Q under the spherical blow-up (with respect to P) is always a point of (widetilde{W}). Moreover, we give an affirmative answer to the problem posed by M. Lassak which says that “Do there exist reduced spherical n-dimensional polytopes (possibly some simplices?) on (mathbb {S}^n), where (nge 3), different from the (1/2^n) part of (mathbb {S}^n?)”.

We study the logarithmic vector bundles associated to arrangements of smooth irreducible curves with small degree on the blow-up of the projective plane at one point. We then investigate whether they are Torelli arrangements, that is, they can be recovered from the attached logarithmic vector bundles.

Let (P_3(x,y)) and (Q_3(x,y)) be polynomials of degree three without constant or linear terms. We characterize the global centers of all polynomial differential systems of the form (dot{x} = y+ P_3(x,y)), (dot{y} =Q_3(x,y)) that are reversible and invariant with respect to the x-axis.

We investigate a class of the nonlinear Schrödinger equation in ( mathbb {R}^N)

where (Nge 3), (lambda >0) and (pin (2,2^*)) with ( 2^*=frac{2 N}{N-2}). Here, (V(x)=V_1(x)) for (x_1>0) and (V(x)=V_2(x)) for (x_1<0), where (V_1,V_2 ) are periodic in each coordinate direction. By providing a splitting Lemma corresponding to non-periodic external potential, we obtain the existence of ground state solution for the above problem. It is worth to mention that the arguments used in this paper are also valid for the Sobolev subcritical problem studied by Dohnal et al. (Commun Math Phys 308:511–542, 2011).

We study the topology of all possible subsums of the generalized multigeometric series (k_1f(x)+k_2f(x)+dots +k_mf(x)+dots + k_1f(x^n)+dots +k_mf(x^n)+dots ,) where (k_1, k_2, dots , k_m) are fixed positive real numbers and f runs along a certain class of non-negative functions on the unit interval. We detect particular regions of this interval for which this achievement set is, respectively, a compact interval, a Cantor set and a Cantorval.

In the paper, the authors intrinsically observe that Hoffman’s combinatorial identity discovered in 1992 and Genčev’s combinatorial identities extended in 2024 can be reformulated in terms of complete Bell polynomials. They provide alternative proofs and offer some elementary generalizations of these combinatorial identities.