Concentrated Winding Multiphase Permanent Magnet Machine Design and Electromagnetic Properties – Case Axial Flux Machine

Concentrated winding permanent magnet machines and their electromagnetic properties are studied in this doctoral thesis. The thesis includes a number of main tasks related to the application of permanent magnets in concentrated winding open slot machines. Suitable analytical methods are required for the first design calculations of a new machine. Concentrated winding machines differ from conventional integral slot winding machines in such a way that adapted analytical calculation methods are needed. A simple analytical model for calculating the concentrated winding axial flux machines is provided. The next three main design tasks are discussed in more detail in the thesis. The magnetic length of the rotor surface magnet machines is studied, and it is shown that the traditional methods have to be modified also in this respect. An important topic in this study has been to evaluate and minimize the rotor permanent magnet Joule losses by using segmented magnets in the calculations and experiments. Determination of the magnetizing and leakage inductances for a concentrated winding machine and the torque production capability of concentrated winding machines with different pole pair numbers are studied, and the results are compared with the corresponding properties of integral slot winding machines. The thesis introduces a new practical permanent magnet motor type for industrial use. The special features of the machine are based on the option of using concentrated winding open slot constructions of permanent magnet synchronous machines in the normal speed ranges of industrial motors, for instance up to 3000 min-1, without excessive rotor losses. By applying the analytical equations and methods introduced in the thesis, a 37 kW 2400 min-1 12-slot 10-pole axial flux machine with rotor-surfacemounted magnets is designed. The performance of the designed motor is determined by experimental measurements and finite element calculations.

Cultural and textual properties in the translation and interpretation of allusions. An analysis of allusions in Dorothy L. Sayers’ detective novels translated into Finnish in the 1940s and the 1980s

Kulttuuriset ja tekstuaaliset tekijät alluusioiden kääntämisessä ja tulkinnassa. Alluusiot Dorothy L. Sayersin 1940- ja 1980-luvuilla suomennetuissa salapoliisiromaaneissa Väitöskirja käsittelee alluusioiden kääntämistä ja tulkintaa. Alluusio on intertekstuaalinen viittaus, jonka tulkitsemiseen tarvitaan implisiittistä tietoa tutuksi oletetusta viittauskohteesta. Käännösongelma alluusiosta tulee, mikäli kohdekulttuurin lukijat eivät tunne viittauskohdetta eivätkä voi päätellä alluusion merkitystä. Tutkimus pyrkii kuitenkin uuden analyysimenetelmän avulla osoittamaan, että vieraat alluusiot eivät välttämättä johda tulkintaongelmiin. Väitöskirja jakautuu kahteen osaan: analyysimenetelmän kehittämiseen (luvut 1-5) sekä tapaustutkimukseen (luvut 6-7). Kehitetyn menetelmän avulla pystytään analysoimaan aikaisempaa tarkemmin, millaisia tulkintamahdollisuuksia alluusiot tarjoavat eri lukijakunnille ja miten lähdetekstin alluusioiden kulttuuriset ja tekstuaaliset piirteet korreloivat käännösstrategioiden kanssa. Tapaustutkimus selvittää, millaisia tulkintamahdollisuuksia Dorothy L. Sayersin 1940- ja 1980-luvuilla suomennettujen salapoliisiromaanien alluusiot tarjosivat aikansa suomalaisille lukijoille. Tavoitteena on myös hahmottaa, miten suomentajien käännösratkaisut ja alluusioiden tulkintamahdollisuudet liittyvät toisaalta lähdetekstin alluusioiden piirteisiin ja toisaalta kohdekulttuurin kontekstiin. Tapaustutkimus tarjoaa näin uutta tietoa suomennoskirjallisuuden ja salapoliisiromaanien historiasta. Analyysimenetelmä määrittelee aikaisempaa alluusioita ja intertekstuaalisuutta käsitelleen tutkimuksen pohjalta ne kulttuuriset ja tekstuaaliset piirteet, jotka vaikuttavat alluusioiden kääntämiseen ja tulkintaan. Kulttuurisessa mielessä alluusio voi olla tietylle lukijakunnalle tuttu tai tuntematon. Tekstuaalisia tekijöitä ovat alluusion muodon ja tyylin tunnusmerkillisyys sekä alluusion pintamerkityksen koherenttius uudessa tekstikontekstissa, ilman tietoa viittauskohteesta. Alluusioiden tulkinnassa on perinteisesti erotettu toisaalta allusiivinen tulkintamahdollisuus, jossa alluusio on lukijoille tuttu ja yhdistettävissä viittauskohteeseensa, toisaalta kulttuuritöyssy, jonka muodostaa lukijoille tuntematon ja pintamerkitykseltään inkoherentti alluusio. Tutkimuksessa määritellään kulttuuristen ja tekstuaalisten tekijöiden perusteella lisäksi kaksi muuta mahdollisuutta. Pseudo-allusiivisessa tulkinnassa tuntematon alluusio erottuu ympäröivästä tekstikontekstista tyyliltään ja on koherentti ainakin kuvaannollisessa mielessä ilman viittauskohdettaan. Ei-allusiivisessa tulkinnassa taas vieras alluusio sulautuu kontekstiin sekä muodoltaan että merkitykseltään niin, ettei lukija edes huomaa mahdollista alluusiota. Tulkintamahdollisuuksien jakauma antaa yleiskuvan siitä, miten tietty lukijakunta pystyi tulkitsemaan tekstin alluusioita. Lisäksi analyysi tarkastelee lähdetekstin ja käännöksen välillä tulkintamahdollisuuksissa tapahtuneita muutoksia sekä niiden vaikutusta tulkinnan vaatimaan vaivannäköön (effort) ja alluusion funktioihin. Tapaustutkimus perehtyy Sayers-suomennosten kulttuurikonteksteihin tarkastelemalla salapoliisiromaanien asemaa suomalaisessa kirjallisuusjärjestelmässä, suomennoksilta odotettuja piirteitä sekä suomentajien ammattikuvaa, taustoja ja työoloja. Tulosten perusteella alluusioiden kääntäminen oli vaativa tehtävä sekä 1940- että 1980-luvun suomentajille. Lähdetekstien alluusioista 60–70 prosenttia oli todennäköisesti kohdelukijoille vieraita. Molempina aikakausina suomennoksilta odotettiin silti sekä kielellistä sujuvuutta että lähdetekstin merkitysten välittämistä. 1940-luvun suomentajien tehtävää vaikeutti lisäksi mm. se, että suomentaminen oli enimmäkseen sivutoimista ja englanti oli harvoin parhaiten hallittu vieras kieli. Nämä olosuhteet lienevät vaikuttaneet etenkin vähäarvoisena pidetyn salapoliisikirjallisuuden suomennoksiin. 1980-luvulla suomentajien aikataulut olivat realistisempia, englannin taidot parempia ja päätoiminen suomentaminen mahdollista. Myös salapoliisiromaanien arvostus oli lisääntynyt. Sekä 1940- että 1980-luvun suomennoksissa kohdelukijoille vieraitakin alluusioita oli usein säilytetty, mikäli ne olivat koherentteja ilman viittauskohdettaan. Sen sijaan vieraita ja pintamerkitykseltään epäselviä alluusioita oli muokattu tai poistettu. Kuitenkin 1980-luvun suomentajat säilyttivät lähdetekstin alluusioita useammin ja tarkemmin kuin 1940-luvun suomentajat. Varsinkin poisjättämistä esiintyi 1940-luvun suomennoksissa enemmän. Alluusioiden tulkintamahdollisuudet olivat kaikissa käännöksissä muuttuneet sikäli, että melko harvat suomennetut alluusiot olivat enää kohdelukijoiden tunnistettavissa. Toisaalta myös kulttuuritöyssyt olivat harvinaisia. Erot 1940- ja 1980-luvun suomennosten välillä näkyivätkin pseudo-allusiivisissa ja ei-allusiivisissa tulkintamahdollisuuksissa. 1980-luvun suomennoksissa vieraat alluusiot oli johdonmukaisesti säilytetty niin, että käännetyt alluusiot voitiin tulkita pseudo-alluusioiksi. Sen sijaan 1940-luvun suomennoksissa vieraita alluusioita oli usein muokattu tai jätetty pois tavalla, joka johti ei-allusiiviseen tulkintaan. Kohdelukijoiden kannalta 1980-luvun suomennettujen alluusioiden tulkitseminen lienee vaatinut jonkin verran enemmän vaivaa. Toisaalta pseudo-allusiivisten käännösten pohjalta oli useimmiten mahdollista rakentaa koherentti tulkinta, ja monesti ne jopa välittivät samankaltaisia funktioita kuin lähdetekstin alluusiot. 1940-luvun suomennosten muokkaukset ja poistot periaatteessa helpottivat tulkintaa, mutta mahdollisia kulttuuritöyssyjä esiintyi edelleen, jopa kääntäjän tekemien muutosten seurauksena. 1940-luvun suomennoksissa myös käännettyjen alluusioiden funktiot olivat muuttuneet enemmän lähdetekstin alluusioihin nähden. Kaiken kaikkiaan 1980-luvun suomennokset olivat lähempänä oman aikansa hyvän käännöksen piirteitä. Toisaalta alluusioiden muokkaaminen sai 1940-luvun suomennokset muistuttamaan enemmän perinteistä arvoituksen ratkaisuun keskittyvää salapoliisiromaania, joten tältä osin ne lienevät vastanneet kohdelukijoiden odotuksia. Kulttuurikontekstin vaikutus siis näkyi sekä käännösstrategioissa että käännettyjen alluusioiden tulkintamahdollisuuksissa. Tutkimustuloksissa korostui kuitenkin myös se, että lähdetekstin alluusion pintamerkitys saattaa vaikuttaa käännösratkaisuihin. Lisäksi käännetyt pseudo-alluusiot saattavat välittää samankaltaisia funktioita kuin lähdetekstin alluusiot. Toisin kuin yleensä on esitetty, kohdelukijoille vieraiden alluusioiden säilyttäminen saattaakin siis olla toimiva ratkaisu.

Magnetic and transport properties of LaMnO3+δ, La1-xCaxMnO3, La1-xCaxMn1-yFeyO3 and La1-xSrxMn1-yFeyO3

Interest to hole-doped mixed-valence manganite perovskites is connected to the ‘colossal’ magnetoresistance. This effect or huge drop of the resistivity, ρ, in external magnetic field, B, attains usually the maximum value near the ferromagnetic Curie temperature, TC. In this thesis are investigated conductivity mechanisms and magnetic properties of the manganite perovskite compounds LaMnO3+, La1-xCaxMnO3, La1-xCaxMn1-yFeyO3 and La1- xSrxMn1-yFeyO3. When the present work was started the key role of the phase separation and its influence on the properties of the colossal magnetoresistive materials were not clear. Our main results are based on temperature dependencies of the magnetoresistance and magnetothermopower, investigated in the temperature interval of 4.2 - 300 K in magnetic fields up to 10 T. The magnetization was studied in the same temperature range in weak (up to 0.1 T) magnetic fields. LaMnO3+δ is the parent compound for preparation of the hole-doped CMR materials. The dependences of such parameters as the Curie temperature, TC, the Coulomb gap, Δ, the rigid gap, γ, and the localization radius, a, on pressure, p, are observed in LaMnO3+δ. It has been established that the dependences above can be interpreted by increase of the electron bandwidth and decrease of the polaron potential well when p is increased. Generally, pressure stimulates delocalization of the electrons in LaMnO3+δ. Doping of LaMnO3 with Ca, leading to La1-xCaxMnO3, changes the Mn3+/Mn4+ ratio significantly and brings an additional disorder to the crystal lattice. Phase separation in a form of mixture of the ferromagnetic and the spin glass phases was observed and investigated in La1- xCaxMnO3 at x between 0 and 0.4. Influence of the replacement of Mn by Fe is studied in La0.7Ca0.3Mn1−yFeyO3 and La0.7Sr0.3Mn1−yFeyO3. Asymmetry of the soft Coulomb gap and of the rigid gap in the density of localized states, small shift of the centre of the gaps with respect to the Fermi level and cubic asymmetry of the density of states are obtained in La0.7Ca0.3Mn1−yFeyO3. Damping of TC with y is connected to breaking of the double-exchange interaction by doping with Fe, whereas the irreversibility and the critical behavior of the magnetic susceptibility are determined by the phase separation and the frustrated magnetic state of La0.7Sr0.3Mn1−yFeyO3.

Effect of pulps fractionation on formation and strength properties of laboratory hand sheets

The objectives of the work were to study the effect of dewatering time varying on formation properties of papersheets, to determine the role of fines fraction in creation of paper with good formation and strength properties of papersheets, and also to study the effect of charge modification of fibers fractionations on formation properties of handsheets. The paper formation is one of the most important structural properties of paper. This property has effect on physical and optical characteristics of paper. In thi work the effect of formation on tensile strength was determined. The formation properties were analyzed by using the AMBERTEC Beta Formation Tester. The PAM addition as a f;locculant agent did some changes in the formation of paper. Paper sheets were also made from different furnishes of both birch and pine pulps. The fibers particles as a fines have great effect on drainability changes. Fines fraction played important role in papermaking. The two kinds of pulps (pine and birch pulps) were also used in this work for investigation of fines role. As it was expected the fines fraction gave positive effect on paper formation, but when fines fraction was added above initial fines content the formation of paper was deteriorated. The effect of paper formation on tensile strength was also determined. In many cases the poor formation of paper had negative effect on strength properties of paper..

Dynamical and structural properties of dendrimer macromolecules

This work is devoted to the study of the dynamical and structural properties of dendrimers. Different approaches were used: analytical theory, computer simulation results and experimental NMR studies. The theory of the relaxation spectrum of dendrimer macromolecules was developed. Relaxation processes which are manifest in the local orientational mobility of dendrimer macromolecules were established and studied in detail. Theoretical results and conclusions were used for experimental studies of carbosilane dendimers.

Growth, magnetic and transport properties of InSb and II-IV-As2 semiconductors doped with manganese

This thesis is devoted to growth and investigations of Mn-doped InSb and II-IV-As2 semiconductors, including Cd1-xZnxGeAs2:Mn, ZnSiAs2:Mn bulk crystals, ZnSiAs2:Mn/Si heterostructures. Bulk crystals were grown by direct melting of starting components followed by fast cooling. Mn-doped ZnSiAs2/Si heterostructures were grown by vacuum-thermal deposition of ZnAs2 and Mn layers on Si substrates followed by annealing. The compositional and structural properties of samples were investigated by different methods. The samples consist of micro- and nano- sizes clusters of an additional ferromagnetic Mn-X phases (X = Sb or As). Influence of magnetic precipitations on magnetic and electrical properties of the investigated materials was examined. With relatively high Mn concentration the main contribution to magnetization of samples is by MnSb or MnAs clusters. These clusters are responsible for high temperature behavior of magnetization and relatively high Curie temperature: up to 350 K for Mn-doped II-IV-As2 and about 600 K for InMnSb. The low-field magnetic properties of Mn-doped II-IV-As2 semiconductors and ZnSiAs2:Mn/Si heterostructures are connected to the nanosize MnAs particles. Also influence of nanosized MnSb clusters on low-field magnetic properties of InMnSb have been observed. The contribution of paramagnetic phase to magnetization rises at low temperatures or in samples with low Mn concentration. Source of this contribution is not only isolated Mn ions, but also small complexes, mainly dimmers and trimmers formed by Mn ions, substituting cation positions in crystal lattice. Resistivity, magnetoresistance and Hall resistivity properties in bulk Mn-doped II-IV-As2 and InSb crystals was analyzed. The interaction between delocalized holes and 3d shells of the Mn ions together with giant Zeeman splitting near the cluster interface are respond for negative magnetoresistance. Additionally to high temperature critical pointthe low-temperature ferromagnetic transition was observed Anomalous Hall effect was observed in Mn doped samples and analyzed for InMnSb. It was found that MnX clusters influence significantly on magnetic scattering of carriers.

Fractionation of Hen Egg and Oat Lipids with supercritical Fluids Chemical and functional Properties of Fractions

Hen eggs and oats (Avena Sativa) are important materials for the food industry. Today, instead of merely satisfying the feeling of hunger, consumers are asking for healthier, biologically active and environmentally friendly products. The growing awareness of consumers’ increasing demands presents a great challenge to the food industry to develop more sustainable products and utilise modern and effective techniques. The modification of yolk fatty acid composition by means of feed supplements is well understood. Egg yolk phospholipids are polar lipids and are used in several applications including food, cosmetics, pharmaceuticals, and special nutrients. Egg yolk phospholipids are excellent emulsifiers, typically sold as mixtures of phospholipids, triacylglycerols, and cholesterol. However, highly purified and characterised phospholipids are needed in several sophisticated applications. Industrial fractionation of phospholipids is usually based on organic solvents. With these fractionation techniques, some harmful residues of organic solvents may cause problems in further processing. The objective of the present study was to investigate the methods to improve the functional properties of eggs, to develop techniques to isolate the fractions responsible for the specific functional properties of egg yolk lipids, and to apply the developed techniques to plant-based materials, too. Fractionation techniques based on supercritical fluids were utilised for the separation of the lipid fractions of eggs and oats. The chemical and functional characterisation of the fractions were performed, and the produced oat polar lipid fractions were tested as protective barrier in encapsulation processes. Modifying the fatty acid compositions of egg yolks with different types of oil supplements in feed had no affect on their functional or sensory properties. Based on the results of functional and sensory analysis, it is evident that eggs with modified fatty acid compositions are usable in several industrial applications. These applications include liquid egg yolk products used in mayonnaise and salad dressings. Egg yolk powders were utilised in different kinds of fractionation processes. The precipitation method developed in this study resembles the supercritical anti-solvent method, which is typically used in the pharmaceutical industry. With pilot scale supercritical fluid processes, non-polar lipids and polar lipids were successfully separated from commercially produced egg yolk powder and oat flakes. The egg and oat-based polar lipid fractions showed high purities, and the corresponding delipidated fractions produced using supercritical techniques offer interesting starting materials for the further production of bioactive compounds. The oat polar lipid fraction contained especially digalactosyadiacylglycerol, which was shown to have valuable functional properties in the encapsulation of probiotics.

Heart rate variability in young adults - Reference values and associations with cardiometabolic risk factors and vascular properties. The Cardiovascular Risk in Young Finns Study.

Background: The function of the autonomic nervous system (ANS) can be evaluated with heart rate variability (HRV). Decreased HRV is associated with aging, the male sex, increased heart rate, and overall increased cardiometabolic risk. It has been hypothesized that early atherosclerotic vascular changes and ANS function are related. Aims: The aims were to assess reference values on HRV in young adults, and examine associations with HRV and cardiometabolic risk factors and metabolic syndrome (MetS) and to study relations between HRV and ultrasonographically measured vascular properties. Participants and methods: The present thesis is part of the Cardiovascular Risk in Young Finns Study. The thesis is based on the follow-up study in 2001, when the study individuals were 24-39 years of age. HRV data were available on 1 956 individuals. Results: HRV was inversely associated with age and heart rate (for all p<0.001). Highfrequency HRV (HF) was higher, and low-frequency HRV (LF) lower in women than men (p<0.0001 for both). MetS was associated with 11% decreased HF and 12% increased LF/HF-ratio in women, and 8% decreased HF and 4% increased LF/HF-ratio in men. Carotid artery distensibility was independently associated with HF and total HRV (for both p<0.05). Conclusions: The reference values in young adults were generated. Decreased HRV was associated with age, the male sex and increased heart rate. Women had higher HF and lower LF variability than men. MetS was related to decrease in HRV. The observed associations between carotid elasticity and HRV, supports the hypothesis that reduction in carotid elasticity may lead to decrease in autonomic cardiac control.

Ultraviolet light protection and weathering properties of wood-polypropylene composites

The main aim of this thesis is to study the effect of pigments on the weathering properties of wood-polypropylene composites (WPC). The studied properties are color change, water absorption, thickness swelling and Charpy impact strength. The impact of weathering and UV exposure on WPCs was studied by using pigments and minerals as protective agents. The study shows that the pigments and/or mineral fillers can be used to improve the weathering properties of WPCs. The effect of pigments was found to vary with the type of pigment and the method of weathering. The black pigment, an inorganic carbon black master-batch, was found to be the most effective one in reduction of the discoloration of WPCs. By preventing discoloration, and further reducing the degradation of the surface of the WPC, the pigments were found to reduce the decrease in the impact strength after weathering. As well as UV protection, the moisture resistance is a significant factor affecting the durability of WPCs. The addition of mineral fillers was found to improve the moisture-related properties, such as water absorption and thickness swelling, of WPC significantly. According to the findings, addition of pigments and mineral fillers to wood-polypropylene composites appears to be beneficial: color stability and moisture resistance can be enhanced especially in outdoor weathering. The combined effect of black pigment (carbon black master-batch) and wollastonite as a mineral filler was found to bring about the most effective properties against weathering.

First-principles study on electronic and structural properties of Cu(In/Ga)Se alloys for solar cells

Thin-film photovoltaic solar cells based on the Cu(In1−xGax)Se2 (CIGS) alloys have attracted more and more attention due to their large optical absorption coefficient, long term stability, low cost, and high efficiency. Modern theoretical studies of this material with first-principles calculations can provide accurate description of the electronic structure and yield results in close agreement with experimental values, but takes a large amount of calculation time. In this work, we use first-principles calculations based on the computationally affordable meta- generalized gradient approximation of the density-functional theory to investigate electronic and structural properties of the CIGS alloys. We report on the simulation of the lattice parameters and band gaps, as a function of chemical composition. The obtained results were found to be in a good agreement with the available experimental data.

Monolithic zirconium dioxide as a full contour restorative material: With special emphasis on the optical and mechanical properties

Full contour monolithic zirconia restorations have shown an increased popularity in the dental field over the recent years, owing to its mechanical and acceptable optical properties. However, many features of the restoration are yet to be researched and supported by clinical studies to confirm its place among the other indirect restorative materials This series of in vitro studies aimed at evaluating and comparing the optical and mechanical properties, light cure irradiance, and cement polymerization of multiple monolithic zirconia material at variable thicknesses, environments, treatments, and stabilization. Five different monolithic zirconia materials, four of which were partially stabilized and one fully stabilized were investigated. The optical properties in terms of surface gloss, translucency parameter, and contrast ratio were determined via a reflection spectrophotometer at variable thicknesses, coloring, sintering method, and after immersion in an acidic environment. Light cure irradiance and radiant exposure were quantified through the specimens at variable thicknesses and the degree of conversion of two dual-cure cements was determined via Fourier Transform Infrared spectroscopy. Bi-axial flexural strength was evaluated to compare between the partially and fully stabilized zirconia prepared using different coloring and sintering methods. Surface characterization was performed using a scanning electron microscope and a spinning disk confocal microscope. The surface gloss and translucency of the zirconia investigated were brand and thickness dependent with the translucency values decreasing as the thickness increased. Staining decreased the translucency of the zirconia and enhanced surface gloss as well as the flexural strength of the fully stabilized zirconia but had no effect on partially stabilized zirconia. Immersion in a corrosive acid increased surface gloss and decreased the translucency of some zirconia brands. Zirconia thickness was inversely related to the amount of light irradiance, radiant exposure, and degree of monomer conversion. Type of sintering furnace had no effect on the optical and mechanical properties of zirconia. Monolithic zirconia maybe classified as a semi-translucent material that is well influenced by the thickness, limiting its use in the esthetic zones. Conventional acid-base reaction, autopolymerizing and dual-cure cements are recommended for its cementation. Its desirable mechanical properties give it a high potential as a restoration for posterior teeth. However, close monitoring with controlled clinical studies must be determined before any definite clinical recommendations can be drawn.

The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.