18 research outputs found
Counting solutions without zeros or repetitions of a linear congruence and rarefaction in b-multiplicative sequences
Consider a strongly -multiplicative sequence and a prime . Studying its
-rarefaction consists in characterizing the asymptotic behaviour of the sums
of the first terms indexed by the multiples of . The integer values of the
"norm" -variate polynomial
where is a primitive -th root of unity, and
determine this asymptotic behaviour. It will
be shown that a combinatorial method can be applied to The method enables deducing functional relations
between the coefficients as well as various properties of the coefficients of
, in particular for
This method provides relations between binomial coefficients. It gives new
proofs of the two identities
and (the
-th Lucas number). The sign and the residue modulo of the symmetric
polynomials of can also be obtained. An algorithm for
computation of coefficients of is
developed.Comment: 31 pages, 2 figures. Published in Journal de Theorie des Nombres de
Bordeau
Raréfaction dans les suites b-multiplicatives
On Ă©tudie une sous-classe des suites b-multiplicatives rarefiĂ©es avec un pas de rarĂ©faction p premier, et on trouve une structure asymptotique avec un exposant alphain]0,1[ et une fonction de rarĂ©faction continue pĂ©riodique. Cette structure vaut pour les suites qui contiennent des nombres complexes du disque unitĂ© (section 1.1), et aussi pour des systĂšmes de numĂ©ration avec b chiffres successifs positifs et nĂ©gatifs (section 1.2). Ce formalisme est analogue Ă celui dĂ©crit (pour le cas particuler de la suite de Thue-Morse) par Gelfond; Dekking; Goldstein, Kelly, Speer; Grabner; Drmota, Skalba et autres. Dans la deuxiĂšme partie, largement indĂ©pendante, on Ă©tudie la rarĂ©faction dans les suites composĂ©es de -1,0 et +1. On se restreint davantage au cas oĂč b engendre le groupe multiplicatif modulo p. Cette hypothĂšse est conjecturĂ©e (Artin) d'ĂȘtre vraie pour une infinitĂ© de nombres premiers. Les constantes qui apparaissent s'expriment alors comme polynĂŽmes symĂ©triques des P(zeta^j) oĂč P est un polynĂŽme Ă coefficients entiers, zeta est une racine primitive p-iĂšme de l'unitĂ©, parcourt les entiers de 1 Ă p-1 (ce lien est explicitĂ© dans la section 1.3). On dĂ©finit une mĂ©thode pour Ă©tudier les valeurs de ces polynĂŽmes symĂ©triques, basĂ©e sur la combinatoire, notamment sur le problĂšme de comptage des solutions des congruences et des systĂšmes linĂ©aires modulo p avec deux conditions supplĂ©mentaires: les rĂ©sidus modulo p utilisĂ©s doivent ĂȘtre non nuls et diffĂ©rents deux Ă deux. L'importance est donnĂ©e Ă la diffĂ©rence entre les nombres de soluions de deux congruences qui ne diffĂšrent que du terme sans variable. Le cas des congruences de la forme Ă©quivaut Ă un rĂ©sultat connu. Le mĂ©moire (section 2.2) lui donne une nouvelle preuve qui en fait une application originale de la formule d'inversion de Möbius dans le p.o.set des partitions d'un ensemble fini. Si au moins deux coefficients distincts sont prĂ©sents, on peut classer les rĂ©ponses associĂ©es Ă toutes les congruences possibles qui ont un ensemble fixe de coefficients (de taille d), dans un tableau qu'on va appeler un "simplexe de Pascal fini". Ce tableau est une fonction delta:N^d->Z restreinte aux points de somme des coordonnĂ©es infĂ©rieure Ă p (un simplexe), avec deux propriĂ©tĂ©s: l'Ă©quation rĂ©cursive de Pascal y est vĂ©rifiĂ©e partout sauf les points oĂč la somme des coefficients est multiple de p (qui seront appelĂ©s les "sources" et forment un sous-rĂ©seau de l'ensemble des points entiers), et les valeurs en-dehors du simplexe induites par l'Ă©quation sont nulles (c'est dĂ©montrĂ©, en rĂ©utilisant la mĂ©thode prĂ©cĂ©dente, dans la section 2.3 et en partie 2.4). On dĂ©crit un algorithme (section 2.4) qui consiste en applications successives de l'Ă©quation dans un ordre prĂ©cis, qui permet de trouver l'unique fonction delta qui vĂ©rifie les deux conditions. On applique ces rĂ©sultats aux suites b-multiplicatives (dans la section 2.5). On montre aussi que le nombre de sources ne dĂ©pend que de la dimension du simplexe d et de la longueur de son cĂŽtĂ© p. On formule la conjecture (partie 2.6) qu'il serait le plus petit possible parmi les tableaux de forme d'un simplexe de la dimention fixe et taille fixe qui vĂ©rifient les mĂȘmes conditions. On montre un premier rĂ©sultat sur les systĂšmes de deux congruences linĂ©aires (section 2.5.4), et on montre (section 1.4) un lien avec une mĂ©thode de Drmota et Skalba pour prouver l'absence de phĂ©nomĂšne de Newman (dans un sens prĂ©cis), dĂ©crit initialement pour la suite de Thue-Morse et tout p tel que b engendre le groupe multiplicatif modulo p, et gĂ©nĂ©ralisĂ© (section 1.4) Ă la suite (-1)^{nombre de chiffres 2 dans l'Ă©criture en base 3 de n} appelĂ©e "++-". Cette problĂ©matique est riche en problĂšmes d'algorithmique et de programmation. DiffĂ©rentes sections du mĂ©moire sont illustrĂ©es dans l'Annexe. La plupart de ces figures sont inĂ©dites.The primary object of study is a subclass of b-multiplicative sequences, p-rarefied which means that the subsequence of terms of index multiple of a prime number p is taken. The sums of their initial terms have an asymptotic structure described by an exponent alphain]0,1[ and a contnous periodic "rarefaction function". This structure is valid for sequences with complex values in the unit disc, in both cases of the usual numerating system (section 1.1) and one with b successive digits among which there are positive and negative (section 1.2). This formalism is analogous to the formalism for the Thue-Morse sequence in texts by Gelfond; Dekking; Goldstein, Kelly, Speer; Grabner; Drmota, Skalba and others. The second, largely independent, part concerns rarefaction in sequences with terms in -1,0 or 1. Most results concern the case where b is a generator of the multiplicative group modulo p. This condition has been conjectured to be valid for infinity of primes, by Artin. The constants which are important, can be written as symmetric polynomials of P(zeta^j) where zeta is a primitive p-th root of unity, P is a polynomial with integer coefficients and j runs through the numbers from 1 to p-1 (section 1.3). The text describes a combinatorics-based method to study the values of these symmetric polynomials, where the combinatorial problem is as follows. Count the solutions of a linear congruence or a system modulo p, which satisfy a condition: the values of variables must be different from each other and from zero. Importance is attached to the difference between the numbers of solutions of two congruences that differ only in the free term. For the congruences of the form this problem reduces to a well-known result. The text (section 2.2) gives an original proof of it, using the Möbius inversion formula in the p.o.set of partitions of a finite set. If at least two distinct coefficients are present, we can fix a set of coefficients (of size d) and put the answers corresponding to all possible linear congruences into an array that will be called "finite Pascal's triangle". It is a function delta:N^d->Z restricted to inputs with the sum of coordinates smaler than p (a simplex), and it has two properties. A recursive equation similar to the equation of Pascal holds everywhere except the points where the sum of coefficients is a multiple of p (a sublattice of Z^d the points of which are called "sources"); the values induced by this equation beyond the simplex are zeroes (section 2.3 and part of 2.4). An algorithm that finds the unique function delta satisfying these condiditions is described (section 2.4). It consists in successive applications of the equation in a precise order. These results are then applied to the b-multiplicative sequences (section 2.5). We also prove that the number of sources depends only on the dimention d and the size p of the simplex. We conjecture (section 2.6) that this number is the smallest possible for all numerical arrays of the same dimention and size that satisfy the same conditions. A first result about the systems of two linear congruences is proved (section 2.5.4). It is shown how these systems are related to a method by Drmota and Skalba of proving the absence of Newman's phenomenon (in a precise sence) initially described for the Thue-Morse sequence and for a prime p such that 2 is a generator of the multiplicative group modulo p, then extended to the sequence (-1)^{number of digits 2 in the ternary extension of n} called "++-". These questions generate many algorithmic and programming problems. Several sections link to illustration situated in the Annexe. Most of these figures are published for the first time.SAVOIE-SCD - Bib.Ă©lectronique (730659901) / SudocGRENOBLE1/INP-Bib.Ă©lectronique (384210012) / SudocGRENOBLE2/3-Bib.Ă©lectronique (384219901) / SudocSudocFranceF
Diffuse neutron scattering in relaxor ferroelectric PbMg1/3Nb2/3O3
High energy resolution neutron spin-echo spectroscopy has been used to
measure intrinsic width of diffuse scattering discovered earlier in relaxor
ferroelectric crystals. The anisotropic and transverse components of the
scattering have been observed in different Brillouin zones. Both components are
found to be elastic within experimental accuracy of 1 eV. Possible
physical origin of the static-like behavior is discussed for each diffuse
scattering contribution.Comment: Submitted to the "Physical Chemistry and Chemical Physics"
(Proceedings of the QENA2004
The rarefaction phenomenon in b-multiplicative sequences.
On Ă©tudie une sous-classe des suites b-multiplicatives rarefiĂ©es avec un pas de rarĂ©faction p premier, et on trouve une structure asymptotique avec un exposant αâ]0,1[ et une fonction de rarĂ©faction continue pĂ©riodique. Cette structure vaut pour les suites qui contiennent des nombres complexes du disque unitĂ© (section 1.1), et aussi pour des systĂšmes de numĂ©ration avec b chiffres successifs positifs et nĂ©gatifs (section 1.2). Ce formalisme est analogue Ă celui dĂ©crit (pour le cas particuler de la suite de Thue-Morse) par Gelfond; Dekking; Goldstein, Kelly, Speer; Grabner; Drmota, Skalba et autres. Dans la deuxiĂšme partie, largement indĂ©pendante, on Ă©tudie la rarĂ©faction dans les suites composĂ©es de -1,0 et +1. On se restreint davantage au cas oĂč b engendre le groupe multiplicatif modulo p. Cette hypothĂšse est conjecturĂ©e (Artin) d'ĂȘtre vraie pour une infinitĂ© de nombres premiers. Les constantes qui apparaissent s'expriment alors comme polynĂŽmes symĂ©triques des P(ζ^j) oĂč P est un polynĂŽme Ă coefficients entiers, ζ est une racine primitive p-iĂšme de l'unitĂ©, j parcourt les entiers de 1 Ă p-1 (ce lien est explicitĂ© dans la section 1.3). On dĂ©finit une mĂ©thode pour Ă©tudier les valeurs de ces polynĂŽmes symĂ©triques, basĂ©e sur la combinatoire, notamment sur le problĂšme de comptage des solutions des congruences et des systĂšmes linĂ©aires modulo p avec deux conditions supplĂ©mentaires: les rĂ©sidus modulo p utilisĂ©s doivent ĂȘtre non nuls et diffĂ©rents deux Ă deux. L'importance est donnĂ©e Ă la diffĂ©rence entre les nombres de soluions de deux congruences qui ne diffĂšrent que du terme sans variable. Le cas des congruences de la forme xâ+xâ+...+xâ=i mod p Ă©quivaut Ă un rĂ©sultat connu. Le mĂ©moire (section 2.2) lui donne une nouvelle preuve qui en fait une application originale de la formule d'inversion de Möbius dans le p.o.set des partitions d'un ensemble fini. Si au moins deux coefficients distincts sont prĂ©sents, on peut classer les rĂ©ponses associĂ©es Ă toutes les congruences possibles qui ont un ensemble fixe de coefficients (de taille d), dans un tableau qu'on va appeler un "simplexe de Pascal fini". Ce tableau est une fonction ÎŽ:N^dâZ restreinte aux points de somme des coordonnĂ©es infĂ©rieure Ă p (un simplexe), avec deux propriĂ©tĂ©s: l'Ă©quation rĂ©cursive de Pascal y est vĂ©rifiĂ©e partout sauf les points oĂč la somme des coefficients est multiple de p (qui seront appelĂ©s les "sources" et forment un sous-rĂ©seau de l'ensemble des points entiers), et les valeurs en-dehors du simplexe induites par l'Ă©quation sont nulles (c'est dĂ©montrĂ©, en rĂ©utilisant la mĂ©thode prĂ©cĂ©dente, dans la section 2.3 et en partie 2.4). On dĂ©crit un algorithme (section 2.4) qui consiste en applications successives de l'Ă©quation dans un ordre prĂ©cis, qui permet de trouver l'unique fonction delta qui vĂ©rifie les deux conditions. On applique ces rĂ©sultats aux suites b-multiplicatives (dans la section 2.5). On montre aussi que le nombre de sources ne dĂ©pend que de la dimension du simplexe d et de la longueur de son cĂŽtĂ© p. On formule la conjecture (partie 2.6) qu'il serait le plus petit possible parmi les tableaux de forme d'un simplexe de la dimention fixe et taille fixe qui vĂ©rifient les mĂȘmes conditions. On montre un premier rĂ©sultat sur les systĂšmes de deux congruences linĂ©aires (section 2.5.4), et on montre (section 1.4) un lien avec une mĂ©thode de Drmota et Skalba pour prouver l'absence de phĂ©nomĂšne de Newman (dans un sens prĂ©cis), dĂ©crit initialement pour la suite de Thue-Morse et tout p tel que b engendre le groupe multiplicatif modulo p, et gĂ©nĂ©ralisĂ© (section 1.4) Ă la suite (-1)^{nombre de chiffres 2 dans l'Ă©criture en base 3 de n} appelĂ©e "++-". Cette problĂ©matique est riche en problĂšmes d'algorithmique et de programmation. DiffĂ©rentes sections du mĂ©moire sont illustrĂ©es dans l'Annexe. La plupart de ces figures sont inĂ©dites.The primary object of study is a subclass of b-multiplicative sequences, p-rarefied which means that the subsequence of terms of index multiple of a prime number p is taken. The sums of their initial terms have an asymptotic structure described by an exponent αâ]0,1[ and a contnous periodic "rarefaction function". This structure is valid for sequences with complex values in the unit disc, in both cases of the usual numerating system (section 1.1) and one with b successive digits among which there are positive and negative (section 1.2). This formalism is analogous to the formalism for the Thue-Morse sequence in texts by Gelfond; Dekking; Goldstein, Kelly, Speer; Grabner; Drmota, Skalba and others. The second, largely independent, part concerns rarefaction in sequences with terms in -1,0 or 1. Most results concern the case where b is a generator of the multiplicative group modulo p. This condition has been conjectured to be valid for infinity of primes, by Artin. The constants which are important, can be written as symmetric polynomials of P(ζ^j) where ζ is a primitive p-th root of unity, P is a polynomial with integer coefficients and j runs through the numbers from 1 to p-1 (section 1.3). The text describes a combinatorics-based method to study the values of these symmetric polynomials, where the combinatorial problem is as follows. Count the solutions of a linear congruence or a system modulo p, which satisfy a condition: the values of variables must be different from each other and from zero. Importance is attached to the difference between the numbers of solutions of two congruences that differ only in the free term. For the congruences of the form xâ+xâ+...+xâ=i mod p this problem reduces to a well-known result. The text (section 2.2) gives an original proof of it, using the Möbius inversion formula in the p.o.set of partitions of a finite set. If at least two distinct coefficients are present, we can fix a set of coefficients (of size d) and put the answers corresponding to all possible linear congruences into an array that will be called "finite Pascal's triangle". It is a function ÎŽ:N^dâZ restricted to inputs with the sum of coordinates smaler than p (a simplex), and it has two properties. A recursive equation similar to the equation of Pascal holds everywhere except the points where the sum of coefficients is a multiple of p (a sublattice of Z^d the points of which are called "sources"); the values induced by this equation beyond the simplex are zeroes (section 2.3 and part of 2.4). An algorithm that finds the unique function delta satisfying these condiditions is described (section 2.4). It consists in successive applications of the equation in a precise order. These results are then applied to the b-multiplicative sequences (section 2.5). We also prove that the number of sources depends only on the dimention d and the size p of the simplex. We conjecture (section 2.6) that this number is the smallest possible for all numerical arrays of the same dimention and size that satisfy the same conditions. A first result about the systems of two linear congruences is proved (section 2.5.4). It is shown how these systems are related to a method by Drmota and Skalba of proving the absence of Newman's phenomenon (in a precise sence) initially described for the Thue-Morse sequence and for a prime p such that 2 is a generator of the multiplicative group modulo p, then extended to the sequence (-1)^{number of digits 2 in the ternary extension of n} called "++-". These questions generate many algorithmic and programming problems. Several sections link to illustration situated in the Annexe. Most of these figures are published for the first time
Raréfaction dans les suites b-multiplicatives
The primary object of study is a subclass of b-multiplicative sequences, p-rarefied which means that the subsequence of terms of index multiple of a prime number p is taken. The sums of their initial terms have an asymptotic structure described by an exponent αâ]0,1[ and a contnous periodic "rarefaction function". This structure is valid for sequences with complex values in the unit disc, in both cases of the usual numerating system (section 1.1) and one with b successive digits among which there are positive and negative (section 1.2). This formalism is analogous to the formalism for the Thue-Morse sequence in texts by Gelfond; Dekking; Goldstein, Kelly, Speer; Grabner; Drmota, Skalba and others. The second, largely independent, part concerns rarefaction in sequences with terms in -1,0 or 1. Most results concern the case where b is a generator of the multiplicative group modulo p. This condition has been conjectured to be valid for infinity of primes, by Artin. The constants which are important, can be written as symmetric polynomials of P(ζ^j) where ζ is a primitive p-th root of unity, P is a polynomial with integer coefficients and j runs through the numbers from 1 to p-1 (section 1.3). The text describes a combinatorics-based method to study the values of these symmetric polynomials, where the combinatorial problem is as follows. Count the solutions of a linear congruence or a system modulo p, which satisfy a condition: the values of variables must be different from each other and from zero. Importance is attached to the difference between the numbers of solutions of two congruences that differ only in the free term. For the congruences of the form xâ+xâ+...+xâ=i mod p this problem reduces to a well-known result. The text (section 2.2) gives an original proof of it, using the Möbius inversion formula in the p.o.set of partitions of a finite set. If at least two distinct coefficients are present, we can fix a set of coefficients (of size d) and put the answers corresponding to all possible linear congruences into an array that will be called "finite Pascal's triangle". It is a function ÎŽ:N^dâZ restricted to inputs with the sum of coordinates smaler than p (a simplex), and it has two properties. A recursive equation similar to the equation of Pascal holds everywhere except the points where the sum of coefficients is a multiple of p (a sublattice of Z^d the points of which are called "sources"); the values induced by this equation beyond the simplex are zeroes (section 2.3 and part of 2.4). An algorithm that finds the unique function delta satisfying these condiditions is described (section 2.4). It consists in successive applications of the equation in a precise order. These results are then applied to the b-multiplicative sequences (section 2.5). We also prove that the number of sources depends only on the dimention d and the size p of the simplex. We conjecture (section 2.6) that this number is the smallest possible for all numerical arrays of the same dimention and size that satisfy the same conditions. A first result about the systems of two linear congruences is proved (section 2.5.4). It is shown how these systems are related to a method by Drmota and Skalba of proving the absence of Newman's phenomenon (in a precise sence) initially described for the Thue-Morse sequence and for a prime p such that 2 is a generator of the multiplicative group modulo p, then extended to the sequence (-1)^{number of digits 2 in the ternary extension of n} called "++-". These questions generate many algorithmic and programming problems. Several sections link to illustration situated in the Annexe. Most of these figures are published for the first time.On Ă©tudie une sous-classe des suites b-multiplicatives rarefiĂ©es avec un pas de rarĂ©faction p premier, et on trouve une structure asymptotique avec un exposant αâ]0,1[ et une fonction de rarĂ©faction continue pĂ©riodique. Cette structure vaut pour les suites qui contiennent des nombres complexes du disque unitĂ© (section 1.1), et aussi pour des systĂšmes de numĂ©ration avec b chiffres successifs positifs et nĂ©gatifs (section 1.2). Ce formalisme est analogue Ă celui dĂ©crit (pour le cas particuler de la suite de Thue-Morse) par Gelfond; Dekking; Goldstein, Kelly, Speer; Grabner; Drmota, Skalba et autres. Dans la deuxiĂšme partie, largement indĂ©pendante, on Ă©tudie la rarĂ©faction dans les suites composĂ©es de -1,0 et +1. On se restreint davantage au cas oĂč b engendre le groupe multiplicatif modulo p. Cette hypothĂšse est conjecturĂ©e (Artin) d'ĂȘtre vraie pour une infinitĂ© de nombres premiers. Les constantes qui apparaissent s'expriment alors comme polynĂŽmes symĂ©triques des P(ζ^j) oĂč P est un polynĂŽme Ă coefficients entiers, ζ est une racine primitive p-iĂšme de l'unitĂ©, j parcourt les entiers de 1 Ă p-1 (ce lien est explicitĂ© dans la section 1.3). On dĂ©finit une mĂ©thode pour Ă©tudier les valeurs de ces polynĂŽmes symĂ©triques, basĂ©e sur la combinatoire, notamment sur le problĂšme de comptage des solutions des congruences et des systĂšmes linĂ©aires modulo p avec deux conditions supplĂ©mentaires: les rĂ©sidus modulo p utilisĂ©s doivent ĂȘtre non nuls et diffĂ©rents deux Ă deux. L'importance est donnĂ©e Ă la diffĂ©rence entre les nombres de soluions de deux congruences qui ne diffĂšrent que du terme sans variable. Le cas des congruences de la forme xâ+xâ+...+xâ=i mod p Ă©quivaut Ă un rĂ©sultat connu. Le mĂ©moire (section 2.2) lui donne une nouvelle preuve qui en fait une application originale de la formule d'inversion de Möbius dans le p.o.set des partitions d'un ensemble fini. Si au moins deux coefficients distincts sont prĂ©sents, on peut classer les rĂ©ponses associĂ©es Ă toutes les congruences possibles qui ont un ensemble fixe de coefficients (de taille d), dans un tableau qu'on va appeler un "simplexe de Pascal fini". Ce tableau est une fonction ÎŽ:N^dâZ restreinte aux points de somme des coordonnĂ©es infĂ©rieure Ă p (un simplexe), avec deux propriĂ©tĂ©s: l'Ă©quation rĂ©cursive de Pascal y est vĂ©rifiĂ©e partout sauf les points oĂč la somme des coefficients est multiple de p (qui seront appelĂ©s les "sources" et forment un sous-rĂ©seau de l'ensemble des points entiers), et les valeurs en-dehors du simplexe induites par l'Ă©quation sont nulles (c'est dĂ©montrĂ©, en rĂ©utilisant la mĂ©thode prĂ©cĂ©dente, dans la section 2.3 et en partie 2.4). On dĂ©crit un algorithme (section 2.4) qui consiste en applications successives de l'Ă©quation dans un ordre prĂ©cis, qui permet de trouver l'unique fonction delta qui vĂ©rifie les deux conditions. On applique ces rĂ©sultats aux suites b-multiplicatives (dans la section 2.5). On montre aussi que le nombre de sources ne dĂ©pend que de la dimension du simplexe d et de la longueur de son cĂŽtĂ© p. On formule la conjecture (partie 2.6) qu'il serait le plus petit possible parmi les tableaux de forme d'un simplexe de la dimention fixe et taille fixe qui vĂ©rifient les mĂȘmes conditions. On montre un premier rĂ©sultat sur les systĂšmes de deux congruences linĂ©aires (section 2.5.4), et on montre (section 1.4) un lien avec une mĂ©thode de Drmota et Skalba pour prouver l'absence de phĂ©nomĂšne de Newman (dans un sens prĂ©cis), dĂ©crit initialement pour la suite de Thue-Morse et tout p tel que b engendre le groupe multiplicatif modulo p, et gĂ©nĂ©ralisĂ© (section 1.4) Ă la suite (-1)^{nombre de chiffres 2 dans l'Ă©criture en base 3 de n} appelĂ©e "++-". Cette problĂ©matique est riche en problĂšmes d'algorithmique et de programmation. DiffĂ©rentes sections du mĂ©moire sont illustrĂ©es dans l'Annexe. La plupart de ces figures sont inĂ©dites
Analysis of phase waves in the ECoG data
Subdural ECoG data recorded from the matrix of electrodes during syllable pronunciation are analyzed by the method of circular-linear regression. Phase waves in 1D electrode arrays and in the whole 2D set of electrodes are detected, and their spatial organization and temporal evolution are studied. Phase portraits of wave vectors indicate the presence of sources, sinks, and saddle points. The analysis of temporal evolution of phase portraits shows that they changed more at the beginning of syllable pronunciation. Furthermore, wave sources were more stable in their localization during the pronunciation. Overall, in spite of large variability of phase portraits, they represent some characterization of the dynamics of electric potential in the cerebral cortex
Optimal Measurement Times for Observing a Brownian Motion over a Finite Period Using a Kalman Filter
International audienceThis article deals with the optimization of the schedule of measures for observing a random process in time using a Kalman filter, when the length of the process is finite and fixed, and a fixed number of measures are available. The measure timetable plays a critical role for the accuracy of this estimator. Two different criteria of optimality of a timetable (not necessarily regular) are considered: the maximal and the mean variance of the estimator. Both experimental and theoretical methods are used for the problem of minimizing the mean variance. The theoretical methods are based on studying the cost function as a rational function. An analytical formula of the optimal instant of measure is obtained in the case of one measure. Its properties are studied. An experimental solution is given for a particular case with n > 1 measures