The purpose of this paper is to derive minimal sufficient statistics for the balanced incomplete block design and the group divisible partially balanced incomplete block design when the Eisenhart Model III (mixed model) is assumed. The results are identical to Hultquist and Graybill's (1965) and Hirotsu's (1965) for the same model without interaction, except for the addition of a statistic, ∑ijY2ij•.

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.

The most significant challenge for next wireless generation is to work opportunistically on the spectrum without a fixed spectrum allocation. Cognitive Radio (CR) is the candidate technology to utilize spectrum white space, which requires the CR to change its operating channel as the white space moves. In a CR ad-hoc network, each node could tune to a different channel; as a result, it cannot communicate with other nodes. This different tuning is due to the difficulty of maintaining Common Control Channel (CCC) in opportunistic spectrum network, and keeping the nodes synchronized in ad-hoc network. The CR ad-hoc network requires a protocol to match tuning channels between ad-hoc nodes, namely, rendezvous channels. In this thesis, two distributed Medium Access Control (MAC) protocols are designed that provide proper rendezvous channel without CCC or synchronization. The Balanced Incomplete Block Design (BIBD) is used in both protocols to provide our protocols a method of rendezvous between CR ad-hoc nodes. In fact, the BIBD guarantees there is at least one common element between any two blocks. If the channels are assigned to the BIBD elements and the searching sequence to the BIBD block, there is a guarantee of a rendezvous at least in one channel for each searching sequence. The first protocol uses a single-BIBD sequence and a multi-channel sensing. Alternatively, the second protocol uses a multi-BIBD sequence and a single-channel sensing. The single-sequence protocol analysis is based on the discrete Markov Chain. At the same time, the sequence structure of the BIBD in a multi-sequence protocol is used to define the Maximum Time to Rendezvous (MTTR). The simulation results confirm that the protocols outperform other existing protocols with respect to Time to Rendezvous (TTR), channel utilization, and network throughput. In addition, both protocols fairly distribute the network load on channels, and share the channels fairly among network nodes. This thesis provides straight forward and efficiently distributed MAC protocols for the CR ad-hoc networks.

The most significant challenge for next wireless generation is to work opportunistically on the spectrum without a fixed spectrum allocation. Cognitive Radio (CR) is the candidate technology to utilize spectrum white space, which requires the CR to change its operating channel as the white space moves. In a CR ad-hoc network, each node could tune to a different channel; as a result, it cannot communicate with other nodes. This different tuning is due to the difficulty of maintaining Common Control Channel (CCC) in opportunistic spectrum network, and keeping the nodes synchronized in ad-hoc network. The CR ad-hoc network requires a protocol to match tuning channels between ad-hoc nodes, namely, rendezvous channels. In this thesis, two distributed Medium Access Control (MAC) protocols are designed that provide proper rendezvous channel without CCC or synchronization. The Balanced Incomplete Block Design (BIBD) is used in both protocols to provide our protocols a method of rendezvous between CR ad-hoc nodes. In fact, the BIBD guarantees there is at least one common element between any two blocks. If the channels are assigned to the BIBD elements and the searching sequence to the BIBD block, there is a guarantee of a rendezvous at least in one channel for each searching sequence. The first protocol uses a single-BIBD sequence and a multi-channel sensing. Alternatively, the second protocol uses a multi-BIBD sequence and a single-channel sensing. The single-sequence protocol analysis is based on the discrete Markov Chain. At the same time, the sequence structure of the BIBD in a multi-sequence protocol is used to define the Maximum Time to Rendezvous (MTTR). The simulation results confirm that the protocols outperform other existing protocols with respect to Time to Rendezvous (TTR), channel utilization, and network throughput. In addition, both protocols fairly distribute the network load on channels, and share the channels fairly among network nodes. This thesis provides straight forward and efficiently distributed MAC protocols for the CR ad-hoc networks.