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To paraphrase Pete Rose, "If it weren't for bad luck, the Wildcats wouldn't have any luck at all." Northwestern began the 2003-2004 season with a legitimate Big 10 post player in Sarah Kwasinski and a supporting cast of decent, young players. They had every reason to expect a more successful season than they had seen in many years. But a string of injuries and player suspensions left the Wildcats with just 8 players, many of whom were inexperienced with suspect basketball skills. June Olkowski's squad came into Mackey Arena on Thursday night with a 0-3 Big 10 record. It was apparent from the opening tip that they expected to depart West Lafayette with a 0-4 mark in the conference. The Wildcats played like ‘fraidy cats in the first half, and, as the Boilers rang up the points, they could do little in response. Half way through the first, the score was 22-4 in the home team's favor. In a complete departure from any other game, Curry used a platoon pattern when she substituted. The entire starting team sat at the 15-minute mark to be replaced by 5 fresh players. Every started then returned to the court at the 5 minute mark. The Boilers' level of play remained high no matter which squad was on the court. When the half was over, 10 Purdue players had played exactly 10 minutes each and the score was 43-12.
Olkowski's half time speech might have been as simple as telling her players that since they had made the trip all the way down to Lafayette and they had brought their uniforms, after all, they might consider playing some basketball. Meanwhile, Purdue's starters appeared to feel as if their night was finished. The Wildcats came out with much more fire than they exhibited in the first half, and began the second stanza with a 7-0 run. With 3 minutes elapsed Curry inserted her "second team." The bench players brought their original intensity and a Lawless three pointer ended the Boiler drought. Despite the Wildcat's increased intensity, the two teams basically traded baskets until the end of the contest. The final score was 71-41.
The Wildcats spent the entire game in a zone defense that sagged more than a 70-year-old pole dancer. The Boilers respond by taking a large number of three point shots, and, as one might expect when they shoot with no defensive pressure, the team connected on 50% of their attempts. The Boilermaker offense appeared to be geared towards speeding up the game by scoring out of the half court after allowing the shot clock to get to single digits. As a result of this patience, 20 of the Vegas Gold and Black's 27 made baskets were assisted. The other 7 were the result of steals and break away lay-ups. Overall the Boilers shot 42% from the floor.
Northwestern had trouble bringing the ball up the court against any of the Boiler guards, and Curry's charges were credited with an incredible 17 steals on the game.
The Wildcat's shooting woes could pretty much be equally credited to Purdue's defensive pressure and their poor shooting. The Boilers allowed Northwestern to shoot up three point attempts, and the purple and white's 20% effort from downtown certainly validated that strategy. Overall the visitors shot 37% from the floor on the game.
This game perfectly illustrated the importance (or lack thereof) of dominating the glass in any given contest. The Wildcats won the battle of the boards 35-30. The number of shot attempts can actually explain the disparity in rebounding numbers. The Wildcats went 17-46 on the night. Thus, only 29 rebounds were available to be snagged on the defensive end. When viewed that way, the fact the Boilers grabbed 19 of them doesn't seem so bad. In the one bright spot for the Wildcats, Sarah Kwasinski nabbed a game-high 10 rebounds. Shereka Wright led the Boilers with 6.
As one might expect from a game played primarily on the perimeter, the fouls were few and far between. Purdue only attempted a total of 9 free throws. Seven were made for a 78% mark on the night. Katie Gearlds was perfect from the line going 2-2.
Purdue recorded 20 assists and 9 turnovers. A team just doesn't do much better than that. This mark is particularly remarkable when one considers that no starter logged more than 16 minutes on the night. The Boilermakers dictated every aspect of the contest, and were able to set the pace and tone of the game at all times.
The starters got the game off on the right foot, and played as a controlled, efficient unit. They appeared to be concentrating more on defense than offense when the game started, but took advantage of the easy scoring opportunities presented to them when Northwestern failed to keep up with ball reversal and skip passes. For much of the game the starters played at perhaps 80% of their usual intensity, and they should have plenty in their tank for the big showdown against Minnesota on Sunday. Perhaps the most amazing feature of the game was that, for the first time in her Purdue career, Shereka Wright was not fouled in the act of shooting during the game and did not attempt any free throws.
While every other player‘s intensity waxed and waned, Erin Lawless was giving 100% at all times. One suspects this is her nature, and that she just isn't capable of ramping it down or doing anything less than going all out. The stat. which best illustrates her intensity is the fact that she attempted 5 of Purdue's 9 free throws. This speaks to her willingness to take the ball to the hole aggressively. Erin demonstrated her three point shooting ability hitting 2 of 3 from “downtown.” Erin also demonstrated suburb passing and decision making as she'd find a player for an assist if her shot was too heavily contested. In an excellent outing, Erin recorded a game-high 16 points (5-8, 2-3 3 pt.ers, 4-5 FT), 4 rebounds, a game-high 5 assists, and a steal to one turnover.
Katie Gearlds' shooting touch returned and she appeared to very comfortable on the court. After the contest Katie reportedly said that the game was fun again, and that certainly was the impression she gave on the court. Katie was able to demonstrate her versatility and basketball sense, as she'd post up down low to score over smaller players, then get out in the passing lanes to grab the steals that led to transition baskets. Katie ended the evening with a total of 11 points (4-9, 1-2 3 pt.er, 2-2 FT), 3 rebounds, 2 assists, 7 (seven!!) steals, and a rebound.
Ashley Mays played an excellent game, easily her best effort as a Boilermaker. Northwestern's shaky dribbling skills played right into Ashley's quick hands, and time after time she was able to make a steal in the half court for an easy Boiler transition bucket. In one post game write-up, Purdue's dominance was credited to the Boiler's "pressing defense." In truth, however, it was a pressing defense of one. Purdue did not run full court presses or traps (it would have been extremely uncharacteristic for Curry to employ such strategies against such an overmatched foe); all they did was allow Ashley to pick her opponent clean. Ashley ended the evening with a total of 6 points (3-6), 3 rebounds, a blocked shot, 4 steals, and 1 turnover.
Sharika Webb ran the point when the bench "shift" was in the game. Sharika did a capable job, and her excellent court vision allows her to get the ball to the right person at the right time. More than any other time this year, she looked to score and took her share of shots, connecting on 30% of her attempts. Defensively she was extremely effective pressuring the ball in the open court. In all Sharika recorded a total of 8 points (3-10, 2-5 3 pt.er), 4 assists, and 2 steals to 1 turnover.
Carol Duncan played a solid game. The role she is taking on makes her "disappear" on the court at times, as she is doing most of her work off the ball. Suffice it to say, when the Boilers are defending tough and in a good offensive flow overall, Carol is doing her job well. Carol recorded 4 points (2-5), 4 rebounds, an assist and a steal.
Sabrina Keys has returned from California and was on the court for the final 3 minutes of the contest. In that time she attempted a shot and committed a turnover.
Kristy Curry must have been the happiest person in Mackey arena. Curry was able to give her starters a rest, but, unlike a total night off, they did get a tune up and were able to break a sweat. The weak opponent also allowed her to give the bench players the minutes they needed to establish a rhythm and play within the flow of the game. This, in turn, allowed them to succeed and to build their confidence in game situations. Last but certainly not least, nobody got hurt - one of the prime worries in any blow out.
Along with the teams, the officials had a very easy time of it on Thursday. Both teams seemed very happy to play a finesse game, and very little contact was made on either end of the court. Although there was no doubt at least one infraction-free game played somewhere during the history of NCAA women's basketball, this game's grand total of 15 personal fouls called must one of the recent finalists for fewest whistles in a contest.
The official attendance was given as 8073, and the lower bowl of Mackey appeared to be about three-quarter's full. It's pretty darn hard to get whipped up over a 30-point blowout, and by and large the crowd was quite. One benefit of playing the bench for extended minutes was that it did keep the crowd engaged as they seemed happy to cheer on the times "their girls" made a noteworthy play.
The Boilers go from the playing the worst team in the Big 10 to one of the best in the country when they take on Minnesota on Sunday. The Gophers are the only undefeated D1 team right now and are ranked sixth in the country. They achieved this record against less than stellar competition, however, so it's not clear just how good the team is really is. The Boilers are about to find out Sunday afternoon. If Purdue is to have any realistic hope of winning the Big 10 regular season crown, they will need to secure a victory. One hopes they are up to the challenge. | http://oldgoldfreepress.com/columnist/Capri_Small/columns/133.shtml |
PUEBLO - Colorado State University-Pueblo Mathematics Professor Janet Barnett has received funding through a major grant from the National Science Foundation as part of a national, seven-university collaboration to develop, test, and publish innovative new materials for teaching undergraduate mathematics.
The Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) program will replace standard classroom lectures with a collection of “primary source projects” (PSP) that directly engage students with the mathematics they are studying. Each PSP will focus on a particular mathematical concept or procedure as it was developed by a historic mathematician. Students read source documents by the original author, and through a series of exercises that are woven throughout the project, develop a fuller understanding of the mathematics they are studying as they react to the historical source, organize their thoughts about the mathematical ideas in the source, and rediscover groundbreaking ideas for themselves.
Barnett’s collaborators in TRIUMPHS include faculty from the University of Colorado - Denver, New Mexico State University, Central Washington University, Florida State University, Ursinus College in Pennsylvania, and Xavier University in Ohio. The team received a total of $1.25 million, most of which will be used for the project’s research component and to train faculty in the method. CSU-Pueblo’s share of the five-year award is $100,000.
According to Barnett, teaching with original source material is not really a new idea.
“The humanities and social sciences do this all the time. For example, students learn philosophy by actually reading great philosophical minds like Plato and Descartes. But in mathematics, we don’t have students read Newton or Gauss.,” she said. “Instead, we give them modern equations and theorems without any motivation as to how or why they were developed. Students miss out on a big part of the picture that they need to deeply understand the mathematics that they are learning.”
Barnett’s belief in the benefits of this teaching method is based on her experience using PSPs with her students over the last decade. She has written seven PSPs with prior support from the NSF, delivered presentations about the method at regional, national, and international conferences, and published several journal articles about it.
“The difference in how well students come to understand the mathematics when it’s placed in its historical context is simply astounding,” Barnett said. She said most math textbooks simply present mathematical ideas in a distilled form that is removed from the questions that motivated their initial development. In contrast, original sources place these ideas in the context of the problem the author wished to solve, and the setting in which the work was done. PSP exercises also enhance students’ verbal and deductive skills, while helping them to develop important mathematical habits of mind such as solving problems, making conjecture, and creating definitions.
The TRIUMPHS team will create and test at least 50 new PSPs of different lengths, on topics ranging from trigonometry and pre-calculus to advanced calculus and abstract algebra. Barnett and her collaborators also will provide training in the use of PSPs to more than 100 faculty members and doctoral students all across the country. More than 50 faculty members from 31 geographically and institutionally diverse institutions already have committed to using PSPs in their classes, including five faculty members at CSU-Pueblo.
TRIUMPHS also will complete an evaluation-with-research study in order to provide the team with feedback throughout the award period, while simultaneously contributing to the research base in STEM (Science, Technology, Engineering and Mathematics) education.
“The research component of the grant is especially exciting, since it will allow us to study the impact of this approach on student learning with more than just anecdotal evidence,” Barnett said.
Colorado State University - Pueblo is a regional, comprehensive university emphasizing professional, career-oriented, and applied programs. Displaying excellence in teaching, celebrating diversity, and engaging in service and outreach, CSU-Pueblo is distinguished by access, opportunity, and the overall quality of services provided to its students. | https://www.csupueblo.edu/news/2015/09-02-triumphs-grant.html |
---
abstract: 'Let $p$ be a prime number and $r\ge 0$ an integer. In this paper, we prove that there exists an anti-equivalence between the category of weak $({\varphi},\hat{G})$-modules of height $\le r$ and a certain subcategory of the category of Galois stable ${\mathbb}{Z}_p$-lattices in potentially semi-stable representations with Hodge-Tate weights in $[0,r]$. This gives an answer to a Tong Liu’s question about the essential image of a functor on weak $({\varphi},\hat{G})$-modules. For a proof, following Liu’s methods, we construct linear algebraic data which classify lattices in potentially semi-stable representations.'
author:
- 'Yoshiyasu Ozeki[^1]'
title: 'Lattices in potentially semi-stable representations and weak $({\varphi},\hat{G})$-modules'
---
Introduction
============
Let $K$ be a complete discrete valuation field of mixed characteristics $(0,p)$ with perfect residue field. We take a system of $p$-power roots $(\pi_n)_{n\ge 0}$ of a uniformizer $\pi$ of $K$ such that $\pi_0=\pi$ and $\pi^p_{n+1}=\pi_n$. We denote by $G_K$ and $G_{K_n}$ absolute Galois groups of $K$ and $K_n:=K(\pi_n)$, respectively.
For applications to interesting problems such as modularity liftings, it is useful to study an integral version of Fontaine’s $p$-adic Hodge theory, which is called integral $p$-adic Hodge theory. It is important in integral $p$-adic Hodge theory to construct “good” linear algebraic data which classify $G_K$-stable ${\mathbb}{Z}_p$-lattices in semi-stable, or crystalline, ${\mathbb}{Q}_p$-representations of $G_K$ with Hodge-Tate weights in $[0,r]$. Nowadays various such linear algebraic data are constructed; for example, so called Fontaine-Laffaille modules, Wach modules and Breuil modules. It is one of the obstructions for the use of these algebraic data that we can not use them without restrictions on the absolute ramification index $e$ of $K$ and (or) $r$. In [@Li3], based on a Kisin’s insight [@Ki] for a classification of lattices in semi-stable representations, Tong Liu defined notions of $({\varphi},\hat{G})$-modules and weak $({\varphi},\hat{G})$-modules. He constructed a contravariant fully faithful functor $\hat{T}$ from the category of weak $({\varphi},\hat{G})$-modules of height $\le r$ into the the category of free ${\mathbb}{Z}_p$-representations of $G_K$. It is the main theorem of [*loc. cit.*]{} that, without any restriction on $e$ and $r$, $\hat{T}$ induces an anti-equivalence between the category of $({\varphi},\hat{G})$-modules of height $\le r$ and the category of lattices in semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ with Hodge-Tate weights in $[0,r]$. In the end of [*loc. cit.*]{}, he posed the following question:
\[question\] What is the essential image of the functor $\hat{T}$ on [*weak*]{} $({\varphi},\hat{G})$-modules?
He showed that, if a representation of $G_K$ corresponds to a weak $({\varphi},\hat{G})$-module of height $\le r$, then it is semi-stable over $K_n$ for some $n\ge 0$ and has Hodge-Tate weights in $[0,r]$. However, the converse does not hold in general. In this paper, we give an answer to Question \[question\]. Denote by $m_0$ the maximum integer such that $K$ contains $p^{m_0}$-th roots of unity. For any non-negative integer $n$, we denote by ${\mathcal}{C}^r_n$ the category of free ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which satisfy the following property; there exists a semi-stable ${\mathbb}{Q}_p$-representation $V$ of $G_K$ with Hodge-Tate weights in $[0,r]$ such that $T\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p$ is isomorphic to $V$ as representations of $G_{K_n}$. Our main result is as follows.
\[Main1\] The essential image of the functor $\hat{T}$ is ${\mathcal}{C}^r_{m_0}$.
Therefore, we conclude that $\hat{T}$ induces an anti-equivalence between the category of weak $({\varphi},\hat{G})$-modules of height $\le r$ and the category ${\mathcal}{C}^r_{m_0}$.
The crucial part of our proof is to show the relation $${\mathcal}{C}^r_{m_0}\subset {\mathcal}{C}^r\subset {\mathcal}{C}^r_m$$ where ${\mathcal}{C}^r$ is the essential image of the functor $\hat{T}$ and $m$ is the maximum integer such that the maximal unramified extension of $K$ contains $p^m$-th roots of unity (cf. Lemma \[Lem:Main1’\]). We have two keys for our proof of this statement. The first one is Proposition \[Main2\], which gives a relation between weak $({\varphi},\hat{G})$-modules and “finite height” representations. For the proof, following the method of Liu’s arguments of [@Li3] and [@Li4], we construct certain linear data which classifies lattices in potentially semi-stable representations. This is a direct generalization of the main result of [@Li3] (the idea for our proof is essentially due to Liu’s previous works). The second one is Proposition \[Thm2\]; it says that the $G_{K_n}$-action of a finite height representation of $G_K$ which is semi-stable over $K_n$ extends to a $G_K$-action which is semi-stable over $K$.\
The author thanks Akio Tamagawa who gave him useful advice in the proof of Lemma \[lastlemma\] in the case where $p$ is odd and $m_0=0$. This work is supported by JSPS KAKENHI Grant Number 25$\cdot$173.
[**Notation :**]{} For any topological group $H$, a free ${\mathbb}{Z}_p$-representation of $H$ (resp. a ${\mathbb}{Q}_p$-representation of $H$) is a finitely generated free ${\mathbb}{Z}_p$-module equipped with a continuous ${\mathbb}{Z}_p$-linear $H$-action (resp. a finite dimensional ${\mathbb}{Q}_p$-vector space equipped with a continuous ${\mathbb}{Q}_p$-linear $H$-action). We denote by ${\mathrm}{Rep}_{{\mathbb}{Z}_p}(H)$ (resp. ${\mathrm}{Rep}_{{\mathbb}{Q}_p}(H)$) the category of them. For any field $F$, we denote by $G_F$ the absolute Galois group of $F$ (for a fixed separable closure of $F$).
Preliminary
===========
In this section, we recall some results on Liu’s $({\varphi},\hat{G})$-modules and related topics. Throughout this paper, let $p\ge 2$ be a prime number. Let $K$ be a complete discrete valuation field of mixed characteristics $(0,p)$ with perfect residue field $k$. Let $L$ be a finite extension of $K$. Take a uniformizer $\pi_L$ of $L$ and a system of $p$-power roots $(\pi_{L,n})_{n\ge 0}$ of $\pi_L$ such that $\pi_{L,0}=\pi_L$ and $\pi^p_{L,n+1}=\pi_{L,n}$. We denote by $k_L$ the residue field of $L$. Put $L_n=L(\pi_{L,n}), L_{\infty}=\cup_{n\ge 0}L_n$ and define $\hat{L}$ to be the Galois closure of $L_{\infty}$ over $L$. We denote by $H_L$ and $\hat{G}_L$ the Galois group of $\hat{L}/L_{\infty}$ and $\hat{L}/L$, respectively. We denote by $K^{{\mathrm}{ur}}$ and $L^{{\mathrm}{ur}}$ maximal unramified extensions of $K$ and $L$, respectively. Note that we have $L^{{\mathrm}{ur}}=LK^{{\mathrm}{ur}}$.
Let $R={\varprojlim}{\mathcal{O}}_{\overline{K}}/p$, where ${\mathcal{O}}_{\overline{K}}$ is the integer ring of $\overline{K}$ and the transition maps are given by the $p$-th power map. We write $\underline{\pi_L}:=(\pi_{L,n})_{n\ge 0}\in R$. Let ${\mathfrak{S}}_{L}:=W(k_L)[\![u_L]\!]$ be the formal power series ring with indeterminate $u_L$. We define a Frobenius endomorphism ${\varphi}$ of ${\mathfrak{S}}_L$ by $u_L \mapsto u_L^p$ extending the Frobenius of $W(k_L)$. The $W(k_L)$-algebra embedding $W(k_L)[u_L]\hookrightarrow W(R)$ defined by $u_L\mapsto [\underline{\pi_L}]$ extends to ${\mathfrak{S}}_L\hookrightarrow W(R)$ where $[\ast]$ is the Teichmüller representative.
We denote by ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}$ the category of ${\varphi}$-modules ${\mathfrak{M}}$ over ${\mathfrak{S}}_L$ which satisfy the following:
- ${\mathfrak{M}}$ is free of finite type over ${\mathfrak{S}}_L$; and
- ${\mathfrak{M}}$ is of height $\le r$ in the sense that ${\mathrm}{coker}(1\otimes {\varphi}\colon {\mathfrak{S}}_L\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}\to {\mathfrak{M}})$ is killed by $E_L(u_L)^r$.
Here, $E_L(u_L)$ is the minimal polynomial of $\pi_L$ over $W(k_L)[1/p]$, which is an Eisenstein polynomial. We call objects of this category [*Kisin modules of height $\le r$ over ${\mathfrak{S}}_L$*]{}. We define a contravariant functor $T_{{\mathfrak{S}}_L}\colon {\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}\to {\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_{L_{\infty}})$ by $$T_{{\mathfrak{S}}_L}({\mathfrak{M}}):=
{\mathrm}{Hom}_{{\mathfrak{S}}_L,{\varphi}}({\mathfrak{M}}, W(R))$$ for an object ${\mathfrak{M}}$ of ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}$. Here a $G_{L_\infty}$-action on $T_{{\mathfrak{S}}_L}({\mathfrak{M}})$ is given by $(\sigma.g)(x)=\sigma(g(x))$ for $\sigma\in G_{L_\infty}, g\in T_{{\mathfrak{S}}_L}({\mathfrak{M}}), x\in {\mathfrak{M}}$.
\[Kisin\] The functor $T_{{\mathfrak{S}}_L}\colon {\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}\to {\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_{L_{\infty}})$ is exact and fully faithful.
Let $S_L$ be the $p$-adic completion of $W(k_L)[u_L, \frac{E_L(u_L)^i}{i!}]_{i\ge 0}$ and endow $S_L$ with the following structures:
- a continuous ${\varphi}_{W(k_L)}$-semilinear Frobenius ${\varphi}\colon S_L\to S_L$ defined by $u_L\mapsto u_L^p$.
- a continuous $W(k_L)$-linear derivation $N\colon S_L\to S_L$ defined by $N(u_L)=-u_L$.
- a decreasing filtration $({\mathrm}{Fil}^iS_L)_{i\ge 0}$ on $S_L$. Here ${\mathrm}{Fil}^iS_L$ is the $p$-adic closure of the ideal generated by $\frac{E_L(u_L)^j}{j!}$ for all $j\ge i$.
The embedding ${\mathfrak{S}}_L\hookrightarrow W(R)$ defined above extends to ${\mathfrak{S}}_L\hookrightarrow S_L\hookrightarrow A_{{\mathrm}{cris}}$ and $S_L[1/p]\hookrightarrow B^+_{{\mathrm}{cris}}$. We take a primitive $p$-power root $\zeta_{p^n}$ of unity for $n\ge 0$ such that $\zeta^p_{p^{n+1}}=\zeta_{p^n}$. We set $\underline{{\varepsilon}}:=(\zeta_{p^n})_{n\ge 0}\in R$ and $t:=-{\mathrm}{log}([\underline{{\varepsilon}}])\in A_{{\mathrm}{cris}}$. For any integer $n\ge 0$, let $t^{\{n\}}:=t^{r(n)}\gamma_{\tilde{q}(n)}(\frac{t^{p-1}}{p})$ where $n=(p-1)\tilde{q}(n)+r(n)$ with $\tilde{q}(n)\ge 0,\ 0\le r(n) <p-1$ and $\gamma_i(x)=\frac{x^i}{i!}$ the standard divided power. Now we denote by $\nu\colon W(R)\to W(\overline{k})$ a unique lift of the projection $R\to \overline{k}$, which extends to a map $\nu \colon B^+_{{\mathrm}{cris}}\to W(\overline{k})[1/p]$. For any subring $A\subset B^+_{{\mathrm}{cris}}$, we put $I_+A={\mathrm}{Ker}(\nu\ {\mathrm}{on}\ B^+_{{\mathrm}{cris}})\cap A$.
We define a subring ${\mathcal}{R}_L$, containing $S_L$, of $B^+_{{\mathrm}{cris}}$ as below: $${\mathcal}{R}_L:=\left\{\sum^{\infty}_{i=0} f_it^{\{i\}}\mid f_i\in S_L[1/p]\
{\mathrm}{and}\ f_i\to 0\ {\mathrm}{as}\ i\to \infty\right\}.$$ Furthermore, we define ${\widehat{\mathcal{R}}}_L:={\mathcal}{R}_L\cap W(R)$. We see that $S_L$ is not $G_L$-stable under the action of $G_L$ in $B^+_{{\mathrm}{cris}}$. However, ${\mathcal}{R}_L, {\widehat{\mathcal{R}}}_L, I_+{\mathcal}{R}_L$ and $I_+{\widehat{\mathcal{R}}}_L$ are $G_L$-stable. Furthermore, they are stable under Frobenius in $B^+_{{\mathrm}{cris}}$. By definition $G_L$-actions on them factor through $\hat{G}_L$.
For an object ${\mathfrak{M}}$ of ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}$, the map ${\mathfrak{M}}\to {\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}$ defined by $x\mapsto 1\otimes x$ is injective. By this injection, we often regard ${\mathfrak{M}}$ as a ${\varphi}({\mathfrak{S}}_L)$-stable submodule of ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}$.
\[def:Liu\] A [*weak $({\varphi},\hat{G}_L)$-module of height $\le r$ over ${\mathfrak{S}}_L$*]{} is a triple $\hat{{\mathfrak{M}}}=({\mathfrak{M}},{\varphi},\hat{G}_L)$ where
- $({\mathfrak{M}},{\varphi})$ is an object of ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}$,
- $\hat{G}_L$ is an ${\widehat{\mathcal{R}}}_L$-semilinear continuous $\hat{G}_L$-action on ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$,
- the $\hat{G}_L$-action commutes with ${\varphi}_{{\widehat{\mathcal{R}}}_L}\otimes {\varphi}_{{\mathfrak{M}}}$, and
- ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})^{H_L}$.
Furthermore, we say that $\hat{{\mathfrak{M}}}$ is a [*$({\varphi},\hat{G}_L)$-module of height $\le r$ over ${\mathfrak{S}}_L$*]{} if $\hat{{\mathfrak{M}}}$ satisfies the additional condition;
- $\hat{G}_L$ acts on ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}/I_+{\widehat{\mathcal{R}}}_L({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})$ trivially.
We always regard ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}$ as a $G_L$-module via the projection $G_L\twoheadrightarrow \hat{G}_L$. We denote by ${}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$ (resp. ${{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$) the category of weak $({\varphi},\hat{G}_L)$-modules of height $\le r$ over ${\mathfrak{S}}_L$ (resp. the category of $({\varphi},\hat{G}_L)$-modules of height $\le r$ over ${\mathfrak{S}}_L$).
We define a contravariant functor $\hat{T}_L\colon {}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to
{\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_L)$ by $$\hat{T}_L(\hat{{\mathfrak{M}}})=
{\mathrm}{Hom}_{{\widehat{\mathcal{R}}}_L,{\varphi}}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}, W(R))$$ for an object $\hat{{\mathfrak{M}}}=({\mathfrak{M}},{\varphi},\hat{G}_L)$ of ${}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$. Here a $G_L$-action on $\hat{T}_L(\hat{{\mathfrak{M}}})$ is given by $(\sigma.g)(x)=\sigma(g(\sigma^{-1}x))$ for $\sigma\in G_L, g\in \hat{T}_L(\hat{{\mathfrak{M}}}),
x\in {\widehat{\mathcal{R}}}_L\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$.
\[importantremark\] We should remark that notations $L_n,{\mathfrak{S}}_L, {\widehat{\mathcal{R}}}_L, {{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L},\dots $ above [*depend on the choices of a uniformizer $\pi_L$ of $L$ and a system $(\pi_{L,n})_{n\ge 0}$ of $p$-power roots of $\pi_L$*]{}. Conversely, if we fix the choice of $\pi_L$ and $(\pi_{L,n})_{n\ge 0}$, such notations are uniquely determined.
\[Thm:Liu\] (1) ([@Li3 Theorem 2.3.1 (1)]) Let $\hat{{\mathfrak{M}}}=({\mathfrak{M}},{\varphi},\hat{G}_L)$ be an object of ${}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$. Then the map $$\theta\colon T_{{\mathfrak{S}}_L}({\mathfrak{M}}) \to \hat{T}_L(\hat{{\mathfrak{M}}})$$ defined by $\theta(f)(a\otimes x):=a{\varphi}(f(x))$ for $a\in {\widehat{\mathcal{R}}}_L$ and $x\in {\mathfrak{M}}$, is an isomorphism of representations of $G_{L_\infty}$.
\(2) ([@Li3 Theorem 2.3.1(2)]) The contravariant functor $\hat{T}_L$ gives an anti-equivalence between the following categories:
- The category of $({\varphi},\hat{G}_L)$-modules of height $\le r$ over ${\mathfrak{S}}_L$.
- The category of $G_L$-stable ${\mathbb}{Z}_p$-lattices in semi-stable ${\mathbb}{Q}_p$-representations with Hodge-Tate weights in $[0,r]$.
\(3) ([@Li3 Theorem 4.2.2]) The contravariant functor $\hat{T}_L\colon {}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to
{\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_L)$ is fully faithful. Furthermore, its essential image is contained in the category of $G_L$-stable ${\mathbb}{Z}_p$-lattices in potentially semi-stable ${\mathbb}{Q}_p$-representations of $G_L$ which are semi-stable over $L_n$ for some $n\ge 0$ and have Hodge-Tate weights in $[0,r]$.
\[Rem:Liu\] Put $m={\mathrm}{max}\{i\ge 0 ; \zeta_{p^i}\in L^{{\mathrm}{ur}}\}$. We claim that any ${\mathbb}{Q}_p$-representation of $G_L$ which is semi-stable over $L_n$ for some $n\ge 0$ is always semi-stable over $L_m$.
In the former half part of the proof of [@Li3 Theorem 4.2.2], a proof of this claim with “$m={\mathrm}{max}\{i\ge 0 ; \zeta_{p^i}\in L\}$” is written. Unfortunately, there is a gap in the proof. In the proof, the assumption that the extension $L(\zeta_n,\pi_{L,n})/L$ is totally ramified is implicitly used (p. $133$, between $l.\ 14$ and $l.\ 21$ of [@Li3]). However, this condition is not satisfied in general. So we need a little modification. Put $m={\mathrm}{max}\{i\ge 0 ; \zeta_{p^i}\in L^{{\mathrm}{ur}}\}$ as the beginning. Denote by $\widehat{L^{{\mathrm}{ur}}}$ the completion of $L^{{\mathrm}{ur}}$. We remark that the completion of the maximal unramified extension of $L_n$ is just $\widehat{L^{{\mathrm}{ur}}}(\pi_{L,n})$. Let $V$ be a ${\mathbb}{Q}_p$-representation of $G_L$ which is semi-stable over $L_n$ for some $n\ge 0$. Then $V$ is semi-stable over $\widehat{L^{{\mathrm}{ur}}}(\pi_{L,n})$. We remark that the proof of [@Li3 Theorem 4.2.2] exactly holds at least under the assumption that the residue field of the base field is algebraically closed. (We need only the first paragraph of [*loc. cit.*]{} here.) Thus we know that $V$ is semi-stable over $\widehat{L^{{\mathrm}{ur}}}(\pi_{L,m})$ and thus we obtain the claim.
Now we restate Theorem \[Main1\] with the above setting of notation and give a further result. Fix the choice of a uniformizer $\pi_K$ of $K$ and a system $(\pi_{K,n})_{n\ge 0}$ of $p$-power roots of $\pi_K$, and define notations $K_n, {{\mathrm}{Mod}}^{r,\hat{G}_K}_{/{\mathfrak{S}}_K},\dots $ with respect to them. Recall that $m_0$ (resp. $m$) is the maximum integer such that $K$ (resp. $K^{{\mathrm}{ur}}$) contains $p^{m_0}$-th (resp. $p^m$-th) roots of unity. We note that the inequality $m_0\le m$ always holds. For any non-negative integer $n$, we denote by ${\mathcal}{C}^r_n$ the category of free ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which satisfy the following property; there exists a semi-stable ${\mathbb}{Q}_p$-representation $V$ of $G_K$ with Hodge-Tate weights in $[0,r]$ such that $T\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p$ is isomorphic to $V$ as representations of $G_{K_n}$.
Our goal in this paper is to show the following:
\[Main1’\] The essential image of the functor $\hat{T}_K\colon {}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_K}_{/{\mathfrak{S}}_K}\to {\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_K)$ is ${\mathcal}{C}^r_{m_0}$.
As an immediate consequence of the above theorem, we obtain
The functor $\hat{T}_K$ induces an anti-equivalence ${}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_K}_{/{\mathfrak{S}}_K}\overset{\sim}{\to} {\mathcal}{C}^r_{m_0}$.
For later use, we end this section by describing the following proposition.
\[totst\] Let $L$ be a finite totally ramified extension of $K$. Then the restriction functor from the category of semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ into the category of semi-stable ${\mathbb}{Q}_p$-representations of $G_L$ is fully faithful.
In view of the theory of Fontaine’s filtered $({\varphi}, N)$-modules, the result immediately follows from calculations of elementary linear algebras.
Proof of Main Theorem
=====================
Our main goal in this section is to give a proof of Theorem \[Main1’\]. In the first three subsections, we prove the following lemma, which plays an important role in our proof.
\[Lem:Main1’\] Denote by ${\mathcal}{C}^r$ the essential image of $\hat{T}_K\colon {}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_K}_{/{\mathfrak{S}}_K}\to {\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_K)$. Then we have ${\mathcal}{C}^r_{m_0}\subset {\mathcal}{C}^r \subset {\mathcal}{C}^r_m$.
Clearly, Theorem \[Main1’\] follows immediately from the lemma if $m_0=m$. However, the condition $m_0=m$ is not always satisfied. Before starting a main part of this section, we give some remarks about this condition.
\[m0=m\] (1) If $k$ is algebraically closed, then $m_0=m$.
\(2) If $K(\zeta_{p^{m_0+1}})/K$ is ramified, then $m_0=m$.
\(3) Suppose that $\zeta_p\in K$ (resp. $\zeta_4\in K$) if $p$ is odd (resp. $p=2$). Then $\hat{K}$ is totally ramified over $K$ if and only if $m_0=m$.
The assertion (1) and (2) is clear. We prove (3). If $m_0 < m$, then $K(\zeta_{p^m})$ is a non-trivial unramified extension of $K$ and thus the extension $\hat{K}/K$ is not totally ramified. Conversely, suppose that $\hat{K}/K$ is not totally ramified. Then there exists an integer $n\ge 0$ such that $K(\zeta_{p^n},\pi_n)/K$ is not totally ramified. This implies so is $K(\zeta_{p^n},\pi_n)/K(\pi_n)$. We may suppose $n\ge m$. Since ${\mathrm}{Gal}(K(\zeta_{p^n},\pi_n)/K(\pi_n))$ is isomorphic to ${\mathbb}{Z}/p^{n-m_0}{\mathbb}{Z}$ (here we need the assumption $\zeta_p\in K$ (resp. $\zeta_4\in K$) if $p$ is odd (resp. $p=2$)), any subfield of $K(\zeta_{p^n},\pi_n)/K(\pi_n)$ is of the form $K(\zeta_{p^l},\pi_n)$ for $m_0\le l\le n$. Thus there exists an integer $m_0\le l_0\le n$ such that $K^{{\mathrm}{ur}}(\pi_n)\cap K(\zeta_{p^n},\pi_n)=K(\zeta_{p^{l_0}},\pi_n)$. We have $\zeta_{p^{l_0}}\in K^{{\mathrm}{ur}}(\pi_n)\cap K^{{\mathrm}{ur}}(\zeta_{p^n})$. Since $\zeta_p\in K$ (resp. $\zeta_4\in K$) if $p$ is odd (resp. $p=2$), we have also $K^{{\mathrm}{ur}}(\pi_n)\cap K^{{\mathrm}{ur}}(\zeta_{p^n})=K^{{\mathrm}{ur}}$. This implies $l_0\le m$. Since the residue field extension corresponding to $K(\zeta_{p^n},\pi_n)/K(\pi_n)$ is non-trivial, the extension $K(\zeta_{p^{l_0}},\pi_n)/K(\pi_n)$ is non-trivial extension and thus so is $K(\zeta_{p^m},\pi_n)/K(\pi_n)$. This implies $1<[K(\zeta_{p^m},\pi_n):K(\pi_n)]=[K(\zeta_{p^m}):K]$ and hence $m_0<m$.
\[rem:m0<m\] The condition $m_0=m$ is not always satisfied. Here are some examples.
\(1) Suppose $p>2$. Set $\alpha:=(2+p)^{1/(p-1)}, \beta:=(-p)^{1/(p-1)}$ and $K:={\mathbb}{Q}_p(\alpha \beta)$. The field $K$ is totally ramified over ${\mathbb}{Q}_p$ since the minimal polynomial of $\alpha\beta$ over ${\mathbb}{Q}_p$ is an Eisenstein polynomial $X^{p-1}-(2+p)(-p)$. It is well-known that ${\mathbb}{Q}_p(\beta)={\mathbb}{Q}_p(\zeta_p)$. The extension $K(\zeta_p)/K$ is not totally ramified since so is ${\mathbb}{Q}_p(\alpha)/{\mathbb}{Q}_p$ and $p>2$ (note that the residue class of $\alpha$ is not contained in ${\mathbb}{F}_p$). Now we take any odd prime $p$ such that the extension ${\mathbb}{Q}(\alpha)/{\mathbb}{Q}$ is unramified (e.g., $p=3,5,7,\dots $). Then $K(\zeta_p)/K$ is an unramified extension. This implies that $m_0=0 < m$. (Moreover, we see that $m=1$.)
\(2) Suppose $p=2$ and set $K:={\mathbb}{Q}_2(\sqrt{-5})$. Then $K(\zeta_4)/K$ is unramified extension of degree $2$, and thus $m_0=1<m$. (Moreover, we see that $m=2$.)
\(3) Let $K'$ be a finite extension of ${\mathbb}{Q}_p$ such that it contains $p$-th roots of unity and $K'(\zeta_{p^{\infty}})/K'$ is a totally ramified extension. Let $K''$ be an unramified ${\mathbb}{Z}_p$-extension of $K'$. We denote by $K'_{(n)}$ and $K''_{(n)}$ the unique degree-$p^n$-subextensions of $K'(\zeta_{p^{\infty}})/K'$ and $K''/K'$, respectively. Explicitly, the field $K'_{(n)}$ coincides with $K'(\zeta_{p^{m'_0+n}})$ where $m'_0=\max \{i\ge 0 \mid \zeta_{p^i}\in K'\}$. If we denote by $M_{(n)}$ the composite field of $K'_{(n)}$ and $K''_{(n)}$, then we have isomorphisms $${\mathrm}{Gal}(M_{(n)}/K')\simeq
{\mathrm}{Gal}(K'_{(n)}/K')\times {\mathrm}{Gal}(K''_{(n)}/K')
\simeq {\mathbb}{Z}/p^n{\mathbb}{Z}\times {\mathbb}{Z}/p^n{\mathbb}{Z}$$ Let $K$ be the subfield of $M_{(n)}/K'$ which corresponds to the group of diagonal components of ${\mathrm}{Gal}(M_{(n)}/K')\simeq
{\mathbb}{Z}/p^n{\mathbb}{Z}\times {\mathbb}{Z}/p^n{\mathbb}{Z}$ via Galois theory. We consider $m_0$ and $m$ for this $K$. Since $K\cap K'_{(n)}=K'$, we know $m_0=m'_0$. On the other hand, since $M_{(n)}=KL_{(n)}=K(\zeta_{p^{m'_0+n}})$ and the extension $M_{(n)}/K$ is unramified, we have $m\ge m_0'+n=m_0+n$.
Lattices in potentially semi-stable representations {#3.1}
---------------------------------------------------
In this subsection we define a notion of $({\varphi},\hat{G}_L,K)$-modules which classifies lattices in potentially semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ which are semi-stable over $L$.
\[def\] A [*$({\varphi},\hat{G}_L,K)$-module of height $\le r$ over ${\mathfrak{S}}_L$*]{} is a pair $(\hat{{\mathfrak{M}}}, G_K)$ where
- $\hat{{\mathfrak{M}}}=({\mathfrak{M}},{\varphi},\hat{G}_L)$ is an object of ${{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$,
- $G_K$ is a $W(R)$-semilinear continuous $G_K$-action on $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$,
- the $G_K$-action commutes with ${\varphi}_{W(R)}\otimes {\varphi}_{{\mathfrak{M}}}$, and
- the $W(R)$-semilinear $G_L$-action on $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}(\simeq W(R)\otimes_{{\widehat{\mathcal{R}}}_L} ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}))$ induced from the $\hat{G}_L$-structure of $\hat{{\mathfrak{M}}}\in{{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$ coincides with the restriction of the $G_K$-action of (2) to $G_L$.
If $(\hat{{\mathfrak{M}}}, G_K)$ is a $({\varphi},\hat{G}_L,K)$-module of height $\le r$ over ${\mathfrak{S}}_L$, we often abuse notations by writing $\hat{{\mathfrak{M}}}$ for $(\hat{{\mathfrak{M}}}, G_K)$ for simplicity. We denote by ${{\mathrm}{Mod}}^{r,\hat{G}_L,K}_{/{\mathfrak{S}}_L}$ the category of $({\varphi},\hat{G}_L,K)$-modules of height $\le r$ over ${\mathfrak{S}}_L$.
We define a contravariant functor $\hat{T}_{L/K}\colon {{\mathrm}{Mod}}^{r,\hat{G}_L,K}_{/{\mathfrak{S}}_L}\to
{\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_K)$ by $$\hat{T}_{L/K}(\hat{{\mathfrak{M}}})=
{\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}, W(R))$$ for an object $\hat{{\mathfrak{M}}}$ of ${{\mathrm}{Mod}}^{r,\hat{G}_L,K}_{/{\mathfrak{S}}_L}$ with underlying Kisin module ${\mathfrak{M}}$. Here a $G_K$-action on $\hat{T}_{L/K}(\hat{{\mathfrak{M}}})$ is given by $(\sigma.g)(x)=\sigma(g(\sigma^{-1}x))$ for $\sigma\in G_K, g\in \hat{T}_{L/K}(\hat{{\mathfrak{M}}}),
x\in W(R)\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$. Note that we have natural isomorphisms $$\begin{aligned}
{\mathrm}{Hom}_{{\widehat{\mathcal{R}}}_L,{\varphi}}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}, W(R))
&\overset{\sim}{\rightarrow}
{\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi}, {\widehat{\mathcal{R}}}_L}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}), W(R))\\
&\overset{\sim}{\rightarrow}
{\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi}, {\mathfrak{S}}_L}{\mathfrak{M}}, W(R)).\end{aligned}$$ Thus we obtain $$\label{eta}
\eta\colon
\hat{T}_L(\hat{{\mathfrak{M}}})\overset{\sim}{\longrightarrow} \hat{T}_{L/K}(\hat{{\mathfrak{M}}}).$$ This is $G_L$-equivariant by the condition (4) of Definition \[def\]. In particular, $\hat{T}_{L/K}(\hat{{\mathfrak{M}}})\otimes_{{\mathbb}{Z}_p} {\mathbb}{Q}_p$ is semi-stable over $L$ by Theorem \[Thm:Liu\] (2).
The goal of the rest of this subsection is to prove the following theorem.
\[thm1\] The contravariant functor $\hat{T}_{L/K}$ induces an anti-equivalence between the following categories:
- The category of $({\varphi},\hat{G}_L,K)$-modules of height $\le r$ over ${\mathfrak{S}}_L$.
- The category of $G_K$-stable ${\mathbb}{Z}_p$-lattices in potentially semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ which are semi-stable over $L$ and have Hodge-Tate weights in $[0,r]$.
The above theorem follows by essentially the same arguments of Liu ([@Li3], [@Li4]), but we write a proof here for the sake of completeness. Before a proof, we recall Liu’s comparison morphisms between $({\varphi},\hat{G}_L)$-modules and representations associated with them. Furthermore, we define its variant for $({\varphi},\hat{G}_L,K)$-modules.
Let $\hat{{\mathfrak{M}}}=({\mathfrak{M}},{\varphi},\hat{G}_L)$ be a weak $({\varphi},\hat{G}_L)$-module of height $\le r$ over ${\mathfrak{S}}_L$. By identifying $\hat{T}_L(\hat{{\mathfrak{M}}})$ with ${\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi}, {\widehat{\mathcal{R}}}_L}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}), W(R))$, we define a $W(R)$-linear map $$\hat{\iota}_L\colon W(R)\otimes_{{\widehat{\mathcal{R}}}_L} ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}) \to
W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_L(\hat{{\mathfrak{M}}})$$ by the composite $W(R)\otimes_{{\widehat{\mathcal{R}}}_L} ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})
\to
{\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_L(\hat{{\mathfrak{M}}}),W(R))\overset{\sim}{\to}
W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_L(\hat{{\mathfrak{M}}})$. Here, the first arrow is defined by $x\mapsto (f\mapsto f(x), \forall{f}\in \hat{T}_L(\hat{{\mathfrak{M}}}))$ and the second is a natural isomorphism. Also, for a $({\varphi},\hat{G}_L,K)$-module $\hat{{\mathfrak{M}}}$ of height $\le r$ over ${\mathfrak{S}}_L$, we define a natural $W(R)$-linear map $$\hat{\iota}_{L/K}\colon W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}\hookrightarrow
W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_{L/K}(\hat{{\mathfrak{M}}})$$ by a similar way. Let ${\mathfrak}{t}$ be an element of $W(R)\smallsetminus pW(R)$ such that ${\varphi}({\mathfrak}{t})=pE_L(u_L)E_L(0)^{-1}{\mathfrak}{t}$. Such ${\mathfrak}{t}$ is unique up to units of ${\mathbb}{Z}_p$.
\[comparison\] (1) ([@Li3 Proposition 3.1.3]) The map $
\hat{\iota}_L
$ as above is injective, which preserves Frobenius and $G_L$-actions. Furthermore, we have ${\varphi}({\mathfrak}{t})^r(W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_L(\hat{{\mathfrak{M}}}))
\subset {\mathrm}{Im}\ \hat{\iota}_L$.
\(2) The map $
\hat{\iota}_{L/K}
$ as above is injective, which preserves Frobenius and $G_K$-actions. Furthermore, we have ${\varphi}({\mathfrak}{t})^r(W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_{L/K}(\hat{{\mathfrak{M}}}))
\subset {\mathrm}{Im}\ \hat{\iota}_{L/K}$.
\(3) Let $\hat{{\mathfrak{M}}}$ be a $({\varphi},\hat{G}_L,K)$-module of height $\le r$ over ${\mathfrak{S}}_L$ with underlying Kisin module ${\mathfrak{M}}$. Then the following diagram is commutative:
$
\displaystyle \xymatrix{
W(R)\otimes_{{\widehat{\mathcal{R}}}_L} ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})
\ar@{^{(}->}^{\hat{\iota}_L}[rr] \ar[d]_{\wr}
& &
W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_L(\hat{{\mathfrak{M}}})
\\
W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}\ar@{^{(}->}^{\hat{\iota}_{L/K}}[rr]
& &
W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_{L/K}(\hat{{\mathfrak{M}}})
\ar[u]_{W(R)\otimes \eta^{\vee}}^{\wr}
}$
Here, the left vertical arrow is a natural isomorphism and $\eta$ is defined in (\[eta\]).
The commutativity of (3) is clear by construction, and the rest assertions follow by essentially the same proof as [@Li3 Proposition 3.1.3].
In the rest of this subsection, we denote by ${\mathrm}{Rep}^{r,L\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$ the full subcategory of ${\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_K)$ appeared in Theorem \[thm1\]. The isomorphism $\eta$ shows below.
The functor $\hat{T}_{L/K}$ has values in ${\mathrm}{Rep}^{r,L\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$.
Next we show the fully faithfulness of the functor $\hat{T}_{L/K}$.
The functor $\hat{T}_{L/K}$ is fully faithful.
Let $\hat{{\mathfrak{M}}}$ and $\hat{{\mathfrak{M}}}'$ be $({\varphi},\hat{G}_L,K)$-modules of height $\le r$ over ${\mathfrak{S}}_L$ with underlying Kisin modules ${\mathfrak{M}}$ and ${\mathfrak{M}}'$, respectively. Take any $G_K$-equivariant morphism $f\colon \hat{T}_{L/K}(\hat{{\mathfrak{M}}})\to \hat{T}_{L/K}(\hat{{\mathfrak{M}}}')$. By the map $\eta$, we identify $\hat{T}_{L/K}(\hat{{\mathfrak{M}}})$ and $\hat{T}_{L/K}(\hat{{\mathfrak{M}}}')$ with $\hat{T}_L(\hat{{\mathfrak{M}}})$ and $\hat{T}_L(\hat{{\mathfrak{M}}}')$, respectively. Since $\hat{T}_L$ is fully faithful, there exists a unique morphism ${\mathfrak}{f}\colon \hat{{\mathfrak{M}}}'\to \hat{{\mathfrak{M}}}$ of $({\varphi},\hat{G}_L)$-modules of height $\le r$ over ${\mathfrak{S}}_L$ such that $\hat{T}_L({\mathfrak}{f})=f$. It is enough to show that ${\mathfrak}{f}$ is in fact a morphism of $({\varphi},\hat{G}_L,K)$-modules, that is, $W(R)\otimes {\mathfrak}{f}\colon W(R)\otimes_{{\varphi}, {\mathfrak{S}}_L}{\mathfrak{M}}'\to W(R)\otimes_{{\varphi}, {\mathfrak{S}}_L}{\mathfrak{M}}$ is $G_K$-equivariant. Consider the following diagram:
$
\displaystyle \xymatrix{
W(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}\ar@{^{(}->}^{\hat{\iota}_{L/K}}[rr]
& &
W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_{L/K}(\hat{{\mathfrak{M}}})
\\
W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}'
\ar@{^{(}->}^{\hat{\iota}_{L/K}}[rr]
\ar[u]^{W(R)\otimes {\mathfrak}{f}}
& &
W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}_{L/K}(\hat{{\mathfrak{M}}}')
\ar[u]^{W(R)\otimes f^{\vee}}
}$
We see that the above diagram is commutative. Since $W(R)\otimes f^{\vee}$ and two horizontal arrows above are $G_K$-equivariant, so is $W(R)\otimes {\mathfrak}{f}$.
\[ess1\] The functor $\hat{T}_{L/K}\colon {{\mathrm}{Mod}}^{r,\hat{G}_L,K}_{/{\mathfrak{S}}_L}\to
{\mathrm}{Rep}^{r,L\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$ is essentially surjective if $L$ is a Galois extension of $K$.
To show this lemma, we recall arguments of [@Li4 §2]. Suppose $L$ is a (not necessary totally ramified) Galois extension of $K$. Let $T$ be an object of ${\mathrm}{Rep}^{r,L\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$. Put $d={\mathrm}{rank}_{{\mathbb}{Z}_p}T$. Take a $({\varphi},\hat{G}_L)$-module $\hat{{\mathfrak{M}}}=({\mathfrak{M}},{\varphi},\hat{G}_L)$ over ${\mathfrak{S}}_L$ such that $\hat{T}_L(\hat{{\mathfrak{M}}})= T|_{G_L}$. We consider the map $\hat{\iota}_L\colon W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}\hookrightarrow W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_L(\hat{{\mathfrak{M}}})
=W(R)\otimes_{{\mathbb}{Z}_p} T^{\vee}$. By the same argument as the proof of [@Li4 Lemma 2.3.1], we can check the following
\[claim\] $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$ is stable under the $G_K$-action via $\hat{\iota}_L$.
We include (a main part of) the proof in [*loc. cit.*]{} of this lemma here since we will use this argument again in the next subsection (cf. the proof of Theorem \[Main2\]). By [@Br], we know that ${\mathcal}{D}:=S_L[1/p]\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$ has a structure of a Breuil module[^2] which corresponds to $V|_{G_L}$, where $V:=T\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p$. In particular, we have a monodromy operator $N_{{\mathcal}{D}}$ on ${\mathcal}{D}$. Set $D:={\mathcal}{D}/I_+S_L[1/p]{\mathcal}{D}$. There exists a unique ${\varphi}$-compatible $W(k_L)$-linear section $s\colon D\hookrightarrow {\mathcal}{D}$. Breuil showed in [*loc. cit.*]{} that $N_{{\mathcal}{D}}$ preserves $s(D)$ and thus we can define $\tilde{N}:=N_{{\mathcal}{D}}|_{s(D)}\colon s(D)\to s(D)$. Then the $G_L$-action on $B^+_{{\mathrm}{st}}\otimes_{S_L[1/p]}s(D)(=B^+_{{\mathrm}{st}}\otimes_{{\widehat{\mathcal{R}}}_L} ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}))$ induced from the $\hat{G}_L$-structure on $\hat{{\mathfrak{M}}}$ is given by $$g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))
\otimes \tilde{N}^i(x)$$ for any $g\in G_L, a\in B^+_{{\mathrm}{st}}$ and $x\in s(D)$. Here, $\underline{{\varepsilon}}(g):=g(\underline{\pi_L})/\underline{\pi_L}\in R^{\times}$. Set $$\bar{D}:=\left\{\sum^{\infty}_{i=0} \gamma_i({\mathfrak}{u})\otimes \tilde{N}^i(x)
\mid x\in s(D) \right\}
\subset B^+_{{\mathrm}{st}}\otimes_{W(k_L)[1/p]}s(D)$$ where ${\mathfrak}{u}:={\mathrm}{log}([\underline{\pi_L}])\in B^+_{{\mathrm}{st}}$. This is a ${\varphi}$-stable $W(k_L)[1/p]$-vector space of dimension $d$. Setting the monodromy $N_{B^+_{{\mathrm}{st}}}$ on $B^+_{{\mathrm}{st}}$ by $N({\mathfrak}{u})=1$, we equip $B^+_{{\mathrm}{st}}\otimes_{W(k_L)[1/p]}s(D)$ (resp. $B^+_{{\mathrm}{st}}\otimes_{{\mathbb}{Q}_p}V^{\vee}$) with a monodromy operator $N$ by $N:=N_{B^+_{{\mathrm}{st}}}\otimes 1_{s(D)}$ (resp. $N:=N_{B^+_{{\mathrm}{st}}}\otimes 1_{V^{\vee}}$). Then it is easy to see that $\bar{D}$ is stable under $N$. On the other hand, we have a natural $G_K$-equivariant injection $\iota\colon B^+_{{\mathrm}{st}}\otimes_{W(k_L)[1/p]} D_{{\mathrm}{st}}(V)
\hookrightarrow B^+_{{\mathrm}{st}}\otimes_{{\mathbb}{Q}_p} V^{\vee}
$ where $D_{{\mathrm}{st}}(V):=(B^+_{{\mathrm}{st}}\otimes_{{\mathbb}{Q}_p} V^{\vee})^{G_L}$ is a filtered $({\varphi},N)$-module over $L$. (Here we remark that $D_{{\mathrm}{st}}(V)$ is equipped with a natural $G_K$-action since $L/K$ is Galois.) Since $G_L$ acts on $\bar{D}$ trivially (cf. §7.2 of [@Li1]), the image of $\bar{D}$ under the injection $
B^+_{{\mathrm}{st}}\otimes_{W(k_L)[1/p]}s(D)=
B^+_{{\mathrm}{st}}\otimes_{{\widehat{\mathcal{R}}}_L} ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})
\overset{\hat{\iota}_{L,B}}{\hookrightarrow}
B^+_{{\mathrm}{st}}\otimes_{{\mathbb}{Q}_p}V^{\vee}
$ is equal to $\iota(D_{{\mathrm}{st}}(V))$. Here, $\hat{\iota}_{L,B}:=B^+_{{\mathrm}{st}}\otimes \hat{\iota}_L$, which is compatible with Frobenius and monodromy operators. Hence we have an isomorphism $\hat{i}\colon D_{{\mathrm}{st}}(V)\overset{\sim}{\longrightarrow} \bar{D}$ which makes the following diagram commutative:
$
\displaystyle \xymatrix{
D_{{\mathrm}{st}}(V)
\ar_{\wr}^{\hat{i}}[d]
& \subset &
B^+_{{\mathrm}{st}}\otimes_{W(k_L)[1/p]} D_{{\mathrm}{st}}(V)
\ar@{^{(}->}^{\qquad \quad \iota}[rr]
& &
B^+_{{\mathrm}{st}}\otimes_{{\mathbb}{Q}_p} V^{\vee}
\ar@{=}[d]
\\
\bar{D}
& \subset &
B^+_{{\mathrm}{st}}\otimes_{W(k_L)[1/p]}s(D)
\ar@{^{(}->}^{\qquad \hat{\iota}_{L,B}}[rr]
& &
B^+_{{\mathrm}{st}}\otimes_{{\mathbb}{Q}_p}V^{\vee}
}$
Note that $\hat{i}$ is compatible with Frobenius and monodromy operators. We identify $D_{{\mathrm}{st}}(V)$ with $\bar{D}$ by $\hat{i}$.
Let $e_1,\dots, e_d$ be a $W(k_L)[1/p]$-basis of $D$, and define a matrix $\bar{N}\in M_d(W(k_L)[1/p])$ by $\tilde{N}(s(e_1),\dots ,s(e_d))=(s(e_1),\dots ,s(e_d))\bar{N}$. Put $\bar{e}_j=\sum^{\infty}_{i=0} \gamma_i({\mathfrak}{u})\otimes \tilde{N}^i(s(e_j))$ for any $j$. Then $\bar{e}_1,\dots ,\bar{e}_d$ is a basis of $D_{{\mathrm}{st}}(V)=\bar{D}$. An easy calculation shows that the monodromy $N$ on $D_{{\mathrm}{st}}(V)=\bar{D}$ is represented by $\bar{N}$ for this basis, that is, $N(\bar{e}_1,\dots ,\bar{e}_d)=(\bar{e}_1,\dots ,\bar{e}_d)\bar{N}$. We define a matrix $A_g\in GL_d(W(k_L)[1/p])$ by $g(\bar{e}_1,\dots ,\bar{e}_d)
=(\bar{e}_1,\dots ,\bar{e}_d)A_g$ for any $g\in G_K$. Since the $G_K$-action on $D_{{\mathrm}{st}}(V)=\bar{D}$ is compatible with $N$, we have the relation $A_gg(\bar{N})=\bar{N}A_g$. Consequently, we have $$\label{action}
g(s(e_1),\dots ,s(e_d))=(s(e_1),\dots ,s(e_d)){\mathrm}{exp}(-\lambda_g\bar{N})A_g$$ in $B^+_{{\mathrm}{st}}\otimes_{{\mathbb}{Q}_p}V^{\vee}$, where $\lambda_g:={\mathrm}{log}([g(\underline{\pi_L})/\underline{\pi_L}])\in B^+_{{\mathrm}{cris}}$. This implies that $B^+_{{\mathrm}{cris}}\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}=B^+_{{\mathrm}{cris}}\otimes_{W(k_L)[1/p]} s(D)$ is stable under the $G_K$-action via $\hat{\iota}_{L,B}$. Now Lemma \[claim\] follows by an easy combination of Proposition \[comparison\] (1) and [@Li3 Lemma 3.2.2] (cf. the first paragraph of the proof of [@Li4 Lemma 2.3.1]).
We continue to use the same notation as above. By Lemma \[claim\], we know that $\hat{{\mathfrak{M}}}$ has a structure of an object of ${{\mathrm}{Mod}}^{r,\hat{G}_L,K}_{/{\mathfrak{S}}_L}$ with the property that the map $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}\overset{\hat{\iota}_L}{\hookrightarrow}
W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_L(\hat{{\mathfrak{M}}})
=W(R)\otimes_{{\mathbb}{Z}_p} T^{\vee}$ is $G_K$-equivariant. Let $\eta\colon \hat{T}_L(\hat{{\mathfrak{M}}})
\overset{\sim}{\longrightarrow} \hat{T}_{L/K}(\hat{{\mathfrak{M}}})$ be the isomorphism defined in (\[eta\]). By Proposition \[comparison\] (3), we know that $W(R)\otimes \eta^{\vee}$ induces an isomorphism $\hat{\iota}_L(W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}})\overset{\sim}{\longrightarrow}
\hat{\iota}_{L/K}(W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}})$, which is $G_K$-equivariant. Since ${\varphi}({\mathfrak{t}})^r(W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_L(\hat{{\mathfrak{M}}}))$ (resp. ${\varphi}({\mathfrak{t}})^r(W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_{L/K}(\hat{{\mathfrak{M}}}))$) is contained in $(\hat{\iota}_L(W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}))$ (resp. $(\hat{\iota}_{L/K}(W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}))$), we known that the map ${\varphi}({\mathfrak{t}})^r(W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_L(\hat{{\mathfrak{M}}}))
\overset{\sim}{\longrightarrow}
{\varphi}({\mathfrak{t}})^r(W(R)\otimes_{{\mathbb}{Z}_p} \hat{T}^{\vee}_{L/K}(\hat{{\mathfrak{M}}}))$ induced from $W(R)\otimes \eta^{\vee}$ is $G_K$-equivariant. Thus so is $\eta\colon T=\hat{T}_L(\hat{{\mathfrak{M}}})\overset{\sim}{\longrightarrow}
\hat{T}_{L/K}(\hat{{\mathfrak{M}}})$.
Let $\hat{e}_1,\dots ,\hat{e}_d$ be a ${\mathfrak{S}}_K$-basis of ${\varphi}^{\ast}{\mathfrak{M}}$, which is also an $S_K[1/p]$-basis of ${\mathcal}{D}$. Denote by $e_i$ the image of $\hat{e}_i$ under the projection ${\mathcal}{D}\twoheadrightarrow D$. Then $e_1,\dots ,e_d$ is a $W(k)[1/p]$-basis of $D$. For these basis, we see that the matrix $A_g\in GL_d(W(k_L)[1/p])$ as above is in fact contained in $GL_d(W(k_L))$ by Lemma \[ess1\]. (However, we never use this fact in the present paper.)
The functor $\hat{T}_{L/K}\colon {{\mathrm}{Mod}}^{r,\hat{G}_L,K}_{/{\mathfrak{S}}_L}\to
{\mathrm}{Rep}^{r,L\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$ is essentially surjective for any finite extension $L$ of $K$.
Let $T$ be an object of ${\mathrm}{Rep}^{L\mathchar`-{\mathrm}{st},r}_{{\mathbb}{Z}_p}(G_K)$. Let $L'$ be the Galois closure of $L$ over $K$ (and fix the choice of a uniformizer of $L'$ and a system of $p$-power roots of it; see Remark \[importantremark\]). Since we have already shown Theorem \[thm1\] for $\hat{T}_{L'/K}$, we know that there exists a $({\varphi},\hat{G}_{L'},K)$-module $\hat{{\mathfrak{M}}}'$ over ${\mathfrak{S}}_{L'}$ such that $\hat{T}_{L'/K}(\hat{{\mathfrak{M}}}')\simeq T$ as representations of $G_K$. On the other hand, we have a unique $({\varphi},\hat{G}_L)$-module $\hat{{\mathfrak{M}}}$ such that $T\simeq \hat{T}_L(\hat{{\mathfrak{M}}})$ as representations of $G_L$ since $T$ is semi-stable over $L$. We denote by ${\mathfrak{M}}'$ and ${\mathfrak{M}}$ underlying Kisin modules of $\hat{{\mathfrak{M}}}'$ and $\hat{{\mathfrak{M}}}$, respectively. By [@Li5 Theorem 3.2.1] and Proposition \[comparison\] (3), the image of $W(R)\otimes_{{\varphi},{\mathfrak{S}}_{L'}} {\mathfrak{M}}'$ under $\hat{\iota}_{L'/K}$ is equal to that of $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$ under $\hat{\iota}_L$. Hence we have a ${\varphi}$-equivariant isomorphism $W(R)\otimes_{{\varphi},{\mathfrak{S}}_{L'}} {\mathfrak{M}}' \simeq W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$. We define a $G_K$-action on $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$ by this isomorphism. Then $\hat{{\mathfrak{M}}}$ has a structure of $({\varphi},\hat{G}_L,K)$-module over ${\mathfrak{S}}_L$ so that $\hat{\iota}_L\colon W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}\hookrightarrow
W(R)\otimes_{{\mathbb}{Z}_p} T^{\vee}$ is $G_K$-equivariant. Since $\hat{T}_{L/K}(\hat{{\mathfrak{M}}})
={\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}},W(R))
\simeq {\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}_{L'}}{\mathfrak{M}}',W(R))
=\hat{T}_{L'/K}(\hat{{\mathfrak{M}}}')=T$ as representations of $G_K$, we have done.
${\mathcal}{C}^r_{m_0}\subset {\mathcal}{C}^r$ {#3.2}
----------------------------------------------
We prove the relation ${\mathcal}{C}^r_{m_0}\subset {\mathcal}{C}^r$ in the assertion of Lemma \[Lem:Main1’\]. At first, fix the choices of a uniformizer $\pi_K$ of $K$ and a system $(\pi_{K,n})_{n\ge 0}$ of $p$-power roots of $\pi_K$, and define notations $K_n, {\mathfrak{S}}_K, {{\mathrm}{Mod}}^{r,\hat{G}_K}_{/{\mathfrak{S}}_K},\dots $ with respect to them (see also Remark \[importantremark\]). We also consider notations ${\mathfrak{S}}_{K_n}, S_{K_n},\dots $ with respect to the uniformizer $\pi_{K_n}:=\pi_{K,n}$ of $K_n$ and the system $(\pi_{K,n+m})_{m\ge 0}$ of $p$-power roots of $\pi_{K_n}$. Note that we have ${\mathfrak{S}}_K\subset {\mathfrak{S}}_{K_n}$, $S_K\subset S_{K_n}$ and $E_{K_n}(u_{K_n})=E_K(u_K)$ with the relation $u^{p^n}_{K_n}=u_K$.
To show the relation ${\mathcal}{C}^r_{m_0}\subset {\mathcal}{C}^r$, it follows from Lemma 2.1.15 of [@Ki] that it suffices to show the following.
\[Main2\] Let $T$ be a free ${\mathbb}{Z}_p$-representation of $G_K$ which is semi-stable over $K_n$ for some $n\le m_0$ and $T|_{G_{K_{\infty}}}\simeq T_{{\mathfrak{S}}_K}({\mathfrak{M}})$ for some ${\mathfrak{M}}\in {\mathrm}{Mod}^r_{/{\mathfrak{S}}_K}$. Then there exists a (unique) weak $({\varphi},\hat{G}_K)$-module $\hat{{\mathfrak{M}}}$ of height $\le r$ over ${\mathfrak{S}}_K$ such that $\hat{T}_K(\hat{{\mathfrak{M}}})\simeq T$.
Let $T, n$ and ${\mathfrak{M}}$ be as in the statement. Note that $K_n$ is a now Galois extension of $K$ for such $n$, and note also that ${\mathfrak{M}}_n:={\mathfrak{S}}_{K_n}\otimes_{{\mathfrak{S}}_K}{\mathfrak{M}}$ is a Kisin module of height $\le r$ over ${\mathfrak{S}}_{K_n}$. By Theorem \[thm1\], there exists a $({\varphi},\hat{G}_{K_n},K)$-module $\hat{{\mathfrak{N}}}$ over ${\mathfrak{S}}_{K_n}$ such that $T\simeq \hat{T}_{K_n/K}(\hat{{\mathfrak{N}}})$. Denote by ${\mathfrak{N}}$ the underlying Kisin module of $\hat{{\mathfrak{N}}}$. Since $T_{{\mathfrak{S}}_{K_n}}({\mathfrak{M}}_n)$ is isomorphic to $T_{{\mathfrak{S}}_{K_n}}({\mathfrak{N}})$, we may identify ${\mathfrak{N}}$ with ${\mathfrak{M}}_n$. Thus ${\mathfrak{M}}_n$ is equipped with a structure of a $({\varphi},\hat{G}_{K_n},K)$-module $\hat{{\mathfrak{M}}}_n$ over ${\mathfrak{S}}_{K_n}$ such that $T\simeq \hat{T}_{K_n/K}(\hat{{\mathfrak{M}}}_n)$. Putting ${\varphi}^{\ast}{\mathfrak{M}}={\mathfrak{S}}_K\otimes_{{\varphi},{\mathfrak{S}}_K}{\mathfrak{M}}$, we know that $G_K({\varphi}^{\ast}{\mathfrak{M}})$ is contained in $W(R)\otimes_{{\varphi},{\mathfrak{S}}_{K_n}}{\mathfrak{M}}_n=W(R)\otimes_{{\varphi},{\mathfrak{S}}_K}{\mathfrak{M}}$. We claim that $G_K({\varphi}^{\ast}{\mathfrak{M}})$ is contained in ${\mathcal}{R}_{K}\otimes_{{\varphi},{\mathfrak{S}}_K}{\mathfrak{M}}$. Admitting this claim, we see that ${\mathfrak{M}}$ has a structure of a weak $({\varphi},\hat{G}_K)$-module of height $\le r$ over ${\mathfrak{S}}_K$ which corresponds to $T$, and hence we finish a proof.
Put ${\mathcal}{D}_n=S_{K_n}[1/p]\otimes_{{\varphi},{\mathfrak{S}}_{K_n}}{\mathfrak{M}}_n$ and ${\mathcal}{D}=S_K[1/p]\otimes_{{\varphi},{\mathfrak{S}}_K}{\mathfrak{M}}$. Let $\hat{e}_1,\dots ,\hat{e}_d$ be a ${\mathfrak{S}}_K$-basis of ${\varphi}^{\ast}{\mathfrak{M}}$, which is an $S_{K_n}[1/p]$-basis of ${\mathcal}{D}_n$ and an $S_K[1/p]$-basis of ${\mathcal}{D}$. Denote by $e_i$ the image of $\hat{e}_i$ under the projection ${\mathcal}{D}\twoheadrightarrow {\mathcal}{D}/I_+S_K[1/p]=:D$. Then $e_1,\dots ,e_d$ is a $W(k)[1/p]$-basis of $D$. By [@Br Proposition 6.2.1.1], we have a unique ${\varphi}$-compatible section $s\colon D\hookrightarrow {\mathcal}{D}$ of the projection ${\mathcal}{D}\twoheadrightarrow D$. Since ${\mathcal}{D}=S_K[1/p]\otimes_{W(k)[1/p]} s(D)$, there exists a matrix $X\in GL_d(S_K[1/p])$ such that $(\hat{e}_1,\dots ,\hat{e}_d)=(s(e_1),\dots ,s(e_d))X$. Now we extend the $G_K$-action on $W(R)\otimes_{{\varphi},{\mathfrak{S}}_{K_n}} {\mathfrak{M}}_n$ to $B^+_{{\mathrm}{cris}}\otimes_{W(k)[1/p]} s(D)
=B^+_{{\mathrm}{cris}}\otimes_{W(R)} (W(R)\otimes_{{\varphi},{\mathfrak{S}}_{K_n}} {\mathfrak{M}}_n)$ by a natural way. Take any $g\in G_K$ and put $\lambda_g={\mathrm}{log}([g(\underline{\pi_{K_n}})/\underline{\pi_{K_n}}])$. We see that $\lambda_g$ is contained in ${\mathcal}{R}_{K}$. Recall that $K_n$ is now a totally ramified Galois extension over $K$. By (\[action\]), we have $g(s(e_1),\dots ,s(e_d))=(s(e_1),\dots ,s(e_d)){\mathrm}{exp}(-\lambda_g\bar{N})A_g$ for some nilpotent matrix $\bar{N}\in M_d(W(k)[1/p])$ and some $A_g\in GL_d(W(k)[1/p])$. Therefore, we obtain $g(\hat{e}_1,\dots ,\hat{e}_d)
=(\hat{e}_1,\dots ,\hat{e}_d)X^{-1}{\mathrm}{exp}(-\lambda_g\bar{N})A_gg(X)$. Since the matrix $X^{-1}{\mathrm}{exp}(-\lambda_g\bar{N})A_gg(X)$ has coefficients in ${\mathcal}{R}_{K}$, we have done.
We remark that, for any semi-stable ${\mathbb}{Q}_p$-representation $V$ of $G_{K_n}$ with Hodge-Tate weights in $[0,r]$, there exists a Kisin module ${\mathfrak{M}}_n\in {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{K_n}}$ such that $V|_{G_{K_{\infty}}}$ is isomorphic to $T_{{\mathfrak{S}}_{K_n}}({\mathfrak{M}}_n)\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p$ (cf. [@Ki Lemma 2.1.15]). The above proposition studies the case where ${\mathfrak{M}}_n$ descends to a Kisin module over ${\mathfrak{S}}_K$, but this condition is not always satisfied. An example for this is given in the proof of Proposition \[prop:rem\].
${\mathcal}{C}^r \subset {\mathcal}{C}^r_m$ {#3.3}
-------------------------------------------
Next we prove the relation ${\mathcal}{C}^r \subset {\mathcal}{C}^r_m$ in the assertion of Lemma \[Lem:Main1’\]. The key for our proof is the following proposition.
\[Thm2\] The restriction functor ${\mathrm}{Rep}_{{\mathbb}{Q}_p}(G_K)\to {\mathrm}{Rep}_{{\mathbb}{Q}_p}(G_{K_n})$ induces an equivalence between the following categories:
- The category of semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ with Hodge-Tate weights in $[0,r]$.
- The category of semi-stable ${\mathbb}{Q}_p$-representations $V$ of $G_{K_n}$ with the property that $V|_{G_{K_{\infty}}}$ is isomorphic to $T_{{\mathfrak{S}}_K}({\mathfrak{M}})\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p$ for some ${\mathfrak{M}}\in {\mathrm}{Mod}^r_{/{\mathfrak{S}}_K}$.
The result below immediately follows from the above proposition.
\[Main3\] Let $T$ be a free ${\mathbb}{Z}_p$-representation of $G_K$ which is semi-stable over $K_n$ for some $n\ge 0$. Then the following conditions are equivalent:
- $T|_{G_{K_{\infty}}}$ is isomorphic to $T_{{\mathfrak{S}}_K}({\mathfrak{M}})$ for some ${\mathfrak{M}}\in {\mathrm}{Mod}^r_{/{\mathfrak{S}}_K}$.
- There exists a semi-stable ${\mathbb}{Q}_p$-representation $V$ of $G_K$ with Hodge-Tate weights in $[0,r]$ such that $T\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p$ is isomorphic to $V$ as representations of $G_{K_{n'}}$ for some $n'\ge 0$.
\[Rem:func\] In the statement of Corollary \[Main3\], we can always choose $n'$ to be $n$. In addition, for a given $T$, $V$ is uniquely determined up to isomorphism. Furthermore, the association $T\mapsto V$ is functorial. These follow from Proposition \[totst\].
Combining this corollary with Theorem \[Thm:Liu\] (3) and Remark \[Rem:Liu\], we obtain the desired relation ${\mathcal}{C}^r \subset {\mathcal}{C}^r_m$. Therefore, it suffices to show Proposition \[Thm2\]. We begin with the following two lemmas.
\[exercise1\] For any $i\ge 0$, we have a canonical decomposition $${\mathrm}{Fil}^iS_{K_n}=\bigoplus^{p^n-1}_{j=0}u^j_{K_n}{\mathrm}{Fil}^iS_K.$$
Exercise.
\[exercise2\] Let ${\mathfrak{M}}$ be a Kisin module of height $\le r$ over ${\mathfrak{S}}_K$.
\(1) ${\mathfrak{M}}_n:={\mathfrak{S}}_{K_n}\otimes_{{\mathfrak{S}}_K}{\mathfrak{M}}$ is a Kisin module of height $\le r$ over ${\mathfrak{S}}_{K_n}$ (with Frobenius ${\varphi}_{{\mathfrak{M}}_n}:={\varphi}_{{\mathfrak{S}}_{K_n}}\otimes {\varphi}_{{\mathfrak{M}}}$).
\(2) Let ${\mathcal}{M}:=S_K\otimes_{{\varphi},{\mathfrak{S}}_K}{\mathfrak{M}}$ and ${\mathcal}{M}_n:=S_{K_n}\otimes_{{\varphi},{\mathfrak{S}}_K}{\mathfrak{M}}=S_{K_n}\otimes_{{\varphi},{\mathfrak{S}}_{K_n}}{\mathfrak{M}}_n$. Define ${\mathrm}{Fil}^i{\mathcal}{M}
:=\{x\in {\mathcal}{M} \mid (1\otimes {\varphi}_{{\mathfrak{M}}})(x)\in {\mathrm}{Fil}^iS_K\otimes_{{\mathfrak{S}}_K} {\mathfrak{M}}\}$ and ${\mathrm}{Fil}^i{\mathcal}{M}_n
:=\{x\in {\mathcal}{M}_n \mid (1\otimes {\varphi}_{{\mathfrak{M}}})(x)\in {\mathrm}{Fil}^iS_{K_n}\otimes_{{\mathfrak{S}}_{K}} {\mathfrak{M}}\}
=\{x\in {\mathcal}{M}_n \mid (1\otimes {\varphi}_{{\mathfrak{M}}_n})(x)\in {\mathrm}{Fil}^iS_{K_n}\otimes_{{\mathfrak{S}}_{K_n}} {\mathfrak{M}}_n \}$. Then the natural isomorphism $S_{K_n}\otimes_{S_K}{\mathcal}{M}\overset{\sim}{\rightarrow} {\mathcal}{M}_n$ induces an isomorphism $S_{K_n}\otimes_{S_K}{\mathrm}{Fil}^i{\mathcal}{M}\overset{\sim}{\rightarrow} {\mathrm}{Fil}^i{\mathcal}{M}_n$.
The assertion (1) follows immediately by the relation $E_K(u_K)=E_{K_n}(u_{K_n})$. In the rest of this proof we identify $S_{K_n}\otimes_{S_K}{\mathcal}{M}$ with ${\mathcal}{M}_n$ by a natural way. We show that $S_{K_n}\otimes_{S_K}{\mathrm}{Fil}^i{\mathcal}{M}={\mathrm}{Fil}^i{\mathcal}{M}_n$. The inclusion $S_{K_n}\otimes_{S_K}{\mathrm}{Fil}^i{\mathcal}{M}\subset {\mathrm}{Fil}^i{\mathcal}{M}_n$ follows from an easy calculation. We have to prove the opposite inclusion. Let $e_1,\dots ,e_d$ be an ${\mathfrak{S}}_K$-basis of ${\mathfrak{M}}$ and define a matrix $A\in M_d({\mathfrak{S}}_K)$ by ${\varphi}_{{\mathfrak{M}}}(e_1,\dots ,e_d)=(e_1,\dots ,e_d)A$. We put $e^{\ast}_i=1\otimes e_i\in {\varphi}^{\ast}{\mathfrak{M}}$ for any $i$. Then $e^{\ast}_1,\dots ,e^{\ast}_d$ is an $S_{K_n}$-basis of ${\mathcal}{M}_n$. Take $x=\sum^d_{k=1}a_ke^{\ast}_k\in {\mathrm}{Fil}^i{\mathcal}{M}_n$ with $a_k\in S_{K_n}$. Since $(1\otimes {\varphi}_{{\mathfrak{M}}})(x)$ is contained in ${\mathrm}{Fil}^iS_{K_n}\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$, we see that the matrix $$X:=A\begin{pmatrix}
a_1\\\rotatebox{90}{\dots}\\ a_d
\end{pmatrix}$$ has coefficients in ${\mathrm}{Fil}^iS_{K_n}$. By Lemma \[exercise1\], each $a_k$ can be decomposed as $\sum^{p^n-1}_{j=0}u^j_{K_n}a^{(j)}_k$ for some $a^{(j)}_k\in S_K$. Writing $A=(a_{lk})_{l,k}$ and $X={}^{{\mathrm}{t}}(x_1,\dots ,x_d)$, we have $$x_l=\sum^d_{k=1}a_{lk}a_k=\sum^{p^n-1}_{j=0}u^j_{K_n}\sum^d_{k=1} a_{lk}a_k^{(j)}.$$ By Lemma \[exercise1\] again, we obtain that $\sum^d_{k=1} a_{lk}a_k^{(j)}\in {\mathrm}{Fil}^iS_K$. If we put $x_{(j)}=\sum^d_{k=1}a_k^{(j)}e^{\ast}_k\in {\mathcal}{M}$, we have $$(1\otimes {\varphi}_{{\mathfrak{M}}})(x_{(j)})=\sum^d_{l=1}(\sum^d_{k=1} a_{lk}a_k^{(j)})e_l,$$ which is contained in ${\mathrm}{Fil}^iS_K\otimes_{{\mathfrak{S}}_K}{\mathfrak{M}}$. Therefore, each $x_{(j)}$ is contained in ${\mathrm}{Fil}^i{\mathcal}{M}$. Since $x=\sum^{p^n-1}_{j=0}u^j_{K_n}x_{(j)}$, we obtain the fact that $x$ is contained in $S_{K_n}\otimes_{S_K}{\mathrm}{Fil}^i{\mathcal}{M}$.
We proceed a proof of Proposition \[Thm2\]. For simplicity, we denote by ${\mathbf}{R}_1$ (resp. ${\mathbf}{R}_2$) the former (resp. latter) category appeared in the statement of Proposition \[Thm2\]. It is well-known (cf. [@Ki Lemma 2.1.15]) that the essential image of ${\mathbf}{R}_1$ under the restriction functor ${\mathrm}{Rep}_{{\mathbb}{Q}_p}(G_K)\to {\mathrm}{Rep}_{{\mathbb}{Q}_p}(G_{K_n})$ is contained in ${\mathbf}{R}_2$. Furthermore, the restriction functor ${\mathbf}{R}_1\to {\mathbf}{R}_2$ is fully faithful since $K_n$ is totally ramified over $K$. Thus it suffices to show the essential surjectivity of the restriction functor ${\mathbf}{R}_1\to {\mathbf}{R}_2$. Let $V$ be a semi-stable ${\mathbb}{Q}_p$-representations $V$ of $G_{K_n}$ with the property that $V|_{G_{K_{\infty}}}$ is isomorphic to $T_{{\mathfrak{S}}_K}({\mathfrak{M}})\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p$ for some ${\mathfrak{M}}\in {\mathrm}{Mod}^r_{/{\mathfrak{S}}_K}$. Set $T:=T_{{\mathfrak{S}}_K}({\mathfrak{M}})$ and take any $G_{K_n}$-stable ${\mathbb}{Z}_p$-lattice $T'$ in $V$ such that $T\subset T'$. There exists a $({\varphi},\hat{G}_{K_n})$-module $\hat{{\mathfrak{N}}}$ of height $\le r$ over ${\mathfrak{S}}_{K_n}$ such that $T'\simeq \hat{T}_{K_n}(\hat{{\mathfrak{N}}})$. Put ${\mathfrak{M}}_n={\mathfrak{S}}_{K_n}\otimes_{{\mathfrak{S}}_K}{\mathfrak{M}}$, which is a Kisin module of height $\le r$ over ${\mathfrak{S}}_{K_n}$. Since the functor $T_{{\mathfrak{S}}_{K_n}}$ from ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{K_n}}$ into ${\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_{K_{\infty}})$ is fully faithful, we obtain a morphism ${\mathfrak{N}}\to {\mathfrak{M}}_n$ which corresponds to the inclusion map $T\hookrightarrow T'$. We note that it is injective and its cokernel ${\mathfrak{M}}_n/{\mathfrak{N}}$ is killed by a power of $p$ since $T'/T$ is $p$-power torsion. Set ${\mathcal}{D}_n
:=S_{K_n}[1/p]\otimes_{{\mathfrak{S}}_{K_n}}{\mathfrak{N}}\simeq S_{K_n}[1/p]\otimes_{{\mathfrak{S}}_{K_n}}{\mathfrak{M}}_n$, ${\mathcal}{D}:=S_K[1/p]\otimes_{{\mathfrak{S}}_K}{\mathfrak{M}}$, ${\mathcal}{N}:=S_{K_n}\otimes_{{\mathfrak{S}}_{K_n}}{\mathfrak{N}}$, ${\mathcal}{M}_n:=S_{K_n}\otimes_{{\mathfrak{S}}_{K_n}}{\mathfrak{M}}_n$ and ${\mathcal}{M}:=S_K\otimes_{{\mathfrak{S}}_K}{\mathfrak{M}}$. We define filtrations ${\mathrm}{Fil}^i{\mathcal}{N}$, ${\mathrm}{Fil}^i{\mathcal}{M}_n$ and ${\mathrm}{Fil}^i{\mathcal}{M}$ as Lemma \[exercise2\] (2). Note that ${\mathcal}{D}_n$ has a structure of a Breuil module which corresponds to $V$. In particular, we have a Frobenius ${\varphi}_{{\mathcal}{D}_n}$, a monodromy operator $N_{{\mathcal}{D}_n}$ and a decreasing filtration $({\mathrm}{Fil}^i{\mathcal}{D}_n)_{i\in {\mathbb}{Z}}$ on ${\mathcal}{D}_n$. It is a result of [@Br §6] that we can equip $D:={\mathcal}{D}_n/I_+S_{K_n}[1/p]{\mathcal}{D}_n$ with a structure of filtered $({\varphi},N)$-module over $K_n$ which corresponds to $V$. Now we recall the definition of this structure and also define some additional notations for later use. The Frobenius ${\varphi}_D$ and the monodromy $N_D$ on $D$ is defined by ${\varphi}_D:={\varphi}_{{\mathcal}{D}_n}\ {\mathrm}{mod}\ I_+S_{K_n}[1/p]{\mathcal}{D}_n$ and $N_D:=N_{{\mathcal}{D}_n}\ {\mathrm}{mod}\ I_+S_{K_n}[1/p]{\mathcal}{D}_n$. We denote by $f_{\pi_n}$ and $f_{\pi}$ the natural projections ${\mathcal}{D}_n\twoheadrightarrow {\mathcal}{D}_n/{\mathrm}{Fil}^1S_{K_n}{\mathcal}{D}_n$ and ${\mathcal}{D}\twoheadrightarrow {\mathcal}{D}/{\mathrm}{Fil}^1S_K{\mathcal}{D}$, respectively. There is a unique ${\varphi}$-compatible section $s\colon D\hookrightarrow {\mathcal}{D}$ of the projection ${\mathcal}{D}\twoheadrightarrow {\mathcal}{D}/ I_+S_K[1/p]{\mathcal}{D}\simeq D$. Note that the composite $D\overset{s}{\hookrightarrow} {\mathcal}{D}\hookrightarrow {\mathcal}{D}_n$, which is also denoted by $s$, is a section of the projection ${\mathcal}{D}_n\twoheadrightarrow {\mathcal}{D}_n/ I_+S_{K_n}[1/p]{\mathcal}{D}_n=D$. Since the composite $D\overset{s}{\to} {\mathcal}{D}_n\overset{f_{\pi_n}}{\to} {\mathcal}{D}_n/{\mathrm}{Fil}^1S_{K_n}{\mathcal}{D}_n$ (resp. $D\overset{s}{\to} {\mathcal}{D}\overset{f_{\pi}}{\to} {\mathcal}{D}/{\mathrm}{Fil}^1S_K{\mathcal}{D}$) maps a basis of $D$ to a basis of ${\mathcal}{D}_n/{\mathrm}{Fil}^1S_{K_n}{\mathcal}{D}_n$ (resp. ${\mathcal}{D}/{\mathrm}{Fil}^1S_K{\mathcal}{D}$), we obtain an isomorphism $D_{K_n}:=K_n\otimes_{W(k)[1/p]} D\overset{\sim}{\rightarrow}
{\mathcal}{D}_n/{\mathrm}{Fil}^1S_{K_n}{\mathcal}{D}_n$ (resp. $D_K:=K\otimes_{W(k)[1/p]} D\overset{\sim}{\rightarrow}
{\mathcal}{D}/{\mathrm}{Fil}^1S_K{\mathcal}{D}$). By this isomorphism, we identify $D_{K_n}$ (resp. $D_K$) with ${\mathcal}{D}_n/{\mathrm}{Fil}^1S_{K_n}{\mathcal}{D}_n$ (resp. ${\mathcal}{D}/{\mathrm}{Fil}^1S_K{\mathcal}{D}$). Then the filtration $({\mathrm}{Fil}^iD_{K_n})_{i\in {\mathbb}{Z}}$ on $D$ over $K_n$ is given by ${\mathrm}{Fil}^iD_{K_n}=f_{\pi_n}({\mathrm}{Fil}^i{\mathcal}{D}_n)$. We note that the filtered $({\varphi},N)$-module $D$ over $K_n$ defined above is weakly admissible since $V$ is semi-stable (see [@CF §3.4] for the definition of weakly admissibility).
Let ${\mathcal{O}}_{K}$ and ${\mathcal{O}}_{K_n}$ be rings of integers of $K$ and $K_n$, respectively. We note that there exists a canonical isomorphism $K_n\otimes_{{\mathcal{O}}_{K_n}} f_{\pi_n}({\mathrm}{Fil}^i{\mathcal}{M}_n)
\simeq K_n\otimes_{{\mathcal{O}}_{K_n}} f_{\pi_n}({\mathrm}{Fil}^i{\mathcal}{N})$ since we have $p^c{\mathrm}{Fil}^i{\mathcal}{M}_n\subset {\mathrm}{Fil}^i{\mathcal}{N}\subset {\mathrm}{Fil}^i{\mathcal}{M}_n$ as submodules of ${\mathcal}{D}_n$, where $c\ge 0$ is an integer such that ${\mathfrak{M}}_n/{\mathfrak{N}}$ is killed by $p^c$. On the other hand, the canonical isomorphism $S_{K_n}[1/p]\otimes_{S_K[1/p]} {\mathcal}{D}
\simeq {\mathcal}{D}_n$ induces an isomorphism $S_{K_n}\otimes_{S_K} {\mathrm}{Fil}^i{\mathcal}{M}
\simeq {\mathrm}{Fil}^i{\mathcal}{M}_n$ (cf. Lemma \[exercise2\] (2)), and it gives an isomorphism ${\mathcal{O}}_{K_n}\otimes_{{\mathcal{O}}_K}f_{\pi}({\mathrm}{Fil}^i{\mathcal}{M})\simeq
f_{\pi_n}({\mathrm}{Fil}^i{\mathcal}{M}_n)$. Furthermore, it follows from [@Li2 Corollary 3.2.3] that a natural isomorphism ${\mathcal}{N}[1/p]\simeq {\mathcal}{D}_n$ preserves filtrations, where ${\mathrm}{Fil}^i({\mathcal}{N}[1/p]):=
({\mathrm}{Fil}^i{\mathcal}{N})[1/p]$. This induces $K_n\otimes_{{\mathcal{O}}_{K_n}} f_{\pi_n}({\mathrm}{Fil}^i{\mathcal}{N})\simeq
{\mathrm}{Fil}^iD_{K_n}$. (Here, we remark that the argument of §3.2 of [*loc. cit.*]{} proceeds even for $p=2$.) Therefore, if we define a decreasing filtration $({\mathrm}{Fil}^iD_K)_{i\in {\mathbb}{Z}}$ on $D_K$ by ${\mathrm}{Fil}^iD_K:=K\otimes_{{\mathcal{O}}_K} f_{\pi}({\mathrm}{Fil}^i{\mathcal}{M})$, then we have a canonical isomorphism $$\label{canonical}
K_n\otimes_K {\mathrm}{Fil}^iD_K
\simeq {\mathrm}{Fil}^iD_{K_n}.$$ Note that we know $D_K={\mathrm}{Fil}^0D_K\supset {\mathrm}{Fil}^1D_K
\supset \cdots \supset {\mathrm}{Fil}^{r+1}D_K=0$. Now we recall that $D$ is weakly admissible as a filtered $({\varphi},N)$-module over $K_n$. It follows from (\[canonical\]) that $D$ is also weakly admissible as a filtered $({\varphi},N)$-module over $K$, and hence the action of $G_{K_n}$ on $V$ extends to $G_K$ so that it is semi-stable over $K$. Therefore, we showed that the restriction functor ${\mathbf}{R}_1\to {\mathbf}{R}_2$ is essentially surjective and this finishes a proof of Proposition \[Thm2\].
${\mathcal}{C}^r_{m_0}= {\mathcal}{C}^r$ {#3.4}
----------------------------------------
Now we are ready to complete a proof of Theorem \[Main1’\]. We put $K_{p^{\infty}}=\bigcup_{i\ge 0} K(\zeta_{p^i})$ and $G_{p^{\infty}}={\mathrm}{Gal}(K_{\infty}K_{p^{\infty}}/K_{p^{\infty}})\subset \hat{G}_K$. We fix a topological generator $\tau$ of $G_{p^{\infty}}$. We start with the following lemma.
\(1) The field $K_{p^{\infty}}\cap K_{\infty}$ coincides with $K$ or $K_1$.
\(2) If $(p,m_0)\not=(2,1)$, then $K_{p^{\infty}}\cap K_{\infty}=K$.
\(3) If $m\ge 2$, then $K_{p^{\infty}}\cap K_{\infty}=K$.
The assertions (1) and (2) are consequences of [@Li2 Lemma 5.1.2] and [@Li3 Proposition 4.1.5], and so it is enough to show (3). We may assume $p=2$. Assume that $K_{p^{\infty}}\cap K_{\infty}\not=K$. Then we have $K_{p^{\infty}}\cap K_{\infty}=K_1$ by (1). Since $K_1$ is contained in $K_{p^{\infty}}$, we have $K_1\subset K(\zeta_{2^{\ell}})$ for $\ell>m$ large enough. Since $m\ge 2$, the extension $K(\zeta_{2^{\ell}})/K(\zeta_{2^m})$ is cyclic and thus there exists only one quadratic subextension in it. By definition of $m$, the extension $K(\zeta_{2^{m+1}})/K(\zeta_{2^m})$ is degree $2$. Since the extension $K_1/K$ is totally ramified but $K(\zeta_{2^m})/K$ is unramified, we see that the extension $K_1(\zeta_{2^m})/K(\zeta_{2^m})$ is also degree $2$. Therefore, we have $K_1(\zeta_{2^m})=K(\zeta_{2^{m+1}})$, and then we have $\pi_1=x\zeta_{2^{m+1}}+y$ with $x,y\in K(\zeta_{2^m})$. Let $\sigma$ be a non-trivial element in ${\mathrm}{Gal}(K(\zeta_{2^{m+1}})/K(\zeta_{2^m}))$. We have $-\pi_1=\sigma(\pi_1)=x\sigma(\zeta_{2^{m+1}})+y=-x\zeta_{2^{m+1}}+y$. Hence $\pi_1=x\zeta_{2^{m+1}}$ and we have $v(\pi_1)=v(x)$. Here, $v$ is a valuation of $K(\zeta_{2^{m+1}})$ normalized by $v(K^{\times})={\mathbb}{Z}$, and we see $v(\pi_1)=1/2$. Since the extension $K(\zeta_{2^m})/K$ is unramified, we have $v(x)\in {\mathbb}{Z}$ but this is a contradiction.
If $(p,m_0)=(2,1)$ and $m=1$, we have $m_0=m$ and then Theorem \[Main1’\] follows immediately from Lemma \[Lem:Main1’\]. Hence we may assume $(p,m_0)\not=(2,1)$ or $m\ge 2$. Under this assumption, the above lemma implies $K_{p^{\infty}}\cap K_{\infty}=K$. In particular, we have $\hat{G}=G_{p^{\infty}} \rtimes H_K$ with the relation $g\sigma=\sigma^{\chi(g)}g$ for $g\in H_K$ and $\sigma\in G_{p^{\infty}}$. Here, $\chi$ is the $p$-adic cyclotomic character. Let $\hat{{\mathfrak{M}}}=({\mathfrak{M}},{\varphi},\hat{G}_K)$ be an object of ${}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_K}_{/{\mathfrak{S}}_K}$ and put $T=\hat{T}_K(\hat{{\mathfrak{M}}})$. Our goal is to show that $T$ is an object of ${\mathcal}{C}^r_{m_0}$. We put ${\mathcal}{D}=S_{K}[1/p]\otimes_{{\varphi},{\mathfrak{S}}_K} {\mathfrak{M}}$ and $D={\mathcal}{D}/I_+S_K[1/p]{\mathcal}{D}$. Let $s\colon D\hookrightarrow {\mathcal}{D}$ be a ${\varphi}$-equivariant $W(k)[1/p]$-linear section of the projection ${\mathcal}{D}\twoheadrightarrow D$ as before, and take a basis $e_1,\dots ,e_d$ of $s(D)$. In ${\mathcal}{R}_K\otimes_{W(k)[1/p]} s(D)={\mathcal}{R}_K\otimes_{{\varphi},{\mathfrak{S}}_K} {\mathfrak{M}}$, the $\tau$-action with respected to the basis $e_1,\dots ,e_d$ is given by $\tau(e_1,\dots e_d)=(e_1,\dots ,e_d)A(t)$ for some matrix $A(t)\in GL_d(W(k)[1/p][\![t]\!])$. Moreover, we have $\hat{G}_K(s(D))\subset ({\mathcal}{R}_K\cap W(k)[1/p][\![t]\!])\otimes_{W(k)[1/p]} s(D)$ by [@Li1 Lemma 7.1.3]. Here are two remarks. The first one is that, the $a$-th power $A(t)^a$, a matrix with coefficients in $W(k)[1/p][\![t]\!]$, of $A(t)$ is well-defined for any $a\in {\mathbb}{Z}_p$. This is because the Galois group $G_{p^{\infty}}=\tau^{{\mathbb}{Z}_p}
\subset \hat{G}_K$ acts continuously on ${\mathcal}{R}_K\otimes_{W(k)[1/p]} s(D)$. The second one is that, for any $g\in H_K$, we have $A(\chi(g)t)=A(t)^{\chi(g)}$ by the relation $g\tau=\tau^{\chi(g)}g$. In particular, we have $$\label{Keyrel}
A(0)^{\chi(g)-1}=I_d.$$ Here, $I_d$ is the identity matrix. With these notation, it follows from the second paragraph of the proof of [@Li3 Theorem 4.2.2] that $T\otimes_{{\mathbb}{Z}_p} {\mathbb}{Q}_p$ is semi-stable over $K_{\ell}$ if $A(0)^{p^{\ell}}=I_d$.
\[lastlemma\] Let the notation be as above. Then we have $A(0)^{p^{m_0}}=I_d$.
First we consider the case where $p$ is odd. Since $H_K$ is canonically isomorphic to ${\mathrm}{Gal}(K_{p^{\infty}}/K)$, the image of the restriction to $H_K$ of the $p$-adic cyclotomic character $\chi\colon \hat{G}_K\to {\mathbb}{Z}_p^{\times}$ is equal to $$\chi(\hat{G}_K)=C\times (1+p^n{\mathbb}{Z}_p)$$ where $n$ is a positive integer and $C\simeq {\mathrm}{Gal}(K(\zeta_p)/K)$ is a finite cyclic group of order prime-to-$p$.
[*The case where $m_0\ge 1$:*]{} In this case, it is an easy exercise to check the equality $n=m_0$ and hence we can choose $g\in H_K$ such that $\chi(g)=1+p^{m_0}$. Thus the result follows by (\[Keyrel\]).
[*The case where $m_0=0$:*]{} In this case, $C$ is non-trivial and hence there exists an element $g\in H_K$ such that $x:=\chi(g)-1$ is a unit of ${\mathbb}{Z}_p$. By (\[Keyrel\]), we have $A(0)^x=I_d$, and then we obtain $A(0)=I_d$.
Next we consider the case where $p=2$.
[*The case where $m_0\ge 2$:*]{} This case is clear since we have $\chi(H_K)=\chi(\hat{G}_K)=1+2^{m_0}{\mathbb}{Z}_2$.
[*The case where $m_0=1$:*]{} In this case, $\chi\ {\mathrm}{mod}\ 4$ is not trivial. Hence there exists $g\in H_K$ such that $\chi(g)=3+4x$ for some $x\in {\mathbb}{Z}_2$. By (\[Keyrel\]), we have $A(0)^{2+4x}=I_d$. Since $1+2x$ is a unit of ${\mathbb}{Z}_2$, this gives the desired equation $A(0)^2=I_d$.
By the above lemma, we obtain the fact that $T\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p$ is semi-stable over $K_{m_0}$. On the other hand, we have already shown that ${\mathcal}{C}^r$ is a subcategory of ${\mathcal}{C}^r_{m}$. Thus there exists a semi-stable ${\mathbb}{Q}_p$-representation $V$ of $G_K$ whose restriction to $G_{K_m}$ is isomorphic to $T\otimes_{{\mathbb}{Z}_p} {\mathbb}{Q}_p$. Moreover, Proposition \[totst\] implies that $V$ and $T\otimes_{{\mathbb}{Z}_p} {\mathbb}{Q}_p$ are isomorphic as representations of $G_{K_{m_0}}$ since they are semi-stable over $K_{m_0}$. Therefore, we conclude that $T$ is an object of the category ${\mathcal}{C}^r_{m_0}$. This is the end of a proof of Theorem \[Main1’\].
Conclusions and more
--------------------
###
We summarize our results here. For any finite extension $L/K$, we denote by ${\mathrm}{Rep}^{r,L\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$ the category of free ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which is semi-stable over $L$ with Hodge-Tate weights in $[0,r]$. We define ${\mathcal}{C}^r_n$ to be the category of free ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which satisfies the following property: there exists a semi-stable ${\mathbb}{Q}_p$-representation $V$ of $G_K$ with Hodge-Tate weights in $[0,r]$ such that $T\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p$ is isomorphic to $V$ as representations of $G_{K_n}$. By definition ${\mathcal}{C}^r_n$ is a full subcategory of ${\mathrm}{Rep}^{r,K_n\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$. Put $m_0={\mathrm}{max}\{i\ge 0 \mid \zeta_{p^i}\in K \}$ and $m={\mathrm}{max}\{i\ge 0 \mid \zeta_{p^i}\in K^{{\mathrm}{ur}} \}$. We have ${\mathrm}{Rep}^{r,K_m\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)
=\bigcup_{n\ge 0} {\mathrm}{Rep}^{r,K_n\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$, ${\mathcal}{C}^r_m=\bigcup_{n\ge 0} {\mathcal}{C}^r_n$ (see Remark \[Rem:Liu\]). Results of [@Li3] and this note give the following diagram (here, “$\subset$” implies an inclusion):
$\displaystyle \xymatrix{
&
& {\mathcal}{C}^r_m \ar^{\subset \quad \qquad}[r]
& {\mathrm}{Rep}^{r,K_m\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)
\\
{}_{{\mathrm}{w}}{{\mathrm}{Mod}}^{r,\hat{G}_K}_{/{\mathfrak{S}}_K}
\ar^{\sim}@{->}_{\hat{T}_K}[rr]
&
& {\mathcal}{C}^r_{m_0} \ar^{\cup}[u]. \ar^{\subset \quad \qquad}[r]
& {\mathrm}{Rep}^{r,K_{m_0}\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K) \ar^{\cup}[u]
\\
{{\mathrm}{Mod}}^{r,\hat{G}_K}_{/{\mathfrak{S}}_K}
\ar^{\cup}[u] \ar^{\sim}_{\hat{T}_K}[rr]
&
&
{\mathcal}{C}^r_{0} \ar^{\cup}[u]. \ar@{=}[r]
&
{\mathrm}{Rep}^{r,K\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K) \ar^{\cup}[u]
}$
###
We give a few remarks for the above diagram. Clearly, all the categories in the middle and right vertical lines are same if $m=0$. On the other hand, if $m \ge 1$, inclusion relations between them are described as follows:
\[prop:rem\] Suppose $m\ge 1$.
\(1) Suppose $1\le n\le m$. Then the category ${\mathcal}{C}^r_n$ is strictly larger than ${\mathcal}{C}^r_{n-1}$. In particular, the category ${\mathrm}{Rep}^{r,K_{n}\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$ is strictly larger than ${\mathrm}{Rep}^{r,K_{n-1}\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$.
\(2) Suppose $n,r\ge 1$. Then the category ${\mathrm}{Rep}^{r,K_n\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$ is strictly larger than ${\mathcal}{C}^r_n$.
\(3) Suppose $n\ge 0$. Then we have ${\mathcal}{C}^0_n={\mathrm}{Rep}^{0,K_n\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$.
\(1) Let $T$ be the induced representation of the rank one trivial ${\mathbb}{Z}_p$-representation of $G_{K_n(\zeta_{p^n})}$ to $G_K$, which is an Artin representation. The splitting field of $T$ is $K_n(\zeta_{p^n})$. Since $n\le m$, the extension $K_n(\zeta_{p^n})/K_n$ is unramified. Thus $T$ is crystalline over $K_n$. On the other hand, $T$ is not crystalline over $K_{n-1}$ since the extension $K_n(\zeta_{p^n})/K_{n-1}$, the splitting field of $T|_{K_{n-1}}$, is not unramified. (This finishes a proof of the latter assertion.) Let $\rho_T\colon G_K\to GL_{{\mathbb}{Z}_p}(T)\simeq GL_d({\mathbb}{Z}_p)$ be the continuous homomorphism associated with $T$, where $d$ is the ${\mathbb}{Z}_p$-rank of $T$. By the assumption $n\le m$, we know that $K(\zeta_{p^n})\cap K_n=K$ and thus we can define a continuous homomorphism $\rho_{T'}\colon G_K\to GL_d({\mathbb}{Z}_p)$ by the composite $G_K\twoheadrightarrow {\mathrm}{Gal}(K(\zeta_{p^n})/K)\simeq {\mathrm}{Gal}(K_n(\zeta_{p^n})/K_n)
\overset{\rho_T}{\hookrightarrow} GL_d({\mathbb}{Z}_p)$. Let $T'$ be the free ${\mathbb}{Z}_p$-module of rank $d$ equipped with a $G_K$-action by $\rho_{T'}$. Then $T'$ is isomorphic to $T$ as representations of $G_{K_n}$ and furthermore it is crystalline over $K$. It follows that $T$ is an object of ${\mathcal}{C}^r_n$.
\(2) Since $m\ge 1$, we know that $L:=K(\zeta_p)$ is an unramified extension of $K$. Thus $\pi_K$ is a uniformizer of $L$. Consider notations ${\mathfrak{S}}_L, S_L,\dots $ (resp. ${\mathfrak{S}}_{L_1}, S_{L_1},\dots$) with respect to the uniformizer $\pi_K$ (resp. $\pi_{K,1}$) of $L$ (resp. $L_1$) and the system $(\pi_{K,n})_{n\ge 0}$ (resp. $(\pi_{K,n+1})_{n\ge 0}$). Let ${\mathfrak{M}}$ be the rank-$2$ free Kisin module over ${\mathfrak{S}}_{L_1}$ of height $1$ given by ${\varphi}(e_1,e_2)=(e_1,e_2)\begin{pmatrix}1 & u_{L_1} \\ 0 & E_{L_1}(u_{L_1})\end{pmatrix}$, where $\{e_1,e_2\}$ is a basis of ${\mathfrak{M}}$. Since ${\mathfrak{M}}$ is of height $1$, there exists a $G_{L_1}$-stable ${\mathbb}{Z}_p$-lattice $T$ in a crystalline ${\mathbb}{Q}_p$-representation of $G_{L_1}$, coming from a $p$-divisible group over the integer ring of $L_1$. We see that $\tilde{T}:={\mathrm}{Ind}^{G_K}_{G_{L_1}}T$ is crystalline over $L_1$. Since $L_1$ is unramified over $K_1$, $\tilde{T}$ is in fact crystalline over $K_1$. Furthermore, $\tilde{T}$ does not come from Kisin modules over ${\mathfrak{S}}_K$ (that is, $\tilde{T}|_{G_{K_{\infty}}}$ is not isomorphic to $T_{{\mathfrak{S}}_K}({\mathfrak{N}})$ for any Kisin module ${\mathfrak{N}}$ over ${\mathfrak{S}}_K$). To check this, it suffices to show that $\tilde{T}$ does not come from Kisin modules over ${\mathfrak{S}}_L$. Essentially, this has been already shown in [@Li3 Example 4.2.3]. Therefore, Corollary \[Main3\] implies that $\tilde{T}$ is not an object of ${\mathcal}{C}^r_n$.
\(3) We may suppose $n\le m$. Take any object $T$ of ${\mathrm}{Rep}^{0,K_n\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$. Since $T$ has only one Hodge-Tate weight zero, the condition $T|_{G_{K_n}}$ is semi-stable implies that $T|_{G_{K_n}}$ is unramified. Thus if we denote by $K_T$ the splitting field of $T$, then $K_TK_n$ is unramified over $K_n$.
First we consider the case where $K_T$ contains $\zeta_{p^n}$. In this case, we follow the idea given in the proof of (1). Denote by $K'$ the maximum unramified subextension of $K_TK_n$ over $K$. Since $K_T$ contains $\zeta_{p^n}$, $K_TK_n/K$ is a Galois extension and hence $K'/K$ is also Galois. Furthermore, it is not difficult to check that the equality $K_TK_n=K'K_n$ holds. Let $\rho_T\colon G_K\to GL_{{\mathbb}{Z}_p}(T)\simeq GL_d({\mathbb}{Z}_p)$ be the continuous homomorphism associated with $T$, where $d$ is the ${\mathbb}{Z}_p$-rank of $T$, and define a continuous homomorphism $\rho_{T'}\colon G_K\to GL_d({\mathbb}{Z}_p)$ by the composite $G_K\twoheadrightarrow {\mathrm}{Gal}(K'/K)\simeq {\mathrm}{Gal}(K_TK_n/K_n)
\overset{\rho_T}{\hookrightarrow} GL_d({\mathbb}{Z}_p)$. Let $T'$ be the free ${\mathbb}{Z}_p$-module of rank $d$ equipped with a $G_K$-action by $\rho_{T'}$. Then $T'$ is isomorphic to $T$ as representations of $G_{K_n}$ and furthermore, $T'$ is crystalline over $K$. It follows that $T$ is an object of ${\mathcal}{C}^0_n$.
Next we consider the general case. Denote by $T_0$ the induced representation of the rank one trivial ${\mathbb}{Z}_p$-representation of $G_{K(\zeta_{p^n})}$ to $G_K$. We define a free ${\mathbb}{Z}_p$-representation $\tilde{T}$ of $G_K$ by $\tilde{T}:=T\oplus T_0$. The splitting fields of $\tilde{T}$ and $T_0$ are equal to $K_{\tilde{T}}:=K_T(\zeta_{p^n})$ and $K(\zeta_{p^n})$, respectively. The representations $\tilde{T}$ and $T_0$ are objects of ${\mathrm}{Rep}^{0,K_n\mathchar`-{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$. Moreover, the above argument implies that $\tilde{T}$ and $T_0$ are contained in ${\mathcal}{C}^0_n$. Therefore, there exist objects $\tilde{{\mathfrak{M}}}$ and ${\mathfrak{M}}_0$ of ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_K}$ such that $T_{{\mathfrak{S}}_K}(\tilde{{\mathfrak{M}}})=\tilde{T}|_{G_{K_{\infty}}}$ and $T_{{\mathfrak{S}}_K}({\mathfrak{M}}_0)=T_0|_{G_{K_{\infty}}}$. Now we recall that the functor $T_{{\mathfrak{S}}_K}$ is fully faithful. If we denote by ${\mathfrak}{f}\colon {\mathfrak{M}}_0\to \tilde{{\mathfrak{M}}}$ a (unique) morphism of ${\varphi}$-modules over ${\mathfrak{S}}_K$ corresponding to the natural projection $\tilde{T}\twoheadrightarrow T_0$, then we obtain a split exact sequence $0\to {\mathfrak{M}}_0\overset{{\mathfrak}{f}}{\to} \tilde{{\mathfrak{M}}}\to {\mathfrak{M}}\to 0$ of ${\varphi}$-modules over ${\mathfrak{S}}_K$. Here, ${\mathfrak{M}}$ is the cokernel of ${\mathfrak}{f}$, which is a finitely generated ${\mathfrak{S}}_K$-module. Since ${\mathfrak{M}}$ is a direct summand of $\tilde{{\mathfrak{M}}}$, it is a projective ${\mathfrak{S}}_K$-module. This implies that ${\mathfrak{M}}$ is a free ${\mathfrak{S}}_K$-module. (Note that, for a finitely generated ${\mathfrak{S}}_K$-module, it is projective over ${\mathfrak{S}}_K$ if and only if it is free ${\mathfrak{S}}_K$ by Nakayama’s lemma.) Furthermore, ${\mathfrak{M}}$ is of height $0$ and hence it is an object of ${\mathrm}{Mod}^0_{/{\mathfrak{S}}_K}$. Since the functor $T_{{\mathfrak{S}}_K}$ is exact, we obtain $T_{{\mathfrak{S}}_K}({\mathfrak{M}})
={\mathrm}{ker}(T_{{\mathfrak{S}}_K}(\tilde{{\mathfrak{M}}})\overset{T_{{\mathfrak{S}}_K}({\mathfrak}{f})}{\longrightarrow} T_{{\mathfrak{S}}_K}({\mathfrak{M}}_0))
={\mathrm}{ker}(\tilde{T}\twoheadrightarrow T_0)=T$. Therefore, $T$ is an object of ${\mathcal}{C}^0_n$ by Corollary \[Main3\].
###
Assume that $m\ge 1$. Let $n\ge 1$ be an integer and $T$ an object of the category ${\mathcal}{C}^r_n$. By definition of ${\mathcal}{C}^r_n$, we have a (unique) semi-stable ${\mathbb}{Q}_p$-representation $V_T$ of $G_K$ with the property that it is isomorphic to $T\otimes_{{\mathbb}{Z}_p} {\mathbb}{Q}_p$ as representations of $G_{K_n}$. It is not clear whether $T$ is stable under the $G_K$-action of $V_T$ for any $T$ or not. Such a stability problem of Galois actions may sometimes cause obstructions in integral theory, and so the following question should be naturally considered.
\[question2\] Let the notation be as above. Does the $G_K$-action of $V_T$ preserves $T$ for any $T$?
We end this paper by showing an answer to this question.
\(1) If $r=0$, then Question \[question2\] has an affirmative answer.
\(2) If $r\ge 1$, then Question \[question2\] has a negative answer.
\(3) Let the notation be as above. Suppose $e(r-1)<p-1$ where $e$ is the absolute ramification index of $K$. If $T$ is potentially crystalline, then the $G_K$-action of $V_T$ preserves $T$. Moreover, any $G_{K_{\infty}}$-stable ${\mathbb}{Z}_p$-lattice of $V_T$ is stable under the $G_K$-action.
\(1) (This is a special case of (3).) The result easily follows from the fact that $T$ as in the question is unramified in this case, and that $G_{K_n}$ and the inertia subgroup of $G_K$ generate $G_K$.
\(2) Our goal is to construct an example which gives a negative answer to the question. First we consider the case where $1\le n\le m_0$. Let $E_{\pi}$ be the Tate curve over $K$ associated to $\pi$. Choose a basis $\{{\mathbf}{e},\ {\mathbf}{f}\}$ of the $p$-adic Tate module $V=V_p(E_{\pi})$ of $E_{\pi}$ such that the $G_K$-action on $V$ with respective to this basis is given by $$g\mapsto
\begin{pmatrix}
\chi(g) & c(g)\\
0 & 1
\end{pmatrix}.$$ Here, $\chi\colon G_K\to {\mathbb}{Z}^{\times}_p$ is the $p$-adic cyclotomic character and $c\colon G_K\to {\mathbb}{Z}_p$ is a map defined by $g(\pi_{K,{\ell}})=\zeta^{c(g)}_{p^{\ell}}\pi_{K,{\ell}}$ for any $g\in G_K$ and $\ell\ge 1$. Let $T_0$ be the free ${\mathbb}{Z}_p$-submodule of $V$ generated by $p^n{\mathbf}{e}$ and ${\mathbf}{f}$. This is $G_{K_n}$-stable but not $G_K$-stable in $V$. Now we put $T={\mathrm}{Ind}^{G_K}_{G_{K_n}}T_0$ and choose a set $S\subset G_K$ of representatives of the quotient $G_K/G_{K_n}$. Since $K_n/K$ is Galois, $T|_{G_{K_n}}$ is of the form $\oplus_{\sigma\in S}\ T_{0,\sigma}$. Here, $T_{0,\sigma}$ is just $T_0$ as a ${\mathbb}{Z}_p$-module and is equipped with a $\sigma$-twisted $G_{K_n}$-action, that is, $g.x:=(\sigma^{-1} g \sigma)(x)$ for $g\in G_{K_n}$ and $x\in T_{0,\sigma}$. We define elements ${\mathbf}{e}_{\sigma}$ and ${\mathbf}{f}_{\sigma}$ of $T_{0,\sigma}$ by ${\mathbf}{e}_{\sigma}:=p^n{\mathbf}{e}$ and ${\mathbf}{f}_{\sigma}:={\mathbf}{f}$. We define $V_{0,\sigma}:=T_{0,\sigma}\otimes_{{\mathbb}{Z}_p} {\mathbb}{Q}_p$ and extend the $G_{K_n}$-action on $V_{0,\sigma}$ to $G_K$ by $$g({\mathbf}{e}_{\sigma},{\mathbf}{f}_{\sigma})=({\mathbf}{e}_{\sigma},{\mathbf}{f}_{\sigma})
\begin{pmatrix}
\chi(g) & c(\sigma^{-1}g\sigma)/p^n\\
0 & 1
\end{pmatrix}$$ for $g\in G_K$. By definition the $G_K$-action on $V_{0,\sigma}$ does not preserve $T_{0,\sigma}$. It is not difficult to check that $V_{0,\sigma}$ is a semi-stable ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights $\{0,1\}$. If we put $V_T=\oplus_{\sigma\in S} V_{0,\sigma}$, then we have the followings:
- $V_T$ is semi-stable over $K$ with Hodge-Tate weights $\{0,1\}$,
- the natural isomorphism $V_T\simeq T\otimes_{{\mathbb}{Z}_p} {\mathbb}{Q}_p$ is compatible with $G_{K_n}$-actions, and
- the $G_K$-action on $V_T$ does not preserve $T$.
This gives a negative answer to Question \[question\] in the case $1\le n\le m_0$.
Next we consider a general case. We may suppose $n=m$. Put $K'=K(\zeta_{p^m})$ and $K'_m=K_mK'$. Then $K'$ is an unramified Galois extension of $K$ and ${\mathrm}{max}\{i\ge 0 \mid \zeta_{p^i}\in K' \}=m$. Thus the above argument shows that there exists a free ${\mathbb}{Z}_p$-representation $T'$ of $G_{K'}$ and a semi-stable ${\mathbb}{Q}_p$-representation $V_{T'}$ of $G_{K'}$ with Hodge-Tate weights $\{0,1\}$ which satisfies the followings:
- there exists an isomorphism $V_{T'}\simeq T'\otimes_{{\mathbb}{Z}_p} {\mathbb}{Q}_p$ of $G_{K'_m}$-representations, and
- the $G_{K'}$-action on $V_{T'}$ does not preserve $T'$.
We regard $T'$ as a ${\mathbb}{Z}_p$-lattice of $V_{T'}$. We define $T:={\mathrm}{Ind}^{G_K}_{G_{K'}}T'$ and $V_T:={\mathrm}{Ind}^{G_K}_{G_{K'}}V_{T'}$. Note that $T$ is naturally regarded as a ${\mathbb}{Z}_p$-lattice of $V$. By definition, the $G_{K'}$-action on $V_T$ does not preserve $T$. In particular, the same holds also for the $G_K$-action. Since $K'/K$ is unramified, we see that $V_T$ is semi-stable over $K$. Furthermore, by Mackey’s formula, we have natural isomorphisms $T\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p
\simeq {\mathrm}{Ind}^{G_{K_m}}_{G_{K'_m}}(T'\otimes_{{\mathbb}{Z}_p}{\mathbb}{Q}_p)
\simeq {\mathrm}{Ind}^{G_{K_m}}_{G_{K'_m}}V_{T'}
\simeq V_T
$ of representations of $G_{K_m}$. Therefore, we conclude that Question \[question2\] has a negative answer for any $n\ge 1$.
\(3) This is a special case of [@Oz Corollary 4.20].
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Tong Liu, *Lattices in filtered $({\varphi},N)$-modules*, J. Inst. Math. Jussieu 2, Volume 11, Issue 03 (2012), 659-693.
Tong Liu, *Compatibility of Kisin modules for different uniformizers*, preprint, arXiv:1302.1888
Yoshiyasu Ozeki, *On Galois equivariance of homomorphisms between torsion potentially crystalline representations*, preprint, arXiv:1304.2095v3
[^1]: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPAN. e-mail: [[email protected]]{} Supported by JSPS KAKENHI Grant Number 25$\cdot$173
[^2]: We do not describe the definition of Breuil modules in this note. See §6.1 of [@Br] for axioms of Breuil modules.
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Abstract: The scope of this review is to revise recent advances of the cell-based therapies of liver diseases with an emphasis on cell donor’s and patient’s age. Regenerative medicine with cell-based technologies as its integral part is focused on the structural and functional restoration of tissues impaired by sickness or aging. Unlike drug-based medicine directed primarily at alleviation of symptoms, regenerative medicine offers a more holistic approach to disease and senescence management aimed to achieve restoration of homeostasis. Hepatocyte transplantation and organ engineering are very probable forthcoming options of liver disease treatment in people of different ages and vigorous research and technological innovations in this area are in progress. Accordingly, availability of sufficient amounts of functional human hepatocytes is crucial. Direct isolation of autologous hepatocytes from liver biopsy is problematic due to related discomfort and difficulties with further expansion of cells, particularly those derived from aging people. Allogeneic primary human hepatocytes meeting quality standards are also in short supply. Alternatively, autologous hepatocytes can be produced by reprogramming of differentiated cells through the stage of induced pluripotent stem cells. In addition, fibroblasts and mesenchymal stromal cells can be directly induced to undergo advanced stage hepatogenic differentiation. Reprogramming of cells derived from elderly people is accompanied by the reversal of age-associated changes at the cellular level manifesting itself by telomere elongation and the U-turn of DNA methylation. Cell reprogramming can provide high quality rejuvenated hepatocytes for cell therapy and liver tissue engineering. Further technological advancements and establishment of national and global registries of induced pluripotent stem cell lines homozygous for HLA haplotypes can allow industry-style production of livers for immunosuppression-free transplantation.
Keywords: cell aging, rejuvenation, liver engineering, liver cell therapy
Introduction
No liver pathology occurs specifically at advanced age, but disease progression, its incidence, and the patient’s reaction to drugs and medical manipulations may differ in younger and older people. It should be recognized though, that chronologic age is an arbitrary characteristic of a person. Much of human aging research concerns older individuals. However, senescence does not necessarily start at an advanced age. Recently, it has been shown that differences in the pace of aging can be readily detected in people in their late 30s.1 This study quantitatively evaluated physiological deterioration across multiple organ systems – cardiovascular, renal, hepatic, immune, and others – in 38-year-old people and found that already at this stage a substantial proportion of young individuals were aging more rapidly than their peers and were showing lower physical fitness, cognitive decline, regression of renal, heart and liver function, and self-reported ill health. Accordingly, not just chronological, but biological age clearly is a very important factor to be considered when new methods of treatment of the people of any age are developed. Regenerative medicine based primarily on stem cell biology research is a new approach to disease treatment largely focused on the correction of age- and pathology-related malfunction of organs and systems by the enhancement of tissue regeneration and substitution of dysfunctional and senescent cells, tissues, and organs.
Liver pathology is a serious health problem worldwide because of high morbidity and mortality in end-stage liver disease associated primarily though not solely with growing cirrhosis and cancer incidence in at least half-a-billion patients with persistent hepatitis B and C infection and in tens of millions of alcoholics.2–4 Allogeneic liver transplantation can save lives and improve health, but only in 20%–30% of patients due to donor shortage, medical contraindications, and social and economic reasons.5,6 In general, present day therapeutic techniques do not consistently ensure patient’s recovery or even the upholding of the status quo. Cell-based technologies of regenerative medicine using autologous or allogeneic hepatocytes or cells capable of hepatogenic differentiation offer a novel approach to better management and diagnostics of liver disease. Indeed, cell transplantation is by far less invasive and expensive compared to transplantation of whole liver or its part, while organ engineering has the potential to solve the problem of liver donor shortage. In addition, human hepatocytes are used in extracorporeal liver support systems and their long-term cultures are gradually substituting animal experiments in drug testing and in vitro disease modeling. Accordingly, availability of sufficient amounts of functional human hepatocytes becomes imperative.
Aging presents additional and not yet fully understood challenges in both hepatocyte manufacture and clinical or experimental applications. The response of the recipient body to cell therapy or a tissue engineering construct transplantation depends on his or her age. Biological aging is a universal phenomenon occurring at different levels of organization including cellular level. During the lifetime of multicellular organisms, cells undergo changes and develop signs of senescence. They also find themselves in a distorted microenvironment.
Mechanisms of cellular senescence have been only partly disclosed and include free radical damage to the inner machinery of the cell, loss of telomerase activity resulting in the shortening of telomeres and cell cycle arrest, accumulation of DNA modifications, alterations of mitochondria functions, and others.7–9 The above-listed mechanisms are of course inter-related. For example, oxidative free radical stress causes nuclear DNA and mitochondria damage. In principle, age-related changes can induce altered reactions of cultured cells taken from older donors to differentiation and dedifferentiation stimuli.10 However, some data accumulated so far suggest that is not always the case. Neither replicative aging in vitro nor cell donor’s advanced age prevents adult differentiated cells, for example, skin fibroblasts from dedifferentiation into induced pluripotent stem cells (iPSCs), while redifferentiation of iPSCs can yield fibroblasts with the characteristics of juvenile proliferating cells, not initial senescent fibroblasts, thus demonstrating the reversibility of aging at the cellular level.11,12 Therefore, cell therapy and tissue engineering followed by tissue and organ transplantation may offer not only relief from liver and other diseases but also a method of rejuvenation, at least at the tissue and organ level. Cell transplantation (cell therapy) and tissue and organ engineering present a novel and yet not fully explored and tested approach to disease treatment. However, it seems to have bright prospects because it is based on high quality and well subsidized fundamental and applied research in cell biology, molecular biology, biology of aging, and related fields.
This review is aimed to expose age-related aspects of human hepatocytes preparation by differentiation of iPSCs obtained from cells taken from adult donors or by transdifferentiation of mesenchymal stem cells (MSCs) and fibroblasts. It also depicts some of the prospective problems associated with the application of cell-based therapies to the treatment of patients with a special focus on age-related issues.
Generation of iPSCs from differentiated cells taken from donors of various ages
Reprogramming of somatic cells into iPSCs presents a unique opportunity to obtain autologous pluripotent cells for cell therapy, tissue engineering, drug testing, and disease modeling. Originally, human iPSCs have been generated from skin fibroblasts of an adult person by transfecting them with a set of four genes highly expressed in early embryogenesis – Oct4, Sox2, Klf4 and c-Myc – often called Yamanaka factors and abbreviated as OSKM.13 Earlier, the same battery of genes had been used to induce pluripotency in mouse embryonic and adult fibroblasts14 suggesting fundamental similarities of the mechanisms of pluripotency induction across the species. Significantly, the induction of pluripotency occurs in a stochastic manner and initially the percentage of reprogrammed cells was quite low.
Studies examining the impact of somatic cell donor age upon the efficiency of reprogramming to pluripotency in mice demonstrated lower reprogramming frequency in cells derived from older animals.15–17 Bone marrow cells from 23-month-old mice transfected with Yamanaka factors generated five times less colonies positive for the stem cell marker alkaline phosphatase compared to cells isolated from 2-month-old animals. Moreover, in older mice, reprogramming took twice as long than in younger ones.
Unlike data from mouse experiments, studies with human cells produced conflicting results and did not show clear impact of cell donor’s age on reprogramming efficacy. Remarkably, iPSCs could be obtained from the fibroblasts of ≥100 year-old people.18 These “centenarian” iPSCs expressed pluripotency markers and were actually pluripotent being able to differentiate into the derivatives of the three germ layers – ectoderm, endoderm, and mesoderm. Using four Yamanaka factors, Somers et al19 derived 100 cell lines from fibroblasts donated by people aged 8–64 years. Reprogramming efficacy was 0.1%–1.5% and did not correlate with donor’s age. All the resultant cells expressed pluripotency markers and robustly differentiated along the endoderm lineage route. In contrast, Sharma et al,20 also using OSKM array, found that skin fibroblasts obtained from donors aged 50–85 years reprogrammed with substantially lower efficacy than cells derived from younger 0–18-year-old donors, but not from 20–49-year-old donors. The nature of these inconsistencies is not fully understood. They may be associated with differences in many parameters characterizing the reprogrammed cells including their proliferative potential or culture conditions.
Since the pioneering works of Takahashi et al,13,14 where OSKM cassette was delivered to fibroblasts by retroviral vectors, other combinations of reprogramming factors and different gene delivery vehicles supplemented by specific iRNAs, proteins and biologically active small molecules have been successfully tested to convert somatic cells to pluripotent state.21 All these approaches provide iPSCs and many of them improve the yield of pluripotent cells. On the other hand, iPSCs produced by different methods may be not quite identical hindering the evaluation of the impact of different parameters including age.
Lapasset et al used a cocktail of six factors (traditional OSKM cassette plus Nanog and Lin28) to effectively reprogram aging cells obtained by prolonged in vitro passaging and showing all signs of replicative senescence, as well as cells derived from people of very advanced (92–101 years) age.11 All the resulting iPSC clones were positive for pluripotency markers Tra1-60 and SSEA-4 and were able to differentiate into the ectoderm, endoderm, and mesoderm derivatives. In addition, they were characterized by low expression levels of inhibitors of cyclin-dependent kinases p16INK4A and p21CIP1 blocking cell cycle progression from phase G1 to phase S, elongated telomeres which did not shorten during passaging, and normal mitochondrial metabolism similar to that of the embryonic stem cells. The authors of the paper argue that neither replicative aging in vitro nor cell donor’s advanced age prevent cells from dedifferentiation into iPSCs. Redifferentiation of iPSCs yielded fibroblasts with the characteristics of juvenile proliferating cells, once more demonstrating the reversibility of aging at the cellular level. These data have been confirmed using a more extended panel of age-dependent cell characteristics including telomere length, mitochondrial function, heterochromatin loss, state of the nuclear lamina, DNA repair machinery, and percentage of aging cells in the population to evaluate differences in the reprogramming of fibroblasts obtained from young and old donors.12 Age-related changes of each of the described parameters observed in fibroblasts taken from older people were eliminated after pluripotency induction and redifferentiation.
The reprogramming efficacy and pace depend upon the levels of stem cell-related genes expression in the initial somatic cells. For example, reprogramming of keratinocytes goes two times faster and 100 times more effectively compared to skin fibroblasts.22 Unlike fibroblasts, keratinocytes have stem cell-related genes expression pattern similar to embryonic stem cells. Two independent studies demonstrated that mouse neural stem/progenitor cells expressing high levels of endogenous Sox2 and c-Myc can be effectively reprogrammed by just two (Oct4 + Klf4 or Oct4 + c-Myc)23 or three (Oct4, Klf4, c-Myc)24 factors. These data suggest higher reprogramming efficacy of stem/progenitor cells compared to adult fully differentiated cells.25
This hypothesis is supported by significant experimental evidence, including data on the reprogramming of B lymphocytes.26,27 Forced expression of Yamanaka factors proved to be insufficient to reprogram B cells to iPSCs, even in case of very efficient exogene transfer when all B cells overexpress OSKM. Pluripotency induction occurred only after overexpression of the OSKM cassette was supplemented with overexpression of the CEBPα transcription factor or knockdown of its suppressor Pax5. Unlike mature cells, B lymphocyte progenitors were easily reprogrammed just by introduction of classical Yamanaka battery.
Abramovich et al were the first to suggest as early as in 2008 that reprogramming of adult somatic cells to iPSCs is accompanied by the reversal of the indications of cell senescence resulting in “rejuvenation” at the cellular level.28 Later, this point of view concerning human cells became predominant.11,12,29,30 It was further supported by demonstrating telomere elongation,31 mitochondria rejuvenation up to the state characteristic of the embryonic stem cells,32 and enhanced DNA reparation capacity33,34 in iPSCs, though in different iPSC lines these changes may be pronounced to a varying extent. In the course of reprogramming to iPSCs, somatic cells undergo profound modification allowing them to revert to the state of “stemness”. Fully reprogrammed iPSCs are similar (though not identical) to embryonic stem cells regarding their gene expression profile35,36 with elevated activity of genes responsible for pluripotence and cell renewal (Oct4, Nanog, Sox2, Lin28, Zic3, Fgf4, Tdgf1, and Rex1) and low activity of genes related to tissue specialization. Their DNA methylation profile also reverts to embryonic state. However, iPSCs may retain remnants of the pattern of DNA methylation characteristic of differentiated cells from which they were derived, the so-called “epigenetic memory”, making them and embryonic stem cells somewhat different.37 The impact of the epigenetic memory on the process of somatic cell reprogramming to iPSCs, stability of iPSCs characteristics during passaging, and the results of redifferentiation have been repeatedly discussed in the literature.38–40 Possibly, epigenetic memory is retained merely at early passages, while full reprogramming including its loss may be a continuing process.
Hence, in humans, cell donor’s age is not crucial for the reprogramming of somatic cells to iPSCs and redifferentiation of the latter. These processes are to a greater extent influenced by other factors including tissue origin of donor cells, extent of their differentiation, presence of hereditary and somatic mutations, nongenetic donor pathologies, and number of iPSCs passages before redifferentiation.
Hepatogenic differentiation of iPSC
Adult human liver tissue contains multiple cell types among which hepatocytes are the prevailing species comprising 70%–80% of the total cell number. Hepatocytes and one other type of liver tissue cells, biliary cells, differentiate from bipotent progenitors belonging to the endoderm lineage and are called hepatoblasts.41 Most other liver cells, including endothelium, stellate cells, and resident macrophages usually designated as Kupffer cells are derived from the mesoderm.
Hepatocytes carry out the majority of crucially important liver functions including glucose metabolism, synthesis, storage and degradation of other metabolites, detoxification of poisonous substances and drugs, bile production, and others. For a long time, primary hepatocytes isolated from human liver biopsies or autopsies served as the “golden standard” in drug metabolism research during preclinical drug testing.42 Consequently, they were a very intensely studied cell type and their production was a top priority. As shown in the Cell therapy and tissue engineering in the treatment of liver diseases section of this article, primary hepatocytes were also tested in cell therapy of liver pathology. Unluckily, primary human hepatocytes have very restricted in vitro proliferation capacity making them virtually inaccessible in quantities sufficient for extensive preclinical research and liver tissue reconstruction.43 In addition, in vitro manipulations can deprive primary human hepatocytes from some of their essential properties causing changes as significant as cytochrome P450 inactivation.44 These limitations shaped further efforts to improve the methods of functional human hepatocytes production, iPSCs obviously being one of the most convenient starting materials for the delivery of both autologous and allogeneic human hepatocytes.
The development of the methods of hepatocytes production from iPSCs started soon after the introduction of iPSC technology. Quite logically, the related protocols were based on procedures initially applied to embryonic stem cells and comprised several steps more or less closely reproducing the stages of hepatocyte differentiation throughout the ontogenesis. Each step started with application of a certain set of growth and transcription factors. Song et al were among the first to introduce a four-step differentiation protocol of the hepatogenic differentiation of human iPSCs starting with iPSC differentiation into the definitive endoderm followed by the stages of hepatocytic specification, hepatoblast expansion, and hepatocyte maturation.45 Three-step protocols also including consecutive stimulation of iPSC differentiation into definitive endoderm, immature hepatocytes, and mature hepatocytes induced by three different sets of growth and transcription factors followed soon.46,47
As indicated above, at the first stage of hepatogenic differentiation, embryonic stem cells or iPSCs are converted into the definitive endoderm. Activin A, a member of the activin family of the transforming growth factor beta (TGF-beta) superfamily, is one of the key factors shaping this process. It has been demonstrated that activin A participates in maintaining the “stemness” via SMAD-dependent activation of pluripotency factors such as Oct-4, Nanog, Nodal, and others.48,49 On the other hand, activin A is known to inhibit cell growth and proliferation by stimulating the transcription of the cell cycle inhibitors p15, p21, and p27KIP150,51 and to enhance differentiation through inhibition of c-Myc translation.52 Furthermore, the Nodal/Wnt signal pathway was shown to play a significant role in human embryonic stem cell differentiation along the endodermal route.53 Nodal is an activin analog also belonging to the TGF-beta superfamily. Activin/Nodal signaling regulates the transcription of Smad2/3 factor essential for differentiation to the definitive endoderm.54 It is also capable of activating many other genes in the course of endoderm development during embryogenesis.50
Activin A promotes differentiation into the definitive endoderm in both mouse55 and human56 embryonic stem cells. However, it has been suspected for a long time that the effects of activin A on human embryonic stem cell differentiation strongly depend on its concentration.57 Low (5 ng/mL) activin A concentrations contribute to the maintenance of undifferentiated state of embryonic stem cells58 and iPSCs59 probably due to the induction of Oct-4, Nanog, Nodal, Wnt3, fibroblast growth factor (FGF)-2, and FGF-8 and suppression of bone morphogenic protein (BMP) signaling, while high concentrations (25–100 ng/mL) promote their differentiation into the definitive endoderm in a dose-dependent manner.55,60 Consequently, activin A concentration of about 100 ng/mL is routinely used at the first stage of most newer iPSC hepatogenic differentiation protocols.61,62 In addition to activin A, other factors including Wnt3a combined with hepatocyte growth factor (HGF),61 B27 serum-free supplement initially designed for maintenance of hippocampal and cortical neurons in culture,62 and LY294002, the specific inhibitor of phosphatidylinositol-3-phosphatase,63 are used to enhance the efficacy of differentiation at stage 1.
During the first phase of hepatogenic differentiation, iPSC cultures undergo substantial morphological and molecular transformation.46,61–63 Dissociation of intercellular contacts prompts formation of more loosely packed associations of spiky shaped cells expressing definitive endoderm markers FoxA2, GATA4, and Sox17 instead of compact clusters typical for iPSCs. The efficacy of endodermal differentiation lies within the 60%–80% range as judged by the emergence of cells expressing definitive endoderm markers.45,46
The second stage of the hepatogenic differentiation of human iPSCs embraces the specification of the definitive endoderm and the start of hepatogenic differentiation per se. Stage 2 can be initiated by the addition of a cocktail comprising HGF, FGF-2/-4, and BMP-2/-4.62,64 The first of the cited publications presents data for human, while the second for porcine iPSC differentiation. Earlier, it has been shown that BMP-4 and FGF-2 play a pivotal role in the process of hepatocytic specification in mouse embryos.65,66 HGF is also crucial for normal liver development. HGF knock-out mice failed to acquire normal liver architecture during embryogenesis and suffered from the disintegration of liver parenchyma.67 HGF and its c-MET receptor are essential for cell proliferation, survival, motility, tissue invasion, and morphogenesis.68 Some authors along with the cocktail of growth factors described above induced the second hepatogenic differentiation stage with chemical substances such as dimethyl sulfoxide and/or beta-mercaptoethanol.61,64
During stage 2, cells continued to proliferate and went through further morphological modification acquiring spindle-like or polygonal epithelium-like shape. Expression of the definitive endoderm markers gradually decreased while the expression of hepatoblast markers hepatocyte nuclear factor 4alpha (HNF-4alpha) and alpha-fetoprotein gradually increased.46,61,62 At this point, 80% of cells were positive for HNF-4alpha.46
During the last phase of the hepatogenic differentiation of iPSCs, hepatoblasts are converted to functionally active hepatocyte-like cells. Oncostatin M (OSM), a member of the cytokine IL-6 family, is the key factor used to guide cells through this stage. In fetal liver, OSM is synthesized by the hematopoietic cells. In the middle of the gestation period, OSM together with glucocorticoids, HGF, and Wnt supports the differentiation of hepatocytes.69–71 In vitro OSM induces the maturation of mouse fetal hepatoblasts manifested by the expression of glucose-6-phosphatase and tyrosine aminotransferase and the accumulation of glycogen.69 OSM stimulates the metabolic maturation of hepatocytes through the activation of gp130 receptor and JAK/Stat3 signal pathway.72 It also assists the morphological maturation of hepatocytes through K-ras activation and formation of E-cadherin-based adherence junctions between fetal cells,73 as well as through the expression of claudin-2 which participates in the establishment of tight junctions enhancing the paracellular barrier function and regulating the permeability of the paracellular zones.74 During hepatogenic differentiation of mouse iPSCs, OSM induces hepatocyte maturation by induction of glucocorticoid production.75 The role of glucocorticoids is further stressed by older works reporting that rat hepatocytes can be maintained in functionally active state and switched from the synthesis of alpha-fetoprotein to albumin production by the synthetic glucocorticoid dexamethasone.76,77
In the course of maturation, hepatocytes acquired features characteristic of mature cells: cuboidal form, numerous vacuoles, and vesicles in the cytoplasm, elevated cytoplasm-to-nucleus ratio, and noticeable well-defined nucleolus. Besides, hepatocyte-like cells derived from iPSCs started to express markers of mature hepatocytes, such as albumin, alpha-1-antitripsin, cytokeratin/s including cytokeratine-8, -18, and -19, different enzymes including cytochrome P450 complex and glutathione S-transferase, and transporters P-glycoprotein 3 and multidrug-resistance protein 1. Studies of the functional maturity showed that these cells secrete high quantities of albumin, utilize urea, and accumulate glycogen.46,47,62,64 The main stages of in vitro iPSC hepatic differentiation are summarized in Table 1.
To our knowledge, no one has studied the effects of the age of the donor of cells used to produce hepatocytes via iPSCs upon the quality of the ensuing hepatocytes. However, as shown in the Generation of iPSCs from differentiated cells taken from donors of various ages section, in humans, cell donor’s age is not pivotal for redifferentiation of iPSCs to fibroblasts and the quality of the resultant fibroblasts. Hence, age may not be a major factor in hepatocyte production via iPSCs as well, though of course this issue has to be addressed experimentally.
Hepatogenic transdifferentiation of mesenchymal stem cells and skin fibroblasts
Two preceding sections describe a two-step procedure of in vitro hepatocyte derivation via the stage of induced pluripotency. At step one, differentiated cells, for example, fibroblasts are converted to pluripotency and at step two, the ensuing pluripotent cells undergo hepatogenic differentiation. As will be shown in this section, probably there are ways of one-step derivation of hepatocytes from adult tissue cells. Bone marrow and other tissues contain minute quantities of pluripotent stem cells throughout life. Many of those cells are of mesodermal origin. They can be isolated from bone marrow, adipose tissue, skin, and stroma of virtually all internal organs along with multipotent (ie, capable of differentiation to different cell types within mesodermal lineage) mesenchymal cells and differentiated fibroblasts and maintained in MSC cultures where they comprise less than 1% of total cell counts. Tissue pluripotent stem cells can be induced to one-step differentiation into the ectoderm and endoderm derivatives including hepatocytes. It is also possible that some partly or even fully differentiated cells can be induced to transdifferentiate into cells of different lineages without full dedifferentiation to embryonic stem cell-like condition.
Since it has been suggested that blood-borne cells of bone marrow origin may be the source of stem cells for liver regeneration,78–82 different marrow cells have been studied as candidates for this role and proved to be able to undergo at least partial hepatogenic differentiation. These include major cell types, namely, MSC78 and hematopoietic cells,81 as well as rare cell species coisolated with stromal cells comprising the so-called multipotent adult progenitor cells82 and very small embryonic-like stem cells.83 It should be noted that since multipotent adult progenitor cells are able to transdifferentiate across the lineage borders (from mesoderm to endoderm), they of course should be named “pluripotent”, not “multipotent”.
Transdifferentiation of MSCs into ectoderm and endoderm derivatives has been repeatedly demonstrated both in vitro and in vivo. Petersen et al79 experimenting with a rat model of liver injury were the first ones to prove formation of oval cells believed to be hepatic stem/progenitor cells from the bone marrow. Schwartz et al82 showed the ability of bone marrow cells to convert into the functional liver cells. Several research groups demonstrated that nonmarrow MSCs isolated from different tissues also can be induced to acquire hepatocyte-like morphology and express hepatocyte marker genes.84–86
Importantly, only a fraction of cultured MSC undergoes differentiation. This fraction probably includes pluripotent stem cells, preexisting in the culture, such as multipotent adult progenitor cells and/or very small embryonic-like stem cells. The other option is genuine transdifferentiation of progenitor cells committed to differentiation into mesodermal lineage derivatives or even of terminally differentiated fibroblasts, also present in the culture. The capacity of hepatogenic differentiation was revealed in MSCs from such sources as adipose tissue,87–89 Wharton’s jelly,90,91 umbilical cord blood,92,93 tooth pulp,94 and many others.
Though evidence concerning molecular pathways involved in MSC transdifferentiation into hepatocytes has been accumulating for more than a decade, it is still limited. MSC transdifferentiation is a complex process regulated by intracellular and external signals. Exact molecular events involved in the conversion of different cell types constituting heterogenous MSC populations may vary in detail. However, activin A, FGF, BMPs, HGF, and OSM are the main external signals switching and maintaining the conversion of any cell type in MSC cultures.41,95,96
Many protocols of MSC transdifferentiation into functionally active hepatocytes have been suggested. The most effective employ consecutive addition of growth factors and other biologically active molecules to the culture medium to achieve partial in vitro reconstruction of the changes in differentiating cell microenvironment occurring in vivo. The most frequently used protocol includes two stages. At stage 1, hepatogenic differentiation is induced by FGF and HGF, while at stage 2 hepatocyte maturation is promoted by OSM.88,90,97,98 The efficacy of hepatogenic differentiation can be enhanced by the addition of dexamethasone, nicotinamide, or insulin–transferrin–selenium. Differentiation is usually carried out in serum-free conditions,88,99 but some effective protocols utilize low serum media.98 The efficiency of hepatocyte formation is higher if cells are initially at the postmitotic stage of the cell cycle achieved in highly confluent (80%–100%) cultures100 or by inhibition of cell proliferation by the addition of epidermal growth factor (EGF) and FGF-2.98
Hepatogenic differentiation of MSCs can be monitored by a standard set of methods. Differentiation-associated changes include morphological transformation, modification of the pattern of expressed genes, and the onset of functional activities characteristic of hepatocytes. In the course of morphological transformation, MSCs acquire polygonal shape, granulated cytoplasm, and tight intercellular contacts.90,97,101 Emergent hepatocyte-like cells start expressing specific marker proteins such as alpha-fetoprotein, albumin, cytokeratine-18 and -19, HNF-4alpha and HNF-1alpha, and others.88,91,102 Alpha-fetoprotein is expressed at the onset of differentiation, while later its expression goes down. Hepatocyte-like cells also express such enzymes as CYP7A1, CYP1A1, CYP2C9, CYP3A4, and NADPH-cytochrome P450 reductase involved in the metabolism of drugs and xenobiotics and synthesis of stearic acid and bile acids.97 It should be noted that in some cases, hepatocyte marker protein expression may be not a very reliable sign of hepatogenic differentiation of MSC. As shown by Campard et al,90 MSCs isolated from the umbilical cord and maintained in standard culture conditions constitutively express hepatocyte markers albumin, alpha-fetoprotein, connexin 32, and cytokeratine-8, -18, and -19.
Microarray analysis of gene expression profile of hepatocytes derived by transdifferentiation of adipose tissue MSCs demonstrated its similarity to the expression profile of adult human hepatocytes.103 Comparison of full-genome expression profiles of adipose tissue-derived MSCs cultivated for 4 weeks in pro-hepatogenic conditions and naïve cells showed activation of genes associated with liver-specific functions including protein metabolism, regulation of the innate immune response, and toxin biodegradation.104 Moreover, mesenchymal line-specific genes were downregulated and epithelium-specific genes upregulated confirming transformation of mesenchymal cells to epithelial state typical for most cells of internal organ parenchyma including hepatocytes.
Functional activity analysis of hepatocyte-like cells derived from MSCs demonstrated secretion of albumin and urea, accumulation of intracellular glycogen and low density lipoprotein uptake.92–94
MSCs from different tissues may be a very convenient starting material for autologous hepatocyte production because of relative simplicity and low cost of the isolation, expansion, and differentiation procedures. Animal experiments have demonstrated that autologous, allogenic, and even xenogenic hepatocyte-like cells from MSCs are able to integrate into adult liver parenchyma.90,105–107
MSCs reside in the stromal and vascular portion of all organs and tissues and can be easily isolated and maintained in vitro due to their plastic adherence and ability to proliferate in conventional culture media.108–111 Remarkably, fibroblasts, though supposed to be fully differentiated cells, can be isolated and cultured in exactly the same conditions. Analysis of morphology, expression of surface markers, and differentiation potential of human MSCs isolated from different sources and “fibroblasts” from skin and liver stroma demonstrated coincidence of many parameters of MSCs and fibroblast cultures.109,112–115 All studied cultures of MSCs and fibroblasts were heterogeneous and, importantly, contained cell subpopulations of varying size differing by the ability to differentiate within (into mesoderm derivatives) and across (into ectoderm and endoderm derivatives) the borders of their primary germ layer. Not surprisingly, some cells showed signs of mesenchymal–epithelial transition revealing the presence of pluripotent cells.115 In vitro cultures of “MSCs” and “fibroblasts” seem to contain the same cell types, but in different proportions: MSC cultures comprise more stem/progenitor cells and fibroblast cultures are stuffed primarily with differentiated cells. Apparently, fibroblast cultures should be similar to MSC cultures as a feasible source of cells for hepatogenic transdifferentiation.
Despite the diversity of tissues from which MSCs and fibroblast cultures for hepatocyte production can be obtained, skin is probably the most suitable source just because of its accessibility. Cultured skin fibroblast-like cells contain subpopulations capable of differentiation within and out of the boundaries of the mesodermal lineage: into adipocytes, osteoblasts, chondrocytes, smooth muscle cells, neurons, astrocytes, and insulin-producing cells.109,112,116–120 Cultures of skin plastic adherent cells contain cells originating from different structures forming this complex organ including dermis, adipose tissue, hair follicles, sweat and sebaceous glands, and others. It should be noted that fibroblast-like cells derived from skin specimens taken from the skull are partly of ectoderm origin because during embryogenesis in this area dermis is formed from the neural crest. The wide assortment of cell types present in the cultures of skin plastic adherent cells ensures the broad spectrum of their possible differentiation routes. Hepatogenic differentiation of some of those cell species has been actually demonstrated.
For instance, Huang et al121 isolated the so-called foreskin-derived fibroblast-like stromal cells (FDSCs) from human foreskin. FDSCs were able to differentiate into adipocytes, osteoblasts, smooth muscle cells, and Schwann cells. They formed spheroids if maintained in Dulbecco’s Modified Eagle’s Medium (DMEM)-F12 with the addition of EGF and FGF-2, while in the absence of growth factors they were growing as a plastic-adherent culture displaying fibroblast-like morphology. Cells from both suspension and plastic-adherent cultures expressed similar spectra of MSC markers being CD90-, CD105-, CD29-, CD44-, SH3-, SH4-, and CD73-positive and CD45- and CD34-negative. Adherent FDSCs expressed less of such embryonic stem cell markers as Oct-4 and E-cadherin, although two types of cultures expressed comparable levels of other embryonic stem cell markers SSEA-1 and SSEA-4. Unlike adhesive cells, spheroid-forming FDSCs expressed the neural crest stem cell marker neurotrophin receptor p75 NTR and hepatocyte markers alpha-fetoprotein and c-Met growth factor suggesting the presence of pluripotent cells and/or cells committed to differentiate along ectodermal or endodermal route. However, not just suspension culture cells but also adhesive cells could be induced to undergo hepatogenic differentiation.122 In the course of transformation, they expressed hepatocyte markers alpha-fetoprotein, albumin, cytokeratine-18 and -19, and CYP3A4. Both cell types got through morphological changes acquiring epithelium-like shape and became able to accumulate glycogen and low density lipoprotein uptake. FDSCs retained differentiation potential for at least 15 passages and after freeze/thaw procedures.
The so-called skin-derived progenitors are another kind of skin cells with high differentiation potential capable of hepatogenic differentiation.116,123 They display fibroblast- like morphology, are cultivated in the presence of EGF and FGF-2, express MSC markers CD29, CD44, CD90, and CD105, and do not express CD14, CD34, CD45, and CD68.124 Unlike FDSCs, skin-derived progenitors do not express p75 NTR.125 In the course of the in vitro differentiation into hepatocytes, skin-derived progenitors consecutively expressed markers of the early and middle phases of hepatogenic differentiation in vivo: CK18, HNF-4alpha, and HNF-1alpha, while no morphological changes occurred. The latter manifested themselves at a more advanced stage when cells acquired polygonal cuboid shape and started to produce albumin.126
Lysy et al127 compared human skin fibroblasts and bone marrow MSCs by their ability to differentiate into the mesoderm (osteoblasts and adipocytes) and endoderm (hepatocytes) derivatives. Skin fibroblasts expressed the pattern of surface markers typical for MSC, had classical fibroblast morphology, and were able to differentiate into osteoblasts and adipocytes confirming their mesodermal origin. After induction of hepatogenic differentiation, both skin fibroblasts and bone marrow MSC acquired hepatocyte-like morphology, started to express liver-specific genes at the transcriptional and translational levels, and accumulated urea. However, some differences between two cultures were revealed. Fibroblasts accumulated less urea than MSCs. Gene expression analysis showed that after hepatogenic differentiation, fibroblasts still remained in the state of mesenchymal–epithelial transition. Finally, fibroblasts retained the capacity for hepatogenic differentiation during three passages, while MSCs during eight passages.
Cell therapy and tissue engineering in the treatment of liver diseases
Human hepatocytes derived from adult cells by reprogramming via iPSCs or by direct transdifferentiation can be used to treat liver pathology applying one of the two existing practical approaches – transplantation of the suspension of hepatocytes (cell therapy) or ex vivo fabrication of the whole liver or its part followed by total or partial surgical substitution of the patient’s liver (tissue/organ engineering) (Figure 1). Both approaches are still in their infancy, but the development of a much simpler method of cell therapy was initiated earlier and diverse cell types and transplantation routes have already been tested in preclinical animal experiments and clinical trials. Initially, hepatocytes tested in preclinical and clinical studies were primary hepatocytes isolated from liver biopsy or autopsy samples, while trials with in vitro processed cells began later.
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Figure 1 Methods of in vitro production of human hepatocytes and biomedical applications utilizing cultured human hepatocytes.
Primary hepatocyte transplantation proved quite effective in the experimental setup128–130 and in the treatment of some liver metabolic disorders, but not acute liver failure or chronic liver disease.131–135 Liver tissue engraftment and participation in its de novo formation is likely to be the major mechanism providing the beneficial effects of primary hepatocytes transplantation. However, other mechanisms, such as paracrine action stimulating tissue regeneration are also involved.
The very limited success of primary hepatocyte transplantation in humans is at least partly related to the inaccessibility of the proper cellular material. Direct isolation of autologous hepatocytes from patient’s liver biopsy is associated with patient’s inconvenience and difficulties obtaining sufficient quantities of viable cells. Therefore, in the clinical context, the only human primary hepatocytes available are poor quality allogeneic cells derived from livers unsuitable for organ transplantation. Lack of reliable sources of primary human hepatocytes remains a major obstacle for their use. Some animal experiments demonstrated effective engraftment of xenogeneic hepatocytes into the liver tissue.136,137 However, the suggested transplantation of animal hepatocytes into humans will hardly be adopted in the near future because of safety concerns.
Further research is underway to improve hepatocyte transplantation methods. Recently, experiments carried out in mice showed very high regeneration-promoting activity of the so-called hybrid periportal hepatocytes residing in portal triads of healthy liver and capable of replenishing the entire chronically damaged parenchyma.138 Importantly, hybrid periportal hepatocytes checked in three disease models exhibited unmatched regeneration-promoting activity, but never originated cancer cells. Now, characterization of analogous human cells and testing their liver regeneration capacity is on the agenda of regenerative hepatology.
Transplantation of MSC by itself produces relief in animal models of liver diseases85,139 and in patients with liver pathology.140–142 Mechanisms underlying the beneficial effects of MSC transplantation may include in situ transdifferentiation of transplanted cells into hepatocytes, paracrine stimulatory action upon resident liver stem cells, or immunomodulation. Liver pathology-related symptoms may also be attenuated by the transplantation of hematopoietic cells143 or their derivatives such as macrophages.144
However, the major part of preclinical and clinical research is conducted with MSCs due to their unique features, such as low immunogenicity, affinity to the sites of ischemia, inflammation or trauma, and ability to modulate immune responses.
Hence, several cell types were tried out in cell therapy of liver diseases. Hepatocytes may well not be the most effective cells, but only autologous hepatocytes fit such applications as liver tissue engineering, underscoring the significance of the development of the methods of their production from adult human cells, either by induction of pluripotency followed by hepatogenic differentiation or direct transdifferentiation of adult stem cells or even terminally differentiated cells.
Liver tissue engineering is still in the initial phase of technological development. The approaches tested so far in animals include obtaining decellularized organ scaffold followed by its repopulation with hepatocytes and other liver cells and bioprinting.135 During decellularization, cells and immunogenic molecules are removed by perfusion of the organ with detergent- and enzyme-containing solutions leaving the extracellular matrix scaffold reproducing organ architecture and detailed 3D microstructure providing adequate framework for blood vessels and biliary ducts.145 The extracellular matrix scaffold can be repopulated by hepatocytes and other liver cells in a bioreactor providing a “neo-organ”. Recellularized liver graft was first successfully transplanted into the rat by Uygun et al.146 The methods were further elaborated147,148 and later included the use of humanized model of porcine liver scaffold and human cells.149
Bioprinting using elements of extracellular matrix and live cells allows precise formation of elaborate 3D tissue and organ structures, including vascular network. This method was successfully applied to make metabolically active 3D hepatic tissue constructs.150,151 At present, 3D bioprinted fragments of liver tissue do not survive for more than a few days. However, this method seems to have great potential for its further development.
New approaches to liver tissue engineering are being developed and important improvements introduced. For example, microencapsulation of hepatocytes before their engraftment into the tissue constructs facilitates long-time survival of functionally intact cells.152 Scaffold-free methods of the assembly of large tissue fragments or whole organs are aimed to totally exclude the use of allogenic biomaterials in organ manufacturing process.153
In addition to donor organ shortage, liver transplantation is restrained by the need to control adverse immune reactions. The use of autologous cells to construct a new organ for transplantation can solve both problems, but has two serious disadvantages. Firstly, building an individual organ for each patient will be very costly. Secondly, it will take at least several months to make a liver or its part from autologous cells and this is not suitable for patients with acute liver failure. Fortunately, recent research developments give hope to produce organs for immunosuppression-free transplantation using allogenic cells. The HLA specificity of an individual is determined by two coexpressed haplotypes, each represented by an HLA-A, HLA-B, and HLA-DR gene. Due to the influence of a number of internal and environmental factors, the combinations of these three genes are not random and there are statistically preferable patterns. It is possible to select homozygous donors with statistically prevalent HLA gene combinations that match substantial numbers of vastly heterozygous potential cell and organ transplantation recipients. Based on these assumptions, Nakatsuji et al calculated that 30 homozygous iPSC lines derived from donors selected from 15,000 Japanese individuals would match 82.2% of potential Japanese recipients, while 50 lines originating from donors selected from 24,000 individuals would raise the score to 90.7%.154 The corresponding figures for UK are the following: 150 selected homozygous HLA-typed volunteers could match 93% of the UK population.155 These findings provided the basis for the idea of a global iPSC lines registry.156 The success of cell therapy of liver diseases and liver tissue engineering depends upon many factors including the age of the cell donor and recipient. As already noted above, biological aging occurs at different levels including cellular level. Throughout lifetime, hepatocytes, fibroblasts, MSCs, hematopoietic cells, macrophages, and other cells mentioned in this review undergo mutually dependent age-related modifications, such as free radical damage to the inner machinery of the cell, shortening of telomeres, accumulation of DNA modifications, alterations of mitochondria functions, and others.7 In stem cells, these changes result in replicative senescence first described in cultivated fibroblasts and widely regarded as a universal tumor-suppressive mechanism.157 Replicative senescence manifests itself by permanent cell cycle arrest, telomere shortening, telomerase-reverse transcriptase dysfunction, irreparable DNA damage, metabolic shift from Krebs cycle toward glycolysis, and finally to cell death. It leads to partial eradication of resident and stem cell populations including those in bone marrow and liver, decline in tissue regeneration capacity, and ultimately causes reduction of parenchymal cell numbers and functional impairment in most organs and tissues including liver.
Biopsies taken from older individuals presumably contain more cells with somatic mutations, chromosome modifications, and fewer stem cells. However, in cell culture, most damaged cells die and cells with short telomeres do not proliferate. This makes the clonal composition of cultured cells population different from clonal composition of cells in originating tissue. Due to elimination of damaged cells, populations of cultured cells seem to become “younger”, at least at first passages. Research in this field is impeded by the lack of an accurate quantitative method of evaluation of the biological age of cells and tissues. The value of the currently prevalent indicator, telomere length, is disputed because it undergoes age-related changes at a different pace in different tissues.158 Recently, Horvath introduced a novel method of evaluation of cell and tissue age based on the studies of cytosine-5 methylation within CpG dinucleotides, also called DNA methylation.159 In this and following studies, DNA methylation showed an excellent correlation with chronological age of cell or tissue donor in most human cell types and tissues, including liver and hepatocytes.159,160 It should be noted that both telomere length and DNA methylation assays show that pluripotency induction in somatic cells is accompanied by the rejuvenation of cells converted into iPSC and shifting of both parameters to the values characteristic of embryonic stem cells.31,159 The situation with rejuvenation of directly transdifferentiated somatic cells is less clear and should be reassessed using more accurate criteria including epigenetic approaches like Horvath’s DNA methylation test.
Conclusion
There is no simple solution to the problems of liver pathology, age-related liver pathology, or age-related diseases in general. Modern medicine still relays primarily on the use of xenobiotic drugs to relieve symptoms. As pointed out by Richard F Walker back in 2006:
[…] a proactive, holistic approach intended to delay onset or avoid development of age-related disease is more logical than a reactive, symptomatic approach.161
Since 2006, there is little change in clinical medicine but huge progress in fundamental research laying the basis for the transformation of the paradigm.
Remarkably, classical donor organ transplantation, including liver transplantation, displays distinct features of a holistic tactic because replacement of heart, kidney, or liver provides coordinated normalization of a number of crucially important homeostasis parameters. Cell-based technologies including cell therapy and tissue/organ engineering offer further advancement of transplantation methodology with much better control over the quality of transplanted material. In addition, research in this field, particularly stem cell and differentiation/dedifferentiation studies, deliver better understanding of aging and age-linked pathology. There are still many questions to be answered and many basic and technical problems to be solved. But it is already clear that cell-based therapies will play an increasingly important role in the development of novel methods of the management of age-related issues. Studies of age-related aspects of the control of liver diseases using cell-based technologies are in their initial phase, but the prospects of cell therapy and organ engineering utilizing rejuvenated hepatocytes produced from somatic cells via iPSC or by direct transdifferentiation seem quite promising.
Production of liver or liver lobes for transplantation is not likely in near future, but achievable in a longer prospective. Autologous or HLA-matched human iPSCs are probably the most convenient source of cell material. Currently, this technological trend attracts investment and in some countries enjoys support from national and local governments. Concerted studies in the field of iPSC technology-based human tissue and organ engineering currently conducted in Japan already deliver results. Masayo Takahashi group of the RIKEN Center for Developmental Biology in Kobe, Japan, was the first to carry out a clinical study of an iPSC-based technology using retinal pigment epithelium cells obtained by differentiation of iPSC derived from autologous dermal fibroblasts.162 Transplantation of retinal pigment epithelium cell sheet into the subretinal space of an elderly woman with age-related moist macular degeneration did not produce serious adverse effects and resulted in partial vision restoration.163 Serious efforts are focused on the engineering of kidneys and methods designed in these studies may be applied to engineering of livers as well.164
At present, California Institute of Regenerative Medicine Human Pluripotent Stem Cell Repository, the largest human iPSC bank, holds just 300 human iPSC lines which is of course very far from quantities needed to provide cellular material for immunosuppression-free therapy. However, several companies started commercial iPSC and derivative production and biobanking. Among these, ReproCELL, Kanagawa, Japan, offers hepatocytes produced from iPSC lines. There are a number of national and international initiatives to create cell repositories and registers big enough to provide starting material to develop immunosuppression-free transplantation technologies.
Transplantation of hepatocytes, MSC, or hematopoietic cells delivered encouraging results in many animal models and some clinical trials. Unfortunately, not enough has been done to reveal the differences in the reactions of younger and older animals or patients with liver pathology to cell transplantation. This is a very important subject for further studies. Taking into account that aging starts early in life and manifests itself at varying chronological age,1 every successful clinical tactic aimed to reverse age-related changes actually contributes to geriatrics. However, to accurately evaluate these approaches, better ways of age assessment at the cellular and tissue levels are needed. The recently introduced method of age estimation at the epigenomic level159 supplements the traditional practice based on telomere length measurements and the combination of those two approaches may provide a more accurate measure of cell and tissue senescence and its reversal. It fully relates to liver pathology management. Liver is an organ with a complex tissue architecture which is difficult to reproduce ex vivo. However, existing approaches including repopulation of decellularized cadaveric human liver scaffolds with hepatocytes and other cells and bioprinting have good chances to be transferred to practical medicine within a decade or two,135,164 while new developments will be arriving.
Acknowledgment
This work was supported by the Russian Science Foundation (grant number 14-15-00648).
Disclosure
The authors report no conflicts of interest in this work.
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abstract: 'We develop an open-system dynamical theory of the Casimir interaction between coherent atomic waves and a material surface. The system — the external atomic waves — disturbs the environment — the electromagnetic field and the atomic dipole degrees of freedom — in a non- local manner by leaving footprints on distinct paths of the atom interferometer. This induces a non-local dynamical phase depending simultaneously on two distinct paths, beyond usual atom-optics methods, and comparable to the local dynamical phase corrections. Non-local and local atomic phase coherences are thus equally important to capture the interplay between the external atomic motion and the Casimir interaction. Such dynamical phases are obtained for finite-width wavepackets by developing a diagrammatic expansion of the disturbed environment quantum state.'
author:
- 'François Impens$^{1,2}$, Claudio Ccapa Ttira$^{2}$, Ryan O. Behunin$^{3,4,5}$, and Paulo A. Maia Neto$^{2}$'
date:
-
-
title: 'Dynamical local and non-local Casimir atomic phases'
---
INTRODUCTION
============
The interplay between the internal atomic dynamics and the electromagnetic (EM) field retardation, brought to light by the pioneering work of Casimir and Polder [@CasimirPolder], is crucial to understand the atom-surface dispersive interaction in the long-distance limit (see [@Intravaia] for a recent review). In contrast, the effect of the external atomic motion on the dispersive interaction is almost always discarded. Notable exceptions are the quantum friction effects resulting from the shear relative motion between two material surfaces [@QuantumFrictionPP] or between an atom and a surface [@QuantumFrictionAP; @Scheel09].
Since the usual atomic velocities are strongly non-relativistic, one might expect the dynamical corrections to the dispersive atom-surface interaction to be very small. Because of their high sensitivity, atom interferometers [@Cronin09; @Kasevich07] are ideal systems for probing such small corrections. There is a growing interest in developing atom interferometers able to probe surface interactions. Measurements of the van der Waals atom-surface interaction with standard atom interferometry have already been achieved [@Cronin04; @CroninVigue; @Lepoutre11], while optical-lattice atom interferometry offers even more promising perspectives to measure the Casimir-Polder interaction in the long-distance regime [@FORCAGpapers].
From a fundamental point of view, the coherent atomic waves evolving in the vicinity of a material surface constitute a particularly rich open quantum system: the external atomic waves, playing the role of the system, interact with an environment involving both long-lived (atomic dipole) and short-lived (EM field) degrees of freedom (dofs). In this paper, we develop an open-system theory of atom interferometers in the vicinity of a material surface. We show that the atomic motion relative to the surface along the interferometer paths gives rise to a non-local dynamical phase correction associated to pairs of paths rather to individual ones as in usual interferometers. In contrast to the local dynamical phase contributions, the non-local dynamical phases may be distinguished from other quasi-static phase contributions in a multiple-path atom interferometer [@MultiplePathAtomInterferometer] since they violate additivity [@NonAdditiveCasimir].
Preliminary results for extremely narrow wavepackets were derived in a previous letter [@DoublePath] from the influence functional [@FeynmanVernon] capturing the net effect of the environment on the atomic center of mass (external) dynamics [@Ryan10; @Ryan11]. The atomic phases were then calculated in terms of closed-time path integrals [@CalzettaHu].
Here we use instead standard perturbation theory to investigate the more realistic case of finite-width wavepackets, allowing us to connect with the van der Waals interferometer experiments [@CroninVigue]. We explicitly calculate the disturbance of the environment [@SAI90] produced by the interaction with the external dofs in the atom interferometer. Since the perturbation is of second-order, the changes of the environment state involves two atomic “footprints”, which can be left either on the same path, or on distinct paths. Provided that the dipole memory time is longer than the time it takes for light to propagate between the two arms, the diagrams for which the atomic waves have “one foot on each path” yield cross non-local phase contributions. For atoms flying parallel to the plate, these cross contributions cancel each other exactly. Otherwise, the differential atomic motion between the two interferometer arms brings into play an asymmetry between the cross-talk diagrams, thanks to the finite velocity of light and the breaking of the translational invariance by the surface. The resulting non-local phase contribution is of the same order of magnitude of the dynamical local corrections. Non-local phase coherences are thus required in a consistent description of dynamical effects in Casimir atom interferometry.
Our formalism also allows for the analysis of the decoherence effect in interferometers [@Barone; @Hackermueller04; @Breuer01; @Lamine06] in the presence of a conducting plane [@CasimirDecoherence; @Sonnentag07]. The analysis of the path-dependent disturbance of the environment provides a clear-cut approach to the derivation of decoherence [@SAI90], which was employed in the derivation of the dynamical Casimir decoherence for neutral macroscopic bodies [@Dalvit00]. Alternatively, the decoherence effect can be obtained from the modulus of the complex influence functional [@Mazzitelli03], which depends on the imaginary part of the environment-induced phase shift. However, here we focus on the real part of the Casimir phase shift, which has been measured experimentally for neutral atoms [@CroninVigue], in contrast with the loss of contrast in the fringe pattern, which has been probed only in the case of charged particles [@Sonnentag07]. Environment-induced phase shifts were also considered in the context of geometrical phases for spin one-half systems [@GeometricPhase].
We shall proceed as follows. In Sec. \[sec:local dynamical Casimir phases\], we develop a local dynamical theory of Casimir atom interferometers, inspired by the atom-optical $ABCD$ formalism [@BordeABCD], and show its consistency with the standard phase obtained from the dispersive potential in the quasi-static limit. In the following sections, we go beyond this heuristic treatment by considering the disturbance of the environment quantum state by the interaction with the external atomic waves, first in the simpler case of point-like wave-packets in \[sec:Feynman diagrams\] and then for finite-width wave-packets in \[sec:finite width\]. This treatment reveals the appearance of dynamical non-local atomic phase coherences in addition to the local contributions already obtained in Sec. \[sec:local dynamical Casimir phases\]. Explicit results for the case of a perfectly-reflecting plane surface are derived in Sec. \[section:relativistic expansion\] and concluding remarks are presented in Sec. \[section:conclusion\].
LOCAL DYNAMICAL THEORY OF CASIMIR PHASES {#sec:local dynamical Casimir phases}
========================================
In this section, we develop a local theory of a Mach-Zehnder atom interferometer in interaction with a material surface (see Fig. \[fig:atom interferometer\] for a typical example). In contrast to the idealized point-like model discussed in Ref.[@DoublePath], the derivation below fully captures the influence of the wave-packet finite width, making our discussion relevant for atom interferometers with large wave-packets, such as those employed in the recent experiments reported in Refs.[@CroninVigue].
In the usual closed-system approach, the atom-surface interaction phase is given by the integration of an external dispersive potential taken at the instantaneous atomic position. Obviously, this standard approach is completely quasi-static – the potential seen by the atoms depends only on their instantaneous position distribution, but not on their velocity. Here, we perform instead a first-principle derivation of this phase based on the interaction energy stored within the quantum dipole and EM field dofs. While capturing non-trivial local relativistic corrections, this treatment yields predictions in agreement with the standard dispersive potential approach when considering the quasi-static limit.
![(color online). Atom interferometer interacting with a conducting plate at $z=0$ during the time $T,$ with the arm $k=1$ parallel to the plate (distance $z_0$) and the arm $k=2$ flying away with a normal velocity $v_{\perp}.$ []{data-label="fig:atom interferometer"}](fig1){width="8.5cm"}
The atomic wave-function is initially a coherent superposition $| \psi_E (0) \rangle = \frac {1} {\sqrt{2}} \left( |\psi_{E}^1(0) \rangle + | \psi_{E}^2(0) \rangle \frac {} {} \right) $ of two wave-packets with the same central position but with different initial momenta. These wave-packets will follow two distinct paths $k=1,2$ as illustrated in Fig. \[fig:atom interferometer\]. The relative phase between these two wave-packets, which determines the local atomic probability function $p(\mathbf{r},t)=|\psi_E(\mathbf{r},t)|^2$, contains contributions from the atom-surface interaction as well as additional ones independent of the surface.
As in Ref. [@DoublePath], we extend the atom-optics $ABCD$ formalism [@BordeHouches; @BordeABCD; @AtomLaserABCD] by including the symmetrized [@DalibardRocCohen] interaction energy $U^{\rm int}_k(t)$ between the atomic dipole and the EM field within the action phase associated to the external atomic propagation along path $k.$ The atom-surface interaction, assumed weak enough to leave unaltered the shape of the atomic wave-packets during the propagation, results merely in atomic phase shifts.
We evaluate $U^{\rm int}_k(t)$ using linear response theory [@WylieSipe], i.e to lowest order in perturbation theory, and then obtain the local Casimir phase $\varphi^{\rm loc}_k= - \frac {1} {\hbar} \int_0^T dt U_k^{\rm int,S}(t)$ along a given path $k$ by picking the surface-dependent contribution $ U_k^{\rm int,S}(t)$ to the total interaction energy. The key ingredient in our derivation is the introduction of an “on-atom field” operator $\hat{\mathbf{E}}(\hat{\mathbf{r}}_a),$ for which the field argument is the atomic position operator $\hat{\mathbf{r}}_a$ instead of a classical position $\mathbf{r}_k(t)$ taken along the central atomic path $k$.
In the Heisenberg picture, the dipole and the on-atom electric field operators can be expressed as the sum of an unperturbed free-evolving part, defined as $\hat{\mathbf{O}}^f(t)= \exp \left( i \hat{H}_0 t/\hbar \right) \hat{\mathbf{O}}(0) \exp \left( -i \hat{H}_0 t/\hbar \right) $ with the free Hamiltonian $\hat{H}_0=\hat{H}_E+\hat{H}_D+\hat{H}_F$ including the external ($H_E$), internal ($H_D$) and EM field ($H_F$) dofs, and of a contribution $\hat{\mathbf{O}}^{in}(t)$ induced by the atom-field coupling $\hat{H}_{AF} = - \hat{\mathbf{d}} \cdot \hat{\mathbf{E}} (\hat{\mathbf{r}}_a).$ To describe the mutual influence between the atomic dipole and the ‘on-atom’ EM field [@WylieSipe], we introduce temporal correlation functions for the corresponding operators. We also introduce four-point correlation functions for the quantized electric field as discussed below.
Precisely, the dipole and field fluctuations are captured by symmetric correlation functions (also refered to as Hadamard Green’s functions) of the free-evolving operators $\hat{\mathbf{O}}^f=\hat{\mathbf{d}}^f(t), \hat{\mathbf{E}}(\hat{\mathbf{r}}_a)^f(t), \hat{\mathbf{E}}^f(\mathbf{r},t)$ ($\{...\}$ denotes the anti-commutator): $$\begin{aligned}
\label{eq:Hadamard Green functions}
G^{H}_{\hat{\mathbf{O}}, \: ij}(x;x') = \frac 1 \hbar \langle \{ \hat{O}_i^{f}(x), \hat{O}_j^{f}(x') \} \rangle. \label{Hadamard}
\end{aligned}$$ For the dipole and on-atom field operators $\hat{\mathbf{O}}=\hat{\mathbf{d}}, \hat{\mathbf{E}}(\hat{\mathbf{r}}_a)$ the arguments in (\[Hadamard\]) are two instants $(x;x') \equiv (t,t')$. For the electric field operator $\hat{\mathbf{O}}=\hat{\mathbf{E}},$ these arguments are two four-vectors $(x;x') \equiv (\mathbf{r},t;\mathbf{r}',t').$
The linear susceptibilities (polarizability for the dipole), generically written as retarded Green’s functions, describe the linear response of field and dipole to dipole and field perturbations, respectively: $$\begin{aligned}
\label{eq:retarded Green functions}
G^{R}_{\hat{\mathbf{O}}, \: ij}(x;x') = \frac {i} {\hbar} \theta(t-t') \langle [ \hat{O}_i^{f}(x), \hat{O}_j^{f}(x') ] \rangle
\end{aligned}$$ with $\theta(t-t') $ denoting the Heaviside step function.
Note that the on-atom field Green’s functions as defined by (\[eq:Hadamard Green functions\]) and (\[eq:retarded Green functions\]) are still quantum operators in the Hilbert space corresponding to the atomic external dofs, since the average is taken over the EM field dofs only. We now take the average $\langle \: \mathcal{G}_{\hat{\mathbf{E}}(\hat{\mathbf{r}}_a)}^{R,H}(t,t') \: \rangle_k $ over the external quantum state $ | \psi^k_{E} \rangle $ corresponding to the single atomic wave-packet $k$. We express the result in terms of the atomic wave-functions $\psi_E^k(\mathbf{r},t)=\langle \mathbf{r} | e^{ - \frac i \hbar \hat{H}_E t} | \psi_E^k(0) \rangle, $ of the external atomic propagator $$\label{eq:definition atomic propagator}
K(\mathbf{r},t;\mathbf{r}',t')= \langle \mathbf{r} | e^{- \frac {i} {\hbar} H_E (t-t')} | \mathbf{r}' \rangle,$$ and of the electric field Green’s functions. For this purpose, we switch to the Schrödinger picture with respect to the external atomic dofs: $\hat{\mathbf{E}}(\hat{\mathbf{r}}_a)(t) = e^{\frac {i} {\hbar} H_E t} \hat{\mathbf{E}}(\hat{\mathbf{r}},t) e^{-\frac {i} {\hbar} H_E t} $ with $\hat{\mathbf{r}} = \hat{\mathbf{r}}_a(0)$ the atomic position operator, and $\hat{\mathbf{E}}(\mathbf{r},t)$ the quantized electric field (Heisenberg-evolved with respect to the Hamiltonian $H_F$) at the classical position $\mathbf{r}$ and time $t$. Using closure relations for the external atomic dofs, one obtains $$\begin{aligned}
\label{eq:on atom Green function wide atomic wave packets}
\langle \mathcal{G}_{\hat{\mathbf{E}}(\hat{\mathbf{r}}_a)}^{R(H)}(t',t) \rangle_k & = & \! \iint \! d^3\mathbf{r} d^3\mathbf{r}'
\psi_E^{k *}(\mathbf{r},t) K(\mathbf{r},t;\mathbf{r}',t') \psi_E^k(\mathbf{r}',t') \nonumber \\
& & \qquad \qquad \times \mathcal{G}_{\hat{\mathbf{E}}}^{R(H)}(\mathbf{r},t;\mathbf{r}',t') \, .
\end{aligned}$$
It is necessary to identify the physically relevant contributions of the field response (and fluctuations) as far as the atom-surface interaction is concerned. By isotropy of the atomic dipole, only the trace of the electric field Green’s functions $\mathcal{G}^{R (H)}_{\hat{\mathbf{E}}}(x;x') \equiv \sum_i G^{R (H)}_{\hat{\mathbf{E}} \: i i}(x;x')$ (with the sum performed on the Cartesian index $i=1,2,3$) is needed to obtain the interaction energy. $\mathcal{G}^{R (H)}_{\hat{\mathbf{E}}}(x;x')$ is the sum of free-space and scattering contributions: $$\label{0S}
\mathcal{G}^{R (H)}_{\hat{\mathbf{E}}}(x;x') = \mathcal{G}^{R (H),0}_{\hat{\mathbf{E}}}(x;x') +
\mathcal{G}^{R (H),S}_{\hat{\mathbf{E}}}(x;x')$$ By symmetry the free-space contributions $ \mathcal{G}^{R (H),0}_{\hat{\mathbf{E}}}({\bf r},t;{\bf r}',t') $ depends only on $|{\bf r}-{\bf r}'|$ and $t-t'$ [@Heitler], whereas the scattering contribution $ \mathcal{G}^{R (H),S}_{\hat{\mathbf{E}}}({\bf r},t;{\bf r}',t')$ can be written in terms of the image of the source point ${\bf r}'$ in the particular case of a planar perfectly-reflecting surface discussed in Sec. IV. More specifically, the free-space retarded Green’s function $\mathcal{G}^{R,0}_{\hat{\mathbf{E}}}({\bf r},t;{\bf r}',t') $ represents the direct propagation from ${\bf r}'$ to $\bf r$ and does not depend on the distance to the material surface, whereas the scattering contribution $\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}({\bf r},t;{\bf r}',t') $ corresponds to the propagation with one reflection at the surface.
When replacing (\[0S\]) into (\[eq:on atom Green function wide atomic wave packets\]), the average on-atom field Green’s functions also split into free-space and scattering contributions, and only the latter contributes to the atom-surface interaction energy $U_k^{\rm int,S}(t)$ and hence to the local Casimir phase $\varphi^{\rm loc}_k.$ The latter is derived by following steps similar to those employed for point-like wave-packets and using expression (\[eq:on atom Green function wide atomic wave packets\]) with the field Green’s function replaced by the scattering contribution $\mathcal{G}^{R (H),S}_{\hat{\mathbf{E}}}({\bf r},t;{\bf r}',t'):$ $$\begin{aligned}
\label{eq:local phase general wide atomic packets}
\varphi^{\rm loc}_k \!& = & \!\frac {1} {4} \! \iint_{0}^{T} \! d t dt' \! \iint \! d^3\mathbf{r} d^3\mathbf{r}' \! \psi_E^{k *}(\mathbf{r},t) \! K(\mathbf{r},t;\mathbf{r}',t')\! \psi_E^k(\mathbf{r}',t') \nonumber \\
& \times & \left[ g_{\hat{d}}^H(t,t') \mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},t;\mathbf{r}',t')
+ g_{\hat{d}}^R(t,t') \: \mathcal{G}_{\hat{\mathbf{E}}}^{H,S}(\mathbf{r},t;\mathbf{r}',t') \right]. \nonumber \\
\end{aligned}$$ with $ g_{\hat{d}}^{R(H)}(t,t') $ representing any diagonal component of the isotropic atomic dipole Green’s function $G^{R(H)}_{\hat{\mathbf{d}}, \: ii}(t,t').$ The two contributions appearing in (\[eq:local phase general wide atomic packets\]) correspond to the separate physical effects responsible for the atom-surface dispersive interaction: radiation reaction and field fluctuations [@Meschede90; @Mendes]. The former, proportional to the field retarded Green’s function, dominates in the van der Waals un-retarded short-distance limit and is of particular relevance in the following sections. Physically, it represents the self-interaction between the fluctuating dipole at time $t$ and position $\bf r$ with its own electric field, produced at an earlier time $t'$ and position ${\bf r}',$ after bouncing off the material surface. This interpretation provides an indication that a cross non-local interaction might also exist, with the field produced at one wave-packet component propagating to a different wave-packet component, as discussed in detail in the following sections.
As a first check of (\[eq:local phase general wide atomic packets\]), we consider the limit of very narrow wave-packets in order to compare with Ref. [@DoublePath]. We assume that the wave-packet width is much shorter than the relevant EM field wave-lengths, and then approximate the position arguments of the Green’s functions $\mathcal{G}_{\hat{\mathbf{E}}}^{(R)H,S}(\mathbf{r},t;\mathbf{r}',t')$ by the central atomic positions $\mathbf{r}_k(t)$ and $\mathbf{r}_k(t')$ taken along the trajectory $k$ at the respective times $t,t'.$ In this case, we can isolate the atomic propagation integral $\psi_E^k(\mathbf{r},t)= \int d^3 \mathbf{r}' K(\mathbf{r},t;\mathbf{r}',t')\! \psi_E^k(\mathbf{r}',t')$ in (\[eq:local phase general wide atomic packets\]) and find $$\begin{aligned}
\label{eq:local phase general expression narrow wavepackets}
\varphi^{\rm loc}_k \! & \approx & \!\frac {1} {4} \! \iint_{0}^{T} \! d t dt' \!
\left[ \frac {} {} g_{\hat{d}}^H(t,t') \mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(r_k(t) , r_k(t') ) \right. \\
& \: & \left. \qquad \qquad \qquad \qquad + g_{\hat{d}}^{R}(t,t') \: \mathcal{G}_{\hat{\mathbf{E}}}^{H,S} (r_k(t),r_k(t') ) \frac {} {} \right]. \nonumber\end{aligned}$$ in agreement with Ref. [@DoublePath].
A second, more important limiting case of Eq. (\[eq:local phase general wide atomic packets\]), corresponds to its quasi-static limit. We also assume thermal equilibrium for the dipole and EM field dofs, and consider long interaction times (stationary regime). In this case, the dipole and electric field Green’s functions depend only on the time difference $\tau=t-t'$ and not on the individual times. The retarded Green’s functions $\mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\tau;\mathbf{r}', 0)$ is non-zero only for a time delay $\tau$ equal to the time it takes for a photon to travel from the source position $\mathbf{r}'$ to the position $\mathbf{r}$ after one reflection at the surface. These durations are, in usual experimental conditions, much shorter than the time scales associated with the external atomic motion. In the quasi-static limit, we treat the external atomic motion as completely “frozen” during the time delay $\tau=t-t'$. In other words, we take $t' := t$ in the external atomic propagator and wave-functions. In this limit, the former simplifies to $K(\mathbf{r},t;\mathbf{r}',t) = \delta (\mathbf{r}-\mathbf{r}')$. The resulting expression can be directly compared with the formula for the dispersive atom-surface potential $V_{\rm Cas}({\bf r})$ [@WylieSipe] as detailed in the Appendix. We then find that the local phase becomes a time integral of the dispersive potential taken at the instantaneous atomic position weighted by the external probability density: $$\label{eq:nonrelativstic limit 5 wide}
\varphi^{\rm loc }_k \approx - \frac {1} {\hbar} \int_{0}^{T} d t \int d^3\mathbf{r} \: |\psi_E^k(\mathbf{r},t)|^2 \: V_{\rm{Cas}}(\mathbf{r}).$$
The quasi-static expression (\[eq:nonrelativstic limit 5 wide\]) was employed as the theoretical model for comparison with experiments [@Cronin04; @CroninVigue; @Lepoutre11]. On the other hand, our more general result (\[eq:local phase general wide atomic packets\]) allows for non-equilibrium [@Ryan11; @Antezza] and non-stationary regimes which cannot be described by the more standard expression (\[eq:nonrelativstic limit 5 wide\]). Explicit results for the dynamical corrections to order ${\bf \dot r}_k(t)/c$ were derived in Ref. [@NonAdditiveCasimir] in the case of very narrow atomic packets flying close to a perfectly-reflecting planar surface. Note, however, that we also find non-local atomic phase corrections to order ${\bf \dot r}_k(t)/c.$ Thus, a full quantum open system approach, to be developed in the next sections, is required to assess the first-order dynamical correction in a consistent way.
NON-LOCAL DYNAMICAL CASIMIR ATOMIC PHASES {#sec:Feynman diagrams}
=========================================
From now on, we no longer model the effect of surface interactions as a local phase shift imprinted on each external atomic wave-packet. We consider instead the evolution of the full quantum state describing the external atomic waves, atomic dipole and EM field. In the discussion to follow, we will refer respectively to the dipole and EM field dofs as the “environment” and to the external atomic waves as the “system”. We consider the case of point-like wave-packets in this section, so as to introduce our method in a simpler setting, thus paving the way for the discussion of finite-width wave-packets in the following sections.
We describe here how the quantum state of the environment is affected by the propagation of the external atomic waves. Because it involves the center-of-mass position operator $\hat{\mathbf{r}}_a$, the dipolar Hamiltonian $\hat{H}_{AF}= - \hat{\mathbf{d}} \cdot \hat{\mathbf{E}} (\hat{\mathbf{r}}_a) $ operates on the environment in a manner which depends on the path followed by the atoms. Thus, such a Hamiltonian acts as a “which-path” marker, leaving an atomic “footprint” on the dipole and EM field quantum states. The phase contribution is of second order in the dipolar interaction Hamiltonian. A Feynman-diagram expansion shows that these footprints actually contain cross terms, involving the two coherent components of the external atomic state propagating on two distinct arms of the interferometer (see Fig. \[new\]). As discussed in detail below, such terms reflect a non-local disturbance of the environment operated at different times by the system. In addition to a loss of contrast in the fringe pattern, such perturbation also induces a non-local double-path atomic phase coherence. We derive here both the local and non-local phases resulting from the influence of the environment. The local phase shifts obtained below correspond exactly to the atom-surface interaction phases and derived in the previous section for finite-width and point-like wave-packets, respectively, whereas the non-local phases cannot be derived from the interaction energy along the different paths taken separately.
![(color online). Double-path footprint left on the environment (dipole + EM field) by the external atomic state through the dipolar interaction $\hat{H}_{AF}.$ []{data-label="new"}](fig2){width="8.5cm"}
In Ref. [@DoublePath], we have briefly outlined an alternative approach, based on the influence functional, which captures the effect of the environment on very narrow atomic waves as a complex phase which can also be recast as a stochastic phase [@Ryan11]. This method leads to the same final results we derive in this section. The equivalence between the two points of views illustrates an important property of open systems [@SAI90]: its evolution is equally well described by considering the accumulation of a stochastic phase, or by analyzing the trace left by the system onto the quantum state of the environment.
Atomic interferences in presence of an environment
--------------------------------------------------
Inspired by Ref. [@SAI90], we calculate the time evolution of the full quantum state, which is initially given by $| \psi (0) \rangle = \frac {1} {\sqrt{2}} \left( | \psi_E^1(0) \rangle + | \psi_E^2(0) \rangle \right) \otimes | \Psi_{DF}(0) \rangle,$ where $| \Psi_{DF}(0) \rangle= | \psi_D(0) \rangle \! \otimes \! | \psi_F(0) \rangle $ denotes the initial environment (internal dipole and EM field) quantum state. We discard the influence of the atom-surface interaction on the external atomic motion (prescribed atomic trajectories), which is a very good approximation in usual experimental conditions [@CroninVigue]. In this section, we assume, for simplicity, that the wave-packet width is much smaller than the relevant field wavelengths (more general results are derived in the following sections). Thus, the interaction is described by the Hamiltonians $ \hat{H}_{AF}(\mathbf{r}_k(t))= - \hat{\mathbf{d}} \cdot \hat{\mathbf{E}} (\mathbf{r}_k(t)) $ parametrized by the wave-packet trajectories represented by the four-vectors $r_k(t)\equiv (\mathbf{r}_k(t),t)$ with $k=1,2$, and acting only on the dipole and EM field Hilbert spaces [@foot_Rontgen]. We work in the interaction picture and the transformed time-dependent interaction Hamiltonian reads $$\label{eq:interaction Hamiltonian Heisenberg}
\hat{\widetilde{H}}_{AF}(r_k(t)) = e^{ \frac {i} {\hbar} ( \hat{H}_D + \hat{H}_F) t } \left(-\hat{\mathbf{d}} \cdot \hat{\mathbf{E}} (\mathbf{r}_k(t)) \right) e^{ - \frac {i} {\hbar} ( \hat{H}_D + \hat{H}_F) t }.$$ At time $t=T$, the full quantum state reads $$\begin{aligned}
| \psi(T) \rangle & =& \frac {1} {\sqrt{2}} | \psi_E^1(T) \rangle \! \otimes \! \mathcal{T} e^{ - \frac {i} {\hbar} \int_0^T dt \hat{\widetilde{H}}_{ AF}(r_1(t)) } | \Psi_{DF}(0) \rangle
\nonumber \\
& + & \frac {1} {\sqrt{2}} | \psi_E^2(T) \rangle \! \otimes \! \mathcal{T} e^{ - \frac {i} {\hbar} \int_0^T dt' \hat{\widetilde{H}}_{ AF}(r_2(t))} \! | \Psi_{DF}(0) \rangle,\nonumber \\
\label{psi}\end{aligned}$$ where $\mathcal{T} $ denotes the time-ordering operator.
Since the dipole and EM field states are not measured in the experiment, we calculate the external reduced density operator $\rho= {\rm Tr}_{DF} \left(| \psi(T) \rangle \langle \psi(T) |\right).$ When replacing (\[psi\]) into this equation, the cross (interference) term represents the external atomic coherence, which we evaluate in the position representation: $$\label{eq:definition coherence rho12}
\rho_{12}({\bf r},{\bf r}';T) = \frac12\,\langle {\bf r}| \psi_E^1(T) \rangle \langle \Psi_{DF}^2(T)|\Psi_{DF}^1(T)\rangle \langle \psi_E^2(T) | {\bf r}' \rangle$$ Thus, the interference term $\rho_{12}^{(0)}=\frac12 \psi_E^2(\mathbf{r}',T)^*\, \psi_E^1(\mathbf{r},T) $ is now multiplied by the scalar product of the disturbed environment states $$\begin{aligned}
\label{eq:environment quantum states product}
\langle \Psi_{DF}^2(T) | \Psi_{DF}^1(T) \rangle & \equiv &e^{i \Phi_{12}}. \end{aligned}$$ The complex phase $\Phi_{12}$ captures the environment effect on the external interference term accumulated over the interaction time $T$: $$\begin{aligned}
\label{eq:environment quantum states product}
e^{i \Phi_{12}} &= &
\langle \Psi_{DF}(0) | \widetilde{\mathcal{T}} e^{ \frac {i} {\hbar} \int_0^T dt \hat{\widetilde{H}}_{ AF}(r_2(t))} \nonumber \\
&& \times \mathcal{T} e^{ - \frac {i} {\hbar} \int_0^T dt \hat{\widetilde{H}}_{ AF}(r_1(t))} \! | \Psi_{ DF}(0) \rangle\end{aligned}$$ with $\widetilde{\mathcal{T}} $ denoting the anti-time-ordering operator (earlier-time operators on the left).
In general the final environmental quantum states have a scalar product smaller than unity $ |\langle \Psi^2_{DF }(T)
| \Psi^1_{DF}(T) \rangle| =e^{-{\rm Im} \Phi^{E}_{12}} < 1,$ leading to an attenuation of the interferometer fringe pattern. In this case, the full quantum state $|\psi(T)\rangle$ given by (\[psi\]) is entangled, indicating the transfer of which-path information on the atomic motion to the environment. The resulting decoherence has been theoretically studied [@CasimirDecoherence] and measured [@Sonnentag07] for charged particles close to a material surface. Here we focus on the complementary effect that is also present in the general formula (\[eq:environment quantum states product\]) for the complex phase $\Phi_{12}.$ In addition to the loss of fringe visibility, the coupling with the dipole and EM field dofs also leads to a displacement of the interference fringes, corresponding to the real part ${\rm Re}\,\Phi_{12}, $ which we analyze in more detail in the remaining part of this paper.
Diagrammatic expansion of the environment-induced phase
-------------------------------------------------------
As in the previous section, we follow a linear response approach and treat the dipolar coupling as a small perturbation. Thus, we perform a diagrammatic expansion of the time-ordered (and anti-time-ordered) exponentials appearing in the the formula (\[eq:environment quantum states product\]) for the environment-induced complex phase $\Phi_{12}$ . We focus on the lowest-order diagrams yielding a finite phase. Special care is required, since the dipolar coupling Hamiltonians $\hat{\widetilde{H}}_{AF}(r_k(t))$ taken at different times do not commute. We calculate $\Phi_{12}$ to first order in the atomic polarizability, allowing us to approximate $e^{i \Phi_{12}} \simeq 1+i \Phi_{12} .$ This is a valid approximation as long as the distance between the atom and the plate is much larger than the atomic size (this assumption also justifies the electric dipole approximation).
It follows from (\[eq:environment quantum states product\]) that first-order diagrams are proportional to ($\langle ... \rangle_0$ denoting the average over the intial environment state $| \Psi_{DF}(0) \rangle$) $$\pm \frac {i} {\hbar} \int_0^T dt \, \langle\, \hat{\mathbf{d}}(t) \cdot \hat{\mathbf{E}}({\bf r}_k(t)) \,\rangle_0.$$ and as a consequence vanish since the the atom has no permanent dipole moment.
Thus, we focus on second-order diagrams, which are quadratic in the EM field and dipole operators. There are two different ways to build second-order diagrams from Eq. (\[eq:environment quantum states product\]): one can either take two interactions pertaining to the same time-ordered (or anti-time-ordered) exponential, or one may take one interaction from each exponential. Diagrams of the first kind correspond to a sequence of interactions along the same path, and are referred to as “single-path” (SP) diagrams. Diagrams of the second kind involve simultaneously two distinct paths, and are thus called “double-path” (DP) diagrams. The two contributions sum up to give the complex environment-induced phase $\Phi_{12}= \Phi^{\rm SP}_{12}+\Phi^{\rm DP}_{12} $.\
### Phase contribution of local single-path diagrams
We consider first the two possible SP diagrams, beginning with the diagram arising from the time-ordered exponential evaluated along the path $1$ in the r.-h.-s. of (\[eq:environment quantum states product\]), whose contribution reads: $$\Phi^{\rm{SP}}_{1 } = \! \frac {i} {\hbar^2} \! \int_0^T dt \! \int_0^t dt' \! \sum_{i,j}\langle \: \hat{d}_i(t) \hat{d}_j(t') \hat{E}_i(r_1(t)) \hat{E}_j(r_1(t')) \: \rangle_0$$ where we sum over the Cartesian indices $i,j=1,2,3$. In order to express the phase $\Phi^{\rm{SP}}_{1 }$ in terms of dipole and electric field Green’s functions (\[Hadamard\],\[eq:retarded Green functions\]), we write the product of dipole (or electric field) operators at distinct times (or space-time points) as the half sum of their commutator and anti-commutator. As in Sec. II, these contributions can be expressed in terms of the scalar dipole $g^{R(H)}_{\hat{d}}(t,t')$ and the trace of the electric field Green’s function $\mathcal{G}^{R(H)}_{\hat{\bf E}}(x;x').$ For the latter we take only the scattering contribution $\mathcal{G}^{R(H),S}_{\hat{\bf E}}(x;x')$ \[see Eq. (\[0S\])\] and then find that ${\rm Re}\, \Phi^{\rm{SP}}_1 = \varphi^{\rm loc}_1$ is precisely the local phase obtained in Sec. \[sec:local dynamical Casimir phases\] for point-like wave-packets.
An analogous SP diagram comes from the anti-time ordered exponential along path $2$ in the r.-h.-s. of (\[eq:environment quantum states product\]), yielding a similar contribution $\Phi^{\rm{SP}}_{2}$ to the complex phase. The reversed time-ordering leads to an additional minus sign in front of each retarded dipole and electric field Green’s functions appearing in the expression for the complex phase. Since ${\rm Re}\,\Phi^{\rm{SP}}_{2}$ contains an odd number of retarded Green’s functions, we find ${\rm Re}\, \Phi^{\rm{SP}}_2 = -\varphi^{\rm loc}_2$ with the local phase $\varphi^{\rm loc}_2$ given again by . Thus, the total contribution of single-path diagrams has a real part $$\label{eq:phase SP}
{\rm Re}\,\Phi_{12}^{\rm SP} = \varphi^{\rm loc}_1-\varphi^{\rm loc}_2$$ Since ${\rm Re}\,\Phi_{12}$ represents the phase coherence of path 1 with respect to path 2, it must be anti-symmetric with respect to the interchange of the two paths. This property is clearly satisfied by the local contribution (\[eq:phase SP\]), and will also hold for the non-local double-path contribution discussed in the following. On the other hand, the imaginary part ${\rm Im}\,\Phi_{12},$ representing decoherence, must be symmetric with respect to the interchange, with both local path contributions being positive and thus leading to an attenuation of fringe pattern. This property is also satisfied by the result derived from (\[eq:environment quantum states product\]) since ${\rm Im}\,\Phi_{12}$ contains an even number of retarded Green’s functions.
In short, the local approach developed in Section \[sec:local dynamical Casimir phases\] provides the correct expressions for the real part of the single-path contributions to the complex phase $\Phi_{12}.$ However, it is unable to yield even the single-path contributions to the imaginary part of $\Phi_{12},$ which represents the decoherence effect. More importantly, the local theory also misses all double-path phase contributions, which we discuss in the remaining part of this section.
### Phase contribution of the non-local double-path diagram
We investigate here the double-path diagram, which involve a product of linear terms issued from both the time-ordered and anti-time-ordered exponentials in the r.-h.-s. of (\[eq:environment quantum states product\]): $$\begin{aligned}
i \Phi^{\rm{DP}}_{12 } & = & \! \Biggl\langle \:\sum_{i,j} \left( \frac {i} {\hbar} \! \int_0^T dt' \hat{d}_i(t') \hat{E}_i(r_2(t')) \right) \nonumber \\
& \: & \times \left( \frac {-i} {\hbar} \int_0^T dt \hat{d}_j(t) \hat{E}_j(r_1(t)) \right) \: \: \Biggr\rangle_0
\label{DP_pre}\end{aligned}$$ As previously, we express the product of two dipole and EM field operators as the half sum of their commutators and anti-commutators. After summing over the Cartesian indices $i,j$ and discarding the contributions from the free-space electric field Green’s functions, we find for the real part $\phi^{\rm{DP}}_{12 } \equiv {\rm Re}\,\Phi^{\rm{DP}}_{12 }$
$$\begin{aligned}
\nonumber
\phi^{\rm{DP}}_{12 } \! & \! = \! & \! \frac {1} {4} \! \! \iint_0^T \! dt' dt \left[ \frac {} {} g_{\hat{d}}^H(t,t') \left( \mathcal{G}_{\hat{\mathbf{E}}}^{R, S} (r_1(t),r_2(t')) - \mathcal{G}_{\hat{\mathbf{E}}}^{R, S} (r_2(t),r_1(t')) \right) + g_{\hat{d}}^{R}(t,t') \left( \mathcal{G}_{\hat{\mathbf{E}}}^{H, S} (r_1(t),r_2(t')) -
\mathcal{G}_{\hat{\mathbf{E}}}^{H, S} (r_2(t),r_1(t')) \right) \right]\\
\label{eq:double path phase}\end{aligned}$$
As required for consistency, the r.-h.-s. of (\[eq:double path phase\]) is anti-symmetrical under the interchange of the two paths, since $\phi^{\rm{DP}}_{12 }$ represents a contribution to the relative phase of path 1 with respect to path 2. Remarkably, this relative phase contribution depends simultaneously on the two distinct paths of the atom interferometer and cannot be split into separate contributions from paths 1 and 2.
The non-negligible contribution to the non-local phase $\phi^{\rm{DP}}_{12 }$ actually comes entirely from the term proportional to $ g_{\hat{d}}^H(t,t')$ in Eq. (\[eq:double path phase\]), which accounts for the long-lived atomic dipole fluctuations. Eq. (\[eq:double path phase\]) shows that the non-local phase results from the asymmetry between the cross self-interactions involving different wave-packets — the fluctuating dipole interacting with the electric field sourced by itself at a different location [@DoublePath].
DYNAMICAL CASIMIR PHASES FOR FINITE-SIZE WAVE-PACKETS {#sec:finite width}
=====================================================
The previous derivation of the dynamical Casimir phases for point-like atomic wave-packets highlighted the basic physical mechanisms behind the appearance of a non-local double-path Casimir phase. However, usual experimental conditions in Casimir interferometry [@Cronin04; @CroninVigue; @Lepoutre11] do not match this assumption, since the width of the atomic wave-packets are of the same order of the atom-surface distances. In this section, we present a derivation of the dynamical local and non local Casimir phases for finite-width wave-packets.
As in the previous section, we consider the interaction picture. However, we no longer consider the interaction Hamiltonian as parametrized by well-defined atomic trajectories. Instead, we now evolve the interaction Hamiltonian with respect to the external atomic dofs associated to the Hamiltonian $\hat{H}_E$, i.e. the time-dependent interaction Hamiltonian can be expressed as a function of the free-evolving dipole $\hat{\mathbf{d}}(t)$, free-evolving electric field $\hat{\mathbf{E}} ( \mathbf{r},t )$ and initial time position operator $\hat{\mathbf{r}}_a$ as $\hat{\widetilde{H}}_{AF}(t)=e^{ \frac {i} {\hbar} \hat{H}_E t } \left[-\hat{\mathbf{d}}(t) \cdot \hat{\mathbf{E}} ( \hat{\mathbf{r}}_a,t ) \right] e^{ -\frac {i} {\hbar} \hat{H}_E t } $. Again, we consider the coherence of the reduced density matrix $\rho_{12}(\mathbf{r},\mathbf{r}',t)$ between the two wave-packets $\psi_E^1(\mathbf{r},t)$ and $\psi_E^2(\mathbf{r}',t)$, related to the free-evolving density matrix coherence $\rho^0_{12}(\mathbf{r},\mathbf{r}';T)=\frac12\psi_E^1(\mathbf{r},t)\psi_E^{2*}(\mathbf{r}',t)$ by $\rho_{12}(\mathbf{r},\mathbf{r}';T)=\rho^0_{12}(\mathbf{r},\mathbf{r}';T)e^{i \phi_{12} (\mathbf{r},\mathbf{r}',T)}$. For a small interaction phase $\phi_{12} (\mathbf{r},\mathbf{r}';T)$, a first-order Taylor expansion yields $ \phi_{12} (\mathbf{r},\mathbf{r}';T) \simeq (-i) \delta \rho_{12}(\mathbf{r},\mathbf{r}';T) / \rho^0_{12}(\mathbf{r},\mathbf{r}';t)$. We have introduced the difference between the free and interacting density matrix coherences $\delta \rho_{12}(\mathbf{r},\mathbf{r}';T)=\rho_{12}(\mathbf{r},\mathbf{r}';T)-\rho^0_{12}(\mathbf{r},\mathbf{r}';T)$, determined below in terms of second-order dipolar interaction diagrams. We also define the average interaction phase coherence $\phi_{12} (T)\equiv \iint d^3 \mathbf{r} d^3 \mathbf{r}' |\psi_E^{1}(\mathbf{r},T)|^2 |\psi_E^{2}(\mathbf{r}',T)|^2 \phi_{12} (\mathbf{r},\mathbf{r}';T)$, equivalently expressed as $$\label{eq:definition average phase}
\phi_{12} (T)= -2i \int d^3 \mathbf{r} \int d^3 \mathbf{r}' \psi_E^{1*}(\mathbf{r},T) \psi_E^{2}(\mathbf{r}',T) \;\delta\rho_{12}(\mathbf{r},\mathbf{r}';T).$$ At the time $T$, the reduced density matrix can be formally expressed as $$\begin{aligned}
\nonumber
\rho_{12}(\mathbf{r},\mathbf{r}';T) & = & \frac12\langle \psi_{DF}(0) | \otimes \langle \psi_E^2(0) | \widetilde{\mathcal{T}} \left[ e^{ \frac {i} {\hbar} \int_0^T dt \hat{\widetilde{H}}_{ AF}(t) } \right] \\
& \: & \times \left( e^{ \frac i \hbar H_E T} | \mathbf{r}'\rangle \langle \mathbf{r} | e^{- \frac i \hbar H_E T} \otimes {\bf 1}_{DF}\right)
\label{eq:atomic density wide nonperturbative1}
\\
& \: & \times \mathcal{T} \left[ e^{ - \frac {i} {\hbar} \int_0^T dt' \hat{\widetilde{H}}_{ AF}(t') } \right]|\psi_{E}^1(0) \rangle \otimes |\psi_{DF}(0) \rangle \nonumber\end{aligned}$$
Let us first investigate the SP paths terms, which correspond to contributions to $\delta \rho_{12}(\mathbf{r},\mathbf{r}',T)$ arising from quadratic terms issued from the same time-ordered (or anti-time ordered) exponential. One considers without loss of generality the SP phase associated with path $1$, which yields the contribution: $$\begin{aligned}
& & \delta\rho_{12}^{\rm SP1}(\mathbf{r}_1,\mathbf{r}_2;T) =
\frac {i} {2\hbar^2} \psi_E^{2*}(\mathbf{r}_2,T) \sum_{i,j=1}^3 \int_0^T dt \int_0^{t} dt' \nonumber \\
& \times & \int d^3\mathbf{r}\int d^3\mathbf{r}' K(\mathbf{r}_1,T;\mathbf{r},t) K(\mathbf{r},t;\mathbf{r}',t') \psi_E^1( \hat{\mathbf{r}}',t') \nonumber \\
& \times & \langle \widetilde{\psi}_{DF}(0) | \hat{d}_i(t) \hat{E}_i ( \mathbf{r},t ) \hat{d}_j(t') \hat{E}_j ( \mathbf{r}',t' ) |\widetilde{\psi}_{DF}(0) \rangle \nonumber
\end{aligned}$$ When taking the average of $\delta\rho_{12}^{\rm SP1}$, one recognizes an integral involving the external atomic propagator , leading to the Casimir phase obtained previously with the local theory.
On the other hand, one derives the DP phase from Eq. by considering the diagrams composed of linear terms issued from both the time-ordered and anti-time ordered exponentials: $$\begin{aligned}
\label{eq:atomic probability function4}
& & \delta\rho_{12}^{\rm DP}(\mathbf{r},\mathbf{r}';T) = \frac12 \sum_{i,j=1}^3
\int d^3 \mathbf{r} \int d^3 \mathbf{r}' \langle \psi_{DF}(0) | \nonumber \\ & \: & \left[
\frac {i} {\hbar} \int_0^T dt' \psi_{E}^{2*}(\mathbf{r}',t')
K (\mathbf{r}',t'; \mathbf{r}_2,T) \hat{d}_i(t') \hat{E}_i ( \mathbf{r}',t' ) \right] \nonumber \\
& \: & \left[ - \frac {i} {\hbar} \int_0^T dt \hat{d}_j(t) \hat{E}_j ( \mathbf{r},t )
K (\mathbf{r}_1,T;\mathbf{r},t) \psi_{E}^1(\mathbf{r},t) \right] |\psi_{DF}(0) \rangle. \nonumber
\end{aligned}$$ The averaging procedure yields a double-path phase which depends simultaneously on the histories of the two wave-functions corresponding to each interferometer arm. As in Section \[sec:Feynman diagrams\], we express the bilinear averages of the dipole and field operators in terms of Hadamard and retarded Green’s functions: $$\begin{aligned}
\label{eq:DP phase wide atomic packet general}
\phi^{\rm{DP}}_{12 }(T) & = & \frac {1} {4} \iint_0^T dt dt' \iint d^3 \mathbf{r} d^3 \mathbf{r}' \left| \psi_{E}^1(\mathbf{r},t) \right|^2 |\psi_{E}^{2}(\mathbf{r}',t')|^2 \nonumber \\
&\times & \left[ \frac {} {} g_{\hat{d}}^H(t,t') \left( \mathcal{G}_{\hat{\mathbf{E}}}^{R, S} (\mathbf{r},t;\mathbf{r}',t' ) - \mathcal{G}_{\hat{\mathbf{E}}}^{R, S} (\mathbf{r}',t';\mathbf{r},t ) \right) \right. \nonumber \\
& + & \left. g_{\hat{d}}^{R}(t,t') \left( \mathcal{G}_{\hat{\mathbf{E}}}^{H, S} (\mathbf{r},t;\mathbf{r}',t' ) -
\mathcal{G}_{\hat{\mathbf{E}}}^{H, S} (\mathbf{r}',t;\mathbf{r},t' ) \right) \frac {} {} \right] \nonumber \\\end{aligned}$$
If one considers that the electric field Green’s functions are uniform over the width of atomic wave-packets, one obviously retrieves the nonlocal DP phase of Section \[sec:Feynman diagrams\] obtained in the narrow atomic wave-packet limit. In order to highlight the dependence of the DP phase on the dynamical atomic motion, we Taylor expand the advanced time wave-function $ | \psi_{E}^k(\mathbf{r},t) |^2 \simeq | \psi_{E}^k(\mathbf{r},t') |^2 + \frac {\partial} {\partial t} | \psi_{E}^k(\mathbf{r},t') |^2\tau $ in Eq. (\[eq:DP phase wide atomic packet general\]). This is an excellent approximation since the time $\tau=|{\bf r}-{\bf r}'|/c$ corresponds to the light propagation between the dipole and its image, and is thus extremely short compared to the typical time scale of the external atomic motion. As before, we assume a stationary regime and write $g^{R,H}_{\hat{d}}(\tau)\equiv
g^{R,H}_{\hat{d}}(t'+\tau,t')$. Using the conservation of the atomic probability, one can express the DP phase (\[eq:DP phase wide atomic packet general\]) in terms of the probability current $\mathbf{j}^k(\mathbf{r},t) = \mbox{Re} \left[ \psi_E^{k*}(\mathbf{r},t) \frac {\hbar} {i m} \nabla \psi_E^k(\mathbf{r},t) \right]$: $$\begin{aligned}
\nonumber
& & \phi^{\rm{DP}}_{12 }(T) = \frac {1} {4 } \sum_{i=1}^3 \int_0^T dt' \int_0^{T-t'} d \tau \iint d^3 \mathbf{r} d^3 \mathbf{r}' \\
& \times & \left( \frac {} {} j_i^1(\mathbf{r},t') | \psi_{E}^2(\mathbf{r}',t') |^2
- j_i^2(\mathbf{r},t') | \psi_{E}^1(\mathbf{r}',t') |^2 \frac {} {} \right) \: \tau \nonumber \\
& \times & \left( \frac {} {} g_{\hat{d}}^H(\tau) \frac {\partial \mathcal{G}_{\hat{\mathbf{E}}}^{R, S} (\mathbf{r},t'+\tau;\mathbf{r}',t' )} {\partial r_i} \right. \nonumber \\
& \: & \left. \frac {} {}+ g_{\hat{d}}^{R}(\tau) \frac {\partial \mathcal{G}_{\hat{\mathbf{E}}}^{H, S} (\mathbf{r},t'+\tau;\mathbf{r}',t' )} {\partial r_i} \right). \label{eq:DP phase current probability}
\end{aligned}$$ The non-local DP phase is thus a dynamical phase correction, with the current density giving the probability density evolution during the very short electromagnetic propagation time $\tau.$ In the next section, we investigate in greater detail the phases acquired by wide wave-packets flying close to a planar perfectly-reflecting surface.
NON-LOCAL DYNAMICAL CORRECTIONS TO THE VAN DER WAALS PHASE FOR A PLANE SURFACE {#section:relativistic expansion}
==============================================================================
In this section, we derive explicit results for the non-local dynamical contributions to the Casimir phase, working at the leading order in $v/c$ ($v$ denotes the magnitude of the atomic center-of-mass velocity). Starting from the general results of Sec. \[sec:finite width\], we describe such corrections for wide atomic packets interacting with a perfectly-reflecting planar surface, located at $z=0.$ Moreover, we shall consider specifically the short-distance van der Waals (vdW) regime probed by the experiments [@Cronin04; @CroninVigue; @Lepoutre11], which corresponds to a stronger atom-surface interaction (thus yielding larger dynamical phase corrections) than the long-distance Casimir-Polder limit. As discussed in Section \[sec:local dynamical Casimir phases\], at these distances the dominant dynamical vdW phase contributions come from the electric field response to dipole fluctuations. The experiments were performed for wide atomic wave-packets filling in the gap between the central trajectory and the conducting plate [@Cronin04; @CroninVigue; @Lepoutre11]. In this case, we show here that the non-local DP phase is enhanced with respect to the result for point-like packets [@DoublePath] by a logarithmic factor.
We take a Mach-Zehnder atom interferometer in the half-space $z>0$ close to the material surface at $z=0$ as illustrated by Fig. \[fig:atom interferometer\]. The two central atomic trajectories share the same velocity component parallel to the plate, but have arbitrary normal velocities: $$\label{eq:atomic trajectories}
\mathbf{r}_k(t)= \mathbf{r}_{0 / \! \! /} (t)+ z_k(t) \,\hat{\mathbf{z}}, \quad k=1,2.$$ The results to follow can be extended to discuss dynamical vdW phase corrections resulting from atomic interactions with a grating as in Refs. [@Cronin04; @CroninVigue; @Lepoutre11].
Electric field and dipole Green’s functions
-------------------------------------------
It is necessary, at this stage, to have at hand explicit expressions for the dipole and electric field Green’s functions. As discussed in Section II, the electric field Green’s functions is decomposed as the sum of free and scattering contributions. Only the latter is relevant for the derivation of the Casimir phases induced by the surface. We first derive the field Green’s functions in Fourier space by writing the electric field operator as a sum over normal modes, taking due account of the perfectly-reflecting surface at $z=0.$ We then derive both the known result for the free-space Green’s function [@Heitler] as well as the scattering contribution $$\begin{aligned}
\label{G_sca}
\mathcal{G}_{\hat{\mathbf{E}}}^{R, S}(x,x') & =
& \frac {\theta(\tau)} {2 \pi\epsilon_0} \frac {\partial^2} {\partial z \partial z'} \left( \frac {\delta ( \tau-|\mathbf{r}-\mathbf{r}'_{\rm I}|/c )} {|\mathbf{r}-\mathbf{r}'_{\rm I}|} \right)
\end{aligned}$$ As expected $ \mathcal{G}_{\hat{\mathbf{E}}}^{R, S}(x,x') $ depends on the time difference $\tau=t-t'$ only and not on the individual times. It is written in terms of the propagation distance $|\mathbf{r}-\mathbf{r}'_{\rm I}|$ between the point $\bf{r}$ and the image $\mathbf{r}'_{\rm I}=(x',y',-z')$ of the source point ${\bf r}'=(x',y',z')$ with respect to the plane surface. Assuming the EM field to be in thermal equilibrium, the electric field Hadamard Green’s function $ \mathcal{G}_{\hat{\mathbf{E}}}^{H, S} (x,x')$ can be obtained from the retarded one thanks to the fluctuation-dissipation theorem.
In order to obtain the dipole Green’s functions, we model the internal atomic degrees of freedom as an harmonic oscillator with a transition frequency $\omega_0$ (and wave-length $\lambda_0$) and assume the atom to be in its ground state. The Hadamard dipole Green’s function is then proportional to the static atomic polarizability $\alpha(0):$ $$\label{eq:dipole Hadamard Green}
g^{H}_{\hat{d}}(t,t')= \alpha(0)\, \omega_0 \,\cos[\omega_0(t-t')].$$
Nonlocal dynamical phases
-------------------------
We consider the limit of wide atomic packets with a well-defined momentum, which is well-suited to describe the dispersion effects associated to the finite width of the atomic packets propagating nearby the plate. In this limit, one may take the probability current involved in the DP path phase as $\mathbf{j}^k(\mathbf{r},t) \simeq |\psi_E^{k}(\mathbf{r},t)|^2 \mathbf{v}_k(t)$ where $\mathbf{v}_k(t)={\bf\dot r}_k(t)$ is a classical velocity [@RemarkWidePacketApprox]. Since the DP phase depends sharply on the distance between the atoms and the conductor and not on their lateral position above this surface, the extension of the atomic wave-packets in the direction $O_z$ normal to the conducting surface is much more critical than the extension of the atomic packets along the directions $O_x,O_y$ parallel to the conductor. Thus, one can safely use one-dimensional atomic wave-packets $\psi_E^{1,2}(z,t)$ in order to model dispersion effects in the nonlocal DP phase acquired by wide atomic beams.
We first model the atomic wave-functions by a step-wise distribution centered on the classical atomic trajectories of time-independent width, i.e. we take $|\psi_{k}^E(z,t)|^2=1/w$ for $z_k(t)-w/2 < z < z_k(t)+w/2$ and zero for $|z-z_k(t)| \geq w/2$ – with a width $w$ such that $w \leq 2 z_0$ where $z_0=z_1(0)=z_2(0)$ is the initial distance between the atomic wave-packet centers and the plate. Naturally, such description is a simple approximation, and a modelling in terms of Gaussian wave-packets would be more accurate. Nevertheless, this approach should yield the correct qualitative picture and has the advantage of giving analytical expressions regarding the dependence of the DP phase towards the wave-packet width.
We calculate the DP phase in the short-distances vdW regime and take $g_d^H(\tau) \approx g_d^H(0) = \omega_0 \alpha(0)$ \[see (\[eq:dipole Hadamard Green\])\]. We consider the linear trajetories , and assume that the distance between the central trajectory endpoints is much larger than the initial altitude $z_0$, yielding the saturation limit of the DP phase [@DoublePath]. Using the step wave-functions in Eq., one obtains an expression for the DP phase taking into account the finite atomic packet extension: $$\phi^{\rm DP}_{12}(z_0,w) = - \frac {3\pi} {\lambda_0} \frac {\alpha(0)} {4\pi \epsilon_0} \frac{1}{w^2} \log \left( 1 - \frac {w^2} {4 z_0^2} \right)$$ When taking the limit $w \ll z_0$ in this expression, one retrieves the DP phase obtained in [@DoublePath] for classical trajectories. On the other hand, the phase $\phi^{\rm DP}_{12}(z_0,w)$ diverges when the wave-packet width $w$ approaches $2 z_0$, i.e. when the edge of the atomic wave-function becomes close to the plate. This suggests that a greater care is needed to evaluate this phase when considering atomic wave-functions which do not vanish at the plate boundary, where the vdW potential becomes infinite.
Indeed, the divergence above is a consequence of our perturbative approach, jointly with the the small phase approximation $e^{i \phi^{\rm DP}_{12}} \simeq 1+ i\phi^{\rm DP}_{12}$, which obviously breaks down at the close vicinity of the plate (dispersion interaction models in general are valid only for distances much larger than the atomic length scale). Fortunately, this divergence can be easily cured, since such contributions lead to quickly oscillating complex exponentials which in fact barely affect the average vdW phase [@Cronin04; @Lepoutre11]. To make our argument more precise, we reintroduce these exponentials in our derivation of the average dynamical phase $\Phi^{\rm{DP}}_{12 }(T)$: $$\begin{aligned}
|A| e^{i \Phi^{\rm{DP}}_{12 }(T)} & = & \int d z^0_1 d z^0_2 |\psi_E^1(z^0_1,0)|^2 |\psi_E^2(z^0_2,0)|^2 \label{eq:nonperturbative DP phase} \\
& \: & \qquad \qquad \qquad \qquad \times e^{i \phi_{12}^{\rm DP}(z^0_1,z^0_2,T)} \nonumber\end{aligned}$$ with the phase $$\begin{aligned}
& & \phi_{12}^{\rm DP}(z^0_1,z^0_2,T) = \frac 1 4 \int_0^T dt' \int_0^{T-t'} d\tau g_{\hat d}^H(\tau) \tau \nonumber \\
& \: & \times \left( v_{1 \: z}(t') - v_{2 \: z}(t') \right) \frac {\partial} {\partial z} \mathcal{G}^R_E \left( \frac {} {} z_1(t') \mathbf{\hat z} ,
t'+\tau;z_2(t')\mathbf{\hat z},t' \frac {} {} \right) \nonumber\end{aligned}$$ and $z_k(t')=z^0_k+\int_0^{t'} dt'' v_{k \: z}(t'').$ We have omitted the common displacement of the atomic wave-packets parallel to the plate on both trajectories thanks to the translational invariance of the field Green’s function along this direction. Using the vdW regime and the saturation limit, and following Ref. [@DoublePath], one finds $$\label{eq:phi DP origin z01 z02}
\phi_{12}^{\rm DP}(z^0_1,z^0_2,T) = \frac {3 \pi } {\lambda_0 } \left( \frac {\alpha(0)} {4 \pi \epsilon_0 } \right) (z^0_1+z^0_2)^{-2}$$.
Eqs.(\[eq:nonperturbative DP phase\],\[eq:phi DP origin z01 z02\]) are the starting point of the derivation to follow. We consider initial atomic wave-functions filling in the gap between the central atomic position and the material surface, taking again a step wave-function approach with this time $w=2 z_0$.
Under the above approximations and following the averaging procedure of Refs. [@Cronin04; @Lepoutre11], one derives the average DP phase $ \tan \phi^{\rm DP}_{12}(w)= I_s/ I_c $ with $I_s = (w_c^2/ 2 w^2) \int_{w_c^2 /w^2}^{+\infty} d \phi \phi^{-2} \sin ( \phi )$ and $I_c = (w_c^2/ 2 w^2) \int_{w_c^2 /w^2}^{+\infty} d \phi \phi^{-2} \cos ( \phi ).$ We have introduced a critical length scale associated with the DP phase $w_c=[\frac {3 \pi } {\lambda_0} \left(\alpha(0)/(4 \pi \epsilon_0)\right)]^{1/2}.$ The distance $r_{\alpha}=[\alpha(0)/(4 \pi \epsilon_0)]^{1/3}$ represents the atomic length scale and is of the order of the Angström. Thus, the length $w_c = \sqrt{3 \pi} r_{\alpha} (r_{\alpha}/\lambda_0) ^{1/2}$ is always several orders of magnitude smaller than any experimentally achievable atomic packet width $w$. Thus, one may keep only the lowest-order quadratic terms in the small parameter $w_c/w$, taking $I_c \simeq 1$ and $$\phi^{\rm DP}_{12}(w) = \frac {3 \pi } {\lambda_0} \left(\frac {\alpha(0)} {4 \pi \epsilon_0}\right) \frac {1} {w^2} \ln \left( \frac {w} {w_c} \right) + O\left(\frac {w_c^4} {w^4}\right)$$ A comparison with the results for point-like packets following identical central trajectories [@DoublePath] shows that wide atomic beams experience an enhancement of the DP phase by a factor $ \ln \left( \frac {w} {w_c} \right).$ Considering $\:^{87} \mbox{Rb}$ atoms and a wave-packet width $w = 40 \: {\rm nm}$ (and thus $z_0 = w_0/2 = 20 \: {\rm nm}$) compatible with the parameters used in the Casimir experiments [@Cronin04; @CroninVigue; @Lepoutre11] for the wave-packets, one obtains a DP phase $\phi^{\rm DP}_{12 \: w} \simeq 3 \times 10^{-6} \: {\rm rad},$ corresponding to an enhancement of roughly one order of magnitude.
CONCLUSION {#section:conclusion}
==========
Using standard perturbation theory, we have addressed dynamical corrections, arising from the external motion, to the Casimir phase acquired by neutral atoms interacting with a material surface. A careful description of retardation effects, combined with the atomic motion, reveals the appearance of a non-local atomic phase coherence, which involves simultaneously a pair of atomic paths instead of a single atomic trajectory as usual in atom optics.
By construction, the non-local phase for a given pair of paths must be anti-symmetric with respect to the interchange of the two paths in the pair. In fact, it results from the difference between the EM propagation distances from one path to the other one after one reflection at the surface. Thus, it vanishes when the two path motions with respect to the plate are symmetrical (as for instance in the case of trajectories parallel to plate). In other words, the symmetry between the two paths is broken by the velocity components normal to the surface and the non-local phase is proportional the difference between the two velocity components of a given pair.
In a previous work [@DoublePath], we had obtained a preliminary estimation of the non-local double-path phase for point-like atomic wave-packets using an independent and less intuitive method based on the influence functional. Here we have obtained these dynamical Casimir phases by keeping track of the quantum state of the environment – the EM field and the atomic dipole degrees of freedom. This treatment provides us with an interesting open-system interpretation of this double-path atomic phase coherence, by showing that it results from a non-local disturbance of the environment by a coherent superposition of external atomic waves propagating across two distinct atomic paths. The approach developed here also corresponds to more realistic experimental conditions, since it takes into account the atomic dispersion in position around the central path, which is relevant for the estimation of the vdW phase [@CroninVigue]. The corresponding general expressions, written in terms of Green’s functions for the field and atomic internal dofs, and of the atomic probability current and wave-functions, are in principle valid for arbitrary geometries and non-equilibrium conditions. We have also derived explicit analytical results for a perfectly-reflecting planar surface in the short-distance regime. In this regime, our treatment reveals a significant enhancement of the non-local DP phase acquired by wide atomic packets with respect to our previous estimation based simply on classical atomic trajectories.
Both the local and non-local dynamical atomic Casimir phases are first-order relativistic corrections arising from the external atomic motion, and thus of similar magnitude. This shows that the relativistic corrections to the Casimir phase are intrinsically non-local.
The authors are grateful to Reinaldo de Melo e Souza for stimulating discussions. This work was partially funded by CNRS (France), CNPq, FAPERJ and CAPES (Brazil).
QUASI-STATIC LIMIT OF THE LOCAL ATOMIC PHASE
============================================
Here, we assume that the field is in thermal equilibrium, and we consider the regime of long atom-surface interaction times, namely we take an atomic time-of-flight $T$ above the conductor much larger than the atomic dipole or field correlation time scales. In this regime, we show that the non-relativistic contribution to the local Casimir phase of Section \[sec:local dynamical Casimir phases\] reduces to the standard phase arising from a dispersive (Casimir) potential. Taking the quasi-static limit of Eq. , one obtains $$\begin{aligned}
\label{phi_local_qs}
\varphi^{\rm loc}_k & \approx & \! \frac 1 4 \int_{0}^{T} \! d t' \int \! d^3\mathbf{r} |\psi_E^k(\mathbf{r},t)|^2 \\
& & \times \int_0^t d\tau \left[ \frac {} {} g_{\hat{d}}^H(\tau) \mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r};\tau)
+ g_{\hat{d}}^R(\tau) \: \mathcal{G}_{\hat{\mathbf{E}}}^{H,S}(\mathbf{r},\mathbf{r};\tau) \frac {} {} \right] \nonumber\end{aligned}$$ We have assumed that the dipole and field fluctuations are stationary in order to write $g_{\hat{d}}^{R(H)}(\tau) \equiv g_{\hat{d}}^{R(H)}(t+\tau,t)$ and $\mathcal{G}_{\hat{\mathbf{E}}}^{R(H),S}(\mathbf{r},\mathbf{r};\tau) \equiv \mathcal{G}_{\hat{\mathbf{E}}}^{R(H),S}(\mathbf{r},t+\tau;\mathbf{r},t) $.
In the equation above, we focus on the integral over the delay $\tau$, whose bounds can be extended to infinity in the regime of large atom-surface interaction times. Using the Parseval-Plancherel relation, we express the local phase in the Fourier domain as follows $$\begin{aligned}
\label{eq:nonrelativstic limit3}
& & \varphi^{\rm loc }_k \approx \frac {1} {8 \pi} \int_{0}^{T} d t \int \! d^3\mathbf{r} |\psi_E^k(\mathbf{r},t)|^2 \\
& \times & \int d \omega \left( \! g_{\hat{d}}^R(\omega) \: \mathcal{G}_{\hat{\mathbf{E}}}^{H,S *}(\mathbf{r},\mathbf{r};\omega) + \mathcal{G}_{\hat{\mathbf{E}}}^{R,S}(\mathbf{r},\mathbf{r}; \omega) g_{\hat{d}}^{H *}(\omega) \right) \nonumber
\end{aligned}$$ The Fourier transform of the Green’s function is defined as: $$\begin{aligned}
g^{R(H)}_{\hat{d}}(\omega) = \int_{-\infty}^{+\infty} d\tau g_{\hat{d}}^{R(H)}(\tau) e^{i \omega \tau} \nonumber \end{aligned}$$ and likewise for $\mathcal{G}^{R(H),S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega).$
Our next step is to express the dispersive potential as a similar frequency integral. We assume that the electric field and dipole dofs are at thermal equilibrum at temperature $\Theta.$ One starts with the general expression derived in Ref. [@WylieSipe]: $$\label{eq:potentialCPgeneralexpression}
V_{\rm{Cas}}(\mathbf{r}) \! = \! - \frac {\hbar} {2 \pi} \! \int_{0}^{+\infty} \! \! \! \! d \omega \, \coth \left( \frac {\hbar \omega} {2 k_B \Theta} \right)
{\rm{Im}} \left[ \! g^R_{\hat{d}}(\omega) \mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega) \! \right]$$ where $k_B$ is the Boltzmann constant. In order to cast (\[eq:potentialCPgeneralexpression\]) in the form of Eq. , we use the fluctuation-dissipation theorem (FDT): $$\begin{aligned}
g^H_{\hat{d}}(\omega) \! & \! = \! & \! 2 \coth \left( \frac {\hbar \omega} {2 k_B \Theta} \right) \mbox{Im} \left[ g^R_{\hat{d}}(\omega) \right] \\
G^{H,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega) \! & \! = \! & \! 2 \coth \left( \frac { \hbar \omega} {2 k_B \Theta} \right) \mbox{Im} \left[ G^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega) \right] \nonumber\end{aligned}$$ Using these relations, we rewrite as $$\begin{aligned}
\label{eq:potential Fourier expression2}
V_{\rm{Cas}}(\mathbf{r}) \! & \! = \! & \! - \frac {\hbar} {4 \pi} \! \int_{0}^{+\infty} \! \! d \omega \left\{ G^H_{\hat{d}}(\omega) \mbox{Re} \left[ \mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega) \right] \nonumber \right. \\
&\: & \qquad \qquad \: \left. + \mbox{Re} \left[ g^R_{\hat{d}}(\omega) \right] \mathcal{G}^{H,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega) \right\}\end{aligned}$$ Then, we use the parity of the Green’s functions with respect to the frequency $\omega$ in order to extend the lower bound of the integral in (\[eq:potential Fourier expression2\]) to $-\infty$. Note that $g^{(R,H)}_{\hat{d}}(-\omega)=g^{(R,H)*}_{\hat{d}}(\omega)$ since the Green’s functions $g^{(R,H)}_{\hat{d}}(t,t')$ are real. In addition, the FDT shows that $g^{H}_{\hat{d}}(\omega)$ is real. Similar relations hold for the electric field Green’s functions $\mathcal{G}^{(R,H),S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega)$. One then derives $$\begin{aligned}
\label{eq:potential Fourier expression3}
V_{\rm{Cas}}(\mathbf{r}) \! & \! = \! & \! - \frac {\hbar} {8 \pi} \! \int \! d \omega \left( g^H_{\hat{d}}(\omega) \mbox{Re} \left[\mathcal{G}^{R,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega) \right] \right. \\
& & \qquad \qquad \qquad \left. + \mbox{Re} \left[ g^R_{\hat{d}}(\omega) \right] \mathcal{G}^{H,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega) \right) \nonumber\end{aligned}$$ We can add $ g^H_{\hat{d}}(\omega) \mbox{Im} \left[ \mathcal{G}^{R,S *}_{\hat{\mathbf{E}}} \left(\mathbf{r},\mathbf{r};\omega\right) \right]$ and $ \mbox{Im} \left[g^{R*}_{\hat{d}}(\omega)\right] \mathcal{G}^{H,S}_{\hat{\mathbf{E}}} \left(\mathbf{r},\mathbf{r};\omega\right)$ to the integrand in (\[eq:potential Fourier expression3\]) since they are odd functions of $\omega:$ $$\begin{aligned}
\label{eq:potential Fourier expression4}
V_{\rm{Cas}}(\mathbf{r}) \! & = & \! \frac {-\hbar} {8 \pi} \! \int \! d \omega \! \left( \! g^H_{\hat{d}}(\omega) \mathcal{G}^{R,S *}_{\hat{\mathbf{E}}} \! \left(\mathbf{r},\mathbf{r};\omega\right) \right. \\
& \: & \qquad \qquad \qquad \qquad \left. + \mathcal{G}^{H,S}_{\hat{\mathbf{E}}}(\mathbf{r},\mathbf{r};\omega) g^{R*}_{\hat{d}} \! (\omega) \! \right) \nonumber \end{aligned}$$ By inspection of Eqs. and (\[eq:potential Fourier expression4\]), we conclude that the local Casimir phase in the quasi-static limit takes the standard form of an atomic Casimir phase [@CroninVigue].
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W. Heitler, *The Quantum Theory of Radiation*, (Dover, New York, 1954), ch. II; C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg *Photons and Atoms: Introduction to Quantum Electrodynamics*, (Wiley, New York, 1989), ch. III.
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In principle we also need the Röntgen interaction term $-{\bf d}\cdot \dot{\bf r}_k \times {\bf B}$ [@Scheel09] in order to have the complete correction to first-order in $\dot{\bf r}_k.$ However, one can show that the Röntgen contribution for short atom-surface distances is much smaller than the dynamical contribution arising from the electric dipolar Hamiltonian calculated here.
| |
Discharging a capacitor through a resistor provides a voltage curve that drops extremely fast at first and then changes very slowly towards the end of the discharge.
Video below shows how I build this circuit on a breadboard.
- Time constant is usually indicated by it’s initials (TC) or the greek letter tau ( τ ).
- RC time constant is determined by the capacitance of the capacitor in farads times the resistance in ohms. τ = RC
- First time constant changes from the starting voltage to about 63% of the final voltage. Each time constant after that changes about 63% of the remaining voltage. After 5 time constants, the voltage is practically completely changed, and therefore the capacitor is usually declared as being fully charged or discharged after 5 time constants.
Video
Quick discharging capacitor RC time constant circuit schematic to breadboard build and oscilloscope
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Updated video and diagram:
A 1,000µF (same as 0.001F) capacitor with a 1,000Ω resistor has a time constant of 1 second. Therefore it will take about 5 seconds to go from 5 volts to 0V.
Discharging capacitor component RC time constant curve oscilloscope measured
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Today I am doing a tutorial on how to make this really pretty square knot bracelet, which is great for stacking up as arm candy and also the most gorgeous idea to gift to your best friends with Friendship Day right around the corner!
This may look complicated at first glance but believe me, follow these instructions and you should be just fine.
You are going to need three strings to make this bracelet. I am using the colors lavender and pink. The length of the middle string should be around 2-3 times your wrist measurement, and the length of the other two strings should be around 5-7 times your wrist measurement. I am going to refer to the middle string (the pink one) as the base string.
To start with, leave out a couple of inches of the base string and tape it at one end. Now keep the base string loose and tape it on the other end too.
Take the other two strings and tape it with the starting end (the end in which we have left a couple of inches sticking out) of the base string as shown in the image above.
I'll repeat again, only the base string needs to be taped at both the ends, the other two strings are only going to be taped on one end.
Now let's move on to how to make the square knots.
Here I have started from the left string. You can start from any side you want to.
1. Place the left string under the base string.
2. Then the same left string over the right string.
3. Now take the right string over the base string and into the loop.
4. Pull on both the strings at the same time and tighten the knot.
Now we'll repeat the same process, this time starting from the right string.
2. Then over the left string.
3. Now the left string over the base string and into the loop.
Again, pull on both the strings at the same time and tighten the knot.
Keep on alternating the sides after each knot.
Repeat the process until you reach the desired length. When you have reached the length that you want, cut the two strings that you were working with (DO NOT CUT THE BASE STRING) as closely and neatly as possible and apply glue to the edges.
Now you can just tie a simple knot at both the ends of your base string and your bracelet is ready, or you can go one step further and make an adjustable closure for the bracelet.
1. Overlap the ends of base string.
2. Secure the ends with two pieces of tape.
3. Take a string about 8 inches long or more, of the color you want the adjustable closure to be. Now treat the two ends of the base strands that have been taped together as a single base strand and start making square knots again.
4. Make the adjustable closure of the desired length. Here I have made three square knots.
Cut the excess string and secure the edges with glue.
To secure the raw edges, instead of glue you can even use clear nail-paint.
This post took a ridiculous amount of time to put together. If you still think that this is going to be too difficult for you, get three strings and just give it a try. This pattern may look elaborate but trust me, it is really, really easy to make. Plus it is so much fun!
Hope you liked this easy and fun DIY! Let me know if you would like to see more of such posts.
P.S. I'll be back to check for typos after a while. :P My mind has stopped working. For now I'll get as far from my laptop as possible. Please let me know if you have any queries.
Thanks Shivani! I just recently came across your YT page! Awesome videos hun!
Oh do try this technique Roanna, it is so much fun!
Lovely DIY Karishma. I am so gonna try it. I used to make colourful friendship bands in school, this DIY brought back all those happy memories.
this is just too cool! loved the pics.
so cute... I would love to made bands for my friends..it add a personal touch. thanks for sharing it. | http://www.gingersnapsxoxo.com/2013/07/diy-square-knot-bracelets-for.html |
Simulation of complex physical systems described by nonlinear partial differential equations are central to engineering and physical science, with applications ranging from weather[44, 8] and climate [46, 40] , engineering design of vehicles or engines , to wildfires and plasma physics . Despite a direct link between the equations of motion and the basic laws of physics, it is impossible to carry out direct numerical simulations at the scale required for these important problems. This fundamental issue has stymied progress in scientific computation for decades, and arises from the fact that an accurate simulation must resolve the smallest spatiotemporal scales. A paradigmatic example is turbulent fluid flow , underlying simulations of weather, climate, and aerodynamics. The size of the smallest eddy is tiny: for an airplane with chord length of 2 meters, the smallest length scale (the Kolomogorov scale) is . Classical methods for computational fluid dynamics (CFD), such as finite differences, finite volumes, finite elements and pseudo-spectral methods, are only accurate if fields vary smoothly on the mesh, and hence meshes must resolve the smallest features to guarantee convergence. For a turbulent fluid flow, the requirement to resolve the smallest flow features implies a computational cost scaling like , where , with and the typical velocity and length scales and the kinematic viscosity. A tenfold increase in leads to a thousandfold increase in the computational cost. Consequently, direct numerical simulation (DNS) for e.g. climate and weather are impossible. Instead, it is traditional to use smoothed versions of the Navier Stokes equations [39, 38] that allow coarser grids while sacrificing accuracy, such as Reynolds Averaged Navier-Stokes (RANS) [14, 1], and Large-Eddy Simulation (LES) [49, 28]. For example, current state-of-art LES with mesh sizes of to million has been used in the design of internal combustion engines , gas turbine engines [54, 20], and turbo-machinery . Despite promising progress in LES over the last two decades, there are severe limits to what can be accurately simulated. This is mainly due to the first-order dependence of LES on the sub-grid scale (SGS) model, especially for flows whose rate controlling scale is unresolved .
Here, we introduce a method for calculating the accurate time evolution of solutions to nonlinear partial differential equations, while using an order of magnitude coarser grid than is traditionally required for the same accuracy. This is a novel type of numerical solver that does not average unresolved degrees of freedom, but instead uses discrete equations that give pointwise accurate solutions on an unresolved grid. We discover these algorithms using machine learning, by replacing the components of traditional solvers most affected by the loss of resolution with learned alternatives. As shown in Fig.1(a), for a two dimensional direct numerical simulation of a turbulent flow, our algorithm maintains accuracy while using coarser resolution in each dimension, resulting in a fold improvement in computational time with respect to an advanced numerical method of similar accuracy. The model learns how to interpolate local features of solutions and hence can accurately generalize to different flow conditions such as different forcings and even different Reynolds numbers [Fig. 1(b)]. We also apply the method to a high resolution LES simulation of a turbulent flow and show similar performance enhancements, maintaining pointwise accuracy on LES simulations using times coarser grids with fold computational speedup. There has been a flurry of recent work using machine learning to improve turbulence modeling. One major family of approaches uses ML to fit closures to classical turbulence models based on agreement with high resolution direct numerical simulations (DNS) [31, 17, 36, 9]
. While potentially more accurate than traditional turbulence models, these new models have not achieved reduced computational expense. Another major thrust uses “pure” ML, aiming to replace the entire Navier Stokes simulation with approximations based on deep neural networks[24, 30, 11, 53, 33, 19]. A pure ML approach can be extremely efficient, avoiding the severe time-step constraints required for stability with traditional approaches. Because these models do not include the underlying physics, they often struggle to enforce constraints, such as conservation of momentum and incompressibility. While these models often perform well on data from the training distribution, they often struggle with generalization. For example, they perform worse when exposed to novel forcing terms. A third approach, which we build upon in this work, uses ML to correct errors in cheap, under-resolved simulations [48, 52, 41]. These models borrow strength from the coarse-grained simulations. In this work we design algorithms that accurately solve the equations on coarser grids by replacing the components most affected by the resolution loss with better performing learned alternatives. We use data driven discretizations [5, 56] to interpolate differential operators onto a coarse mesh with high accuracy [Fig. 1(c)]. We train the solver inside a standard numerical method for solving the underlying PDEs as a differentiable program, with the neural networks and the numerical method written in a framework (JAX ) supporting reverse-mode automatic differentiation. This allows for end-to-end gradient based optimization of the entire algorithm, similar to prior work on density functional theory , molecular dynamics and fluids [48, 52]. The methods we derive are equation specific, and require training a coarse resolution solver with high resolution ground truth simulations. Since the dynamics of a partial differential equation are local, the high resolution simulations can be carried out on a small domain. The models remains stable during long simulations and has robust and predictable generalization properties, with models trained on small domains producing accurate simulations on larger domains, with different forcing functions and even with different Reynolds number. Comparison to pure ML baselines shows that generalization arises from the physical constraints inherent in the formulation of the method.
Ii Background
ii.1 Navier-Stokes
Incompressible fluids are modeled by the Navier-Stokes equations:
|(1a)|
|(1b)|
where is the velocity field, the external forcing, and
denotes a tensor product. The densityis a constant, and the pressure is a Lagrange multiplier used to enforce (1b). The Reynolds number dictates the balance between the convection (first) or diffusion (second) terms in the right hand side of (1a). Higher Reynolds number flows dominated by convection are more complex and thus generally harder to model; flows are considered “turbulent” if . Direct numerical simulation (DNS) solves (1) directly, whereas large eddy simulation (LES) solves a spatially filtered version. In the equations of LES, is replaced by a filtered velocity and an sub-grid term is added to the right side of (1a), with the sub-grid stress defined as . Because is un-modeled, solving LES also requires a choice of closure model for as a function of . Numerical simulation of both DNS and LES further requires a discretization step to approximate the continuous equations on a grid. Traditional discretization methods (e.g., finite differences) converge to an exact solution as the grid spacing becomes small, with LES converging faster because it models a smoother quantity. Together, discretization and closure models are the two principle sources of error when simulating fluids on coarse grids [48, 18].
ii.2 Learned solvers
Our principle aim is to accelerate DNS without compromising accuracy or generalization. To that end, we consider ML modeling approaches that enhance a standard CFD solver when run on inexpensive to simulate coarse grids. We expect that ML models can improve the accuracy of the numerical solver via effective super-resolution of missing details. Because we want to train neural networks for approximation inside our solver, we wrote a new CFD code in JAX
, which allows us to efficiently calculate gradients via automatic differentiation. Our CFD code is a standard implementation of a finite volume method on a regular staggered mesh, with first-order explicit time-stepping for convection, diffusion and forcing, and implicit treatment of pressure; for details see the appendix. The algorithm works as follows: in each time-step, the neural network generates a latent vector at each grid location based on the current velocity field, which is then used by the sub-components of the solver to account for local solution structure. Our neural networks are convolutional, which enforces translation invariance and allows them to be local in space. We then use components from standard standard numerical methods to enforce inductive biases corresponding to the physics of the Navier Stokes equations, as illustrated by the light gray boxes in Fig.1(c): the convective flux model improves the approximation of the discretized convection operator; the divergence operator enforces local conservation of momentum according to a finite volume method; the pressure projection enforces incompressibility and the explicit time step operator forces the dynamics to be continuous in time, allowing for the incorporation of additional time varying forces. “DNS on a coarse grid” blurs the boundaries of traditional DNS and LES modeling, and thus invites a variety of data-driven approaches. In this work we focus on two types of ML components: learned interpolation and learned correction. Both focus on the convection term, the key term in (1) for turbulent flows.
ii.2.1 Learned interpolation (LI)
In a finite volume method, denotes a vector field of volume averages over unit cells, and the cell-averaged divergence can be calculated via Gauss’ theorem by summing the surface flux over the each face. This suggests that our only required approximation is calculating the convective flux on each face, which requires interpolating from where it is defined. Rather than using typical polynomial interpolation, which is suitable for interpolation without prior knowledge, here we use an approach that we call learned interpolation based on data driven discretizations . We use the outputs of the neural network to generate interpolation coefficients based on local features of the flow, similar to the fixed coefficients of polynomial interpolation. This allows us to incorporate two important priors: (1) the equation maintains the same symmetries and scaling properties (e.g., rescaling coordinates ) as the original equations, and (2) as the mesh spacing vanishes, the interpolation module retains polynomial accuracy so that our model performs well in the regime where traditional numerical methods excel.
ii.2.2 Learned correction (LC)
An alternative approach, closer in spirit to LES modeling, is to simply model a residual correction to the discretized Navier-Stokes equations [(1)] on a coarse-grid [52, 48]. Such an approach generalizes traditional closure models for LES, but in principle can also account for discretization error. We consider learned correction models of the form , where LC is a neural network and is the uncorrected velocity field from the numerical solver on a coarse grid. Modeling the residual is appropriate both from the perspective of a temporally discretized closure model, and pragmatically because the relative error between and in a single time step is small. LC models have fewer inductive biases than LI models, but they are simpler to implement and potentially more flexible. We also explored LC models restricted to take the form of classical closure models (e.g., flow-dependent effective tensor viscosity models), but the restrictions hurt model performance and stability.
ii.2.3 Training
The training procedure tunes the machine learning components of the solver to minimize the discrepancy between an expensive high resolution simulation and a simulation produced by the model on a coarse grid. We accomplish this via supervised training where we use a cumulative point-wise error between the predicted and ground truth velocities as the loss function
The ground truth trajectories are obtained by using a high resolution simulation that is then coarsened to the simulation grid. Including the numerical solver in the training loss ensures fully “model consistent” training where the model sees its own outputs as inputs [18, 36, 52], unlike typical a priori training where simulation is only performed offline. As an example, for the Kolmogorov flow simulations below with Reynolds number , our ground truth simulation had a resolution of cells along each spatial dimension. We subsample these ground truth trajectories along each dimension and time by a factor of . For training we use trajectories of sequential time steps each, starting from different random initial conditions. To evaluate the model, we generate much longer trajectories (tens of thousands of time steps) to verify that models remain stable and produce plausible outputs. We unroll the model for 32 time steps when calculating the loss, which we find improves model performance for long time trajectories , in some cases using gradient checkpoints at each model step to reduce memory usage .
Iii Results
We take a utilitarian perspective on model evaluation: simulation methods are good insofar as they demonstrate accuracy, computational efficiency and generalizability. In this case, accuracy and computational efficiency require the method to be faster than the DNS baseline, while maintaining accuracy for long term predictions; generalization means that although the model is trained on specific flows, it must be able to readily generalize well to new simulation settings, including to different forcings and different Reynolds numbers. In what follows, we first compare the accuracy and generalizability of our method to both direct numerical simulation and several existing ML-based approaches for simulations of two dimensional turbulence flow. In particular, we first consider Kolmogorov flow , a parametric family of forced turbulent flows obeying the Navier-Stokes equation [(1)], with forcing , where the second term is a velocity dependent drag preventing accumulation of energy at large scales . Kolmogorov flow produces a statistically stationary turbulent flow, with flow complexity controlled by a single parameter, the Reynolds number Re.
iii.1 Accelerating DNS
The accuracy of a direct numerical simulation quickly degrades once the grid resolution cannot capture the smallest details of the solution. In contrast, our ML-based approach strongly mitigates this effect. Figure 2 shows the results of training and evaluating our model on Kolmogorov flows at Reynolds number . All datasets were generated using high resolution DNS, followed by a coarsening step.
iii.1.1 Accuracy
The scalar vorticity field is a convenient way to describe a two-dimensional incompressible flows . Accuracy can be quantified by correlating vorticity fields,111In our case the Pearson correlation reduces to a cosine distance because the flows considered here have mean velocity of . between the ground truth solution and the predicted state . Fig. 2 compares the learned interpolation model () to fully resolved DNS of Kolmogorov flow () using an initial condition that was not included in the training set. Strikingly, the learned discretization model matches the pointwise accuracy of DNS with a times finer grid. The eventual loss of correlation with the reference solution is expected due to the chaotic nature of turbulent flows; this is marked by a vertical grey line in Fig. 2(b), indicating the first three Lyapunov times. Fig. 2 (a) shows the time evolution of the vorticity field for three different models: the learned interpolation matches the ground truth () more accurately than the baseline, whereas it greatly outperforms a baseline solver at the same resolution as the model (). The learned interpolation model also produces a similar energy spectrum to DNS. With decreasing resolution, DNS cannot capture high frequency features, resulting in an energy spectrum that “tails off” for higher values of . Fig. 2 (c) compares the energy spectrum for learned interpolation and direct simulation at different resolutions after time steps. The learned interpolation model accurately captures the energy distribution across the spectrum.
iii.1.2 Computational efficiency
The ability to match DNS with a times coarser grid makes the learned interpolation solver much faster. We benchmark our solver on a single core of Google’s Cloud TPU v4, a hardware accelerator designed for accelerating machine learning models that is also suitable for many scientific computing use-cases [10, 55, 32]. The TPU is designed for high throughput vectorized operations, and extremely high throughput matrix-matrix multiplication in low precision (bfloat16). On sufficiently large grid sizes ( and larger), our neural net makes good use of matrix-multiplication unit, achieving 12.5x higher throughput in floating point operations per second than our baseline CFD solver. Thus despite using 150 times more arithmetic operations, the ML solver is only about 12 times slower than the traditional solver at the same resolution. The gain in effective resolution in three dimensions (two space dimensions and time, due to the Courant condition) thus corresponds to a speedup of .
iii.1.3 Generalization
In order to be useful, a learned model must accurately simulate flows outside of the training distribution. We expect our models to generalize well because they learn local operators: interpolated values and corrections at a given point depend only on the flow within a small neighborhood around it. As a result, these operators can be applied to any flow that features similar local structures as those seen during training. We consider three different types of generalization tests: (1) larger domain size, (2) unforced decaying turbulent flow, and (3) Kolmogorov flow at a larger Reynolds number. First, we test generalization to larger domain sizes with the same forcing. Our ML models have essentially the exact same performance as on the training domain, because they only rely upon local features of the flows.(see Appendix E and Fig. 5). Second, we apply our model trained on Kolmogorov flow to decaying turbulence, by starting with a random initial condition with high wavenumber components, and letting the turbulence evolve in time without forcing. Over time, the small scales coalesce to form large scale structures, so that both the scale of the eddies and the Reynolds number vary. Figure 3 shows that a learned discretization model trained on Kolmogorov flows can match the accuracy of DNS running at times finer resolution. A standard numerical method at the same resolution as the learned discretization model is corrupted by numerical diffusion, degrading the energy spectrum as well as pointwise accuracy. Our final generalization test is harder: can the models generalize to higher Reynolds number where the flows are more complex? The universality of the turbulent cascade [44, 26, 27] implies that at the size of the smallest eddies (the Kolmogorov length scale), flows “look the same” regardless of Reynolds number when suitably rescaled. This suggests that we can apply the model trained at one Reynolds number to a flow at another Reynolds number by simply rescaling the model to match the new smallest length scale. To test this we construct a new dataset for a Kolmogorov flow with . The theory of two-dimensional turbulence implies that the smallest eddy size decreases as , implying that the smallest eddies in this flow are that for original flow with . We therefore can use a trained model at by simply halving the grid spacing. Fig. 4 (a) shows that with this scaling, our model achieves the accuracy of DNS running at times finer resolution. This degree of generalization is remarkable, given that we are now testing the model with a flow of substantially greater complexity. Fig. 4 (b) visualizes the vorticity, showing that higher complexity is captured correctly, as is further verified by the energy spectrum shown in Fig. 4 (c).
iii.2 Comparison to other ML models
Finally, we compare the performance of learned interpolation to alternative ML-based methods. We consider three popular ML methods: ResNet (RN) , Encoder-Processor-Decoder (EPD) [6, 45] architectures and the learned correction (LC) model introduced earlier. These models all perform explicit time-stepping without any additional latent state beyond the velocity field, which allows them to be evaluated with arbitrary forcings and boundary conditions, and to use the time-step based on the CFL condition. By construction, these models are inivariant to translation in space and time, and have similar runtime for inference (varying within a factor of two). To evaluate training consistency, each model is trained 9 times with different random initializations on the same Kolmogorov
dataset described previously. Hyperparameters for each model were chosen as detailed in AppendixG, and the models are evaluated on the same generalization tasks. We compare their performance using several metrics: time until vorticity correlation falls below to measure pointwise accuracy for the flow over short time windows, the absolute error of the energy spectrum scaled by to measure statistical accuracy for the flow over long time windows, and the fraction of simulated velocity values that does not exceed the range of the training data to measure stability. Fig. 5 compares results across all considered configurations. Overall, we find that learned interpolation (LI) performs best, although learned correction (LC) is not far behind. We were impressed by the performance of the LC model, despite its weaker inductive biases. The difference in effective resolution for pointwise accuracy ( vs upscaling) corresponds to about a factor of two in run-time. There are a few isolated exceptions where pure black box methods outperform the others, but not consistently. A particular strength of the learned interpolation and correction models is their consistent performance and generalization to other flows, as seen from the narrow spread of model performance for different random initialization and their consistent dominance over other models in the generalization tests. Note that even a modest effective coarse-graining in resolution still corresponds to a computational speed-up. In contrast, the black box ML methods exhibit high sensitivity to random initialization and do not generalize well, with much less consistent statistical accuracy and stability.
iii.3 Acceleration of LES
Finally, up until now we have illustrated our method for DNS of the Navier Stokes equations. Our approach is quite general and could be applied to any nonlinear partial differential equation. To demonstrate this, we apply the method to accelerate LES, the industry standard method for large scale simulations where DNS is not feasible. Here we treat the LES at high resolution as the ground truth simulation and train an interpolation model on a coarser grid for Kolmogorov flows with Reynolds number according to the Smagorinsky-Lilly model SGS model . Our training procedure follows the exact same approach we used for modeling DNS. Note in particular that we do not attempt to model the parameterized viscosity in the learned LES model, but rather let learned interpolation model this implicitly. Fig. 6 shows that learned interpolation for LES still achieves an effective upscaling, corresponding to roughly speedup.
Iv Discussion
In this work we present a data driven numerical method that achieves the same accuracy as traditional finite difference/finite volume methods but with much coarser resolution. The method learns accurate local operators for convective fluxes and residual terms, and matches the accuracy of an advanced numerical solver running at 8– finer resolution, while performing the computation 40– faster. The method uses machine learning to interpolate better at a coarse scale, within the framework of the traditional numerical discretizations. As such, the method inherently contains the scaling and symmetry properties of the original governing Navier Stokes equations. For that reason, the methods generalize much better than pure black-box machine learned methods, not only to different forcing functions but also to different parameter regimes (Reynolds numbers). What outlook do our results suggest for speeding up 3D turbulence? In general, the runtime for efficient ML augmented simulation of time-dependent PDEs should scale like
|(2)|
where is the cost of ML inference per grid point, is the cost of baseline numerical method, is the number of grid points along each dimension of the resolved grid, is the number of spatial dimensions and is the effective coarse graining factor. Currently, , but we expect that much more efficient machine learning models are possible, e.g., by sharing work between time-steps with recurrent neural nets. We expect the decrease in effective resolution discovered here to generalize to 3D and more complex problems. This suggests that speed-ups in the range of – may be possible for 3D simulations. Further speed-ups, as required to capture the full range of turbulent flows, will require either more efficient representations for flows (e.g., based on solution manifolds rather than a grid) or being satisfied with statistical rather than pointwise accuracy (e.g., as done in LES modeling). In summary, our approach expands the Pareto frontier of efficient simulation in computational fluid dynamics, as illustrated in Fig. 1(a). With ML accelerated CFD, users may either solve expensive simulations much faster, or increase accuracy without additional costs. To put these results in context, if applied to numerical weather prediction, increasing the duration of accurate predictions from 4 to 7 time-units would correspond to approximately 30 years of progress . These improvements are possible due to the combined effect of two technologies still undergoing rapid improvements: modern deep learning models, which allow for accurate simulation with much more compact representations, and modern accelerator hardware, which allows for evaluating said models with a remarkably small increase in computational cost. We expect both trends to continue for the foreseeable future, and to eventually impact all areas of computationally limited science.
Acknowledgement
We thank John Platt and Rif A. Saurous for encouraging and supporting this work and for important conversations, and Yohai bar Sinai, Anton Geraschenko, Yi-fan Chen and Jiawei Zhuang for important conversations.
Appendix A Direct numerical simulation
Here we describe the details of the numerical solver that we use for data generation, model comparison and the starting point of our machine learning models. Our solver uses a staggered-square mesh in a finite volume formulation: the computational domain is broken into computational cells where the velocity field is placed on the edges, while the pressure is solved at the cell centers. Our choice of real-space formulation of the Navier-Stokes equations, rather than a spectral method is motivated by practical considerations: real space simulations are much more versatile when dealing with boundary conditions and non-rectangular geometries. We now describe the implementation details of each component.
Convection and diffusion
We implement convection and diffusion operators based on finite-difference approximations. The Laplace operator in the diffusion is approximated using a second order central difference approximation. The convection term is solved by advecting all velocity components simultaneously, using a high order scheme based on Van-Leer flux limiter . For the results presented in the paper we used explicit time integration using Euler discretization. This choice is motivated by performance considerations: for the simulation parameters used in the paper (high Reynold number) implicit diffusion is not required for stable time-stepping, and is approximately twice as slow, due to the additional linear solves required for each velocity component. For diffusion dominated simulations implicit diffusion would result in faster simulations.
Pressure
To account for pressure we use a projection method, where at each step we solve the corresponding Poisson equation. The solution is obtained using either a fast diagonalization approach with explicit matrix multiplication
or a real-valued fast Fourier transform (FFT). The former is well suited for small simulation domains as it has better accelerator utilization, while FFT has best computational complexity and performs best in large simulations. For wall-clock evaluations we choose between the fast diagonalization and FFT approach by choosing the fastest for a given grid.
Forcing and closure terms
We incorporate forcing terms together with the accelerations due to convective and diffusive processes. In an LES setting the baseline and ground truth simulations additionally include a subgrid scale model that is also treated explicitly. We use the Smagorinsky-Lilly model ((3)) where is the grid spacing and :
|(3)|
Appendix B Datasets and simulation parameters
In the main text we introduced five datasets: two Kolmogorov flows at and , Kolmogorov flow with on a larger domain, decaying turbulence and an LES dataset with Reynolds number . Dataset generation consisted of three steps: (1) burn-in simulation from a random initial condition; (2) simulation for a fixed duration using high resolution solver; (3) downsampling of the solution to a lower grid for training and evaluation. The burn-in stage is fully characterized by the burn-in time and initialization wavenumber
which represent the discarded initial transient and the peak wavenumber of the log-normal distribution from which random initial conditions were sampled from. The maximum amplitude of the initial velocity field was set tofor forced simulations and
in the decaying turbulence, which was selected to minimize the burn-in time, as well as maintain standard deviation of the velocity field close to. The initialization wavenumber was set to . Simulation parameters include simulation resolution along each spatial dimension, forcing and Reynolds number. The resulting velocity trajectories are then downsampled to the save resolution. Note that besides the spatial downsampling we also perform downsampling in time to maintain the Courant–Friedrichs–Lewy (CFL) factor fixed at , standard for numerical simulations. Parameters specifying all five datasets are shown in Table A1. We varied the size of our datasets from time slices at the downsampled resolution for decaying turbulence to slices in the forced turbulence settings. Such extended trajectories allow us to analyze stability of models when performing long-term simulations. We note that when performing simulations at higher Reynolds numbers we used rescaled grids to maintain fixed time stepping. This is motivated to potentially allow methods to specialize on a discrete time advancements. When reporting the results, spatial and time dimensions are scaled back to a fixed value to enable direct comparisons across the experiments. For comparison of our models with numerical methods we use the corresponding simulation parameters while changing only the resolution of the underlying grid. As mentioned in the Appendix A, when measuring the performance of all solvers on a given resolution we choose solver components to maximize efficiency.
Appendix C Learned interpolations
To improve numerical accuracy of the convective process our models use psuedo-linear models for interpolation [5, 56]. The process of interpolating to is broken into two steps:
-
Computation of local stencils for interpolation target .
-
Computation of a weighted sum over the stencil.
Rather using a fixed set of interpolating coefficients (e.g., as done for typical polynomial interpolation), we choose from the output of a neural network additionally constrained to satisfy , which guarantees that the interpolation is at least first order accurate. We do so by generating interpolation coefficients with an affine transformation on the unconstrainted outputs of the neural network, where is the null-space of the constraint matrix (a matrix of ones) of shape and is an arbitrary valid set of coefficients (we use linear interpolation from the nearest source term locations). This is a special case of the procedure for arbitrary polynomial accuracy constraints on finite difference coefficients described in [5, 56]. In this work, we use a patch centered over the top-right corner of each unit-cell, which means we need unconstrained neural network outputs to generate each set of interpolation coefficients.
Appendix D Neural network architectures
All of the ML based models used in this work are based on fully convolutional architectures. Fig. A1 depicts our three modeling approaches (pure ML, learned interpolation and learned correction) and three architecturs for neural network sub-components (Basic ConvNet, Encoder-Process-Decoder and ResNet). Outputs and inputs for each neural network layer are linearly scaled such that appropriate values are in the range . For our physics augmented solvers (learned interpolation and learned correction), we used the basic ConvNet archicture, with (8 interpolations that need 15 inputs each) for learned interpolation and for learned correction. Our experiments found accuracy was slightly improved by using larger neural networks, but not sufficiently to justify the increased computational cost. For pure ML solvers, we used EPD and ResNet models that do not build impose physical priors beyond time continuity of the governing equations. In both cases a neural network is used to predict the acceleration due to physical processes that is then summed with the forcing function to compute the state at the next time. Both EPD and ResNet models consist of a sequence of CNN blocks with skip-connections. The main difference is that the EPD model has an arbitrarily large hidden state (in our case, size 64), whereas the ResNet model has a fixed size hidden state equal to 2, the number of velocity components.
Appendix E Details of accuracy measurements
In the main text we have presented accuracy results based on correlation between the predicted flow configurations and the reference solution. We reach the same conclusions based on other common choices, such as squared and absolute errors as shown in Fig. A2 comparing learned discretizations to DNS method on Kolmogorov flow with Reynolds number . As mentioned in the main text, we additionally evaluated our models on enlarged domains while keeping the flow complexity fixed. Because our models are local, this is the simplest generalization test. As shown in Fig. A3 (and Fig. 5 in the main text), the improvement for larger domains is identical to that found on a smaller domain.
Appendix F Details of overview figure
The Pareto frontier of model performance shown in Fig. 1(a) is based on extrapolated accuracy to an larger domain. Due to computational limits, we were unable to run the simulations on the grid for measuring accuracy, so the time until correlation goes below was instead taken from the domain size, which as described in the preceding section matched performance on the domain. The learned interpolation model corresponds to the depicted results in Fig. 2, whereas the and models were retrained on the same training dataset for coarser or finer coarse-graining.
Appendix G Hyperparameter tuning
All models were trained using Adam optimizer. Our initial experiments showed that none of the models had a consistent dependence on the optimization parameters. The set that worked well for all models included learning rate of , and . For each model we performed a hyperparameter sweep over the length of the training trajectory used to compute the loss, model capacity such as size and number of hidden layers, as well as a sweep over different random initializations to assess reliability and consistency of training.
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When you are using the Scrum framework, a Sprint cycle involves development and QA. At the end of the Sprint the tasks worked upon and tested are showcased and released.
Typically, for a team of 3 to 4 developers there would be 1 QA resource. What are the developers expected to do when QA is happening? Since the number of developers is much higher than the number of QA testers, bug fixes get done very quick and developers are left with nothing to do towards the end of the Sprint.
What is expected of the developers during this QA testing while following Scrum? | https://pm.stackexchange.com/questions/16361/what-are-developers-expected-to-do-during-testing-in-the-latter-half-of-each-spr/16371 |
A comparison of the efficacy of a Newcastle disease (ND), inactivated, experimental vaccine (NDIEV) and a ND, inactivated, commercial vaccine (NDICV) was under taken. Broiler chickens were divided into 5 groups of 112 birds each. All groups, except group 5 which served as unvaccinated controls, were given a live ND vaccine, strain B1 and different inactivated ND vaccines as follows. Groups 1 and 3 were given a NDIEV at 1 and 10 days old. And groups 2 and 4 were given a NDICV at 1 and 10 days old. When 21, 28, and 35 days old, 28 birds from each group were challenged with a ND virus. Sera was collected for HI antibodies, body weight, feed intake, morbidity, mortality, feed conversion rate and economic performance were observed during 21-31, 28-38, and 35-45 day periods. Results showed that HI antibody titers of NDICV were much better compared with those of a NDIEV. HI antibody titers of birds vaccinated at 10 days of age were higher than those of birds vaccinated at 1 day old. Morbidity, mortality, feed conversion rate, and economic performance of group 1 was worse than those of the other vaccinated groups.
Publisher
Faculty of Veterinary Science, Chulalongkorn University
First Page
51
Last Page
58
Recommended Citation
Pakpinyo, Somsak; Sasipreeyajan, Jiroj; and Chansiripornchai, Niwat
(2003)
"AN INACTIVATED NEWCASTLE DISEASE VACCINE PART II: THE EFFICACY OF AN INACTIVATED, EXPERIMENTAL VACCINE ON THE PREVENTION OF NEWCASTLE DISEASE,"
The Thai Journal of Veterinary Medicine: Vol. 33:
Iss.
1, Article 5. | https://digital.car.chula.ac.th/tjvm/vol33/iss1/5/ |
Liquid crystals (LCs) are becoming increasingly important for applications in the terahertz frequency range. A detailed understanding of the spectroscopic parameters of these materials over a broad frequency range is crucial in order to design customized LC mixtures for improved performance. We present the frequency dependent index of refraction and the absorption coefficients of the nematic liquid crystal 5CB over a frequency range from 0.3 THz to 15 THz using a dispersion-free THz time-domain spectrometer system based on two-color plasma generation and air biased coherent detection (ABCD). We show that the spectra are dominated by multiple strong spectral features mainly at frequencies above 4 THz, originating from intramolecular vibrational modes of the weakly LC molecules.
©2012 Optical Society of America
1. Introduction
A relatively new but remarkably fast growing research topic is the investigation and characterization of the terahertz (THz) spectral properties of liquid crystals (LCs) [1–10]. These studies are spurred by an increased demand for new fast and cost-efficient switchable devices , to manipulate the THz radiation for various applications [12–14] including THz communication [15,16]. As the properties of LCs are typically optimized for applications in the visible part of the electromagnetic spectrum, a customized design of new LC mixtures targeted for applications in the THz range, however, requires detailed knowledge of the dielectric properties over a broad frequency range. Among the most comprehensively investigated LC materials in this frequency range are 4’-pentyl-4-cyanobiphenyl (5CB) and other members of the CB family. These relatively simple structured LCs are found in many recent LC mixtures. Therefore, an extensive database exists for these molecules for the optical, near- and mid-IR frequency range down to approximately 400 cm−1, i.e. approximately 12 THz. In contrast, only very few reports based on Fourier transform infrared (FTIR) spectroscopy data accessing the far-infrared or THz frequency range below 10 THz, exist . The sparse published data indicate that the spectra of most members of the CB family molecules are richly structured by various strong resonances. Polarization-dependent spectra have until now not been obtained with this measurement scheme, and in particular the lower frequency part suffers from a considerable uncertainty.
THz time domain-spectroscopy (THz-TDS) offers a versatile alternative to FTIR for the recording of the polarization dependent absorption as well as index of refraction in the terahertz frequency range . Therefore, significantly more data has become available recently, reporting these parameters for various members of the CB family [19,20]. However, these data are typically limited to frequencies below approximately 2.5 THz. At these low frequencies, nearly no spectroscopic features are observed. In some few cases, the rising slope of a strong feature at higher frequencies can be observed. The most interesting, richly structured frequency band, containing the lowest vibrational modes of the systems and thus bridging the featureless low frequency THz data and the IR data, still remains unexplored.
Recently, a new emission and detection scheme based on photo mixing in air plasma has been demonstrated and applied for ultrabroadband spectroscopy. This new technology already has had a significant impact on THz applications, as it combines a broadband detection scheme with remote sensing technology. As the spectra of many materials such as proteins , explosives , but also liquid crystals typically exhibit the most pronounced characteristic signatures in the frequency range above 2 THz, this technique enables to really fully exploit the detection and identification potential of THz technologies.
Here we present the refractive indices ne and no for extraordinary and ordinary polarization, respectively; as well as the absorption coefficients for both polarizations over the extended frequency range between approximately 200 GHz to 15 THz. The data was recorded using an air biased coherent detection (ABCD) system. We compare our data with previous work in the lower THz frequency range. Several very strong, specific signatures, which vary mainly in their intensity with respect to the polarization, are observed.
2. Liquid crystal material
The liquid crystal investigated here is 4’-pentyl-4-cyanobiphenyl (5CB). In contrast to many other commercial LCs, the structure of 5CB is simple and well known, making it an ideal candidate for scientific studies. It consists of a biphenyl core, a cyano group (CN) on one side and an alkyl chain (C5H11) on the other side. The structure of 5CB is shown in Fig. 1 . The molecule is liquid crystalline at room temperature.
Fig. 1
3. Experimental procedure
The spectra are recorded using a spectroscopy system based on two-color laser plasma THz generation and air biased coherent detection (ABCD) . A schematic of the system is shown in Fig. 2 , and a general description of the air plasma based generation and detection can be found in . Specifically, THz transients are generated by focusing a beam of 1mJ, 35-fs pulses at 800 nm center wavelength and frep = 1 kHz repetition frequency together with the second harmonic of the beam, generated collinearly with the fundamental in a 100 µm thick β-BBO crystal, to form a plasma. A half wave plate for 400 nm rotates the polarization of the 400 nm light to match that of the 800 nm fundamental. A combination of four wave mixing due to the third-order nonlinearity of the air and nonlinear currents in the plasma driven by the laser field [25,26] generates ultrabroadband THz transients, limited in duration and bandwidth only by the duration of the pump laser pulses. The generation efficiency depends critically on the phase difference between the fundamental and second harmonic of the pump beam . We optimize the phase difference by adjusting the distance between the BBO crystal and the focal region.
Fig. 2
The generated THz beam is collimated and refocused by a pair of off-axis paraboloidal mirrors to an intermediate focal plane, where the sample is placed. After the intermediate focus, two additional off-axis paraboloidal mirrors recollimate, guide and focus the beam to the ABCD unit. ABCD, as described elsewhere , is used for detection of the temporal profile of the THz transient, by slow scanning of a delay stage in the 800-nm wavelength probe beam path while recording the intensity of THz-induced second harmonic generation in the detection region with a photomultiplier tube. An applied DC voltage across the detection region serves as optical bias, thereby enabling field-resolved detection of the THz transient.
Figure 3 represents (a) a typical reference pulse and (b) its corresponding amplitude spectrum (b). The highest accessible frequency of the system is approximately 40 THz and it has a signal-to-noise ratio of 55dB at 5THz. The THz pulses are polarized linearly. There has been one polarizer employed before the cell, but none after the cell. As the detector is polarization sensitive, it itself acts as a polarizer.
Fig. 3
The nematic liquid crystals are filled in a specifically designed cell consisting of two 2 mm thick TOPAS windows (Fig. 4 ). Metal wires between the windows act as both electrodes and spacers to cell thickness of d = 500 µm. The lateral distance between the wires, i.e., the width of the cell is 30 mm. A peak voltage of 1.5 kV at a modulation frequency of 1 kHz is applied to align the LCs. The cell can be rotated by 90° in order to perform measurements with the LCs aligned parallel or orthogonal to the polarization of the THz beam. The cell is kept at ambient temperature 22°C and no active temperature stabilization is applied.
Fig. 4
4. Experimental results
The 5CB was prepared at MUT Warsaw and stored under dry nitrogen atmosphere. It has the following phase transitions: Cr 23 N 37.2 Iso, purity >99.95% tested by GC-FID. Figure 5(a) shows the temporal trace of a reference pulse propagating through the empty sample cell compared to the pulses propagating through the loaded cell with the LCs aligned parallel to the polarization of the THz beam and rotated by 90°, respectively. The corresponding amplitude spectrum of the three pulses is shown in Fig. 5(b), together with the reference spectrum, recorded after transmission through the empty cell. The strong absorption band seen in the reference spectrum near 15 THz is due to absorption in the TOPAS polymer. Pronounced spectroscopic features are observed over the entire frequency range, in particular between 4 to 6 THz and above approximately 11 THz. The frequency-dependent absorption coefficient α(ν) and index of refraction n(ν) are calculated from the complex ratio of the spectra of the THz transient transmitted through a filled cell and an empty cell, respectively.
Fig. 5
The frequency dependent refractive indices no and ne for ordinary and extraordinary polarization are shown in Fig. 6(a) . For comparison, the data of an earlier measurement using a time-domain spectrometer with lower bandwidth is also shown. This earlier work in the lower THz frequency range suggested an increasing birefringence with increasing frequency. Our data clearly shows that the average birefringence remains relatively constant at approximately ∆n ≈0.14 above 2 THz and over the entire frequency range up to 15 THz, with local variations in the vicinity of a resonance. For the low frequencies, an almost perfect overlap of the data acquired by the two different systems based on completely different technologies is observed.
Fig. 6
Figure 6(b) shows the absorption coefficient αo and αe of 5CB for ordinary and extraordinary polarization, respectively, together with the previously obtained THz-TDS data for low frequencies. Again, a very good agreement between the latter and the new ABCD data is observed. A monotonous increase in the absorption coefficient is observed at low frequencies. Strong spectroscopic features dominate the spectra in particular between 4 and 6 THz and above 11 THz, with an area of relatively low absorption and small dichroism in between. Interestingly, a similar general behavior was observed in soft glasses where the low-frequency region is dominated by a universal absorption observed in disordered materials while localized, ordered vibrational modes appear above a characteristic frequency known as the Ioffe-Regel transition frequency . This crossover behavior indicates that vibrational modes observed at higher frequencies are localized to small volumes within the sample. As an indicative estimate, we observe an approximate cross-over frequency between universal absorption and discrete absorption lines of 1 THz. Using a sound velocity of 1500 m/s (1.5 nm/ps) , the Ioffe-Regel crossover acoustic wavelength is 1.5 nm, indicating that vibrational order will be present only at length scales comparable or shorter than that. In a disordered system the linear absorption coefficient α(ω) can be described as the product of the vibrational density of states g(ω) and the coupling coefficient between the electromagnetic wave and the vibrations C(ω), so that α(ω) = C(ω)g(ω). For uncorrelated charge fluctuations, as have been observed in glassy systems in the low THz range C(ω) is constant with frequency. We observe a linear frequency dependence of the absorption coefficient at low frequencies, indicating that the density of states is proportional to frequency. Thus, we can interpret the absorption spectrum as an indicator of a largely localized vibrational response of the individual molecules of the liquid crystal, in contrast to a global, phonon-like behavior observed in molecular crystals with a high degree of order .
The intensities of the absorption features are strongly dependent on the respective polarization, which gives already a first indication on the orientation of the dipole moment of the corresponding molecular vibration. For example, the features at approximately 12 and 14 THz are of comparable intensity, whereas the intensity of the feature at approximately 4.5 THz is significantly more pronounced in the absorption spectrum with the cells aligned in ordinary polarization geometry. It is expected that anticipated extended studies on different derivatives of the CB family, where individual atoms or small molecular groups are selectively exchanged, will provide further insight in the origin of these feature and thus the molecular vibrations.
When heated, the LC material approaches the phase transition temperature (nematic/isotropic) and gradually loses its anisotropic properties. Thus, e.g. the two absorption curves for extraordinary and ordinary polarization approach each other and overlap in the isotropic phase. We would expect a significant redshift and broadening of the resonances with increasing temperature, as a direct result of the increased thermally induced disorder and softening of anharmonic vibrational ladders.
5. Calculations
In order to deduce some information on the origin of the observed absorption features, we have calculated the absorption spectra of a 5CB molecule using density functional theory. The calculations were performed using the Gaussian 09 electronic structure program . Both the geometry optimization and the frequency calculations of 5CB were carried out using BVP86 functional [35,36] with def2-SVP basis set . The calculation is based on a single molecule at 0 K. Thus, the calculation offers no information about intermolecular interactions, including the influence of disorder. However, this isolated-molecule approach to the calculation of the vibrational spectrum of the liquid crystal is justified by our experimental indication of localization of the vibrational modes to individual molecules as well as by the lack of strong intermolecular hydrogen bonds in the LC matrix . For simplicity, the polarization dependence has not been considered here. Therefore, while the calculations give some insight in the origin of the molecular vibrations that can be expected in this frequency range, they are not used as a reference for assigning the individual modes observed in the experimental spectrum.
The calculated vibrational frequencies and their intensities (blue bars) are shown in Fig. 7 . The discrete frequency values are broadened using Gaussian profiles, in order to facilitate a comparison with the experimental data. The widths are set to mimic the experimentally observed line widths. Similarly to the experimental spectrum, strong bands below 6 THz and a second set of stronger bands above 12 THz are observed. Although a direct assignment is not possible with this preliminary theoretical data, it is worth to note that even at low frequencies prominent intramolecular vibrations occur. Below 3 THz we determine a discrepancy between the observed and calculated spectra. We attribute the difference to the disorder-induced coupling between the electromagnetic field and the vibrational density of states at low frequencies, which is not considered in the calculation. A further refinement of the calculations is thus expected to allow for a direct assignment of at least some of the experimentally observed modes.
Fig. 7
6. Conclusion
We have shown that by using a dispersion-free, air photonics-based THz-TDS system, the polarization dependent spectra of 5CB between approximately 300 GHz and 12 THz can be recorded. At low frequencies (< 2 THz) we see strong indications of disorder-induced, universal absorption, indicating that the vibrational response is largely due to individual, molecular vibrational modes, and we observe several strong resonance features over the entire frequency range above approximately 2.5 THz, i.e. above the frequency band which has previously been accessed with conventional THz time-domain spectroscopy. The potential of ultrabroadband THz-TDS to bridge the spectroscopic gap between the THz and IR frequency bands is thus further emphasized. The results are expected to increase the knowledge on the low-frequency vibrations giving rise to the observed spectral signatures. A detailed understanding of the THz dielectric properties of these materials is expected to allow for optimizing the custom designed LC mixtures for applications in modulation devices for THz radiation.
Acknowledgments
We thank Dr. David G. Cooke (McGill University) for assistance with air photonics. Nico Vieweg thanks the German Academic Exchange service (DAAD) for funding. The Carlsberg Foundation and H. C. Ørsteds Foundation are acknowledged for partial financial support.
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36. J. P. Perdew, “Density-functional approximation for the correlation energy of the inhomogeneous electron gas,” Phys. Rev. B 33, 8822–8824, Erratum: B 34, 7406 (1986). | https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-20-27-28249&id=246811 |
Human skill is referred to a manager’s ability to work efficiently as a group member and to develop cooperative effort within a group of subordinates. A human skill of a manager is greatly dependent on their ability to motivate staff to perform to their best capabilities. They will facilitate the growth of the team through the phases of a team forming, storming, norming, performing and adjourning. The ability to see the enterprise as a whole refers to conceptual
Abstract The employees in an HR department contribute to a company’s growth by maintaining focus on the organization's objectives and business strategies, maintaining a healthy work environment between company policies and individuals, and focusing on recruiting, maintaining, and utilizing an effective work force, which organizations cannot survive without. . In order to incorporate equality and consistency, it is important for an organization’s human resource department to create a well thought-out interview structure, enabling them to ask the same or similar questions to elicit the desired information. Overall, having a balance between focus on the skillset demanded by the role and the interpersonal style demanded by the organizational culture will create a strong foundation for the recruitment process. Once an employee has been hired, it is the responsibility of the organization’s human resource department to train and inform them of the guidelines and expectations of the company.
Rather, it is a combination of qualities and behaviors. Some people are born with the instinctive ability to motivate others and the ability to manage, but anyone can ascertain how to be successful in a management career. What is management? The dictionary defines management as " the act or art of managing, the conducting or supervising of something (as a business), judicious use of means to accomplish an end, the collective body of those who manage or direct an enterprise ". (Merriam-Webster) There is an enormous distinction pertaining to being a manager and being a "good" manager.
The ability to relate to your co-workers, inspire others to participate, and mitigate conflict with co-workers is essential given the amount of time spent at work each day. Leadership/Management Skills. While there is some debate about whether leadership is something people are born with, these skills deal with your ability to take charge and manage your co-workers. Multicultural Sensitivity/Awareness. There is possibly no bigger issue in the workplace than diversity, and job-seekers must demonstrate a sensitivity and awareness to other people and cultures.
Where Employee Performance Evaluation Fits In some form, most organizations have an overall plan for business success. The employee performance evaluation process, including goal setting, performance measurement, regular performance feedback, employee recognition, and documentation of employee progress, ensures this success. The performance evaluation process—done with care and understanding—helps employees see how their jobs and expected contributions fit within the bigger picture of their organization. The more effective employee performance evaluation processes accomplish these goals and have additional benefits. Documented employee performance evaluations are communication tools that ensure the supervisor and her reporting staff members are clear about the requirements of each employee’s job.
Rachael Jamison What is professionalism? It is the level of excellence or competence that is expected of a professional. Professionalism is a quality that is a great asset for any individual to have, and a person that is professional should always act with integrity, be trustworthy, be respectful of others, and always conduct their business in the appropriate manor Businesses expect a certain level of professionalism from their employees during work hours and when on company property. Workplace etiquette is important for making first impressions and maintaining healthy relationships with superiors and peers. Communication, behavior, and appearance are three are the crucial factors that make a great professional employee, as these are characteristics that can contribute to a company in its goals to be profitable.
Why should organisations implement work life balance – is it a critical business criteria or just “the right thing to do”? MEANING AND DESCRIPTION OF
Functional Areas of Business A manager is a person or persons’ whom helps and guides others to accomplish a goal. Normally, managers fall under an organization or business. A manager helps others by motivating and guiding them. The reason they are put in that position is to make sure others, either individually or as a team, work to get certain job or goal accomplished. The functional areas of business are the different aspects of business that make up a company.
From that I will look at Louis Vuitton, an example of a business that has implemented team work into their production successfully, as a way of outlining clear benefits that can be gained from using team work to carry out a firm’s work. One of the biggest advantages of teamwork is that it can bring together several complimentary skills, more useful than any sole employee, working individually. This helps to ensure a combination of strengths, especially if the team has been chosen carefully, and means you can get a good range of abilities, fields of expertise and personality types, so for every situation there should be at least one person who can deal with it. This is particularly useful for any potential problems that could arise, with a broader selection of people at hand to deal with the problem; it becomes easier to deal with the problem more practically and more efficiently. Furthermore, when working in a team, it allows for the workforce to utilise their strengths more efficiently and this therefore gets more out of the employees.
In addition to that, leader of the organization should guarantee such an environment in which employees can contribute to the innovative production by stating their ideas. And thereby idea generation can be stimulated (Manders, et al., 2015). The principle of involvement of people indicates that all employees at the all levels of the organization are considered as potential contributors for the organization. Giving more responsibility to the employees and right to join certain decision making processes help employees to understand their contribution in the organization. Moreover, they feel eager to show their ideas freely since their ideas are considered as valuable (Santos-Vijande & Alvarez-Gonzales, 2007). | https://www.antiessays.com/free-essays/Do-You-Think-Programs-Like-Equilibrium-31884.html |
Research shows that school attendance and student achievement is directly correlated. Imagine how hard it is to miss a day or two of algebra and then have to get back on track as the whole class is moving forward knowing information that you missed!
We understand that students sometimes are ill or have genuine reasons to miss school, such as doctor or therapists visits, required court appearances, funerals, and so forth.
State law and district attendance policy allow the parent to write notes to document and excuse up to 10 days of student absence for illness or excusable reasons per year; 5 days in the first term (August-January) and 5 days in the second term (January-June). Beyond those 10 days, if a student has a serious reason to miss additional days, the school principal can review parent requests to excuse up to 5 more days per year, for a total of 15 days. That is a lot of time out of school especially when students have to make up the missed work and keep up with the new work. Students do not receive credit for work made up for unexcused absences which impact grades.
Beyond 15 days per year, only doctor/therapist or court notes are accepted to excuse absences. It is very important to document all days of absence with a note which must be turned in at school even if the reason for absences does not allow the day to be excused. It is important to note that family vacations are not excusable days. Questions about attendance should be directed to the attendance clerk at your child's school.
When students begin to accrue "unexcused absences," the district is required to monitor the student's attendance. Calls are made to the home on the day of absence. Letters are sent home when the students begin to have more unexcused absences. When the unexcused days total 10 in a 90-day period or 5 in a 30-day period, the school counselor has to hold a Student Study Team (SST) meeting to talk with the parent/guardian about why the student is missing school and make a plan to get the unexcused absences to stop. Remember, students can easily fall behind in learning when they miss days of school!
If the unexcused absences do not stop, the district is required to refer the student and family for additional services. Failure to have a child attend school is a law violation. In very severe cases, the parent/guardian may be called into court to explain the situation to a judge and can be placed on probation. No one wants to see this happen, so it is very important that the home and school work together toward success for the students! | https://flaglerschools.com/cms/One.aspx?portalId=1363814&pageId=7625306 |
The utility model discloses a magnetic type department post board, which relates to the technical field of post boards and comprises a magnet base body, a magnetic mother set is arranged on one side of the magnet base body, a first magnetic sticker is attracted on one side of the magnet base body, a department name block is arranged on one side of the first magnetic sticker, and a second magnetic sticker is arranged on the other side of the first magnetic sticker. The magnetic attraction type department post board comprises a magnetic attraction mother set, a second magnetic attraction post is attracted to one side of the magnetic attraction mother set, a photo module is arranged on one side of the second magnetic attraction post, a third magnetic attraction post is magnetically attracted to one side of the magnetic attraction mother set and located below the second magnetic attraction post, and a name module is arranged on one side of the third magnetic attraction post. According to the department post board, the magnet base body, the magnetic attraction mother set, the first magnetic attraction post and the photo module are arranged, so that the convenience of content replacement of the department post board is greatly improved, meanwhile, the manufacturing cost of content replacement is reduced, replacement and replacement can be carried out at the same time according to actual needs, and the convenience of the department post board is improved by replacing the photo module, the name module and the department name block. And changing the photo, the name and the department name. | |
VLAN is also known as “Virtual Local Area Network”. The basic purpose of a VLAN is to make sure that there is no physical restriction on the connection of layer 2 switches. These networks hold the ability to span different switches without having any physical restriction. It also adds an extra layer of security by separating the physical networks from the logical ones.It is extremely essential that all the components of a VLAN, both physical and virtual, work perfectly to ensure the smooth flow of processes in the network. However, there are still a number of problems which are encountered during this process. Following, you will find some of the most common problems and ways to troubleshoot them.
Faulty Physical Connections
We need to keep in mind that although the VLAN makes use of a number of virtual connections and components, there are still certain physical connections which need to be maintained. Firstly, whenever you experience any problem, you need to check that the LED lights on physical switches are blinking. This would ensure that the layer 1, the physical layer, is working properly and is not going through any problem. The next step which you need to do is to check whether the cables are connected properly. You should also make sure that the cables are in a good condition and are not worn or cut from any position.
Another common problem which is observed is the late receiving and sending of data on the interface of the switch. This problem might also have another origin, but it is commonly associated with faulty physical components. The issue due to which this problem occurs could be a faulty cable or NIC. In this regard, you might need to consider changing any of these to check whether the problem or resolved or it still pertains. If you find that the problem is not resolved by doing any of these steps, then you should realize that the problem is more technical and move towards other issues to identify the problem.
Duplex Mismatch or Congestion
If you realize that there are a number of collisions on the interface and the transfer of data has become slow, then you should consider having a look at the issue of Duplex Mismatch. A Duplex is referred to as a point-to-point communication between two devices which can communicate with each other in either both directions or just one direction. You should make sure that the both of the devices operate on the same duplex so that the connection remains active.
Another reason why these collisions might occur is due to congestion. When a lot of traffic is being transmitted through a single point, then there is going to be a lot of traffic which would restrict the speed of data transfer. Under such conditions, you should make sure that the data is being routed properly and there is no issue in the process of routing.
Lack of communication between two hosts
If you observe that two devices or hosts in the VLAN are not able to communicate, the first question which comes to your mind should be whether they are in the same VLAN or not? You should be aware of the fact that two hosts can only communicate with each other if they are in the same VLAN. However, if they are not, then there is a need for a router to be in between. This router will enable a communication link to be established between the devices in these VLANs. In this regard, you should insert a router in between to enable them to communicate.
This problem of communication can be caused due to one other reason. You should be aware of the fact that one VLAN has only one IP subnet. In this regard, it needs to be ensured that every host or device in a VLAN has the same subnet address. If two hosts on the VLAN have different subnet addresses due to any reason, they will not be able to communicate in any way. Hence, you should check for their subnet addresses and ensure that they have not been changed due to any reason.
Lack of Communication between a host and a switch
If you observe that the process of communication in a VLAN is not taking place properly, then you should consider the possibility of a faulty connection between a host and the switch. There are certain instances where the host is not able to connect with the switch at all. This problem arises because the host is not in the same subnet in which the switch is in the VLAN. Hence, you should make sure that both of them are in the same subnet.
Unable to create a VLAN
When you create a VLAN, you might come across a number of issues without realizing you fault. This is because there are some factors which are tricky enough to be considered. Whenever you fail to create a VLAN, you should have a look at you VLAN ID. You should ensure that you are not using a VLAN ID which has been reserved. This is an issue which many people overlook not considering the consequences. You should keep in mind that the VLAN IDs from 3968 to 4047 as well as 4094 have been reserved for internal use and cannot be used for any other purpose.
Problem of Missing VLANs
Each port is assigned to a particular VLAN to enable it to be a part of that network. However, there have been issues where the port on a switch becomes inactive. This is because the VLAN which was assigned to the port have been deleted due to some reason. If this happens, you should make use of the “show VLAN” command to search through the table of available LANs to see if any of them has been deleted. If it has been deleted, you will have to create and add it again.
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- What skills CCNA Routing and Switching exam checks? | https://www.actualtests.com/guide-how-to-troubleshoot-vlan-problems.htm |
Referencing my last post, a few questions have come to my attention as we begin this season, one being the title of this post. In light of public exposure to genealogy, through shows like Who Do You Think You Are?, many have been discussing the realistic amount of time it takes to produce the outcomes illustrated on a network show.
Most recently a blog post on Ancestry.com revealed the 1000 hours of research behind the pursuit of Cindy Crawford’s roots. In addition, much of the highlighted research was research completed long before the inception of the show. One point that I did not see mentioned was the financial cost of those 1000 hours. In the real world hiring a genealogist to complete 1000 hours of research may cost anywhere from $20,000 to $100,000 and that is not including travel expenses associated with on-site research. For most of us, this is unrealistic. Besides, if you are going to send someone to research your roots I would hope that someone would be you! Who wants to sit on the sidelines?
So this begs the question, how much is it going to cost to get into the game? The short answer is that it will cost both time and money. The amount it will cost will depend on you and your circumstances.
In my last post I mentioned scheduling your game. I hope that you have committed time to this pursuit. It just might be the most rewarding trophy you place on your shelf or the shelves of your descendants. It doesn’t have to be 1000 hours in a few months. Small gains can still make a first down. Consistent progress may not only help you find family history, but make memories that become your family’s history of the future.
As far as the financial cost, it all depends on your choices. Be forewarned: the pursuit of one’s genealogy and family history has become big business. Nevertheless, one can pursue genealogy and family history with little, if any, additional cost :-) Gathering records and photographs in your possession and interviewing family members and others who knew your family cost no more than your time. Access to a computer, the internet, scanners, and subscription sites may be as close as your nearest Family History Center. On the web, your favorite search engine may list interesting leads and some sources. Be aware that the search algorithm of the different providers may reveal different results. Sometimes great material is missed if you limit your query to one search database.
So, how much is it going to cost to get into the game? It all depends. There is no doubt that such a pursuit comes from discretionary funds and this amount varies from person to person. I would recommend that a set amount be put aside each month. It’s part of the discipline of the game.
Every game has limits. In football, there are four quarters, each lasting 15 minutes. With the exception of a possible tie at the end of a fourth quarter, the game is over when the clock runs out. Know the limits of your game, but don’t let these limits block you. Tackle your limits, whether in time or money, with innovations that provide new paths to success. As the quarterback of your team it is your responsibility to read the defense of the opposing team (limits) and make the necessary adjustments. The goal is to gain yardage for a first down and ultimately a touchdown!
My best to you this coming week…Cheering you on from the bleachers :-)
[After all these considerations, if you decide that you would like to handoff your research project to an assistant coach contact me. Together we can come up with a winning strategy to find your elusive ancestors.]
Copyright ©2013 Lynn Broderick and the Single Leaf. All Rights Reserved. | https://thesingleleaf.com/tag/cost-of-doing-family-history/ |
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abstract: 'This paper is devoted to study the gravitational charged perfect fluid collapse in the Friedmann universe models with cosmological constant. For this purpose, we assume that the electromagnetic field is so weak that it does not introduce any distortion into the geometry of the spacetime. The results obtained from the junction conditions between the Friedmann and the Reissner-Nordstr$\ddot{o}$m de-Sitter spacetimes are used to solve the field equations. Further, the singularity structure and mass effects of the collapsing system on time difference between the formation of apparent horizons and singularity have been studied. This analysis provides the validity of Cosmic Censorship Hypothesis. It is found that the electric field affects the area of apparent horizons and their time of formation.'
author:
- |
M. Sharif [^1] and G. Abbas [^2]\
Department of Mathematics, University of the Punjab,\
Quaid-e-Azam Campus, Lahore-54590, Pakistan.
title: '**Gravitational Charged Perfect Fluid Collapse in Friedmann Universe Models**'
---
[**Keywords:**]{} Electric field; Gravitational collapse; Cosmological constant; Friedmann models.\
[**PACS:**]{} 04.20.-q; 04.40.Dg; 97.10.CV
Introduction
============
Gravitational collapse of a massive star is the result of its self gravity. It occurs when the internal nuclear fuel of the star fails to supply sufficiently high pressure to counter-balance gravity. Gravitational collapse is one of the most important problems in general relativity. According to the singularity theorems [@1] there exist spacetime singularities in generic gravitational collapse. It has been an interesting problem to determine the nature of spacetime singularity. The cosmic censorship hypothesis (CCH) [@2] says that singularities appearing in gravitational collapse are always clothed by the event horizon.
The final fate of gravitational collapse of the massive star depends upon the choice of initial data and equation of state. Many efforts have been made to check its credibility but no final conclusion is drawn. For this purpose, Virbhadra et al. [@3] introduced a new theoretical tool using the gravitational lensing phenomena. Also, Virbhadra and Ellis [@4] studied the Schwarzschild black hole lensing and found that the relativistic images would confirm the Schwarzschild geometry close to the event horizon. The same authors [@5] analyzed the gravitational lensing by a naked singularity and classified it as weak naked singularity and strong naked singularity. In a recent paper [@6], Virbhadra used the gravitational lensing phenomena to find the improved form of the CCH.
Oppenheimer and Snyder [@7] studied dust collapse for the first time and showed that singularity is neither locally or globally naked. This means that they found black hole as a final fate of the dust collapse. Eardely and Smarr [@8] found that inhomogeneous model undergoes to gravitational collapse by forming a singularity that can be either locally or globally naked.
There has been a growing interest to study gravitational collapse in the presence of perfect fluid and other general physical form of the fluid. Misner and Sharp [@9] extended the pioneer work for the perfect fluid. Vaidya [@10] and Santos [@11] used the idea of outgoing radiation of the collapsing body and also included the dissipation in the source by allowing the radial heat flow. Markovic and Shapiro [@12] generalized the pioneer work with positive cosmological constant. Lake [@13] extended it for both positive and negative cosmological constant. Sharif and Ahmad [@14]-[@17] extended spherically symmetric gravitational collapse with positive cosmological constant for perfect fluid. The same authors [@18] have also investigated plane symmetric gravitational collapse using junction conditions which has been extended to spherically symmetric gravitational collapse [@19].
The behavior of electromagnetic field in strong gravitational field has been the subject of interest for the researchers over the past decades. According to Thirukkanesh and Maharaj [@20], the inclusion of electromagnetic field in gravitational collapse predicts that the gravitational attraction is counterbalanced by the Coulomb repulsive force along with the pressure gradient. Sharma et al. [@21] have concluded that electromagnetic field affects the value of red-shift, luminosity and mass of the relativistic compact objects. Nath et al. [@22] have studied the gravitational collapse of non-viscous, heat conducting fluid in the presence of electromagnetic field. They concluded that electromagnetic field reduces pressure and favors the formation of naked singularity.
Recently, we have studied the effect of electromagnetic field on the gravitational collapse by taking spherically symmetric spacetime as interior region and Reissner-Nordstr$\ddot{o}$m as exterior region of the star [@23]. The present article investigates the previous work by taking the Friedmann universe models in the interior of star. In order to preserve the generic properties of the Friedmann universe models in the presence of electromagnetic field, we follow [@24; @25] and assume that electromagnetic field is weak relative to matter, i.e., if $E^2$ is the electromagnetic field contribution in the system then $E^2<<\rho$. The main objectives of this work are the following:
- To study the physical interpretation of electromagnetic field and cosmological constant on gravitational collapse in the Friedmann universe models.
- To see the validity of CCH in this framework.
The plan of the paper is as follows: In the next section, the junction conditions are given. We discuss the solution of the Einstein-Maxwell field equations in section **3**. The apparent horizons and their physical significance are presented in section **4**. Section **5** presents the singularity analysis. We conclude our discussion in the last section.
The geometrized units (i.e., the gravitational constant $G$=1 and speed of light in vacuum $c=1$ so that $M\equiv\frac{MG}{c^2}$ and $\kappa\equiv\frac{8\pi G}{c^4}=8\pi$) are used. All the Latin and Greek indices vary from 0 to 3, otherwise, it will be mentioned.
Junction Conditions
===================
We derive conditions for the smooth matching of two regions (interior and exterior of a star) on the surface of discontinuity. For this purpose, we assume that $\Sigma$ be a timelike $3D$ hypersurface which divides two $4D$ manifolds $V^-$ and $V^+$ respectively. The interior manifold is taken as the Friedmann model $$\label{1}
ds_-^2=dt^2-a(t)^2[d\chi^2-f^2(d\theta^2+\sin\theta^2d\phi^2)],$$ where $f_k (\chi)$ is defined as $$f(\chi)=\begin{cases}
\sin\chi, &k=1\\
\chi, &k=0,\\
\sinh\chi, &k=-1,
\end{cases}$$ $k=1,0,-1$ correspond to closed, flat and open models respectively. $\chi (0\leq\chi<\infty$ for open and closed but $0\leq\chi<\pi$ for flat) is the hyper-spherical angle and $a(t)$ is the scale factor. Further, $\chi$ is related to radial coordinate $r$ as follows: $r=\sin\chi$ (closed), $r=\chi$ (flat) and $r=\sinh \chi$ (open). The Reissner-Nordstr$\ddot{o}$m de-Sitter spacetime is taken as the exterior manifold $$\label{2}
ds_+^2=ZdT^2-\frac{1}{Z}dR^2-R^2(d\theta^2+\sin\theta^2d\phi^2),$$ where $$\label{3}
Z(R)=1-\frac{2M}{R}+\frac{Q^2}{R^2}-\frac{\Lambda}{3}R^2,$$ $M$ and $\Lambda$ are constants and $Q$ is the charge.
The junction conditions are given as follows [@26]:
1. The continuity of first fundamental form over $\Sigma$ gives $$\label{4}
(ds^2_-)_{\Sigma}=(ds^2_+)_{\Sigma}=ds^2_{\Sigma}.$$
2. The continuity of second fundamental form (extrinsic curvature) over $\Sigma$ yields $$\label{5}
[K_{ij}]=K^+_{ij}-K^-_{ij}=0, \quad(i,j=0,2,3)$$ where $K_{ij}$ is the extrinsic curvature defined as
$$\label{6}
K^{\pm}_{ij}=-n^{\pm}_{\sigma}(\frac{{\partial}^2x^{\sigma}_{\pm}}
{{\partial}{\xi}^i{\partial}{\xi}^j}+{\Gamma}^{\sigma}_{{\mu}{\nu}}
\frac{{{\partial}x^{\mu}_{\pm}}{{\partial}x^{\nu}_{\pm}}}
{{\partial}{\xi}^i{\partial}{\xi}^j}),\quad({\sigma},
{\mu},{\nu}=0,1,2,3).$$
Here $\xi^0= t$, $\xi^2=\theta$, $\xi^3= \phi$ are the corresponding parameters on ${\Sigma }$, $x^{\sigma}_{\pm}$ stand for coordinates in $V^{\pm}$, the Christoffel symbols $\Gamma^{\sigma}_{{\mu}{\nu}}$ are calculated from the interior or exterior spacetimes and $n^{\pm}_{\sigma}$ are the components of outward unit normals to ${\Sigma}$ in the coordinates $x^{\sigma}_{\pm}$.
The equation of hypersurface in terms of interior spacetime $V^-$ coordinates is $$\label{8}
f_-(\chi,t)=\chi-\chi_{\Sigma}=0,$$ where $\chi_{\Sigma}$ is a constant as $\Sigma$ is a comoving surface forming the boundary of interior matter. Also, the equation of hypersurface in terms of exterior spacetime $V^+$ coordinates is given by $$\label{9}
f_+(R,T)=R-R_{\Sigma}(T)=0.$$ When we make use of Eq.(\[8\]) in Eq.(\[1\]), the metric on $\Sigma$ takes the form $$\label{10}
(ds_-^2)_\Sigma={dt^2-a(t)^2f(\chi_\Sigma)(d\theta^2+\sin\theta^2d\phi^2)}.$$ Also, Eqs.(\[9\]) and (\[2\]) yield $$\label{11}
(ds_+^2)_\Sigma=[Z(R_\Sigma)-\frac{1}{Z(R_\Sigma)}
(\frac{dR_\Sigma}{dT})^2]dT^2-R_\Sigma^2(d\theta^2+\sin\theta^2d\phi^2),$$ where we assume that $$\label{12}
Z(R_\Sigma)-\frac{1}{Z(R_\Sigma)} (\frac{dR_\Sigma}{dT})^2>0$$ so that T is a timelike coordinate. From Eqs.(\[4\]), (\[10\]) and (\[11\]), it follows that $$\begin{aligned}
\label{13}
R_\Sigma=(af)_\Sigma,\\\label{14}
[Z(R_\Sigma)-\frac{1}{Z(R_\Sigma)}
(\frac{dR_\Sigma}{dT})^2]^{\frac{1}{2}}dT=dt .\end{aligned}$$ Also, from Eqs.(\[8\]) and (\[9\]), the outward unit normals in $V^-$ and $V^+$, respectively, are given by $$\begin{aligned}
\label{15}
n^-_\mu&=&(0,a(t),0,0),\\
\label{16} n^+_\mu&=&(-\dot{R}_\Sigma,\dot{T}, 0,0).\end{aligned}$$
The components of extrinsic curvature $K^\pm_{ij}$ become $$\begin{aligned}
\label{17}
K^-_{00}&=&0,\\
\label{18}
K^-_{22}&=&\csc^2{\theta}K^-_{33}=({ff'}{a})_\Sigma,\\
\label{19}
K^+_{00}&=&(\dot{R}\ddot{T}-\dot{T}\ddot{R}-\frac{Z}{2}\frac{dZ}{dR}\dot{T}^3
+\frac{3}{2Z}\frac{dZ}{dR}\dot{T}\dot{R}^2)_\Sigma,\\
\label{20} K^+_{22}&=&\csc^2{\theta}
K^+_{33}=(ZR\dot{T})_{\Sigma},\end{aligned}$$ where dot and prime mean differentiation with respect to $t$ and $\chi$ respectively. From Eq.(\[5\]), the continuity of extrinsic curvature gives $$\begin{aligned}
\label{21}
K^+_{00}&=&0,\\
\label{22} K^+_{22}&=&K^-_{22}.\end{aligned}$$ Using Eqs.(\[17\])-(\[22\]) along with Eqs.(\[3\]), (\[13\]) and (\[14\]), the junction conditions become $$\begin{aligned}
\label{23}
\dot{(f')}_\Sigma=0,\\
\label{24} M=(\frac{af}{2}-\frac{\Lambda}{6}(af)^3+\frac{Q^2}{2af}
+\frac{{a\dot{a}^2}}{2}{f}^3-\frac{a}{2}{f}{f'}^2)_{\Sigma}.\end{aligned}$$ Equations (\[13\]), (\[14\]), (\[23\]) and (\[24\]) provide the necessary and sufficient conditions for the smooth matching of the two regions over $\Sigma$.
Solution of the Einstein Field Equations
========================================
In this section, we solve the Einstein field equations with cosmological constant for the Friedmann models containing the charged perfect fluid as the source of gravitation. The Einstein field equations with cosmological constant are given by ${\setcounter{equation}{0}}$ $$\label{25}
G_{\mu\nu}-{\Lambda}g_{\mu\nu}=\kappa(T_{\mu\nu}+T^{({em})}_{\mu\nu}).$$ The energy-momentum tensor for perfect fluid is $$\label{26}
{T_{{\mu}{\nu}}={({\rho}+p)}u_{\mu}u_{\nu}-pg_{\mu\nu}},$$ where $\rho$ is the energy density, $p$ is the pressure and $u_\mu=\delta^0_\mu$ is the four-vector velocity in co-moving coordinates. $T^{({em})}_{\mu\nu}$ is the energy-momentum tensor for the electromagnetic field given by $$\label{27}
T^{(em)}_{{\mu}{\nu}}=\frac{1}{4{\pi}}(-g^{{\delta}{\omega}}
F_{{\mu}{\delta}}F_{{\nu}{\omega}}+\frac{1}{4}g_{{\mu}{\nu}}
F_{{\delta}{\omega}}F^{{\delta}{\omega}}).$$ With the help of Eqs.(\[26\]) and (\[27\]), Eq.(\[25\]) takes the form $$\label{28}
R_{{\mu}{\nu}}=8\pi[({\rho}+p)u_{\mu}u_{\nu}
+\frac{1}{2}(p-{\rho})g_{{\mu}{\nu}}
+T^{({em})}_{{\mu}{\nu}}-\frac{1}{2}g_{{\mu}{\nu}}T^{({em})}]
-{\Lambda}g_{{\mu}{\nu}}.$$
Now we solve the Maxwell’s field equations $$\begin{aligned}
\label{29}
F_{\mu\nu}&=&\phi_{\nu,\mu}-\phi_{\mu,\nu},\\\label{30}
F^{\mu\nu}_{}{;\nu}&=&4{\pi}J^{\mu},\end{aligned}$$ where $\phi_{\mu}$ is the four potential and $J^{\mu}$ is the four current. Since the charge is at rest in this system, the magnetic field will be zero. Thus we can choose the four potential and four current as follows $$\begin{aligned}
\label{31}
\phi_{\mu}=({\phi}(t,r),0,0,0),\quad J^{\mu}={\sigma}u^{\mu},\end{aligned}$$ where $\sigma$ is charge density. Using Eqs.(\[29\]) and (\[31\]), the non-zero components of the field tensor are given as follows: $$\label{33}
F_{01}=-F_{10}=-\frac{\partial\phi}{\partial {\chi}}.$$ Also, from Eqs.(\[30\]), (\[31\]) and (\[33\]), we have $$\begin{aligned}
\label{34}
\frac{\partial^2\phi}{\partial {\chi}^2}+2
\frac{f'}{f}=4{\pi}{\sigma} a^2,\\ \label{35}
a\frac{\partial^2\phi}{\partial {\chi
\partial{t}}}+{\dot{a}}\frac{\partial\phi}{\partial {\chi}} =0.\end{aligned}$$
Integration of Eq.(\[36\]) implies that $$\label{36}
\frac{\partial{\phi}}{\partial {\chi}}= \frac{1}{af^2} q(\chi),$$ where $q(\chi) = 4{\pi} \int^{\chi}_0\sigma{a^3f^2d{\chi}}$, is the total charge distribution in the interior spacetime. This amount of charge is the consequence of law of conservation of charge, i.e., $J^\mu_{; \mu}=0$. It is clear that Eq.(\[35\]) is identically satisfied by Eq.(\[36\]). The electromagnetic field intensity is given by $$\label{37}
E=\frac{q}{(af)^2}.$$ Equations (\[36\]) and (\[37\]) yield $$\label{38}
\frac{\partial{\phi}}{\partial {\chi}}= aE.$$ Using Eqs. (\[33\]) and (\[38\]), we get
$$\label{39}
F_{01}=-F_{10}=-aE.$$
The non-zero components of $T^{(em)}_{{\mu}{\nu}}$ and its trace free form turn out to be $$\begin{aligned}
T^{(em)}_{{0}{0}}&=&\frac{1}{8{\pi}}E^2 ,\quad
T^{(em)}_{{1}{1}}=-\frac{1}{8{\pi}}E^2 a^2 ,\quad
T^{(em)}_{{2}{2}}=\frac{1}{8{\pi}}E^2(af)^2,\\
T^{(em)}_{{3}{3}}&=&T^{(em)}_{{2}{2}}\sin^2\theta,\quad
T^{(em)}=0.\end{aligned}$$ When we use these values, the field equations (\[28\]) for the interior spacetime takes the form $$\begin{aligned}
\label{42}
R_{00}&=&-3\frac{\ddot{a}}{a}=4\pi(\rho+3p)
+E^2-{\Lambda},\\
\label{43} R_{11}&=&-\frac{\ddot{a}}{a}-2 \frac{\dot{a}^2}{a^2}
+\frac{2}{{a}^2}\frac{f''}{f}={4\pi}(p-\rho)+E^2-{\Lambda} ,\\
\label{44} R_{22}&=&-\frac{\ddot{a}}{a}-(\frac{\dot{a}}{a})^2
+\frac{1}{a^2}[\frac{f''}{f}+(\frac{f'}{f})^2- \frac{1}{f^2}]
={4\pi}(p-\rho)-E^2-{\Lambda} ,\\
\label{45} R_{33}&=&{\sin}^2{\theta}R_{22},\end{aligned}$$
We would like to mention here that all the results are valid for $E^2 <<\rho$ and hence for stiff matter $(\rho=p),~E^2<<p$. Integrating Eq.(\[23\]) with respect to $t$, it follows that $$\label{46}
f'=W,$$ where $W=W(\chi)$ is an arbitrary function of $\chi$. The energy conservation equation $$\label{47}
T^{\nu}_{{\mu};{\nu}}=0$$ for the perfect fluid with the interior metric shows that pressure is a function of $t$ only, i.e., $$\label{48}
p=p(t).$$ Using the values of $f'$ and $p$ from Eqs.(\[46\]) and (\[48\]) in Eqs.(\[42\])-(\[44\]), it follows that $$\label{49}
2\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^2+\frac{(1-W^2)}{(af)^2}
=\Lambda+{E^2}-8\pi p(t).$$ We consider $p$ as a polynomial in $t$ as given by [@15] $$\label{50}
p(t)=p_c(\frac{t}{T})^{-s},$$ where $T$ is the constant time introduced in the problem due to physical reason by re-scaling of $t$, $p_c$ and $s$ are positive constants. Further, for simplicity, we take $s=0$ so that $$\label{51}
p(t)=p_c.$$ Now Eq.(\[49\]) gives $$\label{52}
2\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^2+\frac{(1-W^2)}{(af)^2}
=\Lambda+{E^2}-{8\pi}p_c.$$ For the static charges $E$ is taken as time independent [@31], so integration of above equation with respect to $t$, yields $$\label{53}
{\dot{a}}^2=\frac{W^2-1}{f^2}+(\Lambda+{E^2}-{8\pi}p_c)\frac{a^2}{3}+2\frac{m}{af^3},$$ where $m=m(\chi)$ is an arbitrary function of $\chi$ and is related to the mass of the collapsing system. Substituting Eqs.(\[46\]), (\[53\]) into Eq.(\[42\]), we get $$\label{54}
m'=\frac{2E'E}{3}(af)^3+{{a}^3{f'}{f^2}}[4\pi(p_c+{\rho})+2{E^2}].$$
For physical reasons, we assume that $(p_c+{\rho})\geqslant0$. Integrating Eq.(\[54\]) with respect to $\chi$, we obtain $$\label{55}
m(\chi)=4\pi{a^3}\int^{\chi}_0({\rho}+{p_c}){f'}{f^2}d{\chi}+2\int^{\chi}_0
E^2{f'}{f^2}d{\chi}+\frac{2}{3}a^3\int^{\chi}_0{E'E}f^3d
{\chi}+m_0,$$ where $m_0$ is taken equal to zero because of finite distribution of matter at the origin. The function $m(\chi)$ must be positive because $m(\chi)<0$ implies negative mass which is not physical. Using Eqs.(\[46\]) and (\[53\]) into the junction condition Eq.(\[24\]), it follows that $$\label{56}
M=\frac{Q^2}{2af}+m+\frac{1}{6}(\Lambda+E^2-{8\pi}p_c)(af)^3.$$ The total energy $\tilde{M}(\chi,t)$ at time $t$ inside the hypersurface $\Sigma$ can be evaluated by using the definition of mass function with the contribution of electromagnetic field for the Friedmann model, which is given by $$\label{58}
\tilde{M}(\chi,t)=\frac{1}{2}(af)(1+({\dot{a}f})^2-{f'}^2) +
\frac{q^2}{2af}.$$ Replacing Eqs.(\[46\]) and (\[53\]) in Eq.(\[58\]), we obtain $$\label{59}
\tilde{M}(r,t)=m(r)+(\Lambda+{E^2}-{8\pi}p_c)\frac{(af)^3}{6}+
\frac{q^2}{2af}.$$ From Eqs.(\[56\]) and (\[59\]), it can be found that $\tilde{M}(r,t)=^{\Sigma}M$ if and only if $q=Q$. This result provides the necessary and sufficient conditions for the continuity of mass in the interior and exterior regions over boundary surface $\Sigma$.
Now we take $(\Lambda+{E^2}-{8\pi}p_c)>0$ such that $E^2<<{8\pi}p$ and assume that $$\label{60}
W(\chi)=1.$$ In order to obtain the analytic solutions in closed form, we use Eqs.(\[46\]), (\[53\]) and (\[60\]) so that $$\label{61}
(af)=(\frac{6m}{\Lambda+{E^2}-{8\pi}p_c})^\frac{1}{3}{\sinh^\frac{2}{3}\alpha(\chi,t)}$$ where $$\label{62}
\alpha(\chi,t)=\frac{\sqrt{3(\Lambda+{E^2}-{8\pi}p_c)}}{2}[t_s(\chi)-t)].$$ Here $t_s(\chi)$ is an arbitrary function of $\chi$ and is related to the time of formation of singularity.
Apparent Horizons
=================
In this section, we discuss the formation of apparent horizons. The boundary of two trapped spheres whose outward normals are null is used to find the apparent horizons. Moreover, we discuss the the physical significance of apparent horizons i.e., area of apparent horizons, time difference between apparent horizons and singularity etc. For the interior spacetime, we find the boundary of two trapped spheres whose outward normals are null as follows: ${\setcounter{equation}{0}}$ $$\label{63}
g^{\mu\nu}(af)_{,\mu} (af)_{,\nu}=\dot{({af})}^2-({f'})^2=0.$$ Using Eqs.(\[46\]) and (\[53\]) in this equation, we get $$\label{64}
(\Lambda+{E^2}-{8\pi}p_c)(af)^3-3(af)+6m=0.$$ When $\Lambda=8\pi p_c-{E^2}$, we have $(af)=2m$. This is called Schwarzschild horizon. For $m=p_c=K=0$, we have $(af)=\sqrt{\frac{3}{\Lambda}}$, which is called de-Sitter horizon. Equation (\[64\]) can have the following positive roots.\
\
**Case (i)**: For $3m<\frac{1}{\sqrt{(\Lambda+{E^2}-{8\pi}p_c)}}$, we obtain two horizons $$\begin{aligned}
\label{65}
(af)_c&=&\frac{2}{\sqrt{(\Lambda+{E^2}-{8\pi}p_c)}}\cos\frac{\varphi}{3},\\
\label{66} (af)_b&=&\frac{-1}{\sqrt{(\Lambda+{8\pi}{E^2}-p_c)}}
(\cos\frac{\varphi}{3}-\sqrt{3}\sin\frac{\varphi}{3}),\end{aligned}$$ where $$\label{67}
\cos\varphi=-3m{\sqrt{(\Lambda+{E^2}-{8\pi}p_c)}}.$$ If we take $m=0$, it follows from Eqs.(\[65\]) and (\[66\]) that $(af)_c=\sqrt{\frac{3}{(\Lambda+{E^2}-{8\pi}p_c)}}$ and $(af)_b=0$. $(af)_c$ and $(af)_b$ are called cosmological horizon and black hole horizons respectively. For $m\neq0$ and $\Lambda\neq{8\pi}p_c-{E^2}$, $(af)_c$ and $(af)_b$ can be generalized [@27] respectively.\
\
**Case (ii):** For $3m=\frac{1}{\sqrt{(\Lambda+{E^2}-{8\pi}p_c)}}$, there is only one positive root which corresponds to a single horizon i.e., $$\label{68}
(af)_c=(af)_b=\frac{1}{\sqrt{(\Lambda+{E^2}-{8\pi}p_c)}}=(af)_{cb}.$$ This shows that both horizons coincide. The range for the cosmological and black hole horizon can be written as follows $$\label{69}
0\leq (af)_{b} \leq \frac{1}{\sqrt{(\Lambda+{E^2}-{8\pi}p_c)}}
\leq (af)_{c} \leq \sqrt{\frac{3}{(\Lambda+{E^2}-{8\pi}p_c)}} .$$ The black hole horizon has its largest proper area ${4\pi}(af)^2=\frac{4\pi}{(\Lambda+{E^2}-{8\pi}p_c)}$ and cosmological horizon has its area between $\frac{4\pi}{(\Lambda+{E^2}-{8\pi}p_c)}$ and $\frac{12\pi}{(\Lambda+{E^2}-{8\pi}p_c)}$.\
\
**Case (iii):** For $3m>\frac{1}{\sqrt{(\Lambda+{E^2}-{8\pi}p_c)}}$, there are no positive roots and consequently there are no apparent horizons.
We now calculate the time of formation of the apparent horizon using Eqs.(\[61\]) and (\[64\]) $$\label{70}
t_n=t_s-\frac{2}{\sqrt{3(\Lambda+{E^2}-{8\pi}p_c})}\sinh^{-1}
(\frac{(af)_n}{2m}-1)^{\frac{1}{2}}, \quad(n=1,2).$$ This implies that $$\label{71}
\frac{(af)_n}{2m}=\cosh^{2}\alpha_n,$$ where $\alpha_n(r,\chi)=\frac{\sqrt{3(\Lambda+{E^2}-{8\pi}p_c)}}{2}[t_s(\chi)-t_n)]$. Equations (\[61\]) and (\[68\]) give $(af)_{c}\geq (af)_{b}$ and $t_{b} \geq t_{c}$ respectively. The inequality $t_{b} \geq
t_{c}$ indicates that the cosmological horizon forms earlier than the black hole horizon. This condition confirms the formation of black hole.
The time difference between the formation of cosmological horizon and singularity and the formation of black hole horizon and singularity can be found as follows. Using Eqs.(\[65\])-(\[67\]), it follows that $$\begin{aligned}
\label{72}
\frac{d(\frac{(af)_c}{2m})}{dm}&=&\frac{1}{m}(-\frac{\sin\frac{\varphi}{3}}{\sin\varphi}
+\frac{3\cos\frac{\varphi}{3}}{\cos\varphi})<0,\\
\label{73}\frac{d(\frac{(af)_b}{2m})}{dm}
&=&\frac{1}{m}(-\frac{\sin\frac{(\varphi+4\pi)}{3}}{\sin\varphi}
+\frac{3\cos\frac{(\varphi+4\pi)}{3}}{\cos\varphi})>0.\end{aligned}$$ The time difference between the formation of singularity and apparent horizons is $$\label{74}
\tau_n=t_s-t_n.$$ It follows from Eq.(\[71\]) that $$\label{75}
\frac{d\tau_n}{d(\frac{Y_n}{2m})}
=\frac{1}{\sinh\alpha_n\cosh\alpha_n{\sqrt{3(\Lambda+{E^2}-{8\pi}p_c)}}}.$$ Using Eqs.(\[72\]) and (\[75\]), we get $$\begin{aligned}
\label{76}
\frac{d\tau_1}{dm}=\frac{d\tau_1}{d(\frac{(af)_c}{2m})}\frac{d(\frac{(af)_c}{2m})}{dm}
=\frac{1}{m{\sqrt{3(\Lambda+{E^2}-{8\pi}p_c)}}\sinh\alpha_1\cosh\alpha_1}\nonumber\\
\times(-\frac{\sin\frac{\varphi}{3}}{\sin\varphi}
+\frac{3\cos\frac{\varphi}{3}}{\cos\varphi})<0 .\end{aligned}$$ This means that time interval between the formation of cosmological horizon and singularity is decreased with the increase of mass. Similarly, from Eqs.(\[73\]) and(\[75\]), we get $$\begin{aligned}
\label{77}
\frac{d\tau_2}{dm}= \frac{1}{m{\sqrt{3
(\Lambda+{E^2}-{8\pi}p_c)}}\sinh\alpha_2\cosh\alpha_2}\nonumber\\
\times(-\frac{\sin\frac{(\varphi+4\pi)}{3}}{\sin\varphi}
+\frac{3\cos\frac{(\varphi+4\pi)}{3}}{\cos\varphi})>0.\end{aligned}$$ This indicates that time difference between the formation of black hole horizon and singularity is increased with the increase of mass.
Singularity Analysis
====================
The Riemann tensor is used to determine whether a singularity is essential or removable. If the curvature becomes infinite at certain point, then the singularity will be essential otherwise removable. Many scalars can be constructed from the Riemann tensor but symmetry assumption can be used to find only a finite number of independent scalars. Some of these are $$R_1=R=g^{ab}R_{ab},\quad R_2=R_{ab}R^{ab},\quad
R_3=R_{abcd}R^{abcd},\quad R_4=R^{ab}_{cd}R_{ab}^{cd}.$$ Here, we give the analysis for the first invariant commonly known as the Ricci scalar. For the Friedmann model, it is given as
$$\label{79}
R=\frac{-3a\ddot{a}f^2+2f''f-3\dot{a}^2f-1+f'^2}{a^2f}.$$
By definition $a>0$ and $\frac{\dot{a}}{a}>0$ [@28], it follows that curves of $a(t)$ versus $t$ must be concave downward and must reach $a(t)=0$ at some finite time in the past. Let us recall this time $t=0$ at which $R=\infty$. In cosmology, extrapolation of the universe expansion backwards in time yields an infinite density at finite past. Also, if the strong energy condition [@1] is satisfied, i.e., $\rho+p\geq 0$ and $(\rho+3p)\geq 0$ then $a=0$ at $t=0$ which implies the divergence of scalar curvature polynomial where $\rho\rightarrow\infty$. This is spacelike singularity usually called *big bang singularity* or *initial singularity* [@29].
Summary and Conclusion
======================
In this paper, we have analyzed the charged perfect fluid collapse with positive cosmological constant in the Friedmann models. For this purpose, we have found junction conditions between the Friedmann models and the Reissner-Nordstr$\ddot{o}$m de-Sitter spacetime. The junction conditions provide the gate way for the exact solution of the field equations with interior spacetime (Friedmann models). The solution of the field equations helps to discuss the dynamics of the collapsing system as follows:
The acceleration parameter $\ddot{a}/a$, given by Eq.(\[14\]), will be zero, positive or negative for $4\pi(\rho+3p)+E^2-{\Lambda}=0,~4\pi(\rho+3p)+E^2-{\Lambda}<0$ or $4\pi(\rho+3p) +E^2-{\Lambda}>0$ respectively. The variation of the scale factor $a(t)$ helps to describe the different stages of matter in the Friedmann models of the universe [@30]. If the scale factor $a(t)$ is decreasing, i.e., $\dot{a}(t)<0$ there will be collapsing (contracting) phase. For increasing scale factor i.e., $\dot{a}(t)>0$ we have the expanding phase while the point where $\dot{a}(t)=0$ corresponds to bounce point. Consequently, the Hubble parameter will be $H<0,~H>0$ and $H=0$ for collapsing, expanding and bouncing phases respectively. Also, we can conclude the following:
- The Newtonian force and acceleration of matter have the same value over the hypersurface $\Sigma$, i.e., ($-\frac{m}{(af)^2}+(\Lambda+{E^2}-{8\pi}p_c)\frac{(af)}{3})_\Sigma$ (see [@23] for detail). In this case, the repulsive force can only be generated if $\Lambda>({8\pi}p_c-{E^2})$ such that [${8\pi}p_c>>{E^2}$]{} over the entire range of the collapsing sphere. In the case of charged perfect fluid collapse with Tolman-Bondi spacetime [@23] there is no restriction on matter and electromagnetic field then the results are valid only for $\Lambda>({8\pi}p_c-{E^2})$ such that ${8\pi}p_c>{E^2}$. It is clear that in the first case the cosmological constant attains higher value than the later case. Thus the cosmological constant plays an effective role to slow down the collapse in the present case than previous one. In other words, isotropy and homogeneity of matter causes to introduce resistance against collapse in the presence of charge.
- Since the cosmological constant $\Lambda$ is affected by pressure and electromagnetic field, we can say that electromagnetic field reduces the effects of $\Lambda$ as compared to perfect fluid case by putting the restriction on $\Lambda$. Hence electromagnetic field increases the gravitational collapse as it decreases the repulsive force produced by $\Lambda$.
- Two physical horizons (cosmological and black hole horizons) are found whose area is decreased by cosmological constant and electromagnetic field. It follows from Eq.(\[70\]) that both horizons form earlier than singularity, so singularity is covered (back hole) and CCH seems to be valid in this case.
- Time difference between the formation of apparent horizon and singularity is decreased by electromagnetic field. Thus we can say that singularity must form earlier than the apparent horizons. Hence electromagnetic field favors the formation of naked singularity. But such situation can never occur because electromagnetic field does not play the dominant role in this case.
- It is found that the time difference between the formation of cosmological (black) horizon and singularity is decreasing (increasing) function of mass of the collapsing system.
[**Acknowledgment**]{}
We would like to thank the Higher Education Commission, Islamabad, Pakistan for its financial support through the [*Indigenous Ph.D. 5000 Fellowship Program Batch-IV*]{}.
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[^1]: [email protected]
[^2]: [email protected]
| |
---
abstract: 'We present two-loop results for the quark condensate in an external magnetic field within chiral perturbation theory using coordinate space techniques. At finite temperature, we explore the impact of the magnetic field on the pion-pion interaction in the quark condensate for arbitrary pion masses and derive the correct weak magnetic field expansion in the chiral limit. At zero temperature, we provide the complete two-loop representation for the vacuum energy density and the quark condensate.'
author:
- |
Christoph P. Hofmann$^a$\
\
\
\
title: Chiral Perturbation Theory Analysis of the Quark Condensate in a Magnetic Field
---
Introduction {#Intro}
============
The quark condensate – order parameter of spontaneous chiral symmetry breaking – is a crucial quantity in particle physics. It comes with no surprise that the relevant literature is extensive. Here we focus on the properties of the quark condensate in an external constant magnetic field. Our calculation within the framework of chiral perturbation theory (CHPT) goes up to two-loop order, but in contrast to the available CHPT-studies – see Refs. [@SS97; @AS00; @Aga00; @Aga01; @AS01; @CMW07; @Aga08; @Wer08; @And12a; @And12b] – we use a coordinate-space representation for the pion propagators and the associated kinematical functions. Other references, also dealing with the quark condensate in a magnetic field, are based upon lattice QCD [@EMS10; @EN11; @BBEFKKSS12a; @BBKKP12; @BBEFKKSS12b; @BBCCEKPS12c; @BBCKS14; @IMPS14; @EMNS18; @EGKKP19], feature analytical studies relying on the Nambu-Jona-Lasinio model [@GR11; @AA13; @FCMPS14; @FCLFP14; @FCP14; @FIPQ14; @ZFL16], or comprise yet other models and methods [@NK11; @FR11; @BEK13; @OS13; @CFS14; @OS14a; @HPS14; @MP15].
In a recent article, Ref. [@Hof19], the present author has pointed out that – in the chiral limit – the two published one-loop series for the finite-temperature quark condensate in a weak magnetic field, independently derived by different authors, are erroneous. The proper series at one-loop order has been established in Ref. [@Hof19] – one of our goals in the actual study is to review the situation at two-loop order. Indeed, errors also occur here. We clarify the situation by providing the correct weak magnetic field expansion of the finite-temperature quark condensate in the chiral limit. One of the advantages of our coordinate-space approach is that it allows for a transparent derivation of the various limits that have to be taken in the calculation: chiral limit ($M \to 0$) and weak magnetic field limit ($|qH| \ll T^2$).
Apart from straightening these issues, we also investigate the impact of the magnetic field on the pion-pion interaction in the quark condensate for arbitrary pion masses. At finite temperature, the interaction constitutes up to ten percent as compared to the leading noninteracting pion gas contribution, and is most pronounced in the chiral limit. When the magnetic field increases, the finite-temperature quark condensate (sum of one- and two-loop contribution at fixed temperature and pion mass) grows monotonically. The effect is again most pronounced in the chiral limit.
Using the dressed pions as pertinent degrees of freedom, the low-temperature series of the quark condensate is characterized by a $T^2$-contribution that refers to the dressed but non-interacting pions, while interaction effects emerge at order $T^4$. In the chiral limit and in weak magnetic fields, the series at [**order $\mathbf T^2$**]{} – organized by the expansion parameter $\epsilon = |qH|/T^2$ ($q$ is the electric charge of the pion) – involves a leading square-root term $\propto \sqrt{\epsilon}$, a term linear in $\epsilon$, followed by a half-integer power $\epsilon^{3/2}$ and a logarithmic contribution $\epsilon^2 \ln \epsilon$. The remaining contributions involve even powers of $\epsilon$. At [**order $\mathbf T^4$**]{} the series exhibits the same structure, with the exception that a term linear in $\epsilon$ is absent – in contrast to what has been reported in the literature.
Finally, we provide the two-loop representation for the QCD vacuum energy density and the zero-temperature quark condensate. The representation involves nonanalytic contributions in the form of logarithms, as well as Gamma and Polygamma functions that depend nontrivially on the ratio between magnetic field and pion mass. In contrast to previous studies we provide the full two-loop representation – not merely the terms that are induced by the nonzero magnetic field.
The article is organized as follows. The two-loop CHPT evaluation is briefly reviewed in Sec. \[CHPT\] to set the basis for the subsequent analysis. In Sec. \[quarkCondensate\] we explore the quark condensate at finite and zero temperature for arbitrary pion masses – in particular also for the physical pion masses – in presence of a magnetic field. In the same section we furthermore compare our findings with the literature and point out errors in the published results. Finally, Sec. \[conclusions\] contains our conclusions. More technical issues are presented in three appendices. In Appendix \[appendixA\] we discuss in detail the two-loop CHPT evaluation at zero temperature. While Appendix \[appendixB\] is devoted to the chiral limit in nonzero magnetic fields at $T$=0, in Appendix \[appendixC\] we consider the same situation at finite temperature which boils down to an analysis of the various kinematical functions required.
Chiral Perturbation Theory Evaluation {#CHPT}
=====================================
The relevant low-energy excitations in two-flavor chiral perturbation theory[^1] are the three pions that are incorporated in the SU(2) matrix $U(x)$ as $$U(x) =\exp(i \tau^i \pi^i(x)/F) \, , \qquad i=1,2,3 \, .$$ Here $\tau^i$ are the Pauli matrices and $F$ stands for the tree-level pion decay constant. While $\pi^0$ describes the neutral pion[^2], the charged pions correspond to the linear combinations $$\pi^\pm = \frac{1}{\sqrt{2}} \Big( \pi^1 \pm i\pi^2 \Big) \, .$$ The Euclidean leading-order (order $p^2$) effective Lagrangian is given by $$\label{L2}
{\cal L}^2_{eff} = \mbox{$ \frac{1}{4}$} F^2 Tr \Big[ {(D_{\mu} U)}^\dagger (D_{\mu} U) - M^2 (U + U^\dagger) \Big] \, ,$$ where $M$ is the tree-level pion mass. In the covariant derivative, $$D_{\mu} U = \partial_\mu U + i [Q,U] A^{EM}_\mu \, ,$$ the quantity $Q$ is the charge matrix of the quarks, i.e., $Q=diag(2/3,-1/3)e$, while the magnetic field $H$ enters via the gauge field $A^{EM}_\mu=(0,0,-H x_1,0)$. As illustrated in Fig. \[figure1\], a two-loop calculation of the free energy density in addition involves the subleading pieces ${\cal L}^4_{eff}$ and ${\cal L}^6_{eff}$ of the effective Lagrangian.
![Chiral perturbation theory diagrams for the QCD free energy density up to order $p^6$. Vertices from ${\cal L}^2_{eff}$ (filled circles), as well as vertices from ${\cal L}^4_{eff}$ and ${\cal L}^6_{eff}$ (denoted by the numbers $4$ and $6$) contribute. The lines refer to the thermal pion propagators.[]{data-label="figure1"}](AF3Dfig1.eps){width="15cm"}
The set of terms proportional to four pion fields generated by the leading piece ${\cal L}^2_{eff}$ – as required for the evaluation of the two-loop diagram 6A – are $$\begin{aligned}
{\cal L}^2_{\{4\}} & = & \frac{1}{3 F^2} \, \pi^0 \partial_{\mu} \pi^0 \Big( \partial_{\mu} \pi^+ \pi^- + \partial_{\mu} \pi^- \pi^+ \Big)
- \frac{1}{3 F^2} \, \partial_{\mu} \pi^0 \partial_{\mu} \pi^0 \pi^+ \pi^- \nonumber \\
& & - \frac{1}{3 F^2} \, \pi^0 \pi^0 \partial_{\mu} \pi^+ \partial_{\mu} \pi^-
- \frac{1}{3 F^2} \, \pi^+ \pi^- \partial_{\mu} \pi^+ \partial_{\mu} \pi^- \nonumber \\
& & + \frac{1}{6 F^2} \, \Big(\partial_{\mu} \pi^+ \pi^- \partial_{\mu} \pi^+ \pi^- + \partial_{\mu} \pi^- \pi^+ \partial_{\mu} \pi^- \pi^+\Big)
\, .\end{aligned}$$ Other pieces from ${\cal L}^2_{eff}$ needed for our calculation are terms with two (diagram 4A) or zero (diagram 2) pion fields, $$\begin{aligned}
{\cal L}^2_{\{2\}} & = & \mbox{$ \frac{1}{2}$} \partial_{\mu} \pi^0 \partial_{\mu} \pi^0 + \partial_{\mu} \pi^+ \partial_{\mu} \pi^-
+ \mbox{$ \frac{1}{2}$} M^2 \pi^0 \pi^0 + M^2 \pi^+ \pi^- \, , \nonumber \\
{\cal L}^2_{\{0\}} & = & - F^2 M^2 \, .\end{aligned}$$
As for the subleading piece ${\cal L}^4_{eff}$, we use the representation given in Eq. (D.2) of Ref. [@Sch03]. The relevant terms for our calculation are those that contain two (diagram 6B) or zero (diagram 4B) pion fields, $$\begin{aligned}
{\cal L}^4_{\{2\}} & = & l_3 \frac{M^4}{F^2} \, \pi^0 \pi^0 + 2l_3 \frac{M^4}{F^2} \, \pi^+ \pi^-
+ (4l_5 - 2l_6 ) \frac{{|qH|}^2}{F^2} \, \pi^+ \pi^- \, , \nonumber \\
{\cal L}^4_{\{0\}} & = & -(l_3 + h_1) M^4 + 4 h_2 {|qH|}^2 \, .\end{aligned}$$ The quantities $l_3, l_5, l_6, h_1, h_2$ are next-to-leading order (NLO) low-energy effective constants.
Finally, following Ref. [@BCE00], the terms from ${\cal L}^6_{eff}$ contributing to the tree-level diagram 6C read $${\cal L}^6_{\{0\}} = - 16 (c_{10} + 2 c_{11}) M^6 - 8 c_{34} M^2 |qH|^2 \, ,$$ where $c_{10}, c_{11}, c_{34}$ are next-to-next-to-leading order (NNLO) low-energy effective constants.
It is convenient to divide the free energy density into two pieces as $$z = z_0 + z^T \, ,$$ where $z_0$ contains all $T$=0 contributions (vacuum energy density), and $z^T$ involves the finite-temperature part – both terms depend on the magnetic field. Before addressing the $T$=0 case, we quote the result for the finite-temperature piece which has been derived within the CHPT coordinate-space approach up to two-loop order in Ref. [@Hof20]: $$\begin{aligned}
\label{fedPhysicalM}
z^T & = & - g_0(M^{\pm}_{\pi},T,0) -\mbox{$ \frac{1}{2}$} g_0(M^0_{\pi},T,0)- {\tilde g}_0(M^{\pm}_{\pi},T,H) \nonumber \\
& & + \frac{M^2_{\pi}}{2 F^2} \, g_1(M^{\pm}_{\pi},T,0) \, g_1(M^0_{\pi},T,0)
- \frac{M^2_{\pi}}{8 F^2} \, {\Big\{ g_1(M^0_{\pi},T,0) \Big\}}^2 \nonumber \\
& & + \frac{M^2_{\pi}}{2 F^2} \, g_1(M^0_{\pi},T,0) \, {\tilde g}_1(M^{\pm}_{\pi},T,H) + {\cal O}(p^8) \, .\end{aligned}$$ The kinematical Bose functions are defined as $$\begin{aligned}
\label{boseFunctions}
g_0({\cal M},T,0) & = & T^4 \, {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \rho^{-3} \, \exp\Big( -\frac{{\cal M}^2}{4 \pi T^2}
\rho \Big) \Bigg[ S\Big( \frac{1}{\rho} \Big) -1 \Bigg] \, , \nonumber \\
g_1({\cal M},T,0) & = & \frac{T^2}{{4 \pi}} \, {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \rho^{-2} \, \exp\Big( -\frac{{\cal M}^2}{4 \pi T^2}
\rho \Big) \Bigg[ S\Big( \frac{1}{\rho} \Big) -1 \Bigg] \, , \nonumber \\
{\tilde g}_0(M^{\pm}_{\pi},T,H) & = & \frac{T^2}{{4 \pi}} \, |qH| {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \rho^{-2} \,
\Bigg( \frac{1}{\sinh(|qH| \rho /4 \pi T^2)} - \frac{4 \pi T^2}{|qH| \rho} \Bigg) \nonumber \\
& & \times \, \exp\Big( -\frac{{(M^{\pm}_{\pi})}^2}{4 \pi T^2} \rho \Big) \Bigg[ S\Big( \frac{1}{\rho} \Big) -1 \Bigg] \nonumber \\
{\tilde g}_1(M^{\pm}_{\pi},T,H) & = & \frac{1}{16 \pi^2} \, |qH| {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \rho^{-1} \,
\Bigg( \frac{1}{\sinh(|qH| \rho /4 \pi T^2)} - \frac{4 \pi T^2}{|qH| \rho} \Bigg) \nonumber \\
& & \times \, \exp\Big( -\frac{{(M^{\pm}_{\pi})}^2}{4 \pi T^2} \rho \Big) \Bigg[ S\Big( \frac{1}{\rho} \Big) -1 \Bigg] \, ,\end{aligned}$$ and $S(z)$ stands for the Jacobi theta function, $$S(z) = \sum_{n=-\infty}^{\infty} \exp(- \pi n^2 z) \, .$$ Note that ${\tilde g}_0$ and ${\tilde g}_1$ explicitly depend on the magnetic field through the hyperbolic sine and that they involve the mass $M^{\pm}_{\pi}$, i.e., the masses of the charged pions in a magnetic field given by $$\label{chargedPionMass}
{(M^{\pm}_{\pi})}^2 = M^2_{\pi} + \frac{{\overline l}_6 - {\overline l}_5}{48 \pi^2} \, \frac{{|qH|}^2}{F^2} \, .$$ The mass $\cal M$ in $g_0$ and $g_1$, according to Eq. (\[fedPhysicalM\]), can either represent $M^{\pm}_{\pi}$ or $M^0_{\pi}$, where the latter is the mass of the neutral pion in a magnetic field, $$\label{neutralPionMass}
{(M^0_{\pi})}^2 = M^2_{\pi} + \frac{M^2}{F^2} \, K_1 \, ,$$ and $K_1$ denotes the integral $$\label{intK1}
K_1 = \frac{|qH|}{16 \pi^2} \, {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \, \rho^{-1} \, \exp\Big( -\frac{M^2_{\pi}}{|qH|} \rho \Big) \,
\Big( \frac{1}{\sinh(\rho)} - \frac{1}{\rho} \Big) \, .$$ The kinematical functions $g_0$ and $g_1$ hence implicitly depend on the magnetic field through the neutral and charged pion masses. Finally, the mass $M_{\pi}$ is the renormalized NLO pion mass in zero magnetic field, $$\label{Mpi}
M^2_{\pi} = M^2 - \frac{{\overline l}_3}{32 \pi^2} \, \frac{M^4}{F^2} + {\cal O}(M^6) \, .$$ The quantities ${\overline l}_3, {\overline l}_5, {\overline l}_6$ are renormalized NLO low-energy effective constants – details are provided in Appendix \[appendixA1\].
We now address the zero-temperature part in the free energy density[^3]. Apart from the temperature-independent tree-level graphs 2, 4B and 6C, we also have $T$=0 contributions from the loop graphs. This is because the thermal propagators for the pions, $$\begin{aligned}
\label{ThermalPropagator}
G^{\pm}(x) & = & \sum_{n = - \infty}^{\infty} \Delta^{\pm}({\vec x}, x_4 + n \beta) \, , \nonumber \\
G^0(x) & = & \sum_{n = - \infty}^{\infty} \Delta^0({\vec x}, x_4 + n \beta) \, , \qquad \beta = \frac{1}{T} \, ,\end{aligned}$$ contain a zero-temperature piece associated with $n$=0. In Appendix \[appendixA2\] we process these $T$=0 contributions and show that all UV-divergences cancel. The final result for the renormalized the vacuum energy density at order $p^6$ then amounts to $$\begin{aligned}
\label{freeEDp6ZeroT}
z^{[6]}_0 & = & \frac{3{\overline l}_3 ({\overline c}_{10} + 2 {\overline c}_{11})}{1024 \pi^4} \, \frac{M^6}{F^2}
- \frac{({\overline l}_6 - {\overline l}_5){\overline c}_{34}}{768 \pi^4} \, \frac{{|qH|}^2 M^2}{F^2} \nonumber \\
& & - \frac{{\overline l}_3}{32 \pi^2} \, \frac{M^4}{F^2} \, K_1
+ \frac{({\overline l}_6 - {\overline l}_5)}{48 \pi^2} \, \frac{{|qH|}^2}{F^2} \, K_1 \, .\end{aligned}$$ The quantities ${\overline l}_i$ and ${\overline c}_i$ are the renormalized NLO and NNLO effective constants, defined in Appendix \[appendixA1\].
The full vacuum energy density also includes the zero-temperature pieces of order $p^4$ and $p^2$, $$z_0 = z^{[6]}_0 + z^{[4]}_0 + z^{[2]}_0 \, ,$$ which are (see Ref. [@Hof19] for $z^{[4]}_0$), $$\begin{aligned}
\label{freeEDp4ZeroT}
z^{[4]}_0 & = & \frac{M^4}{64 \pi^2} \, \Big({\overline l_3} - 4{\overline h_1} - \frac{3}{2}\Big) + \frac{{|qH|}^2}{96 \pi^2} \,
( {\overline h_2} - 1) \nonumber \\
& & - \frac{{|qH|}^2}{16 \pi^2} {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \rho^{-2} \Big( \frac{1}{\sinh(\rho)} - \frac{1}{\rho} + \frac{\rho}{6}
\Big) \, \exp\!\Big( -\frac{M^2}{|qH|} \rho \Big) \, , \nonumber \\
z^{[2]}_0 & = & - F^2 M^2 \, .\end{aligned}$$ The subleading contributions $z^{[4]}_0$ and $z^{[6]}_0$ as displayed above, i.e., the renormalized expressions, are independent of the renormalization scale $\mu$. This is a nontrivial consistency check of our calculation. We now turn to the quark condensate which is the main subject of the present investigation.
Quark Condensate in a Magnetic Field {#quarkCondensate}
====================================
The quark condensate is given by the derivative of the free energy density with respect to the quark mass[^4] $$\langle {\bar q} q \rangle = \frac{\mbox{d} z}{\mbox{d} m} \, .$$ At zero temperature it corresponds to the vacuum expectation value $$\langle 0 | {\bar q} q | 0 \rangle = \frac{\mbox{d} z_0}{\mbox{d} m}
= -\frac{{\langle 0 | {\bar q} q | 0 \rangle}_0}{F^2} \, \frac{\mbox{d} z_0}{\mbox{d} M^2} \, .$$ Note that we have used the leading-order Gell-Mann–Oakes–Renner relation $$M^2 = -\frac{m}{F^2 } \, {\langle 0 | {\bar q} q | 0 \rangle}_0 \, ,$$ where the quantity ${\langle 0 | {\bar q} q | 0 \rangle}_0$ is the quark condensate at $T$=0 (and zero magnetic field) in the chiral limit – as indicated by the lower index “0”. The purely finite-temperature part in the quark condensate amounts to $${\langle {\bar q} q \rangle}^T = - \frac{\mbox{d} P}{\mbox{d} m}
= \frac{{\langle 0 | {\bar q} q | 0 \rangle}_0}{F^2} \, \frac{\mbox{d} P}{\mbox{d} M^2} \, .$$ Up to the sign, the pressure is nothing but the finite-temperature piece in the free energy density, $$P = -z^T \, .$$ In the representation of $z^T$, Eq. (\[fedPhysicalM\]), we have used the NLO renormalized pion mass $M_{\pi}$ instead of $M$. The connection between the two quantities is given by Eq. (\[Mpi\]). For the quark condensate we then obtain $$\langle {\bar q} q \rangle = \frac{{\langle 0 | {\bar q} q | 0 \rangle}_0}{F^2} \,
\Bigg\{\! -\frac{\mbox{d} z_0}{\mbox{d} M^2_{\pi}} + \frac{\mbox{d} P}{\mbox{d} M^2_{\pi}} \Bigg\} \Bigg( 1 - \frac{ M^2_{\pi}}{32 \pi^2 F^2} \,
(2 {\overline l}_3 -1) \Bigg) \, .$$ In the parenthesis we have replaced $M^2$ by $M^2_{\pi}$ which is legitimate at the order we are operating.
It should be pointed out that the zero-temperature quark condensate at order $p^4$, according to Eq. (\[freeEDp4ZeroT\]), involves the NLO effective constant $\overline h_1$ which depends on the renormalization convention (see Ref. [@GL84]). No such ambiguities due to NLO effective constants $\overline h_i$ are introduced in the zero-temperature quark condensate at order $p^6$, according to Eq. (\[freeEDp6ZeroT\]). Likewise, the finite-temperature part of the quark condensate is also free of such renormalization ambiguities.
Finite-Temperature Quark Condensate
-----------------------------------
In order to make powers of temperature in the quark condensate manifest, instead of operating with the Bose functions $g_r$ and ${\tilde g}_r$, we now work with the dimensionless functions $h_r$ and ${\tilde h}_r$ defined as $$\label{conversion}
h_0 = \frac{g_0}{T^4} \, , \quad {\tilde h}_0 = \frac{{\tilde g}_0}{T^4} \, , \qquad
h_1 = \frac{g_1}{T^2} \, , \quad {\tilde h}_1 = \frac{{\tilde g}_1}{T^2} \, , \qquad
h_2 = g_2 \, , \quad {\tilde h}_2 = {\tilde g}_2 \, .$$ With the expression for $z^T$, Eq. (\[fedPhysicalM\]), the finite-temperature part of the quark condensate takes the form $$\frac{{\langle {\bar q} q \rangle}^T}{{\langle 0 | {\bar q} q | 0 \rangle}_0} \, {\Bigg( 1 - \frac{ M^2_{\pi}}{32 \pi^2 F^2} \,
(2 {\overline l}_3 -1)\Bigg)}^{-1}
= - \Big\{ \frac{q_1}{F^2} T^2 + \frac{q_2}{F^4} T^4 + {\cal O}(T^6) \Big\} \, .$$ The respective coefficients, $$\begin{aligned}
q_1 & = & h_1(M^{\pm}_{\pi},T,0) + \mbox{$ \frac{1}{2}$} a_0 h_1(M^0_{\pi},T,0) + {\tilde h}_1(M^{\pm}_{\pi},T,H) \, , \nonumber \\
q_2 & = & + \mbox{$ \frac{1}{2}$} h_1(M^{\pm}_{\pi},T,0) h_1(M^0_{\pi},T,0)
+ \mbox{$ \frac{1}{2}$} h_1(M^0_{\pi},T,0) {\tilde h}_1(M^{\pm}_{\pi},T,H) \nonumber \\
& & - \mbox{$ \frac{1}{8}$} h_1(M^0_{\pi},T,0) h_1(M^0_{\pi},T,0)
- \mbox{$ \frac{1}{2}$} \frac{m^2}{t^2} h_1(M^0_{\pi},T,0) h_2(M^{\pm}_{\pi},T,0) \nonumber \\
& & - \mbox{$ \frac{1}{2}$} a_0 \frac{m^2}{t^2} h_1(M^{\pm}_{\pi},T,0) h_2(M^0_{\pi},T,0)
- \mbox{$ \frac{1}{2}$} a_0 \frac{m^2}{t^2} {\tilde h}_1(M^{\pm}_{\pi},T,H) h_2(M^0_{\pi},T,0) \nonumber \\
& & + \mbox{$ \frac{1}{4}$}a_0 \frac{m^2}{t^2} h_1(M^0_{\pi},T,0) h_2(M^0_{\pi},T,0)
- \mbox{$ \frac{1}{2}$} \frac{m^2}{t^2} h_1(M^0_{\pi},T,0) {\tilde h}_2(M^{\pm}_{\pi},T,H) \, ,\end{aligned}$$ depend in a nontrivial way on the ratios between pion masses, magnetic field and temperature. The NLO mass correction $a_0$ is $$a_0 = \frac{\mbox{d} {(M^0_{\pi})}^2 }{\mbox{d} M^2_{\pi}} = 1 + \frac{K_1}{F^2}
+ \frac{M^2_{\pi}}{F^2} \, \frac{\mbox{d} K_1}{\mbox{d} M^2_{\pi}} \, ,$$ with the integral $\mbox{d} K_1/\mbox{d} M^2_{\pi}$ given by $$\frac{\mbox{d} K_1}{\mbox{d} M^2_{\pi}} = - \frac{1}{16 \pi^2} \, {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \, \exp\Big( -\frac{M^2_{\pi}}{|qH|}
\rho \Big) \, \Big( \frac{1}{\sinh(\rho)} - \frac{1}{\rho} \Big) \, .$$ The coefficient $q_1$ refers to the free pion gas contribution of order $T^2$, while the coefficient $q_2$ captures the pion-pion interaction that emerges at order $T^4$ in the finite-temperature quark condensate.
To asses the magnitude of the interaction, in Fig. \[figure2\], we plot the dimensionless ratio $$\label{xiquark}
\xi_{q}(t,m,m_H) = \frac{q_2 T^2}{q_1 F^2}$$ that measures the effect of the pion-pion interaction in the quark condensate relative to the free pion gas contribution. The dimensionless quantities $t, m$, and $m_H$, $$t = \frac{T}{4 \pi F} \, , \qquad m = \frac{M_{\pi}}{4 \pi F} \, , \qquad m_H = \frac{\sqrt{|qH|}}{4 \pi F} \, ,$$ that we use in the figures, capture temperature, pion mass, and strength of the magnetic field relative to the chiral symmetry breaking scale $\Lambda_{\chi} \approx 4 \pi F \approx \, 1 GeV$. The quantities $t, m$, and $m_H$ must be small since chiral perturbation theory is a low-energy effective theory. Inspecting Fig. \[figure2\] – where we have chosen $T= 108 \, MeV$ and $T= 215 \, MeV$ as well as $m, m_H \le 0.4$ – one notices that the interaction is largest in the chiral limit ($m \to 0$) when no magnetic field is present or when the magnetic field becomes stronger. The effect of the interaction is not tiny – rather it may constitute up to about ten percent relative to the leading free pion gas contribution.
In Fig. \[figure3\], we depict the sum of one- and two-loop contribution, i.e., the dimensionless quantity $$- \Big( q_1 + q_2 \frac{T^2}{F^2} \Big) \, ,$$ for the same two temperatures $T= \{ 108 \, MeV, 215 \, MeV \}$, or, $t= \{ 0.1, 0.2 \}$. As the plots indicate – at fixed $M_{\pi}$ and temperature – the finite-temperature quark condensate increases when the magnetic field grows. The effect is most pronounced in the chiral limit ($m \to 0$).
Let us examine the real world, where the pion masses are fixed at their physical values $M_{\pi} = 140 \, MeV$ ($m = 0.130$)[^5]. In Fig. \[figure4\], on the LHS, we plot the ratio $\xi_{q}$ as a function of temperature and magnetic field strength. The effect of the pion-pion interaction is less than ten percent in the parameter range $t, m_H \le 0.25$ ($T \le 269 \, MeV, \sqrt{|qH|} \le 269 \, MeV$) we are considering. Finally, on the RHS of Fig. \[figure4\], we depict the sum of one- and two-loop contribution in the quark condensate for the same parameter domain. One observes that the finite-temperature quark condensate slightly increases when the strength of the magnetic field grows while temperature is held constant. This effect however is small.
In the chiral limit, the finite-temperature quark condensate reduces to $$\begin{aligned}
\label{quarkCondensateTChiralLimit}
\frac{{\langle {\bar q} q \rangle}^T}{{\langle 0 | {\bar q} q | 0 \rangle}_0} & = &
- \frac{T^2}{F^2} \, \Bigg\{ \frac{1}{24} + h_1(M_H,T,0) - \Big( \frac{|qH| \ln 2}{32 \pi^2 F^2} \Big) \, h_1(0,T,0)
+ {\tilde h_1}(M_H,T,H) \Bigg\} \nonumber \\
& & + \frac{T^4}{24 F^4} \, \Bigg\{ \frac{1}{48} - h_1(M_H,T,0) - {\tilde h}_1(M_H,T,H) \Bigg\} + {\cal O}( T^6) \, .\end{aligned}$$ The mass $M_H$ depends on the magnetic field, $$M^2_H = \frac{{\overline l}_6 - {\overline l}_5}{48 \pi^2} \, \frac{{|qH|}^2}{F^2} \, ,$$ and corresponds to the charged pion mass in the chiral limit. The mass of the neutral pion, on the other hand, tends to zero in the chiral limit[^6].
We now address the question of how the quark condensate in the chiral limit behaves in weak magnetic fields. In this limit – implemented by $|qH| \ll T^2$ – we have to expand the kinematical functions $h_1(M_H,T,0)$ and ${\tilde h}_1(M_H,T,H)$ in Eq. (\[quarkCondensateTChiralLimit\]) in the magnetic-field dependent mass $M_H$, which leads to $$\begin{aligned}
\label{expansionMH}
h_1(M_H,T,0) & = & h_1(0,T,0) - \alpha \epsilon^2 h_2(0,T,0) + \frac{\alpha^2 \epsilon^4}{2!} \, h_3(0,T,0) + {\cal O}(h_4) \, , \nonumber \\
{\tilde h}_1(M_H,T,H) & = & {\tilde h}_1(0,T,H) - \alpha \epsilon^2 {\tilde h}_2(0,T,H)
+ \frac{\alpha^2 \epsilon^4}{2!} \, {\tilde h}_3(0,T,H) + {\cal O}( {\tilde h}_4) \, ,\end{aligned}$$ with $$\alpha = \frac{{\overline l}_6 - {\overline l}_5}{3} \, t^2 \, , \qquad t = \frac{T}{4 \pi F} \, .$$ The structure of this infinite series of kinematical functions is analyzed in Appendix \[appendixC\]. While the functions $h_1(0,T,0)$ and ${\tilde h}_1(0,T,H)$ are well-defined, it should be pointed out that for $r=2,3,4, \dots$, the functions $h_r(0,T,0)$ and ${\tilde h}_r(0,T,H)$ generate various types of divergences in the weak magnetic field expansion parameter $\epsilon$. The notation $h_2(0,T,0), {\tilde h}_2(0,T,H), h_3(0,T,0), {\tilde h}_3(0,T,H), \dots$ is therefore symbolic: it is understood that these functions contain inverse powers of $\epsilon$ as well as logarithms $\ln \epsilon$. These pieces – according to Eq. (\[expansionMH\]) – are then multiplied by even powers of $\epsilon$, in such a way that all divergences ultimately disappear in the quark condensate, as we show in Appendix \[appendixC\]. The outcome is the following series for the finite-temperature two-loop quark condensate in the chiral limit and in weak magnetic fields: $$\begin{aligned}
\label{condensateMySeries}
\frac{{\langle {\bar q} q \rangle}^T}{{\langle 0 | {\bar q} q | 0 \rangle}_0} & = & - \frac{1}{8 F^2} \, T^2
+ \frac{1}{F^2} \, \Bigg\{ \frac{|I_{\frac{1}{2}}|}{8 \pi^{3/2}} \, \sqrt{\epsilon}
-\frac{\ln 2}{16 \pi^2} \, \epsilon \nonumber \\
& & - \frac{\sqrt{2} -4}{8} \, \gamma \, \zeta(\mbox{$ \frac{3}{2}$})\, \epsilon^{3/2}
+ \frac{\gamma}{4 \pi} \, \epsilon^2 \ln \epsilon
+ {\cal O}(\epsilon^2) \Bigg\} \, T^2 \nonumber \\
& & - \frac{1}{384 F^4} \, T^4 + \frac{1}{F^4} \, \Bigg\{ \frac{|I_{\frac{1}{2}}|}{192 \pi^{3/2}} \, \sqrt{\epsilon} \nonumber \\
& & - \frac{\sqrt{2} -4}{192} \, \gamma \, \zeta(\mbox{$ \frac{3}{2}$}) \, \epsilon^{3/2}
+ \frac{\gamma}{96 \pi} \, \epsilon^2 \ln \epsilon
+ {\cal O}(\epsilon^2) \Bigg\} \, T^4 + {\cal O}(T^6) \, .\end{aligned}$$ Recall that $\epsilon$, $$\epsilon = \frac{|qH|}{T^2} \, ,$$ is the relevant expansion parameter, while the other quantities are $$\begin{aligned}
I_{\frac{1}{2}} & = & {\int}_{\!\!\! 0}^{\infty} \, d\rho \rho^{-1/2} \Big( \frac{1}{\sinh(\rho)} - \frac{1}{\rho} \Big) \approx -1.516256 \, ,
\nonumber \\
\gamma & = & \frac{{\overline l}_6 - {\overline l}_5}{12 \pi} \, t^2 \, , \qquad t = \frac{T}{4 \pi F} \, .\end{aligned}$$ The first two lines of Eq. (\[condensateMySeries\]) refer to one-loop order ($\propto T^2$), while the remaining two lines represent two-loop corrections ($\propto T^4$). In the chiral limit, the series for the finite-temperature quark condensate in weak magnetic fields is thus characterized by square-root terms $\propto \sqrt{\epsilon}$, a term linear in $\epsilon$, followed by half-integer powers $\epsilon^{3/2}$ and logarithmic contributions of the form $\epsilon^2 \ln \epsilon$. The remaining contributions involve even powers of $\epsilon$. Notice that the leading corrections – proportional to $\sqrt{\epsilon}$ – come with a positive sign: in the chiral limit, as already illustrated by Fig. \[figure3\], the finite-temperature quark condensate grows if the magnetic field is switched on.
The published results in Refs. [@Aga00; @Aga01; @AS01; @Aga08; @And12a; @And12b] do not quite agree with the above representation. The correct series at one-loop order has been derived and discussed in Ref. [@Hof19]. The two-loop contribution in nonzero magnetic fields, displayed in the second brace of Eq. (\[condensateMySeries\]), again differs from the published two-loop result, Eq.(5.8) of Ref. [@And12b]: the term $$\frac{5 \sqrt{|qH|} T^3}{1536 \pi F^4}$$ in Eq.(5.8) of Ref. [@And12b] should rather read $$\frac{\sqrt{|qH|} T^3}{192 \pi^{3/2} F^4} \, |I_{\frac{1}{2}}| \, .$$ The numerical discrepancy is $$\frac{5}{1536 \pi} \approx 0.00103616 \, , \qquad \frac{|I_{\frac{1}{2}}|}{192 \pi^{3/2}} \approx 0.00141823 \, .$$ Moreover, a term linear in $\epsilon$ in the second brace of Eq. (\[condensateMySeries\]) does not emerge in our expansion – contradicting the result announced in Ref. [@And12b]. It should be emphasized that the series provided in the literature is restricted to linear order in $\epsilon$, while we have analyzed the full structure of the weak magnetic field expansion of the finite-temperature quark condensate in the chiral limit up to two loops.
Zero-Temperature Quark Condensate
---------------------------------
We now turn to the quark condensate at zero temperature: $$\langle 0 | {\bar q} q | 0 \rangle = -\frac{{\langle 0 | {\bar q} q | 0 \rangle}_0}{F^2} \, \frac{\mbox{d} z_0}{\mbox{d} M^2} \, .$$ Recall that ${\langle 0 | {\bar q} q | 0 \rangle}_0$ is the quark condensate at $T$=0, $H$=0 and $M$=0. On the basis of the representations Eqs. (\[freeEDp6ZeroT\]) and (\[freeEDp4ZeroT\]) for the vacuum energy density, we derive $$\begin{aligned}
\label{quarkCondensatep6ZeroT}
\frac{\langle 0 | {\bar q} q | 0 \rangle}{{\langle 0 | {\bar q} q | 0 \rangle}_0} & = &
1 - \frac{{\overline l}_3 - 4{\overline h}_1}{32 \pi^2} \, \frac{M^2}{F^2}
- \frac{K_1}{F^2}
+ \frac{3 {\overline l}_3}{1024 \pi^4} \, \frac{M^4}{F^4}
- \frac{9 {\overline l}_3 ({\overline c}_{10}+2{\overline c}_{11})}{1024 \pi^4} \, \frac{M^4}{F^4} \nonumber \\
& & - \frac{{\overline l}_6 - {\overline l}_5}{768 \pi^4} \, \frac{{|qH|}^2}{F^4}
+ \frac{({\overline l}_6 - {\overline l}_5) {\overline c}_{34}}{768 \pi^4} \, \frac{{|qH|}^2}{F^4}
- \frac{1}{32 \pi^2} \, \frac{M^2}{F^4} \, K_1 \nonumber \\
& & + \frac{{\overline l}_3}{16 \pi^2} \, \frac{M^2}{F^4} \, K_1
+ \frac{{\overline l}_3}{32 \pi^2} \, \frac{M^4}{F^4} \, \frac{\mbox{d} K_1}{ \mbox{d} M^2}
- \frac{({\overline l}_6 - {\overline l}_5)}{48 \pi^2} \, \frac{{|qH|}^2}{F^4} \, \frac{\mbox{d} K_1}{ \mbox{d} M^2} \, .\end{aligned}$$ The explicit expressions for $K_1$ and $\mbox{d} K_1/\mbox{d} M^2$, $$\begin{aligned}
K_1 & = & \frac{M^2}{16 \pi^2} - \frac{M^2}{16 \pi^2} \, \ln \frac{M^2}{2|qH|} + \frac{|qH|}{8 \pi^2} \,
\ln \Gamma \! \Big( \frac{M^2}{2|qH|} + \frac{1}{2} \Big) - \frac{|qH|}{16 \pi^2} \, \ln 2\pi \, , \nonumber \\
\frac{\mbox{d} K_1}{\mbox{d} M^2} & = & \frac{1}{16 \pi^2} \, \ln \frac{|qH|}{M^2} + \frac{1}{16 \pi^2} \,
\Psi \! \Big( \frac{M^2}{2|qH|} + \frac{1}{2} \Big) + \frac{\ln 2}{16 \pi^2} \, ,\end{aligned}$$ are derived in Appendix \[appendixB\]. The series for the quark condensate is organized according to ascending powers of $M^2$ and $|qH|$ – both quantities count as order $p^2$. The respective coefficients depend in a nontrivial manner on the ratio $M^2/|qH|$ and involve renormalized NLO and NNLO effective constants. Let us compare our result with the literature.
The focus of the two-loop CHPT calculation presented in Ref. [@Wer08], was to determine the shift in the zero-temperature quark condensate caused by an external (electro)magnetic field. Our expression, Eq. (\[quarkCondensatep6ZeroT\]), goes beyond the literature since we have derived the whole two-loop representation for the quark condensate – not just the terms induced by the magnetic field.
To analyze the chiral limit of the zero-temperature quark condensate in finite magnetic fields, we invoke the behavior of the NLO and NNLO effective constants. According to Appendix \[appendixA\] we have $$\begin{aligned}
{\overline l}_3, {\overline l}_5, {\overline l}_6 & \propto & \ln M^2 \, , \nonumber \\
{\overline c}_{34} & \propto & \ln M^2 \, , \nonumber \\
{\overline c}_{10} + 2 {\overline c}_{11} & \propto & \ln M^2 \, ,\end{aligned}$$ i.e., the renormalized NLO and NNLO effective constants explode in the limit $M \to 0$. But note that in the quark condensate these constants are multiplied by powers of $M^2$ such that the chiral limit is in fact unproblematic. While some terms in Eq. (\[quarkCondensatep6ZeroT\]) hence disappear in the chiral limit, only the following two terms, $$\frac{({\overline l}_6 - {\overline l}_5)}{768 \pi^4} \, {\overline c}_{34} \, \frac{{|qH|}^2}{F^4}
- \frac{({\overline l}_6 - {\overline l}_5)}{768 \pi^4} \, \frac{{|qH|}^2}{F^4}
\ln\Big( \frac{|qH|}{M^2} \Big) \, ,$$ need special consideration, as they both explode in the chiral limit. However, writing the NNLO effective constant ${\overline c}_{34}$ as $${\overline c}_{34} = \ln\Big( \frac{\Lambda^2_{34}}{M^2_{\pi}} \Big) \, ,$$ where $\Lambda_{34}$ is the renormalization group invariant scale associated with ${\overline c}_{34}$, the two terms can be merged such that the zero-temperature quark condensate in nonzero magnetic fields is well-defined in the chiral limit, taking the form $$\begin{aligned}
\label{quarkCondensatep6ZeroTchiralLimit}
\frac{\langle 0 | {\bar q} q | 0 \rangle}{{\langle 0 | {\bar q} q | 0 \rangle}_0} & = &
1 + \frac{\ln 2}{16 \pi^2} \, \frac{|qH|}{F^2}
- \frac{({\overline l}_6 - {\overline l}_5)}{768 \pi^4} \, \frac{{|qH|}^2}{F^4}
\ln\Big( \frac{|qH|}{\Lambda^2_{34}} \Big)
- \frac{({\overline l}_6 - {\overline l}_5)}{768 \pi^4} \, \frac{{|qH|}^2}{F^4} \nonumber \\
& & - \frac{({\overline l}_6 - {\overline l}_5)}{768 \pi^4} \, \frac{{|qH|}^2}{F^4} \,
\Bigg( \frac{\Gamma'(\mbox{$ \frac{1}{2}$})}{\Gamma(\mbox{$ \frac{1}{2}$})} + \ln 2 \Bigg) \, .\end{aligned}$$ Notice that the $\ln M^2$-dependence in the combination ${\overline l}_6 - {\overline l}_5$ cancels, and we can write $${\overline l}_6 - {\overline l}_5 = \ln\Big( \frac{\Lambda^2_6}{\Lambda^2_5} \Big) \, ,$$ where $\Lambda_5$ and $\Lambda_6$ are the respective renormalization group invariant scales associated with the NLO effective constants ${\overline l}_5$ and ${\overline l}_6$.
Conclusions
===========
We have explored the behavior of the quark condensate subjected to an external magnetic field within the framework of chiral perturbation theory. Unlike previous two-loop evaluations by other authors, we have used a coordinate space representation.
Regarding the finite-temperature quark condensate in the chiral limit and in weak magnetic fields, we have pointed out various errors that have occurred in the literature and have provided the correct series. At order $T^2$ – and in terms of the expansion parameter $\epsilon = |qH|/T^2$ – the leading contribution is proportional to $\sqrt{\epsilon}$, followed by a term linear in $\epsilon$, a half-integer power $\epsilon^{3/2}$ and a logarithmic contribution $\epsilon^2 \ln \epsilon$. The remaining contributions involve even powers of $\epsilon$. At order $T^4$ the pattern repeats itself with the exception that a term linear in $\epsilon$ does not occur.
Leaving the weak magnetic field limit, we have investigated the impact of the magnetic field on the quark condensate at finite temperature. Emphasis was put on the effect of the pion-pion interaction which constitutes up to about ten percent for arbitrary pion masses but also in the real world where $M_{\pi} = 140 \, MeV$. The interaction is largest in the chiral limit. The finite-temperature quark condensate (sum of one- and two-loop contribution) at fixed temperature and fixed pion mass grows monotonically when the magnetic field strength increases. Again, the effect is most pronounced in the chiral limit.
Finally we have derived the two-loop representation for the QCD vacuum energy density and the quark condensate at zero temperature. We have complemented earlier studies by other authors, by providing the full two-loop representation, i.e., not just the terms that emerge on account of the nonzero magnetic field.
A natural – but highly nontrivial – step is to extend the present analysis to the three-loop level, in analogy to the three-loop analysis in zero magnetic field given in the pioneering article [@GL89], based on a coordinate space representation of CHPT. Corresponding work is in progress.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author gratefully acknowledges H. Leutwyler and J. Bijnens for correspondence. Special thanks to J. Bijnens for sharing unpublished results on the order-$p^6$ zero-temperature quark condensate.
Order-$p^6$ Free Energy Density at $T$=0 {#appendixA}
========================================
Low-Energy Effective Constants at NLO and NNLO {#appendixA1}
----------------------------------------------
The aim of the present appendix is to discuss the renormalization group running of the NLO and NNLO effective constants $l^r_i$ and $c^r_i$ in some detail, and then to provide a definition of the renormalized NNLO effective constants ${\overline c}_i$ – in analogy to the definition of the renormalized NLO quantities ${\overline l}_i$.
The NNLO effective constants $c_i$ that appear in ${\cal L}^6_{eff}$, are defined in Ref. [@BCE00] as $$\label{definitionci}
c_i = \frac{{(c \mu)}^{2(d-4)}}{F^2} \, \Bigg\{ c^r_i - \gamma^{(2)}_i \Lambda^2 - \gamma^{(1)}_i \Lambda - \gamma^{(L)}_i \Lambda \Bigg\} \, ,$$ with $$\Lambda = \frac{1}{16 \pi^2} \, \frac{1}{d-4} \, , \qquad \ln c = -\mbox{$ \frac{1}{2}$} \Big[ \ln 4 \pi + \Gamma'(1) +1 \Big] \, .$$ The quantities $\gamma^{(1)}_i, \gamma^{(2)}_i$ are pure numbers and the $c^r_i$ are the renormalized running NNLO effective constants. For the definition of the NLO effective constants $l_i$ that appear in ${\cal L}^4_{eff}$, on the other hand, we adopt the definition given in the original Ref. [@GL84], $$\label{definitionli}
l_i = l^r_i + \gamma_i \lambda \, ,$$ where $$\begin{aligned}
\lambda & = & \mbox{$ \frac{1}{2}$} \, (4 \pi)^{-\frac{d}{2}} \, \Gamma(1-{\mbox{$ \frac{1}{2}$}}d) \mu^{d-4} \nonumber \\
& & = \frac{\mu^{d-4}}{16{\pi}^2} \, \Bigg[ \frac{1}{d-4} - \mbox{$ \frac{1}{2}$} \{ \ln{4{\pi}} + {\Gamma}'(1) + 1 \}
+ {\cal O}(d\!-\!4) \Bigg] \, .\end{aligned}$$ The $\gamma_i$ are pure numbers and the $l^r_i$ are the renormalized running NLO effective constants. The definition, Eq. (\[definitionli\]), can be rewritten as $$l_i = l^r_i + {(c \mu)}^{d-4} \gamma_i \Lambda \, .$$ Note that the $\gamma_i$ also show up in $\gamma^{(L)}_i$, Eq. (\[definitionci\]), in the form of $$\gamma^{(L)}_i = \sum_j \gamma^{(L)}_{ij} {(c \mu)}^{-(d-4)} l^r_j \, ,$$ where the coefficients $\gamma^{(L)}_{ij}$ are again pure numbers.
Since the $c_i$ do not depend on the renormalization scale $\mu$, one concludes that the renormalization group running of the NNLO effective constants $c^r_i$ is $$\label{relationcr}
\mu \frac{\mbox{d} c^r_{i}}{\mbox{d} \mu} = - 2(d-4) c^r_i + \frac{\gamma^{(1)}_i}{8 \pi^2} + \frac{\gamma^{(L)}_i}{16 \pi^2} \, .$$ In the above derivation we have used the fact that the NLO effective constants $l^r_i$ themselves obey the running $$\label{relationlr}
\mu \frac{\mbox{d} l^r_i}{\mbox{d} \mu} = - \frac{\gamma_i}{16 \pi^2} \, {(c \mu)}^{d-4} \, ,$$ which follows from the fact that the $l_i$ do not depend on $\mu$. Furthermore, with the Weinberg consistency condition [@Wei79], $$-2 \gamma^{(2)}_i + \sum_j \gamma^{(L)}_{ij} \gamma_j = 0 \, ,$$ a divergence linear in $\Lambda$ has been eliminated in Eq. (\[relationcr\]).
Instead of the NLO quantities $l^r_i$ that depend on the renormalization scale, alternatively one often uses the NLO effective constants ${\overline l}_i$ that are $\mu$-independent. The connection between the two is [@GL84] $$\label{definitionliBar}
l^r_i = \frac{\gamma_i}{32 \pi^2} \Big( {\overline l}_i + \ln \frac{M^2}{\mu^2} \Big) \, .$$ Let us transfer this connection to NNLO. The specific NNLO effective constants that appear in the vacuum energy density are $c_{10}, c_{11}$ and $c_{34}$, where the last one only matters when a magnetic field is present. Following Ref. [@BCE00] – but using the convention (\[definitionli\]) – it reads $$c_{34} = \frac{{(c \mu)}^{2(d-4)}}{F^2} \, c^r_{34} + \frac{l^r_5 - \mbox{$ \frac{1}{2}$} l^r_6}{F^2} \, \lambda \, .$$ Explicitly, the running of $c^r_{34}$ is given by $$\frac{\mbox{d} c^r_{34}}{\mbox{d} \mu^2} = - \frac{l^r_5 - \mbox{$ \frac{1}{2}$} l^r_6}{32 \pi^2 \mu^2} \, .$$ In analogy to the above definition for the NLO constants ${\overline l}_i$, Eq. (\[definitionliBar\]), that is based on the running (\[relationlr\]), we define the renormalized NNLO effective constant ${\overline c}_{34}$ as $$\label{c34Definition}
c^r_{34} = \frac{{\overline l}_6 - {\overline l}_5}{6144 \pi^4} \, {\overline c}_{34}
+ \frac{{\overline l}_6 - {\overline l}_5}{6144 \pi^4} \, \ln \frac{M^2}{\mu^2} \, .$$ Note that we have used $$\gamma_5 = - \frac{1}{6} \, , \qquad \gamma_6 = - \frac{1}{3} \, .$$ Since $c^r_{34}$ does not depend on $M$, we conclude $$\frac{\mbox{d} {\overline c}_{34}}{\mbox{d} M^2} = - \frac{1}{M^2} \, .$$ The NNLO constant ${\overline c}_{34}$ hence obeys the same simple relation as the NLO constants ${\overline l}_i$, $$\frac{\mbox{d} {\overline l}_i}{\mbox{d} M^2} = - \frac{1}{M^2} \, .$$
Next we consider the NNLO effective constants $c_{10}$ and $c_{11}$ that arise in the tree-level contribution $z_{6C}$ in the absence of the magnetic field. They are defined as (see Ref. [@BCE00]) $$\begin{aligned}
c_{10} & = & \frac{{(c \mu)}^{2(d-4)}}{F^2} \, c^r_{10} + \frac{3}{64 F^2}\, \lambda^2
-\frac{1}{F^2} \Big( \mbox{$ \frac{3}{16}$} l^r_3 + \mbox{$ \frac{1}{16}$} l^r_7 \Big) \lambda \, , \nonumber \\
c_{11} & = & \frac{{(c \mu)}^{2(d-4)}}{F^2} \, c^r_{11} - \frac{9}{128 F^2} \, \lambda^2
+ \frac{1}{F^2} \, \Big( \mbox{$ \frac{9}{32}$} l^r_3 + \mbox{$ \frac{1}{32}$} l^r_7 \Big) \lambda \, .\end{aligned}$$ Note that in the linear combination $c_{10} + 2c_{11}$ – as it appears in the vacuum energy density at order $p^6$ – the dependence on $l^r_7$ cancels and we are left with $$c_{10} + 2c_{11} = \frac{{(c \mu)}^{2(d-4)}}{F^2} \, (c^r_{10} + 2 c^r_{11}) - \frac{3}{32 F^2}\, \lambda^2
+ \frac{3}{8 F^2} \, l^r_3 \lambda \, .$$ Since the $c_i$ do not depend on the renormalization scale $\mu$, we conclude $$\frac{\mbox{d} ( c^r_{10} + 2c^r_{11} )}{\mbox{d} \mu^2} = - \frac{3 l^r_3}{256 \pi^2} \, \frac{1}{\mu^2} \, .$$ Equivalently, by making the replacement $l^r_3 \to {\overline l}_3$, $$l^r_3 = \frac{\gamma_3}{32 \pi^2} \Big( {\overline l}_3 + \ln \frac{M^2}{\mu^2} \Big) \, , \qquad \gamma_3 = -\frac{1}{2} \, ,$$ we can write $$\frac{\mbox{d} ( c^r_{10} + 2c^r_{11} )}{\mbox{d} \mu^2} = \frac{3 {\overline l}_3}{16384 \pi^4} \, \frac{1}{\mu^2}
+ \frac{3}{16384 \pi^4} \, \frac{1}{\mu^2} \, \ln \frac{M^2}{\mu^2} \, .$$ This leads us to the definition of the renormalized combination ${\overline c}_{10} + 2{\overline c}_{11}$ as $$\label{c10c11Definition}
c^r_{10} + 2 c^r_{11} = - \frac{3 {\overline l}_3}{16384 \pi^4} \, ( {\overline c}_{10} + 2 {\overline c}_{11} )
- \frac{3 {\overline l}_3}{16384 \pi^4} \, \ln \frac{M^2}{\mu^2}
- \frac{3}{32768 \pi^4} \, {\Big(\ln \frac{M^2}{\mu^2}\Big)}^2 \, .$$ By construction, the linear combination ${\overline c}_{10} + 2{\overline c}_{11}$ is independent of $\mu$, much like ${\overline c}_{34}$ and the ${\overline l}_i$. Because the expression $c^r_{10} + 2 c^r_{11}$ does not depend on $M$, we also conclude $$\frac{\mbox{d} ({\overline c}_{10} + 2{\overline c}_{11})}{\mbox{d} M^2} = - \frac{1}{M^2} + \frac{1}{M^2} \,
\frac{{\overline c}_{10} + 2{\overline c}_{11}}{{\overline l}_3} \, .$$
Isolating UV-divergences {#appendixA2}
------------------------
Here we focus on the zero-temperature contributions in the free energy density that emerge at order $p^6$ due to the three diagrams $6A$-$C$ displayed Fig. \[figure1\]. The unrenormalized expressions that contain both $T$=0 and finite-temperature pieces are $$\begin{aligned}
z_{6A} & = & \frac{M^2}{2 F^2} \, G^{\pm}_1 G^0_1 - \frac{M^2}{8 F^2} \, G^0_1 G^0_1 \, , \nonumber \\
z_{6B} & = & (4l_5 - 2l_6) \frac{{|qH|}^2}{F^2} \, G^{\pm}_1 + 2 l_3 \frac{M^4}{F^2} \, G^{\pm}_1 + l_3 \frac{M^4}{F^2} \, G^0_1 \, ,
\nonumber \\
z_{6C} & = & -16(c_{10} + 2c_{11}) M^6 - 8 c_{34} {|qH|}^2 M^2 \, ,\end{aligned}$$ where $G^{\pm}_1$ and $G^0_1$ are the thermal pion propagators evaluated at the coordinate origin $x$=0, $$G^{\pm}_1 = G^{\pm}(0) \, , \qquad G^0_1 = G^0(0) \, .$$ Inserting the decomposition of thermal propagators into zero-temperature and finite-temperature pieces (defined in Eq. (\[boseFunctions\])) $$\begin{aligned}
& & G^{\pm}_1 = \Delta^{\pm}(0) + {\tilde g}_1(M,T,H) + g_1(M,T,0) \, , \nonumber \\
& & G^0_1 = \Delta^0(0) + g_1(M,T,0) \, ,\end{aligned}$$ and using the representations of the zero-temperature propagators $\Delta^{\pm}(0)$ and $\Delta^0(0)$, $$\Delta^{\pm}(0) = 2 M^2 \lambda + K_1 \, , \qquad \Delta^0(0) = 2 M^2 \lambda\, ,$$ with $K_1$ and $\lambda$ as $$\begin{aligned}
K_1 & = & \frac{{|qH|}^{\frac{d}{2}-1}}{{(4 \pi)}^{\frac{d}{2}}} \, \, {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \, \rho^{-\frac{d}{2}+1}
\, \exp\Big( -\frac{M^2}{|qH|} \rho \Big) \, \Big( \frac{1}{\sinh(\rho)} - \frac{1}{\rho} \Big) \, , \nonumber \\
\lambda & = & \mbox{$ \frac{1}{2}$} \, (4 \pi)^{-\frac{d}{2}} \, \Gamma(1-{\mbox{$ \frac{1}{2}$}}d) M^{d-4} \nonumber \\
& & = \frac{M^{d-4}}{16{\pi}^2} \, \Bigg[ \frac{1}{d-4} - \mbox{$ \frac{1}{2}$} \{ \ln{4{\pi}} + {\Gamma}'(1) + 1 \}
+ {\cal O}(d\!-\!4) \Bigg] \, ,\end{aligned}$$ we obtain $$\begin{aligned}
\label{fedT0NotRenormalized}
z^{0}_{6A} & = & \frac{3 M^6}{2 F^2} \, \lambda^2 + \frac{M^4}{F^2} \, K_1 \lambda \, , \nonumber \\
z^{0}_{6B} & = & 6 l_3 \frac{M^6}{F^2} \, \lambda + 2 l_3 \frac{M^4}{F^2} \, K_1 + (8l_5 - 4l_6) \frac{{M^2 |qH|}^2}{F^2} \, \lambda
+ (4l_5 - 2l_6) \frac{{|qH|}^2}{F^2} \, K_1 \, , \nonumber \\
z^{0}_{6C} & = & -16(c_{10} + 2c_{11}) M^6 - 8 c_{34} {|qH|}^2 M^2 \, .\end{aligned}$$ The upper index $"0"$ signals that we are considering the $T$=0 part only.[^7] To isolate the UV-divergences in this unrenormalized expression, we use the conventions for the NLO and NNLO effective constants $l_i$ and $c_i$, respectively, that we have provided in Appendix \[appendixA1\]. One finds that in the sum of the three diagrams, all UV-divergences disappear and the renormalized order-$p^6$ vacuum energy density takes the form $$\begin{aligned}
z^{[6]}_0 & = & z^{0}_{6A} + z^{0}_{6B} + z^{0}_{6C} \nonumber \\
& = & \frac{3{\overline l}_3 ({\overline c}_{10} + 2 {\overline c}_{11})}{1024 \pi^4} \, \frac{M^6}{F^2}
- \frac{({\overline l}_6 - {\overline l}_5) {\overline c}_{34}}{768 \pi^4} \, \frac{{|qH|}^2 M^2}{F^2} \nonumber \\
& & - \frac{{\overline l}_3}{32 \pi^2} \, \frac{M^4}{F^2} \, K_1
+ \frac{({\overline l}_6 - {\overline l}_5)}{48 \pi^2} \, \frac{{|qH|}^2}{F^2} \, K_1 \, .\end{aligned}$$ The above representation is renormalization-scale independent. This constitutes a nontrivial check of our calculation.
Analysis of the Integral $K_1$ {#appendixB}
==============================
To analyze the free energy density and the quark condensate in the chiral limit, we must have a closer look at the dimensionally regularized integral $K_1$, $$\begin{aligned}
\label{integralK1}
K_1 & = & \frac{{|qH|}^{\frac{d}{2}-1}}{{(4 \pi)}^{\frac{d}{2}}} \, \, {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \, \rho^{-\frac{d}{2}+1}
\, \exp\Big( -\frac{M^2}{|qH|} \rho \Big) \, \Big( \frac{1}{\sinh(\rho)} - \frac{1}{\rho} \Big) \, .\end{aligned}$$ To this end we first consider the integral $I_2$, defined in (A1) of Ref. [@Hof19] as $$\begin{aligned}
I_2 & = & - \frac{{|qH|}^{\frac{d}{2}}}{{(4 \pi)}^{\frac{d}{2}}} {\int}_{\!\!\! 0}^{\infty} d\rho \rho^{-\frac{d}{2}}
\Big( \frac{1}{\sinh(\rho)} - \frac{1}{\rho} \Big) \, \exp\!\Big( -\frac{M^2}{|qH|} \rho \Big) \, , \nonumber \\
& = & - \frac{{|qH|}^3}{96 \pi^2 M^2} - \frac{{|qH|}^{\frac{d}{2}}}{{(4 \pi)}^{\frac{d}{2}}} {\int}_{\!\!\! 0}^{\infty} d\rho \rho^{-\frac{d}{2}}
\Big( \frac{1}{\sinh(\rho)} - \frac{1}{\rho} + \frac{\rho}{6} \Big) \, \exp\!\Big( -\frac{M^2}{|qH|} \rho \Big) \, .\end{aligned}$$ Comparing these representations, one concludes $$\begin{aligned}
\label{derivativesI2}
K_1 & = & \frac{\mbox{d} I_2}{\mbox{d} M^2} \, , \nonumber \\
\frac{\mbox{d} K_1}{\mbox{d} M^2} & = & \frac{{\mbox{d}}^2 I_2}{{(\mbox{d} M^2)}^2} \, .\end{aligned}$$ Using the property of the Riemann zeta function $$\lim_{s \to 1} \zeta(s,q) = \frac{1}{s-1} - \frac{\Gamma'(q)}{\Gamma(q)} \, ,$$ where $$\zeta(s,q) = \sum^{\infty}_{n=0} \frac{1}{{(q+n)}^s} \, ,$$ the second relation in Eq. (\[derivativesI2\]) yields[^8] $$\frac{\mbox{d} K_1}{\mbox{d} M^2} = \frac{1}{16 \pi^2} \, \ln \frac{|qH|}{M^2} + \frac{1}{16 \pi^2} \,
\Psi \! \Big( \frac{M^2}{2|qH|} + \frac{1}{2} \Big) + \frac{\ln 2}{16 \pi^2} \, ,$$ where $\Psi(x)$ is the Polygamma function $$\Psi(x) = \frac{\Gamma'(x)}{\Gamma(x)} \, .$$ The expression for $K_1$ is obtained by integration, $$K_1 = \frac{M^2}{16 \pi^2} - \frac{M^2}{16 \pi^2} \, \ln \frac{M^2}{2|qH|} + \frac{|qH|}{8 \pi^2} \,
\ln \Gamma \! \Big( \frac{M^2}{2|qH|} + \frac{1}{2} \Big) + C(|qH|) \, .$$ The integration constant $C(|qH|)$ can be determined by setting $M$=0 in the equation above and in the original representation, Eq. (\[integralK1\]). One identifies $$C(|qH|) = - \frac{|qH|}{16 \pi^2} \ln 2 \pi \, .$$ While $K_1$ appears in the free energy density, the derivative $\mbox{d} K_1/\mbox{d} M^2$ is relevant in the quark condensate.
Bose Functions in the Chiral Limit {#appendixC}
==================================
The finite-temperature representation of the quark condensate in the chiral limit, Eq. (\[quarkCondensateTChiralLimit\]), features an infinite series of kinematical Bose functions $g_r$ and ${\tilde g}_r$ that has to be resummed because of the weak magnetic field expansion Eq. (\[expansionMH\]). This is the main focus of the present appendix. The aim is to provide explicit expressions up to order $\epsilon^2 \ln \epsilon $ in the finite-temperature quark condensate.
We first consider the second type of functions[^9] $$\begin{aligned}
\label{ABC}
{\tilde g_r}(M^{\pm}_{\pi}, T, H) & = & \frac{\epsilon}{{(4 \pi)}^{r+1}} T^{d-2r} \, {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \, \rho^{-\frac{d}{2}+r}
\exp \Big( \frac{- {(M^{\pm}_{\pi})}^2}{4 \pi T^2} \rho \Big) \nonumber \\
& & \times \Big( \frac{1}{\sinh(\epsilon \rho/4 \pi)} - \frac{4 \pi}{\epsilon \rho} \Big) \, \Big[ S\Big(\frac{1}{\rho} \Big) -1 \Big] \, .\end{aligned}$$ The crucial point is that – in the chiral limit – the mass $M^{\pm}_{\pi}$ of the charged pions does not tend to zero. Rather, according to Eq. (\[chargedPionMass\]), a magnetic-field dependent mass term survives the chiral limit, $$M^2_H = \frac{{\overline l}_6 - {\overline l}_5}{48 \pi^2} \, \frac{{|qH|}^2}{F^2}
= \frac{16 \pi^2}{3} \, ({\overline l}_6 - {\overline l}_5) t^4 F^2 {\epsilon}^2 \, ,$$ with $$t = \frac{T}{4 \pi F} \, .$$ The pertinent expansion parameter in the weak magnetic field limit $|qH| \ll T^2 $ is $$\epsilon = \frac{|qH|}{T^2} \, .$$ To isolate divergences in the kinematical functions ${\tilde g}_r$ (where $r=0,1,2, \dots$) that arise in the limit $\epsilon \to 0$ ($T$ held fixed while $H \to 0$), we decompose ${\tilde g_r}(M_H, T, H)$ into two pieces, $$\begin{aligned}
\label{decompInt}
{\tilde g_r}(M_H, T, H) & = & \frac{\epsilon \, T^{d-2r}}{{(4 \pi)}^{r+1}} \, {\int}_{\!\!\! 0}^1 \mbox{d} \rho \, \rho^{-\frac{d}{2}+r}
e^{- \gamma \, \epsilon^2 \rho} \, \Big( \frac{1}{\sinh(\epsilon \rho/4 \pi)} - \frac{4 \pi}{\epsilon \rho} \Big) \,
\Big[ S\Big(\frac{1}{\rho} \Big) -1 \Big] \nonumber \\
& & + \frac{\epsilon \, T^{d-2r}}{{(4 \pi)}^{r+1}} \, {\int}_{\!\!\! 1}^{\infty} \mbox{d} \rho \, \rho^{-\frac{d}{2}+r}
e^{- \gamma \, \epsilon^2 \rho} \, \Big( \frac{1}{\sinh(\epsilon \rho/4 \pi)} - \frac{4 \pi}{\epsilon \rho} \Big) \,
\Big[ S\Big(\frac{1}{\rho} \Big) -1 \Big] \nonumber \\
& = & I_a + I_b \, ,\end{aligned}$$ where $$\gamma = \frac{{\overline l}_6 - {\overline l}_5}{12 \pi} \, t^2 \, .$$ The first integral $I_a$ exists for integer $r = 0,1,2, \dots$. Taylor expanding the integrand in the parameter $\epsilon$, we obtain a series with ascending even powers of $\epsilon$ for $r = 0,1,2, \dots$, $${\alpha}_1 {\epsilon}^2 + {\alpha}_2 {\epsilon}^4 + {\alpha}_3 {\epsilon}^6 + {\cal O}({\epsilon}^8) \, .$$ The explicit coefficients are irrelevant for our purposes because the respective terms do not contribute to the quark condensate at the accuracy we are interested in (up to $\epsilon^2 \ln \epsilon $ in the finite-temperature quark condensate). In particular, no $\epsilon$-divergences come from here.
We thus examine the second integral $I_b$ in Eq. (\[decompInt\]) that we process by using the Jacobi identity $$S\Big( \frac{1}{z} \Big) = \sqrt{z} \, S(z) \, .$$ We then obtain the three integrals $$\begin{aligned}
\label{OneTwoThree}
I_b & = & \frac{\epsilon \, T^{d-2r}}{{(4 \pi)}^{r+1}} \, {\int}_{\!\!\! 1}^{\infty} \mbox{d} \rho \, \rho^{r-\frac{d}{2}+\frac{1}{2}}
e^{- \gamma \epsilon^2 \rho} \, \Big( \frac{1}{\sinh(\epsilon \rho/4 \pi)} - \frac{4 \pi}{\epsilon \rho} \Big) \, \Big[ S(\rho) -1
\Big] \nonumber \\
& & + \frac{\epsilon \, T^{d-2r}}{{(4 \pi)}^{r+1}} \, {\int}_{\!\!\! 1}^{\infty} \mbox{d} \rho \, \rho^{r-\frac{d}{2}+\frac{1}{2}}
e^{- \gamma \epsilon^2 \rho} \, \Big( \frac{1}{\sinh(\epsilon \rho/4 \pi)} - \frac{4 \pi}{\epsilon \rho} \Big)
\nonumber \\
& & - \frac{\epsilon \, T^{d-2r}}{{(4 \pi)}^{r+1}} \, {\int}_{\!\!\! 1}^{\infty} \mbox{d} \rho \, \rho^{r-\frac{d}{2}}
e^{- \gamma \epsilon^2 \rho} \, \Big( \frac{1}{\sinh(\epsilon \rho/4 \pi)} - \frac{4 \pi}{\epsilon \rho} \Big) \nonumber \\
& = & I_{b1} + I_{b2} + I_{b3} \, .\end{aligned}$$ The first one – $I_{b1}$ – exists for integer $r = 0,1,2, \dots$. Taylor expanding the integrand and then integrating term by term we get a series of the form $${\beta}_1 {\epsilon}^2 + {\beta}_2 {\epsilon}^4 + {\beta}_3 {\epsilon}^6 + {\cal O}({\epsilon}^8) \, .$$ Again, the coefficients are irrelevant at the accuracy we are interested in. To isolate potential $\epsilon$-divergences in $I_{b2}$, we write the integration limits as $$\begin{aligned}
\label{intLimits}
I_{b2} & = & \frac{\epsilon \, T^{d-2r}}{{(4 \pi)}^{r+1}} \, {\int}_{\!\!\! 0}^{\infty} \mbox{d} \rho \, \rho^{r-\frac{d}{2}+\frac{1}{2}}
e^{- \gamma \, \epsilon^2 \rho} \, \Big( \frac{1}{\sinh(\epsilon \rho/4 \pi)} - \frac{4 \pi}{\epsilon \rho} \Big) \nonumber \\
& & - \frac{\epsilon \, T^{d-2r}}{{(4 \pi)}^{r+1}} \, {\int}_{\!\!\! 0}^{1} \mbox{d} \rho \, \rho^{r-\frac{d}{2}+\frac{1}{2}}
e^{- \gamma \, \epsilon^2 \rho} \, \Big( \frac{1}{\sinh(\epsilon \rho/4 \pi)} - \frac{4 \pi}{\epsilon \rho} \Big) \, .\end{aligned}$$ The first expression can be integrated analytically, $$I^{[1]}_{b2} = T^{d-2r} \Gamma( r - \mbox{$\frac{3}{2}$} ) \Bigg[ -\frac{\gamma^{\frac{3}{2}-r}}{{(4\pi)}^r} \, \epsilon^{3-2r}
+ 2^{-r-\frac{5}{2}} \pi^{-\frac{3}{2}} (2r - 3) \zeta(r-\mbox{$\frac{1}{2}$}, \mbox{$ \frac{1}{2}$} + 2\pi \gamma \epsilon) \,
\epsilon^{\frac{3}{2}-r} \Bigg]\, ,$$ where the generalized Riemann zeta function is defined as $$\zeta(s,a) = \sum^{\infty}_{k=0} \, \frac{1}{{(k+a)}^s} \, .$$ One notices that the integral $I^{[1]}_{b2}$ (for integer $r \ge 2$) leads to $\epsilon$-divergences in the functions ${\tilde g}_r$, namely $$\label{leadingDivergences}
{\tilde g}_r \propto \frac{1}{\epsilon^{2r-3}} \, , \frac{1}{\epsilon^{r-\frac{3}{2}}} \, .$$ As it turns out, these are indeed the leading divergences in the Bose functions ${\tilde g}_r$. With the second expression – $I^{[2]}_{b2}$ – in Eq. (\[intLimits\]) we proceed as before: Taylor expanding the integrand again gives rise to a series displaying even $\epsilon$-powers whose respective coefficients are of no concern to us.
Finally, we analyze the remaining third integral $I_{b3}$ in Eq. (\[OneTwoThree\]). Regularizing it with $N \gg 1$, $$I_{b3} = \lim_{N \to \infty} - \frac{\epsilon \, T^{d-2r}}{{(4 \pi)}^{r+1}} \, {\int}_{\!\!\! 1}^N \mbox{d} \rho \, \rho^{r-\frac{d}{2}}
e^{- \gamma \, \epsilon^2 \rho} \, \Big( \frac{1}{\sinh(\epsilon \rho/4 \pi)} - \frac{4 \pi}{\epsilon \rho} \Big) \, ,$$ the substitution $z = \ln (\epsilon u)$ – for the specific case $r$=2 (and $d$=4) – leads to $$I_{b3}(r=2) = \lim_{N \to \infty} \, -\frac{\epsilon^{-4 \pi \gamma \epsilon}}{16 \pi^2} \, {\int}_{\!\!\! u_0}^{u_N} \mbox{d} u
\, u^{-1-4 \pi \gamma \epsilon} \Big( \frac{1}{\sinh(\ln \epsilon u)} - \frac{1}{\ln \epsilon u} \Big) \, ,$$ with $$u_0 = \frac{e^{\frac{\epsilon}{4 \pi}}}{\epsilon} \, , \qquad u_N = \frac{e^{\frac{N}{4 \pi}}}{\epsilon} \, .$$ The integral can be performed analytically, $$\label{thirdAnalytic}
I_{b3}(r=2) = \frac{1}{16 \pi^2} \, \Bigg\{ {\cal B}\Big( e^{-\frac{N}{2\pi}}; \mbox{$ \frac{1}{2}$} + 2 \pi \gamma \epsilon, 0 \Big)
- {\cal B}\Big( e^{-\frac{\epsilon}{2\pi}}; \mbox{$\frac{1}{2}$} + 2 \pi \gamma \epsilon, 0 \Big)
- {\cal E}(- \gamma \epsilon^2) + {\cal E}(- \gamma N \epsilon) \Bigg\} \, ,$$ where the incomplete beta function and the exponential integral function, respectively, are defined as $$\begin{aligned}
{\cal B}(z;a,b) & = & {\int}_{\!\!\! 0}^z \mbox{d} x \, x^{a-1} {(1-x)}^{b-1} \, , \nonumber \\
{\cal E}(z) & = & - {\int}_{\!\!\! -z}^{\infty} \mbox{d} x \, \frac{e^{-x}}{x} \, .\end{aligned}$$ Expanding $I_{b3}$ in $\epsilon$, one notices that only the second and third expression in Eq. (\[thirdAnalytic\]) lead to $\epsilon$-divergences. Concretely, we obtain a logarithmic divergence, $$I_{b3}(r=2) = -\frac{1}{16 \pi^2} \ln \epsilon + {\cal O}(\epsilon^0) \, .$$ Collecting results, the divergences in the function ${\tilde g}_2$ in the weak magnetic field limit are $${\tilde g}_2 = -\frac{1}{16 \pi^{\frac{3}{2}} \sqrt{\gamma}} \, \frac{1}{\epsilon}
-\frac{\sqrt{2}-4}{32 \pi} \, \zeta(\mbox{$\frac{3}{2}$}) \, \frac{1}{\sqrt{\epsilon}}
-\frac{1}{16 \pi^2} \ln \epsilon + {\cal O}(\epsilon^0) \, .$$
The quark condensate in the chiral limit, according to Eqs. (\[quarkCondensateTChiralLimit\]) and (\[expansionMH\]), features the series $${\cal S}[\tilde g] = - {\hat c} \epsilon^2 {\tilde g}_2 + \frac{{\hat c}^2 \epsilon^4}{2!} \, {\tilde g}_3
- \frac{{\hat c}^3 \epsilon^6}{3!} \, {\tilde g}_4 + {\cal O}({\tilde g}_5) \, ,$$ where $${\hat c} = 4 \pi T^2 \gamma \, , \qquad \gamma = \frac{{\overline l}_6 - {\overline l}_5}{12 \pi} \, t^2 \, .$$ According to Eq. (\[leadingDivergences\]), the leading divergence in the functions ${\tilde g}_r$ is proportional to $\epsilon^{3-2r}$. Therefore each term in the above series gives rise to a contribution linear in $\epsilon$. All these terms have to be taken into account at the order we are operating. The series can be resummed with the result $$\label{resumTildeg}
{\cal S}[\tilde g] = \frac{\sqrt{2}-1}{2 \sqrt{\pi}} \, \sqrt{\gamma} \, \epsilon \, T^2 \, .$$ The quark condensate in the chiral limit – see Eqs. (\[quarkCondensateTChiralLimit\]) and (\[expansionMH\]) – furthermore involves the other type of Bose functions $g_r(M,T,0)$. The structure of the expansion in the mass parameter $M$ for the specific function $g_0(M,T,0)$ has been analyzed in Refs. [@GL89; @Hof10] with the outcome $$\begin{aligned}
g_0(M,T,0) & = & T^4 \, \Bigg[ \frac{\pi^2}{45} \, - \, \frac{1}{12}
\frac{M^2}{T^2} \, + \, \frac{1}{6 \pi} \frac{M^3}{T^3}
\, + \, \frac{ (2\gamma_E -\mbox{$\frac{3}{2}$} )}{32{\pi}^2} \frac{M^4}{T^4} \,
+ \frac{1}{32{\pi}^2} \, \frac{M^4}{T^4} \, \ln \frac{M^2}{16 \pi^2 T^2}
\nonumber \\
& & + 2 \pi^{3/2} \, \sum_{n=3}^{\infty} \frac{(-1)^n}{n!} \,
\Big( \frac{M}{2 \pi T} \Big)^{2n} \, \Gamma(n-\mbox{$\frac{3}{2}$}) \,
\zeta(2n-3) \Bigg] \quad (T \gg M) \, .\end{aligned}$$ With the recursion relation $$g_{r+1} = - \frac{\mbox{d} g_r}{\mbox{d} M^2} \, ,$$ the series for any other $g_r$ with $r =1,2,3, \dots$ can be derived.
In our case of interest, the relevant mass in these functions is $M_H$, $$M^2_H = \frac{{\overline l}_6 - {\overline l}_5}{48 \pi^2} \, \frac{{|qH|}^2}{F^2}
= \frac{16 \pi^2}{3} \, ({\overline l}_6 - {\overline l}_5) t^4 F^2 {\epsilon}^2 \, ,$$ i.e., the mass of the charged pion that survives the chiral limit. We then find that the leading $\epsilon$-divergences in these functions are $$g_r = \frac{(2r-5)!! \, \gamma^{3/2-r}}{2^{3r-2} \pi^{r-1/2}} \, \epsilon^{3-2r} \, , \qquad r=2,3,\dots \, .$$ The series of kinematical functions $g_r$, as it occurs in the quark condensate, $${\cal S}[g] = - {\hat c} \epsilon^2 g_2 + \frac{{\hat c}^2 \epsilon^4}{2!} \, g_3 - \frac{{\hat c}^3 \epsilon^6}{3!} \, g_4 + {\cal O}(g_5)
\, ,$$ hence yields an infinite number of terms that are all linear in $\epsilon$. Resumming, we obtain $${\cal S}[g] = \frac{1 - \sqrt{2}}{2 \sqrt{\pi}} \, \sqrt{\gamma} \, \epsilon \, T^2 \, .$$ This just cancels the contribution from ${\cal S}[\tilde g]$, Eq. (\[resumTildeg\]), such that there are no terms linear in $\epsilon$ in the quark condensate coming from here. The logarithmic contributions, however, that are present both in $g_2$ and ${\tilde g}_2$ do not cancel: in the sum we have $$g_2 + {\tilde g}_2 = \frac{1}{16 \pi^2} \ln \epsilon + {\cal O}(\epsilon^{-1/2}) \, ,$$ giving rise to a contribution $\epsilon^2 \ln \epsilon$ in the quark condensate.
Finally, the $\epsilon$-expansion for the functions $g_1(0,T,0)$ and ${\tilde g}_1(0,T,H)$ that also appear in the quark condensate, Eqs. (\[quarkCondensateTChiralLimit\]) and (\[expansionMH\]), has been provided in Refs. [@GL89; @Hof19]. For completeness we quote the result, $$\begin{aligned}
g_1(0, T, 0) & = & \frac{1}{12} \, T^2 \, , \\
{\tilde g_1}(0, T, H) & = & - \Bigg\{ \frac{|I_{\frac{1}{2}}|}{8 \pi^{3/2}} \sqrt{\epsilon}
- \frac{\ln 2}{16 \pi^2} \, \epsilon
+\frac{\zeta(3)}{384 \pi^4} \, \epsilon^2
- \frac{7 \zeta(7)}{98 304 \pi^8} \, \epsilon^4
+ {\cal O}(\epsilon^6) \Bigg\} \, T^2 \, , \nonumber\end{aligned}$$ with $$\label{I12}
I_{\frac{1}{2}} = {\int}_{\!\!\! 0}^{\infty} \, \mbox{d} \rho \rho^{-1/2} \Big( \frac{1}{\sinh(\rho)} - \frac{1}{\rho} \Big) \approx -1.516256 \, .$$
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[^1]: For reviews of chiral perturbation theory see, e.g., Refs. [@Leu95; @Sch03].
[^2]: Although the Pauli matrix associated with the neutral pion is $\tau^3$, we will denote the neutral pion field as $\pi^0$ in view of its zero charge.
[^3]: To the best of our knowledge, the complete CHPT two-loop representation for the QCD vacuum energy density – containing magnetic-field dependent as well as $H$-independent terms – is not available in the literature.
[^4]: Throughout the study we work in the isospin limit $m = m_u = m_d$.
[^5]: For the tree-level pion decay constant we use the value $F = 85.6 \, MeV$ reported in Ref. [@Aoki20]. Note that in the isospin limit – and in zero magnetic field – the masses of the neutral and the charged pions are identical.
[^6]: See Eqs. (\[chargedPionMass\]) and (\[neutralPionMass\]).
[^7]: The finite-temperature contribution $z^T$ is given by Eq. (\[fedPhysicalM\]).
[^8]: The physical limit $d \to 4$ is straightforward and does not pose any problems.
[^9]: It should be noted that the functions ${\tilde g_r},g_r$ – up to temperature powers – coincide with the functions ${\tilde h_r},h_r$. The conversion is given by Eq. (\[conversion\]).
| |
Bob and Ron Copper of the Copper Family were traditional, unaccompanied folk singers from Rottingdean, near Brighton.
The Copper family have been folk singers for generations. Aside from the Copper family there are almost no recorded examples of harmonic part-singing in English traditional folk song.
The Coppers came to the attention of Kate Lee, one of the founders of the Folk Song Society (later the English Folk Dance and Song Society), who knew she had found something special. James ‘Brasser’ Copper (1845-1924) and his brother Thomas (c.1847-c.1936) were made honorary members of the society, and in 1899 ‘Brasser’ wrote down the songs he knew.
‘Brasser’ had two sons, John (c.1879-1952) and Jim (1882-1954). In 1936, Jim wrote a further volume of songs. Jim had two children, Joyce (1910-?) and Bob (1915-2004). John’s son was Walter Ronald, known as Ron (c.1913-1979).
Together, Jim, John, Ron and Bob sang at numerous venues including the Royal Albert Hall, and received wider public attention followed the broadcast of a six-part television series Song Hunter, presented by Alan Lomax and featuring Jim, Bob and Ron. In 1963 the English Folk Dance and Song Society released an LP of their unaccompanied singing that has had a tremendous influence on the folk song revivals of Britain, Ireland and the United States. The album collected together a number of recordings of Bob and Ron Copper, made by the renowned folksong collector Peter Kennedy during the late 1950s.
Bob wrote several books about the family and its songs, beginning with the widely acclaimed A Song For Every Season in 1971.The accompanying 4-LP set (now a collector’s item) found Bob and Ron singing alongside Bob’s daughter Jill and son John.
The songs that generations of the Copper family learnt to sing whether at home, in the fields or in the pub at the end of the day have been passed down through the decades. According to Bob ‘many of the songs, through being the favourite of a particular singer, would become known as “his song” and no one else would dream of singing it unless the recognized singer was not in the present company. A singer’s repertoire, therefore, was like a little window into his character, for he accumulated his songs through natural selection”. Whether sweet, sad, gallant or gay they ‘would give some indication of the man himself’.
After Ron died in 1979, Jill’s husband Jon was introduced to the core line-up, and some of Bob’s grandchildren began to appear with the group. The six grandchildren now also appear independently as The Young Coppers, singing the same family repertoire. Bob Copper died in 2004, a few days after receiving an MBE. In an obituary by Ken Hunt in The Independent newspaper, Bob Copper was described as “England’s most important traditional folk-singer”.
About this Contributor
- Given name
- Bob and Ron
- Family name
- Copper
- Nationality
- British
- Contributor type
- Artist
- Minutes created
- Christmas #5, Collection 3 #54
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Discover a treasure trove of music education resources created by the experts at Minute of Listening. | https://www.minuteoflistening.org/contributor/bob-and-ron-copper/ |
In nautical navigation the relative bearing of an object is the clockwise angle from the heading of the vessel to a straight line drawn from the observation station on the vessel to the object.
The relative bearing is measured with a pelorus or other optical and electronic aids to navigation such as a periscope, sonar system, and radar systems. Since World War II, relative bearings of such diverse point sources have been and are calibrated carefully to one another. The United States Navy operates a special range off Puerto Rico and another on the west coast to perform such systems integration. Relative bearings then serve as the baseline data for converting relative directional data into true bearings (N-S-E-W, relative to the Earth's true geography). By contrast, Compass bearings have a varying error factor at differing locations about the globe, and are less reliable than the compensated or true bearings.
The measurement of relative bearings of fixed landmarks and other navigational aids is useful for the navigator because this information can be used on the nautical chart together with simple geometrical techniques to aid in determining the position of the vessel and/or its speed, course, etc.
The measurement of relative bearings of other vessels and objects in movement is useful to the navigator in avoiding the danger of collision.
- Example: The navigator on a ship observes a lighthouse when its relative bearing is 45° and again when it is 90°. He now knows that the distance from the ship to the lighthouse is equal to the distance travelled by the vessel between both observations. | http://www.let.rug.nl/~gosse/termpedia2/termpedia.php?language=dutch_general&density=7&link_color=000000&termpedia_system=perl_db&url=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FRelative_bearing |
Data Mining - MTAT.03.183
Starting from Autumn 2018 the Data Mining course will be renamed into:
LTAT.02.002 Introduction to Data Science (Sissejuhatus andmeteadusesse).
Poster session
The poster session will be held on Jan 8, 14:15-17:00 in the atrium of Paabel (Ülikooli 17). At least one person should be present from each team!
Exams
- Dec 18 at 10:15-13:00, room 111 (max 35 students)
- Dec 18 at 10:15-13:00, room 405 (max 25 students)
- Jan 16 at 10:15-13:00, room 403 (max 40 students)
- Jan 16 at 10:15-13:00, room 404 (max 40 students)
- Jan 16 at 10:15-13:00, room 405 (max 40 students)
The resit time for those who need it will be:
- Jan 29 at 10:15-13:00, room 405.
Consultations
Consultations are on Thursdays and the first consultation is on 14th of September. Consultations are voluntary and can be used to come and ask questions, for example if you did not understand something in the lecture or what needs to be done in a particular task in homework.
Course info:
- Lectures: Monday 10:15, Liivi 2-111
- Practice Sessions:
- Group 1: Monday 12:15 - 14:00 (402, Meelis)
- Group 2: Monday 14:15 - 16:00 (122, Mari-Liis)
- Group 3: Monday 12:15 - 14:00 (405, Dima)
- Group 4: Monday 14:15 - 16:00 (405, Dima)
- Consultations: Thursday 16:00-17:00, Liivi 2-405
- Note the change compared to what was announced in the first lecture!
Contacts:
- Course forum: Piazza
We will use Piazza for questions and discussions. In the forum you can post questions (also anonymously) about homeworks or course organization etc. And we can keep the discussion separate for different topics. You should all receive a welcome e-mail that invites you to piazza - don't ignore it and register there (it is sent to your address that is in the SIS). If you somehow don't get the e-mail you can register here (just mark that you are a student and press "Join Classes"). Then you have to fill some information about yourself, which is a little annoying, but do it anyways. The home page of the course forum is here and you can click on Q&A to get to the forum part. That's it.
- Homework submission and grades: Gradescope
You will be added to Gradescope using your e-mail address provided in the SIS. You can submit your homework pdf's there and also receive your grades.
- Lecturer: Meelis Kull ([email protected])
- Teaching Assistants: Mari-Liis Allikivi ([email protected]) and Dmytro Fishman ([email protected])
Grading and requirements:
The grade is calculated from the total number of points (max 100). The points can be earned as follows:
- Homeworks (44 points): there will be 11 homeworks, each worth 4 points;
- Group project and presentation at the poster session (20 points);
- Written exam (36 points);
- Additional points can be earned from bonus tasks within homeworks;
- Attending at least 8 out of 11 practice sessions is compulsory: after missing 3 practice sessions each additional missed practice session results in losing 5 points.
In order to pass the course the student must get at least 50% from homeworks (threshold 22 points), at least 50% from the project (threshold 10 points) and at least 50% from the exam (threshold 18 points). | https://courses.cs.ut.ee/2017/DM/fall/Main/HomePage |
Q:
Determine the primes p for which $6\in Q_p$(where $Q_p$ denotes quadratic residue)
Background This is from Jones Elementary Number theory problem 7.12. I only want the case where $6\in Q_p$ to be worked although the original problem asks for -3,5,6,7,10,169.
Using Legendre's notation and the Quadratic reciprocity law I get:
\begin{align}
\textrm{If }P \equiv & 1 \mod 4&\\
\left(\frac{2}{p}\right)\left(\frac{3}{p}\right) &=\\
\left(\frac{p}{2}\right)\left(\frac{p}{3}\right) &\implies p \equiv 1 \mod 2 \\
&\quad \quad \;\; p \equiv 1 \mod 3 \\
\hline
Thus&\implies p\equiv 1 \mod 6\\
\end{align}
$$
\left.
\begin{array}{l}
p\equiv 1 \mod 4\\
p \equiv 1 \mod 6
\end{array}
\right\}\bbox[5px,border:1px solid red]{p \equiv 1,13 \mod 24}
$$
\begin{align}
\textrm{If }P \equiv & 3 \mod 4&\\
\left(\frac{2}{p}\right)\left(\frac{3}{p}\right) &=\\
\left(\frac{p}{2}\right)\left(-\frac{p}{3}\right) &\implies p \equiv 1 \mod 2 \\
&\quad \quad \;\; p \equiv 2 \mod 3 \\
\hline
Thus&\implies p\equiv 5 \mod 6\\
\end{align}
$$
\left.
\begin{array}{l}
p\equiv 3 \mod 4\\
p \equiv 5 \mod 6
\end{array}
\right\}\bbox[5px,border:1px solid red]{p \equiv 11,23 \mod 24}
$$
The text solution is $p \equiv \pm1 \text{ or } \pm 5 \mod 24$. What did I do wrong?
A:
The Quadratic Reciprocity law only applies to odd primes $p,q$, and $2$ is not an odd prime. In fact we have
$$\left(\frac{2}{p}\right)=(-1)^{(p^2-1)/8}$$
which does not just depend on the value of $p$ mod $4$. So you will have to split the problem up into more cases.
| |
Newly found ancient road to serve tourism
Newly found ancient road to serve tourism
KASTAMONU
Works have been initiated to bring the ancient road, discovered after 1,700 years, to tourism in the ancient city of Pompeiopolis, located in the northern province of Kastamonu’s Taşköprü district.
Excavations carried out in the ancient city of Pompeiopolis, one of the largest cities in Anatolia during the Roman period, continue under the presidency of the Kastamonu Museum Directorate. The excavations in the ancient city, located in the Zımbıllı Tepe of Taşköprü district, the capital of the historical Paphlagonia region, are headed by Tayyar Gürdal.
Gürdal said that they are working to save the ancient road that they discovered after 1,700 years during the recent excavations and that it has not been used for 800 years.
“Pompeipolis was an important city. It takes its name from its founder, Pompeius Magnus, one of the most important political figures in Rome’s transition from republican to imperial rule. Mithridates, the Pontus King of the region, was the last commander to resist the Romans. When all the kingdoms in Anatolia accepted the sovereignty of Rome, Mithridates did not accept this sovereignty and rebelled,” he said.
Speaking about the excavation works, Gürdal said: “We focused on two areas. One is the city’s theater and the other is the city’s residential area and a highly qualified Roman Villa. During the Roman Villa works, we discovered a road that was connected to Sinop through the main road of the city and connected to Anatolia in the north on the east-west line, which provides access to the city from the main road. This road goes to the villa area from the main road and extends from there to the center of the city. Our recent work is carried out in this newly discovered road and villa. The villa was discovered in the second or third century B.C. but continued to be used until the 10th or 12th century B.C. Its floor tiles are mostly mosaic, we uncovered most of them, they are under protection. Other parts will be unearthed in future excavations.”
Gürdal said that a large part of the theater of Pompeipolis was destroyed and added, “Excavation is very expensive there. That’s why we stopped the work in the theater after we found the road. When working in theater, we saw that almost nine out of 10 parts of the theater were destroyed. There are only three rows of seats. We saw that all the remaining rows of seats had been dismantled. Only the infrastructure survives. We were thinking about where these blocks were, and we found them during the roadworks. We saw that most of the pavement blocks of the road were made of the theater’s seating rows and stair blocks.”
Stating that the excavations in the Pompeipolis will continue throughout the year, he said that they need a good sponsor. “How long it will take is directly related to the budget. If we work with very large budgets, we will finish it in a shorter time, but even if we work with very large budgets, there is at least 100 years of work here,” Güldal said. | |
Uncontrolled observation occurs in a natural situation without the impact of exterior or external control. The observer does not prepare ahead of time, but is concerned with day-to-day events and socio-cultural issues. It investigates some of our own circumstances. The researcher simply watches and takes notes.
An uncontrolled observation can be as simple as making notes during a conversation with someone. You should try to get a sense of what is going on in the person's mind and how they are feeling by watching their body language. You can also ask questions to find out more information - for example, you could say "tell me about your friend" or "how did that event affect you?". There are no right or wrong ways to do an observation - just be aware of what you are doing and why you are doing it.
Uncontrolled observations can be useful for finding facts or data about a topic - for example, you might watch people at a party to find out who likes what music. You can also use uncontrolled observations to understand people better - for example, by watching how someone reacts to something upsetting then trying to work out what is causing them to act this way.
Uncontrolled observations can help scientists make generalizations about groups of people or things.
The observer sees things as they are in unrestrained observation. For example, you may use uncontrolled observation to collect data for a report on a factory's current working conditions. Some pointers for effective personal observation: Concentrate on what you're looking at. Don't think about anything else. Be aware of your feelings and don't get upset if something unpleasant is seen by others when being observed.
Controlled observation, naturalistic observation, and participatory observation are all types of observation. Controlled observations are conducted by researchers who determine what variables need to be controlled in order to ensure that their results can be generalized to the population being studied. For example, if they were studying students' math scores, the researcher would control for grade level in the study design. Naturalistic observations are conducted by observers as they go about their daily lives. The observer records what happens without influencing or being influenced by those events. Finally, participatory observations involve participants helping to identify topics for investigation and collecting data during live situations rather than in laboratory studies.
Observation is important for scientists to understand people's behaviors and how they affect our environment. Scientists use observations to infer relationships between factors such as age and behavior, as well as predict future behavior. For example, a scientist might observe how often teenagers drive after drinking alcohol to see if there is a correlation between drinking and driving.
Scientists also use observations to collect data about rare events or things not easily measured otherwise. For example, an investigator may want to know how many children are affected by depression so he or she can make an informed decision about whether or not to start a research project aimed at reducing pediatric depression.
Observation only, no action. Naturalistic observation: observing activity in a more or less natural situation without intervening. The investigator does not influence or control the circumstance. The investigator did not instigate or create the circumstance. Observations are important for scientists to understand how things work out in nature and may help them explain what is happening behind closed doors. For example, an astronomer could make an observation of a solar eclipse to learn about Earth's atmosphere.
Intervention means taking action to change something about the situation. Interventions can be physical (such as throwing water on a fire) or mental (for example, thinking happy thoughts). In science, interventions should be chosen to avoid bias. For example, if you were studying plants and wanted to see which ones would grow best in well-drained soil, it would be inappropriate to choose only those plants that were able to support their own weight. The only fair way to study this question would be to weigh all the plants and then select the ones that needed little or no watering.
Bias is when results are influenced by factors such as prejudices or preferences. Bias can be explicit (such as when someone chooses plants because they believe they will grow better with less water) or implicit (when people act without being aware of their biases). Avoiding bias is important because true findings depend on random chance events occurring by luck rather than by design.
Observation of Nature This approach entails monitoring and researching individuals' spontaneous behavior in natural settings. The researcher merely documents what they notice in any manner they can. Without a system, the researcher notes all important behavior in unstructured observations. For example, a researcher might note that a bird has a red head by simply recording "red head." There is no way to measure or compare features like this with other birds, so it is not considered scientific data.
Naturalists classify organisms into groups (for example, birds) and then further into families (e.g., songbirds). They also make drawings and/or photographs of each species they observe. With these aids, naturalists can more accurately identify their specimens later with laboratory tests or online databases. Modern scientists use tools such as geolocators to study how far and why animals migrate.
In addition to classification systems, naturalists organize data by creating tables or diagrams that show the range of behaviors observed from one organism to another or over time. For example, one could record the number of times per hour that a bird sings during daylight hours and night using a simple tally sheet. One could also estimate the age of a bird by watching which branches it eats first - young birds eat lower branches while older birds eat higher ones. Quantitative data are measured repeatedly for many subjects and objects. | https://escorpionatl.com/what-is-uncontrolled-observation-method |
Description for Operation Ebola: Surgical Care during the West African Outbreak (Operation Health) Paperback. Sheku, Sherry M. Wren Editor(s): Wren, Sherry M.; Kushner, Adam L. Series: Operation Health. Num Pages: 120 pages, 16, 16 black & white halftones. BIC Classification: 1HFD; 3JMG; MBD; MBN; MNC. Category: (U) Tertiary Education (US: College). Dimension: 229 x 152 x 9. Weight in Grams: 181. | https://www.kennys.ie/other-categories/medical-profession/operation-ebola-surgical-care-during-the-west-african-outbreak-operation-health |
Q:
Etymology of witness in Hebrew
What are the origins of the Hebrew word pronounced "ed," meaning "witness?" It is spelled ayin daled. Is it related to the ayin daled portion of the word "Le-ad," which means "forever?" Does it also mean a contract and evidence? Is the origin Aramaic?
A:
Gesenius's Hebrew and Chaldee Lexicon states that it is the present participle of a root ayin-waw-dalet, meaning 'return' or 'repeat', with an Arabic cognate `āda -- the idea apparently being that a witness is one who 'repeats' what he saw. It's presumably not an Aramaic borrowing, since it occurs already in Genesis. Gesenius lists the meaning 'testimony' in addition to 'witness'. In modern Hebrew, though, it only means 'witness'; 'testimony' is a related noun, edut.
As for whether it's related to `ad 'eternity', Gesenius thinks not: he derives the latter from a different root ayin-dalet-he. It seems conceivable that the two roots could be related further back, though; at least, one could certainly imagine a semantic relationship between 'return' and 'eternity'.
A:
TL;DR
עד "witness" is probably derived from עוד ʕ-w-d "to admonish, affirm, turn about", whereas עד "forever" is probably derived from the unrelated עדה ʕ-d-h "pass on, advance".
Explanation
First, there's no need to insert a ל to עד "witness" to form לעד when עד on its own is already a homonym meaning "forever". The ל is a preposition that would yield something like "unto eternity".
Incidentally, another homonym is עד "filthy", derived from a root related to setting a period of time, seemingly connected to the menstrual cycle. I presume Biblical hermeneuticists aren't keen to hang all these meanings on the same peg.
So how do we determine which homonyms are polysemous, i.e. extensions of a single root? One way when we have roots of only two letters is to find the three-letter root that it comes from. (This is an assumption made when studying Hebrew that can be questioned, but no need to here.) Roots where we can't see the third consonant are "defective".
There are a few ways to "repair" a defective root, i.e. determine the third consonant. One is to look for a weak consonant that's likely to disappear: the vowel-like ו vav, ה heth, or י yod. One might come before, between, or after the two letters we can see, as in the common קום q-w-m "to rise". One set of reconstructions would thus be יעד y-ʕ-d or עיד ʕ-y-d or עדי ʕ-d-y.
There are also cases where a letter is commonly dropped from the beginning, e.g. the נ in נגד n-g-d "to tell" or the ל in לקח l-q-kh "to take", which are often missing in conjugations. But these cases are rare. And as I mentioned, looking for a lost ל at the start is pointless since the ל in לעד is a preposition, not part of the verbal root.
We can also look for a consonant that naturally assimilates into another. This typically happens with geminate roots, i.e. roots with two of the same consonant, such as רעע r-ʕ-ʕ "to do evil". That would let us reconstruct maybe עדד ʔ-d-d.
Once you have some candidates you go look for attestations in your dictionary or the text itself. In the text, you can find parallel conjugations. For example, we know that "middle-weak" roots are conjugated a certain way. Do we have any examples of a verbal עד that are conjugated like קום? Or we know that geminate verbs are conjugated another way. Do we have any examples of a verbal עד that are conjugated like רעע ? And so on. Also key is to compare related languages in which the "defective" consonant has not disappeared. Is there an Arabic, Aramaic, Assyrian, or Akkadian root, for example, that corresponds to a hypothetical עדי ?
Having found viable candidates, you then try to match up meanings to establish a reasonable semantic derivation, which is where some disagreement might come in. Sometimes it isn't a purely semantic argument but the vowels of the noun can also give hints as to root it came from.
Something like this has been done for many obscure words in the Bible, and to make a long story short, at least one source relates the different words spelled עד to the roots I noted at the start.
| |
Q:
Playing Cards: Formula for match probability with card draw rules
This is a simplification of a larger problem I have been working on, calculating probabilities based upon a characteristic, with rules for selection based upon a second characteristic. I believe if I can, with your help, determine the correct calculation method for this problem I can then expand it to my larger problem. Thanks in advance!
Setup:
With a deck of 8 cards: ( Ace Clubs Black [ACb], Ace Diamonds Red [ADr], Ace Hearts Red [AHr], Ace Spades Black [ASb], King Clubs Black [KCb], King Diamonds Red [KDr], King Hearts Red [KHr], and King Spades Black [KSb].
Therefore: the deck is (ACb, ADr, AHr, ASb, KCr, KDr KHr, KSb) arranged in fair random order.
Draw Rules:
Draw, without replacement, up to 4 cards. Stop drawing as soon as you have drawn 2 Aces or have drawn the 4th card.
Question:
What is the probability P(A) of drawing exactly 2 black cards and the KHr card? Extra card, if drawn, must be red.
Expected result (derived from counting of the exhaustive list of possibilities):
$$P(A) = {296\over 1680} = {37\over 210}\approx 0.17619$$
I need help developing generalized approach, as the underlying problem I am trying to solve has $_{416}P_6 \approx 5.0E6$ permutations, so full list enumeration and counting is not practical.
Here's how I have been calculating it, but I have gotten stuck.
[EDIT:]
Upon reflection it seems each of the 5 hand options could each be treated like a pick from bags problem:
E.g. You have 2 bags (A, K).
Bag A contains 2 black beads and 2 red beads. Bag K contains 2 black beads, 1 red beads, and one green bead.
For hand "AAK", draw 2 from bag A and 1 from bag K. What is the probability P(G2) that what you draw is B, B, G? $$P(G_2) = \left(\frac{2}{4}\right)\left ( \frac{1}{4} \right ) = 0.125$$
$$P(B_2) = \left(\frac{240}{1680}\right)$$
Then multiply this probability with the P(B2) to get the final hand probability.
$$P(H_2) = P(B_2) * P(G_2) = \left(\frac{240}{1680}\right)*\left(\frac{1}
{8}\right) = \left(\frac{240}{13440}\right) \approx 1.785714E-2$$ and then sum all together $$P(A) = \sum_{i=0}^5 P(G_i)P(B_i)$$
Does that seem correct?
[/EDIT:]
Known permutation facts:
Total Permutations of 4 cards: $T_p = _8P_4 = 1680$
Possible hands drawn, sorted by rank, and their permutations (items in brackets are assumption that the undrawn cards can be any of the remaining):
AA: $B_1 = _4P_2 * [_6P_2] * 1 = 360$ Hands: AAxx
AAK: $B_2 = _4P_2 * _4P_1 * [_5P_1] * 2 = 240$ Hands: AKAx, KAAx
AAKK: $B_3 = _4P_2 * _4P_2 * 3 = 432$ Hands: AKAK, AKKA, KKAA
AKKK: $B_4 = _4P_1 * _4P_3 * 4 = 384$ Hands: AKKK, KAKK, KKAK, KKKA
KKKK: $B_5 = _4P_4 * 1 = 24$ Hands: KKKK
Probability of the hand n: $P(B_n) = {B_n\over T_P}$
Permutations of desired outcome H (of 4 black pick 2, of 1 KHr pick 1, of 3 red pick 1, all permutations): $H_P = \binom{4}{2}\binom{1}{1}\binom{3}{1}*(_4P_4) = 432$
And this is where I am having difficulty:
Permutations based upon drawn hand Bn, where for nCr $\binom{n}{r}$ when r=n nCr = 1, when r>n, nCr = 0.
For B1 [AA]: $H_1 = \binom{2}{2}\binom{2}{2}\binom{0}{1}\binom{1}{1}\binom{5}{2} * (_4P_4) = 0$ Of the 2 Aces, Pick 2 and of those, pick from the 2 black. Of the 0 Kings, pick 1... Impossible
For B2 [AAK]: $H_2 = \binom{2}{2}\binom{1}{1}\binom{5}{1} * (_4P_4) = 120$
All Permutations 4P4
For B3 [AAKK]: $H_3 = \left(\binom{2}{2}\binom{1}{1}\binom{1}{1} + \binom{2}{1}\binom{2}{1}\binom{2}{1}\binom{1}{1} \right) * (_4P_4) = 168$
Pick 2 black from the aces, 1 KHr, and one Kr. OR Pick 1 black from the aces, 1 red from the aces, 1 black from the kings, and 1 KHr. All permutations 4P4.
For B4 [AKKK]: $H_4 = \left(\binom{2}{1}\binom{2}{1}\binom{1}{1}\binom{1}{1} + \binom{2}{1}\binom{2}{2}\binom{1}{1}\binom{1}{1}\right) * (_4P_4) = 144$
Pick 1 black from the 2 for Aces, pick 1 black for the 2 for kings, pick 1 from KHr, and 1 from the remaining Kr OR pick 1 from the 2 red Aces, 2 from the 2 black kings, 1 from KHr and 1 Kr. All permutation 4P4
For B5 [KKKK]: $H_5 = \binom{2}{2}\binom{1}{1}\binom{1}{1} * (_4P_4) = 24$ Of the kings, pick 2 of the 2 blacks, of the remaining kings, pick the 1 KHr, of the remaining 1, pick 1. All permutations 4P4.
The confusion I have is how to go from the Hn value combined with P(Bn) to derive the expected value for P(A). If it is $P(B_n|H_n)$ then I don't know the divisor for $B_n$
As a check: The list enumerated quantities of Hn|Bn are:
$H_1 = 0$
$H_2 = 20$
$H_3 = 108$
$H_4 = 144$
$H_5 = 24$
Thank you for assistance. This is a re-ask of this problem, as my last attempt at the question was down voted twice without comment, so I have completely re-written it. If you do feel this is worthy of a down vote, can I please ask for the favor of a comment of advice on how to clarify the scenario to make it more acceptable. Thanks.
A:
I have resolved my confusion and developed the universal formula to solve this problem, so I am closing this question
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For over 300 years, monsoon has been considered as a gigantic land-sea breeze of regional scale, but now it is considered as a global system over all continents but Antarctica. This new develoment in modern climatology, however, has not yet been responded by paleo-climatology.
Prof. Pinxian Wang from Tongji University, Shanghai, reviews the geological evolution of the global monsoon and its impact, showing that the global monsoon exists through all geological history since at least 600 million years ago. It covaries with various geological cycles including those caused by the geometric changes of the Earth's orbits. The 20,000-year precessional cycle of the global monsoon, for example, is responsible for the collapse of several Asian and African ancient cultures at ~ 4000 years ago. The same cyclicity is seen in the chemical composition of the air, such as methane concentration and isotope composition of air-bubbles captured in ice cores.
Now Wang found that the long-term cycles in the oceanic carbon reservoir also has a global monsoon origin. This 400,000-year cyclicity related to "long eccentricity" of the Earth's orbit, is best seen in carbon isotope compositions of calcite test of foraminifera, a single-cell animal in the ocean. The rhythmic changes in oceanic carbon reservoir were likened to "heartbeat" of the Earth system. This cyclicity becomes longer since 1.6 million years ago, displaying a kind of "arrhythmia" in the Earth system, probably resulting from the growth of the Arctic ice. Although the mechanism of how monsoon drives oceanic carbon cycle remains unclear, the monsoon-related long-term cyclicity should not be overlooked in carbon-cycle modeling for long-term climate prediction.
"It is an authoritative review", said Prof. Andre Berger, University of Louvain, in his commentary, "and probably also the first in which the monsoon issues are reviewed in a global scale through a so long geological history….I totally agree with Wang's argumentation about paying more attention to the importance of the tropical forcing in modulating the Earth's climate system". The geological evolution of the global monsoon is a new topic attracting growing interest from both modern and paleo-climatologic communities. An international symposium on global monsoon was organized by the PAGES (Past Global Changes) project in Shanghai in 2008, and the next symposium is scheduled in 2010.
References:
Wang, P., 2009. Global monsoon in a geological perspective. Chinese Science Bulletin, 54(7): 1113-1136
Berger,A.,2009. Monsoon and general circulation system. Chinese Science Bulletin, 54(7): 1111-1112
Wang, P., Tian, J., Cheng, X., et al., 2004. Major Pleistocene stages in a carbon perspective: The South China Sea record and its global comparison. Paleoceanography, 19, PA4005, doi: 10.1029/2003PA000991
P X Wang | EurekAlert!
Further information:
http://www.tongji.edu.cn
Further reports about: > Earth system > Earth's magnetic field > Earth's orbits > Global monsoon > carbon isotope compositions > carbon reservoir > gigantic land-sea breeze > global ocean > long-term carbon cycles > modern climatology > paleo-climatology
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Physicists working with Roland Wester at the University of Innsbruck have investigated if and how chemical reactions can be influenced by targeted vibrational excitation of the reactants. They were able to demonstrate that excitation with a laser beam does not affect the efficiency of a chemical exchange reaction and that the excited molecular group acts only as a spectator in the reaction.
A frequently used reaction in organic chemistry is nucleophilic substitution. It plays, for example, an important role in in the synthesis of new chemical...
Optical spectroscopy allows investigating the energy structure and dynamic properties of complex quantum systems. Researchers from the University of Würzburg present two new approaches of coherent two-dimensional spectroscopy.
"Put an excitation into the system and observe how it evolves." According to physicist Professor Tobias Brixner, this is the credo of optical spectroscopy.... | https://www.innovations-report.com/html/reports/earth-sciences/global-monsoon-drives-long-term-carbon-cycles-ocean-132401.html |
- Heat the oven to 350°F (180°C). Lline 2-3 baking sheets with parchment paper and set aside.
- In the work bowl of a stand mixer fitted with the paddle attachment, combine the butter and granulated sugar together until light and fluffy, scraping down the sides of the bowl as necessary.
- Reduce the speed to low and add in the egg yolks, vanilla extract, and salt and mix to incorporate. Scrape down the bowl and the paddle attachment.
- Add in all of the flour and mix over low speed until all of the flour has been incorporated, scraping down the paddle attachment and the bowl as necessary.
- Fit a large piping bag with a large star tip.
- Transfer the dough into the piping bag and pipe the dough into the shape of your choice. Depending on the size of your piping bag, you may need to add the dough to the bag in batches. Take care not to make the cookies too large–keep them to a maximum length of 2 inches (5 cm) if you are making oval/oblong shaped cookies. Space the cookies 1 inch (3 cm) apart on the baking sheets.
- Bake the cookies for 11-13 minutes, or until they are pale golden at the edges. Rotate the baking sheets halfway through baking.
- Cool the cookies completely on the baking sheets set onto cooling racks. Once cooled, you can store them in a container or a disposable aluminum cake pan covered lightly with aluminum foil.
- To decorate: Line 2 baking sheets with parchment paper.
- Melt the chocolate melting wafers according to the package directions.
- Dip half of every cookie into the chocolate, letting the excess drip off. Pour the toppings of your choice over the chocolate while it is still wet.
- Carefully place the decorated cookies on the parchment-lined baking sheet and allow the chocolate to set completely before plating or storing the cookies (it will go from shiny to opaque).
Notes
- A stand mixer fitted with the paddle attachment is essential for making the dough for these cookies, as it is thick and dense.
- Use a large piping bag and the largest star tip you can find (I used size 1M) so that the dough slides out easily.
- Run your hands under very warm water to help warm the dough in the piping bag just enough to slide out easily.
- You may want to use a sharp paring knife or scissors to cut the dough away from the piping bag as you press out dough since it is sticky and may not break off easily on its own.
- The cookies will keep for 1 week stored in a covered container or tin. | https://www.flaviasflavors.com/cookies-crackers/smitten-kitchens-bakery-style-butter-cookies/print/4299/ |
Ronan, Colin: THE SHORTER SCIENCE AND CIVILISATION IN CHINA. Volume 4. Cambridge, 1994. xv, 334 pp. 247-388 b/w illustrations (including many plates). Tables, bibliography, index. 24x16 cm. Cloth.
GBP 20.00
The shortened version of Joseph Needham's monumental work is still a substantial and informative contribution to knowledge, although 'prepared with a general non-scientific readership in mind'. This volume covers the main sections of Volume IV, part 2 of its original: Engineers: their status, tools and materials; Basic mechanical principles and types of machines; Land transport; Clockwork; and Windmills and aeronautics.
Subjects: Science
Item 564 in List 204.
Record produced by Hanshan Tang Books, www.hanshan.com. | https://hanshan.com/r/RONS41.HTM |
What should be adjusted first alkalinity or pH?
You should test alkalinity first because it will buffer pH. Your reading should be in the range of 80 to 120 parts per million (ppm). If you need to increase the alkalinity, add an increaser. To lower it, you’ll add a sodium bisulfate.
How do you increase alkalinity?
The industry standard has always been to use sodium bicarbonate (baking soda) to raise total alkalinity and sodium carbonate (soda ash) to raise pH — the exception being if both total alkalinity and pH are low.
Does adding alkalinity raise pH?
Alkalinity Increaser is sodium bicarbonate (also called sodium hydrogen carbonate). It raises Total Alkalinity, and pH which is too low. Since it has a pH of only 8.3, it will generally have a lesser effect on pH. Diluted in water, Alkalinity Increaser will not raise pH above its normal range.
How do I adjust the alkalinity in my pool?
To raise the TA level (and not the pH too), you add Baking Soda (Sodium Bicarbonate). Baking Soda will also raise the pH of the pool water slightly. If you need to raise both the pH and TA, then use Soda Ash until the pH comes to the proper level, then use Baking Soda to make further adjustments to the TA if needed.
How soon can you swim after adding alkalinity?
20 minutes
What to add to pool if alkalinity is low?
of baking soda per 10,000 gallons of water will raise alkalinity by about 10 ppm. If your pool’s pH tested below 7.2, add 3-4 pounds of baking soda. If you’re new to adding pool chemicals, start by adding only one-half or three-fourths of the recommended amount.
What causes alkalinity to drop in a pool?
There is a perfectly natural reason for this: evaporation and agitation of your water leads to a drop in the total alkalinity. … This is what is called the water degassing phenomenon. The removal of this dissolved CO2 from the water will have a direct impact on the total alkalinity, and will cause its value to fall.
Why is my alkalinity low?
When the pH level becomes unbalanced, the phenomenon is called pH bounce and can cause low alkalinity in the pool, as the case may be. With low alkalinity, the standard amount of chlorine added to your pool would be useless. It would only be useful when more than the usual amount is added for standard results.
Will pH down lower alkalinity?
You start by adding muriatic acid as needed in the pool. This will lower both the pH and total alkalinity. … In other words, if you lower a pH of 8.3 to the recommended level using muriatic acid, the total alkalinity will also reduce but will still remain on the higher side.
What is the difference between alkalinity and pH?
Water alkalinity and pH are not the same. Water pH measures the amount of hydrogen (acid ions) in the water, whereas water alkalinity is a measure of the carbonate and bicarbonate levels in water. … For all water sources, it is the alkalinity that actually determines how much acid to use, not the pH.
What happens if pH is too low in pool?
Low pH water will cause etching and deterioration of plaster, grout, stone, concrete and tiling. Any vinyl surfaces will also become brittle, which increases risk of cracks and tears. All of these dissolved minerals will hold in the solution of your pool water; which can result in staining and cloudy pool water. | https://poolbuilderstx.com/important-about-swimming-pools/how-to-increase-ph-and-alkalinity-in-pool.html |
Q:
spherical mean of solution of the helmholtz equation
I'm stuck with this problem.
Given a domain $\Omega \subset \mathbb{R}^3$ where the function $u$ satisfies:
$u_{xx} +u_{yy}+u_{zz} + k^2 u = 0$, I am asked to find the spherical mean over the sphere $\{(x, y, z)\in \mathbb{R}^3; ||(x-x_0, y-y_0, z-z_0)||=R\} \subset \Omega$.
I obviously thought of trying to adapt the mean value property for harmonic functions but to no avail. I then sought to find a general form of the solution using separation of variables. This way I believe that, if using spherical coordinates, the azimuthal factor of the solution is proportional $\Psi(\psi)=e^{in\psi}$, which means that when integrating out the azimuthal angle in
$\int u d\sigma=-1/k^2\int \nabla^2 u d\sigma $ I will get zero by periodicity. I feel this solution is probably wrong. Any help would be appreciated. Thanks!
A:
Let $v(r)$ be the mean on the sphere of radius $r$. Then $v'(r)$ is the mean of the normal derivative. Hence, $4\pi r^2 v'(r)$ is the flux of $\nabla u$ out of the sphere. By the divergence theorem, this flux is equal to the integral of $\Delta u$ over the ball bounded by the sphere. The latter integral is $$\int_{B_r}\Delta u = -k^2\int_{B_r} u = -4\pi k^2\int_0^r s^2 v(s)\,ds$$
Thus,
$$r^2 v'(r) = -k^2\int_0^r s^2 v(s)\,ds \tag{1}$$
Differentiate $v$ to get an ODE:
$$ r^2 v''(r) +2r v'(r)+k^2 r^2 v(r) =0 \tag{2}$$
Looks like the Bessel equation, except for the factor of $2$, which makes it a spherical Bessel equation: unlike the ordinary one, (2) has a rather elementary solution (sinc).
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Find fZ and fP for the common-mode response of the differential amplifier as shown if rx = 250 Ω, Cμ = 0.5 pF, REE = 25 MΩ, CEE = 1 pF, and RC = 50 kΩ.
Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes!*
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About Greyville Racecourse
The Greyville Racecourse surrounds the Royal Durban Golf Club's Championship course, which provides a picture of green grass and foliage throughout the year framed in the background by the many hotels of Durban's Golden Mile. This right-handed track has a circumference of 2 800 metres, and a run-in of 500 metres and no straight for sprints.
Unusual features include the golf course in the centre of the track and the Drill Hall. The construction of these subways resulted in the course having an uphill section from the 2 400 metre post up to the 1 800 metre position, thereafter a gentle downward slope for about 800 metres, followed by an uphill section from the 1000 metre mark into the straight. In 1996, history was made when it became the first track in the country to successfully install floodlights, enabling the course to accommodate night racing at all its midweek meetings.
Accommodation near Greyville Racecourse
Sun1 Durban
Avg per night
The Concierge Boutique Bungal... | https://www.safarinow.com/destinations/greyville/Racecourses/Greyville-Racecourse.aspx |
Q:
ray tracer objects stretch when off center
I am writing a ray tracer program for my computer graphics class. So far I only have spheres implemented and a shadow ray. The current problem is that when i move my sphere off center it stretches. here is the code that i use to calculate if a ray is intersecting a sphere:
bool Sphere::onSphere(Ray r)
{
float b = (r.dir*2).innerProduct(r.pos + centre*-1);
float c = (r.pos + centre*-1).innerProduct(r.pos + centre*-1) - radius*radius;
return b*b - 4*c >= 0;
}
here is the code that i use to spawn each ray:
for(int i = -cam.width/2; i < cam.width/2; i++)
{
for(int j = -cam.height/2; j < cam.height/2; j++)
{
float normi = (float)i;
float normj = (float)j;
Vector pixlePos = cam.right*normi + cam.up*normj + cam.forward*cam.dist + cam.pos*1;
Vector direction = pixlePos + cam.pos*-1;
direction.normalize();
Vector colour = recursiveRayTrace(Ray(pixlePos, direction), 30, 1, 0);
float red = colour.getX()/255;
float green = colour.getY()/255;
float blue = colour.getZ()/255;
fwrite (&red, sizeof(float), 1, myFile);
fwrite (&green, sizeof(float), 1, myFile);
fwrite (&blue, sizeof(float), 1, myFile);
}
}
recursiveRayTrace:
Vector Scene::recursiveRayTrace(Ray r, float maxDist, int maxBounces, int bounces)
{
if(maxBounces < bounces)
return Vector(0,0,0);
int count = 0;
for(int i = 0; i < spheres.size(); i++)
{
if(spheres.at(i).onSphere(r))
{
Vector colour(ambiant.colour);
for(int j = 0; j < lights.size(); j++)
{
Vector intersection(r.pos + r.dir*spheres.at(i).getT(r));
Ray nRay(intersection, lights.at(i).centre + intersection*-1);
colour = colour + lights.at(i).colour;
}
return colour;
}
}
return Vector(0,0,0);
}
What i get is an sphere that is stretched in the direction of the vector from the center to the center of the circle. I'm not looking for anyone to do my homework. I am just having a really hard time debugging this on. Any hints are appreciated :) Thanks!
Edit: cam.dist is the distance from the camera to the view plane
A:
The stretching is actually a natural consequence of perspective viewing and it is exaggerated if you have a very wide field of view. In other words moving the camera back from your image plane should make it seem more natural.
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CROSS-REFERENCE TO RELATED APPLICATION
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
DETAILED DESCRIPTION OF THE INVENTION
EXAMPLES
[Manufacture of Supply Roller]
Example 1
[Evaluation]
This application claims the benefit under 35 U.S.C. §119 (a) of Korean Patent Application No. 10-2006-0079806, filed on Aug. 23, 2006, in the Korean Intellectual Property Office, the entire disclosure of which is hereby incorporated by reference in its entirety.
1. Field of the Invention
The present invention relates to a supply roller of a developing device for an image forming apparatus, and to a method of manufacturing the supply roller. More particularly, the present invention relates to a supply roller of a developing device for an image forming apparatus, which is small and exhibits excellent toner supply properties while preventing occurrence of a ghost phenomenon and a toner-filming phenomenon which cause a deterioration in the image quality. The invention is further directed to a method of manufacturing the image forming apparatus and the supply roller.
2. Description of the Related Art
In image forming apparatuses, a latent image is formed on a photoreceptor using a light scanner, a toner is supplied from a developing device having toner composition to form an image, and the latent image is developed with toner to form an image. Image forming apparatuses comprise a plurality of rollers, which are necessary to perform various operations. Among these rollers, a supply roller supplies toner from the developing device to the other components such as the photoreceptor.
FIG. 1
The general image forming operation in the image forming apparatus is described with reference to as follows.
FIG. 1
is a view schematically illustrating a standard image forming apparatus.
16
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18
First, a charging roller charges a photoreceptor , and an electrostatic latent image to be developed is formed on the charged photoreceptor by a laser scanning unit (LSU) .
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15
A supply roller supplies toner from inside the developing device to a developing roller . The toner supplied to the developing roller is thinned to a uniform thickness by a toner layer control apparatus , and at the same time is charged due to high friction caused by interaction between the developing roller and the toner layer control apparatus .
15
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19
When the toner passing through the toner layer control apparatus comes into contact with the photoreceptor , the latent image formed on the photoreceptor is developed. The developed toner is transferred onto print paper by a transfer roller , and then completely fixed onto the print paper so that an image is formed.
11
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17
If the toner formed on the photoreceptor remains after printing, the toner is cleaned by a cleaning blade . The toner separated from the photoreceptor by the cleaning blade is collected separately to be removed later.
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13
The toner in the supply roller in the image forming apparatus has a constant charge to mass ratio (Q/M) in association with the interaction between the developing roller and the toner layer control apparatus . The supply roller supplies the toner to the developing roller, and recovers remaining toner not used for the development of the latent image in the developing device.
The supply roller is usually formed from a polyurethane foam or silicone foam. Polyurethane foam has a lower hardness and price than silicone foam.
Recently, image forming apparatuses have been developed with reduced size, longer lifespan, low temperature fixing properties, and a capability of forming glossy images. In particular, color image forming apparatuses have been required to be increasingly miniaturized, because the color image forming apparatuses comprise developing devices containing different color toners.
However, in order to miniaturize the image forming apparatus, each constituent must be miniaturized, but problems arise when miniaturizing the supply roller. When the outer diameter of the supply roller is small, the toner supply properties are reduced and the ghost phenomenon and toner-filming phenomenon occur.
The ghost phenomenon is observed when toner is supplied by a small supply roller to cause unstable electrification of the toner, and as a result, a residual image is formed on the final image unintentionally.
Additionally, the toner-filming phenomenon means that a gap portion of the supply roller is filled with fine toner particles so that the supply roller has inferior supply properties. This problem caused by the toner-filming phenomenon stands out more clearly due to the miniaturization of the supply roller.
Therefore, supply rollers are required which can be manufactured in a small size in response to the need for the miniaturization of image forming apparatuses, and can overcome the above problems caused by miniaturization.
Exemplary embodiments of the present invention address at least the above problems and/or disadvantages and provide at least the advantages described below. Accordingly, an exemplary aspect of the present invention is to provide a supply roller of a developing device for an image forming apparatus, which is small and exhibits excellent toner supply properties while preventing the occurrence of the ghost phenomenon and toner-filming phenomenon that causes deterioration in image quality and the invention is also directed to a method of manufacturing the supply roller.
3
3
In order to achieve the above-described aspects of the present invention, a supply roller of a developing device for an image forming apparatus is provided, which includes a shaft; and a conductive resilient member enclosing an outer circumference of the shaft. The conductive resilient member has a density of about 60 kg/mto 120 kg/m, and an outer diameter of about 8.0 mm to 10.0 mm.
In an exemplary implementation, the shaft has an outer diameter of about 4.0 mm to 6.0 mm.
Additionally, the conductive resilient member is formed from a composition which comprises a polyurethane, a conductive additive, a blowing agent, and a surfactant.
In an exemplary implementation, the polyurethane is prepared by reacting a polyol and a polyisocyanate in the presence of a catalyst. The catalyst is selected from among organometallic compounds, amine-based compounds, and mixtures thereof. In one embodiment, the monomer components are reacted in the presence of the catalyst and the blowing agent to form a polyurethane foam structure suitable for use in forming the supply roller of the invention.
In an exemplary implementation, the organometallic compounds used as a catalyst comprise at least one metal selected from the group consisting of tin, lead, iron, and titanium.
In an exemplary implementation, the amine-based compounds used as a catalyst comprise a tertiary amine.
In an exemplary implementation, the conductive additive of the conductive resilient member comprises a compound having a hydroxyl group on its end, and comprises a polyalkylene glycol.
Additionally, the polyalkylene glycol is selected from the group consisting of a polyethylene glycol, a polypropylene glycol, a polytetramethylene glycol, a polyethylene glycol-polypropylene glycol copolymer, a ring-opening adduct of bisphenol A ethylene oxide, and a ring-opening adduct of bisphenol A propylene oxide.
In an exemplary implementation, the polyalkylene glycol has a molecular weight of about 300 to about 3,000.
3
3
In an exemplary implementation, the conductive additive of the conductive resilient member further comprises at least one salt selected from among alkali metal salts and alkaline earth metal salts, in addition to the polyalkylene glycol. In an exemplary implementation, the blowing agent comprises either water or a halogenated alkane. One suitable halogenated alkane is trichlorofluoromethane. Typically the blowing agent is included in an amount to produce a polyurethane product having a density of about 60 kg/Mto 120 kg/MIn an exemplary implementation, a silicone surfactant is used as the surfactant.
3
3
According to one exemplary aspect of the present invention, a method of manufacturing a supply roller of a developing device for an image forming apparatus, the method comprises preparing a conductive resilient member from a composition comprising a polyurethane, a conductive additive, a blowing agent and a surfactant; cutting the resulting conductive resilient member into a cylindrical shape, forming a shaft-shaped hole in the center of the conductive resilient member; and pushing a shaft through the hole, and heating, and adhering the conductive resilient member and the shaft. The conductive resilient member has a density of about 60 kg/mto 120 kg/mand an outer diameter of about 8.0 mm to 10.0 mm.
In an exemplary implementation, the shaft has an outer diameter of about 4.0 mm to 6.0 mm.
These and other aspects of the invention will become apparent form the following detailed description of the invention which in conjunction with the annexed drawings disclose various embodiments of the invention.
Certain exemplary embodiments of the present invention will now be described in greater detail with reference to the accompanying drawings.
The matters defined in the description such as a detailed construction and elements are provided to assist in a comprehensive understanding of the embodiments of the invention and are merely exemplary. Accordingly, those of ordinary skill in the art will recognize that various changes and modifications of the embodiments described herein can be made without departing from the scope and spirit of the invention. Also, descriptions of well-known functions and constructions are omitted for clarity and conciseness.
FIG. 2
13
13
13
13
13
a
b
a.
is a perspective view illustrating a supply roller of a developing device usable in an image forming apparatus according to an exemplary embodiment of the present invention. The supply roller includes a shaft and a conductive resilient member enclosing an outer circumference of the shaft
13
b
3
3
The conductive resilient member in one preferred embodiment has a density of about 60 kg/mto 120 kg/m, and an outer diameter of about 8.0 mm to 10.0 mm.
1
3
13
13
13
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13
b
a
b
b
a
a.
The conductive resilient member as prepared is formed or cut into a cylindrical shape to have a desired outer diameter. In order to insert the shaft into the conductive resilient member , a shaft-shaped hole is formed in the center of the conductive resilient member . The shaft desirably has an outer diameter of about 4.0 mm to 6.0 mm, and thus the shape of the hole should correspond to the outer diameter of the shaft
13
13
13
a
b
When the hole is formed, the shaft is pushed into and through the conductive resilient member , and the supply roller is manufactured following predetermined steps.
13
a
The shaft may be any shaft usable in manufacturing the roller, but desirably has an outer diameter of about 4.0 mm to 6.0 mm.
13
a
The shaft is desirably made of metal, and metal alloy containing metals such as aluminum, iron and/or nickel.
13
b
The conductive resilient member is formed from a molding composition which comprises a polyurethane, a conductive additive, a blowing agent, and a surfactant. In one embodiment of the invention the conductive resilient member is produced by molding a composition comprising polyurethane-forming monomer components, at least one conductive additive additive, a blowing agent and a surfactant. The composition is reacted to a polyurethane foam containing the conductive additive.
In this exemplary embodiment of the present invention, the polyurethane is obtained by mixing a compound containing at least two active hydrogens and a compound containing at least two isocyanate groups with additives in the presence of a catalyst, and a blowing, and curing the mixture to harden the composition and form the conductive resilient product.
For the compound containing the at least two active hydrogens, a polyol may be used. Examples of suitable polyols include a polyether polyol, a polyester polyol, and a polyetherester polyol having a terminal hydroxyl group on its end, but are not necessarily limited thereto. Additionally, a denatured polyol such as an acryl-denatured polyol or a silicone-denatured polyol can be used as the polyurethane used in the supply roller.
For the compound containing the at least two isocyanate groups, a polyisocyanate may be used. Examples of suitable polyisocyanates include toluene dilsocyanate (TDI), 4,4-diphenylmethane diisocyanate (MDI) and mixtures thereof, but are not necessarily limited thereto. Additionally, a denatured polyisocyanate can be used as the polyisocyanate.
The polyurethane is desirably prepared by reacting the polyol and the polyisocyanate in the presence of the catalyst. The catalyst is desirably selected from among organometallic compounds, amine-based compounds, and mixtures thereof.
The type and amount of the catalyst used are decided by taking into consideration the blowing properties, reaction time, increase in the ventilation rate of a polyurethane foam, and minimization of the density deviation.
The organometallic compounds used as the catalyst comprise at least one metal selected from the group consisting of tin, lead, iron, and titanium. It is preferable that the amine-based compounds used as the catalyst comprise a tertiary amine.
More desirably, the catalyst is selected from among a tertiary amine and a tin catalyst.
The conductive additive desirably comprises a compound having a terminal hydroxyl group on its end, and a polyalkylene glycol. In addition to the polyalkylene glycol, the conductive additive further comprises at least one salt selected from the group consisting of alkali metal salts and alkaline earth metal salts.
The polyalkylene glycol may comprise condensates of a linear or branched ethylene glycol, a propylene glycol, a tetramethylene glycol, 1,3-butadiol, 1,4-butadiol, neopentyl glycol, 1,6-hexanediol, and bisphenol A. In other words, the polyalkylene glycol may comprise a polyethylene glycol, a polypropylene glycol, a polytetramethylene glycol, a polyethylene glycol-polypropylene glycol copolymer, a ring-opening adduct of bisphenol A ethylene oxide, and a ring-opening adduct of bisphenol A propylene oxide.
Additionally, a polyester diol such as a polyadipate diol, a polycarbonate diol, and a polycaprolactone diol may be used as the compound having a hydroxyl group on its end.
The polyalkylene glycol or the polyalkylene diol desirably has a molecular weight of about 300 to 3,000. If the molecular weight is less than 300, the unreacted materials in the resulting polyurethane foam migrate to the surface, and if the molecular weight is equal to or higher than 3,000, the high viscosity of the polyalkylene glycol may inhibit the formation of the polyurethane foam.
The metal salts usable as the conductive additive according to the exemplary embodiment of the present invention may include perchlorate, chlorate, hydrochlorate, bromate, oxo acid salt, fluoroborate, sulfate, ethylsulfate, carboxylate, and sulfonate of the alkali metals and the alkaline earth metals, but are not necessarily limited thereto. Desirably, the metal salts may be lithium perchlorate.
Examples of the alkali metal salt are selected from the group consisting of lithium, sodium, potassium, rubidium, and cesium salts, but are not necessarily limited thereto. Desirably, lithium salts may be used.
Additionally, examples of the alkaline earth metal salts are selected from the group consisting of beryllium, magnesium, calcium, strontium, barium and radium salts, but are not necessarily limited thereto.
100
In the exemplary embodiments of the present invention, the amount of the conductive additive having a terminal hydroxyl group on its end to be added is about 3 phr (parts per hundred rubber) to about 100 phr based on the amount of the polyol. If the amount is equal to or lower than 3 phr, sufficient conductivity is not provided to the resulting polyurethane. If the amount is equal to or higher than phr, the resulting polyurethane foam disintegrates and the cells are irregularly formed.
The blowing agent forms bubbles in the polyurethane, which helps to form the foam. The blowing agent usable in the exemplary embodiment of the present invention may comprise any blowing agent usable in blowing the polyurethane.
The blowing agent may be either water or a low-boiling point material such as a halogenated alkane. Examples of the halogenated alkane may include trichlorofluoromethane, but desirably water is used as the blowing agent.
The surfactant improves miscibility by reducing surface tension, causes the bubbles generated by the blowing agent to be of a uniform size, and stabilizes the blowing agent by controlling the cell structure of the polyurethane foam.
Desirably, a silicon surfactant can be used as the surfactant.
The surfactant is desirably added in an amount in the range of about 0.1 phr to about 5 phr based on the amount of polyol added in order to form the polyurethane. When the amount of the surfactant is equal to or less than 0.1 phr, the proper functioning of the surfactant cannot be guaranteed, and when the amount of the surfactant is equal to or higher than 5 phr, properties such as its compression set, may be reduced.
In the exemplary embodiments of the present invention, a method of manufacturing a supply roller of a developing device for an image forming apparatus comprises preparing a conductive resilient member comprising a polyurethane, a conductive additive, a blowing agent and a surfactant; cutting the conductive resilient member into a cylindrical shape, and forming a shaft-shaped hole in the center of the conductive resilient member; and pushing a shaft into and through the hole, heating, and adhering the conductive resilient member and the shaft.
3
3
The conductive resilient member prepared by the manufacturing method according to the exemplary embodiment of the present invention has a density of about 60 kg/mto about 120 kg/mand an outer diameter of about 8.0 mm to about 10.0 mm.
Additionally, the shaft has an outer diameter of about 4.0 mm to about 6.0 mm.
First, a conductive resilient member was prepared. GP-3000 (manufactured by KOREA POLYOL Co., Ltd., containing 54 mgKOH/g of a hydroxy group) and KE-848 (manufactured by KOREA POLYOL Co., Ltd, containing 30 mgKOH/g of a hydroxyl group) as a polyester polyol, were combined with water as a blowing agent, a silicone surfactant as a surfactant, a catalyst, and a compound containing polyethylene glycol and lithium perchlorate as a conductive additive having a terminal hydroxyl group on its end, to obtain a pre-mixed polyol.
The conductive additive having a terminal hydroxyl group on its end was obtained in the following manner. To a methyl ethyl ketone solvent, were added 100 g of a polyethylene glycol having a molecular weight of 500 and 10 g of lithium perchlorate, and the resulting mixture was reacted at a temperature of 50° C. to 80° C. for 16 to 20 hours. This reaction was monitored using a Fourier Transform-Infrared Spectroscope (FT-IR), and the methyl ethyl ketone solvent was distilled off under a reduced pressure of 30 to 5 mmHg to obtain a conductive additive.
Toluene diisocyanate (TDI) as a polyisocyanate was added to the prepared pre-mixed polyol, and then agitated at 2000 rpm. The resulting mixture was injected into a mold, and then dried in a forced air convection oven at 60° C. for 20 minutes to prepare a conductive resilient member.
The prepared conductive resilient member was cut into a cylindrical shape, and a shaft-shaped hole was then formed longitudinally in the center of the cylindrical column. A metal shaft, wound with a hot melt sheet, was pushed into the hole. The conductive resilient member and the shaft were attached to each other by heating in a forced air convection oven at 120° C. for 30 minutes. The adhered conductive resilient member was polished by a polisher, and both ends of the conductive resilient member were then cut. By this process, a supply roller was manufactured.
A supply roller was prepared in the supply roller manufacturing method described above, using the following quantities of each component.
Content
Composition
(phr)
Polyol:
GP-3000 (manufactured by KOREA POLYOL
80.0
Co., Ltd.)
KE-848 (manufactured by KOREA POLYOL
20.0
Co., Ltd.)
Polyisocyanate: TDI
105.0
Catalyst:
Stannous octoate
0.3
Triethylamine
0.2
Blowing agent: Water
4.0
Surfactant: Silicone surfactant
1.5
Conductive additive
35.0
The volume resistance and density of the supply roller manufactured in Example 1 were measured as follows.
(1) Resistance: the supply roller was mounted in a Jig, conductive shafts of 200 g were put on both ends of an upper part of the supply roller, −100 V of a direct current (DC) voltage was applied to the shaft, and the roller was rotated at a certain speed (for example, 30 rpm) to measure the electric current. The measured current value was converted to a resistance value using the following Equation.
Resistance (Ω)=Voltage (V)/Electric current (A)
(2) Density: the weight of the conductive resilient member having a width of 300 mm, a length of 300 mm, and a thickness of 50 mm was measured.
3
3
Density (kg/m)=Weight (kg)/Volume (m)
3
The supply roller manufactured in Example 1 had a volume resistance of 0.5 MΩ and a density of 100 kg/m.
Supply rollers were manufactured by changing the outer diameter of the supply roller, the outer diameter of the shaft and the density of the conductive resilient member, and images were formed using each of the manufactured supply rollers to measure image quality.
(1) Outer Diameter of Supply Roller
3
When a supply roller having a volume resistance of 0.5 MΩ was manufactured in which the density of the conductive resilient member was 100 kg/mand the outer diameter of the shaft was 6.0 mm, the outer diameter of the supply roller was changed to 7.0 mm, 8.0 mm, 9.0 mm, 10.0 mm, and 12.0 mm to measure image quality.
When the supply properties of the supply roller and the ghost phenomenon occurring on images were evaluated by the naked eye as the criteria for the image quality, the results were recorded using ◯ to represent “Excellent”, Δ to represent “Good”, and × to represent “Poor.” A toner was inserted into a gap of the conductive resilient member to block the gap when printing images for a long time period, so that the supply properties of the supply roller were reduced. The ghost phenomenon occurs when a residual image is generated on a formed image due to a difference in the charge amount of the toner. Accordingly, the image quality was evaluated by the naked eye by determining whether the ghost phenomenon occurred or whether the supply properties were reduced.
The results of the evaluation are shown in Table 1.
TABLE 1
Outer diameter (mm)
7.0
8.0
9.0
10.0
12.0
Supply
X
Δ
Δ
◯
Δ
properties
Ghost
X
◯
◯
◯
◯
phenomenon
(2) Outer Diameter of Shaft
3
3
0
When a supply roller having a volume resistance of 0.5 MΩ and an outer diameter of 9.0 mm was manufactured, in which the conductive resilient member had the density of 100 kg/m, shafts with an outer diameter of . mm, 4.0 mm, 5.0 mm, 6.0 mm, and 7.0 mm were used to measure the image quality.
When the supply properties of the supply roller and the ghost phenomenon occurring on images were evaluated by the naked eye as the criteria for the image quality, the results were recorded using ◯ to represent “Excellent”, Δ to represent “Good”, and × to represent “Poor.” The results of the evaluation are shown in Table 2.
TABLE 2
Outer diameter (mm)
3.0
4.0
5.0
6.0
7.0
Supply
X
◯
◯
Δ
X
properties
Ghost
X
◯
◯
◯
◯
phenomenon
(3) Density of Conductive Resilient Member
3
3
3
3
3
3
When a supply roller having a volume resistance of 0.5 MΩ was manufactured to have a shaft with an outer diameter of 6.0 mm, and a supply roller with an outer diameter of 9.0 mm, the density of the conductive resilient member was set at 40 kg/m, 60 kg/m, 80 kg/m, 100 kg/m, 120 kg/m, and 140 kg/musing a method such as changing the content of the composition, and the image quality was then measured. In order to change the density, a method was used in which the contents of a polyol and a polyisocyanate were changed, or an amount of the mixture injected into a mold was varied while maintaining the ratio of the total content of the composition.
When the supply properties of the supply roller and the ghost phenomenon occurring on images were evaluated by the naked eye as the criteria for the image quality, the results were recorded using ◯ to represent “Excellent”, Δ to represent “Good”, and × to represent “Poor.” The results of the evaluation are shown in Table 3.
TABLE 3
Density (kg/m<sup>3</sup>)
40
60
80
100
120
140
Supply
◯
◯
Δ
Δ
Δ
X
properties
Ghost
X
Δ
Δ
◯
◯
◯
phenomenon
Referring to Table 1, a greater outer diameter of the supply roller corresponded to superior supply properties and prevention of the ghost phenomenon. However, if the outer diameter of the supply roller is too large, it is difficult to miniaturize the image forming apparatus.
Accordingly, when the outer diameter of the supply roller was 7.0 mm, the supply properties were reduced and the ghost phenomenon was obvious. When the outer diameter of the supply roller was 12.0 mm, the ghost phenomenon was less obvious. The ghost phenomenon occurred due to a difference in the charge amount of the toner by increasing the nip portion between the developing roller and the supply roller to increase the toner stress. The increased nip portion allowed the load of a toner cartridge to be increased, resulting in image deviations caused by such factors as jitter. Additionally, as it is necessary to miniaturize the image forming apparatus, particularly the developing device, high quality images can be formed using a small supply roller having an outer diameter of 8.0 mm to 10.0 mm.
Referring to Table 2, when the outer diameter of the shaft was 3.0 mm, the quality of the formed image was reduced when the above test was observed. In this case, the diameter of the shaft was very small, the shaft was bent, and thus the toner supply properties were reduced and the ghost phenomenon was obvious.
However, when the outer diameter of the shaft was in the range of 4.0 mm to 5.0 mm, the quality of the formed image was excellent as observed in the above test. When the outer diameter of the shaft was in the range of more than 6.0 mm to 7.0 mm, the outer diameter of the shaft was increased to reduce the thickness of the conductive resilient member because the outer diameter of the supply roller remained constant. Accordingly, the toner supply properties were reduced due to the occurrence of the toner filming phenomenon. Therefore, when the outer diameter of the supply roller remained constant, it was most desirable that the outer diameter of the shaft was in the range of 4.0 mm to 6.0 mm.
3
3
3
3
Referring to Table 3, when the conductive resilient member had a density of 40 kg/m, the image quality was reduced due to the ghost phenomenon occurring on images. When the conductive resilient member had a density of 140 kg/m, the supply properties were reduced due to the occurrence of the toner filming phenomenon. Therefore, it was most suitable that the density of the conductive resilient member was in the range of 60 kg/mto 120 kg/m.
As described above, the exemplary embodiments of the present invention provide a supply roller of a developing device for an image forming apparatus, which can be manufactured in a compact size and which exhibits excellent toner supply properties while preventing occurrence of ghost phenomenon and toner-filming phenomenon causing deterioration in image quality.
Additionally, since the toner-filming phenomenon is prevented, the lifespan of the supply roller can be guaranteed to be longer.
The foregoing embodiments and advantages are merely exemplary and are not to be construed as limiting the present invention. The present teaching can be readily applied to other types of apparatuses. Also, the description of the embodiments of the present invention is intended to be illustrative, and not to limit the scope of the claims, and many alternatives, modifications, and variations will be apparent to those skilled in the art.
BRIEF DESCRIPTION OF THE DRAWINGS
The above aspects and features of the present invention will be more apparent by describing certain exemplary embodiments of the present invention with reference to the accompanying drawings, in which:
FIG. 1
is a view schematically illustrating a general image forming apparatus; and
FIG. 2
is a perspective view illustrating a supply roller of a developing device usable in an image forming apparatus according to an exemplary embodiment of the present invention. | |
Vegetarian Soups and Salad
Craving something heartier?
Use fresh, seasonal produce to make light and healthy soups and salad recipes.
Perfect for any occasion be it Holiday party or any gatherings.
Follow these vegetarian salads (Such as Quinoa black bean corn salad, Southwest Salad, Wild Rice Salad, and many more) and soup recipes (Like Lasagna Soup, Vegan Mushroom Stroganoff, Carrot tomato soup, etc.) for a comforting meal in a bowl.
Instant Pot Detox Vegetable Soup
Instant Pot Tomato Basil Soup
Instant Pot Stuffed Pepper Soup
Instant Pot Creamy Potato Leek Soup
Instant Pot Broccoli Cheese Soup
Spinach Soup
Instant Pot 15 Bean Soup
Fajeto
Balela Salad | Mediterranean chickpea salad recipe
AS SEEN IN: | https://www.cookingcarnival.com/category/soup-salad/ |
Q:
JSON & Javascript - Dynamic DOM manipulation
I'm learning JSON, and DOM Manipulation. I've been trying to get my HTML code to dynamically update w/ information from a JSON file.
<div class="towns">
<div class="franklin">
<h3 class="town-title">Franklin</h3>
<p class="f-motto">ssd</p>
<p class="f-year">1232</p>
<p class="f-pop">123123123</p>
<p class="f-rain">1213</p>
</div>
<div class="greenville">
<h3 class="town-title">Greenville</h3>
<p class="g-motto">ssd</p>
<p class="g-year">1232</p>
<p class="g-pop">123123123</p>
<p class="g-rain">1213</p>
</div>
<div class="springfield">
<h3 class="town-title">Springfield</h3>
<p class="s-motto">ssd</p>
<p class="s-year">1232</p>
<p class="s-pop">123123123</p>
<p class="s-rain">1213</p>
</div>
</div>
This is the Javascript i'm using to retrieve data and also update the DOM.
<script>
var requestURL = 'https://byui-cit230.github.io/weather/data/towndata.json';
var request = new XMLHttpRequest();
request.open('GET', requestURL);
request.responseType = 'json';
request.send();
request.onload = function() {
var stats = request.response;
populateHeader(stats);
showHeroes(stats);
}
function populateHeader (jsonOBJ) {
var motto = jsonOBJ['towns'][0]['motto'];
document.querySelector('.f-motto').innerHTML = motto;
}
</script>
As of right now I know i could statically update the 'motto' by putting in a number in jsonOBJ['towns'][number]; but, I was wanting to do something like this:
var n = 0;
var motto = jsonOBJ['towns'][n]['motto'];
and set all classes to .motto instead of .__-motto
i've tried doing a for loop to dynamically change variable 'n'; but, i've only gotten the same motto across all three elements. Any help would be awesome.
Thanks!
json code:
{
"towns" : [
{
"name": "Franklin",
"motto": "Where you will grow!",
"yearFounded": 1788,
"currentPopulation": 30458,
"averageRainfall": 21,
"events" : [
"March 4: March to the Drum of Donuts",
"September 5 - 11: Founder Days",
"December 1 - 26: Christmas in the Heart"
]
},
{
"name": "Greenville",
"motto": "Green is our way of life.",
"yearFounded": 1805,
"currentPopulation": 33458,
"averageRainfall": 25,
"events" : [
"February 10-12: Greenbration",
"May 8 - 18: Greenville Founder Days",
"June 20: Verde and Valiant Day",
"November 15-16: Greensome Gathering"
]
},
{
"name": "Placerton",
"motto": "Positive Placement in Placerton.",
"yearFounded": 1946,
"currentPopulation": 512,
"averageRainfall": 39,
"events" : [
"July 4: A Blaze of Glory",
"October 20: Fall through Fall"
]
},
{
"name": "Springfield",
"motto": "Where everyone is lifted.",
"yearFounded": 1826,
"currentPopulation": 17852,
"averageRainfall": 17,
"events" : [
"January 8: Spring into Winter",
"April 10-20: Celebration of Life",
"July 31-Aug 15: Dog Days of Summer Festival"
]
}
]
}
A:
Try a forEach to get your n.
//use querySelectorAll to get all the html elements in one go
var mottos = document.querySelectorAll('.f-motto');
//forEach passes in each element and it's index for each iteration
jsonOBJ['towns'].forEach(function(town, index){
mottos[index].innerHTML = town.motto;
});
| |
Vegetable Buddha Bowls with Carrot Ginger Dressing
“Buddha bowl” is a name for an entire meal, usually vegan, in a bowl. Buddha bowls often start with a grain, have a plant protein of some kind, and lots and lots of vegetables. The Carrot Ginger Dressing in this bowl is not only a delicious and vibrant dressing, it also has powerful anti-inflammatory powers thanks to both the ginger and carrots!
Course
Dinner
,
Main Course
Cuisine
Modern
Difficulty
Medium
Browse Category
Vegan & Vegetarian
Duration
30-60 min
Diet
Plant-Based
,
Vegan
,
Vegetarian
Cooking Technique
Pressure Cook
,
Sauté
Main Ingredient
Avocado
,
Broccoli
,
Brown Rice
,
Carrot
,
Cucumber
,
Ginger
,
Red Cabbage
Keyword
Buddha bowl recipe
,
instant pot Buddha bowl recipe
,
instant pot recipe
,
instant pot vegan recipe
,
instant pot vegetables
,
instant pot vegetarian recipes
,
pressure cooker recipes
,
vegan
,
vegan recipes
Servings
Prep Time
4
servings
20
minutes
Cook Time
26
minutes
Servings
Prep Time
4
servings
20
minutes
Cook Time
26
minutes
Ingredients
1
cup
short-grain brown rice
1 1/4
cups
vegetable stock
divided
2 1/2
tbsp
extra-virgin olive oil
2 1/2
tbsp
apple cider vinegar
2
large carrots
peeled and thinly sliced, divided
1
tbsp
fresh ginger
peeled and chopped
1
tbsp
fresh lime juice
1/4
tsp
pure stevia powder
3/4
tsp
toasted sesame oil
1/8
tsp
salt
1 1/2
cups
frozen shelled edamame
thawed
1 1/2
cups
broccoli florets
thinly sliced
4
cups
red cabbage
thinly sliced
1
medium cucumber
thinly sliced
2
medium avocados
peeled, pitted, and thinly sliced
2
tbsp
sesame seeds
2
scallions
thinly sliced
Instructions
Place the rice and
1 cup
stock in the inner pot. Secure the lid.
Press the
Manual or Pressure Cook
button and adjust the time to
24 minutes
.
While the rice is cooking, make the dressing. In a powerful blender, blend the olive oil, vinegar, half of the carrot slices, ginger, lime juice, stevia, sesame oil, and salt until the mixture is super smooth. Set aside.
When the timer beeps, let
pressure release naturally
until float valve drops and then unlock lid. Use a fork to fluff the rice. Press the
Cancel
button.
Press the
Sauté
button and add
1/4 cup
stock with the edamame and broccoli. Gently stir and then let them cook until warm, about
2 minutes
.
To assemble the Vegetable Buddha Bowls, spoon one quarter of the rice mixture into each of the four bowls. Add one quarter each of the remaining carrot slices, cabbage, cucumber, and avocado slices to each bowl, keeping the ingredients separated. Sprinkle one quarter of the sesame seeds and scallions over each bowl and drizzle with one quarter of the Carrot Ginger Dressing. | https://recipes.instantpot.com/recipe/vegetable-buddha-bowls-with-carrot-ginger-dressing/print/ |
Proofs of the basic mathematical principals of ECE Theory are offered here. These proofs are proofs of Riemann and Cartan geometry, and are important to study and master. They are as follows and give much more detail than that found in any textbook that I know of.
An understanding of ECE theory can only be obtained if these proofs are mastered. This study will also allow the reader to see how the harasser cell perpetrates its fraud by use of deliberate errors. Without knowledge of Cartan and Riemann geometry the reader is left in a perpetual void, or state of uncertainty, and this is what the fraudsters thrive on. This scientific fraud is similar to forgery of a painting in the art world. Spanish translation of the proofs and flow charts can be found in the Spanish Section.
Proof of the anti-symmetry of the connection: This is a student level exercise in Riemann geometry. By definition the commutator of co-variant derivatives is anti-symmetric in its indexes.
Proof that curvature implies torsion and vice versa: This is definitive proof two, giving details which are left out by Carroll in his chapter three. It shows that spacetime torsion is always present in any spacetime in any dimension, irrespective of any other assumption such as metric compatibility or tetrad postulate. This proof cannot be "changed" by invoking mysterious symbolism as does Rodrigues, whose professional reputation has self-destructed as a consequence of years of misrepresentation of geometry and arrogant bombast. Similarly Bruhn and Jadczyk, or anyone who tries to misrepresent Riemann or Cartan geometry. There is no further need to read the stuff turned out by these people, but it must always be pointed out that they are misrepresenting mathematics and corrupting science. Editors who cite their stuff do not know what they are doing.
Proof of the tetrad postulate: The tetrad postulate is the very fundamental requirement that the complete vector field be independent of the way in which it is defined by its components and basis elements. It has been used since 1925 and is used in any proof of Cartan geometry. Suddenly and mysteriously, the tetrad postulate started to become "debatable" as soon as I started to use it. If others such as Carroll use it, it is OK, but if my colleagues and I use it, it is not OK. So this is blatant fraud perpetrated by the same well known people for years. In definitive proof three I will give the proof given by Carroll in his chapter three, and again give more detail than Carroll, in fact complete detail of the proof. With a bit of effort and practice, these proofs are not difficult for trained mathematicians (e.g. A level students). Physicists and chemists are normally expected to have an A level in mathematics and to do university undergraduate courses in mathematics. So there is really no excuse in saying that these proofs cannot be understood. That is the kind of thing the fraudsters thrive on.
Self checking proof of the Cartan Bianchi identity: This was first given in paper 15 of the ECE source papers about five years ago (www.aias.us) and no genuine mathematician has objected to that proof. This is hardly surprising because it is used in standard student courses in Cartan geometry. Later proofs of the identity were given in papers 99 ff. as overviewed on the ECE Sci Topics site (now also available on www.aias.us by clicking on "Myron Evans"). The homogeneous field equations of dynamics and electrodynamics are based directly on this identity, first given by Cartan in about 1925, and taught ever since. These are given in vector format in the ECE engineering model, which has been coded up. Using this model, patents have been written and applied for. In my shorthand notation the identity is
D ^ T := R ^ q := q ^ R ------------------ (1)
and brings out the fact that torsion (T) is linked ineluctably to curvature (R). So we can see that if torsion is omitted as in the now obsolete standard model then something is bound to go wrong. That something was discovered in papers 93, 95, and 120 for example using computer algebra. Again, no genuine mathematician has objected to those papers, written by four authors in total. This identity is probably Cartan's most elegant theorem, because it shows that the cyclic sum of three curvature tensors is identically equal to the same cyclic sum of the definitions of the same three curvature tensors. Eq. (1) is a most elegant expression of this result of geometry. In order to arrive at this result, the tetrad postulate is used as always in Cartan
Self checking proof of the Cartan Evans dual identity: This will be the proof of the Cartan Evans dual identity, the basis of the inhomogeneous field equations of dynamics and electrodynamics in ECE theory. It was used in papers 93, 95 and 120 to show that the Einstein field equation and all its solutions violate geometry, a catastrophe for the standard model. Since 2007, there have been no genuine objections to this work, it is based on the use of computer algebra. Any attempted "refutation" of this standard student level work is mathematical fraud. It is of great importance to reveal this corruption of science and these five proofs are all that is needed. No attempt should be made to read the garbage being thrown at geometry by the standard fringe, as Prof. Dunning-Davies points out, this is personal baiting, a violation of ethics, those of communication and mathematics.
Example of relabeling of summation indices: This is known as relabeling of summation indices. Sometimes these are known as dummy indices. This procedure occurs in many proofs of Riemann and Cartan geometry.
New General Condition for any metric: Horst Eckardt and I feel that this looks like a useful new result which can be used to test metrics from the Einstein equation. It simplifies the metric compatibility condition to one where the ordinary partial derivative can be used. The whole of Cartan's geometry can be developed in this way.
Basic Hypothesis of Gravitational Physics Flow Chart This is the second ECE hypothesis which leads to an economic description of all planar orbits, including those of galaxies, and links in to the highly developed subject of angular momentum theory (P. W. Atkins, "Molecular Quantum Mechanics", many editions, M. W. Evans and J.- P. Vigier "The Enigmatic Photon" on www.aias.us Omnia Opera, and M. W. Evans (ed.) "Modern Non-Linear Optics" (2001), two reviews of which are on the Omnia Opera). This means that the anti-symmetric connection can also be developed in as many ways as angular momentum theory can be developed, revealing after one hundred years of incorrect gravitational theory the true meaning of spacetime connection. It is the spinning of spacetime. | http://www.aias.us/index95e8.html?goto=showPageByTitle&pageTitle=Definitive_Mathematical_Proofs_of_ECE_Theory |
How to Register for Classes
It is the student’s responsibility to stay on track with graduation requirements, and to make sure the Faculty Adviser approves course selections prior to registering for each semester. Schedules should be reviewed again if changes are made.
Hybrid Online students please note some of the information below will only pertain to you once you arrive on-campus.
The GSAS Academic Calendar indicates the general dates that students can register. The system will allow students to register only during specific registration appointment times on Student Services Online (SSOL) (accessed with UNI and password). “REG APPTS” reveals the designated times.
The Columbia Course Directory and Vergil offer key information about each class including the course size, prerequisites, and if there are any restrictions. For example, if a course is not open to students in GSAS, it would not be open to an MA Statistics student. For further details about registration procedures, students may refer to the GSAS Registration page.
Before registering, the onus is on the student to reach out to their assigned Faculty Adviser to have the individual course list approved (in person or via email).
Registration in Graduate School of Arts and Sciences is a two-part process that consists of registering for both individual classes and also for a full or fractional Residence Unit. A Residence Unit (RU) is a registration category that determines the amount of tuition.
HERE is a video that describes how to use SSOL for registration. HERE is a video about using Vergil for planning your courses and schedule.
Residence Units
Below are Residence Unit Categories for each of the MA Programs. For more information about Residence Units, see also Tuition & Fees and the GSAS Website.
Full-time students must be registered for a Full Residence Unit (RU) in the first two semesters and an Extended Residence Unit (ER) in their final semester. A student who is registered for a full RU or ER is considered to have full time status, regardless of the number of courses. Both RU and ER allow any number of points up to and including twenty points. ER is required after completing a total of TWO RUs. Always check with ISSO about international and visa-related issues.
Domestic students have the option of being full-time or part-time. One RU is considered full-time. Part-time allows for QR (Quarter Residence Unit) or HR (Half Residence Unit). ALL MA Program students are required to complete two RUs to graduate.
After completing a total of two RUs, all students must register for ER. One can register for ER for more than one semester (e.g., Summer Term and Fall Term).
|Statistics On-campus and Hybrid Online/On-Campus Categories|
|Full Residence Unit (RU)||20 credits maximum (RESI G0001, call # 99991)|
|Extended Residence (ER)||20 credits maximum (EXRS G0001, call # 77771) Required after completion of two Residence Units|
|Summer ER||20 credits maximum (EXRS UN0001, call # 10009) Taken during the summer semester|
|Half Residence Unit (HR)||Three courses maximum (RESI G0002, call # 99992)|
|Quarter Residence Unit (QR)||Two courses maximum (RESI G0003, call # 99993)|
Registration Guidelines
For registration guidelines please review:
How many courses?
- The number of classes taken must be approved by the Faculty Adviser.
- To remain in Good Academic Standing (GPA at 3.0 and above), it is recommended to take no more than four classes per semester, especially in the first semester.
- Students on F-1 visa must register for a full RU in their first two semesters and ER after completing two RUs.
- A full RU or ER allows any number of points up to and including twenty points.
- Part-time students may register for a QR or HR, as long as the program is completed within four years and a total of two RUs have been fulfilled.
- Most full-time students should plan for three full semesters. Completing the MA Program in two semesters requires faculty adviser approval.
Which classes to take?
- Here are details about the Statistics MA Program requirements.
- Here are approved electives.
- Here is information about Cross Registration.
- Here are electives grouped according to interest area, such as finance, data science, public health, insurance, or the Ph.D. track.
- Here is a list of courses that will not count for the MA Degree in Statistics.
What if a class is “restricted?”
- This means that it is closed to MA Statistics Students.
- Look up the course in the Columbia Directory and review the section “Open To.”
- Some courses open to additional students on a later date. This is often found in the “Notes” section of the course.
- Here is information about cross registration into other schools.
What if a class is full?
- SSOL is the primary source for all registration including the waitlists.
- The student should register as soon as possible to get on the electronic waitlist.
- Waiting is the next step. Eventually, the student’s request to join the class will either be accepted or denied.
- Students should not email the instructor to bypass others on the waitlist.
- There is no need to reach out to a Faculty Adviser except in a case where a class is required for graduation in the last semester of the program.
- This is a helpful Guide for waitlists.
- Questions may be sent to [email protected].
- It is important to have a back-up plan, such as an alternative course or a different section that is still open.
- It is not permissible to remain on the waitlist if you are accepted into another section of the same course.
- If accepted into a desired section, please drop all other sections to open those waitlists to others.
Residence Unit (RU) Status
- Always review RU status.
- Questions about tuition may be sent to [email protected].
- If the total number of points goes over twenty, there will be extra charges.
- If a student needs to change to a lower RU, this must be done manually by the student. It does not happen automatically when one registers for fewer credits.
- Statistics students in the MA Program may not register for greater than fifteen points at one time. Special permission must be obtained to register for more than fifteen.
Courses must have normal grading to count.
- Any course with P/F or R Grade does not count toward graduation.
- The four CORE REQUIRED courses MAY NOT be taken Pass/Fail.
How to add or drop a class after registration:
- The first two weeks of the semester is called the “Change of Program Period” because students may add and drop classes in SSOL without penalty.
- Your schedule should be set at the end of the “Change of Program Period.”
- Refer to the Academic Calendar for add/drop and other deadlines.
- No classes may be dropped after the LAST DAY TO DROP. There are no exceptions.
- No classes may be changed to R or Pass/Fail after the DEADLINE to change to PASS/FAIL or R.
- The deadlines for half- semester courses are different, to remain in proportion to the deadlines for full-semester courses:
- Half-semester courses must be dropped within the first two weeks.
- The deadline for Pass/Fail or for R credit is the first day of the fifth week. (Remember that core classes may not be taken for Pass/Fail or R credit. Nor may they be taken over to receive a higher grade.)
- HERE is information about the post-Change of Program Period.
- Students who stop attending a course without officially dropping will be assigned a grade.
The Registration Adjustment Form (RAF)
The Registration Adjustment Form may NOT be used to enter a class that is noted as RESTRICTED in your SSOL schedule. The Registration Adjustment Form (RAF) may be used only under these special circumstances:
- During cross registration when the other school has no special procedure. The form should be brought to GSAS Main Office located at 107 Low Library with the signature and UNI of one’s Faculty Adviser, the instructor, and possibly an appropriate administrator of the other school. An email print-out approval attached to the RAF will count for a signature.
- When needing to sign-up for greater than the allowed fifteen credits for MA Statistics Students.
- To drop the last or only class. A special request must be made to the Office of Student Affairs at 107 Low Library. An application for a Leave of Absence or Withdrawal is required based upon the particular situation.
Under all circumstances, the RAF gets handed in to 107 Low Library (GSAS).
Important Links for Registration
- University Registrar– University Registrars Office
- Registration–GSAS instructions for registration
- Directory of Classes or Vergil – either can be used to find out more information about a class including cap and if there are any restrictions.
- GSAS Academic Calendar
- Registration Adjustment Form– This is for special cases to add/drop that are not accessible via SSOL.
- Request for Transcripts
- Student Financial Services: Billing and Payments
- FAQs-Registration
Cross Registration
Cross Registration is always at the discretion of the school, the program and the course instructor. The Statistics Department does not have any influence on allowing you into a cross registered class. Before attempting to cross register, MA Students must check in with their Faculty Adviser to ensure that it is a class they can take for credit.
Some graduate schools have their own cross registration processes. Please review the links below for information on how to cross register for those graduate schools. For schools not listed below, reach out to the specific department and inquire about their cross registration options and process.
Admittance into any course is not guaranteed. All courses are subject to availability.
Below are links for how to cross register into the following schools:
- Computer Science .
- Industrial Engineering and Operations Research .
- Columbia Business School
- Most courses have a prerequisite of Capital Markets “FINC B8306/ MATH G4076” or “waiver exam”.
- Email [email protected].
- Data Science Institute, please email [email protected] regarding course access through their internal lottery and survey process.
MA students may register for courses at TC, as long as they get permission from TC and their adviser.
Transfer Credit
Courses taken in Statistics at Columbia University prior to entering the M.A. program may count toward the M.A. degree if approved by the Program Director and upon final review by the Graduate School. According to the policy of the Statistics Department, no more than 4 classes (and a maximum of 1 RU), taken at Columbia prior to admission in the M.A. program may be applied towards the M.A. degree. No Columbia course may be applied to more than one degree.
When transfer credit is granted, concomitant Residence Units may also be credited toward fulfilling the two RU requirement for the M.A. Degree. Here is more information about transfer credit.
A student may receive Transfer Credit by following this procedure:
- There is a Transfer Credit form on the GSAS Website. The top of this form must be filled out with student signature on the appropriate line.
- The courses should be listed in the blank area provided.
- “Previous Institution” should be Columbia University, i.e., where the courses were taken.
- The form may be given or emailed to the Program Coordinator who will pass it to the Director for approval.
- The Program Director will determine the number of points and RUs to be credited.
- Once the form has been approved and signed by the Director, it can be submitted to GSAS in 107 Low.
Courses Taken for R Grade or Pass/Fail
During the semester there are important dates to remember including the LAST DAY to DROP COURSES and the LAST DAY for R Grade & PASS/FAIL. These deadlines are strictly followed.*
- Courses taken for R or Pass/Fail do not count towards the MA Statistics degree.
- Core classes may not be taken for R or P/F.
- Click HEREfor further details on the “Pass/Fail” grade.
- Not all courses are eligible for either P/F or R.
- For P/F, check in SSOL to see if the course is eligible for Pass/Fail and select.
- R grades must be approved by the course instructor. You must make an arrangement with the instructor.
- Click HERE for further details on the “R” grade.
The Faculty Adviser should be notified before the status of a course is changed.
- Students are not allowed to re-register for a course to improve letter grade. (The only exception would be for a Grade of F which must be approved by the Program Director or the MA Program Executive Director, as well as the Graduate School of Arts & Sciences.)
- Review REQUIREMENTS of the MA Degree.
- Review GOOD ACADEMIC STANDING.
*Deadlines for half-semester courses are different from full-semester courses. Half-semester courses must be dropped within the first two weeks. The deadline to change the grade of a half-semester course to Pass/Fail is the first day of the fifth week of the course. Core courses may NOT be taken for Pass/Fail or R Credit.
Summer Registration
Summer Session classes are offered to students in the MA Statistics Program through the School of Professional Studies. The Columbia University Directory contains a link to view Statistics courses broken down by semester.
- Because the School of Professional Studies administers the Summer Session, please see the Registration and Calendars pages for details about registration policies and deadlines.
- In addition to registering for classes, an MA Statistics student must also register for the appropriate Residence Unit.
- If the course counts for graduation, a full-time student must register for ER.
- If the course will not count for graduation, please visit the GSAS website.
- Please make sure to review the GSAS information on how to register for courses as well as summer tuition HERE.
Questions about registering for Summer courses can be sent to: [email protected].
Continuous Registration and Leave of Absence
All students, full-time and part-time, are required to register each Fall and Spring semester until all degree requirements have been completed or until the deadline for completing the degree has been reached. See Continuous Registration and Satisfactory Academic Progress. Students who must interrupt studies for compelling reasons need to request a Leave of Absence by filling out the Form on this PAGE. GSAS will respond to the request.
GSAS allows part-time students to take up to four years to complete the degree. All part-time students must be registered continuously for each Fall and Spring semester. A Leave of Absence Form must be filled out if a part-time student does not take class during any Fall or Spring semester. Please review the information HERE.
International students must complete the Masters Degree within three semesters of full-time registration. A Leave of Absence may interfere with the ability to qualify for OPT. Please contact ISSO for details.
Students who wish to return from any approved leave of absence – whether personal, medical, or military – need to complete the Return from a Leave of Absence Form at least a few weeks prior to returning to the program. For more information on how one gets reinstated into the program after an unofficial leave, please refer to this link: Reinstatement. | http://stat.columbia.edu/ma-programs/current-students/registration/ |
Jinshanling section lives up to its special beauty among China Great Wall by being slightly in ruins. A travel to Jinshanling will give you the vivid image of what the primitive walls looked like in ancient China. Jinshanling Great Wall, located some 140 kilometers to the northeast of Beijing, starts from the Wangjinglou Tower in the east and ends at Longyukou in the west and stretches about 10 kilometers. There are 5 main passes and 67 either one tiered or two-tiered watchtowers, in this section with enemy towers on it every 100 meters. There are some windows for shooting arrows on the first floor. Roofs of the towers were used to store weapons and hay, and also could be bedrooms for soldiers. The big Jinshan Watchtower, small Jinshan Watchtower, Wangjinglou Tower, Taochun Tower and Wall for Preventing Horses are the highlights of the trip to Jinshanling section.
A trip to the best preserved part of the Great Wall of China with many original features will worth your traveling. Jinshanling tours take good advantag of the excellent condition of its wall and the majesty of its watchtowers. From afar, it conjures the image of a dragon lying across the mountains and it is easy to imagine the kind of awe it would have inspired in ancient times. For many Great Wall travel experts, it is the ultimate expression of Great Wall construction to be seen anywhere in China and it is revered for its strategic significance and aesthetic majesty. There is a three-kilometer (two miles) section of the Wall was rebuilt and is bathed after dark in colored light, making a splendid "Night Great Wall". In addition, an eight hundred meters (0.5 miles) long cable was built to entertain the tourists. Anyone who travel to the Jinshanling Great Wall should not miss them. | http://www.itourbeijing.com/china-great-wall/jinshanling-travel-guide.htm |
chorobe replied to gr5's topic in Help, Tips & TricksI am also trying to print straight on the glass so I can have that glassy shiny surface. I typically have used a glue stick which works great for adhesion but the glue marks are on the ugly side. Some of you here have said the PVA wood glue solution is just as shiny while some say it is more of a dull or matte finish. I don't have wood glue on me now, but I have tried with regular school glue (diluted to about a 2 to 1 ratio) and that did not adhere well at all plus the surface was matte. If I want matte, I'd just as well print straight on to the painter's tape without any glue. So those of you who do use the PVA solution are you really getting mirror finish with it? I just wanted to make sure before I buy some of the glue. | https://community.ultimaker.com/profile/349073-chorobe/ |
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Finishing the Reset3:24 with Ben Deitch
In this video we'll finish resetting the game by shuffling the deck and dealing out each of the tableau piles!
-
0:00
Getting back to our Reset game function, we still need to shuffle the deck and
-
0:04
deal out each of the tableau files.
-
0:07
Shuffling the deck is easy.
-
0:08
Remember, we already created a function for it in our deck class.
-
0:13
So let's add a line below where we reset our foundation piles and
-
0:17
then type deck.reset to reset our deck to have all 52 cards and shuffle it.
-
0:23
The last thing we need to do is set up each of those tableau piles.
-
0:27
To do this, we need to loop through each of the piles and
-
0:30
make sure that each pile has the right number of cards.
-
0:33
Specifically, the first pile should have one card, the second pile should have two,
-
0:38
and so on, until we get to the seventh pile which should have seven.
-
0:42
An easy way for us to do this is to make use of another extension function on
-
0:47
the array class called forEachIndexed.
-
0:50
Let's add a couple of lines below where we reset our deck and
-
0:54
then type TableauPiles.forEachIndexed.
-
0:58
ForEachIndexed still loops through the entire array just like forEach does.
-
1:03
But with forEachIndexed we also get access to which index we're evaluating.
-
1:09
So since the action for forEachIndexed takes in both the index and
-
1:14
the item, we can't get away with using the it keyword.
-
1:19
With the forEach function, the action only requires one parameter.
-
1:24
So we're allowed to omit it and column will declare it for us as it.
-
1:29
But with forEachIndexed, the action requires both the index and the item.
-
1:34
So we're not allowed to omit our parameters this time.
-
1:37
All right, let's add a line after the arrow.
-
1:41
And then let's create a variable to represent the cards that need to go
-
1:45
in this tableau pile.
-
1:46
Let's call it cardsInPile.
-
1:49
So val cardsInPile, and
-
1:52
since we initialize a tableau with a mutable list of cards,
-
1:57
let's give this variable a type of MutableList of cards.
-
2:03
Then let's set this equal to a new array, and
-
2:06
before we get to the array let's go ahead and convert it to MutableList.
-
2:11
Perfect, getting back to the array, for the size let's pass in i + 1,
-
2:17
which will end up being the numbers 1 through 7.
-
2:21
Just what we need.
-
2:22
Next, for the second parameter to our array, the initializer function,
-
2:27
let's add brackets to make a lambda expression and then,
-
2:31
inside our lambda expression, we just need to return the next card in the deck.
-
2:36
So let's type deck.drawCard, and there we go.
-
2:42
Now let's add a line, And
-
2:46
then let's set this tableauPile equal to a new tableauPile with those cards in it.
-
2:52
Let's type tableauPiles, add index i, and
-
2:57
set it equal to a new TableauPile with those cards in it.
-
3:03
Nice, now, for each TableauPile, we'll draw a card i + 1 times,
-
3:09
and then use those cards to replace the old TableauPile pile with the new one.
-
3:15
And once we're done, we'll be ready to start playing solitaire.
-
3:18
In the next video, we'll start adding some of the functionality to let us
-
3:22
actually play a game of solitaire. | https://teamtreehouse.com/library/finishing-the-reset |
Typical red brick house located in an idyllic community in the Lüneburger Heather´s southern region.
This two apartment, family house offers two completely separated accommodations with separate entrances. The apartments are large, light-flooded and peaceful. The house is only occupied by vacation guests.
With both apartments combined you could lodge up to 10 guests in 10 beds. There would be three bathrooms at your disposal. The apartments have been recently renovated and completely outfitted with everything you may need.
The neighborhood is very friendly and helpful.
Just a one minute drive from the A7 Autobahn/Highway, from the Schwarmstedt exit. 3 amusement parks are within a 30 minute drive away, various destinations of interest, the Hannover airport (20 min.), Hannover (30 min.), Hamburg (60 min.), Bremen (45 min.), Berlin (3.5 hours)
Directly on the Aller-Leine-Bike-Route
Train station is in Schwarmstedt, 3.4 km away. Swimming lake 1 km Free Parking spots for 6 cars, two garage parking spots for a fee of 10 € each
Free bike parking, if required in a secure building. | https://zedernweg.de/en/ |
What kind of savings can a company expect from a Jackson Gainsharing Plan?
Follow the instructions on this sheet to calculate the savings (assuming a 10% increase in productivity), typically realized by Jackson Gainsharing Plan users. This is a calculation worksheet
only
and none of the information is saved. This information
will not
be used by W.M. Jackson & Co., Inc. for any purpose whatsoever. It is simply an exercise to illustrate potential savings. After you have filled in the fields, click on the calculate button. When it has calculated the savings and redisplayed your information, you may print this form out or enter new data for new calculations. Simply click the clear button when finished.
Example
First estimate the value of sales for your company:
$
(A)
$10,000,000
Now, estimate and accumulate the total labor costs for your company:
Direct Labor:
$
$1,450,000
Indirect Labor:
$
$500,000
Vacation Pay:
$
$200,000
Holiday Pay:
$
$100,000
Insurance Benefits:
$
$400,000
Pension Plan:
$
$300,000
Other Benefits:
$
$50,000
Total:
$
(B)
$3,000,000
Divide Item (B) by Item (A) to get a percentage:
(C)
30%
A 10% increase in productivity would be 10% of Item (C):
(D)
3%
The savings or increased profits for our company would be Item (D) multiplied by item (A): | http://wmjackson.com/savings.php |
Pleasant Valley, Utah – “You want to be safe?” asked the owner of Pleasant Valley Farm.
“You’re going to be safer here,” he added.
The owners of the farm were in the midst of a massive flock of ducks, geese, chickens and turkeys that they were expecting to sell to a New York-based poultry company.
But after more than three months of waiting, the ducks, chickens, geysers and turrets were finally released.
The ducks and geese were released in the middle of the night in an area close to the property line of the dairy farm, and the turkeys were released at 10am, just before a snowstorm had fallen.
“We are a small, quiet farm, but it’s a large flock of birds,” said the farm’s owner, Chris Leavitt, a retired mechanical engineer.
The birds were being released to the public for a few hours to avoid crowds.
“There are a lot of people on the road here,” Leavitty said.
“They’ll probably make a big noise, and we want to make sure they don’t, so we don’t have to wait for people to get on the roads.”
A flock of geese at the farm.
Image copyright Facebook/Leavitt FarmChris Leavitte, owner of the Pleasantville Farm, has been on the fence for nearly three months.
He is a retired mechanic who worked on a number of the farms in his career, including the one that he owns.
He said he has had a number a security guards on duty at the fence, and that he has not seen any problems with the birds.
“If you don’t do it right, they’re going up to you, they’ll be chasing you, and they’re not going to let you go,” he said.
The farm is about 1,000 kilometres (800 miles) north-west of Salt Lake City.
“It’s been an incredible experience,” said Leavittle, who had hoped to sell the ducks and turns to a turkey processor.
“I really don’t understand how anyone can take their safety and security so seriously, especially in a place like this.”‘
You’re not getting in trouble for it’The farm owner said he would take any requests from anyone who wanted to buy the birds, and he had already received dozens of emails from people wanting to purchase them.
“You’re a good citizen, so you’re not coming into this farm and saying, ‘I want to buy this duck and I want to take it to the processing plant’,” Leavititt said.’
You can be a little bit of a pest’The ducks, which weigh between 50 and 100 kilograms, were all sold in large crates to people who wanted them for their ducks, he said, adding that he was hoping to sell a couple of dozen of them.
He hoped that a company would be interested in buying the birds for him.
“The birds are worth a lot more than just money.
They’re beautiful animals,” he explained.”
People can buy these for a very, very nice price, and you can be kind of a little pest, but you’re still a good, hardworking person.”
Leavitte said that if he received any requests for more than 20 turkeys, he would consider it.
“One person bought two turkeys and they are beautiful, and I would consider that an awesome donation,” he told Al Jazeera.
“Some people have bought turkeys for the ducks.
I think the ducks are worth about $100 each.”‘
I would be glad to give a duck to anybody’But Leavitto said that he would not sell any more turkeys to people.
“That’s a personal decision,” he acknowledged.
“My job is to protect the birds.”
The duck farm at Pleasant Valley farm.
The owner of Leaviterve is hoping that a poultry processor will buy some of the ducks from him, and if they do, he will donate the birds to the farmer for the farm to sell.
“Our ducks have been here a long time,” he lamented. | https://farokhi-co.com/archives/90 |
Q:
Can a proper subgroup of the multiplicative group of a finite field form an arithmetic progression.
To rule out special cases, here the proper subgroup should not be $1$, and the length of arithmetic progression is finite and at least $3$. If we can permute a subgroup to form an AP, the it also meets the requirement.
Arithmetic progression in this context is just $\{a+nd| n\in \mathbb{N}， n\le N\}\subset \mathbb{F}$, where $\mathbb{F}$ is the given finite field. I tried some cases like $\mathbb{Z}/(p)$, and didn't find any.
So my question is whether we have infinitely many finite fields qualified or the proposition is generally false?
A:
The only such progressions are the "obvious" ones, namely:
$$S = \{1\}, \quad \{-1,1\} \ \text{when $\mathrm{char}(k) > 2$}, \quad \{1,2,3,4,\ldots,p-1\}.$$
The first coincides with the last when $\mathrm{char}(k) = 2$, and the second coincides with the last when $\mathrm{char}(k) = 3$.
Let $p$ denote the characteristic of $k$, let $n$ denote the length of the subgroup/arithmetic progression $S$, and let $q = |k|$. The multiplicative subgroup $k^{\times}$ of a finite field is cyclic, so if $|S| = n$, then
$$S = \{1,\zeta,\zeta^2,\ldots,\zeta^{n-1}\}$$
where $\zeta$ is a primitive $n$th root of unity.
Some Reductions: If $k$ has characteristic $p$, then in any arithmetic progression $a, a+d,a+2d,\ldots$, the initial term $a$ and the $p+1$st term $a+pd=a$ coincide, and thus $n = |S| \le p$. Moreover, the order $|S|$ divides the order of $k^{\times}$ which is $q-1$ where $q$ a power of $p$. Thus $(n,p) = 1$, and hence $n < p$. It follows that if $p = 2$ then $n = 1$ and so $S = \{1\}$, and if $p = 3$ then either $n = 1$ and $S = \{1\}$ or $n = 2$ and $S = \{1,-1\}$.
The argument: We may assume that $p > 3$, $2 < n \le p - 1$, and $(n,p) = 1$.
Suppose that the common difference of the arithmetic progression is $d \in k$. Since $1 \in S$, the set $S$ also must have the form:
$$S = \{1+id \ | \ i \in [-a,n-1-a]\}$$
for some $0 \le a \le n-1$. But now we find that
$$\sum_{x \in S} x = \sum_{i=-a}^{n-1-a} 1 + i d
= \frac{n(2-d-2ad+dn)}{2}.$$
but we also have
$$\sum_{x \in S} x = 1 + \zeta + \ldots + \zeta^{n-1} = \frac{\zeta^n-1}{\zeta - 1} = 0$$
since $\zeta^n = 1$. From these two evaluations we deduce (noting that $n \ne 0$ because $(n,p) = 1$) that
$$(1+2a-n)d = 2.$$
Since $k$ has characteristic different from $2$, it follows that $d$ and $(1+2a-n)$ are both units, and hence that
$$d = \frac{2}{1+2a-n}$$
But now we compute that
$$\sum_{x \in S} x^2 = 1 + \zeta^2 + \ldots + \zeta^{2(n-1)} = \frac{\zeta^{2n} - 1}{\zeta^2 - 1} = 0$$
where we assume that $n > 2$ so $\zeta^2 - 1 \ne 0$, and
$$\sum_{x \in S} x^2 = \sum_{n=-a}^{n-1-a} (1 + i d)^2
= \frac{n (n^2 - 1)}{3 (1 + 2 a - n)^2},$$
where we use that $k$ has characteristic $p > 3$ so there are no issues with the RHS. Now combining these two formulae, we deduce that $n(n+1)(n-1) = 0$. Since $1< n < p$ and $p$ is prime, this can happen only when $n = p-1$. The elements of order $p-1$ in $k$ are precisely the non-zero roots of $x^p - x = 0$, or equivalently the primitive subfield $\mathbf{F}_p$. Thus we have found all such examples.
| |
- Digital capture of the attendance of workers employed under the Mahatma Gandhi National Rural Employment Guarantee Scheme (MGNREGS) has been made universal by the Centre from January 1.
- The Union government, arguing for transparency and accountability in May 2021, had started a pilot project to capture attendance via a mobile application, the National Mobile Monitoring System (NMMS).
- From May 16, 2022 capturing attendance via the app was made compulsory for all worksites with 20 or more workers. This required uploading two time-stamped and geotagged photographs of the workers.
- The job fell on the mates/supervisors, who are paid only marginally more than the unskilled workers. There were widespread complaints over the lack of technical support, the necessity to own a smartphone, paying for an Internet connection, and issues with erratic Internet connectivity.
- In the latest order, dated December 23, the Ministry has ordered that digitally capturing attendance is now mandatory for all worksites, regardless of the number of workers engaged, and will be applicable from January 1, 2023.
- This directive comes even as many complaints and loopholes pointed out earlier by users have not been plugged yet.
- Siraj Dutta, who is associated with the NREGA SangharshMorcha in Jharkhand, points out three major problems with the new system.
- The app-based attendance system carries forward the problem with electronic muster rolls, which replaced the paper muster rolls and was in use before the NMMS was introduced.
- Here, the muster roll has to be generated based on demand and therefore, no worker can come and join at the worksite. And if out of the 10 workers on the electronic muster roll, only two turn up, usually the worksite is not opened, therefore, in a way, denying them work too.
- “The second big problem is the two-time stamped photographs. Often, the workers may finish their work but are forced to return to the worksite for the second photograph,” Mr. Dutta said.
- The endless conditions placed on MGNREGS workers themselves, many activists feel, is enough to dissuade them from relying on the scheme, thus failing its basic purpose.
- “Every time they [the government] have brought in a technology-based solution, they claim it will remove corruption. Are they saying now that they are capturing attendance via a mobile application, there will be no corruption
- These are blatant methods to reduce the spread and effectiveness of the programme,” Nikhil Dey, founder member of the Mazdoor Kisan Shakti Sangathan, said. | https://currentaffairs.chinmayaias.com/centre-mandates-universal-digital-capturing-of-mgnregs-attendance/ |
Description from IMDB:
“Fraternal twins set out to rid Boston of the evil men operating there while being tracked down by an FBI agent.”
After they are nearly killed in the aftermath of a bar fight, brothers Connor (Sean Patrick Flanery) and Murphy (Norman Reedus) MacManus use their multi-lingual abilities to learn of an upcoming meeting of the Russian mob and decide to take justice into their own hands. When the deed is done, their friend Rocco (David Della Rocco), a peon in the Italian Mafia, shows up and it seems he was set up. He decides to help the brothers take down the syndicate as a form of revenge. Meanwhile, the brothers are dogged by virtuoso F.B.I. Agent, Paul Smecker (Willem Dafoe), who slowly loses his nerve as each killing baffles him until he discovers who the vigilantes are and must decide whether he will help them.
The cinematography and story structure are probably the most redeeming qualities of The Boondock Saints. Each crime scene in the film is presented after the introduction of how they began. After establishing the setting and situation, the film cuts to the detectives going over the crime scene and re-creating what happened. Duffy also makes use of slow motion to emphasize certain moments in the story and heighten the dramatic effect.
That is around where the good in the movie ends. The MacManus brothers seem to derive their actions and aesthetic on what will look cool. Placing pennies over the eyes of the dead, saying a long prayer after killing mafiosos, and dressing alike all seem to stem from Duffy sitting down, saying, “Wouldn’t it look cool if…” and then shooting it on film. Part of this is inherent in the “family prayer” that the brothers recite at the end of their killing; it is reminiscent of the Ezekiel 25:17 line spoken by Samuel L. Jackson’s character in the 1992 film, Pulp Fiction, whose character admits that he initially thought it was some cold stuff to say before he killed someone.
I found that I continued to have issues with the film as I watched. The fake accents (both leads are American actors) sometimes drop during scenes, especially for Reedus, which pulled me out of the film and was something I noticed from the very first viewing. There isn’t any character development between the brothers and Rocco; they stay at the same level throughout the movie which leaves them flat and doesn’t allow any emotional connection unless you happen to be as dissatisfied with the justice system as they are. There are some funny moments, such as getting lost and arguing in an air duct and an incident that befalls the cat that belongs to Rocco’s girlfriend, but they are overshadowed by the movie’s odd treatment of women and minorities.
There is a racist joke included in the film and, while it introduces the animosity between Rocco and Vincenzo (Ron Jeremy. Yes…THAT Ron Jeremy), it really doesn’t serve much purpose besides gaining a few guffaws from certain viewers. I also question the entire reason that Smecker’s character is a homosexual, and can only reach the conclusion that it is so the viewer will more readily believe he is comfortable dressing in drag to infiltrate the mafia boss’s house near the end of the film. In fact, this movie is filled with machismo and the only female characters are defined by violence and weakness; Rocco points a gun at his girlfriend and her friend, the stripper who faints during one of the hits is sexually assaulted by Rocco (and it’s played off as a joke), the wife of a man they are going to kill is knocked out with a stun gun, and Smecker is knocked unconscious by Il Dulce (Billy Connolly) because he thinks Smecker is a woman and won’t kill him as a rule.
I’m not sure how many times I have watched The Boondock Saints since I first heard about it eight years ago. I will admit that, at the time, I was heavily influenced by it. The masculinity, the cool pea coats, and sense of fraternity all appealed to me, but that has changed as the years went by and I studied film. This is considered a cult film, with a fanbase that defends it to the end of the earth; I once considered myself a fan, but I fear I no longer can after this review. Let me be clear; I had no intention of lambasting it so viciously, but there were such glaring issues that I could not ignore. If you need something on in the background of your Saint Paddy’s Day party (and the group is mostly comprised of men), then this is a good film for you; if not, best to avoid it.
Verdict: 2 fraternal action movies out of 5
Recommended for: 17-year-olds, those seeking the smallest mention of Irish culture in film, fans of cool-looking movies, and those who enjoy Fight Club (1999) (I only mention this because my copy of The Boondock Saints (1999) also included the Edward Norton and Brad Pitt film).
Not recommended for: Those seeking accurate depictions of Irish people, females, those easily offended by cursing, or those who want substance with their spectacle.
The images featured in this post can be found through the hyperlinks below. | https://perpetuallypastdue.com/2018/03/16/boondock-saints-1999-review/?like_comment=1687&_wpnonce=49790e6443 |
It’s a joyous and magical experience to watch a drizzle of chickpea water turn into a fluffy, sweet meringue mix! Ideally, use a stand mixer with a whisk attachment for this recipe as in our experience hand-whisks don’t have the power to get the chickpea water to stiff peaks.
MAKES 18
140ml aquafaba (the drained water from 1 x 400g tin chickpeas)
½ tsp cream of tartar 100g caster sugar 2–3 bananas
25g dark chocolate
FOR THE CARAMEL SA UCE
150g caster sugar
120ml full-fat coconut milk a pinch of salt
½ tsp dairy-free butter
FOR THE CASHEW CR EAM
150g cashews
600ml full-fat coconut milk 2 tbsp icing sugar
1 tsp vanilla extract
½ banana
Stand mixer | Line 3 baking sheets with parchment paper
| Preheat oven to 180°C | Frying pan | Small saucepan | Liquidiser
Pour the aquafaba into the mixer | Turn the mixer on to high and leave it running | Add the cream of tartar and continue to beat | After 2 minutes add the caster sugar, one spoonful at a time | Beat on high for 10–15 minutes | It’s ready when the aquafaba has magically transformed into a thick, meringue-like mixture that won’t fall off a spoon turned upside down
Spoon the meringue mixture on to the lined baking sheets to make nests about 8cm wide, no more than 1.5cm high and smooth on top, leaving 2cm between them | You should end up with about 18 nests (you can draw 8cm circles on the parchment paper, then flip over the paper and use them as templates)
Put the trays in the oven and immediately reduce the heat to 100°C
| Bake for 2 hours, then turn off the heat, leave the door closed and let the meringues cool completely, preferably overnight | Cooling the meringues overnight in the oven allows them to set properly and reduces the chances of them cracking due to sudden changes in temperature. | https://www.oceanroadmagazine.com.au/mini-banoffee-meringues/ |
Snail Bob realizing it’s his grandfather’s birthday. Now your goal is to get him safely there. The animation is relatively similar to the first episode, instead of a construction site the scenery is an enchanted forest.
You encounter many different types of forest creatures, like ants, caterpillars, and even a scary robotic snail that wants to steal Bob’s shell.
Snail Bob 2 Game Online
Snail Bob 2 game uses the same tool dynamic but, the added forest characters seem to increase the pleasure of game-play. It keeps the puzzles refreshing and exciting as opposed to frustrating, which can occur with the first episode.
Snail Bob 2 version has an increased number of levels from twenty to twenty-five, which increases the length of game-play, so that’s another plus. When you get to the final level of this series, you can watch Bob giving his grandfather’s present and joining the birthday party. | https://www.sonsaur.com/snail-bob-2/ |
TECHNICAL FIELD
BACKGROUND ART
CITATION LIST
Patent Literature
SUMMARY OF THE INVENTION
DESCRIPTION OF EMBODIMENTS
First Exemplary Embodiment
Second Exemplary Embodiment
Third Exemplary Embodiment
Fourth Exemplary Embodiment
INDUSTRIAL APPLICABILITY
REFERENCE MARKS IN THE DRAWINGS
The present invention relates to an electronic device that includes an angular velocity sensor to achieve the functions of allowing the user to read electronic books, displaying images, and reproducing music or videos.
FIG. 9
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is an external view of conventional electronic device . Electronic device includes right body part , left body part , opening and closing shaft connecting right and left body parts and openably and closably, and acceleration-and-angular velocity sensor . Right body part includes right-side LCD unit , and left body part includes left-side LCD unit to display books, images, etc. Electronic device further includes a controller (not shown), which detects the opening and closing angle between right body part and left body part according to an acceleration or angular velocity detected by acceleration-and-angular velocity sensor . The controller also determines the direction in which a large acceleration is generated when electronic device is opened or closed, and then performs a process to turn pages forward or backward (cf. Patent Literature 1).
PLT 1: Japanese Unexamined Patent Publication No. 2009-217415
The electronic device of the present invention includes a housing having a display, an angular velocity sensor, an acceleration sensor, and a controller. The angular velocity sensor detects an angular velocity around the X axis parallel to the display. The acceleration sensor detects an acceleration along the Z axis, which is perpendicular to the display and orthogonal to the X axis. The controller performs a first process when the angular velocity sensor detects a positive angular velocity first and then detects a negative angular velocity, and the acceleration sensor detects an acceleration along the Z axis. The controller, on the other hand, performs a second process when the angular velocity sensor detects a negative angular velocity first and then detects a positive angular velocity, and the acceleration sensor detects an acceleration along the Z axis.
The above configuration allows different processes to be performed depending on the order of occurrence of an angular velocity in the positive direction and an angular velocity in the negative direction when the housing is rotated. The above configuration also ensures detection of the rotation of the housing, allowing the first and second processes to be performed only when the detection is ensured. As a result, the user can operate the electronic device accurately with one hand in the environment where an acceleration or an angular velocity can occur due to vibration.
In recent years, portable electronic devices such as mobile phones, electronic book readers, and tablet terminals are becoming popular which allow the user to read books, display images, and reproduce music and videos. Such a portable electronic device is required to allow the user to operate it accurately with one hand in the environment where an acceleration or an angular velocity can occur due to vibration while, for example, he/she is walking with the other hand holding a bag, or riding in a train with the other hand hanging on to a strap.
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FIG. 9
In electronic device shown in , however, pages are turned forward or backward based on the opening and closing angle and acceleration. Therefore, in the environment where an acceleration or an angular velocity can occur due to vibration, the user cannot accurately operate electronic device with one hand.
The electronic device developed to solve the aforementioned problem will now be described with reference to drawings. In these embodiments, the same components as in the preceding embodiments are denoted by the same reference numerals, and thus a detailed description thereof may be omitted in the subsequent embodiments.
FIG. 1
FIG. 2
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shows electronic device according to a first exemplary embodiment of the present invention. is a block diagram of electronic device . Electronic device includes housing having display , angular velocity sensor , and controller . Angular velocity sensor detects an angular velocity around the X axis which is parallel to display . Controller performs a first process or a second process depending on the output of angular velocity sensor . More specifically, controller performs the first process when angular velocity sensor detects positive angular velocity first and then detects negative angular velocity . Controller , on the other hand, performs the second process when angular velocity sensor detects negative angular velocity first and then detects positive angular velocity . Note that when the user is holding electronic device in one hand, positive angular velocity is in a clockwise direction, whereas negative angular velocity is in a counterclockwise direction.
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Thus, controller can perform different processes depending on the order of occurrence of an angular velocity in the positive direction (positive angular velocity ) and an angular velocity in the negative direction (negative angular velocity ) when housing is rotated. This allows the user to accurately operate electronic device with one hand in the environment where an acceleration or an angular velocity can occur due to vibration.
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Electronic device further includes acceleration sensor which can detect an acceleration along the Y axis. Controller performs the first process when acceleration sensor detects an acceleration in the positive direction of the Y axis, and angular velocity sensor detects a positive angular velocity first and then detects a negative angular velocity. Controller , on the other hand, performs the second process when acceleration sensor detects an acceleration in the negative direction of the Y axis, and angular velocity sensor detects a negative angular velocity first and then detects a positive angular velocity. This control can further improve the accuracy of operation of electronic device .
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Specific examples of electronic device include mobile phones, electronic book readers, tablet terminals, and other portable electronic devices allowing the user to read electronic books (hereinafter, books), displaying images, and reproducing music (musical compositions) or videos.
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When electronic device has the function of displaying books for the user to read, the first process may be to turn pages of a book forward, and the second process may be to turn pages of the book backward. When electronic device is displaying the front cover of a book, the first process may be to display the next book, and the second process may be to display the preceding book.
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When electronic device has the function of displaying one of a plurality of sequential images at a time, the first process is to display the image following the currently displayed image, and the second process is to display the image preceding the currently displayed image. Alternatively, when an image is displayed partially, the first process may be to display an undisplayed right side, and the second process may be to display an undisplayed left side.
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When electronic device has the function of reproducing a plurality of sequential musical compositions or videos, the first process is to reproduce the next musical composition or video, and the second process is to reproduce the preceding musical composition or video. Alternatively, the first process may be to fast-forward, and the second process may be to rewind (review).
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The content of electronic books, images, musical compositions, etc. is stored in a storage unit connected to controller but not shown. Alternatively, the content may be stored in an external storage unit connected via wiring and/or wirelessly to storage unit . The external storage unit may be connected to controller via the Internet and wiring and/or wirelessly.
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FIG. 1
The following is a specific description of a control using angular velocity sensor and acceleration sensor . Angular velocity sensor and acceleration sensor are configured to output angular velocities and accelerations, respectively, around the axes corresponding to the X, Y, and Z axes shown in . Such sensors are disclosed, for example, in Japanese Unexamined Patent Publication Nos. 2010-230346 and H11-352143. It is not, however, necessary to use a three-axis type sensor as long as the sensor meets the use application of each exemplary embodiment.
FIGS. 3A and 3B
FIG. 1
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show the measurement results of angular velocity sensor and acceleration sensor , respectively. More specifically, these graphs show the outputs of angular velocity sensor and acceleration sensor around the X, Y, and Z axes shown in .
FIG. 3A
FIG. 3A
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In , the horizontal axis represents time, and the vertical axis represents angular velocity. Note that time increases from right to left. shows fluctuations in angular velocity around the X axis, in angular velocity around the Y axis, and in angular velocity around the Z axis.
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FIG. 3A
A “first operation” is defined as follows. The user holds housing in the right hand with display facing upward (in the positive direction of the Z axis); rotates housing in the positive direction (clockwise direction) of the X axis first; and then rotates it in the negative direction (counterclockwise direction) to return it to the original position. As shown in , the “first operation” is performed three times at times t, t, and t as shown by waveforms A, B, and C, respectively.
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FIG. 3A
A “second operation” is defined as follows. The user holds housing in the right hand with display facing upward (in the positive direction of the Z axis); rotates housing in the negative direction (counterclockwise direction) of the X axis first; and then rotates it in the positive direction (clockwise direction) to return it to the original position. As shown in , the “second operation” is performed three times at times t, t, and t as shown by waveforms D, E, and F, respectively.
FIG. 3A
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As shown in , in the “first operation”, the positive angular velocity is detected first and then the negative angular velocity is detected. In the “second operation”, the negative angular velocity is detected first and then the positive angular velocity is detected. The reason for this is that when the user operates electronic device provided with display , he/she inevitably turns display face up in the end in order, for example, to read a book displayed thereon.
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It is possible to perform a predetermined process by only using the angular velocity in the case that housing is rotated in a single direction, either positive or negative. More specifically, the first process can be performed when an angular velocity in the positive direction is detected, and the second process can be performed when an angular velocity in the negative direction is detected. In this case, however, when the user is operating electronic device with one hand, while walking with the other hand holding a bag or riding in a train with the other hand hanging on to a strap, controller may falsely recognize the angular velocity caused by walking vibration or train vibration. As a result, the first or second process may be executed mistakenly.
FIG. 3A
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As shown in , however, the user's intended operation always involves turning display face up. Therefore, executing a process by considering this returning operation can prevent false (unwanted) operation due to walking or a train.
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Thus, when angular velocity sensor detects a positive angular velocity first and then detects a negative angular velocity, controller determines that the “first operation” has been done purposefully by the user, and performs a first process. When, on the other hand, angular velocity sensor detects a negative angular velocity first and then detects a positive angular velocity, controller determines that the “second operation” has been done purposefully by the user, and performs a second process. These determinations prevent false operation due to the angular velocity caused by walking vibration or train vibration, allowing the user to accurately operate electronic device with one hand.
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Controller may have a predetermined threshold for the angular velocity. If the absolute value of an angular velocity is equal to the threshold, controller determines that the angular velocity has been detected, thereby further reducing the influence of the angular velocity caused by walking vibration or train vibration.
FIG. 3B
FIG. 3A
FIG. 3B
FIGS. 3A and 3B
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In , the horizontal axis represents time, and the vertical axis represents acceleration. In the same manner as in , time increases from right to left. shows fluctuations in acceleration along the X axis, along acceleration along the Y axis, and along acceleration along the Z axis. The time in the horizontal axis is the same between .
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Acceleration along the Y axis shows waveforms A, B, and C that indicate accelerations in the positive direction at the times t, t, and t, respectively, in the “first operation”. Acceleration further shows waveforms D, E, and F that indicate accelerations in the negative direction at the times t, t, and t, respectively, in the “second operation”.
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When the user performs the “first operation”, it is very difficult to rotate housing while keeping it in the same position. Therefore, the user inevitably moves housing in the positive direction along the Y axis. Similarly, when performing the “second operation”, the user inevitably moves housing in the negative direction along the Y axis. As a result, the waveforms shown in are generated.
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Making use of these features of the human body movement allows electronic device to select the operation more accurately. More specifically, assume that acceleration sensor detects an acceleration in the positive direction of the Y axis, and angular velocity sensor detects a positive angular velocity first and then detects a negative angular velocity. In this case, controller determines that the “first operation” has been done purposefully by the user, and performs the first process. Assume, on the other hand, that acceleration sensor detects an acceleration in the negative direction of the Y axis, and angular velocity sensor detects a negative angular velocity first and then detects a positive angular velocity. In this case, controller determines that the “second operation” has been done purposefully by the user, and performs the second process. Thus, the “first operation” and the “second operation” can be distinguished from each other using acceleration, and also using angular velocity. This can prevent false operation, allowing electronic device to select operations more accurately.
FIG. 4
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is a process flowchart in which controller performs a first process or a second process based on angular velocity sensor . In Step S, the control is started. In Step S, angular velocity sensor detects a positive or negative angular velocity around the X axis. When no positive or negative angular velocity around the X axis is detected, the process returns to the starting point SP to restart detection of an angular velocity again.
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When a positive angular velocity is detected in Step S, the process proceeds to Step S where controller determines whether or not a negative angular velocity around the X axis is detected within a predetermined time. When the negative angular velocity around the X axis is detected within the predetermined time, the process proceeds to Step S where controller performs a first process. After this, the process returns to the starting point SP. When the negative angular velocity around the X axis is not detected within a predetermined time, the process directly returns to the starting point SP.
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When, on the other hand, a negative angular velocity is detected in Step S, the process proceeds to Step S where controller determines whether or not a positive angular velocity around the X axis is detected within a predetermined time. When the positive angular velocity around the X axis is detected within a predetermined time, the process proceeds to Step S where controller performs a second process. After this, the process returns to the starting point SP. When the positive angular velocity around the X axis is not detected within the predetermined time, the process directly returns to the starting point SP.
FIG. 5
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is a process flowchart in which controller performs a first process or a second process based on angular velocity sensor and acceleration sensor . In Step S, the control is started. In Step S, acceleration sensor detects a positive or negative acceleration. When no acceleration is detected, the process returns to the starting point SP to restart detection of an acceleration.
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When a positive acceleration is detected in Step S, the process proceeds to Step S where controller determines whether or not angular velocity sensor detects a positive angular velocity around the X axis. When the positive angular velocity around the X axis is not detected, the process returns to the starting point SP; otherwise, the process proceeds to Step S.
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In Step S, controller determines whether or not a negative angular velocity around the X axis is detected within a predetermined time. When the negative angular velocity is detected within the predetermined time, the process proceeds to Step S where a first process is performed. After this, the process returns to the starting point SP. When the negative angular velocity is not detected within a predetermined time, the process directly returns to the starting point SP.
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When a negative acceleration is detected in Step S, the process proceeds to Step S where controller determines whether or not a negative angular velocity around the X axis is detected. When the negative angular velocity around the X axis is not detected, the process returns to the starting point SP; otherwise, the process proceeds to Step S.
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In Step S, controller determines whether or not a positive angular velocity around the X axis is detected within a predetermined time. When the positive angular velocity is detected within the predetermined time, the process proceeds to Step S where a second process is performed. After this, the process returns to the starting point SP. When the positive angular velocity is not detected within the predetermined time, the process directly returns to the starting point SP.
FIGS. 4 and 5
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The predetermined times in are determined to allow the user to complete a series of operations which begins with rotating electronic device in the positive or negative direction and ends with rotating it in the opposite direction to return it to the original position. The predetermined times are preferably 0.1 seconds or more, which allows preventing false operation due to external disturbance such as impact that the user does not expect. It is also preferable that the predetermined times be within 2 seconds, which allows distinguishing between a first “first operation” and a second “first operation”, and also distinguishing between a first “second operation” and a second “second operation”.
FIG. 4
FIG. 5
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In , the predetermined time in Step S may be different from that in Step S. Similarly, in , the predetermined time in Step may be different from that in Step S. This provides the user with a different tactile feel between the “first operation” and the “second operation”.
FIGS. 4 and 5
Alternatively, the predetermined times in may be configured to be capable of being set by the user. In this case, each user can individually adjust the time required to recognize the “first operation” and the “second operation”, thereby obtaining a comfortable operability.
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FIG. 1
FIG. 1
In the present exemplary embodiment, the “first operation” and the “second operation” indicate lateral rotation of electronic device (rotation around the X axis shown in ) done by the user. These operations, however, may alternatively be longitudinal rotation of electronic device (rotation around the Y axis shown in ) to perform the first or second process.
FIGS. 1
FIGS. 6A and 6B
FIGS. 1 and 2
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In the present exemplary embodiment, a control using an acceleration along the Z axis will now be described with reference to , , A, and B. show comparative experimental results when the user holds housing in the right hand and in the left hand, respectively. Note that electronic device in the present exemplary embodiment has the same basic configuration as that in the first exemplary embodiment described with reference to .
FIG. 6A
FIGS. 3A and 3B
FIG. 6B
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shows angular velocity and acceleration when the user performs the “first operation” and the “second operation” while holding housing in the right hand as in . shows angular velocity and acceleration when the user performs the “first operation” and the “second operation” while holding housing in the left hand.
FIGS. 6A and 6B
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As understood from , the order of occurrence of positive and negative angular velocities around the X axis, and the direction of generation of acceleration along the Y axis are the same regardless of the hand in use. More specifically, as regards angular velocity around the X axis, in the “first operation”, a positive angular velocity is detected first and then a negative angular velocity is detected regardless of the hand in use. In the “second operation”, a negative angular velocity is detected first and then a positive angular velocity is detected regardless of the hand in use. As regards acceleration along the Y axis, in the “first operation”, an acceleration in the positive direction is detected, whereas in the “second operation”, an acceleration in the negative direction is detected regardless of the hand in use.
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In contrast, acceleration along the Z axis has a different waveform depending on the hand in use. When the user holds housing in the right hand, acceleration along the Z axis decreases only slightly in the “first operation”, but greatly decreases in the “second operation”. When the user holds housing in the left hand, acceleration along the Z axis greatly decreases in the negative direction in the “first operation”, but decreases only slightly in the “second operation”. The amount of rotation differs depending on whether the user is rotating housing in his/her hand toward or away from his/her body. This seems to be the reason for the above-described waveforms of acceleration along the Z axis.
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In the initial state with display facing upward, the Z-axis direction of electronic device coincides with the direction of gravity. At this moment, the gravitational acceleration is at its maximum. The farther housing is rotated around the X axis from this state, the larger the angle is between the Z-axis direction of electronic device and the direction of gravity. This results in a decrease in the acceleration along the Z axis detected by acceleration sensor .
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Because of the structure of the human arm, the amount of rotation is small when the user rotates housing away from his/her body (the “first operation” when holding it in the right hand, the “second operation” when holding it in the left hand). The rotation away from his/her body corresponds to the “first operation” when the user holds housing in the right hand, and corresponds to the “second operation” when the user holds housing in the left hand. The small amount of rotation makes a small angle between the Z-axis direction of electronic device and the direction of gravity, thereby only slightly reducing acceleration along the Z axis. In contrast, the amount of rotation is larger when the user rotates housing toward his/her body than when the user does it away from his/her body. The rotation toward his/her body corresponds to the “second operation” when the user holds housing in the right hand, and corresponds to the “first operation” when the user holds housing in the left hand. This large amount of rotation makes a large angle between the Z-axis direction of electronic device and the direction of gravity, thereby greatly decreasing acceleration along the Z axis.
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As described above, controller can determine which hand the user has used to operate electronic device from the difference in the change of acceleration along the Z axis due to the structure of the human arm.
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Thus, controller can detects the “first operation” or the “second operation” by using angular velocity around the X axis and acceleration along the Y axis around the X axis. Furthermore, when detecting the “first operation”, controller can determine it to be an operation done by the left hand if the change in acceleration along the Z axis is below a predetermined threshold. If the change is not below the predetermined threshold, i.e. the change is equal to or more than the predetermined threshold, controller can determine it to be an operation done by the right hand.
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Similarly, when detecting the “second operation”, controller can determine it to be an operation done by the right hand if acceleration along the Z axis is below the predetermined threshold. If the acceleration is not below the predetermined threshold, i.e. the acceleration is equal to or more than the predetermined threshold, controller can determine it to be an operation done by the left hand.
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Especially when electronic device has game functions, controller can provide different operations in games by determining which hand the user is using to operate it. In a baseball or golf game, for example, controller can determine the dominant hand of the user from the hand used for the operation, and provide batting and pitching operations according to his/her dominant hand.
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FIGS. 1
FIG. 7
FIGS. 1 and 2
In the present exemplary embodiment, another control using acceleration along the Z axis will be described with reference to , , and . shows comparative experimental results indicating the difference between the case where the user rotates housing in the right hand too far back in the “first operation” and the case where the user performs the “second operation”. Note that electronic device in the present exemplary embodiment has the same basic configuration as that in the first exemplary embodiment described with reference to .
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In the “first operation”, the user rotates electronic device from the starting position shown in S in a clockwise direction as shown in S, and rotates it back to the original position as shown in S. If rotating electronic device too far back as shown in S, the user rotates it back again as shown in S. These operations shown in S to S are represented by waveforms to , respectively, of angular velocity around the X axis.
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In the “second operation”, the user rotates electronic device from the starting position shown in S in a counterclockwise direction as shown in S, and rotates it back to the original position as shown in S. These operations shown in S to S are represented by waveforms to , respectively, of angular velocity around the X axis.
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The waveform resulting from the action of rotating back (S) to the action of rotating back again (S) in the “first operation” is substantially identical to the waveform resulting from the starting position (S) to the action of rotating back (S) in the “second operation”. Therefore, it is very difficult to distinguish between these waveforms. As a result, controller may falsely recognize the “first operation” performed by the user as the “second operation”, thereby performing the second process.
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However, the use of acceleration along the Z axis can discriminate between the case where electronic device is rotated too far back in the “first operation” and the case where the “second operation” is performed. As understood from , while waveform shows a slight decrease in acceleration along the Z axis in the “first operation”, waveform shows a large decrease in the “second operation”. The reason for this seems to be the difference in the amount of rotation depending on whether the user rotates housing purposefully or rotates it too far back unintentionally.
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Consequently, even when a negative angular velocity is detected first and then a positive angular velocity is detected, if the change in acceleration along the Z axis is not below the predetermined threshold, controller can determine that housing has been rotated too far back in the “first operation” and does not perform the second process.
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Thus, even when angular velocity sensor detects a positive angular velocity first and then detects a negative angular velocity, if the change in acceleration along the Z axis is below the predetermined threshold, controller performs the process occurring immediately before the detection of the negative angular velocity preferentially over the first process. Similarly, even when angular velocity sensor detects a negative angular velocity first and then detects a positive angular velocity, if the change in acceleration along the Z axis is below the predetermined threshold, controller performs the process occurring immediately before the detection of the positive angular velocity preferentially over the second process. These controls prevent false operation due to rotating electronic device too far back, thereby accurately detecting operations performed by the user with one hand.
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In the second and third exemplary embodiments, controller performs the first process when angular velocity sensor detects positive angular velocity first and then detects negative angular velocity , and acceleration sensor detects a change in acceleration along the Z axis. Controller , on the other hand, performs the second process when angular velocity sensor detects negative angular velocity first and then detects positive angular velocity , and acceleration sensor detects a change in acceleration along the Z axis. Thus, using the output of acceleration sensor in addition to the output of angular velocity sensor ensures detection of the “first operation” and the “second operation”.
FIGS. 3A
FIGS. 3B
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In , A, B, and referred to in the first to third exemplary embodiments, a positive value is output as a positive angular velocity, and a negative value is output as a negative angular velocity. Alternatively, however, a negative value may be output as a positive angular velocity, and a positive value may be output as a negative angular velocity because the polarity of the output signal of angular velocity sensor is arbitrarily assigned. Similarly to the case of angular velocity, in , A, B, and , a positive value is output as a positive acceleration, and a negative value is output as a negative acceleration. Alternatively, however, a negative value may be output as a positive acceleration, and a positive value may be output as a negative acceleration because the polarity of the output signal of acceleration sensor is arbitrarily assigned.
FIG. 8A
FIG. 8B
FIG. 8A
is a plan view of an electronic device according to a fourth exemplary embodiment of the present invention. is a block diagram of the electronic device shown in .
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FIGS. 1 and 2
Electronic device includes strain sensors R and L on the right and left sides, respectively, of housing in addition to the configuration of electronic device shown in . The outputs of strain sensors R and L are fed to controller in the same manner as the outputs of angular velocity sensor and of acceleration sensor . Except for this feature, electronic device has the same basic configuration as electronic device .
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Strain sensors R and L are disposed in positions subjected to finger pressure or thumb pressure when the user holds electronic device . Assume that the user presses strain sensor R to create a strain when the user performs the “first operation”. In this case, controller detects this strain, and when, for example, the first process is to turn pages forward, the user can jump a plurality of pages forward at a time. Assume, on the other hand, that the user presses strain sensor L to create a strain when the user performs the “second operation”. In this case, controller detects this strain, and when, for example, the second process is to turn pages backward, the user can return a plurality of pages at a time. In other cases, a plurality of contents can be forwarded or returned in the selection of content such as images, musical compositions, and videos. In addition, the amount of content to be forwarded or returned can be increased or decreased depending on the magnitude of the strain. Note that this process can be performed without acceleration sensor .
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If acceleration or angular velocity is accidentally applied to electronic device while the user is carrying it in a bag with the power switch on, the first or second process may be performed without the user's knowledge. In contrast, when the user is holding electronic device in his/her hand, strain sensors R and L are pressed, and detect generation of a strain having a reference value. If the outputs of strain sensors R and L are the reference value or greater, controller determines that electronic device is held in the user's hand. Therefore, it is preferable that controller be configured to perform the first or second process if receiving an output based on an angular velocity around the X axis from angular velocity sensor in this state. This control prevents the first or second process from being performed without the user's knowledge when, for example, electronic device is in a bag with the power switch on.
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Furthermore, while the user is rotating electronic device around the X axis, the strains applied to strain sensors R and L are changed. Of strain sensors R and L, the lower one in position is subjected to more gravitational acceleration than the higher one. As a result, the higher one has a smaller strain, and the lower one has a larger strain. When, for example, electronic device is rotated in a counterclockwise direction, strain sensor L has a larger output, and strain sensor R has a smaller output. Therefore, it is preferable that controller be configured to calculate the difference in change from the reference value between the respective strain sensors (the difference value), and that controller perform the first or second process when the absolute value of a difference of the difference values is equal to or more than the predetermined threshold. This control ensures the detection of rotation done by the user.
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Alternatively, controller may be configured to determine the direction of rotation depending on whether the difference value is positive or negative. This determination can be made without angular velocity sensor , but using both improves the accuracy of determining the direction of rotation.
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Strain sensors R and L can detect comparatively as small a strain as is generated by finger pressure or thumb pressure, which is several tens of grams per square centimeter. One such strain sensor is disclosed in Japanese Unexamined Patent Publication No. 2007-085993.
FIG. 8A
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In , strain sensors R and L are disposed so as to project from housing , buy may alternatively be formed in the same plane as the side surfaces of housing . Further alternatively, housing may cover strain sensors R and L as long as finger pressure or the like can reach these sensors via the side surfaces of housing . In this case, all or part of housing may be made of a deformable material.
FIG. 8A
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In , strain sensors R and L are disposed on each side surface of housing ; alternatively, however, a plurality of strain sensors may be disposed on each side surface of housing . Disposing a plurality of strain sensors on each side allows detection of strain distribution. When the user holds electronic device in his/her hand, if housing has a width fitting the palm, the thumb is placed on one of the side surfaces of housing , and at least two of the fingers are placed on the other side. In this case, while strain is concentrated in one region on the side where the thumb is placed, strain is dispersed in two or more regions on the other side. Detecting such strain distribution on each side surface allows controller to determine which hand the user is using to hold electronic device , thereby providing advantageous effects similar to those of the second exemplary embodiment. In the case of using a strain sensor capable of detecting strain distribution, it is sufficient to use a single strain sensor R and a single strain sensor L.
The electronic device of the present invention allows the user to operate it accurately with one hand in the environment where an acceleration or an angular velocity can occur due to vibration, and therefore, is useful as an electronic device that allows the user to read books, displays images and reproduces music or videos.
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, electronic device
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display
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housing
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angular velocity sensor
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acceleration sensor
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controller
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positive angular velocity
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negative angular velocity
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angular velocity around the X axis
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A, B, C, D, E, F waveform
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angular velocity around the Y axis
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angular velocity around the Z axis
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acceleration along the X axis
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acceleration along the Y axis
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A, B, C, D, E, F waveform
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acceleration along the Z axis
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, , , , , , , , , waveform
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R, L strain sensor
BRIEF DESCRIPTION OF DRAWINGS
FIG. 1
is a perspective view of an electronic device according to a first exemplary embodiment of the present invention.
FIG. 2
FIG. 1
is a block diagram of the electronic device shown in .
FIG. 3A
FIG. 1
shows output waveforms of an angular velocity sensor in the electronic device shown in .
FIG. 3B
FIG. 1
shows output waveforms of an acceleration sensor in the electronic device shown in .
FIG. 4
FIG. 1
is a process flowchart of the electronic device shown in .
FIG. 5
FIG. 1
is another process flowchart of the electronic device shown in .
FIG. 6A
shows output waveforms of an angular velocity sensor and an acceleration sensor in an electronic device according to a second exemplary embodiment of the present invention.
FIG. 6B
shows output waveforms of the angular velocity sensor and the acceleration sensor in the electronic device according to the second exemplary embodiment of the present invention.
FIG. 7
shows output waveforms of an angular velocity sensor and an acceleration sensor in an electronic device according to a third exemplary embodiment of the present invention.
FIG. 8A
is a plan view of an electronic device according to a fourth exemplary embodiment of the present invention.
FIG. 8B
FIG. 8A
is a block diagram of the electronic device shown in .
FIG. 9
is a perspective view of a conventional electronic device. | |
This assignment requires that you log onto the Securities and Exchange Commission’s web site (www.sec.gov). Once you are on the web site, scroll down to the section labeled “Staff Interpretations†and then click on the link titled “Staff Accounting Bulletins.†Then, on the next screen you should scroll down until you see the link to “SAB No. 99â€, dated August 12, 1999. Click on the SAB 99 link. Use this document to answer the questions noted below. If you have any trouble with the above instructions, try this address to locate SAB No. 99: http://www.sec.gov/interps/account/sab99.htm.
Read SAB No. 99 to answer these questions:
? Who is responsible for issuing Staff Accounting Bulletins?
? What types of companies does SAB No.99 apply to?
? What is the main purpose of SAB No. 99 (e.g., Why did the SEC deem it necessary to issue the SAB?)?
? What does SAB No. 99 conclude about the use of a numerical threshold in establishing materiality? How should such a threshold be used?
? Briefly summarize in your own words the FASB Statement of Financial Accounting Concepts No. 2 definition of “materiality.â€
? In assessing the “total mix†of information surrounding a misstatement, what does the Staff of the SEC believe management and an auditor should consider?
? Give two examples from SAB No. 99 of a quantitatively immaterial misstatement that might be deemed “material†from a qualitative perspective.
? Assume that you found a misstatement in inventory that you believe materially overstates inventory and total assets. Then, assume that you found a second misstatement that you believe materially understates accounts receivable and total assets by approximately the same amount as the inventory misstatement. How does SAB No. 99 address management’s and the auditor’s ability to let the two misstatements offset one another without either misstatement being corrected in the financial statements?
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The Uyo Zonal Office of the Economic and Financial Crimes Commission (EFCC) has arrested 22 suspected internet fraudsters in Imo state.
The suspects are aged between 20 and 41 years.
They were arrested by the commission’s operatives at various locations following intelligence report according to the anti-graft’s Spokesperson, Dele Oyewale.
In a statement on Thursday in Uyo, Oyewole said the suspects were apprehended during a 3-hour raid on Monday, September 8, 2020, at Ugwuma town and Egbu, Owerri North.
The arrested suspects are; Chinonso Anozie Frank, Ogadibo Victor Chikeluba, Clinton Njoku Onyekachi, Okoro Edwin Chigozie, Anozie Chibuzor Harold, Anyaeriuba Canice, Ogbuehi Anderson, Okere Nonye, Njoku Paul Chibuike, Ndalaka Felix Chukwuebuka and Anele Chibuike Timothy.
READ ALSO: JUST IN: Gunmen Attack Abuja Community, abduct 20 resident
Others are; Chidi Kalu, Uparai Daniel, Ubochi Christian Ikechukwu, Udoka Obi, Chiagozie Morris Obiano, Moses Akaedite, Henry Princewill Chisom, Prince Ibe, Ugwuegbu Fredrick Ekenna, William Peter Obinna and Udoka Ihiekwe Kelechi.
Oyewale disclosed that several cars and gadgets were recovered from the suspects including a black Lexus E5 330 with registration number NWA 866 AW; a silver-coloured Toyota Corolla LE with registration number, FST 520 GJ; A silver-coloured Toyota Camry with registration number, KRD 687 GF and a blue Toyota Corolla LE with registration number, RSH 889 TD.
Others items recovered during the raid include a silver coloured Lexus E5 with registration number, MMA 745 AC; a black Toyota Venza with registration number, KWL 134 AV; a grey coloured Mercedes ML350 Jeep with registration number, GWA 587 EP and silver coloured Toyota Highlander with registration number, BWR 912 JV.
Also recovered from them are 15 laptops and 20 sophisticated mobile phones.
The EFCC spokesman said that the suspects will be charged to court once investigations are concluded. | https://www.nextedition.com.ng/efcc-arrests-22-suspected-internet-fraudsters-in-imo |
Wallace may be a small town – its population is just shy of 800 – but they’re looking to make a mark on the summer music festival scene. The first fest was named 2012’s Best Blues event by the Inland Empire Blues Society and welcomed nearly 2,000 attendees.
This year’s festival should prove to be even more successful: Not only are there more scheduled performers, but two new outdoor stages have been added to the mix. There will also be a beer garden set up near the main stage, with even more music going on in between the headlining artists’ sets.
From early Friday evening to late Saturday night, 21 different musical acts will be performing at various venues throughout town.
The main stage will house most of the festival’s out-of-town acts, including Arkansas native Seth Freeman, North Carolina’s Nikki Hill, and Ian Siegal, who hails from England and has won several British Blues Awards.
Several bands from the 2012 festival are returning for this year’s festivities, including Sammy Eubanks, the Fat Tones, and Anita Royce and the High Rollers. Other regional acts include Bakin’ Phat, Big Mumbo Blues Band and Boise’s John Németh.
The 2013 festival is being expanded by a day to accommodate a Sunday morning pancake feed at the Elks Lodge, where the Sara Brown Band, first performing on Saturday night, will make an encore appearance. | http://www.spokesman.com/stories/2013/jul/11/bigger-broader-bluesier/?photos |
# 1968 British Hard Court Championships
The 1968 British Hard Court Championships was a combined men's and women's tennis tournament played on outdoor clay courts at The West Hants Club in Bournemouth in England. It was the first tournament in the Open Era of tennis. The tournament was held from 22 April to 27 April 1968. Ken Rosewall and Virginia Wade won the first open singles titles while the men's team of Roy Emerson and Rod Laver and the women's team of Christine Truman Janes and Nell Truman won the first open doubles titles.
## First tournament of the Open Era
The 1968 British Hard Court Championships (BHCC) hold the distinction of being the first open era tennis tournament. Prior to this tournament professional players were banned by the International Lawn Tennis Federation (ILTF) from competing in tournaments, including the Grand Slams, which were organized by the ILTF and its national organizations. Although all players, amateurs and professionals, were allowed to compete at the 1968 BHCC the players who were part of the World Championship Tennis (WCT) circuit did not participate. Players from the rival National Tennis League (NTL) did enter and in the men's singles event made up the first six seeds. The tournament started on 22 April at 1:43 p.m. when John Clifton served and won the first point of the open era. Clifton lost his first-round match to Owen Davidson who thus became the first winner of an open era tennis match. Ken Rosewall won the men's singles title, taking home $2,400, while runner-up Rod Laver received $1,200. Their final was suspended in the second set due to rain and was finished the following day. Virginia Wade won the women's singles title, defeating Winnie Shaw in the final, but did not take home the winner's prize of $720 as she was still an amateur at the time of the tournament. She subsequently became the first amateur to win a title in the Open Era. Christine Janes and her sister Nell Truman became the first winners of an open tennis event by winning the women's doubles title. The tournament was considered a success and attracted almost 30,000 visitors. The young British player Mark Cox went down in tennis history by becoming the first amateur player to beat a professional, when he defeated 39-year-old American Pancho Gonzales in five sets in a second-round match that lasted two and a quarter hours.
## Finals
### Men's singles
Ken Rosewall defeated Rod Laver 3–6, 6–2, 6–0, 6–3
### Women's singles
Virginia Wade defeated Winnie Shaw 6–4, 6–1
### Men's doubles
Roy Emerson / Rod Laver defeated Andrés Gimeno / Pancho Gonzales 8–6, 4–6, 6–3, 6–2
### Women's doubles
Christine Truman Janes / Nell Truman defeated Fay Toyne-Moore / Anette du Plooy 6–4, 6–3
### Mixed doubles
Virginia Wade / Bob Howe defeated Fay Toyne-Moore / Jimmy Moore 6–4, 6–3 | https://en.wikipedia.org/wiki/1968_British_Hard_Court_Championships |
The utility model discloses a multi-layer self-heating system packaging container. A plurality of container bodies can be overlapped and combined; the container upper cover is mounted at the upper part of the upper-layer container body; the container base is positioned at the bottom of the lower-layer container body; wherein a plurality of groups of downward inclined buckling blocks are arranged on the inner side of each container body, a tin foil inner layer with a through hole is arranged in the container body through the buckling blocks, soda lime is arranged in a space below the tin foil inner layer, and an inner bag used for packaging articles is arranged above the tin foil inner layer. When cooked food in the container needs to be eaten, the upper container cover is taken down, a certain amount of water is added into the container body, then the upper container cover is covered, soda lime and water are mixed and react, and heat is released to heat the inner bag; after heating fora certain time, taking out the container upper cover and the inner bag, taking out the tin foil inner layer, pouring out substances such as soda lime and the like, and putting the inner bag back intothe container body for opening the bag for eating. | |
“Over 80” and “Driving with Excess Blood Alcohol” are the same thing. Specifically, they are a criminal charge, and refer to an accused person who has operated a motor vehicle with more than 80 milligrams of alcohol in their blood. We often hear people say, “I wasn’t drunk. How could they charge me with Driving with Excess Blood Alcohol?”Being charged with a single count of “Driving with Excess Blood Alcohol” does not mean that a person was impaired or drunk. Rather, it simply means that they drove with more than the permissible amount of alcohol in their system. A person can look and feel perfectly sober, yet still blow over the legal limit.
How do I know if I’m being investigated for “Over 80”?
The investigation starts with an individual being stopped. An individual may be stopped as a result of a RIDE program or an investigation relating to a Highway Traffic Act (H.T.A.). If the officer suspects that the driver is operating a motor vehicle with alcohol in their body, they will demand that the driver provide a sample of their breath into an “Approved Screening Device” at the roadside. In this circumstance, the police do not necessarily believe that the driver is “impaired”, rather they have a suspicion that the driver has consumed alcohol and is operating a motor vehicle. If the driver fails the approved screening device, then the officer has grounds to arrest the driver and take them back to the station to provide further samples of their breath into a breathalyzer. Once at the station, the arrested party will be required to provide two samples of their breath into a breathalyzer. If both of these samples of the accused’s blood alcohol concentration (B.A.C.) are over 80 milligrams of alcohol per 100 millilitres of blood, then the individual will be charged with “Driving with Excess Blood Alcohol/Over 80”
How many drinks do I have to have to be “over 80”
Often, we are asked, “How many drinks does it take to be over the legal limit?” Our answer is the same every time: it depends. “Blood Alcohol Concentration” (BAC) depends on a number of factors such as gender, height, weight, race etc. There are a number of apps and charts on the internet that allow users to input the number, volume and type of drinks consumed over a certain period of time, in order to calculate “projected” B.A.C. However, one should be cautious, as these methods of calculation are not always accurat
It is not that difficult for some people to attain a B.A.C. in excess of the legal limit. A “standard drink” which you purchase at a bar has 13.5 milligrams of alcohol. In other words, 12 ounces (341 ml) of beer or cider with 5 per cent alcohol, 5 ounces (140 ml) of wine with 12 per cent alcohol or 1.5 ounces (43 ml) of liquor (such vodka, gin, or whiskey) with 40 per cent alcohol, all contain 13.5 milligrams of alcohol. Things like drinking a lot of water, having a coffee, eating, or “walking it off” do not decrease your B.A.C. Decreasing your B.A.C. simply takes time.
At Passi & Patel, our professional criminal lawyers in Mississauga, Milton, and Brampton understand legal issues can be stressful. Please call us at 905-459-0004 or e-mail us at [email protected] to schedule a free consultation. | https://www.passipatel.com/what-does-over-80-mean/ |
SOS Bag Making Machine with twisted rope handle Curioni SUN Model : SUN 540 Mfg year : 1998 Size range Bag width : 220 – 540 mm Bag width (with handle) : 240 – 540 mm Bottom width : 80 – 200 mm Tube length : 400 – 780 mm Max reel diam. 1500 mm Paper Core diam. 76 mm – Standard * Nordson Hotmelt : - a) Longitudinal (tube) b) Handle (patch / rope handle) + Cold Glue Unit All machines or equipment is offered as it is where is with all faults without warranty and is subject to prior sales. All information is given by the source, which deemed reliable. | https://www.paperindex.com/product-details/sos-bag-making-machine-with-twisted-rope-handle/68853 |
Q:
Joining table to union of two tables?
I have two tables: orders and oldorders. Both are structured the same way. I want to union these two tables and then join them to another table: users. Previously I only had orders and users, I am trying to shoehorn oldorders into my current code.
SELECT u.username, COUNT(user) AS cnt
FROM orders o
LEFT JOIN users u
ON u.userident = o.user
WHERE shipped = 1
AND total != 0
GROUP BY user
This finds the number of nonzero total orders all users have made in table orders, but I want to this in the union of orders and oldorders. How can I accomplish this?
create table orders (
user int,
shipped int,
total decimal(4,2)
);
insert into orders values
(5, 1, 28.21),
(5, 1, 24.12),
(5, 1, 19.99),
(5, 1, 59.22);
create table users (
username varchar(100),
userident int
);
insert into users values
("Bob", 5);
Output for this is:
+----------+-----+
| username | cnt |
+----------+-----+
| Bob | 4 |
+----------+-----+
After creating the oldorders table:
create table oldorders (
user int,
shipped int,
total decimal(4,2)
);
insert into oldorders values
(5, 1, 62.94),
(5, 1, 53.21);
The expected output when run on the union of the two tables is:
+----------+-----+
| username | cnt |
+----------+-----+
| Bob | 6 |
+----------+-----+
Just not sure where or how to shoehorn a union into there. Instead of running the query on orders, it needs to be on orders union oldorders. It can be assumed there is no intersect between the two tables.
A:
You just need to union this way:
SELECT u.username, COUNT(user) AS cnt
FROM
(
SELECT * FROM orders
UNION
SELECT * FROM oldorders
) o
LEFT JOIN users u ON u.userident = o.user
WHERE shipped = 1
AND total != 0
GROUP BY user;
First get the combined orders using UNION between orders and oldorders table.
The rest of the work is exactly same what you did.
SEE DEMO
Note:
Left join doesn't make sense in this case. Orders for which the users don't exist then you will get NULL 0 as output. This doesn't hold any value.
If you want <user,total orders> for all users including users who might not have ordered yet then you need to change the order of the LEFT JOIN
| |
Conservationists are outraged over the Federal Government's draft plans for 44 Australian marine parks, saying it almost halves the protections set five years ago.
Under the new proposal, mid-water trawling will be allowed and more areas will be opened to commercial and recreational fishing.
Marine park zones: Category 2012 2017 Green (high level protection) 36% 20% Yellow: (allows for sustainable use, but protects seafloor) 24% 43% Blue: (allows for sustainable use) 40% 37% Source: Parks Australia
Green zones, which offer the highest protection, will be reduced but yellow zones, which allow for sustainable use but protect the seafloor, will be increased.
The plans cover Commonwealth waters off the coast of New South Wales, Queensland, Western Australia, South Australia and the Northern Territory starting about five kilometres offshore.
In 2012, the then-Gillard government expanded the network of marine reserves but they were suspended from operation under then-prime minister Tony Abbott, who ordered an independent review.
The new draft plan, aimed at striking a balance between commercial fishing operations and protecting the environment, was released in July.
Removal of Geographe Bay protection 'inexplicable'
No-take fishing zones set up to limit fishing in Geographe Bay off the West Australian coast have been scrapped under the new plan.
About 30,000 humpback whales move down the WA coast to migrate, proving a major drawcard for tourists
The new plan scraps no-take fishing zones in WA's Geographe Bay. ( ABC News: Louisa Rebgetz )
Marine campaigner Adrian Meder has worked along the south-west WA coast for 20 years and described the plans as a "bit of a clanger".
"It's sort of inexplicable that the Government would propose getting rid of the most meaningful protection in our bay," Mr Meder said.
Environmentalist Michelle Grady from Pew Charitable Trusts echoed the concerns.
The Federal Government's draft plan affects 44 Australian marine parks. ( ABC News: Louisa Rebgetz )
"We're very surprised and we can't work it out. This is completely contrary to the science it is completely contrary to public consultation," she said.
She leads an alliance of 25 environment groups called Save Our Marine Life who oppose the plans.
"We're very concerned that Australia's marine parks are proposed to have half of the sanctuary zone protection removed under these plans," Ms Grady said.
"That would be a precedent in Australia and a global precedent."
Federal National Parks director Sally Barnes said the latest plans protected biodiversity but also limited the impact on local industries.
The whales in Geographe Bay are a major drawcard for tourists. ( ABC News: Louisa Rebgetz )
She said describing the plans as a "cut back" were false.
"There have been proposals for different configurations of zoning but there are not any zoning restrictions or regulations in place in most of the Commonwealth waters," Ms Barnes said.
"When we get them operational, we'll have one of the largest areas of protection in no-take zones in the world.
"The same number of conservation features are protected in green zones in the plans released today as those in 2012."
'No real protection' for Coral Sea reefs
The Coral Sea off Queensland, considered the jewel in the crown of the 2012 network, is a key sticking point, and is home to more than 40 reefs which are a haven for marine life.
Tuna caught in the Coral Sea off Queensland. ( ABC News: Mick Fanning )
Diving operator Craig Stephen said the Osprey, Holmes and Flinders reefs in the Coral Sea were crucial areas, particularly for sharks.
"None of these reefs have been offered any real protection," Mr Stephen said.
"There's [an] amount of protection at Osprey reef, which is an iconic reef, but the other reefs for the most part have been left exposed to extractive practices.
"There is no balance — the balance is in favour of extractive practices."
Fisherman Pavo Walker relies on fish caught in the Coral Sea to keep his business going. ( ABC News: Mick Fanning )
But not everyone is against the changes proposed in the draft plans and fisher Pavo Walker is among those who welcomes them.
He owns the largest wild-caught tuna fishing operation in Australia, based at Mooloolaba on Queensland's Sunshine Coast, and relies entirely on the Coral Sea to survive.
Mr Walker said the 2012 plans would cripple the industry.
"The 2012 … model would have seen a complete end to our fishery and our industry and we are talking about a world gold standard sustainable fishery that would have been closed for emotional reasons, not scientific," Mr Walker said.
Mr Walker uses a longline with thousands of baited hooks that are left at sea for several hours to catch large tuna and swordfish.
Pavo Walker's crew at Mooloolaba with fresh tuna caught in the Coral Sea. ( ABC News: Mick Fanning )
The bulk of the catch is exported to Japan and the United States.
"When you start putting boxes all around the ocean and our gear drifts into them it becomes unfishable," Mr Walker said.
He said Australia imports a large proportion of fish from overseas markets where there were not the same sustainability standards.
Ms Barnes said areas that needed protection would be protected.
"When these plans go into place we will have reduced commercial fishing in selected areas where we think the conservation values are so high that we need to protect them from any extraction and that's an area about the size of Victoria approximately," Ms Barnes said.
It will be Federal Environment Minister Josh Frydenberg who will have the final say before taking the plans to Federal Parliament, but getting it through the Senate may prove a challenge.
The draft management plans are open for consultation until September 20.
| |
The federal government’s new draft marine park plans are based on an unsubstantiated premise: that protection of Australia’s ocean wildlife is consistent with activities such as fishing and oil and gas exploration.
Under the proposed plans, there would be no change to the boundaries of existing marine parks, which cover 36% of Commonwealth waters, or almost 2.4 million square kilometres. But many areas inside these boundaries will be rezoned to allow for a range of activities besides conservation.
The plans propose dividing marine parks into three types of zones:
- Green: “National Park Zones” with full conservation protection
- Yellow: “Habitat Protection Zones” where fishing is allowed as long as the seafloor is not harmed
- Blue: “Special Purpose Zones” that allow for specific commercial activities.
Crucially, under the new draft plans, the amount of green zones will be almost halved, from 36% to 20% of the marine park network, whereas yellow zones will almost double from 24% to 43%, compared with when the marine parks were established in 2012.
The government has said that this approach will “allow sustainable activities like commercial fishing while protecting key conservation features”.
But like the courtiers told to admire the Emperor’s non-existent new clothes, we’re being asked to believe something to be true despite strong evidence to the contrary.
The Emperor’s unrobing
The new plans follow on from last year’s release of an independent review, commissioned by the Abbott government after suspending the previous network of marine reserves implemented under Julia Gillard in 2012.
Yet the latest draft plans, which propose to gut the network of green zones, ignore many of the recommendations made in the review, which was itself an erosion of the suspended 2012 plans.
The extent of green zones is crucial, because the science says they are the engine room of conservation. Fully protected marine national parks – with no fishing, no mining, and no oil and gas drilling – deliver far more benefits to biodiversity than other zone types.
The best estimates suggest that 30-40% of the seascape should ideally be fully protected, rather than the 20% proposed under the new plans.
Partially protected areas, such as the yellow zones that allow fishing while protecting the seabed, do not generate conservation benefits equivalent to those of full protection.
While some studies suggest that partial protection is better than nothing, others suggest that these zones offer little to no improvement relative to areas fully open to exploitation.
Environment minister Josh Frydenberg has pointed out that, under the new plans, the total area zoned as either green or yellow will rise from 60% to 63% compared with the 2012 network. But yellow is not the new green. What’s more, yellow zones have similar management costs to green zones, which means that the government is proposing to spend the same amount of money for far inferior protection. And as any decent sex-ed teacher will tell you, partial protection is a risky business.
What do the draft plans mean?
Let’s take a couple of examples, starting with the Coral Sea Marine Park. This is perhaps the most disappointing rollback in the new draft plan. The green zone, which would have been one of the largest fully protected areas on the planet, has been reduced by half to allow for fishing activity in a significantly expanded yellow zone.From http://www.environment.gov.au/marinereservesreview/reports and https://parksaustralia.gov.au/marine/management/draft-plans/
This yellow zone would allow the use of pelagic longlines to fish for tuna. This is despite government statistics showing that around 30% of the catch in the Eastern Tuna and Billfish fishery consists of species that are either overexploited or uncertain in their sustainability, and the government’s own risk assessment that found these types of fishing lines are incompatible with conservation.
What this means, in effect, is that the plans to establish a world-class marine park in the Coral Sea will be significantly undermined for the sake of saving commercial tuna fishers A$4.1 million per year, or 0.3% of the total revenue from Australia’s wild-catch fisheries.
Contrast this with the A$6.4 billion generated by the Great Barrier Reef Marine Park in 2015-16, the majority of which comes from non-extractive industries.
This same erosion of protection is also proposed in Western Australia, where the government’s draft plan would reduce green zones by 43% across the largest marine parks in the region.http://www.environment.gov.au/marinereservesreview/reports and https://parksaustralia.gov.au/marine/management/draft-plans/
Again, this is despite clear evidence that the fishing activities occurring in these areas are not compatible with conservation. Such proposals also ignore future pressures such as deep-sea mining.
The overall effect is summarised neatly by Frydenberg’s statement that the government’s plans will:
…increase the total area of the reserves open to fishing from 64% to 80% … (and) make 97% of waters within 100 kilometres of the coast open for recreational fishing.
Building ocean resilience
Science shows that full protection creates resilience by supporting intact ecosystems. Fully protected green zones recover faster from flooding and coral bleaching, have reduced rates of disease, and fend off climate invaders more effectively than areas that are open to fishing.
Green zones also contribute indirectly to the blue economy. They help support fisheries and function as “nurseries” for fish larvae. For commercial fisheries, these sanctuaries are more important than ever in view of the declines in global catches since we hit “peak fish” in 1996.
Of course it is important to balance conservation with sustainable economic use of our oceans. Yet the government’s new draft plan leaves a huge majority of Australia’s waters open to business as usual. It’s a brave Emperor who thinks this will protect our oceans.
So let’s put some real clothes on the Emperor and create a network of marine protection that supports our blue economy and is backed by science. | https://dailybulletin.com.au/the-conversation/31653-australia%EF%BF%BD%EF%BF%BD%EF%BF%BDs-new-marine-parks-plan-is-a-case-of-the-emperor-s-new-clothes |
False positive bronchoalveolar lavage galactomannan: Effect of host and cut-off value.
Bronchoalveolar lavage galactomannan (BAL-GM) is a mycological criterion for diagnosis of probable invasive aspergillosis (IA) per European Organization for Research and Treatment of Cancer/Mycoses Study Group (EORT-MSG) consensus criteria, but its real-world positive predictive value (PPV) has not been well-studied. Our aim was to estimate the PPV of BAL-GM in a contemporary cohort of patients with positive BAL-GM. We identified consecutive patients with ≥1 positive BAL-GM value (index ≥ 0.5) at Brigham and Women's Hospital from 11/2009 to 3/2016. We classified patients as having no, possible, probable, or proven IA, excluding BAL-GM as mycological criterion. We studied 134 patients: 54% had hematologic malignancy (HM), and 10% were solid organ transplant (SOT) recipients. A total of 42% of positive (≥0.5) BAL-GM results were falsely positive (PPV 58%). The number of probable IA cases was increased by 23% using positive BAL-GM as mycologic criterion alone. PPV was higher in patients with HM or SOT (P < 0.001) and with use of higher thresholds for positivity (BAL-GM ≥ 1 vs 1-0.8 vs 0.8-0.5: P = 0.002). 42% of positive BAL-GM values were falsely positive. We propose a critical reassessment of BAL-GM cutoff values in different patient populations. Accurate noninvasive tests for diagnosis of IA are urgently needed.
| |
Discipline:
Language Arts Subject:
Oral Communication Grade:
6
Southern United States and MexicoIn the Desert
Habitat
Where my animal lives: Mexico and Southern United States
Giant Kangaroo Rat (Dipodomys ingens)
Name: Kiara Ms. Bias 2nd hour
Fun Facts - GrolierFun Facts - World BookPictures - WikipediaVideos - Arkive Facts - ArkiveFacts - Image
My Sources
Their total size is 31.2-34.8.Their tail Size is 15.7-19.4. They can jump 6.5 feet or more with hop.They leave the nest after 4 or 5 weeks.They only grow 15 inches.They use the front feet to stuff food in their mouth.
Interesting facts
Level of Endangerment: Endangered
Why I'm endangered
How we are saving them
My plan to save them: | https://edu.glogster.com/glog/giant-kangaroo-rat/26ogsdok53o?=glogpedia-source |
Taxes hit the 99% the hardest
The past few years have seen a multitude of protests on income inequality, decrying how the 1% makes so much more than the 99% and how this injustice creates vast inequality, social divisions and growing class anger.
Taxes — local, state and federal — fall disproportionately hardest on people with smaller incomes. But the whole structure of the tax system in the United States is so complicated and so fragmented that it is hard to see what is going on.
On Jan. 14, the Institute on Taxation and Economic Policy released its fifth “Who Pays?” report on state and local taxes. Its conclusion is stark: On average, the tax rate for the poorest 20 percent of U.S. residents is twice that of the richest 1 percent. (itep.org/whopays)
The way many states have configured their tax system — where the percentage of income paid in taxes doesn’t depend on one’s income and the state depends on consumption taxes — makes this problem worse.
There are 10 states where the poorest 20 percent pay up to seven times as much of their income in taxes as the 1%. Washington state is the most regressive, followed by Florida, Texas, South Dakota, Illinois, Pennsylvania, Tennessee, Arizona, Kansas and Indiana.
Another egregious aspect of the U.S. tax system is that income obtained from wealth goes under the rubric of capital gains and is more lightly taxed than income earned by working. Almost all wealth income belongs to the top 10 percent.
Many of the attempts to make these tax systems fairer and less burdensome have helped, but at the cost of making an already complicated system more so. Calls for simplifying it generally lead to putting more of a burden on the lowest paid.
Other taxes on the poor
Not everyone may pay federal income taxes, but they certainly pay payroll taxes on all their income — that is, Social Security and Medicare taxes. However, someone making more than $500,000 a year pays these taxes on only about 20 percent of their income, since these taxes are capped at $118,500 in 2015.
There is a federal excise tax, currently $0.184, on every gallon of diesel oil and gasoline sold in the U.S.
Allowing mortgage interest payments to be a tax deduction, a major benefit for homeowners, is another way the federal tax system works against poorer people, many of whom rent because they don’t have the money to buy.
The 50 states and the District of Columbia each have their own tax code, and each state has its own city and local governments, some of which have their own tax codes.
Each state also has its own excise taxes on fuel, cigarettes, alcohol and tobacco, which affect low-wage earners more than the better off, since they are flat taxes. Sales taxes are another form of taxation with a disproportionate effect on low-wage earners. | https://www.workers.org/2015/01/18001/ |
Reference checks are an imperative part of the recruitment process, says Manisha Maligaspe, Oceania Transaction Advisory Services Recruitment Lead for EY. “Genuine reference checking is about whether the candidate actually worked at the place they said they did and carried out the tasks they said they were doing,” she says. “Essentially, it’s a means of verifying the candidate’s skills and experiences they represented in their resume and throughout the assessment process and interviews.”
While reference checks can be time consuming, thoroughly checking out a candidate before you offer them a position is likely to save you time in the long run and potentially avoid any issues later on.
The difference between personal and professional references
A personal reference is someone who has not worked with the candidate and who can discuss the individual’s values, characteristics and personality. A professional reference is usually a former employer, client, colleague or supervisor who can recommend the candidate’s work ethic, skills and attitude.
“In a perfect world, the candidate would provide a previous manager or supervisor that they have reported to directly,” says Maligaspe. “A professional reference check, ideally where the candidate has reported to that individual, is usually an accurate confirmation of the candidate’s employability and skills.”
Employers might ask for a personal reference if the candidate is young or has not had much experience, but Maligaspe says it’s usually best to ask candidates to provide the name of at least one professional referee.
How to get around a ‘no reference policy’
A ‘no reference policy’ is when an organisation has a rule not to give verbal and / or written references about current or former employees.
Hirers can ask candidates to check with other recent employers to see if they are willing to give a reference or ask for a character reference from an ex-colleague or former manager at the previous employer in lieu of a formal reference. “You could even ask for a candidate to provide performance review feedback in lieu of a reference check,” says Maligaspe.
Can candidates legally ask to see notes from a reference check?
“If the prospective employer has obtained referee reports for the unsuccessful job applicant, then the applicant is entitled to access those referee reports,” says Kelly Godfrey, Principal Solicitor with Employment Lawyers Australia.
The reason for this is that an inaccurate referee’s report can affect a candidate’s employment opportunities. “As such, a job applicant has the right to access and correct any personal information that is held about them,” Godfrey adds.
What you can and can't say about former employees
References are a tricky legal area, says Godfrey. “If you are asked to provide a reference and it is not favourable to the employee, it may be easier to decline and provide a statement of service instead.”
A statement of service sets out the employee’s commencement date, finish date, sometimes the reason for termination, position titles held and may briefly describe the duties the employee undertook. “The statement of service gives no assessment of how well the former employee performed in the role,” says Godfrey.
“If you decide to provide a reference, good or bad, to reduce the legal risks involved, you should ensure that there is objective evidence to support the statements you make.”
Best practice reference checks
Talk to the right people
“For the employer to get an accurate picture of how the candidate performs in a professional environment, the reference check provider has to ideally be a supervisor or manager – those who can speak knowledgeably about the applicant’s performance, communication, cultural fit and work ethic,” says Maligaspe.
Protect the candidate’s privacy
“The employer should ask the candidate whether they have informed their previous employer of their job search otherwise it would be a very awkward situation,” advises Maligaspe.
Ask the right questions and be specific
“Employers should implement behavioural-based interviewing in a reference check situation,” says Maligaspe. “Such as, “Tell me about a time when Joe went above and beyond to complete a critical task”. Questions like this provide in-depth insight into a candidate’s interpersonal skills, adaptability and overall work ethic. | https://insightsresources.seek.com.au/what-you-need-to-know-about-reference-checks |
Ignaz Pleyel (1757-1831) was at one time the most famous composer in the world. The popularity of his music eclipsed that of even his teacher Haydn and publishers vied to bring out his latest works as soon as they were finished. Some 2000 separate prints of Pleyel works had appeared by 1800 and his fame extended to every corner of Europe and as far afield as North America. Pleyel's career as a composer spanned less than thirty years with the majority of his works composed in the 1780s. He founded a successful publishing house in Paris in the mid-1790s and later began manufacturing keyboard instruments. With increasing demands on his time from his business concerns Pleyel's productivity as a composer dropped sharply and he ceased composing around 1805. Among the authentic chamber works the duos occupy a particularly interesting place. The Six Duos for Violin & Violoncello (Ben 501-506) appeared in over twenty editions in Pleyel's lifetime, the first in 1787, and in a variety of scorings. Although their unusual instrumentation undoubtedly contributed to the popularity of the Duos, their musical qualities - evidenced by the large number of contemporary arrangements - probably played a greater role. The works abound in attractive and distinctive melodic material; Pleyel writes idiomatically for both instruments and his part-writing is varied and interesting. | https://www.artaria.com/products/pleyel-ignaz-three-duos-for-violin-cello-benton-501-503-ae408 |
The second heat wave of summer arrives: where will it affect and how long will the high temperatures last?
High temperatures will rise this weekend in several points of the Peninsula. The State Meteorological Agency (AEMET) alerts of the arrival of a new heat wave, the second this summer.
The heat and the coastal phenomena mark the forecast of the weather for this day, which put on alert six provinces. The heat will activate warnings in Cordoba, Jaen, Seville and Badajoz while coastal phenomena will activate in A Coruña and Girona.
The heat wave is scheduled for the weekend and will affect the whole country, including the Canary Islands -it will be the first heat wave for the archipelago-. High temperatures could reach 40 degrees at some points and the heat wave will last until the middle of next week or “maybe more”, according to the spokesman of the Aemet, Ruben del Campo.
Temperatures will range on average between 35ºC and may exceed 40ºC in some areas. For example, 35 degrees maximum are expected in large areas of the Peninsula from Saturday, and more than 40 degrees in the valleys of the Tagus, Guadiana and Guadalquivir, which could reach 42 degrees.
The 40ºC will also be exceeded in the Ebro Valley, Douro area, western Castile and León and southern Galicia.
For the Canary archipelago it will be the first heat wave this summer. There it is expected that temperatures will increase from Friday. | https://www.bayradio.fm/2022/07/07/the-second-heat-wave-of-summer-arrives-where-will-it-affect-and-how-long-will-the-high-temperatures-last/ |
Please use this identifier to cite or link to this item:
http://publications.jrc.ec.europa.eu/repository/handle/JRC114229
|Title:||Barium ferrite magnetic nanoparticles labeled with 223Ra: a new potential magnetic radiobioconjugate for targeted alpha therapy|
|Authors:||BILEWICZ ALEKSANDER; CEDROWSKA EDYTA; GAWEDA W; BRUCHERTSEIFER FRANK; MORGENSTERN ALFRED|
|Citation:||JOURNAL OF LABELLED COMPOUNDS & RADIOPHARMACEUTICALS vol. 62 no. S3 p. 103|
|Publisher:||WILEY-BLACKWELL|
|Publication Year:||2019|
|JRC N°:||JRC114229|
|ISSN:||0362-4803 (online)|
|URI:||http://publications.jrc.ec.europa.eu/repository/handle/JRC114229|
|DOI:||10.1002/jlcr.3724|
|Type:||Articles in periodicals and books|
|Abstract:||223Ra, as radium chloride, is the first commercially and widely used α-radiopharmaceutical. It is easily obtained from the 227Ac/223Ra generator. However, 223Ra is used only for treatment of bone metastases derived from primary prostate and breast cancers. Unfortunately, the lack of an appropriate bifunctional ligand for radium was the reason why 223Ra has not yet found application in receptor targeted therapy. Because Ra2+ and Ba2+ are nearly identical cations in our studies we propose to use barium ferrite (BaFe12O19) nanoparticles as multifunctional carriers for 223Ra radionuclide for targeted α therapy. Barium hexaferrite nanoparticles labelled with 223Ra were synthesized by a modified autoclave method described by Drofenik et al . The reaction mixture of FeCl3, BaCl2 and 223RaCl2 was alkalized with NaOH solution. Next, the reaction mixture was stirred in autoclave at 210oC for 6 h. Obtained radioactive, magnetic [223Ra]BaFe12O19 nanoparticles were washed with distilled water and hydrochloric acid (0.001 M HCl). Obtained magnetic BaFe12O19 nanoparticles were characterized by transmission emission microscopy and dynamic light scattering. The diameter of synthesized nanoparticles was ~20 nm and the determined magnetization of nanoparticles in room temperature was about 42 emu/g. Yield of labelling was about 70% (for 100 kBq 223Ra). Stability of the obtained radioactive nanoparticles was tested in various biological solutions: 0.01M PBS, 0.9% NaCl and in human blood serum. It is confirmed that 223Ra was highly retained inside nanoparticles in every tested solution. Only about 20% of 211Pb (recoiled decay product of 223Ra) was found in solution. In order to synthesize a radiobioconjugate having affinity to HER2 receptors, the monoclonal antibody trastuzumab was conjugated to the obtained barium ferrite nanoparticles. Firstly, the surface of barium ferrite nanoparticles was modified with 3-phosphonopropionic acid linker using a method described by Mohapatra et al , and then, the monoclonal antibodies were coupled to the barium ferrite nanoparticles using the carbodiimide chemistry. Synthesized bioconjugate was characterized by thermogravimetric analysis, dynamic light scattering and were tested for stability in biological fluids. The obtained [223Ra]BaFe12O19-CEPA-trastuzumab radiobioconjugate almost quantitatively retains 223Ra and majority of the daughter products. In-vitro biological studies indicate that [223Ra]BaFe12O19-CEPA-trastuzumab exhibits high affinity and cytotoxicity to the to the SKOV3 ovarian cell line.|
|JRC Directorate:||Nuclear Safety and Security|
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Items in repository are protected by copyright, with all rights reserved, unless otherwise indicated. | https://publications.jrc.ec.europa.eu/repository/handle/JRC114229 |
Sustainability and society
Tilburg University's sustainability research includes issues in economics, ethics, and law. Researchers focus on, for instance, Corporate Social Responsibility, sustainable development, environmental economics, climate change and sustainable investment.
Experts and their expertise
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Reyer GerlaghProfessor
#energy economics #environmental economics #climate change #emission tradings
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Jonathan VerschuurenProfessor
#climate change #environmental law # European Environmental Law
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Saskia LavrijssenProfessor
#energy markets #consumer behaviour #European Law
Prof. Martijn Groenleer about SMILE: Together towards zero-energy neighborhoods
News about our research on sustainability and society
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1.6 million for interdisciplinary project ‘Conflict in Transformations’10th January 2022
Urban Europe has awarded a grant of 1.2 million euros for the interdisciplinary research proposal ‘Conflict in Transformations’, which will be topped up by the research consortium partners with 0.4 million euros. The project studies whether and how political and legal institutions suppress or actively use conflict for transformation towards more sustainable cities.
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Social responsibility central theme of winners Tilburg University Challenge 202110th December 2021
The Tilburg University Challenge 2021 has awarded all three prizes for the best business plan proposed by student teams to socially responsible ideas. During a livestream event on December 9th the Ideation Award went to Anouk van Anrooij and Daniel Gelsing for their healthy and fruity drink that is also sustainable. The Business Ready Award and the Audience Award went to Freek van Litsenburg, who, with Bigger Picture Clothing, sells clothing that is both sustainable in production and has a large social impact.
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Reactions to Climate Change Summit COP26: Too little, too late, small steps only17th November 2021
Will the climate change deal of COP26 get us anywhere? Tilburg researchers agree that the results do not match the urgency of the problems, that are getting bigger and bigger. After examining the small print, they have discovered a few bright spots: procedures have been agreed to keep countries more focused. However, the Dutch government and the EU need to make every effort, and soon, too. The problem of climate change can only be solved collectively.
Collaborations
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Academic Collaborative Center ‘Widespread Prosperity in the Region’
Science and practice are joining forces: a new Academic Collaborative Center 'Widespread Prosperity in the Region' to more explicitly and structurally connect science and practice at the regional level for a number of important social tasks. People, Planet and Profit!More information
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Community Klimaat en Energietransitie
Om de klimaatverandering af te remmen en de energietransitie te versnellen, moeten energie- en klimaatdoelen vertaald worden in concrete acties op internationaal, Europees, nationaal en regionaal niveau. Samen met maatschappelijke partners werkt Tilburg University aan het realiseren van die acties.More information
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Climate and Energy Transition
Within Tilburg University, various disciplines collaborate on energy and climate research. The researchers do this together with the government and civil society organizations.More information
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Tilburg Sustainability Center
TSC wants to help deal with the challenges of a globalizing world economy, where climate change and resource scarcity require innovative and interdisciplinary approaches and where both policymakers and corporations find it increasingly important to achieve growth in a responsible and sustainable manner.More information
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Want to stay informed about this topic and other scientific developments at Tilburg University? Sign up for our press releases and choose the topics you find interesting. | https://www.tilburguniversity.edu/research/topics/sustainability-and-society |
WASHINGTON - The U.S. economy grew at its fastest pace in
almost two years in the third quarter, while business spending
was stronger than previously estimated, pointing to some
underlying strength that should be sustained. (USA-ECONOMY/
(WRAPUP 1), expect by 1400 GMT/9 AM ET, by Lucia Mutikani, 500
words)
S&P cuts EU's AAA rating, European officials dismiss move
BRUSSELS - Credit agency Standard & Poor's cuts its triple-A
rating of the European Union by one notch, saying it has
concerns about how the bloc's budget is financed, a view EU
leaders and other officials dismiss as misguided.
(EUROPE-CREDITRATING/S&P (UPDATE 3), moved, 500 words)
European stocks edge up, euro dips on S&P downgrade
PARIS - European shares inch up with riskier assets still in
demand following a broadly neutral shift in U.S. monetary
policy, and the euro dips after rating agency S&P downgrades the
European Union from triple-A. (MARKETS-GLOBAL/ (WRAPUP 6),
moved, by Blaise Robinson, 700 words)
LONDON - Over the past decade, shareholders of major blue
chip firms have received more in tax credits from the government
than they lost through their firms' corporate income tax. This
means that, in effect, the UK government is subsidising them to
own shares. (TAX-BRITAIN/CREDITS (INSIGHT, PIX, GRAPHIC), By Tom
Bergin, moved, 1,500 words)
See also: TAX-BRITAIN/CREDITS (FACTBOX)
INVESTMENT
Investors boost cash as they round off a bumper stocks year
LONDON - Leading global investors boosted cash levels this
month to the highest since July 2012 as they wrap up a year that
has produced double-digit gains in equities, Reuters polls show.
(FUNDS-POLL/GLOBAL, moved, by Natsuko Waki, 80 words)
Cash-rich firms, investors waiting for each other to spend
LONDON - Companies and investors are sitting on trillions of
dollars of cash as they head into 2014, waiting for each other
to start spending. (INVESTMENT-CASH (ANALYSIS), moved, by
Natsuko Waki, 70 words)
Buoyant stock markets lift year's share offerings by 24 pct
LONDON/NEW YORK/HONG KONG - This year has been the biggest
for equity fundraising globally since 2010, thanks to improving
confidence among companies on the back of the strong investor
demand for stocks, according to Thomson Reuters data. (GLOBAL
EQUITYCAPITALMARKETS/Q4, moved, by Kylie MacLellan, Olivia Oran
and Elzio Barreto, 500 words)
+ See also:
- INVESTMENTBANKING FEES/, moved, by Clare Hutchison, 400
words
China nears moment of truth on IPO reform
SHANGHAI - China's plan to build confidence in domestic
stock markets, and turn around their reputation as financial
casinos, will depend on a regulatory gamble paying off next
year. (CHINA-IPO/ (ANALYSIS), moved, by Pete Sweeney, 1,000
words)
ECONOMY
Bernanke's 11th hour pivot smoothes path for Yellen at Fed
SAN FRANCISCO/NEW YORK - By ensuring the Federal Reserve
begins trimming its massive bond-buying stimulus before a more
hawkish contingent of voters comes on board next year, Fed
Chairman Ben Bernanke has greased the skids politically for his
successor, Janet Yellen. (USA FED/YELLEN, moved, by Ann Saphir
and Jonathan Spicer, 700 words)
BOJ keeps massive stimulus while Fed begins tapering
TOKYO - The Bank of Japan keeps monetary policy steady and
maintains its view that the economy is recovering moderately,
encouraged by growing signs that the benefits of its massive
stimulus are spreading through broader sectors of the economy.
(JAPAN-ECONOMY/BOJ (UPDATE 1), moved, by Leika Kihara and
Stanley White, 700 words)
Portugal pledges alternative route to cutting deficit
LISBON - Portugal's government pledges to find alternative
fiscal measures to reduce the budget deficit after the
Constitutional Court delivers another blow to plans to reduce
state spending and smoothly exit a bailout in mid-2014.
(PORTUGAL-COURT, by Axel Bugge, expect by 1500 GMT,/10 AM ET,
550 words)
Spain reviews power auctions after price hike threat
MADRID - Spain is looking for a new method to set
electricity prices after a double-digit spike at an wholesale
auction threatens a painful rise in household bills, the prime
minister says. (SPAIN-POWER/HIKE (UPDATE 1), expect by 1400
GMT/9 AM ET, by Jose Elias Rodriguez and Tracy Rucinski, 500
words)
South Africa's biggest union cuts political ties with ANC
JOHANNESBURG - South Africa's biggest union will not support
the ruling ANC in elections next year, its general secretary
says, in a blow to President Jacob Zuma, whose political support
with the working class is eroding fast. (SAFRICA-NUMSA/ANC
(UPDATE 2), moved, by Peroshni Govender, 700 words)
RESOURCES
Copper squeeze flares, more battles seen between bulls/bears
LONDON/NEW YORK - One investor amasses huge position in
London copper contracts, stoking worries about a market squeeze
amid a shortage of copper stocks and despite efforts on other
commodities exchanges to tighten rules on speculative trading.
(COPPER-SQUEEZE/, expect by 1500 GMT/10 AM ET, by Eric Onstad
and Josephine Mason, 950 words
LONDON - BAE Systems faces investor worries over its growth
prospects after the United Arab Emirates pulls out of talks to
buy 60 Eurofighter Typhoon combat jets, in a blow to the UK
government which has pushed hard to land the $9.8 billion deal.
(BAE-UAE/UPDATE 1), moved, by Paul Sandle and Sarah Young, 500
words)
Trending Stories
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Painting for Madelen is a positive meditative process of expressing her natural creative energy into abstract form.
"It is the feeling of aliveness and joy, something that does things to or for me".
The colors represent emotions, with different layers of texture, growing until it finally explodes into a radiant visual moment of completeness.
"It is an art in itself to know when the work is complete".
It can take several weeks until she finishes a painting, until satisfied is truly done. You find a fusion of sharpness and softness coming into her work. She has a determination to excude the living process of life in her art, to catch the moment of perfection in its contrast.
Each work is infused with boundless energy, curiosity and joy for life that is 100% pure Madelen. | https://artbyleek.com/art-by-leek/about/ |
Along with the effort toward developing EQ, many have begun to investigate the notion that the brain may have intelligences which go further than just our IQ and EQ.
Pseudo-artistic play; no interest in the creation afterwards Pseudo-artistic play; no interest in the creation afterwards Full creation of art; calculation of artistic effects; intent to preserve, discuss and appreciate the work after it has been created Notes 1 Empathy is the ability to be affected by the state of another individual or creature.
This is seen in bodily mimicry as well as emotional contagion. Emotional empathy has its roots in bodily mimicry, as one individual unconsciously mimics the facial expression of another.
It means that one individual has an idea, a theory, about what another individual believes, perceives or intends to accomplish. The expected brain mass is that required for basic survival tasks.
If the actual brain mass is larger than expected, then the extra mass is available for higher-level cognition. Human EQ is much greater than chimp or bonobo EQ. Bonobos were recognized as a separate species less than years ago and began to be fully documented less than 50 years ago.
Before that time, many ethologists and anthropologists believed that humans are innately violent and aggressive. Morality, it was thought, is a veneer of cooperative sociality on an underlying bestial nature. Now that we know about bonobos, the range of human behavioral potential seems to have expanded.
We recognize that humans have the capacity to live in peace and to defuse conflict proactively with pleasure. In addition, male dominance seemed a natural part of things until the discovery of bonobos; now we see that dominance by females may be equally natural.
Two things stand out from the comparison of species above. First, our difference from chimps and bonobos is a matter of degree, not kind.
There are few, if any, uniquely human traits that chimps or bonobos do not have to a lesser degree. We are embedded in nature and are not a species unique and special. The one trait that seems most unique is the cultural, not biological, innovation of nuclear family pair bonding.
Second, humans have the capacity to amplify the characteristics found in our sibling species. Humans have greater brain size and intelligence, so we can do more effectively all the things our siblings can. For instance, our use of tools and technologies enables us to produce food in more variety and abundance.
Our use of language enables us to communicate more effectively and to perpetuate what we learn through culture. Chimps and bonobos seem to be able to conceptualize that something not happening in the present will happen later, but humans have a greatly enhanced ability to visualize and anticipate the future.Published: Mon, 5 Dec The main purpose of the paper is to prove that the essence of human nature lays primarily in person’s ability to reason; capacity that is uniquely human and allows people to make decisions that would shape their norms of conduct as shown in the works of ancient and modern philosophers.
- The question of human nature and the facets of evil present itself numerous times in the captivating movie, “The Good Son”. This psychological thriller exceeds an audience’s expectation to the portrayal of childhood psychiatric disorders. The medieval philosophers such as Augustine understood human nature to incorporate the metaphysics of form and matter, abilities that are unique to humans, and the character of human rational soul.
As such, the medieval philosophers shared the view that body and soul are fundamentally different.
The relationship between human and nature can be described in different ways; it can be beautiful, cruel or at times puzzling. Human responds to nature in different ways.
Based on their surrounding, humans can simply accept nature, deal with their situation, or make efforts to change it.
Essay Human Origin And The Creation Of Human Nature There are many definitions of the word story, but one definition in particular fits the topic of human origins perfectly; a narration of the events in the life of a person or the existence of a thing, or such events as a subject for narration.
These facets of human nature are a product of genetically coded survival instincts modified by the totality of our environment and expressed as neurochemically-mediated emotions and actions. Reason, reflection and conscious morality are comparatively rare. | https://lomupixetira.urbanagricultureinitiative.com/facets-of-human-nature-essay-40016yc.html |
The image below depicts the most commonly used measurements for fish. For freshwater fish, the measurements that you need to use are total length and girth.
The total length is the maximum length of the fish, with the mouth closed and the tail fin pinched together. The best way to obtain this length is to push the fish’s snout up against a vertical surface with the mouth closed and the fish lying along a tape measure, then pinch the tail fin closed and determine the total length. Do not pull a flexible tape measure along the curve of the fish. These photos show a bass on a measuring board with the mouth held shut.
Conversely, most marine (saltwater) regulations refer to the “fork length,”and scientists often use “standard length,” which is to the end of the fleshy part of the body. “Standard length” has the advantage of not being affected by minor damage to the tail fin, nor does it give too much credit to a fish for the relatively lightweight tail when calculating a fish’s condition.
Girth is best measured with a fabric ruler, like tailors use. It can also be determined by drawing a string around the fish at its widest point marking where the string overlaps and then measuring the distance between the overlapping points on a conventional ruler. The measurement should be taken perpendicular to the length of the fish. This measurement is analogous to measuring the circumference of someone's waist. Knowing the girth is important when trying to certify a fish for a record and provides useful information to biologists about the relative condition of a fish.
Although it cannot be used to certify an official weight, use of the length and girth can give you a good estimate of a bass’ weight. Scientists use a rather complex formula to attain the greatest accuracy. The equation is: Log (weight in grams) = -4.83 + 1.923 x Log (total length in millimeters) + 1.157 x Log (girth in millimeters). A 22-inch-long bass with a girth of 15 inches weighs about 6.0 pounds using this formula.
Fortunately, there are several other easy formulas that you can use, although they are not as accurate, they will give you a rough estimate. A quick, though very rough, estimate of torpedo-shaped fish such as young bass can be obtained by using total length (in inches) squared, times girth (in inches) divided by 1,200. A 22-inch-long bass with a girth of 15 inches weighs about 6.1 pounds using this formula.
Another common option used for estimating bass weights is girth (in inches) squared, times length (in inches) divided by 800. A 22-inch-long bass with a girth of 15 inches weighs about 6.2 pounds using this formula.
These fish measurement methods provide a consistent, well-defined measurement technique. These methods encourage angler compliance with fishery management regulations.
Total Length is now measured from the most forward point of the head, with the mouth closed, to the furthest tip of the tail with the tail compressed or squeezed, while the fish is lying on its side.
Fish regulated by fork length are measured from the tip of the fish jaw or tip of the snout with closed mouth to the center of the fork in the tail.
Stone Crab claws must measure at least 2¾ inches in length measured by a straight line from the elbow to the tip of the lower immovable finger. The forearm (propodus) shall be deemed to be the largest section of the claw assembly that has both a moveable and immovable finger and is located farthest from the body of the crab.
Spiny Lobster must have a minimum carapace length of greater than 3 inches and the measurement must take place in the water. The carapace is measured beginning at the forward edge between the rostral horns, excluding any soft tissue, and proceeding along the middle to the rear edge of the carapace.
For this and more Fishing tips please go to VISIT FLORIDA'S Official Fishing Guide. | https://www.visitflorida.com/en-us/things-to-do/florida-fishing/fishing-tips-how-to-properly-measure-your-fish.html |
WRAP_Polezzi_Fascim_and_Nature_Open_Access_June2014.pdf - Accepted Version - Requires a PDF viewer.
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Official URL: http://dx.doi.org/10.1080/13532944.2014.927355
Abstract
During the period of Fascism, a variety of discourses and representations where attached to colonial landscapes and to their uses. African nature was the subject of diverse rhetorical strategies, which ranged from the persistence of visions of wilderness as the locus of adventure to the domesticating manipulations of an incipient tourist industry aiming to familiarize the Italian public with relatively tame forms of the exotic. Contrasting images of bareness and productivity, primitivism and modernization, resistance to change and dramatic transformation found their way into accounts of colonial territories ranging from scientific and pseudo-scientific reports to children’s literature, from guide books to travel accounts, all of which were sustained not just by written texts but also by iconographic representations.
The article will look at the specific example of accounts of Italian Somalia in order to explore Fascist discourses about colonial nature and its appropriation. Documents examined will include early guidebooks to the colonies, a small selection of travel accounts aimed at the general public, as well as the works of a number of geographers and geologists who were among the most active polygraphs of the period, and whose writings addressed a wide range of Italian readers. | http://wrap.warwick.ac.uk/61852/ |
Willow trees are ornamental trees that have can bring a special look to your garden or landscape. Many features of this tree can interest children, allowing you to teach them the history of willows and their practical medicinal uses.
The history of willow trees goes back to Sumeria, where the tree was a symbol for creation, growth and emerging consciousness. For example, it is symbolic because people may see it as a portrayal of sadness, although sitting beneath the tree is comforting.
The White Willow was a place where individuals would sometimes make wishes. People would approach the tree and tie a loose knot in a limb, making a wish while doing so. Once the wish came true, the person would go back and untie the knot.
Willow bark and its medicinal uses go back to Hippocrates. Chewing on the bark could reduce inflammation and fevers. Headaches, back pain and osteoarthritis are also some ailments for which willow bark can relieve pain since it contains a chemical similar to aspirin. | https://www.ehow.co.uk/facts_7171921_kids-information-willow-trees.html |
(1) I love coffee and drink at least 10 cups a day (mostly decaf). This is one of my favorite mugs.
(2) If you expect to see the makings of 365 really interesting and well composed shots, you have come to the wrong place! I will be happy if there are a dozen keepers in the bunch. | https://www.nawset.com/Galleries/Project365/Project-365/i-7LsMvV3 |
Read several letters from the site you are writing to for an idea of the types of letters that the editors print. Sample letter from the Dallas Morning News: The surgeon general of the U. Public Health Service Commissioned Corps and thus the leading spokesman on matters of public health in the federal government of the United States.
Learn how to write and send effective print and e-mailed letters to editors of various media types, together with examples, that will gain both editorial and reader support.
What is a letter to the editor? Why should you write a letter to the editor?
When should you write a letter to the editor? Should you use e-mail to fax or to writing an editorial letters your letter? How do you write a letter to the editor? How do you get your letter accepted?
You feel strongly about an issue, and you want to let people know what you think. You believe you can even influence people to take some action if you speak your mind. But, you want to reach an audience larger than just your friends or your group membership.
Letters to the editor can be an effective way to get the word out. A letter to the editor is a written way of talking to a newspaper, magazine, or other regularly printed publication. Letters to the editor are generally found in the first section of the newspaper, or towards the beginning of a magazine, or in the editorial page.
They can take a position for or against an issue, or simply inform, or both.
They can convince readers by using emotions, or facts, or emotions and facts combined. Letters to the editor are usually short and tight, rarely longer than words. Using a few carefully placed letters, you can generate plenty of community discussion.
You can also keep an issue going by preventing it from disappearing from the public eye. You can stimulate the interest of the news media and create more coverage for the matters you're working on.
You can also send a "good news" letter to bring recognition to people who deserve it or acknowledge the success of an effort. Letters to the editor are among the most widely read features in any newspaper or magazine.
They allow you to reach a large audience. You can probably think of many more specific reasons why you might want to write to the editor, but here are a few general ones: You are angry about something, and want others to know it You think that an issue is so important that you have to speak out Part of your group's strategy is to persuade others to take a specific action Or you want to: Suggest an idea to others Influence public opinion Educate the general public on a specific matter Influence policy-makers or elected officials directly or indirectly Publicize the work of your group and attract volunteers or program participants When should you write a letter to the editor?
Letters to the editor can be written any time you want to shape public opinion, tell others how you feel about people, programs, or ideas, or just inform the public on a certain issue.
They are a great way to increase awareness of the issues that you or your organization are working for, as well as to advocate for your cause.Writing a letter to the editor or an opinion editorial (op-ed) can be a useful way to share your knowledge about infant-toddler issues with the local community and policymakers.
Write a letter to the editor.
Letters exist to provide a forum for public comment or debate. A letter to the editor is meant to express your opinion or point of view about an article you have read from a news organization or website.
The early modern period was a great golden age of letters and letter writing in many parts of the Atlantic world.
The 18th century in particular saw a flourishing of the epistolary genre across numerous settings. Letter-writing manuals taught elite and commoner alike how to craft a wide variety of.
Are you applying for jobs working as an editor or a writer? Take the time to customize your own letter, being sure to match your qualifications to the requirements listed in the job posting. While organization and strong writing abilities are required for nearly every editorial or writing position, jobs can vary widely in terms of other desired skills.
Writing a letter to the editor can be a cathartic way to express your viewpoint on a subject that you are passionate about. The letters to the editor section of a publication offers a platform for community opinions covering everything from local municipality issues to national politics, but editors.
When writing a letter to the editor of a newspaper or magazine, keep the following tips in mind: Respond quickly. If you read or see something you want to respond to, send your letter by email (or use the news site's online form if they have one) the same day, or by the next day at the latest. | https://cehojolekysac.lausannecongress2018.com/writing-an-editorial-letters-8990vu.html |
In the current circumstances of COVID-19, when our everyday lives are being thrown into a new routine, we seek comfort by returning to the things most familiar to us, in this case: 'Home'.
'Home' isn't just an iconic National Day song—since it was conceived by composer Dick Lee, it has become a beloved classic, occupying an enduring place in the canon of Singaporean songs. The song has endured and taken on different permutations over the years: from a remixed version performed by JJ Lin at the 2004 NDP to a 2011 MINDEF remake that involved 39 Singaporean vocalists.
In light of the coronavirus situation, it has become a song for Singaporeans to rally around: on 14 April, 900 Singaporeans—including homegrown artists such as Nathan Hartono, Shigga Shay, Charlie Lim, and more—came together in a virtual choir to perform the song.
George Leong—known for his music arrangement work on Leslie Cheung's acclaimed hit 'Chase' and Sandy Lam's album, Scars, as well as his heavy involvement with the Xinyao movement—served as the Music Arranger, Producer and Mixing Engineer of this project. Recently, he spoke to Hear65 about the process of gathering all 900 Singaporean voices into a virtual choir cover of 'Home'.
How was the process in putting this together?
I started the project by playing a piano guide. This guide was used to produce a tutorial for the participants to practice. Darius also arranged the Melody, Soprano, Alto , Tenor and Bass Choral parts based on the piano guide. This guide and a score was sent out to the participants to sing and record their voices and video to.
I then finished up the arrangement on Logic Pro X and had Han Oh create a virtual string orchestra from his home studio, Colin Yong dub bass from his home and Daniel Chai dubbed guitars from his home. All done remotely.
Meanwhile, I imported all the music instrument tracks as audio files into Pro Tools and started doing a mix while waiting for the submissions to come in.
I imported the audio tracks batch by batch and started organising them into their Melody, Soprano, Alto, Tenor and Bass sections.
What were some of the challenges?
The submissions started coming in around 5th April. I realised that the audio quality was really bad and there was a lot of background noise on most of the tracks. We started doing the tedious work of de-noising the tracks. There were hundreds of tracks coming in everyday and it was pretty overwhelming.
De-noising works by taking a sample of the ambient noise in the room and then subtracting it from the rest of the track under the vocal recording. It cannot be automated and needs to be painstakingly tweaked track by track to ensure best audio quality.
The tracks were quite messy as everybody seemed to have a different rhythmic interpretation of 'Home'. I got the help of Daniel Wong (Ardant Studio) to align the tracks one by one, phrase by phrase.
The most challenging part was trying to make the vocal tracks which were recorded separately on mobile phones and earpiece microphones to blend together as one choir. It took some time to find the right EQ, compression and adding the right reverb to get the sound of a massive choir singing in a concert hall.
Here is the fader view of the project in mix. There were so many vocal tracks that they had to be pre mixed into stems and re combined together with the music.
What were some of the fun parts?
It is definitely a relief that everything turned out great. It was very exciting for me to watch the video premiere on Facebook and Youtube were there were hundreds of people watching at the same time. Working on this project made me feel very patriotic. Listening to the voices of so many people singing our favourite Singapore song really makes one feel emotional!
If you want to try it out for yourself, here's a tutorial by Darius Lim
What's next for George Leong?
Working on more virtual and remote music projects! That's the best we can do right now in the middle of the Covid Circuit Breaker!
Watch the video below. | https://hear65.bandwagon.asia/articles/how-900-singaporean-voices-were-put-together-for-home-an-interview-with-george-leong |
Meiji Restoration.
Extracts from this document...
Introduction
Meiji Restoration The Tokugawa shogunate or Tokugawa bakufu (a de facto central administration) was a feudal military dictatorship established in 1603 by Tokugawa Ieyasu. After the Battle of Sekigahara in 1600, central authority fell to Ieyasu who received the title of Sei-i-tai-shogun ("Great Barbarian Quelling General"). This was originally an imperial title bestowed on the commander of armed forces employed against the turbulent frontier tribes of the north, i.e., against the indigenous Ainu. Imprisoned in the Tokyo Imperial Palace and virtually under 'house arrest' the emperor was forced into a powerless de-jure role while the Shogunate conceived strategies to permanently retain their power. In an effort to isolate enemies and maintain rule, the Tokugawa initiated a process of land relocation where Han (domains) were divided into fuadi daimyo (friends-direct vassals) and tozama daimyo ('outside' daimyo). One way of ensuring a static society was to reduce social mobility to a minium by rigidly separating the various classes of society and forbidding movement between them. A caste system developed comprising of daimyo (land lords) followed by, samurai (warriors), nomin (peasants), chonin (townspeople/ merchants) and eta (untouchables). The samurai had a hereditary superiority over the other three lower classes and enjoyed the privilege of kirigesute-gomen- the right to cut down with one of their two swords any non-samurai who insulted them. Samurai devoted their body and soul to their master daimyo in accordance with the religious code of honour, the oral code of Bushido. The nomin and chonin formed more than 90% of the population and shouldered the burden of supporting the state through excessive taxation. Strongly encouraged by the bakufu, Confucian teaching treated rulers as uniquely fitted to govern and provided an ethical backing for the demand of absolute loyalty. The re-allotment of fiefs after Sekigahara left samurai without an overlord and many became ronin, literally 'wave men'-masterless warriors. A morbid fear existed in the bakufu that the ronin and hostile daimyo in the southwest would invoke foreign help to create civil disruption and threaten the Tokugawa hegemony. ...read more.
Middle
The need for a military establishment based on western organization and technology, paid from the resources that commerce and industry made available, was necessary to withstand the forces of colonisation that already infected Asia. The Meiji oligarchy, predominantly young junior rank samurai from Satsuma and Choshu, realised progress must be based on imitation so the can "catch up and overtake". The Iwakura Mission in December 1871 (led by Iwakura Tomomi) gathered knowledge from America and Europe to facilitate nationalism, which in turn fed the fires of imperialism. Western ideas ultimately bred insecurity about themselves and although the Restoration was a return to the patterns of antiquity, international competition required the acquisition of modern tools. In 1869, daimyo voluntarily surrendered their lands to the emperor "so that a uniform rule may prevail throughout the empire allowing Japan to rank equally with other nations of the world". The daimyo received compensation and were relieved from supporting their samurai. Disbanded in 1876, samurai were pensioned off, their traditional feudal privilege discontinued and their right to wear distinctive dress and carry two swords was abolished. Their stipends were replaced with government bonds leading to poverty and humiliation at their loss of status though some learnt administration skills and entered the bureaucracy. The liberation of people from feudal restrictions increased geographic mobility. Another blow to samurai pride was the introduction of conscription (1871) implying that any Japanese could acquire the martial virtues regarded for centuries as the attribute of a minority privileged class. A national army raised by universal conscription, in which the peasants provided submissive and disciplined soldiers, indoctrinated the idea of service to the state and reverence for the Emperor. Training and organisation were western style where officers from France and Germany supervised the nucleus of a modern army while the British provided naval instructions. Saigo Takamori of the Satsuma clan proposed that Japan advance her national prestige and provide military employment through a foreign war, particularly against Korea, as means of "giving vent to samurai frustration and energy" (Pyle 1996). ...read more.
Conclusion
Then in 1902 through an Anglo-Japanese alliance came the clearest acknowledgment of Japan's acceptance as an equal by the West. It gave assurance that they could strike at enemies with success. In 1894 Japan engaged in war with China over their mutual interests in Korea, which was won by Japan the following year. The acquisition of Korea was vital as 'dumping ground' for their excess population and resource exploitation. The Sino-Japanese War reinforced their 'might is right' view of international relations and established itself as a colonial power in East Asia. Russian possession of Port Arthur, completion of the Trans-Siberian Railway, development of Vladivostock, and Russian commercial activity in the Korean peninsula resulted a conflict of interests and inevitably the Russ-Japanese War (1904/05). The war in Manchuria was a humiliating defeat for the Russians and the end of Tsarist pretensions in Korea and South Manchuria. The Treaty of Portsmouth in 1905 gave Japan the railway, Liatung and recognised her "paramount interests" in Korea (which it subsequently annexed in 1910). It demonstrated that the policy of modernisation was producing impressive results because for the first time in modern history a non-European nation defeated a European power in a full-scale war. It restored Japan's self-confidence when they turned European skills and ideas against her, elevating Japan to the "peers of western peoples" (Richard Storry). The supply of entrepreneurs, liquidation of feudal restraints, infrastructure of modern state, an indigenous arms industry, a high credit rating and large amounts of capital saved during Edo period gave the Meiji government a superb foundation after taking over these pre-1868 industrial undertakings. However compounded with financial instability, Western belligerence displayed in the Opium Wars initiated rapid changes and forced the Japanese government to emulate their imperialist stance against other nations in the Pacific region. Hence the arrival of the West accelerated predestined technological progress and provided the final impetus towards total modernisation. According to Beasley, "a nationalism rooted in conservative view of society already dissolved much of the tradition it developed to defend. ...read more.
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1950's that required individuals to obtain permission from the police before they could move. Initially this was enforced because of the scarcity of jobs in urban centers. The government's policy changed during the 1970's when urban demand for unskilled workers skyrocketed.
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One industry hit hard by this in was the British Coal industry. 3.1.1. The British Coal Industry The movement of industries away from the UK has been highly visible since the 1980's. The collapse of both the British Coal mining & steel industries can be attributed to the fact that
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Is there any man in his sense that believes that the crowded population of these islands could exist for a single day if we were to cut adrift from the great dependencies which now look to us for protection and which are the natural markets for our trade?
By creating barriers that inhibit these new entrants, all the current companies in the industry will benefit. If you are considering the entry in an industry, you are the phantom competitor. As the new entrant, you must consider how difficult it will be for you to overcome the existing barriers.
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Albudeite -:- The Municipality of Albudeite and Judoc University of Murcia negotiate a collaboration agreement to train young people in the municipality.
The Deputy Mayor of the Municipality of Albudeite, Juan Antonio Cabas Vicente, has received in the consistory to the student association Judoc Junior Communication and Documentation company of the University of Murcia, on the occasion of the proposal of agreement between both entities.
This future agreement will consist of awareness campaigns about the dangers of social networks among the youngest of Albudeite, drug use and complementary training.
In the words of Juan Antonio Cabas Vicente, the signing of this future agreement will amplify the educational offer of Albudeite and, most importantly, will raise awareness among the youth of the municipality about the dangers of drugs and the future consequences of the misuse of social networks in the I presented. | http://portaldealbudeite.com/article.asp?id=117207602 |
The St. Martin Parish Sheriff’s Office is asking for the public’s help in solving a 36-year-old cold case.
On Dec. 5, 1981, at 12:41 a.m., a woman in her mid to late 20s was killed when she was struck by a vehicle on Interstate 10 westbound in Breaux Bridge.
Police were unable to locate any personal identification on the woman or in her belongings.
At the time of her death, she was wearing a gray and black cowl neck Coronet Casuals sweater, Wrangler jeans (size 30-31), white knee socks and red and white Pro-wings jogging shoes.
She had brown hair and brown eyes. She was approximately 5 feet 3 inches tall and weighed 130 to 145 pounds.
The Jane Doe was buried in St. Bernard Cemetery in March 1982, thanks to efforts from Pellerin Funeral Home and St. Bernard Church. Two local residents, Lester Guidry and Sophie Cormier, took it upon themselves to look after her grave until their own deaths. Another unidentified person has now taken over those duties.
Residents have cared for the grave of Jane Doe.(Photo: Submitted photo)
Through the years, law enforcement has made several efforts to identify the woman and notify her next of kin, as well as identify the driver who struck her.
That investigation has yielded some clues.
Based on information from a trucker, authorities believe the woman might be from the Oklahoma City area and was traveling to Texas. She may have been seen at a truck stop at the I-10 Henderson exit at one time.
Police have also learned that some of her clothes were manufactured in Canada. She had one abdominal scar.
The woman had a brown paper bag from Howard’s Supermarket, 950 Ninth Ave., Port Arthur, Texas. The bag included a jar of Noxzema, a piece of chicken wrapped in paper marked ‘Champagne’s’, a pair of size 10 Wrangler misses’ jeans, a short-sleeved plaid Western shirt, various undergarments, maroon and brown knee socks and a medium white nightgown with blue dots.
“We are hoping that through the powers of social media, and the spirit of the Christmas season, that perhaps someone may be able to identify her,” the sheriff’s office said.
Should you have any information as to the identity of this young lady, please contact the St. Martin Parish Sheriff’s Office at (337) 394-3071. | |
Links to pheochromocytoma paraganglioma clinical care guidelines are below.
AAES Guidelines for Adrenalectomy
International Consensus on Initial Screening and Follow-Up of Asymptomatic SDHx Mutation Carriers
CME Activities
Hereditary Pheochromocytoma and Paraganglioma: Strategies for Screening and Surveillance
Metastatic Pheochromocytoma and Paraganglioma
Clinical Vignettes to Ask the Expert: Rare Cases of Pheochromocytoma & Paraganglioma
Hereditary Syndromes with NETs: MEN1/2, VHL Pheochromocytoma /Paraganglioma
Real World Considerations for I-131 MIBG
Identification and Hereditary Management of Pheochromocytoma and Paraganglioma
AACE Master Class: Laboratory Testing in Endocrinology: The Highs and the Lows
Educational Patient Information
Healthcare providers can request a supply of our two informational brochures be sent to you for distribution to your patient population. Send an email to [email protected] with your contact info and address. | https://pheopara.org/clinicians/careguidelines |
Q:
Can this checksum algorithm be improved?
We have a very old, unsupported program which copies files across SMB shares. It has a checksum algorithm to determine if the file contents have changed before copying. The algorithm seems easily fooled -- we've just found an example where two files, identical except a single '1' changing to a '2', return the same checksum. Here's the algorithm:
unsigned long GetFileCheckSum(CString PathFilename)
{
FILE* File;
unsigned long CheckSum = 0;
unsigned long Data = 0;
unsigned long Count = 0;
if ((File = fopen(PathFilename, "rb")) != NULL)
{
while (fread(&Data, 1, sizeof(unsigned long), File) != FALSE)
{
CheckSum ^= Data + ++Count;
Data = 0;
}
fclose(File);
}
return CheckSum;
}
I'm not much of a programmer (I am a sysadmin) but I know an XOR-based checksum is going to be pretty crude. What're the chances of this algorithm returning the same checksum for two files of the same size with different contents? (I'm not expecting an exact answer, "remote" or "quite likely" is fine.)
How could it be improved without a huge performance hit?
Lastly, what's going on with the fread()? I had a quick scan of the documentation but I couldn't figure it out. Is Data being set to each byte of the file in turn? Edit: okay, so it's reading the file into unsigned long (let's assume a 32-bit OS here) chunks. What does each chunk contain? If the contents of the file are abcd, what is the value of Data on the first pass? Is it (in Perl):
(ord('a') << 24) & (ord('b') << 16) & (ord('c') << 8) & ord('d')
A:
MD5 is commonly used to verify the integrity of transfer files. Source code is readily available in c++. It is widely considered to be a fast and accurate algorithm.
See also Robust and fast checksum algorithm?
A:
I'd suggest you take a look at Fletcher's checksum, specifically fletcher-32, which ought to be fairly fast, and detect various things the current XOR chain would not.
A:
You could easily improve the algorithm by using a formula like this one:
Checksum = (Checksum * a + Data * b) + c;
If a, b and c are large primes, this should return good results. After this, rotating (not shifting!) the bits of checksum will further improve it a bit.
Using primes, this is a similar algorithm to that used for Linear congruential generators - it guarantees long periods and good distribution.
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Biomechanics and Gait Analysis presents a comprehensive book on biomechanics that focuses on gait analysis. It is written primarily for biomedical engineering students, professionals and biomechanists with a strong emphasis on medical devices and assistive technology, but is also of interest to clinicians and physiologists. It allows novice readers to acquire the basics of gait analysis, while also helping expert readers update their knowledge. The book covers the most up-to-date acquisition and computational methods and advances in the field. Key topics include muscle mechanics and modeling, motor control and coordination, and measurements and assessments.
This is the go to resource for an understanding of fundamental concepts and how to collect, analyze and interpret data for research, industry, clinical and sport.
- Details the fundamental issues leading to the biomechanical analyses of gait and posture
- Covers the theoretical basis and practical aspects associated with gait analysis
- Presents methods and tools used in the field, including electromyography, signal processing and spectral analysis, amongst others
Please Note: This is an On Demand product, delivery may take up to 11 working days after payment has been received.
1 Introduction to biomechanics 2 Basic biomechanics 3 Advanced biomechanics 4 Why and how we move: the Stickman story 5 Power spectrum and filtering 6 Revisiting a classic: Muscles, Reflexes, and Locomotion by McMahon 7 The basics of gait analysis 8 Gait variability: a theoretical framework for gait analysis and biomechanics 9 Coordination and control: a dynamical systems approach to the analysis of human gait 10 A tutorial on fractal analysis of human movements 11 Future directions in biomechanics: 3D printing
Dr. Nick Stergiou is the Chair of the Department of Biomechanics and the Distinguished Community Research Chair in Biomechanics and Professor as well as the Director of the Biomechanics Research Building and the Center for Research in Human Movement Variability at the University of Nebraska at Omaha where his primary appointment is. He is also a Professor of the Department of Environmental, Agricultural, and Occupational Health of the College of Public Health at the University of Nebraska Medical Center. His research focuses on understanding variability inherent in human movement. Dr. Stergiou's research spans from infant development to older adult fallers, and has impacted training techniques of surgeons and treatment and rehabilitation techniques of pathologies, such as peripheral arterial disease. He has received more 30 million dollars in personal funding from NIH, NASA, NSF, the NIDRR/US Department of Education, and many other agencies and foundations. He has also several inventions and has procured a private donation of $6 million to build the 23,000 square feet Biomechanics Research Building that has opened in August of 2013. This is the first building dedicated to biomechanics research in the world. It is also the first building on his campus exclusively dedicated to research. He is an international authority in the study of Nonlinear Dynamics and has published more than 200 peer reviewed articles. He has written 2 books already, one for CRC Press. | https://www.researchandmarkets.com/reports/4772111/biomechanics-and-gait-analysis |
‘This is really my coming home,’ Amy Denet Deal (formerly Yeung) tells us from her colourful, fabric-strewn studio-shop in Albuquerque’s Old Town. She’s referring to the arc of her life’s journey. Born to a Diné (Navajo) mother but adopted and raised in a non-Native family in Indiana, Amy always knew a bit about her heritage, but only learned the full picture as an adult, after tracking down her biological mother.
At the time, Amy was a high-flying designer based in Los Angeles. She had gone to fashion school, lived around the world and spent years, she says, ‘designing fast fashion clothing destined for landfills’. But something didn’t sit right with all that waste, and she was looking for a new direction. So, one day, she packed up her possessions, waved goodbye to LA and moved to Albuquerque, New Mexico, to work on a new project with purpose.
That project was Orenda Tribe (renamed in 2021 as 4Kinship), Amy’s upcycled and vintage clothing brand that makes ‘sustainably reimagined’ items. ‘We find old things, damaged things, things that are stuck in some warehouse that have been there for 50 years, and then we simply reimagine what they could be through colour or reconfiguration,’ she says. Orenda Tribe’s pieces are eye-catching, bright and unique – from tie-dyed vintage field jackets and turquoise jewellery to Mexican folk dresses and rodeo belt buckles. Amy avoids creating new items from scratch. ‘We’ve got enough stuff already on the planet,’ she says. ‘Let’s just focus on that.’
Not long after launching the brand, Covid-19 ripped through Navajo Nation with a particular ferocity. Indigenous populations have been some of the hardest hit during the pandemic, and Amy has since dedicated huge amounts of time and energy drumming up money through a range of avenues to support her wider community. ‘We’ve raised close to a million in funding, close to a million in masks and close to a million in in-kind donations,’ she says, while pointing out that her former fashion career has been particularly helpful during this time.
‘In my 56 years on the planet, I’ve run a bunch of fashion companies, I’ve been a creative director, a vice- president. But now that I’m back home, I consider myself more a solutionary,’ she says. ‘Learning how to create wealth, how to pivot on a daily basis, how to get things from point A to point B – those are really good skills to come up with solutions during the Covid-19 crisis on Navajo Nation. So, I’m pretty damn happy with the life I had before, because it all got me to this place where I can be of service to those who desperately need help right now.
Indigenous pride
‘We love supporting our fellow Indigenous designers,’ says Amy. Here are five of her top picks.
1. B. Yellowtail – fashion company run by Bethany Yellowtail, a member of the Indigenous Northern Cheyenne Nation.
2. Ginew – the only Native American-owned denim collection, based in Portland, Oregon (and the cover stars of Courier issue 27).
3. OXDX – a Diné-owned clothing label based in Tempe, Arizona.
4. Lilium Orsus – an online fashion shop run by Amy’s daughter Lily, who also models for Orenda Tribe.
5. Thundervoice Hat Co. – a Diné-made, reclaimed, sustainable hat brand.
This article was first published in Courier issue 39, February/March 2021. To purchase the issue or become a subscriber, head to our webshop. | https://mailchimp.com/courier/article/how-i-live-amy-yeung/ |
For readers of Room and The Girls, a dazzling, tenderhearted debut about healing, family, and the exquisite wisdom of children, narrated by a six-year-old boy who reminds us that sometimes the littlest bodies hold the biggest hearts, and the quietest voices speak the loudest.
Squeezed into a coat closet with his classmates and teacher, first grader Zach Taylor can hear gunshots ringing through the halls of his school. A gunman has entered the building, taking nineteen lives and irrevocably changing the very fabric of this close-knit community. While Zach's mother pursues a quest for justice against the shooter's parents, holding them responsible for their son's actions, Zach retreats into his super-secret hideout and loses himself in a world of books and art. Armed with his newfound understanding, and with the optimism and stubbornness only a child could have, Zach sets out on a captivating journey towards healing and forgiveness, determined to help the adults in his life rediscover the universal truths of love and compassion needed to pull them through their darkest hours. Read the reviews.
Get the Reading Group Guide for your Book Club. | https://www.rhiannonnavin.com/only-child |
The Bank of Canada raised its benchmark interest rate by 25 basis points to 4.5 per cent, the highest it’s been since 2007.
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This is the central bank’s eighth consecutive increase in an unprecedented cycle of hiking that began last March when the lending rate stood at 0.25 per cent. Governor Tiff Macklem has been raising rates to rein in decades-high inflation that far-outpaced the bank’s target of two per cent.
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In commentaries on the Bank of Canada’s latest move, most economists said they believe the bank has likely hit the pause button on this cycle of rate increases.
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Here is what they have to say about where rates will go from here.
Benjamin Reitzes, BMO Economics
The bank delivered 25 basis points as expected, and is once again leading the way among global central banks as it’s the first to signal a pause. The shift was a bit more dovish than anticipated, driving Government of Canada bond yields lower and putting the Canadian dollar on the defensive. While policymakers haven’t shut the door on more hikes, the bar for further tightening is quite high. It looks like a March move is off the table barring some wild data. The April policy decision will be more definitive as we’ll have a few employment and CPI (consumer price index) reports by then. BMO’s base case remains that the BoC is on hold through the rest of 2023.
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Charles St-Arnaud, Alberta Central
The key message in today’s decision is that the central bank expects interest rates to remain on hold for some time. However, it clearly states that it remains ready to hike again, if inflation does not ease as expected, continuing to show a strong commitment to restoring price stability. This suggests the BoC could hike later this year if underlying inflationary pressures prove stickier. However, we note that, at this point, the likelihood of further rate hikes is low.
Today’s decision supports our view that the BoC is likely done with its tightening. As such, we believe that interest rates will stay on hold until at least the end of 2023, as we believe that inflation has likely peaked.
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Stephen Brown, Capital Economics
The Bank of Canada accompanied its smaller 25-basis-point hike with new guidance that it intends to hold the policy rate at the current 4.5 per cent while it assesses the impact of the cumulative interest rate increases so far. While the bank did not rule out future rate hikes entirely, the new guidance reinforces our view that the bank’s next move is likely to be a rate cut, albeit not until later this year.
We continue to (believe) that the bank is underestimating how quickly core (inflation) prices will decline, with our forecasts still pointing to a drop in headline inflation to two per cent by the second half of this year. The upshot is that we remain confident that today’s hike will be the last and we see scope for the bank to start cutting interest rates again as soon as the third quarter.
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Andrew Grantham, CIBC Economics
The Bank of Canada hiked rates by a further 25 basis points today, but provided some unexpected guidance that this may be the peak for the current cycle. The 25-basis-point increase, taking the overnight rate to 4.5 per cent, was well anticipated by the consensus. The bank pointed to stronger than expected growth at the end of 2022, a tight labour market and still elevated short-term inflation expectations as reasons for the policy move today. However, the statement also pointed to an easing in the three-month rates of core inflation, and the expectation that overall inflation will come down “significantly” this year due to the energy prices, improvements in supply chains and the lagged effects of higher interest rates.
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Possibly because of greater confidence that inflation is easing, the bank changed its guidance to state that if the economy evolves as it expects then the policy rate will be kept on hold at its current level, although the statement also warned that the bank was willing to raise rates further if needed. The MPR (Monetary Policy Report) projections for GDP growth are set at one per cent this year and 1.8 per cent in 2024, which is little changed relative to October but a bit higher than our own forecasts. Because of that, we suspect that the economy will indeed evolve inline or even a little weaker than the bank suspects, and that today’s hike in interest rates will indeed mark the final one of this cycle.
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Nathan Janzen, RBC Economics
With inflation still running very hot, governor (Tiff) Macklem confirmed that the central bank is more concerned about upside than downside risks to inflation. The pause in interest rate increases is conditional on the outlook evolving as expected and inflation pressures continuing to ease. Still, our GDP forecasts are slightly weaker than the BoC’s projections — we expect a ‘moderate’ recession and higher unemployment this year — so we don’t expect they’ll need to restart. Odds are that this was the last interest rate increase of this hiking cycle and we look for the BoC to now leave the overnight rate at the current 4.5 per cent level for the rest of this year.
James Orlando, TD Economics
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The BoC’s first meeting of 2023 looks to be the last in which it will raise its policy rate. Heading into today, the bank had communicated that it could go either way with today’s decision — deciding between a final hike or a pause. Given the robustness of consumer spending and employment trends, the BoC clearly felt it needed this final hike to solidify the turn in economic momentum.
Looking at the bank’s forecast, the economy is set for a consumer led slowdown, with GDP likely to “stall through the middle of 2023.” Greater conviction in this has also led the BoC to cut its inflation forecast. With the belief that the economy is on the path to price stability, the BoC can now step to the sidelines and let its restrictive policy filter through the economy. Though it does have the option to hike again should inflation prove unco-operative, we are expecting it to hold rates at this level for most of 2023, before cutting at the end of the year to drive a better balance between interest rates being too far in restrictive territory and a weakening economy.
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Tom O’Gorman, Franklin Templeton Canada
As expected, the Bank of Canada raised interest rates by 25 basis points. This met our and the markets’ expectations including the fact that (the Bank of Canada) formally announced a pause. What happens next will depend on how inflation trends going forward. While they could be forced to hike further should inflation not continue to move towards their two per cent target, we do not expect them to cut anytime soon.
Raising rates is a blunt tool with long lags in terms of the impact to the real economy. Recent job numbers have been very strong but there are now signs that the increase in rates is slowing economic activity. The markets may have gotten ahead of themselves by anticipating a potential rate cut in 2023. We don’t expect the BoC to declare victory until they achieve their goals. | https://googlepublishers.com/todays-hike-will-be-the-last-economists-say-bank-of-canada-to-put-interest-rates-on-pause/ |
It's the wicked tradition of springtime -- setting the clock forward. Spring forward? For most of us, it's more like stumbling sideways into daylight-saving time.
This year, the joy occurs before winter has a good chance to thaw. Prepare yourself -- it's this weekend when we reset the alarm clock.
Daylight-saving time begins at 2 a.m. Sunday morning. Losing an hour's sleep isn't easy for an already sleep-deprived nation.
You know the drill: On Monday morning, you hit the snooze too many times, stagger out of bed, grab an extra cup of coffee -- and push yourself into summer mode. But take heart. Those first few mornings don't have to be dire, if you plan ahead. A few strategic steps will help your body adjust quite easily, according to snooze experts with the American Academy of Sleep Medicine.
"Come Monday morning, you might be the only bright-eyed and bushy-tailed employee at the office," said Ralph Downey, III, PhD, medical director of the Sleep Disorders Center at Loma Linda University Medical Center in Loma Linda, Calif., in a news release.
Mother Nature vs. the Alarm Clock
Here's what you're up against: The advent of daylight-saving time is a double-whammy for the human body, says David Glass, PhD, a biological sciences professor at Kent State University in Ohio.
"In the spring, we not only have to get up an hour early -- but we're also fighting the extra 20 or 30 minutes of sleep our bodies naturally want every day," he tells WebMD. "In the fall, the time change is more in line with our internal clock." Are you sabotaging your sleep? Take the quiz.
The body is wired with a sleep-wake cycle that advances a bit every 24 hours, Glass explains. "If I put you in a dimly lit cave, where you didn't know what time it was, you would get up 20 to 30 minutes later every day." Daylight reins in this natural tendency because daylight controls melatonin, a hormone made by the brain that helps regulate our sleep cycles.
But there's more: We've also got "Sunday night syndrome" working against us, says Kenneth Sassower, MD, a staff neurologist in the Sleep Disorders Unit at Massachusetts General Hospital, and neurology instructor at Harvard Medical School.
"Studies show that Sunday nights are the worst nights to fall asleep, even when it's not daylight-saving time," Sassower tells WebMD. "If you've stayed up late, slept in all weekend, by Monday morning you're exhausted. Your body clock is disrupted, so you aren't ready to get up when the alarm goes off."
Continued
Survival Tips
How to offset Monday-morning drag?
- Prepare yourself! Make the time change incrementally beforehand. "Set your alarm clock 15 minutes earlier and earlier for five days or so," Sassower suggests. "It helps. When the time change hits, you're already there. It's the same advice I give to people who are traveling out of the country."
Indeed, daylight-saving time is much like jet lag -- "the older you are, the more difficulty you will have," says Dennis H. Nicholson, MD, director of Sleep Disorders Center at Pomona Valley Hospital Medical Center in California. "It will take one to two days to reprogram."
On Saturday:
- Around midday, get some vigorous exercise. "Exercise helps advance the body clock, just as bright light exposure does," says Glass.
- Don't exercise too late in the day. "Exercise raises your body temperature," explains Nicholson. "People get sleepy as their body temperature goes down, not when it's elevated."
Sunday morning:
- Get up at your regular time -- whether you had a good night's sleep or not. "Don't let yourself sleep in," says Nicholson. "If you stay in bed, your body will never adjust."
- Spend an hour or more outside, preferably in the sunshine. "That's hard for folks to do, but it's very important," Glass says. "Sunlight is especially helpful in advancing your body clock."
- Take a morning walk. After a short night, walking is an easy exercise that will help advance your body clock, says Glass.
Good "sleep hygiene" also helps:
- Don't eat a heavy meal before bedtime.
- Don't drink a lot of caffeine or alcohol.
- Don't nap during the day, or at least keep it brief -- 10 to 15 minutes.
- Stop working on any task an hour before bedtime to calm down.
- Don't discuss emotional issues at bedtime.
- Make sure your sleep environment is comfortable.
- Don't turn lights on at night. Use a small night-light instead.
What About Melatonin?
Taking a melatonin supplement (1 to 3 milligrams) one hour before bedtime might also ease the time change, Glass suggests. However, studies of melatonin have had mixed results.
Melatonin supplements are sold over the counter as dietary supplements and aren't held to the same FDA standards as prescription drugs. Some studies showed that supplements don't help with sleep problems, but others suggested that melatonin might ease jet lag and have a modest effect with insomnia.
Carefully timed daylight exposure works just as well -- helping regulate melatonin that the body naturally produces, Nicholson explains. "When we're exposed to daylight early in the day, the release of melatonin is suppressed. As daylight dims in the evening, melatonin is released. It's daylight that [controls] the sleep cycle."
If you continue having difficulty adjusting to daylight-saving time, call an accredited sleep center or a sleep specialist, Nicholson adds. | https://www.webmd.com/sleep-disorders/features/ease-your-way-to-daylight-saving-time |
Statement following today's meeting of the F1 Commission
The first meeting of the F1 Commission in 2021 was held today in an online format. The FIA and Formula 1 confirm that positive discussions took place on a number of key topics relating to future Sporting, Technical and Financial Regulations as well as the future direction of Power Unit regulations.
2021 Calendar
The group was updated on the "TBC" space on the current version of the 2021 calendar. It is the intention of Formula 1 to fill the position with a race at Portimao in Portugal on the dates already held in the calendar. The final agreement is still subject to contract with the promoter.
Robust COVID-19 protocols enabled Formula 1 to run 17 events in 2020 and will enable us to run a World Championship again in 2021. While changing circumstances may require flexibility, the FIA and Formula 1 are working at all levels from government to local organisation to ensure that the calendar goes ahead as planned.
Proposed Regulation Changes
In order to permit a more equitable distribution of tyre testing during 2021, and taking into consideration the challenges presented to this programme due to the move to 18" tyres and the COVID-19 Pandemic, the FIA, following the request of Pirelli, proposed to increase the number test days allocated for such purpose from 25 to 30. This proposal was accepted unanimously.
Overview of key next Formula 1 generation car and PU objectives and proposed anticipation to 2025
In a significant development for the sport that reflects the unity and collaborative spirit between the FIA, Formula 1 and the teams, a vote on the freeze of Power Unit development was undertaken during the meeting, and the proposal was unanimously agreed by all teams and Power Unit Manufacturers. As such, engine development will be frozen from the start of 2022.
A high-level working group has been established including current and potential Power Unit manufacturers and fuel suppliers.
The definition of the objectives for the next generation of F1 car and Power Unit is of the utmost importance to the FIA and Formula 1, and together with teams and Power Unit manufacturers, there is strong alignment on the overall goals - particularly the need to reduce cost and reach carbon neutrality.
The key objectives for the 2025 Power Unit are:
Environmental Sustainability and social and automotive relevance
Fully sustainable fuel
Creating a powerful and emotive Power Unit
Significant cost reduction
Attractiveness to new Power Unit manufacturers
Environmental Accreditation
The FIA is pleased that Formula 1 and a number of the teams have achieved the highest level of FIA Environmental Accreditation. This was discussed during the Commission, with a glide path outlined for all teams to reach the highest level of Accreditation over the coming years, with this goal being integrated into the Formula 1 Sporting Regulations.
Cost control
As 2021 sees the introduction of a cost cap for the first time in Formula 1, various topics around controlling costs and how this overall objective can be achieved over the coming years were tabled during the meeting.
As part of this, a working group will be created - including the drivers themselves - to discuss the topic of driver and senior team management contracts.
Race weekend format
All teams recognised the major importance of engaging fans in new and innovative ways to ensure an even more exciting weekend format. There was, therefore, broad support from all parties for a new qualifying format at some races, and a working group has been tasked with creating a complete plan with the aim to reach a final decision before the start for the 2021 Championship.
All regulatory changes are subject to approval by the World Motor Sport Council. | |
Pentacle Co-Curates Dance At The Brooklyn Botanic Garden, Starting July 14
While the Garden remains closed, we are sharing special performances by artists whose work connects plants, people, and the planet. In a time when we cannot gather together in the Garden, we are grateful to bring the voices and work of these artists to you virtually.
Each performance will premiere live on BBG's Facebook page, and afterward remain available here and on Facebook to enjoy and share.
Co-Curatorial Partner for all four dance events: Pentacle
Raphael Xavier & Gary Dourdan
Tuesday, July 14, 2020 | 7 p.m. EDT
Performed in front of the weeping beech near Oak Circle in June 2020.
In this work, Xavier explores the intuitive concept of listening, assessing, and adapting through improvisational street dance. Acoustic sound and movement drawn from environment and experience intertwine to create an ambient performance that reflects on growth and maturity. Dourdan accompanies on flute, guitar, and beats.
Raphael Xavier and Gary Dourdan Biographies
A multitalented artist who has forged an exceptional approach to improvisation, Raphael Xavier is a self-taught hip-hop dancer and has practiced breaking since 1983. His work draws upon B-boy culture and his background as a photographer and musical artist. As a sound designer, his understanding of movement and musicality allows him to structure beats, noises, and sounds into captivating music that draws upon emotion and coincides with his choreography. Xavier started choreographing dance with the Brandywine School of Ballet in 1995 and spent years with the renowned hip-hop dance company Rennie Harris Puremovement; he's been recognized as a Pew Fellow, Guggenheim Fellow, and United States Artist Fellow and his solo and ensemble choreographic dance works have been performed worldwide. Originally from Wilmington, Delaware, Xavier currently lives in Philadelphia and is a guest lecturer in dance at Princeton University.
Born in Philadelphia, Gary Dourdan (Kolâde Kouyaté) evolved in a culturally rich environment: His uncle was a saxophonist who played with Platinum Hook and Sister Sledge, his brother was a DJ on WRTI radio, and his father encouraged his children to explore literature and the arts. A graduate of the prestigious Freedom Theatre Performing Arts training program, Dourdan relocated to New York where he spearheaded two musical projects, the eclectic act Rent Money and rock band New Congregation, and also sang for the Bell Café Band. Dourdan is an actor who has appeared on CSI and in the film Alien Resurrection and has performed at the Emmy Awards with Macy Gray and on stage with hip-hop artist DMC at the Live 8 concert in Ontario, Canada. He works closely with multiple nonprofit organizations including the Heart Fund, aimed at bringing "state-of-the-heart" surgery to children in need in areas such as Haiti, India, and West Africa.
Company SBB: Unnatural Contradictions
Tuesday, July 21, 2020 | 7 p.m. EDT
Performed in the Osborne and Woodland Gardens in June 2020.
This film study by Stefanie Batten Bland's Company SBB offers the point of view of a Garden visitor who happens upon three installation solos by Jennifer Payán, Yeman Brown, and Bland herself paying homage to Breonna Taylor, Ahmaud Arbery, and George Floyd. Costumes by Shane Ballard and music by long-term SBB collaborator Paul Damian Hogan.
Stefanie Batten Bland Artist Biography
Jerome Robbins awardee Stefanie Batten Bland is an interdisciplinary global artist who interrogates contemporary and historical culture with work situated at the intersection of dance-theatre and installation. She created her Company SBB in Paris in 2008 and established it in New York City in 2011, when she was in residency at Baryshnikov Arts Center. Known for her unique visual and movement aesthetic and commissioned by numerous global fashion and lifestyle companies, Batten Bland also directs dance cinema films and recently created a virtual global performance for EU Day and a physical performance installation related to climate change, both for the United Nations.
SBB received her MFA in interdisciplinary arts from Goddard College and lives in SoHo with her family. A 2019 fellow for New York University's Center for the Ballet Arts, in 2019 she was named a choreographer for American Ballet Theatre's inaugural Women's Movement initiative and premiered her recent work "Look Who's Coming To Dinner" at La MaMa for FIAF's Crossing the Line Festival. Company SBB will celebrate its tenth U.S. anniversary in the 2021-2022 season with historic and new works.
Dance Heginbotham: You Look Like a Fun Guy
Tuesday, July 28, 2020 | 7 p.m. EDT
Performed on Cherry Esplanade in June 2020.
You Look Like a Fun Guy is based broadly on avant-garde composer John Cage's methods of creation and his commitment to mycology. Dancers Courtney Lopes and Mykel Marai Nairne share a series of identical, short movement phrases; the phrases are performed in an order randomly selected for each performer.
Dance Heginbotham Company Biography
Dance Heginbotham (DH) is a New York-based contemporary dance company committed to supporting, producing, and sustaining the work of choreographer John Heginbotham. With an emphasis on collaboration, DH enriches national and international communities with its unique blend of inventive, thoughtful, and rigorous dance theater works and a mission to move people through dance.
The company is celebrated for its vibrant athleticism, humor, and theatricality. Well-known for his 14-year tenure as a dancer with Mark Morris Dance Group, artistic director John Heginbotham creates work known for its "tight formal structure and inventive movement, bolstered by a disarming wit and strangeness" (New Yorker). In recognition of his unique artistic vision, Heginbotham received the 2014 Jacob's Pillow Dance Award and a 2018 Guggenheim Fellowship.
Development of You Look Like a Fun Guy has been generously supported through creative residencies at the National Center for Choreography at the University of Akron (NCCAkron), where Heginbotham is currently an artist in residence; and White Oak, funded by the Howard Gilman Foundation.
Inspiration for this work: In 1959, John Cage correctly listed the 24 white-spored Agaricus species to win top prize on the Italian game show Lascia o raddoppia. He gave most of the winnings to his artistic and life partner, choreographer Merce Cunningham, specifically for the purchase of a VW bus to transport members of the young Merce Cunningham Dance Company during its early performance tours.
Urban Bush Women: The Artist Journal
Tuesday, August 4, 2020 | 7 p.m. EDT
Performed in the Water Garden in June 2020.
In this sharing of active research, company members Courtney Cook and Love Muwwakkil explore practices for navigating discomfort, healing, and visioning onward that have charted the survival and progression of people of color. Engaging in "Listening, Re-Membering, and Restoring," the dancers cite their bodies, their environment, and the Garden's collection as reservoirs rich with experience and memory.
Urban Bush Women (UBW) galvanizes artists, activists, audiences, and communities through performances, artist development, education, and community engagement. With a groundbreaking performance ensemble at its core and ongoing initiatives including the Summer Leadership Institute (SLI), BOLD (Builders, Organizers & Leaders through Dance), and the Choreographic Center, UBW continues to strengthen the overall ecology of the arts by promoting artistic legacies, projecting the voices of the underheard and people of color, bringing attention to and addressing issues of equity in the dance field and throughout the United States, and providing platforms for culturally and socially relevant experimental art makers. | https://www.broadwayworld.com/bwwdance/article/Pentacle-Co-Curates-Dance-At-The-Brooklyn-Botanic-Garden-Starting-July-14-20200713 |
Authors deserve fairness, not charity
Living under lockdown, with social distancing, was once unimaginable but is now the new normal for most of us. Cinema, theatres and film production were among the first cultural activities to shut down, and they will be the last in line to be up and running again. However, the cultural sector continues to show resilience to mobilise, adapt and outlive the COVID-19 crisis.
The members of the SAA (collective management organisations) are working to ensure that authors’ royalties for the exploitation of their works are distributed on time or even in advance, social funds are mobilised and campaigns have been developed to advocate for authors not to be left behind by governments’ emergency plans. By now, most EU Member States have set funds and support for employment, social security, business continuity and tax alleviations. The European Audiovisual Observatory provides an overview of the national measures in place for the audiovisual sector. However, not all measures are adapted to the specific situation of freelancers in the TV and film sector (as shown by a recent UK study). The European Parliament acknowledged that many self-employed cultural workers, "(…) were struggling well before the outbreak" and have been particularly hit. Indeed, “[their] income streams have been unexpectedly reduced to zero and [they] now have little or no support from the social system” (resolution).
Creators, their representative organisations and policymakers are coming together in different online forums to discuss the COVID-19 impact on the cultural and creative sectors and its policy responses. The conversations are slowly moving from talks about the threats the virus poses, to ways to seize the “opportunities”. Professor Pier Luigi Sacco, spoke at an OECD webinar about the fast accelerating digitalisation the crisis has brought, resulting in new forms of markets and business models and the possibility to redesign the global content ecosystem. And, who are better placed to imagine new innovative solutions than creators themselves, said Philippe Kern, Director of KEA. The cultural and creative sectors play a key role in contributing to the COVID-19 exit-strategy. The sectors have the ability to mobilise the population, create an intercultural dialogue and build trust, concluded Kern.
In COVID-19 confinement, more than ever, people turn to online culture, consuming films, music and other arts. Musicians stream concerts from their home, museums open their exhibitions online and video-on-demand platforms extend their catalogues and release movies that have not premiered in cinemas. While many realised the indispensable value of culture in times of crisis, its online availability for free is taken for granted and the devastating socio-economic impact on creators is overlooked. Jean-Michel Jarre, President of CISAC, made the point at a UNESCO webinar that paying creators and artists for the exploitation of their works should not be based on charity, but on the contrary, it should build on the principle of fairness as set out in the EU Directive on Copyright in the Digital Single Market. Filmmakers’ right to remuneration for the exploitation of their works by video-on-demand platforms is unfortunately not an established principle in too many countries. The increased consumption of films online during the lockdown does not compensate authors for the lost work as it does not generate any revenue to many authors. Netflix’s aid packages to the film sector in some European countries come across as generous, yet is it proportionate to the companies record-high market value and monthly earning of almost €1 billion? (Cineuropa).
The crisis has unveiled the weak protection of authors online, but it has also shown that the cultural and creative sectors are better equipped during the COVID-19 crisis in countries with a robust copyright law and support system. With a solid infrastructure in place, creators can play a vital role “in the healing and recovery process that our societies are going to face in the months and years ahead”. This is a message from 100 organisations to decision-makers that the SAA signed.
Stay safe! | https://www.saa-authors.eu/en/blog/639-authors-deserve-fairness-not-charity |
Learn about the many mission-related programs and support opportunities the JSN provides, in particular with regard to engaging in the Spiritual Exercises.
Mission Formation
Tom Noonan
Director of Ignatian Mission Formation
If you have questions related to mission formation, please contact me.
Updated January 18, 2022
COVID-19 Protocols and Guidelines:
JSN Vaccination Protocol for In-Person Events: In consideration of the common good, only those who are fully vaccinated and have received a CDC or NACI approved booster shot against COVID-19 may attend JSN events and meetings. Given the many places from which our constituents come, we believe this is a faithful response to Jesus’ command to “love our neighbor as ourselves.”
Health Guidelines: The JSN will continue to follow CDC, local area and location-specific guidelines when hosting any event. Specific parameters regarding health and safety will be provided immediately prior to the event as well as on the JSN web site.
Programs
Colloquium
The Colloquium on Jesuit Education is a triennial meeting of Ignatian educators focusing on a pertinent current issue in Jesuit secondary education. To learn more, please visit our Colloquium page.
Ignatian Colleagues Gatherings
In place of the prior sequence of specific role-based cohort meetings we have decided to move toward a broader, thematic structure of conference-wide Ignatian Colleagues Gatherings across a two-year cycle in the areas of academics, co-curricular and student life, leadership, mission and ministry, and school operations and support.
To learn more about our new model for network-wide gatherings, please see our Ignatian Colleagues Gatherings page.
Cohort Connections
The JSN provides ongoing opportunities by which members of our network can engage in synchronous and asynchronous conversations. For more information, please visit our Cohort Connections page.
JSN Spiritual Exercises Mini Grant Program
The purpose of the Jesuit Schools Network (JSN) Spiritual Exercises Mini Grant Program is to foster meaningful engagement in the Spiritual Exercises of Saint Ignatius Loyola by those serving in JSN member schools. The JSN commits to collaborating with member schools as they form women and men to serve a school’s Jesuit mission.
Eligibility
Regular full-time employees of JSN member schools are eligible for financial support as they participate in formative experiences of the Spiritual Exercises. Employees who participate in retreats, seminars, spiritual direction, etc., specific to the Spiritual Exercises may receive up to $300 or ¼ the cost of each session per fiscal year (after applied school funding support as defined by a policy such as the school’s Professional Development Plan or Employee Handbook).
How to Apply
Personal Statement – A candidate should submit a Personal Statement that includes a reflection on how the proposed experience would enhance his or her current or desired ministry in Jesuit secondary or presecondary education. In preparing the statement, he or she should prayerfully consider the Universal Apostolic Preferences, particularly UAP01, “Showing the way to God,” and include that reflection within the essay. The statement should not exceed 350 words.
Experience Description – Candidates should develop and submit a brief description of the proposed experience, including a summary of planned activity, the benefit of the experience to the individual and to the school, and the expected outcomes.
Letter of Good Standing – The member school shall attest to a participant’s employment status in a Letter of Good Standing.
Application Materials – Application materials are received on a rolling basis. Because funds are limited, these mini grants will be awarded in the order in which application materials are received. We cannot guarantee that all requests will be funded.
Please send all materials and letters electronically to:
The JSN Spiritual Exercises Mini Grant Program
c/o Tom Noonan
Director of Ignatian Mission Formation
Email: [email protected]
Resources
The Jesuit Schools Network continuously compiles and curates the following resources related to Mission Formation:
Tom Noonan
Director of Ignatian Mission Formation
If you have questions related to our Colloquium, please contact me. | https://jesuitschoolsnetwork.org/mission-formation/ |
Objective: This study explored using person-centered scheduling with telepsychiatry for rural community geriatric patients. Quantitative research approaches were used to determine the level of satisfaction participants experienced with person-centered scheduling and geriatric telepsychiatry. Method: Quantitative data were collected by using the Zung Self-Rating Anxiety Scale (SAS) before scheduling the first appointment and to assess the intervention’s effectiveness after the telepsychiatry session. Results: Person-centered telepsychiatry scheduling decreased geriatric patients’ anxiety as evidenced by Zung SAS scores. Conclusion: Older adult patients saw telepsychiatry as a viable means of treatment. Future research with geriatrics from different regions is needed.
Implications and future directions include exploring patient responses from different regions such as rural areas vs. urban metropolitan areas. Qualitative data from different age categories, 65 to 75 and over 75 may yield different perspectives. The results of this study are consistent with the benefits of person-centered approaches and the benefits of telepsychiatry.
Key words: Person-centered scheduling, geriatrics, telepsychiatry
Recommended Citation
Long, Jody G.; Wilkerson, Patricia A.; Taylor, Evi; Hall, John H.; and Peters, Christopher
(2018)
"Using Person-centered Scheduling with Geriatric Patients to Reduce Anxiety with Telepsychiatry,"
Contemporary Rural Social Work Journal: Vol. 10
:
No.
1
, Article 12. | https://digitalcommons.murraystate.edu/crsw/vol10/iss1/12/ |
Author of A New Chapter Dana Mrkich is a dynamic Energy Intuitive, Writer, Teacher and Speaker. Having received visions and messages about the era-2012 ‘Shift’ from a young age, Dana’s life focus has been to help people remember who they really are so that together we can create the best possible reality for ourselves and the planet. A speaker at the 2010 Star Knowledge Conference in New Mexico alongside Native American and Mayan Elders, and a panellist at the 2005 World Peace Congress in Italy, Dana’s passion is to share what she knows about these times: we are not experiencing the end of the world, we are experiencing the end of a cycle and the beginning of a new one. Dana offers her empowering Soul Sessions via email (written downloads), phone and Skype to clients all around the world, containing the guidance and healing insights you most need as you step further into your true self and best possible reality. Dana is the creator of e-Course Create a Life you Love (which includes her ChakraShift Meditations). Her globally popular Monthly Visions are featured on Spirit Library. A passionate advocate for truth-based, empowering media, Dana is the former host of radio shows TruthSeeker and Visioning the Dream Awake. Dana holds regular talks and workshops in Australia and the US, and is a popular radio guest on shows in many countries. Dana is committed to humanity’s conscious evolution, and the merging of consciousness with action, in all aspects of social and global life. She has travelled to over 35 countries, visiting sacred sites all over the world, and has a Bachelor’s Degree in Communications majoring in Social Inquiry & Media Production and Journalism.
http://www.danamrkich.com/
Sorry we couldn't complete your registration. Please try again. | http://www.blogtalkradio.com/everydayconnection/2012/07/25/dana-mrkich--a-new-chapter |
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What do a marathon and a divorce have in common? Sounds like the start to a corny joke, am I right? I have run marathons and gone through a divorce, so I feel like I can rightfully use the phrase, “It’s a marathon, not a sprint”.
The pain and discomfort, physically and emotionally—is real for both. The reality is that they are both a long journey, and you can’t get to the end by trying to rush to the finish.
A marathon (that is 26.2 miles or 42.2 kilometers) is something you embark upon voluntarily to challenge yourself. A divorce, however, is not usually a choice that you think you’ll ever be making the day you are saying your wedding vows.
Who gets married telling themselves that in five, ten or twenty years that they want to challenge themselves to get divorced to see how hard it will be and how accomplished they will feel at the end of it?
Continuing with that “challenge yourself” theme, challenging yourself to run a marathon or try something new is admirable. However, not many of us seek out experiences in life that are going to challenge our spirit, cause anxiety, change our life plans unexpectedly, or break our hearts.
Even if divorce is something that you initiated or wanted, it doesn’t mean it’s easy. It’s an end of something you entered into with a person for a reason. Grief will still take place; tears will still be shed, and then healing will begin. That is when the real marathon begins, and you begin to write your next chapter.
You have aid stations to help you get through a marathon, but there aren’t necessarily aid stations to help you get through a divorce…or are there?
I made this statement in the blog I wrote after I ran the NYC marathon in 2019:
You have probably heard people referring to life as a marathon. “Just take it slow, it’s a marathon not a sprint”. Because I have both run a marathon and I’m going through life, I don’t feel like you can accurately compare a “real” marathon to the journey of life. When you are going through life, you don’t have aid stations along the way, you don’t have people cheering you on and ringing cow bells to encourage you to keep going.”
I said running a marathon shouldn’t be compared to life (and I was particularly referring to divorce at that time) BUT, I am going to retract that statement I made a mere two years ago because honestly—I DO think there are similarities.
Not only comparing a divorce to a marathon, but any challenging situation you are presented with in life. You don’t sprint to the finish, you take it slow. You go through the shit along the way, wade through the ugly, learn and grow and holy shit…the feeling of accomplishment after the journey—will leave you speechless.
I also said two years ago that unlike with a marathon, in life you don’t have aid stations along the way or people cheering you on. Well, I call bullshit on myself! Yes, you do!
Aid stations can be a metaphor for you taking time to practice self-care and self-love. And the cheers—if you are seeking therapy, have a life coach or even a good network of family and friend that are supporting you, THEY are your cheering section.
In a marathon you can skip an aid station or tune out the cheers. In life, it’s your choice to run past those metaphorical aid stations or listen to the metaphorical cheers. But I encourage you to stop at them, listen to them, they make the “run” a tiny bit easier.
This weekend I just ran the New York Marathon for the 4th time!
Two years ago when I ran this same marathon, it was one year post divorce and the emotions that it brought about were intense because I was still running my “divorce marathon”. This year, the emotions were even more intense, but the tears that were flowing were tears of happiness not sadness.
If you compare the divorce recovery process or navigating any challenging time in your life to a marathon, remember that people run faster than others. Many times we want to speed up and “run” faster to get to that finish line of our pain or grief. But just like in a marathon, if you start off too fast, you are most likely going to lose your steam…you have to pace yourself.
Take your time when you are navigating your next chapter. One foot in front of the other, one day at a time. Just like the feeling of accomplishment you would have if you crossed the finish line of a marathon, you have that same feeling of accomplishment when you overcome a hardship in your life! Slow and steady my friend.
Remember, You Got This!
XOXO
If you would like to learn more about working with me, CLICK HERE. | https://goodthingsaregonnacome.com/its-a-marathon-not-a-sprint/ |
Is Florida blind to pipelines?
Dan Christensen, Miami Herald, 2 October 2015,
Firm says Gov. Scott’s stake in pipeline project is ‘irrelevant’,:
Environmental groups seek to block $3 billion project
Coalition seeks all correspondences from Scott, his office
Staff says governor’s blind trust shields him from conflict
Lawyers for a company that wants to build a natural gas pipeline in North Florida have told a judge that environmental opponents should be blocked from “presenting evidence or argument” about Gov. Rick Scott’s financial interest in the company.
“Such evidence is irrelevant and the admission of which would be unfairly prejudicial,” attorneys for Sabal Trail Transmission, LLC told Administrative Law Judge Bram D.E. Canter in last week’s filing.
The Florida Department of Environmental Protection (DEP), which is backing the $3 billion Sabal Trail pipeline, filed a similar argument earlier this month when it called the “allegation regarding a conflict of interest … not material to this proceeding.”
Sabal Trail is a joint venture of Spectra Energy Partners and Florida Power & Light parent NextEra Energy. Spectra Energy’s investors have included Scott.
The nonprofit WWALS Watershed Coalition filed for an administrative hearing on Sept. 3 after state regulators said they intended to award Sabal Trail both a permit and rights to drill under riverbeds in order to build the 267-mile stretch of 36-inch underground pipeline in Florida. The nonprofit has asked the judge to deny the permit.
Among the documents WWALS has asked the DEP to produce are all communications from Scott or his executive office about the Sabal Trail project since the governor took office.
The case is proceeding quickly. On Sept. 21, Sabal Trail’s attorneys at the Tallahassee law firm Hopping Green & Sams invoked a law that Scott signed in May 2013 that speeds up the permitting process for the construction of interstate natural gas pipelines.
Under the law, challenges to new pipelines must be heard within 30 days “regardless of whether the parties agree to the summary proceedings.” Before Scott signed the law, natural gas pipelines were specifically excluded from consideration for expedited review.
As the story says, the hearing in WWALS v. Sabal Trail & FL-DEP will be 19-22 October 2015 in Jasper (or Live Oak). You can help WWALS in this legal case by contributing to our IndieGoGo crowdfunding campaign.
And you can still ecomment to FERC, ask the Army Corps of Engineers for a hearing, and oppose the pipeline in other ways.
WWALS’s petition contends the pipeline poses threats to native wildlife, including threatened species, and argues that proposed drilling into the area’s karst limestone to lay pipe could cause new sinkholes to form.
The group, an affiliate of the Waterkeeper Alliance, also raises a potentially explosive political issue: Whether Gov. Scott, as a trustee of the state board that owns the land beneath the rivers, has a conflict of interest because of his investments in Spectra Energy and Williams Company, owner of the Transco pipeline from which Sabal Trail plans to obtain its gas.
“The governor and other public officials are prohibited by state ethics laws from owning stock in businesses subject to their regulation or that do business with state agencies,” the group’s petition says.
The governor’s blind trust is supposed to shield him, and the public, from conflicts of interest by putting his investments under the control of an independent trustee, and keeping them secret. Public officers who put their assets in a qualified blind trust receive immunity from prohibited conflicts of interest.
As FloridaBulldog.org has reported, however, Scott’s trustee is Hollow Brook Wealth Management, run by his longtime business crony Alan Baazar. The blind trust also has proved ineffective in preventing public disclosure of Scott’s assets.
Moreover, Florida’s qualified blind trust law, which Scott signed into law in May 2013, does not contemplate the unique situation that has transpired as Scott has used the law.
The story goes into more detail on all these points.
-jsq
You can join this fun and work by becoming a WWALS member today!
Short Link: | https://wwals.net/2015/10/06/sabal-trail-says-fl-gov-scotts-stake-in-pipeline-project-is-irrelevant/ |
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