京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第18400号 / 情博第515号 / 新制||情||91(附属図書館) / 31258 / 京都大学大学院情報学研究科数理工学専攻 / (主査)准教授 辻本 諭, 教授 中村 佳正, 教授 梅野 健 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

This thesis is based on four papers dealing with two different areas of mathematics.Paper I–III are in combinatorics, while Paper IV is in mathematical physics.In combinatorics, we work with design theory, one of whose applications aredesigning statistical experiments. Specifically, we are interested in symmetric incompleteblock designs (SBIBDs) and triple arrays and also the relationship betweenthese two types of designs.In Paper I, we investigate when a triple array can be balanced for intersectionwhich in the canonical case is equivalent to the inner design of the correspondingsymmetric balanced incomplete block design (SBIBD) being balanced. For this we derivenew existence criteria, and in particular we prove that the residual designof the related SBIBD must be quasi-symmetric, and give necessary and sufficientconditions on the intersection numbers. We also address the question of whenthe inner design is balanced with respect to every block of the SBIBD. We showthat such SBIBDs must possess the quasi-3 property, and we answer the existencequestion for all know classes of these designs.As triple arrays balanced for intersections seem to be very rare, it is natural toask if there are any other families of row-column designs with this property. In PaperII we give necessary and sufficient conditions for balanced grids to be balancedfor intersection and prove that all designs in an infinite family of binary pseudo-Youden designs are balanced for intersection.Existence of triple arrays is an open question. There is one construction of aninfinite, but special family called Paley triple arrays, and one general method forwhich one of the steps is unproved. In Paper III we investigate a third constructionmethod starting from Youden squares. This method was suggested in the literaturea long time ago, but was proven not to work by a counterexample. We show interalia that Youden squares from projective planes can never give a triple array bythis method, but that for every triple array corresponding to a biplane, there is asuitable Youden square for which the method works. Also, we construct the familyof Paley triple arrays by this method.In mathematical physics we work with solitons, which in nature can be seen asself-reinforcing waves acting like particles, and in mathematics as solutions of certainnon-linear differential equations. In Paper IV we study the non-commutativeversion of the two-dimensional Toda lattice for which we construct a family ofsolutions, and derive explicit solution formulas. / Denna avhandling baseras på fyra artiklar som behandlar två olika områden avmatematiken. Artikel I-III ligger inom kombinatoriken medan artikel IV behandlarmatematisk fysik.Inom kombinatoriken arbetar vi med designteori som bland annat har tillämpningardå man ska utforma statistiska experiment.I artikel I undersöker vi när en triple array kan vara snittbalanserad vilket i detkanoniska fallet är ekvivalent med den inre designen till den korresponderandesymmetriska balanserade inkompletta blockdesignen (SBIBD) är balanserad. För dettapresenterar vi nya nödvändiga villkor. Speciellt visar vi att den residuala designentill den korresponderande SBIBDen måste vara kvasi-symmetrisk och ger nödvändigaoch tillräckliga villkor för dess blockskärningstal. Vi adresserar ocksåfrågan om när den inre designen är balanserad med avseende på alla SBIBDensblock. Vi visar att en sådan SBIBD måste ha den egenskap som kallas kvasi-3 ochsvarar på existensfrågan för alla kända klasser av sådana designer.Eftersom snittbalanserade triple arrays verkar vara väldigt sällsynta är detnaturligt att fråga om det finns andra familjer av rad-kolumn designer som hardenna egenskap. I artikel II ger vi nödvändiga och tillräckliga villkor för att enbalanced grid ska vara snittbalanserad och visar att alla designer i en oändlig familjav binära pseudo-Youden squares är snittbalanserade.Existensfrågan för triple arrays är öppen fråga. Det finns en konstruktionsmetodför en oändlig men speciell familj kallad Paley triple arrays och så finns det enallmän metod för vilken ett steg är obevisat. I artikel III undersöker vi en tredjekonstruktionsmetod som utgår från Youden squares. Denna metod föreslogs i litteraturenför länge sedan men blev motbevisad med hjälp av ett motexempel. Vivisar bland annat att Youden squares från projektiva plan aldrig kan ge en triplearray med denna metod, men att det för varje triple array som korresponderartill ett biplan, så finns det en lämplig Youden square för vilken metoden fungerar.Vidare konstruerar vi familjen av Paley triple arrays med denna metod.Inom matematisk fysik arbetar vi med solitoner som man i naturen kan få sesom självförstärkande vågor vilka beter sig som partiklar. Inom matematiken ärde lösningar till vissa ickelinjära differentialekvationer. I artikel IV studerar vi dettvådimensionella Toda-gittret för vilken vi konstruerar en familj av lösningar ochäven explicita lösningsformler.

In this thesis we address the analytic and numerical aspects of isospectral flows. Such flows occur in mathematical physics and numerical linear algebra. Their main structural feature is to retain the eigenvalues in the solution space. We explore the solution of Isospectral flows and their stochastic counterpart using explicit generalisation of Magnus expansion. \par In the first part of the thesis we expand the solution of Bloch--Iserles equations, the matrix ordinary differential system of the form $ X'=[N,X^{2}],\ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),\ N\in \mathfrak{so}(n), $ where $\textrm{Sym}(n)$ denotes the space of real $n\times n$ symmetric matrices and $\mathfrak{so}(n)$ denotes the Lie algebra of real $n\times n$ skew-symmetric matrices. This system is endowed with Poisson structure and is integrable. Various important properties of the flow are discussed. The flow is solved using explicit Magnus expansion and the terms of expansion are represented as binary rooted trees deducing an explicit formalism to construct the trees recursively. Unlike classical numerical methods, e.g.\ Runge--Kutta and multistep methods, Magnus expansion respects the isospectrality of the system, and the shorthand of binary rooted trees reduces the computational cost of the exponentially growing terms. The desired structure of the solution (also with large time steps) has been displayed. \par Having seen the promising results in the first part of the thesis, the technique has been extended to the generalised double bracket flow $ X^{'}=[[N,X]+M,X], \ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),$ where $N\in \textrm{diag}(n)$ and $M\in \mathfrak{so}(n)$, which is also a form of an Isospectral flow. In the second part of the thesis we define the generalised double bracket flow and discuss its dynamics. It is noted that $N=0$ reduces it to an integrable flow, while for $M=0$ it results in a gradient flow. We analyse the flow for various non-zero values of $N$ and $M$ by assigning different weights and observe Hopf bifurcation in the system. The discretisation is done using Magnus series and the expansion terms have been portrayed using binary rooted trees. Although this matrix system appears more complex and leads to the tri-colour leaves; it has been possible to formulate the explicit recursive rule. The desired structure of the solution is obtained that leaves the eigenvalues invariant in the solution space.

Η παρούσα εργασία αφορά στη μελέτη Χαμιλτώνιων συστημάτων Ν μη γραμμικών ταλαντωτών, όπως είναι αυτό των Fermi Pasta και Ulam (FPU), με στόχο την βαθύτερη κατανόηση της δυναμικής των σχεδόν-περιοδικών τροχιών και του ρόλου των αντίστοιχων τόρων στο χώρο φάσεων, καθώς αυξάνουμε την ενέργεια Ε και τον αριθμό βαθμών ελευθερίας Ν του συστήματος. Το βασικό μας αποτέλεσμα είναι ότι υπάρχουν τόροι χαμηλής διάστασης, που προκύπτουν από τη συνέχεια των αντίστοιχων του γραμμικού συστήματος, οι οποίοι ευθύνονται για τις FPU επαναλήψεις και εμποδίζουν την ισοκατανομή της ενέργειας μεταξύ όλων των κανονικών τρόπων ταλάντωσης. Αναλύοντας ευστάθεια αυτών των τόρων, μπορέσαμε να δώσουμε μια πληρέστερη ερμηνεία στο Παράδοξο των FPU, συνδέοντας και συμπληρώνοντας έτσι δύο από τις επικρατέστερες ερμηνείες του εν λόγω φαινομένου. / The present work concerns the study of Hamiltonian systems of N nonlinear coupled oscillators, as it is the one by Fermi Pasta and Ulam (FPU), in order to understand the dynamics of quasi-periodic orbits and the role of their corresponding tori in phase space, as we increase the energy E and the number N of the degrees of freedom. Our fundamental result is that there exist tori of low dimension, that come from the continuation of the corresponding tori of the linear system, which are responsible for the FPU recurrences and prevent the system from equipartition of the energy among all normal modes. By investigating the stability of these tori, we achieved to provide a more complete explanation for the FPU paradox, connecting and supplementing in this way two of the most dominant approaches for this paradox.