scholarly journals Determinants of some pentadiagonal matrices

2021 ◽  
Vol 56 (2) ◽  
pp. 271-286
Author(s):  
László Losonczi ◽  
Keyword(s):  
Explicit Formula ◽  
Main Diagonal

In this paper we consider pentadiagonal \((n+1)\times(n+1)\) matrices with two subdiagonals and two superdiagonals at distances \(k\) and \(2k\) from the main diagonal where \(1\le k \lt 2k\le n\). We give an explicit formula for their determinants and also consider the Toeplitz and “imperfect” Toeplitz versions of such matrices. Imperfectness means that the first and last \(k\) elements of the main diagonal differ from the elements in the middle. Using the rearrangement due to Egerváry and Szász we also show how these determinants can be factorized.

scholarly journals Patterns in Khovanov link and chromatic graph homology

2018 ◽  
Vol 27 (03) ◽  
pp. 1840007
Author(s):  
Radmila Sazdanovic ◽  
Daniel Scofield
Keyword(s):  
Explicit Formula ◽  
Homology Group ◽  
Jones Polynomial ◽  
Main Diagonal ◽  
Khovanov Homology ◽  
The Third ◽  
Graph Homology

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme–Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.

scholarly journals ON MULTIPLE ZETA VALUES OF EXTREMAL HEIGHT

2015 ◽  
Vol 93 (2) ◽  
pp. 186-193 ◽  
Author(s):  
MASANOBU KANEKO ◽  
MIKA SAKATA
Keyword(s):  
Explicit Formula ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Maximal Height ◽  
Zeta Values ◽  
Sum Formula

We give three identities involving multiple zeta values of height one and of maximal height: an explicit formula for the height-one multiple zeta values, a regularised sum formula and a sum formula for the multiple zeta values of maximal height.

scholarly journals Pointwise multiple averages for sublinear functions

2018 ◽  
Vol 40 (6) ◽  
pp. 1594-1618
Author(s):  
SEBASTIÁN DONOSO ◽  
ANDREAS KOUTSOGIANNIS ◽  
WENBO SUN
Keyword(s):  
Large Class ◽  
Explicit Formula ◽  
Pointwise Convergence ◽  
Limit Function ◽  
Invariant Property ◽  
Ergodic Averages ◽  
Order Zero ◽  
Good For ◽  
Sublinear Functions ◽  
Uniquely Ergodic

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

Liquidation risk in the presence of Chapters 7 and 11 of the US bankruptcy code

2014 ◽  
Vol 01 (03) ◽  
pp. 1450023 ◽  
Author(s):  
Bin Li ◽  
Qihe Tang ◽  
Lihe Wang ◽  
Xiaowen Zhou
Keyword(s):  
Diffusion Process ◽  
Explicit Formula ◽  
Firm Value ◽  
The State ◽  
The Us ◽  
Quantitative Understanding ◽  
State Changes ◽  
Bankruptcy Code ◽  
General Time

We aim at quantitatively measuring the liquidation risk of a firm subject to both Chapters 7 and 11 of the US bankruptcy code. The firm value is modeled by a general time-homogeneous diffusion process in which the drift and volatility are level dependent and can be easily adjusted to reflect the state changes of the firm. An explicit formula for the probability of liquidation is established, based on which we gain a quantitative understanding of how the capital structures before and during bankruptcy affect the probability of liquidation.

A unified explicit formula for calculating the maximum embedment loss of deepwater anchors in clay

2021 ◽  
Vol 236 ◽  
pp. 109454
Author(s):  
Haixiao Liu ◽  
Ke Liang ◽  
Jinsong Peng ◽  
Zhong Xiao
Keyword(s):  
Explicit Formula

scholarly journals On multiple zeta values and finite multiple zeta values of maximal height

2018 ◽  
Vol 14 (04) ◽  
pp. 975-987
Author(s):  
Hideki Murahara ◽  
Mika Sakata
Keyword(s):  
Explicit Formula ◽  
Alternative Proof ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Maximal Height ◽  
Zeta Values

An explicit formula for the height-one multiple zeta values (MZVs) was proved by Kaneko and the second author. We give an alternative proof of this result and its generalization. We also prove its counterpart for the finite multiple zeta values (FMZVs).

scholarly journals On the Volume of the Zero Cell of a Class of Isotropic Poisson Hyperplane Tessellations

2014 ◽  
Vol 46 (3) ◽  
pp. 622-642 ◽  
Author(s):  
Julia Hörrmann ◽  
Daniel Hug
Keyword(s):  
Euclidean Space ◽  
Asymptotic Behaviour ◽  
Explicit Formula ◽  
Dimensional Euclidean Space ◽  
Arbitrary Dimensions ◽  
Zero Cell ◽  
Parametric Class

We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result, we obtain an explicit formula for the variance of the volume of the zero cell in arbitrary dimensions. From this formula we deduce the asymptotic behaviour of the volume of the zero cell as the dimension goes to ∞.

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