Real and complex operator norms

Any bounded linear operator between real (quasi-)Banach spaces T : X ® Y has a natural bounded complex linear extension TC : XC ® YC defined by the formula TC(x+iy)=Tx+iTy for x,yÎX, where XC and YC are so called reasonable complexifications of X and Y, respectively. We are interested in the exact relation between the norms of the operators TC and T. This relation can be expressed in terms of the constant gX,Y appearing in the inequality||TC|| £ gX,Y ||T|| considered for all bounded linear operators T : X®Y between (quasi-)Banach spaces. The work on the constant gLp,Lq for 0 &lt; p,q £ ¥, or shortly gp,q, is traced back to M. Riesz, Thorin, Marcinkiewicz, Zygmund, Verbitckii, Krivine, Gasch, Maligranda, Defant and others. In this thesis we try to summarize the results of these authors. We also present some new estimates for gp,q in the case when at least one of the spaces is quasi-Banach as well as in the case when the spaces are supplied with discrete measures. For example, we get that gp,q £ 2 for all 0 &lt; p,q £ ¥. Furthermore we obtain some new results concerning the relation between complex and real norms of the operators between spaces of functions of bounded p-variation and between mixed norm Lebesgue spaces. Looking for the criteria of the equality of real and complex norms of operators from a Banach lattice into the same Banach lattice we find a number of examples of two dimensional Orlicz spaces different from Lebesgue spaces and a simple operator between them with non-equal real and complex norms. We also consider in detail the Clarkson inequality which can be interpreted in terms of a certain operator norm inequality appearing as an example in many parts of the thesis. It turns out that complex norm of this operator can be easily obtained but to find the real one is not so trivial. With the help of the Clarkson inequality we construct an operator between Lebesgue spaces with non-atomic measures which has different real and complex norms. Finally, we consider both complex and real versions of the Riesz-Thorin interpolation theorem in the first quadrant and by using numbers gp,q find, for example, that the real Riesz constant is bounded by 2 for all 0 &lt; p,q £ ¥.Godkänd; 2007; 20070220 (ysko)</p

Simple non-iterative calibration for triaxial accelerometers

For high precision measurements, accelerometers need recalibration between different measurement occasions. In this paper we derive a simple calibration method for triaxial accelerometers with orthogonal axes. Just like previously proposed iterative methods, we compute the calibration parameters (biases and gains) from measurements of the Earth gravity for six different unknown orientations of the accelerometer. However, our method is non-iterative, so there are no complicated convergence issues depending on input parameters, round-off errors etc.The main advantages of our method are that only from the accelerometer output voltages it gives a complete knowledge of whether it is possible, with any method, to recover the accelerometer biases and gains from the output voltages, and when this is possible, we have a simple explicit formula for computing them with a smaller number of arithmetic operations than previous iterative approaches. Moreover, we show that such successful recovery is guaranteed if the six calibration measurements deviate with angles smaller than some upper bound from a natural setup with two horizontal axes. We provide an estimate from below of this upper bound that, for instance, allows 5 degree deviations in arbitrary directions for the Colibrys SF3000L accelerometers in our lab.Validerad; 2011; Bibliografisk uppgift: Article no 125103; 20110503 (grip

Real and complex operator norms between quasi-banach Lp -L q spaces

Relations between the norms of an operator and its complexification as a mapping from Lp to Lq has been recognized as a serious problem in analysis after the publication of Marcel Riesz's work on convexity and bilinear forms in 1926. We summarize here what it is known about these relations in the case of normed Legesgue spaces and investigate the quasinormed case, i. e. we consider all 0 &lt; p,q ≤ 8 . In particular, in the lower triangle, that is, for 0&lt; p≤q≤∞ these norms are the same. In the upper triangle and the normed case, that is, when 1 ≤ q &lt; p ≤ ∞ the norm of the complexification of a real operator is obviously not bigger than 2 times its real norm. In 1977 Krivine proved that the constant 2 can be replaced by √2 . On the other hand, it was suspected that in the case of quasi-normed Lebesgue spaces (0 &lt; q &lt; p ≤ ∞ ) the corresponding constant could be arbitrarily large, but as we will see this is not the case. More precisely, we prove that this constant for quasi-normed Lebesgue spaces is between 1 and 2. Some additional properties and estimates of this constant with some results about the relation between complex and real norms of operators, including those between two-dimensional Orlicz spaces are presented in the first four chapters. Finally, in Chapter 5, we use the results on the estimates of the norms in the proof of the real Riesz-Thorin interpolation theorem valid in the first quadrantValiderad; 2011; 20110317 (andbra)</p

On Clarkson's inequality in the real case

The best constant in a generalized complex Clarkson inequality is Cp,q () = max {21-1/p , 21/q , 21/q -1/p +1/2} which differs moderately from the best constant in the real case Cp,q () = max {21-1/p , 21/q ,Bp,q }, where . For 1 < q < 2 < p < the constant Cp,q () is equal to Bp,q and these numbers are difficult to calculate in general. As applications of the generalized Clarkson inequalities the (p, q)-Clarkson inequalities in Lebesgue spaces, in mixed norm spaces and in normed spaces are presented.Validerad; 2007; 20071010 (evan

Non-iterative calibration for accelerometers with three non-orthogonal axes, reliable measurement setups and simple supplementary equipment

The input-output relationship of an accelerometer depends on parameters that are sensitive to temperature and air humidity. High accuracy field measurements therefore require simple in-field estimation of these parameters.We present an extension of a simple non-iterative six-parameter calibration method for triaxial accelerometers with orthogonal input axes to a nine-parameter method that also handles non-orthogonal axes and cross-axis interference.We derive necessary and sufficient conditions on the accelerometer output that guarantee that the nine parameters can be uniquely determined from the calibration measurements in an idealized scenario with no noise or quantization errors. The method is based on measurements of the Earth gravity with the accelerometer placed at rest in at least nine different orientations.The choice of orientations is important for measurement accuracy. We compare two different setups, one called A090-45, which is based on 90 and 45 degree rotations of the accelerometer and one called A0max sep that has maximized smallest angle between any two of the orientations. For the A090-45 setup we have constructed a simple test equipment for quick positionings of the accelerometer. For the A0max sep setup, a similar equipment is more complicated to construct, but equally simple to use.We have done Monte Carlo simulations with accelerometer orientations deviating at most D degrees from the desired A090-45 or A0max sep and with D ranging from 1◦ to 30◦. For real-world noisy environments and D up to 18◦, our simulations showed slightly smaller errors for the A0max sep than for the A090-45 setup. For noise standard deviation typical for our field measurements, the measurement errors after nine-parameter calibration were about 100 times smaller than those for six-parameter calibration both for the A0max sep setup and, as long as D ≤ 13◦ for the A090-45 setup. For the A090-45 setup, however, we found that combinations of large noise levels and/or large D can makesix-parameter calibration the better choice.Validerad; 2013; Bibliografisk uppgift: Article no 035002 ; 20120328 (grip

Gruvvägsbron I Kiruna : Deformationskapacitet

En bedömning har gjorts av hur stora deformationer Gruvvägsbron i Kiruna kan klara med bibehållen bärförmåga. I projektet har mätningar av brons beteende vid statisk och dynamisk belastning utförts under vinter- respektive sommarförhållanden. Mätningarna har utvärderats och analyserats med FEMprogram för att kunna bedöma brons styvhet och deformationskapacitet. Syftet har varit att undersöka hur mycket bron klarar av i stödförändringar på grund av gruvdriften. Preliminära gränsvärden för stödförändringar har tagits fram.Brons statiska funktionssätt är tillfredsställande. Enligt hittillsvarande mätningar (t o m dec 2010) har endast små markrörelser ägt rum vilket inte nämnvärt påverkar brons funktion.Brons dynamiska funktionssätt är också tillfredsställande. Uppmätta egenmoder och egenfrekvenser är normala för denna typ av förspänd balkbro. Bron är förhållandevis styv och de analytiska och numeriska beräkningar(FEM-analyser) som gjorts tyder på att den har förhållandevis begränsad deformationskapacitet. Vi bedömer att bron klarar följande lokala deformationsskillnad mellan två närliggande pelare: (a) i horisontalled ca 40mm och (b) i vertikalled ca 80 mm. Om tillåtna värden sätts till hälften av de möjliga erhålls:Tillåten differens för rörelse(a) i tvärled mellan en pelares över- och underkant till = ca 20 mm(b) i höjdled mellan två intilliggande stöd till = ca 40 mmVärdena kan komma att revideras (förfinas) i samband med att resultat blir tillgängliga från fortsatta mätningar och observationer av brons beteende.Några markrörelser av betydelse inträffade inte mellan november 2006 och 2008, men under 2009 och 2010 har rörelser skett med några mm. Brons längdändring på grund av temperaturvariationer är dock betydligt större. Bronär ca 3 – 4 cm längre på sommaren än på vintern. Eftersom markrörelserna förväntas bli betydligt större i samband med att gruvans brytningsfront närmar sig bron försågs brons pelare under sommaren2010 med en anordning rdning så att deras läge kan justeras om markrörelserna blirojämna.Godkänd; 2011; 20150519 (elfgren

Signal processing in modal analysis of bridges

Godkänd; 2009; 20101112 (grip

Vibrationsanalys för tillståndsbedömning av byggkonstruktioner : Tillämpningsexempel : (Main results and summary in Swedish. Detailed results in English Appendices.)

I projektet beskrivs och tillämpasnågra metoder för tillståndsbedömningav olika typer av konstruktioner med och utan skador.Tillståndsbedömningen skermed hjälp av vibrationsanalys.Detektering av skador av olika typ och storlekdiskuteras. Randvillkorens betydelse undersöks för att få överensstämmelse mellan förutspådda och uppmätta egenmoder och egenfrekvenser.Med den programvara som utvecklats är det möjligt att detektera sprickor i enkla betongplattor med hjälp av vibrationsmätningar. För mer avancerade konstruktioner finns en potential att vidareutveckla programvaran.Mätningar och analyser presenterasför femkonstruktioner: (1) En betongplatta –Sprickor har kunnat identifieras utifrån vibrationsmätningar och en successivt uppdaterad finit element modell (FEM).(2) En bågbro av betong över Långforsen i Kalix älv–Okända randvillkor har kunnat bestämmasför t ex undergrundens styvhetmedhjälp avuppmätta vibrationer. Detekteringav specifika skadorhar ännu inte kunnat genomföras.(3) En fackverksbro av stålöver Åby älv–Enviss skadedetektering har kunnat genomföras med vibrationsmätningar. Brons sårbarhet för korrosion och lokala mekaniska skador har studerats.(4) En förspänd betongbrotill gruvan i Kiruna–Modeller av bron har arbetats framför obelastatoch brottbelastat tillståndoch jämförts med uppmätta vibrationer med gott resultat.(5) Ett betonghus med nio våningar i Luleå –En FEM-modell av huseti bruksstadiethar tagits fram och resultat från den har jämförts med uppmätta vibrationer med gott resultat.Metoder och resultatpresenteras översiktligti rapportens huvudtext. Detaljerade beskrivningar gesi appendix.I appendix presenterasäven en programvaramed färdiga moduler för de undersökta konstruktionerna. Programmet kan laddas ner och användasfrittIn the project,methods are presented and appliedfor assessment ofdifferent types ofbuilding structures with and without damages.The assessmentis done withhelp ofvibration analysis.Detection of defects of different types and sizes are discussed. The influence of boundary conditions isstudied in order to get agreement between predicted and measured Eigen modes and Eigen frequencies.With the program that has been developed,is it now possible to detect cracks in simple concrete structures with help of measurements of vibrations. For more advanced structures there is a potentialto further develop the program. Measurements and analyses have been made for fivestructures:(1) A concrete slab–Cracks have been detected based on vibrationmeasurementsand a successivelyupdated Finite Element Model (FEM).(2) Aconcrete archbridge at Långforsen in Kalix River-Unknown boundary conditions havebeen possible todetermine with help of vibration measurementsregarding for example foundation properties. Detection of specific damages hasnot yet been possible.(3) A steel truss bridge over Åby River–Some detection of damages hasbeen possible to carry outwith Finite Element Model updatingregarding corrosion and mechanical damages.. (4) A prestressed concrete bridge to the mine in Kiruna-Several models of theundamaged and damagedbridgehave been established and they have been calibrated to measured vibration propertieswith good results.(5) A nine storey concrete building in Luleå–A FEM model has been developedfor the serviceability limit stateand results from it has been compared to measured vibrationswith good results.A general presentation is given ofthe developedmethods and resultsin the main text. Detailed descriptions are provided in Appendices.A program is presented with ready-to-use models for the studied structures. The program may be downloaded and is presented in detail in an Appendix.Dessutom har Zheng Huang, Chao Wang, Niklas Bagge, Jens Häggström, Jonny Nilimaa, Thomas Forsberg,Ola Enokson,Thomas Blanksvärd och Ulf Ohlsson bidragit med material och synpunkteroch som medförfattare i artiklar och konferensbidrag.SBUF (Svenska Byggbranschens Utvecklingsfond

Modelling of Damage and its Use in Assessment of a Prestressed Bridge

A 55 years old and 121.5 m long ve-span prestressed bridge has been subjected toshear failure test in Kiruna, Sweden. This in-situ test is a desirable test to validate andcalibrate the existing nonlinear nite element program for predicting the shear behaviorof reinforced and prestressed concrete structures.Two 3D nite element (FE) models of the Kiruna Bridge are built in commercial soft-ware Abaqus, one using shell-elements and one using a combination of shell and beam ele-ments. Predictions obtained from these two models are well consistent with mode shapesand eigenfrequencies computed from acceleration measurements on the bridge before andafter loading it to failure. Shear-failure test of this bridge performed by Lulea Universityof Technology (LTU) is also simulated using the built-in concrete damage plasticity (CDP)model in Abaqus. The predicted load-displacement curve is in good agreement with themeasurement. Verication of the CDP model is conducted at element and member levelwith two dierent damage parameter evolutions. According to the verication, it indi-cates the damage parameter will aect the predicted shear behavior of reinforced concretestructures and it is not reliable to adopt the CDP model to simulate the shear behaviorof reinforced concrete structures based on the present research.A long term goal is to use use the measured mode shapes, eigenfrequencies and FEmodels for evaluating methods for damage identication. Such methods are important formaintenance of dierent structures, for extending their life span and for better knowledgeof their load carrying capacity. We describe how so-called sparse regularization niteelement method updating (FEMU) methods can be used. We then demonstrate someimportant properties of such methods with simulations on a Kirchho plate. For instance,the simulations suggest that both eigenfrequencies and mode shapes should be used forprecise localization of the damage