On the number of two-dimensional threshold functions

A two-dimensional threshold function of k-valued logic can be viewed as coloring of the points of a k x k square lattice into two colors such that there exists a straight line separating points of different colors. For the number of such functions only asymptotic bounds are known. We give an exact formula for the number of two-dimensional threshold functions and derive more accurate asymptotics.Comment: 17 pages, 2 figure

Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Let $m,n\ge 2$, $m\le n$. It is well-known that the number of (two-dimensional) threshold functions on an $m\times n$ rectangular grid is {eqnarray*} t(m,n)=\frac{6}{\pi^2}(mn)^2+O(m^2n\log{n})+O(mn^2\log{\log{n}})= \frac{6}{\pi^2}(mn)^2+O(mn^2\log{m}). {eqnarray*} We improve the error term by showing that $$ t(m,n)=\frac{6}{\pi^2}(mn)^2+O(mn^2). $

The computational power and complexity of discrete feedforward neural networks

The number of binary functions that can be defined on a set of L vectors in R^N equals 2^L . For L>N the total number of threshold functions in N-dimensional space grows polynomially (2^N(N-1))while the total number of Boolean functions, definable on N binary inputs, growsexponentially ( 2^2^2), and as N increases a percentage of threshold functions in relation to the total number of Boolean functions - goes to zero. This means that for the realization of a majority of tasks a neural network must possess at least two or three layers. The examples of small computational problems are arithmetic functions, like multiplication, division, addition, exponentiation or comparison and sorting. This article analyses some aspects of two- and more than two layers of threshold and Boolean circuits (feedforward neural nets), connected with their computational power and node, edge and weight complexity

Appendix A: “The Basic Postulates of Accounting”

Functional Magnetic Resonance Imaging (fMRI) is a relatively new imaging technique, first reported in 1992, which enables mapping of brain functions with high spatial resolution. Functionally active areas are distinguished by a small signal increase mediated by changes in local blood oxygenation in response to neural activity. The ability to non-invasively map brain function and the large number of MRI scanners quickly made the method very popular, and fMRI have had a huge impact on the study of brain function, both in healthy and diseased subjects. The most common clinical application of fMRI is pre-surgical mapping of brain functions in order to optimise surgical interventions. The clinical fMRI examination procedure can be divided into four integrated parts: (1) patient preparation, (2) image acquisition, (3) image analysis and (4) clinical decision. In this thesis, important aspects of all parts of the fMRI examination procedure are explored with the aim to provide recommendations and methods for prosperous clinical usage of the technique. The most important results of the thesis were: (I) administration of low doses of diazepam to reduce anxiety did not invalidate fMRI mapping results of primary motor and language areas, (II) the choice of visual stimuli equipment can have severe impact on the mapping of visual areas, (III) three-dimensional fMRI imaging sequences did not perform better than two-dimensional imaging sequences, (IV) adaptive spatial filtering can improve the fMRI data analysis, (V) clinical decisions should not be based on activation results from a single statistical threshold

Adaptive space-time model order reduction with dual-weighted residual (MORe DWR) error control for poroelasticity

In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.Comment: 33 pages, 9 figures, 3 table

Resistance Fluctuations in Randomly Diluted Networks

The resistance R(x,x’) between two connected terminals in a randomly diluted resistor network is studied on a d-dimensional hypercubic lattice at the percolation threshold pc. When each individual resistor has a small random component of resistance, R(x,x’) becomes a random variable with an associated probability distribution, which contains information on the distribution of currents in the individual resistors. The noise measured between the terminals may be characterized by the cumulants Mq(x,x’) of R(x,x’). When averaged over configurations of clusters, M¯q(x,x’)~‖x-x’‖ψ̃(q). We construct low-concentration series for the generalized resistive susceptibility, χ(q), associated with M¯q, from which the critical exponents ψ̃(q) are obtained. We prove that ψ̃(q) is a convex monotonically decreasing function of q, which has the special values ψ̃(0)=DB, ψ̃(1)=ζ̃R, and ψ̃(∞)=1/ν. (DB is the fractal dimension of the backbone, ζ̃R is the usual scaling exponent for the average resistance, and ν is the correlation-length exponent.) Using the convexity property and the accepted values of these three exponents, we construct two approximant functions for ψ(q)=ψ̃(q)ν, both of which agree with the series results for all q\u3e1 and with existing numerical simulations. These approximants enabled us to obtain explicit approximate forms for the multifractal functions α(q) and f(q) which, for a given q, characterize the scaling with size of the dominant value of the current and the number of bonds having this current. This scaling description fails for sufficiently large negative q, when the dominant (small) current decreases exponentially with size. In this case χ(q) diverges at a lower threshold p*(q), which vanishes as q→-∞