Instruktivna analiza Chopinove 2. Sonate u b-molu, op. 35, "Marche funebre"

Frédéric François Chopin svojim je skladateljskim opusom zadužio cijeli pijanistički svijet i ostavio trag koji je promijenio shvaćanje klavira kao glazbenog instrumenta. Bio je fasciniran ljudskim glasom te u njegovim skladbama klavir zvuči poput glasa koji pjeva. Kako je i sam bio odličan izvođač, znatno je utjecao na usavršavanje klavirske tehnike. Klavirska sonata, koja je u razdoblju klasike bila jedna od najčešćih glazbenih vrsta, nije zaobišla Chopina, iako sonata u tom vremenu znatno gubi na popularnosti. Tako su prijašnji skladatelji pisali čak i do pedeset sonata za svoga života, dok je u romantizmu i nadolazećim razdobljima taj broj pao na jednoznamenkasti, što objašnjava činjenica da su skladatelji više svoje glazbene zamisli. Chopin je napisao tri sonate za klavir. Prva sonata posvećeni traženju novih formi u kojima bi izrazili u c-molu iz op. 4 njegovo je rano djelo, koje je izdano tek nakon njegove smrti, dok treća sonata u h-molu iz op. 58 spada u njegova kasna djela. U ovom radu analizirat će se Chopinova druga sonata u b-molu iz op. 35 "Marche funèbre". Mnogi su umjetnici analizirali ovo djelo ali tu je uglavnom bilo riječ o formalno-harmonijskoj analizi koja je primarno namijenjena glazbenim teoretičarima ili muzikolozima. Ova je analiza, za razliku od takve, prvenstveno usmjerena pijanistima i pedagozima i problemima s kojim se susreću prilikom sviranja skladbe. Cilj ovog rada je pridonijeti lakšem izvođenju, uvježbavanju i čitanju ovoga djela. Isto tako ovaj rad možda potakne druge pijaniste na pisanje slične literature koja može olakšati posao svladavanja neke nepoznate skladbe.With his music works, Frédéric François Chopin made a huge impact on piano world and left a mark that changed perception of piano as a musical instrument. His fascination with human voice led him to write melodies that sound exactly like singing voice. He was a great performer so he also made a huge impact on piano technique. Piano sonata did not bypass Chopin as a composer despite the fact it was not so popular among composers during his life but it was most common music form in the Classic era. Back then, composers wrote even 50 sonatas during their career, while in romantism that number dropped to one digit number because composers started to look for new form to express their ideas. Chopin wrote 3 piano sonatas. First one in c–minor op. 4 is his early piece and it was published only after his death. Third sonata in b–minor op. 58 belongs to his late works. This work will analyse his second sonata in b-flat minor op. 35 "Marche funèbre". There are lot of analysis that are mainly about sonata form and harmony. This work, on the other hand, is dedicated to pianists and piano teachers and problems that they have while playing. This work's goal is to make practice, interpretation and reading of this piece easier. Also, it might give motivation to other pianists to write similar works of their own

Instruktivna analiza Chopinove 2. Sonate u b-molu, op. 35, "Marche funebre"

Frédéric François Chopin svojim je skladateljskim opusom zadužio cijeli pijanistički svijet i ostavio trag koji je promijenio shvaćanje klavira kao glazbenog instrumenta. Bio je fasciniran ljudskim glasom te u njegovim skladbama klavir zvuči poput glasa koji pjeva. Kako je i sam bio odličan izvođač, znatno je utjecao na usavršavanje klavirske tehnike. Klavirska sonata, koja je u razdoblju klasike bila jedna od najčešćih glazbenih vrsta, nije zaobišla Chopina, iako sonata u tom vremenu znatno gubi na popularnosti. Tako su prijašnji skladatelji pisali čak i do pedeset sonata za svoga života, dok je u romantizmu i nadolazećim razdobljima taj broj pao na jednoznamenkasti, što objašnjava činjenica da su skladatelji više svoje glazbene zamisli. Chopin je napisao tri sonate za klavir. Prva sonata posvećeni traženju novih formi u kojima bi izrazili u c-molu iz op. 4 njegovo je rano djelo, koje je izdano tek nakon njegove smrti, dok treća sonata u h-molu iz op. 58 spada u njegova kasna djela. U ovom radu analizirat će se Chopinova druga sonata u b-molu iz op. 35 "Marche funèbre". Mnogi su umjetnici analizirali ovo djelo ali tu je uglavnom bilo riječ o formalno-harmonijskoj analizi koja je primarno namijenjena glazbenim teoretičarima ili muzikolozima. Ova je analiza, za razliku od takve, prvenstveno usmjerena pijanistima i pedagozima i problemima s kojim se susreću prilikom sviranja skladbe. Cilj ovog rada je pridonijeti lakšem izvođenju, uvježbavanju i čitanju ovoga djela. Isto tako ovaj rad možda potakne druge pijaniste na pisanje slične literature koja može olakšati posao svladavanja neke nepoznate skladbe.With his music works, Frédéric François Chopin made a huge impact on piano world and left a mark that changed perception of piano as a musical instrument. His fascination with human voice led him to write melodies that sound exactly like singing voice. He was a great performer so he also made a huge impact on piano technique. Piano sonata did not bypass Chopin as a composer despite the fact it was not so popular among composers during his life but it was most common music form in the Classic era. Back then, composers wrote even 50 sonatas during their career, while in romantism that number dropped to one digit number because composers started to look for new form to express their ideas. Chopin wrote 3 piano sonatas. First one in c–minor op. 4 is his early piece and it was published only after his death. Third sonata in b–minor op. 58 belongs to his late works. This work will analyse his second sonata in b-flat minor op. 35 "Marche funèbre". There are lot of analysis that are mainly about sonata form and harmony. This work, on the other hand, is dedicated to pianists and piano teachers and problems that they have while playing. This work's goal is to make practice, interpretation and reading of this piece easier. Also, it might give motivation to other pianists to write similar works of their own

Private Key Extension of Polly Cracker Cryptosystems

In 1993 Koblitz and Fellows proposed a public key cryptosystem, Polly Cracker, based on the problem of solving multivariate systems of polynomial equations, which was soon generalized to a Dröbner basis formulation. Since then a handful of improvements of this construction has been proposed. In this paper it is suggested that security, and possibly efficiency, of any Polly Cracker-type cryptosystem could be increased by altering the premises regarding private- and public information

Exponential Time Complexity of the Permanent and the Tutte Polynomial

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting

Instruktivna analiza Chopinove 2. Sonate u b-molu, op. 35, "Marche funebre"

Frédéric François Chopin svojim je skladateljskim opusom zadužio cijeli pijanistički svijet i ostavio trag koji je promijenio shvaćanje klavira kao glazbenog instrumenta. Bio je fasciniran ljudskim glasom te u njegovim skladbama klavir zvuči poput glasa koji pjeva. Kako je i sam bio odličan izvođač, znatno je utjecao na usavršavanje klavirske tehnike. Klavirska sonata, koja je u razdoblju klasike bila jedna od najčešćih glazbenih vrsta, nije zaobišla Chopina, iako sonata u tom vremenu znatno gubi na popularnosti. Tako su prijašnji skladatelji pisali čak i do pedeset sonata za svoga života, dok je u romantizmu i nadolazećim razdobljima taj broj pao na jednoznamenkasti, što objašnjava činjenica da su skladatelji više svoje glazbene zamisli. Chopin je napisao tri sonate za klavir. Prva sonata posvećeni traženju novih formi u kojima bi izrazili u c-molu iz op. 4 njegovo je rano djelo, koje je izdano tek nakon njegove smrti, dok treća sonata u h-molu iz op. 58 spada u njegova kasna djela. U ovom radu analizirat će se Chopinova druga sonata u b-molu iz op. 35 "Marche funèbre". Mnogi su umjetnici analizirali ovo djelo ali tu je uglavnom bilo riječ o formalno-harmonijskoj analizi koja je primarno namijenjena glazbenim teoretičarima ili muzikolozima. Ova je analiza, za razliku od takve, prvenstveno usmjerena pijanistima i pedagozima i problemima s kojim se susreću prilikom sviranja skladbe. Cilj ovog rada je pridonijeti lakšem izvođenju, uvježbavanju i čitanju ovoga djela. Isto tako ovaj rad možda potakne druge pijaniste na pisanje slične literature koja može olakšati posao svladavanja neke nepoznate skladbe.With his music works, Frédéric François Chopin made a huge impact on piano world and left a mark that changed perception of piano as a musical instrument. His fascination with human voice led him to write melodies that sound exactly like singing voice. He was a great performer so he also made a huge impact on piano technique. Piano sonata did not bypass Chopin as a composer despite the fact it was not so popular among composers during his life but it was most common music form in the Classic era. Back then, composers wrote even 50 sonatas during their career, while in romantism that number dropped to one digit number because composers started to look for new form to express their ideas. Chopin wrote 3 piano sonatas. First one in c–minor op. 4 is his early piece and it was published only after his death. Third sonata in b–minor op. 58 belongs to his late works. This work will analyse his second sonata in b-flat minor op. 35 "Marche funèbre". There are lot of analysis that are mainly about sonata form and harmony. This work, on the other hand, is dedicated to pianists and piano teachers and problems that they have while playing. This work's goal is to make practice, interpretation and reading of this piece easier. Also, it might give motivation to other pianists to write similar works of their own

Barn og krigstraumer : en teoretisk studie og en undersøkelse av noen barn i Bosnia og Hercegovina

Problemstilling Avhandlingen omhandler barn med direkte og indirekte krigsopplevelser. Avhandlingen tar opp teori om traume og ulike aspekter av traumatiske opplevelser forårsaket av krig og konsekvenser dette har for barns intellektuelle, emosjonelle og sosiale vansker. Barn utsatt for traumer påvirkes av flere faktorer som deres egne ressurser, familie og nettverket. Avhandlingen tar opp med vekt på skolen. Skolen er en viktig faktor i alle aspekter ved barnets utvikling og liv, fordi skolen er ved siden av foreldre den viktigste oppdrager. Teorier som det settes fokus på i avhandlingen, belyser barnets evne til mestring ved hjelp av resiliencefaktorer. I denne forbindelsen blir det redegjort for resiliencefaktorer som fremmer mestring hos barn utsatt for stress og risiko. Dette var utgangspunkt for avhandlingens problemstilling som er følgende: Hvordan krig påvirker barn? Hva vet vi beskytter barn mot traumer og hva gjør at barn og unge mestrer? Hva kan skolen gjøre for å fremme mestring hos barn med traumatiske opplevelser som følge av krig? Metode For å kunne belyse problemstillingen ble det foretatt teoristudie og en kvalitativ studie. Undersøkelsen ble bygd på teorier om traume, mestring og resiliencefaktorer. Grunnmaterialet i den empiriske undersøkelsen var intervju av en liten gruppe elever som var direkte eller indirekte involvert i krigen i Bosnia og Hercegovina. Andre kilder til informasjon var spørreskjema som vi brukte på elevenes klassestyrere og innsyn i elevenes journal fra et ”detraumatiseringsprosjektet” elevene var allerede involvert i. Spørsmålet vi stilte oss var om funnene fra vår undersøkelse kan forklares ut fra de teoretiske redegjørelsene: om hvordan traumer, mestring, og resiliencefaktorer kan gi seg utslag hos noen barn etter krig, med tanke på skoleutfordringer også. Analysen av intervjuene og spørreskjema gikk ut på å sammenligne resultater av undersøkelsen med teorier. Grunnlaget for analysen av resultatene var analytisk generalisering. Det vil si at hvis resultatene støtter allerede eksisterende teori på området, så styrkes disse teoriene. Resultater/konklusjon Som oppsummering kan vi si at forventningene til undersøkelsen ble innfridd og at resultater fra undersøkelsen kan forklares ut fra teorier. Elevene som var med i undersøkelsen viste kjennetegn på at de er påvirket av traumatiske hendelser under og etter krigen og at resiliencefaktorer som har vært til stede i barnas liv, har gjort dem i bedre stand til å møte skolerelaterte utfordringer. De eksisterende teorier og den empiriske undersøkelsen har hjulpet oss til å forstå at det er viktig å utvikle tiltak på skolen for barn som var/er utsatt for traumatiske belastninger, for å fremme resiliencefaktorer som har vært til stedet i deres liv

Algorithms and theory for polynomial eigenproblems

In this thesis we develop new theoretical and numerical results for matrix polynomials and polynomial eigenproblems. This includes the cases of standard and generalized eigenproblems. Two chapters concern quadratic eigenproblems $(M\lambda^2+D\lambda+K)x=0$, where $M$, $D$ and $K$ enjoy special properties that are commonly encountered in modal analysis. We discuss this application in some detail, in particular the mathematics behind discrete dampers. We show how the physical intuition of a damper that gets stronger and stronger can be mathematically proved using matrix analysis. We then develop an algorithm for quadratic eigenvalue problems with low rank damping, which outperforms existing algorithm both in terms of speed and accuracy. The first part of our algorithm requires the solution of a generalized eigenproblem with semidefinite coefficient matrices. To solve this problem we develop a new algorithm based on an algorithm proposed by Wang and Zhao [SIAM J. Matrix Anal. Appl. 12-4 (1991), pp. 654--660]. The new algorithm computes all eigenvalues in a backward stable and symmetry preserving manner. The next two chapters are about equivalences of matrix polynomials. We show, for an algebraically closed field $\mathbb{F}$, that any matrix polynomial $P(\lambda)\in\mathbb{F}[\lambda]^{n\times m}$, $n\leq m$, can be reduced to triangular form, while preserving the degree and the finite and infinite elementary divisors. We then show that the same result holds for real matrix polynomials if we replace ``triangular'' with ``quasi-triangular,'' that is, block-triangular with diagonal blocks of size $1\times 1$ and $2 \times 2$. The proofs are constructive in the sense that we build up triangular and quasi-triangular matrix polynomials starting the Smith form. In this sense we are solving structured inverse problems. In particular, our results imply that the necessary constraints that make list of elementary divisors admissible for a real square matrix polynomial of degree $\ell$ are also sufficient conditions. For the case of matrix polynomials with invertible leading coefficients, we show how triangular/quasi-triangular forms, as well as diagonal and Hessenberg forms, can be computed numerically. Finally, we present a backward error analysis of the shift-and-invert Arnoldi algorithm for matrices. This algorithm is also of interest to polynomial eigenproblems with easily constructible monic linearizations. The analysis shows how errors from the linear system solves and orthonormalization process affect the Arnoldi recurrence. Residual bounds for linear systems and columnwise backward error bounds for QR factorizations come to play, so we discuss these in some detail. The main result is a set of backward error bounds that can be estimated cheaply. We also use our error analysis to define a sensible condition for ``breakdown,'' that is, a condition for when the Arnoldi iteration should be stopped