Sharp Upper Bound Research Articles

SPECTRAL RADIUS AND AVERAGE 2-DEGREE SEQUENCE OF A GRAPH

In a simple connected graph, the average 2-degree of a vertex is the average degree of its neighbors. With the average 2-degree sequence and the maximum degree ratio of adjacent vertices, we present a sharp upper bound of the spectral radius of the adjacency matrix of a graph, which improves a result in [Y. H. Chen, R. Y. Pan and X. D. Zhang, Two sharp upper bounds for the signless Laplacian spectral radius of graphs, Discrete Math. Algorithms Appl.3(2) (2011) 185–191].

On the Mean-Width of Isotropic Convex Bodies and their Associated Lp-Centroid Bodies

For any origin-symmetric convex body $K$ in $\mathbb{R}^n$ in isotropic position, we obtain the bound: \[ M^*(K) \leq C \sqrt{n} \log(n)^2 L_K ~, \] where $M^*(K)$ denotes (half) the mean-width of $K$, $L_K$ is the isotropic constant of $K$, and $C>0$ is a universal constant. This improves the previous best-known estimate $M^*(K) \leq C n^{3/4} L_K$. Up to the power of the $\log(n)$ term and the $L_K$ one, the improved bound is best possible, and implies that the isotropic position is (up to the $L_K$ term) an almost $2$-regular $M$-position. The bound extends to any arbitrary position, depending on a certain weighted average of the eigenvalues of the covariance matrix. Furthermore, the bound applies to the mean-width of $L_p$-centroid bodies, extending a sharp upper bound of Paouris for $1 \leq p \leq \sqrt{n}$ to an almost-sharp bound for an arbitrary $p \geq \sqrt{n}$. The question of whether it is possible to remove the $L_K$ term from the new bound is essentially equivalent to the Slicing Problem, to within logarithmic factors in $n$.

Life-Span of Solutions to Critical Semilinear Wave Equations

The final open part of the famous Strauss conjecture on semilinear wave equations of the form □u = |u| p , i.e., blow-up theorem for the critical case in high dimensions was solved by Yordanov and Zhang [21], or Zhou [25] independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. Recently, Takamura and Wakasa [18] have obtained the sharp upper bound of the lifespan of the solution to the critical semilinear wave equations, and their method is based on the method in Yordanov and Zhang [21]. In this paper, we give a much simple proof of the result of Takamura and Wakasa [18] by using the method in Zhou [25] for space dimensions n ≥ 2.

Second Hankel Determinant for a Subclass of Alpha Convex Functions

In this paper a sharp upper bound of second Hankel determinant 2 2 4 3 a a − a for the functions belonging to the class M (A, B) α of alpha convex functions is established. The class of alpha convex functions is an extended version of the classes of starlike functions and convex functions. By giving the particular values to alpha, it is easy to obtain the results of starlike and convex functions. The class discussed in this paper is an extended version of the class of alpha convex functions. By giving the particular values to A and B, the result of alpha convex functions can be easily obtained.

Galois points for a normal hypersurface

We study Galois points for a hypersurface $X$ with $\dim {\rm Sing}(X) \le \dim X-2$. The purpose of this article is to determine the set $\Delta(X)$ of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number of Galois points in terms of $\dim X$ and $\dim {\rm Sing}(X)$ if $\Delta(X)$ is a finite set, and prove that $X$ is a cone if $\Delta(X)$ is infinite. To achieve our purpose, we need a certain hyperplane section theorem on Galois point. We prove this theorem in arbitrary characteristic. On the other hand, the hyperplane section theorem has other important applications: For example, we can classify the Galois group induced from a Galois point in arbitrary characteristic and determine the distribution of Galois points for a Fermat hypersurface of degree $p^e+1$ in characteristic $p>0$.

Hankel Determinant for -Valent Alpha-Convex Functions

The objective of the present paper is to obtain the sharp upper bound of for p-valent α-convex functions of the form in the unit disc .

Sharp upper bound for the first eigenvalue

Let \(M\) be a closed hypersurface in a noncompact rank-1 symmetric space \((\overline{\mathbb{M }}, ds^2)\) with \(-4 \le K_{\overline{\mathbb{M }}} \le -1,\) or in a complete, simply connected Riemannian manifold \(\mathbb M \) such that \(0 \le K_\mathbb{M } \le \delta ^2\) or \(K_\mathbb{M } \le k\) where \(k = -\delta ^2\) or 0. In this paper we give sharp upperbounds for the first eigenvalue of laplacian of \(M\).

Bifurcation of isolated closed orbits from degenerated singularity in $\mathbb{R}^{3}$

In this paper, we study the bifurcation of isolated closed orbits from degenerated singularity of $3$-dimensional polynomial system $dx/dt = Q(x)$. For some types of $Q(x)$, we get the lower bound for the number of these isolated closed orbits. In particular cases, an explicit (sometimes sharp) upper bound is obtained. Using these results, we investigate degenerated Hopf bifurcation and give a sufficient condition for the existence of isolated closed orbits. Also we show that the $3$ species model of degree $3$ admits $2$ isolated closed orbits bifurcating from origin.

Wavelet Density Estimation and Statistical Evidences Role for a GARCH Model in the Weighted Distribution

We consider n observations from the GARCH-type model: Z = UY, where U and Y are independent random variables. We aim to estimate density function Y where Y have a weighted distribution. We determine a sharp upper bound of the associated mean integrated square error. We also make use of the measure of expected true evidence, so as to determine when model leads to a crisis and causes data to be lost.

Sharp Upper Bounds for the Laplacian Spectral Radius of Graphs

The spectrum of the Laplacian matrix of a network plays a key role in a wide range of dynamical problems associated with the network, from transient stability analysis of power network to distributed control of formations. LetG=(V,E)be a simple connected graph onnvertices and letμ(G)be the largest Laplacian eigenvalue (i.e., the spectral radius) ofG. In this paper, by using the Cauchy-Schwarz inequality, we show that the upper bounds for the Laplacian spectral radius ofG.

Some inequalities on the skew-spectral radii of oriented graphs

Let G be a simple graph and be an oriented graph obtained from G by assigning a direction to each edge of G. The adjacency matrix of G is and the skew-adjacency matrix of is . The adjacency spectral radius of G and the skew-spectral radius of are defined as the spectral radius of and respectively. In this paper, we firstly establish a relation between and . Also, we give some results on the skew-spectral radii of and its oriented subgraphs. As an application of these results, we obtain a sharp upper bound of the skew-spectral radius of an oriented unicyclic graph. MSC:05C50, 15A18.

Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems

In [1] and [2] upperbounds have been given for the number of large amplitude limit cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. In the current paper we improve the upperbounds from [1] and [2] by one unity, obtaining sharp results. We therefore introduce the 'method of cloning variables' that might be useful in other cyclicity problems.

Polynomial algorithm for sharp upper bound of rainbow connection number of maximal outerplanar graphs

For a finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G , there is a rainbow path connecting them. Rainbow connection number, r c ( G ) , of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2011) [5] proved that computing r c ( G ) is NP-hard and deciding if r c ( G ) = 2 is NP-complete. When edges of G are colored with fixed number k of colors, Kratochvil [6] proposed a question: what is the complexity of deciding whether G is rainbow connected? is this an FPT problem? In this paper, we prove that any maximal outerplanar graph is k rainbow connected for suitably large k and can be given a rainbow coloring in polynomial time.

Adaptive sequential estimation for ergodic diffusion processes in quadratic metric

An adaptive nonparametric procedure is constructed for estimating the unknown drift coefficient in ergodic diffusion processes. A sharp non-asymptotic upper bound (an oracle inequality) is obtained for a quadratic risk. Furthermore, an asymptotic lower bound for the minimax quadratic risk is found that equals to the Pinsker constant. Asymptotic efficiency is proved, that is, the asymptotic quadratic risk of the constructed estimator coincides with this constant.

The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

The final open part of Straussʼ conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang (2006) [18], or Zhou (2007) [21] independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou (1995) [10], the lifespan T ( ε ) of solutions of u t t − Δ u = u 2 in R 4 × [ 0 , ∞ ) with the initial data u ( x , 0 ) = ε f ( x ) , u t ( x , 0 ) = ε g ( x ) of a small parameter ε > 0 , compactly supported smooth functions f and g, has an estimate exp ( c ε − 2 ) ⩽ T ( ε ) ⩽ exp ( C ε − 2 ) , where c and C are positive constants depending only on f and g. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations.

On the (Laplacian) spectral radius of weighted trees with fixed matching number q and a positive weight set

Denote by T ( n , q , w 1 , w 2 , … , w n - 1 ) the set of n-vertex weighted trees with matching number q and fixed positive weight set W n - 1 = { w 1 , w 2 , … , w n - 1 } , where w 1 ⩾ w 2 ⩾ ⋯ ⩾ w n - 1 > 0 . Tan [S.W. Tan, On the sharp upper bound of spectral radius of weighted trees, J. Math. Res. Exposition 29 (2009) 293–301] determined the weighted tree in T ( n , q , w 1 , w 2 , … , w n - 1 ) with the largest adjacent spectral radius , whereas in [S.W. Tan, On the Laplacian spectral radius of weighted trees with a positive weight set, Discrete Math. 310 (2010) 1026–1036] Tan determined the weighted tree in T ( n , q , w 1 , w 2 , … , w n - 1 ) with the largest Laplacian spectral radius. In this paper, we use a unified approach to identify the unique weighted tree in T ( n , q , w 1 , w 2 , … , w n - 1 ) with the largest adjacent spectral radius and largest Laplacian spectral radius, respectively.

Adaptive Wavelet Estimation of a Biased Density for Strongly Mixing Sequences

The estimation of a biased density for exponentially strongly mixing sequences is investigated. We construct a new adaptive wavelet estimator based on a hard thresholding rule. We determine a sharp upper bound of the associated mean integrated square error for a wide class of functions.

Primitive zero-symmetric sign pattern matrices with the maximum base

In [B. Cheng, B. Liu, The base sets of primitive zero-symmetric sign pattern matrices, Linear Algebra Appl. 428 (2008) 715–731], Cheng and Liu studied the bases of primitive zero-symmetric sign pattern matrices. The sharp upper bound of the bases was obtained. In this paper, we characterize the sign pattern matrices with the sharp bound.

On the eccentric connectivity index of a graph

If G is a connected graph with vertex set V , then the eccentric connectivity index of G , ξ C ( G ) , is defined as ∑ v ∈ V deg ( v ) ec ( v ) where deg ( v ) is the degree of a vertex v and ec ( v ) is its eccentricity. We obtain an exact lower bound on ξ C ( G ) in terms of order, and show that this bound is sharp. An asymptotically sharp upper bound is also derived. In addition, for trees of given order, when the diameter is also prescribed, precise upper and lower bounds are provided.

Isosceles Sets

In 1946, Paul Erdős posed a problem of determining the largest possible cardinality of an isosceles set, i.e., a set of points in plane or in space, any three of which form an isosceles triangle. Such a question can be asked for any metric space, and an upper bound ${n+2\choose 2}$ for the Euclidean space ${\Bbb E}^{n}$ was found by Blokhuis. This upper bound is known to be sharp for $n=1$, 2, 6, and 8. We will consider Erdős' question for the binary Hamming space $H_{n}$ and obtain the following upper bounds on the cardinality of an isosceles subset $S$ of $H_{n}$: if there are at most two distinct nonzero distances between points of $S$, then $|S|\leq{n+1\choose 2}+1$; if, furthermore, $n\geq 4$, $n\ne 6$, and, as a set of vertices of the $n$-cube, $S$ is contained in a hyperplane, then $|S|\leq{n\choose 2}$; if there are more than two distinct nonzero distances between points of $S$, then $|S|\leq{n\choose 2}+1$. The first bound is sharp if and only if $n=2$ or $n=5$; the other two bounds are sharp for all relevant values of $n$, except the third bound for $n=6$, when the sharp upper bound is 12. We also give the exact answer to the Erdős problem for ${\Bbb E}^{n}$ with $n\leq 7$ and describe all isosceles sets of the largest cardinality in these dimensions.