This thesis presents a case study of mathematical production scheduling optimization applied on Arla Foods AB’s production of dairy products. The scheduling was performed as a possible remedy for problems caused by overcrowded finished goods warehouse. Based on the scheduling, conclusions were made on whether the existing two-shift production is sufficient or if an additional night shift should be introduced. In parallel, an empirical and theoretical analysis on the perceived effects of night shift work on employees was conducted. For the optimization, mixed integer programming was used to model the production context through a discrete time scheduling lot-sizing model developed in this thesis. The model developed and implemented on Arla Foods AB contributes to the research field through its feature of relatively low complexity enabling scheduling of extensive production systems when applied in industrial contexts where products may be categorized. The thesis concludes that mathematical production scheduling can solve Arla Foods AB’s production problematics and suggests reallocation of the existing shifts for the purpose of reduced costs and acceptable warehouse levels. This reallocation would incur production during inconvenient hours whereas management remedies reducing negative effects of night shift work are identified. / Denna avhandling innefattar en studie av matematisk optimering av produktionsscheman applicerad på Arla Foods ABs produktion av mejeriprodukter. Schemaläggningen utfördes som en möjlig lösning på produktionsproblematik orsakad av överfyllda färdigvarulager. Utifrån de optimerade produktionsschemana drogs slutsatser kring om dagens produktionsstruktur på två skift är tillräcklig eller om introduktion av ett andra nattskift skulle vara fördelaktig. Parallellt med detta presenteras en empirisk och teoretisk studie kring de produktionsanställdas uppfattning kring effekter av att arbeta nattskift. För optimeringen har heltalsoptimering (eng: mixed integer programming) använts för modellering av produktionen genom en produktionsplaneringsmodell med diskret tidsrepresentation (eng: discrete time scheduling lot-sizing model ) som utvecklas i denna avhandling. Denna model, som även appliceras på Arla Foods ABs produktion, presenteras i detalj och karaktäriseras av låg komplexitet vilket möjliggör schemaoptimering av omfattande produktionssystem givet att produktportföljen kan kategoriseras i produktgrupper med liknande egenskaper ur ett produktionsperspektiv. Avhandlingen fastslår att matematisk optimering av produktionsscheman har potential att lösa produktionsproblematiken på Arla Foods AB och föreslår en reallokering av den nuvarande produktionen för minskade kostnader och utjämnade nivåer i färdigvarulager. Produktionsomläggningen skulle innebära produktion under obekväm arbetstid vilket föranleder en analys av initiativ som har potential att minska de negativa effekterna av nattskiftarbete för de produktionsanställda.

Abstract Purpose – The purpose of this thesis is to give the investors a better understanding on how to interpret the costs of funds with performance fee with high-water mark and give some guidelines when comparing funds with different fee structures, i.e. mutual funds and hedge funds. Mathematical approaches – Two mathematical approaches are used in the study. The first approach is to describe the high-water mark contract as a partial differential equation, which has the characteristics of Black-Scholes equation. The second approach is to numerically simulate the evolution of a fund’s value. During the development of the fund’s value the cost of the fees are calculated and discounted. Findings – It is found that the expected cost of the performance fee with high-water mark, vary a lot. An example is when the volatility increases the expected cost of performance fee drastically raises while the management fee is unchanged. Another interesting finding is that the order of when the fees’ are charged affects the expected cost of the performance fee. Conclusion – The guidelines for the investor is to invest in a fund with a performance fee in low volatile markets and a fund with just the management fee in high volatile markets. Another impact is the time step which the high-water mark level is controlled. The investor wants these controls as infrequently as possible. If the controls are done at a daily basis the expected cost of the performance fee is higher than in a monthly control. It is also concluded that the Normanbelopp of a fund with a performance fee should not be trusted. Key-words:

This paper explores how the five largest trading venues in the Nordic region compete after theimplementation of MiFID in November 2007. I investigate: (1) if NASDAQ OMX’s market sharehas increased post the introduction of major changes to its market structure, and (2) how anexchange operator can attract equity share order flow in the near future. By applying event studiesto NASDAQ OMX’s market share over time, I find that introducing a faster trading system andadmitting a high frequency trading firm as a member both have a negative impact on NASDAQOMX’s market share. The reductions in market share can be explained by high frequency tradingfirms’ trading behaviour. Introducing central counterparty clearing has a positive effect onNASDAQ OMX’s market share, which highlights market participants’ appreciation of a securetrading environment, and confirms that it is not only posting the best bid and ask quotes thatattracts order flow to an exchange. It can be concluded that NASDAQ OMX in the future needs toaddress an important trade-off between total turnover and market share, as the two are not alwayspositively correlated.

The recent observed growth in the diffusion of Special Purpose Acquisition Companies phenomena on the U.S stock market may be analyzed from a mathematical standpoint, where different approaches of the Bass Diffusion Model might be utilized. The Bass diffusion model originates from analysis of product diffusion, where only a few applications have been seen by financial scholars. The thesis takes a multi analytical approach to examine the phenomena, where multiple regression analysis and Bayesian statistics are used in the parameter estimation processes. Estimated parameter are applied in three different scenarios of expressing the Bass diffusion model in a discrete time state. By utilizing these different approaches that arise, the study shows that the diffusion of Special Purpose Acquisition Companies Initial Public Offerings in fact can be analyzed from a mathematical standpoint utilizing the Bass diffusion model. Some approaches and scenarios indicate better results in terms of fitting the diffusion, while purposing practical actualities towards the reader and market practitioners. The study further purposes potential modifications that might improve the results of fitting the phenomena

This report presents and discusses proofs of the Malgrange-Ehrenpreis theorem, which states that every non-zero linear partial differential operator with constant coefficients has a fundamental solution. The main topic is explicit formulae, and more specifically, how they can be used to prove the theorem. Two different formulas will be considered in detail and the aim is to provide a fundamental and elementary description of how to prove the Malgrange-Ehrenpreis theorem using those formulas. In addition to the proofs, an example of how to use one of the formulas for the Cauchy-Riemann operator is shown. Finally, the report also contains a chapter discussing a few different notable methods of proof and their historical signifance.

This thesis presents how Tikhonov’s regularization can be used to solve an inverse problem of Helmholtz equation inside of a rectangle. The rectangle will be met with both Neumann and Dirichlet boundary conditions. A linear operator containing a Fourier series will be derived from the Helmholtz equation. Using this linear operator, an expression for the inverse operator can be formulated to solve the inverse problem. However, the inverse problem will be found to be ill-posed according to Hadamard’s definition. The regularization used to overcome this ill-posedness (in this thesis) is Tikhonov’s regularization. To compare the efficiency of this inverse operator with Tikhonov’s regularization, another inverse operator will be derived from Helmholtz equation in the partial frequency domain. The inverse operator from the frequency domain will also be regularized with Tikhonov’s regularization. Plots and error measurements will be given to understand how accurate the Tikhonov’s regularization is for both inverse operators. The main focus in this thesis is the inverse operator containing the Fourier series. A series of examples will also be given to strengthen the definitions, theorems and proofs that are made in this work.

In this thesis, we study integral representations of Herglotz-Nevanlinna functions, that is to say holomorphic functions defined on a product of several copies of the complex upper half-plane having non-negative imaginary part. The manuscript is divided into three parts, beginning with a general introduction followed by two papers. In the general introduction, we familiarize ourselves with the concept of a Herglotz-Nevanlinna function as well as providing a comprehensive introduction into the theory of integral representations for this particular class of functions. Paper I treats exclusively the two-variable case and presents an integral representation of Herglotz-Nevanlinna functions in two complex variables in terms of a real number, two non-negative numbers and a positive Borel measure satisfying two properties. Three properties that hold for the class of measures appearing in such integral representations are also proven. In Paper II, we provide an integral representation for the class of Herglotz-Nevanlinna functions in arbitrarily many complex variables in terms of a real number, a linear term and a positive Borel measure satisfying two properties. Properties of the class of measures appearing in this representation are then discussed in detail as well as alternative descriptions of said class. Finally, a symmetry formula satisfied by Herglotz-Nevanlinna functions is proved at the end.

Pólya and Schur completely characterized all real-rootedness preserving linear operators acting on the standard monomial basis in their famous work from 1914. The corresponding eigenvalues are from then on known as multiplier sequences. In 2009 Borcea and Br\"and\'en gave a complete characterization for general linear operators preserving real-rootedness (and stability) via the symbol. Relying heavily on these results, in this thesis, we are able to completely characterize multiplier sequences for generalized Laguerre bases. We also apply our methods to reprove the characterization of Hermite multiplier sequences achieved by Piotrowski in 2007.

In this thesis, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schroedinger equation with quartic potentials. In the first paper, we consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k > 2 boundary conditions, except for the case d is even and k = d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions. In the second paper, we only consider even polynomial potentials, and show that the spectral determinant for the eigenvalue problem consists of two irreducible components. A similar result to that of paper I is proved for k boundary conditions.

This report investigates the relationship between the yield volatility and the price volatility in the Swedish market. The method given in our report can be used to analyze any market with appropriate data set. We have used a time-series data of interest rate yield curves from Swedish government bonds. The curves are bootstrapped from the bills and bonds. The linear interpolation on these curves results in the nodes i.e. 1Y, 2Y,..., 10Y. We also need prices for instruments. A good choice is to use the synthetic government bonds namely SE GVB 2Y, SE GVB 5Y, and SE GVB 10Y. They are issued every day with maturity 2, 5, and 10 years. We also use the time-series of these bonds. These bonds have a yearly coupon of 6%. We can get zero-coupon values of these bonds by stripping their coupons using the interest rate yield curves. We have time-series data of zero-coupon prices with maturities 2, 5, and 10 years and time-series data of interest rates with the same tenors. We can use our data to calculate their respective volatilities to investigate how they are related to each other.