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.

     

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

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

     

On some new identities for the Fibonomial coefficients

Let F _n be the nth Fibonacci number. The Fibonomial coefficients \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F\) are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with \(\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1\) and \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0\) . In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l.     

An existence principle for solutions to a singular boundary-value problems

The singular boundary-value problem

$ \left\{ {\begin{array}{*{20}{c}} {{u^{\prime\prime}} + g\left( {t,u,{u^{\prime}}} \right) = 0\quad {\text{for}}\quad t \in \left( {0,1} \right),} \hfill \\ {u(0) = u(1) = 0} \hfill \\ \end{array} } \right. $

is studied. The singularity may appear at u?=?0, and the function g may change sign. An existence theorem for solutions to the above boundary-value problem is proposed, and it is proved via the method of upper and lower solutions.

     

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.

     

On some results of Hua in short intervals

In this note, we prove some results of Hua in short intervals. For example, 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 {N/5} } \right| \leqslant U,\left| {p - {{\left( {N/5} \right)}^{\tfrac{1}{k}}}} \right|\leqslant UN - \tfrac{1}{2} + \tfrac{1}{k},j = 1,2,3,4,} \hfill \\ \end{array} } \right. $

where \( U = N\tfrac{1}{2} - \eta + \varepsilon \) with \( \eta = \frac{2}{{\kappa \left( {K + 1} \right)\left( {{K^2} + 2} \right)}} \) and \( K = {2^{k - 1}},k\geqslant 3. \)

     

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.

     

Multiple soliton solutions for a quasilinear Schrödinger equation

Using Morse theory, truncation arguments and an abstract critical point theorem, we obtain the existence of at least three or infinitely many nontrivial solutions for the following quasilinear Schrödinger equation in a bounded smooth domain

$$\left\{ {\begin{array}{*{20}{c}} { - {\Delta _p}u - \frac{p}{{{2^{p - 1}}}}u{\Delta _p}\left( {{u^2}} \right) = f\left( {x,u} \right)\;in\;\Omega } \\ {u = 0\;on\;\partial \Omega .} \end{array}} \right.$$

(0.1)

Our main results can be viewed as a partial extension of the results of Zhang et al. in [28] and Zhou and Wu in [29] concerning the the existence of solutions to (0.1) in the case of p = 2 and a recent result of Liu and Zhao in [21] two solutions are obtained for problem 0.1.

     

On a Composition Preserving Inequalities between Polynomials

The Schur-Szegö composition of two polynomials \(f\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{z^j}} \) and \(g\left( z \right) = \sum\nolimits_{j = 0}^n {{B_j}{z^j}} \), both of degree n, is defined by \(f * g\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{B_j}{{\left( {\begin{array}{*{20}{c}}n \\ j \end{array}} \right)}^{ - 1}}{z^j}} \). In this paper, we estimate the minimum and the maximum of the modulus of f * g(z) on z = 1 and thereby obtain results analogues to Bernstein type inequalities for polynomials.     

Seas of squares with sizes from a Π<Stack><Subscript>1</Subscript><Superscript>0</Superscript></Stack> set

A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most (1+o(1))( _n/2 ⁿ ). Later, Burosch–Demetrovics–Katona–Kleitman–Sapozhenko asked for the number α(n) of such families, and they proved that \({2^{\left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)}} \leqslant \alpha \left( n \right) \leqslant {2^{2\sqrt 2 \left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)\left( {1 + o\left( 1 \right)} \right)}}\) They conjectured that the constant \(2\sqrt 2 \) can be removed in the exponent of the right-hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.     

Quantitative properties of ground-states to an <Emphasis Type="Italic">M</Emphasis>-coupled system with critical exponent in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In this paper, we consider the ground-states of the following M-coupled system:

$$\left\{ {\begin{array}{*{20}{c}}{ - \Delta {u_i} = \sum\limits_{j = 1}^M {{k_{ij}}\frac{{2{q_{ij}}}}{{2*}}{{\left| {{u_j}} \right|}^{{p_{ij}}}}{{\left| {{u_i}} \right|}^{{q_{ij}} - {2_{{u_i}}}}},x \in {\mathbb{R}^N},} } \\{{u_i} \in {D^{1,2}}\left( {{\mathbb{R}^N}} \right),i = 1,2, \ldots ,M,}\end{array}} \right.$$

where \(p_{ij} + q_{ij} = 2*: = \frac{{2N}}{{N - 2}}(N \geqslant 3)\). We prove the existence of ground-states to the M-coupled system. At the same time, we not only give out the characterization of the ground-states, but also study the number of the ground-states, containing the positive ground-states and the semi-trivial ground-states, which may be the first result studying the number of not only positive ground-states but also semi-trivial ground-states.

     

Packing spanning graphs from separable families

Let \(\mathcal{G}\) be a separable family of graphs. Then for all positive constants ? and Δ and for every sufficiently large integer n, every sequence G ₁,..., G _t ∈ \(\mathcal{G}\) of graphs of order n and maximum degree at most Δ such that

$$\left( {{G_1}} \right) + \cdots + e\left( {{G_t}} \right) \leqslant \left( {1 - \epsilon } \right)\left( {\begin{array}{*{20}{c}}n \\ 2 \end{array}} \right)$$

packs into K _n . This improves results of Böttcher, Hladký, Piguet and Taraz when \(\mathcal{G}\) is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyárfás and Lehel (1976) for the case that all trees have maximum degree at most Δ.

The proof uses the local resilience of random graphs and a special multi-stage packing procedure.     

Graphs with small total rainbow connection number

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that \({\text{trc(G) = 3 if}}\left( {\begin{array}{*{20}{c}}{n - 1} \\2\end{array}} \right) + 1 \leqslant \left| {{\text{E(G)}}} \right| \leqslant \left( {\begin{array}{*{20}{c}}n \\2\end{array}} \right) - 1\), and \({\text{trc(G)}} \leqslant {\text{6 if }}\left| {{\text{E(G)}}} \right| \geqslant \left( {\begin{array}{*{20}{c}}{n - 2} \\2\end{array}} \right) + 2\). Next, we investigate the total rainbow connection numbers of graphs G with |V(G)| = n, diam(G) ≥ 2, and clique number ω(G) = n ? s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem.     

Diffraction of light by two superposed parallel ultrasonic waves,having arbitrary frequencies. General symmetry properties; method of solution by series expansion

In this study about the diffraction of light by superposed parallel ultrasonics, with frequency ration ₁:n ₂, we deduce a general symmetry property for the intensities of the diffraction pattern: if the intensities of the ordersn and ?n are equal the phase-difference must be of the form:

$$\delta = \frac{{n_1 - n_2 }}{{n_1 }} \begin{array}{*{20}c} \pi \\ 2 \\ \end{array} + p \frac{\pi }{{n_1 }}$$     

A Pohozaev Identity for the Fractional Hnon System

In this paper, we study the Pohozaev identity associated with a Henon-Lane-Emden system involving the fractional Laplacian:■in a star-shaped and bounded domain Ω for s ∈(0,1). As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritical cases.     

A remark on the existence of entire large and bounded solutions to a (<Emphasis Type="Italic">k</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">k</Emphasis><Subscript>2</Subscript>)-Hessian system with gradient term

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system

$$\left\{ {\begin{array}{*{20}c}{S_{k_1 } \left( {\lambda \left( {D^2 u_1 } \right)} \right) + a_1 \left( {\left| x \right|} \right)\left| {\nabla u_1 } \right|^{k_1 } = p_1 \left( {\left| x \right|} \right)f_1 \left( {u_2 } \right)} & {for x \in \mathbb{R}^N ,} \\{S_{k_2 } \left( {\lambda \left( {D^2 u_2 } \right)} \right) + a_2 \left( {\left| x \right|} \right)\left| {\nabla u_2 } \right|^{k_2 } = p_2 \left( {\left| x \right|} \right)f_2 \left( {u_1 } \right)} & {for x \in \mathbb{R}^N .} \\\end{array} } \right.$$

Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the k _i -Hessian operator, a ₁, p ₁, f ₁, a ₂, p ₂ and f ₂ are continuous functions.

     

The emission spectrum of IBr excited in the presence of Argon

Wavelengths and wavenumbers of the band heads in the region 3915–3540 Å are recorded as obtained from the measurements of the plates taken on a first order 21-feet grating spectrograph. Earlier workers recently reported 40 bands of this system covering the region 3900–3800 Å. All the bands of this system obtained in the present experiments are analysed as involving the³ Π (1) state for lower state. The constants for the lower state are such that they represent well the ΔG (v+1/2) values obtained in the present experiments fromv=0 tov=26 as well as those obtained by Brown fromv=9 tov=43. The vibrational constants of the two states involved are:

$$\begin{gathered} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } } \\ {137 \cdot 8 cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } x_e ^{\prime \prime } } \\ {0 \cdot 571 cm.^{ - 1} } \\ \end{array} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } y_e ^{\prime \prime } } \\ { - 0 \cdot 1156 cm.^{ - 1} } \\ \end{array} \begin{array}{*{20}c} {\omega _e z_e ^{\prime \prime } } \\ {3 \cdot 09 \times 10^{ - 3} cm.^{ - 1} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\omega _e ^{\prime \prime } t_e ^{\prime \prime } } \\ { - 2 \cdot 5 \times 10^{ - 5} cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^\prime } \\ {90 \cdot 1 cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^\prime x_e ^\prime } \\ {0 \cdot 15 cm.^{ - 1} } \\ \end{array} \hfill \\ \end{gathered} $$     

Standard monomials for <Emphasis Type="Italic">q</Emphasis>-uniform families and a conjecture of Babai and Frankl

Let n, k, α be integers, n, α>0, p be a prime and q=p ^α. Consider the complete q-uniform family

$\mathcal{F}\left( {k,q} \right) = \left\{ {K \subseteq \left[ n \right]:\left| K \right| \equiv k(mod q)} \right\}$

We study certain inclusion matrices attached to F(k,q) over the field\(\mathbb{F}_p \). We show that if l≤q?1 and 2l≤n then

$rank_{\mathbb{F}_p } I(\mathcal{F}(k,q),\left( {\begin{array}{*{20}c} {\left[ n \right]} \\ { \leqslant \ell } \\ \end{array} } \right)) \leqslant \left( {\begin{array}{*{20}c} n \\ \ell \\ \end{array} } \right)$

This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.     

A positive solution of a nonlinear Schrdinger system with nonconstant potentials

In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u~3+ β(x)v~2u, x ∈R~N,-?v + V_2(x)v = μ_2(x)v~3+ β(x)u~2v, x ∈R~N,u, v ∈H~1(R~N),where N = 1, 2, 3; V_1(x) and V_2(x) are positive and continuous, but may not be well-shaped; and μ_1(x), μ_2(x)and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), μ_i(x)(i = 1, 2) and β(x) satisfy some additional conditions.     

Energy Scattering of a Generalized Davey–Stewartson System in Three Dimension

In this paper, we study the global scattering result of the solution for the generalized Davey–Stewartson system(i?_tu + Δu = |u|~2u + uv_(x1),(t, x) ∈ R × R~3,-Δv =(|u|~2)_(x1).)The main difficulties are the failure of the interaction Morawetz estimate and the asymmetrical structure of nonlinearity(in particular, the nonlinearity is non-local). To compensate, we utilize the strategy derived from concentration-compactness idea, which was first introduced by Kenig and Merle [Invent.Math., 166, 645–675(2006)].