Let p be an odd prime, and define f(x) as follows: f(x) as the sum from 1 to k of a_i times x raised to the power of (p to the power of (alpha_i+1)) in F_(p to the power of n)[x] where 0 is less than or equal to alpha_1 < alpha_2 < ... < alpha_k where alpha_k is equal to alpha. We consider the exponential sum S(f, n) equal to the sum_(x as x runs over the finite field with (p to the n elements) of zeta_(p to the power of Tr_n (f(x))), where zeta_p equals e to the power of (2i times pi divided by p) and Tr_n is the trace from the finite field with p to the n elements to the finite field with p elements.We provide necessary background from number theory and review the basic facts about quadratic forms over a finite field with p elements through both the multivariable and single variable approach. Our main objective is to compute S(f, n) explicitly. The sum S(f, n) is determined by two quantities: the nullity and the type of the quadratic form Tr_n (f(x)). We give an effective algorithm for the computation of the nullity. Tables of numerical values of the nullity are included. However, the type is more subtle and more difficult to determine. Most of our investigation concerns the type. We obtain "relative formulas" for S(f, mn) in terms of S(f, n) when the p-adic order of m is less than or equal to the minimum p-adic order of the alphas. The formulas are obtained in three separate cases, using different methods: (i) m is q to the s power, where q is a prime different from 2 and p; (ii) m is 2 to the s power; and (iii) m is p. In case (i), we use a congruence relation resulting from a suitable Galios action. For case (ii), in addition to the congruence in case (i), a special partition of the finite field with p to the 2n elements is needed. In case (iii), the congruence method does not work. However, the Artin-Schreier Theorem allows us to compute the trace of the extension from the finite field with p to the pn elements to the fi nite field with p to the n elements rather explicitly.When the 2-adic order of each of the alphas is equal and it is less than the 2-adic order of n, we are able to determine S(f, n) explicitly. As a special case, we have explicit formulas for the sum of the monomial, S(ax to the power of (1+ (p to the power of alpha)).Most of the results of the thesis are new and generalize previous results by Carlitz, Baumert, McEliece, and Hou.

Πολλές εφαρμογές ψηφιακής επεξεργασίας σημάτων (DSP) και πολυμέσων μπορούν να ωφεληθούν από τη χρήση ενός αριθμητικού συστήματος υπολοίπων (RNS). Ανάμεσα στους πιο συχνά χρησιμοποιούμενους διαιρέτες σε τέτοια συστήματα είναι αυτοί της μορφής 2^n - 1 και 2^n + 1, ενώ ανάμεσα στις πιο συχνά χρησιμοποιούμενες λειτουργίες βρίσκονται ο πολλαπλασιασμός και το άθροισμα τετραγώνων. Οι λειτουργίες αυτές προς το παρόν υλοποιούνται με τη χρήση ξεχωριστών μονάδων και συνεχόμενων κύκλων μηχανής. Στην παρούσα εργασία προτείνονται δύο αρχιτεκτονικές για μονάδες οι οποίες μπορούν να εκτελέσουν είτε το X * Y είτε το X^2 + Y^2, ανάλογα με την τιμή ενός σήματος ελέγχου. Εξετάζεται τόσο η modulo 2^n - 1, όσο και η ελαττωμένη κατά ένα modulo 2^n + 1 αριθμητική. / Digital signal processing (DSP) and multimedia applications often profit from the use of a Residue Number System (RNS). Among the most commonly used moduli in such systems are those of 2^n - 1 and 2^n + 1 form and among the most commonly used operations are multiplication and sum-of-squares. These operations are currently performed using distinct design units and consecutive machine cycles. In this paper, we propose two architectures for units that perform either the X * Y or the X^2 + Y^2 operation depending on the value of a control signal. Both modulo 2^n - 1 and diminished-1 2^n + 1 arithmetic is considered.

I denna studie undersöks vilka uppfattningar SUM-elever, elever med särskilda utbildningsbehov i matematik, har av matematikundervisning när de i klassrummet får specialpedagogiskt stöd i ämnet och hur de känner sig när de följer med specialläraren ut från klassrummet. Det övergripande syftet med studien är att belysa inkludering och exkludering ur ett elevperspektiv. Studien har en kvalitativ ansats och har genomförts i form av semistrukturerade intervjuer med åtta SUM-elever från åk 4 och 5. Till grund för analysen används Asp-Onsjös (2006) tre olika aspekter på inkludering; rumslig, social och didaktisk.   Resultatet visar att den inkludering som eleverna anser vara viktigast när de lär sig matematik är den didaktiska inkluderingen, inte den rumsliga eller sociala. Alla elever i studien kände sig socialt inkluderade och vad gäller den rumsliga inkluderingen spelade den inte någon större roll för dessa SUM-elever. Däremot visade elevuppfattningarna att de kände sig didaktiskt exkluderade i matematikklassrummet. Faktorer som påverkar detta är bland annat brist på konkret material, långa arbetspass samt möjlighet att kommunicera matematik för att förstå den. / Abstract This study investigates the perceptions that SEM-students, special education need in mathematics, have when they receive special education in mathematics and how they feel when they leave the classroom to do mathematics with the special education teacher. The overall aim of this study is to highlight the inclusion and exclusion from a student perspective. The study has a qualitative approach and has been implemented in the form of semi-structured interviews with eight students in special education need in mathematics, SEM-students, from grades 4 and 5 in Sweden. The basis for our analysis is three different aspects of inclusion; spatial, social and didactic (Asp-Onsjö, 2006). The results show that the aspect of inclusion that students consider to be most important is didactic inclusion. All the students in the study felt socially included and the spatial inclusion did not play a major role for these SEM- students. The students did articulate that they felt didactical excluded in the classroom. Factors that affected this feeling of exclusion was lack of concrete materials, long working hours and the ability to communicate math to understand it.

Atmospheric corrosion indoors is of great practical importance for the degradation of metals, for example in electronics, military equipment, and cultural heritage items. It involves a wide range of chemical, electrochemical, and physical processes occurring in gas, liquid, and solid phases, and at the interfaces between them. Hence, a molecular understanding of the fundamental interactions during atmospheric corrosion is of utmost importance. Copper is one of the most used metals in electrical contacts, power generators, heat exchangers, etc. and is prone to indoor atmospheric corrosion. Although corrosion and oxidation of copper in the presence of corrosion stimulators is thermodynamically inevitable, there are ways to reduce the kinetics of corrosion and oxidation reactions. Self assembled monolayers (SAMs) of organic molecules, when adsorbed on copper surfaces, have proven to be efficient barriers against copper corrosion. However, understanding at the molecular level of the initial stages of corrosion of SAM covered copper in atmospheric corrosion conditions is lacking. The main reason is the inability of the conventional analytical methods to detect and characterize very thin corrosion products formed during the initial stages (from seconds to days) of atmospheric corrosion of SAM covered copper. To overcome this situation a highly surface sensitive technique, vibrational sum frequency spectroscopy (VSFS), has been utilized in situ and ex situ in this thesis to detect and follow the oxidation of alkanethiol SAM covered copper in dry air as well as to assess the conformational changes of SAM molecules during oxidation.  A very sensitive gravimetric method, quartz crystal microbalance with dissipation monitoring (QCM-D), and a highly sensitive and versatile optical technique, nanoplasmonic sensing (NPS), were combined in situ with VSFS to quantify this very slow oxidation process. This combination allowed the heterogeneity of the oxidation process as well as the mass and the rigidity of the corrosion products to be detected simultaneously. To address indoor atmospheric corrosion conditions where carboxylic acids play an important role we next studied the interaction between SAM covered copper and humidified air, to which formic acid was added. The in situ identification of the corrosion products and their formation kinetics was done using near surface sensitive infrared reflection/absorption spectroscopy (IRAS), and the effect of hydrocarbon chain length in alkanethiol SAMs on their corrosion protection efficiency was investigated. The effect of the anchoring group in the SAMs on their corrosion protection efficiency was studied for hexaneselenol using -SeH as the anchoring group, and the results were compared with its thiol counterpart, hexanethiol, with -SH as the anchoring group. Complementary in situ and ex situ VSFS measurements were performed to assess the quality of the SAMs before, during, and after exposure. It was shown that the SAMs of alkanethiols greatly inhibited the formation of copper (I) oxide and slowed down the formation of other corrosion products, i.e. copper formate and copper hydroxid. This was due to a selective hindrance of the corrosion stimulators, oxygen, water, and formic acid molecules reaching the copper-SAM interface. The corrosion inhibiting effect increased with the hydrocarbon chain length. The SAMs of hexaneselenols, on the other hand, exhibited an accelerated formation of copper (I) oxide, copper formate and copper hydroxide compared to an unprotected surface as a result of the partial removal of hexaneselenol molecules from the copper surface during prolonged exposure. The experience gained in characterizing and quantifying thin copper oxides was further used to explore the influence of gamma (γ) radiation on copper corrosion in anoxic water. This multi-analytical approach included IRAS, cathodic reduction, confocal Raman microscope, atomic force microscopy, scanning electron microscopy, x-ray photoelectron spectroscopy, and inductively coupled plasma - atomic emission spectroscopy. The results clearly showed that copper dissolution as well as the oxide layer thickness increase with gamma radiation under the exposure conditions. / Atmosfärisk korrosion under inomhusförhållanden är av stor praktisk betydelse på grund av dess inverkan på exempelvis vårt kulturarv i museimiljöer, tillförlitligheten hos elektronik i olika industriella sammanhang, eller militär utrustning förvarad i olika förråd. Den atmosfäriska korrosionen styrs av ett brett spektrum av kemiska, elektrokemiska och fysikaliska processer som äger rum i tre faser: atmosfären, den tunna fuktfilmen på objektytan och den fasta fasen, samt i de bägge fasgränserna mellan de tre faserna. För att kunna hitta motmedel mot korrosionen är det av yttersta vikt att öka den molekylära förståelsen för dessa processer. Koppar är en mycket använd metall i elektriska eller elektroniska komponenter, i värmeväxlare eller VVS-sammanhang, som beslag och i en rad olika dekorer. Metallen korroderar eller oxiderar spontant i många korrosiva miljöer, men det finns ett brett spektrum av metoder för att minska korrosions- eller oxidationshastigheten. Monoskikt av tätpackade självassocierande organiska molekyler (engelska: self assembled monolayers, förkortat SAM) adsorberade på kopparytan har visat sig vara effektiva barriärer för kopparkorrosion. Den molekylära insikten i dessa monoskikts funktionssätt för att minska den atmosfäriska korrosionen är dock ännu rätt så begränsad. Den främsta orsaken är oförmågan hos mer etablerade analytiska metoder att kunna karakterisera de ytterst små mängder av korrosionsprodukter som bildas under den atmosfäriska korrosionens inledande skeenden upp till några dagars exponering. Den extremt ytkänsliga och i korrosionssammanhang fortfarande relativt oprövade analysmetoden summafrekvensspektroskopi (engelska: vibrational sum frequency spectroscopy, förkortat VSFS) har därför använts för att under pågående exponering följa det mycket långsamma oxidationsförlopp som uppstår när koppar, skyddat av något organiskt monoskikt, exponeras för torr luft. VSFS har även kunnat användas för att under pågående oxidation följa strukturella förändringar hos monoskiktet. För att kvantifiera en så långsam oxidationsprocess har även en annan extremt masskänslig metod kunnat kombineras med VSFS, en kvartskristallmikrovåg med s.k. dissipationsövervakning, förkortat QCM-D. Ytterligare en i korrosionssammanhang oprövad men lika masskänslig teknik har kunnat kombineras med VSFS. Den metoden besitter än så länge bara ett engelskt namn, nanoplasmonic sensing (NPS). Kombinationen VSFS–QCM-D–NPS har utnyttjats i en serie unika försök, där inte bara de ytterst långsamma oxidationshastigheterna kunnat mätas upp, utan även andra viktiga faktorer såsom graden av heterogenitet i den bakomliggande oxidationsprocessen. För att närma sig en miljö som kan efterlikna korrosiva inomhusförhållanden har atmosfären i nästa steg befuktats och dessutom har låga halter av myrsyra tillsats. Just tillsatsen av karboxylsyror har visat sig generera korrosionsprodukter med en sammansättning som på koppar och vissa andra metaller efterliknar de som bildas under atmosfärisk korrosion inomhus. Identifiering av korrosionsprodukter och deras tillväxthastighet på koppar, skyddat av olika långa tätpackade kolkedjor med en tiolgrupp i ena ändan som binder till kopparsubstratet, har kunnat ske med infraröd reflektions-absorptionsspektroskopi (IRAS) under in situ-förhållanden. Ju längre kolvätekedjor desto större korrosionsinhibieringsförmåga kunde påvisas. När den på koppar förankrade tiolgruppen ersattes med en selenolgrupp blev korrosionsinhibieringsförmågan sämre. Kompletterande mätningar in situ och ex situ  utfördes med hjälp av VSFS för att undersöka kvaliteten på de tätpackade kolvätekedjorna, varvid kunde påvisas att graden av tätpackning hos kolkedjorna försämrades med ökad exponeringstid. Förutom den allmänna nedbromsningen av korrosionshastigheten på koppar blev sammansättningen av bildade korrosionsprodukter på oskyddat koppar en annan än på  koppar skyddat av tioler. I det förra fallet detekterades korrosionsprodukterna koppar(I)oxid, koppar(II)format och koppar(II)hydroxid, under det att ingen koppar(I)oxid påvisades på skyddat koppar, endast små mängder koppar(II)format och koppar(II)hydroxid kunde detekteras. De adsorberade kolkedjorna tycks hindra de korrosionsstimulerande molekylerna vatten, myrsyra och syrgas från att nå kopparytan lika effektivt. När de tiolförankrade kolvätekedjorna ersattes med selenolförankrade kolvätekedjor desorberades en del kolvätekedjor från kopparsubstratet vid längre exponeringstider. Resultatet blev att mängden korrosionsprodukter nu blev signifikant större än på oskyddat koppar, sannolikt på grund av galvanisk korrosion. Erfarenheterna från detta doktorsarbete vad gäller kvantifiering av små mängder kopparoxider har även utnyttjats för att undersöka inverkan av g-strålning på kopparkorrosion i rent vatten. Härvid användes ett multianalytiskt angreppssätt bestående av IRAS, katodisk reduktion, konfokal Ramanmikroskopi, atomkraftsmikroskopi, svepelektronmikroskopi, fotoelektronspektroskopi, samt analys av utlöst mängd koppar i vattenlösningen med induktivt kopplad plasmaatomemissionsspektroskopi. Resultaten visar tydligt att utlösningen av koppar, liksom det bildade oxidskiktets tjocklek, ökar med g-strålningen under rådande exponeringsförhållanden. / <p>QC 20131206</p>

Magistro baigiamajame darbe analizuojami matematinės statistikos tikimybiniai skirstiniai. Darbo tikslas – atlikti nepriklausomų beta atsitiktinių dydžių sandaugos pasiskirstymo analizę. Ši tematika aktuali sudėtingų sistemų patikimumo analizės teorijoje. Darbe aprašyta neaprėžtai dalių tikimybinių skirstinių teorija, kuri naudojama atsitiktinių dydžių sumos pasiskirstymui tirti, ir M-dalių tikimybių pasiskirstymo modelių teorija, naudojama nepriklausomų atsitiktinių dydžių sandaugos pasiskirstymo analizei atlikti. Baigiamajame darbe suformuluotos dvi teoremos, nusakančios, kuriais atvejais daugindami baigtinį skaičių nepriklausomų beta atsitiktinių dydžių su skirtingomis parametrų reikšmėmis gauname beta atsitiktinį dydį, kurio parametrai išreikšti per dauginamųjų parametrus. Šios teoremos įrodytos, kuomet dauginame du nepriklausomus beta atsitiktinius dydžius. Taip pat darbe suformuluota ir įrodyta teorema, nurodanti beta atsitiktinio dydžio logaritmo charakteringosios funkcijos analitinę išraišką. / The mathematical statistics probability distributions are analyzed in this master thesis. The main purpose is to carry out the analysis of independent beta random variables products distribution. This theme is relevant to the reliability analysis of complex systems theory. The theoretics of unlimited divisible probability distributions is described in this master thesis. Following theory is useful to investigate the sums of independent random variables. The theoretics of M-divisible probability distributions is also described in this master thesis. It is useful to investigate the product of independent random variables. Two theorems about the product of finite number of independent beta random variables are formulated in this master thesis. Following theorems tells us, that product is beta random variable again, and its parameters expressions are related with multiplicands parameters. Theorems are proved when we are multiplying two independent beta random variables. Another theorem that is about characteristic function of beta random variable logarithm is formulated and proved in this master thesis.

Very specific protein-salt interactions are involved in a multitude of biological phenomena such as protein folding/stability, enzymatic activity, and signal transduction events. In this work, we used two very simple, protein-mimic model biopolymers to obtain a better understanding of specific cation effects operating in aqueous protein environments. The two biopolymers used were Elastin-like Polypeptides (ELPs) and poly(N-isopropylacrylamide) (PNIPAM). ELPs are an especially an ideal model system as these polypeptides can be easily genetically engineered to observe the effect of specific amino acid residues and peptide chain length on these salt interactions. Both of these biopolymers are also highly thermoresponsive as their aqueous solutions undergo a hydrophobic collapse/aggregation induced phase transition process above a lower critical solution temperature (LCST). Thermodynamic measurements of these biopolymers were carried out under various salt solution conditions. Additionally, both of these biopolymers are suitable for making surface specific spectroscopic measurements. Vibrational sum frequeny spectroscopy (VSFS), a non-linear interface sensitive spectroscopic technique, was employed here to investigate biologically relevant cation interactions which occur at peptide/protein surfaces. First, the LCST response of a non-polar ELP and a neutral biopolymer, PNIPAM, was investigated in the presence of 12 different alkali, alkaline-earth metal and transition metal chloride salts. Even though the salt interactions for uncharged proteins are dominated by anions, subtle specific cation effects were also observed. The results followed a direct Hofmeister series for cations. Most alkali cations are excluded from the polar amide regions of proteins. More polarizable cations, however, can solvate the hydrophobic moieties and somewhat counter the salting-out effect of the chloride anion. More charged and hydrated ions like lithium and divalent cations showed a weak interaction to the amide moiety through their hydration shell. The role of acidic amino acid residues in inducing cation specificities was investigated using an aspartate-rich ELP system. Both thermodynamic and spectroscopic data conclusively proved that the negative charge on protein surfaces is the main driving force for cation partitioning and specificity under physiological relevant concentration regimes. Apparent binding constants of carboxylate moieties with cations were determined. This is the first quantitative and thoroughly systematic study of such biologically relevant cation-carboxylate interactions prevalent in enzyme active sites and protein surfaces.

In this essay we use complex analysis, in particular modern residue calculus, to compute certain Riemann integrals. / I den har uppsatsen använder vi komplex analys, då särskilt modern residykalkyl, for att beräkna vissa Riemann-integraler.

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Studien undersöker hur SUM – elever (elever i Särskilda Utbildningsbehov i Matematik), upplever övergången mellan grundskolans år 9 och gymnasieskolans första år. Syftet med studien är att identifiera de parametrar som SUM – elever uppfattar som framgångsfaktorer avseende organisation, pedagogik och motivation. Studien genomfördes som en kvalitativ undersökning med intervju som metod. Sex olika intervjuer genomfördes och det övergripande resultatet är att SUM – elever efterfrågar en mer varierad undervisning med större fokus på konkret material eller praktiska uppgifter som berör deras vardag. Något helt avgörande för SUM – elevers matematikinlärning är även det förhållningssätt läraren har gentemot dessa elever, för att de ska kunna tillägna sig så mycket av matematisk förståelse som möjligt under deras första år på gymnasieskolan.

This thesis introduces normal factor graphs under a new semantics, namely, the exterior function semantics. Initially, this work was motivated by two distinct lines of research. One line is ``holographic algorithms,'' a powerful approach introduced by Valiant for solving various counting problems in computer science; the other is ``normal graphs,'' an elegant framework proposed by Forney for representing codes defined on graphs. The nonrestrictive normality constraint enables the notion of holographic transformations for normal factor graphs. We establish a theorem, called the generalized Holant theorem, which relates a normal factor graph to its holographic transformation. We show that the generalized Holant theorem on one hand underlies the principle of holographic algorithms, and on the other reduces to a general duality theorem for normal factor graphs, a special case of which was first proved by Forney. As an application beyond Forney's duality, we show that the normal factor graphs duality facilitates the approximation of the partition function for the two-dimensional nearest-neighbor Potts model. In the course of our development, we formalize a new semantics for normal factor graphs, which highlights various linear algebraic properties that enables the use of normal factor graphs as a linear algebraic tool. Indeed, we demonstrate the ability of normal factor graphs to encode several concepts from linear algebra and present normal factor graphs as a generalization of ``trace diagrams.'' We illustrate, with examples, the workings of this framework and how several identities from linear algebra may be obtained using a simple graphical manipulation procedure called ``vertex merging/splitting.'' We also discuss translation association schemes with the aid of normal factor graphs, which we believe provides a simple approach to understanding the subject. Further, under the new semantics, normal factor graphs provide a probabilistic model that unifies several graphical models such as factor graphs, convolutional factor graphs, and cumulative distribution networks.