Slice knots with distinct Ozsváth-Szabó and Rasmussen invariants

As proved by Hedden and Ording, there exist knots for which the Ozsváth-Szabó and Rasmussen smooth concordance invariants, and , differ. The Hedden-Ording examples have nontrivial Alexander polynomials and are not topologically slice. It is shown in this note that a simple manipulation of the Hedden-Ording examples yields a topologically slice Alexander polynomial one knot for which and differ. Manolescu and Owens have previously found a concordance invariant that is independent of both and on knots of polynomial one, and as a consequence have shown that the smooth concordance group of topologically slice knots contains a summand isomorphic to . It thus follows quickly from the observation in this note that this concordance group contains a summand isomorphic to .

     

Clasp-pass move and Vassiliev invariants of type three for knots

Recently it has been proved that if and only if two knots and have the same value for the Vassiliev invariant of type two, then can be deformed into by a finite sequence of clasp-pass moves. In this paper, we determine the difference of the values of the Vassiliev invariant of type three between two knots which can be deformed into each other by a clasp-pass move.

     

The best possibility of the grand Furuta inequality

Let be invertible bounded linear operators on a Hilbert space satisfying , and let be real numbers satisfying Furuta showed that if , then . This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality
and the Ando-Hiai inequality ( ).

In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if , then there exist invertible matrices with which do not satisfy .

     

The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces

Let be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature . If is a compactly supported function of bounded variation on , then satisfies the Sobolev inequality

Conversely, letting be the characteristic function of a domain recovers the sharp form of the isoperimetric inequality for simply connected surfaces with . Therefore this is the Sobolev inequality ``equivalent' to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces.

Under the same assumptions on , if is a closed curve and is the winding number of about , then the Sobolev inequality implies

which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature .

     

The best constant of the Moser-Trudinger inequality on

We consider the best constant of the Moser-Trudinger inequality on under a certain orthogonality condition. Applying Moser's calculation, we construct a counterexample to the sharper inequality with the condition.

     

Equivalence of the wielandt inequality and the kantorovich inequality

This note shows the equivalence of two well-known inequalities: the Wielandt inequality and the Kantorovich inequality.     

Some remarks on the Gronwall integral inequality

In this paper, a generalized Volterra-type integral inequality is developed. On the basis of this inequality, the effect of fractional-order $\omega$ on the application of the integer-order Gronwall integral inequality (IOGII) is discussed. Specially speaking, the IOGII cannot be directly used to reckon the solution of integral inequality with the order . It seems that both the IOGII and the generalized Volterra-type integral inequality can be applied to estimate the solution of integral inequality with the order $\omega \ge 1$, and results are consistent, but this is just a coincidence.     

Harnack inequality for the linearized parabolic Monge-Ampère equation

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation

on parabolic sections associated with , under the assumption that the Monge-Ampère measure generated by satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group , where denotes the group of all invertible affine transformations on .

     

Heisenberg群中一类非凸区域上的Hardy不等式

通过构造向量函数,得到了Heisenberg群中一类非凸区域上的Hardy不等式,从而推广了以前的相关结论.     

The intersection of three spheres in a sphere and a new application of the Sato-Levine invariant

Take transverse immersions such that (1) is an embedding, (2) and is connected, and (3) . Then we obtain three surface-links = (, ) in , where =(1,2,3), (2,3,1), (3,1,2). We prove that, we have the equality where is the Sato-Levine invariant of , if all are semi-boundary links.

     

A CR Poincaré inequality on the complex sphere

The main purpose of this paper is to prove a CR Poincaré inequality with sharp exponent on the sphere in complex space. We use the complex tangential gradient on the sphere instead of the usual Laplace-Beltrami gradient on the sphere.     

An application of the convolution inequality for the Fisher information

A characterization of the normal distribution by a statistical independence on a linear transformation of two mutually independent random variables is proved by using the convolution inequality for the Fisher information.     

On the Chebyshev type inequality for seminormed fuzzy integral

The Chebyshev type inequality for seminormed fuzzy integral is discussed. The main results of this paper generalize some previous results obtained by the authors. We also investigate the properties of semiconormed fuzzy integral, and a related inequality for this type of integral is obtained.     

The logarithmic HLS inequality for systems on compact manifolds

We prove an optimal logarithmic Hardy-Littlewood-Sobolev inequality for systems on compact m-dimensional Riemannian manifolds, for any m?2. We show that a special case of the inequality, involving only two functions, implies the general case by using an argument from the theory of linear programing.     

The best constant of Sobolev inequality on a bounded interval

The best constants Cm,j of Sobolev embedding of Hm(0,a) into Cj[0,a](0?j?m−1) are obtained. Especially, when a=∞, these constants can be represented in a closed form.     

Some extensions of the Kantorovich inequality and statistical applications

Kantorovich gave an upper bound to the product of two quadratic forms, (X′AX) (X′A⁻¹X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants |X′AX| |X′A⁻¹X| where X is n × k matrix such that X′X = I_k. In this paper we determine the bounds for the traces and determinants of matrices of the type X′AYY′A⁻¹X, X′B²X(X′BCX)⁻¹ X′C²X(X′BCX)⁻¹ where X and Y are n × k matrices such that X′X = Y′Y = I_k and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation.     

关于P-凸函数的积分型Jensen不等式

基于P-凸函数的函数凸性,研究了P-凸函数的Jensen型不等式的积分形式,通过定积分的定义计算,得到了P-凸函数的积分型Jensen不等式;利用P-凸函数的一个充要条件,建立了P-凸函数的积分型Jensen不等式的加权形式.     

The full Markov-Newman inequality for Müntz polynomials on positive intervals

For a function defined on an interval let

The principal result of this paper is the following Markov-type inequality for Müntz polynomials. Theorem. Let be an integer. Let be distinct real numbers. Let . Then

where the supremum is taken for all (the span is the linear span over ).

     

A Hilbert-type integral inequality on the fractal space

Based on the local fractional calculus, by using the methods of weight function, a Hilbert-type fractal integral inequality and its equivalent form are given. Their constant factors are proved being the best possible, and its applications are discussed briefly.