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145
Common hyperplane medians for random vectors
 Amer. Math. Mo
, 1988
"... is a cyclic group, we can choose a generator g for K *. Then, for x E D *, x E N * is equivalent to xgx1 E K and, for y E N *, we will have y E Kx if and only if xgx1 = ygy1 We let q = IN*/K *. Now D is a (left) vector space over K and, for a E K, Da is a Klinear operator on D. From the above, w ..."
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is a cyclic group, we can choose a generator g for K *. Then, for x E D *, x E N * is equivalent to xgx1 E K and, for y E N *, we will have y E Kx if and only if xgx1 = ygy1 We let q = IN*/K *. Now D is a (left) vector space over K and, for a E K, Da is a Klinear operator on D. From the above, we have DP =DPm=Da. Thus the minimal polynomial for Da is a divisor of xpmx and necessarily splits in K[X] into distinct first degree factors so that Da is diagonalizable. Applying this to Dg, for Ek = {x: [g, x] = kx}, we have Eo K and D = YEk, where the sum is direct and taken over all k E K with Ek # 0. Now, if x E D *, gx xg kx for some k E K is equivalent to requiring that x belong to N *. Moreover, y E Ek is equivalent to y E Kx. Then each Ek is a Ksubspace of dimension 1 and Ek * ig the coset K *x in N *. Hence, dimKD = q. From the structure of finite fields it follows readily that K is a Galois extension of Z. We can identify N *7K * with a subgroup of G(K/Z) and, if J is the fixed field for N */K *, a E J implies xax1 = a for all x E N*. Then Da is zero on each Ek so that Da = 0 and a E Z. Hence N *7K * G(K/Z), implying dimzK IN*/K*l = q. Combining the results above leads to dimzD = (dimKD)(dimzK) = q2 and dimZC(b) = q for all b E D, b 5 Z. If r = jZj then ID*I = 1 and IC(b)*1 = rq 1 for b 4 Z. Thus, if s is the number of conjugacy classes containing more than one element, the class equation applied to D * gives r 1 = (r 1) + s(rq 1/rq1), which implies r q(q 1) + + r q + 1 must be a divisor of r 1, giving the desired contradiction.
Hyperplanes and the Acquisition of Common Sense Reasoning
, 1993
"... The overall aim of the paper is to demonstrate that, from a machine learning point of view, connectionist networks are not black boxes. Trained networks contain rich and varied internal representations gleaned from training sets. Analysis of these representations can provide useful results concernin ..."
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concerning the generalizability of these networks to novel examples. More particularly in the case of common sense reasoning, hyperplane analysis can demonstrate the adaptive power of connectionist networks when presented with information concerning new entities and the ability of such networks to cope
Nonlinear BlackBox Modeling in System Identification: a Unified Overview
 Automatica
, 1995
"... A nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area with structures based on neural networks, radial basis networks, wavelet networks, hinging hyperplanes, ..."
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Cited by 225 (16 self)
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A nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area with structures based on neural networks, radial basis networks, wavelet networks, hinging hyperplanes
Weighted Grassmannians and stable hyperplane arrangements
, 2008
"... We give a common generalization of (1) Hassett’s weighted stable curves, and (2) HackingKeelTevelev’s stable hyperplane arrangements. ..."
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Cited by 4 (2 self)
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We give a common generalization of (1) Hassett’s weighted stable curves, and (2) HackingKeelTevelev’s stable hyperplane arrangements.
MEASURE FOR FAMILIES OF HYPERPLANES SYSTEMS IN THE AFFINE SPACE
"... ABSTRACT. We study the problem of the measurability of families of khyperplanes having a common fixed point and placed in A2n+1, with n ≥ 3. We prove that for k = 2n or k = 3 and 2n = 3t, for some t ∈ N, the family is not measurable. 1. ..."
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ABSTRACT. We study the problem of the measurability of families of khyperplanes having a common fixed point and placed in A2n+1, with n ≥ 3. We prove that for k = 2n or k = 3 and 2n = 3t, for some t ∈ N, the family is not measurable. 1.
Hinging hyperplane models for multiple predicted variables
 IN SSDBM, 2012
"... Modelbased learning for predicting continuous values involves building an explicit generalization of the training data. Simple linear regression and piecewise linear regression techniques are well suited for this task, because, unlike neural networks, they yield an interpretable model. The hinging ..."
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with r and the result is no longer being compact or interpretable. We propose a generalization of the hinging hyperplane approach for several predicted variables. The algorithm considers all predicted variables simultaneously. It enforces common hinges, while at the same time restoring the continuity
Using Tangent Hyperplanes to Direct Neural Training
"... The trend in the development and adoption of neural training regimes for supervised learning in feedforward neural networks has been to follow gradient approaches already successful in numerical analysis. However, weak direction can be seen to be common to the neural implementation of such approache ..."
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Cited by 2 (0 self)
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The trend in the development and adoption of neural training regimes for supervised learning in feedforward neural networks has been to follow gradient approaches already successful in numerical analysis. However, weak direction can be seen to be common to the neural implementation
BER analysis of Bayesian equalization using orthogonal hyperplanes
, 2003
"... Bayesian symbolbysymbol detection using a finite sequence observation space has been the subject of renewed research interest. The Bayesian transverse equalizer (BTE) and Bayesian decision feedback equalizer (BDFE) are two common Bayesian detectors. It is often difficult to evaluate the biterror ..."
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Bayesian symbolbysymbol detection using a finite sequence observation space has been the subject of renewed research interest. The Bayesian transverse equalizer (BTE) and Bayesian decision feedback equalizer (BDFE) are two common Bayesian detectors. It is often difficult to evaluate the bit
On the General Hyperplane Section of a Projective Curve
, 1998
"... . Here we study cohomological and geometrical properties of the general zerodimensional section of a nonreduced curve C ae P n and (in positive characteristic) of an integral variety. 0. Introduction The main actors in this paper will be zerodimensional subschemes of a projective space P r ..."
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. This paper consists of three parts. The link between the three parts is given by the common main actors. The content of the last part (section 6) is used in the second part for the proof of Theorems 5.1 and 5.4. In the first part of this paper (i.e. in the first three sections ) we study the general
Results 1  10
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145