Vi visar att permutationsstatistikerna antal fall och antal inversioner har identisk fördelning med antal vänsterpunkter och total båglängd för länkade partitioner. Även fördelningen för antal cykler och xpunkter bevisas vara identiska. Vidare tittar vi närmare på Mahonska tal, speciellt Mahonska tal med mönsternotation. / We show that the permutationstatistics descents and inversions have identical distribution with the statistics leftpoints and arclength for linked partitions. We also show that the number of cycles and xed points have an identical distribution. Furthermore we take a closer look at Mahonian numbers, specically Mahonian numbers with pattern notation.

I denna uppsats om analytisk fortsättning så ges en intuitiv inblick hur denna process går till. Av analytisk fortsättning får vi ett sätt att utöka bortom den domän där funktionen från början är definierad. En fortsättning kan fås på flera sätt. Det är framförallt en metod som presenteras i uppsatsen, nämligen analytisk fortsättning längs kurvor. Genom att studera homotopier så får vi ett sätt att se på hur dessa kurvor förhåller sig till varandra. Homotopier ger oss också viktiga definitioner som enkelt sammanhängande område, vilket kommer ha stor betydelse när vi ser på en analytisk fortsättning. När vi studerar hur en analytisk fortsättning påverkas av olika områden så fås en viktig sats, nämligen Monodromisatsen som ger oss ett sätt att bygga analytiska funktioner.

The focus of this master thesis is to develop a model that measures the risk-neutral probability distributionof the future value of a portfolio consisting of options on the S&P 500 index. The cornerstone of the model is an explicit and thorough construction of the local volatility surface. The parametric model of Coleman etal. (1998), with some modifications, is used, representing the local volatility surface through a bicubic spline. The local volatility surface is optimized to be consistent with market data on option prices, futures contracts and Overnight Index Swap (OIS) interest rates. Repricing of options is done through a finite difference method (FDM) approach presented by Andersen and Brotherton-Ratcliffe (1998), using the Crank-Nicholson scheme. An interior point solver is used to minimize the squared pricing error weighted by liquidity in each option. Fast and accurate gradients of the objective function are obtained using the Automatic Differentiation library Autograd.The local volatility surface is constructed for multiple dates and the systematic changes are analyzed through Principal Component Analysis (PCA) of the logarithmic local variance. A stochastic process is assigned to the local volatility surface using the sensitivities towards systematic changes identified through the PCA. Using a Gaussian Kernel Density Estimator, the probability density function (PDF) of the future value of the portfolio is measured. The method requires simulated portfolio values, which are achieved through FDM pricing using simulations of the local volatility surface and the underlying index. The cumulative distribution function (CDF) is finally computed by integration of the PDF. To evaluate the measured probability distribution, a normal CDF inversion of 106 measured out-of-sample CDF values are compared to theoretical normal distribution quantiles with Q-Q plots. The constructed local volatility surface is consistent with market prices to an extent where it is more accurate for more liquid options. It is in most cases realistic with respect to smoothness, but have an unexpectedly large offset from the at-the-money strike level in the skew structure. It is unstable from date to date and also significantly dependent on choice of parameters, limited by computational power, and input data. The unstable construction of the local volatility surface results in measurement noise that cause negative auto correlation in the principal components, which impairs their explanatory ability. The main result show that the shape of the probability distribution is measured accurately, but the standard deviation (or volatility) is overestimated.

<p>The subject of the thesis is foundational aspects of partial functions (Papers 1, 2 & 4) and some choice principles (Papers 3 & 4) in the context of constructive mathematics.</p><p>Paper 1 studies the inversion functions of commutative rings. The foundational problem of having them only partially defined is overcome by extending them to total functions. This cannot be done constructively unless the rings themselves are extended at the same time. We study such extensions, called wheels. It is investigated how identities for wheels relate to identities for commutative rings.</p><p>Paper 2 studies the foundations of partial functions in Martin-Löf's type theory according to the view of subsets as propositional functions, in particular in connection with equivalence relations that the functions are supposed to preserve. The first and second isomorphism theorems of algebra are verified, showing that our approach is flexible enough for some natural mathematical proofs to be carried out.</p><p>Paper 3 shows that the difference between the principles of intensional and extensional choice can be described as the principle of excluded middle plus a certain mild extensionality principle, which follows from the principle that functions are identical if they are identical at every point.</p><p>Paper 4 studies a constructive calculus of indefinite and definite descriptions. It has the property that it can be interpreted straightforwardly in type theory with all terms referring to individuals. In this respect it differs from other constructive calculi of descriptions, which are known to be conservative extensions of description-free calculi but for which descriptions cannot be interpreted as referring to individuals in general.</p><p>The appendix includes a predicative version of Birkhoff's theorem. It states that if a class of algebras is closed under homomorphic images, subalgebras and products and contains a set-indexed family of algebras that satisfies the same identities as the class, then the class can be axiomatized by a set of equations.</p>

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<p>Syftet med uppsatsen är att försöka ta reda på om andraspråkselever har svårare med matematik på grund av att de inte behärskar språket eller om det finns andra orsaker till detta. Hur kan man förebygga och åtgärda detta och hur kan undervisningen anpassas till dessa elever?</p><p>Mina frågeställningar är:</p><p>• Har språket en stor betydelse i matematikundervisning?</p><p>• Hur arbetar lärare respektive speciallärare med elever som har invandrarbakgrund i matematik?</p><p>• Finns det andra orsakar till svårigheterna att förstå matematik?</p><p>Jag har gjort intervjuer med tre lärare som alla har flerårig erfarenhet av att undervisa elever med invandrarbakgrund.</p><p>Resultat visar att den stora svårigheten för de här eleverna, är bristen av språkförståelse och underliggande orsaker, som man bär med sig från sitt liv i ett krigshärjat land. Allt tillsammans skapar en brist på koncentration.</p><p>Slutsatsen blir: För elever med invandrarbakgrund är det så viktigt att öka förståelsen och begreppsinlärningen att känna till vad ordet betyder ger en ingång till problemet. Bildens betydelse förstärker tänkandet tillsammans med ordets innebörd och bilden förstärker vi inlärningen.</p>