Let V and W be two real vector spaces and let &sim be a relation on both V and W. A linear function T : V → W is said to be a linear preserver (respectively strong linear preserver) of &sim if Tx &sim Ty whenever x &sim y (respectively Tx &sim Ty if and only if x &sim y). In this paper we characterize all linear functions T : M_{n,m} → M_{n,k} which preserve or strongly preserve multivariate an...

The aim of this paper is to solve a linear preserver problem on the function algebra C(X). We show that in case X is a first countable compact Hausdorff space, every linear bijection φ : C(X) → C(X) having the property that diam(φ(f)(X)) = diam(f(X)) (f ∈ C(X)) is of the form φ(f) = τ · f ◦ φ+ t(f)1 (f ∈ C(X)) where τ ∈ C, |τ | = 1, φ : X → X is a homeomorphism and t : C(X) → C is a linear func...

for $a,bin m_{nm},$ we say that $a$ is left matrix majorized (resp. left matrix submajorized) by $b$ and write $aprec_{ell}b$ (resp. $aprec_{ell s}b$), if $a=rb$ for some $ntimes n$ row stochastic (resp. row substochastic) matrix $r.$ moreover, we define the relation $sim_{ell s} $ on $m_{nm}$ as follows: $asim_{ell s} b$ if $aprec_{ell s} bprec_{ell s} a.$ this paper characterizes all linear p...

For nearly two decades, the amyloidhypothesis [22] has dominated the field of Alzheimer’s disease (AD) and during that time, massive efforts have focused on the role of amyloidin the pathogenesis of the disease. Given such focus, it is truly unfortunate that it is still unclear whether amyloidis either necessary or sufficient for the frank development of AD or whether amyloidcauses the associat...

<p><span data-preserver-spaces="true">The demands of 21st Century learning are now in the Industrial Revolution Era 4.0 with 4C characters. Educators must develop their creativity to improve quality so that students can be more creative learning. </span><em>Instrumental Analytical Chemistry</em><span data-preserver-spaces="true"> is a course studies modern in...

The famous shortest path tree lemma states that, for any node s in a graph G = (V,E), there is a subgraph on O(n) edges that preserves all distances between node pairs in the set {s}×V . A very basic question in distance sketching research, with applications to other problems in the field, is to categorize when else graphs admit sparse subgraphs that preserve distances between a set P of p node...