On trigonometric-like decompositions of functions with respect to the cyclic group of order n

The cyclic group labeled family of quasi-projection operators is used for investigation of decomposition of functions with respect to the cyclic group of order n . Series of new identities thus arising are demonstrated and new perspectives for further investigation are indicated as for example in the case of q-extended special polynomials

Graded posets zeta matrix formula

The way to arrive at formula of zeta matrix for any graded posets with the finite set of minimal elements is delivered following the first reference. This is being achieved via adjacency and zeta matrix description of bipartite digraphs chains, the representatives of graded posets. The bipartite digraphs elements of such chains amalgamate to form corresponding cover relation graded poset digraphs with corresponding adjacency matrices being amalgamated throughout natural join as special adequate database operation. The colligation of reachability and connectivity with the presented description is made explicit. The special posets encoded via kodags directed acyclic graphs as cobeb posets cover relations digraphs are recognized as an example of differential posets subfamily. As on this night one reminisce anniversary of death of distinguished johann bernoulli the first this sylvester night article is to commemorate this date.Comment: 15 pages, affiliated to The Internet Gian-Carlo Rota Polish Seminar http://ii.uwb.edu.pl/akk/sem/sem_rota.ht

New formulas for Stirling-like numbers and Dobinski-like formulas

Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $[1]$. These extensions naturally encompass the well known $q$- extensions. The indicatory references are to point at a part of the vast domain of the foundations of computer science in arxiv affiliation.Comment: 9 pages, presented at the Gian-Carlo Rota Polish Seminar, http://ii.uwb.edu.pl/akk/sem/sem_rota.ht

On Generalized Clifford Algebra C4(n)C_4^{(n)} and GLq(2;C)GL_q(2;C) Quantum Group

The non commuting matrix elements of matrices from quantum group $GL_q(2;C)$ with $q\equiv \omega$ being the $n$-th root of unity are given a representation as operators in Hilbert space with help of $C_4^{(n)}$ generalized Clifford algebra generators appropriately tensored with unit $2\times 2$ matrix infinitely many times. Specific properties of such a representation are presented.Comment: 12 page

Natural join construction of graded posets versus ordinal sum and discrete hyper boxes

One introduces here the natural join P \os Q of graded posets $< P,\leq_P >$ and $$ with correspondingly maximal and minimal sets being identical as expressed by ordinal sum $P\oplus Q$ apart from other definition and due to that one arrives at a simple proof of the $M{\"{o}}bius$ function formula for cobweb posets. We also quote the other authors explicit formulas for the zeta matrix and its inverse for any graded posets with the finite set of minimal elements from earlier works of the author. These formulas are based on the formulas for cobweb posets and their $Hasse$ diagrams or graphs named $KoDAGs$ which are interpreted as chains of binary complete or universal relations joined by the natural join operation. Natural join of two independent sets is therefore the ordinal sum of this trivially ordered posets represented also by directed biclique named dibiclique and correspondingly by their $Hasse$ diagrams or graphs named $KoDAGs$. Such cobweb posets and equivalently their Hasse diagrams or graphs named $KoDAGs$ are also encoded by discrete hyper-boxes and the natural join operation of such discrete hyper boxes is just cartesian product of them accompanied with projection out of common faces. All graded posets with no mute vertices in their $Hasse$ diagrams which means that no vertex has indegree or outdegree equal zero are natural join of chain of relations and may be at the same time interpreted an $n-ary$ relation, $n \in N \cup \{\infty \}$.Comment: 51 pages, 11 figures, The Internet Gian-Carlo Rota Polish Seminar article http://ii.uwb.edu.pl/akk/sem/sem\_rota.ht

Pascal like matrices - an accessible factory of one source identities and resulting applications

The extension of pascalian like matrices depending on a variable from any field of zero characteristics are shown at work for the first time. Their properties appear to be one source factory of identities and resulting foreseen applicationsComment: 8 page

Crossed product of a C*-algebra by a semigroup of bounded positive linear maps. Interactions

The paper presents a construction of the crossed product of a C*-algebra by a commutative semigroup of bounded positive linear maps generated by partial isometries. In particular, it generalizes Antonevich, Bakhtin, Lebedev's crossed product by an endomorphism, and is related to Exel's interactions. One of the main goals is the Isomorphism Theorem established in the case of actions by endomorphisms.Comment: 20 page

Fibonacci q-gaussian sequences

The summation formula within pascalian triangle resulting in the fibonacci sequence is extended to the $q$-binomial coefficients $q$-gaussian triangles.Comment: 5 pages, 2 figure

The logarithmic fibbinomial formula

Roman logarithmic binomial formula analogue has been found . It is presented here also for the case of fibonomial coefficients which recently have been given a combinatorial interpretation by the present author.Comment: 1 figur

Prefab posets` Whitney numbers

We introduce a natural partial order in structurally natural finite subsets the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers` triangular array are then calculated and the explicit formula for them is provided. Next - in the second construction - we endow the set sums of prefabiants with such an another partial order that their their bell like numbers include fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial newton coefficients and fibonacci numbers onto the infinity of their relatives among which there are also the fibonacci triad sequences and binomial-like coefficients (incidence coefficients included).Comment: 8 pages, 5 figure