The concepts of continuity for set-valued mappings were introduced by G. Bouligand and K. Kuratowski. There are two ways defining differentiability of set-valued mapping. One is defined by classical differentiability theorem and another is defined by normal cone which was introduced by B.S. Mordukhovich. In this thesis, we survey various definitions of continuity and differentiability for set-valued mapping.

The Pacific Crest Trail has become increasingly popular since Clinton Clarke first envisioned such a trail in the 1930’s. By comparing the original motives and experience of the trail to the realities of the trail today, the trail’s fluid narrative becomes apparent. While this narrative is ever changing, over the course of the trail’s history one theme has remained constant – a notably problematic relationship with wilderness rooted in an exaltation of the sublime and post-frontier ideals. This thesis focuses on how the Pacific Crest Trail’s development over the past eighty years has created an experience that, on the surface, is notably different from Clarke’s original vision for the trail, but is still influenced by a perception of wilderness born from a romanticization of nature and a pursuit to preserve the western frontier. Chapter one, The Historic Trail, investigates Clarke’s manners and motives behind promoting the trail. Chapter two, The Popular Trail, examines the visual culture surrounding the trail, from nineteenth century landscape painting to the trail’s presence in social media today. Chapter three, The Trail Community, focuses on the growth of a strong community of hikers, and what this means for the future of the trail.

Since the beginning of the 20th century air power has been seen as an offensive instrument in warfare. During the Cold War, Sweden had one of the largest air forces in the world despite a clearly defensive strategy. Today's Swedish Air Force has been reformed since the Cold War and there are uncertainties about how this has changed the use of air power. The purpose of this study is to investigate what changes occurred in the Swedish Air Force related to offensive and defensive use of air power. This is illustrated by examining and comparing the use of air power in the Air Force during the Cold War and today. The study is based on the theories of air power and illustrates the use of air power from multiple perspectives in order to provide as broad picture of the Air Force as possible. The results demonstrate that a change has occurred where air forces moved more towards working together with other forces over a broader range of applications than during the Cold War.

Until the recent publication of twelve choral compositions, Rebecca Clarke (1886-1979) was known solely as a professional violist and composer of chamber music and art songs. Clarke composed choral music throughout her active period from 1906 to 1944. In 2004, the first study of Clarke's complete compositional output provided an introduction to the choral music, but only covered selected works. The present study traces the development of Clarke's compositional style through chronological analysis of all twelve choral compositions and an incomplete fragment. Clarke's choral music reveals the selection of high quality, expressive texts; exploration of the timbral, registral, and textural potential of unaccompanied choral music; changes in the treatment of all musical elements; the persistent acquisition of new techniques; and reliance on English choral genres including the madrigal, glee, carol, part song and motet. Chapter one establishes Clarke's importance through a survey of publication, criticism, and scholarship. The chapter also examines the society in which Clarke lived and the issues women composers encountered. A biography then reveals that despite obstacles, Clarke tenaciously pursued compositional study, eagerly acquired new techniques, and expressed enthusiasm for each compositional project. Her skill was confirmed by success in competitions and festivals. Throughout her active period, Clarke supported herself as a professional violist who specialized in chamber music, and a busy performing schedule limited her compositional work. Chapter two documents Clarke's formative vocal- and chamber-music experiences and suggests that her thorough knowledge of chamber music influenced her approach to choral composition. The chapter continues with analysis of Clarke's first seven choral works. The first three are well-crafted part songs that demonstrate Clarke's assimilation of basic compositional techniques. The next four show the increasing complexity of Clarke's style that culminates in her mature masterpiece, "He That Dwelleth in the Secret Place of the Most High." Chapter three presents analysis of three choral arrangements, two works for women's voices, and a choral fragment for mixed voices. The last five complete choral compositions confirm elements of Clarke's mature style and demonstrate her interest in exploring the new challenges of choral arranging and writing for women's voices. While Clarke's choral arrangements of her own songs are idiomatic adaptations for unaccompanied, mixed voices, the last three compositions display the diversity of styles Clarke employed in her late works. Chapter four summarizes changes in Clarke's choral style from 1906 to 1944, examines reasons for her obscurity, and raises questions that merit further research. The appendix which follows clarifies Clarke's intentions and illustrates common editorial issues and solutions through comparison of choral manuscripts and published editions.

Framträdande luftmaktsteorier förefaller i regel vara skrivna av stormaktsföreträdare. Ett implicit antagande att dessa teorier förutsätter betydande militär kapacitet och sofistikerad teknologi gör att de uppfattas som ”stormaktsteorier” som likt en klubba slår hårt mot motståndaren, med begränsad relevans för mindre stater som saknar stormakternas omfattande resurser. Luftmakt finns dock också i länder som kan betraktas som småstater. Även småstater kan – teoretiskt – antingen enskilt eller i en småstatsallians genom välriktade nålstick också skapa strategisk effekt med befintliga resurser.   Denna studie utmanar antagandet att luftmaktsteorier kan betraktas som antingen stormakts- eller småstatsteorier. Genom att pröva Robert Papes luftmaktsteori och Shaun Clarkes SPOT-paradigm i såväl stormakts- som småstatsperspektivet på ett gemensamt fall, Georgienkriget 2008 mellan Ryssland och Georgien, skapas ny kunskap om förklaringskraften hos respektive teori. Studien visar att Papes luftmaktsteori kan förklara stormakters användning av luftmakt, men belägg saknas för att den också kan förklara småstaters luftmaktsanvändning. Vidare saknas belägg för att Clarkes SPOT-paradigm kan förklara vare sig småstaters eller stormakters användande av luftmakt i det aktuella fallet. Däremot visar analysen att SPOT-paradigmet förutsätter ett visst mått av kvantitet och kvalitet, även hos småstater, för att omsättas i en skarp operation. Då ett mått på detta inte ges, bör SPOT-paradigmet främst betraktas som ett förhållningssätt till hur strategisk effekt kan skapas med tillgängliga resurser.

There are two types of air forces; the US, and the others. Most of what is written today concerning airpower comes from the pens of large nations. This has led to small nations view of airpower is being characterized by the theories of larger nations. Shaun Clarke questions whether such writings are applicable to the airpowers of the world’s small nations. Clarke’s theory explores the potential of offensive airpower in the context of small sized defense forces. His airpower theory has often been praised with the exception that it’s not empirically substantiated. The aim of this study is to test and analyze the explanatory power of Clarke’s theory on the Yom Kippur War in 1973 and the Second Lebanon War in 2006 with the purpose of creating empirical evidence in support of, or against, Clarke’s theory.   The results of the study are mixed. Clarke’s theory has a relatively high explanatory power about Israel’s success during the Yom Kippur War and low explanatory power during the Second Lebanon War. Finally, this study argues that smaller nations shouldn’t be discouraged from applying offensive airpower in a strategic manner.

Den engelska kompositören Rebecca Clarke (1886–1979) komponerade vokalmusik med stor skicklighet och kvalitet men hennes verk är inte allmänt kända. I detta konstnärliga examensarbete analyseras ett av hennes viktigaste verk för soloröst och piano; The Seal Man (1922). Det är en romans som är baserad på en gammal keltisk legend om en man som är till hälften säl. Studien riktar fokus på Clarkes impressionistiska kompositionsstil och textens relation till musiken. Detta analyseras metodiskt genom instudering av partitur och läsning av författaren John Masefields (1878–1964) text som Clarkes The Seal Man är baserad på. Studiens resultat påvisar Clarkes skicklighet att i sin komposition gestalta olika karaktärer och stämningar i texten genom musikaliska medel samt influenser från Claude Debussy som var en av impressionismens främsta kompositörer. För mig som sångerska har noggrann analys av Clarkes stil och textens relation med musiken varit till stor hjälp för att uppnå min egna konstnärligt mogen interpretation av sången The Seal Man. / <p>Programm:</p><p>Rebecca Clarke - The Seal Man</p><p>Gösta Nystroem - Själ och landskap 1. Vitt land, 2. Önskan, 3. Bara hos den...</p><p>Benjamin Britten - On this Island Op. 11, 1. Let the florid music praise!, 2. Now the leaves are falling fast, 3. Seascape, 4. Nocturne, 5. As it is, plenty</p><p></p><p>Emmi Kauppinen, sopran</p><p>Chiara Schmidt, piano</p>

Sammanfattning: Teorier om luftmakt tar sin historiska utgångspunkt i resursstarka stormakters behov, varför de förutsatt stora resurser och avancerad teknik. För småstater har detta förhållande varit problematiskt då de varit tvungna att använda sig av stormakternas luftmaktsteorier trots att förutsättningar saknades i form av resurser och teknik. Under 2000-talet har några forskare utarbetat luftmaktsteorier utformade för småstater, vars förklaringskraft prövas i föreliggande studie.  Föreliggande studies syfte är att förklara effekten av luftmakt med luftmaktsteorier speciellt framtagna för småstater och teorierna som prövas är Clarkes modell ´Strategic, Persuasion, Oriented, Targeting bombing´ (SPOT) och Pashakhanlous ´The Underdog´s Model´ (UM).   Luftkriget i Ukraina är fallet som valts för att pröva teoriernas förklaringskraft mot ett aktuellt fall.  Ukraina är ett stort land med en stor befolkning, men knappa resurser gör att en landet har en större nytta av att tillämpa en småstatsstrategi än en storstatsstrategi. Studien visar att Ukraina uppnår bra resultat både inom SPOT- och UM:s modellkriterier. Resultatet påvisar att en småstats möjligheter att påverka eller vinna mot en starkare motståndare ökar om teoriernas kriterier tillämpas i sin helhet eller i delar. Studien har inte funnit några nya eller avvikande kriterier för mindre staters framgångar i luftmaktskriget.

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In this work we present a new tool for the convergence analysis of numerical optimization methods. It is based on the concepts of the Clarke derivative and set-valued mappings. Our goal is to apply this tool to minimization problems with non-smooth and noisy objective functions. After deriving a necessary condition for minimizers of such functions, we examine two unconstrained optimization routines. First, we prove new convergence theorems for Implicit Filtering and General Pattern Search. Then we show how these results can be used in practice, by executing some numerical computations. / Master of Science