For a polynomial P(z) of degree n having no zeros in |z| < 1, it was recently proved in [9] that
$$\left| {{z^s}{P^{\left( s \right)}}\left( z \right) + \beta \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{{{2^s}}}P\left( z \right)} \right| \leqslant \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{2}\left( {\left| {1 + \frac{\beta }{{{2^s}}}} \right| + \left| {\frac{\beta }{{{2^s}}}} \right|} \right)\mathop {\max }\limits_{\left| z \right| = 1} \left| {P\left( z \right)} \right|$$
for every β ∈ C with |β| ≤ 1, 1 ≤ s ≤ n and |z| = 1. In this paper, we obtain the Lp mean extension of the above and other related results for the sth derivative of polynomials.
Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the ki-Hessian operator, a1, p1, f1, a2, p2 and f2 are continuous functions.
. As applications, we obtain several sharp q-analysis inequalities of the classical positive integral operators, including the Hardy operator and its adjoint operator, the Hilbert operator, and the Hardy-Littlewood-Pólya operator.
Let (Fn)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.
Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let Tk(V) denote the set of all 2knk possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in Tk(V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log2n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that
Let {Xn}n?≥?1 be a sequence of strictly stationary m-dependent random variableswith EX1 = 0 and \( \mathrm{E}{X}_1^2<\infty \), and let (bn) be an increasing sequence of positive numbers such that bn?↑?∞ and \( {b}_n/\sqrt{n}\downarrow 0\kern0.5em \mathrm{as}\kern0.5em n\to \infty \). We establish a moderate deviation principle of \( {\left({b}_n\sqrt{n}\right)}^{-1}{\sum}_{i=1}^n{X}_i \) under the condition
which is weaker than the classical exponential integrability condition. The results in the present paper weaken the assumptions of Chen [5] and extend partially the results of Eichelsbacher and Löwe [10]. 相似文献
In this paper we consider the random r-uniform r-partite hypergraph model H(n1, n2, ···, nr; n, p) which consists of all the r-uniform r-partite hypergraphs with vertex partition {V1, V2, ···, Vr} where |Vi| = ni = ni(n) (1 ≤ i ≤ r) are positive integer-valued functions on n with n1 +n2 +···+nr = n, and each r-subset containing exactly one element in Vi (1 ≤ i ≤ r) is chosen to be a hyperedge of Hp ∈ H (n1, n2, ···, nr; n, p) with probability p = p(n), all choices being independent. Let
$${\Delta _{{V_1}}} = {\Delta _{{V_1}}}\left( H \right)$$
and
$${\delta _{{V_1}}} = {\delta _{{V_1}}}\left( H \right)$$
be the maximum and minimum degree of vertices in V1 of H, respectively;
$${X_{d,{V_1}}} = {X_{d,{V_1}}}\left( H \right),{Y_{d,{V_1}}} = {Y_{d,{V_1}}}\left( H \right)$$
,
$${Z_{d,{V_1}}} = {Z_{d,{V_1}}}\left( H \right)and{Z_{c,d,{V_1}}} = {Z_{c,d,{V_1}}}\left( H \right)$$
be the number of vertices in V1 of H with degree d, at least d, at most d, and between c and d, respectively. In this paper we obtain that in the space H(n1, n2, ···, nr; n, p),
all have asymptotically Poisson distributions. We also answer the following two questions. What is the range of p that there exists a function D(n) such that in the space H(n1, n2, ···, nr; n, p),
A graph G is vertex pancyclic if for each vertex \({v \in V(G)}\) , and for each integer k with 3 ≤ k ≤ |V(G)|, G has a k-cycle Ck such that \({v \in V(C_k)}\) . Let s ≥ 0 be an integer. If the removal of at most s vertices in G results in a vertex pancyclic graph, we say G is an s-vertex pancyclic graph. Let G be a simple connected graph that is not a path, cycle or K1,3. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K3}, where a divalent path in G is a path whose interval vertices have degree two in G. The s-vertex pancyclic index of G, written vps(G), is the least nonnegative integer m such that Lm(G) is s-vertex pancyclic. We show that for a given integer s ≥ 0,
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. 相似文献
An edge-coloring of a graph G is an assignment of colors to all the edges of G. A gc-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a gc-coloring with k colors is called the gc-chromatic index of G and denoted by \(\chi\prime_{g_{c}}\)(G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\)(G) = δg(G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\). 相似文献
Let C(M) be the space of all continuous functions on M? ?. We consider the multiplication operator T: C(M) → C(M) defined by Tf(z) = zf(z) and the torus
For integers m > r ≥ 0, Brietzke (2008) defined the (m, r)-central coefficients of an infinite lower triangular matrix G = (d, h) = (dn,k)n,k∈N as dmn+r,(m?1)n+r, with n = 0, 1, 2,..., and the (m, r)-central coefficient triangle of G as
It is known that the (m, r)-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array G = (d, h) with h(0) = 0 and d(0), h′(0) ≠ 0, we obtain the generating function of its (m, r)-central coefficients and give an explicit representation for the (m, r)-central Riordan array G(m,r) in terms of the Riordan array G. Meanwhile, the algebraic structures of the (m, r)-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of m and r. As applications, we determine the (m, r)-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach.
In this paper we give a new alternative proof of the local higher integrability in Orlicz spaces of the gradient for weak solutions of quasilinear parabolic equations of p-Laplacian type
for any p > 0. Moreover, we point out that our results are homogeneousregularity estimates in Orlicz spaces and improve the known results for such equations by using some new techniques. Actually, our results can be extended to the global estimates and cover a more general class of degenerate/singular parabolic problems of p-Laplacian type.
where γi>?1, δ0 is Dirac measure at 0, and the coefficients aij form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result:
$\sum_{j=1}^n a_{ij}\int_{\mathbb{R}^2}e^{u_j} dx = 4\pi(2+\gamma _i+\gamma_{n+1-i}), \quad\forall\;1\leq i \leq n.$
This generalizes the classification result by Jost and Wang for γi=0, \(\forall\;1\leq i\leq n\). (ii) We prove that if γi+γi+1+?+γj?? for all 1≤i≤j≤n, then any solution ui is radially symmetric w.r.t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system.
A finite p-group P is called resistant if, for any finite group G having P as a Sylow p-group, the normalizer NG(P) controls p-fusion in G. Let P be a central extension as
We investigate the nonlinear Schrödinger equation iut+Δu+|u|p?1u = 0with 1+ 4/N < p < 1+ 4/N?2 (when N = 1, 2, 1 + 4/N < p < ∞) in energy space H1 and study the divergent property of infinite-variance and nonradial solutions. If \(M{\left( u \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( u \right) \prec M{\left( Q \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( Q \right)\) and \(\left\| {{u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}\left\| {\nabla {u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}{\left\| {\nabla Q} \right\|_2}\), then either u(t) blows up in finite forward time or u(t) exists globally for positive time and there exists a time sequence tn → +∞ such that \({\left\| {\nabla u\left( {{t_n}} \right)} \right\|_2} \to + \infty \). Here Q is the ground state solution of ?(1?sc)Q+ΔQ+Qp?1Q = 0. A similar result holds for negative time. This extend the result of the 3D cubic Schrödinger equation obtained by Holmer to the general mass-supercritical and energy-subcritical case. 相似文献
Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A gc-coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The gc-chromatic index of G, denoted by χ′gc (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the gc-chromatic index equal to δg(G) or δg(G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′gc (G) = δg(G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011. 相似文献
Let G be a graph and k ≥ 2 a positive integer. Let h: E(G) → [0, 1] be a function. If \(\sum\limits_{e \mathrel\backepsilon x} {h(e) = k} \) holds for each x ∈ V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh = {e ∈ E(G): h(e) > 0}. A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G ? I has a fractional k-factor for every independent set I of G. In this paper, we prove that if n ≥ 9k ? 14 and for any subset X ? V (G) we have
Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ?p, β + 1 (Γ) continuously onto \(\mathcal{N}_{p,\beta } (\Gamma )\). The norm of a function in ?p, β (Γ) is defined as
, where q± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ?p, β + 1 (Γ) onto ?p, β(Γ), where ?p, β(Γ) is the direct sum of \(\mathcal{N}_{p,\beta }^ + (\Gamma )\) (Γ) and the space
(Γ) of functions on Γ of the form p(z) = Σk = 0mt(k)Rezk with the parameter m = [μ ? β ? p?1]. The operator I ? 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I ? 2W continuously maps the space ?p, β + 1 × ?p, β + 1(Γ) to the space \(\mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma )\).
On a contour with an inward peak, the obtained representation of the operator I ? 2W and theorems on the boundedness of auxiliary integral operators imply that the images of vector-valued functions from ?p, β + 1 × ?p, β + 1(Γ) have components representable as sums of functions from the spaces \(\mathcal{N}_{p,\beta }^ - (\Gamma )\)(Γ) and ?p, β(Γ). 相似文献