In this short note, we show that the problem of computing recursive teaching dimension (RTD) for a concept class (given explicitly as input) requires nΩ(log⁡n)-time, assuming exponential time hypothesis (ETH). This matches running nO(log⁡n) brute-force algorithm problem.

We prove that certain instances of the iterated matrix multiplication (IMM) family of polynomials with N variables and degree n require NΩ( √ n) gates when expressed as a homogeneous depth-five ΣΠΣΠΣ arithmetic circuit with the bottom fan-in bounded by N1/2−ε. By a depth-reduction result of Tavenas, this size lower bound is optimal and can be achieved by the weaker class of homogeneous depth-fo...

The goal of this paper is the construction of a data-sparse approximation to the Schur complement on the interface corresponding to FEM and BEM approximations of an elliptic equation by domain decomposition. Using the hierarchical (H-matrix) formats we elaborate the approximate Schur complement inverse in an explicit form. The required cost O(NΓ log q NΓ ) is almost linear in NΓ – the number of...

We consider directed graphs where each edge is labeled with an integer weight and study the fundamental algorithmic question of computing the value of a cycle with minimum mean weight. Our contributions are twofold: (1) First we show that the algorithmic question is reducible in O(n2) time to the problem of a logarithmic number of min-plus matrix multiplications of n× n-matrices, where n is the...

We study the one-dimensional discrete quasi-periodic Schrödinger equation −ϕ(n + 1) − ϕ(n − 1) + λV (x + nω)ϕ(n) = Eϕ(n), n ∈ Z We show that for " typical " C 3 potential V , if the coupling constant λ is large, then for most frequencies ω, the Lyapunov exponent is positive for all energies E, and the corresponding eigenfunctions ϕ decay exponentially.

In the frame of a topological model of electromagnetism, it is shown that the Planck-Einstein law E = nω for the energy of monochromatic radiation can be formulated in a cubic cavity in such a way that the number of photons n is a topological invariant, equal to one fourth of the degree of a map between two orbifolds. PACS: 11.10; 03.50; 02.40; 41.10

ROS: Reactive Oxygen Species; RAS: Renin Angiotensin System; PKC-Δ: Protein Kinase C-Δ; NOX: NADPH Oxidase; BK Channels: Ca2+ Activated Potassium Channels; RAAS: Renin Angiotensin Aldosterone System; Caox: Calcium Oxalate; ACE2: Angiotensin-Converting Enzyme 2; Ang II: Angiotensin II; EMT: Epithelial To Mesenchymal Transition; AT1R :Ang II Type 1 Receptor; CKD: Chronic Kidney Disease; L-NAME: N...