173 research outputs found
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 numbers of the second kind and -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 .
These extensions naturally encompass the well known - 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 and Quantum Group
The non commuting matrix elements of matrices from quantum group
with being the -th root of unity are given a
representation as operators in Hilbert space with help of
generalized Clifford algebra generators appropriately tensored with unit 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 and with correspondingly maximal and minimal sets being
identical as expressed by ordinal sum apart from other definition
and due to that one arrives at a simple proof of the 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 diagrams or graphs named
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 diagrams or graphs named . Such cobweb posets and equivalently their
Hasse diagrams or graphs named 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 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 relation, .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
Fibonacci q-gaussian sequences
The summation formula within pascalian triangle resulting in the fibonacci
sequence is extended to the -binomial coefficients -gaussian triangles.Comment: 5 pages, 2 figure
New type Stirling like numbers - an email style letter
The notion of the Fibonacci cobweb poset from [1] has been naturally extended
to any admissible sequence in [2] where it was also recognized that the
celebrated prefab notion of Bender and Goldman [3] - (see also [4,5]) - admits
such an extension so as to encompass the new type combinatorial objects from
[2] as leading examples. Recently the present author had introduced also [6]
two natural partial orders in there: one in grading-natural subsets of
cobweb`s prefabs sets [2] and in the second proposal one endows the set sums of
the so called "prefabiants" with such another partial order that one arrives at
Bell-like numbers including Fibonacci triad sequences introduced by the present
author in [7]. Here we quote the basic observations concerning the new type
Stirling like numbers as they appear in [6]. For more on notation, Stirling
like numbers of the first kind and for proofs - see [6].Comment: 3 page
q-Poisson, q-Dobinski, q-Rota and q-coherent states
q- Dobinski formula may be interpreted as the average of powers of a random
variable X_q with the q- Poisson distribution.Comment: 4 page
A note on V-binomials recurrence for Lucas companion to sequence
The looked for V-binomials recurrence for Lucas companion to sequence
is deliveredComment: 9 page
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