Some results on quasi-symmetric designs with exceptional parameters

U ovoj disertaciji proučavamo kvazisimetrične dizajne s iznimnim parametrima te metode za njihovu konstrukciju s pretpostavljenom grupom automorfizama. Tablica dopustivih iznimnih parametara kvazisimetričnih $2-(v, k, \lambda)$ dizajna s presječnim brojevima x i y je proširena do v = 150 te je time dobiveno ukupno 260 skupova parametara, pri čemu za njih 172 egzistencija kvazisimetričnih dizajna nije poznata. Pomoću Kramer-Mesnerove metode – poznate metode za konstrukciju t-dizajna sa zadanom grupom automorfizama, konstruirani su mnogi dizajni s parametrima $t-(v, k, \lambda)$ odgovarajućih kvazisimetričnih dizajna iz navedene tablice te je time pokazano koliko je problem konstrukcije 2-dizajna bez uvjeta kvazisimetričnosti lakši. U konstrukciji kvazisimetričnih dizajna sa zadanom grupom automorfizama osnovni koraci su generiranje orbita i konstrukcija dizajna iz dobivenih orbita. U disertaciji su opisani algoritmi za generiranje kratkih orbita te svih orbita podskupova zadane veličine nekog skupa pod djelovanjem pretpostavljene grupe automorfizama. Također, opisani su algoritmi na kojima se temelje metode za konstrukciju kvazisimetričnih dizajna, a to su Kramer-Mesnerova metoda, metoda temeljena na traženju klika i metoda temeljena na orbitnim matricama. Korištenjem navedenih algoritama konstruirani su novi kvazisimetrični dizajni s parametrima 2-(28, 12, 11), x = 4, y = 6, zatim 2-(36, 16, 12), x = 6, y = 8 te 2-(56, 16, 6), x = 4, y = 6. Isto tako, utvrđena je egzistencija kvazisimetričnog 2-(56, 16, 18) dizajna s presječnim brojevima x = 4 i y = 8, koja ranije nije bila poznata, te su konstruirana četiri neizomorfna dizajna s tim parametrima. Nadalje, utvrđeno je da samo već poznati kvazisimetrični dizajni s projektivnim parametrima postoje s određenim grupama automorfizama.In this thesis, we study quasi-symmetric designs with exceptional parameters and methods for their construction with prescribed automorphism groups. A $t-(v, k, \lambda)$ design is quasi-symmetric if any two blocks intersect either in x or in y points, for non-negative integers x < y. The classification of quasi-symmetric 2-designs is a difficult open problem and there are many triples $(v, k, \lambda)$ for which existence is unknown. A. Neumaier in [74] defines four classes of quasi-symmetric 2-designs: multiples of symmetric designs, strongly resolvable designs, Steiner designs, and residuals of biplanes. If a quasi-symmetric design or its complement does not belong to any of these classes, he calls it exceptional. In the same paper, he published the first table of admissible exceptional parameters of quasi-symmetric 2-designs with v < 40. Updated and extended tables were published by V. D. Tonchev, A. R. Calderbank, and M. S. Shrikhande. The last table was published in [85] and contains all exceptional parameters with v ≤ 70. The thesis is divided into three chapters. In the first chapter, we give some basic definitions, notations, and results about t-designs, quasi-symmetric t-designs, and quasisymmetric 2-designs in particular. We also explain their important connections with strongly regular graphs and self-orthogonal codes. Furthermore, we describe the known families of quasi-symmetric 2-designs. Finally, at the end of this chapter we update the table of admissible exceptional parameters with new results and extend it to v = 150. The new table contains 260 parameter sets, and for 172 parameter sets the existence of quasi-symmetric designs is unknown. In the second chapter, we explore the construction of 2-designs with prescribed automorphism groups, not necessarily quasi-symmetric. We give some basic concepts and results from group theory, in particular permutation groups, and consider (full) automorphism groups of designs. Furthermore, we describe one of the most widespread methods for the construction of designs with prescribed automorphism groups, the Kramer-Mesner method. Using this method, we construct a lot of designs with exceptional parameters $2-(v, k, \lambda)$ of quasi-symmetric designs. The conclusion is that the construction of 2-designs is much easier without the condition of quasi-symmetry. In the last and most important chapter, we develop computational methods for the construction of quasi-symmetric designs with a prescribed automorphism group G. The v construction is done in two basic steps: firstly, generate the good orbits of G on k-element subsets of a v-element set, and secondly, select orbits comprising blocks of the design. In the first section of this chapter, we give some ideas for selecting automorphism groups. In the second section, we explain algorithms for generating orbits. Depending on the parameters of the design and size of the automorphism group, we use the algorithm for short orbits, the algorithm for all orbits, or generate orbits from orbit matrices. In the third section, we explain methods for the construction of quasi-symmetric designs from the generated orbits. We use the Kramer-Mesner method, a method based on clique search, and a method based on orbit matrices. All these methods had been previously known in design theory. We adapt them for the construction of quasi-symmetric designs and perform the constructions for some feasible parameters and groups. We increase the number of known designs with parameters 2-(28, 12, 11), x = 4, y = 6, 2-(36, 16, 12), x = 6, y = 8, 2-(56, 16, 6), x = 4, y = 6, and establish the existence of quasi-symmetric 2-(56, 16, 18) designs with intersection numbers x = 4 and y = 8. Furthermore, using binary codes associated with newly constructed 2-(56, 16, 6) designs, we find even more quasi-symmetric designs with these parameters. All the new quasi-symmetric 2-(56, 16, 18) designs can be extended to symmetric 2-(78, 22, 6) designs and in this way the number of known symmetric designs with these parameters is significantly increased. In the last section of this chapter, we attempt construction of quasi-symmetric designs with projective parameters and certain automorphism groups, but find only known examples of such designs

Some results on quasi-symmetric designs with exceptional parameters

U ovoj disertaciji proučavamo kvazisimetrične dizajne s iznimnim parametrima te metode za njihovu konstrukciju s pretpostavljenom grupom automorfizama. Tablica dopustivih iznimnih parametara kvazisimetričnih $2-(v, k, \lambda)$ dizajna s presječnim brojevima x i y je proširena do v = 150 te je time dobiveno ukupno 260 skupova parametara, pri čemu za njih 172 egzistencija kvazisimetričnih dizajna nije poznata. Pomoću Kramer-Mesnerove metode – poznate metode za konstrukciju t-dizajna sa zadanom grupom automorfizama, konstruirani su mnogi dizajni s parametrima $t-(v, k, \lambda)$ odgovarajućih kvazisimetričnih dizajna iz navedene tablice te je time pokazano koliko je problem konstrukcije 2-dizajna bez uvjeta kvazisimetričnosti lakši. U konstrukciji kvazisimetričnih dizajna sa zadanom grupom automorfizama osnovni koraci su generiranje orbita i konstrukcija dizajna iz dobivenih orbita. U disertaciji su opisani algoritmi za generiranje kratkih orbita te svih orbita podskupova zadane veličine nekog skupa pod djelovanjem pretpostavljene grupe automorfizama. Također, opisani su algoritmi na kojima se temelje metode za konstrukciju kvazisimetričnih dizajna, a to su Kramer-Mesnerova metoda, metoda temeljena na traženju klika i metoda temeljena na orbitnim matricama. Korištenjem navedenih algoritama konstruirani su novi kvazisimetrični dizajni s parametrima 2-(28, 12, 11), x = 4, y = 6, zatim 2-(36, 16, 12), x = 6, y = 8 te 2-(56, 16, 6), x = 4, y = 6. Isto tako, utvrđena je egzistencija kvazisimetričnog 2-(56, 16, 18) dizajna s presječnim brojevima x = 4 i y = 8, koja ranije nije bila poznata, te su konstruirana četiri neizomorfna dizajna s tim parametrima. Nadalje, utvrđeno je da samo već poznati kvazisimetrični dizajni s projektivnim parametrima postoje s određenim grupama automorfizama.In this thesis, we study quasi-symmetric designs with exceptional parameters and methods for their construction with prescribed automorphism groups. A $t-(v, k, \lambda)$ design is quasi-symmetric if any two blocks intersect either in x or in y points, for non-negative integers x < y. The classification of quasi-symmetric 2-designs is a difficult open problem and there are many triples $(v, k, \lambda)$ for which existence is unknown. A. Neumaier in [74] defines four classes of quasi-symmetric 2-designs: multiples of symmetric designs, strongly resolvable designs, Steiner designs, and residuals of biplanes. If a quasi-symmetric design or its complement does not belong to any of these classes, he calls it exceptional. In the same paper, he published the first table of admissible exceptional parameters of quasi-symmetric 2-designs with v < 40. Updated and extended tables were published by V. D. Tonchev, A. R. Calderbank, and M. S. Shrikhande. The last table was published in [85] and contains all exceptional parameters with v ≤ 70. The thesis is divided into three chapters. In the first chapter, we give some basic definitions, notations, and results about t-designs, quasi-symmetric t-designs, and quasisymmetric 2-designs in particular. We also explain their important connections with strongly regular graphs and self-orthogonal codes. Furthermore, we describe the known families of quasi-symmetric 2-designs. Finally, at the end of this chapter we update the table of admissible exceptional parameters with new results and extend it to v = 150. The new table contains 260 parameter sets, and for 172 parameter sets the existence of quasi-symmetric designs is unknown. In the second chapter, we explore the construction of 2-designs with prescribed automorphism groups, not necessarily quasi-symmetric. We give some basic concepts and results from group theory, in particular permutation groups, and consider (full) automorphism groups of designs. Furthermore, we describe one of the most widespread methods for the construction of designs with prescribed automorphism groups, the Kramer-Mesner method. Using this method, we construct a lot of designs with exceptional parameters $2-(v, k, \lambda)$ of quasi-symmetric designs. The conclusion is that the construction of 2-designs is much easier without the condition of quasi-symmetry. In the last and most important chapter, we develop computational methods for the construction of quasi-symmetric designs with a prescribed automorphism group G. The v construction is done in two basic steps: firstly, generate the good orbits of G on k-element subsets of a v-element set, and secondly, select orbits comprising blocks of the design. In the first section of this chapter, we give some ideas for selecting automorphism groups. In the second section, we explain algorithms for generating orbits. Depending on the parameters of the design and size of the automorphism group, we use the algorithm for short orbits, the algorithm for all orbits, or generate orbits from orbit matrices. In the third section, we explain methods for the construction of quasi-symmetric designs from the generated orbits. We use the Kramer-Mesner method, a method based on clique search, and a method based on orbit matrices. All these methods had been previously known in design theory. We adapt them for the construction of quasi-symmetric designs and perform the constructions for some feasible parameters and groups. We increase the number of known designs with parameters 2-(28, 12, 11), x = 4, y = 6, 2-(36, 16, 12), x = 6, y = 8, 2-(56, 16, 6), x = 4, y = 6, and establish the existence of quasi-symmetric 2-(56, 16, 18) designs with intersection numbers x = 4 and y = 8. Furthermore, using binary codes associated with newly constructed 2-(56, 16, 6) designs, we find even more quasi-symmetric designs with these parameters. All the new quasi-symmetric 2-(56, 16, 18) designs can be extended to symmetric 2-(78, 22, 6) designs and in this way the number of known symmetric designs with these parameters is significantly increased. In the last section of this chapter, we attempt construction of quasi-symmetric designs with projective parameters and certain automorphism groups, but find only known examples of such designs

Some results on quasi-symmetric designs with exceptional parameters

U ovoj disertaciji proučavamo kvazisimetrične dizajne s iznimnim parametrima te metode za njihovu konstrukciju s pretpostavljenom grupom automorfizama. Tablica dopustivih iznimnih parametara kvazisimetričnih $2-(v, k, \lambda)$ dizajna s presječnim brojevima x i y je proširena do v = 150 te je time dobiveno ukupno 260 skupova parametara, pri čemu za njih 172 egzistencija kvazisimetričnih dizajna nije poznata. Pomoću Kramer-Mesnerove metode – poznate metode za konstrukciju t-dizajna sa zadanom grupom automorfizama, konstruirani su mnogi dizajni s parametrima $t-(v, k, \lambda)$ odgovarajućih kvazisimetričnih dizajna iz navedene tablice te je time pokazano koliko je problem konstrukcije 2-dizajna bez uvjeta kvazisimetričnosti lakši. U konstrukciji kvazisimetričnih dizajna sa zadanom grupom automorfizama osnovni koraci su generiranje orbita i konstrukcija dizajna iz dobivenih orbita. U disertaciji su opisani algoritmi za generiranje kratkih orbita te svih orbita podskupova zadane veličine nekog skupa pod djelovanjem pretpostavljene grupe automorfizama. Također, opisani su algoritmi na kojima se temelje metode za konstrukciju kvazisimetričnih dizajna, a to su Kramer-Mesnerova metoda, metoda temeljena na traženju klika i metoda temeljena na orbitnim matricama. Korištenjem navedenih algoritama konstruirani su novi kvazisimetrični dizajni s parametrima 2-(28, 12, 11), x = 4, y = 6, zatim 2-(36, 16, 12), x = 6, y = 8 te 2-(56, 16, 6), x = 4, y = 6. Isto tako, utvrđena je egzistencija kvazisimetričnog 2-(56, 16, 18) dizajna s presječnim brojevima x = 4 i y = 8, koja ranije nije bila poznata, te su konstruirana četiri neizomorfna dizajna s tim parametrima. Nadalje, utvrđeno je da samo već poznati kvazisimetrični dizajni s projektivnim parametrima postoje s određenim grupama automorfizama.In this thesis, we study quasi-symmetric designs with exceptional parameters and methods for their construction with prescribed automorphism groups. A $t-(v, k, \lambda)$ design is quasi-symmetric if any two blocks intersect either in x or in y points, for non-negative integers x < y. The classification of quasi-symmetric 2-designs is a difficult open problem and there are many triples $(v, k, \lambda)$ for which existence is unknown. A. Neumaier in [74] defines four classes of quasi-symmetric 2-designs: multiples of symmetric designs, strongly resolvable designs, Steiner designs, and residuals of biplanes. If a quasi-symmetric design or its complement does not belong to any of these classes, he calls it exceptional. In the same paper, he published the first table of admissible exceptional parameters of quasi-symmetric 2-designs with v < 40. Updated and extended tables were published by V. D. Tonchev, A. R. Calderbank, and M. S. Shrikhande. The last table was published in [85] and contains all exceptional parameters with v ≤ 70. The thesis is divided into three chapters. In the first chapter, we give some basic definitions, notations, and results about t-designs, quasi-symmetric t-designs, and quasisymmetric 2-designs in particular. We also explain their important connections with strongly regular graphs and self-orthogonal codes. Furthermore, we describe the known families of quasi-symmetric 2-designs. Finally, at the end of this chapter we update the table of admissible exceptional parameters with new results and extend it to v = 150. The new table contains 260 parameter sets, and for 172 parameter sets the existence of quasi-symmetric designs is unknown. In the second chapter, we explore the construction of 2-designs with prescribed automorphism groups, not necessarily quasi-symmetric. We give some basic concepts and results from group theory, in particular permutation groups, and consider (full) automorphism groups of designs. Furthermore, we describe one of the most widespread methods for the construction of designs with prescribed automorphism groups, the Kramer-Mesner method. Using this method, we construct a lot of designs with exceptional parameters $2-(v, k, \lambda)$ of quasi-symmetric designs. The conclusion is that the construction of 2-designs is much easier without the condition of quasi-symmetry. In the last and most important chapter, we develop computational methods for the construction of quasi-symmetric designs with a prescribed automorphism group G. The v construction is done in two basic steps: firstly, generate the good orbits of G on k-element subsets of a v-element set, and secondly, select orbits comprising blocks of the design. In the first section of this chapter, we give some ideas for selecting automorphism groups. In the second section, we explain algorithms for generating orbits. Depending on the parameters of the design and size of the automorphism group, we use the algorithm for short orbits, the algorithm for all orbits, or generate orbits from orbit matrices. In the third section, we explain methods for the construction of quasi-symmetric designs from the generated orbits. We use the Kramer-Mesner method, a method based on clique search, and a method based on orbit matrices. All these methods had been previously known in design theory. We adapt them for the construction of quasi-symmetric designs and perform the constructions for some feasible parameters and groups. We increase the number of known designs with parameters 2-(28, 12, 11), x = 4, y = 6, 2-(36, 16, 12), x = 6, y = 8, 2-(56, 16, 6), x = 4, y = 6, and establish the existence of quasi-symmetric 2-(56, 16, 18) designs with intersection numbers x = 4 and y = 8. Furthermore, using binary codes associated with newly constructed 2-(56, 16, 6) designs, we find even more quasi-symmetric designs with these parameters. All the new quasi-symmetric 2-(56, 16, 18) designs can be extended to symmetric 2-(78, 22, 6) designs and in this way the number of known symmetric designs with these parameters is significantly increased. In the last section of this chapter, we attempt construction of quasi-symmetric designs with projective parameters and certain automorphism groups, but find only known examples of such designs

Annular Seals of High Energy Centrifugal Pumps: Presentation of Full Scale Measurement

Prediction of rotordynamic behavior for high energy concentration centrifugal pumps is a challenging task which still imposes considerable difficulties. While the mechanical modeling of the rotor is solved most satisfactorily by finite element techniques, accurate boundary conditions for arbitrary operating conditions are known for journal bearings only. Little information is available on the reactive forces of annular seals, such as neck ring and interstage seals and balance pistons, and on the impeller interaction forces. The present focus is to establish reliable boundary conditions at annular seals. For this purpose, a full scale test machine was set up and smooth and serrated seal configurations measured. Dimensionless coefficients are presented and compared with a state of the art theory

Global sensitivity analysis for stochastic simulators based on generalized lambda surrogate models

Global sensitivity analysis aims at quantifying the impact of input variability onto the variation of the response of a computational model. It has been widely applied to deterministic simulators, for which a set of input parameters has a unique corresponding output value. Stochastic simulators, however, have intrinsic randomness due to their use of (pseudo)random numbers, so they give different results when run twice with the same input parameters but non-common random numbers. Due to this random nature, conventional Sobol' indices, used in global sensitivity analysis, can be extended to stochastic simulators in different ways. In this paper, we discuss three possible extensions and focus on those that depend only on the statistical dependence between input and output. This choice ignores the detailed data generating process involving the internal randomness, and can thus be applied to a wider class of problems. We propose to use the generalized lambda model to emulate the response distribution of stochastic simulators. Such a surrogate can be constructed without the need for replications. The proposed method is applied to three examples including two case studies in finance and epidemiology. The results confirm the convergence of the approach for estimating the sensitivity indices even with the presence of strong heteroskedasticity and small signal-to-noise ratio

Apodized phase mask coronagraphs for arbitrary apertures

Phase masks coronagraphs can be seen as linear systems that spatially redistribute, in the pupil plane, the energy collected by the telescope. Most of the on-axis light must ideally be rejected outside the aperture to be blocked with a Lyot stop, while almost all off-axis light must go through it. The unobstructed circular apertures of off-axis telescopes make this possible but all major telescopes are however on-axis and the performance of these coronagraphs is dramatically reduced by the central obstruction. Their performance can be restored by using an additional optimally designed apodizer that changes the amplitude in the first pupil plane so that the on-axis light is rejected outside the obstructed aperture of the telescope. The numerical optimization model is built by maximizing the apodizer's transmission while setting constraints on the extremum values of the electric field that the Lyot stop does not block. The coronagraphic image is compared to what a non-apodized phase mask coronagraph provides and an analysis is made of the trade-offs that exist between the apodizer transmission and the Lyot stop properties. The existence of a solution and the mask transmission depend on the aperture and the Lyot stop geometries, and on the constraints that are set on the on-axis attenuation. The system throughput is a concave function of the Lyot stop transmission. In the case of a VLT-like aperture, apodizers with a transmission of 0.16 to 0.92 associated with a four-quadrant phase mask provide contrast as low as a few 1e-10 at 1 lambda/D from the star. The system's maximum throughput is 0.64, for an apodizer with an 0.88 transmission and a Lyot stop with a 0.69 transmission. Optimizing apodizers for a vortex phase mask requires computation times much longer than in the previous case, and no result is presented for this mask.Comment: 16 page

Novel Design of a Wideband Ribcage-Dipole Array and its Feeding Network

In this thesis the focus was on the design, fabrication, and tests of the feeding networks individually and within an array system. The array feeding network is a corporate-fed type utilizing equal-split, stepped-multiple sections of the conventional Wilkinson power divider in microstrip form with a unique topology. The feeding network was specifically designed for a broadside relatively small linearly-polarized wideband UHF non-scanning array for directed power applications that uses an array radiator with a new volumetric ribcage dipole configuration. The array has a large impedance bandwidth and consistent front lobe gain over the wide frequency band. Theoretical and experimental results describing the performance of the array feeding network and the array are presented and discussed