Special Kähler–Ricci potentials and Ricci solitons

On a manifold of dimension at least six, let (g, τ) be a pair consisting of a Kähler metric g which is locally Kähler irreducible, and a nonconstant smooth function τ. Off the zero set of τ, if the metric \({\widehat{g}=g/\tau^{2}}\) is a gradient Ricci soliton which has soliton function 1/τ, we show that \({\widehat{g}}\) is Kähler with respect to another complex structure, and locally of a type first described by Koiso, and also Cao. Moreover, τ is a special Kähler–Ricci potential, a notion defined in earlier works of Derdzinski and Maschler. The result extends to dimension four with additional assumptions. We also discuss a Ricci–Hessian equation, which is a generalization of the soliton equation, and observe that the set of pairs (g, τ) satisfying a Ricci–Hessian equation is invariant, in a suitable sense, under the map \({(g,\tau) \rightarrow (\widehat{g},1/\tau)}\) .     

Parametrized Garfunkel–Bankoff inequality and improved Finsler–Hadwiger inequality

This paper deals with a new generalization of the Garfunkel–Bankoff inequality by introducing a parameter, which relaxes the conditions of the Garfunkel–Bankoff inequality. As applications, an improved version of the Finsler–Hadwiger inequality is obtained.     

An improvement for the Trudinger–Moser inequality and applications

In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W^1,n(Rⁿ)$W^{1,n}({\mathbb{R}}^n)$, n?2$n?2$, into the Orlicz space _LΦα$L_{{\Phi }_{\alpha }}$ determined by the Young function _Φα(s)${\Phi }_{\alpha }(s)$ behaving like e^{α|s|n/(n−1)}−1$e^{\alpha {|s|}^{n/(n-1)}}-1$ as |s|→+∞$|s|\to +\infty$. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rⁿ${\mathbb{R}}^n$.     

Moment inequality and Hölder inequality for BSDEs

Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper proves that if g is positively homogeneous in （y, z） and is decreasing in y, then the Moment inequality for BSDEs with generator g holds in general, and if g is positively homogeneous and sub-additive in （y, z）, then the HSlder inequality and Minkowski inequality for BSDEs with generator g hold in general.     

Generalized Moser–Trudinger inequality for unbounded domains and its application

We give a version of the Moser–Trudinger inequality for Orlicz–Sobolev spaces embedded into exponential and multiple exponential spaces on unbounded domains in ${\mathbb R^n, n \geq 2}$ . Applying this result and the Mountain Pass Theorem we study the existence of non-trivial weak solutions to the problem $$\begin{array}{ll}u \in W^1 L^{\Phi}(\mathbb R^n)\quad{\rm and}\\\quad -{\rm div} \left(\Phi ' (|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)+V(x)\Phi'(|u|)\frac{u}{|u|} =f(x,u)\quad{\rm in}\, \mathbb R^n,\end{array}$$ where Φ is a Young function such that the space ${W^1 L^{\Phi}(\mathbb R^n)}$ is embedded into an Orlicz space of the exponential or multiple exponential type, the nonlinearity f(x, t) has the corresponding critical growth and V(x) is a continuous potential.     

A Hardy–Ramanujan-type inequality for shifted primes and sifted sets

Lithuanian Mathematical Journal - We establish an analog of the Hardy–Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we...     

On soliton solutions for Boussinesq–Burgers equations

The periodic wave solutions for Boussinesq–Burgers equations are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.     

Exact periodic kink-wave and degenerative soliton solutions for potential Kadomtsev–Petviashvili equation

Exact periodic kink-wave solution, periodic soliton and doubly periodic solutions for the potential Kadomtsev–Petviashvii (PKP) equation are obtained using homoclinic test technique and extended homoclinic test technique, respectively. It is investigated that periodic soliton is degenerated into doubly periodic wave varying with direction of wave propagation.     

Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

We prove an optimal relative isoperimetric inequality

The Cauchy–Schwarz inequality and the induced metrics on real vector spaces mainly on the real line

It is very well known that the Cauchy–Schwarz inequality is an important property shared by all inner product spaces and the inner product induces a norm on the space. A proof of the Cauchy–Schwarz inequality for real inner product spaces exists, which does not employ the homogeneous property of the inner product. However, it is shown that a real vector space with a product satisfying properties of an inner product except the homogeneous property induces a metric but not a norm. It is remarkable to see that the metric induced on the real line by such a product has highly contrasting properties relative to the absolute value metric. In particular, such a product on the real line is given so that the induced metric is not complete and the set of rational numbers is not dense in the real line.     

Reverse Brascamp–Lieb inequality and the dual Bollobás–Thomason inequality

Archiv der Mathematik - We prove that if $$f:{mathbb {R}}^nrightarrow [0,infty )$$ is an integrable log-concave function with $$f(0)=1$$ and $$F_1,ldots ,F_r$$ are linear subspaces of...     

Hermite–Hadamard inequality for convex stochastic processes

In this note we extend the classical Hermite–Hadamard inequality to convex stochastic processes.     

Rigidity Theorems for Einstein–Thorpe Metrics Dedicated to my parents

We will prove that every Einstein–Thorpe metric on T ⁸ must be flat and that on compact oriented hyperbolic manifolds of dimension 8, every Einstein–Thorpe metric is a hyperbolic metric up to rescalings and diffeomorphisms.     

The Brunn–Minkowski–Firey inequality for nonconvex sets

The definition of Minkowski–Firey Lp$L_p$-combinations is extended from convex bodies to arbitrary subsets of Euclidean space. The Brunn–Minkowski–Firey inequality (along with its equality conditions), previously established only for convex bodies, is shown to hold for compact sets.