Search

This thesis presents three pieces of work. Within the first two thirds of the thesis, we study Arens regularity of Banach algebras. We first study Arens regularity of weighted semigroup algebras that arise from totally ordered semilattices. This is a natural continuation of [24], where they focus on studying Arens regularity of the unweighted case. We provide a sufficient condition for when a weighted semigroup algebra is not strongly Arens irregular and a characterization of Arens regularity of the weighted semigroup algebra. We then focus on three specific totally ordered semilattices, the natural numbers with the minimum operation, the natural numbers with the maximum operation and the integers with the maximum operation to obtain stronger results than those obtained for a generic totally ordered semilattice. Later on, we focus on two different Banach sequence algebras, the James p-algebra and the Feinstein algebra. Amongst other properties, we prove that the Feinstein algebra is Arens regular, which provides a second example of an Arens regular natural Banach sequence algebra that is not an ideal in its bidual, the first one being the remarkable example obtained in [7]. We study whether the James p-algebra is a BSE algebra with a BSE norm, for 1 In the final part of the thesis, we focus on Blaschke products. We study the decomposability of a finite Blaschke product B of degree a potency of 2 into n degree-2 Blaschke products, examining the connections between Blaschke products, the elliptical range theorem, Poncelet theorem, and the monodromy group. We show that if the numerical range of the compression of the shift operator of B a Blaschke product of degree n is an ellipse, then B can be written as a composition of lower-degree Blaschke products that correspond to a factorization of the integer n. We also show that a Blaschke product of degree 2^n with an elliptical Blaschke curve has at most n distinct critical values, and we use this to examine the monodromy group associated with a regularized Blaschke product B. We prove that if B can be decomposed into n degree-2 Blaschke products, then the monodromy group associated with B is the wreath product of n cyclic groups of order 2. Lastly, we study the group of invariants of a Blaschke product B of order 2^n when B is a composition of n Blaschke products of order 2.

In 1940, the Hrvatska revija journal announced that the British Museum in London kept a mediaeval codex mentioning King Trpimir and frescoes in the church of St. George at Putalj. To this day, the controversy over the existence of the codex, frescoes and, moreover, the author of this piece of news has not ended. About three hundred small fragments of frescoes have been found during the archaeological research at Putalj. This paper presents information on the bearer of news about the codex. After reviewing all the discovered fragments, it was established that some of them were from early mediaeval frescoes., Godine 1940. u Hrvatskoj je reviji objavljena vijest da u British Muse- um u Londonu postoji srednjovjekovni kodeks u kojem se spominje kralj Trpimir i freske u crkvi sv. Jurja na Putalju. Sve do danas nije završen prijepor o postojanju kodeksa, fresaka pa, štoviše, i autora te vijesti. Ar- heološkim istraživanjima na Putalju je pronađeno tristotinjak sitnih ulo- maka fresaka. U radu se donose podaci o donositelju vijesti o kodeksu, a potom se donosi pregled svih pronađenih ulomaka na osnovi kojeg je ustanovljeno da među njima ima i onih koji pripadaju ranosrednjovje- kovnim freskama.

El compost dímer [{RuCl2(p-cimè)}2] reacciona amb diferents lligands fosfina donant pas a la formació dels compostos monòmers de ruteni(II) [RuCl2(p-cimè)L] on L = PPh2H, PPh2(CH2)xpy (on x = 0 i 1) i PPh2C∫CR (on R = Ph, tBu i p-Tol). Si se'l fa reaccionar amb la fosfina PPh2C∫CPPh2 s'obté el compost d'estequiometria [{RuCl2(p-cimè)}2PPh2C∫CPPh2] on la fosfina actua com a lligand bidentat pont entre dos àtoms de ruteni. Quan es fan reaccionar els compostos [RuCl2(p-cimè)PPh2(CH2)xpy] amb NaBF4 s'obtenen els compostos [RuCl(p-cimè)PPh2py]BF4 i [RuCl(p-cimè)PPh2CH2py]BF4. En aquests compostos la fosfina actua com a lligand quelant P-N.L'estudi cristal·logràfic efectuat confirma les estructures proposades per als compostos [RuCl2(p-cimè)PPh2H], [RuCl2(p-cimè)PPh2py] i [RuCl2(p-cimè)PPh2py]BF4. El compost [RuCl2(p-cimè)PPh2C∫CPh] cristal·litza formant dos tipus de cristalls diferents amb dues estructures pràcticament idèntiques, amb l'única diferència de l'orientació dels anells fenil units directament al fòsfor en relació a l'anell del p-cimè.Els compostos [RuCl2(p-cimè)PPh2py] i [RuCl(p-cimè)PPh2py]BF4 actuen com a catalitzadors de la hidrogenació catalítica de substrats com olefines i alquins. Els rendiments són comparables als obtinguts amb altres catalitzadors malgrat que la cinètica és més lenta.La reacció de Ru3(CO)12 amb la fosfina PPh2py en excés d'àcids carboxílics dóna lloc als complexos dímers de Ru(I) [Ru2(HCOO)2(CO)4(PPh2py)2] i [Ru2(CH3COO)2(CO)4(PPh2py)2] en els quals el lligand carboxilat actua com a lligand pont i la fosfina actua com a lligand terminal. El dímer [Ru2(HCOO)2(CO)4(PPh2py)2] reacciona amb ditiols formant els corresponents complexos [Ru2(S(CH2)xS)(CO)4(PPh2py)2] (on x = 2 i 3) en els quals el lligand ditiolat actua com a lligand pont i la fosfina com a lligand terminal.La reacció de Ru3(CO)12 amb la fosfina PPh2CH2py en presència d'un excés d'àcids carboxílics dóna lloc als complexos monòmers de Ru(II) [Ru(HCOO)(CO)2(PPh2CH2py)]HCOO i [Ru(CH3COO)(CO)2(PPh2CH2py)]CH3COO en els quals els lligands carboxilat i fosfina actuen com a lligands bidentats. El complex [Ru(HCOO)(CO)2(PPh2CH2py)]HCOO reacciona amb el ditiol HS(CH2)2SH donant lloc al compost [Ru(S(CH2)2S)(CO)2(PPh2CH2py)] en el qual els lligands ditiolat i fosfina actuen com a lligands quelants., El compuesto dímero [{RuCl2(p-cimeno)}2] reacciona con diferentes ligandos fosfina dando lugar a la formación de los compuestos monómeros de rutenio(II) [RuCl2(p-cimeno)L] con L = PPh2H, PPh2(CH2)xpy (x = 0 y 1) y PPh2C∫CR (con R = Ph, tBu y p-Tol). Si se hace reaccionar este dímero con la fosfina PPh2C∫CPPh2 se obtiene el compuesto de estequiometría [{RuCl2(p-cimeno)}2PPh2C∫CPPh2] en el que la fosfina actúa como ligando bidentado puente entre dos átomos de rutenio. Cuando se hacen reaccionar los compuestos [RuCl2(p-cimeno)PPh2(CH2)xpy] con NaBF4 se obtienen los compuestos [RuCl(p-cimeno)PPh2py]BF4 y [RuCl(p-cimeno)PPh2CH2py]BF4. En éstos, la fosfina actúa como ligando quelante P-N.El estudio cristalográfico efectuado confirma las estructuras propuestas para los compuestos [RuCl2(p-cimeno)PPh2H], [RuCl2(p-cimeno)PPh2py] y [RuCl2(p-cimeno)PPh2py]BF4. El compuesto [RuCl2(p-cimeno)PPh2C∫CPh] cristaliza formando dos tipos de cristales diferentes con dos estructuras prácticamente idénticas, con la única diferencia de la orientación de los anillos fenilo unidos directamente al fósforo en relación al anillo del p-cimeno.Los compuestos [RuCl2(p-cimeno)PPh2py] y [RuCl(p-cimeno)PPh2py]BF4 actúan como catalizadores de la hidrogenación catalítica de substratos como olefinas y alquinos. Los rendimientos son comparables a los obtenidos con otros catalizadores aunque la cinética es más lenta.La reacción de Ru3(CO)12 con la fosfina PPh2py en exceso de ácidos carboxílicos da lugar a los complejos dímeros de Ru(I) [Ru2(HCOO)2(CO)4(PPh2py)2] y [Ru2(CH3COO)2(CO)4(PPh2py)2] en los que el ligando carboxilato actúa como ligando puente y la fosfina actúa como ligando terminal. El dímero [Ru2(HCOO)2(CO)4(PPh2py)2] reacciona con ditioles formando los correspondientes complejos [Ru2(S(CH2)xS)(CO)4(PPh2py)2] (x = 2 y 3) en los que el ligando ditiolato actúa como ligando puente y la fosfina como ligando terminal.La reacción de Ru3(CO)12 con la fosfina PPh2CH2py en presencia de un exceso de ácidos carboxílicos da lugar a los complejos monómeros de Ru(II) [Ru(HCOO)(CO)2(PPh2CH2py)]HCOO y [Ru(CH3COO)(CO)2(PPh2CH2py)]CH3COO en los que los ligandos carboxilato y fosfina actúan como ligandos bidentados. El complejo [Ru(HCOO)(CO)2(PPh2CH2py)]HCOO reacciona con el ditiol HS(CH2)2SH formando el compuesto [Ru(S(CH2)2S)(CO)2(PPh2CH2py)] en el qual los ligandos ditiolato y fosfina actúan como ligandos quelantes., The dimeric compound [{RuCl2(p-cymene)}2] reacts with different phosphine ligands to give the monomeric ruthenium(II) compounds [RuCl2(p-cymene)L] with L = PPh2H, PPh2(CH2)xpy (x = 0 and 1) and PPh2C∫CR (with R = Ph, tBu and p-Tol). Reaction with PPh2C∫CPPh2 leads to the compound [{RuCl2(p-cymene)}2PPh2C∫CPPh2], where the ligand acts as a bridge between two ruthenium atoms. When compounds [RuCl2(p-cymene)PPh2(CH2)xpy] react with NaBF4, [RuCl(p-cymene)PPh2py]BF4 and [RuCl(p-cymene)PPh2CH2py]BF4 are obtained. In these compounds, the phosphine ligands act as chelating P-N molecules.The crystallographic study confirms the structures proposed for compounds [RuCl2(p-cymene)PPh2H], [RuCl2(p-cymene)PPh2py] and [RuCl2(p-cymene)PPh2py]BF4. The structure of [RuCl2(p-cymene)PPh2C∫CPh] shows two type of crystalls with practically identical structure. The only difference is the orientation of phenyl rings directly linked to phosphorous.Compounds [RuCl2(p-cymene)PPh2py] and [RuCl(p-cymene)PPh2py]BF4 catalyse the hidrogenation of substrates such as olefines and alkynes. The yields are comparable to those obtained with other catalysts although the kinetics are slower.The reaction of Ru3(CO)12 with PPh2py with an excess of a carboxylic acid leads to the dimeric Ru(I) complexes [Ru2(HCOO)2(CO)4(PPh2py)2] and [Ru2(CH3COO)2(CO)4(PPh2py)2] in which the carboxylate ligands act as a bridge and the phosphine acts as a terminal ligand. The dimeric compound [Ru2(HCOO)2(CO)4(PPh2py)2] reacts with dithiols forming the corresponding complexes [Ru2(S(CH2)xS)(CO)4(PPh2py)2] (x = 2 and 3) in which the dithiolate ligands act as bridging molecules and the phosphine acts as a terminal ligand. The reaction of Ru3(CO)12 with PPh2CH2py in the presence of an excess of carboxylic acids gives the monomeric Ru(II) compounds [Ru(HCOO)(CO)2(PPh2CH2py)]HCOO and [Ru(CH3COO)(CO)2-(PPh2CH2py)]CH3COO in which carboxylate and phosphine act as bidentate ligands. The complex [Ru(HCOO)(CO)2(PPh2CH2py) ]HCOO reacts with 1,2-ethanedithiol forming the compound [Ru(S(CH2)2S)(CO)2(PPh2CH2py)] where dithiolate and phosphine ligands act as chelating ligands.