The largest size of an (s,s + 1)-core partition with parts of the same parity

2020 ◽  
pp. 1-14
Author(s):  
Hayan Nam ◽  
Myungjun Yu
Keyword(s):  
Explicit Formula ◽  
Coprime Integers

Finding the largest size of a partition under certain restrictions has been an interesting subject to study. For example, it is proved by Olsson and Stanton that for two coprime integers [Formula: see text] and [Formula: see text], the largest size of an [Formula: see text]-core partition is [Formula: see text]. Xiong found a formula for the largest size of a [Formula: see text]-core partitions with distinct parts. In this paper, we find an explicit formula for the largest size of an [Formula: see text]-core partition such that all parts are odd (or even).

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 A Lucas–Lehmer approach to generalised Lebesgue–Ramanujan–Nagell equations

2021 ◽  
Author(s):  
Vandita Patel
Keyword(s):  
Computationally Efficient ◽  
Efficient Approach ◽  
Coprime Integers ◽  
Primitive Divisor

AbstractWe describe a computationally efficient approach to resolving equations of the form $$C_1x^2 + C_2 = y^n$$ C 1 x 2 + C 2 = y n in coprime integers, for fixed values of $$C_1$$ C 1 , $$C_2$$ C 2 subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.

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 The number of maximum primitive sets of integers

2021 ◽  
pp. 1-15
Author(s):  
Hong Liu ◽  
Péter Pál Pach ◽  
Richárd Palincza
Keyword(s):  
Related Problem ◽  
Maximum Size ◽  
Multiplicative Error ◽  
Coprime Integers ◽  
Primitive Sets

Abstract A set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = lim n→∞ f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well. We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$ . We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.

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).

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