A note on integral transforms

A note on the comparison between Laplace and Sumudu transforms

A note on $lambda$-Aluthge transforms of operators

Let $A=U|A|$ be the polar decomposition of an operator $A$ on a Hilbert space $mathscr{H}$ and $lambdain(0,1)$. The $lambda$-Aluthge transform of $A$ is defined by $tilde{A}_lambda:=|A|^lambda U|A|^{1-lambda}$. In this paper we show that emph{i}) when $mathscr{N}(|A|)=0$, $A$ is self-adjoint if and only if so is $tilde{A}_lambda$ for some $lambdaneq{1over2}$. Also $A$ is self adjoint if and onl...

A Note on Dichotomies for Metric Transforms

We show that for every nondecreasing concave function ω : [0,∞) → [0,∞) with ω(0) = 0, either every finite metric space embeds with distortion arbitrarily close to 1 into a metric space of the form (X,ω◦d) for some metric d on X, or there exists α = α(ω) > 0 and n0 = n0(ω) ∈ N such that for all n > n0, any embedding of {0, . . . , n} ⊆ R into a metric space of the form (X,ω ◦ d) incurs distorti...

a note on the comparison between laplace and sumudu transforms

Note on Integral Distances

A planar point set S is called an integral set if all the distances between the elements of S are integers. We prove that any integral set contains many collinear points or the minimum distance should be relatively large if |S| is large.