scholarly journals A nonlinear Lazarev–Lieb theorem: L2-orthogonality via motion planning

2020 ◽  
pp. 1-17
Author(s):  
Florian Frick ◽  
Matt Superdock
Keyword(s):  
Lower Bound ◽  
Motion Planning ◽  
Unit Circle ◽  
Smooth Function ◽  
Unit Interval ◽  
Inner Product ◽  
Lower Bounding ◽  
Integrable Functions ◽  
Equivariant Topology ◽  
Norm Bound

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to [Formula: see text] can be simultaneously annihilated in the [Formula: see text] inner product by a smooth function to the unit circle. Here, we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain [Formula: see text]-norm bound. Our proof uses a variety of motion planning algorithms that instead of contractibility yield a lower bound for the [Formula: see text]-coindex of a space.

scholarly journals Best Simultaneous Approximation in Orlicz Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
M. Khandaqji ◽  
Sh. Al-Sharif
Keyword(s):  
Banach Space ◽  
Simultaneous Approximation ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Unit Interval ◽  
Luxemburg Norm ◽  
Integrable Functions ◽  
Best Simultaneous Approximation ◽  
Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

CONSTRUCTIVE ESTIMATION OF APPROXIMATION FOR TRIGONOMETRIC NEURAL NETWORKS

2012 ◽  
Vol 10 (03) ◽  
pp. 1250021
Author(s):  
JIANJUN WANG ◽  
WEIHUA XU ◽  
BIN ZOU
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Lower Bound ◽  
Upper Bound ◽  
Intrinsic Property ◽  
Order Of Approximation ◽  
Integrable Functions ◽  
Hidden Layer ◽  
Hidden Neurons ◽  
Bound Estimation

For the three-layer artificial neural networks with trigonometric weights coefficients, the upper bound and lower bound of approximating 2π-periodic pth-order Lebesgue integrable functions [Formula: see text] are obtained in this paper. Theorems we obtained provide explicit equational representations of these approximating networks, the specification for their numbers of hidden-layer units, the lower bound estimation of approximation, and the essential order of approximation. The obtained results not only characterize the intrinsic property of approximation of neural networks, but also uncover the implicit relationship between the precision (speed) and the number of hidden neurons of neural networks.

scholarly journals The compression of a slant Hankel operator to H2

2003 ◽  
Vol 74 (88) ◽  
pp. 129-136
Author(s):  
Taddesse Zegeye ◽  
S.C. Arora
Keyword(s):  
Unit Circle ◽  
Complex Plane ◽  
Hankel Operator ◽  
Inner Product ◽  
Unilateral Shift

A slant Hankel operator K? with symbol ? in L?(T) (in short L?), where T is the unit circle on the complex plane, is an operator whose representing matrix M = (aij) is given by ai,j = (?,z-2i-j), where (?, ?) is the usual inner product in L2(T) (in short L2). The operator L? denotes the compression of K? to H2(T) (in short H2). We prove that an operator L on H2 is the compression of a slant Hankel operator to H2 if and only if U *L = LU2, where U is the unilateral shift. Moreover, we show that a hyponormal L? is necessarily normal and L? can not be an isometry.

Lower and upper bounds for the linear arrangement problem on interval graphs

2018 ◽  
Vol 52 (4-5) ◽  
pp. 1123-1145
Author(s):  
Alain Quilliot ◽  
Djamal Rebaine ◽  
Hélène Toussaint
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Feasible Solution ◽  
Interval Graph ◽  
Upper Bounds ◽  
Unit Interval ◽  
Lower And Upper Bounds ◽  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

We deal here with theLinear Arrangement Problem(LAP) onintervalgraphs, any interval graph being given here together with its representation as theintersectiongraph of some collection of intervals, and so with relatedprecedenceandinclusionrelations. We first propose a lower boundLB, which happens to be tight in the case ofunit intervalgraphs. Next, we introduce the restriction PCLAP of LAP which is obtained by requiring any feasible solution of LAP to be consistent with theprecedencerelation, and prove that PCLAP can be solved in polynomial time. Finally, we show both theoretically and experimentally that PCLAP solutions are a good approximation for LAP onintervalgraphs.

scholarly journals Biconcave-function characterisations of UMD and Hilbert spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Jinsik Mok Lee
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Complex Banach Space ◽  
Unit Interval ◽  
Martingale Differences ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Image Position ◽  
Umd Banach Spaces

Suppose that X is a real or complex Banach space with norm |·|. Then X is a Hilbert space if and only iffor all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and |x − Y| ≤ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function η: {(x, y) ∈ X × X: |x − y| ≤ 2} → R such that η(0, 0) = 2 andIf the condition η(0, 0) = 2 is eliminated, then the existence of such a function η characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).

scholarly journals Sur les solutions friables de l'équation a+b=c

2013 ◽  
Vol 154 (3) ◽  
pp. 439-463 ◽  
Author(s):  
SARY DRAPPEAU
Keyword(s):  
Lower Bound ◽  
Asymptotic Behaviour ◽  
Recent Work ◽  
Upper Bound ◽  
Smooth Function ◽  
Riemann Hypothesis ◽  
Smooth Solutions ◽  
Compactly Supported ◽  
The One

AbstractIn a recent paper [5], Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Conditionally under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we prove a more precise conditional estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bretèche and Granville in a recent work [2]. We also prove, conditionally under the Generalised Riemann Hypothesis, the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour.

scholarly journals On the achievable rate of bandlimited continuous-time AWGN channels with 1-bit output quantization

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sandra Bender ◽  
Meik Dörpinghaus ◽  
Gerhard P. Fettweis
Keyword(s):  
Lower Bound ◽  
Mutual Information ◽  
Gaussian Noise ◽  
Continuous Time ◽  
Additive White Gaussian Noise ◽  
White Gaussian Noise ◽  
Information Rate ◽  
Achievable Rate ◽  
Zero Crossing ◽  
Lower Bounding

AbstractWe consider a real continuous-time bandlimited additive white Gaussian noise channel with 1-bit output quantization. On such a channel the information is carried by the temporal distances of the zero-crossings of the transmit signal. We derive an approximate lower bound on the capacity by lower-bounding the mutual information rate for input signals with exponentially distributed zero-crossing distances, sine-shaped transition waveform, and an average power constraint. The focus is on the behavior in the mid-to-high signal-to-noise ratio (SNR) regime above 10 dB. For hard bandlimited channels, the lower bound on the mutual information rate saturates with the SNR growing to infinity. For a given SNR the loss with respect to the unquantized additive white Gaussian noise channel solely depends on the ratio of channel bandwidth and the rate parameter of the exponential distribution. We complement those findings with an approximate upper bound on the mutual information rate for the specific signaling scheme. We show that both bounds are close in the SNR domain of approximately 10–20 dB.

scholarly journals Degree estimates of one class cut-offs for improper integrals

2021 ◽  
Vol 2131 (3) ◽  
pp. 032018
Author(s):  
A Pozhidaev ◽  
O Khaustova
Keyword(s):  
Lower Bound ◽  
Gaussian Distribution ◽  
Distribution Density ◽  
Closed Interval ◽  
Concave Function ◽  
Theoretical Studies ◽  
Main Difficulty ◽  
Integrable Functions ◽  
Improper Integrals

Abstract The paper considers a normalized non-integral integral of the first kind with a variable lower bound. In this case the integrand is a generalization of the standard Gaussian distribution density. Such integrals are often called cutoffs or incomplete functions. The purpose of this paper is to obtain power inequalities for this kind of integrals. The necessity of obtaining this type of estimations is due to the fact that incomplete functions have become widespread in applications and theoretical studies. The peculiarity of the results established in the article consists in the fact that arbitrary degrees of a given integral for any value of an argument are evaluated from above not by means, of the value of integrable functions at a certain point, but by the value of the integral in question at some point proportional to this argument. The coefficient of proportionality, a parameter, can take any value from some closed interval. The main difficulty in obtaining these inequalities is that the integrand is a logarithmically concave function, that is, its logarithm is a concave function. The paper also proves that both limits of the closed interval for the parameter cannot be extended. This shows that the obtained estimates are unimprovable.

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