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The rise of the technological age has had a huge impact on society, from the way we connect with people to the way war is waged.  As a result, countries had to adapt their strategies to include modern technology and subsequently an increase in information management. An increase of information both between forces and to the rest of society through different media outputs. During research it was discovered that there is currently a lack of theoretical explanations on information warfare (IW). A theory is only valid if it has a clear connection to empirical data. Therefore, this study aims to verify Robert Leonhard’s theory for the information age in relation to IW. Two cases were examined: Russia and the USA's utilization of IW during the war against Georgia in 2008 and the Invasion of Iraq in 2003. The analysis will be based on the three dimensions of IW: the cognitive, physical and information arena. The result shows that R. Leonhard’s theory is valid in accordance with the USA’s usage of IW and partially Russia’s use of IW. It is therefore possible to draw the conclusion that the study strengthens the theory. Consequently, this research can be used as a steppingstone to future studies.

The increased probability that future battles will take place in cities can leave military organizations unprepared for future conflicts. Urban warfare requires a different tactical approach than battles in other terrain due to its unique complexity. Previous research on urban warfare tends to agree that combined arms contribute to urban tactical success but lacks on explanations for how it carries into effect inside cities.   This study will examine how combined arms can lead to tactical victory, in a theory-testing approach, by analyzing the U.S assaults on Aachen in 1944 and Fallujah in 2004 during Operation Phantom Fury. Since the Swedish Armed Forces can be considered tactically influenced by Robert Leonhards theory on combined arms, the purpose of this study is to examine the theory’s ability to explain tactical success in urban terrain. The results show that an attacking combined arms force with superior firepower can put the defending enemy in a dilemma. However, the dilemma creates unfavorable conditions for the Alcyoneus principle of the theory to explain why the attacking forces succeeded.

This study examines the theory of Robert Leonhard about combined arms and in what extent it can explain the outcome of the Six-day war and the second Lebanon war. Since both the Six-day war and the second Lebanon war involved Israel but the outcome differs, a gap and the need for further studies appears. The theory of combined arms consists of three principles that to some extent needs to be fulfilled in order to optimise the chances of winning. Since some of the Swedish Armed Forces doctrines are based out of this theory, it is of importance to further test the ideas of Leonhard. By testing the explanatory value of the theory, the study aims to create deeper and wider knowledge to the subject which in a broader perspective also contributes to the strengthening of the Swedish Security Politics as well as the Swedish Armed Forces. The result of this multiple case study shows that the theory to a high degree can explain the outcome of the wars. Since few studies have analysed the theory applied on a small state like Israel, the ability to generalize has increased with this study, although further studies are needed.

The Axiom in the studies of military science is that implementing combined arms will result in atactical victory, in maneuver warfare. But there have been historical examples of when that has not been the case, and we in this profession must learn from these defeats. The intention of this study is to examine if Robert Leonhard’s theory of combined arms can explain the tactical defeats of the German forces during the battle of Alam el Halfa and Operation Citadel.  The conclusion of this study is that the German forces implemented combined arms warfare, but still lost. The German struggled during both cases to fully implement the dilemma principle and the alcyoneus principle. The implication of this result leads to the discussion that even if combined arms warfare is implemented, the enemy doesn’t necessarily suffer the effect as explained by Leonhard in the dilemma principle and the alcyoneus principle.

Este estudo se dedica a uma parte pouco conhecida dos trabalhos de Leonhard Euler (1707 - 1783) relacionado à música enquanto uma ciência matemática. Tais trabalhos mostram, em certo sentido, um lado pitagórico do pensador e também algumas contribuições do mesmo à teoria musical. O interesse deste matemático pelo assunto permeia várias obras e épocas de sua vida, mas neste trabalho focamos em três delas: um conjunto de nove cartas que compõem a obra Lettres a une princesse dAllemagne sur divers sujets de physique e de philosophie (1768) e dois artigos, Conjecture sur la raison de quelques dissonances generalement reçues dans la musique (1766) e De harmoniae veris principiis per speculum musicum repraesentatis (1774). Para possibilitar uma melhor compreensão desses textos, faremos uma revisão histórica do tratamento aritmético dado à música desde os tempos clássicos até o Renascimento com enfoque nas principais contribuições que levam à construção da escala da entonação pura ou justa, trabalhada por Euler nos originais estudados. Após a apresentação e análise desses trabalhos eulerianos, especialmente seus diagramas para representar sons e acordes e do seu expoente de um acorde para medir consonâncias, terminaremos essa dissertação refletindo sobre as implicações pedagógicas e históricas, bem como as potencialidades e limitações do uso de fontes originais de determinados mestres do pensamento matemático ocidental, nomeadamente o próprio Euler, na formação de matemáticos, professores e licenciados. / This present research takes a journey into a little know part of Leonhard Eulers works about music as a mathematical science. Those works show, at certain level, his Pythagorean thoughts and also his contributions to music theory. Eulers interest for this subject permeates many of his works through his lifetime (1707 - 1783), but here we focus on three of them: a set of nine letters from the book Lettres a une princesse dAllemagne sur divers sujets de physique e de philosophie (1768), and two articles, Conjecture sur la raison de quelques dissonances generalement reçues dans la musique (1766) and De harmoniae veris principiis per speculum musicum repraesentatis (1774). To allow a better understanding of his ideas, first we put on a historical review of the arithmetic treatment of music since the ancient Greece to the Renaissance Era pointing out the main contributions to the development of the pure just intonation scale, the one used by Euler on the works just mentioned. After presenting and analyzing these Eulers contributions, specially his diagrams to represent sounds and chords to the eyes and the concept of exponent of a chord to measure consonances, we conclude this dissertation thinking about the pedagogical and historical impact, and also the potential and limitations concerning the use of historical sources of the masters of mathematics, to the training of future mathematics and teachers.

Univ., Diss.--Budapest, 2001.

Este estudo se dedica a uma parte pouco conhecida dos trabalhos de Leonhard Euler (1707 - 1783) relacionado à música enquanto uma ciência matemática. Tais trabalhos mostram, em certo sentido, um lado pitagórico do pensador e também algumas contribuições do mesmo à teoria musical. O interesse deste matemático pelo assunto permeia várias obras e épocas de sua vida, mas neste trabalho focamos em três delas: um conjunto de nove cartas que compõem a obra Lettres a une princesse dAllemagne sur divers sujets de physique e de philosophie (1768) e dois artigos, Conjecture sur la raison de quelques dissonances generalement reçues dans la musique (1766) e De harmoniae veris principiis per speculum musicum repraesentatis (1774). Para possibilitar uma melhor compreensão desses textos, faremos uma revisão histórica do tratamento aritmético dado à música desde os tempos clássicos até o Renascimento com enfoque nas principais contribuições que levam à construção da escala da entonação pura ou justa, trabalhada por Euler nos originais estudados. Após a apresentação e análise desses trabalhos eulerianos, especialmente seus diagramas para representar sons e acordes e do seu expoente de um acorde para medir consonâncias, terminaremos essa dissertação refletindo sobre as implicações pedagógicas e históricas, bem como as potencialidades e limitações do uso de fontes originais de determinados mestres do pensamento matemático ocidental, nomeadamente o próprio Euler, na formação de matemáticos, professores e licenciados. / This present research takes a journey into a little know part of Leonhard Eulers works about music as a mathematical science. Those works show, at certain level, his Pythagorean thoughts and also his contributions to music theory. Eulers interest for this subject permeates many of his works through his lifetime (1707 - 1783), but here we focus on three of them: a set of nine letters from the book Lettres a une princesse dAllemagne sur divers sujets de physique e de philosophie (1768), and two articles, Conjecture sur la raison de quelques dissonances generalement reçues dans la musique (1766) and De harmoniae veris principiis per speculum musicum repraesentatis (1774). To allow a better understanding of his ideas, first we put on a historical review of the arithmetic treatment of music since the ancient Greece to the Renaissance Era pointing out the main contributions to the development of the pure just intonation scale, the one used by Euler on the works just mentioned. After presenting and analyzing these Eulers contributions, specially his diagrams to represent sounds and chords to the eyes and the concept of exponent of a chord to measure consonances, we conclude this dissertation thinking about the pedagogical and historical impact, and also the potential and limitations concerning the use of historical sources of the masters of mathematics, to the training of future mathematics and teachers.

Even though battles have been fought in built-up areas for a long time, there have not been any specific theories connected to it. As the world becomes increasingly urbanized, the problem of not having any complete theories with which to analyze military operations in urban terrain becomes bigger. This essay aims to contribute to the field of military theory by investigating if Robert Leonhard’s theory of combined arms can be used in analyzing such operations and to be able to explaining the outcome of them.   The main question posed in this essay is; to what extent can Leonhard’s theory explain the outcome of military operations in urban terrain? The method used to answer this question is a comparative method formed as a case study of two military operations in urban terrain, Operation Vigilant Resolve and Operation Phantom Fury, which both took place in Fallujah during the Iraq war 2004. The intention is to test Leonhard’s theory in both cases by assigning the theory several indicators, which are tested in each case.   The result shows that the indicators appeared almost equally in both cases, even though one of the operations was said to be a success and the other a failure. However during the occasions when the coalition force used combined arms, both operations achieved military effectiveness. The conclusion is that Leonhard’s theory of combined arms cannot be used to explain the outcome of military operations in urban terrain. The theory has not been verified and strengthened by empirical evidence, but weakened, which is the opposite of what should have been the result. Combined arms-theory cannot be used to assimilate lessons learned from military operations in urban terrain.

Neste trabalho, apresentaremos e definiremos os poliedros, seus elementos e vamos diferenciar os poliedros convexos dos não convexos. Será exposta a Relação de Euler (ou Teorema de Euler), teorema a qual diz: Seja um poliedro convexo com A arestas, F faces e V vértices, vale a igualdade V - A + F = 2. Daremos alguns detalhes sobre poliedros não-convexos. Chegaremos à parte mais importante deste trabalho que é definir os poliedros de Platão (ou regulares) e provar a existência de apenas cinco poliedros regulares: o tetraedro, o hexaedro (cubo), o octaedro, o dodecaedro e icosaedro.