On a Congruence Involving Generalized Fibonomial Coefficients

Let (F_n)_n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_F} = \frac{{{F_{n - k + 1}} \cdots {F_{n - 1}}{F_n}}}{{{F_1} \cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±1 (mod 5), then p?\({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a ≥ 1. In 2010, in particular, Kilic generalized the Fibonomial coefficients for

$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_{F,m}} = \frac{{{F_{\left( {n - k + 1} \right)m}} \cdots {F_{\left( {n - 1} \right)m}}{F_{nm}}}}{{{F_m} \cdots {F_{km}}}}$$

. In this note, we generalize Marques, Sellers and Trojovský result to prove, in particular, that if p ≡ ±1 (mod 5), then \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_{F,m}} \equiv 1\) (mod p), for all a ≥ 0 and m ≥ 1.

     

The <Emphasis Type="Italic">p</Emphasis>-adic order of some fibonomial coefficients whose entries are powers of <Emphasis Type="Italic">p</Emphasis>

Let (F _n ) _n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${\left[ {\begin{array}{*{20}{c}} m \\ k \end{array}} \right]_F} = \frac{{{F_{m - k + 1}} \cdots {F_{m - 1}}{F_m}}}{{{F_1} \cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then \(p{\left| {\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]} \right._F}\) for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a, b ≥ 1 and for all prime number p.

     

On the Periodic Logistic Map

In this paper, the famous logistic map is studied in a new point of view. We study the boundedness and the periodicity of non-autonomous logistic map

$${x_{n + 1}} = {r_n}{x_n}\left( {1 - {x_n}} \right),n = 0,1, \ldots ,$$

where {r _n } is a positive p-periodic sequence. The sufficient conditions are given to support the existence of asymptotically stable and unstable p-periodic orbits. This appears to be the first study of the map with variable parameter r.

     

A note on some results of Hua in short intervals

In this paper, we improve the previous results of the authors [G. Lü and H. Tang, On some results of Hua in short intervals, Lith. Math. J., 50(1):54–70, 2010] by proving that each sufficiently large integer N satisfying some congruence conditions can be written as

$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k},} \hfill \\ {\left| {{p_j} - \sqrt {{\frac{N}{5}}} } \right| \leqslant U,\quad \left| {p - {{\left( {\frac{N}{5}} \right)}^{\frac{1}{k}}}} \right| \leqslant U\,{N^{ - \frac{1}{2} + \frac{1}{k}}},\quad j = 1,\,2,\,\,3,\,4,} \hfill \\ \end{array} } \right. $

where U = N ^1/2?η+ε with \( \eta = \frac{1}{{2k\left( {{K^2} + 1} \right)}} \) and K = 2k ^?1, k ? 2.

     

Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =(x_1,x_2,x_3)∈R~(n_1)×R~(n_2)×R~(n_3) and ξ =(ξ_1,ξ_2,ξ_3)∈R~(n_1)×R~(n_2)×R~(n_3). One of our main results is the following:Assume that m(ξ) is a function on R~(n_1+n_2+n_3) satisfying ■ with s_i n_i(1/p-1/2) for 1≤i≤3. Then T_m is bounded from H~p(R~(n_1)×R~(n_2)×R~(n_3) to H~p(R~(n_1)×R~(n_2)×R~(n_3)for all 0 p≤1 and ■ Moreover, the smoothness assumption on s_i for 1≤i≤3 is optimal. Here we have used the notations m_(j,k,l)(ξ)=m(2~jξ_1,2~kξ_2,2~lξ_3)Ψ(ξ_1)Ψ(ξ_2)Ψ(ξ_3) and Ψ(ξ_i) is a suitable cut-off function on R~(n_i) for1≤i≤3, and W~(s_1,s_2,s_3) is a three-parameter Sobolev space on R~(n_1)×R~(n_2)× R~(n_3).Because the Fefferman criterion breaks down in three parameters or more, we consider the L~p boundedness of the Littlewood-Paley square function of T_mf to establish its boundedness on the multi-parameter Hardy spaces.     

Spectrum and the trace formula for a two-dimensional Schrödinger operator in a homogeneous magnetic field

In the space L ₂(?²), we consider the operator

$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$

. We study the spectrum of H and, for V ∈ C ₀ ² (?²), prove the trace formula

$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $

where c ₀ = π ^?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ _k ⁽ⁱ⁾ are the eigenvalues of H.

     

Strong isochronism of a center and a focus for systems with homogeneous nonlinearities

We suggest a new approach to studying the isochronism of the system

${{dx} \mathord{\left/ {\vphantom {{dx} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - y + p_n (x,y),{{dy} \mathord{\left/ {\vphantom {{dy} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = x + q_n (x,y),$

where p _n and q _n are homogeneous polynomials of degree n. This approach is based on the normal form

${{dX} \mathord{\left/ {\vphantom {{dX} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - Y + XS(X,Y),{{dY} \mathord{\left/ {\vphantom {{dY} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = X + YS(X,Y)$

and its analog in polar coordinates. We prove a theorem on sufficient conditions for the strong isochronism of a center and a focus for the reduced system and obtain examples of centers with strong isochronism of degrees n = 4, 5. The present paper is the first to give examples of foci with strong isochronism for the system in question.

     

A continuous model for systems of complexity 2 on simple abelian groups

It is known that if p is a sufficiently large prime, then, for every function f: Z_p → [0, 1], there exists a continuous function f′: T → [0, 1] on the circle such that the averages of f and f′ across any prescribed system of linear forms of complexity 1 differ by at most ∈. This result follows from work of Sisask, building on Fourier-analytic arguments of Croot that answered a question of Green. We generalize this result to systems of complexity at most 2, replacing T with the torus T² equipped with a specific filtration. To this end, we use a notion of modelling for filtered nilmanifolds, that we define in terms of equidistributed maps and combine this notion with tools of quadratic Fourier analysis. Our results yield expressions on the torus for limits of combinatorial quantities involving systems of complexity 2 on Z_p. For instance, let m₄(α, Z_p) denote the minimum, over all sets A ? Z_p of cardinality at least αp, of the density of 4-term arithmetic progressions inside A. We show that limp→∞ m₄(α, Z_p) is equal to the infimum, over all continuous functions f: T₂ →[0, 1] with \({\smallint _{{T^2}}}f \geqslant a\), of the integral

$$\int_{{T^5}} {f\left( {\begin{array}{*{20}{c}}{{x_1}} \\ {{y_1}} \end{array}} \right)} f\left( {\begin{array}{*{20}{c}}{{x_1} + {x_2}} \\ {{y_1} + {y_2}} \end{array}} \right)f\left( {\begin{array}{*{20}{c}}{{x_1} + 2{x_2}} \\ {{y_1} + 2{y_2} + {y_3}} \end{array}} \right).f\left( {\begin{array}{*{20}{c}}{{x_1} + 3{x_2}} \\ {{y_1} + 3{y_2} = 3{y_3}} \end{array}} \right)d{\mu _{{T^5}}}({x_1},{x_2},{y_1},{y_2},{y_3})$$

     

Extracting outer function part from Hardy space function

Any analytic signal fa(e~(it)) can be written as a product of its minimum-phase signal part(the outer function part) and its all-phase signal part(the inner function part). Due to the importance of such decomposition, Kumarasan and Rao(1999), implementing the idea of the Szeg?o limit theorem(see below),proposed an algorithm to obtain approximations of the minimum-phase signal of a polynomial analytic signal fa(e~(it)) = e~(iN0t)M∑k=0a_k~(eikt),(0.1)where a_0≠ 0, a_M≠ 0. Their method involves minimizing the energy E(f_a, h_1, h_2,..., h_H) =1/(2π)∫_0~(2π)|1+H∑k=1h_k~(eikt)|~2|fa(e~(it))|~2dt(0.2) with the undetermined complex numbers hk's by the least mean square error method. In the limiting procedure H →∞, one obtains approximate solutions of the minimum-phase signal. What is achieved in the present paper is two-fold. On one hand, we rigorously prove that, if fa(e~(it)) is a polynomial analytic signal as given in(0.1),then for any integer H≥M, and with |fa(e~(it))|~2 in the integrand part of(0.2) being replaced with 1/|fa(e~(it))|~2,the exact solution of the minimum-phase signal of fa(e~(it)) can be extracted out. On the other hand, we show that the Fourier system e~(ikt) used in the above process may be replaced with the Takenaka-Malmquist(TM) system, r_k(e~(it)) :=((1-|α_k|~2e~(it))/(1-α_ke~(it))~(1/2)∏_(j=1)~(k-1)(e~(it)-α_j/(1-α_je~(it))~(1/2), k = 1, 2,..., r_0(e~(it)) = 1, i.e., the least mean square error method based on the TM system can also be used to extract out approximate solutions of minimum-phase signals for any functions f_a in the Hardy space. The advantage of the TM system method is that the parameters α_1,..., α_n,...determining the system can be adaptively selected in order to increase computational efficiency. In particular,adopting the n-best rational(Blaschke form) approximation selection for the n-tuple {α_1,..., α_n}, n≥N, where N is the degree of the given rational analytic signal, the minimum-phase part of a rational analytic signal can be accurately and efficiently extracted out.     

On the numerical index of real <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(µ)-spaces

We give a lower bound for the numerical index of the real space L _p (µ) showing, in particular, that it is non-zero for p ≠ 2. In other words, it is shown that for every bounded linear operator T on the real space L _p (µ), one has

$\sup \left\{ {|\int {|x{|^{p - 1}}{\rm{sign}}(x)Tx d\mu |:x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {{{M_p}} \over {12{\rm{e}}}}||T||$

where \({M_p} = {\max _{t \in \left[ {0,1} \right]}}{{|{t^{p - 1}} - t|} \over {1 + {t^p}}} > 0\) for every p ≠ 2. It is also shown that for every bounded linear operator T on the real space L _p (µ), one has

$\sup \left\{ {\int {|x{|^{p - 1}}|Tx| d\mu :x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {1 \over {2{\rm{e}}}}||T||$

.

     

An invariant cubature formula of degree nine for a hypercube

In this work, cubature formulas for computation of integrals over the hypercube in R ⁿ

$C_n = \{ x = (x_1 ,x_2 ,...,x_n ) \in R^n | - 1 \leqslant x_i \leqslant 1,i = 1,2,...,n\} $

are constructed using Sobolev?s theorem. These formulas are precise for all polynomials of degree at most nine and are invariant under the group of all orthogonal transformations of the hyperoctahedron

$G_n = \left\{ {x = (x_1 ,x_2 ,...,x_n ) \in R^n |\sum\limits_{i = 1}^n {|x_i |} \leqslant 1} \right\}$

onto itself.

Section 1 contains introduction into the subject and a review of known results. In Sections 2 and 3, we determine parameters of the cubature formula for n = 4 and n = 3, respectively. Numerical results (the nodes and coefficients of the cubature formulas) are presented in Section 4.     

Complementary inequalities to improved AM-GM inequality

Following an idea of Lin, we prove that if A and B are two positive operators such that 0 mI ≤ A ≤m'I≤ M'I ≤ B ≤ MI, then Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2Φ~2(A≠B) and Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2(Φ(A)≠Φ(B))~2 where K(h)=(h+1)~2/4 and h = M/m and Φ is a positive unital linear map.     

On the strong convergence of a general-type Krasnosel’skii–Mann’s algorithm depending on the coefficients

Let H be a Hilbert space, \({(W_n)_{n \in \mathbb{N}}}\) a suitable family of mappings, S a nonexpansive mapping and D a strongly monotone operator. We are interested in the strong convergence of the general scheme

$$x_{n + 1} = \gamma x_{n} + (1 - \gamma)W_{n} (\alpha_{n}S_{x_{n}} + (1 - \alpha_{n})(I - \mu_{n}D)x_{n}),\quad \gamma \in [0, 1),$$

in dependence of the coefficients \({(\alpha_{n})_{n \in \mathbb{N}}}\) and \({(\mu_{n})_{n \in \mathbb{N}}}\) .

     

Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$

, where the smoothing operators are defined by

$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$

.

     

Geometric interpretation of the Euler-Knopp summation method in the problem of inverting the Laplace transform

In this paper, we consider a method for inverting the Laplace transform F(s) = \(\int\limits_0^\infty {e^{ - st} f(t)dt} \), which consists in representing the original function by the Laguerre series

$f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt).} $

(1)

First, we perform a conformal mapping of the plane (s), which depends on parameter ξ. The value of the parameter is determined by the location of the singular points of the given representation. Under this mapping, series (1) takes the form

$f(t) = \frac{{b - \xi }}{b}\exp (\xi t)\sum\limits_{k = 0}^\infty {c_k L_k ((b - \xi )t).} $

It is demonstrated that such inverting scheme is equivalent to applying the Picone-Tricomi method with further acceleration of the rate of convergence of series (1) using the Euler-Knopp nonlinear procedure

$\sum\limits_{k = 0}^\infty {a_k z^k } = \sum\limits_{k = 0}^\infty {A_k (p)\frac{{z^k }}{{(1 - pz)^{k + 1} }},} A_k (p) = \sum\limits_{j = 0}^k {\left( \begin{gathered} k \hfill \\ j \hfill \\ \end{gathered} \right)( - p)^{k - j} a_j } .$

Under this approach, the original function is represented by the series

$f(t) = \exp \left( {\frac{{bpt}}{{p - 1}}} \right)\sum\limits_{k = 0}^\infty {\frac{{A_k (p)}}{{(1 - p)^{k + 1} }}L_k } \left( {\frac{{bpt}}{{1 - p}}} \right),$

where parameters ξ and p are related by the formula p = x/(ξ ? b). Unlike many other methods for summation of series, in the scheme suggested, there is no need to investigate the regularity conditions.

     

On sums of binomial coefficients involving Catalan and Delannoy numbers modulo $$p^2$$

In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime \(p>3\), we determine

$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$

modulo \(p^2\), where \(C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right) \) is the k-th Catalan number and \(C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right) \) is the second-order Catalan numbers of the first kind. And we prove that

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$

where \(D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) \) is the n-th Delannoy number and \(q_p(2)=(2^{{p-1}}-1)/p\) is the Fermat quotient.

     

Boundedness of Hausdorff operators on the power weighted Hardy spaces

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H_(|x|α)~1(R~2) ( -1 ≤α≤0), defined by H_(Φ,A)f(x)=∫R~2Φ(u)f(A(u)x)du,where Φ∈L_loc~1(R~2),A(u) = (α_(ij)(u))_(i,j=1)~2 is a 2×2 matrix, and each α_(i,j) is a measurablefunction.We obtain that HΦ,A is bounded from H_(|x|~α)~1(R~2) ( -1≤α≤0) to itself, if∫R2|Φ(u)‖det A~(-1)(u)|‖A(u)‖~(-α)ln(1+‖A~(-1)(u)‖~2/|det A~(-1)(u)|)du∞.This result improves some known theorems, and in some sense it is sharp.     

On positive solutions of a reciprocal difference equation with minimum

In this paper we consider positive solutions of the following difference equation $$x_{n + 1} = \min \left\{ {\frac{A}{{x_n }},\frac{B}{{x_{n - 2} }}} \right\}, A, B > 0.$$ We prove that every positive solution is eventually periodic. Also, we present here some results concerning positive solutions of the difference equation $$x_{n + 1} = \min \left\{ {\frac{A}{{x_n x_{n - 1} ...x_{n - k} }},\frac{B}{{x_{n - (k + 2)} ...x_{n - (2k + 2)} }}} \right\}, A, B > 0.$$      

On the uniform Kreiss resolvent condition

Let B be a Banach space with norm ‖ · ‖ and identity operator I. We prove that, for a bounded linear operator T in B, the strong Kreiss resolvent condition

$\parallel (T - \lambda I)^{ - k} \parallel \leqslant \frac{M}{{(|\lambda | - 1)^k }}, |\lambda | > 1,k = 1,2, \ldots ,$

implies the uniform Kreiss resolvent condition

$\left\| {\sum\limits_{k = 0}^n {\frac{{T^k }}{{\lambda ^{k + 1} }}} } \right\| \leqslant \frac{L}{{|\lambda | - 1}}, |\lambda | > 1, n = 0,1,2, \ldots .$

We establish that an operator T satisfies the uniform Kreiss resolvent condition if and only if so does the operator T ^m for each integer m ? 2.

     

Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups

The (Nörlund) logarithmic means of the Fourier series is:

$t_n f = \frac{1}{{l_n }}\sum\limits_{k = 1}^{n - 1} {\frac{{S_k f}}{{n - k}}} , where l_n = \sum\limits_{k = 1}^{n - 1} {\frac{1}{k}} $

. In general, the Fejér (C,1) means have better properties than the logarithmic ones. We compare them and show that in the case of some unbounded Vilenkin systems the situation changes.