scholarly journals On the Stanley-Wilf Conjecture for the Number of Permutations Avoiding a Given Pattern

10.37236/1477 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Richard Arratia
Keyword(s):  
Recent Work ◽  
Short Note ◽  
Random Permutations ◽  
Simple Argument ◽  
Sigma 1 ◽  
Wilf Conjecture

Consider, for a permutation $\sigma \in {\cal S}_k$, the number $F(n,\sigma)$ of permutations in ${\cal S}_n$ which avoid $\sigma$ as a subpattern. The conjecture of Stanley and Wilf is that for every $\sigma$ there is a constant $c(\sigma) < \infty$ such that for all $n$, $F(n,\sigma) \leq c(\sigma)^n$. All the recent work on this problem also mentions the "stronger conjecture" that for every $\sigma$, the limit of $F(n,\sigma)^{1/n}$ exists and is finite. In this short note we prove that the two versions of the conjecture are equivalent, with a simple argument involving subadditivity We also discuss $n$-permutations, containing all $\sigma \in {\cal S}_k$ as subpatterns. We prove that this can be achieved with $n=k^2$, we conjecture that asymptotically $n \sim (k/e)^2$ is the best achievable, and we present Noga Alon's conjecture that $n \sim (k/2)^2$ is the threshold for random permutations.

scholarly journals PARAMETRIZING ELLIPTIC CURVES BY MODULAR UNITS

2015 ◽  
Vol 100 (1) ◽  
pp. 33-41 ◽  
Author(s):  
FRANÇOIS BRUNAULT
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Recent Work ◽  
Short Note ◽  
Torsion Subgroup ◽  
Modular Functions ◽  
Open Questions ◽  
Modular Units

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

scholarly journals The Infinite limit of random permutations avoiding patterns of length three

2019 ◽  
Vol 29 (1) ◽  
pp. 137-152 ◽  
Author(s):  
Ross G. Pinsky
Keyword(s):  
Probability Measure ◽  
Metric Space ◽  
The Other ◽  
Random Permutations ◽  
Compact Metric Space ◽  
Sigma 1 ◽  
Limiting Behaviour ◽  
Random Probability

AbstractFor $$\tau \in {S_3}$$, let $$\mu _n^\tau $$ denote the uniformly random probability measure on the set of $$\tau $$-avoiding permutations in $${S_n}$$. Let $${\mathbb {N}^*} = {\mathbb {N}} \cup \{ \infty \} $$ with an appropriate metric and denote by $$S({\mathbb{N}},{\mathbb{N}^*})$$ the compact metric space consisting of functions $$\sigma {\rm{ = }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{ }}$$ from $$\mathbb {N}$$ to $${\mathbb {N}^ * }$$ which are injections when restricted to $${\sigma ^{ - 1}}(\mathbb {N})$$; that is, if $${\sigma _i}{\rm{ = }}{\sigma _j}$$, $$i \ne j$$, then $${\sigma _i} = \infty $$. Extending permutations $$\sigma \in {S_n}$$ by defining $${\sigma _j} = j$$, for $$j \gt n$$, we have $${S_n} \subset S({\mathbb{N}},{{\mathbb{N}}^*})$$. For each $$\tau \in {S_3}$$, we study the limiting behaviour of the measures $$\{ \mu _n^\tau \} _{n = 1}^\infty $$ on $$S({\mathbb{N}},{\mathbb{N}^*})$$. We obtain partial results for the permutation $$\tau = 321$$ and complete results for the other five permutations $$\tau \in {S_3}$$.

scholarly journals A note on superposition of two unknown states using Deutsch CTC model

2016 ◽  
Vol 31 (29) ◽  
pp. 1650170 ◽  
Author(s):  
Sasha Sami ◽  
Indranil Chakrabarty
Keyword(s):  
Recent Work ◽  
Short Note ◽  
Quantum States ◽  
Quantum Protocol

In a recent work, authors prove a yet another no-go theorem that forbids the existence of a universal probabilistic quantum protocol producing a superposition of two unknown quantum states. In this short note, we show that in the presence of closed time-like curves (CTCs), one can indeed create superposition of unknown quantum states and evade the no-go result.

scholarly journals A SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS

2018 ◽  
Vol 98 (3) ◽  
pp. 481-493 ◽  
Author(s):  
MARKUS FAULHUBER
Keyword(s):  
Wigner Distribution ◽  
Short Note ◽  
Gabor Frame ◽  
Gabor Frames ◽  
Simple Argument ◽  
Gabor Systems ◽  
One Dimensional ◽  
Separable Case ◽  
The One ◽  
Frame Set

We give a simple argument which shows that Gabor systems consisting of odd functions of$d$variables and symplectic lattices of density$2^{d}$cannot constitute a Gabor frame. In the one-dimensional, separable case, this follows from a more general result of Lyubarskii and Nes [‘Gabor frames with rational density’,Appl. Comput. Harmon. Anal.34(3) (2013), 488–494]. We use a different approach exploiting the algebraic relation between the ambiguity function and the Wigner distribution as well as their relation given by the (symplectic) Fourier transform. Also, we do not need the assumption that the lattice is separable and, hence, new restrictions are added to the full frame set of odd functions.

scholarly journals Transaction fee mechanism design

2021 ◽  
Vol 19 (1) ◽  
pp. 52-55
Author(s):  
Tim Roughgarden
Keyword(s):  
Mechanism Design ◽  
Recent Work ◽  
Economic Change ◽  
Short Note ◽  
Late Summer ◽  
Variable Size ◽  
Current Transaction ◽  
Tightly Coupled

Demand for blockchains such as Bitcoin and Ethereum is far larger than supply, necessitating a mechanism that selects a subset of transactions to include "on-chain" from the pool of all pending transactions. EIP-1559 is a proposal to make several tightly coupled changes to the Ethereum blockchain's transaction fee mechanism, including the introduction of variable-size blocks and a burned base fee that rises and falls with demand. These changes are slated for deployment in Ethereum's "London fork," scheduled for late summer 2021, at which point it will be the biggest economic change made to a major blockchain to date. This short note provides an overview of recent work by the author that formally investigates and compares the incentive guarantees offered by Ethereum's current transaction fee mechanism and the new mechanism proposed in EIP-1559.

scholarly journals A simple note on the Yoneda (co)algebra of a monomial algebra

2021 ◽  
Vol 73 (2) ◽  
pp. 275-277
Author(s):  
E. Herscovich
Keyword(s):  
Linear Combination ◽  
Short Note ◽  
Simple Argument ◽  
Monomial Algebra

UDC 512.7 If is a monomial -algebra, it is well-known that is isomorphic to the space of (Anick) -chains for . The goal of this short note is to show that the next result follows directly from well-established theorems on -algebras, without computations: there is an -coalgebra model on satisfying that, for and , is a linear combination of , where , and . The proof follows essentially from noticing that the Merkulov procedure is compatible with an extra grading over a suitable category. By a simple argument based on a result by Keller we immediately deduce that some of these coefficients are .

scholarly journals On the Generating Hypothesis in Noncommutative Stable Homotopy

2015 ◽  
Vol 116 (2) ◽  
pp. 301 ◽  
Author(s):  
Snigdhayan Mahanta
Keyword(s):  
Recent Work ◽  
Short Note ◽  
Stable Homotopy ◽  
Triangulated Categories ◽  
Generating Hypothesis ◽  
The Way

Freyd's Generating Hypothesis is an important problem in topology with deep structural consequences for finite stable homotopy. Due to its complexity some recent work has examined analogous questions in various other triangulated categories. In this short note we analyze the question in noncommutative stable homotopy, which is a canonical generalization of finite stable homotopy. Along the way we also discuss Spanier-Whitehead duality in this extended setup.

Why God is Not a Consequentialist

1993 ◽  
Vol 29 (2) ◽  
pp. 239-243 ◽  
Author(s):  
T. D. J. Chappell
Keyword(s):  
Recent Work ◽  
Moral Philosophy ◽  
John Stuart Mill ◽  
Orthodox Christianity ◽  
Simple Argument

Can there be a moral philosophy which combines Christianity and consequentialism? John Stuart Mill himself claimed that these positions were, at the least, not mutually exclusive, and quite possibly even congenial to one another; and some recent work by Christian philosophers in America has resurrected this claim. But there is a simple argument to show that consequentialism and orthodox Christianity are not so much as jointly assertible.

scholarly journals Two more fermionic minimal models

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Justin Kulp
Keyword(s):  
Recent Work ◽  
Short Note ◽  
Minimal Models

Abstract In this short note, we comment on the existence of two more fermionic unitary minimal models not included in recent work by Hsieh, Nakayama, and Tachikawa. These theories are obtained by fermionizing the ℤ2 symmetry of the m = 11 and m = 12 exceptional unitary minimal models. Furthermore, we explain why these are the only missing cases.

What sentimentalists should say about emotion

2019 ◽  
Vol 42 ◽  
Author(s):  
Charlie Kurth
Keyword(s):  
Moral Judgment ◽  
Recent Work ◽  
Relevant Information ◽  
Moral Cognition ◽  
Multilevel Structure

Abstract Recent work by emotion researchers indicates that emotions have a multilevel structure. Sophisticated sentimentalists should take note of this work – for it better enables them to defend a substantive role for emotion in moral cognition. Contra May's rationalist criticisms, emotions are not only able to carry morally relevant information, but can also substantially influence moral judgment and reasoning.

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