scholarly journals Barred Preferential Arrangements

10.37236/2482 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Connor Ahlbach ◽  
Jeremy Usatine ◽  
Nicholas Pippenger
Keyword(s):  
Linear Combination ◽  
Infinite Series ◽  
Combinatorial Identities ◽  
Blocks Of Elements

A preferential arrangement of a set is a total ordering of the elements of that set with ties allowed. A barred preferential arrangement is one in which the tied blocks of elements are ordered not only amongst themselves but also with respect to one or more bars. We present various combinatorial identities for $r_{m,\ell}$, the number of barred preferential arrangements of $\ell$ elements with $m$ bars, using both algebraic and combinatorial arguments. Our main result is an expression for $r_{m,\ell}$ as a linear combination of the $r_k$ ($= r_{0,k}$, the number of unbarred preferential arrangements of $k$ elements) for  $\ell\le k\le\ell+m$. We also enumerate those arrangements in which the sections, into which the blocks are segregated by the bars, must be nonempty. We conclude with an expression of $r_\ell$ as an infinite series that is both convergent and asymptotic.

A BASIS OF THE BASIC $\mathfrak{sl} ({\bf 3}, {\mathbb C})^~$-MODULE

2001 ◽  
Vol 03 (04) ◽  
pp. 593-614 ◽  
Author(s):  
ARNE MEURMAN ◽  
MIRKO PRIMC
Keyword(s):  
Lie Algebras ◽  
Infinite Series ◽  
Vertex Operator ◽  
Combinatorial Identities ◽  
Combinatorial Identity ◽  
Affine Lie Algebras ◽  
Standard Formula ◽  
Colored Partitions ◽  
Level 3 ◽  
Principal Specialization

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers–Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard [Formula: see text]-modules. We obtained several infinite series of new combinatorial identities through the construction of all standard [Formula: see text]-modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper we extend our construction to the basic [Formula: see text]-module and, by using the principal specialization of the Weyl–Kac character formula, we obtain a Rogers–Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky–Wilson's approach for affine Lie algebras of higher ranks, say for [Formula: see text], n ≥ 2, in a way parallel to the next level of complexity seen when passing from the Rogers–Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli ≥7.

Some Infinite Series of 2-Weighing and 3-Weighing Matrices from Hadamard Matrices

2018 ◽  
Vol 9 (12) ◽  
pp. 2165-2168
Author(s):  
Gopal Prajapati ◽  
Mithilesh Kumar Singh
Keyword(s):  
Infinite Series ◽  
Hadamard Matrices ◽  
Weighing Matrices

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

Recognition by Linear Combination of Models

1989 ◽  
Author(s):  
Shimon Ullman ◽  
Ronen Basri
Keyword(s):  
Linear Combination

A new application of quasi monotone sequences and quasi power increasing sequences to factored infinite series

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5105-5109
Author(s):  
Hüseyin Bor
Keyword(s):  
Infinite Series ◽  
Summability Factors ◽  
Monotone Sequences ◽  
Power Increasing Sequences ◽  
Weaker Conditions ◽  
Increasing Sequences

In this paper, we generalize a known theorem under more weaker conditions dealing with the generalized absolute Ces?ro summability factors of infinite series by using quasi monotone sequences and quasi power increasing sequences. This theorem also includes some new results.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

scholarly journals An Accurate Reconstruction of CMB E Mode Signal over Large Angular Scales using Prior Information of CMB Covariance Matrix in ILC Algorithm

2020 ◽  
Author(s):  
Ujjal Purkayastha ◽  
Vipin Sudevan ◽  
Rajib Saha
Keyword(s):  
Power Spectrum ◽  
Linear Combination ◽  
Covariance Matrix ◽  
Large Scale ◽  
Prior Information ◽  
Future Generation ◽  
Temperature Anisotropy ◽  
Satellite Mission ◽  
Angular Power Spectrum ◽  
Accurate Reconstruction

Abstract Recently, the internal-linear-combination (ILC) method was investigated extensively in the context of reconstruction of Cosmic Microwave Background (CMB) temperature anisotropy signal using observations obtained by WMAP and Planck satellite missions. In this article, we, for the first time, apply the ILC method to reconstruct the large scale CMB E mode polarization signal, which could probe the ionization history, using simulated observations of 15 frequency CMB polarization maps of future generation Cosmic Origin Explorer (COrE) satellite mission. We find that the clean power spectra, from the usual ILC, are strongly biased due to non zero CMB-foregrounds chance correlations. In order to address the issues of bias and errors we extend and improve the usual ILC method for CMB E mode reconstruction by incorporating prior information of theoretical E mode angular power spectrum while estimating the weights for linear combination of input maps (Sudevan & Saha 2018b). Using the E mode covariance matrix effectively suppresses the CMB-foreground chance correlation power leading to an accurate reconstruction of cleaned CMB E mode map and its angular power spectrum. We compare the performance of the usual ILC and the new method over large angular scales and show that the later produces significantly statistically improved results than the former. The new E mode CMB angular power spectrum contains neither any significant negative bias at the low multipoles nor any positive foreground bias at relatively higher mutlipoles. The error estimates of the cleaned spectrum agree very well with the cosmic variance induced error.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals Partially constrained internal linear combination: A method for low-noise CMB foreground mitigation

2021 ◽  
Vol 103 (10) ◽  
Author(s):  
Y. Sultan Abylkairov ◽  
Omar Darwish ◽  
J. Colin Hill ◽  
Blake D. Sherwin
Keyword(s):  
Linear Combination ◽  
Low Noise
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