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New Estimates of Numerical Values Related to a Simplex

Author: Alexey Yu. Ukhalov
Mikhail V. Nevskii
Publication venue
Publication date: 01/01/2017
Field of study

Let

n\in {\mathbb N}

and

Q_n=[0,1]^n

. For a nondegenerate simplex

S\subset {\mathbb R}^n

, by

\sigma S

we denote the homothetic copy of~

S

with center of homothety in the center of gravity of

S

and ratio of~homothety

\sigma

. By

\xi(S)

we mean the minimal

\sigma>0

such that

Q_n\subset \sigma S

. By

\alpha(S)

denote the minimal

\sigma>0

such that

Q_n

is~contained in a translate of~

\sigma S

. By

d_i(S)

we denote the

i

th axial diameter of

S

, i.\,e. the maximum length of~the segment contained in

S

and parallel to the

i

th coordinate axis. Formulae for~

\xi(S)

\alpha(S)

d_i(S)

were proved earlier by the first author. Define

\xi_n=\min\{ \xi(S): S\subset Q_n\}.

We always have

\xi_n\geq n.

We discuss some conjectures formulated in the previous papers. One of~these conjectures is the following. For~every

n

, there exists

\gamma>0

, not depending on

S\subset Q_n

, such that an~inequality

\xi(S)-\alpha(S)\leq \gamma (\xi(S)-\xi_n)

holds. Denote by

\varkappa_n

the minimal

\gamma

with such a~property. We prove that

\varkappa_1=\frac{1}{2}

; for

n>1

, we obtain

\varkappa_n\geq 1

. If

n>1

and

\xi_n=n,

then

\varkappa_n=1

. The equality

\xi_n=n

holds if

n+1

is an Hadamard number, i.\,e. there exists an Hadamard matrix of~order

n+1

. This proposition is known; we give one more proof with the direct use of Hadamard matrices. We prove that

\xi_5=5

. Therefore, there exists

n

such that

n+1

is not an Hadamard number and nevertheless

\xi_n=n

. The~minimal

n

with such a property is equal to

5

. This involves

\varkappa_5=1

and also disproves the following previous conjecture of the first author concerning the characterization of Hadamard numbers in terms of~homothety of simplices:

n+1

is an Hadamard number if and only if

\xi_n=n.

This statement is valid only in one direction. There exists a simplex

S\subset Q_5

such that the boundary of the simplex

5S

contains all the vertices of the cube

Q_5

. We describe a one-parameter family of simplices contained in

Q_5

with the property

\alpha(S)=\xi(S)=5.

These simplices were found with the use of numerical and symbolic computations. %Numerical experiments allow to discover Another new result is an inequality \(\xi_6

Crossref

Directory of Open Access Journals

On Optimal Interpolation by Linear Functions on an n-Dimensional Cube

Author: Alexey Yu. Ukhalov
Mikhail V. Nevskii
Publication venue
Publication date: 2018
Field of study

Let

n\in{\mathbb N}

, and let

Q_n

be the unit cube

[0,1]^n

. By

C(Q_n)

we denote the space of continuous functions

f:Q_n\to{\mathbb R}

with the norm

\|f\|_{C(Q_n)}:=\max\limits_{x\in Q_n}|f(x)|,

\Pi_1\left({\mathbb R}^n\right)

--- the set of polynomials of

n

variables of degree

\leq 1

(or linear functions). Let

x^{(j)},

1\leq j\leq n+1,

be the vertices of

n

-dimnsional nondegenerate simplex

S\subset Q_n

. An interpolation projector

P:C(Q_n)\to \Pi_1({\mathbb R}^n)

corresponding to the simplex

S

is defined by equalities

Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right).

The norm of

P

as an operator from

C(Q_n)

C(Q_n)

may be calculated by the formula

\|P\|=\max\limits_{x\in ver(Q_n)} \sum\limits_{j=1}^{n+1} |\lambda_j(x)|.

Here

\lambda_j

are the basic Lagrange polynomials with respect to

S,

ver(Q_n)

is the set of vertices of

Q_n

. Let us denote by

\theta_n

the minimal possible value of

\|P\|.

Earlier, the first author proved various relations and estimates for values

\|P\|

and

\theta_n

, in particular, having geometric character. The equivalence

\theta_n\asymp \sqrt{n}

takes place. For example, the appropriate, according to dimension

n

, inequalities may be written in the form \linebreak

\frac{1}{4}\sqrt{n}

<\theta_n <3\sqrt{n}. If the nodes of the projector

P^*

coincide with vertices of an arbitrary simplex with maximum possible volume, we have

\|P^*\|\asymp\theta_n.

When an Hadamard matrix of order

n+1

exists, holds

\theta_n\leq\sqrt{n+1}.

In the paper, we give more precise upper bounds of numbers

\theta_n

for

21\leq n \leq 26

. These estimates were obtained with the application of maximum volume simplices in the cube. For constructing such simplices, we utilize maximum determinants containing the elements

\pm 1.

Also, we systematize and comment the best nowaday upper and low estimates of numbers

\theta_n

for a concrete \(n.\

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Author Instructions

Author: Instructions Author
Publication venue
Publication date: 04/11/2013
Field of study

Crossref

Cartographic Perspectives (E-Journal - North American Cartographic Information Society, NACIS)

Going Beyond Counting First Authors in Author Co-citation Analysis

Author: Zhao Dangzhi
Publication venue
Publication date: 01/01/2005
Field of study

The present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed

E-LIS

Variations on the Author

Author: Sayad Cecilia
Publication venue
Publication date: 01/01/2016
Field of study

“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship

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Kent Academic Repository

Appropriate Similarity Measures for Author Cocitation Analysis

Author: Waltman L.R.
Eck N.J.P. van
Publication venue
Publication date
Field of study

We provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authorsâ€™ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis

Research Papers in Economics

Linear Interpolation on a Euclidean Ball in Rⁿ

Author: Alexey Yu. Ukhalov
Mikhail V. Nevskii
Publication venue
Publication date: 2019
Field of study

For x^{(0)}\in{\mathbb R}^n, R>0, by

B=B(x^{(0)};R)

we denote a Euclidean ball in

{\mathbb R}^n

given by the inequality

\|x-x^{(0)}\|\leq R

\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}

. Put

B_n:=B(0,1)

. We mean by

C(B)

the space of continuous functions

f:B\to{\mathbb R}

with the norm

\|f\|_{C(B)}:=\max_{x\in B}|f(x)|

and by

\Pi_1\left({\mathbb R}^n\right)

the set of polynomials in

n

variables of degree

\leq 1

, i.e. linear functions on

{\mathbb R}^n

. Let

x^{(1)}, \ldots, x^{(n+1)}

be the vertices of

n

-dimensional nondegenerate simplex

S\subset B

. The interpolation projector

P:C(B)\to \Pi_1({\mathbb R}^n)

corresponding to

S

is defined by the equalities

Pf\left(x^{(j)}\right)=%f_j:=f\left(x^{(j)}\right).

Denote by

\|P\|_B

the norm of

P

as an operator from

C(B)

into

C(B)

. Let us define

\theta_n(B)

as minimal value of

\|P\|

under the condition

x^{(j)}\in B

. In the paper, we obtain the formula to compute

\|P\|_B

making use of

x^{(0)}

R

, and coefficients of basic Lagrange polynomials of

S

. In more details we study the case when

S

is a regular simplex inscribed into

B_n

. In this situation, we prove that

\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},

where

\psi(t)=\frac{2\sqrt{n}}{n+1}\bigl(t(n+1-t)\bigr)^{1/2}+\bigl|1-\frac{2t}{n+1}\bigr|

(0\leq t\leq n+1)

and integer

a

has the form

a=\bigl\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\bigr\rfloor.

For this projector,

\sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1}

. The equality

\|P\|_{B_n}=\sqrt{n+1}

takes place if and only if

\sqrt{n+1}

is an integer number. We give the precise values of

\theta_n(B_n)

for

1\leq n\leq 4

. To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball

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Dispelling the Myths Behind First-author Citation Counts

Author: Zhao Dangzhi
Publication venue
Publication date: 01/01/2006
Field of study

We conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more sophisticated methods

E-LIS

Author Index

Author: Author Index
Publication venue
Publication date
Field of study

Nao informado

koamabayili/VECTRON-author-checklist: VECTRON author checklist

Author: koamabayili
Publication venue
Publication date: 19/04/2022
Field of study

We have done our best to complete the author checklist relating to the use of animals in the hut study. Note that the objective for the hut study was to evaluate the IRS treatment applications for residual efficacy against Anopheles mosquitoes, including the local An. coluzzii mosquito population. Cows were only used to attract mosquitoes into the huts and no tests were carried out directly on the cows. The author checklist is intended for use with studies where experiments are carried out on animals, which is why we have had such difficulty in completing this for the hut study, as many of the questions do not relate to how the cows were used

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