An integration of Euler's pentagonal partition

A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler's recurrence based on the pentagonal numbers, but where the coefficients result from a discrete integration of Euler's coefficients. Both a bijective proof and one based on generating functions show the equivalence of the subject recurrences.Comment: 22 pages, 2 figures. The recurrence investigated in this paper is essentially that proposed in Exercise 5.2.3 of Igor Pak's "Partition bijections, a survey", Ramanujan J. 12 (2006), but casted in a different form and, perhaps more interestingly, endowed with a bijective proof which arises from a construction by induction on maximal part

Investigating the proteomic profile of cocoa beans for understanding the development of cocoa flavour

Cocoa seed storage proteins play an important role in flavour development as aroma precursors are formed from their degradation during fermentation. Major proteins in the beans of Theobroma cacao are the storage proteins belonging to the vicilin and albumin classes. Although both these classes of proteins have been extensively characterised, there is still limited information on the expression and abundance of other proteins present in cocoa beans. This work is the first attempt to characterize the whole cocoa bean proteome by nano-LC-ESI MS/MS analysis using tryptic digests of cocoa bean protein extracts. The results of this analysis showed that over 1000 proteins could be identified using a species-specific Theobroma cacao database. The majority of the identified proteins were involved with metabolism and energy. Albumin and vicilin storage proteins showed the highest intensity values among all detected proteins. A comparison of MS/MS data searches carried out against larger non-specific databases confirmed that using a species-specific database can increase the number of identified proteins, and at the same time reduce the number of false positives. The proteomic profiles of cocoa beans from four genotypes with different genetic background and flavour profiles have also been analysed employing a bottom-up label-free UHPLC-MS/MS approach. From a total of 430 identified proteins, 61 proteins were found significantly differentially abundant among the four cocoa genotypes analysed with a fold change of 2 or more. PCA analysis allowed a clear separation of the genotypes based on their proteomic profiles. Interestingly, proteases which degrade storage proteins during fermentation have been found differentially abundant in some of the genotypes analysed. These proteins are involved in the release of flavour precursors, and therefore might play a key role in the shaping of the final flavour profile. Different genotype-specific levels of other enzymes which generate volatiles compounds that could potentially lead to flavour-inducing compounds have also been detected. Overall, this study shows that UHPLC-MS/MS data can differentiate cocoa bean varieties, and thus might be linked to differences in their flavour profile. Finally, a method to identify and quantify free peptides from fermented cocoa beans by UHPLC-MS/MS analysis has been developed. A total of 155 peptides could be identified and quantified in fermented cocoa beans using this approach. The vast majority of these peptides were associated to vicilin and a 21 kDa albumin, which are the most abundant proteins in cocoa beans. This methodology could be applied to assess the free peptides profiles of cocoa beans at different stage of fermentation

Algebraic Aspects of Families of Fuzzy Languages

We study operations on fuzzy languages such as union, concatenation, Kleene $\star$, intersection with regular fuzzy languages, and several kinds of (iterated) fuzzy substitution. Then we consider families of fuzzy languages, closed under a fixed collection of these operations, which results in the concept of full Abstract Family of Fuzzy Languages or full AFFL. This algebraic structure is the fuzzy counterpart of the notion of full Abstract Family of Languages that has been encountered frequently in investigating families of crisp (i.e., non-fuzzy) languages. Some simpler and more complicated algebraic structures (such as full substitution-closed AFFL, full super-AFFL, full hyper-AFFL) will be considered as well.\ud In the second part of the paper we focus our attention to full AFFL's closed under iterated parallel fuzzy substitution, where the iterating process is prescribed by given crisp control languages. Proceeding inductively over the family of these control languages, yields an infinite sequence of full AFFL-structures with increasingly stronger closure properties