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The middle name is the forgotten, underused part of a person’s identity. It can be used to convey your family heritage or your personal tastes.
The following list has over 120 interesting and creative ideas for what you could use as your son Paxton’s middle name. Middle names are an opportunity to create something unique and meaningful for your child.
What Does Paxton Mean?
Paxton was once a surname, but it has evolved into a boy’s given name over time. In the English language, Paxton is a boy’s name of Latin origin that means “peace town.”
It comes from the phrase Poecc’s settlement a medieval English given phrase that translates as “peaceful farm.” So it means peaceful one, or from the town of peace.
How do You Pronounce Paxton?
Paxton is pronounced p-acks-tuhn
Famous People Called Paxton
- Paxton Baker is a successful businessman from the United States.
- Paxton Fielies is a South African singer and composer who was crowned the winner of Season 13 of Idols.
- Paxton Lynch (born 1994), American football player
- Paxton Mills (1948–2001), American radio broadcaster
- Paxton Whitehead (born 1937), English actor Paxton Whitehead (born 1937), American football player
How Popular is Paxton as a name?
Paxton first appeared in the US popular name charts in 1921 but never really took off in the 20th century, then remaining bottom of the list until the year 2006 when it began to rise dramatically.
Since 2008 it’s been in the top 500 boys names averaging around the 200th spot. In 2020 it ranked number 262.
How to Choose the Perfect Middle Name for Paxton
The most appropriate middle name for Paxton is the one that blends nicely with his first and last names. For example, choose the middle names you want for Paxton from the list below and jot them down on a piece of paper.
Say the names out loud with Paxton at the start. For example, Paxton Preston. Remove any names that are strange, bizarre, or difficult to pronounce.
Step 2 should be repeated, but this time include the last name and listen to how it sounds. Delete any further names that don’t sound suitable.
Examine the initials of the remaining names and identify the ones that work particularly well together. Make certain that it does not sound absurd or harsh. Consider looking for ones that can be given nicknames when they are grouped together.
By the time you are finished with steps 1-4, you will have cut down the selection to a manageable amount of options that will assist you in deciding on the ideal middle name for Paxton.
120 Best Middle Names for Paxton
First Name for Middle Name Paxton
If you have already decided on Paxton as the middle name you might be looking for the perfect first name as Paxton makes a great middle name. We have our top 20 here!
10 French Middle Names to Go with Paxton
10 Irish Middle Names for Paxton
If you like Irish girls names check out our post.
10 Spanish Middle Names to Go with Paxton
10 Italian Middle Names for Paxton
10 Middle Names for Paxton with 1 Syllable
10 Middle Names for Paxton with 2 Syllables
10 Middle Names for Paxton with 3 Syllables
Middle Names for Paxton with the Same Initial
Middle Names for Paxton Starting with A Vowel
Nicknames for Paxton
Paxton is often given the following nicknames:
- Pax
- Ton
- Tonny
- Ax
- Paxie
- Pac
Different Ways to Spell Paxton
- Paxten
- Paxtyn
- Paxtin
- Packston
Similar Sounding Names to Paxton
- Pax
- Daxton
- Preston
- Peyton
- Braxton
- Jaxson
- Princeton
- Payton
Sister Names for Paxton
Brother Names for Paxton
Why give a middle name for Paxton?
There are many reasons to give your child a middle name. A middle name can honor a loved one who has passed away. Middle names can also be chosen to reflect the parents’ ethnic heritage or personal interests, for example, those with Irish heritage could choose to use an Irish middle name.
Whatever your reason, it’s important to choose a name that will be meaningful for you and your child. Not everyone has a middle name, and indeed many people have multiple middle names. I have three, and they were all named after relatives that my parents cared about dearly.
The story behind my middle names matters and I love the fact my middle names represent both my grandmothers and my mother. My own children all have middle names named after their relatives.
Often people can make the middle name a nickname later on or even you can put the initials together to form a new name, for example, Paxton Gunner could become PG!
Research has also indicated that having middle names improves peoples chances of getting a better job in the future!
According to a study published in the European Journal of Social Psychology, people who have middle names are believed to have a high social position and to be intellectually superior.
Take Away For Middle Name Paxton
Middle names are a very practical and traditional choice when naming your child, but it is also one of the most creative decisions you can make! You will get to choose from over 100 middle name ideas that we have provided in this article.
We hope our list helps you find inspiration as well as some peace-of-mind because finding the perfect middle name for Paxton doesn’t need to be difficult or overwhelming at all! | https://mommyandlove.com/middle-names-for-Paxton/ |
EDITOR’S NOTE: Updated at 8:30 a.m. EST (1330 GMT) Nov. 11 after scrubbed launch attempt.
Orbital ATK’s Antares rocket will deliver the commercial Cygnus supply ship to an orbit with an altitude between 117 miles (189 kilometers) and 184 miles (296 kilometers) within about nine minutes of liftoff Saturday from Virginia’s Eastern Shore.
The rocket’s two RD-181 engines will ignite around 3.6 seconds before liftoff from pad 0A at the Mid-Atlantic Regional Spaceport, a complex owned by the state of Virginia at NASA’s Wallops Flight Facility.
Launch is timed for 7:14 a.m. EST (1214 GMT) Saturday.
The first stage’s two RD-181 engines will power up to 864,000 pounds of thrust and burn for 3 minutes, 35 seconds to accelerate the rocket to more than 8,750 mph (3.9 kilometers per second) and an altitude of 61 miles (99 kilometers), then separate from the upper stage’s Castor 30XL motor about six seconds later.
The launch, known as OA-8 in Orbital ATK’s station resupply manifest, will be the second Antares mission using RD-181 engines, which the company ordered from the Russian engine-builder NPO Energomash to replace decades-old Russian-built AJ26 engines blamed for an Antares rocket crash seconds after liftoff in October 2014.
The RD-181 engines performed better than predicted on an Antares launch in October 2016, giving engineers confidence to loosen performance limits for the OA-8 launch. The engines produce more thrust than the AJ26s, and they will be programmed to fire around six seconds longer on the OA-8 launch than on the last Antares flight.
“We flew to a delta velocity threshold,” said Kurt Eberly, Orbital ATK’s Antares program manager, in a press briefing earlier this year. “When we hit that, we shut down the engines. We had a lot of fuel left in the tanks. Now, we’re just going to move that threshold a little higher and burn more of the fuel in the first stage. At that point in the flight regime, the acceleration is pretty high because the stage is pretty light. Most of the propellant is gone, so you actually pick up quite a bit of performance by burning just a few more seconds into that propellant residual in the tanks.”
With the higher performance, the upgraded Antares can carry approximately 300 pounds more cargo than managers initially expected. The lifting of additional conservative flight constraints, coupled with further minor changes to the vehicle, will further raise the Antares rocket’s payload capacity another 300 pounds in the coming years.
Once the first stage finished its job on the OA-8 launch the Antares rocket’s 12.8-foot-diameter (3.9-meter) diameter payload shroud will jettison in two halves at T+plus 4 minutes, 11 seconds. An interstage adapter that connected the first and second stages will separate at T+plus 4 minutes, 16 seconds.
The launcher’s Castor 30XL solid-fueled upper stage will ignite at T+plus 4 minutes, 24 seconds, and generate up to 104,300 pounds of thrust during a burn lasting 2 minutes, 42 seconds. The second stage motor will burn out at T+plus 7 minutes, 6 seconds, then deploy the Cygnus spacecraft at T+plus 9 minutes, 6 seconds.
The spacecraft’s two cymbal-shaped electricity-generating solar arrays will unfurl in a fan-like motion around 90 minutes into the mission, and the ship’s thrusters will begin fine-tuning its approach to the space station with a series of course-correction burns Sunday and Monday, setting up for a laser-guided final approach Tuesday.
Email the author.
Follow Stephen Clark on Twitter: @StephenClark1. | https://spaceflightnow.com/2017/11/11/antares-rockets-launch-timeline-on-the-oa-8-cargo-mission/ |
This guide will provide all of the information you need on the Japanese word kudasai, including its translation, meaning, usage, sentence examples, and more!
What does kudasai (ください) mean?
According to Japanese Particles Master, kudasai (ください) is the imperative form of the verb “kudasaru,” which means “to give” in Japanese. The rough English translation is “please give me.” It can also be used as an auxiliary verb put after another verb to soften its meaning, often translated to simply “please” in English. This word is used by native speakers to ask someone to do something in a polite way. It is often used with other verbs. The hiragana of kudasai is ください.
What are other conjugations of the verb kudasaru?
According to Japanese Verb Conjugator, kudasaru is a verb that means “to give.” In the present tense, the verb has four different conjugations: plain positive, plain negative, polite positive, and polite negative.
The plain positive form is kudasaru. The plain negative form is kudasaranai. The plain polite version is kudasaimasu, and the polite negative version is kudasaimasen. Kudasai is the imperative “plain positive” form of the verb kudasaru.
How is kudasai used?
When someone requests something from another person in Japanese, they use the “te” form of the verb, in which many of the words end in “te” or “de,” according to Kawakawa Learning Studio. This verb form can also be used to link to thoughts or phrases together. When using kudasai, one will use the te form of the verb along with the word kudasai to create a polite request.
For example, NHK states that in the Japanese language, the verb “to eat” is tabemasu. The te form of tabemasu is tabete. Therefore, tabete kudasai means “please eat.” This is the same general structure that is followed when using the word kudasai: a verb in its te form followed by the word kudasai to create a polite request.
This is frequently seen in the common phrase yamete kudasai, which means “please stop it,” or according to Urban Dictionary, with the word “aishite.” “Aishite” roughly translates to “love me.” Therefore, aishite kudasai means “please love me.”
What’s the difference between kudasai and onegaishimasu?
According to Thought Co, both kudasai and onegaishimasu (お願いします) are Japanese words that are used when making requests, and both have rough translations to the word “please” in English. However, the two words have nuanced differences that are important to learn when using the two different words in Japanese.
Kudasai is a more familiar and casual word that is used when one requests something they know they are entitled to. This would be used if someone is requesting something from a friend or sibling, or someone of the same or lower social status to you. Kudasai is used when making a request that involves a specific action with te form verbs. This is similar to the word “tu” in Spanish.
On the other hand, onegaishimasu is more formal than kudasai. It is more polite and is used when requesting a favor, or when requesting something from a superior, elder, or a stranger. People use onegaishimasu when making a request for service, like asking for someone on the phone or giving a destination to an Uber driver. It followers a similar pattern to the word “usted” in Spanish.
How is kudasai commonly used in a sentence?
Below are a few common phrases in which one might use the word “kudasai.” According to Elon, “yoku kiite kudasai” is a common phrase that means “please listen carefully.” A teacher may tell a room full of students “yoku kiite kudasai” when asking them to listen to instructions. Kudasai is often used at the end of a sentence.
According to Matcha JP, “chotto matte kudasai” is another common expression that means “please wait a moment.” Someone might use this if their significant other is walking ahead of them, but they need to stop and tie their shoe.
“Chotto kite kudasai” translates to “can you come with me?” A woman may whisper this to her friend when she is going to the bathroom if she feels unsafe or if she wants to talk to her friend in private.
“Chotto oshiete kudasai” means “could you tell me?” This can be used when asking for directions or other information one does not know. If asking for something from someone from a higher social class, the word onegaishimasu would be used instead.
Many of these phrases use the word “chotto.” According to ThoughtCo, “chotto” roughly translates to “a small amount” and is used when asking for something small – a small moment, or a small piece of information.
“Keisatsu/kyuukyuusha o yonde kudasai” means “please call the police/an ambulance.” These expressions are used in an emergency and used when someone needs immediate assistance. It is similar to the American English, “Call 911.”
Overall, the word kudasai is a Japanese word that means “please” or “please give me” in English. This word can either be used to make requests, or as a way to soften imperative verbs in formal settings to make them more polite. It is often translated as “please.” The more formal version of the word please is onegaishimasu. If you’re going to use the word kudasai in Japanese, ganbatte kudasai – please do your best, do not give up, and hang in there!
Sources: | https://thewordcounter.com/meaning-of-kudasai/ |
---
abstract: 'Consider an arrangement of $n$ congruent zones on the $d$-dimensional unit sphere $S^{d-1}$, where a zone is the intersection of an origin symmetric Euclidean plank with $S^{d-1}$. We prove that, for sufficiently large $n$, it is possible to arrange $n$ equal zones of suitable width on $S^{d-1}$ such that no point belongs to more than a constant number of zones, where the constant depends only on the dimension and the width of the zones. Furthermore, we also show that it is possible to cover $S^{d-1}$ by $n$ equal zones such that each point of $S^{d-1}$ belongs to at most $A_d\ln n$ zones, where the $A_d$ is a constant that depends only on $d$. This extends the corresponding $3$-dimensional result of Frankl, Nagy and Naszódi [@FNN2016]. Moreover, we also examine coverings of $S^{d-1}$ with equal zones under the condition that each point of the sphere belongs to the interior of at most $d-1$ zones.'
address:
- 'Auburn University, 221 Parker Hall, Auburn, AL 36849, U.S.A.'
- 'Department of Geometry, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary'
- 'Department of Geometry, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary'
- 'Department of Geometry, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary'
author:
- 'A. Bezdek'
- 'F. Fodor'
- 'V. Vígh'
- 'T. Zarnócz'
title: On the multiplicity of arrangements of equal zones on the sphere
---
Introduction and Results
========================
A [*plank*]{} in the Euclidean $d$-space ${\mathbb{R}}^d$ is a closed region bounded by two parallel hyperplanes. The width of a plank is the distance between its bounding hyperplanes. The famous plank problem of Tarski [@T1932] seeks the minimum total width of $n$ planks that can cover a convex body $K$ (a compact convex set with non-empty interior).
In this paper we consider a spherical variant of the plank problem which originates from L. Fejes Tóth [@LFT1973]. Following Fejes Tóth, we will call the parallel domain of spherical radius $w/2$ of a great sphere $C$ on the $d$-dimensional unit sphere $S^{d-1}$ a [*spherical zone*]{}, or zone for short. $C$ is the central great sphere of the zone and $w$ is its (spherical) width. For positive integers $d\geq 3$ and $n$, let $w(d,n)$ denote the minimum width of $n$ zones that can cover $S^{d-1}$. Fejes Tóth asked in [@LFT1973], among other questions, what is $w(3,n)$ under the condition that the zones have equal width. He conjectured that in the optimal configuration the central great circles of the zones all go through an antipodal pair of points and they are distributed equally, so in this case $w(d,n)=\pi/n$. The conjecture of Fejes Tóth was verified for $n=3$ (Rosta [@Rosta1972]) and $n=4$ (Linhart [@Linhart1974]). Fodor, Vígh and Zarnócz [@FVZ2016] gave a lower bound for $w(3,n)$ that is valid for all $n$. Very recently, Jiang and Polyanskii [@JP2017] completely solved L. Fejes Tóth’s conjecture by proving for all $d$, that to cover $S^{d-1}$ by $n$ (not necessarily equal) zones, the total width of the zones must be at least $\pi$, and that the optimal configuration is essentially the same as conjectured by L. Fejes Tóth.
Here, we examine arrangements of equal zones on $S^{d-1}$ from the point of view of multiplicity. The multiplicity of an arrangement is the maximum number of zones the points of the sphere belong to. We seek to minimize the multiplicity for given $d$ and $n$ as a function of the common width of the zones. It is clear that for $n\geq d$, the multiplicity of any arrangement with $n$ equal zones is at least $d$ and at most $n$. Notice that in the Fejes Tóth configuration the multiplicity is exactly $n$, that is, maximal.
In particular, if $d=3$ and $n\geq 3$, then the multiplicity of any covering is at least $3$. Our first result is a very slight strengthening of this simple fact for the case when $n\geq 4$.
\[lowerbound\] Let $n\geq 1$ be an integer, and let $S^2$ be covered by the union of $n$ congruent zones. If each point of $S^2$ belongs to the interior of at most two zones, then $n\leq 3$. If, moreover, $n=3$, then the three congruent zones are pairwise orthogonal.
Note that Theorem \[lowerbound\] does not imply that the multiplicity of a covering of $S^2$ with $n\geq 4$ equal zones would have to be larger than $3$. In fact, one can cover $S^2$ with $4$ zones such that the multiplicity is $3$. For this, consider three zones whose central great circles pass through a pair of antipodal points (North and South Poles) and are distributed evenly. Let the fourth zone’s central great circle be the Equator. It is easy to see that the common width can be chosen in such a way that there is no point contained in more than three zones. Also, one can arrange five zones such that the multiplicity is still $3$. We start with the previously given four zones, and take another copy of the zone whose central great circle is the Equator. Now slightly tilt these two zones. It is not difficult to see that the multiplicity of the resulting configuration is $3$. The details are left to the reader.
We further note, see Remark \[remark1\], that the statement of Theorem \[lowerbound\] can probably be extended to all $d\geq 3$. In particular, it certainly holds for $3\leq d\leq 100$.
Now, we turn to the question of finding upper bounds on the multiplicity of arrangements of zones on $S^{d-1}$. Let $\alpha:{\mathbb N}\to (0,1]$ be a positive real function with $\lim_{n\to\infty}\alpha(n)=0$. For a positive integer $d\geq 3$, let $m_d=\sqrt{2\pi d}+1$. Let $k:{\mathbb N}\to\mathbb{N}$ be a function that satisfies the (somewhat technical) limit condition
$$\label{eq:limit}
\limsup_{n\to\infty} \alpha(n)^{-(d-1)}\left (\frac{e\;C_d^*\; n\;\alpha(n)}{k(n)}\right )^{k(n)}=\beta<1.$$
\[thm:ujmain\] For each positive integer $d\geq 3$, and any real function $\alpha(n)$ described above, for sufficiently large $n$, there exists an arrangement of $n$ zones of spherical half-width $m_d\alpha(n)$ on $S^{d-1}$ such that no point of $S^{d-1}$ belongs to more than $k(n)$ zones.
The following statement provides an upper bound on the multiplicity of coverings of the $d$-dimensional unit sphere by $n$ congruent zones.
\[thm:main\] For each positive integer $d\geq 3$, there exists a positive constant $A_d$ such that for sufficiently large $n$, there is a covering of $S^{d-1}$ by $n$ zones of half-width $m_d\frac{\ln n}{n}$ such that no point of $S^{d-1}$ belongs to more than $A_d\ln n$ zones.
Below we list some more interesting special cases according to the size of the function $\alpha(n)$.
With the same hypotheses as in Theorem \[thm:ujmain\], the following statements hold.
- If $\alpha(n)=n^{-(1+\delta)}$ for some $\delta>0$, then $k(n)=const.$. Mo rover, if $\delta>d-1$, then $k=d$.
- If $\alpha(n)=\frac{1}{n}$, then $k(n)=B_d\frac{\ln n}{\ln\ln n}$ for some suitable constant $B_d$.
We note that Theorem \[thm:main\] and an implicit version of Theorem \[thm:ujmain\] were proved by Frankl, Nagy and Naszódi [@FNN2016] for the case $d=3$. They provided two independent proofs, one of which is a probabilistic argument and the other one uses the concept of VC-dimension. We further add that the weaker upper bound of $O(\sqrt{n})$ on the minimum multiplicity of coverings of $S^2$ was posed as an exercise in the 2015 Miklós Schweitzer Mathematical Competition [@KN2015] by A. Bezdek, F. Fodor, V. Vígh and T. Zarnócz (cf. Exercise 7).
Our proofs of Theorems \[thm:ujmain\] and \[thm:main\] are based on the probabilistic argument of Frankl, Nagy and Naszódi [@FNN2016], which we modified in such a way that it works in all dimensions. In the course of the proof we also give an upper estimate for the constant $A_d$ whose order of magnitude is $O(d)$.
Obviously, there is a big gap between the lower and upper bounds for the multiplicity of coverings of $S^{d-1}$ by equal zones. At this time, it is an open problem that the minimum multiplicity of coverings of $S^{d-1}$ by $n$ equal zones is bounded or not, and it also remains unknown whether the multiplicity is monotonic in $n$. Frankl, Nagy and Naszódi conjecture that, in fact, that the multiplicity of coverings of $S^{d-1}$ by $n$ equal zones tends to infinity as $n\to\infty$, cf. Conjectures 4.2 and 4.4 in [@FNN2016].
The multiplicity of coverings of ${\mathbb{R}}^d$ and $S^{d}$ by convex bodies have been investigated in the past. In their classical paper, Erdős and Rogers [@ER1961] proved, using a probabilistic argument, that ${\mathbb{R}}^d$ ($d\geq 3$) can be covered by translates of a given convex body such that the density of the covering is less than $d\log d+d\log\log d+4n$ and no point of space belongs to more than $e(d\log d+d\log\log d+4n)$ translates. Later, Füredi and Kang [@FK2008] gave a different proof of the result of Erdős and Rogers using John ellipsoids and the Lovász Local Lemma. Böröczky and Wintsche [@BV2003] showed that for $d\geq 3$ and $0<\varphi< \pi/2$, $S^d$ can be covered by spherical caps of radius $\varphi$ such that the multiplicity of the covering is at most $400d\ln d$.
Proofs
======
Proof of Theorem \[lowerbound\]
-------------------------------
Assume that $n\geq 3$ and $S^2$ is covered by $n$ congruent zones such that no point of $S^2$ belongs to the interior of more than two zones. Then the $n$ central great circles of the zones divide $S^2$ into convex spherical polygons. As no three or more such great circles can pass through a point of $S^2$, every such polygon has at least three sides.
In contrast to the Euclidean plane, the incircle of every convex spherical polygon is uniquely determined. The inradius of each such polygon is less than or equal to the half-width of the zones.
We will use the following lemma.
\[lemma-incircle\] Every convex spherical polygon with $k>3$ sides and inradius $r$ contains a point $P$ whose distance from at least three sides is less than $r$.
Denote the incircle by $C$ and denote its centre by $O$.
[*Case 1.*]{}
It is easy to see that among the tangent sides there must be two, say $e$ and $f$, which are not adjacent on the boundary of $C$. The extensions of $e$ and $f$ form a spherical $2$-gon. Start moving the centre $O$ along the diagonal of this $2$-gon toward its closest endpoint. Then the distances of $O$ from the extended sides $e$ and $f$ continuously decrease and $O$ eventually gets arbitrarily close to an additional side. When this happens $O$ has a distance from at least three sides less than $r$.
[*Case 2.*]{}
Let $e$ and $f$ be the only two sides tangent to the incircle $C$. Consider again the $2$-gon whose sides are the extensions of $e$ and $f$. Notice that $C$ is also the incircle of this $2$-gon. Thus, moving $O$ along the diagonal towards either of the two endpoints continuously decreases the distances of $O$ from the extended sides $e$ and $f$. At least one of the direction will take $O$ arbitrarily close to an additional side. When it happens, then $O$ has a distance from at least three sides less than $r$.
Lemma \[lemma-incircle\] yields immediately that each spherical polygon determined by the $n$ central great circles of the zones is a spherical triangle. The vertices and sides of these triangular domains form a planar graph $G$ on $S^2$. The number $v$ of vertices is $2{n\choose 2}$, and the number of edges is $2n(n-1)$. By Euler’s formula, the number $f$ of faces (the number of spherical triangles) is $$f=e+2-v=n^2-n+2.$$ Furthermore, the degree of each vertex is four, thus $4v=3f$, which yields that $$n^2-n-6=0.$$ The only positive root of the above quadratic equation is $n=3$.
Let $n=3$, and assume that the central great circles of two zones intersect in the North and South poles of $S^2$. The part of $S^2$ not covered by these two zones is the union of two or four spherical $2$-gons bounded by small circular arcs that are parts of the boundaries of the zones. It is easy to see that if the uncovered part consists of only two such $2$-gons, then there must be a point of $S^2$ which belongs to the interior of all three zones. As the vertices of the uncovered $2$-gons that are on the same hemisphere (say the Northern one) must be on one of the bounding small circles of the third zone, they must be coplanar. This is only satisfied when the first two zones are perpendicular. This finishes the proof of Theorem \[lowerbound\].
\[remark1\] Consider now $n$ equal zones on $S^{d-1}$ such that no point belongs to the interior of more than $d-1$ zones. Then the central great spheres of the zones divide $S^{d-1}$ into convex spherical polytopes similar to the $3$-dimensional case. We note that the argument of Lemma \[lemma-incircle\] can be generalized to arbitrary $d$, only one has to consider $d-1$ cases instead of two; we leave the detailed argument to the interested reader. Thus, the central great spheres of the zones divide $S^{d-1}$ into spherical simplices.
Now, a similar combinatorial analysis can be carried out, with the help of the Euler-Poincaré formula, as in $S^2$. Let $f_{i,d}(n)$ denote the number of $i$-dimensional faces determined by the central great spheres of the $n$ zones for $d\geq 3$ and $n\geq d-1$. We use the conventions: $f_{-1,d}(n)=1$ and $f_{d,d}(n)=1$. As we have seen in the proof of Lemma \[lemma-incircle\], $f_{0,3}=2{n\choose 2}$, $f_{1,3}(n)=2n(n-1)$, and $f_{2,3}=n^2-n+2$.
Then we have the following recursion for $f_{i,d}(n)$ when $d\geq 4$: $$\begin{aligned}
f_{0,d}(n)&=2{n\choose d-1},\\
f_{i,d}(n)&=\frac{n}{d-i-1}f_{i,d-1}(n-1) \quad (1\leq i\leq d-2),\\
f_{d-1,d}(n)&=\frac{2}{d}f_{d-2,d}(n).\end{aligned}$$ As the $n$ central great spheres are in general position, a vertex is incident with exactly $d-1$ of them, which explains the formula for $f_{0,d}(n)$. Since the cells are simplices, counting its facets one gets the identity $2f_{d-2,d}(n)=df_{d-1,d}(n)$. Finally, if $1\leq i\leq d-2$, then consider a fixed central great sphere. The other central great spheres intersect the chosen one in $n-1$ great spheres (of one less dimension) that are in general position. Taking into account that we have $n$ great spheres and that an $i$-dimensional face is incident with exactly $d-i-1$ great spheres, one gets the second formula above.
Now, for a fixed $d$, using the Euler–Poincaré formula, $\sum_{i=-1}^{d}(-1)^{d+1}f_{i,d}(n)=0$ one can obtain a polynomial equation $p(d,n)=0$ of degree at most $d-1$ in $n$. When $n=d$, then $n$ pairwise orthogonal equal zones satisfy all conditions, thus, $n=d$ is always a root of $p(d,n)$. In particular, for $4\leq d\leq 6$, the reduced forms of $p(d,n)$ in which the coefficient of $n^{d-1}$ is $1$ are the following $$\begin{aligned}
p(4,n)&=(n-4)(n+1)n,\\
p(5,n)&=(n-5)(n^3-n^2-2n-8),\\
p(6,n)&=(n-6)(n-2)(n-1)^2n.\end{aligned}$$ Thus, if $d=4$ or $6$, then $n=d$ is the largest root that satisfies our conditions. In the case $d=5$ one can check that $p(5,d)$ has two complex roots and two real roots, one real root is $5$ and the other one is smaller than $5$.
We can now formulate the following conjecture.
\[conj\] Let $d\geq 3$ and $n\geq 1$ be integers, and let $S^{d-1}$ be covered by the union of $n$ congruent zones. If each point of $S^2$ belongs to the interior of at most two zones, then $n\leq d-1$. If, moreover, $n=d$, then the $d$ congruent zones are pairwise orthogonal
By Theorem \[lowerbound\] and the above argument we have proved the first statement of Conjecture \[conj\] for $3\leq d\leq 6$. If $n=d$, then the orthogonality of the zones can be proved essentially the same way as in the proof of Theorem \[lowerbound\]. Furthermore, we have computed the roots of $p(d,n)$ for $7\leq d\leq 100$ by computer (numerically) and observed than in each case the largest real root is $n=d$, which supports our conjecture.
Finally we note that computer calculations suggest that in the case when $d\geq 6$ is even, $$p(d,n)=(n-d)(n-d+5)\prod_{i=0}^{d-4}(n-i).$$
Proof of Theorem \[thm:ujmain\]
-------------------------------
For two points $P,Q\in S^{d-1}$, their spherical distance is the arclength of the shorter unit radius circular arc on $S^{d-1}$ that connects them. We denote the spherical distance by $d_S(P,Q)$.
Let $0<\omega\leq \pi/2$. We say that the points $P_1,\ldots, P_m\in S^{d-1}$ form a [*saturated set*]{} for $\omega$ if the spherical distances $d_S(P_i,P_j)\geq\omega$ for all $i\neq j$ and no more points can be added such that this property holds. Investigating the dependence of $m$ on $d$ and $\omega$ is a classical topic in the theory of packing and covering; for a detailed overview of known results in this direction see, for example, the survey paper by Fejes Tóth and Kuperberg [@FTGK].
It is clear that $m$ is of the same order of magnitude as $\omega^{-(d-1)}$. In the next lemma, we prove a somewhat more precise statement. Although the content of the lemma is well-known, we give a proof because we need inequalities for $m$ with exact constants in subsequent arguments, and also for the sake of completeness. Let $\kappa_d$ denote the volume of the $d$-dimensional unit ball $B^d$.
Let $0<\varepsilon <1$. Then there exists $0<\omega_0\leq \pi/2$ depending on $\varepsilon$ with the following property. Let $0<\omega<\omega_0$, and let $P_1,\ldots, P_m$ be a saturated point set on $S^{d-1}$ such that $d_S(P_i,P_j)\geq\omega$ for $i\neq j$. Then $$\frac{1}{1+\varepsilon}\cdot \frac{d\kappa_d}{\kappa_{d-1}}\cdot \omega^{-(d-1)}\leq m\leq (1+\varepsilon)\cdot \frac{8^{\frac{d-1}{2}}d\kappa_d}{\kappa_{d-1}}\cdot \omega^{-(d-1)}.$$
The following formula is well-known for the surface area $S(t)$ of a cap of height $t$ of $S^{d-1}$, $$\lim_{t\to 0+} S(t)\cdot t^{-\frac{d-1}{2}}=2^{\frac{d-1}{2}}\kappa_{d-1}.$$ Therefore, there exists $0<t_0=t_0(\varepsilon)$ such that for all $0<t<t_0$ it holds that
$$\frac{1}{1+\varepsilon}\cdot 2^{\frac{d-1}{2}}\kappa_{d-1}\leq S(t)\cdot t^{-\frac{d-1}{2}}\leq (1+\varepsilon)\cdot 2^{\frac{d-1}{2}}\kappa_{d-1}.$$ Furthermore, let $0<\omega_0=\omega_0(\varepsilon)$ be such that $t_0=1-\cos\omega_0$.
The spherical caps of (spherical) radius $\omega/2$ centred at $P_1,\ldots, P_m$ form a packing on $S^{d-1}$, and the spherical caps of radius $\omega$ form a covering of $S^{d-1}$. In view of the above inequalities for the surface area of caps, we obtain that for $0<\omega<\omega_0$ it holds that $$m\cdot \frac{1}{1+\varepsilon}\cdot 2^{\frac{d-1}{2}}\kappa_{d-1}(1-\cos \frac\omega 2)^{\frac{d-1}{2}} \leq d\kappa_d\leq m\cdot (1+\varepsilon)\cdot 2^{\frac{d-1}{2}}\kappa_{d-1}(1-\cos \omega)^{\frac{d-1}{2}}.$$ By simple rearrangement we get that $$\frac{1}{1+\varepsilon}\cdot \frac{d\kappa_d}{2^{\frac{d-1}{2}}\kappa_{d-1}(1-\cos\omega)^{\frac{d-1}{2}}}\leq m\leq (1+\varepsilon)\cdot \frac{d\kappa_d}{2^{\frac{d-1}{2}}\kappa_{d-1}(1-\cos\frac\omega 2)^{\frac{d-1}{2}}}$$ Now, we use that for $0<x<1$, it holds that $x^2/4<1-\cos x<x^2/2$, which follow simply from the Taylor series of $\cos x$, and obtain $$\frac{1}{1+\varepsilon}\cdot \frac{d\kappa_d}{\kappa_{d-1}}\cdot \omega^{-(d-1)}\leq m\leq (1+\varepsilon)\cdot \frac{8^{\frac{d-1}{2}}d\kappa_d}{\kappa_{d-1}}\cdot \omega^{-(d-1)}.$$
We denote a spherical zone of (spherical) half-width $t$ by $\Pi(t)$. Simple geometry shows that $$\lim_{t\to 0^+} S(\Pi(t))\cdot t^{-1}=2(d-1)\kappa_{d-1}.$$ Let $\varepsilon>0$. Then there exists $t_1=t_1(\varepsilon)>0$ such that for $0<t<t_1$ the following holds $$(1+\varepsilon)^{-1}\cdot 2(d-1)\kappa_{d-1}\cdot t\leq S(\Pi(t))\leq (1+\varepsilon)\cdot 2(d-1)\kappa_{d-1}\cdot t.$$
Let $\alpha(n)$ be a given positive function with $\lim_{n\to\infty}\alpha(n)=0$. From now on, let $\varepsilon=1$. Let $n$ be sufficiently large. Let $m_d=\sqrt{2\pi d}+1$.
Let $Q_1,\ldots, Q_m$ be a saturated set of points on $S^{d-1}$ such that $d_S(Q_i,Q_j)\geq \alpha(n)/2$ for any $i\neq j$. It follows from Lemma 1 that $$\begin{aligned}
m &\leq 2\cdot \frac{8^{\frac{d-1}{2}}d\kappa_d}{\kappa_{d-1}}\cdot (\alpha(n)/2)^{-(d-1)}\\
&=2\cdot \frac{2^{\frac{d-1}{2}}d\kappa_d}{\kappa_{d-1}}\cdot \alpha(n)^{-(d-1)}\\
&= c_d\; \alpha(n)^{-(d-1)}.\end{aligned}$$
Consider $n$ independent random points from $S^{d-1}$ chosen according to the uniform probability distribution and consider the corresponding spherical zones\
$\Pi_1,\ldots, \Pi_n$ of (spherical) half-width $m_d\alpha(n)$ whose poles are these points. Furthermore, let $\Pi_i^-$ ($\Pi_i^+$) be the corresponding planks of half-width $(m_d-1)\alpha(n)$ ($(m_d+1)\alpha(n)$).
Now, we are going to estimate the probability of the event that there exists a point $p$ on $S^{d-1}$ which belongs to at least $k=k(n)$ zones. The probability that a point $p\in S^{d-1}$ belongs to a spherical plank $\Pi_i^+$ can be estimated from above as follows. $${\mathbb P}(p\in\Pi_i^+)\leq \frac{4(m_d+1)(d-1)\kappa_{d-1}}{d\kappa_d}\;\alpha(n)=C_d^*\; \alpha(n).$$ Note that $C_d^*=O(d)$ as $d\to\infty$.
Then $$\begin{aligned}
&{\mathbb P}(\exists p\in \Pi_{i_1}\cap\dots\cap\Pi_{i_k}:\text{ for some }1\leq i_1<\ldots <i_k\leq n)\\
\leq\ &{\mathbb P}(\exists Q_j\in\Pi^+_{i_1}\cap\dots\cap\Pi^+_{i_k}:\text{ for some }1\leq i_1<\ldots <i_k\leq n)\\
\leq\ &m\cdot{\mathbb P}(Q_1\in\Pi^+_{i_1}\cap\dots\cap\Pi^+_{i_k}:\text{ for some }1\leq i_1<\ldots <i_k\leq n)\\
\leq\ &m\cdot\binom n{k(n)}\left(C_d^*\;\alpha(n)\right)^{k(n)}\\
\leq\ &c_d\;\alpha(n)^{-(d-1)}\binom n{k(n)}\left(C_d^*\;\alpha(n)\right)^{k(n)}\end{aligned}$$
An application of the Stirling-formula (cf. Page 10 of [@FNN2016]) yields that $$\label{binom}
\binom nk\leq C \frac{n^n}{k^k(n-k)^{n-k}}$$ for some suitable constant $C>0$.
Then applying we get that $$\begin{aligned}
&c_d\; \alpha(n)^{-(d-1)}\binom n{k(n)}\left(C_d^*\;\alpha(n)\right)^{k(n)}\notag\\
\leq &c_d\;\alpha(n)^{-(d-1)} \cdot C \frac{n^n (n-k(n))^{k(n)}}{\left(k(n)\right)^{k(n)-n}}
\left(C_d^*\;\alpha(n)\right)^{k(n)}\notag\\
\leq &\tilde{c_d}\;\alpha(n)^{k(n)-d+1}\left (\frac{n}{k(n)}\right )^{k(n)} (e\cdot C_d^*)^{k(n)}\label{eq:final}\notag\\
&=\tilde{c_d}\;\alpha(n)^{-(d-1)} \left (\frac{e\;C_d^*\; n\;\alpha(n)}{k(n)}\right )^{k(n)}.\end{aligned}$$
\[limit\]
By we obtain $$\limsup_{n\to\infty}{\mathbb P}(\exists p\in \Pi_{i_1}\cap\dots\cap\Pi_{i_k}:\text{ for some }1\leq i_1<\ldots <i_k\leq n)<1,$$ therefore the probability of the event that no point of $S^{d-1}$ belongs to at least $k(n)$ zones is positive for sufficiently large $n$. This finishes the proof of Theorem \[thm:ujmain\].
Proof of Theorem \[thm:main\]
-----------------------------
Let $\alpha(n)=\frac{\ln n}{n}$, and let $k(n)=A_d\ln n$, where $A_d$ be a suitable positive constant that satisfies the following equation $$\left (\frac{C_d^*}{x}\right )^{x}=e^{-d-x}.$$ Then $$\begin{aligned}
\eqref{eq:limit}&=\lim_{n\to\infty}\tilde{c_d}\frac{n^{d-1}}{(\ln n)^{d-1}}\cdot n^{A_d}\left (\frac{C_d^*}{A_d}\right )^{A_d\ln n}=0.\label{eq:final}\end{aligned}$$
Furthermore, in this case the probability that an arbitrary fixed point $p$ of $S^{d-1}$ is in $\Pi_i^-$ (for a fixed $i$) is $${\mathbb P}(p\in\Pi_i^-)\geq 2^{-1}\cdot \frac{2(d-1)\kappa_{d-1}}{d\kappa_d}\cdot (m_d-1)\alpha(n).$$ Using the inequality $\frac{\kappa_{d-1}}{d\kappa_d}>\frac{1}{\sqrt{2\pi d}}$ (cf. Lemma 1 in [@BGW1982]), we obtain that $${\mathbb P}(p\in\Pi_i^-)\geq \frac{(m_d-1)(d-1)}{\sqrt{2\pi d}}\cdot \frac{\ln n}{n}=(d-1)\frac{\ln n}{n}$$ Thus, the probability that $\cup_{1}^{n}\Pi_i$ does not cover $S^{d-1}$. $$\begin{aligned}
{\mathbb P}(S^{d-1}\not\subseteq\cup_{1}^{n}\Pi_i) &\leq
{\mathbb P}(\exists Q_j\notin\cup_{1}^{n}\Pi_i^-)\\
&\leq m\cdot {\mathbb P}(Q_1\notin\cup_{1}^{n}\Pi_i^-)\\
&\leq c_d \left (\frac{n}{\ln n}\right )^{d-1}\cdot
\left (1-(d-1)\frac{\ln n}{n}\right )^n\\
&\leq 2c_d \left (\frac{1}{\ln n}\right )^{d-1}\end{aligned}$$ for a large enough $n$. Therefore $$\label{eq:firstcase}
\lim_{n\to\infty}{\mathbb P}(S^{d-1}\not\subseteq\cup_{1}^{n}\Pi_i)=0.$$
Thus, taking into account and , the probability of the event that all $S^{d-1}$ is covered by the zones and no point of $S^{d-1}$ belongs to more than $A_d\ln n$ zones is positive for sufficiently large $n$. This finishes the proof of Theorem \[thm:main\].
We note that $A_d=O(d)$ as $d\to\infty$. Clearly, $A_d$ can be lowered slightly by taking into account all the factors of .
We further note that one can obtain the result of Theorem \[thm:main\] with the help of Theorem 1.5 of [@FNN2016] using the VC-dimension of hypergraphs; for more details we refer to the discussion in [@FNN2016] after Theorem 1.4. However, as this alternate proof is less geometric in nature, we decided to describe the more direct probabilistic proof of Theorem \[thm:main\]. We leave the proof of Theorem \[thm:main\] that uses the VC-dimension to the interested reader. Furthermore, the direct probabilistic argument provides an explicit estimate of the involved constant $A_d$, as well.
Proof of Corollary 1
--------------------
Let $\alpha(n)=\frac{1}{n^{1+\delta}}$ for some $\delta>0$. If $k=k(n)>(d-1)/\delta+d-1$, then $$\begin{aligned}
\eqref{eq:limit}&=\lim_{n\to\infty} n^{(1+\delta)(d-1)}
\left (\frac{e\;C_d^*\;n^{-\delta}}{k}\right )^k\\
&=\lim_{n\to\infty} n^{(1+\delta)(d-1)-\delta k}=0.\end{aligned}$$ This means that in this case, for sufficiently large $n$, we can guarantee that one can arrange $n$ zones of half-width $m_d\alpha(n)$ on $S^{d-1}$ such that no point belongs to more than $k=const.$ zones, and the value of $k$ only depends on $d$ and $\delta$. Moreover, if $\delta>d-1$, then $k=d$ suffices. Of course, in this case the zones cannot cover $S^{d-1}$. This proves i) of Corollary 1.
Now, let $\alpha(n)=\frac{1}{n}$, and let $k(n)=B_d\frac{\ln n}{\ln\ln n}$, where $B_d>\max \{e\;C_d^*, d-1\}$ is a positive constant. Then $$\begin{aligned}
\eqref{eq:limit}&=\lim_{n\to\infty}n^{d-1}\left(\frac{e\;C_d^*\;\ln\ln n}{B_d\;\ln n}\right )^{B_d\frac{\ln n}{\ln\ln n}}\\
&\leq \lim_{n\to\infty} \left(\frac{n^{\frac{(d-1)\ln\ln n}{B_d\ln n}}\ln\ln n}{\ln n}\right )^{B_d\frac{\ln n}{\ln\ln n}}=0,\end{aligned}$$ as $$\begin{aligned}
&\lim_{n\to\infty}\frac{n^{\frac{(d-1)\ln\ln n}{B_d\ln n}}\ln\ln n}{\ln n}\\
=&\lim_{n\to\infty}\exp\left (\frac{d-1}{B_d}\ln\ln n+\ln\ln\ln n-\ln\ln n\right )=0.\end{aligned}$$
This finishes the proof of part ii) of Corollary 1. The above statement is interesting because $\alpha(n)=\frac 1n$ is the smallest order of magnitude for the half-width of the zones for which one can possibly have a covering.
We note that the $d=3$ special case of part ii) of Corollary 1 was explicitly proved by Frankl, Nagy and Naszódi in [@FNN2016] (cf. Theorem 4.1) in a slightly different form both by the probabilistic method and using VC-dimension. We also note that the general $d$-dimensional statement of part ii) of Corollary 1 may also be proved from Theorem 1.6 of [@FNN2016].
Acknowledgements
================
Research of A. Bezdek was partially supported by ERC Advanced Research Grant 267165 (DISCONV). F. Fodor, V. Vígh, and T. Zarnócz were supported by Hungarian National Research, Development and Innovation Office – NKFIH grant 116451. V. Vígh was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
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A new Vivo smartphone Vivo Y32 with model number V2158A was spotted on TENAA. It looks quite similar to the Y33s with a waterdrop notch and triple camera setup on the back. The Y32 also features a flat frame just like the Y33s.
The phone also passed by the 3C agency which revealed it packs 18W charging and confirmed the device will not feature 5G connectivity.
Vivo Y32 specifications (expected)
The Vivo Y32 runs on Android 11 and has a 6.51-inch HD+ (1,600×720 pixels) TFT display and is powered by an octa-core SoC, along with 4GB, 6GB, and 8GB RAM options. The phone also has 64GB, 128GB, and 256GB of onboard storage models.
The camera setup on the Vivo phone is listed with a 13-megapixel primary camera sensor at the back, along with a 2-megapixel secondary sensor. The TENAA listing also shows that there is an 8-megapixel selfie camera sensor at the front. | https://www.nextnewssource.com/vivo-y32-spotted-on-tenaa-database/ |
This topic contains 1 reply, has 2 voices, and was last updated by joe_r82 6 years, 9 months ago.
My cousin wants more than anything else in the whole world right now to become a police detective in a few years. I do not really know much about the process, but am extremely curious.
I guess I understand that you first must become a regular police officer, however I am not even sure how one goes about doing that. And even after you are an officer, what happens next? I mean do you need to pass a test, or is there a nomination process? Maybe it is something such that when a new position as a detective becomes available you need to apply and then interview for the job, with each officer competing on their own career record and merits. If there are any current police detectives out there, or even uniformed officers, it would be great if you could respond to this question.
I just want a bit of insight so that together my cousin and I can have a clearer picture about how to go about all of this and he can decide if he really has the desire and commitment to go through it all once he understands all of the steps involved. Any help received will be most appreciated. Thanks to all who reply.
You are basically right about the steps to becoming a police detective. To my knowledge, there is no department in the country that hires detectives directly from civilians. In other words, you must first become a police officer. Sometimes, departments will look to other police units in different cities or states to fill detective positions (but usually only if they are specialized detective positions), but generally this is a promotion from within type of situation.
So, the first step is becoming a uniformed officer. Your son will need to be at least 21 in order for this. College degrees are not necessary, although it is highly recommended that he have one. Studying criminology or a related discipline in college would be the best thing to do to secure a future detective position. After college, simply apply to the department of his choice. There will be an intensive hiring process, involving a thorough background check and psychological testing and physical testing. So, make sure that during the teenage and college years he keeps his nose clean and generally stays out of trouble.
Once accepted into the force, he will likely be assigned a patrol. The major key now is for him to excel at his job. It would even be very helpful if he developed a specialty or area of interest. Perhaps this could be something like accidents or narcotics. Spending time around current detectives will also help.
Eventually, if he has developed enough of a good reputation within the department, he may be asked to apply for an opening in as a detective. If not, after a few years, he can ask for a transfer to the detective unit and obtain a letter of recommendation from his commanding officer. | https://www.asecurelife.com/forums/topic/steps-to-becoming-a-police-detective/ |
Gracie Allen, who was married to George Burns once said, "Never place a period where God has placed a comma." Never place a period where God has placed a comma. The wedding guest wearing the wrong clothes in the scripture today is an example of what happens when we put a period where God has placed a comma. Too bad for him! What is this story about, you might ask! Well, let's take a look.
The story is really an allegory of salvation history. It is a symbolic attempt to describe for us the salvation process. It is a picture of salvation. The wedding feast represents the age to come. It is eternity, heaven, eternal life. The king throwing the party—that's God. The party is for his Son—the Messiah, the one we call Jesus Christ.
There's an interesting passage in Revelation 19:7-9 that sheds some light on this messianic banquet. John writes: "Let us rejoice and be glad and give him glory! For the wedding of the Lamb has come, and his bride has made herself ready. Fine linen, bright and clean, was given her to wear. (Fine linen stands for the righteous acts of the saints.) Remember this for later. Then the angel said to me, 'Write: "Blessed are those who are invited to the wedding supper of the Lamb!"'"
Now, that means you and I are blessed. We are blessed because we have been invited to this wedding party. You and I have been invited to the eternal celebration. Matthew says in 22:14, "many are invited." I believe the word "many" really means "all." I believe that Jesus died for everyone. And If Jesus died for everyone, He is going to include us all on the invitation list for the eternal party. That is just logical. Why leave someone off the list whom you bled and died for?
Yep, we are all invited.
Jesus also says, "but few are chosen." This statement—"Many are invited, but few are chosen"—is not a prediction of the proportion of the saved to the damned. Matthew is not trying to frighten Christians with the thought that the statistical odds are against them. He is attempting to encourage us to make a vigorous effort to live the Christian life. He is prodding us not to be complacent in our walk with Christ. He is giving us a heads-up that we need to be careful we don't fool ourselves into thinking we have achieved spiritual perfection. He is encouraging us to keep growing. God says, "You are invited—comma." Look how many of the invited guests in this story don't even show up for God's party. Check out this bunch in verse 3 that was invited and flat refuse to show up—period. Then in verse 5 right in the face of the king some of them decided work was more important then the party—period. They decided that their livelihood was more important then their life. Seems what they were bringing in was more important than what they were giving out. They were more interested in protecting what they had than in using what they had to help someone else enjoy the party.
It gets worse in verse 6. This gang of thugs abuses the messengers. The prophets and missionaries who bring the invitation are laughed at, shouted down or dismissed because the invitation means the invitee is required to do something different then they had been used to doing—period.
Problem with our periods, though, is they don't stop God. Those periods only stop the ones who put the period where God has a comma. You see, God is not ready to stop moving. He won't use a period until the end of time. He is still moving. When we lay down a period in our spiritual journey we miss what God is doing next.
Anyway, back to the story. God invites some more folks and they respond to the invitation. In this invitation God invites all kinds of people—some good, some bad. This invitation goes out to some scary characters. He invites folks we wouldn't want in our house. Thieves are invited. Homeless folk, poor people living in trailers are invited. Immigrants from Arab countries get an invite along with people living with HIV and AIDS. God wants them all—us—at his party. So this mob of "undesirables" comes to the party. They come because they know God is better then anything else they could have in this life. They want the best, so they accept God's invitation to celebrate Jesus.
That is, except for one man. Now, this is important. In verse 11 the king notices a man not wearing wedding clothes. The poorly dressed man had heard the invitation—like we have heard it. The man had accepted the invitation—like we have accepted it. But, the man did not respond with a changed lifestyle. He put a period where God placed a comma. The required garment for this party is righteousness. Remember the passage from Revelation? "Fine linen, bright and clean, was given her to wear. (Fine linen stands for the righteous acts of the saints.)" That means living in accordance with the teachings of Jesus Christ. Righteousness is easily explained in one sentence: it is the sentence that is Matthew 28:19—"Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, and teaching them to obey everything I have commanded you."
Wow! Have you worn the garment of righteousness recently? Have you noticed it hanging in your closet lately? When do you remember seeing it last? Did you send it to the cleaners and forget to go get it back? Did it get packed away with the winter stuff? Have you gone and made a disciple lately? Have we been obeying everything Jesus has commanded us? Wow! Have we placed a period where God has only placed a comma?
Hearing the invitation to receive Jesus Christ isn't enough. If we stop there we are placing a period where God has placed a comma. Accepting the invitation to receive Jesus as your Savior isn't enough. If we stop there we are placing a period where God has placed a comma. We are called to put on the garment of righteousness, to obey everything Christ has commanded and make disciples of everyone. We are called to walk the walk, to talk the talk, to put our money where our mouth is, lead by example, carry the torch, beat the drum, tell the story, get our hands dirty, think outside the box, carry our cross, follow Jesus Christ.
If we chose to place a period anywhere in our spiritual journey as individuals or as a church we will be like the man who was thrown out of the eternal party. Today is a great day to erase the periods we have been using and replace them with God's commas. Hey, God has a great party going on and he wants us all in the celebration. Come on! Put on the garment of righteousness and join the party. It's not too late! God is still using commas because he hasn't reached the end of the sentence yet. For the rest of our lives, let's get dressed and go party with Jesus. | http://www.ucc.org/god-is-still-speaking_church-resources_gods-party |
Woman Drinks Nothing but Soda for 16 Years, Discovers It's Not In Fact Good For Her Health
It is not news that drinking soda regularly can make a person unhealthy, but it may shock some that drinking it regularly for years can lead to serious heart problems.
One 31-year-old woman from Monaco said she drank only soda since she was 15-years-old. She had not touched water or any other drink during that time.
It was when she went to the doctor for fainting that she realized how unhealthy the habit was.
During a European Heart Rhythm Association meeting this week, the case was presented.
"The woman, who lives in Monaco, a small country near southern France, was brought to a hospital after she fainted. A blood test showed she had severely low potassium levels. And a test of her heart's electrical activity revealed she had a condition called long QT syndrome, which can cause erratic heart beats," they said.
"The woman did not have a family history of heart or hormone problems. But she told her doctors that, since the age of 15, she had not drunk any water - soda was the only liquid she consumed. She drank about 2 liters of cola daily, she said."
Once she learned her cola habit was likely the cause of her heart problem, she stopped drinking it. Within a week, it fully resolved itself.
"One of the take-home messages is that cardiologists need to be aware of the connection between cola consumption and potassium loss, and should ask patients found to have QT prolongation about beverage habits," Dr. Naima Zarqane said.
Other researchers say they should study the effects of soda on potassium levels, as it often lowers them. | https://cookingpanda.com/blogs/food-news/woman-drinks-nothing-but-soda-for-16-years-discovers-its-not-in-fact-good-for-her-health |
This homemade mustard is an excellent condiment for roasted meats. In an attractive jar it also makes a lovely gift.
Ingredients
- 4 1/2 lbs of white grapes
- 2 lbs baking apples (rennettes if available)
- 2 Large pears
- 1 cup vin santo
- 2 tablespoons vin santo
- 1/4 cup dry mustard
- 4 1/2 ounces candied lemon peel, diced
- 4 8 ounce jars
- 2 Tbsp cinnamon
Details
Preparation
Step 1
1. Crush grapes in a large bowl, cover and refrigerate for 2 days.
2. squeeze juice through a cheesecloth into a bowl and set aside. Discard pulp.
3. Peel, core and slice apples and pears very thinly. Place in a large nonstick saucepan and add 1 cup vin santo. Cook over medium heat for 5 minutes or until wine is absorbed.
4. Add apple/pear mixture to grape juice and cook over low heat until the mixture has a jamlike consistency (about 30 minutes). Let cool completely.
5. In a small saucepan heat 2 Tbsp vin santo and mix in dry mustard. Add to jam. Stir in candied lemon peel.
6. Pour into small jars and sprinkle the top of each with cinnamon. | https://www.keyingredient.com/recipes/14261073/tuscan-mustard-with-vin-santo/ |
BACKGROUND
SUMMARY
DETAILED DESCRIPTION
Large organizations may operate many different physical locations around a city, state, country, or even around the world. In order to provide real-time data and voice connectivity between locations, a number of wide area network (“WAN”) links may be established. The WAN links can be utilized to support data communications between the locations, including audio and/or audio/video sessions made between client devices placed at the locations. For instance, voice over Internet protocol (“VoIP”) calls may be made between clients placed at various locations connected by one or more WAN links. Similarly, video calls might also be made between clients placed at locations connected by WAN links.
In some real-time communications installations, a call admission control (“CAC”) server provides functionality for managing bandwidth utilization. For instance, a CAC server might participate during the setup of a new call to ensure that enough bandwidth is available on the appropriate network links to support the call. If sufficient bandwidth is not available, the CAC server might not allow the call to be completed, or might cause the call to be routed over another network such as the public switched telephone network (“PSTN”) or the Internet.
In order to provide redundancy and high availability in the provision of CAC services such as those described above, it is often necessary for CAC servers to establish connections with one another and to share data. In previous solutions, however, large numbers of network connections are typically required between CAC servers at different geographic locations. Additionally, previous solutions might require the synchronization of large amounts of data between CAC servers. As a result, previous solutions may require significant network overhead to support the connections and synchronization of data for the CAC feature, which can be very expensive especially when the network locations are dispersed over a large geographic area.
It is with respect to these and other considerations that the disclosure made herein is presented.
Technologies are described herein for efficient connection management and synchronization in the provision of CAC services. Through the utilization of the technologies and concepts presented herein, CAC servers that provide CAC services can be inter-connected utilizing fewer network connections than possible in previous solutions. Additionally, the CAC servers can synchronize data in a more efficient fashion than previously possible, thereby utilizing less bandwidth.
According to one aspect presented herein, CAC servers are organized into groups referred to herein as pools. For instance, in one installation a pool of CAC servers might be located in the western United States and configured to provide CAC services for clients located in that area. Another pool of CAC servers might be located in the eastern United States and configured to provide CAC services to clients located in that area and in the mid-western United States. Other geographic configurations might also be utilized.
According to another aspect, each of the CAC servers in a pool of CAC servers establishes a network connection to each of the other CAC servers in the same pool. For instance, in one embodiment each of the CAC servers in a pool establishes and maintains a transmission control protocol (“TCP”) connection to every other CAC server in the same pool. These network connections might be utilized to synchronize data among CAC servers in the same pool (referred to herein as “intra-pool” data synchronization). As an example, one of the CAC servers in a pool may receive session data from a client. Session data is data relating to a particular session, such as data indicating the amount of bandwidth utilized for a session. When such data is received at a CAC server, the CAC server synchronizes the data with each of the other CAC servers in the same pool by way of the network connections established with the other CAC servers.
According to another aspect, each of the CAC servers in a pool are further configured to establish exactly one network connection to a CAC server in another pool of CAC servers. For instance, each of the CAC servers in a first pool might establish exactly one network connection to a CAC server in a second pool. The CAC servers in the second pool also establish exactly one network connection to CAC servers in the first pool. The CAC server to which each connection is established might be selected randomly, using a round-robin approach, or by way of another mechanism. These network connections might be utilized to synchronize data between CAC servers in different pools (referred to herein as “inter-pool” data synchronization).
When data is received at a CAC server that is to be synchronized to another pool, the CAC server transmits the data to a CAC server in the other pool by way of a network connection to a CAC server in the other pool. For instance, status updates might be periodically synchronized by way of an incoming network connection from a CAC server in a different pool. A status update is data identifying the status of one or more network links and may include information such as data identifying a network link, a media type (such as voice or audio), and a current utilization of the network link. Commit data indicating that a portion of the bandwidth of a network link has been committed to a session might also be synchronized between pools and between CAC servers in the same pool in a similar fashion.
This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended that this Summary be used to limit the scope of the claimed subject matter. Furthermore, the claimed subject matter is not limited to implementations that solve any or all disadvantages noted in any part of this disclosure.
The following detailed description is directed to technologies for efficient connection management and synchronization in the provision of CAC services. As discussed briefly above, CAC servers in a pool are fully connected to one another by way of network connections. Session data received at one of the CAC servers in a pool is synchronized to each of the other CAC servers in the pool by way of the connections. Each CAC server in a first pool may also establish a connection with exactly one CAC server in a second pool. Status updates and commit data are synchronized from a CAC server in the first pool to a CAC server in the second pool. Status updates may be synchronized on incoming network connections from the second pool. The commit data may also be synchronized to each of the CAC servers in the second pool. In this manner, fewer inter-pool network connections are required and less data is utilized to synchronize data between pools that in previous solutions.
While the subject matter described herein is presented in the general context of program modules that execute in conjunction with the execution of an operating system and application programs on a computer system, those skilled in the art will recognize that other implementations may be performed in combination with other types of program modules. Generally, program modules include routines, programs, components, data structures, and other types of structures that perform particular tasks or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the subject matter described herein may be practiced with other computer system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, minicomputers, mainframe computers, and the like.
In the following detailed description, references are made to the accompanying drawings that form a part hereof, and which are shown by way of illustration specific embodiments or examples. Referring now to the drawings, in which like numerals represent like elements through the several figures, aspects of various technologies for efficient connection management and synchronization in the provision of CAC services will be described.
FIG. 1
FIG. 1
FIG. 1
100
102
102
111
102
102
Turning now to , details will be provided regarding one embodiment presented herein for efficient connection management and synchronization in the provision of CAC services. In particular, is a network diagram showing an illustrative operating environment along with several components provided according to embodiments presented herein. As shown in , the environment includes a central site A configured for communication with a central site B by way of a WAN link . As discussed briefly above, the central sites A-B may be located in different physical locations around a city, state, country, or even around the world.
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In order to provide data and voice connectivity between the central sites A-B, a WAN link has been established between the two locations. The WAN link can be utilized to support data communications between the central sites A-B, including audio and/or audio/video sessions made between client devices situated at the central sites A-B. For instance, VoIP calls may be made between clients located at the central sites A-B. Similarly, video calls might also be made between clients placed at the central sites A-B. It should be appreciate that the term “call” and “session” may be used interchangeably herein.
FIG. 1
FIG. 1
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As shown in , the central sites A-B include a number of clients A-B (which may be referred to herein collectively as the clients ). For instance, the central site B includes the client B and the central site A includes the client A. The clients A-B are devices that are capable of establishing an audio or video communications session with another client via a network link. For instance, in various embodiments, the clients A-B may be wired or wireless telephones configured for communication over a data communications connection, desktop, laptop, or other types of computers configured with an appropriate software client for audio and/or video communications over a data communications link, and other types of devices. It should be appreciated that while illustrates two central sites A-B and two clients A-B, many more network sites and clients may be utilized. It should also be appreciated that the environment has been shown in a simplified form and that many more network connections and computing systems may be utilized in order to implement the various concepts disclosed herein.
FIG. 1
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As also shown in , each of the central sites includes a server pool . For instance, the central site A has a server pool A and the central site B includes a server pool B. The server pools A-B are installations of one or more server computers configured to provide services to the clients A-B. For instance, according to various embodiments, the server pools A-B may assist the clients A-B during call setup and establishment, name resolution, media conversion, and in other ways. In order to provide this functionality, the servers operating within the server pools A-B may be configured to expose a number of services to the clients A-B.
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In one implementation, the pools A-B include one or more CAC servers A-F (which may be referred to collectively as “the CAC servers ”). As discussed above, a CAC server provides functionality for managing bandwidth utilization. For instance, a CAC server might participate during the setup of a new call between the clients A-B to ensure that enough bandwidth is available on the WAN link to support the call. If sufficient bandwidth is not available, the CAC server might not allow the call to be completed, or might cause the call to be routed over another network such as the PSTN or the Internet. For instance, the CAC servers might receive and respond to bandwidth policy requests and other queries received from the clients A-B. A bandwidth policy request is a request for a specified amount of bandwidth for a new media session.
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FIGS. 2A-9
In one implementation, one of the clients A-B generates a bandwidth policy request prior to the establishment of a media session. If a CAC server determines that the requested amount of bandwidth can be granted, the CAC server may permit the new call to be completed. Thereafter, the clients A and B will establish a media session and begin transmitting media over the WAN link . For instance, if the session established between the client A and the client B is a voice session, the clients A and B will transmit appropriate audio data across the WAN link . The session between the client A and the client B may then continue and be terminated in a conventional fashion. Additional details regarding the operation of the CAC servers will be provided below with respect to the .
FIG. 1
FIGS. 2A-2C
112
110
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112
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112
As illustrated in , each of the CAC servers might also maintain a runtime database, such as the runtime database A maintained by the CAC server A. The runtime databases maintain, among other things, data regarding the current utilization of the network links for which a particular pool is responsible. For instance, the runtime database A is utilized to maintain data regarding one or more network links assigned to the CAC servers A-C. Details regarding the type of data that might be stored in the runtime databases will be provided below with regard to .
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FIGS. 3-9
In order to ensure that each of the CAC servers A-F has access to data for making decisions regarding bandwidth allocation (such as data stored in a runtime database ), the CAC servers A-F are configured to synchronize data with one another. Intra-pool synchronization might be utilized to synchronize data between CAC servers in the same pool, such as the CAC servers A-C. Inter-pool synchronization might be utilized to synchronize data between CAC servers in different pools, such as between the CAC server F in the pool B and the CAC server A in the pool A. Details regarding several embodiments disclosed herein for both intra-pool and inter-pool data synchronization will be provided below with regard to .
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FIG. 1
FIG. 1
It should be appreciated that although six CAC servers A-F have been shown in , more or fewer CAC servers might be utilized in a particular network installation. Additionally, although two pools A-B have been shown in , more or fewer pools might also be utilized. The number of CAC servers and pools might be selected based upon call volume, network topology, geographic considerations, and other factors.
FIGS. 2A-2C
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200
are data structure diagrams showing aspects of several data structures for storing WAN link capacity data, WAN link utilization data, and a client bandwidth allocation according to one embodiment disclosed herein. As discussed briefly above, the data structures A-C may be stored in the runtime database and utilized by the CAC servers . It should be appreciated that the data structures A-C are merely illustrative and that other data structures or types might be utilized to store the data described herein. The data might be stored in a completely different fashion and additional data might also be stored.
FIG. 2A
FIG. 2A
FIG. 2A
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shows a data structure A configured for storing data indicating WAN link capacity. The CAC servers might utilize information regarding the capacity and utilization of a WAN link to determine whether a request for bandwidth may be granted and to provide other functionality. In the illustrative embodiment shown in , the data structure A includes a first field A that identifies the WAN link . The data structure A also includes a field B that identifies a particular media type, such as audio or video. The data structure A also includes a field C that identifies the maximum bandwidth for a particular link and media type. For instance, in the example shown in , data has been provided for a WAN link capable of supporting five mega-bits per second (“Mbps”) of audio and three Mbps of video.
FIG. 2B
FIG. 2B
FIG. 2B
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shows a data structure B utilized to maintain data indicating a current utilization of a WAN link . The data stored in the data structure B might also be utilized by the CAC servers to determine whether a request for bandwidth may be granted and to provide other functionality. The illustrative data structure B shown in includes a field D for storing data identifying the particular WAN link , a field E for storing data identifying a particular media type, and a field F for storing data identifying a current allocation of bandwidth for the link and media type. For instance, in the example shown in , the data structure B includes data indicating that the WAN link currently has four Mbps of audio and two Mbps of video allocated thereto. Accordingly, in view of the example data structures A and B, the CAC servers may conclude that the WAN link has one Mbps available for audio and one Mbps available for video sessions.
FIG. 2C
FIG. 2C
FIG. 2C
200
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shows an illustrative data structure C that may be utilized to store data indicating the amount of bandwidth utilized by each client in an in-progress media session. In the illustrative example shown in , the data structure C includes a field G for storing data identifying each client in an in-progress media session and a field H for storing data indicating the currently allocated bandwidth to each client. In the example shown in , one hundred kilo-bits per second (“kbps”) has been allocated to two clients. It should be appreciated that the data structure C has also been simplified and that other data might also be stored in the data structure C for use by the CAC servers .
FIG. 3
FIG. 3
302
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is a network diagram showing an illustrative network connections made between CAC servers A-C in a pool A of servers according to one implementation. As discussed briefly above, the CAC servers A-C are organized into a group referred to herein as pool A. As also discussed briefly above, the CAC servers A-C in a given server pool A might be assigned various network links for which they are responsible. The CAC servers A-C are then responsible for management of the bandwidth on the assigned network links. For instance, the CAC servers A-C might receive and respond to requests to allocate bandwidth to clients on the assigned networks. Depending upon the number of requests to be processed by the CAC servers , more or fewer CAC servers might be utilized in a particular network installation than shown in .
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FIG. 3
FIGS. 4-5
As also discussed briefly above, intra-pool synchronization may be performed in order to ensure that each of the CAC servers A-C in a pool A has access to the same data utilized to make bandwidth policy discussions. In order to enable this functionality, each of the CAC servers A-C in a server pool A may be fully connected to one another. In order to establish these connections, each of the CAC servers A-C in a pool A establishes a network connection to each of the other CAC servers in the same pool. For instance, in the illustrative embodiment shown in , the CAC server A has established a network connection C with the CAC server B and a network connection A with the CAC server C. Similarly, the CAC server B has established a network connection D with the CAC server A and a network connection F with the CAC server C. The CAC server C has established a network connection B with the CAC server A and network connection E with the CAC server B. As will be discussed in greater detail below, the CAC servers A-C utilize the network connections A-F for intra-pool synchronization. Additional details regarding intra-pool synchronization will be provided below with respect to .
FIG. 4
FIG. 4
110
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402
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110
is a network diagram showing aspects of one embodiment for synchronizing data among CAC servers A-C in a pool A of CAC servers. In the example shown in , a client A has transmitted session data to the CAC server A. As discussed briefly above, session data is data relating to a particular session such as data indicating the amount of bandwidth utilized for a session. As an example, the client A may transmit session data to a CAC server A following establishment of a session that indicates the amount of bandwidth actually utilized for the session.
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110
When the session data is received at the CAC server A, the CAC server synchronizes the session data to the CAC servers B and C. In particular, the connection C is utilized to synchronize the session data with the CAC server B and the connection A is utilized to synchronize the session data to the CAC serve C.
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FIG. 4
When the session data is received by the CAC servers B and C, these CAC servers might update their respective instances of the runtime database to reflect the newly received session data . In this manner, each of the CAC servers A-C continually maintains data that is consistent with the other CAC servers for making bandwidth policy decisions. It should be appreciated that, although not illustrated in , the CAC servers B and C also operate similarly to synchronize session data with to the CAC servers in the same server pool A.
FIG. 5
FIG. 5
500
110
Turning now to , additional details will be provided regarding the embodiments presented herein for efficient connection management and synchronization in the provision of CAC services. In particular, shows a routine that illustrates aspects of the operation of the CAC servers for intra-pool synchronization according to one embodiment disclosed herein.
It should be appreciated that the logical operations described herein are implemented (1) as a sequence of computer implemented acts or program modules running on a computing system and/or (2) as interconnected machine logic circuits or circuit modules within the computing system. The implementation is a matter of choice dependent on the performance and other requirements of the computing system. Accordingly, the logical operations described herein are referred to variously as operations, structural devices, acts, or modules. These operations, structural devices, acts and modules may be implemented in software, in firmware, in special purpose digital logic, and any combination thereof. It should also be appreciated that more or fewer operations may be performed than shown in the figures and described herein. These operations may also be performed in a different order than those described herein.
500
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110
106
FIG. 3
The routine begins at operation , where each of the CAC servers establishes a network connection to every other CAC server in the same pool in the manner described above with respect to . As discussed briefly above, these network connections may comprise TCP connections or other suitable network connections.
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Once the network connections have been established between CAC servers in the same server pool , the routine proceeds from operation to operation . At operation the CAC servers determine whether session data has been received. If session data has not been received, the routine proceeds again to operation where a similar determination is made. If, however, session data has been received, the routine proceeds from operation to operation .
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At operation , the CAC server synchronizes the session data to all of the other CAC servers in the same pool . As discussed above, the network connections from the CAC server that receives the session data to the other CAC servers in the same server pool might be utilized to synchronize the session data . Once the session data has been synchronized to each of the other CAC servers in the same server pool , the routine returns to operation , described above.
FIG. 6
602
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106
Referring now to , a network diagram will be described that shows illustrative network connections made between CAC servers A-C in a first pool A and CAC servers D-G in a second pool B. As discussed briefly above, in one embodiment the CAC servers in each pool are further configured to perform inter-pool synchronization. In this manner, CAC servers located in different pools can synchronize data necessary for making bandwidth policy decisions to one another. This information might then be utilized by a CAC server to respond to bandwidth requests for network links owned by another server pool . Additional details regarding this process will be provided below.
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According to one embodiment, each of the CAC servers are configured to establish exactly one network connection to a CAC server in another pool of CAC servers for use in inter-pool synchronization. For instance, each of the CAC servers A-C might establish exactly one network connection to a CAC server in the server pool B. The CAC servers D-G in the pool B may also establish exactly one network connection to CAC servers A-C in the pool A.
FIG. 6
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In the example shown in , the CAC server A has established a network connection A to the CAC server G, the CAC server B has established a network connection D to the CAC server E, and the CAC server C has established a network connection E to the CAC server E. The CAC server G has established a connection B to the CAC server B, the CAC server F has established a network connection C to the CAC server B, the CAC server E has established a network connection F to the CAC server C, and the CAC server D has established the network connection G to the CAC server C.
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According to various embodiments, the particular CAC server to which a network connection is established might be selected randomly. For instance, a load balancer might be placed in front of the CAC servers D-G that randomly selects a CAC server D-G to which a new connection is to be established. In another embodiment, a round robin approach might be utilized to select a particular CAC server to which a new network connection is to be established. In this example, each of the CAC servers A-D might maintain a list of CAC servers in other pools and utilize this list to select a CAC server for connection utilizing a round robin algorithm. It should be appreciated that other mechanisms might also be utilized to select the particular CAC server to which a network connection will be established.
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FIG. 6
FIG. 6
FIG. 6
As also discussed briefly above, each of the CAC servers are configured to periodically transmit a status update to a CAC server in another server pool. For instance, in the example shown in , the CAC server G is transmitting a status update to the CAC server A. According to one implementation, the CAC servers synchronize data, such as the status update , on an incoming connection. For instance, in the example shown in , the connection A was established by the CAC server A to the CAC server G. Accordingly, from the perspective of the CAC server G, the connection A is an incoming connection. Each of the CAC servers is configured to synchronize data on an incoming network connection in the manner shown in in this manner.
604
604
604
FIG. 7
FIG. 7
As discussed briefly above, the status update is data identifying the status of one or more network links. As will be described below with reference to , the status update might include data identifying a network link, a media type, and a current utilization of the network link. Additional details regarding an illustrative data structure for implementing the status update will be provided below with reference to .
FIG. 7
FIG. 7
FIG. 7
FIG. 7
604
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Turning now to , an illustrative data structure will be described for implementing the status update . The illustrative data structure shown in includes a first field A for identifying a particular network link. The data structure also includes a field B for identifying a particular media type, such as audio or voice, on the corresponding network link. The data structure also includes a field G that includes data indicating the currently utilization of the network link identified in the field A for the media type identified in the field B. In the example shown in , a link is described for which four out of six available Mbps have been utilized for audio and for which two out of an available three Mbps have been utilized for video. It should be appreciated that a status update might include additional information than illustrated in and be formatted in an entirely different fashion.
FIG. 8
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Referring now to , a network diagram will be described showing aspects of one embodiment for synchronizing commit data between two pools of CAC servers according to one embodiment presented herein. As discussed briefly above, commit data is data indicating that a portion of the bandwidth of a particular network link has been committed for a session. A CAC server , such as the CAC server G in the pool B, may synchronize the commit data to another pool A when the CAC server G handles a request for bandwidth for a network connection owned by another server pool A. Because, as discussed above, each of the pools A-B is responsible for processing requests for a particular segment of the network, and because each of the pools A-B is synchronized with the other pool, a bandwidth policy decision for a media session can be made by every CAC server A-G. Previous solutions required propagating a decision request to the concerned owner pool during call setup. Implementations utilizing the technologies presented herein can reduce call setup time by a round trip between locations of the CAC server pools as compared to previous solutions.
FIG. 8
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In the example shown in , the CAC server G has processed a bandwidth policy decision for a media session which resulted in bandwidth being committed for a network link managed by the server pool A. In order to ensure that each of the CAC servers A-G has access to consistent bandwidth information, the CAC server G synchronizes the commit data to the CAC server B via the outgoing connection B. The CAC server B might also synchronize the commit data to the CAC servers D-F in the pool B.
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When the CAC server B receives the commit data , the CAC server B may update its instance of the runtime database . The CAC server B also synchronizes the commit data to the CAC servers A and C. The CAC servers A and C, in turn, update their instances of the runtime database . In this manner, the commit data is synchronized between each of the CAC servers in the pools A-B.
FIG. 9
FIG. 7
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Referring now to , a flow diagram will be described showing an illustrative routine for synchronizing data between CAC servers in a first pool A of CAC servers A-C and CAC servers D-G in a second pool B of CAC servers according to one embodiment disclosed herein. The routine begins at operation , where each of the CAC servers A-G establishes a connection to exactly one other CAC server A-G in another pool. This process was described above with reference to . Once the connections A-G have been established, the routine proceeds from operation to operation .
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FIG. 6
At operation , each of the CAC servers A-G periodically transmits a status update on an incoming network connection. For instance, as discussed above with reference to , the CAC server G may transmit a status update on the incoming network connection A to the CAC server A. The receiving CAC server A then synchronizes the status update to the other CAC servers B-C in the same server pool A.
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FIG. 8
From operation , the routine proceeds to operation where each of the CAC servers A-G make a determination as to whether a commit was performed on behalf of another server pool. If not, the routine proceeds to operation , described above. If so, the routine proceeds from operation to operation where the CAC server performing the commit operation synchronizes the commit data to the proper pool on an outgoing network connection. For instance, as discussed above, with reference to , the CAC server G may receive a commit for bandwidth on a network link assigned to the server pool A. In response thereto, the CAC server G transmits the commit data to the CAC server B on the outgoing network connection B. The CAC server B then synchronizes the commit data to the CAC servers A and C. From operation , the routine proceeds to operation , described above. It should be appreciated that commit data might only be synchronized to another pool when the commit involves network routes that belong to another pool.
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It should be appreciated that, according to embodiments, each client transmits a request during call setup to a CAC server to determine if requested bandwidth is available (multiple paths could be checked for the bandwidth availability). If the bandwidth is available, the client transmits data (called “a commit”) indicating how much bandwidth was actually utilized. In an infrastructure with multiple pools where network links are owned by different pools , the checks are typically answered by the CAC server that receives the check.
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FIG. 1
Since, as described above, each CAC server has the current usage details because of status updates and commits received from the other CAC servers , a check can be answered by any CAC server without contacting the other CAC servers . Later, when the client sends the commit, the commit is transmitted to the various CAC servers as described above. Call setup time may therefore be minimized where a local CAC server answers a bandwidth check. In the embodiment shown in , for instance, the client A might primarily use the network paths owned by the CAC servers in pool A and, similarly, the client B might utilize network paths owned by pool B. As a result, a brief delay in transmitting status updates (status updates may be transmitted periodically as discussed above) to another pool is acceptable.
FIG. 10
FIG. 10
1000
shows an illustrative computer architecture for a computer capable of executing the software components described herein for efficient connection management and synchronization in the provision of CAC services. The computer architecture shown in illustrates a conventional desktop, laptop computer, or server computer and may be utilized to execute the various software components described herein.
FIG. 10
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The computer architecture shown in includes a central processing unit (“CPU”), a system memory , including a random access memory (“RAM”) and a read-only memory (“ROM”) , and a system bus that couples the memory to the CPU . A basic input/output system (“BIOS”) containing the basic routines that help to transfer information between elements within the computer , such as during startup, is stored in the ROM . The computer further includes a mass storage device for storing an operating system , application programs, and other program modules, which will be described in greater detail below.
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The mass storage device is connected to the CPU through a mass storage controller (not shown) connected to the bus . The mass storage device and its associated computer-readable media provide non-volatile storage for the computer . Although the description of computer-readable media contained herein refers to a mass storage device, such as a hard disk or CD-ROM drive, it should be appreciated by those skilled in the art that computer-readable storage media can be any available computer storage media that can be accessed by the computer .
1000
By way of example, and not limitation, computer-readable storage media may include volatile and non-volatile, removable and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules or other data. For example, computer-readable storage media includes, but is not limited to, RAM, ROM, EPROM, EEPROM, flash memory or other solid state memory technology, CD-ROM, digital versatile disks (“DVD”), HD-DVD, BLU-RAY, or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other non-transitory medium which can be used to store the desired information and which can be accessed by the computer .
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FIG. 10
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According to various embodiments, the computer may operate in a networked environment using logical connections to remote computers through a network such as the network . The computer may connect to the network through a network interface unit connected to the bus . It should be appreciated that the network interface unit may also be utilized to connect to other types of networks and remote computer systems. The computer may also include an input/output controller for receiving and processing input from a number of other devices, including a keyboard, mouse, or electronic stylus (not shown in ). Similarly, an input/output controller may provide output to a display screen, a printer, or other type of output device (also not shown in ).
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As mentioned briefly above, a number of program modules and data files may be stored in the mass storage device and RAM of the computer , including an operating system suitable for controlling the operation of a networked desktop, laptop, or server computer. The mass storage device and RAM may also store one or more program modules. In particular, the mass storage device and the RAM may store program modules which, when executed, cause the CAC servers to operate in the manner described above. The mass storage device and RAM may also store other program modules and data, such as the runtime database .
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In general, software applications or modules may, when loaded into the CPU and executed, transform the CPU and the overall computer from a general-purpose computing system into a special-purpose computing system customized to perform the functionality presented herein. The CPU may be constructed from any number of transistors or other discrete circuit elements, which may individually or collectively assume any number of states. More specifically, the CPU may operate as one or more finite-state machines, in response to executable instructions contained within the software or modules. These computer-executable instructions may transform the CPU by specifying how the CPU transitions between states, thereby physically transforming the transistors or other discrete hardware elements constituting the CPU .
Encoding the software or modules onto a mass storage device may also transform the physical structure of the mass storage device or associated computer readable storage media. The specific transformation of physical structure may depend on various factors, in different implementations of this description. Examples of such factors may include, but are not limited to: the technology used to implement the computer readable storage media, whether the computer readable storage media are characterized as primary or secondary storage, and the like. For example, if the computer readable storage media is implemented as semiconductor-based memory, the software or modules may transform the physical state of the semiconductor memory, when the software is encoded therein. For example, the software may transform the states of transistors, capacitors, or other discrete circuit elements constituting the semiconductor memory.
As another example, the computer readable storage media may be implemented using magnetic or optical technology. In such implementations, the software or modules may transform the physical state of magnetic or optical media, when the software is encoded therein. These transformations may include altering the magnetic characteristics of particular locations within given magnetic media. These transformations may also include altering the physical features or characteristics of particular locations within given optical media, to change the optical characteristics of those locations. Other transformations of physical media are possible without departing from the scope and spirit of the present description, with the foregoing examples provided only to facilitate this discussion.
Based on the foregoing, it should be appreciated that technologies for efficient connection management and synchronization in the provision of CAC services have been presented herein. Although the subject matter presented herein has been described in language specific to computer structural features, methodological acts, and computer readable media, it is to be understood that the invention defined in the appended claims is not necessarily limited to the specific features, acts, or media described herein. Rather, the specific features, acts and mediums are disclosed as example forms of implementing the claims.
The subject matter described above is provided by way of illustration only and should not be construed as limiting. Various modifications and changes may be made to the subject matter described herein without following the example embodiments and applications illustrated and described, and without departing from the true spirit and scope of the present invention, which is set forth in the following claims.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is a network diagram showing an illustrative operating environment along with several components provided according to embodiments presented herein;
FIGS. 2A-2C
are data structure diagrams showing aspects of data structures for storing WAN link capacity data, WAN link utilization data, and client bandwidth allocation data according to one embodiment disclosed herein;
FIG. 3
is a network diagram showing illustrative network connections made between CAC servers in a pool of CAC servers according to one implementation;
FIG. 4
is a network diagram showing aspects of one embodiment for synchronizing session data among CAC servers in a pool of CAC servers;
FIG. 5
is a flow diagram showing aspects of one process presented herein for synchronizing data between CAC servers in a pool of CAC servers;
FIG. 6
is a network diagram showing illustrative network connections made between CAC servers in a first pool of CAC servers and CAC servers in a second pool of CAC servers according to one implementation;
FIG. 7
is a data structure diagram showing aspects of a data structure for storing status update data utilized in one embodiment disclosed herein;
FIG. 8
is a network diagram showing aspects of one embodiment for synchronizing commit data between two pools of CAC servers in one embodiment disclosed herein;
FIG. 9
is a flow diagram showing aspects of one process presented herein for synchronizing data between CAC servers in a first pool of CAC servers and CAC servers in a second pool of CAC servers according to one embodiment disclosed herein; and
FIG. 10
is a computer architecture diagram showing an illustrative computer hardware and software architecture for a computing system capable of implementing the embodiments presented herein. | |
Joan's recipes from the Millpond Inn B&B
Because Joan is asked for the recipe for her yogurt constantly, we are adding it here:
Ingredients:
Starter: you need 8 oz of a natural plain yogurt or A freeze dried yogurt product called Yo’gourmet
Milk, any kind or quantity you like. I use whole and make a gallon at a time. The gallon of milk will make about two quarts of drained yogurt
The whole process takes 24 hours
Process:
Cook milk in a large covered pot on a medium low heat till a layer of bubbles covers the top. This is between 180-190 degrees
Drop the heat to your lowest setting, crack the cover, and let it cook for and additional 1-2 hours
Remove cover and let milk cool to 110-115 degrees. NO HOTTER
Break holes in the skim around the edge and in the middle and add starter
If you are using yogurt as a starter, take it out of the frig when you start cooling your milk. This will allow it to warm a little
Wrap the covered pot in 2-4 bath towels to keep it warm and put in a warm spot for 10-12 hours. I put mine on a griddle set at warm. Hot water tanks and boilers also work well
Refrigerate the Yogurt for 2-4 hour.
Drain the yogurt in a colander lined with 2 layers of paper towels, in the Frig. If you don’t want the skim in the yogurt spoon it off before draining and use in smoothie
Reduce the yogurt by half
To make fruit flavored add honey or sweetener to taste, a bit of vanilla and any fresh, frozen fruit or even jam.. Enjoy
Joan's yogurt recipe - It has been called the best in the world. | https://millpondinnbb.com/joan-s-recipes.html |
Q:
Let $f:I_m\to I_n$ be injective. Then $m\leq n$
Please check if my proof is fine or it contains any error!
Theorem: Let $I_m=\{i\in\mathbb N\mid i\leq m\},I_n=\{i\in\mathbb N\mid i\leq n\},\text{ and } f:I_m\to I_n$ be injective. Then $m\leq n$.
Let $T=\{m\in\mathbb N\mid \forall n\in\Bbb N(f:I_m\to I_n\text{ be injective }\implies m\leq n)\}$. Since $0\leq n$ for all $n\in\mathbb N,0\in T$. Assume $k\in T$, then $\forall n\in\Bbb N(f:I_k\to I_n\text{ is injective }\implies k\leq n)$. I will prove $k+1\in T$ as follows.
Let $g:I_{k+1}\to I_n$ be injective. $I_{k+1}$ contains at least two points and $g$ is injective, so does $I_n$, or equivalently $n=t+1$. Let $\tau:I_n\to I_n$ be a mapping that transposes $g(k+1)$ and $n$, and leaves the other elements of $I_n$ fixed. Then $\tau\circ g:I_{k+1}\to I_n$ is injective, and $\tau\circ g(k+1)=n$. As a result, ${\tau\circ g}_{|I_k}I_k\to I_t$ is injective $\implies k\leq t$ [Since the theorem is true for $m=k$] $\implies k+1\leq t+1\implies k+1\leq n$. Thus $k+1\in T$.
By principle of induction, $T=\mathbb N$, and consequently the theorem is proved. $\blacksquare$
A:
A Proof by contradiction:
If possible suppose $m > n$, then $I_m$ must have at least two points which have the same image. (Piegon Hole principle)
And thus, $f: I_m \to I_n$ is not injective, which is a contradiction.
| |
---
abstract: 'This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples.'
address: |
Igor Belegradek\
School of Mathematics\
Georgia Institute of Technology\
Atlanta, GA 30332-0160
author:
- Igor Belegradek
bibliography:
- 'wp-bangalore.bib'
title: '**Topology of open nonpositively curved manifolds**'
---
plus3pt minus3pt plus3pt minus3pt
[^1]
Among many monographs and surveys on aspects of nonpositive curvature, none deals with topology of open complete nonpositively curved manifolds, and this paper aims to fill the void. Most of the material discussed here is not widely-known. A number of questions is posed, ranging from naive to hopelessly difficult. Proofs are supplied when there is no explicit reference, and as always, the author is solely responsible for mistakes.
This survey has a narrow focus and does not discuss topological properties of
- hyperbolic $3$-manifolds [@Thu-3d-notes; @Kap-book; @CanMcC],
- open negatively pinched manifolds [@Bel-invent; @BK-acta],
- non-Riemannian nonpositively curved manifolds [@Dav-book],
- compact nonpositively curved manifolds [@FJO-survey; @Lue-surv-asph],
- higher rank locally symmetric spaces [@Mar-book; @Ebe-book; @BesFei02; @Gel-vol-rk] and their compactifications [@BorJi],
which are covered in the above references. We choose to work in the Riemannian setting which leads to some simplifications, even though many results hold in a far greater generality, and the references usually point to the strongest results available. Special attention is given to rank one manifolds, such as manifolds of negative curvature. Nonpositively curved manifolds are aspherical so we focus on groups of finite cohomological dimension (i.e. fundamental groups of aspherical manifolds), or better yet, groups of type [*F*]{} (i.e. the fundamental groups of compact aspherical manifolds with boundary).
[**Conventions:**]{} unless stated otherwise, manifolds are smooth, metrics are Riemannian, and sectional curvature is denoted by $K$.
[**Acknowledgments:**]{} The author is grateful for NSF support (DMS-1105045). Thanks are due to Jim Davis, Denis Osin, Yunhui Wu, and the referee for correcting misstatements in the earlier version.
Flavors of negative curvature {#sec: flavors of neg curv}
=============================
A [*Hadamard manifold*]{} is a connected simply-connected complete manifold of $K\le 0$. By the Cartan-Hadamard theorem, any Hadamard manifold is diffeomorphic to a Euclidean space. Thus any complete manifold of $K\le 0$ is the quotient of a Hadamard manifold by a discrete torsion-free isometry group (torsion-freeness can be seen geometrically: any finite isometry group of a Hadamard manifold fixes the circumcenter of its orbit, or topologically: a nontrivial finite group has infinite cohomological dimension so it cannot act freely on a contractible manifold).
A Hadamard manifold is a [*visibility*]{} manifold if any two points at infinity can be joined by a geodesic. A [*flat*]{} in $X$ is a convex subset isometric to a Euclidean space. A [*flat half plane*]{} in $X$ is a convex subset isometric to a Euclidean half plane. An [*infinitesimal flat*]{} of a geodesic is the space of parallel Jacobi fields along a geodesic. A geodesic is called [*rank one*]{} if its infinitesimal flat is one-dimensional. A complete manifold of $K\le 0$ [*has rank one*]{} if it contain a complete (i.e. defined for all times) geodesic of rank one.
The following conditions on a Hadamard manifold $X$ represent various manifestations of negative curvature:
- $K$ is bounded above by a negative constant,
- $X$ is Gromov hyperbolic,
- $X$ is a visibility manifold,
- $K<0$,
- $X$ contains no flat half plane,
- $X$ contains a complete geodesic that does not bound a flat half plane.
The implications $(1)\Rightarrow (4)\Rightarrow (5)\Rightarrow (6)$ and $(1)\Rightarrow (2)\Rightarrow (3)\Rightarrow (5)\Rightarrow (6)$ are immediate from definitions except for $(2)\Rightarrow (3)$ which can be found in [@BriHae Lemma VIII.3.2].
\[prop: curv flavors\] Every other implication fails in dimension two.
There are of course analogs of (5) or (6) such as “no flat strip” or “no flat plane”, see also various axioms on [@EbeOni], but the above list is what comes up most often.
Kleiner [@Kle-inf-flat] proved that a Hadamard manifold has rank one if and only if it contains a complete geodesic that does not lie in a two-dimensional flat (see [@Bal-book Proposition IV.4.4] for the case when ${\mbox{Iso}}(X)$ satisfies the duality condition, e.g. contains a lattice). Since (6) is intermediate between the two conditions, it is equivalent to them.
That $(2)\nRightarrow (4)$ follows by doubling along the boundary any nonpositively curved compact surface of negative Euler characteristic whose metric is cylindrical near the boundary; the universal cover is hyperbolic because hyperbolicity is a quasiisometry invariant, but it contains a flat strip.
To show that $(3)\nRightarrow (2)$ recall a consequence of the Gauss-Bonnet theorem, mentioned on [@BGS page 57], that a $2$-dimensional Hadamard manifold $X$ is visibility if and only if for some $p\in X$ the total curvature of every sector bounded by two rays that start at $p$ is infinite. Any smooth nonpositive function $K$ on $[0,\infty)$ can be realized as the curvature of a rotationally symmetric metric $dr^2+f(r)^2d\theta$ on $\mathbb R^2$, namely, $f$ is a unique solution of $f^{\prime\prime}+Kf=0$, $f(0)=1$, $f^\prime(0)=1$. Note that $f(r)\ge r$ by Sturm comparison for ordinary differential equations, and the total curvature equals $2\pi\int_0^\infty Kfdr$. Let ${I_i}$ be a sequence of disjoint compact subintervals of $(0,\infty)$ such that $I_i$ has length $i$, and let $K$ be a smooth negative function on $[0,\infty)$ that equals $-\frac{1}{i}$ on $I_i$. The associated rotationally symmetric metric has infinite total curvature on every sector from the origin, so it is visibility, and it contains arbitrary large regions of curvature $-\frac{1}{i}$, which contain triangles violating $\delta$-hyperbolicity for any $\delta$.
That $(4)\nRightarrow (3)$ is also shown in [@EbeOni Example 5.10]: modifying the argument of the previous paragraph yields a rotationally symmetric metric with $K<0$ and finite total curvature.
To see that $(6)\nRightarrow (5)$ start from a finite volume complete hyperbolic metric on a punctured torus, and modify the metric near the end to a complete metric of $K\le 0$ that is a cylinder outside a compact set. The cylinder lifts to a flat half plane so (5) fails, but any geodesic through a point of $K<0$ satisfies (6).
The other non-implications are formal consequence of the above implications and non-implications.
The non-implications in the above proof are justified by two dimensional examples, and it seems that similar examples in higher dimensions can be produced via iterated warping with $\cosh r$, i.e. replacing $X$ with $\mathbb R\times_{\cosh r} X$, which contains $X$ as a totally geodesic submanifold. For some non-implications we need to insert the warped product as a convex subset in a closed manifold of $K\le 0$ following Ontaneda, see [@Pha-hypersurf]. The warping clearly preserves conditions (1), (4), (5), (6).
Does the warping with $\cosh r$ preserve and ?
One wants to understand the relations among (1)–(6) when ${\mbox{Iso}}(X)$ is large.
If ${\mbox{Iso}}(X)$ is cocompact, then clearly $(1)\Leftrightarrow (4)$, and also $(2)\Leftrightarrow (3)\Leftrightarrow (5)$; in fact, Gromov hyperbolicity is equivalent to uniform visibility for proper CAT($0$) spaces [@BriHae Proposition III.3.1], and visibility is equivalent to the non-existence of a $2$-flat for proper CAT($0$) spaces with cocompact isometry groups [@BriHae Theorem II.9.33]. For $2$-dimensional Hadamard manifolds with cocompact isometry group $(5)\Leftrightarrow (6)$ because if $X$ contains a flat half plane and has cocompact isometry group, then $X$ is isometric to $\mathbb R^2$. On the other hand, $(6)\nRightarrow (5)$ in dimensions $>2$ with examples given by the [*Heintze manifolds*]{}, obtained by chopping of cusps of a finite volume complete hyperbolic manifold, changing the metric near the cusp a metric of $K\le 0$ with totally geodesic flat boundary, and doubling along the boundary; the boundary lifts to a flat of dimension $>1$, while any geodesic through a point of $K<0$ has rank one.
If $\dim(X)=2$ and ${\mbox{Iso}}(X)$ contains a lattice, then either $X$ is isometric to $\mathbb R^2$ or $X$ is visibility [@Ebe-MemAMS-1979 Proposition 2.5] so that $(6)\Leftrightarrow (3)$.
If $\dim(X)>2$, then $(6)\nRightarrow (3)$; indeed Nguyen Phan [@Pha-neg] and Wu [@Wu-transl-length] show that the universal covers of the finite volume manifolds of $K<0$ constructed in [@Fuj-warp] are not visibility, so $(4)\nRightarrow (3)$.
Eberlein showed [@Ebe-lattices-annals] that the ends of a finite volume complete manifold with bounded nonpositive curvature and visibility universal cover are $\pi_1$-injectively embedded. This is not the case for Buyalo’s example [@Buy] of a finite volume complete $4$-manifold of $-1<K<0$; thus its universal cover is not visibility.
Let us compare $(3)$ and $(4)$ in higher dimensions. Non-visibility of $X$ can be checked by studying its two-dimensional totally geodesic submanifolds, cf. [@Pha-neg; @Wu-transl-length], while proving visibility of $X$ gets much harder, and indeed, it is a strong restriction on $X$ with a variety of consequences for the geometry of horoballs, Tits boundary, and isometry groups. By contrast, the condition “$K<0$” is easy to verify but has few implications. Either condition implies that every complete geodesic in $X$ has rank one.
Manifolds of dimensions two and three
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A classification of open connected $2$-manifolds goes back to Ker[é]{}kj[á]{}rt[' o]{} [@Ker-2-mfld]; see [@Ric-2-mfld; @Gol-2-mfld; @Sie-2-mfld] for more recent accounts.
Richards [@Ric-2-mfld Theorem 3] proved that every open surface is obtained from $S^2$ by removing a closed totally disconnected subset, and then removing a finite or countable family of disjoint closed disks and attaching handles or Möbius bands along boundaries of the disks.
Any closed totally disconnected subset $T\subset S^2$ can be moved by an ambient homeomorphism to a subset of the standard Cantor set. (Like any compact totally disconnected metric space, $T$ is homeomorphic to a subset $Q$ of the Cantor set in $S^2$. The homeomorphism type of a planar surface is determined by the homeomorphism type of its the space of ends [@Ric-2-mfld Theorem 1], which for $S^2\setminus T$, $S^2\setminus Q$ are identified with $T$, $Q$, respectively. Hence there is a homeomorphism $S^2\setminus T\to S^2\setminus Q$, which extends to a homeomorphism of the end compactifications $S^2\to S^2$ mapping $T$ to $Q$).
The uniformization theorem equips any connected surface with a constant curvature metric, which can be chosen hyperbolic for open surfaces:
Any open connected $2$-manifold admits a complete metric of constant negative curvature.
Given an open manifold $M$, we equip it with a complete Riemannian metric, and pull the metric back to the universal cover $\widetilde M$, where $\pi_1(M)$ acts isometrically, and hence by preserving the conformal class of the metric. By the uniformization theorem $\widetilde M$ is conformal to the hyperbolic plane $\mathbf H^2$ or to $\mathbb C$. A self-diffeomorphism of $\mathbf H^2$ that preserve the conformal class of the hyperbolic metric is an isometry of $\mathbf H^2$. A self-diffeomorphism of $\mathbb C$ that preserves the conformal class of the Euclidean metric is of the form $z\to az+b$ or $z\to a\bar z+b$, and hence it either is an isometry (i.e. $a\bar a=1$), or its square has a fixed point (in fact, $z\to az+b$ fixes $\frac{b}{1-a}$ and similarly for the square of $z\to a\bar z+b$). The deck-transformation $\pi_1(M)$-action is free, so it preserves the hyperbolic or the Euclidean metric. Thus $M$ admits a complete metric of constant curvature $-1$ or $0$. Finally, a flat open $2$-manifold is $\mathbb R^2$, an annulus, or a Möbius band, so it also admits a complete hyperbolic metric.
Most of what is known on the geometrization of an open $3$-manifold requires it to have finitely generated fundamental group, or better yet to be the interior of a compact manifold.
\[thm: 3-mflds npc\] The interior of any compact aspherical $3$-manifold with nonempty boundary admits a complete metric of $-1\le K\le 0$.
We refer to [@Kap-book] for the terminology used in this proof. Let $N$ be a compact aspherical $3$-manifold. Asphericity implies that $N$ contains no essential $2$-sphere or projective plane. There is a decomposition of $N$ along incompressible $2$-sided tori and Klein bottles into Seifert and atoroidal pieces, see [@BonSie1987], which extends previous proofs by Johannson and Jaco-Shalen to the non-orientable case. Each piece is $\pi_1$-injectively embedded and hence aspherical.
Atoroidal pieces contain no essential annuli or Möbius bands whose boundary circles lie on the components of $\d N$ of zero Euler characteristic (for the orientable case see e.g. [@Hat-3mfld-notes Lemma 1.16], and the non-orientable case follows by passing to the orientation cover for an essential annulus or Möbius band stays essential in the orientation cover, and a virtually Seifert piece is Seifert).
Define the pared manifold structure on every atoroidal piece N by letting its parabolic locus $P$ be the union of boundary components of zero Euler characteristic. Thurston’s hyperbolization theorem [@Kap-book Theorem 1.43] gives $(N,P)$ a geometrically finite hyperbolic structure whose parabolic subgroups are of rank $2$ and bijectively corresponding to components of $P$. Every Seifert piece has a nonpositively curved metric with totally geodesic (flat) boundary by Leeb [@Lee-3mnf-npc]. In the same paper Leeb gives a gluing procedure for identifying rank $2$ cusps and flat boundary components of Seifert pieces. (Leeb assumes that hyperbolic pieces have boundary of zero Euler characteristic but this does not matter since his gluing runs in the cusps, which in our case have rank $2$). The result is a complete nonpositively curved metric on the interior of $N$.
A variation of the above proof identifies $N$ with a compact locally convex $C^1$ manifold of $K\le 0$ with boundary, namely, replace geometrically finite pieces with the ${\varepsilon}$-neighborhoods of their convex cores, where the cusps are chopped off, and the metric at cusps are modified so that their boundaries are flat and totally geodesic. Thus the group $\pi_1(N)$ is CAT($0$), which was first noted in [@Bri-3mfd-npc Theorem 4.3] with a slightly different proof.
More information on $3$-manifold groups can be found in the survey [@AFW-3mflds-grps]; in particular, if $M$ is open aspherical $3$-manifold with finitely generated fundamental group, then the advances on the virtual Haken conjecture imply that $\pi_1(M)$ virtually embeds into a right angled Artin group, which has a number of group-theoretic consequences, e.g. $\pi_1(M)$ is linear over $\mathbb Z$.
By Scott/Shalen compact core theorem any open aspherical $3$-manifold $M$ with finitely generated deformation retracts to a compact codimension zero submanifold (to which Theorem \[thm: 3-mflds npc\] applies) but the topology of $M$ is still a mystery.
A open $3$-manifold is called [*tame*]{} if it is homeomorphic to the interior of a compact manifold. Marden’s Tameness Conjecture predicted tameness of every open complete $3$-manifold with $K\equiv -1$ and finitely generated fundamental group, and it was proved by Agol [@Ago-tame] and Gabai-Calegari [@CalGab-tame]. The following question is still open:
Let $M$ be an open $3$-manifold with a complete metric of $K\le 0$ and finitely generated fundamental group. Is $M$ tame?
The same question is open for manifolds of $K\le -1$ or $-1\le K\le 0$. The answer is yes under any of the following assumptions:
- $M$ admits a complete negatively pinched metric (due to Bowditch [@Bow-tame] who built on proofs of Marden’s Tameness Conjecture by Agol, Calegari-Gabai, and Soma).
- $M$ is a cover of the interior of a compact manifold (as proved by Long-Reid in [@Can-tamness-2008] that combines results of Simon with the proof of Tameness Conjecture; here the assumption that $M$ has $K\le 0$ is not needed).
- $M$ has a complete metric of $-1\le K\le 0$ and $\mathrm{Inj}\,\mathrm{Rad}\to 0$ (as proved by Schroeder [@BGS Appendix 2]).
By contrast, there exits non-tame, open $3$-manifolds with universal cover diffeomorphic to $\mathbb R^3$ and fundamental groups isomorphic to $\mathbb Z$ [@ScoTuc] and $\mathbb Z\ast\mathbb Z$ [@FreGab].
The non-tame fake open solid torus in [@ScoTuc] does not admit a complete metric of $K\le 0$ because $\mathbb R^2$-bundles over $S^1$ are the only open, complete $3$-manifolds of $K\le 0$ with infinite cyclic fundamental group. (The isometric $\mathbb Z$-action in the universal cover must stabilize a geodesic or a horoball, in the former case the nearest point projection to the geodesic descends to an $\mathbb R^2$-bundle over $S^1$, while in the latter case the quotient is the product of $\mathbb R$ and an open surface homotopy equivalent to a circle, which is an $\mathbb R$-bundle over $S^1$).
Towards a rough classification of discrete isometry groups {#sec: rough}
==========================================================
In this section we sketch a higher-dimensional analog of the classification of open complete constant curvature surfaces; details appear in Sections \[sec: non-parabolic\]–\[sec: anchored\]. We refer to [@BGS; @Ebe-book; @BriHae] and Section \[sec: flavors of neg curv\] for background on Hadamard manifolds, adopt the following notations:
- $X$ is a Hadamard manifold with ideal boundary $X(\infty)$,
- ${\mbox{Iso}}(X)$ is the isometry group of $X$,
- $\G$ is a subgroup of ${\mbox{Iso}}(X)$,
and consider the following three classes of complete manifolds of $K\le 0$ of the form $X/G$ where $\G\le{\mbox{Iso}}(X)$ is discrete and torsion-free:
1. $\G$ contains no parabolic elements.
2. $\G$ contains a rank one element.
3. $\G$ fixes a point $\xi\in X(\infty)$, and the associated $\G$-action on the space $L_\xi$ of lines asymptotic to $\xi$ is free and properly discontinuous.
There are severe algebraic restrictions on $\G$ in the cases (1)-(2), and in the case (3) the manifold $X/\G$ is diffeomorphic to the product of $\mathbb R$ and $L_\xi/\G$, where $L_\xi$ is diffeomorphic to a Euclidean space.
$\G$ satisfies (3) if it stabilizes a horoball. (Indeed, $L_\xi$ is equivariantly diffeomorphic to a horosphere, and the discreteness of $\G$ implies that its action on $L_\xi$ is properly discontinuous).
Suppose $X$ has rank one. Does every discrete torsion-free isometry group $\G$ of $X$ satisfy one of the conditions above?
The answer is yes if $X$ is visibility (i.e. any two points of $X(\infty)$ are endpoints of a geodesic, which happens e.g. if $K\le -1$), see Corollary \[cor: visib horoball\].
Without any assumption on $X$ the answer is no, e.g. when $X/\G$ is the product of two finite volume, complete, open, hyperbolic surfaces.
\(a) The classes (1), (2), (3) are clearly not disjoint, e.g. a the cyclic group generated by a translation in $\mathbb R^n$ lies in (1) and (3). On the other hand, any (discrete) subgroup satisfying (2) and (3) is virtually cyclic, see Proposition \[prop: fixed pt discrete rk1 implies virt-Z\]. (b) A given group may be isomorphic to three different isometry groups satisfying (1), (2), (3) respectively. (c) If $\G\le{\mbox{Iso}}(X)$ is a discrete subgroup, then the requirement “$\G$ is isomorphic to a discrete isometry group of a Hadamard manifold that satisfies (3)” does not restrict $\G$ because the isometric $\G$-action on the warped product Hadamard manifold $\mathbb R\times_{e^r}\! X$ stabilizes a horoball. On the other hand, if we fix the dimension, this does become a nontrivial restriction as follows. (d) Define the [*Euclidean action dimension*]{} as the smallest dimension of a Euclidean space on which $\G$ acts smoothly and properly discontinuously, and if the Euclidean action dimension of $\G$ equals $\dim(X)$, then $\G$-action on $X$ cannot satisfy (3) because $L_\xi$ is diffeomorphic to the Euclidean space of lower dimension. See [@BKK02; @BesFei02; @Yoo04; @Des06] for computations of a related invariant called the [*action dimension*]{} which usually equals the Euclidean action dimension.
Groups of non-parabolic isometries {#sec: non-parabolic}
==================================
In this section we discuss groups in the class (1) of Section \[sec: rough\]. An isometry of $X$ is [*elliptic, axial*]{}, or [*parabolic*]{} if the minimum of its displacement function is zero, positive, or not attained, respectively. If $\g$ is a non-parabolic isometry, then the set $\mathrm{Min}(\g)$ of points where the displacement functions attains a minimum splits as $C_\g\times \mathbb R^k$, where $C_\g$ is a closed convex subset with $\g$ acting as the product of the trivial action on $C_\g$ and a translation on $\mathbb R^k$ [@BriHae Theorem II.7.1]. If $\g$ is axial, its axes are precisely the lines $\{x\}\times\mathbb R$, $x\in C_\g$.
Theorems \[thm: abelian semisimple\], \[thm: normalizer of abelian semisimple\], \[thm: semisimple nonembed\](2ab) are part of the flat torus theorem “package” discovered by Gromoll-Wolf [@GroWol-flat-tor] and Lawson-Yau [@LawYau-flat-tor], and generalized in [@BriHae].
\[thm: abelian semisimple\] Let $A\le{\mbox{Iso}}(X)$ be an abelian discrete subgroup that consists of non-parabolic isometries. Set $\mathrm{Min}(A):=\cap_{a\in A}\mathrm{Min}(a)$. Then $\mathrm{Min}(A)$ is a nonempty, closed, convex $A$-invariant subset that splits as $C_A\times\mathbb R^m$ where $A$ acts trivially on $C_A$ and by translations on $\mathbb R^m$; $A$ is finitely generated of rank $\le m\le\dim(X)$; $A$ has finite intersection with each conjugacy class in $\mathrm{Iso}(X)$.
Any periodic abelian discrete subgroup of ${\mbox{Iso}}(X)$ is finite (because it is countable, hence locally finite, and so is a union of finite subgroups whose fixes points set is a descending family of totally geodesics submanifolds of $X$, which has to stabilize by dimension reasons). Thus abelian discrete groups of non-parabolic isometries are finitely generated, which is a key feature of the Riemannian setting. By contrast $\mathbb Q$ can act properly by non-parabolic isometries on a proper CAT($0$) space, which is the product of a simplicial tree and a line [@BriHae Example II.7.13].
\[thm: normalizer of abelian semisimple\] Let $A\le{\mbox{Iso}}(X)$ be an abelian discrete subgroup that consists of axial isometries, and let $N\le{\mbox{Iso}}(X)$ is a subgroup that normalizes $A$. Then $N$ stabilizes $\mathrm{Min}(A)$ and preserves its product decomposition; $A$ is centralized by a finite index subgroup of $N$; $A$ is a virtual direct factor of $N$ if $N$ is finitely generated and $A\le N$.
\(1) is proved in [@BGS Lemma 7.1(1)]. Discreteness of $A$ and (1) implies that $A$-action on $\mathbb R^m$ is properly discontinuous, so $A$ has rank $\le m$, which gives (2). Claim (3) follows from [@BriHae Lemma II.7.17(2)], and (4), (5), (6) are parts of the flat torus theorem [@BriHae Theorem II.7.1].
\[thm: semisimple nonembed\] Let $\G\le\mathrm{Iso}(X)$ be a subgroup without parabolic elements. If $H$ is commensurable to $\G$, then $H$ is isomorphic to a discrete group of non-parabolic isometries of some Hadamard manifold. The following groups do not embed into $\G$: any solvable group that is not virtually abelian; the Baumslag-Solitar group $\langle x,y\,|\, xy^mx^{-1}=y^l\rangle$ with $m\neq\pm l$; $\pi_1(L)$, where $L$ is a closed aspherical $3$-manifold that admits no metric of $K\le 0$.
\(1) is immediate via the induced representation construction [@KapLee-hadgr Theorem 2.3]. To prove (2a) note that $G$ must be polycyclic (combine finite generation of abelian discrete subgroups of non-parabolic isometries with Mal’cev’s theorem that a solvable group whose abelian subgroups are finitely generated is polycyclic). Then proceed by induction on Hirsch length, and use Theorem \[thm: normalizer of abelian semisimple\](6) to split virtual $\mathbb Z$-factors one at a time, see [@BriHae Theorem II.7.16]. Also (2b) follows from Theorem \[thm: abelian semisimple\](3), see [@BriHae Theorem III.$\Gamma$.1.1(iii)]. Finally, to prove (2c) invoke the solution of the virtual Haken conjecture [@Ago-virthak]; thus $L$ has a Haken finite cover, so it admits a metric of $K\le 0$ by the main result in [@KapLee-hadgr Corollaries 2.6-2.7].
1. If a closed aspherical $3$-manifold admits no metric of $K\le 0$, then it is Seifert or graph (the manifold is virtually Haken [@Ago-virthak] so the claim follows from [@Lee-3mnf-npc; @KapLee-hadgr]).
2. Closed aspherical Seifert manifolds that admit no metric of $K\le 0$ are precisely those modelled on Nil, Sol, or $\widetilde{SL}_2(\mathbb R)$ as easily follows from Theorems \[thm: abelian semisimple\](6), \[thm: semisimple nonembed\](2a), and the observation that the Seifert manifolds modelled on $\mathbb R^3$ or $\mathbb H^2\times\mathbb R$ are nonpositively curved.
3. The problem which orientable closed graph manifolds admit metrics of $K\le 0$ was resolved in [@BuyKib-geometr-graph-II; @BuySve-surv] who found several combinatorial criteria on the gluing data. (These papers only consider manifolds with no embedded Klein bottles, but the assumption can be easily removed as was explained to the author by Buyalo).
4. A non-orientable closed $3$-manifold admits a metric of $K\le 0$ if and only if its orientation cover does [@KapLee-hadgr].
5. A closed aspherical graph manifold admits a metric of $K\le 0$ if and only if its fundamental group virtually embeds into a right angled Artin group [@Liu-graph].
6. Kapovich-Leeb used Theorem \[thm: semisimple nonembed\](2c) to give other examples of groups that do not act on Hadamard spaces by non-parabolic isometries [@KapLee-hadgr].
Groups with rank one elements: prelude {#sec: rk1}
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Sections \[sec: rk1\]–\[sec: orbit equiv\] collect what is known on the class (2) of Section \[sec: rough\].
An axial isometry $\g$ has [*rank one*]{} if it has an axis that does not bound a flat half plane. This property can be characterized in terms of the splitting $\mathrm{Min}(\g)\cong C_\g\times\mathbb R^k$, namely, $\g$ has rank one if and only if $C_\g$ is compact.
Any discrete subgroup of ${\mbox{Iso}}(X)$ that normalizes a rank one element is virtually cyclic (in fact, the normalizer preserves the splitting, and hence fixes a point of $C_\g$ and stabilizing the corresponding axis).
Rank one isometries were introduced by Ballmann [@Bal-book Theorem III.3.4], who proved that if $X$ is a rank one and $\G\le{\mbox{Iso}}(X)$ is any subgroup satisfying the duality condition (e.g. a lattice), then $\G$ contains a rank one element. He then used a ping pong argument to find a copy of non-cyclic free group inside $\G$. Various aspects of isometry groups containing rank one elements were further studied in [@BesFuj-geomtop2002; @BalBuy-rk1; @BesFuj-gafa2009; @Ham-rk1; @Ham-rk1-tot-disc; @CapFuj]. In particular, the following is due to [@BesFuj-gafa2009 Proposition 5.11] or [@Ham-rk1 Theorem 1.1(4)]:
\[thm: rk1 element exists\] If $\G\le\mathrm{Iso}(X)$ is a discrete subgroup that contains a rank one element and is not virtually-$\mathbb Z$, then $\G$ contains a non-cyclic free subgroup consisting of rank one elements.
Sisto proved [@Sis-contr-rand Theorem 1.4] that if $\G$ in Theorem \[thm: rk1 element exists\] is finitely generated, then its generic element has rank one, where “generic” roughly means that the probability that a word written in random finite generating set of $G$ represents a rank one element approaches $1$ exponentially with the length of the word.
Acylindrically hyperbolic groups and rank one elements {#sec: acyl hyp}
======================================================
In the last decade it was realized that many groups of geometric origin contain (suitably defined) rank one elements, which allowed for a uniform treatment of such groups and resulted in a host of applications. A crucial notion in these developments is acylindricity which goes back to Sela and Bowditch. In connection with rank one elements different versions of acylindricity were introduced and studied by Bestvina-Fujiwara [@BesFuj-geomtop2002; @BesFuj-gafa2009], Hamenst[ä]{}dt [@Ham-iso-gps], Dahmani-Guirardel-Osin [@DGO], Sisto [@Sis-contr-rand], and most recently Osin showed [@Osi-acy-hyp] that all these approaches are equivalent.
An isometric action of a group $G$ on a Gromov hyperbolic space $(X, d)$ is called
- [*non-elementary*]{} if its limit set consists of $>2$ points,
- [*acylindrical*]{} if for each ${\varepsilon}>0$ there are $R, N$ such that if $d(x, y)\ge R$, then at most $N$ elements $g\in G$ satisfy $d(x, gx)\le{\varepsilon}$ and $d(y, gy)\le {\varepsilon}$.
A group $G$ is [*acylindically hyperbolic*]{} if it admits a non-elementary acylindrical isometric action on a Gromov hyperbolic space.
The class of acylindically hyperbolic groups includes many groups of geometric origin, e.g. any subgroup of a relatively hyperbolic group that is not virtually-cyclic and does not lie is peripheral subgroup, or all but finitely many mapping class groups; see [@Osi-acy-hyp], for other examples.
Of particular importance for this section is the following result of Sisto [@Sis-contr-rand] who actually proves it for any group acting properly and isometrically on a proper CAT($0$) space:
\[thm: sisto rk1\] [**(Sisto)**]{} If $\G\le{\mbox{Iso}}(X)$ is a discrete subgroup that contains a rank one element, then $\G$ is virtually cyclic or acylindrically hyperbolic.
Dahmani-Guirardel-Osin [@DGO] introduced a notion of a [*hyperbolically embedded subgroup of $G$*]{}, and Osin [@Osi-acy-hyp] proved that $G$ is acylindrically hyperbolic if and only if $G$ contains an infinite, proper, hyperbolically embedded subgroup. What Sisto actually showed is that any rank one element $\g\in \G$ lies in a virtually cyclic hyperbolically embedded subgroup $E(\g)$. We omit the definition of a hyperbolically embedded subgroup, and just note that they are almost malnormal by [@DGO Proposition 4.33]:
If $H$ is a hyperbolically embedded subgroup of a group $G$, then $H$ is almost malnormal in $G$, i.e. $H\cap gHg^{-1}$ is finite for all $g\notin H$.
Here are other applications [@DGO; @Osi-acy-hyp], which hold in particular when $G$ is a discrete, non-virtually-cyclic subgroup of ${\mbox{Iso}}(X)$ containing a rank one element.
\[thm: acy hyp list\] If $G$ is acylindrically hyperbolic, then $G$ has a non-cyclic, normal, free subgroup, every countable group embeds into a quotient of $G$, every infinite subnormal subgroup of $G$ is acylindrically hyperbolic, $G$ has no nontrivial finite normal subgroups if and only if every conjugacy class in $G$ is infinite, every s-normal subgroup of $G$ is acylindrically hyperbolic, if $G$ equals the product of subgroups $G_1,\dots, G_k$, then at least one $G_i$ is acylindrically hyperbolic. $G$ is not the direct product of infinite groups. any group commensurable to $G$ is acylindrically hyperbolic. any co-amenable subgroup of $G$ acylindrically hyperbolic.
Proofs of (1)–(4) are in [@DGO Theorem 8.6, Lemma 8.11, Theorem 8.12] (1)–(2), while (5)–(7) are proved in [@Osi-acy-hyp Corollary 1.5, Proposition 1.7, Corollary 7.3], and (8) appears in [@MinOsi-acy--hyp Lemma 3.8].
The claim (9) is due to Osin who communicated to the author the following argument and kindly permitted to include it here. Suppose $G$ is acylindrically hyperbolic and $K\le G$ is not. In the next paragraph we find a non-cyclic free subgroup $F\le G$ with $F\cap K=\{ 1\}$. The group $F$ is not amenable, and its action by left translations on the set $G/H$ is free, so there is no $F$-invariant finitely additive probability measure on $G/K$. The contrapositive of (9) now follows because if $K$ were co-amenable, $G/K$ would admit a $G$-invariant (and hence $F$-invariant) finitely additive probability measure.
To construct $F$ note that [@Osi-acy-hyp Theorems 1.1-1.2] yield a non-elementary acylindrical $G$-action on a Gromov hyperbolic space for which the subgroup $K$ is either elliptic or else virtually-$\mathbb Z$ and contains a loxodromic element (we refer to [@Osi-acy-hyp] for terminology). If $K$ is elliptic, then [@Osi-acy-hyp Theorems 1.1] yields independent loxodromic elements $a,b\in G$. The standard ping-pong argument shows that for some $n\gg 1$, the subgroup $\langle a^n,b^n\rangle$ is free of rank $2$ and all its non-trivial elements are loxodromic. In particular, $\langle a^n,b^n\rangle\cap K=\{ 1\}$. If $K$ is virtually-$\mathbb Z$ and contains a loxodromic element $c$, then [@Osi-acy-hyp Theorems 1.1] gives loxodromic elements $a,b\in G$ such that $a,b,c$ are independent. Again by ping-pong $\langle a^n, b^n, c^n\rangle $ is free of rank $3$ so that $\langle a^n, b^n\rangle\cap K=\{ 1\}$. Thus we get a non-cyclic free subgroup $F=\langle a^n,b^n\rangle$ with $F\cap K=\{ 1\}$, and in fact all nontrivial elements of $F$ are loxodromic.
A subgroup $K\le G$ is [*subnormal*]{} if there are subgroups $G_i\le G$ with $G_0=G$, $G_k=K$, and such that $G_{i}$ is a normal in $G_{i-1}$ for all $i=1,\dots , k$.
If a group $G$ equals the product of subgroups $G_1,\dots, G_k$, one says that $G$ [*boundedly generated by*]{} $G_1,\dots, G_k$.
Two groups are [*commensurable*]{} if they have isomorphic finite index subgroups.
A subgroup $K\le G$ is [*s-normal*]{} if $K\cap gKg^{-1}$ is infinite for each $g\in G$. Thus the Baumslag-Solitar group $B(m,n)=\langle a,b\,|\, ab^m=b^na\rangle$ is not acylindrically hyperbolic, except for $B(0,0)$, because $\langle b\rangle$ is s-normal and not acylindrically hyperbolic.
\[sec: co-amen\] A subgroup $K\le G$ is [*co-amenable*]{} if one of the following holds:
1. every continuous affine $G$-action on a convex compact subset of a locally convex space with a $K$-fixed point has a $G$-fixed point;
2. $\ell^\infty(G/K)$ has a $G$-invariant mean;
3. $G/K$ has a $G$-invariant finitely additive probability measure;
4. the inclusion $K\hookrightarrow G$ induces injections in bounded cohomology in all degrees with coefficients in any dual Banach $G$-module.
The equivalence $\text{(1)}\Leftrightarrow \text{(2)}$ is proved in [@Eym-book], while $\text{(2)}\Leftrightarrow \text{(3)}$ follows from the standard correspondence between means and measures, and $\text{(3)}\Leftrightarrow \text{(4)}$ can be found in [@MonPop-co-amen]. See also [@Moo-thesis] for leisurely discussion of co-amenability.
Here we are mainly interested in examples of non-amenable groups that admit co-amenable subgroups:
- A normal subgroup $N\unlhd\, G$ is co-amenable if and only if $G/N$ is amenable.
- If $K$ is co-amenable in $N$, and in turn $N$ is co-amenable in $G$, then $K$ is co-amenable in $G$.
- the image of a co-amenable subgroup under an epimorphism $G\to \bar G$ is co-amenable.
- If $\theta\co K\to K$ is a monomorphism, and $G:=\langle K,t\,|\, tkt^{-1}=
\theta(k), k\in K\rangle$ is the associated HNN-extension, then $K$ is co-amenable in $G$.
The first three facts above are straightforward, while the last one is due to Monod-Popa [@MonPop-co-amen].
Starting from a group $K$ that is not acylindrically hyperbolic one can use iterated HNN-extensions and extensions with amenable quotient to get many examples of non acylindrically hyperbolic groups. (These constructions preserve finiteness of cohomological dimension if the initial $K$ and every amenable quotient have finite cohomological dimension).
Bounded cohomology and rank one elements
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Bounded cohomology naturally appear in a variety of contexts, see e.g. [@Gro-vol-bd-coh; @Mon-icm]. Of particular interest for our purposes is the [*comparison map*]{} $$\iota (G)\co H^2_b(G;\mathbb R)\to H^2(G;\mathbb R)$$ between the bounded and ordinary cohomology in degree two, which encodes some subtle group-theoretic properties: $\bullet$ **(Johnson) If $G$ is amenable, then $H^p(G;\mathbb R)=0$ for $p>0$ [@Jon-bound-coh; @Nos-bound-coh]. $\bullet$ **(Burger-Monod) $\iota(G)$ is injective if $G$ is the fundamental group of an irreducible, finite volume complete manifold of $K\le 0$, no local Euclidean de Rham factor, and rank $\ge 2$ [@BurMon]. $\bullet$ **(Bavard) Injectivity of $\iota(G)$ is equivalent to vanishing of the stable commutator length on $[G,G]$ [@Bav-commut]. Thus if $\iota(G)$ is non-injective, then there is $g\in [G,G]$ such that the minimal number of commutators needed to represent $g^n$ grows linearly with $n$.******
\[thm: bestv-fuj-rk1\] **(Bestvina-Fujiwara, Osin)If $G$ is acylindrically hyperbolic, then the comparison map $\iota(\G)$ has infinite dimensional kernel.**
This was proved in [@BesFuj-geomtop2002] for a class of groups which according to [@Osi-acy-hyp] coincides with the class of acylindrically hyperbolic groups.
For discrete subgroup $\G\le{\mbox{Iso}}(X)$ with rank one elements the above theorem was first established in [@BesFuj-gafa2009]. Discreteness of $\G$ in Theorem \[thm: bestv-fuj-rk1\] can be weakened to the weak proper discontinuity [@BesFuj-gafa2009], but it cannot be dropped, e.g. the projection of any irreducible lattice $\Lambda\le{\mbox{Iso}}({\mathbf H}^2)\times {\mbox{Iso}}({\mathbf H}^2)$ to either factor acts on the hyperbolic plane isometrically, effectively, and by rank one isometries, but the comparison map $\iota(\Lambda)$ is injective [@BurMon].
Monod-Shalom’s class and rank one elements {#sec: orbit equiv}
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In [@MonSha-jdg2004; @MonSha-orb-ann2006], Monod-Shalom introduced and studied the following class of groups, which they thought of as a cohomological manifestation of negative curvature: Let $\mathcal C_{\text{reg}}$ be the class of countable groups $G$ such that $H^2_b(G; \ell^2(G))\neq 0$, which refers to the bounded cohomology of $G$ with coefficients in the regular representation.
A way to prove that $H^2_b(G; \ell^2(G))\neq 0$ is to show that the corresponding comparison map $H^2_b(G; \ell^2(G))\to H^2(G; \ell^2(G))$ has infinite dimensional kernel, which was done for many “hyperbolic-like” groups in [@MinMonSha]. The following was proved by Hamenst[ä]{}dt [@Ham-bound-coh], and later from a different perspective by Hull-Osin [@HulOsi]:
$\mathcal C_{\text{reg}}$ contains every countable acylindrically hyperbolic group, and hence any discrete subgroup $\G\le{\mbox{Iso}}(X)$ that contains a rank one element and is not virtually-$\mathbb Z$.
For discrete subgroup $\G\le{\mbox{Iso}}(X)$ with rank one elements the above theorem was first established in [@Ham-rk1-tot-disc].
As proved in [@MonSha-orb-ann2006 Chapter 7], examples of groups not in $\mathcal C_{\text{reg}}$ include
- amenable groups,
- products of at least two infinite groups,
- lattices in higher-rank simple Lie groups (over any local field),
- irreducible lattices in products of compactly generated non-amenable groups.
and the class $\mathcal C_{\text{reg}}$ is closed under
- passing to an infinite normal subgroup,
- passing to a co-amenable subgroup,
- measure equivalence.
We refer to [@Fur-surv] for a survey on measure equivalence; e.g. commensurable groups are measure equivalent.
Is every group in $\mathcal C_{\text{reg}}$ acylindrically hyperbolic?
To transition to our next topic, note that non-virtually-cyclic discrete groups with rank one elements never fix a point at infinity:
\[prop: fixed pt discrete rk1 implies virt-Z\] If $\G$ is discrete, fixes a point at infinity, and contains a rank one element, then $\G$ is virtually cyclic.
This follows from [@BesFuj-gafa2009 Section 6] provided $\G$ satisfies the weak proper discontinuity condition, which is implied by discreteness. The idea is that if $g\in\G$ has rank one, then either $\G$ is virtually-$\mathbb Z$, or $\G$ contains another rank one element $h$ such that their axis $A_g$, $A_h$ do not have the same sets of endpoints at infinity. A rank one element fixes precisely two points at infinity, the endpoints of its axis. Since $\G$ has a fixed point, it must be a common endpoint of $A_g$, $A_h$, and this contradicts weak proper discontinuity: there is a subsegment $I$ of $A_g$ and an infinite subset $Q$ of $\G$ such that the distances between the endpoints of $I$ and $g(I)$, $g\in Q$ are uniformly bounded.
The (non-discrete) stabilizer of a boundary point in the hyperbolic plane contains rank one elements without being virtually-$\mathbb Z$.
Groups that fix a point at infinity: prelude {#sec: fix a point at infinity}
============================================
Basic properties of horoballs, horospheres, and Busemann functions can be found in [@BGS; @Ebe-book; @Bal-book]. A [*horoball*]{} in $X$ is the Hausdorff limit of a sequence of metric balls in $X$ with radii going to infinity. A [*horosphere*]{} is the boundary of a horoball. Every point at infinity $\xi$ is represented by a Busemann function $b_\xi\co X\to \mathbb R$, which is determined by $\xi$ up to an additive constant. The fibers of $b_\xi$ are the horospheres centered at $\xi$, and the sublevel sets of $b$ are horoballs centered at $\xi$. The function $b_\xi$ is a $C^2$ Riemannian submersion $X\to\mathbb R$, and in particular, each horosphere is diffeomorphic to the Euclidean space of dimension $\dim(X)-1$.
If $\G$ fixes a point $\xi$ at infinity of $X$, then $\G$ permutes horospheres centered at $\xi$, and associating to $\g\in\G$ the distance by which it moves a horosphere to a concentric one defines a homomorphism $\G\to\mathbb R$, which is in general nontrivial (think of the stabilizer of a point at infinity of the hyperbolic plane).
Let $L_\xi$ be the space of lines in $X$ asymptotic to $\xi$. The geodesic flow towards $\xi$ identifies $X$ with the total space of a principal $\mathbb R$-bundle over $L_\xi$, which is trivial as every horosphere centered at $\xi$ gives a section. In particular, $L_\xi$ has a structure of a smooth manifold diffeomorphic to a horosphere about $\xi$. If $\G$ fixes $\xi$, then it acts smoothly on $L_\xi$.
Recall the condition (3) of Section \[sec: rough\]: $\G$ fixes a point $\xi\in X(\infty)$, and the associated $\G$-action on the space $L_\xi$ of lines asymptotic to $\xi$ is free and properly discontinuous. Under this condition the principal $\mathbb R$-bundle $X\to L_\xi$ descends to an orientable (and hence trivial) real line bundle $X/\G\to L_\xi/\G$, so we get:
If $\G$ satisfies the condition of , then $X/\G$ is diffeomorphic to the product of $\mathbb R$ and $L_\xi/\G$.
The prime example of a group satisfying (3) is a discrete torsion-free subgroup $\G$ that stabilizes a horoball, in which case $\G$ stabilizes every concentric horoball, so that $X$ is $\Gamma$-equivariantly is diffeomorphic to the product of $\mathbb R$ with a horosphere, and (3) follows because $\G$ acts freely and properly discontinuously on $X$. More examples are needed:
Let $\G$ be any discrete torsion-free isometry group of $X$ whose fixed point set at infinity is nonempty.
- Does $\G$ satisfies for some $\xi$?
- If $\G$ satisfies , does $\G$ stabilize a horoball?
- What is the structure of $\G$ if it does not stabilize a horoball?
\(1) An axial isometry does not stabilize a horoball centered at an endpoint of one of its axis, but it can stabilize another horoball (e.g. translation in the plane stabilizes any half plane whose boundary is parallel to the translation axis). (2) Any parabolic isometry stabilizes a horoball by Lemma \[lem: center has parabolic\] but different parabolics in $\G$ can stabilize different horoballs. (3) An elliptic element fixing a point at infinity stabilizes a horoball centered at the point (because it fixes a ray from a fixes point inside $X$ to the fixed point at infinity).
\[quest: no rank one implies grounded\] Let $\G\le{\mbox{Iso}}(X)$ be discrete, containing a parabolic and no rank one elements. What conditions on $X$ ensure that $\G$ fixes a point at infinity?
Recall that the limit set $\Lambda (\G)$ is the set of accumulation points of the $\G$-orbit of a point of $X$. Ballmann-Buyalo [@BalBuy-rk1 Proposition 1.10] gave the following characterization of groups containing rank one elements in terms of the Tits radius of the limits set:
\[prop: ballmann-buyalo, rank 1\] $\G$ contains no rank one element if and only if $\Lambda(\G)$ lies in the Tits ball of radius $\le\pi$ centered at a point of $\Lambda(\G)$.
Combining this characterization with results of Schroeder [@BGS Appendix 3] one can answer Question \[quest: no rank one implies grounded\] when every component of the Tits boundary of $X$ has radius $\le\frac{\pi}{2}$:
\[cor: tits exist rank one\] If every component of $X(\infty)$ equipped with the Tits metric has radius $\le\frac{\pi}{2}$, then a subgroup $\G\le\mathrm{Iso}(X)$ either contains a rank one element or fixes a point at infinity.
Following [@BGS page 220], for a subset $Q\subset X(\infty)$, let $$C_Q=\{z\in Q\,|\, Q\ \text{lies in the Tits ball of radius}\ \frac{\pi}{2}
\ \text{centered at}\ z\}.$$ Lower semicontinuity of the Tits distance implies that if $Q$ is closed in the cone topology on $X(\infty)$, then so is $C_Q$ [@BGS 4.9]. If $C_Q$ is nonempty, then clearly it has Tits diameter $\le\frac{\pi}{2}$.
Apply this to $Q=\Lambda(\G)$, which is a closed $\G$-invariant subset. Since $\G$ has no rank one element, Proposition \[prop: ballmann-buyalo, rank 1\] implies that $\Lambda(\G)$ lies in the Tits ball of radius $\le\pi$ about one of its points, and by our assumption the ball must have radius $\le\frac{\pi}{2}$ (for Tits metric is length so the distance between different components is infinite). Thus $C_{\Lambda(\G)}$ is a closed subset of Tits diameter $\le\frac{\pi}{2}$.
By the main result of [@BGS Appendix 3] any subset of $X(\infty)$ that is closed in the cone topology and has Tits diameter $\le\frac{\pi}{2}$ has a unique [*center*]{}, defined as the center of the closed (Tits) ball of the smallest radius among all balls containing the subset. Let $z_G$ be the unique center of $C_{\Lambda(\G)}$. Since $\Lambda(G)$ is $\G$-invariant, so is $C_{\Lambda(\G)}$, and hence $\G$ fixes $z_G$.
The components of the Tits boundary are points if (and only if) $X$ is visibility [@BGS 4.14], so Corollary \[cor: tits exist rank one\] applies if $X$ is visibility, in which case one can say more:
\[cor: visib horoball\] If $X$ is visibility and $\G$ contains a parabolic element but no rank one elements, then $\G$ contains no axial isometries, $\Lambda (\G)$ is a point, the fixed point set of $\G$ at infinity equals $\Lambda (\G)$, and $\G$ stabilizes every horoball centered at $\Lambda (\G)$.
Since $\G$ contains no rank one elements, Proposition \[prop: ballmann-buyalo, rank 1\] implies that $\Lambda(\G)$ is a point (as the Tits distance between any two distinct points is infinite). Being a visibility space, $X$ has no flat half spaces, so $\G$ contains no axial isometries. The limit set is $\G$-invariant, so $\Lambda(\G)$ is a fixed point of $\G$. If $\G$ fixed any other point, it would also be fixed by the cyclic subgroup generated by a parabolic in $\G$, but the fixed point set of any abelian subgroup containing a parabolic has Tits radius $\le\frac{\pi}{2}$, which again is a single point. Thus $\Lambda(\G)$ is a unique fixed point of $\G$.
By Corollary \[cor: visib horoball\] and Proposition \[prop: fixed pt discrete rk1 implies virt-Z\] any non-virtually-cyclic, discrete isometry group of a visibility manifold that fixes a point at infinity must stabilize a horoball. There is a sizable class of groups to which this applies, e.g. by Corollary \[cor: commut subgr grounded\] it contains the product of any nontrivial torsion-free groups, see [@KarNos04] for more examples.
Groups whose center contains a parabolic {#sec: center with parabolic}
========================================
The following result is implicit in [@BGS Lemma 7.3, 7.8].
\[lem: center has parabolic\] $\G\le{\mbox{Iso}}(X)$ stabilizes a horoball if it has a finite index subgroup $\G_0$ whose center $Z(\G_0)$ contains a parabolic isometry.
Fix a parabolic isometry $z\in Z(\G_0)$, and right coset representatives $g_1,\dots g_k$ of $\G_0$ in $\G$. Then the function $x\to \sum_i d(g_i^{-1}zg_i x, x)$ is convex and $\G$-invariant. Since $x\to d(zx, x)$ does not assume its infimum, neither does the above convex function. Then a limiting process outlined in [@BGS Lemma 3.9], cf. [@BriHae Lemma II.8.26], gives rise via Arzela-Ascoli theorem to a $\G$-invariant Busemann function, whose sublevel sets are $G$-invariant horoballs.
Flat torus theorem [@BriHae Chapter II.7] restricts a discrete isometry group of $X$ whose center consists of axial elements, which can be summarized as follows, see [@Bel-bus].
\[thm: into-center\] Let $G$ be a group with subgroups $H$, $G_0$ such that their centers $Z(H)$, $Z(G_0)$ are infinite, $Z(H)\subseteq Z(G_0)$, the index of $G_0$ in $G$ is finite, and one of the following conditions hold: $Z(H)$ is not finitely generated; any homomorphism $H\to\mathbb R$ is trivial. $H$ is finitely generated, and $Z(H)$ contains a free abelian subgroup that is not a direct factor of any finite index subgroup of $H$.If a discrete subgroup of ${\mbox{Iso}}(X)$ is isomorphic to $G$, then it stabilizes a horoball.
The reader may want to first think through the case when $H=G_0=G$, and then go on to observe that if Theorem \[thm: into-center\] holds for $H$, $G_0$, $G$, then it also does for $H$, $G_0\times K$, $G\times K$ for any group $K$.
\[ex: rationals\] Theorem \[thm: into-center\](1) applies, e.g. to any infinitely generated, torsion-free, countable abelian group of finite rank, such as $(\mathbb Q, +)$, where finiteness of rank ensures finiteness of cohomological dimension [@Bie-book Theorem 7.10].
It is unknown whether there is a group of type [*F*]{} with infinitely generated center. Such a group cannot be elementary amenable [@Bel-bus], or linear over a field of characteristic zero [@AlpSha82 Corollary 5]. Sanity check: there does exist a finitely presented group with solvable word problem whose center contains every countable abelian group [@Oul-center Corollary 3].
\[ex: with center (3)\] (groups of type [*F*]{} to which Theorem \[thm: into-center\](3) applies, see [@Bel-bus]):
1. $H$ is the fundamental group of the total space of any principal circle bundle with non-zero rational Euler class and a finite aspherical cell complex as the base.
2. $H$ is a torsion-free, finitely generated, non-abelian nilpotent group.
3. \[item: Seifert 3-manifold\] $H$ is the fundamental group of any closed orientable Seifert $3$-manifold modelled on $\widetilde{SL}_2(\mathbb R)$.
4. \[item: Sp lattice\] $H$ is the preimage of any torsion-free lattice in $Sp_{2n}(\mathbb R)$ under the universal cover $\widetilde{Sp}_{2n}(\mathbb R)\to Sp_{2n}(\mathbb R)$ for $n\ge 2$.
5. \[item: amalgam over center\] $H$ is the amalgamated product $G_1\ast_A G_2$ where $G_1$, $G_2$ have type [*F*]{} and are finitely generated, $A$ lies in the center of $G_1$, $G_2$ and contains a subgroup that is not a virtual direct factor of $G_1$.
Anchored groups and fixed points at infinity {#sec: anchored}
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Let us discuss algebraic conditions that force an isometry group of $X$ to fix a point at infinity.
If a subgroup $\G\le{\mbox{Iso}}(X)$ contains a parabolic element, stabilizes a closed convex noncompact subset $W\subseteq X$, and fixes a point at infinity of $W$, then we say that $\G$ [*is anchored in*]{} $W$. (Passing to an invariant closed convex subset is essential in some inductive arguments, e.g. in Theorem \[thm: co-amen grounded\]).
\[thm: anchored\] Let $\G\le{\mbox{Iso}}(X)$ be a subgroup and $W$ be a any closed, convex, noncompact $\G$-invariant subset of $X$. Then $\G$ is anchored in $W$ if one of the following holds: $\G$ is abelian and contains a parabolic. $\G$ has a normal subgroup that is anchored in $W$.
\(1) For $W=X$ this is due to Schroeder [@BGS Appendix 3], see also cf. [@FNS]. The general case follows from a result of Caprace-Lytchak [@CapLyt10 Corollary 1.5] that the centralizer of a parabolic isometry of a CAT($0$) space of finite telescopic dimension has a fixed point at infinity. Closed convex subsets of Hadamard manifolds have finite telescopic dimension, see [@CapLyt10 Section 2.1], and a parabolic isometry of $X$ that stabilizes a closed convex subset acts in that subset as a parabolic isometry.
\(2) For $W=X$ this is due to Eberlein [@Ebe-book Proposition 4.4.4], whose proof generalizes to our setting via [@FNS Proposition 5.7] or [@BalLyt Proposition 1.4].
Given a class of Hadamard manifolds $\mathcal C$, we say that a group $G$ is [*clinging in*]{} $\mathcal C$ if for any discrete subgroup $\G\le{\mbox{Iso}}(Y)$ such that $Y\in \mathcal C$ and $\G$ is isomorphic to $G$, and for any $\G$-invariant closed convex noncompact subset $W$ of $Y$, the group $\G$ is anchored in $W$. If $G$ is clinging in the class of all Hadamard manifolds, we simply call $G$ [*clinging*]{}.
In particular, $G$ is clinging in $\mathcal C$ if no such $\G$ exists but we of course are interested in nontrivial examples. Theorem \[thm: clinging\] and Corollary \[cor: elem amen\] below can be found in [@Bel-bus].
\[thm: clinging\] A group $G$ is clinging in $\mathcal C$ if one of the following is true: $G$ has a clinging in $\mathcal C$ normal subgroup, or $G$ is the union of a nested sequence of clinging in $\mathcal C$ subgroups. $G$ is as in . $G$ is virtually solvable and not virtually-$\mathbb Z^k$ for any $k$. a normal abelian subgroup of $G$ contains an infinite $G$-conjugacy class.
\[cor: elem amen\] Let $G$ be a finitely generated, torsion-free group that has a nontrivial, normal, elementary amenable subgroup. Then either $G$ is clinging, or $G$ has a nontrivial, finitely generated, abelian, normal subgroup that is a virtual direct factor of $G$.
Splitting results of Schroeder [@Sch-split-1985] and Monod [@Monod-superrid-jams] give another source of groups fixing points at infinity.
\[thm: commut subgr grounded\] Let $W$ be a closed, convex, noncompact subset of a Hadamard manifold. If a discrete torsion-free isometry group of $W$ contains two commuting subgroups $\G_1$, $\G_2$, then one of them fixes a point at infinity of $W$.
Suppose neither $\G_1$ nor $\G_2$ fixes a point at infinity of $W$, and in particular $\G_1\G_2$ is nontrivial. Since $\G_1$ fixes no point at infinity, it follows from [@Monod-superrid-jams Proposition 27, Remark 39 and Subsection 4.6] that $W$ contains a non-empty minimal convex closed $\G_1$-invariant subset $C_1$, and moreover, the union $C$ of such sets splits as $C_1 \times C_2$ for some bounded convex subset $C_2$ where $\G_1$, $\G_2$ preserve the splitting and act trivially on $C_2$, $C_1$, respectively. (Boundedness of $C_2$ is a key point, so we explain it here: if $C_2$ is unbounded, it contains a ray $s\to r(s)$, so given $x\in C_1$ we get a ray $\{x\}\times r$ in $X$ which is mapped by any $\g\in \G_1$ to an asymptotic ray as $\g$ maps $(x,r(s))$ to $(\g(x), r(s))$; thus $\G_1$ fixes a point at infinity contradicting the assumptions). Since $C_2$ is bounded, $\G_2$ fixes the circumcenter of $C_2$, and hence fixes a point $z_2\in C$. Repeating the same argument with $\G_2$, $\{z_2\}$ in place of $\G_1$, $C_1$ shows that the union $Z$ of minimal convex closed $\G_2$-invariant subsets splits as the product of $\{z_2\}$ and a bounded convex subset, and the splitting is invariant under $\G_1$, $\G_2$. It follows that $\G_1\G_2$ fixes a point of $Z$. This is where we need that $\G_1\G_2$ lies in a torsion-free discrete subgroup, because it implies that $\G_1\G_2$ is trivial.
\[cor: clinging no-parabolic\] If $G_1$ and $G_2$ are groups each containing a subgroup as in , then $G_1\times G_2$ is clinging in the class of all Hadamard manifolds.
Suppose $G_1\times G_2$ is isomorphic to a discrete subgroup $\G\le{\mbox{Iso}}(X)$ stabilizing a closed, convex, noncompact subset $W$. By Theorem \[thm: semisimple nonembed\](2), each factor contains a parabolic, and one of them fixes a point at infinity by Corollary \[thm: commut subgr grounded\], and hence is anchored in $W$. So $\G$ is anchored in $W$ by Theorem \[thm: anchored\](2).
\[cor: commut subgr grounded\] If $G_1$, $G_2$ are nontrivial torsion-free groups, then $G_1\times G_1$ is clinging in the class of Hadamard manifolds containing no flat half planes.
Suppose $G_1\times G_2$ is realized as a discrete isometry group of a Hadamard manifold with no flat half planes, and suppose $W$ is a closed, convex, noncompact invariant subset. Since $G_1\times G_2$ is torsion-free, by Theorem \[thm: commut subgr grounded\] one of the factors fixes a point at infinity of $W$. If say $G_1$ contains a hyperbolic element $h$, then $h$ has rank one as $X$ contains no flat half planes, so the centralizer of $h$ in $G_1G_2$ is cyclic, and also contains $h$ and $G_2$ violating the assumption that $G_2$ is nontrivial. Thus $G_1$ consists of parabolics, and by symmetry so does $G_2$. One of the groups $G_1$, $G_2$ fixes a point at infinity, hence it is anchored in $W$, and so is $G_1G_2$ by Theorem \[thm: anchored\](2).
Results of Caprace-Monod imply:
\[thm: co-amen grounded\] If $H$ is clinging in a class of Hadamard manifolds $\mathcal C$, and $G$ contains $H$ as a co-amenable subgroup, then $G$ is clinging in $\mathcal C$.
Realize $G$ as a discrete isometry group of a Hadamard manifold in $\mathcal C$ stabilizing a closed, convex, noncompact subset $W$. By assumption $H$ contains a parabolic, hence so does $G$. If $G$ does not fix a point at infinity of $W$, then by [@CapMon09 Theorem 4.3] $W$ contains a minimal closed convex $G$-invariant subspace $U$. Note that $U$ has no Euclidean de Rham factor (as the other factor would then be a smaller $\G$-invariant subset). The co-amenability implies that $H$ fixes no point at infinity of $U$ [@CapMon-discrete Proposition 2.1] so it is not clinging in $\mathcal C$.
Burger-Schroeder [@BurSch87] showed that any amenable subgroup $\G\le{\mbox{Iso}}(X)$ either fixes a point at infinity or stabilizes a flat, and this generalizes to actions on proper CAT($0$) spaces by Adams-Ballmann [@AdaBal]. Even more generally, Caprace-Monod [@CapMon-discrete Corollary 2.2] obtained the same conclusion whenever $\G$ contains two commuting co-amenable subgroups (and also gave examples with non-amenable $\G$). To make this result into a source of groups that fix a point at infinity more examples are needed, and with our focus on manifolds one has to answer the following.
Is there a group that contains two commuting co-amenable subgroups, has finite cohomological dimension, and is not virtually solvable?
As mentioned in Section \[sec: fix a point at infinity\], if $\G$ fixes a point at infinity, then $\G$ permutes horospheres centered at the point defining a homomorphism $\G\to\mathbb R$. Thus [*if $\G\le{\mbox{Iso}}(X)$ has no nontrivial homomorphism into $\mathbb R$, then $\G$ stabilizes a horoball if and only if $\G$ fixes a point at infinity.*]{}
Examples of clinging groups with finite abelianization (and hence no nontrivial homomorphisms into $\mathbb R$) are abound, see [@Bel-bus], and there are many such groups of finite cohomological dimension, or even of type [*F*]{} (so they may well be the fundamental groups of complete manifolds of $K\le 0$).
Note that the property of having finite abelianization is inherited by amalgamated products (clearly), and by extensions (due to right exactness of the abelianization functor). Moreover, an extension with a finite quotient often has finite abelianization, e.g. the abelianization of the semidirect product $A\rtimes B$ is $(A^{\mathrm{ab}})_B\times B^{\mathrm{ab}}$, where $(A^{\mathrm{ab}})_B$ is the coinvariants for the $B$-action on $A^{\mathrm{ab}}$.
Homotopy obstructions (after Gromov and Izeki-Nayatani) {#sec: homotopy obstr}
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If $M$ is a complete manifold of $K\le 0$, then $\pi_1(M)$ has finite cohomological dimension. A group has finite cohomological dimension if and only if it is the fundamental group of a manifold whose universal cover is diffeomorphic to a Euclidean space.
Gromov asked [@Gro-asy] whether every countable group of finite cohomological dimension is isomorphic to some $\pi_1(M)$ where $M$ is complete of $K\le 0$. (The question is a good illustration of how little we know about open manifolds of $K\le 0$.) The answer is no due to groundbreaking works of Gromov [@Gro-rand03] and Izeki-Nayatani [@IseNay05] on groups with strong fixed point properties.
These papers combine certain averaging procedures with ideas of harmonic map superrigidity to produce many a group $G$ such that
- any isometric $G$-action on a Hadamard manifold has a fixed point;
- $G$ has type [*F*]{} (i.e. is the fundamental group of a compact aspherical manifold with boundary).
Since the $\pi_1(M)$-action on the universal cover of $M$ is free, it follows that there is no complete manifold $M$ of $K\le 0$ and $\pi_1(M)\cong G$. The methods actually reach far beyond Hadamard manifolds, and apply to isometric $G$-actions on a wide variety of spaces, see [@IseNay-surv; @NaoSil11; @IKN].
Gromov’s examples are certain torsion-free hyperbolic groups produced from a sequence of graphs $\G_n$ whose edges are labeled with words of length $j$ in an alphabet of $d>1$ letters. The words are chosen randomly, and reversing orientation of the edge corresponds to taking inverse of a word. Given the data let $G(\G_n, d, j)$ be the quotient group of $F_d$, the free group on $d$ generators, by the relations corresponding to the cycles in $\G_n$. The main result is that there is a sequence of expander graphs $\G_n$ such that for a large enough $j$ the group $G(\G_n, d, j)$ is torsion-free, hyperbolic, and satisfies (a) with probability $\to 1$ as $n\to\infty$. Like any torsion-free hyperbolic group, $G(\G_n, d, j)$ has type [*F*]{}.
Izeki-Nayatani’s original example is any uniform torsion-free lattice in $PSL_3(\mathbb Q_p)$, which has type [*F*]{} because it acts freely and properly discontinuously on the associated Euclidean building.
The class of groups that satisfy (a) includes any finite group, or more generally any locally finite infinite subgroup such as $\mathbb Q/\mathbb Z$. There are much deeper examples in [@ABJLMS] of finitely presented, infinite, non-torsion-free groups that fix a point for any action by a homeomorphisms on a contractible manifold. None of these groups satisfies (b) as they have nontrivial finite order elements.
\[rmk: refl gr trick\] By Davis’s reflection group trick any group of type [*F*]{} embeds into the fundamental group of a closed aspherical manifold. Thus there is a closed aspherical manifold that is not homotopy equivalent to a complete manifold of $K\le 0$.
Is there a closed aspherical manifold whose fundamental group satisfies ?
Homeomorphism obstructions: exploiting $\mathbb R$ factors {#sec: homeo obstr}
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Call a group $G$ [*reductive*]{} if for any epimorphism $G\to H$ such that $H$ is a discrete, non-cocompact, torsion-free isometry group of a Hadamard manifold stabilizing a horoball, or a totally geodesic submanifold where $H$ acts cocompactly.
\[ex: elem\] (1) Any quotient of a reductive group is reductive. (2) Any finitely generated, virtually nilpotent group is reductive (as follows from Sections \[sec: non-parabolic\], \[sec: center with parabolic\], see [@Bel-bus]).(3) Any irreducible, uniform lattice in the isometry group of a symmetric space of $K\le 0$ and real rank $>1$ is reductive, by the harmonic map superrigidity [@Duc Theorem 1.2], see also [@Bel-bus].
A manifold is [*covered by $\mathbb R^n$*]{} if its universal cover is diffeomorphic to $\mathbb R^n$. Thus any complete $n$-manifold of $K\le 0$ is covered by $\mathbb R^n$.
It is well-known that an open $K(G,1)$ manifold covered by $\mathbb R^n$ exists if and only if $G$ is a countable group of finite cohomological dimension, see e.g. [@Bel-bus].
A manifold is [*covered by $\mathbb R\times\mathbb R^{n-1}$*]{} if it is diffeomorphic to the product of $\mathbb R$ and a manifold covered by $\mathbb R^{n-1}$. For instance, if $G$ is a discrete torsion-free isometry group of a Hadamard manifold that satisfies the condition (3) of Section \[sec: rough\], then $M$ is covered by $\mathbb R\times\mathbb R^{n-1}$. As we saw above (3) can be forced by purely algebraic assumptions on $\pi_1(M)$:
If $M$ is a complete connected manifold of $K\le 0$ such that $\pi_1(M)$ is either clinging with finite abelianization, or satisfies the assumptions of Theorem \[thm: into-center\], then $M$ is covered by $\mathbb R\times\mathbb R^{n-1}$.
A trivial method of producing manifolds that are covered by $\mathbb R^n$ but not covered by $\mathbb R\times\mathbb R^{n-1}$ is to consider any manifold of minimal dimension among all manifolds in its homotopy type that are covered by a Euclidean space, which yields (see [@Bel-bus]):
Any aspherical manifold is homotopy equivalent to a manifold covered by $\mathbb R^n$ but not covered by $\mathbb R\times\mathbb R^{n-1}$.
This method is non-constructive for it is not easy to decide whether a specific open manifold has the minimal dimension in the above sense (see [@BKK02; @BesFei02; @Yoo04; @Des06] for the manifolds of such minimal dimensions).
\[cor: inf gener\] If $G$ is reductive, clinging with finite abelianization, or as in , then any $K(G,1)$ manifold is homotopy equivalent to a manifold that admits no metric of $K\le 0$ and is covered by a Euclidean space.
Corollary \[cor: inf gener\] applies if $G=\mathbb Q$, see Example \[ex: rationals\].
An essential tool in understanding manifolds covered by $\mathbb R\times\mathbb R^{n-1}$ is the recent result of Guilbault [@Gui-prod-R07]: if an open manifold $W$ of dimension $\ge 5$ is homotopy equivalent to a finite complex, then $\mathbb R\times W$ is diffeomorphic to the interior of a compact manifold. Building on this result, the author [@Bel-bus] proved
\[thm-intro: reg nbhd\] Let $W$ be an open $(n-1)$-manifold with $n\ge 5$ that is homotopy equivalent to a finite complex of dimension $k\le n-3$. Then $\mathbb R\times W$ is diffeomorphic to the interior of a regular neighborhood of a $k\!$-dimensional finite subcomplex.
With more work one gets [@Bel-bus] the following applications:
\[thm: R-factor conseq\] Let $L$ be a finite aspherical CW complex such that $G=\pi_1(L)$ is reductive, clinging with finite abelianization, or as in . Suppose that $L$ is homotopy equivalent to a complete $n$-manifold $M$ of $K\le 0$ and $n\ge 5$, and set $l=\dim(L)$. If $l\le n-3$, then $M$ diffeomorphic to the interior of a regular neighborhood of a $k\!$-dimensional finite subcomplex. If $L$ is a closed manifold of dimension $<\frac{2n-2}{3}$, then $M$ is diffeomorphic to the total space of a vector bundle over $L$. If $l<\frac{n}{2}$, then every complete $n$-manifold of $K\le 0$ in the tangential homotopy type of $M$ is diffeomorphic to $M$. If $l\le n-3$, then the tangential homotopy type of $M$ contains countably many open $n$-manifolds that admit no complete metric of $K\le 0$.
Can one strengthen the conclusion “countably many” in the part of to “a continuum of”?
A positive answer is given in [@Bel-bus] under a technical assumption which holds e.g. if $L$ is a closed manifold, or if either $\mathbb Z^3$ or $\mathbb Z\ast \mathbb Z$ does not embed into $G$.
Limiting Theorem \[thm: R-factor conseq\] to certain classes of manifolds of $K\le 0$ may result in enlarging the class of allowable fundamental groups, e.g. applying the theorem to manifolds with visibility universal cover, we can allow $G$ to be the product of any two nontrivial groups.
As an application of Theorem \[thm-intro: reg nbhd\], we get the following characterization of $\mathbb R^n$:
\[cor-intro: Rn\] An open contractible $n\!$-manifold $W$ is homeomorphic to $\mathbb R^n$ if and only if $W\times S^1$ admits a metric of $K\le 0$.
If $n=4$, then “homeomorphic” in Corollary \[cor-intro: Rn\] cannot be upgraded to “diffeomorphic”: if $W$ is an exotic $\mathbb R^4$, then $W\times S^1$ is diffeomorphic to $\mathbb R^4\times S^1$.
Benefits of a lower curvature bound
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Complete manifolds of ${\mbox{Ric}}\ge -(n-1)$ are central to the global Riemannian geometry. For manifolds of $K\le 0$, a lower Ricci curvature bound at a point is equivalent (by standard tensor algebra considerations) to a lower sectional curvature bound at the same point; by rescaling one can always make the bounds equal the curvature of the hyperbolic $n$-space.
In the seminal work [@Gro-vol-bd-coh] Gromov uncovered a relation between the simplicial volume and volume growth, which for complete manifolds with ${\mbox{Ric}}\ge -(n-1)$ is governed by Bishop-Gromov volume comparison. The following can be found in [@Gro-vol-bd-coh p.13, 37].
\[thm: Gromov simp vol\] [**(Gromov)**]{} *Let $W$ be an $n$-manifold such that every component $C_i$ of $\d W$ is compact. If the interior of $W$ is homeomorphic to a complete manifold of ${\mbox{Ric}}\ge -(n-1)$, then $\displaystyle{\sum_i ||C_i||}\le \displaystyle{\liminf_{r\to\infty}}\frac{\mathrm{Vol}\, B_p(r)}{r}$.*
Here $B_p(r)$ is the $r$-ball in $M$ centered at $p$, and $||C_i||$ is the simplicial volume of $C_i$.
If $M$ in Theorem \[thm: Gromov simp vol\] has finite volume, or more generally sublinear volume growth, then each component of $\d W$ has zero simplicial volume.
Does every complete manifold with $-1\le K\le 0$ and sublinear volume growth admit a finite volume metric?
Any infinite cyclic cover a closed connected manifold $L$ of $K\le 0$, i.e. the cover corresponding to the kernel of a surjective homomorphism $\pi_1(L)\to \mathbb Z$, has linear growth.
Study manifolds of $-1\le K\le 0$ with linear volume growth.
Another consequence of a lower curvature bound is the famous Margulis lemma which appeared in [@BGS] for manifolds of $-1\le K\le 0$ following unpublished ideas of Margulis, and in [@FukYam; @KPT] for manifolds of $K\ge -1$. The following version of the Margulis lemma for manifolds of ${\mbox{Ric}}\ge -(n-1)$ is due to Kapovitch-Wilking [@KapWil] with essential ingredients provided by prior works of Cheeger-Colding.
\[thm: wilk-kap margulis\] **(Kapovitch-Wilking) *For each $n$ there are constants $m$ and ${\varepsilon}\in (0,1)$ such that if $p$ is a point of a complete $n$-manifold with ${\mbox{Ric}}\ge -(n-1)$ on $B_p(1)$, then the image of $\pi_1(B_p({\varepsilon}))\to\pi_1(B_p(1))$ has a nilpotent subgroup generated by $n$ elements and of index $\le m$.***
In fact, the nilpotent subgroup in Theorem \[thm: wilk-kap margulis\] has a generated set $\{s_1, \dots, s_n\}$ such that $s_1$ is central and the commutator $[s_i, s_j]$ is contained in the subgroup generated by $s_1, \dots, s_{i-1}$ for each $1<i<j$. Another universal bound on the number of generators of any given $r$-ball is given by
\[thm: wilk-kap fg\] **(Kapovitch-Wilking) *For each $n$, $r$ there is a constant $k$ such that if $p$ is a point in a complete Riemannian manifold $M$ such that $\pi_1(B_p(r))\to\pi_1(M,p)$ is onto and ${\mbox{Ric}}\ge -(n-1)$ on $B_p(4r)$, then $\pi_1(M,p)$ is generated by $\le k$ elements.***
An important feature of the two preceding results is that no curvature control is required outside a compact subset.
Injectivity radius going to zero at infinity
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We say that a subset $S$ of a Riemannian manifold [*has ${Inj}\,{Rad}\to 0$*]{} if and only if for every ${\varepsilon}>0$ the set of points of $S$ with injectivity radius $\ge {\varepsilon}$ is compact; otherwise, we say $S$ [*has ${Inj}\,{Rad}\not\to 0$*]{}.
By volume comparison any finite volume complete manifold of $K\le 0$ has $\mathrm{Inj}\,\mathrm{Rad}\to 0$ [@BGS 8.4].
\[prop: surface with injrad go to zero\] Any finite volume complete real hyperbolic manifold admits a complete metric with $\mathrm{Inj}\,\mathrm{Rad}\to 0$, bounded negative curvature, infinite volume, and sublinear volume growth.
We just do the two dimensional case; the general case is similar. Any end of a finite volume complete hyperbolic surface surface has an annular neighborhood with the metric $dt^2+e^{-2t}d\phi^2$, $t>0$. Modify it to the metric $dt^2+f^2(t)d\phi^2$ where $f$ is a convex decreasing function such that $f(t)=e^{-t}$ for small $t$, and $f(t)=t^{-\a}$, $\a\in (0,1)$ for large $t$. Let $\Sigma_f$ be the resulting complete Riemannian $2$-manifold, and let $\Sigma_f^r$ denote “$\Sigma_f$ with the portion with $t>r$ chopped off”. Now $\Sigma_f$ has $\bullet$ $\mathrm{Inj}\,\mathrm{Rad}\to 0$ because $f$ monotonically decreases to zero,$\bullet$ infinite volume since $\frac{1}{2\pi}{\mbox{Vol}}(\Sigma_f^r)$ grows (sublinearly) as $\int_0^r f(s)ds=\frac{r^{1-\a}}{1-\a}$,$\bullet$ bounded negative sectional curvature because on the annular neighborhood $K=-\frac{f^{\prime\prime}}{f}<0$ which equals $-\frac{\a(\a+1)}{t^2}$ for $t>r$.
Is there a complete manifold of $K\le 0$ and $\mathrm{Inj}\,\mathrm{Rad}\to 0$ that admits no complete finite volume of $K\le 0$? What is the answer in the presence of a lower curvature bound.
Gromov [@Gro-jdg78] pioneered the study of ends of negatively curved manifolds via the critical point theory for distance functions, which was extended by Schroeder [@BGS Appendix 2] as follows:
[**(Schroeder)**]{} \[thm: schroeder\] If $M$ is a complete manifold of $\mathrm{Inj}\,\mathrm{Rad}\to 0$ and $-1\le K\le 0$, then either $M$ is the interior of a compact manifold, or $M$ contains a sequence of totally geodesic, immersed, flat tori with diameters approaching zero.
None of the assumptions in the above theorem can be dropped due to examples of Gromov [@BGS Chapter 11] and Nguyen Phan [@Pha-neg] in which $\pi_1(M)$ is infinitely generated, see Section \[sec: inf hom\].
Find a geometrically meaningful compactification of complete manifolds of $\mathrm{Inj}\,\mathrm{Rad}\to 0$ and $-1\le K\le 0$.
In the locally symmetric case this was accomplished in [@Leu-invent; @Sap-compactif; @Leu-diff-geom-appl], and pinched negatively curved manifolds are naturally compactified by horospheres.
A weak substitute for a geometrically meaningful compactification is given by the following general theorem [@CheGro-chop]:
[**(Cheeger-Gromov)**]{} *\[thm: cheeger-gromov\] For each $n$ there is a constant $c$ such that any complete finite volume $n$-manifold $M$ of $|K|\le 1$ admits an exhaustion by compact smooth codimension zero submanifolds $M_i$ with boundary such that $M_i\subset \mathrm{Int}\,(M_{i+1})$, the norm of the second fundamental form of $\d M_i$ is $\le c$, and ${\mbox{Vol}}\,(\d M_i)\to 0$ as $i\to \infty$.*
The proof constructs a controlled exhaustion function on $M$. For a related work based on different technical tools see [@SchYau-diff-geom-book Theorem I.4.2] and [@Daf-exh; @WanLin-exh].
If $M$ in Theorem \[thm: cheeger-gromov\] is the interior of a manifold $W$ with compact boundary, then considering the components of $\d M_i$ that lie in a collar neighborhood of $\d W$, we conclude that
\[cor: cheeger-gromov\] If $W$ is a manifold with compact connected boundary whose interior admits a complete finite volume metric $g$ of $|K|\le 1$, then $\mathbb R\times\d W$ contains the sequence of compact separating hypersurfaces $H_i$ which are homologous to $\{0\}\times\d W$ and satisfy $\mathrm{MinVol}(H_i)\to 0$ as $i\to\infty$. Moreover, $\d W$ has even Euler characteristic and zero simplicial volume; If $\d W$ is orientable, then its Pontryagin numbers vanish. If $g$ also has $K\le 0$, then the $\ell^2$-Betti numbers of $\d W$ vanish, and hence $\d W$ has zero Euler characteristic.
Projecting onto the $\d W$ factor yields a degree one map $\d M_i\to \d W$ (if $\d W$ is non-orientable, then so is $H_i$, and we get a degree one map of their orientation covers). Thus $\d W$ has zero simplicial volume, which of course we already knew by Theorem \[thm: Gromov simp vol\].
By Chern-Weil theory the Pontryagin numbers $p_I(L)$ of a closed manifold $L$ satisfy $|p_I(L)|\le c_l\, \mathrm{MinVol}(L)$ where $c_l$ is a constant depending only on $l=\dim(L)$ [@Gro-vol-bd-coh]. Thus $p_I(H_i)\to 0$ as $i\to\infty$. Since Pontryagin numbers are oriented cobordism invariant, we conclude that if $\d W$ in Corollary \[cor: cheeger-gromov\] is orientable, then its Pontryagin numbers vanish.
The boundary of a compact manifold has even Euler characteristic [@Dol-book Corollary VIII.8.8], so applying this to the cobordism between $\d W$ and $H_i$ we see that $\chi(\d W)+\chi(H_i)$ is even, and again Chern-Weil theory implies $\chi(H_i)\to 0$ as $i\to\infty$, and the claim follows.
Finally, vanishing of the $\ell^2$-Betti numbers follows from [@CheGro-vndim Theorem 1.2], and their alternating sum equals the Euler characteristic.
Do any of the conclusions hold for complete manifolds with $-1\le K\le 0$ and $\mathrm{Inj}\,\mathrm{Rad}\to 0$.
If $\Sigma_f$ is as in Proposition \[prop: surface with injrad go to zero\], then the Riemannian product $\Sigma_f\times\Sigma_f$ has $\mathrm{Inj}\,\mathrm{Rad}\to 0$ but superlinear volume growth if $0<\a\le\frac{1}{2}$ because for large $r$ the subset $\Sigma_f^r\times\Sigma_f^r$ is sandwiched between concentric balls of radii $r$ and $3r$, and its volume grows superlinearly as $\a\le\frac{1}{2}$. Thus proving that the boundary has zero simplicial volume one requires new ideas beyond Theorem \[thm: Gromov simp vol\].
Negatively curved manifolds with uniform volume bound
=====================================================
For a connected complete Riemannian manifold $M$ we denote by $\widetilde M$ its universal cover with the pullback metric.
Fukaya [@Fuk-finiteness-neg] proved the following result, whose analog for closed manifolds of dimension $\neq 3$ is due to Gromov [@Gro-jdg78]:
\[thm: fukaya finiteness\] **(Fukaya) *Given $V$ and $n\neq 3, 4$, only finitely many of diffeomorphism classes contain open complete $n$-manifolds $M$ such that $K<0$ or $\widetilde M$ is visibility, $K\ge -1$, ${\mbox{Vol}}\,(M)< V$.***
In dimension four Fukaya proved that the class of manifolds satisfying (1)-(3) contains only finitely many homotopy types (the missing ingredient is the weak h-cobordism theorem, which is unknown for h-cobordisms between closed $3$-manifolds).
The theorem fails in dimension three as there are infinitely many (both open and closed) hyperbolic $3$-manifolds with uniformly bounded volume [@Thu-3d-notes]. Taking products with flat tori demonstrates that (1) cannot be replaced with $K\le 0$, even though the optimal curvature condition is unclear.
Is true with replaced by “$\widetilde M$ has rank one”, or “$\widetilde M$ contains no flat half planes”?
The proof in [@Gro-jdg78; @Fuk-finiteness-neg] established an upper diameter bound in terms of volume, and then applies Cheeger’s finiteness theorem (if $M$ is open the diameter bound is for a compact domain $D$ such that $M\setminus D$ is the interior of an h-cobordism). The strategy fails if one merely assumes that $\widetilde M$ has rank one by the following example: Chop off a cusp of a finite volume complete real hyperbolic manifold, and modify the metric to have totally geodesic flat boundary and $K\le 0$. Then double along the boundary, which gives a finite volume complete rank one manifold of $K\le 0$ and volume bounded roughly by $2{\mbox{Vol}}(M)$, but its diameter can be chosen arbitrary large by chopping deeper into the cusp.
How does the number of diffeomorphism types of manifolds $M$ in grows with $n$ and $V$?
In the locally symmetric case the above question was extensively studied, see [@Gel-vol-growth] and references therein.
Non-aspherical ends of nonpositively curved manifolds
=====================================================
If a (not necessarily connected) manifold $B$ is diffeomorphic to the boundary of a connected, smooth (not necessarily compact) manifold $W$, then we say that $B$ [*bounds*]{} $W$.
Any aspherical manifold $B$ bounds a noncompact aspherical manifold, namely $B\times [0,1)$, and in fact, the universal cover of $B\times (0,1)$ is a Euclidean space. Note that $B\times (0,1)$ admits a complete metric of $K\le 0$ if $B$ is an infranilmanifold [@BK-GAFA], or if $B$ itself admits a complete metric of $K\le 0$. On the other hand, if $\pi_1(B)$ contains a subgroup with strong fixed point properties as in Section \[sec: homotopy obstr\], then $B\times (0,1)$ admits no complete metric of $K\le 0$. Our ignorance is illustrated by the following
Does every closed aspherical manifold bounds a manifold whose interior admits a finite volume complete metric of $K\le 0$?
In this section we discuss similar matters when $B$ is closed and not aspherical. We focus on easy-to-state results and refer to [@BelPha-enp] for a complete account.
Boundaries of compact manifolds with a complete metric of $K\le 0$ on the interior could be quite diverse:
(1) The total space of any vector bundle a closed manifold of $K\le 0$ admits a complete metric of $K\le 0$ [@And-vb].(2) Complete finite volume locally symmetric manifold of $K\le 0$ and $\mathbb Q$-rank $\ge 3$ are interiors of compact manifolds with non-aspherical boundary.(3) A complete manifold $M$ of $K\le 0$ is [*convex-cocompact*]{} if it deformation retracts onto a compact locally convex subset; such $M$ is the interior of a compact manifold whose boundary is often non-aspherical.
There seem to be no simple description of closed manifolds that bound aspherical ones, and some obstructions are summarized below. In order for $B$ to bound an aspherical manifold, a certain covering space of $B$ must bound a contractible manifold. In formalizing how this restricts the topology of $B$, the following definition is helpful: given a class of groups $\mathcal Q$, a group is [*anti*]{}–$\mathcal Q$ if it admits no nontrivial homomorphism into a group in $\mathcal Q$. Clearly, the class of anti–$\mathcal Q$ groups is closed under extensions, quotients, and any group generated by a family of anti–$\mathcal Q$ subgroups is anti–$\mathcal Q$.
Let $\mathcal A_n$ denote the class of fundamental groups of aspherical $n$-manifolds. See [@BelPha-enp] for examples of anti–$\mathcal A_n$ groups in such as:
1. Any group generated by a set of finite order elements.
2. Any irreducible lattice in the isometry group of a symmetric space of rank $\ge 2$ and dimension $>n$ [@BesFei02].
The following summarizes some obstructions that prevent a manifold from bounding an aspherical one.
\[thm: intro-asp\] If $B$ bounds an aspherical, non-contractible $n$-manifold, and $\pi_1(B)$ is anti–$\mathcal A_n$, then $B$ is noncompact, parallelizable, its $\mathbb Z$-valued intersection form of vanishes, and its $\mathbb Q/\mathbb Z$-valued torsion linking form vanishes.
The following manifolds do not bound aspherical ones:
1. The connected sum of lens spaces, because it is a closed manifold whose fundamental group is anti–$\mathcal A_n$.
2. The product of any manifold with $CP^k$ with $k\ge 2$.
3. The connected sum of any manifold and the product of two closed manifolds whose fundamental groups are anti–$\mathcal A_n$.
4. The product of a punctured $3$-dimensional lens space and a closed manifold whose fundamental group is anti–$\mathcal A_n$.
5. Any manifold that contains the manifold in (2)–(4) as an open subset.
Let $\mathcal{NP}_n$ denote the class of the fundamental groups of complete $n$-manifolds of $K\le 0$; of course $\mathcal{NP}_n\subseteq\mathcal{A}_n$. Examples of anti-$\mathcal{NP}_n$ groups of type [*F*]{} discussed in Section \[sec: homotopy obstr\] immediately imply (see [@BelPha-enp]):
\[thm: anti-NP-Gromov\] There is a closed non-aspherical manifold that bounds a manifold whose interior is covered by a Euclidean space; bounds no manifold whose interior has a complete metric of $K\le 0$.
Other obstructions come from results of Section \[sec: homeo obstr\]. Given groups $I$, $J$ and a class of groups $\mathcal Q$ we say that [*$I$ reduces to $J$ relative to $\mathcal Q$*]{} if every homomorphism $I\to Q$ with $Q\in\mathcal Q$ factors as a composite of an epimorphism $I\to J$ and a homomorphism $J\to Q$. Here we are mainly interested in groups that reduce relative to $\mathcal{NP}_n$ to the groups from the parts (2)-(3) of Example \[ex: elem\], which have finite virtual cohomological dimension.
\[thm-intro: codim >2\] Let $n\ge 6$, let $G$ be a group from of virtual cohomological dimension $\le n-3$, and let $B$ be a closed $(n-1)$-manifold such that $\pi_1(B)$ reduces to $G$ relative to $\mathcal{NP}_n$. If $B$ bounds a manifold $N$ such that $\mathrm{Int}(N)$ admits a complete metric of $K\le 0$, then there is a closed manifold $L$ of dimension $\le n-3$ such that
- $L$ is either an infranilmanifold, or an irreducible, locally symmetric manifold of $K\le 0$ and real rank $\ge 2$.
- $N$ is the regular neighborhood of a PL-embedded copy of $L$.
- If $N$ is not diffeomorphic to the product of a compact manifold and a closed interval, then $L$ admits a metric of $K\le 0$ and $N$ is the total space of a linear disk bundle over $L$.
In [@BelPha-enp] we give examples of manifolds $B$ that cannot bound a manifold $N$ as in the above theorem such as
Let $B$ be the total space of a linear $S^k$ bundle over a closed non-flat infranilmanifold such that $k\ge 3$ and the rational Euler class of the bundle is nonzero. Then $B$ does not bound a manifold whose interior admits a complete metric of $K\le 0$.
A boundary component of a manifold is [*incompressible*]{} if its inclusion induces injections on all homotopy groups. Reductive groups were defined in Section \[sec: homeo obstr\].
\[thm: tors-free quot\] Let $B$ be a closed $(n-1)$-manifold such that $\pi_1(B)$ is reductive and any nontrivial quotient of $\pi_1(B)$ in the class $\mathcal{NP}_n$ has cohomological dimension $n-1$. If $B$ bounds a manifold $N$ such that $\mathrm{Int}(N)$ admits a complete metric of $K\le 0$, then $B$ is incompressible in $N$.
\[ex: tf general\] Theorem \[thm: tors-free quot\] applies to whenever $\pi_1(B)$ is isomorphic to $\pi_1(L)$ where $L$ is an $(n-1)$-manifold of from Theorem \[thm-intro: codim >2\](1) and $\pi_1(L)$ has no proper torsion-free quotients. Examples of such $L$ include any higher rank, irreducible, locally symmetric manifolds of $K\le 0$ (thanks to the Margulis Normal Subgroup Theorem), as well as certain infranilmanifolds, see [@BelPha-enp].
Given a compact boundary component $B$ of a manifold $N$, an [*end $E$ of $\mathrm{Int}(N)$ that corresponds to $B$*]{} is the intersection of ${\mbox{Int}}(N)$ with a closed collar neighborhood of $B$; note that $E$ is diffeomorphic to $[1,\infty)\times B$.
\[thm: tg\] Let $B$ be a closed connected manifold that bounds a manifold $N$, and let $E$ be an end of $\mathrm{Int}(N)$ corresponding to $B$. If $\pi_1(B)$ is reductive, and $\mathrm{Int}(N)$ admits a complete metric of $K\le 0$ and $\mathrm{Inj}\,\mathrm{Rad}\to 0$ on $E$, then $B$ is incompressible in $N$.
\[ex: sphere bundle\] If $B$ is the total space of a linear $S^k$ bundle with $k\ge 2$ over a manifold $L$ as in \[thm-intro: codim >2\](1), then $B$ does not bound a manifold whose interior admits a complete metric of $K\le 0$ and $\mathrm{Inj}\,\mathrm{Rad}\to 0$.
Riemannian hyperbolization (after Ontaneda) {#sec: ontaneda}
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A recent work of Ontaneda [@Ont-smth-hyp] allows to dramatically expand the list of known finite volume complete manifolds of $-1\le K<0$ of dimensions $>3$. Unlike earlier examples, Ontaneda’s method assembles a manifold of $K\le -1$ in a lego-like fashion from identical blocks according to a combinatorial pattern specified by a cube complex structure on any given manifold. Each block is a compact real hyperbolic manifold with corners, where every boundary face is totally geodesic and the faces’s combinatorial pattern is that of a cube. This process results in a singular metric, which Ontaneda is able to smooth into a complete Riemannian metric of $K\le -1$, provided the block’s faces have a sufficiently large normal injectivity radii; such blocks exist.
The idea of building locally CAT($0$) manifold out of identical blocks is due to Gromov [@Gro-hypgr] who came up with several hyperbolization procedures turning a simplicial complex into a locally CAT($0$) cubical complex. Gromov’s ideas were developed and made precise in [@DavJan-jdg-hyperb; @DJW], cf. [@Dav-book], and furthermore Charney-Davis [@ChaDav] developed the [*strict hyperbolization*]{} that turns a cubical complex into a piecewise polyhedral complex whose faces are the blocks of the previous paragraph. A key feature of the procedures is that the link of every vertex of the hyperbolized polyhedral complex is a subdivision of the corresponding link in the original complex, which has two consequences:
- The hyperbolization(s) turns a manifold triangulation into a locally CAT($0$) manifold with a cubical complex structure;
- The strict hyperbolization turns a locally CAT($0$) cubical complex (or manifold) into a locally CAT($-1$) polyhedral complex (manifold, respectively).
Trivial exceptions aside, the resulting piecewise Euclidean (or piecewise hyperbolic) metric is non-Riemannian. Smoothing the piecewise hyperbolic metric into a Riemannian metric of $K\le -1$ in [@Ont-smth-hyp] is a technological tour de force.
1. The cube complex fed into Ontaneda’s construction need not be locally CAT($0$), so the resulting manifolds of $K\le -1$ in [@Ont-smth-hyp] are a priori not homeomorphic to locally CAT($-1$) manifolds of [@ChaDav].
2. Charney-Davis [@ChaDav] describe a canonical smoothing of their manifolds (but not of the metrics), yet this smoothing is not necessarily equal to the smooth structure in [@Ont-smth-hyp] even when there is a face preserving homeomorphism between the two manifolds.
3. Taking the normal injectivity radius of the block’s faces large enough, one can make the sectional curvature of the resulting manifold arbitrary close to $-1$.
4. If one starts with a closed manifold, then Ontaneda’s procedure yields a closed manifold of $K\le -1$, but if the initial manifold is open, its triangulation contains infinitely many simplices, so the resulting Riemannian manifold of $K\le -1$ has infinite volume.
In order to produce finite volume examples, Ontaneda relativizes the construction as follows: Start with a closed manifold $B$ that bounds a compact manifold $W$, cone off the boundary, apply the strict hyperbolization, and remove the cone point. The result is a compact manifold with boundary, and Ontaneda gives it a certain smooth structure in which the boundary becomes diffeomorphic to $B$. Topological properties of the resulting smooth manifold $R_{W,B}$ mirror those of $W$, see a summary in [@Bel-hyper-polyh], and in particular by varying $W$, one finds $R_{W,B}$’s such that
- $R_{W,B}$ has a nontrivial rational Pontryagin class if $\dim(W)\ge 4$,
- $H^*(R_{W,B})$ contains a subring isomorphic to the cohomology ring of a given finite CW complex of dimension $<\dim(W)/2$, see [@Ont-smth-hyp],
- $\pi_1(R_{W,B})$ surjects on a given finitely presented group if $\dim(W)\ge 4$.
Moreover $R_{W,B}$ enjoys the following properties:
- $R_{W,B}$ is orientable if so is $W$,
- the inclusion of each component of $B$ into $R_{W,B}$ is $\pi_1$-injective [@DJW],
- $R_{W,B}$ is aspherical if and only if each component of $B$ is aspherical [@DJW],
- $\pi_1(R_{W,B})$ is hyperbolic relative to the images of the fundamental groups of components of $B$ [@Bel-hyper-polyh].
The piecewise hyperbolic metric on $R_{W,B}$ is singular and incomplete near the removed cone point, but again when the normal injectivity radius of the block’s faces large enough, Ontaneda is able to smooth the metric away from a punctured neighborhood of the cone point, while on that neighborhood the metric has to be constructed by an ad hoc method depending on $B$. The following result is implicit in [@Ont-smth-hyp].
\[thm-intro: ont\] [**(Ontaneda)**]{} Let $B$ be the boundary of a compact $n$-manifold and suppose that if $n\ge 6$, then any h-cobordism from $B$ to another manifold is a product; $\mathbb R\times B$ admits a complete metric $g$ of sectional curvature within $[-1, 0)$ such that $(-\infty, 0\,]\times B$ has finite $g$-volume, and $g=dr^2+e^{2r} g_{_B}$ on $[c,\infty)\times B$ for some $c>0$ and a metric $g_B$ on $B$. Then $B$ bounds a compact smooth manifold whose interior admits a complete metric of finite volume and sectional curvature in $[-1,0)$.
Condition (ii) implies that each component of $B$ is aspherical, and hence has torsion-free fundamental group. The Whitehead Torsion Conjecture, which is true for many groups of geometric origin [@BarLue-borelconj; @BFL], predicts that all torsion-free groups have zero Whitehead torsion. If the conjecture is true for the fundamental group of each component of $B$, then (i) holds. Condition (ii) has been checked in a number of cases in [@BK-GAFA; @Pha-sol; @Bel-ewp] implying:
\[thm: sol class B\] A manifold $B$ bounds a compact manifold whose interior admits a complete metric of finite volume and sectional curvature in $[-1,0)$ if $B$ is a closed $3$-dimensional $Sol$ manifolds [@Pha-sol], or $B$ bounds a compact manifold, and belongs to the class $\mathcal B$ [@Bel-ewp].
Here $\mathcal B$ is the smallest class of closed manifolds of positive dimension such that
- $\mathcal B$ contains each infranilmanifold [@BK-GAFA], each closed manifold of $K\le 0$ with a local Euclidean de Rham factor of positive dimension, and every circle bundle of type ;
- $\mathcal B$ is closed under products, disjoint unions, and products with any compact manifold of $K\le 0$.
An orientable circle bundle [*has type* ]{} if its base is a closed complex hyperbolic $n$-manifold whose holonomy representation lifts from $PU(n,1)$ to $U(n, 1)$, and if the Euler class of the bundle equals $-m\frac{\omega}{4\pi}$ for some nonzero integer $m$, where $\omega$ is the Kähler form of the base. For example, every nontrivial orientable circle bundle over a genus two orientable closed surface has type (K), see [@Bel-ewp].
Consider the class of closed manifolds $B$ to which applies. Does the class contain every circle bundle over a closed manifold of $K<0$? Every closed infrasolvmanifold? Is the class closed under products?
Topology of known manifolds of bounded negative curvature
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In dimensions $>3$ most (if not all) known examples of complete manifolds of $K\le 0$ come from combining
- locally symmetric metrics of $K\le 0$ arising from arithmetic or reflection groups,
- iterated and multiple warped products, building on the seminar work of Bishop-O’Neill on singly warped products of $K\le 0$ [@BisON-warp].
Ontaneda’s work [@Ont-smth-hyp] illustrates how the two methods combine: the block is an arithmetic real hyperbolic manifold with corners, and smoothing the metric involves sophisticated warped product considerations.
Prior examples of finite volume complete manifold of $K\le 0$ that are not locally symmetric were typically produced by starting with a locally symmetric complete manifold of $K\le 0$ and performing one of the following operations, which can be combined or iterated:
- doubles (Heintze, see [@Sch-warp]) and twisted doubles [@Ont-double03],
- branched covers [@MosSiu; @GroThu; @ForSch90; @Zhe96; @Ard-thesis00; @Der05; @FO-branch06],
- cusp closing [@Sch-cusp-close; @BuyKob-cusp-close; @HumSch; @And-cusps-einst],
- cut and paste a tubular neighborhood of a totally geodesic submanifold of codimension $1$, or dimensions $0$ or $1$, see [@FJO-survey] for a survey,
- remove a family of totally geodesic submanifolds of codimension two, which results in an incomplete metric that in some cases can be modified to a complete metric of $K\le 0$ [@Fuj-warp; @AbrSch; @Buy; @Bel-ch-warp; @Bel-rh-warp].
With few general results available, it makes sense to study topology of known examples in more detail. For the rest of the section let $M$ be an open connected complete finite volume manifold of $-1\le K<0$. The prior discussions gives
- (Section \[sec: acyl hyp\]) $\pi_1(M)$ is acylindrically hyperbolic because $M$ has finite volume and $K<0$,
- (Theorem \[thm: schroeder\]) $M$ is the interior of a compact manifold $N$, which is uniquely determined up to attaching an h-cobordism to the boundary.
- (Corollary \[cor: cheeger-gromov\]) $\d N$ has zero simplicial volume and Euler characteristic, and if $\d N$ is orientable, $\d N$ has zero Pontryagin numbers.
We would get a lot more information if $\pi_1(M)$ were hyperbolic relative to a collection of easy-to-understand peripheral subgroups (mainly because relatively hyperbolic groups inherit many properties from their peripheral subgroups, see below). Here is a prototypical example:
If $M$ is negatively pinched (i.e $K$ is bounded between two negative constants), then $\pi_1(N)$ is hyperbolic relatively the fundamental groups of the components of $\d N$ [@Far-rel-hyp; @Bow-rel-hyp] which are virtually nilpotent [@BGS].
One might expect that $\pi_1(M)\cong\pi_1(N)$ is always hyperbolic relative to the fundamental groups of components of $\d N$. This idea runs into difficulties because $\d N$ need not be $\pi_1$-incompressible [@Buy], but when it works it does so to great effect, and become a major source of information about $\pi_1(M)$.
We refer to [@Osi-rel-hyp] for background on relatively hyperbolic groups; as a part of the definition we require that relatively hyperbolic groups are finitely generated and not virtually cyclic, and their peripheral subgroups are infinite and proper.
\[thm: co-Hopf\] **(Belegradek) *If $W$ is a compact aspherical manifold of dimension $\ge 3$ such that the components of $\d W$ are aspherical and $\pi_1$-injectively embedded, and $\pi_1(W)$ is hyperbolic relatively to the fundamental groups of the components of $\d W$, then***
(a) *if $\pi_1(W)$ splits nontrivially as amalgamated product or HNN extension over a subgroup $K$, then $K$ is acylindrically hyperbolic.*
(b) *$\pi_1(W)$ is co-Hopf, i.e. its injective endomorphisms are surjective.*
(c) *$\mathrm{Out}(\pi_1(W))$ is finite if for every component $B$ of $\d W$ the group $\pi_1(B)$ is either not relatively hyperbolic or has a relatively hyperbolic structure whose peripheral subgroups are not relatively hyperbolic.*
Mayer-Vietoris sequence in the group cohomology applied to the fundamental group of the double of $W$ along $\d W$ can be used to show that $\pi_1(W)$ can only split over a non-elementary subgroup proving (a), which in turn implies (b)–(c) by results of Dru[ţ]{}u-Sapir [@DruSap-split]. An alternative proof of (b) is immediate from the fact that $(W,\d W)$ has positive simplicial norm [@MinYam]. Details can be found in [@Bel-hyper-polyh].
The fundamental group of a closed aspherical manifold with zero simplicial volume is not relatively hyperbolic (as noted in [@BelHru] this follows from [@MinYam]). Thus if $W$ is as in Theorem \[thm: co-Hopf\] and $\mathrm{Int}(W)$ admits a complete metric finite volume of ${\mbox{Ric}}\ge -(n-1)$, then $\d W$ has zero simplicial volume by Theorem \[thm: Gromov simp vol\], so $\mathrm{Out}(\pi_1(W))$ is finite.
Proving relative hyperbolicity was the main goal of the author’s work in [@Bel-hyper-polyh; @Bel-ch-warp; @Bel-rh-warp; @BelHru] where it was accomplished in the following cases:
\[ex: rel hyp fin vol\] (1) If $R_{W,B}$ is from Section \[sec: ontaneda\], then $\pi_1(R_{W,B})$ is hyperbolic relative to the fundamental groups of components of $B$ [@Bel-hyper-polyh]. If $B$ is as in Theorem \[thm: sol class B\], then $R_{W,B}$ admits a complete finite volume metric of $-1\le K<0$. (2) If $V$ is a closed manifold of $K<0$, and $S$ is an embedded, codimension two, compact, totally geodesic submanifold, then $\pi_1(V\setminus S)$ is hyperbolic relative to the fundamental groups of boundary components of a tubular neighborhood of $S$ in $V$ [@BelHru]. A finite volume complete metric of $-1\le K<0$ on $V\setminus S$ was constructed in [@Bel-ch-warp; @Bel-rh-warp] when either $V$ is real hyperbolic, or $V$ and $S$ are complex hyperbolic.(3) Let $V$ be a closed manifold of $K<0$, and $S$ be an immersed, codimension two, compact, totally geodesic submanifold whose preimage to the universal cover $\widetilde V$ of $V$ is [*normal*]{} in the sense of Allcock and is “sparse” in the sense that and two disjoint components are sufficiently separated. With these assumptions $\pi_1(V\setminus S)$ is hyperbolic relative to the fundamental group of the boundary components of a regular neighborhood of $S$ in $V$ [@BelHru]. If $V$ is real hyperbolic, then $V\setminus S$ admits a complete finite volume metric of $-1\le K<0$ by [@AbrSch].
A version of the claim in Example \[ex: rel hyp fin vol\](2)-(3) holds when $V$ is an open complete finitely volume negatively pinched manifold.
Without the assumptions that the preimage of $S$ to $\widetilde V$ is normal and “sparse”, it seems unlikely that $\pi_1(V\setminus S)$ in Example \[ex: rel hyp fin vol\](3) is hyperbolic relative to some easy-to-understand peripheral subgroups, so we ask:
Let $V$ be a finite volume complete negatively pinched manifold, and $S$ be an immersed, codimension two, compact, totally geodesic submanifold. Is $\pi_1(V\setminus S)$ acylindrically hyperbolic?
If $G$ is a finitely generated group that is hyperbolic relatively to a finite family of peripheral subgroups, then $G$ inherits the following properties of its peripheral subgroups:
1. solvability of the word problem [@Far-rel-hyp; @Osi-rel-hyp] (which is a litmus test for decency of a group).
2. solvability of conjugacy problem [@Bum-conj].
3. being fully residually hyperbolic [@Osi-fill; @GroMan-fill]; here given a class of groups $\mathcal C$, a group $G$ is [*fully residually*]{} $\mathcal C$ if any finite subset of $G$ can be mapped injectively by a homomorphism of $G$ onto a group in $\mathcal C$.
4. being biautomatic [@Reb].
5. finiteness of asymptotic dimension [@Osi-asy-dim]
6. rapid decay property [@DruSap-rapid-decay]
7. Tits alternative: a subgroup of a relatively hyperbolic group that does not contain a non-abelian free subgroup is [*elementary*]{}, i.e. virtually-$\mathbb Z$, finite, or contained in a peripheral subgroup [@Tuk-tits].
If $V$ and $S$ are as in Example \[ex: rel hyp fin vol\](2), then the peripheral subgroups in the relatively hyperbolic groups structure on $\pi_1(V\setminus S)$ are the fundamental groups of circle bundles over components of $S$. The peripheral subgroups have solvable word and conjugacy problems, have finite asymptotic dimension and rapid decay property, are biautomatic, residually hyperbolic, and their non-virtually-abelian groups contain free nonabelian subgroups, see [@Bel-ch-warp; @Bel-rh-warp]. Hence all these properties are inherited by $\pi_1(V\setminus S)$.
The fact that most real hyperbolic manifolds are a-Kähler [@Gro-asy] was used in [@Bel-hyper-polyh] to show
$R_{W,B}$ is not homeomorphic to an open subset of a Kähler manifold of real dimension $\ge 4$.
In fact, “not homeomorphic” can be replaced with “not proper homotopy equivalent” under a mild assumption on the strict hyperbolization block [@Bel-hyper-polyh].
Zoo of finite volume rank one manifolds {#sec: inf hom}
=======================================
In this section we discuss examples and structure of connected open complete finite volume manifolds of $K\le 0$ and rank one, with a particular focus on manifolds of $K<0$.
\[thm: rk1 acyl hyp\] If $V$ is a complete finite volume manifold of rank one, then $\pi_1(V)$ is acylindically hyperbolic.
The universal cover $\widetilde V$ has rank one, and Ballmann [@Bal-book] proved that a lattice in the isometry group of a rank one Hadamard manifold contains a rank one element that lies in a noncyclic free subgroup, so Sisto’s Theorem \[thm: sisto rk1\] applies.
Does every complete rank one manifold with $K\le 0$ and $\mathrm{Inj}\,\mathrm{Rad}\to 0$ have acylindically hyperbolic fundamental group?
Kapovitch-Wilking’s version of the Margulis lemma only calls for $K\ge -1$ on a ball of radius $1$, and that curvature bound can always be achieved by rescaling. Since acylindically hyperbolic groups are not virtually nilpotent, Theorem \[thm: wilk-kap margulis\] immediately implies:
If ${\varepsilon}$ is the constant in , then a finite volume complete rank one manifold $M$ contains no point $p$ such that $K\ge -1$ on $B_p(1)$ and the inclusion $B_p({\varepsilon})\hookrightarrow M$ is $\pi_1$-surjective.
Thus if $\pi_1(M)$ is “concentrated” on an ${\varepsilon}$-ball, then $K$ blows up near that ball.
Known examples of finite volume complete manifolds of $K<0$ that admit no finite volume metric of $-1\le K\le 0$ are based on Theorem \[thm: Gromov simp vol\] that a compact boundary component of a manifold whose interior has a complete metric of ${\mbox{Ric}}\ge-(n-1)$ has zero simplicial volume. Indeed, Nguyen Phan [@Pha-neg] proved
**(Nguyen Phan) *In each dimension $\ge 3$ there exists a finite volume complete manifold of $K<0$ that is the interior of a compact manifold whose boundary admits a real hyperbolic metric.***
Recall that a closed real hyperbolic manifold has a positive simplicial volume. The proof of the above theorem is by explicit construction; alternatively, it follows from Ontaneda’s Theorem \[thm-intro: ont\] by inserting in the cusp the warped product $\mathbb R\times_{e^r} B$ where $B$ is any closed manifold of $K\le 0$ with nonzero simplicial volume. In fact, this argument proves:
**(Ontaneda) *If a closed manifold of $K\le 0$ is diffeomorphic to the boundary of a compact manifold, then it is diffeomorphic to the boundary of a compact manifold whose interior admits a complete finite volume metric of $K\le -1$.***
There are closed manifolds of $K\le 0$ that are not homeomorphic to the boundaries of compact manifolds. One source of such examples is evenness of the Euler characteristic of the boundary of a compact manifold [@Dol-book Corollary VIII.8.8]. Examples of closed manifold of $K\le 0$ with odd Euler characteristic include suitable closed non-orientable hyperbolic surfaces, Mumford’s complex hyperbolic surface of Euler characteristic $3$ [@HerPau Proposition2.2], or their products.
Iterated gluing along totally geodesic boundaries sometimes yields finite volume $M$ with infinitely generated fundamental group. This phenomenon was discovered by Gromov [@BGS Chapter 11] who produced such $3$-dimensional graph manifolds with $K\in [-1, 0]$ and no local Euclidean de Rham factor. Other examples with infinitely generated fundamental groups, due to Nguyen Phan [@Pha-neg], are finite volume manifolds with $K\le -1$ and infinitely many ends, appearing in all dimensions $\ge 2$.
A related idea of Nguyen Phan [@Pha-neg] uses infinite cyclic covers of closed manifolds of $K<0$ to produce two-ended finite volume manifolds of $K<0$:
\[thm: cyclic cover\] **(Nguyen Phan) *If $\widehat L$ is an infinite cyclic cover of a closed manifold $L$ of $K<0$ of dimension $l\ge 3$, then $\widehat L$ admits a complete finite volume metric metric of $K<0$.***
Here $\widehat L\to L$ is the covering that corresponds to the kernel of any epimorphism $\pi_1(L)\to\mathbb Z$. A most famous example is when $\widehat L$ corresponds to the fiber group in a closed hyperbolic $3$-manifolds that fibers over a circle.
(of $\widehat L$ with infinitely generated fundamental group) Suppose that $\pi_1(L)$ surjects onto a noncyclic free group $F_r$. The kernel of any epimorphism $F_r\to\mathbb Z$ is infinitely generated, so hence so is the kernel of the composite $\pi_1(L)\to F_r\to\mathbb Z$.
Is $\pi_1(\widehat L)$ always infinitely generated when $\dim(\widehat L)>3$?
If $\dim(\widehat L)\ge 6$ and $\widehat L$ is homotopy equivalent to a finite cell complex (or more generally is finitely dominated), then $L$ (smoothly) fibers over a circle, and the fiber is a closed aspherical manifold whose inclusion into $L$ corresponds homotopically to the covering $\widehat L\to L$. Indeed, Siebenmann’s version [@Sie-totwh] of Farrell’s fibering obstruction lies in the group $\mathrm{Wh}(\pi_1(L))$, which is zero by [@FJ-dynI].
Another restriction on $\pi_1(\widehat L)$ is that its outer automorphism group is infinite (as easily follows from the fact that $\pi_1(\widehat L)$ has trivial centralizer in $\pi_1(L)$). Combining with the previous remark we see that if $\dim(\widehat L)\ge 6$ and $\widehat L$ is finitely dominated, then $\widehat L$ is homotopy equivalent to a closed aspherical manifold of dimension $\ge 5$ whose fundamental group has infinite outer automorphism group and embeds into the hyperbolic group $\pi_1(L)$; it seems such a closed aspherical manifold cannot exist, so we ask:
Can $\widehat L$ ever be finitely dominated when $\dim(\widehat L)>3$?
\[ex: phan surface cross R\] In dimension three Theorem \[thm: cyclic cover\] gives a finite volume metric of $K<0$ on the product of a closed hyperbolic surface with $\mathbb R$, and interestingly, the metric can be chosen so that the corresponding nonuniform lattice contains no parabolics, see [@Pha-neg].
Is there a nonuniform lattice in the isometry group of a Hadamard manifold of dimension $>3$ that contains no parabolics?
Borel-Serre [@BorSer74] compute the $\mathbb Q$-rank of a finite volume complete locally symmetric $n$-manifold $V$ as $n-{\mbox{cd}}(\pi_1(V))$, where ${\mbox{cd}}$ denotes the cohomological dimension; alternatively, $\mathbb Q$-rank equals the dimension of an asymptotic cone of $V$ [@Hat-asy-cone], cf. [@JiMcP], and $\mathbb Q$-rank can also be defined in terms of flats in $V$ [@Mor-book]. If $V$ has rank one (i.e. contains a rank one geodesic), then the $\mathbb Q$-rank of $V$ equals $1$. One wonders whether any of these relations between ${\mbox{cd}}$, asymptotic cone, and absence of $2$-dimensional flats extend to finite volume complete manifolds of rank one.
Is the asymptotic cone of a complete finite volume rank one manifold of $K\le 0$ always a tree?
\[quest: values of cd\] What values does $\mathrm{cd}$ take on the fundamental groups of open complete finite volume $n$-manifolds of $K\le -1$, rank one, or $K<0$? Is $\mathrm{cd}$ always equal $n-1$?
To better understand the above question let us relate bounds on ${\mbox{cd}}$ with the fundamental group at infinity of an open aspherical $n$-manifold $M$. First note that if ${\mbox{cd}}(\pi_1(M))\le n-2$, then $M$ is one-ended [@Sie-collar Proposition 1.2] but the converse fails (think of the open Möbius band). In the simplest case when $M$ is the interior of a compact manifold, we get the following clean statement:
Let $N$ be a compact aspherical $n$-manifold with boundary. Then $\mathrm{cd}(\pi_1(M))\le n-2$ if and only if $\d N$ is connected and the inclusion $\d N\hookrightarrow N$ is $\pi_1$-surjective.
Clearly ${\mbox{cd}}(\pi_1(M))\le n-1$. Now Poincaré-Lefschetz duality in the universal cover [@Bro-book Corollary VIII.8.3] implies that ${\mbox{cd}}(\pi_1(M))=n-1$ if and only if the boundary of the universal cover of $N$ is not connected. The latter is equivalent to “either $\d N$ is not connected or $\d N\hookrightarrow N$ is not $\pi_1$-surjective” by elementary covering space considerations.
Question \[quest: values of cd\] should be compared with Nguyen Phan’s Example \[ex: phan surface cross R\] of a finite volume manifold of $K<0$ that is quite small homologically, and perhaps there are even smaller examples. We finish with the following tantalizing
Given $n>2$, does the interior of the $n$-dimensional handlebody admit a complete finite volume metric of $K\le 0$?
The interior of an odd-dimensional handlebody admits no complete finite volume metric of $-1\le K\le 0$. (The boundary of an odd-dimensional handlebody with $g$ handles has Euler characteristic $2-2g$ while under our geometric assumptions the Euler characteristic vanishes by Theorem \[cor: cheeger-gromov\] and $g\neq 1$ because $\mathbb Z$ is not acylindrically hyperbolic).
[^1]: *2010 Mathematics Subject classification. Primary 53C20. * Keywords: nonpositive curvature, discrete group, open manifold, ends, negative curvature, finite volume, rank one.**
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Q:
Completing the square of $(x+a)(x+b)$
The problem is simple, to complete the square of $(x+a)(x+b)$. My calculations yield
$$\left(x+\frac{a+b}{2}\right)^2-\frac{(a+b)^2}{4}+ab,$$
But the textbook's answer is different ("problem 361", at the bottom of the page):
$$\left(x+\frac{a+b}{2}\right)^2-\frac{(a-b)^2}{4}$$
Did I do anything the wrong way?
$$(x+a)(x+b)=x^2+xb+ax+ab=x^2+x(a+b)+ab=$$
$$=\left(x^2+2*\frac{a+b}{2}*x+\left(\frac{a+b}{2}\right)^2\right)-\left(\frac{a+b}{2}\right)^2+ab=$$
$$=\left(x+\frac{a+b}{2}\right)^2-\frac{(a+b)^2}{4}+ab$$
A:
You did nothing wrong.
Note that
$$-\frac{(a+b)^2}{4}+ab=\frac{-(a+b)^2+4ab}{4}=\frac{-a^2+2ab-b^2}{4}=-\frac{(a-b)^2}{4}$$
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FIELD OF THE INVENTION
BACKGROUND OF THE INVENTION
OBJECTS OF THE INVENTION
SUMMARY OF THE INVENTION
DETAILED DISCRIPTION OF THE INVENTION
EXAMPLES
Example-1
Design of Soil Nailing for Stabilisation of Vertical Cut Slopes for Construction of Road Under the Approach Embankment of Bridge by Box Pushing Technique at West End Approach of Old Yamuna Bridge No. 249, Delhi Shahadra Section
Field Investigations, Design and Construction Methodology
Design for Soil Nailing
Construction Methodology
General Arrangements
Arrangements for Soil Nailing
Technical Challenges Overcome
Example-2
Box Pushing Technique with Soil Nailing at Apsara Border (without Retaining Wall)
Example-3
Large Size Water Pipe Line Pushing Below Sand Embankment Near Yamuna Bridge, Delhi
Advantages of the Invention
Present invention relates to the method of repeated de-stabilisation and stabilisation of vertical cut slope of highly collapsible sandy soil by ‘Soil Nailing Technique’ for the construction of underpass below the rail/road traffic through tunneling process. The underpass is constructed below the railway track where 250 to 300 trains passing over a day which required uninterrupted railway track having zero mistake zone in Delhi.
Rapid growth in population, industries, infrastructure development in the urban area tremendously increased the traffic volume which resulted the traffic congestion on the roads led to shortage of road space at ground level, therefore, there is need to create extra space above or below the ground level to meet out this demand. Construction of elevated roads/railways disturb the traffic system, however, the underground structures like multilevel roads or underpasses, road tunnels, metro systems do not disturb the surrounding. In this regards the other scopes of underground structures like; Malls, multilevel basements, water supply, flood water storage tunnels, sewers, cable tunnels, substations, air raid shelters, and storage facilities, etc tremendously increased in the city. To meet out the current traffic and other demands, civil engineers have been using valuable underground space beneath the urban areas. All these underground structures involve huge construction cost time and manpower not only this, these structures also required special construction skill. Moreover, all these structures associated with foundations. Therefore, the stability of these structures is at most important. The stability of these structure is mainly depends upon the foundation soil and vertical stability of side walls. In nature soil is generally exist in heterogeneous state, it is not necessary that all the time soil condition may suit the structural requirements. The inadequate stability of slope can be improved by suitable ground improvement techniques. There are many methods available for ground improvement. Soil Nailing Technique has proved a safe and economical solution (10% to 30%) if we compare with the other method of stabilisations.
In general, the requirement of shallow depth tunnel is more because of usability, availability of land and project costs. It has been analysed and found that stability of shallow depth tunnel are very less as compared to deep tunnels. The shallow depth tunnels have been constructed using cut and cover techniques, which have often proved highly disruptive requiring road closures and property demolition. At shallow depth the natural arching properties of ground do not develop. With the advancement of civil engineering in globe, the pre-cast technology had proved a time and cost saving technique for the construction of civil engineering projects. In-pre cast technology, the project is usually constructed in steps followed part to the whole. Therefore, the scope of underground construction using pre-cast panel is gaining popularity. Therefore, our research emphasised on stepwise stabilisation of soil inside and outside of precast box to be pushed through the jacking technique into the existing soil masses.
References may be made to U.S. Pat. No. 4,009,579, wherein Delbert M. Patzner et. al provide a method for constructing tunnels and underpasses quickly and inexpensively, with conventional readily available equipment, without disrupting existing constructions and without interrupting or delaying service thereon. The method comprises the inserting a plurality of longitudinal support members side by side through the ground beneath the existing structure followed by excavating a longitudinal increment of the ground beneath the support members. Thereby, installing tunnel forming precast sections beneath the support members in place of each longitudinal increment of excavated ground to support which the support members, repeat the excavation and placement of tunnel forming section till the full length of tunnel is covered.
References may be made to patent U.S. Pat. No. 4,139,320 wherein, a process for excavating and constructing a tunnel with the help of an excavating device excavator equipped with a screw conveyor has been provided. This process is directed to excavating and constructing a tunnel under a railway or a road on a bank or on level land in the direction transverse to the railway or road. In this process, pits are dug on both sides (entrance and exit end). In this respect, a hollow casing unit of a box shape is coupled to the rear end of the excavator equipped with a screw conveyor. As the excavator advances or digs forward a given distance, another casing unit is in turn coupled to the rear end of the preceding casing unit, and then such a step is repeated, until the excavator goes out of the wall of another pit. The sand and soil inside the outer wall of the tunnel are excavated and removed, after which reinforcing steel bars and a mold are placed along the inner surface of the wall of the tunnel. Concrete is then poured into the hollow casing units themselves as well as between the mold and the wall. Thus, the hollow casing units form an integral part of the wall of a tunnel, as an outer wall.
References may be made to patent U.S. Pat. No. 4,405,260 wherein a method of constructing underpass across railway and highway without affecting normal traffic thereof, the steps of excavating a traction ditch on one side of the road foundation and a launching ditch on the other; building a traction wall with traction holes therein against the road foundation in the traction ditch; and sequentially tracing a precast box culvert one after another through perforating, anchoring and jack driving according to the construction line until a predetermined configuration is completed thereat. Subsequently, build pier foundations, supports, and a bridging beam; arrange shell pipes; place PC steel reinforcements; and, after a certain curing period, perform pre-stress operations in the precast box culverts of the structure and grout cement mortar therein. Finally, excavate the earth volume under the structure and finish the road surface of the underpass for opening to traffic.
The previous patents/intentioned revealed that, soil stabilisation part inside the tunnel boundary has not been covered by any of the investigators. All the above said methods used for generalized soil conditions. The present invention is fruitfully worked for all collapsible soil/generalized soil conditions and any kind of loading conditions. The stabilisation of soil using ‘Soil Nailing Technique’ will be viable solution for such kind of underpass constructions.
In this patent application, step wise stabilisation of soil slope inside and outside of the tunnel and construction of underpass is explained in a simplified way irrespective of soil type/conditions.
To carry out this task, detailed field and laboratory investigations were carried out and all the relevant data pertaining to the project was collected. Based on our previous experience of handling projects of underground construction and the problems of slope stability in hilly terrain in landslides prone areas, it was decided to adopt ‘Soil Nailing Technique’ for the stabilisation of vertical cut slope in sandy strata to facilitate the box pushing through sand, while maintaining the movement of train without any interruption.
Soil Nailing is a relatively new construction technique used in Europe and America but very little work in this regard is carried out in India. Soil nailing consists of reinforcing the soil mass by introducing a series of thin elements called Nails to resist tension, bending and shear stresses. The reinforcing elements are made of steel round bars called as Nails. Nails are installed sub-horizontally or horizontally into the soil mass in pre-bored holes, which are grouted along their full length to form “Grouted Nails” or simply driven into the ground, called as “Driven Nails”. The nails or metallic reinforcement, which are installed horizontally into the soil mass improve the shear strength and resist bending and tensile stresses developed in soils under loading. This technique is generally recommended to stabilise cut slopes, which are cohesive in nature and under static overburden pressure. However, under this project, the technique is conceptualised for stabilising pure sandy soil of collapsible nature under heavy dynamic loading. The concept was initially tested in a small scale laboratory model studies. Based on the observations a design was developed for a large scale live project with heavy dynamic loads with the help of nails and supportive plates. The bending and shear stresses were checked at different locations in the entire soil mass to prevent failure due to shear and surface erosion.
Though it was quite difficult to replicate the field conditions in the model test, nevertheless, the model studies provided a great insight to understand the behaviour of mass movement of sandy soil under heavy and dynamic loads with and without soil nailing. On the basis of the model studies, strategy for design and construction methodology of the project was formulated. The technique helped in successfully pushing the three boxes and creating an underpass in a record period of time without any kind of problem and this has resulted to open the bye-pass road much before the commencement of the commonwealth games.
This technique was tried first time in the world for such a kind of project in zero tolerance zones.
Main object of the present invention is to provide de-stabilisation and stabilisation of vertical cut slope of highly collapsible soil mass by ‘Soil Nailing Technique’ used for construction of rail/road underpass beneath the rail/road traffic without disrupting the live condition of traffic. Another object of the present invention is to provide inexpensively and safely construction of underpass under highly loaded and zero tolerance where 250 to 300 trains passing over that track.
Yet another object of the present invention is to permit the safe construction of tunnel/underpass beneath continuously used rail road traffic without the use of alternate route of the rail track.
Yet another object of the present invention is to prevention of sudden collapse of sandy soil in dynamic loading conditions.
i. marking position of the box on the face of the retaining wall or embankment;
ii. dismantling the retaining wall above the marked position of the box and providing temporary support by shuttering plates having holes for pre decided position of nails to be driven in the vertical face;
iii. nailing the soil mass by using grouted nails and optionally driven nail above the marked position of the box;
iv. again dismantling the retaining wall, placing the shuttering plates with pre drilled position of nails and inserting only the driven nails from top to bottom of box pushing area;
v. leaving complete nail system for period in the range of 8 to 12 hrs to mobilize the friction of the nails;
vi. bringing box close to the soil nailed wall face;
vii. loosening the one top row of shuttering plates, excavating the soil for 30 to 40 cm depth;
viii. repeating step (vii) till the entire rows of the nails are covered for 30 to 40 cm depth followed by pushing the box in excavated area of 30 to 40 cm depth;
ix. pushing the nails into the soil mass and again tightening of shuttering plates;
x. again repeating step (vii) to (ix) till 50% of the box pushing length;
xi. cutting the nails in the range of 25 to 35 cm to create the space for box pushing when the one first/pointed end of the nails will touch with other end of retaining wall followed by placing vertical nails in order to increase the stability of cut slope;
xii. again repeating step (vii) to (ix) till complete insertion of box for making underpass.
In an embodiment of the present invention, thickness of the shuttering plate used is in the range of 3 to 5 mm.
Accordingly, present invention provides a process for making underpass through railway track or road without service interruption in stepwise de-stabilisation and stabilisation of highly collapsible soil mass by soil nailing technique and the said process comprising the steps of:
In another embodiment of the present invention, length of the grouted nail and driven nails in step (iii) is equal to the length of the underpass.
In yet another embodiment of the present invention, diameter of the grouted nail used is in the range of 90 to 110 mm.
In yet another embodiment of the present invention, diameter of the driven nail used is in the range of 25 to 32 mm.
In yet another embodiment of the present invention, length of the driven nail used in step (iv to xii) is optimised with height of the vertical cut height in 0.7H wherein H is the height of vertical cut slope.
In yet another embodiment of the present invention, it should be ensured that all the driven nails should be have an extra length of at least 40 cm outside the shuttering plate.
In the present invention, the term “De-Stabilisation” means, disturb the existing soil stability by the application of external forces which reduces the shear strength of soil and tends to failure. The main requirement of “De-Stabilisation” is to create space, where the precast box is to be pushed by cutting soil strata below the track which resulted development of instability in the existing soil strata.
The term “Stabilisation” means, the improvement of the engineering properties of the soil either by the addition of some admixture or by soil reinforcement. In the present invention soil reinforcement using Soil Nailing Technique has been used, which increases the shear strength of the soil mass. This technique results in the considerable increase in the frictional resistance of soil, leading to the improved shear strength and load carrying capacity of the soil mass.
Present invention provides stepwise repeated de-stabilisation and stabilisation of highly collapsible soil mass by ‘soil nailing technique’ used for construction of underpass through Road or Rail embankment confined with or without Retaining walls.
The various options are involved for considering different type of problems, in first case, Railway embankment is constructed by using two retaining walls sandy soil is used as a backfilled material. The underpass is to be constructed by using precast boxes which is to be pushed through these retaining walls with jacking technique. The most typical problem is de-stabilization of backfilled compacted sand by dismantling of retaining walls for creating a space for pushing of box and again stabilisation of sand inside and surrounding to the box.
An innovative technique is investigated for this kind of problem and also same can be used for similar kind of projects of de-stabilisation and stabilisation of soil for underpass construction in all kind of soils.
Geotechnical investigation of site is to be carried out where the underpass is to be constructed.
Evaluate index and engineering properties of soil up to 1.5 times of the B. The total depth of investigation (H+1.5B), where, B and H are the width and depth of foundation/Box.
Compute the safe bearing capacity of soil at foundation level.
Analyses of vertical cut slopes and evaluate Factor of Safety (FOS) for stability of slope;
If, required FOS of cut slope does not meet the requirement of the project then adopt any ground improvement technique. Soil Nailing has proved a viable solution for De-Stabilisation and Stabilisation of soil for such kind of projects.
Design the soil Nailing Technique with respect to soil parameters.
To check the efficacy of Nails, initial field pull out test be conducted and it is mandatory also. Prior to execution of nails system, actual design can be changed according to friction factor obtained from Pull out test, if required.
The most effective dia 25 mm to 32 mm and length 0.7H of the driven nails.
If the loading intensity is very high above the box, pull out test on Grouted Nails be conducted the most effective size of Grouted Nail using for steel (d=25 mm), hole dia (4d=100 mm). The ratio is to be maintained “d” and ‘4d’ for nail and hole dia respectively.
After having friction factor of nail, the complete nails system can be designed with the above said dia range of nails or by using available software like; GEO4.
The work of Soil Nailing be started from the top of the retaining wall and gradually proceed towards bottom of the retaining wall.
The position of the box is to be marked on the face of the retaining wall.
Due to inherent properties of sand, the vertical face will be stable up to 40-50 cm in height.
The retaining wall is to be dismantled up to 40-50 cm in height.
To provide a temporary support (shuttering Plate-3 mm thick) to the vertical face after the dismantling of retaining wall. A designed nail dia (+5 mm) and spacing is to be drilled in plate to facilitate the driving of Nails directly through these designed holes.
The pointed shoes of the nails are to fabricated, which will be ease in the pushing of the Nail. The driven Nail of 25 mm dia of having 3 m length can be easily pushed manually
Temporary support to protect the surface erosion, shuttering plate of central hole of designed spacing can be used.
The suitable scaffolding arrangement is to be made simultaneously for soil nailing
Adequate numbers of drilling machines are to be deployed for installation of Nails as per the site conditions.
The parapet wall face of the retaining wall is removed till the soil strata appears and immediately pre decided position of hole in the shuttering plate fabricated with 3 mm steel plate be paced to protect the surface erosion.
Now the ballast/soil in the dismantled portion be graded/swiped, to make the slope 2.0(H):1.0(V) so as to retain the soil and ballast at its position. The first row of grouted nails be installed as this juncture. The process is to be repeated till the required rows of grouted nails are inserted as per design system of Nails.
The retaining wall is further dismantled, the shuttering plates with pre drilled position of nails be placed for temporary protections of surface erosion.
The suitable size of the plate can be fixed by as per the spacing and height of cut slope. (One plate should cover minimum two rows and two columns of designed nails).
This process will be continued till all the rows of driven nail shown in the design scheme are inserted.
It should be ensured that all the driven nails should be have an extra length of at least 40 cm outside the shuttering plate.
By following the above said now, the complete retaining wall is now converted into soil Nailed wall.
It may be ensured that no disruption of train movement should be there during this entire process of nail driving and wall dismantling.
The box is to be brought very close to the Soil Nailed wall face.
The anchor system is to be left without any disturbance for minimum eight hours so that required friction on nails is mobilised.
After eight hours, the top shuttering plates (one row) be loosened and soil behind that plate up to a depth of 30 cm to be excavated/removed. The excavation of soil and removal of the excavated soil will be done subsequently. The excavated soil will be removed by manually or mechanical arrangement.
This procedure of loosening of plate, excavation of soil, removal of soil, further pushing of same Nail into the soil mass and again tightening of shuttering plate with the nail be followed till the entire rows of the nails are covered.
Excavation of the soil face is to be undertaken in such a manner, that the excavated face should have the same slope of the cutting blade (fitted with the box).
It should be ensured that the loosed shuttering plates may be immediately tightened at new position towards the soil mass, so that it supports the new soil face.
All the above said procedure be repeated for each subsequent pushing of box.
As per the optimised design, Nails are designed in varying length.
In order to create the space for box pushing, the nails are to be pushed in subsequent stages, a stage will come when the one first/pointed end of the nails will touch with other end of retaining wall. In such case, the required pushing length (i.e. 30 cm approx) of the nail to be cut from (0.7H initial length) to create the space for box pushing.
The cutting of Nail length in subsequent stages lead to shortage of designed Nail length resulted the instability to the cut slope.
In order to increase the stability of cut slope, extra vertical nails of same dia up to the bottom of slope (minimum=0.9H) be placed from vertical.
The vertical Nails are to be placed prior to 50% of length of pushing.
FIGS. 3(
a
h
3
The complete schematic procedures for wall dismantling, placing of shuttering plate, driving of nails, excavations of soil, pushing of box are shown in ) to ().
Stepwise procedure involved in the de-stabilisation and stabilisation of embankment Sandy soil confined with Retaining walls is as follow:
In case of no retaining walls, Nails are directly placed in the collapsible soil and improve the stability of slope. All the above said steps be strictly followed except dismantling of walls.
TABLE 1
CHECK FOR BEARING CAPACITY OF NAILS
Verification of nails bearing capacity
Nail
Inserted
B.cap.
Nail force Computed B.cap.
No.
[kN]
[kN]
[kN]
1
117.00
0.00
8.25
2
121.50
0.00
17.13
3
56.00
0.74
8.05
4
57.20
2.43
10.16
5
57.60
3.02
12.67
6
57.60
3.61
15.59
7
32.00
3.59
10.29
8
32.00
4.16
11.91
9
32.00
3.54
13.53
10
24.00
4.69
10.30
11
24.00
4.66
11.72
12
24.00
6.71
13.14
13
24.00
7.45
14.57
14
24.00
5.12
15.99
15
24.00
5.01
17.41
16
24.00
5.09
18.83
17
24.00
5.16
20.25
18
24.00
5.24
21.67
19
24.00
5.31
23.09
20
24.00
5.39
24.51
21
24.00
5.46
25.93
22
24.00
5.54
27.35
23
24.00
4.20
28.42
24
24.00
4.24
29.49
Computed bearing capacity is determined for driven (not grouted) nails. The real bearing capacity is significantly higher and the bearing capacity of nails is acceptable.
TABLE 2
CHECK FOR EXTERNAL STABILITY
Check for overturning stability:
Resisting moment Mres = 0.9 * 8790.37 = 7911.34 kNm/m
Overturning moment Movr = 466.96 kNm/m
Wall for overturning is ACCEPTABLE
Check for slip:
Resisting lateral force Hres = 0.9 * 581.44 = 523.29 kN/m
Active lateral force Hact = 163.63 kN/m
Wall for slip is ACCEPTABLE
Forces acting at the center of the footing bottom:
Overall moment M = −3607.29 kN/m
Normal force N = 1173.17 kN/m
Shear force Q = 163.63 kN/m
Bearing capacity of foundation soil check:
Eccentricity of normal force e = 0.00 cm
Maximum allowable eccentricity e, allow = 265.32 cm
Eccentricity of the normal force is ACCEPTABLE
Stress at the footing bottom Sigma = 145.92 kPa
Bearing capacity of foundation soil Rd = 300.00 kPa
Bearing capacity of foundation soil is ACCEPTABLE
TABLE 3
CHECK FOR INTERNAL STABILITY
SLIP SURFACE AFTER OPTIMIZATION
Angle of slip surface = 11.00 degr.
Origin of slip surface = 8.50 m
Gravitational force = 830.61 kN/m
Overall force transmitted by nails behind sl. surf. = 140.08 kN/m
Driving forces on slip surface (grav. force) = 158.49 kN/m
Driving forces on slip surface (pressure) = 291.62 kN/m
Resiting forces on slip surface (soil) = 466.77 kN/m
Resisting forces on slip surface (nails) = 137.50 kN/m
Stability factor Fh/Fm = 1.34 > 1.25
Stability of slip surface is acceptable.
Stability Analysis of End Portion: Stage when Nails Cannot be Driven Further (4 m from Exit End+Retaining Wall Thickness from Far End).
It was found during the time of design that the embankment is unstable even with nails when the box is reached at a location, where the width of fill is less than 4.5 m as at this stage the horizontal nails were not able to provide sufficient stability. In such cases, the lengths of the nails are keep on reducing, a unstable condition is creating at or 4.5 m before the exit end. In view of the instability of the embankment at this end, vertical nails were designed. In order to increase the factor of safety pressure grouting with cement is also suggested. The factors of safety for different condition are indicated in tables given on next page.
TABLE 4
STABILITY ANALYSIS OF END PORTION
F.O.S of an embankment with
Surcharge
Horizontal & Vertical
(kN/m)
Horizontal nails
nails
80
1.02
1.16
100
0.97
1.10
125
0.92
1.04
TABLE 5
SPACING OF VERTICAL NAILS
Horizontal Spacing of
Distance from
vertical Nails
horizontal (m)
(m)
Length (m)
0.3
0.5
7.0
1.3
0.5
7.0
2.3
0.5
7.0
3.3
0.5
7.0
Following examples are given by way of illustration and therefore should not, be construed to limit the scope of the invention.
During the recently concluded Commonwealth games, it was proposed to construct a bye pass road from ISBT (Kashmeri Gate-Delhi) to ITO to decongest the existing ring road traffic, which traverses through the Yamuna Bazar, Shantivan and Rajghat to connect ITO Bridge. In order to construct the proposed bye-pass, named as “Salimgarh Fort to Velodrome Road”, it was necessary to cross the existing Shahadra-Old Delhi railway line, which was constructed on an embankment about 15 m high adjacent to old Yamuna Bridge popularly known in Delhi as Steel Bridge (Loha Pul). The upper portion of the steel bridge is being used for the rail movement and lower one is being used by road traffic. This railway bridge is considered as life line of Delhi as more than 350 trains cross this bridge, which include Rajdhani, Shatabdi and several express and goods train. The railway bridge along with the approach embankment was constructed about 135 years ago by British Engineers. During the preliminary investigation carried out by the railway authorities, it was found that the high approach embankment is made up of pure sand and is confined between the two stone masonry retaining walls.
In order to cross this railway track, there were two options; either to construct a′ flyover over the existing railway line or to construct an underpass below the existing railway line. The construction of a flyover over the existing railway line was ruled out by the hard pressed authorities i.e., Delhi PWD and Indian Railway due to the exorbitant cost, problems of land acquisition and time constraint at the time of Commonwealth Games.
It was therefore decided to construct an underpass. It was further decided that technique of “Box Pushing” which is now gaining momentum in various civil engineering projects dealing with under ground projects be adopted for the construction of an underpass.
Normally in box pushing technique, the precast box is pushed below the existing ground by making vertical cuts in the ground, and subsequently the box is pushed in the soil using hydraulic Jacks and simultaneously removing the soil inside the box. The precast boxes are fitted with cutting shoes of the required size all along the face of the box. These cutting shoes facilitate the driving of box into the soil mass. This technique is quite successful in soils having cohesion as such soils can stand in vertical position without external support for considerable time.
In view of the extensive volume of traffic likely to use the bye pass, it was proposed to provide three precast RCC boxes below the existing railway track to facilitate free flow of traffic. The dimensions of two boxes as per the available space and geometry were worked out to be 12.1 m×7.35 m each and the remaining one has a dimension of 10.6 m×5.6 m.
To accomplish this task, it was required to remove the retaining wall made up of stone masonry on both sides of the embankment to facilitate the box pushing. However, the biggest challenge in this project was to retain the dry sandy strata in vertical position without collapse under the dynamic loads caused by moving trains, after the demolition of the retaining wall, so that the box can be gradually pushed in the sand. The additional challenge was to keep the train movement operational without interruption during the period of box pushing. The railway authorities and contracting agency had no clue to carry out the project work under such a typical situation. The railway interacted with several agencies to suggest a suitable methodology to carry out the work, but most of them have indicated that it is not feasible. The literature survey on the topic has also not revealed that work of such a nature is being carried out anywhere in the world.
To carry out this task, detailed field and laboratory investigations were carried out and all the relevant data pertaining to the project was collected. Based on our previous experience of handling projects of underground construction and the problems of slope stability in hilly terrain in landslides prone areas, it was decided after lot of deliberations with railway authorities to adopt ‘Soil Nailing Technique’ for the stabilisation of vertical cut slope in sandy strata to facilitate the box pushing through sand, while maintaining the movement of train without any interruption.
Under this project, the soil nailing technique is conceptualised for stabilising pure sandy soil of collapsible nature under heavy dynamic loading. The concept was initially tested in a small scale laboratory model studies. Based on the observations a design was developed for a large scale live project with heavy dynamic loads with the help of nails and supportive plates. The bending and shear stresses were checked at different locations in the entire soil mass to prevent failure due to shear and surface erosion.
Though it was quite difficult to replicate the field conditions in the model test, nevertheless, the model studies provided a great insight to understand the behaviour of mass movement of sandy soil under heavy and dynamic loads with and without soil nailing. On the basis of the model studies, strategy for design and construction methodology of the project was formulated. Complete design and construction methodology for the proposed technique was provided by CRRI. The technique helped in successfully pushing the three boxes and creating an underpass in a record period of time without any kind of problem and this has resulted to open the bye-pass road much before the commencement of the commonwealth games.
Since the bridge and the approach embankment and other adjoining structures were constructed about 135 years ago, no data related to the wall structure and soil fill behind the retaining wall was available with the site engineers.
In the absence of records, cross-section of retaining wall was explored by adopting GPR technique. The GPR study showed that the retaining walls have a battered face towards earth side having thickness more than 2 m.
The underpass was to be constructed at a location, where rail level was about 9.2 m above the natural ground level and the embankment is contained in between two long rubble stone retaining walls. There were two main lines, i.e., North and South bound tracks and the width between the retaining wall is 15 m.
0
Backfill material behind the retaining wall from natural ground level up-to the rail bed was uniformly graded fine sand (Cohesion, C=0 and angle of internal friction ø=29). Below the natural ground level, there is conglomerate soil up to 2 m depth and thereafter the strata consist of fine sand up to 6 m depth.
In order to create an underpass, it was decided that Box pushing technique would be adopted.
The estimated pushing length worked to be about 22 m. The precast box segments were required to be pushed in highly unstable cohesion-less strata. Also rubble stone masonry wall on reception and exit ends of the box were required to be dismantled, which would expose unsupported earth face of 8 m height prone to collapse. The cohesion-less soil strata was to be stabilised by adopting suitable technique prior to taking up pushing operation.
After the thorough investigation of the site condition and keeping in view the project requirement, CRRI team proposed the use of soil nailing technique for the stabilisation of sandy soil under the heavy dynamic loads caused by rail movement.
The design of soil nail system was carried out using software GEO 4 available at CRRI. which is generally used to evaluate slope stability problems of high embankments. A system of grouted and driven nails was considered in the design. The input parameters considered in the design were “External loading due to railway track including ballast—110 kN/m per track” The detailed results of nailing design is presented in Annexure-I to Annexure-III (as suggested by railway officials), “Geometry of the cut slope” considered and the “back fill soil properties”.
In order to carry out the analysis and to design a suitable configuration of nails to be grouted or nailed into the soil mass, so that the entire mass of sand confined between the two retaining walls be maintained in vertical position even after removing the two side walts, it was essential to determine the apparent coefficient of friction (f*) between in-situ soil and nail for design of nail network for the stabilisation of vertical cut. The in-situ pull-out tests were conducted at the site by driving the nails through the thick stone masonry wall. Two different methods of nailing were adopted viz. i) driven nail-32 mm, ii) perforated pipe nail-89 mm dia with perforation of 12 mm @ 50 mm c/c in staggered manner. Six pull out tests were conducted on grouted nails and eight tests were conducted on driven nails at different locations. The results of Pull out test were used in design analysis and computation.
FIG. 2
FIG. 2
In order to finalise the spacing, length and diameter for driven and anchored nails to keep the soil in vertical position after the removal of wall, a suitable scheme was designed using GEO 4 software. The nailing scheme indicating details of spacing of the nails required for stabilising the cut face at the underpass location is given in Table 1 and configuration of nails is shown in . One row of grouted nails as in the top and 22 rows of driven nails below the grouted nails have been considered. For grouted nails, diameter is taken as 100 mm and length of the nails is kept 15 m shown in . For driven nails, two different diameters of the nails were considered—32 mm and 28 mm. The inclination of the nails has been considered at zero degree with respect to horizontal (i.e., Nails to be driven horizontally) and nail heads are to be anchored with plate. Length of the driven nails varied according to their location.
Based on the above Nailing scheme, the stability of vertical sand reinforced with nails was checked for both External and Internal Stability. The computer output and the relevant analysis for verification of nail bearing capacity (pull out strength), verification of entire wall (global stability) and stability of slip surface after optimisation of iterations.
TABLE 1
Proposed Design Scheme of Soil nailing for Box Pushing
Effective
Dia.
Depth
Length
Origi-
Of
Type
from
of
nal
S.
Nail
of
rail
Spacing (m)
Nail
length
No
(mm)
nail
top (m)
Verical
Horizontal
(m)
of nail
1
100**
Grouted
1.3
—
0.5
15
15
2
32*
Driven
1.55
0.3
0.4
15
15
3
32*
nails
1.75
0.2
0.3
15
15
4
32*
2
0.25
0.3
15
15
5
32*
2.3
0.3
0.3
15
15
6
32
2.6
0.3
0.3
8
8.3
7
32
2.9
0.3
0.3
8
8.3
8
32
3.2
0.3
0.3
8
8.3
9
28
3.6
0.4
0.3
6
6.3
10
28
4
0.4
0.3
6
6.3
11
28
4.4
0.4
0.3
6
6.3
12
28
4.8
0.4
0.3
6
6.3
13
28
5.2
0.4
0.3
6
6.3
14
28
5.6
0.4
0.3
6
6.3
15
28
6
0.4
0.3
6
6.3
16
28
6.4
0.4
0.3
6
6.3
17
28
6.8
0.4
0.3
6
6.3
18
28
7.2
0.4
0.3
6
6.3
19
28
7.6
0.4
0.3
6
6.3
20
28
8
0.4
0.3
6
6.3
21
28
8.4
0.4
0.3
6
6.3
22
28
8.7
0.3
0.3
6
6.3
23
28
9
0.3
0.3
6
6.3
**these nails should be grouted/driven up to another side of the retaining wall.
*indicates the nail comes under box top cover. Nails should cut at regular intervals during box pushing. The length of the nails mentioned above is effective lengths. The total length of the nails should be kept 30 cm extra for the movement or driving of nails.
In order to push the box below the railway line, it was required to remove the random rubble masonry at the first instance. Since the soil is cohesion less with very little or no shear strength under unconfined state, it was proposed to remove the wall in small segments and simultaneously retain the backfill by inserting nails at suitable space and retaining the face of the unsupported mass with a facia panel. Since the executing agency has no prior experiencing of executing works of such a nature, it was proposed by the railway engineers to push the smaller box first. The step wise procedure for undertaking the work as suggested and followed at site is described below.
A number of girders were provided below the sleepers at regular intervals. These girders were allowed to rest on one side on the retaining wall/box with pulley arrangement and on the other side on soil/ballast.
The suitable scaffolding arrangement was made simultaneously for soil nailing.
Adequate numbers of drilling machines were deployed for installation of Nails.
The work of Soil Nailing started from the top of the retaining wall and gradually proceeded towards bottom of the retaining wall.
The position of the boxes was marked on the face of the retaining wall.
The parapet wall face of the retaining wall was removed till the soil strata appears.
Now the ballast/soil in the dismantled portion was graded/swiped to make the slope 2.0(H):1.0(V) so as to retain the soil and ballast at its position. The first row of grouted nails was installed as this juncture. The process was repeated till the two rows of grouted nails were inserted.
The wall was further dismantled and the nails as shown in the table 1 were inserted. The shuttering plates were also provided on the nail heads to retain the soil temporarily. The size of shuttering plate was approximately 50×50×3 mm.
It may be noted that nails up to the sixth row from top were driven up to the full length i.e., up to 15 m. The next three rows were driven up to 8 m. The nails within or inside the box were initially driven up to only 6 m. The nails were pushed gradually inside the box as the box was advanced slowly with the help of hydraulic jacks fitted behind the box. The aluminium strips were provided at top of the box to minimise the friction between box roof and the soil.
This process continued till all the rows of driven nail shown in the design scheme are inserted.
By following this system, it was possible to retain the entire soil mass with the help of nails and plates. It may be noted that no disruption of train has occurred during this entire process of nail driving and wall dismantling.
After the removal of the entire retaining wall on one side, the box was brought very close to the shuttering plates.
It was ensured that all the driven nails should be having an extra length of at least 40 cm outside the shuttering plate.
The anchor system was left without any disturbance for minimum eight hours so that required friction on nails is mobilised.
FIGS. 3(
a
h
3
After eight hours, the top shuttering plates (one row) were loosened and soil up to a depth of 30 cm was removed. Similarly the plates and soil in the inside portion of box was removed up to the bottom of box. It created a space of approximately 30 cm for the box to be pushed immediately. The procedure of wall dismantling and driving of nails has been depicted schematically with the help of ) to ().
Excavation of the soil face was undertaken in such a manner, that the excavated face is having the same slope of the cutting blade (fitted with the box).
Shuttering plates were immediately tightened after excavation and driving the nails, so that it supports the soil face.
Now the box pushing operation was started and box was pushed for a distance of about 30 cm or less. In this manner, the box was further pushed inside the fill.
This process was continued until the box has been pushed to about 8 m from the exit side retaining wall. When it was not possible to push the nails further inside the soil due to the obstruction caused by the exit side wall, the nails were cut.
The slope studies indicated that in the end portion the soil mass was once again found to be non stable with only horizontal nails and therefore it was decided to go for vertical nails also in the remaining portion.
Vertical driven nails were inserted near the exit side retaining wall as shown in the design (Annexure IV). This work is to be completed at least 20 days before the box reaches to a distance of 8 m from exit side retaining wall.
The driven nails of 15 in length (touched with the other retaining wall) which were covering or in front of the thickness of the box at the top level were trimmed off with gas welding at every 30 cm increments after excavation.
This procedure has helped in taking the box up to the exit side. After reaching the other side of the wall, the other side of the wall was dismantled and box was pushed further. The entire process as described above has facilitated in creating an underpass below the rail embankment. The stepwise procedure as discussed was also followed for the remaining two boxes, which were of larger size and finally an underpass were constructed with a very simple and innovative technology as suggested by CRRI.
The biggest challenge in this project was to suggest and design a system to retain the collapsible sandy strata in vertical position under the dynamic loads caused by moving trains, after the demolition of the retaining wall, so that the box can be gradually pushed inside the sand, to create an underpass.
The additional challenge was to develop a methodology for box pushing, so that the train movement remain operational without interruption during the period of box pushing.
During the entire period of construction, CRRI team remained on site and guided the engineers of the railway and the contractor on day to day basis. Since the work of such a nature was carried out for the first time in the country, minor modifications in the design and construction methodology as per the site conditions were to be made from time to time and the same were duly checked and verified by the CRRI scientists using the available software against all possible mode of failures.
In addition to the above, there were a number of site specific problems during the period of construction; such as convergence of nails at several locations, development of piping phenomenon in sand and collapsing of sand at some locations, which were immediately overcome/tackled due to the vast experience and knowledge of soil mechanics principles of the team of scientists dealing with the project work.
After the successful completion of box pushing at old Yamuna Bridge, AFCONS (a multinational construction company) on the recommendations of the Northern Railway approached CRRI to give a complete design and construction methodology for creating an underpass below railway line at Apsara border, which is very close to Delhi-Shahibabad border. This project was more challenging than the previous one and here length of underpass was more than the previous one. Here again on the same concept of soil Nailing, CRRI provided a complete design and construction methodology, which was successfully implemented on this project as well. Few photograph of this site are given here.
This project work was given to CRRI by Larsen and Tubro (L& T). Here again using the technique of soil nailing, a large size of water pipe line was pushed below at 12 m high sand railway embankment retained between two retaining wall. Few photographs of the same are given below.
Installation of Soil Nailing Technique is quite simple and fast.
Length of Nail can be coped/curtailed with site constraints and variations in ground conditions encountered during construction
The equipment required for the construction of Soil Nailing is very simple and light.
This technique can be easily be mobilised at cramped and difficult sites
The soil. Nailing techniques performs well even in seismically zones.
There could be time and cost savings about 10 to 30 percent when we compared with other earth retaining techniques.
Soil Nailing Technique requires a very less space to implement.
BRIEF INSCRIPTION OF THE DRAWINGS
FIG. 1
represents general Lay Out of Box;
FIG. 2
a
Configuration of Nail Design Scheme;
FIG. 2
b
represents embankment with horizontal nails;
FIG. 2
c
represents embankment with horizontal and vertical nails;
FIG. 3(
a
) Sand confined with two retaining walls;
FIG. 3(
b
) Dismantling of parapet wall;
FIG. 3(
c
) Replacement of wall by Nails and plate support;
FIG. 3(
d
) Further Dismantling of wall;
FIG. 3(
e
) Further removal of wall and simultaneous support by Nails and plates;
FIG. 3(
f
) Total removal of retaining wall with Nails and Plates ready for Box pushing;
FIG. 3
g
() Process showing Nails pushing, backfill removal and Box pushing;
FIG. 3(
h
) Process showing Nails pushing, backfill removal and further Box pushing;
FIG. 4
Pipe pushed below sand embankment using soil nailing technique. | |
This set of Data Structure Multiple Choice Questions & Answers (MCQs) focuses on “Queue Operations”.
1. If the elements “A”, “B”, “C” and “D” are placed in a queue and are deleted one at a time, in what order will they be removed?
A) ABCD
B) DCBA
C) DCAB
D)ABDC
Explanation: The FIFO method is used in the queue. i.e. the “First in, First Out” technique. As a result, the elements should be omitted in the following order: ABCD.
2. A data structure in which elements can be inserted or deleted at/from both ends but not in the middle is?
A) Queue
B) Circular queue
C) Dequeue
D) Priority queue
Explanation: We can insert and remove elements from both ends in dequeuer. For insertion and deletion of elements in the queue, we can use the first in, first out theory. In a priority queue, the element with the lowest priority will be removed.
3. A normal queue, if implemented using an array of size MAX_SIZE, gets full when?
A) Rear = MAX_SIZE – 1
B) Front = (rear + 1)mod MAX_SIZE
C) Front = rear + 1
D) Rear = front
Explanation: When Rear = MAX SIZE – 1, there isn’t enough space in the queue for the elements to be included. As a consequence, the queue fills up.
4. Queues serve major role in ______________
A) Simulation of recursion
B) Simulation of arbitrary linked list
C) Simulation of limited resource allocation
D) Simulation of heap sort
Explanation: The stack data structure is used to simulate recursion. Linked lists are used to simulate arbitrary linked lists. Since the first entered data needs to be given first priority during resource allocation, simulations of resource allocation use queues. Heap data structure is used to simulate heap sort.
5. Which of the following is not the type of queue?
A) Ordinary queue
B) Single ended queue
C) Circular queue
D) Priority queue
Explanation: There are often two ends of a queue. As a consequence, a single-ended queue is not the sort of queue to use.
6. A linear list of elements in which deletion can be done from one end (front) and insertion can take place only at the other end (rear) is known as _____________
A) Queue
B) Stack
C) Tree
D) Linked list
Explanation: Queue is a linear array of elements in which deletion occurs at the front and insertion occurs at the back. In stack, the last element entered will be removed first.
7. The data structure required for Breadth First Traversal on a graph is?
A) Stack
B) Array
C) Queue
D) Tree
Explanation: BFS, or Breadth First Search Traversal, takes the starting vertex first, followed by adjacent unvisited vertices. To add further unvisited vertices to the graph, the first vertex that was added as an unvisited adjacent vertex list will be considered once more. The First In First Out theory must be observed to achieve the first unvisited vertex. The FIFO theory is used in the queue.
8. A queue follows __________
A) FIFO (First In First Out) principle
B) LIFO (Last In First Out) principle
C) Ordered array
D) Linear tree
Explanation: The first element added to the queue would be removed first, according to the FIFO principle.
9. Circular Queue is also known as ________
A) Ring Buffer
B) Square Buffer
C) Rectangle Buffer
D) Curve Buffer
Explanation: Ring Buffer is another name for Circular Queue. A circular queue is a linear data structure in which the first and last positions are connected to form a circle. It takes the shape of a bell.
A queue is a linear structure in which operations are carried out in a given order. First in, first out is the order (FIFO). Any queue of customers for a resource where the customer who arrived first is served first is a good example of a queue. Stacks and queues vary in how they are extracted. A queue is an abstract data form that is widely used in computing. It is based on the first-in-first-out (FIFO) principle. Queues are useful for situations where you want things to happen in the order they were called but the machine can’t keep up. | https://quicklyread.com/queue-operations-questions-and-answers/ |
Senator Erik Simonson (DFL-Duluth) today announced a new $70,000 investment in student safety for Duluth Public Schools. The funds are part of $25 million in school safety grants for “facility upgrades” at public schools statewide approved May 2018 by the Legislature and Governor Mark Dayton. The Minnesota Department of Education (MDE) received a total of 1,187 completed grant applications, requesting $255.5 million – more than 10 times the available amount of funding.
“I am pleased that Duluth schools are receiving new public dollars to improve student safety,” said Sen. Simonson. “At the same time, it is important to acknowledge that our schools are chronically underfunded and have been for many years. When applications for school safety improvements outnumber by ten-fold the amount of funding lawmakers provided, it shows that we as a state are falling far short of meeting needs communicated by our local school officials.”
Duluth Public Schools applied for school safety grants at 14 of its schools. MDE awarded Denfeld High School $61,566 for secure building entrances and communication, and Rockridge Academy $9,415 for secure building entrances. A total of 123 schools statewide received public dollars via the grant program.
“Appropriating $25 million to fortify buildings only scratches the surface of the school safety conversation, which I expect to remain front and center when a new Legislature and Governor return to our State Capitol in January,” added Sen. Simonson. “Increasing the number of licensed school counselors, psychologists, and social workers in our schools needs to be a major part of that conversation.”
How the school safety grant application process worked
According to MDE, school districts were allowed to submit separate grant applications for each building. Due to the large number of applications received, high-priority projects submitted on the first day were assigned random numbers to determine the order of funding up to the available $25 million. The final grant award values will be determined after recipients get complete bids from contractors. Schools were able to apply with qualifying projects for up to $500,000 per building. MDE, in consultation with the Minnesota Department of Public Safety’s Minnesota School Safety Center, determined grant eligibility based on project priority, with half of the grant funds available to schools outside of the 11-county metropolitan area. High-priority projects included improvements to exterior entry security, such as access controls, keyless entry systems, door locking and glass integrity, door alarm systems, and structure changes to main entrances. Additions or improvements to communication systems, such as electronic emergency notification systems for staff and first responders, were also considered high-priority projects. | https://senatedfl.mn/senator-erik-simonson-announces-new-investment-in-student-safety-for-duluth-public-schools/ |
Rambla de Catalunya 122,
Barcelona
08008
Rambla de Catalunya 122,Barcelona 08008
No one has favorited this theater yet
Inaugurated in 1908 as Sala Merce with 200 seats, changed its name to Cine Atlantico in 1936 and the seating capacity was increased to 250 (all seats were on a single floor). It specialized in showing family films, especially Disney and lasted as a cinema until 16th January 1987 when its closed with the Walt Disney film “Basil, The Great Mouse Detective”. It was demolished in August 1988 with only the facade being retained. A hotel was built on the site.
Contributed by elmorovivo
Want to be emailed when a new comment is posted about this theater? | http://cinematreasures.org/theaters/53588 |
TECHNICAL FIELD
BACKGROUND
SUMMARY
DETAILED DESCRIPTION
The First Embodiment
The Second Embodiment
The Third Embodiment
The present disclosure relates to the field of mobile communication, and in particular, to a method and system for transmitting a position reference signal.
th
Orthogonal Frequency Division Multiplexing (OFDM) technology is a multicarrier modulation communication technology in essence and is one of the core technologies for the 4generation mobile communication. A multipath channel of an OFDM has a characteristic of frequency selective fading in a frequency domain. In order to overcome the frequency selective fading, a channel is divided into a plurality of OFDM subchannels in the frequency domain, wherein the frequency spectral characteristic of each subchannel is approximately flat, and all the subchannels are orthogonal to each other, therefore frequency spectrums of the subchannels are allowed to be overlapped with each other, thereby not only the problem of selective fading is overcome, but also the utilization degree of the frequency spectrum resource is improved.
rd
A Long Term Evolution (LTE) system is an important program of the 3Generation Partnership Project (3GPP). When the LTE system adopts a normal cyclic prefix, a time slot includes 7 uplink/downlink symbols and has a length of 7 uplink/downlink symbols, and when the LTE system adopts an extended cyclic prefix, a time slot includes 6 uplink/downlink symbols and has a length of 6 uplink/downlink symbols.
FIG. 1
A Resource Element (RE) is a subcarrier on an OFDM symbol, 12 contiguous subcarriers and 7 contiguous OFDM symbols constitute a downlink Resource Block (RB) which is 180 kHz in the frequency domain and has a time length of a normal time slot in a time domain, as shown in . When the LTE system performs resource allocation, a resource block is taken as a basic unit for allocation. Wherein when an extended cyclic prefix is adopted, the number of contiguous OFDM symbols comprising an RB is 6.
0
1
2
3
1
FIG. 2(
FIG. 2(
FIG. 2(
FIG. 2(
a
b
a
b
The LTE system supports the MIMO application of four antennae, and the corresponding antenna port #, antenna port #, antenna port # and antenna port # respectively adopt a method of full bandwidth Cell-Specific Reference Signals (CRSs). When the cyclic prefix is a normal cyclic prefix, the positions of these cell-specific reference signals in a physical resource block are shown in ). When the cyclic prefix is an extended cyclic prefix, the positions of these cell-specific reference signals in a physical resource block are shown in ). In ) and ), the horizontal coordinate represents a sequence number of a subframe on an OFDM symbol.
In addition, a UE-specific reference signal is also provided, which is transmitted only in the time-frequency domain position where a UE-specific Physical Downlink Shared Channel (PDSCH) is located. Wherein the functions of the cell-specific reference signal include quality measurement for downlink channel and downlink channel estimation, i.e., the quality measurement and demodulation for a downlink channel.
A base station needs to measure the position of User Equipment (UE) in a cell, so that it can perform effective configuration and scheduling for the UE. At present, the CRS is adopted for measuring the terminal, thus some limitations exist as follows:
(1) CRS sequence repeats in each frame, so the mutual correlation is poor;
(2) when the transmission is performed with two antennae, the maximum multiplexing factor is 3, and the interference between the adjacent cells is large;
(3) the CRS power is semi-static configured, so the positioning performance is limited.
At present, a solution used to solve the above problems is to position the UE by transmitting a Position Reference Signal (PRS), and thus ensure the positioning precision of the UE. However, in the existing technologies, only the physical resource which uses a resource block as a unit to transmit a position reference signal is defined, and the positions in all the resource blocks where the position reference signal is transmitted are the same, whereas as for how to transmit the position reference signal, such as transmitting the resource block index of the position reference signal, the specific time-frequency position in the resource block, and the sequence of the reference signal and the like. No specific solution has been provided yet.
For this reason, it is urgent to provide a specific method for transmitting a position reference signal in the industry to ensure the positioning precision of the UE.
The technical problem to be solved in the present disclosure is to provide a method for transmitting a position reference signal to ensure the positioning precision of UE.
In order to solve the above technical problem, the present disclosure provides a method for transmitting a position reference signal, the method comprises the following:
frequency domain positions for transmitting a position reference signal are n physical resource blocks, and the value of n is obtained according to signaling;
time domain positions for transmitting the position reference signal are remaining orthogonal frequency division multiplexing symbols in a subframe except for orthogonal frequency division multiplexing symbols for transmitting a physical downlink control channel and orthogonal frequency division multiplexing symbols for transmitting a cell-specific reference signal; and
the position reference signal is transmitted using the frequency domain position and the time domain position.
Preferably, the method may further comprise a process of determining orthogonal frequency division multiplexing symbols used for transmitting the position reference signal in the frequency domain position and the time domain position; wherein the process may specifically comprise the following:
0
1
2
i
N−1
i
i
th
th
th
an array A of an N×N common sequence is determined, wherein A=[a, a, a, . . . , a, . . . , a], both columns and rows are numbered respectively from 0, N elements different with each other are included in the array A, the value of each element is an integer ranged from 0 to N−1, and arepresents that an element in the arow of the icolumn is 1, and elements in the other positions in the icolumn are 0;
ID
0
1
2
i
N−1
ID
i
(i+h)mod N
i
(i+h)mod N
cell
cell
h=X
N,p
X/N
b
a
+p
N,i=
. . . ,N−
p=X
N,h
X/N
b
a
+p
N,i=
. . . ,N−
when the identity of a cell is N, an index of the subframe for transmitting the cell-specific reference signal is determined as SubframeIndex, then an N×N array B=[b, b, b, . . . , b, . . . , b] corresponding to the cell Nis:
mod =floor(),=()mod 0,1,2, 1; ormod =floor(),=()mod 0,1,2, 1;
ID
ID
cell
cell
wherein x mod y represents an operation for calculating a remainder, floor (x) represents a rounding down operation, and X=Nor X=N+SubframeIndex;
the number of the orthogonal frequency division multiplexing symbols for transmitting the position reference signal in the subframe is determined to be n, and the first n columns or the first n rows of the array B are chosen, or the last n columns or the last n rows of the array B are chosen;
1
there is a one-to-one corresponding correlation between the chosen n columns or n rows and the n orthogonal frequency division multiplexing symbols, and a position where the element in each column or each row is located corresponds to a position of a subcarrier where the position reference signal is located on a corresponding orthogonal frequency division multiplexing symbol in each physical resource block for transmitting the position reference signal.
Preferably, the physical resource blocks may be n physical resource blocks which are discrete at equal intervals or n contiguous physical resource blocks.
Preferably, the n physical resource blocks which are discrete at equal intervals may be numbered with r, r+k, r+2×k, . . . , r+(n−1)×k respectively, wherein r may represent the starting position of the n physical resource blocks which are discrete at equal intervals, and k may represent an interval between two adjacent physical resource blocks.
Preferably, when the resource block corresponding to a current downlink bandwidth is m and all the physical resource blocks may be numbered from 0, r=0, k=└m/n┘; wherein └x┘ may represent a rounding down operation.
Preferably, the n contiguous physical resource blocks may be n contiguous physical resource blocks starting from a low frequency, or n contiguous physical resource blocks with zero frequency as the center, n contiguous physical resource blocks with high frequency as a cut-off or n contiguous physical resource blocks obtained according to a notification of the signaling.
Preferably, all the available physical resource blocks may be numbered from 0 according to an order of from low frequency to high frequency, and the last one may be numbered with r; and
the n contiguous physical resource blocks starting from the low frequency may be n contiguous physical resource blocks numbered from 0 to n−1;
as to the n contiguous physical resource blocks with the zero frequency as their center, a zero-frequency subcarrier may be located at the center of the n contiguous physical resource blocks, and the n contiguous physical resource blocks may comprise 12 n contiguous subcarriers, i.e., 6 n subcarriers of the low frequency adjacent to zero frequency and 6 n subcarriers of the high frequency adjacent to zero frequency;
the n contiguous physical resource blocks with high frequency as a cut-off may be n contiguous physical resource blocks numbered from r−n+1 to r.
Preferably, the number n and the starting position of the physical resource blocks may be obtained according to one or two signaling.
Preferably, when the resource block corresponding to a current downlink bandwidth may be m, then the values of n may be 1, 5, 10 and 20; or 6, 12, 25 and 50; or 10, 20, 40 and m; or 2, 5, 10 and 20; or 5, 10, 20 and 40; or └m/6┘, └m/4┘, └m/2┘ and m; or └m/12┘, └m/6┘, └m/3┘ and m; wherein └x┘ represents a rounding down operation.
Preferably, in a Multimedia Broadcasting Single Frequency Network (MBSFN) subframe, the time domain positions for transmitting the position reference signal may be 10 contiguous orthogonal frequency division multiplexing symbols which are from the third one to the last one in the MBSFN subframe.
Preferably, in a non-MBSFN subframe for transmitting the position reference signal, the number of the orthogonal frequency division multiplexing symbols for transmitting the physical downlink control channel may be 2, and an antenna port of a base station may be 4 or 2.
Preferably, when a system adopts a normal cyclic prefix, the time domain positions for transmitting the position reference signal may be the third, fourth, sixth and seventh orthogonal frequency division multiplexing symbols, as well as the tenth, eleventh, thirteenth and fourteenth orthogonal frequency division multiplexing symbols in the non-MBSFN subframe; or
the time domain positions for transmitting the position reference signal may be the third, fourth, sixth and seventh orthogonal frequency division multiplexing symbols, as well as the ninth, tenth, eleventh, thirteenth and fourteenth orthogonal frequency division multiplexing symbols in the non-MBSFN subframe.
Preferably, when a system adopts an extended cyclic prefix, the time domain positions for transmitting the position reference signal may be the third, fifth and sixth orthogonal frequency division multiplexing symbols, as well as the ninth, eleventh and twelfth orthogonal frequency division multiplexing symbols in the non-MBSFN subframe; or,
the time domain positions for transmitting the position reference signal may be the third, fifth and sixth orthogonal frequency division multiplexing symbols, as well as the eighth, ninth, eleventh and twelfth orthogonal frequency division multiplexing symbols in the non-MBSFN subframe.
Preferably, in each physical resource block for transmitting the position reference signal, only one subcarrier on the orthogonal frequency division multiplexing symbol for transmitting the position reference signal may be used for transmitting the position reference signal.
Preferably, the position reference signal may be a pseudo-random sequence, which is first mapped to a corresponding physical resource block in the frequency domain and then mapped to the corresponding physical resource block in the time domain.
In order to solve the above technical problem, the present disclosure further provides a system for transmitting a position reference signal, the system comprises a transmitting unit configured to transmit a position reference signal using a frequency domain position and a time domain position; wherein
the frequency domain positions for transmitting the position reference signal are n physical resource blocks, and the value of n is obtained according to signaling;
the time domain positions for transmitting the position reference signal are remaining orthogonal frequency division multiplexing symbols in a subframe except for orthogonal frequency division multiplexing symbols for transmitting a physical downlink control channel and orthogonal frequency division multiplexing symbols for transmitting a cell-specific reference signal.
Preferably, the system may further comprise a determining unit configured to determine an orthogonal frequency division multiplexing symbols used for transmitting the position reference signal in the frequency domain position and the time domain position; wherein the determining may specifically comprises the following:
0
1
2
i
N−1
i
i
th
th
th
an array A of an N×N common sequence is determined, wherein A=[a, a, a, . . . , a, . . . , a], both columns and rows are numbered from 0, N elements different with each other are included in the array A, a value of each element is an integer ranged from 0 to N−1, and arepresents that an element in the arow of the icolumn is 1, and elements in the other positions in the icolumn are 0;
ID
0
1
2
i
N−1
ID
i
(i+h)mod N
i
(i+h)mod N
cell
cell
h=X
N,p
X/N
b
a
+p
N,i=
. . . ,N−
p=X
N,h
X/N
b
a
+p
N,i=
. . . ,N−
when the identity of a cell is N, an index of the subframe for transmitting the cell-specific reference signals are determined as SubframeIndex, then an N×N array B=[b, b, b, . . . , b, . . . , b] corresponding to the cell Nmay be:
mod =floor(),=()mod 0,1,2, 1; ormod =floor(),=()mod 0,1,2, 1;
ID
ID
cell
cell
wherein x mod y represents an operation for calculating a remainder, floor (x) represents a rounding down operation, and X=Nor X=N+SubframeIndex;
the number of the orthogonal frequency division multiplexing symbols for transmitting the position reference signal in the subframe is determined to be n, and the first n columns or the first n rows of the array B are chosen, or the last n columns or the last n rows of the array B are chosen;
1
there is a one-to-one corresponding correlation between the chosen n columns or n rows and the n orthogonal frequency division multiplexing symbols, and the position where the element in each column or each row is located corresponds to the position of a subcarrier where the position reference signal is located on a corresponding orthogonal frequency division multiplexing symbol in each physical resource block for transmitting the position reference signal.
In order to solve the above technical problems, the present disclosure also provides a method for transmitting a position reference signal, the method comprises:
a corresponding index value is allocated to a combination of a period and its corresponding subframe offset which are used for transmitting a position reference signal, and a corresponding correlation of the combination and the corresponding index value is established;
frequency domain positions for transmitting the position reference signal are n physical resource blocks, and the value of n is obtained according to signaling; time domain positions for transmitting the position reference signal are remaining orthogonal frequency division multiplexing symbols in a subframe expect for orthogonal frequency division multiplexing symbols for transmitting a physical downlink control channel and orthogonal frequency division multiplexing symbols for transmitting a cell-specific reference signal;
the position reference signal is transmitted according to the established corresponding correlation, the allocated index value, the frequency domain position and the time domain position.
Preferably, the method may further comprise the following:
the combination and the corresponding index value and the corresponding correlation are respectively stored in a base station and a terminal;
after configuring the period and the corresponding subframe offset of the position reference signal, the base station determines the index value according to the corresponding correlation and transmits the index value to the terminal.
Preferably, the method may further comprise the following:
the terminal obtains the period and the corresponding subframe offset of the position reference signal which are configured by the base station according to the received index value and the corresponding correlation, and receives the position reference signal transmitted by the base station according to the obtained period and the subframe offset.
Preferably, the period may include {16, 32, 64, 128} ms or {16, 32, 64, 128, OFF} ms; wherein OFF represents that positioning function is turned off;
when the period is 16 ms, a value of the subframe offset is an integer ranged from 0 to 15;
when the period is 32 ms, a value of the subframe offset is an integer ranged from 0 to 31;
when the period is 64 ms, a value of the subframe offset is an integer ranged from 0 to 63;
when the period is 128 ms, a value of the subframe offset is an integer ranged from 0 to 127;
when the period is OFF, the subframe offset is a default value.
Preferably, when the period is 16 ms, combinations of the period and the corresponding subframe offset may be {16, 0}, {16, 1}, {16, 2}, . . . , {16, 14}, {16, 15}, and the corresponding index values may be respectively 0 to 15 in sequence;
when the period is 32 ms, combinations of the period and the corresponding subframe offset may be {32, 0}, {32, 1}, {32, 2}, . . . , {32, 30}, {32, 31}, and the corresponding index values may be respectively 16 to 47 in sequence;
when the period is 64 ms, combinations of the period and the corresponding subframe offset may be {64, 0}, {64, 1}, {64, 2}, . . . , {64, 62}, {64, 63}, and the corresponding index values may be respectively 48 to 111 in sequence;
when the period is 128 ms, combinations of the period and the corresponding subframe offset may be {128, 0}, {128, 1}, {128, 2}, . . . , {128, 126}, {128, 127}, and the corresponding index values may be respectively 112 to 239 in sequence;
when the period is OFF, a combination of the period and the corresponding subframe offset may be {OFF, default value}, and the corresponding index value may be 240.
In order to solve the above technical problems, the present disclosure also provides a system for transmitting a position reference signal, the system comprises a transmitting unit configured to transmit a position reference signal according to an established corresponding correlation, an allocated index value, a frequency domain position and a time domain position; wherein
a corresponding index value is allocated for a combination of a period and corresponding subframe offset which are used for transmitting a position reference signal, and a corresponding correlation is established for the combination and the corresponding index value;
the frequency domain positions for transmitting the position reference signal are n physical resource blocks, and the value of n is obtained according to signaling; the time domain positions for transmitting the position reference signal are the remaining orthogonal frequency division multiplexing symbols in a subframe except for orthogonal frequency division multiplexing symbols for transmitting a physical downlink control channel and orthogonal frequency division multiplexing symbols for transmitting a cell-specific reference signal.
Preferably, the system may further comprise a storing unit and an index value transmitting unit; wherein
the storing unit is configured to respectively store the combination and the corresponding index value and the corresponding correlation in a base station and a terminal;
the index value transmitting unit is configured to determine the index value according to the corresponding correlation and transmit the index value to the terminal.
Preferably, the system may further comprise a receiving unit, which is used by the terminal for obtaining the period and the corresponding subframe offset of the position reference single which are configured by the base station according to the received index value and the corresponding correlation, and used for receiving the position reference signal transmitted by the base station according to the obtained period and subframe offset.
Compared with the relating technologies, the technical solution of the present disclosure makes the time-frequency positions for transmitting the position reference signal in the adjacent cells different, thereby reducing the interference between cells, ensuring the positioning precision of the UE and Improving the overall performance of the system. In addition, the present disclosure also provides a technical solution for a transmitting period of a position reference signal and the subframe offset when the position reference signal is transmitted.
In the following description, the present disclosure will be described in detail in combination with the accompanying drawings and embodiments, so that those skilled in the art can fully understand the realization process that how the present disclosure solves the technical problem using the technical means and achieves the technical effect, and can implement the present disclosure according to the realization process.
In the technical solution of the present disclosure, frequency domain positions for transmitting a position reference signal are n physical resource blocks, wherein the value of n is obtained according to a notification of the signaling. Here, the above physical resource block only represents the frequency domain position of the position reference signal.
The n physical resource blocks can be n physical resource blocks which are discrete at equal intervals, i.e., the n physical resource blocks correspond to the n physical resource blocks numbered with r, r+k, r+2×k, . . . , r+(n−1)×k, wherein r represents the starting position number of the n physical resource blocks, and k represents an interval between two adjacent physical resource blocks. The resource block corresponding to the current downlink bandwidth is set to be m, all the physical resource blocks are numbered from 0, and then, r=0, and k=└m/n┘, wherein └x┘ represents a rounding down operation.
The n physical resource blocks can also be n contiguous physical resource blocks, such as, n contiguous physical resource blocks starting from low frequency, or can be n contiguous physical resource blocks with zero frequency as the center, or can also be n contiguous physical resource blocks with high frequency as a cut-off. The value of n is obtained according to a notification of signaling, and the signaling overhead can be 2 bits.
2
RB
RB
RB
DL
DL
DL
The n contiguous physical resource blocks can also be n contiguous physical resource blocks obtained according to a notification of signaling; and the starting position of the n contiguous physical resource blocks can be obtained together according to the signaling and can also be obtained according to other signalings. When one signaling is adopted for notifying the number (namely n) and the starting position of the physical resource blocks, the signaling overhead is ┌log(N(N+1)/2)┐, wherein Nrepresents the number of the resource blocks corresponding to the downlink bandwidth.
The corresponding correlation is as follows: for example, all the available physical resource blocks are numbered from 0 according to an order of from low frequency to high frequency, the last number is r, and then,
the n contiguous physical resource blocks starting from low frequency are n contiguous physical resource blocks numbered from 0 to n−1;
as to the n contiguous physical resource blocks with zero frequency as the center, a zero frequency subcarrier is located at the center of the n contiguous physical resource blocks, and the n contiguous physical resource blocks comprise 12 n contiguous subcarriers, i.e., 6 n subcarriers of low frequency adjacent to zero frequency and 6 n subcarriers of high frequency adjacent to zero frequency;
the n contiguous physical resource blocks with high frequency as a cut-off are n contiguous physical resource blocks which are numbered from r−n+1 to r with r as the cut-off number of the physical resource block.
For example, if the resource blocks corresponding to the current downlink bandwidth is m, then the values of n corresponding to the signaling of 2 bits are 1, 5, 10 and 20; or 6, 12, 25 and 50; or 10, 20, 40 and m; or 2, 5, 10 and 20; or 5, 10, 20 and 40; or └m/6┘, └m/4┘, └m/2┘ and m; or └m/12┘, └m/6┘, └m/3┘ and m.
Time domain positions for transmitting the position reference signal are remaining OFDM symbols in a subframe, except for OFDM symbols for transmitting a physical downlink control channel and OFDM symbols for transmitting a cell-specific reference signal.
Further, in a Multimedia Broadcasting Single Frequency Network (MBSFN) subframe, the time domain positions for transmitting the position reference signal are ten contiguous OFDM symbols which are from the third OFDM signal to the last OFDM symbol in the MBSFN subframe.
In a general subframe (non-MBSFN subframe), when the system adopts a normal cyclic prefix, the time domain positions for transmitting the position reference signal are the third, fourth, sixth and seventh OFDM symbols, as well as the tenth, eleventh, thirteenth and fourteenth OFDM symbols in the subframe; or
the time domain positions for transmitting the position reference signal are the third, fourth, sixth and seventh OFDM symbols, as well as the ninth, tenth, eleventh, thirteenth and fourteenth OFDM symbols in the subframe.
In a general subframe (non-MBSFN subframe), when the system adopts an extended cyclic prefix, the time domain positions for transmitting the position reference signal are the third, fifth and sixth OFDM symbols, as well as the ninth, eleventh and twelfth OFDM symbols in the subframe; or
the time domain positions for transmitting the position reference signal are the third, fifth and sixth OFDM symbols, as well as the eighth, ninth, eleventh and twelfth OFDM symbols in the subframe.
That is to say, in the general subframe (non-MBSFN subframe), the number of the OFDM symbols for transmitting the physical downlink control channel is 2 in a subframe for transmitting the position reference signal, and the current antenna port of the base station is 4 antenna port; or
in the general subframe (non-MBSFN subframe), the number of the OFDM symbols for transmitting the physical downlink control channel is 2 in a subframe for transmitting the position reference signal, and the current antenna port of the base station is 2 antenna port.
Further, in each resource block for transmitting the position reference signal, only one subcarrier on an OFDM symbol for transmitting the position reference signal is used for transmitting the position reference signal.
Further, the sequence of the position reference signal is a pseudo-random sequence, which is first mapped to a corresponding physical resource block in the frequency domain and then mapped to the corresponding physical resource block in the time domain.
An index value is allocated to each combination of a period and corresponding subframe offset which are used for transmitting the position reference signal, a corresponding correlation is established between the index value and the combination of the period and the corresponding subframe offset which are used for transmitting the position reference signal, and the corresponding correlation is stored on both sides of a base station and a terminal, and the index value and the combination of the period and the subframe offset are also stored.
After configuring for the terminal the period and the corresponding subframe offset which are used for transmitting the position reference signal, the base station can determine corresponding index value according to the information of the corresponding correlation and then transmits the index value to the terminal. The terminal can obtain the period and the subframe offset which are used for transmitting the position reference signal and configured by the base station according to the stored corresponding correlation and the index value, and can finish the receiving of the position reference signal according to the obtained period and subframe offset.
The period of transmitting the position reference signal can be {16, 32, 64, 128} ms, and can also be {16, 32, 64, 128, OFF} ms, wherein OFF represents that the positioning function is turned off, i.e., positioning is not performed.
When the period of transmitting the position reference signal is 16 ms, the value of the subframe offset for transmitting the position reference signal is an integer ranged from 0 to 15, there are 16 combinations of the constituted {period, subframe offset}, i.e., {16, 0}, {16, 1}, {16, 2}, . . . , {16, 14} and {16, 15}, and the corresponding index values are respectively 0 to 15 in sequence.
When the period of transmitting the position reference signal is 32 ms, the value of the subframe offset for transmitting the position reference signal is an integer ranged from 0 to 31, there are 32 combinations of the constituted {period, subframe offset}, i.e., {32, 0}, {32, 1}, {32, 2}, . . . , {32, 30} and {32, 31}, and the corresponding index values are respectively 16 to 47 in sequence.
When the period of transmitting the position reference signal is 64 ms, the value of the subframe offset for transmitting the position reference signal is an integer ranged from 0 to 63, there are 64 combinations of the constituted {period, subframe offset}, i.e., {64, 0}, {64, 1}, {64, 2}, . . . , {64, 62} and {64, 63}, and the corresponding index values are respectively 48 to 111 in sequence.
When the period of transmitting the position reference signal is 128 ms, the value of the subframe offset for transmitting the position reference signal is an integer ranged from 0 to 127, there are 128 combinations of the constituted {period, subframe offset}, i.e., {128, 0}, {128, 1}, {128, 2}, . . . , {128, 126} and {128, 127}, and the corresponding index values are respectively 112 to 239 in sequence.
If the period of transmitting the position reference signal also includes OFF, the corresponding subframe offset can be a default value, the index value corresponding to the constituted combination of {period, subframe offset}, i.e., {OFF, default value} is 240.
The specific corresponding correlation between the index value and the combination of the period and the subframe offset which are used for transmitting the position reference signal is shown in Table 1 or Table 2:
TABLE 1
Index Value I<sub>PRS</sub>
Period (ms)
Subframe Offset
0-15
16
I<sub>PRS</sub>
16-47
32
I<sub>PRS </sub>−16(0-31)
48-111
64
I<sub>PRS </sub>−48(0-63)
112-239
128
I<sub>PRS </sub>−112(0-127)
240
OFF
Default Value-
241-255
Reserved
Reserved
TABLE 2
Index Value I<sub>PRS</sub>
Period (ms)
Subframe Offset
0-15
16
I<sub>PRS</sub>
16-47
32
I<sub>PRS </sub>−16(0-31)
48-111
64
I<sub>PRS </sub>−48(0-63)
112-239
128
I<sub>PRS </sub>−112(0-127)
240-255
Reserved
Reserved
The period and the subframe offset which are used for transmitting the position reference signal can be respectively notified by adopting different signalings.
In addition, the position reference signal can be sent on h contiguous downlink subframes, and the value of h can be 1, 2, 4 or 6 and is obtained according to signaling.
The base station transmits to the terminal configuration information related to the position reference signal and then transmits the position reference signal in the corresponding time-frequency position, and then the terminal detects the position reference signal according to the received configuration information and finishes positioning according to the detected position reference signal.
In the following description, the technical features of the present disclosure will be clearly described by using mathematical expressions.
0
1
2
i
N−1
i
i
th
th
th
If an array A of an N×N common sequence is assumed, and A=[a, a, a, . . . , a, . . . , a], both columns and rows are numbered from 0, N elements different with each other are included in the array A, the value of each element is an integer ranged from 0 to N−1, wherein arepresents that the element in the arow of the icolumn is 1, and elements in the other positions in the icolumn are 0.
ID
0
1
2
i
N−1
ID
i
(i+h)mod N
i
(i+h)mod N
cell
cell
h=X
N
p
X/N
b
a
+p
N,i=
. . . ,N−
p=X
N
h
X/N
b
a
+p
N,i=
. . . ,N−
When the identity of a cell is N, an index of the subframe for transmitting the reference signal is SubframeIndex, and then an N×N array B=[b, b, b, . . . , b, . . . , b] corresponding to the cell Nis:
mod formula (1)=floor() formula (2)=()mod 0,1,2, 1 formula (3)ormod formula (4)=floor() formula (5)=()mod 0,1,2, 1 formula (6)
wherein
X mod y represents a reminder operation;
floor (X) represents a rounding down operation;
ID
ID
cell
cell
X=N, or X=N+SubframeIndex.
On each OFDM symbol for transmitting the position reference signal, only one subcarrier is used for transmitting the position reference signal data, and as to the position of the subcarrier for transmitting the position reference signal in the resource block, it is generated by the array B.
For example, if subcarriers in a resource block are numbered from 0 to 11, according to the number n of the OFDM symbols for transmitting the position reference signal in the subframe, the first n columns or the first n rows of the array B are chosen, or the last n columns or the last n rows of the array B are chosen.
1
1
th
th
th
th
th
th
th
th
There is one-to-one corresponding correlation between the chosen n columns or n rows and the n OFDM symbols, and the position where the element in each column or each row is located corresponds to the position of a subcarrier on a corresponding OFDM symbol in the resource block, where the position reference signal is located. That is, in the chosen array, if the element is assumed to be located in the jrow in the icolumn (or in the jcolumn in the irow), the position reference signal is accordingly located on the jsubcarrier on the corresponding OFDM symbol k in the icolumn (or on the jsubcarrier on the corresponding OFDM symbol k in the irow) in the chosen array.
In the embodiment, a position reference signal configured through signaling is sent in the full bandwidth, that is, the position reference signal is transmitted on each resource block, and the time-frequency position of the position reference signal in each resource block is the same.
In an MBSFN subframe, the time domain positions for transmitting the position reference signal are ten contiguous OFDM symbols in the MBSFN subframe which are from the third OFDM symbol to the last OFDM symbol in the subframe.
ID
ID
ID
ID
i
(i+h)mod N
0
1
2
N−1
cell
cell
cell
cell
h=N
p
N
b
a
+p
N,i=
. . . ,N−
B=[b
,b
,b
, . . . ,b
When the array A is [1, 3, 7, 4, 5, 2, 10, 9, 12, 8, 6, 11], the cell identity Nis 1, N=12, X=N, and then,
mod 12;=floor(/12);=()mod 0,1,2, 1;i.e., ];
it is obtained that:
B=[3, 7, 4, 5, 2, 10, 9, 12, 8, 6, 11, 1].
FIG. 3
FIG. 3
1
In the MBSFN subframe, the first 10 columns in the array are chosen, the specific positions of the corresponding position reference signals in the resource block is shown in , wherein the symbol T represents the position of the subcarrier where the position reference signal is located. The abscissa shown in represents the sequence number of the subframe on the OFDM symbol.
In the embodiment, a position reference signal configured through signaling is sent in the full bandwidth, that is, the position reference signal is sent on each resource block, and the time-frequency position of the position reference signal in each resource block is the same.
When the system adopts a normal cyclic prefix, the time domain positions for transmitting the position reference signal are the third, fourth, sixth and seventh OFDM symbols, as well as the tenth, eleventh, thirteenth and fourteenth OFDM symbols in the subframe.
ID
ID
ID
ID
i
(i+h)mod N
0
1
2
N−1
cell
cell
cell
cell
h=N
p
N
b
a
+p
N,i=
. . . ,N−
B=[b
,b
,b
, . . . ,b
When the array A is [1, 3, 7, 4, 5, 2, 10, 9, 12, 8, 6, 11], the cell identity Nis 1, N=12, X=N, and then,
mod 12;=floor(/12);=()mod 0,1,2, 1;i.e., ];
it is obtained that:
B=[3, 7, 4, 5, 2, 10, 9, 12, 8, 6, 11, 1].
FIG. 4
FIG. 4
1
When the system adopts a normal cyclic prefix, the first 8 columns in the array are chosen, and the specific positions of the corresponding position reference signals in the resource block is shown in . The abscissa shown in represents the sequence number of the subframe on the OFDM symbol.
In the embodiment, a position reference signal configured through signaling is sent in the full bandwidth, that is, the position reference signal is sent on each resource block, and the time-frequency position of the position reference signal in each resource block is the same.
When the system adopts an extended cyclic prefix, the time domain positions for transmitting the position reference signal are the third, fifth and sixth OFDM symbols, as well as the ninth, eleventh and twelfth OFDM symbols in the subframe.
ID
ID
ID
ID
i
(i+h)mod N
0
1
2
N−1
cell
cell
cell
cell
h=N
p
N
b
a
+p
N,i=
. . . ,N−
B=[b
,b
,b
, . . . ,b
When the array A is [1, 3, 7, 4, 5, 2, 10, 9, 12, 8, 6, 11], the cell identity Nis 1, N=12, X=N, and then,
mod 12;=floor(/12);=()mod 0,1,2, 1;i.e., ];
it is obtained that:
B=[3, 7, 4, 5, 2, 10, 9, 12, 8, 6, 11, 1].
FIG. 5
FIG. 5
1
When the system adopts an extended cyclic prefix, the first six columns in the array are chosen, and the specific positions of the corresponding position reference signals in the resource block are shown in . The abscissa in represents the sequence number of the subframe on the OFDM symbol.
The value of the common sequence A can be [1, 3, 7, 4, 5, 2, 10, 9, 12, 8, 6, 11], and can also be [1, 2, 5, 10, 12, 7, 8, 11, 4, 6, 3, 9], [1, 2, 4, 8, 5, 10, 9, 7, 3, 6] or [1, 2, 8, 11, 10, 4, 7, 12, 5, 3, 9, 6].
It needs to state that the above three embodiments only describe the time-frequency positions in a resource block without referring to the position of the physical resource block, and the physical resource block is taken as full bandwidth for description, therefore, the selection modes of the above-mentioned three physical resource blocks are not involved.
The present disclosure provides a system for transmitting a position reference signal, and the system comprises a transmitting unit configured to transmit a position reference signal according to a frequency domain position and a time domain position.
Wherein the frequency domain positions for transmitting the position reference signal are n physical resource blocks, and the value of n is obtained according to signaling. The time domain positions for transmitting the position reference signal are the remaining orthogonal frequency division multiplexing symbols in a subframe except for orthogonal frequency division multiplexing symbols for transmitting a physical downlink control channel and orthogonal frequency division multiplexing symbols for transmitting a cell-specific reference signal.
Preferably, the system further comprises a determining unit configured to determine an orthogonal frequency division multiplexing symbol used for transmitting the position reference signal in the frequency domain position and the time domain position; wherein the determining specifically comprise the following:
0
1
2
i
N−1
i
i
th
th
th
an array A of an N×N common sequence is determined, wherein A=[a, a, a, . . . , a, . . . , a], both columns and rows are numbered from 0, N elements different with each other are included in the array A, the value of each element is an integer ranged from 0 to N−1, and arepresents that the element in the arow of the icolumn is 1, and the elements in the other positions in the icolumn are 0;
ID
0
1
2
i
N−1
ID
i
(i+h)mod N
i
(i+h)mod N
cell
cell
h=X
N,p
X/N
b
a
+p
N,i=
. . . ,N−
p=X
N,h
X/N
b
a
+p
N,i=
. . . ,N−
when the identity of a cell is N, an index of the subframe for transmitting the cell-specific reference signal is determined as SubframeIndex, then an N×N array B=[b, b, b, . . . , b, . . . , b] corresponding to the cell Nis:
mod =floor(),=()mod 0,1,2, 1; ormod =floor(),=()mod 0,1,2, 1;
ID
ID
cell
cell
wherein x mod y represents an operation for calculating a remainder, floor (X) represents a rounding down operation, and X=Nor X=N+SubframeIndex;
the number of the orthogonal frequency division multiplexing symbols for transmitting the position reference signal in the subframe is determined to be n, and then the first n columns or the first n rows of the array B are chosen, or the last n columns or the last n rows of the array B are chosen;
1
There is one-to-one corresponding correlation between the chosen n columns or n rows and the n orthogonal frequency division multiplexing symbols, and the position where the element in each column or each row is located corresponds to the position of a subcarrier where the position reference signal is located on a corresponding orthogonal frequency division multiplexing symbol in each physical resource block for transmitting the position reference signal.
The present disclosure also provides a system for transmitting a position reference signal, and the system comprises a transmitting unit configured to transmit a position reference signal according to an established corresponding correlation, an allocated index value, a frequency domain position and a time domain position.
Wherein a corresponding index value is allocated for a combination of a period and corresponding subframe offset which are used for transmit the position reference signal, and a corresponding correlation for the combination and the corresponding index value is established; the frequency domain positions for transmitting the position reference signal are n physical resource blocks, and the value of n is obtained according to signaling; the time domain positions for transmitting the position reference signal are the remaining orthogonal frequency division multiplexing symbols in a subframe expect for orthogonal frequency division multiplexing symbols for transmitting a physical downlink control channel and orthogonal frequency division multiplexing symbols for transmitting a cell-specific reference signal.
Preferably, the system further comprises a storing unit and an index value transmitting unit. Wherein the storing unit is configured to store the combination and the corresponding index value and the corresponding correlation respectively in a base station and a terminal. The index value transmitting unit is configured to determine the index value according to the corresponding correlation and transmit the index value to the terminal.
Preferably, the system further comprises a receiving unit, which is used by the terminal for obtaining the period and the corresponding subframe offset of the position reference signal which are configured by the base station according to the received index value and the corresponding correlation, and used for receiving the position reference signal sent by the base station according to the obtained period and subframe offset.
The above description is only the illustrating embodiments of the present disclosure, which is not used to limit the present disclosure. To those skilled in the art, various modifications and changes can be made from the present disclosure. Any modification, equivalent substitution and improvement etc, made within the spirit and principle of the present disclosure, shall be within the protection scope of the appended claims of the present disclosure.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is a schematic diagram illustrating a physical resource block of an LTE system with system bandwidth of 5 MHz in the relating technologies;
FIG. 2(
FIG. 2(
a
b
) and ) are schematic diagrams illustrating the positions of a cell-specific reference signal of an LTE system in a physical source block in the relating technologies; and
FIG. 3
FIG. 5
to are schematic diagrams illustrating specific positions of carriers in a resource block where position reference signals are located according to the first embodiment to the third embodiment of the present disclosure. | |
Ammonia is considered an important potential energy carrier in the energy mix of a decarbonized shipping fleet. Ammonia (NH3) combustion emits water and nitrogen but no carbon compounds. Green ammonia produced from harvesting wind or solar energy is therefore a potential climate-neutral fuel.
DNV’s carbon risk framework has been adapted to study the feasibility of ammonia when used as ship fuel in a joint industry project (JIP) between DNV, TotalEnergies, Samsung Heavy Industries and a major Asian energy transport company. The JIP analysed a very large crude carrier (VLCC) design assumed to operate on a Middle East to Far East Asia route at a speed of 13 knots.
The existing VLCC specifications were studied to determine what should be done to minimize the costs and effort of a potential future retrofit when switching from LNG to ammonia. This assessment was based on the DNV draft rules for ammonia as fuel. The ship was assumed to begin operating in 2024 and have a 20-year lifespan. The study also looked at safety, environmental compatibility, costs and the overall business case for the individual fuel and operating scenarios.
For an ammonia-ready newbuild, the design arrangements must account for both LNG and ammonia. For example, ammonia requires larger tanks to achieve the same level of autonomy – which has implications for the superstructure – and a stronger support structure underneath. Other design properties, such as tank and piping materials, should be chosen with a potential retrofit for ammonia in mind. Ultimately the owner and yard must decide jointly about the desired degree of ammonia-readiness.
There are five key considerations for a conversion and retrofit from dual-fuel (DF) LNG to DF ammonia on a vessel designed to be “ammonia-ready”. The LNG fuel gas supply system (FGSS) must be replaced entirely. Modifications are necessary on the tank system, main engine, safety systems, gensets and boilers. The required technology is not fully available at present (2021).
The energy density of ammonia is considerably lower than that of LNG, almost half in terms of volume, limiting the vessel’s autonomy. The bunkering strategy of the vessel must be reassessed.
This diagram reflects the IMO carbon emission reduction trajectories: a minimum alignment curve, which aims to achieve a 70% reduction in carbon intensity by 2050 relative to 2008 (aligned with the current IMO 2050 target), and a curve aiming to achieve net-zero carbon emissions by 2050. (AER: CO2 emissions per vessel capacity divided by distance sailed.)
The study examined two operating profiles and two fuel strategies for the ship under investigation, which is equipped with a dual-fuel engine and will operate on LNG as long as compliance allows. To prevent the vessel from becoming non-compliant, it will eventually be converted to ammonia. Two strategies for alignment with CO2 reduction scenarios are compared: Progressive introduction of bio-LNG Retrofit to operate on ammonia (for either main engine or main engine and auxiliaries) LNG fuel will keep the vessel compliant for 17 years in a minimum alignment scenario. Conversion to ammonia will not be economically credible at this vessel age and progressive incorporation of bio-LNG will most likely be preferred. When aiming for net-zero emissions by 2050, LNG fuel will be compliant with this more ambitious trajectory for the first ten years of operation and then conversion to ammonia or bio-LNG are the two options.
Various other key market players may influence the picture by imposing their own carbon-reduction schemes. The EEOI-based decarbonization trajectory used by the Sea Cargo Charter places greater emphasis on the operating profile and time spent in laden condition and appears more challenging than the minimum alignment trajectory. (EEOI: Annual CO2 emissions divided by the product of distance sailed in laden condition and actual cargo on board.)
This comparison of the total cost of ownership (TCO) for the scenarios investigated in the study is speculative considering the uncertainties regarding fuel prices, fuel availability and other factors. Nevertheless, based on current assumptions the key influential factor for TCO is fuel cost. The “net zero by 2050” scenarios come with higher overall costs. The LNG with progressive bio-LNG option is found to be the least costly solution. A future CO2 tax will further increase the TCO without changing the overall conclusions dramatically.
Key influential factors for the business case include fuel availability and prices, a potential future CO2 tax, the slope of the decarbonization trajectory, and retrofitting expenditures (CAPEX). In the present study, the gradual mixing-in of bio-LNG has a lower TCO than the ammonia option. However, a variety of factors could shift the picture and make ammonia appear more feasible. It is important to bear in mind the uncertainties affecting the study, in particular fuel price developments, fuel taxation, fuel availability in the relevant trading area, the regulatory decarbonization trajectory that will be applied, and the shipowner’s own decarbonization ambitions.
The additional costs to enable the VLCC to run on LNG (rather than conventional fuel oil) were about US$20m. On top of this, an additional approximately US$2m was needed at the newbuilding stage to make the vessel ready for a future retrofit to ammonia. In the study, the most cost-effective compliance strategy in all cases was to operate on LNG with increasing amounts of bio-LNG blended in. There is a way towards 2050, but the “right” direction is very ship- and trade-specific, and a range of assumptions need to be established.
Get regular tanker insights! | https://www.dnv.com/expert-story/maritime-impact/Ammonia-ready-VLCC-design-as-a-decarbonization-option.html |
On 17th January 2006, the City of London resolved to grant consent to increase the height of Heron Tower by 4 floors, taking the structural height to 203m and the total height to 242m.
Heron Tower replaces two buildings, Bishops House and Kempson House, built in 1976 and 1960 respectively.
The building's cladding covers an area of 40,000 sq m along with 31,150 sq m of glazing.
10 of the 18 lifts are glass and double-decker and travel at speeds of up to 7 metres (23 feet) per second.
Planning consent was granted in 2002 for a 42-storey tower, reaching a structural height of 183m with a mast taking the total height to 222m. The consent provided for 63,105sq m of floorspace.
The tower is home to around 4,300 workers.
The lifts are entirely custom built for the tower to reflect its design; the operating panels display a legend of the tower showing passengers exactly where they are at any given point.
Heron Tower is constructed from steel and is clad with a stainless steel curtain wall which is naturally ventilated on the east and west sides. The south side incorporates photovoltaic glass which is cooled by recycled air, thus reducing energy consumption.
There is a 70,000 litre aquarium in the entrance lobby.
There are three concourse levels which include retail areas and entrance foyers.
Levels 41 to 46 are dedicated to plant.
Heron Tower is constructed from around 8,500 pieces of structural steel.
The building is topped with a communications mast.
Office facilities are arranged around a north facing atrium and are stacked in units of twelve, each of three stories.
Net office space amounts to 40,836 square metres (439,552 square feet).
Offices are situated on levels 2 through to 37 whilst levels 38 to 40 incorporate a bar, conference facilities and a public restaurant situated on level 39, which is served by two dedicated high-speed glass lifts. | https://www.emporis.de/buildings/101374/110-bishopsgate-london-united-kingdom |
TECHNICAL FIELD
BACKGROUND ART
[Patent Document 1] Japanese Unexamined Patent Application Publication No. 52-16997
[Patent Document 2] Japanese Unexamined Patent Application Publication No. 3-283320
[Patent Document 3] Japanese Unexamined Patent Application Publication No. 49-114389
[Patent Document 4] Japanese Examined Patent Application Publication No. 1-8698
DISCLOSURE OF INVENTION
BEST MODES FOR CARRYING OUT THE INVENTION
Example 1
Example 2
INDUSTRIAL APPLICABILITY
3
The present invention relates to a useful method of producing a NbSn superconducting wire material and to a Nb-based rod material for producing a superconducting wire material, the rod material being used as a raw material in the production method.
An example of an application in which a superconducting wire material has been in practical use is its use as a superconducting magnet used in a high-resolution nuclear magnetic resonance (NMR) analyzer. The higher the magnetic field generated by the superconducting magnet, the higher the resolution of the NMR analyzer. Therefore, recently, there has been an increase in the magnetic field generated by such a superconducting magnet.
3
3
As a superconducting wire material used for such a superconducting magnet for high-magnetic-field generation, a NbSn wire material has been practically used. A bronze process is mainly employed for producing this NbSn superconducting wire material.
3
3
FIG. 1
FIG. 1
2
1
2
2
2
In the bronze process, a composite material for producing a NbSn superconducting wire material that is schematically shown in is used. In this composite material, a plurality of (in , seven) core members made of Nb or a Nb-based alloy are embedded in a Cu—Sn-based alloy (bronze) matrix . These core members are subjected to wire drawing, thereby reducing the diameter thereof. Thus, the core members are formed into filaments. A plurality of the composite materials including the filaments of the core members and the bronze are bundled to form a group of wire materials. Copper (stabilizing Cu) for stabilization is arranged on the outer surface of the group of wire materials, and wire drawing is then performed. After the wire drawing has been performed, the group of wire materials is subjected to a heat treatment (diffusion heat treatment) at about 600° C. or higher and 800° C. or lower, thereby forming a NbSn compound layer at the interface between the filaments and the matrix.
3
In addition to the above bronze process, as a process of producing a NbSn superconducting wire material, a tube process, an internal diffusion process, a powder process, and the like are also known.
3
3
FIG. 2
4
3
5
6
4
3
Among these processes, in the tube process, a composite material for producing a NbSn superconducting wire material that is schematically shown in is used. In this composite material, a core member made of Sn or a Sn-based alloy is arranged in a tube (pipe member) made of Nb or a Nb-based alloy. This composite material is inserted in a Cu pipe , as needed; subjected to a diameter-reducing process such as wire drawing; and then heat-treated. Accordingly, a diffusion reaction between Nb and Sn occurs, thus producing NbSn (for example, Patent Document 1). Furthermore, from the viewpoint of workability, a Cu pipe may be arranged between the core member and the Nb tube (for example, Patent Document 2).
3
3
FIG. 3
8
7
9
7
8
8
9
In the internal diffusion process, a composite material for producing a NbSn superconducting wire material that is schematically shown in is used. In this composite material, a core member made of Sn or a Sn-based alloy is embedded at the central part of a base material made of Cu or a Cu-based alloy, and a plurality of (in the figure, 15) core members made of Nb or a Nb-based alloy are arranged in the base material and around the core member . This composite material is subjected to wire drawing and then heat-treated. Accordingly, Sn in the core member diffuses and reacts with Nb in the core members , thus producing NbSn (for example, Patent Document 3).
3
3
FIG. 4
11
10
10
11
10
In the powder process, a composite material for producing a NbSn superconducting wire material that is schematically shown in is used. This composite material is produced by a step of forming a powder core part by filling a sheath (pipe member) made of Nb or a Nb-based alloy with a raw material powder containing at least Sn (for example, a Ta—Sn-based powder), and a step of further inserting the sheath and the powder core part into a Cu billet (not shown). This composite material is subjected to a diameter-reducing process such as extruding or wire drawing to formed into a wire material. Subsequently, the wire material is wound around a magnet or the like and then heat-treated. Accordingly, a NbSn superconducting phase is formed from the inner surface side of the sheath .
FIGS. 2 to 4
For convenience of explanation, a single-core composite material is shown in . However, in practical use, a multi-core composite material in which a plurality of single cores are arranged in a Cu matrix is generally used.
3
3
3
8
9
FIG. 3
FIG. 3
Furthermore, it has been proposed that, in producing a superconducting wire material using the above composite material, elements such as Ti, Ta, Zr, and Hf are added to the NbSn phase. It is believed that the addition of these elements in a NbSn superconducting wire material improves superconducting characteristics of the superconducting wire material at high magnetic fields, compared with a NbSn superconducting wire material not containing these elements. For example, Patent Document 4 describes that adding Ti to the Sn metal core (core member in ) in an amount of 30 atomic percent or less and adding Ti to the Nb metal cores (core members in ) in an amount of 5 atomic percent or less can improve the critical current density Jc of the superconducting wire material in an external magnetic field of 15 T (Tesla) or more.
In the production of the superconducting wire material, since a diameter-reducing process such as extruding or wire drawing is performed for a composite material used as a precursor of the superconducting wire material, a wire material having a circular cross-sectional shape is generally used as the composite material. Furthermore, in some cases, after the composite material is worked to a certain degree, hexagonal drawing, which is a drawing for changing the cross-sectional shape of the composite material to a hexagon, is performed. Several or several hundreds of the raw materials having a hexagonal cross section are combined to form a multi-core composite wire material, and wire drawing is further performed for this composite wire material. In addition, in the case where workability is degraded during drawing, intermediate annealing may be performed. Thus, wire drawing of the composite material is performed until the diameter of the composite material, which is about several tens to several hundreds millimeters before the wire drawing, is reduced to several microns.
In such wire drawing with a high working ratio, it is necessary that the cross-sectional shape of the raw material be uniformly changed by the wire drawing. In each of the above processes, Nb or a Nb-based alloy is used as a raw material (a pipe member or a core member). When wire drawing with a high working ratio is performed for this raw material, a phenomenon in which the circular cross section of the raw material made of Nb or a Nb-based alloy in the composite material cannot be maintained and is changed to a cross section having the shape of a rhombus or a rectangle may occur. Furthermore, as described above, in the case where elements such as Ti, Ta, Zr, and Hf are added to Nb or a Nb-based alloy used as a raw material in order to improve the characteristics of the final superconducting wire material, the addition of these elements degrades workability instead, and thus, the above phenomenon may occur more easily.
The above phenomenon causes breaking of a wire material in the course of drawing. Alternatively, the above phenomenon may cause problems such as a decrease in the critical current density (Jc), a decrease in the n-value (a value used as an indicator showing the sharpness of the transition from the superconducting state to the normal conducting state), and an increase in the AC loss in the final superconducting wire material.
In view of the above circumstances, hitherto, the occurrence of the above problems has been prevented by adjusting the drawing ratio so as not to change the cross-sectional shape of the raw material, that is, by preparing a raw material for drawing having a small cross-sectional area in advance and working the raw material with a relatively low working ratio. However, the production efficiency in this method is extremely low. Accordingly, it has been desired to establish a technique in which a satisfactory working can be realized without deformation even when a raw material for drawing having a large cross-sectional area is used.
3
The present invention has been made in order to meet the above desire. It is an object of the present invention to provide a Nb-based rod material which is used for producing a NbSn superconducting wire material and in which workability of Nb or a Nb-based alloy can be satisfactory, and to provide a useful method of producing a superconducting wire material which exhibits satisfactory superconducting characteristics (in particular, the critical current density and the n-value) using the Nb-based rod material.
To achieve this object, the present invention provides a Nb-based rod material used for producing a superconducting wire material, wherein the Nb-based rod material is formed to be columnar or substantially columnar by casting a raw material of the Nb-based rod material using a casting mold having a circular or substantially circular cross-sectional shape, and by hot-working or cold-working with a working apparatus whose cross-sectional shape is a circular or substantially circular shape.
In the Nb-based rod material made of Nb or a Nb-based alloy, the Nb-based rod material is preferably formed so that a circular cross-sectional shape or a substantially circular cross-sectional shape is maintained through the steps of the hot-working or the cold-working.
Furthermore, in the Nb-based rod material made of Nb or a Nb-based alloy, more preferably, for example, the following requirements are satisfied: (a) the crystal grain size of the rod material is in the range of 5 to 100 μm (and more preferably, in the range of 5 to 50 μm); (b) the concentration of at least one type of element selected from the group consisting of carbon, nitrogen, oxygen, and hydrogen is 200 ppm or less; (c) the rod material contains at least one type of element selected from the group consisting of Ti, Ta, Zr, and Hf in an amount in the range of 0.1 to 20 mass percent; and (d) the rod material contains Nb in an amount of 70 mass percent or more.
Furthermore, the present invention provides a production method for achieving the above object including a first step of forming a composite material for producing a superconducting wire material by combining a hot-worked or cold-worked columnar or substantially columnar Nb-based rod material with Cu or a Cu-based alloy and Sn or a Sn-based alloy, or a Cu—Sn-based alloy; a second step of forming a precursor wire material for producing a superconducting wire material by reducing the diameter of the combined composite material for producing a superconducting wire material to form a wire material; and a third step of forming a superconducting phase by heat-treating the precursor wire material for producing a superconducting wire material.
In this production method, for example, when a composite material for producing a superconducting wire material is formed by combining the columnar or substantially columnar Nb-based rod material with a Cu—Sn-based alloy, a bronze process or an internal diffusion process can be employed. Alternatively, when a composite material for producing a superconducting wire material is formed by processing a columnar or substantially columnar Nb-based rod material into a cylindrical or substantially cylindrical shape, and then combining the Nb-based rod material with Cu or a Cu-based alloy and Sn or a Sn-based alloy, a powder process or a tube process can be employed.
The present inventors have studied from various angles the cause of uneven deformation of Nb or a Nb-based alloy (an alloy containing at least Nb in an amount of 70 mass percent or more), which is used as a member of a composite material for producing a superconducting wire material, generated by wire-drawing the composite material. As a result, present inventors have found that a specific aggregate texture is formed in accordance with the history of the production process, and this aggregate texture is a cause of the uneven deformation. It is believed that since Nb or a Nb-based alloy is not easily recrystallized, the above phenomenon of the formation of the specific aggregate texture significantly occurs. Furthermore, even if Nb or the Nb-based alloy is recrystallized, the recrystallized aggregate texture tends to deform the circular cross-sectional shape before wire drawing to a rectangular or rhombic cross-sectional shape.
In a stage of casting, Nb or a Nb-based alloy is formed as a cast slab having a circular or rectangular cross-sectional shape. In the subsequent stage of workings (hot working and cold working), the cross-sectional shape is changed to a rectangle, a rhombus, or an ellipse. Finally, Nb or the Nb-based alloy is provided as a raw material for a composite material having a circular cross section or a rectangular cross section. In the raw material produced by such a process, corner portions of the cross-sectional shape (four portions in the case of a rectangle) are significantly deformed, and a specific aggregate texture is significantly developed at the portions. These portions having the developed aggregate texture are in a state in which the shape of the material is not easily changed. It is believed that this makes it difficult to perform uniform working in the subsequent wire drawing stage, and thus, the cross-sectional shape of the material becomes a distorted shape.
Consequently, the present inventors have conducted intensive studies on an aggregate texture that can prevent uneven deformation. As a result, it has been found that when a raw material has an aggregate texture which is axially symmetric with respect to the center in the cross section, the cross-sectional shape is not changed to a rectangle or a rhombus even in the later stage of wire drawing, and the drawing can be continued while maintaining a circular shape or a substantially circular shape. Note that the term “substantially circular shape” includes not only a shape that is not a perfect circle but is approximately a circle, but also a hexagonal cross-sectional shape.
An aggregate texture that is desired in the present invention is an axially symmetric structure. In order to obtain such an aggregate texture, casting is performed using a casting mold having a circular or a substantially circular cross section in the stage of casting, and working is performed using a working apparatus having a circular or a substantially circular cross section.
More specifically, it has been found that, through a production process, when a process during which the cross section of a material is constantly subjected to axial symmetry (a process during which a circular cross section of a material is maintained) is performed, the above preferable aggregate texture is developed. In particular, in the case where workings (hot working and cold working) are performed, the above finding does not merely mean that the final cross-sectional shape is a circular or substantially circular shape, but means that the material is preferably processed so as to maintain a circular shape or substantially circular shape in all the steps. Note that the hot working in the present invention includes hot rolling, hot forging, and the like. The cold working in the present invention includes cold rolling, cold forging, and the like.
In the Nb-based rod material made of Nb or a Nb-based alloy of the present invention, the average crystal grain size thereof is preferably in the range of 5 to 100 μm, and more preferably in the range of 5 to 50 μm. This crystal grain size affects the workability. When the average crystal grain size is less than 5 μm, work hardening significantly occurs, and thus, cracking easily occurs during wire drawing.
On the other hand, as the average crystal grain size increases, the workability (ductility) becomes more satisfactory. However, when the average crystal grain size exceeds 100 μm, the surface property is degraded (irregularities are easily formed on the surface). Consequently, when such a Nb-based rod material is formed into a composite material, a deformation resistance with an adjacent member increases, and uniform working may become difficult. When the size (diameter) of a casting mold is small, a sufficient working ratio is not ensured, and the grain size may be larger than the above range. In such a case, upset forging in which compression is performed in the longitudinal direction may be performed.
In the Nb or Nb-based alloy rod of the present invention, carbon, nitrogen, oxygen, hydrogen, and other elements are contained as inevitable impurities. These are elements forming an interstitial solid solution (interstitial elements). If an excessive amount of these elements are contained, work hardening excessively occurs, and thus forming and working may become difficult. Therefore, the total concentration of these elements is preferably 200 ppm or less. On the other hand, the lower limit of the concentration of these elements is not particularly determined, but this concentration is preferably 20 ppm or more. The superconducting wire material to be produced is a composite material of the Nb or Nb-based alloy rod and copper or a copper alloy. Accordingly, if the content of these elements is excessively small, a difference in the deformation resistance between the Nb or Nb-based alloy rod and copper or a copper alloy disposed at the periphery of the rod is excessively large. This difference in the deformation resistance induces uneven deformation, such as sausaging and a ribbon-shaped deformation, during the formation of the composite material and may cause degradation of characteristics.
The average crystal grain size can be controlled by processings such as casting and rolling and by an adjustment by annealing. The concentration of the above impurities can be reduced by, for example, decreasing the degree of vacuum during melting of the alloy, or melting repeatedly in a high vacuum atmosphere.
The Nb or Nb-based alloy rod contains at least one type of element selected from the group consisting of Ti, Ta, Zr, and Hf in an amount in the range of 0.1 to 20 mass percent, as needed. These elements are effective in improving superconducting characteristics (in particular, the critical current density Jc) of the final wire material. In order to achieve this effect, the content of the above elements is preferably 0.1 mass percent or more. However, a content exceeding 20 mass percent degrades the workability.
In producing the superconducting wire material using the above-described Nb or Nb-based alloy rod, a uniform working can be performed in which a substantially circular cross-sectional shape of Nb or the Nb-based alloy, which is formed into filaments, can be maintained, and the current distribution in the cross section can be uniform. As a result, the critical current density Jc and the n-value can be improved. Furthermore, an appropriate control of the crystal grain size as described above can decrease the contact resistance with a member adjacent to the Nb or Nb-based alloy and suppress coupling between the filaments, thus decreasing the AC loss in the superconducting wire material.
3
The NbSn-based superconducting wire material including the above Nb-based alloy rod for producing a superconducting wire material can be produced in accordance with a known method. For example, the production of the superconducting wire material is preferably performed by a method including the following steps (a) to (c):
(a) a step of forming a composite material for producing a superconducting wire material by combining a hot-worked or cold-worked columnar or substantially columnar Nb-based rod material with Cu or a Cu-based alloy and Sn or a Sn-based alloy, or a Cu—Sn-based alloy;
(b) a step of forming a precursor wire material for producing a superconducting wire material by reducing the diameter of the combined composite material for producing a superconducting wire material to form a wire material; and
(c) a step of forming a superconducting phase by heat-treating the precursor wire material for producing a superconducting wire material.
FIGS. 1 and 3
FIGS. 2 and 4
In this method of producing a superconducting wire material, by combining a columnar or substantially columnar Nb-based rod material with, for example, Cu or a Cu-based alloy and Sn or a Sn-based alloy, or a Cu—Sn-based alloy, the composite materials for producing a superconducting wire material shown in can be formed, and these composite materials can be used for the bronze process or the internal diffusion process. In addition, by processing a columnar or substantially columnar Nb-based rod material into a cylindrical or substantially cylindrical shape, and then combining the Nb-based rod material with Cu or a Cu-based alloy and Sn or a Sn-based alloy, the composite materials for producing a superconducting wire material shown in can be formed, and these composite materials can be used for a powder process or a tube process. In the powder process, a powder mainly composed of Sn (for example, a Ta—Sn powder) is used as the Sn-based alloy. That is, such a powder is also included in the Sn-based alloy to be combined.
The present invention will now be described more specifically using examples. However, the present invention is not limited to the examples below. The present invention can be carried out by adding an appropriate modification within the scope that can be fitted with the points described above or below, and such modifications are also included in the technical scope of the present invention.
Niobium (Nb) rods are produced by casting using a cylindrical casting mold having an inner diameter of 300 mm and then rolled under the following condition (A) or condition (B) until the final diameter of the rods is reduced to 14 mm.
(A) 4-pass hot rolling is performed using a rolling mill in which the rolling cross-sectional shape is a circle, and 4-pass hot rolling is then performed using a rolling mill in which the rolling cross-sectional shape is an ellipse.
(B) 4-pass hot rolling is performed using a rolling mill in which the rolling cross-sectional shape is a circle.
In both a Nb rod obtained under condition (A) (hereinafter referred to as “Nb rod A”) and a Nb rod obtained under condition (B) (hereinafter referred to as “Nb rod B”), the concentration of inevitable impurities can be reduced by controlling conditions for melting by means of an electron beam (EB). More specifically, the intensity of the beam, the cross-sectional area of the beam, the output of the beam and the number of times of melting, the degree of vacuum during melting, and the like are controlled. Thereby, among the inevitable impurities, the concentration of carbon (C) is reduced to 30 ppm, the concentration of nitrogen (N) is reduced to 20 ppm, the concentration of oxygen (O) is reduced to 20 ppm, and the concentration of hydrogen (H) is reduced to 10 ppm. According to the measurement results of the average crystal grain size after rolling, both the Nb rod A and the Nb rod B have an average crystal grain size of 100 μm.
FIG. 1
Each of the Nb rods A and B has an outer diameter of 14 mm and a length of 200 mm. Composite materials in which seven Nb rods A or seven Nb rods B are embedded in a Cu-15 mass % Sn-0.3 mass % Ti alloy having an outer diameter of 67 mm are prepared (refer to ). Each of these composite materials undergoes extruding and wire drawing to produce a wire material (hexagonal single-core wire material) having a regular hexagonal cross section with sides of 2 mm.
3
3
3
These hexagonal single-core wires are cut so as to have a predetermined length, and 673 of the wires are bundled. A Nb diffusion barrier layer having a thickness of 1.5 mm is provided in a Cu tube having an inner diameter of 68 mm and an outer diameter of 160 mm. The bundle of the hexagonal single-core wires is arranged inside the Cu tube, thus producing a multi-core composite material. By performing extrusion and wire drawing of this composite material, a precursor wire material for producing a superconducting wire material having a final wire diameter of 0.3 mm is produced. Subsequently, NbSn-formation heat treatment for the composite material is performed at 700° C. for 100 hours to produce a NbSn superconducting wire material. The critical current density Jc, the n-value, and the AC loss are measured using the NbSn superconducting wire material.
[Measurement of Critical Current Density Jc]
A sample (superconducting wire material) is energized in an external magnetic field of 18 T in liquid helium, and the generated voltage is measured by a four-probe method. A current value when the measured voltage matches with a predetermined value (a voltage value at which an electric field of 0.1 μV/cm is generated) is measured as a critical current Ic. The critical current density Jc is determined by dividing the measured critical current Ic by the cross-sectional area corresponding to a non-Cu portion in the cross-sectional area of the wire material.
[Measurement of N-Value]
An (Ic-V) curve is obtained by the same measurement as the measurement for the critical current. In this curve, the value of n (i.e., “n-value”) is determined as a slope of a curve in which both data of Ic and V between 0.1 μV/cm and 1.0 μV/cm are shown by the logarithm. More specifically, the relationship between the current and the voltage is empirically represented by an approximate expression of expression (1) below. The v-value is determined on the basis of this expression.
V=Vc
Iop/Ic
n
() (1)
Here, Iop represents the operation current of a magnet, Ic represents the critical current of a wire material, and Vc represents a reference voltage defining Ic.
[Measurement of Ac Loss]
A magnetization curve is measured by a pick-up coil method in a state in which the external magnetic field is swept in the range of ±3T in liquid helium. The area of this magnetization curve is measured as the AC loss.
The results of the above measurements are shown in Table 1.
TABLE 1
Critical current
Ac loss
Nb rod
density Jc (A/mm<sup>2</sup>)
n-Value
(mJ/cc) ± 3T
A
165
21
236
B
195
29
195
These results show that the superconducting wire material produced using the Nb rod B has a satisfactory critical current density Jc and a satisfactory n-value, and in addition, the AC loss of this superconducting wire material is low.
Niobium (Nb)-7.5 mass % Ta alloy rods are produced by casting using a cylindrical casting mold having an inner diameter of 300 mm and then rolled under the following condition (C) or condition (D) until the final diameter is reduced to 55 mm.
(C) 4-pass hot rolling is performed using a rolling mill in which the rolling cross-sectional shape is a circle, and 4-pass hot rolling is then performed using a rolling mill in which the rolling cross-sectional shape is a rectangle (Nb-based alloy rod C).
(D) 4-pass hot rolling is performed using a rolling mill in which the rolling cross-sectional shape is a circle (Nb-based alloy rod D).
In both a Nb rod obtained under condition (C) (hereinafter referred to as “Nb-based alloy rod C”) and a Nb rod obtained under condition (D) (hereinafter referred to as “Nb-based alloy rod D”), the concentration of inevitable impurities can be reduced by controlling conditions for melting by means of EB. More specifically, the concentration of C is reduced to 20 ppm, the concentration of N is reduced to 20 ppm, the concentration of O is reduced to 30 ppm, and the concentration of H is reduced to 10 ppm. According to the measurement results of the average crystal grain size after rolling, both the Nb-based alloy rod C and the Nb-based alloy rod D have an average crystal grain size of 150 μm.
Next, a perforation process of the Nb-based alloy rods C and D is performed, thus producing pipe members each having an outer diameter of 55 mm, an inner diameter of 30 mm, and a length of 150 mm.
A Ta powder and a Sn powder are weighed such that the atomic ratio of Ta:Sn is 6:5, and the powders are mixed with a V-blender for about 30 minutes. The mixed powder (base powder) thus obtained is heat-treated at 950° C. in vacuum for 10 hours and then crushed. Furthermore, 5 mass percent of a Cu powder and 25 mass percent of Sn powder are added to the base powder, thereby allowing a new mixed powder to be prepared.
FIG. 4
A plurality of composite materials are prepared by filling the new mixed powder inside each of the pipe members (refer to ). These composite materials are inserted into a Cu billet having an outer diameter of 65 mm and an inner diameter of 30 mm. The Cu billet then undergoes extruding and wire drawing, thus forming a wire material having a regular hexagonal cross section with sides of 4 mm, i.e., a hexagonal single-core wire material.
The hexagonal single-core wires thus obtained are cut so as to have a predetermined length, and 163 of the cut wires are bundled. This bundle is arranged inside a Cu tube having an outer diameter of 65 mm and an inner diameter of 58 mm, thus producing a multi-core composite material. By performing extrusion and wire drawing of the composite material, a precursor wire material for producing a superconducting wire material having a final wire diameter of 1.2 mm is produced.
3
3
3
Subsequently, NbSn-formation heat treatment for the precursor wire material is performed at 650° C. for 250 hours to produce a NbSn superconducting wire material. The critical current density Jc, the n-value, and the AC loss are measured as in Example 1 using the NbSn superconducting wire material. The results are shown in Table 2.
TABLE 2
Critical current
Ac loss
Nb alloy rod
density Jc (A/mm<sup>2</sup>)
n-Value
(mJ/cc) ± 3T
C
230
35
1,410
D
345
43
1,024
These results show that the superconducting wire material produced using the Nb-based alloy rod D has a satisfactory critical current density Jc and a satisfactory n-value, and in addition, the AC loss of this superconducting wire material is low.
In another example, four forging dies having a circular forging cross-sectional shape with different diameters are prepared. Forging is performed in which a Nb rod is repeatedly roundly pressed so that the diameter of the rod is decreased stepwise, using the dies in descending order of size. Thus, a Nb rod is produced. The resulting circular Nb rod produced by forging with dies each having a circular cross section has better characteristics than a forged Nb rod produced by reducing the diameter so as to have a circular shape by pressing with a flat die while a Nb material is rotated.
The Nb-based alloy rods in Example 2 contain Ta. Alternatively, the addition of, for example, Ti, Zr, or Hf to the Nb-based alloy rods is also effective. The content of Nb is preferably 80 mass percent or more.
3
The present invention provides a Nb-based rod material in which anisotropy is eliminated to enable satisfactory uniform working. In addition, the present invention provides a method of producing a NbSn superconducting wire material which has an excellent critical current density and a large n-value and which can generate a high magnetic field by using the rod material as a raw material. The superconducting wire material thus produced is useful for the realization of, for example, an NMR magnet, a magnet for an accelerator, and a magnet for nuclear fusion which are compact and which can be produced at a low cost.
BRIEF DESCRIPTION OF DRAWINGS
FIG. 1
is a cross-sectional view that schematically shows a composite material used in a bronze process.
FIG. 2
is a cross-sectional view that schematically shows a composite material used in a tube process.
FIG. 3
is a cross-sectional view that schematically shows a composite material used in an internal diffusion process.
FIG. 4
is a cross-sectional view that schematically shows a composite material used in a powder process. | |
Q:
Finding the Laplace transform of the solution of the given IVP
Find the the Laplace transform $Y(s)$ of the solution of the given initial value problem
$$y''+y=\begin{cases}t & 0 < t < 1 \\ 0 & 1 < t < \infty \end{cases}$$
$$y(0)=0$$
$$y'(0)=0$$
What i tried
The laplace transform becomes
$$Y(S)+sy(0)+y'(0)=\begin{cases}\frac{1}{s^2} & 0 < \frac{1}{s^2} < 1 \\ 0 & 1 < \frac{1}{s^2} < \infty \end{cases}$$ Since $$y(0)=0$$
$$y'(0)=0$$ the equation becomes
$$Y(S)=\begin{cases}\frac{1}{s^2} & 0 < \frac{1}{s^2} < 1 \\ 0 & 1 < \frac{1}{s^2} < \infty \end{cases}$$ AM i correct. COuld anyone please explain. Thanks
A:
The Laplace Transform of the derivative of a function $y(t)$ with initial value $y(o)$ is $sY(s)-y(0)$ while the Laplace Transform of the second derivative of a function $y(t)$ with initial values $y(o)$ and $y'(0)$ is $s^2Y(s)-sy(0)-y'(0)$.
So, taking the Laplace Transform of the RHS of the ODE yields
$$(s^2+1)Y(s)-sy(o)-y'(0)$$
For the RHS, the Laplace Transform is given by
$$\int_0^{1} te^{-st}dt=-\frac{e^{-s}}{s}-\frac{e^{-s}-1}{s^2}$$
Equating sides reveals that
$$(s^2+1)Y(s)-sy(o)-y'(0)=-\frac{e^{-s}}{s}-\frac{e^{-s}-1}{s^2}$$
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The purpose of this workshop is to develop ideas that will further define the problem space, the key problems and the critical questions that need to be answered to make progress toward understanding, developing, and evaluating of Trustworthy Algorithmic Decision-Making.
There will be 4-5 parallel sessions during each time set aside for breakouts on the agenda. Each parallel session will focus on one of the high-level themes that emerge from the note-taking and affinity diagramming during the first half of Day 1. Workshop participants will rotate through the different themes, working on a different theme during each scheduled breakout session. Each breakout session has a specific focus: Brainstorm, Synthesize, “How might we…”, and Problem Statement, such that a small group will be working on each theme during each stage, and then hand off their work to the group working on that theme during the next stage. At each stage, the groups will be randomized so that everyone gets to meet, work with and bounce ideas off of new people.
Each parallel breakout session is 1 hour and 15 minutes long; the last 15 minutes should be spent capturing and documenting for the next group. Don’t forget to do quick introductions first thing during each breakout session!
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The goal of this activity is to build on the idea generation in the previous phase, and identify the big ideas and key concepts related to the overarching theme. The main output of this phase is at least 3-5 “insight statements” about problems that need to be understood better and/or solved, along with text to describe each insight.
The goal of this activity is to expand on the insight statements, and rephrase them as questions that need to be answered. This transforms the thinking about the insights into opportunities for future research and design activities. The main output of this phase is one question per insight statement, along with notes captured from the discussion.
The goal of this activity is to select one problem statement from the candidates produced in the previous session, and further describe it. The main output is a presentation about it that you will deliver to all of the workshop participants after lunch on Day 2.
The presentation should cover the following: | http://trustworthy-algorithms.org/breakouts.html |
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The change in potential energy of the pole vaulter will be equal to her change in kinetic energy.
Assume the ground to be the position of zero potential energy. Then the equation for the conservation of energy is
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---
author:
- Victor Chulaevsky$^1$
title: |
UNIVERSITE DE REIMS CHAMPAGNE-ARDENNE\
Laboratoire de Mathématiques\
UMR 6056 5 truecm A SIMPLE EXTENSION OF STOLLMANN’S LEMMA\
TO CORRELATED POTENTIALS
---
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————————————————————————————————————————
$^1$ [**Permanent address:**]{} Victor Tchoulaevski\
Université de Reims, Laboratoire de Mathématiques\
Moulin de la Housse - B.P. 1039 - 51687 Reims cedex 2\
France E-Mail: [email protected]
1 truecm
[**A SIMPLE EXTENSION OF STOLLMANN’S LEMMA\
TO CORRELATED POTENTIALS** ]{}
1 truecm [**Victor Chulaevsky**]{}\
Département de Mathématiques, UMR CNRS 6056\
Université de Reims, Moulin de la Housse - B.P. 1039\
51687 Reims cedex 2, France\
E-mail: [email protected]
We propose a fairly simple and natural extension of Stollmann’s lemma to correlated random variables, described earlier in [@Ch]. This extension allows (just as the original Stollmann’s lemma does) to obtain Wegner-type estimates even in some problems of spectral analysis of random operators where the Wegner’s lemma is inapplicable (e.g. for multi-particle Hamiltonians).
To the best of author’s knowledge, such an extension seems to be original. However, the author will appreciate any reference to articles or preprints where similar results are proved.
Introduction
============
The regularity problem for the limiting distribution of eigen-values of infinite dimensional self-adjoint operators appears in many problems of mathematical physics. Specifically, consider a lattice Schrödinger operator (LSO, for short) $H:\ell^2(\Z^d)\to \ell^2(\Z^d)$ given by $$(H\psi)(x) = \sum_{y:\, |y-x|=1} \,\, \psi(y) + V(x)\psi(x); \; x,y\in\Z^d.$$ For each finite subset $\Lam\subset \Z^d$, let $E_j^\Lam, j=1,\dots, |\Lam|$, be eigen-values of $H$ with Dirichlet b.c. in $\Lam$. Consider the family of finite sets $\Lam_L = [-L,L]^d \cap \Z^d$ and define the following quantity (which does not necessarily exist for an arbitrary LSO): $$k(E)
= \lim_{L\to\infty} \frac{1}{(2L+1)^d} \card \left\{ j:\,E_j^{\Lam_L}\leq E \right\}.$$ If the above limit exists, $k(E)$ is called the limiting distribution function (LDF) of e.v. of $H$. One can easily construct various examples of the function $V:\Z^d\to \R$ (called potential of the operator $H$) for which the LDF does not exist. One can prove the existence of LDF for periodic potentials $V$, but even in this, relatively simple situation existence of $k(E)$ is not a trivial fact.
However, one can prove existence of $k(E)$ in a large class of [*ergodic random*]{} potentials. Namely, consider an ergodic dynamical system $(\Omega,\cF,\db P, \{T^x, \, x\in\Z^d\})$ with discrete time $\Z^d$ and a mesurable function (sometimes called a hull) $v:\Omega\to\R$. Then we can introduce a family of sample potentials $$V(x,\omega) = v(T^x\omega), \; x\in\Z^d,$$ labeled by $\omega\in\Omega$. Under the assumption of ergodicity of $\{T^x\}$, the quantity $$k(E,\omega)
= \lim_{L\to\infty} \frac{1}{(2L+1)^d} \card \left\{ j:\,E_j^{\Lam_L}(\omega)\leq E \right\}$$ is well-defined $\db P$-a.s. Moreover, $k(E,\omega)$ is $\db P$-a.s. independent of $\omega$, so its value taken for a.e. $\omega$ is natural to take as $k(E)$. In such a context, $k(E)$ is usually called [*integrated density of states*]{} (IDS, for short). It admits an equivalent definition: $$k(E) = \esm{ (f,\Pi_(-\infty,E](H(\omega)f)},$$ where $f\in\ell^2(\Z^d)$ is any vector of unit norm, and $\Pi_(-\infty,E](H(\omega)$ is the spectral projection of $H(\omega)$ on $(-\infty,E]$. The reader can find a detailed discussion of the existence problem of IDS in excellent monographs by Carmona and Lacroix [@CL] and by Pastur and Figotin [@PF1].
It is not difficult to see that $k(E)$ can be considered as the distribution function of a normalized measure, i.e. probability measure, on $\R$. If this measure $dK(E)$, called measure of states, is absolutely continuous with respect to Lebesgue measure $dE$, its density (or Radon–Nikodim derivative) $dK(E)/dE$ is called the density of states (DoS). In physical literature, it is customary to neglect the problem of existence of such density, for if $dK(E)/dE$ is not a function, then “it is simply a generalized function”. However, the real problem is not terminological. The actual, explicit estimates of the probabilities of the form $$\pr{\exists \, \text{ eigen-value } E^{\Lam_L}_j\in(a,a+\eps)}$$ for LSO $H_{\Lam_L}$ in a finite cube $\Lam_L$ of size $L$, for small $\eps$, often depend essentially upon the existence and the regularity properties of the DoS $dk(E)/dE$.
Apparently, the first fairly general result relative to existence and boundedness of the DoS is due to Wegner [@W]. Traditionally referred to as Wegner’s [*lemma*]{}, it certainly deserves to be called [*theorem*]{}.
Assume that $\{V(x,\omega), x\in\Z^d\}$ are i.i.d. r.v. with bounded density $p_V(u)$ of their common probability distribution: $\|p_V\|_\infty = C < \infty$. Then the DoS $dk(E)/dE$ exists and is bounded by the same constant $C$.
The proof can be found, for example, in the monograph [@CL].
This estimate and some of its generalizations have been used in the multi-scale analysis (MSA) developed in the works by Fröhlich and Spencer [@FS], Fröhlich, Spencer, Martinelli ans Scoppola [@FMSS], von Dreifus and Klein [@DK1], [@DK2], Aizenman and Molchanov [@AM], and in a number of more recent works where the so-called Anderson Localization phenomenon has been observed. Namely, it has been proven that all e.f. of random lattice Schrödinger operators decay exponentially at infinity with probability one (for $\db P$-a.e. sample of random potential $V(\omega)$). Von Dreifus and Klein [@DK2] proved an analog of Wegner estimate and used it in their proof of localization for Gaussian and some other correlated (but non-deterministic) potentials. The author of these lines recently proved, in a joint work with Yu. Suhov [@ChS], an analog of Wegner estimate for a system of two or more interacting quantum particles on the lattice under the assumption of analyticity of the probability density $p_V(u)$, using a rigorous path integral formula by Molchanov (see a detailed discussion of this formula in the monograph [@CL]). In order to relax the analyticity assumption in a multi-particle context, V.C. and Yu. Suhov later used ([@ChS2]) a more general and flexible result guaranteeing existence and boundedness of the DoS: the Stollmann’s lemma, which we discuss below.
In the present work, we propose a fairly simple and natural extension of Stollmann’s lemma to correlated, but still non-deterministic random fields generating random potentials. [*To the best of author’s knowledge, such an extension seems to be original, although very simple. However, the author will appreciate any reference to published papers or preprints where the same or similar result was mentioned and proved.*]{} Our main motivation here is to lay out a way to interesting applications to localization problems for multi-particle systems.
Stollmann’s lemma for product measures
======================================
Recall the Stollmann’s lemma and its proof for independent r.v. Let $m\geq 1$ be a positive integer, and $J$ an abstract finite set with $|J| (=\card J) = m$. Consider the Euclidean space $\db R^J \cong \R^m$ with standard basis $(e_1, \dots, e_m)$, and its positive quadrant $$\db R^J_+ = \myset{ q\in\db R^J:\, q_j\geq 0, \,\, j=1, 2, \dots, m }.$$ For any measure $\mu$ on $\db R$, we will denote by $\mu^m$ the product measure $\mu
\times \dots \times \mu$ on $\db R^J$. Furthermore, for any probability measure $\mu$ and for any $\eps>0$, define the following quantity: $$s(\mu, \eps) = \sup_{a\in\db R} \,\, \int_{a}^{a+\eps} d\mu(t)$$ and assume that $s(\mu,\eps)$ is finite. Furthermore, let $\mu^{m-1}$ be the marginal probability distribution induced by $\mu^m$ on $q'=(q_2, \dots, q_m)$.
Let $J$ be a finite set with $\,|J| = m$. Consider a function $\Phi:\, \db R^J\to\db R$ on $\R^J$ which we will identify with $\R^m$. Function $\Phi$ is called $J$-monotonic if it satisfies the following conditions: for any $r\in\db R^m_+$ and any $q\in \db R^m$, $$\label{Monoton1}
\Phi(q+r)\geq \Phi(q);$$ moreover, for $e=e_1 + \dots + e_m\in \db R^m$, for any $q\in\db R^m$ and for any $t>0$ $$\label{Monoton2}
\Phi(q + t \cdot e) - \Phi(q) \geq t.$$
It is convenient to introduce the notion of $J$-monotonic operators considered as quadratic forms. In the following definition, we use the same notations as above.
Let $\cH$ be a Hilbert space. A family of self-adjoint operators $B:\cH \times \R^J \to \cH$ is called $J$-monotonic if, for any vector $f\in\cH$ with $\|f\|=1$, the function $\Phi_f: \R^J \to \R$ defined by $$\Phi_f(q) = (B(q)f, f)$$ is monotonic.
In other words, the quadratic form $Q_{B(q)}(f):= (B(q)f,f)$ as function of $q\in\R^J$ is non-decreasing in any $q_j$, $j=1, \dots, |J|$, and $$(B(q+t\cdot e)f, f) - (B(q)f,f) \geq t\cdot \|f\|^2.$$
By virtue of the variational principle for self-adjoint operators, if an operator family $H(q)$ in a finite-dimensional Hilbert space $\cH$ is $J$-monotonic, then each eigen-value $E_k^{B(q)}$ of $B(q)$ is a $J$-monotonic function.
If $H(q), \, q\in\R^J$, is a $J$-monotonic operator family in Hilbert space $\cH$, and $H_0:\cH\to\cH$ is an arbitrary self-adjoint operator, then the family $H_0 + H(q)$ is also $J$-monotonic.
This explains why the notion of monotonicity is relevant to spectral theory of random operators. Note also, that this property can be easily extended to physically interesting examples where $\cH$ has infinite dimension, but $H(q)$ have, e.g., compact resolvent, as in the case of Schrödinger operators in a finite cube with Dirichlet b.c. and with bounded potential, so the respective spectrum is pure point, and even discrete.
\[Stollmann\] Let $J$ be a finite index set, $|J|=m$, $\mu$ be a probability measure on $\db R$, and $\mu^m$ be the product measure on $\R^J$ with marginal measures $\mu$. If the function $\Phi:\, \db R^J \to \db R$ is $J$-monotonic, then for any open interval $I\subset \db R$ we have $$\mu^m\myset{ q:\, \Phi(q) \in I } \leq m \cdot s(\mu, |I|).$$
Let $I=(a,b)$, $b-a=\eps>0$, and consider the set $$A = \myset{ q:\, \Phi(q) \leq a }.$$ Furthermore, define recursively sets $A^\eps_j$, $j=0, \dots, m$, by setting $$A^\eps_0 = A, \; A^\eps_j = A^\eps_{j-1} + [0,\eps]e_j :=
\, \myset{ q+te_j:\, q\in A^\eps_{j-1}, \,\, t\in[0,\eps] }.$$ Obviously, the sequence of sets $A^\eps_j$, $j=1, 2, ...$, is increasing with $j$. The monotonicity property implies $$\myset{ q:\, \Phi(q) < b } \subset A^\eps_m.$$ Indeed, if $\Phi(q) < b$, then for the vector $q':=q- \eps\cdot e$ we have by (2): $$\Phi(q') \leq \Phi(q' + \eps \cdot e) - \eps
= \Phi(q) - \eps \leq b - \eps \leq a,$$ meaning that $q'\in \myset{ \Phi \leq a} = A$ and, therefore, $$q = q' + \eps\cdot e \in A^\eps_m.$$ Now, we conclude that $$\myset{ q:\, \Phi(q) \in I } = \myset{ q:\, \Phi(q) \in (a,b) }$$ $$= \myset{ q:\, \Phi(q) < b } \setminus \myset{ q:\, \Phi(q) \leq a }
\subset A^\eps_m \setminus A.$$ Furthermore, $$\mu^m\myset{ q:\, \Phi(q) \in I } \leq
\mu^m\left( A^\eps_m \setminus A \right)$$ $$= \mu^m\left( \bigcup_{j=1}^m\left( A^\eps_j \setminus A^\eps_{j-1} \right)
\right)
\leq \sum_{j=1}^m \mu^m\left( A^\eps_j \setminus A^\eps_{j-1} \right).$$ For $q'\in\db R^{m-1}$, set $$I_1(q') = \myset{ q_1\in\db R:\, (q_1,q')\in A^\eps_1\setminus A}.$$ By definition of set $A^\eps_1$, this is an interval of length not bigger than $\eps$. Then we have $$\label{onestep}
\mu^m( A^\eps_1 \setminus A) = \int d\mu^{m-1}(q') \, \int_{I_1} d\mu(q_1)
\leq s(\mu, \eps).$$ Similarly, we obtain for $j=2, \dots, m$ $$\mu^m( A^\eps_j \setminus A^\eps_{j-1}) \leq s(\mu,\eps),$$ yielding $$\mu^m\myset{ q:\, \Phi(q) \in I } \leq
\sum_{j=1}^m \mu^m( A^\eps_j \setminus A^\eps_{j-1})
\leq m \cdot c(\mu, \eps). \;\;\;\mathQED$$
Now, taking into account the above Remark 1, Stollmann’s theorem yields immediately the following estimate.
Let $H_\Lam$ be an LSO with random potential $V(x;\om)$ in a finite box $\Lam\subset\Z^d$ with Dirichlet b.c., and $\Sigma(H_\Lam)$ its spectrum, i.e. the collection of its eigen-values $E^{(\Lam)}_j$, $j=1, \dots, |\Lam|$. Assume that r.v. $V(x;\cdot)$ are i.i.d. with marginal distribution function $F_V$ satisfying $$s(\eps) = \sup_{a\in \R} \,\, (F_V(a+\eps) - F_V(a) ) < \infty.$$ Then $$\pr{ \dist(\Sigma(H_\Lam(\om), E) \leq \eps } \leq |\Lam|^2 s(\eps).$$
Extension to multi-particle systems
===================================
Results of this section have been obtained by the author and Y. Suhov [@ChS].
Let $N> 1$ and $d\geq 1$ be two positive integers and consider a random LSO $H=H(\om)$ which can be used, in the framework of tight-binding approximation, the as the Hamiltonian of a system of $N$ quantum particles in $\Z^d$ with random external potential $V$ and interaction potential $U$. Specifically, let $x_1, \ldots, x_N\in\Z^d$ be positions of quantum particles in the lattice $\Z^d$, and $\ux = (x_1, \ldots, x_N)$. Let $\{V(x;\om), \, x\in\Z^d\}$ be a random field on $\Z^d$ describing the external potential acting on all particles, and $U:\, (x_1,\ldots, x_N) \mapsto \R$ be the interaction energy of the particles. In physics, $U$ is usually to be symmetric function of its $N$ arguments $x_1, \ldots, x_N\in\Z^d$. We will assume in this section that the system in question obeys either Fermi or Bose quantum statistics, so it is convenient to assume $U$ to be symmetric. Note, however, that the results of this section can be extended, with natural modifications, to more general interactions $U$. Further, in [@ChS] $U$ is assumed to be finite-range interaction: $$\supp \, U \subset \{\ux:\, \max (|x_j-x_k|\leq r)\}, \; r<\infty.$$ Such an assumption is required in the proof of Anderson localization for multi-particle systems, however, it is irrelevant to the Wegner–Stollmann estimate we are going to discuss below.
Now, let $H$ be as follows: $$(H(\om) f)(\ux) = \sum_{j=1}^N \, \left( \Delta^{(j)} + V(x_j;\om)\right)
+ U(\ux),$$ where $\Delta^{(j)}$ is the lattice Laplacian acting on the $j$-th particle, i.e. $$\Delta^{(j)} = \truc{\one}{}{1} \otimes \ldots \otimes \truc{\Delta}{}{j} \otimes
\ldots \otimes \truc{\one}{}{N}$$ acting in Hilbert space $ \ell^2(\Z^{Nd})$. For any finite “box” $$\Lam = \Lam^{(1)} \times \ldots \times \Lam^{(N)} \subset \Z^{Nd}$$ one can consider the restriction, $H_\Lam(\om)$, of $H(\om)$ on $\Lam$ with Dirichlet b.c. It is easy to see that the potential $$W(\ux) = \sum_{j=1}^N \, V(x_j;\om) + U(\ux)$$ is no longer an i.i.d. random field on $\Z^{Nd}$, even if $V$ is i.i.d. Therefore, neither Wegner’s nor Stollmann’s [*estimate*]{} does not apply [*directly*]{}. But, in fact, Stollmann’s lemma [*does*]{} apply to multi-particle systems, virtually in the same way as to single-particle ones.
Assume that r.v. $V(x;\cdot)$ are i.i.d. with marginal distribution function $F_V$ satisfying $$s(\eps) = \sup_{a\in \R} \,\, (F_V(a+\eps) - F_V(a) ) < \infty.$$ Then $$\pr{ \dist(\Sigma(H_\Lam(\om), E) \leq \eps } \leq |\Lam| \cdot M(\Lam)\cdot s(\eps),$$ with $$M(\Lam) = \sum_{j=1}^N \, \card\, \Lam^{(j)} .$$
Fix $\Lam$ and consider the union of all lattice points in $\Z^d$ which belong to the single-particle projections $\Lam^{(j)}$, $j=1, \dots, N$: $$\cX(\Lam) = \bigcup_{j=1}^N \, \Lam^{(j)} \subset \Z^d.$$ Now we can apply Stollmann’s lemma to $H_\Lam$ by taking the index set $J = \cX(\Lam)$ and auxiliary probability space $\R^J$. Indeed, the random potential $\hat V(\ux;\om) := V(x_1;\om) + \dots + V(x_N;\om)$ can be re-written as follows: $$V(x_1;\om) + \dots + V(x_N;\om)
= \sum_{y\in \cX(\Lam)} c(\ux, y) V(y;\om)$$ with integer coefficients $c(\ux,y)$ such that
$$\label{multi}
c(\ux, y) \geq 0, \; \sum_{y\in \cX(\Lam)} c(\ux, y) = N.$$
For example, if $N=2$, one can have either $V(x_1,\om) + V(x_2;\om)$ with $x_1\neq x_2$, in which case we have $$c(\ux, y) =
\begin{cases}
1, & \text{ if } y=x_1 \text{ or } y=x_2 \\
0, & otherwise
\end{cases}$$ or $V(x_1;\om) + V(x_1;\om)=2V(x_1;\om)$ for “diagonal” points $(x_1,x_2)$, where $$c(\ux, y) =
\begin{cases}
2, & \text{ if } y=x_1 \\
0, & otherwise
\end{cases}$$ In any case, as shows (\[multi\]), random potential at $\ux\in\Lam$ is a linear function of one or more coordinates in the auxiliary space $\R^J$ growing at rate $\geq Nt\geq t$ along the principal diagonal $\{q_1 = q_2 = \dots = q_{|J|} = t\in\R\}$. Hence, the operators of multiplication by $\hat V(\ux;\om)$ form a $J$-monotonic family, and, by virtue of Remark 2, the same holds for $H = H_0 + U + \hat V(\om)$, just as in the single-particle case (and even “better”, for $N>1$ !). By Theorem 2, this implies immediately the estimate $$\pr{ \dist(\Sigma(H_\Lam(\om), E) \leq \eps } \leq |\Lam|^2 s(\mu,\eps). \qquad \mathQED$$
It is not difficult to see that the same argument, with obvious notational modifications, applies to Fermi and Bose lattice quantum systems, i.e. to restrictions of $H$ to the subspaces of symmetric (Bose case) or anti-symmetric (Fermi case) functions of $N$ arguments $x_1, \dots, x_N$ on $(\Z^d)^{N}$.
Extension to correlated random variables
========================================
Now let $\mu^m$ be a measure on $\db R^m$ with marginal distributions of order $m-1$, $$\mu^{m-1}_j(q'_{\neq j}) = \mu^{m-1}_j(q_1, \dots, q_{j-1}, q_{j+1}, \dots, q_m),
\; j=1, \dots, m,$$ and conditional distributions $\mu^{1}_j(q_j \cnd q'_{\neq j})$ on $q_j$ given all $q_k, k\neq j$. For every $\eps>0$, define the following quantity: $$C_1(\mu^m, \eps) = \max_{j} \,\sup_{a\in\db R} \,\,
\int d\mu^{m-1}(q'_{\neq j}) \, \int_{a}^{a+\eps} d\mu(q_1|q'_{\neq j})$$ and assume that $C_1(\mu,\eps)$ is finite: $$\label{CondA}
\max_{j} \,\sup_{a\in\db R} \,\,
\int d\mu^{m-1}(q'_{\neq j}) \, \int_{a}^{a+\eps} d\mu(q_1|q'_{\neq j}) < \infty.$$
As a simple sufficient condition of finiteness of $C_1(\mu,\eps)$, one can use, e.g., a uniform continuity (but not necessarily [**absolute continuity**]{} !) of the single-point conditional distributions, $$\max_{j}\,\sup_{ q'_{\neq j} } \, \sup_{a\in\db R} \,\,
\int_{a}^{a+\eps} d\mu(q_j|q'_{\neq j}) \leq C_2(\mu^m,\eps) < \infty$$ or even the existence and uniform boundedness of the [**density** ]{} $p(q_j|q'_{\neq j})$ of these conditional distributions: $$\sup_{q_j\in\db R} p(q_j|q'_{\neq j}) \leq C_3(\mu^m,\eps).$$
In applications to localization problems, the aforementioned continuity moduli $C_1(\mu^m,\eps)$, $ C_2(\mu^m,\eps)$, $C_3(\mu^m,\eps)$ need to decay not too slowly as $\eps\to 0$. A power decay of order $O(\eps^\beta)$ with $\beta>0$ is certainly sufficient, but it can be essentially relaxed. For example, it suffices to have an upper bound of the form $$C_1\left(\mu^m, e^{-L^{\beta}}\right) \leq Const \cdot L^{-B},$$ uniformly for all sufficiently large $L>0$ with some (arbitrarily small) $\beta>0$ and with $B>0$ which should sufficiently big, depending on the specific spectral problem.
Using notations of the previous section, one can formulate the following generalization of Stollmann’s lemma.
\[ExtStollmann\] Let $\Phi:\, \db R^J \to \db R$, $\R^J \cong \R^m$, be a $J$-monotonic function and $\mu^m$ a probability measure on $\R^m\cong \R^J$ with $C_1(\mu^m,\eps)<\infty$. Then for any interval $I\subset \db R$ of length $|I|=\eps>0$, we have $$\mu^m\myset{ q:\, \Phi(q) \in I } \leq m \cdot C_1(\mu, \eps).$$
We proceed as in the proof of Stollmann’s lemma and introduce in $\db R^m$ the sets $A = \myset{ q:\, \Phi(q) \leq a }$ and $A^\eps_j$, $j=0, \dots, m$. Here, again, we have $$\myset{ q:\, \Phi(q) \in I } = \subset A^\eps_m \setminus A$$ and $$\mu^m\myset{ q:\, \Phi(q) \in I }
\leq \sum_{j=1}^m \mu^m\left( A^\eps_j \setminus A^\eps_{j-1} \right).$$ For $q'_{\neq 1}\in\db R^{m-1}$, we set $$I_1(q'_{\neq 1}) = \myset{ q_1\in\db R:\, (q_1,q'_{\neq 1})\in A^\eps_1\setminus A}.$$ Furthermore, we come to the following upper bound which generalizes (\[onestep\]): $$\label{onestepgen}
\mu^m( A^\eps_1 \setminus A) = \int d\mu^{m-1}(q') \, \int_{I_1} d\mu(q_1|q')
\leq C_1(\mu, \eps).$$ Similarly, we obtain for $j=2, \dots, m$ $$\mu^m( A^\eps_j \setminus A^\eps_{j-1}) \leq C_1(\mu,\eps),$$ yielding $$\mu^m\myset{ q:\, \Phi(q) \in I } \leq
\sum_{j=1}^m \mu^m( A^\eps_j \setminus A^\eps_{j-1})
\leq m \cdot C_1(\mu, \eps). \;\;\;\mathQED$$
Application to Gaussian random fields
=====================================
Let $V(x,\omega), x\in\Z^d$, $d\geq 1$, be a regular stationary Gaussian field of zero mean on the lattice $\Z^d$. The regularity implies that the field $V(\cdot, \omega)$ is non-deterministic, i.e. the conditional probability distribution of $V(0,\cdot)$ given $\{V(y), y\neq 0\}$ is Gaussian with strictly positive variance. In other terms, the r.v. $V(0,\cdot)$, considered as a vector in the Hilbert space $\cH_{V,\Z^d}$ generated by linear combinations of all $V(x,\cdot)$, $x\in\Z^d$, with the scalar product $$(\xi,\eta) = \esm{\xi\, \eta},$$ does not belong to the subspace $\cH_{V,\Z^d \setminus \{0\}}$: $$\| V(0,\cdot) - \Pi_{\cH_{V,\Z^d \setminus \{0\}}} V(0,\cdot) \|^2
= \tilde \sigma_0^2 >0,$$ where $$\Pi_{\cH_{V,\Z^d}} \xi
= \esm{\xi \, \Big|\, V(x,\cdot), x\in\Z^d \setminus \{0\}}.$$ Furthermore, for any subset $\Lam \subseteq \Z^d \setminus \{0\}$, $$\| V(0,\cdot) - \Pi_{\cH_{V, \Lam}} V(0,\cdot) \|^2 \geq \tilde \sigma_0^2,$$ since $$\cH_{V,\Lam} \subset \cH_{V,\Z^d \setminus \{0\}}.$$ Therefore, the conditional variance of $V(0,\cdot)$ given any non-zero number of values of $V$ outside $x=0$ is bounded from below by $\tilde \sigma_0^2$. Respectively, the conditional probability density of $V(0,\cdot)$, for any such nontrivial condition is uniformly bounded by $(2\pi\tilde \sigma_0^2)^{-1/2} <\infty$. Now a direct application of Lemma \[ExtStollmann\] leads to the following statement.
Let $\Lam \subset \Z^d$ be a finite subset of the lattice, and $\Lam' \subset \Z^d \setminus \Lam$ any subset disjoint with $\Lam$ ($ \Lam'$ may be empty). Consider a family of LSO $H_{\Lam}(\om)$ with Gaussian random potential $V(\om)$ in $\Lam$, with Dirichlet b.c. on $\pt \Lam$. Then for any interval $I\subset \db R$ of length $\eps>0$, we have $$\pr{ \Sigma( H_{\Lam}) \cap I \neq \emptyset\,|\, V(y,\cdot), y \in \Lam'}
\leq C(V) \, |\Lam|^2 \, \eps,$$ where the constant $C(V)<\infty$ whenever the Gaussian field $V$ is non-deterministic.
Application to Gibbs fields with continuous spin
================================================
Apart from Gaussian fields, there exist several classes of random lattice fields for which the hypothesis of Lemma \[ExtStollmann\] can be easily verified. For example, conditional distributions of Gibbs fields are given explicitly in terms of their respective interaction potentials. Specifically, consider a lattice Gibbs field $s(x,\omega)$ with bounded continuous spin, $$s: \Omega \times \Z^d \to \cS = [a,b] \subset \db R$$ generated by a short-range, bounded, two-body interaction potential $u(\cdot,\cdot)$. The spin space is assumed to be equipped with the Lebesgue measure $ds$. In other words, consider the formal Hamiltonian $$H(s) = \sum_{x\in \Z^d} h(x)
+ \sum_{x\in \Z^d} \sum_{|y-x|\leq R} u_{|x-y|}(s(x), u(y)),$$ where $h:\cS\to \db R$ is the self-energy of a given spin. The interaction potentials $u_{|x-y|}(s(x),s(y))$ vanish for $|x-y|>R$ and are uniformly bounded: $$\max_{l\leq R}\sup_{s,t\in\cS} |u_{l}(s,t)| <\infty.$$ Then for any lattice point $x$ and any configuration $s' = s'_{\neq x}$ of spins outside $\{x\}$, the [*single-site*]{} conditional distribution of $s(x)$ given the external configuration $s'$ admits a [*bounded density*]{} $$p(s_x \,|\,s'_{\neq x})
= \frac{ e^{-\beta U(s_x | s'})}{ \Xi(\beta, s') }
= \frac{ e^{-\beta U(s_x | s'})}{ \int_{\cS} e^{-\beta U(t | s'}) \, dt}$$ with $$U(s_x | s') := \sum_{y:\, |y-x|\leq R} u_{|x-y|}(s_x,s'_y)$$ satisfying the upper bound $$|U(s_x | s')| \leq (2R+1)^d \sup_{s,t\in \cS} |u_{l}(s,t)| <\infty.$$ A similar property is valid for sufficiently rapidly decaying long-range interaction potentials, for example, under the condition $$\label{GibbsSummable}
\sup_{s,t\in\cS} \, |u_{|y|}(s,t)| \leq \frac{Const }{ |y|^{d+1+\delta}}, \; \delta>0.$$ as well as for more general, but still uniformly summable many-body interactions. Here is one possible Wegner–Stollmann-type result concerning such random potentials.
Let $\Lam \subset \Z^d$ be a finite subset of the lattice, $\Lam' \subset \Z^d \setminus \Lam$ any subset disjoint with $\Lam$ ($ \Lam'$ may be empty), and let $s(x,\omega)$ be a Gibbs field in $\Lam$ with continuous spins $s\in\cS=[a,b]$ generated by a two-body interaction potential $u_l(s,t)$ satisfying condition (\[GibbsSummable\]), with any b.c. on $\Z^d \setminus \Lam$. Consider a LSO $H_{\Lam}$ with random potential $V(x,\omega)= s(x,\omega)$. Then for any interval $I\subset \db R$ of length $\eps>0$, we have $$\pr{ \Sigma( H_{\Lam}) \cap I \neq \emptyset\,|\, V(y,\cdot), y \in \Lam'}
\leq C(V) \, |\Lam|^2 \, \eps,\; C(V)<\infty.$$
In the case of unbounded spins and/or interaction potentials, the uniform boundedness of conditional single-spin distributions does not necessarily hold, since the energy of interaction of a given spin $s(0)$ with the external configuration $s'$ may be arbitrarily large (depending on a particular form of interaction) and even [*infinite*]{}, if $s'(y) \to \infty$ too fast. In such situations, our general condition (\[CondA\]) may still apply, provided that rapidly growing configurations $s'$ have sufficiently small probability, so that the outer integral in the r.h.s. of (\[CondA\]) converges.
Conclusion
==========
Wegner–Stollmann-type estimate of the density of states in finite volumes is a key ingredient of the MSA of spectra of random Schrödinger (and some other) operators. The proposed simple extension of Stollmann’s lemma shows that a very general assumption on correlated random fields generating potential rules out an abnormal accumulation of eigen-values in finite volumes. This extension applies also to multi-particle systems [@ChS2].
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| |
POTTSVILLE — Parents dropping off their children Monday for the first day of school at John S. Clarke Elementary Center said they were happy the students were returning, but they have worries.
“We’re hopeful we can make it through the school year,” said Angela Bowers, of Pottsville, her 5-year old daughter, Aspen Bowers, a kindergartner, standing beside her. Both were wearing face masks.
Elementary center students, and those at the two other schools comprising the Pottsville Area School District, returned to class Monday, arriving by bus or family vehicle. Acting Elementary Center Principal Michael Maley and Acting Superintendent Jared Gerace greeted them.
“We’re really excited to have the kids back,” Maley said, adding that it’s been 170 days since students were in the building.
Along with schools statewide, John S. Clarke was ordered closed March 13 to curb the spread of the coronavirus. The school to which the students returned has been prepared for social distancing. Small red squares with K, 1, 2, 3, 4 line the front hallway and entrance on 16th Street, red dots with white footprints on them; and one-way green arrows are pasted on hallway floors.
In classrooms, desks are spaced out, while plexiglass shields stand in front of teachers’ desks and at the main office. Special subjects such as library, art, music and gym will be done in classrooms. However, Maley said, while the weather is warm, nearby playgrounds and fields will be used for recess and classes, along with the playground area and other outdoor areas at the school. Students won’t be allowed to use the playground equipment, he said.
At the elementary school, students can eat in the cafeteria, but must be served meals by staff, and sit on table benches adorned with spray-painted Crimson Tide logos. Additional seating is also available on the cafeteria stage.
Districtwide, face masks are required inside school buildings and, while busing is provided, parents are urged to transport their students to school.
The district opened Monday under a blended format, where students are divided into groups, Learning Group A and Learning Group B, and come into the buildings alternately for in-person instruction two days a week and participating in online instruction three days a week. All students take part in virtual learning Wednesdays to allow for the school buildings to be cleaned.
Parent worries
Chad Harig, of Pottsville, said the start of school is exciting, but as a single father of three, he wishes the school were open full time. His eldest child, Sophia Herb, 6, is in first grade at the school, while his son, Robert Harig, 4, is in pre-kindergarten at Pottsville Child Development Center and his youngest son, Thomas Harig, 2, hasn’t started school.
As for Bowers, she said she is worried about how she and her husband, Dave Bowers, will handle the upcoming school year while working opposite shifts. Along with her daughter, she has a son, Davin Bowers, who started seventh grade at D.H.H. Lengel Middle School on Monday.
Maley said elementary students choosing to learn virtually have their first day Wednesday and will complete introductory assignments, while students in the two learning groups who are not in school are also completing introductory assignments virtually.Gerace said as of Monday, 2,404 students are enrolled, of which approximately 87% are doing blended learning and 12% are doing all virtual instruction. At the same time last year, there were 2,513 students in the district.
He said he thinks some of the decline is due to students enrolling in other districts and families moving away. A few students chose cyber-charter and charter schools for the school year, Gerace added. | http://www.republicanherald.com/news/education/mixed-thoughts-from-parents-as-pottsville-area-elementary-students-start-school/article_323bdb59-81c5-5658-88c8-307b32689e0d.html |
Littlehampton Brick Co is one of Adelaide's most acclaimed manufacturers of quality clay house bricks, fire bricks, ivory bricks, textured cream & old red bricks and pavers.
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There are currently no reviews of Littlehampton Clay Bricks and Pavers, so be the first to write one! | https://www.localbusinessguide.com.au/business/littlehampton-clay-bricks-and-pavers/ |
Five men with handguns rob Kernersville store
KERNERSVILLE, N.C. — Police say a convenience store was robbed by five men who were each armed with a handgun early Tuesday in Kernersville.
Police said each of the five suspects were individually armed with a handgun when they entered the Quality Mart convenience store at 1580 N.C. 66 South around 12:30 a.m.
The suspects demanded money from the clerks and ran away from the store after receiving an undisclosed amount of cash.
Police said a K-9 led them south to an adjoining apartment complex, where they believe the suspects had a waiting vehicle.
One suspect was described as speaking with a Spanish accent. At the time of the robbery, he was wearing a yellow sweatshirt with a toboggan and ski mask.
The other suspects were described as wearing gloves, dark hooded sweatshirts, and white “homemade” hoods or face masks.
No other descriptions were available.
Anyone with information about the incident is asked to call Kernersville Police. | https://myfox8.com/2012/04/24/five-men-with-handguns-rob-kernersville-store/ |
Accounting:Activity based costin and High-low method.
See attached file for charts and multiple answer choices.
1. The controller of JoyCo has requested a quick estimate of the manufacturing supplies needed for the month of July when production is expected to be 470,000 units. Below are actual data from the prior three months of operations.
Production in units Manufacturing supplies
March 450,000 $723,060
April 540,000 $853,560
May 480,000 $766,560
Using these data and the high-low method, what is the best estimate of the cost of manufacturing supplies that would be needed for July? (Assume that this activity is within the relevant range.)
2. The following is Addison Corporation's contribution format income statement for last month:
What is the company's margin of safety in dollars? The company has no beginning or ending inventories. A total of 20,000 units were produced and sold last month.
Sales $1,000,000
Variable expenses $ 700,000
Contribution margin $300,000
Fixed Expenses $180,000
Net operating income $120,000
3. Dilloo Company uses an activity-based costing system with three activity cost pools. The company has provided the following data concerning its costs and its activity based costing system:
The "Other" activity cost pool consists of the costs of idle capacity and organization-sustaining costs.
You have been asked to complete the first-stage allocation of costs to the activity cost pools.
How much cost, in total, should NOT be allocated to orders and products in the second stage of the allocation process if the activity-based costing system is used for internal decision-making?
4. The contribution margin ratio of Lukasiewicz Corporation's only product is 62%. The company's monthly fixed expense is $297,600 and the company's monthly target profit is $37,200.
Required:
Determine the dollar sales to attain the company's target profit.
Solution Summary
The problem deals with accounting topics: High-low method of cost estimation and Activity based costing. | https://brainmass.com/business/financial-accounting-bookkeeping/accounting-activity-based-costin-and-high-low-method-342613 |
Welcome Class of 2023!
September 6, 2019
Pictured: Incoming undergraduate students at the New Student Welcome on the Long Island campus.
How will you create your opportunity at New York Institute of Technology? This was the question university leaders asked the Class of 2023 during New Student Welcome on September 4.
The day’s events were part of Week of Welcome, a weeklong celebration to cheer on new and returning students, and encourage them to take advantage of all that the university has to offer.
New students participated in a number of activities on both campuses, including one where they created their own New York Tech bear. Two students at the New Student Welcome in New York City show off their creations.
Optimism echoed during welcome sessions in New York City and Long Island, where the students of the Class of 2023 were encouraged to leave their comfort zones in pursuit of their passions. Tiffani Blake, M.D., M.Ed., interim dean of students, led the class in energetic chants of “it’s time” and “we can do this,” as they prepared to begin the next chapter of their academic careers. New York Institute of Technology President Hank Foley, Ph.D., empowered students to develop a growth mindset and stressed the importance of positive thinking with the word ‘yet.’ He emphasized that while students may not be good at something yet, there is always the opportunity to improve. Before calling on students to share their goals for the year, Provost and Vice President for Academic Affairs Junius Gonzales, M.D., M.B.A., imparted advice from members of last year’s incoming class, who are now sophomores. Their messages underscored that a sense of belonging comes directly from a willingness to engage in one’s community, which can spark growth both in and out of the classroom.
Undergraduate Class of 2023 Fast Facts:
- The major with the most students: B.S. Life Sciences
- Most common country for non-U.S. students: China
- Age of the oldest student: 49
- Age of the youngest student: 16
The momentum continued throughout the week with a number of events including a campus scavenger hunt in Long Island and Student Involvement Fair in New York City, as well as barbeques, luncheons, and social events on both campuses.
The festivities will last through the end of the month with Homecoming and Family Weekend on September 20 through 22. | https://www.nyit.edu/box/features/welcome_class_of_2023 |
Protein contains nine essential amino acids necessary for building and repairing body tissue, and is a major component of enzymes, hormones, and antibodies. Protein foods include egg whites (the highest quality protein), chicken or turkey (white meat is leanest), fish, cheeses (cottage cheese is the best), milk and yogurt
(non-fat is best), legumes (combined with a grain to have all nine amino acids), and nuts. Cheeses (except cottage cheese) and nuts should be consumed in moderation due to their high calories. Protein recommendations are 10-35% of your caloric intake, or approximately 0.8-1.0 gram of protein per kilogram of body weight per day. Protein sources contain 4 calories per gram.
Formula to calculate your protein requirement:
Weight in pounds:_____divided by 2.2 =________weight in kilograms
Weight in kilograms:_____X 0.8 (or 1.0) =_______protein requirements per day
We recommend eating a protein source with each meal to be sure you receive your daily allowance. A serving size of meat/poultry/fish is about the size of your palm and the thickness of a deck of cards. Serving sizes for milk, cottage cheese, yogurts are 1/2 cup. Again, non-fat or low fat are recommended. Cheeses and nuts should be avoided or used sparingly if weight loss is your goal. A serving size of cheese is one ounce, about the size of the tip of your thumb. Two tablespoons of peanut butter is 190 calories, and 1 oz. (about 1/2 cup) of almonds or peanuts is 170 calories. | https://myweightloss.website/2018/08/16/protein/ |
Q:
Unty Canvas not representative of actual layout
I have a canvas on the screen which for some reason is very small compared to my scene. I have a major issue placing UI components in the correct place on the canvas so that they appear in the correct place on the actual game screen.
If you take a look at the 2 attached screen grabs it may make more sense.
the issue I have is that when I run this on my 1080p tv the text does not get positioned correctly at all. if I place it as in the first image then it ends up 8 squares from the top and 15 squares from the right, if I place it as in the second image then it appears 12 squares from the top and 21 from the right.....
I just want the text to be near the top corner.
How do I get the edit screen and actual game to match positioning?
A:
Ok, along with some pointers from Catwood above I worked out the issue. I had not set the canvas to stretch with screen!
So it was placing the text correctly it was just that the canvas was staying a fixed size rather than filling the screen.
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Linear relationships are examined in practically every discipline of the natural and social sciences. Using the statistics functions of the TI-83+/TI-84+ it is relatively easy to investigate this relationship by performing a linear regression on two variables, such as the following list:
|X||1||3||5||7||9|
|Y||10||31||55||73||94|
You can also find this tutorial for the TI-89/TI-92 Plus/Voyage 200 here.
Suppose X represents the number of workers in a certain factory and Y represents the number of widgets they are able to produce in an hour. Obviously, the more workers this factory has, the more widgets they will be able to produce. We can use a linear regression to determine the exact nature of this relationship.
From the home screen, press, , and then . This sets up the calculator’s built-in list editor.
Next, press
Now enter the X data into L1 and Y data into L2 by using the arrow buttons to select a cell, then pressing , typing in the corresponding number, and pressing again to confirm. The lists should automatically scale as you add more data.
When done, press , , to select LinReg(ax+b).
Press to confirm. The calculator will display your regression equation.
This display means that our regression equation is Y = 10.5X+.1. Using this equation, we can say that we would expect X=4 workers to produce around Y=44 widgets, even though we have no actual data collected for X=4.
If your calculator does not already, you can set it to display some correlation coefficients by pressing 2nd 0 to get to the catalog screen, then, since alpha-lock is automatically on, press
Press to paste it and again to confirm. Now re-run the linear regression and we get two more statistics:
Little r is the coefficient of correlation, which tells how closely the data is correlated to the line. r² is the coefficient of determination, and represents the percentage of variation in data that is explained by the linear regression. These numbers are extremely common in elementary statistics.
Every time your calculator runs a regression, it stores the most recent regression equation in the variable RegEq. To access this variable, press. This is extremely helpful when you want to graph your regression line, for example when comparing to a plot of the original data. | https://www.calcblog.com/performing-a-linear-regression-on-the-ti-83-or-ti-84/ |
Ruffles are an add-on technique that can be built onto any of the base techniques. You will need to know the Basic Frame technique before beginning this tutorial.
In French Beading, loops of beads can be sewn around the edges of a petal to make Ruffled edges.
I’m going to show you two ways to make ruffles: the traditional method, and a new method I developed which I have suitably named “ruffle-as-you-go”. Both methods work just as well, so pick whichever way is easiest for you to work with.
Traditional Ruffle Method
For this exercise, use 24 gauge (.5 mm) wire for the petal, 28 gauge (.315 mm) wire for the ruffles, with <5 grams of size 11/0 seed beads. Usually I choose 28 gauge wire that matches either the ruffle or the petal beads. I will be using different colors in this tutorial so you can make sense of the pictures a little easier.
The traditional way to make French Beaded ruffles involves cutting a long length of thin wire (usually 28 gauge) and sewing ruffles around the edges of a petal after you’ve wrapped and completed all the regular rows in a petal.
A pattern that uses ruffles will look something like this:
Make 1: 9 row BF, 10 bead BR, RB RT.
- Reduce to two bottom wires.
- Add 7-bead ruffles around the petal, skipping 4 beads between.
- Use 24 gauge (.5 mm) wire to construct a Basic Frame using 10 beads for the Basic Row, and 9 rows total. The shape is Round Bottom, Round Top. After completing the rows, tie off the working wire and remove it from the petal.
- Cut a length of 28 gauge wire approximately 1 1/2″ ft long (~ 45 cm). Attach this wire to the petal by wrapping it around the Bottom Wire.
- Move the 28 gauge ruffle wire into the starting position by inserting the end into the back of the petal, between the two outer rows of beads. (Photo 1)
- Pull the wire all the way through and secure it between the bottom wire and the first bead on the outer row. (Photo 2)
5. Add 7 beads to the ruffle wire. (Photo 3)
6. Count 4 beads on the outer row and insert the end of the ruffle wire into the back of the petal between the two outer rows. (Photo 4)
7. Pull the wire all the way through, securing it between the 4th and 5th beads. (Photo 5)
8. Add 7 more beads to the ruffle wire. Count 4 more beads along the outer row and secure the second ruffle by looping the wire around. (Photo 6)
9. Continue adding more 7-bead ruffles along the outer edge of the petal, skipping 4 beads between them. Every two to three ruffles, wrap the wire around twice for extra security.
NOTE: As you go along, you may notice that it gets harder and harder to get the wire between beads. Or you may notice that your outer row on the petal is starting to bulge outward. This happens because there’s only so much extra space between beads for the wire to take up. To make more space for more wire wraps, we need to crush a bead. Use some type of flat-nosed pliers to carefully pinch one of the beads in the outer row of the petal (Photo 7). Please do this carefully so bits of glass don’t fly everywhere, and so you don’t accidentally nick the wire. This will remove the bead and give more space in the row between beads. Only do this if you absolutely have to. Another way is to plan ahead while making the petal and simply reduce the number of beads in that row, leaving bare wire space in the row. This method is best for large leaves and petals, when you know for certain that you will need extra space. (Example shown in Photo 8.) You may or may not need extra space in smaller petals – like the one from this sample pattern – so I usually use the first method with them.
10. A few ruffles away from the end, count the number of beads you have left. You may not have a number that is divisible by 4. To fix that issue, simply adjust the bead count between ruffles. For example: I had 10 beads left in my row, so I skipped 5 beads between instead of 4. (Photo 9) Try to keep the adjusted number near the original count so the difference in ruffle size won’t be very obvious.
11. After securing the last ruffle between the last bead and the bottom wire, tie off the ruffle wire by wrapping it around the petal stem wire. The finished ruffled petal is shown in Photo 10.
There is a large variety of of ruffles that can be made this way simply by altering the bead count between ruffles, and the number of beads per ruffle.
Photo 11 shows a petal with 7-bead ruffles spaced every 5 beads.
Photo 12 shows a petal that begins with short 5-bead ruffles spaced every 2 beads that grow incrementally up to 11-bead ruffles as you get to the top of the petal, then back down to 5-bead ruffles at the bottom.
Sometimes you may need to start the ruffles somewhere other than the bottom of the petal.
In this case, you will lace the ruffle tail wire in a few rows at the starting position to secure the end of the wire (Photo 13), then make the ruffles and lace the ending tail wire in a couple rows to secure the other end (Photo 14).
Ruffle-as-you-go
If you completed my Beginner Course, you probably remember Lace-as-you-go. Now I’m going to show you Ruffle-as-you-go. As the name implies, you’ll be making the ruffles along the outer row before wrapping the row onto the frame wires. With this method you will not have to cut any long lengths of wire. You will also not have to leave off beads while making rows in order to leave room for wire wraps between them, or crush beads if there isn’t enough space. But it is a little awkward, especially the first few times you do it.
We’ll work with a new sample pattern, using the same materials as before.
Make 1: 9 row Basic Frame, 6 bead Basic Row, Round Bottom – Pointed Top.
- Add 5-bead ruffles every 2 beads along the outer row.
- Reduce to two bottom wires.
- On 24 gauge wire, construct a Basic Frame petal using 6 beads for the Basic Row. Wrap rows 2-7 with a round bottom and round top.
- Estimate the beads needed for the last two rows by looping a beaded wire around the petal (Photo 15). After the beads and wire for the last two rows, measure around 4 inches (10 cm) of extra working wire, then cut from the spool. Make a small loop in the end of the working wire to prevent the beads from sliding off.
3. String the ruffle beads onto the spool of 28 gauge wire. For this exercise you’ll need around 5 1/2 inches (14 cm) of bead strung. Leave the wire attached to the spool.
4. Wrap the starting tail of the ruffle wire around the petal stem wire, then move the ruffle wire into the starting position by wrapping it once around the petal working wire in front of the beads. (Photo 16)
5. Count out 2 beads on the petal working wire, and 5 beads on the ruffle wire. Slide the rest of the beads further down their wires and out of the way. Loop the ruffle wire once around the working wire just after the 2 spacer beads. Move 2 more beads down the working wire, and 5 more beads down the ruffle wire. Make sure you are looping tightly. (Photo 17)
6. If spaces form between beads, just push them together as you move along making more ruffles. (Photo 18)
7. Continue making 5-bead ruffles, remembering to wrap twice every two or three ruffles for extra security. (Photo 19)
8. Measure the row carefully, making sure you’ve pushed all the beads and ruffles as close together as you can. Then wrap just the petal working wire around the top wire. (Photo 20)
9. Continue making ruffles around the last row of beads. As you get close to the end of the row, check carefully to see if you’ll need to alter the bead count. For mine I needed to have 3 beads between and 6 beads in the ruffle to make a ruffle of the same height that also reached the bottom wire.
10. After the last ruffle, wrap both the petal working wire and the ruffle wire around the bottom wire then clip and remove them from the petal. Clip and fold the top wire. The finished petal is shown in Photo 21.
© 2018 Lauren Harpster / Bead & Blossom. The images and written instructions are copyright protected. This tutorial may not be printed and distributed for personal gain or teaching classes, but feel free to print a copy for personal use. The images may not be uploaded onto other websites. If you would like to share the tutorial, please do so with a link! | https://beadandblossom.com/learn-french-beading-free-tutorials/ruffles/ |
Starbucks baristas have several responsibilities,greet customers, take orders, prepare handcrafted drinks according to customer specification, cash handling. Baristas are also responsible for cleaning and organization of the store, while maintaining an outstanding customer service.
Job Type: Part-time
Pay: $9.00 per hour
Schedule:
- Day shift
- Monday to Friday
- Night shift
- On call
- Overtime
- Weekends
Supplemental Pay:
- Tips
Work authorization:
- United States (Required)
Shifts:
- Morning (Preferred)
- Mid-Day (Preferred)
- Evening (Preferred)
Work Location:
- One location
Work Remotely: | https://www.myfitjob.com/job/barista-and-cashier-starbucks-downtown-tampa-channel-district-tampa-fl-33602/ |
The utility model relates to a roasting storehouse of furnace charges for smelting a steel alloy, which belongs to the technical field of roasting equipment of the furnace charges for smelting the steel alloy, and is used for roasting the alloy furnace charges during a smelting process. According to the technical scheme, the roasting storehouse is fixed at the upper part of a bracket; a roasting beam is mounted in the roasting storehouse; an automatic bell-type discharge valve is mounted at the top of the roasting storehouse; a fuel gas supplying and replacing device, a combustion air device and a cooling water device are respectively connected with the roasting beam; a dust removing device is mounted at the upper part of the roasting storehouse; a feeding and discharging device is mounted at a discharge port of the roasting storehouse; and a weighing device is positioned below the roasting storehouse and opposite to the feeding and discharging device. The inner wall of the roasting storehouse is of a composite structure formed by lightweight fireproofing materials and cast iron plates, so that the inner wall of the roasting storehouse is good in abrasion resistance and heat preservation; a plurality of combustion nozzles and air regulators are mounted in the water-cooled roasting beam, so that the even heating is realized, i.e., the roasting efficiency is improved; and the automation degree and the production safety can be improved by adopting the fuel gas supplying and replacing device. The roasting storehouse provided by the utility model is convenient to operate and maintain and can meet the requirement of production. | |
I have been a little slow getting on board with quinoa. Actually that's not entirely true as I used to make Noah a quinoa and strawberry porridge which he adored!
But given how nutritious it is I have made a concerted effort to buy and cook some this week and have had great results.
Firstly for dinner I made the quinoa salad pictured above and it was insanely delicious. Seriously! I could eat a big bowlful of this every day of the week. I drew my inspiration from this Quirky Cooking recipe but just used the ingredients I had.
I am planning on making a batch of this (or a variation depending on what we have in the fridge) most weeks to have as my weekday lunches. The possibilities are endless with different combinations of meat, veg and herbs.
With some leftover cooked quinoa I made a very simple porridge which we al had this morning for breakfast. I prepared it in the thermomix but just a pot on the stove would work just as well. To 2 cups of cooked quinoa I added enough nut milk to cover, 2 tbs rice malt syrup and 1/2 ts cinnamon and let it simmer for 5 mins or so until the milk was about half absorbed. 2 of my 3 boys had this with me for breakfast today and loved it so this is a keeper as well.
There is still a little quinoa leftover and I am planning on using it in some bliss balls (recipe to come). | http://www.mywholefoodfamily.com/2015/03/quinoa-salad-with-sausage-and-roast.html |
Your mission:
- Develop new products, services and business models to extend our portfolio strategy and significantly stimulate sustainable and profitable growth;
- Nurture, collect and evaluate innovation ideas in close collaboration with both customers and partners as well as internal resources;
- Maintain and evolve the existing innovation process to ensure follow on business through new and existing client relationships.
Your responsibilities:
- Create consistent innovation strategy and roadmap and continuously identify innovation opportunities;
- Nurture both internal and external idea creation by involving internal colleagues as well as external customer and market views;
- Evaluate business cases for new ideas after market analysis, competitive research, ecosystem elaboration and market acceptance monitoring;
- Challenge existing products, services, processes, procedures; develop new approaches and business models and act as the main innovation driver for well-selected projects;
- Continuously coordinate a systematic idea creation between Management, Product Management, Engineering and Sales;
- Close monitoring of trending technologies, evolving markets and identifying high-potential start-ups in these markets;
- Develop product positioning and pricing in close relationship with Product Management and Utimaco Management;
- Become a trusted advisor to the Utimaco Group;
- Drive innovation for released products together with the Product Management;
- Provide information on innovation topics, associated business cases and its benefit to internal and external stakeholders;
- Coordinate Go-to-Market-Strategy for new products in close relationship with the Product Manager and the Utimaco Management.
Your profile
- Technical university degree and professional experience within a software manufacturer in the IT industry;
- At least 5+ years professional experience in innovation management or a related role;
- Profound experience in identifying opportunities through quantitative and qualitative research as well as developing business cases for ideas, new products and/or services;
- Telco experience (market, equipment, protocols networks) is a clear plus;
- Strong understanding of customer and market requirements combined with business acumen and capacity to articulate competitive value to different stakeholders;
- Able to use project management methodologies and run projects through all stages of the project lifecycle;
- Visionary, enthusiastic, creative entrepreneur with excellent communication and presentation skills in English and German;
- Mobile and able to travel up to 30%.
We offer
- A true global presence: in Europe, Americas, and Asia;
- A leader in one of the hottest market: Cybersecurity;
- Growing at fast pace, gaining market share over our competitors;
- An open and friendly corporate culture characterized by a constructive and cooperative relationship;
- Utimaco benefit package;
- The professional and personal support through targeted further education opportunities.
Do you feel addressed?
Then we look forward to receiving a meaningful application stating the earliest possible starting date and your salary expectations. | https://www.utimaco.com/careers/offer/job/177-innovation-manager-m-f-x/?tx_utimacodvinci_jobs%5Blang%5D=en&cHash=1ad75fc9c158765720e84f9689006706 |
Q:
Problem with particular proof regarding infinite total variation of Brownian motion
I have some problems with a proof from the last page of this pdf:
Brownian motion has infinite total variation.
Could we say that variance is exactly $\frac{c_{1}}{n}$ for some constant $c_{1}$?
How Var of $V_{n}$ is calculated? Is it just a sum of variances of every absolute increment? That is every two absolute increments are independent, just like increments of B-motion withouth moduli?
And most of all how to apply Tchebychev’s inequality? I see that some kind of strange squaring took place before Tchebychev could be used, but that's that. I do not see how to get greater equal instead of less equal, and other things.
A:
Since the increments $X_{t_n}-X_{t_{n-1}},\ldots,X_{t_1}-X_{t_0}$ are independent, we know that $|X_{t_n}-X_{t_{n-1}}|,\ldots,|X_{t_1}-X_{t_0}|$ are independent. Therefore, the variance of the sum $$V_n = \sum_{j=1}^n |X_{t_j}-X_{t_{j-1}}|$$ is indeed, by Bienaymé's formula, given by the sum of variances: $$\text{var} \, (V_n) = \sum_{j=1}^n \text{var} \, (|X_{t_j}-X_{t_{j-1}}|).$$ We can even calculate the variance explicitly, but the important point for the proof is that the variance is uniformly bounded, i.e. $$\sup_{n \in \mathbb{N}} \text{var} \, (V_n)<\infty.$$
Tschebysheff's inequality states that $$\mathbb{P}(|X-\mathbb{E}X| \geq r) \leq \frac{1}{r^2} \text{var} \, (X)$$ for any $X \in L^2$ and $r>0$. This implies $$\mathbb{P}(|X-\mathbb{E}(X)|<r) \geq 1- \frac{\text{var}(X)}{r^2} \tag{1}.$$ Since $V_n \geq 0$, we have $$\begin{align*} \mathbb{P}\left(V_n- c_0 \sqrt{n} \geq - \frac{1}{2} c_0 \sqrt{n} \right) &= \mathbb{P}\left(-V_n+c_0 \sqrt{n} \leq \frac{1}{2} c_0 \sqrt{n} \right) \\ &\stackrel{V_n \geq 0}{=} \mathbb{P}\left( |-V_n+c_0 \sqrt{n}| \leq \frac{1}{2} c_0 \sqrt{n} \right). \end{align*}$$ Applying $(1)$ with $r = \frac{1}{2} c_0 \sqrt{n}$ and $X = - V_n$ gives the claimed inequality.
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Many people questioned about the benefits of Mathematics during our childhood days. Topology in all its many ramifications may have been the greatest progress area in twentieth-century mathematics; it consists of point-set topology , set-theoretic topology , algebraic topology and differential topology Particularly, situations of modern-day topology are metrizability concept , axiomatic set concept , homotopy principle , and Morse principle Topology also includes the now solved Poincaré conjecture , and the nonetheless unsolved areas of the Hodge conjecture Other results in geometry and topology, including the 4 shade theorem and Kepler conjecture , have been proved only with the help of computers.
The time period utilized mathematics also describes the professional specialty during which mathematicians work on practical issues; as a career focused on sensible problems, applied mathematics focuses on the “formulation, research, and use of mathematical fashions” in science, engineering, and other areas of mathematical apply.
An instance of an intuitionist definition is “Mathematics is the mental activity which consists in finishing up constructs one after the opposite.” 37 A peculiarity of intuitionism is that it rejects some mathematical concepts thought of legitimate according to different definitions.
Mathematical language can be difficult to understand for newcomers because even common phrases, corresponding to or and only, have a extra precise which means than they have in everyday speech, and different phrases equivalent to open and subject seek advice from particular mathematical ideas, not coated by their laymen’s meanings.
The examine of area originates with geometry – specifically, Euclidean geometry , which mixes area and numbers, and encompasses the well-identified Pythagorean theorem Trigonometry is the department of mathematics that deals with relationships between the edges and the angles of triangles and with the trigonometric capabilities. | https://www.terryjohnsonsflamingos.com/fundamental-mathematics.html |
Calculate the probability of positive for tweets about health care reform (target=hcr) from your 20 tweets plus the first 30 tweets annotated by jbaldridge in HCR Tweets - Set3. As in part B, calculate the probability based only on positive and negative tweets (about hcr).
Dataset 1: You collect 1000 tweets about The Flaming Lips and label each of them as positive or negative, finding that 600 of them are positive. The word best is in 110 positive tweets and 10 negative ones. Use this to compute the probability of best given positive and and the probability of best given negative.
Dataset 2: You ask 100 people at The Flaming Lips concert whether they would rate the show positively or negatively, and 80 of them respond positively. Use this to compute the probability of positive and the probability of negative.
In a dataset of tweets annotated for polarity, there were 125 tweets annotated as neutral, 149 as positive, and 304 as negative.
I create a subjectivity classifier that identifies tweets as objective (neutral) or subjective (either positive or negative). It labels 148 tweets as objective and the rest as subjective. Of the 148, only 97 were actually objective. | http://lnc-f12.utcompling.com/assignments/hw3-classification |
Q:
How can we show that this nonnegative symmetric bilinear form is closable?
Let
$(E,\mathcal E)$ be a measurable space
$\mu$ be a measure on $(E,\mathcal E)$ and $$\mu f:=\int f\:{\rm d}\mu$$ for Borel measurable $f:\mathbb R\to\mathbb R$ with $f\ge0$ or $\mu|f|<\infty$
$\mathcal A_0$ be a subspace of $\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ closed under multiplication and dense in $L^p(\mu)$ for all $p\ge1$
$\Gamma$ be a bilinear symmetric operator on $\mathcal A_0$ and $$\Gamma(f):=\Gamma(f,f)\;\;\;\text{for }f\in\mathcal A_0$$
Assume
$\forall f\in A_0:\exists c\ge0:\forall g\in\mathcal A_0:\left|\mu\Gamma(f,g)\right|\le c\left\|g\right\|_{L^2(\mu)}$
$\mu\Gamma(f^2,g)+2\langle\Gamma(f),g\rangle_{L^2(\mu)}=2\mu\Gamma(fg,f)$ for all $f,g\in\mathcal A_0$
$\Gamma(f)\ge0$ for all $f\in\mathcal A_0$
By 1., there is a unique linear symmetric operator $(\mathcal A_0,L)$ on $L^2(\mu)$ with $$\langle Lf,g\rangle_{L^2(\mu)}=-\mu\Gamma(f,g)\;\;\;\text{for all }f,g\in\mathcal A_0.\tag1$$ By 2., $$L(fg)=2\Gamma(f,g)+fLg+gLf\;\;\;\text{for all }f,g\in\mathcal A_0.\tag2$$
Question 1: How can we show that $${\Gamma(f,g)}^2\le\Gamma(f)\Gamma(g)\tag3$$ for all $f,g\in\mathcal A_0$? By 3., $$0\le\Gamma(f+\lambda g)=\Gamma(f)+2\lambda\Gamma(f,g)+\lambda^2\Gamma(g)\;\;\;\text{for all }\lambda\in\mathbb R.\tag4$$ If $\Gamma$ would be positive definite, then we could choose $$\lambda:=-\frac{\Gamma(f,g)}{\Gamma(g)}$$ and conclude. Are we able to prove positive definiteness of $\Gamma$? If not, is there an other way to show $(3)$?
Let $$\mathcal E(f,g):=\mu\Gamma(f,g)\;\;\;\text{for }f,g\in\mathcal A_0.$$ By $(3)$, $${\mathcal E(f,g)}^2\le\mathcal E(f)\mathcal E(g)\;\;\;\text{for all }f,g\in\mathcal A_0.\tag5$$
Assume
$\mu(Lf)=0$ for all $f\in\mathcal A_0$
By 4., $$\mathcal E(f,g)=-\langle f,Lg\rangle_{L^2(\mu)}=-\langle Lf,g\rangle_{L^2(\mu)}\;\;\;\text{for all }f,g\in\mathcal A_0.\tag6$$
Question 2: How can we show that $\mathcal E$ is closable? Let $(f_n)_{n\in\mathbb N}$ be $\mathcal E$-Cauchy with $$\left\|f_n\right\|_{L^2(\mu)}\xrightarrow{n\to\infty}0.\tag7$$ By $(6)$ and $(5)$, $$0\le\mathcal E(f_n)\le\left\|Lf_m\right\|_{L^2(\mu)}\left\|f_n\right\|_{L^2(\mu)}+{\mathcal E(f_n-f_m)}^{\frac12}{\mathcal E(f_n)}^{\frac12}\;\;\;\text{for all }m,n\in\mathbb N.\tag8$$ My problem is that $\mathcal E(f_n)$ is occuring on the right-hand side of $(8)$. Why can we nevertheless conclude $\mathcal E(f_n)\xrightarrow{n\to\infty}0$?
A:
Gerw basically answered your questions, but I think it's worth summarizing.
The Cauchy-Schwarz inequality
$$
a(x,y)^2\leq a(x,x)a(y,y)
$$
holds for every symmetric positive semidefinite bilinear form $a\colon V\times V\to \mathbb{R}$. You have already shown this in the case $a(y,y)>0$. If $a(y,y)=0$, then
$$
a(x+\lambda y,x+\lambda y)=a(x,x)+2\lambda a(x,y)
$$
is a nonnegative affine function of $\lambda$, hence $a(x,y)=0$.
If $T$ is a symmetric linear operator in $H$ such that $\langle Tx,x\rangle\geq 0$ for all $x\in D(T)$, then the bilinear form
$$
a(x,y)=\langle Tx,y\rangle
$$
is closable. Let $(x_n)$ be an $a$-Cauchy sequence such that $\|x_n\|_H\to 0$. With the same computation as in the question, one gets
$$
a(x_n,x_n)\leq\|Tx_m\|_H\|x_n\|_H+a(x_n-x_m,x_n-x_m)^{1/2}a(x_n,x_n)^{1/2}.
$$
The first term goes to zero as $n\to\infty$, while $a(x_n-x_m,x_n-x_m)$ is small for large $m,n$ and $a(x_n,x_n)$ is bounded (since $(x_n)$ is $a$-Cauchy). Thus $a(x_n,x_n)\to 0$.
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Manufactured by GE Automation and Controls, the IC698CHS009 is a front or rear mounted rack case with nine module slots. This rack is capable of housing various RX7i and Series 90-70 modules and CPUs. Each module connector on the rack backplane is spaced 0.8 inches or 20.3 millimeters apart with each slot able to accommodate one single-width module. Double width modules consume two rack backplane slots each. Slot zero on the backplane is reserved for the power supply, and slot one is intended for the programmable logic controller CPU. The remaining backplane slots are available for a variety of single or double width combinations.
The IC698CHS009 comes with the following features and abilities: slot sensing for input/output modules, with no DIP switches or jumpers necessary for module addressing; automatic daisy chaining of interrupt acknowledge and bus grant signals, with no slot jumpers necessary; J2 connectors with speeds up to 64 bits per cycle; RX7i power supply compatibility; and cooling fan compatibility.
RX7i racks are regarded as “open equipment,” so they must be installed in an IP54 safety enclosure. The dimensions for this rack are as follows: The height is 11.15 inches (283 mm). The width is 12.6 inches (320 mm). The depth is 7.25 inches (184 mm). If a cooling fan is to be installed, an additional 9 inches (23 cm) space between racks is necessary.
A cooling fan is necessary if the following modules are installed: IC698CPE020, IC698CRE020, and IC698PSA350. There are three cooling fan assemblies available, each intended for a different power source. The IC697ACC621 fan is intended for 120 VAC, the IC697ACC624 fan is intended for 240 VAC, and the IC697ACC644 fan is intended for 24VDC.
To shield ground RX7i modules and devices, ensure both top and bottom faceplate screws are firmly connected to the encasement. For Series 90-70 modules and devices, a ground clip will make contact with the conductive rail when installed. For safety grounding, connect the studs mounted on the side of the IC698CHS009 to the earth ground with an AWG #12 wire.
To reduce the impact of electrical magnetic interference (EMI), ensure that the power supply is securely mounted, connect the GND terminal on the power supply to the terminal on either side of the rack using an AWG #12 wire with a ring terminal and star washer, and connect the rack GND terminal to the earth ground.
|Module Type:||Standard Rack|
|Mounting Location:||Rear|
|Number of Slots:||9 Single Width, 5 Double Width|
|Rack Slot Size:||0.8 inch|
|Dimensions:||11.15 x 12.6 x 7.25 in. (H x W x D)|
|Power Supply:||RX7i Power Supply in Slot 0|
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Number of Slots:
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Slots 1 through 8 are 0.8" (20.3mm) wide. (The CPU is installed in slot 1.)
Slot 9 is 1.6” wide. If a single width module is installed in slot 9, it is recommended that a single-width filler faceplate be used to close the extra width opening.
Slot 0 (power supply slot) is 2.4" (61.0mm) wide.
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Maximum Current (from RX7i power supplies)
100 watt supply: +5V
+12V
-12V
350 watt supply: +5V
+12V
-12V
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20 amps Total output power –100W 2 amps
1 amps
60 amps Total output power – 350W 12 amps
4 amps
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I/O References
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User configurable with programming/configuration software
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Dimensions
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Height Width Depth (Note that all Series 90-70 11.15" 12.6" 7.5" modules extend 1.7" (43 mm) 283mm 320mm 190mm beyond front of rack.)
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VME
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System designed to support VME64
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* For environmental specifications and compliance to standards (for example, FCC or European Union Directives), refer to Appendix A of the PACSystems RX7I Installation Manual, GFK-2223.
Troubleshooting IC698CHS009
Troubleshooting information is available on IC698CHS009's website page; it also includes a datasheet user-manual and a wiring diagram.
For more information on the IC698CHS009 Standard PACSystems 9-Slot Wall (Rear) Mount Rack of the GE Fanuc RX7i Series, please see the Datasheet User-Manual.
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Description
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Catalog Number
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Rack - 9 slots, rear mount
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IC698CHS009
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Rack - 9 slots, front mount
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IC698CHS109
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Rack Fan Assembly (required for CPE020/CRE020, PSA350), 120 VAC
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IC697ACC621
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Rack Fan Assembly (required for CPE020/CRE020, PSA350), 230 VAC
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IC697ACC624
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Rack Fan Assembly (required for CPE020/CRE020, PSA350), 24 VDC
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IC697ACC644
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Gasketed filler faceplate, single-width
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IC698ACC735
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Gasketed filler faceplate, double-width
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IC698ACC720
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Note: For Conformal Coat option, or Low Temperature Testing option please consult the factory for price and availability.
Common related search terms: ConfigurationDatasheet, Emerson, Manual, PDF, PLC, Price, Repair, Specification, Troubleshoot, User-Manual, Wiring,
PDF Supply Company, LLC. sells surplus and refurbished products, which are sourced through independent channels. All warranties and support, if applicable, are with PDF Supply Company, LLC. and not the manufacturer; manufacturer warranties do not apply. PDF Supply Company, LLC. is not an authorized distributor or representative for the listed manufacturers and makes no representations as to any quality control performed by any listed manufacturers on the products. The products listed on this website may vary as to their country of origin; the accessories, and other items included with the product; and the language used on the packaging, the parts, and any related instructions or printed material related to the products. PDF Supply Company, LLC. does not sell or license software that may be needed to operate certain hardware and customers must obtain any necessary software licensing, maintenance, or upgrades from the manufacturer or other authorized source. This website is not sanctioned or approved by any manufacturer or other authorized source. This website is not sanctioned or approved by any manufacturer or tradename listed. Designated trademarks, brand names and brands appearing herein are the property of their respective owners. | https://www.pdfsupply.com/automation/ge-fanuc/rx7i-pacsystem/IC698CHS009 |
Salmon is excellent for seafood enthusiasts. It is a source of low-calorie proteins and omega-3s. It also provides minerals like potassium, and if you are looking to lose some pounds, it is excellent for weight control. You may decide to have it alone, but it will be better with a compatible side dish.
What is salmon?
Salmon is a common type of food that most people categorize as oily fish. It is highly nutritious, and it is also rich in cholesterol whose level may vary depending on the salmon species. Like many other individuals, I like salmon because it doesn’t have the typical fishy taste.
Instead, it has a subtle refreshing flavor that may differ according to the recipe that you decide to follow when cooking it. You can add ingredients of your choice. These may include red peppers, mayonnaise, and Italian parsley. Additionally, you also get to choose if you will grill, roast, fry, or smoke them.
The fact that it is like slightly flavored meat allows you to serve it with a variety of side dishes. In case you are having problems deciding which one to choose, below are some that you can try out.
What will go well with salmon?
There are many recipes that you can pick to go with the healthy fish. Make them simple to ensure that they don’t overpower the salmon’s taste.
Remember to make sure that they are different from the main dish. For instance, you should not serve your salmon with another type of fish; it cannot work. You must always strive to strike a balance to ensure that the meal provides all the nutrients that your body requires.
Garlicky Roasted Broccoli
Your salmon can pair well with roasted garlic broccoli. Olive oil would be perfect to use, and you can roast it until it is golden to get a more delicious taste. Consider adding some pepper and salt. Some sugar will caramelize the broccoli and remove any bitterness that it may have developed during roasting.
Butter Roasted Sweet Potatoes
This side dish is great for salmon because of its crisp skin and buttery, creamy softness inside. You will need sweet potatoes, salt, and butter. Adding some coconut oil will achieve a faintly nutty flavor that will make the fish taste better. Roasting the sweet potatoes will take about 20 minutes.
Homemade cabbage slaw
Cabbage slaw is easy to make. It has a refreshing, pleasant taste that perfectly complements your buttery salmon. You can choose to shred/grate some carrots and add them to it. Other ingredients that would make it better include Dijon mustard to make it spicy and more flavorsome, champagne or red wine vinegar for fresh dressing as well as mayonnaise to make the dressing a bit creamy.
With carrots and cabbage, you do not have to add sugar. The side dish is already sweet enough, and you shouldn’t overdo it.
Wilted escarole with lemon and garlic
Heating the escarole a little will mellow out their naturally bitter taste. Remember to use some oil, garlic, salt, and pepper to bring out the vegetable’s best flavors. Cooking it requires only a few minutes, and then you can serve it with lemon wedges, together with the salmon. If you cannot find escarole, you may use romaine instead.
Sesame Peanut Cucumber Salad
There is no doubt that this salad can be great to have with salmon, especially on a hot summer day. It is light but extremely rich in flavors. For the dressing, consider using ginger, honey, vinegar, sesame oil, and a little lime juice. If you want the salad’s taste to be bolder, you can refrigerate it for 30 minutes.
Seaweed Salad
Your salmon will pair incredibly well with this salad, which is delicious and pretty easy to make. With its mineral and vitamin richness, it ensures that your meal is balanced. Even though the salmon brings a considerable amount of protein to the table, you can also think about adding some silken tofu for the same.
Wild Rice Pilaf
Make some delicious wild rice pilaf to serve with your salmon. The dish originated from India. Because the rice and the grain will cook at different times, you can prepare them separately to get the best taste of each. To achieve an extra fluffiness, you can rinse the rice before sautéing. It also gets rid of excess starch.
Citrus Olive Couscous
This recipe will be tastier to have with salmon. To bring out a refreshingly sharp flavor, consider using some green marinated olives. Add enough salt and pepper to taste, and if you want a fragrant aroma, you can also include lemon thyme leaves. Remember that this dish needs fresh orange juice to contrast with the salty taste of the olives.
Simple Swiss Chard Pasta
Pasta enthusiasts can consider making this side dish, which only takes 20 minutes to make, for eating with salmon. The Swiss chard adds crunch to the recipe, and you can sauté it using canned tomatoes. For more flavor, use chopped walnuts or pine nuts and pecorino; but they are optional.
Warm Vegetable Curry
Vegetable curry, which is warm and has a variety of spices like cumin, cinnamon, and fennel seeds, will be a delicious side dish for salmon. It will take 30 or fewer minutes to make it. If you want to enjoy it maximally, make sure that you don’t let it go cold before you serve it.
Marinated white beans
This is a great replacement for a raw vegetable side dish with vinegar dressing and herbs. Do not hesitate to add garlic, fresh herbs and soak them in shallot-infused olive oil. If you want the beans to be better, consider cooking them one or two days ahead.
Having salmon is beneficial to your health because it is rich in minerals and proteins. To relish the fish more, think about getting side dishes for it. There are many choices that you can go for as long as you like them. These include salads, rice, slaws, and potatoes. As you choose them, confirm that they help you meal to be more balanced. | https://www.southendformaggio.com/what-to-serve-with-salmon/ |
As NASA’s priorities change over time, missions are phased out or cancelled. NASA may choose to end support of successful programs after they have completed their original mission in order to support new scientific and exploratory goals. Some programs end due to technology failures or unjustifiable costs. Each past program, however, has led to lessons learned. The past programs presented below are important due to their contributions to not only NASA’s future space goals, but also advances here on earth.
SPACE SHUTTLE PROGRAM
The Space Shuttle, as humanity's first reusable spacecraft, demonstrated breakthrough manned flight technology and was essential to the construction of the International Space Station (ISS). The space shuttle fleet - Columbia, Challenger, Discovery, Atlantis, and Endeavour - flew 135 missions, in order to repair satellites, conduct cutting-edge research, and build the International Space Station. The Space Shuttle program flew its last flight in July 2011, after more than thirty years of operations. The program not only advanced robotics and how we build and fly airplanes, but also set the groundwork for the development of commercial space vehicles - such as the SpaceX Falcon 9, with its Crew Dragon capsule - that carries astronauts to ISS, and possibly beyond. Developed under the NASA Commercial Crew Program, this NASA/industry partnership provides transport services to and from low-Earth orbit and the ISS - allowing NASA to focus on building spacecraft and rockets for deep space missions.
CONSTELLATION PROGRAM
With an ultimate plan to use the moon to prepare for future human and robotic missions to Mars and other destinations, Constellation was a program to develop a spacecraft (Orion) and booster vehicles to replace the Space Shuttle – and eventually send astronauts back to the Moon for the first time in 50 years. After determining that the program was over budget, behind schedule, and lacking critical technologies, funding was cut for Constellation in 2010, culminating in the eventual cancellation of the program. While the Constellation program to service the ISS is being replaced with the Commercial Crew Program, NASA’s work was allowed to proceed on the Orion capsule and Orion had its first flight test in 2014. In 2017 Orion became part of the current crewed lunar exploration program, Artemis.
ASTEROID REDIRECT MISSION (ARM)
Introduced in 2013, NASA’s Asteroid Redirect Mission (ARM) called for sending a robotic spacecraft to a near Earth asteroid to grab a boulder a few meters across and return it to cislunar space. Astronauts flying on an Orion spacecraft would then visit the boulder, perform studies and collect samples for return to Earth. ARM struggled to win support, particularly in Congress, where some members felt the mission was not relevant to NASA’s long-term goal of sending humans to Mars in the 2030s. While ARM was phased out in 2017, NASA emphasized that many of the central technologies in development for that mission, such as solar electric propulsion, will continue, as they constitute vital capabilities needed to advance NASA’s human path to Mars. Work on ARM’s solar electric propulsion technology, as well as other ARM elements, will continue, as NASA notes that “the capabilities that we were developing in ARM, were not mission-specific,” [but are] “applicable to a wide variety of missions.”
Updated October 2020 by Tina Allen
Comments are closed. | https://mrr.dawnbreaker.com/portals/space/past-programs/ |
Forums › ACCA Forums › ACCA MA Management Accounting Forums › Holding and Order cost
- This topic has 3 replies, 2 voices, and was last updated 10 months ago by John Moffat.
- AuthorPosts
- March 19, 2022 at 12:36 pm #651524sarbrinaParticipant
- Topics: 22
- Replies: 49
- ☆☆
Hi Mr. Maffot,
I got this question from a different website and im struggling to arrive at their answer. Please help me understand why.
Q.) Budgeted sales of X for December are 18,000 units. At the end of the production process for X, 10% of
production units are scrapped as defective. Opening inventories of X for December are budgeted to be
15,000 units and closing inventories will be 11,400 units. All inventories of finished goods must have
successfully passed the quality control check. What is the production budget for X for December?
My answer for holding cost is $4000 but their answer is $4500. And ordering cost, their answer is $400000.
Please explain how it is done.March 20, 2022 at 7:40 am #651536John MoffatKeymaster
- Topics: 56
- Replies: 51583
- ☆☆☆☆☆
The question as you have typed it asks how many units will be produced and makes no mention of holding costs or ordering costs. They will budget on producing 16,000 units.
Given that there is no information about costs given in the question I have absolutely no idea where you are getting $4,000 from.March 20, 2022 at 1:54 pm #651549sarbrinaParticipant
- Topics: 22
- Replies: 49
- ☆☆
Oh! My apologies Mr. Maffot, I have typed out the wrong question. My bad. The question is:
Q.)Bakers & Sweets Plc are known for brownies for which raising agent is required. Annual demand of raising agent is 20,000 kilograms. The demand is considerably even through the year. The cost to place each order is Rs. 50 where is cost to hold one kilo of raising agent is Rs. 2.If each order is of 3,000 units and buffer stock is maintained at 1,000 units around the year then:
What is the combined total annual costing of holding and ordering the raising agent?
This is the right question.March 21, 2022 at 8:13 am #651575John MoffatKeymaster
- Topics: 56
- Replies: 51583
- ☆☆☆☆☆
I am surprised that your book does not show the workings for the answer. You should be using a Revision Kit from one of the ACCA Approved Publishers – they do show the workings 🙂
The holding cost per year is (3,000/2 + 1,000) x Rs 2 = Rs 5,000. It seems as though the answer in your book is wrong.
The ordering cost per year is 20,000/3,000 orders x Rs 50 = Rs 333.33.
Again, either your book has made a mistake (or you have typed out the question wrongly. Are you sure that the demand is 20,000 per year and not 20,000 per months?)
In future you must ask in the Ask the Tutor Forum if you want me to answer. This forum is for students to help each other.
- AuthorPosts
- You must be logged in to reply to this topic. | https://opentuition.com/topic/holding-and-order-cost/ |
The beautiful neo-Byzantine Romanesque St. Clement church built in 1918 in the Lincoln Park neighborhood of Chicago is one of a relatively small number of churches of this style in the US. Inspired by the iconic Hagia Sofia in modern day Istanbul, Turkey, St. Clement was designed by the same St. Louis architect, Thomas P. Barnett, as the larger Cathedral Basilica of St. Louis on which it was modeled. Like the Cathedral Basilica, St. Clement was conceived as a blend of eastern and western styles which is the hallmark of the Byzantine Romanesque style and traces back to the crossroads of eastern and western cultures of the ancient Christian city of Constantinople (Istanbul). The church also houses a beautiful and impressive collection of mosaic artwork, most prominently viewed inside the central dome and in the apse dome – a replica of that in the ancient church of the same patronage: San Clemente in Rome. The parish is home to several thousand parishioners and households, and boasts an extremely active and robust young membership.
The exterior stonework of the church is sturdy and handsomely-proportioned, with the finer details reserved for a select number of column capitals and decorative bands, particularly at important locations such as the triumphal entry portal. Similar to many other churches in the Chicago area, the interior utilizes rich faux marble finishes on plaster as a backdrop to the many rich mosaics inlays, which include both vivid icon imagery and decorative elements replete with ancient Christian symbolism. An interesting historical detail is the inclusion of eight female saints, two on each of the four primary pilasters supporting the dome just below the pendentives. This was a major statement for the time period. The brilliant stained glass – particularly the three spectacular rose windows – adds even more color to the interior, but simultaneously achieves balance and brilliance with the entirety of the interior imagery. Whereas some more well-known churches in the Chicago area such as St. John Cantius – this year voted 'Most Beautiful Church in America' in a poplar poll – favor ornate detail on virtually every surface, St. Clement's careful placement of areas of restraint in decoration and contrasting levels of detail provide a welcome harmony and visual hierarchy that is difficult to achieve. The effect is breathtaking. For those who might evaluate using the classical model, the church embodies completeness, proportion, and clarity.
A visit to Lincoln Park, which houses the Zoo and Conservatory, definitely warrants a quick trip down the street to see this magnificent church. In a city where many of the ethnic shifts in once-thriving neighborhoods have led to under-maintained and sometimes shuttered churches, it is wonderful to see this architectural gem still proudly serving its thriving population and so wonderfully representing the union of heaven and earth in the liturgy.
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This article discusses the phonological system of Standard Macedonian (unless otherwise noted) based on the Prilep-Bitola dialect. For discussion of other dialects, see Macedonian dialects. Macedonian possesses five vowels, one semivowel, three liquid consonants, three nasal stops, three pairs of fricatives, two pairs of affricates, a non-paired voiceless fricative, nine pairs of voiced and unvoiced consonants and four pairs of stops.
Contents
Vowels
|Front||Central||Back|
|Close||i||u|
|Mid||ɛ||(ə)||ɔ|
|Open||a|
Schwa
The schwa is phonemic in many dialects (varying in closeness to [ʌ] or [ɨ]) but its use in the standard language is marginal. When writing a dialectal word and keeping the schwa for aesthetic effect, an apostrophe is used; for example, ⟨к’смет⟩, ⟨с’нце⟩, etc. When spelling aloud, each consonant is followed by the schwa. The individual letters of acronyms are pronounced with the schwa in the same way: ⟨МПЦ⟩ ([mə.pə.t͡sə]). The lexicalized acronyms ⟨СССР⟩ ([ɛs.ɛs.ɛs.ɛr]) and ⟨МТ⟩ ([ɛm.tɛ]) (a brand of cigarettes), are among the few exceptions.
Vowel length
Vowel length is not phonemic. Vowels in stressed open syllables in disyllabic words with stress on the penult can be realized as long, e.g. ⟨Велес⟩ [ˈvɛːlɛs] (listen) 'Veles'. The sequence /aa/ is often realized phonetically as [aː]; e.g. ⟨саат⟩ /saat/ [saːt] 'colloq. hour'.
Consonants
|Labial||Dental||Alveolar||Palatal||Velar|
|Nasal||m||n̪1||ɲ|
|Plosive||voiceless||p||t̪||c||k|
|voiced||b||d̪||ɟ||ɡ|
|Affricate||voiceless||t̪͡s̪||t͡ʃ|
|voiced||d̪͡z̪||d͡ʒ|
|Fricative||voiceless||f||s̪||ʃ||x|
|voiced||v||z̪||ʒ|
|Approximant||ɫ̪1||j|
|Trill||r1|
|Lateral||l||ʎ|
^1 The alveolar trill (/r/) is syllabic between two consonants; for example, ⟨прст⟩ [ˈpr̩st] 'finger'. The dental nasal (/n/) and dental lateral (/ɫ/) are also syllabic in certain foreign words; e.g. ⟨њутн⟩ [ˈɲutn̩] 'newton', ⟨Попокатепетл⟩ [pɔpɔkaˈtɛpɛtɫ̩] 'Popocatépetl', etc.
The labiodental nasal [ɱ] occurs as an allophone of /m/ before /f/ and /v/ (e.g. ⟨трамвај⟩ [ˈtraɱvaj] 'tram'). The velar nasal [ŋ] similarly occur as an allophone of /n/ before /k/ and /ɡ/ (e.g. ⟨англиски⟩ [ˈaŋɡliski] 'English'). The latter realization is avoided by some speakers who strive for a clear, formal pronunciation.
Phonological processes
At morpheme boundaries (represented in spelling) and at the end of a word (not represented in spelling), voicing opposition is neutralized.
Stress
The word stress in Macedonian is antepenultimate, meaning it falls on the third from last syllable in words with three or more syllables, and on the first or only syllable in other words. This is sometimes disregarded when the word has entered the language more recently or from a foreign source. The following rules apply:
- Disyllabic words are stressed on the second-to-last syllable.
For example, ⟨дете⟩ [ˈdɛtɛ] 'child', ⟨мајка⟩ [ˈmajka] 'mother' and ⟨татко⟩ [ˈtatkɔ] 'father'.
- Trisyllabic and polysyllabic words are stressed on the third-to-last syllable.
For example, ⟨планина⟩ [ˈpɫanina] 'mountain', ⟨планината⟩ [pɫaˈninata] 'the mountain' and ⟨планинарите⟩ [pɫaniˈnaritɛ] 'the mountaineers'.
Exceptions include:
- Verbal adverbs (i.e. words suffixed with ⟨-ќи⟩): e.g. ⟨викајќи⟩ [viˈkajci] 'shouting', ⟨одејќи⟩ [ɔˈdɛjci] 'walking'.
- Foreign loanwords: e.g. ⟨клише⟩ [kliˈʃɛ] 'cliché', ⟨генеза⟩ [ɡɛˈnɛza] 'genesis', ⟨литература⟩ [litɛraˈtura] 'literature', ⟨Александар⟩ [alɛkˈsandar], 'Alexander' (Possibly based on hellenised variations of indigenous Bryges and/or Enchele naming conventions), etc. | https://www.wikiyy.com/en/Macedonian_phonology |
We will release a steel ball from rest down a smooth angled plane upon which it moves with acceleration of 0,5 m·s−2. Then it moves to a level plane. In total, it travels 20 meters in 12 seconds. For how long did it move on the angled plane?
Neglect friction of the environment and the plane.
List of known information
a = 0,5 m·s−2 ball’s acceleration s = 20 m total travelled distance t = 12 s total time of balls movement t1 = ? (s) time of ball’s movement on the angled plane
Hint 1: Distance and speed of ball on angled plane
Draw a picture and mark all needed quantities.
What distance will the ball travel on the angled plane and what speed will it reach?
Hint 2: Ball’s movement on level plane
What is the ball’s movement on the level plane?
What is its speed during this movement?
What distance does it travel?
Hint 3: Total distance of the ball
Express the total distance the ball has travelled.
Solve the quadratic equation that you get for time t1 and figure out which root of the equation fits the task’s conditions.
Overall Solution
Picture and marking of quantities: | http://physicstasks.eu/2226/steel-ball |
Boiling is not needed to get this water vapor. … As you increase the temperature of the water, there are more and more water particles that have enough energy to leave the water phase and become water vapor. So the water vapor pressure will increase with the temperature of the water (this is important).
Does boiling increase pressure?
At higher elevations, where the atmospheric pressure is much lower, the boiling point is also lower. The boiling point increases with increased pressure up to the critical point, where the gas and liquid properties become identical.
How is boiling affected by pressure?
Pressure Affects the Boiling Point
When atmospheric pressure increases, the boiling point becomes higher, and when atmospheric pressure decreases (as it does when elevation increases), the boiling point becomes lower. Pressure on the surface of water tends to keep the water molecules contained.
Does water boil at lower pressure?
When atmospheric pressure is lower, such as at a higher altitude, it takes less energy to bring water to the boiling point. Less energy means less heat, which means water will boil at a lower temperature at a higher altitude.
Why does boiling point increase with pressure?
The boiling point of a liquid is directly affected by atmospheric pressure. This is the pressure exerted by the weight of the air molecules above the liquid. In an open system this is called atmospheric pressure. The greater the pressure, the more energy required for liquids to boil, and the higher the boiling point.
Why do bubbles form in boiling water?
Boiling begins near the source of heat. When the pan bottom becomes hot enough, H2O molecules begin to break their bonds to their fellow molecules, turning from sloshy liquid to wispy gas. The result: hot pockets of water vapor, the long-awaited, boiling-up bubbles.
Which boils faster water or alcohol Why?
As alcohol evaporates at a much faster rate compared with water due to its lower boiling temperature (82 compared to 100 degrees C), it is able to carry away more heat from the skin. This means for a given amount of time much more alcohol evaporates than water.
What happened to the temperature of water while it is boiling?
When boiling occurs, the more energetic molecules change to a gas, spread out, and form bubbles. … In addition, gas molecules leaving the liquid take away heat energy. Therefore the temperature of the liquid remains constant during boiling. For example, water will remain at 100ºC while boiling.
Why does boiling happen?
As a liquid is heated, its vapor pressure increases until the vapor pressure equals the pressure of the gas above it. … In order to form vapor, the molecules of the liquid must overcome the forces of attraction between them. The temperature of a boiling liquid remains constant, even when more heat is added.
What increases boiling point?
Large molecules have more electrons and nuclei that create van der Waals attractive forces, so their compounds usually have higher boiling points than similar compounds made up of smaller molecules.
How can you make water boil faster?
Truth: Hot water boils faster.
If you’re in a hurry, turn your tap to the hottest setting, and fill your pot with that hot tap water. It’ll reach boiling a bit faster than cold or lukewarm water. You can also get the water even hotter by using your electric kettle.
Does salt boil water faster?
In fact, adding salt does the very opposite of making water boil faster. Instead, it makes it take longer for the water to boil! The salt actually increases the boiling point of the water, which is when the tendency for the water to evaporate is greater than the tendency for it to remain a liquid on a molecular level.
What liquid has the highest boiling point?
Explanation: Acetone 56.0 ∘C .
At what pressure does water boil at room temperature?
The normal boiling point of the liquid is the temperature at which the liquid boils at one atmosphere of external pressure. For water the normal boiling point is 100C (212F). In our experiment, the water in our flask has a particular vapor pressure at room temperature.
Why does water boil faster at lower pressure?
It also impacts the boiling point of water: the temperature at which liquid water begins turning to vapor, which occurs when its vapor pressure equals the atmospheric pressure. At a higher elevation, the lower atmospheric pressure means heated water reaches its boiling point more quickly—i.e., at a lower temperature. | https://houseofherby.com/boiling/does-boiling-water-increase-pressure.html |
City leaders in Destin say they are nearing a solution for bill property owners for trash pick-up.
They’re calling the latest proposal a great compromise. Under the plan, waste management would handle the billing and collection for $6 dollars a year per customer. The annual rate will run home owners about $257 dollars, an 11% decrease in their bill.
The first reading for both proposals will take place next Monday, with the final vote set for December 5, 2011. | http://www.wjhg.com/news/newschannel7today/headlines/Destin_Trash_Pick-up_133882223.html?site=full |
What properties of light are determined by wavelength?
1 Answer
The properties of light we can determine from the wavelength are the frequency and Energy - however, observing the source of the wavelength does lead to a host of other information.
Explanation:
The wave equation is
so knowing the wavelength we can determine the frequency and the Energy of the photon.
However, if the source of the light is known a whole set of information can be garnered from it.
e.g. the temperature and possible composition of the source - using the spectral line corresponding to the particular wavelength we are observing.
velocity of the source - using either Doppler law (or Hubble law in case of stars).
Distance from the source - by the inverse square law - a technique used in astronomy where similar stars in near and distant galaxies allow us to find the distance to those distant galaxies.
What material is present between the source and the detector (Absorption spectroscopy).
just to state a few. | https://socratic.org/questions/what-properties-of-light-are-determined-by-wavelength |
Sampler Info is a utility node you can use to get all kinds of information useful for building shader networks. The job of Sampler Info is to give you information about the each point on a surface as it is being "sampled", that is, calculated for rendering purposes.
Sampler Info can give you information about a point's position in space, its orientation and tangency, and its location relative to the camera.
Many of the attributes for this node provide values in "camera coordinate space". This is the local object space of the camera. Each camera (in its own space) is located at the point 0, 0, 0. It is looking straight along the negative Z axis, and the positive Y axis is pointing up.
Point Camera gives you the position of the point being sampled, in camera coordinate space.
Point Obj gives you the position of the point being sampled, in the object's local coordinate space.
Point World gives you the position of the point being sampled, in world coordinate space.
Normal Camera gives you the surface normal at the point being sampled, in camera coordinate space.
The "surface normal" is a vector that points directly away from the surface (at right angles to it) at that point.
U V Coord gives you the U and V surface coordinates of the point being sampled.
Ray Direction gives you a vector that points from the point being sampled to the camera position, in camera coordinate space.
Tangent U Camera is the surface tangent in the U direction at the point being sampled, in camera coordinate space.
Tangents are not well-defined for polygon objects.
Tangent V Camera is the surface tangent in the V direction at the point being sampled, in camera coordinate space.
Pixel Center gives you the location of the pixel in the final image that corresponds to the point being sampled.
For example, let's say you are rendering an image at a resolution of 200 x 200. Then 0, 0 is the position of the lower-left corner of the image, and 200, 200 is the position of the upper-right corner. If we are sampling a point in the middle of the image, its Pixel Centre value would be approximately 100, 100.
Flipped Normal tells you which side of a surface is being sampled. Most shaders treat both sides of a surface the same, but you may want to have them different.
For example, if you were creating an image of money, you would want single flat surfaces with one image on one side, and a different image on the other side.
Facing Ratio gives you a number between 0 and 1 that tells you if the point being sampled is facing towards or away from the camera. A value of 1 means that it is facing the camera head-on. A value of 0 means that is is facing 90 degrees from the camera.
In mathematical terms, Facing Ratio is the cosine of the angle between Ray Direction and Normal Camera. | http://download.autodesk.com/global/docs/maya2014/en_us/Nodes/samplerInfo.html |
Fifth Harmony on 7/27: The group share details of their 'vulnerable' new album
Fifth Harmony has a new album, and Fifth Harmony can’t talk about it. Or anyhow, they can’t go too deep into specifics. The pop girl-group behind last year’s “Worth It” and the new hit single “Work From Home” has a new album hitting on May 20, 2016. The title of the album is 7/27 – a reference to the date Fifth Harmony formed, back when the five singers were contestants on the American version of The X-Factor. Beyond the title, though, 7/27 is shrouded behind a veil of NSA-level secrecy, because everything in pop culture is J.J. Abrams now.
But that doesn’t stop the singers from teasing the album a little bit. “We’ve had a lot more creative say,” says Normani Kordei, who explains how the singers had greater input into this second album than the tunes on their debut LP Reflection. “It’s more personal than the last.”
Her bandmate, Camila Cabello, agrees. “I think on the Reflection album, and since we got off of X-Factor, we tried to establish what the sound was for Fifth Harmony. What best represented us. With our first album, you heard the 808 urban-leaning drums as a skeleton in some of the songs, like ‘Boss’ and ‘Worth It.’ You can hear that now in ‘Work From Home,’ and you’re gonna be hearing an expansion of that.”
“People will get a deeper insight into who we are,” says Ally Brooke. “We talk about topics from love to heartbreak to confusion about someone. The main thing for all of us was having a ballad, a slower song. Finally, they get to hear more of a vulnerable side to us! We didn’t really have that in Reflection, and that’s something we all fought for.”
“There’s a song on here that’s from our last album cycle,” says Lauren Jauregui. “It was thrown out, gone. Dinah and I kept bringing it up, and finally they gave in.” Why did they have to fight for the song? Lauren smiles. “It’s, like, politics.”
One thing that isn’t changing: The name “Fifth Harmony,” which was actually the third name for the group formerly known as LYLAS and 1432. “You take control of your name,” says Dinah Jane. “The way you carry your name is the vibe people get off it.”
To continue reading more about Fifth Harmony, pick up the new issue of Entertainment Weekly, on newsstands Friday, or buy it here – and subscribe now for more exclusive interviews and photos, only in EW. | https://ew.com/article/2016/03/25/fifth-harmony-727-interview-1/ |
---
abstract: 'Let $F^\lambda_{\sigma} [G]$ be a crossed product of a group $G$ and the field $F$. We study the Lie properties of $F^\lambda_{\sigma} [G]$ in order to obtain a characterization of those crossed products which are upper (lower) Lie nilpotent and Lie $(n,m)$-Engel.'
address:
- ' A. BovdiInstitute of Mathematics, University of Debrecen,P.O. Box 12, H-4010 Debrecen, Hungary'
- ' A. Grishkov Departamento de Matemática, (IME-USP), Rua do Matao, 1010 - Cidade Universitária, CEP 05508-090, Sao Paulo - SP, Brasil'
author:
- Adalbert Bovdi and Alexander Grishkov
bibliography:
- 'Bovdi\_Grishkov\_final.bib'
title: Lie properties of crossed products
---
Introduction
============
Let $G$ be a group and let $Aut(F)$ be the group of automorphisms of a field $F$. Assume the mapping $ \sigma: G\to Aut(F)$ and the twisting function $ \lambda: G\times G\to U(F) $ satisfy the following conditions: $$\label{E:1}
\lambda(a, bc)\lambda (b, c)=\lambda (ab, c)\lambda (
a,b)^{\sigma(c)}$$ and $$\label{E:2}
\alpha ^{\sigma(a)\cdot \sigma(b)}= \alpha^{\sigma(ab)}$$ for all $a, b, c\in G$ and $\alpha(a,b)\in F$. The twisting function $\lambda$ is also called a [**factor system**]{} of the group $G$ over the field $F$ relative to $\sigma$. It is a $2$-cocycle of the group of units $U(F)$ of $F$ with the natural $G$-module structure (in the cohomology group of $G$ in $U(F)$).
Assign to every $g\in G$ a symbol $\widetilde{g}$, and let $$\textstyle
F^\lambda_{\sigma}[G]=\{\sum_{g\in G}\widetilde{g}\alpha_g \mid
\alpha_g\in F\}$$ be the set of all formal sums with finitely many nonzero coefficients $\alpha_g$. Two elements $\textstyle x=\sum_{g\in G}
\widetilde{g}\alpha_g \quad\text{and} \quad y=\sum_{g\in
G}\widetilde{g}\beta_g $ from $F^\lambda_{\sigma} [G]$ are equal if and only if $\alpha_g =\beta_g$ for all $g\in G$. On the set $F^\lambda_{\sigma} [G]$ addition and multiplication are defined as follows:
- $\sum_{g\in G}\widetilde{g}\alpha_g + \sum_{g\in
G}\widetilde{ g}\beta_g=\sum_{g\in G}
\widetilde{g}(\alpha_g+\beta_g);$
- $\widetilde{
g}\widetilde{h}=\widetilde{gh}\lambda (g, h)$, where $\lambda$ is the twisting function;
- $\alpha \widetilde{g}=
\widetilde{g} \alpha^{\sigma(g)}$.
The product of arbitrary elements $x$ and $y$ is determined by distributivity. It is easy to check that $F^\lambda_{\sigma}
[G]$ is an associative ring which is called a [**crossed product**]{} of the group $G$ over the field $F$. If $\sigma$ is a trivial mapping then we shall denote this ring by $F^\lambda [G]$ and it is a [**twisted group algebra**]{}.
From (\[E:1\]) and (\[E:2\]) follows that $$\lambda (h, 1)^{\sigma(1)}= \lambda (h, 1)= \lambda (1, 1), \qquad
\lambda (1, h)=\lambda (1, 1)^{\sigma(h)}.$$ Clearly,$\widetilde{1}\cdot \lambda (1, 1)^{-1}$is the identity element of the ring $F^\lambda_{\sigma} [G]$ and $$\label{E:3}
\widetilde{g}^{-1}=\lambda (g^{-1}, g)^{-1}\lambda (1,
1)^{-1}\widetilde{g^{-1}}= \widetilde{g^{-1}}\lambda (g,
g^{-1})^{-1}\lambda (1, 1)^{-1}.$$ For the twisted group algebra $F^\lambda [G]$ we can always assume without loss of generality, that the twisting function is normalized, that is $$\lambda (h, 1)= \lambda (1, 1)= \lambda (1, h)=1,$$ so $\widetilde{1}$ is its identity element.
In the theory of ordinary group rings the Lie properties play an important role. Group algebras with the many “good” Lie properties were described during the 70’s using the theory on polynomial identities. Later these results were applied to the study of the group of units. In most cases the Lie structure reflects very well the characteristics of the group of units, there is a close relationship between their properties of them. Here we describe the structure of those crossed products which are upper (lower) Lie nilpotent and Lie $(n,m)$-Engel. We generalize results of Passi, Passman and Sehgal [@pps] which were obtained for the group algebras.
Preliminaries Results
=====================
Let $F^\lambda [G]$ be a twisted group algebra with normalized twisting function $\lambda$. Then $$W=\{w\in G\mid \lambda(g,w)=\lambda(w,g)=1 \;\;\text{for all}\;\;
g\in G\}$$ is a subgroup of $G$. Indeed, if $w_1,w_2, w\in W$ and $g\in G$, then $$\begin{split}
\widetilde{(w_1w_2)}\widetilde{g}&=\widetilde{w_1}\widetilde{w_2}\widetilde{g}=\widetilde{w_1}(\widetilde{w_2g})=\widetilde{w_1w_2g};\\
\widetilde{g}\widetilde{(w_1w_2)}&=\widetilde{g}\widetilde{w_1}\widetilde{w_2}=
\widetilde{(gw_1)}\widetilde{w_2}=\widetilde{gw_1w_2}.
\end{split}$$ So $\lambda(w_1w_2,g)=\lambda(g,w_1w_2)=1$ and $w_1w_2\in W.$ Furthermore, by (\[E:1\]) and (\[E:3\]), it is easy to check that $$\begin{split}
\widetilde{w^{-1}}\widetilde{g}=\widetilde{w}^{-1}\widetilde{g}
=(\widetilde{g}^{-1}\widetilde{w})^{-1}&=\lambda(g,g^{-1})(\widetilde{g^{-1}}\widetilde{w})^{-1}
=\lambda(g,g^{-1})(\widetilde{g^{-1}w})^{-1}\\
&=\lambda(g,g^{-1})\lambda(g^{-1}w,w^{-1}g)^{-1}\widetilde{w^{-1}g}\\
&=\lambda(g^{-1},w)^{-1}\lambda(w,w^{-1}g)^{-1}\widetilde{w^{-1}g}
= \widetilde{w^{-1}g}.
\end{split}$$ Similarly, $\widetilde{g}\widetilde{w^{-1}}= \widetilde{gw^{-1}}$, and this shows that $w^{-1}\in W$, so $W$ is a subgroup.
Let $H$ be a normal subgroup of $G$ such that $H\subseteq W$ and denote by ${\bf I}(H)$ the ideal of $F^\lambda[G]$ generated by the elements $\widetilde{h}-\widetilde{1}$ with $h \in H$. It is easy to see that if $\{u_i\}$ is a transversal of $H$ in $G$ then the elements of the form $\widetilde{u_i}(\widetilde{h}-\widetilde{1})$ with $h\ne 1$ constitute an $F$-basis of the ideal ${\bf I}(H)$.
Define a new function: $\mu: G/H \times G/H \to U(F)$ the following way: Choose two representatives $g_i=u_kh_1$ and $g_j=u_lh_2$ with $h_i\in H\subseteq W$ of the cosets of $H$ in $G$. Then by (\[E:1\]) in $F^\lambda[G]$ we get $$\begin{split}
\lambda(g_i,g_j)&=\lambda(u_kh_1,u_lh_2)\\
&=\lambda(u_k,h_1)^{-1}\lambda(h_1,u_lh_2)\lambda(u_k,h_1u_lh_2)=\lambda(u_k,h_1u_lh_2)
\end{split}$$ and $h_1u_lh_2=u_lh$ for suitable $h\in H$. It is easy to check that $$\begin{split}
\lambda(u_k,h_1u_lh_2)&= \lambda(u_k,u_lh)\\
&=\lambda(u_l,h)^{-1}\lambda(u_k,u_l)\lambda(u_ku_l,h)=\lambda(u_k,u_l).
\end{split}$$ We proved that the twisting function $\lambda$ satisfies the condition $\lambda(g_i,g_j)=\lambda(u_k,u_l)$ and it can define the function $$\label{E:4}
\mu(g_iH,g_jH)=\lambda(g_i,g_j),$$ which is a twisting function of $G/H$. Of course, any element of $F^\lambda [G]$ has the form $\sum_iu_ix_i$ with $x_i\in F^\lambda
[H]$ and $x_i+ {\bf I}(H)=\lambda_i + {\bf I}(H)$ for suitable $\lambda_i\in F$. Now it is easy to see that $F^\lambda [G]/{\bf
I}(H)$ is an $F$-algebra with a basis consisting of the images $\widetilde{u_i}$ of the coset representatives $u_i$ of $H$ in $G$. We obtain the isomorphism: $$\label{E:24}
F^\lambda [G]/{\bf I}(H)\cong F^\mu [G/H].$$ Unlike ordinary group algebras, the twisted group algebra $F^\lambda [G]$ does not have a natural group basis. If the algebra $F^\lambda [G]$ has an $F$-basis $\widetilde{G}=\{\widetilde{g}\mid g\in G\}$ such that for each $g$ of $G$ there exists an element $d_g\in F$ such that the elements of the set $\{d_g\widetilde{g}\mid g\in G\}$ form a group basis for $F^\lambda [G]$, then $F^\lambda [G]$ is called [**untwisted**]{}. In this situation $F^\lambda [G]$ is isomorphic to $FG$ via this diagonal change of the basis. In addition, $F^\lambda [G]$ is called [**stably untwisted**]{} if there exists an extension $K$ of the field $ F$ such that $K^\lambda [G]$ is untwisted.
We use a criteria from [@passm] to verify that a twisted group algebra is untwisted or stably untwisted.
[@passm]\[L:1\] Let $F^\lambda [G]$ be a twisted group algebra.
- $F^\lambda [G]$ is untwisted if and only if there exists an $F$-algebra homomorphism $F^\lambda [G]\to F$.
- $F^\lambda [G]$ is stably untwisted if and only if there exists an $F$-algebra homomorphism of $F^\lambda [G]$ into a commutative $F$-algebra $R$.
Recall that $\{u_i\}$ is a transversal of $H$ in $G$ and each element from $G$ can be written uniquely in the form $g=zu_i$ with $z\in H$.
\[L:5\] Let $F^\lambda [G]$ be a twisted group algebra and assume that the twisting function $\lambda$ of the normal subgroup $H$ of $G$ satisfies the condition $\lambda(h_1,h_2)=1$ for all $h_1,h_2\in H$. Then the algebra $F^\lambda [G]$ can be realized alternatively as a twisted group algebra $F^\tau [G]$ with the following diagonal change of the basis by $$\overline{g}= \begin{cases}
\widetilde{g}, & \quad \text{if }\quad g\in H;\\
\lambda(z,u_j)\widetilde{g} & \quad \text{if }\quad g=zu_j\in G\setminus
H,\quad z\in H,
\end{cases}$$ with twisting function $\tau$. This twisting function has the property $\tau(h,g)=1$ for all $h\in H$ and $g\in G$.
If $\tau(g,h)=1$ for all $h\in H$ and $g\in G$, then $$F^\tau [G]/{\bf I}(H)\cong F^\tau [G/H].$$
[**Proof**]{}. It is easy to see that if $g=h_1u_j$ then $$\begin{split}
\tau(h,g)\overline{hg}&=\overline{h}\cdot
\overline{h_1u_j}=\widetilde{h}\cdot
\widetilde{h_1u_j}\lambda(h_1,u_j)\\
&= \widetilde{h} \cdot\widetilde{h_1}\cdot\widetilde{u_j}=
\widetilde{hh_1}\cdot\widetilde{u_j}=
\lambda(hh_1,u_j)\widetilde{hg}=\overline{hg}.
\end{split}$$ This yields $\tau(h,g)=1$.
If $\tau(g,h)=1$ for all $h\in H$ and $g\in G$, then we can apply (\[E:24\]) to obtain the lemma.
The twisted group algebras satisfying polynomial identities have been classified in [@pass] and this result was modified in [@liu] for the stably untwisted case.
[@liu] \[L:11\] If $F^\lambda[G]$ is a twisted group algebra of positive characteristic $p$ with a polynomial identity of degree $n$ then $G$ has a subgroup $A$ of finite index such that $F^\lambda[A]$ is stably untwisted, the commutator subgroup $A'$ of $A$ is a finite $p$-group and $|G:A||A'|$ is bounded by a fixed function of $n$.
Let $F^\lambda[G]$ be a twisted group algebra. For each $h\in G$ of order $k$ we have $$\label{E:8}
\textstyle\widetilde{h}^{k}=\prod_{i=1}^{k-1}\lambda(h^i,h)\cdot
\widetilde{1}.$$ It is convenient to say that $\mu(h)=\prod_{i=1}^{k-1}\lambda(h^i,h)$is the [**twist**]{} of $\widetilde{h}$ which plays an important role in the study of $F^\lambda[G]$. An $p$-element $u$ of $G$ is called an [**untwisted $p$-element**]{} if the order of $u$ coincides with the order of $\widetilde{u}\gamma$ for some $\gamma\in F$.
\[L:2\] Let $F^\lambda [G]$ be a twisted group algebra of positive characteristic $p$ such that the commutator ideal $F^\lambda
[G]^{[2]}=[F^\lambda [G],F^\lambda [G]]F^\lambda [G]$ is a nil ideal and let $a,b\in G$.
- If $\widetilde{a}$ is a $p$-element, then $a$ is also a $p$-element, its order coincides with the order of $\widetilde{a}$ and its twist is $\mu(a)=1.$
- If $a$ and $b$ are untwisted $p$-elements, and the orders of $\widetilde{a}\gamma_1$ and $\widetilde{b}\gamma_2$ coincide with the order of $a$ and $b$ respectively for some $\gamma_i\in F$ then $ab$ is also a $p$-element, the order $p^l$ of $ab$ coincides with the order of $\widetilde{a}\widetilde{b}\gamma_1\gamma_2$ and $$\mu(ab)=(\gamma_1\gamma_2\lambda(a,b))^{-p^l}.$$
- The group commutator $(a,b)$ is an untwisted $p$-element for all $a,b\in G$ and if $p^m$ is the order of $(a,b)$, then $$\bigl(\lambda(a,b)^{-1}\lambda(b,b^{-1}ab)\lambda(a,(a,b)
\lambda((b,a),(a,b))^{-1}\bigr)^{-p^m}=\mu((a,b)).$$
[**Proof**]{}. $(i)$Let $\widetilde{a}$ be an element of order $p^t$ and let $k$ be the order of $a$. Then (\[E:8\]) yields that $\widetilde{a}^{k}=\mu(a)\widetilde{1}$, so $\widetilde{a}^{kp^t}=\mu(a)^{p^t}\widetilde{1}=\widetilde{1}$. It follows that the twist $\mu(a)$ of $\widetilde{a}$ satisfies the condition $\mu(a)^{p^{t}}=1$. But in the field $F$ of characteristic $p$ this is possible only if $\mu(a)=1$. Hence the order of $a$ coincides with the order of $\widetilde{a}$ and $\mu(a)=1$.
$(ii)$First note that if $F^\lambda [G]^{[2]}$ is a nil ideal, then $$xy\equiv yx \pmod{F^\lambda [G]^{[2]}},\qquad (xy)^{n}\equiv x^ny^n
\pmod{F^\lambda [G]^{[2]}}$$ for all $x,y\in F^\lambda [G]$ and for all $n$. It follows that the nilpotent elements of $F^\lambda [G]$ form an ideal $N$ and $F^\lambda [G]/N$ is commutative.
Let $a$ and $b$ be untwisted $p$-elements, and assume that the orders of $\widetilde{a}\gamma_1$ and $\widetilde{b}\gamma_2$ coincide with the order of $a$ and $b$ respectively for some $\gamma_i\in F.$ Since $\widetilde{a}\gamma_1-\widetilde{1}\in N$ and $\widetilde{b}\gamma_2-\widetilde{1}\in N$, we have $$\widetilde{a}\widetilde{b}\gamma_1\gamma_2-\widetilde{1}=(\widetilde{a}\gamma_1-\widetilde{1})
(\widetilde{b}\gamma_2-
\widetilde{1})+(\widetilde{a}\gamma_1-\widetilde{1})+(\widetilde{b}\gamma_2-\widetilde{1})\in
N,$$ because the nilpotent elements of $F^\lambda [G]$ form an ideal. It follows that $$(\widetilde{a}\widetilde{b}\gamma_1\gamma_2-\widetilde{1})^{p^{l_1}}=0$$ for some $l_1$, which shows that $\widetilde{a}\widetilde{b}\gamma_1\gamma_2$ has order $p^l$. Now $(\widetilde{ab}\lambda(a,b)\gamma_1\gamma_2)^{p^l}=\widetilde{1}
$ implies that $ab$ has order $p^m$ and, by (\[E:8\]), we obtain that $$(\widetilde{ab}\lambda(a,b)\gamma_1\gamma_2)^{p^m}=
\mu(ab)\cdot(\lambda(a,b)\gamma_1\gamma_2)^{p^m}\cdot\widetilde{1}$$ which yields $(\mu(ab)\cdot(\lambda(a,b)\gamma_1\gamma_2)^{p^m})^{p^{l-m}}=1$. Then $m=l$ and $$\mu(ab)(\lambda(a,b)\gamma_1\gamma_2)^{p^m}=1.$$ Hence $ab$ has order $p^l$ and $\mu(ab)= (\lambda(a,b)\gamma_1\gamma_2)^{-p^m}$.
$(iii)$ Obviously, the Lie commutator $[\widetilde{a},\widetilde{b}]$ belongs to the nil ideal $F^\lambda [G]^{[2]}$ for all $a,b\in G$ and the identity $$\label{E:9}
[\widetilde{a},\widetilde{b}]=\widetilde{a}^{-1}\widetilde{b}^{-1}
((\widetilde{a},\widetilde{b})-\widetilde{1})$$ ensures that $(\widetilde{a},\widetilde{b})-\widetilde{1}$ is nilpotent. Then $(\widetilde{a},\widetilde{b})$ has order $p^m$ and an easy computation shows that $$\label{E:10}
(\widetilde{a},\widetilde{b})=\widetilde{(a,b)}\chi((a,b))$$ for some $ \chi((a,b))$ of $F$. The argument of the proof of (i) states that $p^m$ is the order of $(a,b)$ and $$\label{E:11}
(\widetilde{a},\widetilde{b})^{p^m}=\mu((a,b))
\chi((a,b))^{p^m}\widetilde{1}=\widetilde{1}.$$ Moreover, from (\[E:3\]) and (\[E:10\]) follows that $$\begin{split}
\chi((a,b)) =&\widetilde{a}^{-1}\widetilde{b}^{-1}
\widetilde{a}\cdot\widetilde{a^{-1}ba}\lambda(b, (b,a))
\lambda((b,a),(a,b))^{-1}\\
=&\widetilde{a}^{-1}\widetilde{b}^{-1}\widetilde{ba}
\lambda(a,a^{-1}ba)\lambda(b,(b,a)) \lambda((b,a),(a,b)^{-1}\\
=&\lambda(b,a)^{-1}\lambda(a,a^{-1}ba)\lambda(b,(b,a))
\lambda((b,a),(a,b))^{-1}.
\end{split}$$
\[C:3\] If $F^\lambda[G]$ is a twisted group algebra of positive characteristic $p$ such that $F^\lambda [G]^{[2]}$ is a nil ideal then the group commutator $(a,b)$ is an untwisted $p$-element for any $a,b\in G$ and the product of untwisted $p$-elements of $G$ is also an untwisted $p$-element.
Upper and lower Lie nilpotent\
crossed products
===============================
Let $R$ be an associative ring. The lower Lie central series in $R$ is defined inductively as follows: $$\gamma_1(R)=R, \quad \gamma_2(R)=[\gamma_1(R), R], \ldots,
\gamma_n(R)=[\gamma_{n-1}(R), R], \ldots .$$ The two-sided ideal $ R^{[n]}=\gamma_n(R)R$ of $R$ is called the [**$n$-th lower Lie power**]{} of $R$. By Gupta and Levin [@gupt], these ideals satisfy the conditions $$\label{E:5}
R^{[m]}R^{[n]}\subseteq R^{[n+m-2]},\qquad (n, m\geq 2).$$
Let us define by induction a second set of ideals in $R$: $$R^{(1)}=R, \quad R^{(2)}=[R^{(1)}, R]R, \ldots,
R^{(n)}=[R^{(n-1)}, R]R, \ldots$$ The ideal $R^{(n)} $ is called [**the $n$-th upper Lie power**]{} of $R$ and these ideals have the property $$\label{E:6}
R^{(n)}R^{(m)}\subseteq R^{(n+m-1)}, \qquad (n, m\geq 1).$$
Recall that $R$ is called [**upper Lie nilpotent**]{} if $R^{(n)}=0$ for some $n$. Similarly, the ring $R$ with $R^{[m]}=0$ for some $m$ is called [**lower Lie nilpotent**]{}; these classes of rings are different.
First assume that $R$ is lower Lie nilpotent and let $R^{[t]}=0$. For $k\geq 3$ we choose an $l$ such that $l(k-1)+2\geq t$. Then (\[E:5\]) forces $ (R^{[k]})^l= R^{[l(k-1)+2]}=0, $ so $R^{[k]}$ $(k\geq 3)$ is a nilpotent ideal. Note that every element of $R^{[2]}$ has the form $$x=[a_1,b_1]r_1+[a_2,b_2]r_2+\cdots+[a_s,b_s]r_s$$ for some $a_i,b_i,r_i\in R$ and $$\label{E:7}
\begin{split}
([a_j,b_j]r_j)^2=&r_j[a_jb_j,b_j,a_j]r_j+[a_jb_j,b_j,a_j]r_j\\
&+[a_j,b_j,a_j]b_jr_j+ [a_jb_j,b_j,r_j][a_jb_j,b_j]r_j.
\end{split}$$
Let $R$ be of characteristic $p$. Then, by Brauer’s formula, $$x^p=([a_1,b_1]r_1)^p+([a_2,b_2]r_2)^p+\cdots+([a_s,b_s]r_s)^p+z$$ for suitable $z=[c_1,d_1]+[c_2,d_2]+\cdots+[c_q,d_q]\in [R,R]$ and $c_i,d_i\in R$. Now (\[E:7\]) ensures that $([a_j,b_j]r_j)^p\in R^{[3]}$, because $R^{[3]}$ is an ideal. Since the elements $[c_i,d_i]$ and $[c_j,d_j]$ commute modulo $R^{[3]}$, Brauer’s formula implies that $$x^{p^2}=([c_1,d_1])^p+([c_2,d_2])^p+\cdots+([c_q,d_q])^p\in
R^{[3]}.$$ There exists $s$ such that $x^{p^s}=0$ for every $x\in R^{[2]}$, because $R^{[3]}$ is a nilpotent ideal, so the ideal $R^{[2]}$ is nil.
Similar results are valid for upper Lie nilpotent rings $R$.
\[L:3\] Let $F^\lambda [G]$ be a twisted group algebra of positive characteristic $p$ such that either$F^\lambda [G]^{[m]}=0$, or $F^\lambda [G]^{(m)}=0$.If $p^t\geq m$, then
- $b^{p^t}$ is a central element of $G$ for any $b\in
G$;
- for $q\ne p$ each $p$-element $a\in G$ commutes with any $q$-element $c\in G$ and $$\lambda(a,c)=\lambda(c,a).$$
[**Proof.**]{}For $a$ and $b$ of $G$ the well known Lie commutator formula confirms $$[\widetilde{a},\widetilde{b},p^t]=[\widetilde{a},
\underbrace{\widetilde{b},\widetilde{b},\ldots,
\widetilde{b}}_{p^t}]=[\widetilde{a},\widetilde{b}^{p^t}]=0.$$ Hence $\widetilde{a}\widetilde{b}^{p^t}=\widetilde{b}^{p^t}\widetilde{a}$, this yields $ab^{p^t}=b^{p^t}a$ and $\lambda(a,b^{p^t})=\lambda(b^{p^t},a)$ for all $a\in G$. Since any $q$-element $c$ of $G$ can be written as $c=b^{p^t}$ for some $b$, the desired assertion follows.
We present two different proofs for the next result.
\[L:7\] If $F^\lambda [G]$ is a twisted group algebra of $char(F)=p$ such that either $F^\lambda [G]^{[p^t]}=0$ or $F^\lambda
[G]^{(p^t)}=0$, then $G'$ is a finite $p$-group.
[**Proof**]{}. The first proof of the lemma uses the theory of polynomial identities. As we showed before $F^\lambda[G]$ satisfies a polynomial identity, and by Lemma \[L:11\], the group $G$ has a normal subgroup $A$ of finite index such that $F^\lambda[A]$ is stably untwisted and the commutator subgroup $A'$ of $A$ is a finite $p$-group.
We start with some facts which will be used freely.
1\. If $g,h\in G$ and $(g,h)=1$, then $[\widetilde{g},\widetilde{h}]=0$. Indeed, $
[\widetilde{g},\widetilde{h}]=\widetilde{gh}(\lambda(g,h)-\lambda(h,g))$ is nilpotent which is possible only if $\lambda(g,h)-\lambda(h,g)=0.$
2\. If $h$ is central in $G$, then $\widetilde{h}$ is a central element of $F^\lambda [G]$.
3\. By Lemma \[L:2\], $(a,b)$ is a $p$-element for all $a,b\in
G$ and
$$\label{E:18}
(\widetilde{a},\widetilde{b})=\widetilde{(a,b)}\chi((a,b)).$$
Now let $F^\lambda [G]$ be upper (lower) Lie nilpotent and assume that $A$ is abelian. If $P$ is the maximal $p$-subgroup of $A$, then $A=P\times Q$ for a suitable central $p'$-subgroup $Q$, because $G'$ is $p$-group by Lemma \[L:2\]. Moreover, $P$ belongs to the $FC$-center of $G$ and assume that $C_P(g)$ has infinite index in $P$ for some $g$ of $G$. For brevity, put $\chi((g,g_i))=\pi_i$. Clearly, $(g,g_1)\ne 1$ for suitable $g_1\in P$ and, using the fact that $F^\lambda[A]$ is commutative, we have $$[\widetilde{g},\widetilde{g_1}]=\widetilde{g}\widetilde{g_1}(\widetilde{1}-(\widetilde{g_1},\widetilde{g}))
=\widetilde{g}\widetilde{g_1}(\widetilde{1}-\widetilde{(g_1,g)}\pi_1).$$ Since the subset $\{(h, g)\mid h\in P\}$ of $P$ is infinite, there exists $g_2\in P$ such that $$(\widetilde{1}-\widetilde{(g_2,g)}\pi_2)(\widetilde{1}-\widetilde{(g_1,g)}\pi_1)\ne
0.$$ Clearly,$[\widetilde{g_1},\widetilde{g_2}]=0$ and $$\begin{split}
[\widetilde{g},\widetilde{g_1},\widetilde{g_2}]=
[\widetilde{g}\widetilde{g_1}(\widetilde{1}-\widetilde{(g_1,g)}\pi_1),\widetilde{g_2}]
&=[\widetilde{g},\widetilde{g_2}]\widetilde{g_1}(\widetilde{1}-\widetilde{(g_1,g)}\pi_1)\\
&=\widetilde{g}\widetilde{g_2}\widetilde{g_1}(\widetilde{1}-\widetilde{(g_2,g)}
\pi_2)(\widetilde{1}-\widetilde{(g_1,g)}\pi_1).
\end{split}$$ As before, it is easy to see that for each $n$ there exist $g_1,g_2,\ldots, g_n$ in $P$ such that $$(\widetilde{1}-\widetilde{(g_n,g)}\pi_n)(\widetilde{1}-\widetilde{(g_{n-1},g)}\pi_{n-1})
\cdots(\widetilde{1}-\widetilde{(g_1,g)}\pi_1)\ne 0$$ and $$\begin{split}
[\widetilde{g},&\widetilde{g_1},\ldots,\widetilde{g_n}]=\\
&=\widetilde{g}\widetilde{g_n}\widetilde{g_{n-1}}\cdots\widetilde{g_1}(\widetilde{1}-\widetilde{(g_n,g)}\pi_n)
(\widetilde{1}-\widetilde{(g_{n-1},g)}\pi_{n-1})\cdots(\widetilde{1}-\widetilde{(g_1,g)}\pi_1).
\end{split}$$ Clearly $[\widetilde{g},\widetilde{g_1},\ldots,\widetilde{g_n}]\ne
0$ for each $n$, contradicting to the assumption that $F^\lambda
[G]$ is upper (lower) Lie nilpotent. Thus for any $g$ of $G$ the centralizer $C_P(g)$ has finite index in $P$. But this imply that $C_G(g)$ has finite index in $G$, because $Q$ is a central subgroup and $A$ has finite index. Therefore we can suppose below that $G$ is an $FC$-group.
Assume that the $p$-group $G'$ is infinite. Then $P$ is infinite, $b_1=(g_1,g_2)\ne 1$ for suitable $g_1, g_2$ and $$\begin{split}
[\widetilde{g_1},\widetilde{g_2}]=\widetilde{g_2}\widetilde{g_1}((\widetilde{g_1},\widetilde{g_2})-\widetilde{1})
&=\widetilde{g_2}\widetilde{g_1}(\widetilde{(g_1,g_2)}-\widetilde{1})\\
&=\widetilde{g_2}\widetilde{g_1}(\chi((g_1,g_2))\widetilde{b_1}-\widetilde{1}).
\end{split}$$ Now we claim that there exists a sequence $\{g_i\}$ with the following properties: $ g_{2n+1}$, $g_{2n+2}$ and $b_{2n+1}=(g_{2n+1}, g_{2n+2})$ are elements of $$C_G(\{g_1,g_2,\ldots,g_{2n}, b_1,b_3,\ldots,
b_{2n-1} \})$$ and $$(g_{2n+1},g_{2n+2})=b_{2n+1}\notin\langle b_1,b_3,\ldots,
b_{2n-1}\rangle.$$ Indeed, assume that the sequence $g_1,g_2,\ldots,g_{2n}$ is given. The subgroup $$C= C_G(\{g_1, g_2, \ldots, g_{2n},b_1,b_3,\ldots, b_{2n-1} \})$$ of the $FC$-group $G$ has finite index. Neumann’s result [@neu] asserts that in an $FC$-group with infinite commutator subgroup $G'$ the commutator subgroup of the subgroup $C$ is also infinite. Then $C'$ is a group with infinite commutator subgroup and it contains elements $g_{2n+1}$ and $g_{2n+2}$ such that $$b_{2n+1}=(g_{2n+1}, g_{2n+2}) \not\in \langle b_1, b_3, \ldots,
b_{2n-1} \rangle,$$ because $G'$ is a locally finite $p$-subgroup. Recall that if $(a,b)=1$, then $[\widetilde{a},\widetilde{b}]=0$ and we put $\pi_{2n+1} =\chi((g_{2n+1}, g_{2n+2}))$. Now the properties of the sequence imply that $$[\widetilde{g_1}, \widetilde{ g_2} \widetilde{g_3}] =
[\widetilde{g_{1}},\widetilde{ g_2}]\widetilde{g_3}=
\widetilde{g_3}\widetilde{g_2}\widetilde{g_1}(\pi_1\widetilde{b_1}-\widetilde{1}),$$ and $$\begin{split}
[\widetilde{g_1}, & \widetilde{ g_2} \widetilde{g_3}, \widetilde{
g_4} \widetilde{g_5}, \ldots, \widetilde{g_{2n}}
\widetilde{g_{2n+1}}]=\\
& = \widetilde{g_{2n+1}}\widetilde{ g_{2n}}\cdots
\widetilde{g_{2}} \widetilde{g_{1}}
(\pi_{2n-1}\widetilde{b_{2n-1}}-\widetilde{1})\cdots
(\pi_3\widetilde{b_3}-\widetilde{1})
(\pi_1\widetilde{b_1}-\widetilde{1}).
\end{split}$$ Clearly, this is nonzero for each $n$, contradicting that $F^\lambda [G]$ is upper (lower) Lie nilpotent. Therefore $G'$ is a finite $p$-group, as claimed.
Finally, let $A'$ be of order $p^t$. Our assertion is valid for $t=0$ and assume its truth for $t-1$. Lemma \[L:3\] says that $b^{p^t}$ belongs to the center of $G$ for any $b\in G$. It follows that any conjugacy class of $G$, which belongs to $A'$, has $p$-power order or it is central. This yields that $A'$ has central subgroup $L=\gp{c}$ of order $p$ and by Lemma \[L:2\] there exists $\gamma\in F$ such that $\widetilde{c}\gamma$ has order $p$. Then $F^\lambda [G]$ can be realized in a second way as a twisted group algebra $F^\tau [G]$ with the following diagonal change of the basis $$\overline{g}= \begin{cases}
\widetilde{c}^i\gamma^i, & \text{if }\quad g=c^i;\\
\widetilde{g} & \text{if }\quad g\in G\setminus L
\end{cases}$$ with twisting function $\tau$. Since $\overline{c^i}$ is central, Lemma \[L:5\] asserts that $$F^\tau [G]/{\bf I}(L)\cong F^\tau [G/L].$$ Of course, $F^\tau [G/L]$ is upper (lower) Lie nilpotent, so we can apply induction to obtain that $G'$ is a finite $p$-group.
[*The second proof for $char(F)>2$*]{}.For any $a,b,c \in G$ the Lie commutator $[\widetilde{a},\widetilde{b}, \widetilde{c}]$ belongs to the nilpotent ideal $F^\lambda_\sigma [G]^{[3]}$ and let $x$ be a non-zero fixed element from the annihilator of $F^\lambda_\sigma
[G]^{[3]}$. Clearly, $$\label{E:13}
\widetilde{a}\widetilde{b}
\widetilde{c}x-\widetilde{b}\widetilde{a} \widetilde{c}x=
\widetilde{c}\widetilde{a}\widetilde{b}x-
\widetilde{c}\widetilde{b} \widetilde{a}x,$$ and without loss of generality we can assume that $1 \in
Supp(x)$. Assume that $Supp(x)=\{x_1=1, x_2,\ldots,x_l\}$. It is convenient to distinguish the following cases:
[**1.**]{}$Supp(\widetilde{c}\widetilde{a}\widetilde{b}x)\cap
Supp(\widetilde{c}\widetilde{b} \widetilde{a}x)$ is not empty. Then $cabx_i=cbax_j$ for suitable $i$ and $j$, so the commutator $(a,b)$ can be written as $x_jx_i^{-1}$.
[**2.**]{}$Supp(\widetilde{c}\widetilde{a}\widetilde{b}x)\cap
Supp(\widetilde{c}\widetilde{b} \widetilde{a}x)$is an empty set. The length of the right side of (\[E:13\]) is $2l$ and thus the length of the left one is also $2l$. This means that $$Supp(\widetilde{a}\widetilde{b} \widetilde{c}x)\cap
Supp(\widetilde{b}\widetilde{a} \widetilde{c}x)=\oslash.$$ Now assume that$Supp(\widetilde{a}\widetilde{b}
\widetilde{c}x)\cap
Supp(\widetilde{c}\widetilde{a}\widetilde{b}x)$ and $Supp(\widetilde{a}\widetilde{b} \widetilde{c}x)\cap
Supp(\widetilde{b}\widetilde{a} \widetilde{c}x)$are not empty sets. There exist $x_i,x_j,x_t,x_r\in Supp(x)$ such that $abcx_i=cabx_j$,$abcx_t=cbax_r$ and the commutator $(a,b)$ coincides with an element of the form $x_{i_1}^{\pm 1}x_{i_2}^{\pm
1}x_{i_3}^{\pm 1}x_{i_4}^{\pm 1}.$ Similar statement is valid if $$Supp(\widetilde{b}\widetilde{a}
\widetilde{c}x)\cap
Supp(\widetilde{c}\widetilde{a}\widetilde{b}x)\quad \text{ and
}\quad Supp(\widetilde{b}\widetilde{a} \widetilde{c}x)\cap
Supp(\widetilde{b}\widetilde{a} \widetilde{c}x)$$ are not empty sets. It remains to consider one of the following two subcases:
[**2.1**]{} $\widetilde{a}\widetilde{b} \widetilde{c}x=
\widetilde{c}\widetilde{a}\widetilde{b}x$ and $\widetilde{b}\widetilde{a} \widetilde{c}x=
\widetilde{c}\widetilde{b}\widetilde{a}x$. Then the commutators $(ab,c)$ and $(ba,c)$ can be written as $x_jx_i^{-1}$.
[**2.2**]{}$\widetilde{a}\widetilde{b} \widetilde{c}x=
-\widetilde{c}\widetilde{b}\widetilde{a}x$ and $ \widetilde{c}\widetilde{a}\widetilde{b}x=-
\widetilde{b}\widetilde{a} \widetilde{c}x$. This yields that $$abcx_i=cbax_j\quad \text{and}\quad (ab,c)=x_jx_i^{-1}.$$
Now, if $p\ne 2$, we put $c=b^{-1}$ and then $(ab,c)=(a,b)^{-1}$. The foregoing immediately implies that there are only finitely many group commutators of the form $(a,b)$ with $a,b\in G$; each commutator $(a,b)$ is a $p$-element which has only a finite number of conjugates. Then, as well known, $G'$ is a finite $p$-group.
\[T:1\] Let $F^\lambda_{\sigma} [G]$ be a crossed product of a group $G$ and the field $F$ of characteristic $0$ or $p$. Then
- Any upper (lower) Lie nilpotent crossed product $F^\lambda_{\sigma} [G]$ is a twisted group algebra.
- The twisted group algebra $F^\lambda[G]$ is lower (upper) Lie nilpotent if and only if one of the following condition holds:
- $F^\lambda [G]$ is a commutative algebra (i.e $G$ is abelian and the twisting function is symmetric).
- $char(F)=p$, $G$ is a nilpotent group with commutator subgroup of $p$-power order and the untwisted $p$-elements of $G$ form a subgroup. Moreover, for any $a,b\in G$ the group commutator $(a,b)$ is an untwisted $p$-element and $\bigl(\lambda(a,b)^{-1}\lambda(b,b^{-1}ab)\lambda(a,(a,b)
\lambda((b,a),(a,b))^{-1}\bigr)^{-p^m}=\mu((a,b)), $ where $p^m$ is the order of $(a,b)$.
[**Proof.**]{}Let $F^\lambda_{\sigma} [G]$ be a lower (upper) Lie nilpotent crossed product of characteristic 0 or $p$ and let $H=ker \sigma.$ The twisted group algebra $F^\lambda [H]$ is a subring of the crossed product $F^\lambda_{\sigma} [G]$ and, by [@bb], for every non-zero ideal $I$ of $F^\lambda_{\sigma}
[G]$ we have $$\label{E:12}
F^\lambda [H]\cap I \ne 0.$$ Recall that if $char(F)=p$ and $\Delta(G)$ has no element of order $p$, then Theorem 3.5 from [@mpass] states that $F^\lambda_{\sigma} [G]$ is a semiprime ring. For $char F=0$ according to Corollary 6 from [@pas] the algebra $F^\lambda[H]$ is semiprime and (\[E:12\]) ensures that $F^\lambda_{\sigma} [G]$ is also semiprime. It follows that $
F^\lambda_\sigma [G]^{[3]}=0$ and (\[E:7\]) implies that $([a,b]r)^2=0$ for all $a,b,r\in F^\lambda_{\sigma} [G]$. Clearly, $[a,b]F^\lambda_{\sigma} [G]$ is a nilpotent ideal, but this is possible only if $F^\lambda_\sigma [G]^{[2]}=0$ and then $F^\lambda_\sigma [G]$ is a commutative twisted group algebra, as required.
Finally assume that $p$ divides the order of some element of $\Delta(G)$ and $F^\lambda_\sigma [G]$ is a noncommutative crossed product. Then the Lie commutator $
[\widetilde{a},\widetilde{1}\alpha]=\widetilde{a}(\alpha-\alpha^{\sigma(a)})
$ belongs to the nil ideal $F^\lambda_\sigma [G]^{[2]}$ for every $\alpha\in F$ and the element $\alpha-\alpha^{\sigma(a)}$ of the field $F$ is zero, because it is nilpotent. Thus $\sigma$ is trivial and so the crossed product $F^\lambda_\sigma [G]$ is a twisted group algebra.
Let $F^\lambda [G]$ be a twisted group algebra such that $char(K)=p$ and $F^\lambda [G]^{[p^t]}=0$. By Lemma \[L:2\], the commutator subgroup $G'$ is a $p$-group and Lemma \[L:7\] forces that $G'$ is finite. Furthermore, Lemma \[L:3\] says that $b^{p^t}$ belongs to the center of $G$ for every $b\in G$. So the quotient $G/C$ of $G$ by the center $C$ is a $p$-group of finite exponent with finite commutator subgroup. Clearly, the orders of those conjugacy classes of the group $G/C$, which are contained in the finite normal $p$-subgroup $(G/C)'$, are $p$-powers. Hence $(G/C)'$ has a nontrivial central subgroup $L$ and thus $G/C$ is a nilpotent group. Remark that Lemma \[L:2\] confirms the remaining statements.
The converse statement was proved in [@bk], so the proof is complete.
Using the notation of untwisting, Theorem \[T:1\] can be formulated as
\[C:1\] Let $F^\lambda [G]$ be a twisted group algebra of a group $G$ and a field $F$ of characteristic $0$ or $p>0$. The algebra $F^\lambda[G]$ is lower (upper) Lie nilpotent if and only if one of the following conditions holds:
- $F^\lambda [G]$ is commutative;
- $char(F)=p$, $G$ is a nilpotent group such that $G'$ is a finite $p$-group and $F^\lambda[G]$ is stably untwisted.
[**Proof.**]{} Let $F^\lambda [G]$ be a noncommutative lower (upper) Lie nilpotent algebra. By Theorem \[T:1\], $char(F)=p$, $G$ is a nilpotent group and $G'$ has $p$-power order. As we remarked before, the nilpotent elements of $F^\lambda [G]$ form an ideal $N$ and $F^\lambda [G]/N$ is commutative. By Lemma \[L:2\] for any $g$ of $G'$ we can choose $\gamma_g\in F$ such that the order of $\widetilde{g}\gamma_g$ coincides with the order of $g$. Then $\widetilde{g}\gamma_g-\widetilde{1}$ is nilpotent and belongs to the radical $J(F^\lambda [G'])$ of $F^\lambda [G']$. So $J(F^\lambda [G'])$ is nilpotent of codimension 1 and from $$J(F^\lambda [G'])\cdot F^\lambda [G]= F^\lambda [G]\cdot
J(F^\lambda [G'])$$ follows that $J(F^\lambda [G'])\cdot F^\lambda [G]$ is a nilpotent ideal. For all $a,b\in G$ we have $$\label{E:21}
[\widetilde{a},\widetilde{b}]=\widetilde{a}^{-1}\widetilde{b}^{-1}
((\widetilde{a},\widetilde{b})-\widetilde{1})=\widetilde{a}^{-1}\widetilde{b}^{-1}(\widetilde{(a,b)}\chi((a,b))-
\widetilde{1}),$$ and if $p^m$ is the order of the group commutator $(a,b)$ then by Lemma 4 we obtain $$\bigl(\widetilde{(a,b)}\chi((a,b))-\widetilde{1}\big)^{p^m}=\mu((a,b))\chi((a,b))^{p^m}-\widetilde{1}=0.$$ Consequently $[\widetilde{a},\widetilde{b}]\in J(F^\lambda
[G'])\cdot F^\lambda [G]$, which implies that $$F^\lambda [G]/J(F^\lambda
[G'])\cdot F^\lambda [G]$$ is a commutative algebra. Now Lemma \[L:1\] states that $F^\lambda [G]$ is stably untwisted.
Conversely, if $F^\lambda [G]$ is stably untwisted, then there exists an extension $K$ of the field $F$ such that $K^\lambda
[G]$ is isomorphic to the ordinary group algebra $KG$ via a diagonal change of basis and the result for $KG$ has been already known. Since $F^\lambda [G]$ is a subalgebra of $K[G]$, it is lower (upper) Lie nilpotent.
$(n,m)$-Engel crossed products
==============================
Let $R$ be an associative ring and let $n,m$ be fixed positive integers. If $$[a,\underbrace{b^m,b^m,\ldots,b^m}_n]=0$$ for all elements $a,b\in R$, then $R$ is called [**$(n,m)$-Engel.**]{}
Clearly, an $(n,m)$-Engel ring satisfies the polynomial identity $$[x,\underbrace{y^m,y^m,\ldots,y^m}_n].$$ Let $p^t$ be the smallest positive integer such that $n\leq p^t$ and let $m=p^lr$ with $(p,r)=1$. If $F\gp{x,y}$ is a noncommutative polynomial ring with indeterminates $x$ and $y$ over a field $F$ of characteristic $p$, then $$\label{E:14}
\begin{split}
[x,\underbrace{y^m,y^m,\ldots,y^m}_{p^t}]&=[x,\underbrace{y^{p^lr},y^{p^lr},\ldots,y^{p^lr}}_{p^t}]\\
&=[x,y^{p^{l+t}r}]= [x,\underbrace{y^r,y^r,\ldots,y^r}_{p^{l+t}}].
\end{split}$$ Therefore, if $R$ is an $(n,m)$-Engel ring of $char(R)=p>0$, then it is $(p^{l+t},r)$-Engel ring, too.
\[T:2\] Let $F^\lambda_{\sigma} [G]$ be a crossed product of a group $G$ and a field $F$ of positive characteristic $p$.
- Any $(n,m)$-Engel crossed product $F^\lambda_{\sigma}
[G]$ is a twisted group algebra.
- If $F^\lambda [G]$ is an $(n,m)$-Engel twisted group algebra, then either $F^\lambda [G]$ is commutative, or the following conditions hold:
- $G$ has a normal subgroup $B$ of a finite index such that commutator subgroup $B'$ has $p$-power order, the $p$-Sylow subgroup $P/B$ of $G/B$ is a normal subgroup, $G/P$ is a finite abelian group of an exponent that divides $m$ and $P$ is a nilpotent subgroup.
- the untwisted $p$-elements of $G$ form a subgroup and for all $a,b\in G$ the commutator $(a,b)$ is an untwisted $p$-element such that $\bigl(\lambda(a,b)^{-1}\lambda(b,b^{-1}ab)\lambda(a,(a,b)
\lambda((b,a),(a,b))^{-1}\bigr)^{-p^m}=\mu((a,b))$, where $p^m$ is the order of $(a,b)$. Moreover, $F^\lambda[B]$ is stably untwisted and $|G:B||B'|$ is bounded by a fixed function of $n$ and $m$.
[**Proof.**]{} Let $F^\lambda_\sigma[G]$ be an $(n,m)$-Engel crossed product of characteristic $p>0$. Then we can apply Theorem 3 of Kezlan [@kez] which states that $F^\lambda_\sigma [G]^{[2]}$ is a nil ideal. Hence the nilpotent elements of $F^\lambda_\sigma [G]$ form an ideal $N$ and $F^\lambda_\sigma [G]/N$ is commutative. Clearly,$[\widetilde{a},\widetilde{1}\cdot
\alpha]=\widetilde{a}(\alpha-\alpha^{\sigma(a)})$belongs to the nil ideal $F^\lambda_\sigma [G]^{[2]}$ for every $\alpha\in
F$. It follows that the element $\alpha-\alpha^\sigma(a)$ of the field $F$ is zero, because it is nilpotent. So $\sigma$ is trivial and hence $F^\lambda_\sigma [G]=F^\lambda [G]$ is a twisted group algebra.
First assume that $G$ has no element of order $p$. By Corollary 2 from [@passm], the nil ideal $F^\lambda[G]^{[2]}$ is zero. So the twisted group algebra $F^\lambda [G]$ is commutative.
Now let $G$ be a noncommutative group with a $p$-element. Without loss of generality, by (14), we can assume that $(m,p)=1$. Lemma \[L:2\] ensures that every group commutator and their products are $p$-elements. Therefore $G'$ is a $p$-group. Choose the smallest positive integer $p^t$ with $n\leq p^t$ and let $a,
b\in G$. The Lie commutator formula ensures that $$[\widetilde{a},\widetilde{b}^m,p^t]=[\widetilde{a},
\underbrace{\widetilde{b}^m,\widetilde{b}^m,\ldots,
\widetilde{b}^m}_{p^t}]=[\widetilde{a},\widetilde{b}^{mp^t}]=0.$$ This yields $\widetilde{a}\widetilde{b}^{mp^t}=\widetilde{b}^{mp^t}\widetilde{a}$. Hence $b^{mp^t}$ is central for any $b\in G$. By Lemma \[L:11\],$G$ has a normal subgroup $A$ of finite index such that commutator subgroup $A'$ has $p$-power order. Our aim is to prove that if $B$ is the subgroup generated by $A$ and the center of $G$, then $F^{\lambda}[B]$ is stably untwisted. Of course, $B'$ is a finite $p$-group, so the ideal $I=J(F^\lambda
[B'])\cdot F^\lambda [B]$ is nilpotent, because the nilpotent elements of $F^\lambda [G]$ form an ideal. Indeed, if $p^r$ is the order of the commutator $(c,d)$ of $B$, then Lemma \[L:2\] ensures that $((\widetilde{c},\widetilde{d})-1)^{p^r}=0$and $$(\widetilde{c},\widetilde{d})-1\in J(F^\lambda [B'])\cdot
F^\lambda [B].$$ Then (\[E:21\]) asserts that$[\widetilde{c},\widetilde{d}]\in J(F^\lambda [B'])\cdot
F^\lambda [B]$,and $ F^\lambda [B]/J(F^\lambda
[B'])\cdot F^\lambda [G]$is a commutative algebra. Now, by Lemma \[L:1\], $F^\lambda_{\sigma} [B]$ is stably untwisted, as required.
Now $b^{mp^t}$ is central for any $b\in G$ and it belongs to $B$ and the $p$-Sylow subgroup $P/B$ of $G/B$ is normal, because the commutator subgroup of $G$ is a $p$-group. Then, by the Schur-Zassenhaus’s theorem (6.2.1 theorem of [@gor]), $G/B$ is a semidirect product of the finite $p$-Sylow subgroup $P/B$ and a finite abelian group $M/B$ whose exponent divides $m$. Since $B^{mp^t}$ is a central subgroup, there remains to show that $P/B^{mp^t}$ is nilpotent. To prove this we proceed by induction on the order of the commutator subgroup $L$ of $B/B^{mp^t}$. First suppose that $L'=\gp{1}$. The abelian group $B/B^{mp^t}$ is a direct product of the $p$-subgroup $D$ and a subgroup $H$ whose exponent divides $m$. The finite $p$-group $P/B$ acts by conjugation on these subgroups. Of course for every $d\in
P/B^{mp^t}$ and $h\in H$ the $p'$-element $d^{-1}hd$ coincides with $h(h,d)$ and the commutator $(h,d)$ is a $p$-element. Therefore, the action of $P/B$ on $H$ is trivial and $P/B$ acts as a finite $p$-group on $D$.
Define the subgroups$D_1=(P/B^{mp^t},D)$ and $D_j=(P/B^{mp^t},D_{j-1})$for $j>1$. By Lemma V.4.1 [@sh], this sequence of subgroups is such that $D_l=\gp{1}$ for some $l$. But the finite $p$-group $P/B$ is nilpotent, so a suitable term $K_s$ of the lower central series of the group $K=P/B^{mp^t}$ is contained in $B/B^{mp^t}$ and $$K_{s+1}=(K,K_s)\subseteq (K,B/B^{mp^t})= (K,D\times H)
\subseteq (K,D)=D_1,$$ because $H$ is central in $P/B^{mp^t}.$ Similarly, we conclude that $K_{s+i}\subseteq D_i$ and the group $P/B^{mp^t}$ must be nilpotent, because $D_l=1$.
Finally, assume that $L'\not=\gp{1}$. The orders of those conjugacy classes of the group $P/B^{mp^t}$, which are contained in the finite normal $p$-subgroup $L'$, are $p$-powers. Hence $L'$ has a nontrivial central subgroup, and by induction on the order of $L'$, we see that $P/B^{mp^t}$ is nilpotent. Consequence, $P$ is nilpotent as asserted.
\[C:2\] The twisted group algebra $F^\lambda [G]$ of positive characteristic $p$ is $n$-Engel if and only if either $F^\lambda
[G]$ is commutative, or the following conditions hold:
- $G$ is a nilpotent group with a normal subgroup $B$ of a finite $p$-power index, $B'$ is a finite $p$-group and $F^\lambda[B]$ is stably untwisted;
- the untwisted $p$-elements of $G$ form a subgroup, the commutator $(a,b)$ is an untwisted $p$-element for all $a,b\in G$ and $$\label{E:15}
\bigl(\lambda(a,b)^{-1}\lambda(b,b^{-1}ab)\lambda(a,(a,b)
\lambda((b,a),(a,b))^{-1}\bigr)^{-p^m}=\mu((a,b)),$$ where $p^m$ is the order of $(a,b)$.
[**Proof.**]{} There remains to prove only the sufficiency of these conditions, and as before, we can assume that $F$ is an algebraically closed field of characteristic $p$.
Let $G$ be a nilpotent group with a normal abelian subgroup $B$ of a finite $p$-power index. Then for any $a,b$ of $B$ the twist of $\widetilde{(a,b)}$ is equal to 1. So (\[E:15\]) asserts that $F^\lambda [B]$ is a commutative algebra. Now we try to adapt the method of the proof of Theorem V.6.1 from [@sh]. For this we need additional information about the nilpotent groups $G$ with a normal abelian subgroup $B$ of index $p^s$ and about the twisted group algebras. Certainly, $(B,G^{p^s})=1$ and Lemma V.6.2 from [@sh] implies that $(B, G)^{p^m}=1$ for some $m\geq s$. It follows that $$(G^{p^{s+m}},G) \subseteq (B^{p^m},G)=\gp{1}.$$ Hence, if $t=s+m$, then $g^{p^t}$ is central in $G$ and (\[E:15\]) confirms that $[\widetilde{g^{p^t}},\widetilde{h}]=0$ for any $h\in G$. So $\widetilde{g}^{p^t}$ is central. Now for every $y=\sum_{g\in G}c_g\widetilde{g}$ of $F^\lambda [G]$ we have that $\sum_{g\in G} c_g^{p^t} \widetilde{g}^{p^t}$is central in $F^\lambda [G]$ and $$\textstyle
y^{p^t}= \sum_{g\in G} c_g^{p^t} \widetilde{g}^{p^t} + y_1,$$ for suitable $ y_1\in [F^\lambda [G],F^\lambda [G]]$. For all $a,b\in G$ we have $$\label{E:22}
[\widetilde{a},\widetilde{b}]=\widetilde{a}^{-1}\widetilde{b}^{-1}
((\widetilde{a},\widetilde{b})-\widetilde{1})=\widetilde{a}^{-1}\widetilde{b}^{-1}
(\widetilde{(a,b)}\chi((a,b))-\widetilde{1})$$ and if $p^m$ is the order of $(a,b)$ then by (\[E:15\]), $$\bigl(\widetilde{(a,b)}\chi((a,b))-\widetilde{1}\big)^{p^m}=\mu((a,b))\chi((a,b))^{p^m}-\widetilde{1}=0.$$ Consequently $[F^\lambda [G],F^\lambda [G]] \subseteq
F^\lambda[G]\cdot J(F^\lambda [G'])$. Since the subgroup $D=(G,
B)$ is normal in $G$ and $B/D$ is central in $G/D$ of index$|G/D:B/D|=p^s$,it follows that $G'/D$ is a finite group of order $p^l$. By Theorem 1.6 [@passr], we have $$(F^\lambda[G]\cdot J(F^\lambda [G'])^{p^l}\subseteq
J(F^\lambda[D])\cdot F^\lambda [G].$$ It follows that$\textstyle y_1^{p^l}= \sum_ {i=1}^{p^s} z_i
\widetilde{t_i}$,where $t_1=1, t_2,\ldots,t_{p^s}$ is a transversal of $B$ in $G$. Furthermore, $$z_i\in F^\lambda[B]\cap J(F^\lambda[D])\qquad (i=1, 2,\ldots, p^s)$$ are nilpotent elements with nilpotency index at most $p^m$ and each of these elements commute, because $B$ is abelian. The inner automorphisms of $\widetilde{G}$ induces a finite group of automorphisms $T$ on $\widetilde{B}$. The action of $T$ on $z_1,
z_2,\ldots,z_{p^s}$ produces only finitely many images and denote by $L$ the subring of the commutative algebra $F^\lambda[B]$ generated by these images. Of course $L$ is nilpotent, its nilpotency index is at most$p^r=p^{m+s+1}|T|$which does not depend on $L$. Clearly, $y_1^{p^{l+r}}=0$, and by the foregoing, $$\textstyle y^{p^{t+l+r}}= \sum_{g\in G} c_g^{p^{t+l+r}}
\widetilde{g}^{p^{t+l+r}}$$ is central in $FG$. By the Lie commutator identity we obtain that $$[x,y,p^{t+l+r}]=[x,y^{p^{t+l+r}}]=0$$ and $FG$ is Lie $p^{t+l+r}$-Engel.
Finally, let $B'$ be of order $p^t$. Our assertion is valid for $t=0$, assume its truth for $t-1$. The normal subgroup $B'$ of a nilpotent group contains a central cyclic subgroup $L=\gp{c\mid\,c^p=1}$. Now we take $F^\lambda G$ and make the following change of basis: $$\overline{g}= \begin{cases}
\widetilde{g}, quad & \text{if }\quad g\in G\setminus L;\\
\widetilde{c^i}\sqrt{{\mu(c)}^{-i}} \quad & \text{if }\quad g=c^i\in
L.
\end{cases}$$ Then $ \rho (c^i,c^j)= 1$ and $\overline{c}^p=\widetilde{1}$. By Lemma \[L:5\], $F^\lambda[G]$ can be realized in a second way as a twisted group ring with new basis $\{\overline{\overline{g}}\}$ and a twisting function $\tau (g,h)$ which satisfies $\tau(c^k,g)=1$. Clearly,$\tau(c^k,g)=\tau(g,c^k)=1$.By (\[E:24\]), we have that $$F^\lambda G/{\bf I}(L)\cong F^\tau [G/L]$$ and by (\[E:4\]),$ \tau(g_iH,g_jH)=\lambda(g_i,g_j)$.Hence the twisting function $\tau(a,b)$ satisfies the required conditions, and by the inductive hypothesis, $F^\mu [G/L]$ is $p^m$-Engel for some $m$. It follows that $[x,y,p^m]=[x,y^{p^m}]\in {\bf I}(L)$for all $x,y\in
F^\lambda G$, so $[x,y^{p^m}]=(\overline{c}-1)z$for some $z\in F^\lambda G$. Since $\overline{c}$ is central, we have $$\textstyle
[x,y^{p^m},p]= [x,\underbrace{y^{p^m},y^{p^m},\ldots
,y^{p^m}}_p]\in (\overline{c}-1)^p F^\lambda G =0.$$ This implies that $[x,y,p^{m+1}]=0$ and the proof is complete.
| |
BACKGROUND OF THE INVENTION
The present invention generally relates to toys, and more particularly to a whirling toy which can be played alone or competitively between two or more players.
An object of the present invention is to provide a whirling toy which is simple in construction and economical to manufacture.
It is another object of the present invention to provide a whirling toy which can be played alone or competitively between two or more players.
It is still another object of the present invention to provide a whirling toy which can be tailored to provide long periods of recreation for young children, while it can also be played with high degrees of sophistication requiring great coordination, dexterity, and knowledge of physical principles.
It is yet another object of the present invention to provide a whirling toy which includes streamers or the like for stabilizing the whirling toy during its rotary movements prior to release into its trajectory.
It is a further object of the present invention to provide a whirling toy which includes sound producing means for assuring that when the toy is played competitively all the players have the same advantage.
It is still a further object of the present invention to provide a whirling toy which is easy and convenient to play, and which enables the player to become proficient with practice.
It is yet a further object of the present invention to provide a whirling device of the type generally described above which is rugged in construction.
SUMMARY OF THE INVENTION
In order to achieve to above objects, as well as others which will become apparent hereafter, a whirling toy in accordance with the present invention comprises a weighted object. An elongate member is provided having a predetermined length and attached at one end thereof to said weighted object. In this manner, holding said elongate member at the other end thereof permits whirling of said weighted object in a generally vertical plane about a radius approximately equal to said predetermined length and permits release of said weighted object in a generally upward direction. Indication means are advantageously provided for determining the distance of said weighted object after its return to the ground from the point of original release. The weighted object may include sound producing means associated therewith which is responsive to the flow of air proximate to said weighted object while the same is being whirled in the air. In this manner, said weighted object produces an audible sound during whirling.
BRIEF DESCRIPTION OF THE DRAWINGS
A more complete appreciation of the invention and many of the attendant advantages thereof will be readily obtained by reference to the following detailed description when considered in connection with the accompanying drawings wherein:
FIG. 1 is a side elevational view of a whirling toy in accordance with the present invention, as shown at a moment of its rotary motion in a vertical plane when the object and the string by which it is held by the user are in a horizontal plane with the hand of the user, this figure showing two possible trajectories for the weighted object when the same is released at two different moments of time during its whirling motion;
FIG. 2 is a schematic representation of a possible trajectory of the weighted object when released substantially at the center of a target pattern, showing the position where the object lands when it falls to the ground;
FIG. 3 illustrates an alternate embodiment of the weighted object of the whirling toy which includes means for producing an audible sound when the whirling velocity reaches a predetermined threshold value;
FIG. 4 is an enlarged cross-sectional view of a sound-producing means of the type which can be used in the weighted object shown in FIG. 3;
FIG. 5 illustrates the details of a threshold velocity detecting means used in conjunction with the whirling toy shown in FIG. 3, the velocity sensing element being in the nature of a cover biased by a coil spring;
FIG. 6 is similar to FIG. 5, except that the cover is biased by a helical spring; and
FIG. 7 is similar to FIGS. 5 and 6, except that the cover is biased by a leaf spring.
DETAILED DESCRIPTION
Referring now to the figures, in which the same reference numerals are used to designate the identical or similar parts throughout, and first referring to FIG. 1, the whirling toy in accordance with the present invention is generally identified by the reference numeral 10.
The whirling toy 10 includes a weighted object 12. The actual weight or shape of the weighted object 12 is not critical. However, for aerodynamic reasons, the weighted object should preferably be streamlined, for reasons which will become apparent hereafter.
An elongate member, which is in the nature of a spring 14, is attached at one end thereof to the weighted object 12. For this purpose, any string connecting means 16 can be used. In order to stabilize the weighted object during whirling at high speeds, the weighted object is advantageously provided with at least one streamer 18 attached to the weighted object substantially as shown. In this manner, the streamers generally follow the path of the weighted object along the path of travel thereof.
The weighted object may comprise, for example, a small sack filled with sand. However, as noted, this is not critical and any suitable weighted object can be used.
In use, the user 20 grips the other end of the string 14 at a point O and whirls the object along a circular path in a horizontal plane. The weighted object 12 thus whirls about a radius approximately equal to a predetermined length r of the string 14. Release of the weighted object, as will be discussed in connection with FIG. 2, permits the weighted object to be cast in a generally upward direction, at a velocity which is a function of the length r and the whirling speed.
Referring to FIG. 2, there is shown a target pattern 22 comprising concentric circles C1, C2 and C3 of the type normally used in target practice or shooting in conjunction with other target games. In using the whirling toy, the user or player stands in the center of the target pattern 22 at the point designated by "C". The user whirls the weighted object 12 as discussed above and suggested in FIG. 1 about a radius r at the point O. Preferably, the user whirls the weighted object 12 such that the point O coincides as closely as possible to the center C of the target pattern 22.
Referring to FIGS. 1 and 2, it will be evident that when the weighted object is released at the moment when the object and its string 14 are in a common horizontal plane with the hand of the user 20, the weighted object 12 will continue to move in a substantially vertical direction at an angle &agr;.sub.1 approximately equal to 90° with the horizontal. The velocity v.sub.1 is a function of how rapidly the weighted object is rotated or the angular velocity &ohgr; and the length of the string r since v=&ohgr;r. Absent wind velocities, the weighted object will climb to a predetermined height which is a function of v.sub. 1 and return along its path of ascent to land at or very near to the point C of original release. However, releasing the weighted object 12 at the aforementioned ideal moment requires considerable coordination and timing which, however, can be developed with practice.
Referring to FIG. 2, it will be seen that when the weighted object is whirled about the point O and released at an angle &agr;.sub. 2, it rises along a trajectory to a point P which is at a maximum height h. sub. o above the ground. Once it reaches this maximum height, the object begins to descent along the path shown. Under these circumstances, the maximum height is given by the following expression: ##EQU1## where v.sub. o is the initial velocity, &agr; is the angle of release with respect to the horizontal plane, and g represents the effect of gravity. An important expression for purposes of the game of the present invention, as will be more fully apparent hereafter, is the expression for the horizontal range R, shown as d.sub.o in FIG. 2. This is the distance travelled by the object along the horizontal direction from the point of release at C when the object is propelled upwardly at an angle other than 90° to the horizontal at&agr;=90°, sin 2. alpha.=0 so that R=0!, in the absence of wind disturbances. This expression is: ##EQU2## As will be seen, therefore, the horizontal range or displacement along the ground is proportional to the square of the initial velocity as well as proportional to the sine of twice the angle . alpha.. Stated otherwise, for the same angle &agr;, the greater the initial velocity or the more rapidly the object is whirled, the greater will be the range. On the other hand, for the same initial velocities, the range increases as the object is cast at angles which deviate more from the vertical direction. Based on this last expression (2), if the object is cast precisely in the vertical direction so that &agr;=90. degree., it doesn't much matter what the initial velocity is since the object will simply return along the path of rise and fall at the point of release. However, if &agr; is any angle other than 90°, there are two factors, as aforementioned, which must be considered in using the toy, namely the initial velocity and the angle of release.
Since it is very rare when there are no wind currents at all, more advanced players may also wish to take into account the effects of wind velocity. Since linear displacement along the horizontal direction is simply a function of the wind velocity as well as the time that the object is in the air, the errors attributed to wind will be minimized when the time that the object is in the air is also minimized. However, as will be discussed hereafter, a lower limit may be set on the time by requiring that the object be cast into the air at a minimum initial velocity. This will also therefore result in the object rising and falling through the air during a predetermined time interval. In this connection, it may also be noted that the effects of wind velocity may either aid or aggravate initial errors during release of the object. Clearly, referring to FIG. 2, if the object is cast at an angle &agr;. sub.o and would normally land a distance d.sub.o from the initial point C, in the absence of wind velocity, the distance d.sub.o would be increased in the presence of wind moving from left to right as viewed in FIG. 2, while the distance d.sub.o would be decreased in the presence of a wind movement from right to left as viewed in FIG. 2.
As suggested, the object of the game is to whirl the object 12 at a relatively high speed and release the same in a vertical direction. In most instances, especially with those untrained, the object will not move along a vertical direction as represented by the velocity v.sub.1 in FIG. 1, but will be cast off at a slight angle from the vertical, as suggested by the velocity v.sub.2. Under these circumstances, the object will not fall back to the initial starting point C in FIG. 2, but will be propelled along a horizontal range or distance which is a function of factors or parameters which can be controlled by the player.
Once the object falls to the ground, it can be measured in relation to the starting point C either by the use of a target pattern 22 or by simply using a measuring device of any convenient type, such as a tape measure 24.
When playing alone, of course, the player can simply practice and cast the object at any initial velocity v.sub.o that he or she wishes. However, because the horizontal range is a function of the square of the initial velocity, it would clearly give a player in competition an advantage if he could cast his object at a smaller initial velocity. To assure that all players have the same advantage, and that all the objects cast in competition have at least a minimum initial velocity V. sub.o, the weighted object 12 is advantageously provided with means for producing an audible sound upon attaining the minimum velocity. The specific means for achieving or producing such an audible signal is not critical for purposes of the present invention, and any such means can be used which will provide an audible signal when a player whirls the object 12 at a fast enough speed.
Referring to FIG. 3, there is shown a modified embodiment or construction of the weighted object and is designated by the reference numeral 10'. While the weighted body, as suggested, can comprise any suitable mass such as a small bag filled with sand, the embodiment of FIG. 3 is illustrative of a construction suitable for producing the audible sounds as proposed above. Thus, the weighted object 10' has a body 26 provided at the trailing end thereof with a plurality of fins 28 which serve as the stabilizing means. Of course, any other stabilizing means may be utilized, such as the streamers 18 of FIG. 1.
The weighted object 10' is provided at the leading or other end thereof with a dome or head portion 30 which is shown generally spherically shaped and is advantageously made of an elastomeric soft material. In this way, if the weighted object is inadvertently propelled towards a bystander or if the weighted object hits the bystander as it descents from its trajectory, the injury to the bystander will be avoided.
The dome 30 is provided with a central opening or air inlet 32 which is adapted to admit air during forward movement of the weighted object. A plurality of circumferentially spaced air outlet slots or vents 34 are provided for allowing the air which enters into the air opening or inlet 32 to escape. In this manner, there is permitted a flow of air through the body of the weighted object 10', and generally, the faster that the weighted object 10' is whirled, the greater is the speed of air movement through the internal openings and cavities in the weighted object. Therefore, providing a sound producing means within the weighted object 10' which responds to the passage of air proximate thereto produces an audible sound.
Referring to FIG. 4, there is shown, by way of example only, one type of sound producing means which can be used in conjunction with the weighted object 10' of FIG. 3. The sound producing means, which is generally designated in FIG. 4 by the reference numeral 36, is in the nature of a reed device of the type which is used, for example, in accordions. The sound producing means 36 includes a sound board 38 which includes air inlet openings or air holes 40. Provided laterally of the sound board 38 are one or more slotted plates 42 which are provided with slots 44 to produce air outlet openings 46. At the far end, there is provided a sound post 48 which forces the air entering through the air inlet openings 40 to exit through the air outlet openings 46, as indicated by the arrows in FIG. 4. The air, between the time that it enters through the air inlet openings 40 and leaves through the air outlet openings 46 moves through air passages 50 in which there are located reeds 52 associated with each slot 44 or air outlet opening 46. The reeds may be screwed or riveted over the slots 44 by means of connecting means 54. The reeds themselves may be made of watch-spring steel or brass, and are permitted to vibrate freely in their slots so that they are at times referred to as "free reeds". To minimize air consumption, the slots may be closed by flexible flaps or leather members 56 which are formed so as to normally cover or close the slots 44, and only open in response to air pressures of predetermined value. In this sense, the members 56 serve as velocity detectors which open the slots 44 when the weighted object 10' reaches a predetermined speed so as to enable the reeds 52 to become actuated and emit an audible sound. Naturally, by controlling the dimensions and flexibility of the members 56, the threshold velocity may be selected at which the audible sounds will be emitted.
When the sound producing means 36 shown in FIG. 4 is placed within the weighted object 10' of FIG. 3, the air entering through the central inlet opening 32 is admitted through the air inlet openings 40 of the sound producing means, and the air which leaves the air outlets 46 of the sound producing means is permitted to escape through the air outlet slots 34 in the body 26.
Another possible approach for controlling the threshold velocity at which a sound signal is emitted is to provide suitable covers for the central air inlet opening 32 of the weighted object 10', and have these covers open at the selected velocities so as to permit air passage to pass the reeds. Below the predetermined velocities, the covers remain closed and no air is emitted into the sound producing means. Thus, in FIG. 5, the dome 30 is shown to house a sound producing means 36 of the type shown in FIG. 4 (shown in phantom outline), with the central opening 32 being normally closed by cover 58. The cover is supported by a resilient biasing spring 60. When the whirling toy reaches the desired velocity, sufficient air pressure upon the cover 58, as represented by the arrow in FIG. 5, is sufficient to overcome the restoring forces of the spring 60, and the cover 58 is deflected downwardly or inwardly into the dome 30. At such time, air is permitted to flow through the sound producing means 36 with the resultant emission of audible signals, as described above.
In FIG. 6, a variant of the embodiment of FIG. 5 is shown, wherein instead of a coil spring as shown in FIG. 5, the cover 58' is supported by a helical spring 60' which, likewise, deflects inwardly in response to sufficient air pressures acting on the cover. In FIG. 7, the springs shown in FIGS. 5 and 6 are replaced by a leaf-type spring 60" which supports the cover 58".
The operation of the embodiment shown in FIGS. 5, 6 and 7 are generally the same as described above in connection with FIGS. 3 and 4. However, where a separate cover is provided for the central opening 32 which responds to predetermined air pressures and, therefore velocities, it may be possible to dispense with the velocity deflectors 56 since both the deflectors 56 and the covers serve essentially the same function. Both regulate the amount of air which is permitted to flow past the reeds and, to that extent, prevent the generation of audible signals until such time that the whirling toy is rotated at a high enough speed and, therefore, has attained a sufficiently linear velocity.
As will be evident from the above description, the whirling toy 10 can be used alone for entertainment or amusement by even very young children, while it can be used by older children and adults both alone or in competition with others in what amounts to a game requiring great skill, including coordination and timing, as well as the familiarity with the basic laws of physics. When used competitively, means are advantageously provided for accurately measuring the performance of each player and, therefore, comparing the performances between players. Such measuring means may consist of any suitable device, two such means being shown in FIG. 2 to include a target pattern and a simple measure. Further, and particularly in competition, since performance is proportional to the speed at which the toy is whirled and released into its trajectory, means are advantageously provided for assuring that each person competing releases the toy at a minimum velocity as above noted. This assures that all the players have an equal advantage and, in a sense, it assures that the initial velocities of the whirling toys of all the players is substantially constant. Accordingly, based on the above equation for horizontal range R, once the initial velocities are constant, the only variable that remains is the angle &agr; so that the more vertically a player propels his toy, the closer it will return to its initial point of departure or closer to the point C in FIG. 2. The contest, then, is reduced to the ability of the players to release their whirling toys just at the right moment when the toy is travelling in a vertical direction.
In addition to serving as stabilizers, the streamers 18 can be decorative and made out of multi-colored strips of paper or plastic. When the toy is propelled upwardly or descends, the streamers give the appearance of a trail of fire or smoke, such as may be associated with a rocket or falling meteor.
The length r of the string 14 is not critical, as above suggested. However, since the linear velocity v is a function of the product of the whirling speed or angular velocity &ohgr; and the length r, it is clear that for the same rotational speed, the greater the string length r, the greater is the velocity v. In competition, therefore, all players should use the same string lengths, except that different lengths can be used if handicaps are desired. Of course, where the toys are to be used by young children, the string lengths may have to be shorter since a toy 10 whirled in a vertical plane by a short shild may otherwise hit the ground.
While the principles of the present invention have been described in terms of specific embodiments, clearly, the invention is not intended to be limited to the presently preferred embodiments described. A person skilled in the art may modify or change the constructions or applications from the teachings of the principles of the present invention without departing from the spirit and the scope thereof. | |
Are you healed 4 weeks after C-section?
How long does it take to recover from a C-section? It takes about six weeks to recover from a C-section, but each person’s timeline will be different. An incision — typically a horizontal cut made in your lower abdomen — can take weeks to heal.
What should I do after one month of C-section?
People can speed up their recovery from a C-section with the following methods:
- Get plenty of rest. Rest is vital for recovery from any surgery.
- Ask for help. Newborns are demanding.
- Process your emotions.
- Take regular walks.
- Manage pain.
- Watch for signs of infection.
- Fight constipation.
- Get support for breastfeeding.
Can I bend after one month of C-section?
Lifting more than your baby, stretching, straining and deep bending are not recommended until about 4-6 weeks post-delivery OR until you are able to do these movements with no pain or strain and your incision feels like it has healed.
Is it normal to have pain 4 weeks after C-section?
If you notice these signs, see a health care provider immediately. The healing process also varies from person to person. Some people may experience tenderness and discomfort for up to eight weeks after a C-section.
Can I climb stairs after 1 month C-section?
Doctors, traditionally, have advised women to avoid stairs after a C-section. But Kathryn Houston, a clinical instructor of obstetrics and gynecology at the University of California, San Francisco, shrugs off that recommendation. “Stairs are fine as long as you take them slowly,” she says.
How long does it take for internal stitches to heal after C-section?
In most cases, you’ll recover easily and quickly (within 6 to 8 weeks) and have just a small scar. Sometimes, you can do everything right and still have complications.
How do I know my C-section is healing?
The biggest outcome predictor is how other scars on your body have healed. While many women will see their c-section scar thin out and gradually fade in color over time, some scars will protrude and remain reddish or purple for longer.
When does C-section heal internally?
It takes 4 to 6 weeks to recover from a C-section “The uterus, abdominal wall, and skin need to heal after a C-section. The initial healing occurs within 4 to 6 weeks postpartum,” says Malavika Prabhu, MD, a specialist of maternal-fetal medicine at New York-Presbyterian and Weill Cornell Medicine.
How long does it take to heal internally after C-section?
Why does my C-section scar hurt after a month?
Some women feel pain, restriction, or a pulling sensation on or around their scar months or even years after surgery. This is normally due to the build-up of scar tissue which can stick to muscles or even organs and cause pain.
How do I know my C-section scar is healing?
By two weeks, your scar should look and feel much better. That said, it can take anywhere from six weeks to three months before you’re fully healed.
How do you know C-section opened inside?
The internal C-section incision on the uterus can also open or rupture….These include:
- severe abdominal pain.
- vaginal bleeding.
- dizziness.
- low blood pressure.
- a fever.
- painful urination.
- painful bowel movements.
- severe constipation or the inability to have a bowel movement. | https://pfeiffertheface.com/are-you-healed-4-weeks-after-c-section/ |
How do you tell what edition a Pokemon card is?
How do you tell what edition a Pokemon card is?
It's actually quite simple. On the card, usually on the left of the Pokémon information, there should be a label that says “1st edition.” In some other sets, there can also be a little sphere with a “1” and the letters above it. If you see that information, then the card you own is a first edition version.
How can you tell if your Pokemon cards are worth money?
Look in the bottom right corner of the card to find the rarity symbol, next to the card number: A circle means the card is common, while a diamond marks uncommon cards. These are easy to find, and not usually worth much unless the card was printed in 1999 or 2000.
What do the signs on Pokemon cards mean?
A tiny symbol on the bottom right-hand corner will let you know the rarity of a card. A circle on your card means it's common, a diamond indicates that your card is uncommon, and a star means it's rare.
How do you know if your Pokemon card is shadowless? | https://boardgamestips.com/pokemon-trading-card-game/how-do-you-tell-what-edition-a-pokemon-card-is/ |
Yesterday morning I left a handle to set and went for a stroll along a part of the canal I haven't fished for many years. The sun had gone in when I set off so the fleece was required. By the time I had gone as far as I felt like going the fleece had to be removed. It was a glorious June morning. The air was still, the canal calm and fish were basking. Small roach were cruising about slowly just under the surface and every now and then I'd disturb something larger from the margins where the lily pads are starting to reach the surface.
One of these margin baskers revealed itself. To my great surprise it was a koi carp of around five pounds. I've been convinced that there are carp in this canal for quite a while. I've seen swirls and other surface activity which has been ever so carpy over the years, and as they have been in the main canal for a long time there's no reason why some couldn't have found their way through the locks. One koi isn't proof positive, but it makes me think there might well be more.
I retraced my steps, checking out likely eel swims and watching the wildlife. There were a few damsels brought out of hiding by the sun. Not very active as they were still warming up after the cool overnight rain. Common blues and blue-tailed. Passing through an overgrown part of the path with ash and alder on either side I first heard then spotted a female blackcap. A couple of buzzards had been soaring on high earlier on. With sunshine, blue skies and fluffy clouds I couldn't wait for evening to get the eel rods out again.
Back home the glue had set so the rings where whipped on before lunch, and the first coat of varnish applied after I had eaten. Then it was time to sort the gear out again and re-rig the eel rods. The decision had been made to go with braid this time. I'd also been browsing eel rigs on the internet and come up with an idea.
There isn't much info about eel fishing to be found. Only a few rigs seem to get used, and they all seem to involve far too much clutter. Rather like a lot of carp anglers, eel fishers seem to spend a lot of time not catching anything. They use this time to think up new ways to incorporate as much junk into their rigs as possible!
Although I have always disliked the Dyson rig for pike fishing it gets good press from eel anglers. I could see it's value for fishing later in the season when the bottom weed gets thicker. Rather than thinking of it as a paternoster variant, I now look at it as an off-bottom leger rig. Looking at one of the rig diagrams I found gave me another idea though.
The evening was starting to cool when I arrived at my chosen swim. The float rod was set up. Quickly done thanks to it being a two piece carp waggler rod I keep rigged with my trusty old Abu 501.
A few casts close in over some maggots produced nothing, but the first chuck across, tight to the reeds, saw the float bob and dip then slide away. The first baiter was swung to hand and popped in the landing net serving as a keepnet.
It didn't take me long to add three more so I put the float rod away and got out the eel rods at half eight. One fished a roach head and was cast across to where the water starts to deepen. The second rod put my new idea into practice.
The lead was removed from the run ring and tied to a length of nylon about two and a half feet long. A loop was tied in the other end of this link and clipped to the run ring. This rig was baited with a single lob worm. My thinking is that I can now cast the lead into the far bank reeds (where they are quite sparse) without fear of the hook snagging a stem while presenting a bait close to the outer edge of the reeds. Obviously I'll not be casting the lead deep into the reeds, but if I do overcast slightly I can easily pull it towards me, again without fear of the hook catching. I didn't find out if the plan was successful.
An hour after casting the eel baits out a mist began to form over the canal. Dew was condensing on the rods and banksticks. It was getting chilly and damp. I was glad I'd put my waterproof overtrousers on and remembered my woolly hat. Jackdaws croaked their way to roost in loose flocks. Sedge warblers sang their evening song accompanied by blackbirds staking out their territories for the night. The light faded, celebratory jubilee fireworks boomed and sparkled in the distance, the moon rose full and low in the clear sky. It was getting chillier still.
Ten o'clock is the magic hour at the moment and true to form the right hand bobbin dropped off and the braid peeled from the spool at five minutes to. Not for the first time my strike was fruitless. This has been the story of all my eel fishing over the years. I can get eels to take my baits, but connecting with them is a hit and miss affair. I recast hoping that the eel, or another, would return.
By ten thirty I was not only chilly, but also wishing I'd fetched a flask. Mist over the water on a moonlit night is the sort of image you see in fishing books as presaging a memorable catch. It's all lies. Cold and thirsty I packed up. Roll on the muggy evenings of high summer. | http://blog.lumbland.co.uk/2012/06/chilling.html |
Electrical circuit can be connected in two basic ways, in series or in parallel.
In a series circuit, all the components are connected one after the other in one single path.
Figure shows a series circuit where three bulbs, L1 , L2 and L3 are connected to a switch and a cell.
In a parallel circuit, all the components are connected with their corresponding ends joined together at common points to form separate and parallel paths. These paths are called branches.
Figure shows a parallel circuit, where the three bulbs, L1 , L2 and L3 and their respective switches are connected with their ends joined at two common points before the ends are connected to a cell.
The brightness of each bulb in a series circuit is equally the same since the same current flows through each bulb.
The brightness of the bulbs in a parallel circuit is brighter than those in a series circuit with the same number of bulbs.This is because the bulbs in the parallel circuit draw as much current as a single bulb.
Household wiring circuits operating devices such as a lamp, air conditioner, water heater, fan and washing machine are connected in parallel to the mains supply. Current flow from the mains supply to these circuits are controlled by a fuse box.
In the fuse box or a consumer unit, a circuit breaker is connected in series to each circuit.
There is a main switch in the fuse box which is connected in series to the mains supply.
Therefore, in case of an emergency or repair work, the connection between the mains supply and the household wiring circuits can be broken by turning off the main switch.
How do you calculate the total resistance of a parallel circuit?
Table gives the summary of the comparison between a series circuit and a parallel circuit.
In a parallel circuit, the current has more than one path to flow. The current from the source splits into separate branches. Therefore, in a parallel circuit, the current leaving and returning to the source is the sum of the currents in the separate paths, where I = I1 = I2 + I3 = I4.
In a parallel circuit, the potential drop across each component in the circuit is the same as they share the same two points, P and Q. The potential difference across a component (3 V) is the potential difference of any other component connected in between. Therefore, the potential difference across the separate branches of a parallel circuit are the same, where V = V1 = V2.
3. When a bulb in a series circuit has blown up, the other bulbs would not be able to light up. 3. When a bulb in a parallel circuit has blown up, the other bulbs would still be able to light up.
Most of the electric components in our household appliances are connected in parallel circuits. There are some which are connected in combinations of series and parallel circuits.
Figure shows a hairdryer with two switches, A and B.
Figure shows the circuit diagram of the hairdryer. The fan and the resistor, R are connected in series, while the fan and the heating element are connected in parallel.
This is an example of an application of combining series and parallel circuits together.
When the main switch is closed, the fan is switched on and the air blown out from the hairdryer is cold. Switch A is used to control the heating element. When it is closed, the heating element is turned on and the air blown out is hot.
The speed of the fan can be controlled by connecting a resistor, R in series with the fan. The speed of the fan is in slow mode when the main switch is closed.
When switch B is closed, the current bypasses resistor R and flows straight to the fan. This will increase the voltage across the fan and the speed of the fan can be increased.
Aim: To identify series and parallel circuits.
All the electrical circuits as shown in Table are set up.
The circuits are switched on and the brightness of the bulbs are observed and compared.
In each circuit, one of the bulbs is removed and what happens to the other bulb is observed.
The circuits in 1 and 4 are connected in series. These circuits have only one path for the charges to flow from one terminal to another terminal of the battery.
The circuits in 2 and 3 are connected in parallel. These circuits have more than one continuous path or branch for the charges to flow from one terminal to another terminal of the battery.
The bulbs in the parallel circuits light up brighter as compared to the bulbs in the series circuits. The effective resistance in a parallel circuit is much smaller and the current that passes through each bulb is larger.
e series circuit is broken when one of the bulbs is removed and current cannot continue to pass through the circuit.
When one of the bulbs is removed from a parallel circuit, the other bulb still lights up. This shows that the broken circuit in one branch will not affect the circuit in other branches.
Aim: To compare the current, I and potential difference, V in series and parallel circuits.
The electrical circuit is set up as shown in above Figure.
The brightness of the bulbs in the circuit is observed. The readings of the ammeters and voltmeters are recorded in Table 1.
One of the bulbs is removed and what happens to the other bulb in the circuit is observed. The readings of the ammeters and voltmeters are recorded in Table 2.
The bulbs, batteries, ammeters and voltmeters are now connected as shown in Figure.
The brightness of the bulbs is observed. The readings of the ammeters and voltmeters are recorded in Table 1.
One of the bulbs is removed and what happens to the other bulb is observed. The new readings of the ammeters and voltmeters are recorded in Table 2.
In the series circuit, the current that passes through each bulb is the same. The potential difference across each bulb is also the same as the bulbs are similar. The sum of the potential difference across each bulb is the same as the potential difference across the battery.
In the parallel circuit, the bulbs are connected parallel to the battery and share the same two points, therefore the potential difference across each bulb is the same as the potential difference across the battery. The total current that flows in the circuit is the sum of the current that passes through each bulb in separate paths.
In the series circuit, the circuit will break off if one of the bulbs is removed. The ammeter reading is zero as no current passes through it. The potential difference across each bulb is also zero as no current passes through it.
In the parallel circuit, only the path in which the bulb has been removed will break off. The current can still flow through the other path. The potential difference across the bulb is still the same as the potential difference across the battery.
The bulbs in the parallel circuit light up brighter as compared to the bulbs in the series circuit. This is because in a parallel circuit, the potential difference across each bulb is very much higher as compared to the potential difference of each bulb in a series circuit. A bulb that lights up brighter indicates that the current that passes through it is larger. | https://www.aplustopper.com/series-parallel-circuits/ |
The TD's Ignition and Low Petrol Warning lamps are certain to come up quite often on TD forums. The reason for this is due to the existence of some resistance wire wrapped around the sockets of original TD indicators. This is further brought about by the use of a variety of bulbs in these sockets, e.g., 12 volt bulbs and 2.5 volt bulbs. Hopefully, this page will help to answer some of the questions. The approach that I have taken is to make a series of measurements on my 1952 TD. A digital multimeter (DMM) and Ohm's Law provide virtually all of the answers. For those who may not be familiar with it, Ohm's Law is the vary basic relationship of voltage, current and resistance in electrical circuits, V=I*R. When dealing with power (watts) it is common to see P (watts)=V (volts) * I (amps), or P=V^2/R, or P=I^2*R.
I have reason to believe that the ignition and fuel lamp sockets in my 52TD are still original. The wiring in car was in very poor shape when I began the restoration in 1988, but it appeared that none of the components on the instrument panel had ever been replaced. The only components that I had to replace were the Inspection Sockets. When I removed the components from the panel in preparation for repainting it I noted that the Ignition and Petrol Light sockets were covered with a shiny black wrapper of some sort. My 'restoration' philosophy has been to retain functional components whenever possible, so the sockets, lenses and bulbs were reinstalled just as they were found, even though the green lens has faded almost to being transparent.
The bulb in my Petrol Warning Light is labeled "EDISWAN 2.5V .2A FNG". I measured the resistance of the bulb at about 1.2 ohms. This is a 'cold' reading. From ohm's law, the operating value for a 2.5V/.2A bulb is about 12.5 ohms (R=V/I). The bulb in the Ignition Warning Light merely says '12V'.
Wiring descriptions and terminology used in the following discussion are taken from wiring diagrams found in Section N of The M.G. Midget (Series "TD) and (Series "TF") Workshop Manual, particularly the diagram on page N.23.
The Petrol Warning Light is wired between the A4 terminal of the fuse box and the Petrol Tank Unit where it is closed to earth when the petrol gets low. I removed the appropriate green wire from the A4 terminal and reconnected it through a digital ammeter. I then went under the car and removed the green w/black wire from the sending unit at the tank and connected it to ground. The reading on the ammeter was 152mA, i.e., .152 A. The battery voltage was 12.22 volts, so ohm's law says that the circuit resistance is about 12.22/.152 ~ 80.4 ohms.
Next step was to disconnect the battery and convert the DMM into a digital ohmmeter. I connected the meter to the green line that connects to the A4 terminal and then probed the innards of the Petrol Warning Light socket. There was continuity (0 ohms) between the green wire and the center terminal of the lamp socket and no connection to the outer terminal of the socket, i.e., the internal spring. I then connected the meter to the (disconnected) end of the cable at the Petrol Tank Unit. Probing the lamp socket showed no connection to the inner terminal of the socket, but a measurement of 70.2 ohms was found to the outer (spring) bulb contact. So, the resistance of the socket wiring is about 70 ohms. Add another 10 ohms for the warm light bulb and the operating current of about 150mA is just about what one might expect.
How about the Ignition Warning Light? The only printing on the bulb in my Ignition Warning Light says "12V". The table on page N.18 of the Workshop Manual says that the bulb should be a #987, 12V, 2.2W. Applying ohm's law (R=E2/P) will give us the bulb's operating resistance, in this case (12*12/2.2) ~ 65 ohms.
A similar test setup was used. Please note that the Ignition Warning Light is connected between the A3 terminal of the fusebox and the 'D' terminal of the Control Box, the smaller of the two yellow wires. I removed the yellow wire from terminal 'D' and connected it through the DMM (as an ammeter) and then turned on the key (after reconnecting the battery). The reading on the ammeter was 108 mA (.108A). Again, ohm's law says that the circuit resistance, R= V/I, or 12.22V/.108A ~ 113 ohms.
The battery was then disconnected and the DMM was converted into an ohmmeter. The yellow lead from the D terminal was found to go directly to the center pin of the Ignition Warning Light socket (bulb removed). I then measured the resistance from terminal A3 of the fusebox to the outer (spring) portion of the socket and found 70.5 ohms! Sounds familiar.
Subtracting this 70.5 ohms from the circuit resistance of 113 ohms indicates a lamp resistance of about 43 ohms. This would mean that the lamp is operating at about .5 watts, since P=I*I*R, (.108*.108*43~.5).
So, it seems that an unmolested, original, warning lamp socket probably has about 70 ohms of resistance wire wrapped around it. The Ignition Warning Lamp should last forever since it's operating at less than half of its designed voltage at its highest stress level. Using non-resistor sockets and 12v bulbs can work fine, but will be much brighter (and hotter). Unlike some later MG models, the Ignition Warning Light is not a necessary component of the charging system in a TD. There is also an indication that the 12v warning bulbs are the same as the ones used for instrument illumination. To check this out I took one of my instrument illumination bulbs, labeled "12V 2.2W" and repeated the current draw tests with this bulb. The current through the Petrol Warning Light dropped from 145 mA to 100.2 mA. There was a noticeable decrease in brightness. The bulb was then swapped in the Ignition Warning Light and the current was seen to drop from 108.8mA to 100.6mA. Again, there was a noticeable decrease in brightness, but it was still quite useable. If one wants to emulate the wired socket with an unwired one and a discreet resistor it would just be a matter of obtaining a suitable resistor and wiring it in. The actual value of the resistor isn't critical. Standard resistor values in the neighborhood are 68 ohms and 75 ohms. But, just make certain that it's rating is at least 2 watts.
BTW, the Brown & Gammons catalog lists a bulb #GLB987 for both the warning lamps and "dash & instrument illumination". Think I'll start looking for some miniature screw base 2.5 volt lamps. The G.E. Lighting Catalog lists a #14 bulb specified as a 2.5v, 1 watt, miniature screw base, G-3.5 lamp for 2 D-cell flashlights. It sounds good, but it doesn't work. I bought a package of #14 bulbs from Radio Shack and tried one in my Ignition Light. It was barely visible. The bulb package says that it's a 300mA lamp. I'm looking for a source of 2.5v, .2A, G-3.5, miniature screw base bulbs.
A Unipart package, labeled 'Rover recommended', Auto Bulb, GLB 987 lists the bulb as 12 V 2.2W. The bulbs sold by Abingdon Spares are called '987'. | http://www.ttalk.info/Tech/Indicators.htm |
Victorville is located in the state of California and has a lot of culture to offer as well as great sights and interesting destinations. So if you’re planning a trip to Victorville, you’ve come to the right place!
Here you can find different housings and hotels around Victorville
Just type in your destination and get many different suggestions.
Vacation in Victorville
Victorville, a city located in the Mojave Desert of California, may not be the first place that comes to mind when you think of a vacation destination. However, this city has a lot to offer visitors, especially those who enjoy spending time outdoors.
There are several parks located in Victorville, such as Green Tree Golf Course, Victor Valley Museum, and Mojave Narrows Regional Park. Green Tree Golf Course is a great place to spend a day playing golf with friends or family. The Victor Valley Museum offers visitors a chance to learn about the history of the area and see some interesting exhibits. Mojave Narrows Regional Park is a beautiful place to go for a hike or a picnic.
If you are looking for a place to stay while you are visiting Victorville, there are plenty of hotels and motels to choose from. For those who want a more unique experience, there are also several bed and breakfasts located in the city.
Whether you are looking for a place to golf, hike, or just relax, Victorville is a great vacation destination. With its many parks and hotels, you are sure to find everything you need for a perfect getaway.
Sights in Victorville
Victorville is a city located in the Victor Valley of southwestern San Bernardino County, California. Names after Jacob Nash Victor, a rancher who helped settle the area, it was founded in 1895. The City of Victorville covers an area of Victorville is the Victor Valley’s largest city and one of the Inland Empire’s leading economic centers.
According to the 2010 census, the city had a population of 115,903, up from 102,659 in 2000. The estimated population as of July 1, 2013 was 122,122.
Victorville is the commercial hub of the Victor Valley, a high desert area located in the Mojave Desert. The city is exposes to strong hybrid winds which can produce damaging gusts over 80mph certain times of the year.
The main airport in the area is Southern California Logistics Airport, which used to be George Air Force Base. It serves as a general aviation facility for both scheduled and charter flights.
There are many historical sites located in or near Victorville. The Mojave Narrows Regional Park contains Historic Britton Dam, which was built in 19181921, and is on the National Register of Historic Places. The Old Stone House, believed to be the oldest standing structure in the Victor Valley, was built in 1860. It is now a California State Historical Landmark.
The Ghost Town of Greenwater was a mining town that flourished in the early 1900s. It is now a popular spot for hiking, picnicking, and offroad vehicle riding.
Other popular attractions in Victorville include the Mojave River Forks Regional Park, the California Route 66 Museum, zoos, and golf courses.
History of Victorville
Victorville is a city located in the Victor Valley of southwestern San Bernardino County, California.incorporated in 1962. The city is named after the California Southern Railroad’s founder, Jacob Nash Victoor.
The Mojave Trails National Monument is located partially within the city limits.
The city of Victorville is bordered by Apple Valley on the east, Hesperia on the south, and Adelanto on the west.
The area now known as Victorville was originally inhabited by the Serrano people. The Mojave Trail ran through the area, and was used by Native Americans and settlers alike.
In 1885, a man named Jacob Nash Victor established a railroad hub in the area, which led to rapid growth. The town was officially incorporated in 1962.
Today, Victorville is a thriving community with a population of over 115,000. The city is home to a number of businesses and industries, and is a popular destination for shopping, dining, and entertainment.
Other vacation destinations in the United States: | https://www.tourism.de/vacation-in-victorville/ |
The vertices of a 3-4-5 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
Hence E is the correct answer.
Hit Kudos if you like what you see!
USC Marshall MBA.PM (2019 Intake)Class of 2022 Calling all applicants! | https://gmatclub.com/forum/the-vertices-of-a-3-4-5-right-triangle-are-the-centers-of-three-mutual-291680.html |
Technical Field
The present invention relates to a novel benzylamide compound, method for producing the same, and miticide containing the compound.
Background Art
Due to the emergence of mites resistant to miticides in recent years as a result of long-term use of miticides, it has become difficult to accomplish control by use of known miticides.
Under such circumstances, there has been an urgent demand for the development of new types of miticides that are expected to achieve excellent miticidal activity.
5
1-20
For example, Patent Literature (PTL) 1) discloses a compound represented by following Formula (A):
wherein R represents substituted or unsubstituted C alkyl, substituted or unsubstituted amino, N-containing heterocycles, or the like, and reports that this compound exhibits miticidal activity.
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5
However, in PTL 1, mainly, urea compounds are produced, and although the amide compounds where R is alkyl, haloalkyl, aryl, or cycloalkyl are also produced, no amide compounds where R is benzyl is disclosed. In addition, PTL 1 nowhere discloses that the above compound (A) exhibits ovicidal activity.
Citation List
Patent Literature
Japanese Patent Application Laid-open No. 2011-042611
PTL 1:
Summary of Invention
Technical Problem
An object of the present invention is to provide a novel benzylamide compound or a salt thereof that exhibits miticidal activity.
Another object of the present invention is to provide a method for producing the benzylamide compound or the salt thereof.
A further object of the present invention is to provide a new type of miticide containing the benzylamide compound or the salt thereof.
Solution to Problem
Advantageous Effects of Invention
The present inventors conducted extensive research to achieve the above objects, and succeeded in synthesizing a compound represented by the following Formula (1) or a salt thereof that has miticidal activity. The present inventors have conducted further research based on the above findings. The present invention has thereby been accomplished.
More specifically, the present invention is defined in the independent claims, to which reference should now be made.
The benzylamide compound or the salt thereof according to the present invention achieves an excellent miticidal effect with a small amount thereof.
With the present invention, the benzylamide compound and the salt thereof can simply be produced with an excellent yield.
Additionally, with the present invention, a new type of miticide containing the benzylamide compound or the salt thereof according to the present invention can be provided.
Description of Embodiments
1. benzylamide compound or a salt thereof
2. Method for producing a benzylamide compound and a salt thereof
Step 1
Step 1A (when Y is a leaving group)
Step 1B (when Y is a hydroxyl group)
Step 1C
Step 2
Step 3
Method for producing a sulfide compound represented by Formula (1-1) or a salt thereof
Production route 1 (Step 4)
Production Route 2
Production Route 3
Production Route 3A (when Y is a leaving group)
_
Step 3B (when Y is a hydroxyl group)
Production Route 3C
Production Route 4
Step 5
Pest-Controlling Agent
Examples
Production Example 1:
Preparation of N-(2-fluoro-4-methylphenyl)-2-(4-(trifluoromethoxy) phenyl) acetamide (4-14)
Production Example 2:
Preparation of 5-(2-(4-(trifluoromethoxy) phenyl) acetamide)-4-fluoro-2-methylbenzene-1-sulfonyl chloride (5-14)
Production Example 3:
Preparation of N-(2-fluoro-5-mercapto-4-methylphenyl)-2-(4-(trifluoromethoxy) phenyl) acetamide (6-14)
Example 1:
Preparation of N-(5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylphenyl)-2-(4-(trifluoromethoxy) phenyl) acetamide (1A-14)
Production Example 4:
Preparation of N-(2-fluoro-4-methylphenyl) acetamide
Production Example 5:
Preparation of 5-acetamido-4-fluoro-2-methylbenzene-1-sulfonyl chloride
Production Example 6:
Preparation of N-(2-fluoro-5-mercapto-4-methylphenyl) acetamide
Production Example 7:
Preparation of N-(5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylphenyl) acetamide
Production Example 8:
Preparation of 5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylaniline
Example 2:
Preparation of N-(5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylphenyl)-2-phenylacetamide (1A-1)
Example 3 (comparative example):
Preparation of N-(5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylphenyl)-2-(2-chlorophenyl) acetamide (1A-3)
_
Example 4 (comparative example):
Preparation of N-(5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylphenyl)-2-(2,5-dichlorophenyl) acetamide (1A-4)
Example 5:
Preparation of 2-(4-(ethylthio)phenyl)-N-(2-fluoro-4-methyl-5-((2,2,2-trifluoroethyl)thio)phenyl)acetamide (1B-1)
Example 6:
Preparation of N-(2-fluoro-4-methyl-5-((2,2,2-trifluoroethyl)thio)phenyl)-2-(4-(propylthio)phenyl)acetamide (1B-2)
Example 7:
Preparation of N-(2-fluoro-4-methyl-5-((2,2,2-trifluoroethyl)thio)phenyl)-2-(4-(isopropylthio)phenyl) acetamide (1B-3)
Example 8:
Preparation Example 1: Emulsions
Preparation Example 2: Wettable powders
Preparation Example 3: Granules
Preparation Example 4: Dusts
Preparation Example 5: Flowable preparations
Test Example 1 (Miticidal test on Two-Spotted Spider Mites)
Test Example 2 (Ovicidal test on Two-Spotted Spider Mites)
Throughout the entire specification, a singular expression should be understood as encompassing the concept thereof in the plural form unless specifically noted otherwise. Thus, singular articles (e.g., "a", "an" and "the") should also be understood as encompassing the concept thereof in the plural form unless specifically noted otherwise. Further, the terms used herein should be understood as being used in the meaning that is commonly used in the art, unless specifically noted otherwise. Thus, unless defined otherwise, all terminologies and scientific technical terms that are used herein have the same meaning as the terms commonly understood by those skilled in the art to which the present invention pertains. In case of a contradiction, the present specification (including the definitions) takes precedence.
1
2
3
4
5
6
7
8
9
10
11
The present invention is directed to a compound represented by Formula (1):
or a salt thereof (hereinafter sometimes referred to as "benzylamide compound (1) of the present invention" or "compound (1)"),
wherein R, R, R, R, R, R, R, R, R, R, R, X and n are as defined in claim 1.
Next, the terms in the present specification are described below.
In the present specification, the number of substituents of a group defined by "optionally substituted" or "substituted" is not particularly limited if it is substitutable, and is one or plural. In addition, unless otherwise indicated, the description for each group is also applied when the group is one part of or a substituent on other groups.
1-6
"C alkyl" means a linear or branched, saturated hydrocarbon group having one to six carbon atoms.
2-6
"C alkenyl" means a linear or branched, unsaturated hydrocarbon group having two to six carbon atoms and containing one to three double bonds.
2-6
"C alkynyl" means a linear or branched, unsaturated hydrocarbon group having two to six carbon atoms and containing one triple bond.
3-8
"C cycloalkyl" means a cyclic alkyl having three to eight carbon atoms, and includes those cyclic alkyl having a partially bridged structure.
1-6
1-6
1-6
1-6
"C alkoxy" refers to a "C alkyloxy group", and the "C alkyl" moiety is defined the same as the above-described "C alkyl".
"Aryl" means a monocyclic or polycyclic aromatic hydrocarbon.
"Heterocyclic" means a saturated, unsaturated, or aromatic heterocyclic group which has at least one of nitrogen, oxygen, phosphorus and/or sulfur atoms in the ring and may be bonded at any substitutable position.
The following shows specific examples of each group as used in this specification.
Examples of halogen include, for instance, fluorine, chlorine, bromine, and iodine.
1-6
1-6
Examples of C alkyl include, for instance, methyl, ethyl, n-propyl, isopropyl, n-butyl, isobutyl, sec-butyl, tert-butyl, n-pentyl, n-hexyl, and other C straight-chain or branched-chain alkyl.
1-6
1-6
Examples of C haloalkyl include, for instance, fluoromethyl, chloromethyl, bromomethyl, iodomethyl, difluoromethyl, trifluoromethyl, 2,2,2-trifluoroethyl, pentafluoroethyl, 3,3,3-trifluoropropyl, 4,4,4-trifluorobutyl, heptafluoroisobutyl, and other C straight-chain or branched-chain alkyl substituted with 1 to 9, and preferably 1 to 5, halogen atoms.
1-6
1-6
Examples of C alkoxy include, for instance, methoxy, ethoxy, n-propoxy, isopropoxy, n-butoxy, isobutoxy, sec-butoxy, tert-butoxy, and other C straight-chain or branched-chain alkoxy.
1-6
1-6
Examples of C haloalkoxy include, for instance, fluoromethoxy, chloromethoxy, bromomethoxy, iodomethoxy, difluoromethoxy, trifluoromethoxy, 2,2,2-trifluoroethoxy, pentafluoroethoxy, 3,3,3-trifluoropropoxy, 4,4,4-trifluorobutoxy, heptafluoroisobutoxy, and other C straight-chain or branched-chain alkoxy substituted with 1 to 9, preferably 1 to 5, halogen atoms.
1-6
1-6
1-6
1-6
Examples of C alkoxy C alkyl include, for instance, methoxymethyl, ethoxymethyl, n-propoxymethyl, isopropoxymethyl, n-butoxymethyl, isobutoxymethyl, sec-butoxymethyl, tert-butoxymethyl, methoxyethyl, ethoxyethyl, methoxy-n-propyl, methoxy-n-butyl, and other alkoxyalkyl in which C straight-chain or branched-chain alkyl is substituted with C straight-chain or branched-chain alkoxy.
1-6
1-6
Examples of C haloalkoxy C alkyl include, for instance, fluoromethoxymethyl, chloromethoxymethyl, bromomethoxymethyl, iodomethoxymethyl, difluoromethoxymethyl, trifluoromethoxymethyl, 2,2,2-trifluoroethoxymethyl, and other straight-chain or branched-chain alkoxyalkyl substituted with 1 to 9, preferably 1 to 5, halogen atoms.
3-8
Examples of C cycloalkyl include, for instance, cyclopropyl, cyclobutyl, cyclopentyl, cyclohexyl, cycloheptyl, and cyclooctyl.
3-8
1-6
Examples of C cycloalkyl C alkyl include, for instance, cyclopropylmethyl, cyclobutylethyl, cyclopentylmethyl, and cyclohexylmethyl.
1-6
1-6
Examples of C alkylcarbonyl include, for instance, methylcarbonyl (acetyl), ethylcarbonyl (propionyl), n-propylcarbonyl (butyryl), isopropylcarbonyl (isobutyryl), n-butylcarbonyl (valeryl), isobutylcarbonyl (isovaleryl), sec-butylcarbonyl, tert-butylcarbonyl, and other C straight-chain or branched-chain alkylcarbonyl groups.
1-6
1-6
Examples of C haloalkylcarbonyl include, for instance, fluoromethylcarbonyl, chloromethylcarbonyl, bromomethylcarbonyl, iodomethylcarbonyl, dichloromethylcarbonyl, trichloromethylcarbonyl, difluoromethylcarbonyl, trifluoromethylcarbonyl, chlorodifluoromethylcarbonyl, bromodifluoromethylcarbonyl, dichlorofluoromethylcarbonyl, 2,2,2-trichloroethylcarbonyl, 2,2,2-trifluoroethylcarbonyl, pentafluoroethylcarbonyl, and other C straight-chain or branched-chain alkylcarbonyl substituted with 1 to 9, and preferably 1 to 5, halogen atoms.
Examples of arylcarbonyl include, for instance, benzoyl, tert-butylbenzoyl, and substituted or unsubstituted benzoyl group; 1-naphthoyl, 2- naphthoyl, and substituted or unsubstituted naphthoyl group.
Examples of aryloxycarbonyl include, for instance, phenoxycarbonyl, 4-diaminophenoxycarbonyl, 4-fluorophenoxycarbonyl, 4-tert-butylphenoxycarbonyl, and substituted or unsubstituted phenoxycarbonyl group; 1-naphthoxycarbonyl, 2-naphthoxycarbonyl, and substituted or unsubstituted naphthoxycarbonyl group.
1-6
1-6
Examples of C alkoxycarbonyl include, for instance, methoxycarbonyl, ethoxycarbonyl, n-propoxycarbonyl, isopropoxycarbonyl, n-butoxycarbonyl, isobutoxycarbonyl, sec-butoxycarbonyl, tert-butoxycarbonyl, and other C straight-chain or branched-chain alkoxycarbonyl groups.
1-6
1-6
Examples of C haloalkoxycarbonyl include, for instance, fluoromethoxycarbonyl, chloromethoxycarbonyl, bromomethoxycarbonyl, iodomethoxycarbonyl, dichloromethoxycarbonyl, trichloromethoxycarbonyl, difluoromethoxycarbonyl, trifluoromethoxycarbonyl, 2,2,2-trifluoroethoxymethyl, pentafluoroethoxycarbonyl, 3,3,3-trifluoropropoxycarbonyl, 4,4,4-trifluorobutoxycarbonyl, heptafluoroisopropoxycarbonyl, and other C straight-chain or branched-chain alkoxycarbonyl substituted with 1 to 9, preferably 1 to 5, halogen atoms.
1-6
1-6
Examples of cyano C alkyl include, for instance, cyanomethyl, cyanoethyl, cyano-n-propyl, cyano-isopropyl, cyano-n-butyl, cyano-isobutyl, cyano-sec-butyl, cyano-tert-butyl, cyano-n-hexyl, and other C straight-chain or branched-chain alkyl substituted with a cyano group.
1-6
1-6
Examples of cyano C alkoxy include cyanomethoxy, cyanoethoxy, cyano-n-propoxy, cyano-isopropoxy, cyano-n-butoxy, cyano-iso-butoxy, cyano-sec-butoxy, cyano-tert-butoxy, cyano-hexyloxy, and other C straight-chain or branched-chain alkoxy substituted with a cyano group.
2-6
Examples of C alkenyl include, for instance, vinyl, allyl, 2-butenyl, 3-butenyl, and 1-methylallyl.
2-6
Examples of C haloalkenyl include, for instance, 2,2-dichlorovinyl, 2,2-dibromovinyl, 2,2-difluorovinyl, 2,2-dibromovinyl, 3,3-difluoro-2-allyl, 4,4-difluoro-3-butenyl, and 4,4,4-trifluoro-2-butenyl.
2-6
Examples of C alkynyl include, for instance, ethynyl, 2-propynyl (propargyl), 1-methyl-2-propynyl, 1-butynyl, 2-butynyl, and 3-butynyl.
2-6
Examples of C haloalkynyl include, for instance, fluoroethynyl, bromoethynyl, chloroethynyl, iodoethynyl, and 3,3,3-trifluoro-1-propynyl.
1-6
1-6
Examples of C alkylsulfonyl include, for instance, methylsulfonyl, ethylsulfonyl, n-propylsulfonyl, isopropylsulfonyl, n-butylsulfonyl, isobutylsulfonyl, sec-butylsulfonyl, tert-butylsulfonyl, and other C straight-chain or branched-chain alkylsulfonyl groups.
1-6
1-6
Examples of C haloalkylsulfonyl include, for instance, fluoromethylsulfonyl, chloromethylsulfonyl, bromomethylsulfonyl, iodomethylsulfonyl, dichloromethylsulfonyl, trichloromethylsulfonyl, difluoromethylsulfonyl, trifluoromethylsulfonyl, chlorodifluoromethylsulfonyl, bromodifluoromethylsulfonyl, dichlorofluoromethylsulfonyl, 2,2,2-trichloroethylsulfonyl, 2,2,2-trifluoroethylsulfonyl, pentafluoroethylsulfonyl, and other C straight-chain or branched-chain alkylsulfonyl substituted with 1 to 9, and preferably 1 to 5, halogen atoms.
1-6
1-6
Examples of C alkylsulfinyl include, for instance, methylsulfinyl, ethylsulfinyl, n-propylsulfinyl, isopropylsulfinyl, n-butylsulfinyl, isobutylsulfinyl, sec-butylsulfinyl, tert-butylsulfinyl, and other C straight-chain or branched-chain alkylsulfinyl groups.
1-6
1-6
Examples of C haloalkylsulfinyl include, for instance, fluoromethylsulfinyl, chloromethylsulfinyl, bromomethylsulfinyl, iodomethylsulfinyl, dichloromethylsulfinyl, trichloromethylsulfinyl, difluoromethylsulfinyl, trifluoromethylsulfinyl, chlorodifluoromethylsulfinyl, bromodifluoromethylsulfinyl, dichlorofluoromethylsulfinyl, 2,2,2-trichloroethylsulfinyl, 2,2,2-trifluoroethylsulfinyl, pentafluoroethylsulfinyl, and other C straight-chain or branched-chain alkylsulfinyl substituted with 1 to 9, and preferably 1 to 5, halogen atoms.
1-6
1-6
Examples of C alkylthio include, for instance, methylthio, ethylthio, n-propylthio, isopropylthio, n-butylthio, isobutylthio, sec-butylthio, tert-butylthio, and other C straight-chain or branched-chain alkylthio.
1-6
1-6
Examples of C haloalkylthio include, for instance, fluoromethylthio, chloromethylthio, bromomethylthio, iodomethylthio, dichloromethylthio, trichloromethylthio, difluoromethylthio, trifluoromethylthio, chlorodifluoromethylthio, bromodifluoromethylthio, dichlorofluoromethylthio, 2,2,2-trichloroethylthio, 2,2,2-trifluoroethylthio, pentafluoroethylthio, and other C straight-chain or branched-chain alkylthio substituted with 1 to 9, and preferably 1 to 5, halogen atoms.
3-8
Examples of C cycloalkylsulfonyl include, for instance for instance, cyclopropylsulfonyl, cyclobutylsulfonyl, cyclopentylsulfonyl, and cyclohexylsulfonyl.
3-8
Examples of C cycloalkylsulfinyl include, for instance, cyclopropylsulfinyl, cyclobutylsulfinyl, cyclopentylsulfinyl, and cyclohexylsulfinyl.
3-8
Examples of C cycloalkylthio include, for instance, cyclopropylthio, cyclobutylthio, cyclopentylthio, and cyclohexylthio.
3-8
1-6
Examples of C cycloalkyl C alkylsulfonyl include, for instance, cyclopropylmethylsulfonyl, 2-cyclopropylethylsulfonyl, 3-cyclopropylpropylsulfonyl, and cyclohexylmethylsulfonyl.
3-8
1-6
Examples of C cycloalkyl C alkylsulfinyl include, for instance, cyclopropylmethylsulfinyl, 2-cyclopropylethylsulfinyl, 3-cyclopropylpropylsulfinyl, and cyclohexylmethylsulfinyl.
3-8
1-6
Examples of C cycloalkyl C alkylthio include, for instance, cyclopropylmethylthio, 2-cyclopropylethylthio, 3-cyclopropylpropylthio, and cyclohexylmethylthio.
1-6
1-6
1-6
1-6
Examples of C alkoxy C alkylsulfonyl include, for instance, methoxymethylsulfonyl, ethoxymethylsulfonyl, n-propoxymethylsulfonyl, isopropoxymethylsulfonyl, n-butoxymethylsulfonyl, sec-butoxymethylsulfonyl, tert-butoxymethylsulfonyl, methoxyethylsulfonyl, and other alkoxyalkylsulfonyl in which C straight-chain or branched-chain alkylsulfonyl is substituted with C straight-chain or branched-chain alkoxy.
1-6
1-6
1-6
1-6
Examples of C alkoxy C alkyl sulfinyl include, for instance, methoxymethylsulfinyl, ethoxymethylsulfinyl, n-propoxymethylsulfinyl, isopropoxymethylsulfinyl, n-butoxymethylsulfinyl, sec-butoxymethylsulfinyl, tert-butoxymethylsulfinyl, 2-methoxyethylsulfinyl, and other alkoxyalkylsulfinyl in which C straight-chain or branched-chain alkylsulfinyl is substituted with C straight-chain or branched-chain alkoxy.
1-6
1-6
1-6
1-6
Examples of C alkoxy C alkylthio include, for instance, methoxymethylthio, ethoxymethylthio, n-propoxymethylthio, isopropoxymethylthio, n-butoxymethylthio, sec-butoxymethylthio, tert-butoxymethylthio, 2-methoxyethylthio, and other alkoxyalkylthio in which C straight-chain or branched-chain alkylthio is substituted with C straight-chain or branched-chain alkoxy.
2-6
Examples of C alkenyloxy include, for instance, vinyloxy, 1-propenyloxy, isopropenyloxy, allyloxy, 2-butenyloxy, 3-butenyloxy, and 1-methylallyloxy.
2-6
Examples of C haloalkenyloxy include, for instance, 2,2-dichlorovinyloxy, 2,2-dibromovinyloxy, 2,2-difluorovinyloxy, 2,2-dibromovinyloxy, 3,3-difluoro-2-allyloxy, 4,4-difluoro-3-butenyloxy, and 4,4,4-trifluoro-2-butenyloxy.
2-6
Examples of C alkynyloxy include, for instance, ethynyloxy, 2-propynyloxy, 1-methyl-2-propynyloxy, 1,1-dimethyl-2-propynyloxy, 1-butynyloxy, 2-butynyloxy, and 3-butynyloxy.
2-6
Examples of C haloalkynyloxy include, for instance, fluoroethynyloxy, bromoethynyloxy, chloroethynyloxy, iodoethynyloxy, and 3,3,3-trifluoro-1-propynyloxy.
1-6
1-6
Examples of C alkylsulfonyloxy include, for instance, methylsulfonyloxy, ethylsulfonyloxy, n-propylsulfonyloxy, isopropylsulfonyloxy, n-butylsulfonyloxy, isobutylsulfonyloxy, sec-butylsulfonyloxy, tert-butylsulfonyloxy, and other C straight-chain or branched-chain alkylsulfonyl groups.
1-6
1-6
Examples of C haloalkylsulfonyloxy include, for instance, fluoromethylsulfonyloxy, chloromethylsulfonyloxy, bromomethylsulfonyloxy, iodomethylsulfonyloxy, dichloromethylsulfonyloxy, trichloromethylsulfonyloxy, difluoromethylsulfonyloxy, trifluoromethylsulfonyloxy, chlorodifluoromethylsulfonyloxy, bromodifluoromethylsulfonyloxy, dichlorofluoromethylsulfonyloxy, 2,2,2-trichloroethylsulfonyloxy, 2,2,2-trifluoroethylsulfonyloxy, pentafluoroethylsulfonyloxy, and other C straight-chain or branched-chain alkylsulfonyloxy substituted with 1 to 9, and preferably 1 to 5, halogen atoms.
1-6
1-6
Examples of C alkylsulfinyloxy include, for instance, methylsulfinyloxy, ethylsulfinyloxy, n-propylsulfinyloxy, isopropylsulfinyloxy, n-butylsulfinyloxy, isobutylsulfinyloxy, sec-butylsulfinyloxy, tert-butylsulfinyloxy, and other C straight-chain or branched-chain alkylsulfinyloxy groups.
1-6
1-6
Examples of C haloalkylsulfinyloxy include, for instance, fluoromethylsulfinyloxy, chloromethylsulfinyloxy, bromomethylsulfinyloxy, iodomethylsulfinyloxy, dichloromethylsulfinyoxy, trichloromethylsulfinyloxy, difluoromethylsulfinyloxy, trifluoromethylsulfinyloxy, chlorodifluoromethylsulfinyloxy, bromodifluoromethylsulfinyloxy, dichlorofluoromethylsulfinyloxy, 2,2,2-trichloroethylsulfinyloxy, 2,2,2-trifluoroethylsulfinyloxy, pentafluoroethylsulfinyloxy, and other C straight-chain or branched-chain alkylsulfinyloxy substituted with 1 to 9, and preferably 1 to 5, halogen atoms.
1-6
1-6
Examples of substituted or unsubstituted amino include, for instance, amino, monoalkylamino, dialkylamino, and monoacylamino. Examples of the alkyl include C alkyl mentioned above. Examples of the acyl include C alkoxycarbonyl, haloalkoxycarbonyl, and arylcarbonyl mentioned above.
Examples of aryl include, for instance, phenyl, 1-naphthyl, and 2-naphthyl.
1-6
Examples of aryl C alkyl include, for instance, benzyl, phenylethyl, and phenyl-n-propyl.
Examples of aryloxy include, for instance, phenoxy, 1-naphthyloxy, and 2-naphthyloxy.
1-6
Examples of aryl C alkoxy include, for instance, benzyloxy, phenoxyethoxy, phenoxy-n-propoxy, phenyl-n-butoxy, 1-naphthylmethoxy, and 2-naphthylmethoxy.
Examples of arylsulfonyl include, for instance, phenylsulfonyl, 1-naphthylsulfonyl, and 2-naphthylsulfonyl.
Examples of arylsulfinyl include, for instance, phenylsulfinyl, 1-naphthylsulfinyl, and 2-naphthylsulfinyl.
Examples of arylthio include, for instance, phenylthio, 1-naphthylthio, and 2-naphthylthio.
1-6
Examples of aryl C alkylsulfonyl include, for instance, benzylsulfonyl, phenylethylsulfonyl, phenyl-n-propylsulfonyl, phenyl-n-butylsulfonyl, 1-naphthylmethylsulfonyl, and 2-naphthylmethylsulfonyl.
1-6
Examples of aryl C alkylsulfinyl include, for instance, benzylsulfinyl, phenylethylsulfinyl, phenyl-n-propylsulfinyl, phenyl-n-butylsulfinyl, 1-naphthylmethylsulfinyl, and 2-naphthylmethylsulfinyl.
1-6
Examples of aryl C alkylthio include, for instance, benzylthio, phenylethylthio, phenyl-n-propylthio, phenyl-n-butylthio, 1-naphthylmethylthio, and 2-naphthylmethylthio.
All the Aryls mentioned above may optionally be further substituted. Examples of the number of substituents include, for instance, 1 to 20 (preferably 1 to 10, and more preferably 1 to 5).
Examples of a heterocyclic group include, for instance, thienyl, furyl, tetrahydrofuryl, dioxolanyl, dioxanyl, pyrrolyl, pyrrolinyl, pyrrolidinyl, oxazolyl, isoxazolyl, oxazolinyl, oxazolidinyl, isoxazolinyl, thiazolyl, isothiazolyl, thiazolinyl, thiazolidinyl, isothiazolinyl, pyrazolyl, pyrazolidinyl, imidazolyl, imidazolinyl, imidazolidinyl, oxadiazolyl, oxadiazolinyl, thiadiazolinyl, triazolyl, triazolinyl, triazolidinyl, tetrazolyl, tetrazolinyl, pyridyl, dihydropyridyl, tetrahydropyridyl, piperidyl, oxazinyl, dihydroxazinyl, morpholino, thiazinyl, dihydrothiazinyl, thiamorpholino, pyridazinyl, dihydropyridazinyl, tetrahydropyridazinyl, hexahydropyridazinyl, oxadiazinyl, dihydrooxadiazinyl, tetrahydrooxadiazinyl, thiadiazolyl, thiadiazinyl, dihydrothiadiazinyl, tetrahydrothiadiazinyl, pyrimidinyl, dihydropyrimidinyl, tetrahydropyrimidinyl, hexahydropyrimidinyl, pyrazinyl, dihydropyrazinyl, tetrahydropyrazinyl, piperazinyl, triazinyl, dihydrotriazinyl, tetrahydrotriazinyl, hexahydrotriazinyl, tetrazinyl, dihydrotetrazinyl, indolyl, indolinyl, isoindolyl, indazolyl, quinazolinyl, dihydroquinazolyl, tetrahydroquinazolyl, carbazolyl, benzoxazolyl, benzoxazolinyl, benzisoxazolyl, benzisoxazolinyl, benzothiazolyl, benzisothiazolyl, benzisothiazolinyl, benzimidazolyl, indazolinyl, quinolinyl, dihydroquinolinyl, tetrahydroquinolinyl, isoquinolinyl, dihydroisoquinolinyl, tetrahydroisoquinolinyl, pyridoindolyl, dihydrobenzoxazinyl, cinnolinyl, dihydrocinnolinyl, tetrahydrocinnolinyl, phthalazinyl, dihydrophthalazinyl, tetrahydrophthalazinyl, quinoxalinyl, dihydroquinoxalinyl, tetrahydroquinoxalinyl, purinyl, dihydrobenzotriazinyl, dihydrobenzotetrazinyl, phenothiazinylfuranyl, benzofuranyl, chromanyl, and benzothienyl.
These heterocyclic groups include those substituted at any substitutable position with an oxo or thioketone group.
1-6
Examples of heterocyclic C alkyl include, for instance, 2-pyridylmethyl, 3-pyridylmethyl, 2-pyrazinylmethyl, pyrimidinylmethyl, and 2-quinolinylmethyl.
Examples of heterocyclicoxy include, for instance, 2-pyridyloxy, 3-pyridyloxy, 2-pyrazinyloxy, pyrimidinyloxy, and 2-quinolinylmethyloxy.
All the heterocyclics mentioned above may optionally be further substituted. Examples of the number of substituents include, for instance, 1 to 20 (preferably 1 to 10, and more preferably 1 to 5).
5
6
R and R, taken together with the carbon atom to which they bond, may bond to each other to form a 3- to 8-membered ring via or not via at least one heteroatom.
3-8
Examples of hetero atom in the specification include, for instance, an oxygen atom, a sulfur atom, a nitrogen atom. Examples of 3- to 8-membered ring include: for instance, cyclopropane, cycloheptane, and other C cycloalkyls; tetrahydropyran, piperidine, and other heterocyclics.
1-6
1-6
1-6
1-6
1-6
1-6
1-6
1-6
3-8
3-8
1-6
1-6
1-6
1-6
1-6
1-6
1-6
2-6
2-6
2-6
2-6
1-6
1-6
1-6
1-6
1-6
1-6
3-8
3-8
3-8
3-8
1-6
3-8
1-6
3-8
1-6
1-6
1-6
1-6
1-6
1-6
1-6
2-6
2-6
2-6
2-6
1-6
1-6
1-6
1-6
5
1-6
1-6
1-6
1-6
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1-6
1-6
1-6
1-6
1-6
1-6
1-6
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1-6
Examples of "substituents" for the above substituted groups include:for instance, the halogen, nitro, cyano, hydroxyl, formyl, C alkyl, C haloalkyl, C alkoxy, C haloalkoxy, C alkoxy C alkyl, C haloalkoxy C alkyl, C cycloalkyl, C cycloalkyl C alkyl, C alkylcarbonyl, C haloalkylcarbonyl, arylcarbonyl, aryloxycarbonyl, C alkoxycarbonyl, C haloalkoxycarbonyl, C cyanoalkyl, C cyanoalkoxy, C alkenyl, C haloalkenyl, C alkynyl, C haloalkynyl, C alkylsulfonyl, C haloalkylsulfonyl, C alkylsulfinyl, C haloalkylsulfinyl, C alkylthio, C haloalkylthio, C cycloalkylsulfonyl, C cycloalkylsulfinyl, C cycloalkylthio, C cycloalkyl C alkylsulfonyl, C cycloalkyl C alkylsulfinyl, C cycloalkyl C alkylthio, C alkoxy C alkylsulfonyl, C alkoxy C alkylsulfinyl, C alkoxy C alkylthio, C alkenyloxy, C haloalkenyloxy, C alkynyloxy, C haloalkynyloxy, C alkylsulfonyloxy, C haloalkylsulfonyloxy, C alkylsulfinyloxy, C haloalkylsulfinyloxy, carboxyl, OCN, SCN, SF, substituted or unsubstituted amino, aryl, aryl C alkyl, aryloxy, aryl C alkoxy, arylsulfonyl, arylsulfinyl, arylthio, aryl C alkylsulfonyl, aryl C alkylsulfinyl, aryl C alkylthio, heterocyclic, heterocyclic C alkyl, and heterocyclic oxy. Of these, preferable substituents are halogen, nitro, C alkyl, C haloalkyl, C alkoxy, C haloalkoxy, C alkylsulfonyl, C haloalkylsulfonyl, C alkylsulfinyl, C haloalkylsulfinyl, C alkylthio, C haloalkylthio, substituted or unsubstituted amino, aryl, and heterocyclic, and more preferable substituents are fluorine, chlorine, nitro, methyl, ethyl, trifluoromethyl, methoxy, and trifluoromethoxy.
1-6
1-6
1-6
1-6
1-6
1-6
Preferable substituted aryl groups are halogen-substituted aryl, C alkyl-substituted aryl, C haloalkyl-substituted aryl, halogen and C haloalkyl-substituted aryl, C alkoxy-substituted aryl, C haloalkoxy-substituted aryl, and C alkylthio-substituted aryl. More preferable substituted aryl groups are chlorine-substituted aryl, fluorine-substituted aryl, trifluoromethyl-substituted aryl, chlorine- and trifluoromethyl-substituted aryl, trifluoromethoxy-substituted aryl, and methoxy-substituted aryl, and methylthio-substituted aryl.
1-6
1-6
1-6
1-6
1-6
Preferable substituted heterocyclic groups are halogen-substituted heterocyclic, C alkyl-substituted heterocyclic, C haloalkyl-substituted heterocyclic, C alkoxy-substituted heterocyclic, C haloalkoxy-substituted heterocyclic, and C alkylthio-substituted heterocyclic. More preferable substituted heterocyclic groups are chlorine-substituted heterocyclic, fluorine-substituted heterocyclic, trifluoromethyl-substituted heterocyclic, trifluoromethoxy-substituted heterocyclic, methoxy-substituted heterocyclic, and methylthio-substituted heterocyclic.
The salts of the compounds represented by Formula (1) may be any type of salts as long as they are agriculturally acceptable. Examples of the salts include a hydrochloride salt, a sulfate salt, a nitrate salt, and other inorganic acid salts; an acetate salt, a methanesulfonic acid salt, and other organic acid salts; a sodium salt, a potassium salt, and other alkali metal salts; a magnesium salt, a calcium salt, and other alkaline earth metal salts; dimethylammonium, triethylammonium, and other quaternary ammonium salts.
X represents oxygen or sulfur.
Symbol n represents an integer of 0 to 2.
1
Among compounds (1) of the present invention, a preferable compound is a compound in which R is trifluoromethyl or trifluoroethyl.
4
4
1-6
2-6
2-6
1-6
Among compounds (1) of the present invention, a preferable compound is a compound in which R is hydrogen, C alkyl, C alkenyl, C alkynyl, or C haloalkyl, and a more preferable compound (1) is a compound in which R is hydrogen, methyl, ethyl, n-propyl, n-butyl, 3,3,3-trifluoro-n-propyl, heptafluoroisopropyl, or propargyl.
5
6
5
6
1-6
Among compounds (1) of the present invention, a preferable compound is a compound in which R and R are identical or different and each represent hydrogen, halogen, or C alkyl, and a more preferable compound (1) is a compound in which R and R are hydrogen, fluorine, methyl, isopropyl, or tert-butyl.
9
9
9
1-6
1-6
1-6
1-6
1-6
1-6
1-6
1-6
1-6
1-6
1-6
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1-6
1-6
1-6
1-6
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1-6
2
Among compounds (1) of the present invention, a preferable compound is a compound in which R is hydrogen, halogen, nitro, cyano, C alkyl, C haloalkyl, C alkoxy, C haloalkoxy, C alkylsulfonyl, C haloalkylsulfonyl, C alkylsulfinyl, C haloalkylsulfinyl, C alkylthio, C haloalkylthio, substituted or unsubstituted amino, substituted or unsubstituted aryl, substituted or unsubstituted heterocyclic, or substituted or unsubstituted heterocyclic oxy; a more preferable compound is a compound in which R is hydrogen, halogen, nitro, C alkyl, C haloalkyl, C alkoxy, C haloalkoxy, C alkylsulfonyl, C haloalkylsulfonyl, C alkylsulfinyl, C haloalkylsulfinyl, C alkylthio, C haloalkylthio, substituted or unsubstituted amino, substituted or unsubstituted aryl, or substituted or unsubstituted heterocyclic; and a further more preferable compound (1) is a compound in which R is hydrogen, fluorine, chlorine, bromine, nitro, methyl, ethyl, n-propyl, isopropyl, tert-butyl, n-pentyl, trifluoromethyl, methoxy, ethoxy, n-propoxy, trifluoromethoxy, trifluoromethylsulfonyl, trifluoromethylthio, methylsulfonyl, methylthio, NH, phenyl, 2-chlorophenyl, 4-chlorophenyl, 3,4-dichlorophenyl, 2-fluorophenyl, 3-fluorophenyl, 4-fluorophenyl, 3,5-difluorophenyl, 2-chloro-4-fluorophenyl, 2-trifluoromethylphenyl, 3-trifluoromethylphenyl, 4-trifluoromethylphenyl, 3-chloro-5-trifluoromethylphenyl, 4-chloro-3-trifluoromethylphenyl, 2-trifluoromethoxyphenyl, 3-trifluoromethoxyphenyl, 4-trifluoromethoxyphenyl, 4-methoxyphenyl, 2-methylthiophenyl, 5-trifluoromethyl-2-pyridyl, or 5-pyrimidyl, ethylthio, n-propylthio, isopropylthio, difluoromethylthio, 4-phenylphenyl, 4-cyanophenyl, 3-chlorophenyl, 2,3,4-trichlorophenyl, 3-trifluoromethoxyphenyl, 2,2,2-trifluoroethylthio, 2-(methylthio)-phenyl, 2,3-dichlorophenyl, 2,3,4-trifluorophenyl, 2,4-dichlorophenyl, 2,5-dichlorophenyl, 3,4-difluorophenyl, 3-(benzo[d][1,3]dioxol-5-yl)phenyl, 3,5-dichlorophenyl, 4-(ethylthio)-phenyl, 4-acetylphenyl, or 4-(dimethylamino)-phenyl.
1
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
2
Alternatively, among compounds (1) of the present invention, a preferable compound is a compound in which R is - CHC(X) (X) (X) wherein X, X, and X are identical or different and each represent halogen; a more preferable compound (1) is a compound in which X, X, and X are identical or different and each represent fluorine, chlorine, or bromine; a further more preferable compound (1) is a compound in which X, X, and X are identical or different and each represent fluorine or chlorine; and a most preferable compound (1) is a compound in which X, X, and X are fluorine.
2
Among compounds (1) of the present invention, a preferable compound is a compound in which R is methyl.
3
3
3
3
Among compounds (1) of the present invention, a preferable compound is a compound in which R is halogen; a more preferable compound is a compound in which R is fluorine, chlorine, or bromine; a further more preferable compound is a compound in which R is fluorine or chlorine; and a most preferable compound is a compound in which R is fluorine.
4
4
4
4
1-6
1-6
Among compounds (1) of the present invention, a preferable compound is a compound in which R is hydrogen, C alkyl, or C haloalkyl; a more preferable compound (1) is a compound in which R is hydrogen, methyl, or ethyl; a more preferable compound (1) is a compound in which R is hydrogen or methyl; and a most preferable compound (1) is a compound in which R is hydrogen.
5
6
5
6
5
6
5
6
5
6
Among compounds (1) of the present invention, a preferable compound is a compound in which R and R are hydrogen or halogen; a more preferable compound is a compound in which R and R are identical or different and each represent hydrogen, fluorine, chlorine, or bromine; a further more preferable compound is a compound in which R and R are identical or different and each represent hydrogen, fluorine, or chlorine; a still further more preferable compound is a compound in which R and R are identical or different and each represent hydrogen or fluorine; and a most preferable compound (1) is a compound in which R and R are hydrogen.
9
4
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6
4
5
6
4
5
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4
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4
5
6
2
Among compounds (1) of the present invention, a preferable compound is a compound in which R is -L-CH-C(X) (X) (X) wherein L is a single bond, oxygen, or sulfur and X, X, and X are identical or different and each represent hydrogen or halogen; a more preferable compound (1) is a compound in which L is oxygen or sulfur and X, X, and X are identical or different and each represent halogen; a further more preferable compound (1) is a compound in which L is oxygen or sulfur and X, X, and X are identical or different and each represent fluorine, chlorine, or bromine; and a still further more preferable compound (1) is a compound in which L is oxygen or sulfur and X, X, and X are identical or different and each represent fluorine or chlorine.
Among compounds (1) of the present invention, a preferable compound is a compound in which X is oxygen.
Among compounds (1) of the present invention, a preferable compound is a compound in which n is 0.
7
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10
11
9
9
12
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15
16
12
13
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16
9
1-6
1-6
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3-8
3-8
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1-6
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3-8
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3
8
3-8
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2
6
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5
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1
6
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Compounds (1) of the present invention in which R, R, R, and R are hydrogen are preferably those wherein R is a group of the formula:
wherein * is the point of attachment to the carbon adjacent to R; R, R, R, R, and R are hydrogen, halogen, nitro, cyano, hydroxyl, formyl, C alkyl, C haloalkyl, C alkoxy, C haloalkoxy, C alkoxy C alkyl, C haloalkoxy C alkyl, C cycloalkyl, C cycloalkyl C alkyl, C alkylcarbonyl, C haloalkylcarbonyl, arylcarbonyl, aryloxycarbonyl, C alkoxycarbonyl, C haloalkoxycarbonyl, C cyanoalkyl, C cyanoalkoxy, C alkenyl, C haloalkenyl, C alkynyl, C haloalkynyl, C alkylsulfonyl, C haloalkylsulfonyl, C alkylsulfinyl, C haloalkylsulfinyl, C alkylthio, C haloalkylthio, C cycloalkylsulfonyl, C cycloalkylsulfinyl, C- cycloalkylthio, C cycloalkyl C alkylsulfonyl, C cycloalkyl C alkylsulfinyl, C cycloalkyl C alkylthio, C alkoxy C alkylsulfonyl, C alkoxy C alkylsulfinyl, C alkoxy C alkylthio, C alkenyloxy, C haloalkenyloxy, C alkynyloxy, C- haloalkynyloxy, C alkylsulfonyloxy, C haloalkylsulfonyloxy, C alkylsulfinyloxy, C haloalkylsulfinyloxy, carboxyl, OCN, SCN, SF, substituted or unsubstituted amino, aryl, aryl C alkyl, aryloxy, aryl C alkoxy, arylsulfonyl, arylsulfinyl, arylthio, aryl C alkylsulfonyl, aryl C alkylsulfinyl, aryl C- alkylthio, heterocyclic, heterocyclic C alkyl, or heterocyclic oxy, all of which may optionally be further substituted; a more preferable compound is a compound in which R, R, R, R, and R are identical or different and each represent hydrogen, halogen, cyano, C haloalkyl, C haloalkoxy, or C alkylthio; and a further more preferable compound is a compound in which R is phenyl, 2-fluorophenyl, 2-chlorophenyl, 2-trifluoromethoxyphenyl, 2-(methylthio)phenyl, 2,3-dichlorophenyl, 2,4-dichlorophenyl, 2-chloro-4-fluoro-phenyl, 2,3,4-trifluorophenyl, 2,3,4-trichlorophenyl, 3-fluorophenyl, 3-chlorophenyl, 3-trifluoromethylphenyl, 3-trifluoromethoxyphenyl, 3,4-difluorophenyl,3,4-dichlorophenyl, 3,5-difluorophenyl, 3,5-dichlorophenyl, 3-trifluoromethyl-4-chloro-phenyl, 3-chloro-5-trifluoromethyl-phenyl, 3,4,5-trifluorophenyl, 4-fluorophenyl, 4-chlorophenyl, 4-bromophenyl, 4-methylphenyl, 4-methoxyphenyl, 4-trifluoromethylphenyl, 4-trifluoromethoxyphenyl, 4-cyanophenyl, 4-phenylphenyl, 4-acetylphenyl, 4-(dimethylamino)phenyl, 4-(methylthio)phenyl, or 4-(ethylthio)phenyl.
When the compound (1) has isomers such as optical isomers, stereoisomers, regioisomers,
any of the isomers and mixtures thereof are included within the scope of the compound (1). For example, when the compound (1) has optical isomers, the optical isomer separated from a racemic body is also included within the scope of the compound (1). Each of such isomers may be obtained as a single compound by known synthesis and separation means (e.g., concentration, solvent extraction, column chromatography, recrystallization, etc.).
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No limitations are placed on the method for producing a benzylamide compound (1) (compound (1-1) and compound (1-2)) according to the present invention, and the benzylamide compound (1) can be produced by Steps 1 to 5 represented by Reaction Scheme 1 below:
wherein R, R, R, R, R, R, R, R, R, R, R, X, and n are as defined in claim 1.
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An amide compound (hereinafter may be referred to as "compound (4)") represented by Formula (4) can be produced by reacting an aniline compound (hereinafter may be referred to as "compound (2)") represented by Formula (2)) with a benzylcarbonyl compound (hereinafter may be referred to as "compound (3)") represented by Formula (3) (Reaction Scheme 2):
wherein R, R, R, R, R, R, R, R, R, R, X are as defined in claim 1.
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1-6
Y represents a leaving group or a hydroxyl group, and examples of the leaving group include: halogen such as chlorine, bromine, and iodine; substituted or unsubstituted C alkyl sulfonate; and substituted or unsubstituted aryl sulfonate. Examples of the substituent include the aforementioned substituents such as the halogen and the C haloalkyl.
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A phenylacetamide compound (hereinafter may be referred to as "compound (4)") represented by Formula (4) can be produced by reacting the aniline compound (hereinafter may be referred to as "compound (2)) represented by Formula (2)) with a benzylcarbonyl compound (hereinafter may be referred to as "compound (3A) represented by Formula (3A) (Reaction Scheme 3):
wherein R, R, R, R, R, R, R, R, R, R, and X are as defined in claim 1. Y' represents a leaving group.
Examples of the benzylcarbonyl compound (3A) include, for instance, phenylacetyl chloride, phenylacetyl bromide, and other substituted or unsubstituted phenylacetyl halides; and ethyl phenylacetate, methyl phenylacetate, and other substituted or unsubstituted phenylacetic acid esters.
A used ratio of the aniline compound (2) and the benzylcarbonyl compound (3A) in the reaction therebetween is not particularly limited and thus can appropriately be selected from a wide range. Relative to 1 mole of the aniline compound (2), typically approximately 1 to 5 moles of the benzylcarbonyl compound (3A) and preferably approximately equimolar to 1.2 moles thereof is used.
The aforementioned reaction can be performed under absence or presence of a base. Among the above, the reaction is performed preferably under the presence of the base. As the base, a conventionally known base can widely be used, and examples of the base include: sodium carbonate, potassium carbonate, cesium carbonate, potassium bicarbonate, sodium bicarbonate, and other alkali metal carbonates; sodium hydroxide, potassium hydroxide, and other alkali metal hydroxides; alkali metal hydrides such as sodium hydride and potassium hydride, and other inorganic bases; sodium methoxide, sodium ethoxide, potassium tert-butoxide, and other alkali metal alkoxides; pyridine, triethylamine, diethylamine, dimethylamine, methylamine, imidazole, benzimidazole, diisopropylethylamine, 4-dimethylamine pyridine, piperidine, and other organic bases. Any separate one of these bases or a combination of two or more types thereof is used.
Relative to 1 mole of the aniline compound (2), typically approximately 1 to 10 moles of the base and preferably approximately 1 to 5 moles thereof may excessively be used. When triethylamine or pyridine as organic base is used, it can be used in large excess to serve also as a reaction solvent.
The aforementioned reaction is performed in an appropriate solvent or without any solvent. When the aforementioned reaction is carried out in the solvent, no limitations are placed on the solvent as long as the solvent is inactive with respect to the aforementioned reaction. Examples of such a solvent include: n-hexane, cyclohexane, n-heptane, and fatty acid or alicyclic hydrocarbon-based solvents; benzene, chlorobenzene, toluene, xylene, and other aromatic hydrocarbon-based solvents; methylene chloride, 1,2-dichloroethane, chloroform, and carbon tetrachloride, and other halogenated hydrocarbon-based solvents; diethyl ether, tetrahydrofuran (THF), 1,4-dioxane, and other ether-based solvents; methyl acetate, ethyl acetate, and other esters solvents; acetonitrile; N,N-dimethylformamide (DMF) and other amide-based solvents; and dimethyl sulfoxide and other sulfoxide-based solvents. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited, and is typically within a range between -10°C and a boiling point of the solvent used and preferably 0 to 25°C. Reaction time varies depending on, for example, the reaction temperature, and the reaction typically ends in approximately 0.5 to 24 hours.
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As another method for obtaining the phenylacetamide compound (4), the compound (4) can be produced by reacting the aniline compound (2) with a phenylacetic acid compound (hereinafter may be referred to as "compound (3B)") represented by Formula (3B) (Reaction Scheme 4):
wherein R, R, R, R, R, R, R, R, R, R, and X are as defined in claim 1.
A used ratio of the aniline compound (2) and the phenylacetic acid compound (3B) in the reaction therebetween is not particularly limited and thus can appropriately be selected from a wide range. Relative to 1 mole of the aniline compound (2), typically approximately 1 to 5 moles of the phenylacetic acid compound (3B) and preferably approximately equimolar to 1.2 moles thereof is used.
The aforementioned reaction can be performed under absence or presence of a condensing agent. Among the above, the aforementioned reaction is preferably performed under the presence of the condensing agent. As the condensing agent, a conventionally known condensing agent can be used, and examples of the condensing agent include 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide (EDCI HCl), 1-hydroxybenzotriazole (HOBT), 1-[bis(dimethylamino) methylene]-1H-1,2,3-triazolo [4, 5-b] pyridinium-3-oxide hexafluorophosphate (HATU), bis (2-oxo-3-oxazolidinyl) phosphine acid chloride (BOP-C1), and propylphosphonic acid anhydride (T3P). Any separate one of these condensing agents or a combination of two or more types thereof is used.
Relative to 1 mole of the aniline compound (2), typically 1 to 10 moles of the condensing agent and preferably approximately 1 to 3 moles thereof can excessively be used.
The aforementioned reaction is performed in an appropriate solvent or without any solvent. When the aforementioned reaction is carried out in the solvent, no limitations are placed on the solvent as long as the solvent is inactive with respect to the aforementioned reaction. Examples of such a solvent include: fatty n-hexane, cyclohexane, n-heptane, and other acid or alicyclic hydrocarbon-based solvents; benzene, chlorobenzene, toluene, xylene, and other aromatic hydrocarbon-based solvents; methylene chloride, 1,2-dichloroethane, chloroform, carbon tetrachloride, and other halogenated hydrocarbon-based solvents; diethyl ether, THF, and 1,4-dioxane, and other ether-based solvents; methyl acetate, ethyl acetate, and other esters solvents; acetonitrile; DMF and other amide solvents; and dimethyl sulfoxide and other sulfoxide-based solvents. Any one of these solvents can be used alone or a combination of two or more types of the solvents can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited and is typically within a range between -10°C and a boiling point of the solvent used and preferably within a range between -5°C and the boiling point of the solvent. Reaction time varies depending on, for example, the reaction temperature, and the reaction typically ends in approximately 0.25 to 24 hours.
Note that as a method for producing the phenylacetamide compound (4), a phenylacetic acid halide compound (3C) obtained by reacting the phenylacetic acid compound (3B) with a halogenation reagent can be used as a raw material.
The aforementioned reaction can be performed under presence of a base. As the base, any of the same bases as those described above can be used, and preferable examples of the base include triethylamine, pyridine, di-isopropylamine, 4-diisopropylethylamine, 4-dimethylamine pyridine, lutidine, and other organic bases, and this base can also much excessively be used to be also used as a reaction solvent.
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Examples of the halogenation reagent include, for instance, POCl, POBr, SOCl, SOCl, oxalyl chloride.
Relative to 1 mole of the aniline compound (2), typically 1 to 10 moles of the halogenation reagent and preferably approximately 1 to 5 moles thereof can be used.
The aforementioned reaction is performed in an appropriate solvent or without any solvent. When the aforementioned reaction is carried out in the solvent, no limitations are placed on the solvent as long as the solvent is inactive with respect to the aforementioned reaction. As such a solvent, the aforementioned solvents are listed. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited and is typically within a range between -10°C and a boiling point of the solvent used and preferably within a range between -5°C and the boiling point of the solvent. Reaction time varies depending on, for example, the reaction temperature, and the reaction typically ends in approximately 0.25 to 24 hours.
The aniline compound (2), the benzylcarbonyl compound (3A), the phenylacetic acid compound (3B), and phenylacetic acid halide compound (3C) in Step 1 used as starting materials in Step 1 are known compounds or compounds that can easily be produced by a known method.
The compound (4) obtained by the method shown in Step 1 is easily isolated from a reaction mixture to be purified by use of typical isolation means and purification means, for example, filtration, solvent extraction, distillation, recrystallization, column chromatography, etc.
After end of the reaction, the compound (4) can be provided for next reaction without being isolated from the reaction system.
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A sulfonyl chloride compound (hereinafter may be referred to as "compound (5)") represented by Formula (5) can be produced by chlorosulfonating the amide compound (4) (Reaction Scheme 5):
wherein R, R, R, R, R, R, R, R, R, R, and X are as defined in claim 1.
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A reagent used for the chlorosulfonation is not particularly limited, and for example, include chlorosulfonic acid. When using chlorosulfonic acid, the step can be carried out in one step. For the chlorosulfonation, a two-step method including sulfonation and then chlorination can be used. The sulfonyl chloride compound (5) can be produced by reacting the amide compound with a sulfonation reagent to produce an HOSO-substituted amide compound and then reacting the HOSO-containing amide compound with a chlorination agent.
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The reagent used for the sulfation is not particularly limited, and for example, chlorosulfonic acid, sulfuric acid are provided. Examples of the chlorinating agent used for the chlorination include, for instance, chlorine, POCl, SOCl, SOCl, and oxalyl chloride.
When the chlorosulfonic acid is used, a used ratio between the amide compound (4) and the chlorosulfonic acid in the reaction therebetween is not particularly limited and can appropriately be selected from a wide range. Relative to 1 mole of the amide compound (4), typically approximately 1 to 50 moles of chlorosulfonic acid and preferably approximately 1 to 20 moles thereof is used.
When the sulfonation reagent and the chlorinating agent are used, a used ratio between the sulfonation reagent and the chlorinating agent in the reaction between the amide compound (4) and the sulfonation reagent is not particularly limited and can appropriately be selected from a wide range. Relative to 1 mole of the amide compound (4), typically approximately 1 to 50 moles of the sulfonation reagent and preferably approximately 1 to 20 moles thereof is used. A used ratio between the two in the reaction between the amide compound (4) and the chlorinating agent is not particularly limited, and can appropriately be selected from a wide range. Relative to 1 mol of the amide compound (1), typically approximately 1 to 50 moles of the sulfuric acid and preferably 1 to 20 moles thereof is used.
The aforementioned reaction is performed in an appropriate solvent or without any solvent. When the aforementioned reaction is carried out in the solvent, no limitations are placed on the solvent as long as the solvent is inactive with respect to the aforementioned reaction. As examples of such a solvent, the same solvents as those described above are listed. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited, and is typically within a range between -20°C and a boiling point of the solvent used, preferably -10°C to 150°C, and more preferably 0 to 100°C. Reaction time varies depending on, for example, the reaction temperature and the reaction typically ends in approximately 0.25 to 24 hours.
The sulfonyl chloride compound (5) obtained by the method shown in Step 2 is easily isolated from a reaction mixture to be purified by use of typical isolation means and purification means, for example, filtration, solvent extraction, distillation, recrystallization, column chromatography, etc.
After end of the reaction, the sulfonyl chloride compound (5) can be provided for next reaction without being isolated from the reaction system.
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A thiol compound (hereinafter may be referred to as "compound (6)") represented by Formula (6) can be produced by reacting the sulfonyl chloride compound (5) with a reducing agent (Reaction Scheme 6):
wherein R, R, R, R, R, R, R, R, R, R, and X are as defined in claim 1.
A used ratio between the sulfonyl chloride compound (5) and the reducing agent in the reaction therebetween is not particularly limited and can appropriately be selected from a wide range. Relative to 1 mole of the sulfonyl chloride compound (5), typically approximately 1 to 50 moles of the reducing agent and more preferably approximately 1 to 20 moles thereof is used.
As the reducing agent, any of conventionally known reducing agents can widely be used, and examples of the reducing agent include: triphenylphosphine and other phosphorous compounds; reducing agents containing metal and acid such as zinc and acid, tin (II) and acid, and iron and acid; and reducing agent, red phosphorus, iodine, dichlorodimethylsilane-zinc-dimethylacetamide, lithium aluminum hydride, and other specific reducing agents. Examples of the acid include acetic acid and other organic acids; and hydrochloric acid, sulfuric acid, and other inorganic acids.
The aforementioned reaction is performed in an appropriate solvent. No limitations are placed on the solvent as long as the solvent is inactive with respect to the reaction. As examples of such a solvent, the same solvents as those described above are listed. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited and is typically within a range between -20°C and a boiling point of the solvent used, preferably -10°C to 150°C, and more preferably 20 to 120°C. Reaction time varies depending on, for example, the reaction temperature and the reaction typically ends in approximately 0.25 to 24 hours.
The thiol compound (6) obtained by the method shown in Step 3 is easily isolated from a reaction mixture to be purified by use of typical isolation means and purification means, for example, filtration, solvent extraction, distillation, recrystallization, column chromatography, etc.
After end of the reaction, the thiol compound (6) can be provided for next reaction without being isolated from the reaction system.
Examples of the method for producing the sulfide compound represented by Formula (1-1) include, but are not limited to, a production route 1, a production route 2, a production route 3, and a production route 4, described below.
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A sulfide compound (1-1) can be produced by reacting the thiol compound (6) with an alkyl reagent (hereinafter may be referred to as "alkyl reagent (7)) represented by Formula (7) (Reaction Scheme 7):
wherein R, R, R, R, R, R, R, R, R, R, R, and X are as defined in claim 1, and G represents a leaving group.
As examples of the leaving group, the same leaving groups as those described above are listed.
A used ratio between the thiol compound (6) and the alkyl reagent (7) in the reaction therebetween is not particularly limited and can appropriately be selected from a wide range. Relative to 1 mole of the thiol compound (6), typically approximately 1 to 10 moles of the alkyl reagent (7) and preferably approximately 1 to 5 moles thereof is used.
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Examples of the alkyl reagent (7) include, for instance, methyl iodide, ethyl bromide, and other C alkyl halides; trifluoromethyl iodide, trifluoromethyl bromide, trifluoroethyl iodide, trifluoroethyl bromide, and other C haloalkyl halides.
The aforementioned reaction can be performed under presence of a base. Among the above, the aforementioned reaction is preferably performed under the presence of the base. As examples of the base, conventionally known bases can widely be used, and any of the same bases as those described above can be used.
Relative to 1 mole of the thiol compound (6), typically 1 to 10 moles of the base and preferably approximately 1 to 3 moles thereof can be used. When triethylamine or pyridine as organic base is used, it can be used in large excess to serve also as a reaction solvent.
The aforementioned reaction can be performed by further adding a radical starting agent. Examples of the radical starting agent include, for instance, sulfurous acid, a sulfurous acid salt, Rongalit (product name, sodium-formaldehyde-sulfoxylate), and other sulfurous acid adducts. The base and the radical starting agent can be used in combination.
When the radical starting agent is used, as an additive amount thereof, relative to 1 mole of the thiol compound (6), typically 0.1 to 10 moles of the radical starting agent and preferably approximately 0.1 to 5 moles thereof can be used.
The aforementioned reaction is performed in an appropriate solvent. Examples of the solvent include: n-hexane, cyclohexane n-heptane, and other fatty acid or alicyclic hydrocarbon-based solvents; benzene, chlorobenzene, toluene, xylene, and other aromatic hydrocarbon-based solvents; methylene chloride, 1,2-dichloroethane, chloroform, carbon tetrachloride, and other halogenated hydrocarbon-based solvents; diethyl ether, THF, 1,4-dioxane, and other ether-based solvents; methyl acetate, ethyl acetate, and other ester-based solvents; acetonitrile; DMF, N,N-dimethylacetamide, N-methyl-2-pyrolidone, and other amide-based solvents; dimethyl sulfoxide and other sulfoxide-based solvents; alcohol-based solvents such as sulfolane, methanol, ethanol, isopropyl alcohol, aprotic polar solvents; and water. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited, and is typically within a range between -20°C and a boiling point of the solvent used, preferably -10°C to 60°C, and more preferably 0 to 50°C. Reaction time varies depending on, for example, the reaction temperature and the reaction typically ends in approximately 0.25 to 24 hours.
The sulfide compound (1-1) obtained by the method shown in Step 4 is easily isolated from a reaction mixture to be purified by use of typical isolation means and purification means, for example, filtration, solvent extraction, distillation, recrystallization, column chromatography, etc.
After end of the reaction, the sulfide compound (1-1) can be provided for next reaction without being isolated from the reaction system.
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A sulfide compound (hereinafter may be referred to as "compound (1-1b)") represented by Formula (1-1b) can be produced by reacting a sulfide compound (hereinafter may be referred to as "compound (1-1a)") represented by Formula (1-1a) with a compound (hereinafter may be referred to as "compound (7') represented by Formula (7'): R4'-G (Reaction Scheme 8):
wherein R, R, R, R, R, R, R, R, R, R and X are as defined in claim 1, and R represents formyl, C alkyl, C haloalkyl, C alkoxy, C haloalkoxy, C alkoxy C alkyl, C haloalkoxy C alkyl, C cycloalkyl, C cycloalkyl C alkyl, C alkyl carbonyl, C haloalkyl carbonyl, C alkoxycarbonyl, C haloalkoxycarbonyl, arylcarbonyl, aryloxy carbonyl, C alkenyl, C haloalkenyl, C alkynyl, C haloalkynyl, C alkylsulfonyl, C haloalkylsulfonyl, C alkylsulfinyl, C haloalkylsulfinyl, C alkylthio, C haloalkylthio, aryl, aryl C alkyl, arylsulfonyl, arylsulfinyl, arylthio, and heterocyclic, and these groups may optionally be further substituted. G represents a leaving group.
As examples of the leaving group, the leaving groups as those described above are listed.
A used ratio between the sulfide compound (1-1a) and the compound (7') in the reaction therebetween is not particularly limited and can appropriately be selected from a wide range. Relative to 1 mole of the former, typically approximately 1 to 10 moles of the latter and preferably approximately equimolar to 5 moles thereof is used.
The aforementioned reaction can be performed under presence of a base. Among the above, the aforementioned reaction is preferably performed under the presence of the base. As the base, conventionally known bases can be used and any of the same bases as those described above can be used.
Relative to 1 mole of the sulfide compound (1-1a), a stoichiometric amount of the base or an excessive amount thereof over the aforementioned amount can be used. Preferably one to ten times of the base and more preferably one to five times thereof may excessively be used. When triethylamine, pyridine, or like an organic base is used, it can be used in large excess to serve also as a reaction solvent.
The aforementioned reaction is performed in an appropriate solvent. Examples of the solvent include: n-hexane, cyclohexane, n-heptane, and other fatty acid or alicyclic hydrocarbon-based solvents; benzene, chlorobenzene, toluene, xylene, and other aromatic hydrocarbon-based solvents; methylene chloride, 1,2-dichloroethane, chloroform, carbon tetrachloride, and other halogenated hydrocarbon-based solvents; diethyl ether, THF, 1,4-dioxane, and other ether-based solvents; methyl acetate, ethyl acetate, and other esters solvents; acetonitrile; DMF, N,N-dimethylacetamide, N-methyl-2-pyrolidone, and other amide-based solvents; dimethyl sulfoxide and other sulfoxide-based solvents; alcohol-based solvents such as sulfolane, methanol, ethanol, and isopropyl alcohol and other aprotic polar solvents; and water. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited and is typically within a range between -20°C and a boiling point of the solvent used, preferably -10°C to 60°C, and more preferably 20 to 50°C. Reaction time varies depending on, for example, the reaction temperature and the reaction typically ends in approximately 0.25 to 24 hours.
The sulfide compound (1-1b) obtained by the method shown in Step 4 is easily isolated from a reaction mixture to be purified by use of typical isolation means and purification means, for example, filtration, solvent extraction, distillation, recrystallization, column chromatography, etc.
After end of the reaction, the sulfide compound (1-1b) can be provided for next reaction without being isolated from the reaction system.
The sulfide compound (1-1) can be produced in accordance with not only what have been mentioned above but also production routes 3, 4, and 5.
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The sulfide compound (1-1a) can be produced by reacting an aniline compound (hereinafter may be referred to as "compound (8)") with a phenylacetic acid compound (3) (Reaction Scheme 9):
wherein R, R, R, R, R, R, R, R, R, R, X, and Y are as defined in claim 1.
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The sulfide compound (1-1a) can be produced by reacting the aniline compound (8) with a benzylcarbonyl compound (3A) (Reaction Scheme 10):
wherein R, R, R, R, R, R, R, R, R, R, and X are as defined in claim 1, and Y' represents a leaving group.
Examples of the benzylcarbonyl compound (3A) include, for instance, the same compounds as those of Step 1A.
WO2007/131680
The aniline compound (8) used as a starting material can be produced according to methods described in .
A used ratio between the aniline compound (8) and the benzylcarbonyl compound (3A) in the reaction therebetween is not particularly limited and thus can appropriately be selected from a wide range. Relative to 1 mole of the former, typically aproximately 1 to 5 moles of the latter and preferably approximately equimolar to 1.2 moles thereof is used.
The aforementioned reaction can be performed under absence or presence of a base. Among the above, the aforementioned reaction is preferably performed under the presence of the base. As examples of the base, any of the same bases as those shown in Step 1 above can be used. Any separate one of these bases or a combination of two or more types thereof is used.
Relative to 1 mole of the aniline compound (8), a stoichiometric amount of the base or an excessive amount thereof over the aforementioned amount can excessively be used.
Preferably one to five times of the base may excessively be used. When triethylamine or pyridine as organic base is used, it can be used in large excess to serve also as a reaction solvent.
The aforementioned reaction is performed in an appropriate solvent or without any solvent. When the aforementioned reaction is carried out in the solvent, any of the same solvents as those shown in Step 1 above can be used. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited and is typically within a range between -20°C and a boiling point of the solvent used and preferably 0 to 50°C. Reaction time varies depending on, for example, the reaction temperature and the reaction typically ends in approximately 0.5 to 24 hours.
The aniline compound (8) used as a starting material is a known compound or a compound that can easily be produced by a known method.
The sulfide compound (1-1a) is easily isolated from a reaction mixture to be purified by use of typical isolation means and purification means, for example, filtration, solvent extraction, distillation, recrystallization, column chromatography, etc.
After end of the reaction, the sulfide compound (1-1a) can be provided for next reaction without being isolated from the reaction system.
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As another method for obtaining the phenylacetamide compound (1-1a), the compound (1-1a) can be produced by reacting the aniline compound (8) with a phenylacetic acid compound (3B) (Reaction Scheme 11):
wherein R, R, R, R, R, R, R, R, R, R, and X are as defined in claim 1.
A used ratio between the aniline compound (8) and the phenylacetic acid compound (3B) in the reaction therebetween is not particularly limited and thus can appropriately be selected from a wide range. Relative to 1 mole of the former, typically approximately 1 to 5 moles of the latter and preferably equimolar to 1.2 moles thereof is used.
The aforementioned reaction can be performed under absence or presence of a condensing agent. Among the above, the aforementioned reaction is preferably performed under the presence of the condensing agent. As examples of the condensing agent, the same condensing agents as those shown in Step 1B are listed. Any separate one of these condensing agents or a combination of two or more types thereof is used.
Relative to 1 mole of the aniline compound (8), a stoichiometric amount of the condensing agent or an excessive amount thereof over the aforementioned amount can be used. Preferably approximately one to five times of the condensing agent may excessively be used.
The aforementioned reaction can be performed under absence or presence of a base. Among the above, the aforementioned reaction is preferably performed under the presence of the base. As the base, any of the same bases as those shown in Step 1 above can be used. Any separate one of these bases or a combination of two or more types thereof is used.
Relative to 1 mole of the aniline compound (8), a stoichiometric amount of the base or an excessive amount thereof over the aforementioned amount can be used. Preferably approximately 1 to 5 times of the base can excessively be used. When triethylamine, pyridine, or like an organic base is used, it can be used in large excess to serve also as a reaction solvent.
The aforementioned reaction is performed in an appropriate solvent or without any solvent. When the aforementioned reaction is carried out in the solvent, any of the same solvents as those shown in Step 1 above can be used. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited and is typically within a range between -20°C and a boiling point of the solvent used and preferably 0 to 25°C. Reaction time varies depending on, for example, the reaction temperature and the reaction typically ends in approximately 0.5 to 24 hours.
Note that as a method for producing the phenylacetamide compound (1-1a), a phenylacetic acid halide compound (3C) obtained by reacting the phenylacetic acid compound (3B) with a halogenation reagent can be used as a material.
The aforementioned reaction can be performed under presence of a base. As the base, any of the same bases as those described above can be used, and preferable examples of the base include triethylamine, pyridine, di-isopropylamine, 4-diisopropylethylamine, 4-dimethylamine pyridine, lutidine, and other organic bases. The bases can much excessively be used to be also used as reaction solvents.
3
3
2
2
2
Examples of the halogen reagent include, for instance, POCl, POBr, SOCl, SOCl, and oxalyl chloride.
Relative to 1 mole of the aniline compound (2), typically 1 to 10 moles of the halogenation reagent and preferably approximately 1 to 5 moles thereof can be used.
The aforementioned reaction is performed in an appropriate solvent or without any solvent. When the aforementioned reaction is carried out in the solvent, no limitations are placed on the solvent as long as the solvent is inactive with respect to the aforementioned reaction. As examples of such a solvent, the aforementioned solvents are listed. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited and is typically within a range between 10°C and a boiling point of the solvent used and preferably within a range between -5°C and the boiling point of the solvent. Reaction time varies depending on, for example, the reaction temperature, and the reaction typically ends in approximately 0.25 to 24 hours.
The sulfide compound (1-1a) is easily isolated from a reaction mixture to be purified by use of typical isolation means and purification means, for example, filtration, solvent extraction, distillation, recrystallization, column chromatography, etc.
After end of the reaction, the sulfide compound (1-1a) can be provided for next reaction without being isolated from the reaction system.
1
2
3
4
5
6
7
8
9
10
11
The sulfide compound (1-1) can be produced by reacting a sulfide compound (hereinafter may be referred to as "compound (9)") with an amide compound (hereinafter may be referred to as "compound (10)") represented by Formula (10) (Reaction Scheme 12) :
wherein R, R, R, R, R, R, R, R, R, R, R, and X are as defined in claim 1, and Z represents a leaving group.
A used ratio between the sulfide compound (9) and the amide compound (10) in the reaction therebetween is not particularly limited and can appropriately be selected from a wide range. Relative to 1 mole of the former, typically approximately 1 to 10 moles of the latter and preferably approximately equimolar to 5 moles thereof is used.
The aforementioned reaction can be performed under absence or presence of a base. Among the above, the aforementioned reaction is preferably performed under the presence of the base. As the base, any of the same bases as those shown in Step 1 above can be used. Any separate one of these bases or a combination of two or more types thereof is used.
Relative to 1 mole of the aniline compound (9), typically 1 to 10 moles of the base and preferably approximately 1 to 5 moles thereof is used.
The aforementioned reaction is performed in an appropriate solvent or without any solvent. When the aforementioned reaction is carried out in the solvent, any of the same solvents as those shown in the Step 1 above can be used. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited and is typically within a range between -10°C and a boiling point of the solvent used and preferably between -0°C and the boiling point of the solvent. Reaction time varies depending on, for example, the reaction temperature and the reaction typically ends in approximately 0.5 to 24 hours.
EP3002279
WO2012/176856
The sulfide compound (9) used as a starting material can be produced according to methods described in and .
The sulfide compound (1-1) is easily isolated from a reaction mixture to be purified by use of typical isolation means and purification means, for example, filtration, solvent extraction, distillation, recrystallization, column chromatography, etc.
After end of the reaction, the sulfide compound (1-1) can be provided for next reaction without being isolated from the reaction system.
1
2
3
4
5
6
7
8
9
10
11
A benzylamide compound (hereinafter may be referred to as "compound (1-2)") represented by Formula (1-2) can be produced by reacting a sulfide compound represented by Formula (1-1) with an oxidizing agent (Reaction Scheme 13):
wherein R, R, R, R, R, R, R, R, R, R, R, X, and n' are as described in claim 1.
A used ratio between the benzylamide compound (1-1) and the oxidizing agent in the reaction therebetweeen is not particularly limited and can appropriately be selected from a wide range. Relative to 1 mole of the former, typically approximately 1 to 10 moles of the latter and preferably approximately equimolar to 5 moles thereof is used.
The aforementioned reaction can be performed under presence of the oxidizing agent. As the oxidizing agent, any of known oxidizing agents can be used as long as the oxidizing agent can achieve oxidization of sulfide into sulfoxide, and examples of the oxidizing agent include a combination of: performic acid, peracetic acid, pertrifluoroacetic acid, perbenzoic acid, m-chloroperbenzoic acid (mCPBA), o-carbonylperbenzoic acid, and other peracids; hydrogen peroxide, t-butylhydroperoxide, cumene hydroperoxide, and other alkyl hydroperoxides; and titanium tetraisopropoxide and other titanium tetraalkoxides dichromate, sodium bichromate, potassium bichromate, and other dichromate salts; and permanganic acid, sodium permanganate, potassium permanganate, and other permanganates. Any separate one of these oxidizing agents or a combination of two or more types thereof is used.
Relative to 1 mole of the benzylamide compound (1-1), a stoichiometric amount of the oxidizing agent or an excessive amount thereof over the aforementioned amount can excessively be used. Preferably one to ten times of the oxidizing agent and more preferably approximately one to five times thereof may be used.
The aforementioned reaction can further be performed by adding a catalyst.
The aforementioned reaction is performed in an appropriate solvent. Examples of the solvent include: n-hexane, cyclohexane, n-heptane, and other fatty acid or alicyclic hydrocarbon-based solvents; benzene, chlorobenzene, toluene, xylene, and other aromatic hydrocarbon-based solvents; methylene chloride, 1,2-dichloroethane, chloroform, carbon tetrachloride, and other halogenated hydrocarbon-based solvents; diethyl ether, THF, 1,4-dioxane, and other ether-based solvents; methyl acetate, ethyl acetate, and other esters solvents; acetonitrile; DMF, N,N-dimethylacetamide, N-methyl-2-pyrolidone, and other amide-based solvents; dimethyl sulfoxide and other sulfoxide-based solvents; alcohol-based solvents such as sulfolane, methanol, ethanol, isopropyl alcohol, and other aprotic polar solvents. Any one of these solvents can be used alone or a combination of two or more types thereof can be used when necessary.
Reaction temperature for the aforementioned reaction is not particularly limited, and is typically within a range between -20°C and a boiling point of the solvent used, preferably -10°C to 60°C, and more preferably 20 to 50°C. Reaction time varies depending on, for example, the reaction temperature, and the reaction typically ends in approximately 0.25 to 24 hours.
The sulfide compound (1-2) obtained by the method shown in Step 5 is easily isolated from a reaction mixture to be purified by use of typical isolation means and purification means, for example, filtration, solvent extraction, distillation, recrystallization, chromatography, etc.
Each compound (1) obtained after the completion of the reactions shown in Reaction Scheme 1 to Reaction Scheme 13 may be easily isolated from the reaction mixture and purified by known isolation and purification techniques, such as filtration, solvent extraction, distillation, recrystallization, and column chromatography.
When compound (1) has regioisomers, each regioisomer may be separated by a usual separation step, such as silica gel chromatography.
Compound (1) of the present invention may be used as an active ingredient of a pest-controlling agent. Examples of pest-controlling agents include agents (agricultural and horticultural insecticide, miticides, nematicides, or soil insecticides) for controlling pests, mites, nematode, or soil pests that all cause problems in the agricultural and horticultural fields; and animal-ectoparasite-controlling agents (e.g., pulicide, ixodicide, and pedivulicideon).
For use as an active ingredient of a pest-controlling agent, it is possible to use compound (1) of the present invention as is with no additional components. However, it is usually preferable to use the compound by combining with a solid carrier, liquid carrier, or gaseous carrier (propellant), and optionally with a surfactant and other adjuvants for pharmaceutical preparation, and formulating the resulting mixture into various forms such as oil solutions, emulsions, wettable powders, flowable preparations, granules, dusts, aerosols, fumigants, or the like, according to known preparation methods.
Compound (1) of the present invention is usually contained in these formulations in a proportion of 0.01 to 95 wt%, and preferably 0.1 to 50 wt%.
Examples of solid carriers usable in the formulations include solid carriers in a fine powder or granular form, such as clay (e.g., kaolin clay, diatomaceous earth, synthetic hydrated silicon dioxide, bentonite, Fubasami clay, and acid clay), talc, ceramic, other inorganic minerals (e.g., celite, quartz, sulfur, active carbon, calcium carbonate, and hydrated silica), and chemical fertilizers (e.g., ammonium sulfate, ammonium phosphate, ammonium nitrate, urea, and ammonium chloride).
N,N-
N
N
Examples of liquid carriers include water, alcohols (e.g., methanol and ethanol), ketones (e.g., acetone and methylethylketone), aromatic hydrocarbons (e.g., benzene, toluene, xylene, ethylbenzene, and methylnaphthalene), aliphatic hydrocarbons (e.g., hexane, cyclohexane, kerosene, and light oil), esters (e.g., ethyl acetate and butyl acetate), nitriles (e.g., acetonitrile and isobutyronitrile), ethers (e.g., diisopropyl ether and dioxane), acid amides (e.g., dimethylformamide and ,-dimethylacetamide), halogenated hydrocarbons (e.g., dichloromethane, trichloroethane, and carbon tetrachloride), dimethylsulfoxide, soybean oil, cottonseed oil, and vegetable oils.
Examples of gaseous carriers include butane gas, LPG (liquefied petroleum gas), dimethyl ether, and carbon dioxide gas.
Examples of surfactants include alkyl sulfates, alkyl sulfonates, alkylaryl sulfonates, alkyl aryl ethers, polyoxyethylene adducts thereof, polyethylene glycol ethers, polyhydric alcohol esters, and sugar alcohol derivatives.
Examples of adjuvants for pharmaceutical preparation include fixing agents, dispersants, and stabilizers.
Examples of the fixing agents and dispersants include casein, gelatin, polysaccharides (e.g., starch, gum arabic, cellulose derivatives, and alginic acid), lignin derivatives, bentonite, sugars, and water-soluble synthetic polymers (e.g., polyvinyl alcohol, polyvinyl pyrrolidone, and polyacrylic acids).
Examples of stabilizers include PAP (acidic isopropyl phosphate), BHT (2,6-di-tert-butyl-4-methylphenol), BHA (mixture of 2-tert-butyl-4-methoxyphenol and 3-tert-butyl-4-methoxyphenol), vegetable oils, mineral oils, fatty acids, and fatty acid esters.
For the pest-controlling agent of the present invention, it is preferable to use compound (1) as is, or by diluting it with water. The pest-controlling agent of the present invention may be used by mixing with, for example, other pest-controlling agents, such as known insecticides, nematicides, acaricides, fungicides, herbicides, plant-growth-controlling agents, synergists, soil conditioners, and animal feeds, or it may be used simultaneously with these agents without mixing.
The amount of the pest-controlling agent of the invention is not limited, and may be suitably selected from a wide range according to various conditions such as the concentration of active ingredient, the form of preparation, type of disease or pest to be treated, type of plant, severity of disease, time for application, method for application, chemicals to be used in combination (insecticide, nematicide, miticide, fungicide, herbicide, plant growth control agent, synergist, soil conditioner, etc.), and amount and type of fertilizer.
2
2
When used as a pesticide, compound (1) of the present invention is usually used in an amount of 0.01 to 500 g/100 m, and preferably 1 to 200 g/100 m.
2
2
When used as a miticide, compound (1) of the present invention is usually used in an amount of 0.1 to 500 g/100 m, and preferably 1 to 200 g/100 m.
When the emulsion, wettable powder, or flowable preparation is used by diluting with water, the concentration is 0.1 to 1,000 ppm, and preferably 1 to 500 ppm. The granules or dusts can be used as is without dilution.
Compound (1) of the present invention is characterized by having a particularly excellent miticidal activity and a broad spectrum of activity.
Compound (1) of the present invention is effectively used as an agricultural and horticultural insecticide, miticide, nematicide, or a soil insecticide. Specifically, compound (1) of the present invention is effective for controlling pests, such as green peach aphids, cotton aphids, and other aphids; diamondback moths, cabbage armyworms, common cutworms, codling moths, bollworms, tobacco budworms, gypsy moths, rice leafrollers, smaller tea tortrix moths, Colorado potato beetles, cucurbit leaf beetles, boll weevils, plant hoppers, leafhoppers, scales, bugs, whiteflies, thrips, grasshoppers, anthomyiid flies, scarabs, black cutworms, cutworms, ants, and agricultural pest insects; slugs, snails, and other gastropods; rat mite, cockroaches, houseflies, house mosquitoes, and other hygiene-harming insects; angoumois grain moths, adzuki bean weevils, red flour beetles, mealworms, and other stored-grain insects; casemaking clothes moths, black carpet beetles, subterranean termites, and other clothes-harming insects and house- and household-harming insects,
mites, such as two-spotted spider mites, carmine spider mites, citrus red mites, Kanzawa spider mites, European red mites (fruit tree spider mites), broad mites, pink citrus rust mites, bulb mites, and other plant-parasitic mites; Tyrophagus putrescentiae, Dermatophagoides farinae, Chelacaropsis moorei, and other house dust mites, and
soil pests, such as root-knot nematodes, cyst nematodes, root-lesion nematodes, white-tip nematode, strawberry bud nematode, pine wood nematode, and other plant parasitic nematodes; pill bugs, sow bugs, and other isopods.
The pest-controlling agent of the present invention is also effective for controlling various pests resistant to chemicals such as organophosphorus agents, carbamate agents, synthetic pyrethroid agents, and neonicotinoid agent.
As used herein, "or" is used when "at least one or more" matters listed in the sentence can be used.
As described above, the present invention has been explained while showing preferred embodiments to facilitate understanding. Hereinafter, the present invention is described in more detail with reference to the following Production Examples and Examples.
3
To a solution of 2-fluoro-4-methylaniline (2-14; 1.1 g, 8.79 mmol, 1 equiv.) and 2-(4-(trifluoromethoxy) phenyl) acetic acid (3b-14; 2.12 g, 9.67 mmol, 1.1 equiv.) in pyridine (10 ml) slowly added POCl (1.6 ml, 17.58 mmol, 2 equiv.) at 0 °C. The reaction was further maintained at the same temperature for 15 minutes. The reaction mixture was then quenched into ice and the product was then extracted with ethyl acetate. The combined organic layer was washed by 1N HCl solution followed by brine solution, dried over sodium sulfate, filtered and concentrated under reduced pressure to get 2.20 g of the crude product 4-14 as yellow solid. The crude product thus obtained was further used as such without any purification.
1
3
H NMR (CDCl): 8.10 (t, J = 8.6 Hz, 1H), 7.39-7.37 (m, 2H), 7.25-7.23 (m, 3H), 6.92-6.85 (m, 2H), 3.75 (s, 2H), 2.29 (s, 3H).
Chlorosulfonic acid (14.0 g, 120 mmol, 18 equiv.) was added to N-(2-fluoro-4-methylphenyl)-2-(4-(trifluoromethoxy) phenyl) acetamide (4-14; 2.20 g, 6.72 mmol, 1 equiv.) at a temperature below 50 °C. The reaction mixture was then stirred at room temperature overnight. The reaction mixture was then quenched into ice, the product was then extracted with ethyl acetate. The combined organic layer was washed by distilled water, dried over sodium sulfate, filtered and concentrated under reduced pressure to get 2.60 g of the crude product 5-14 as black viscous oil. The crude product thus obtained was further used as such without any purification.
1
3
H NMR (CDCl): 9.08 (d, J = 7.6 Hz, 1H), 7.39-7.36 (m, 2H), 7.25-7.24 (m, 3H), 7.12 (d, J = 10.8 Hz, 1H), 3.79 (s, 2H), 2.71 (s, 3H).
To a mixture of 5-(2-(4-(trifluoromethoxy) phenyl) acetamide)-4-fluoro-2-methylbenzene-1-sulfonyl chloride (5-14; 2.60 g, 6.11 mmol, 1 equiv.) in toluene (20 ml) was added triphenyl phosphine (4.8 g, 18.35 mmol, 3 equiv.) at room temperature. The reaction was then heated to 100 °C for 3 hours. The reaction mixture was cooled to room temperature and all the volatiles were distilled out by rotary evaporator. The crude product thus obtained was purified by column chromatography on silica gel with a mixture of ethyl acetate and n-hexane as an eluent to obtain 1.0 g of the title compound 6-14 as an off white solid.
1
3
H NMR (CDCl): 8.26 (d, J = 7.6 Hz, 1H), 7.38-7.36 (m, 2H), 7.25-7.21 (m, 3H), 6.87 (d, J = 10.8 Hz, 1H), 3.74 (s, 2H), 3.30 (s, 1H), 2.25 (s, 3H).
To a cooled mixture of N-(2-fluoro-5-mercapto-4-methylphenyl)-2-(4-(trifluoromethoxy) phenyl) acetamide (6-14; 1.00 g, 2.78 mmol, 1 equiv.) in DMF (10 ml) was added cesium carbonate (0.90 g, 2.78 mmol, 1 equiv.) followed by sodium formaldehyde sulfoxylate (0.33 g, 2.78 mmol, 1 equiv.). To this mixture was then added slowly trifluoroethyl iodide (0.639 g, 3.06 mmol, 1.1 equiv.) at 0 °C and the resulting mixture was then stirred at room temperature for 6 hours. The reaction mixture was then poured into distilled water and extracted with dichloromethane. The combined organic layer was washed with distilled water, dried over sodium sulfate, filtered and concentrated under reduced pressure to obtain a crude product. The crude product thus obtained was purified by column chromatography on silica gel with a mixture of ethyl acetate and n-hexane as an eluent to obtain 0.95 g of the title compound 1A-14 as a pale yellow solid.
To a mixture of 2-fluoro-4-methylaniline (5.50 g, 43.95 mmol, 1 equiv.) in chloroform (30 ml), a solution of acetic anhydride (4.49 g, 43.95 mmol, 1 equiv.) in chloroform (20 ml) was slowly added at 0 °C. The reaction mixture was then stirred at room temperature for 3 hours. The reaction mixture was then quenched into sodium bicarbonate solution and the product was extracted with dichloromethane. The combined organic layer was washed by sodium bicarbonate solution followed by distilled water, dried over sodium sulfate, filtered and concentrated under reduced pressure to get 5.92 g of the crude product as white solid. The crude product thus obtained was further used as such without any purification.
1
3
H NMR (CDCl): δ 8.14-8.10 (m, 1H), 7.25 (bs, 1H), 6.93-6.88 (m, 2H), 2.31 (s, 3H), 2.20 (s, 3H).
Chlorosulfonic acid (20.56 g, 176.46 mmol, 5 equiv.) was slowly added to N-(2-fluoro-4-methylphenyl) acetamide (5.90 g, 35.29 mmol, 1 equiv.) keeping the temperature of the reaction mixture below 50 °C. The resulting mixture was then heated to 70 °C for 4 hours. After cooling to room temperature, the reaction mixture was then poured carefully into ice, the precipitate was filtered, washed well with distilled water and dried to get 7.3 g of crude product as light brown solid. The crude product thus obtained was further used as such without any purification.
1
3
H NMR (CDCl): δ 9.09 (d, J = 7.6 Hz, 1H), 7.48 (bs, 1H), 7.14 (d, J = 10.8 Hz, 1H), 2.72 (s, 3H), 2.25 (s, 3H).
To a mixture of 5-acetamido-4-fluoro-2-methylbenzene-1-sulfonyl chloride (7.00 g, 26.34 mmol, 1 equiv.) in glacial acetic acid (60 ml) was portion-wise added zinc dust (34.44 g, 526.80 mmol, 20 equiv.) at room temperature. The resulting mixture was then refluxed for 4 hours. After cooling to room temperature, the reaction mixture was diluted with distilled water and ethyl acetate and filtered through celite bed. The organic layer was washed well by distilled water, dried over sodium sulfate, filtered and concentrated under reduced pressure to get 3.64 g of the crude product as pale yellow solid. The crude product thus obtained was further used as such without any purification.
1
3
H NMR (CDCl): δ 8.25 (d, J = 7.6 Hz, 1H), 7.29 (bs, 1H), 6.89 (d, J = 11.6 Hz, 1H), 3.34 (bs, 1H), 2.26 (s, 3H), 2.20 (s, 3H).
To a cooled mixture of N-(2-fluoro-5-mercapto-4-methylphenyl) acetamide (3.10 g, 15.56 mmol, 1 equiv.) in DMF (30 ml) was added cesium carbonate (5.07 g, 15.56 mmol, 1 equiv.) followed by sodium formaldehyde sulfoxylate (1.84 g, 15.56 mmol, 1 equiv.). To this mixture was then added slowly trifluoroethyl iodide (3.27 g, 15.56 mmol, 1 equiv.) and the resulting mixture was then stirred at room temperature for 6 hours. The reaction mixture was then poured into distilled water and extracted with dichloromethane. The combined organic layer was washed with distilled water, dried over sodium sulfate, filtered and concentrated under reduced pressure to get crude product. The crude product thus obtained was purified by column chromatography on silica gel with a mixture of ethyl acetate and n-hexane as an eluent to obtain 2.90 g of the title compound as an off white solid.
1
3
H NMR (CDCl): δ 8.49 (d, J = 8.0 Hz, 1H), 7.29 (bs, 1H), 6.96 (d, J = 11.6 Hz, 1H), 3.42-3.35 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H), 2.21 (s, 3H).
To a mixture of N-(5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylphenyl) acetamide (2.20 g, 7.82 mmol, 1 equiv.) in ethanol/water (30 ml/4 ml) was added concentrated HCl (30 ml). The resulting mixture was then refluxed for 6 hours. After cooling to room temperature, all volatiles were removed by vacuum distillation and pH of the residue was then made basic by slow addition of 1N NaOH solution. The product was then extracted with ethyl acetate. The combined organic layer was then washed with distilled water followed by brine solution, dried over sodium sulfate, filtered and concentrated under reduced pressure to get crude product as a brown oil. The crude product thus obtained was further used as such without any purification.
1
3
H NMR (CDCl): δ 6.98 (d, J = 9.2 Hz, 1H), 6.86 (d, J = 11.6 Hz, 1H), 3.64 (bs, 2H), 3.32-3.25 (q, J = 9.6 Hz, 2H), 2.36 (s, 3H).
3
To a cooled solution of 5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylaniline (0.10 g, 0.42 mmol, 1 equiv.) in chloroform (10 ml), triethylamine (0.046 g, 0.46 mmol, 1.1 equiv.) was added followed by slow addition of 2-phenylacetyl chloride (0.068 g, 0.44 mmol, 1.05 equiv.). The resulting mixture was then stirred at room temperature for 14 hours. The reaction mixture was then poured into NaHCO solution and the product was extracted by dichloromethane. The combined organic layer was then washed with distilled water followed by brine solution, dried over sodium sulfate, filtered and concentrated under reduced pressure to get 0.125 g of title product as an off white solid.
3
To a cooled solution of 5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylaniline (0.05 g, 0.21 mmol, 1 equiv.) in dichloromethane (10 ml), triethylamine (0.042 g, 0.42 mmol, 2.0 equiv.) was added followed by slow addition of 2-(2-chlorophenyl) acetyl chloride (0.04 g, 0.21 mmol, 1 equiv.). The resulting mixture was then stirred at room temperature for 14 hours. The reaction mixture was then poured into NaHCO solution and the product was extracted by dichloromethane. The combined organic layer was then washed with distilled water followed by brine solution, dried over sodium sulfate, filtered and concentrated under reduced pressure to get crude product. The crude product thus obtained was purified by column chromatography on silica gel with a mixture of ethyl acetate and n-hexane as an eluent to obtain 0.07 g of the title compound as a brown solid.
3
To a cooled mixture of 5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylaniline (0.05 g, 0.21 mmol, 1 equiv.) and 2-(2,5-dichlorophenyl)acetic acid (0.05 g, 0.25 mmol, 1.2 equiv.) in pyridine (3 ml), POCl (0.08 g, 0.52 mmol, 2.5 equiv.) was added very slowly. After few minutes, the reaction mixture was poured into ice and the product was extracted with ethyl acetate. The combined organic layer was then washed with 1N HCl followed with distilled water, dried over sodium sulfate, filtered and concentrated under reduced pressure to get crude product. The crude product thus obtained was purified by column chromatography on silica gel with a mixture of ethyl acetate and n-hexane as an eluent to obtain 0.023 g of the title compound as a light yellow solid.
3
To a cooled mixture of 5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylaniline (0.20 g, 0.835 mmol, 1 equiv.) and 2-(4-(ethylthio)phenyl)acetic acid (0.186 g, 1.021 mmol, 1.2 equiv.) in pyridine (3 ml), POCl (0.08 g, 5.348 mmol, 6.4 equiv.) was added very slowly. After few minutes, the reaction mixture was poured into ice and the product was extracted with ethyl acetate. The combined organic layer was then washed with 1N HCl followed with distilled water, dried over sodium sulfate, filtered and concentrated under reduced pressure to get crude product. The crude product thus obtained was purified by column chromatography on silica gel with a mixture of ethyl acetate and n-hexane as an eluent to obtain 0.18 g of the title compound as a yellow solid.
3
To a cooled mixture of 5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylaniline (0.20 g, 0.835 mmol, 1 equiv.) and 2-(4-(propylthio)phenyl)acetic acid (0.327 g, 1.556 mmol, 1.8 equiv.) in pyridine (3 ml), POCl (0.08 g, 5.348 mmol, 6.4 equiv.) was added very slowly. After few minutes, the reaction mixture was poured into ice and the product was extracted with ethyl acetate. The combined organic layer was then washed with 1N HCl followed with distilled water, dried over sodium sulfate, filtered and concentrated under reduced pressure to get crude product. The crude product thus obtained was purified by column chromatography on silica gel with a mixture of ethyl acetate and n-hexane as an eluent to obtain 0.13 g of the title compound as a yellow solid.
3
To a cooled mixture of 5-(2,2,2-trifluoroethylthio)-2-fluoro-4-methylaniline (0.20 g, 0.835 mmol, 1 equiv.) and 2-(4-(isopropylthio)phenyl)acetic acid (0.155 g, 0.737 mmol, 0.8 equiv.) in pyridine (3 ml), POCl (0.08 g, 5.348 mmol, 6.4 equiv.) was added very slowly. After few minutes, the reaction mixture was poured into ice and the product was extracted with ethyl acetate. The combined organic layer was then washed with 1N HCl followed with distilled water, dried over sodium sulfate, filtered and concentrated under reduced pressure to get crude product. The crude product thus obtained was purified by column chromatography on silica gel with a mixture of ethyl acetate and n-hexane as an eluent to obtain 0.12 g of the title compound as a yellow solid.
The compounds shown in Tables 1 to 4, other than the compounds obtained in Examples 1 to 7, were produced by methods similar to the methods described in Examples 1 to 7 or methods described in the description.
1
Tables 2 and 4 show H-NMR data of the thus obtained compounds of the present invention. Compounds of comparative examples are labelled with an asterisk (*) in Tables 1 and 3.
3
3
3
2
2
2
Table 1
<chemistry id="chem0018" num="0018"><img id="ib0018" file="IMGB0018.TIF" wi="77" he="26" img-content="chem" img-format="GIF" /></chemistry>
<b>S. No.</b>
<b>R<sup>2</sup></b>
<b>R<sup>3</sup></b>
<b>R<sup>4</sup></b>
<b>R<sup>5</sup></b>
<b>R<sup>6</sup></b>
<b>R<sup>7</sup></b>
<b>R<sup>8</sup></b>
<b>R<sup>9</sup></b>
<b>R<sup>10</sup></b>
<b>R<sup>11</sup></b>
<b>X</b>
<b>n</b>
1A-1
Me
F
H
H
H
H
H
H
H
H
O
0
1A-2
Me
F
H
H
H
H
H
Cl
H
H
O
0
1A-3∗
Me
F
H
H
H
Cl
H
H
H
H
O
0
1A-4∗
Me
F
H
H
H
Cl
H
H
Cl
H
O
0
1A-5∗
Me
F
H
H
H
H
Cl
Cl
H
H
O
0
1A-6∗
Me
F
H
H
H
H
Cl
H
Cl
H
O
0
1A-7∗
Me
F
H
H
H
Cl
Cl
H
H
H
O
0
1A-8∗
Me
F
H
H
H
Cl
H
Cl
H
H
O
0
1A-9∗
Me
F
H
H
H
H
Cl
H
H
H
O
0
1A-10
Me
F
H
H
H
H
H
t-Bu
H
H
O
0
1A-11∗
Me
F
H
H
H
Cl
H
H
H
Cl
O
0
1A-12
Me
F
H
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-13
Me
F
H
H
H
H
H
F
H
H
O
0
1A-14
Me
F
H
H
H
H
H
OCF<sub>3</sub>
H
H
O
0
1A-15∗
Me
F
H
H
H
F
H
F
H
F
O
0
1A-16∗
Me
F
H
H
H
Br
H
OMe
H
H
O
0
1A-17∗
Me
F
H
H
H
H
F
H
H
H
O
0
1A-18∗
Me
F
H
H
H
F
H
H
H
H
O
0
1A-19∗
Me
F
H
H
H
H
OCF<sub>3</sub>
H
H
H
O
0
1A-20∗
Me
F
H
H
H
F
H
H
H
F
O
0
1A-21∗
Me
F
H
H
H
Me
H
H
H
H
O
0
1A-22∗
Me
F
H
H
H
Me
H
Me
H
Me
O
0
1A-23
Me
F
H
H
H
H
H
Br
H
H
O
0
1A-24
Me
F
H
H
H
H
H
OMe
H
H
O
0
1A-25
Me
F
H
H
H
H
H
OEt
H
H
O
0
1A-26
Me
F
H
H
H
H
H
O-<i>n</i>-Pr
H
H
O
0
1A-27
Me
F
H
H
H
H
H
3,4-Cl<sub>2</sub>-Ph
H
H
O
0
1A-28
Me
F
H
H
H
H
H
4-OCF<sub>3</sub>-Ph
H
H
O
0
1A-29
Me
F
H
H
H
H
H
4-CF<sub>3</sub>-Ph
H
H
O
0
1A-30
Me
F
H
H
H
H
H
<i>i</i>-Pr
H
H
O
0
1A-31
Me
F
H
H
H
H
H
NH<sub>2</sub>
H
H
O
0
1A-32
Me
F
H
H
H
H
H
NO<sub>2</sub>
H
H
O
0
1A-33
Me
F
H
H
<i>i</i>-Pr
H
H
Cl
H
H
O
0
1A-34
Me
F
H
Me
Me
H
H
Cl
H
H
O
0
1A-35
Me
F
H
H
H
H
H
Cl
H
H
O
1
1A-36
Me
F
H
H
H
H
H
Cl
H
H
O
2
1A-37
Me
F
H
H
H
H
H
OCF<sub>3</sub>
H
H
O
1
1A-38
Me
F
H
H
H
H
H
OCF<sub>3</sub>
H
H
O
2
1A-39
Me
F
H
H
H
H
H
CF<sub>3</sub>
H
H
O
1
1A-40
Me
F
H
H
H
H
H
CF<sub>3</sub>
H
H
O
2
1A-41
Me
F
Me
H
H
H
H
Cl
H
H
O
0
1A-42
Me
F
Me
H
H
H
H
OCF<sub>3</sub>
H
H
O
0
1A-43
Me
F
Me
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-44
Me
Br
H
H
H
H
H
Cl
H
H
O
0
1A-45
Me
F
Et
H
H
H
H
Cl
H
H
O
0
1A-46
Me
F
H
H
H
H
H
<i>n</i>-Pr
H
H
O
0
1A-47
Me
F
H
H
H
H
H
SCF<sub>3</sub>
H
H
O
0
1A-48
Me
F
Et
H
H
H
H
OCF<sub>3</sub>
H
H
O
0
1A-49
Me
F
Me
H
H
H
H
SCF<sub>3</sub>
H
H
O
0
1A-50
Me
Br
Me
H
H
H
H
Cl
H
H
O
0
1A-51
Me
Br
Me
H
H
H
H
OCF<sub>3</sub>
H
H
O
0
1A-52
Me
Br
Me
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-53
Me
Br
H
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-54
Me
Br
H
H
H
H
H
OCF<sub>3</sub>
H
H
O
0
1A-55
Me
Cl
Et
H
H
H
H
Cl
H
H
O
0
1A-56
Me
Cl
Me
H
H
H
H
OCF<sub>3</sub>
H
H
O
0
1A-57
Me
Cl
Me
H
H
H
H
Cl
H
H
O
0
1A-58
Me
Cl
Et
H
H
H
H
OCF<sub>3</sub>
H
H
O
0
1A-59
Me
Cl
H
H
H
H
H
OCF<sub>3</sub>
H
H
O
0
1A-60
Me
Cl
H
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-61
Me
Cl
H
H
H
H
H
Cl
H
H
O
0
1A-62
Me
Cl
H
H
H
H
H
SCF<sub>3</sub>
H
H
O
0
1A-63
Me
F
H
H
H
H
H
4-Cl-Ph
H
H
O
0
1A-64
Me
F
H
H
H
H
H
5-CF<sub>3</sub>-2-Py
H
H
O
0
1A-65
Me
F
H
H
H
H
H
Ph
H
H
O
0
1A-66
Me
F
H
H
H
H
H
3,5-F<sub>2</sub>-Ph
H
H
O
0
1A-67
Me
F
H
H
H
H
H
2-SMe-Ph
H
H
O
0
1A-68
Me
F
H
H
H
H
H
2-Cl-Ph
H
H
O
0
1A-69
Me
F
H
H
H
H
H
Me
H
H
O
0
1A-70
Me
F
H
H
H
H
H
Et
H
H
O
0
1A-71
Me
F
H
H
H
H
H
n-pent
H
H
O
0
1A-72
Me
F
Me
H
H
H
H
SMe
H
H
O
0
1A-73
Me
Br
Me
H
H
H
H
SCF<sub>3</sub>
H
H
O
0
1A-74
Me
Br
Me
H
H
H
H
SMe
H
H
O
0
1A-75
Me
Br
H
H
H
H
H
SCF<sub>3</sub>
H
H
O
0
1A-76
Me
F
H
H
H
H
H
SMe
H
H
O
0
1A-77
Me
Br
H
H
H
H
H
SMe
H
H
O
0
1A-78
Me
F
Et
H
H
H
H
SMe
H
H
O
0
1A-79
F
Me
H
H
H
H
H
Cl
H
H
O
0
1A-80
F
Me
H
H
H
H
H
OCF<sub>3</sub>
H
H
O
0
1A-81
F
Me
H
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-82
Me
Cl
Me
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-83
Me
Cl
Et
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-84
F
Me
H
H
H
H
H
SCF<sub>3</sub>
H
H
O
0
1A-85
Me
Br
Et
Me
H
H
H
Cl
H
H
O
0
1A-86
Me
F
Me
Me
H
H
H
Cl
H
H
O
0
1A-87
Me
F
Et
Me
H
H
H
Cl
H
H
O
0
1A-88
Me
Br
Me
Me
H
H
H
Cl
H
H
O
0
1A-89
Me
Br
H
Me
H
H
H
Cl
H
H
O
0
1A-90
Me
F
H
Me
H
H
H
Cl
H
H
O
0
1A-91
Me
F
<i>n</i>-Pr
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-92
Me
F
<i>n</i>-Bu
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-93
Me
F
Me
H
H
H
H
3,4-Cl<sub>2</sub>-Ph
H
H
O
0
1A-94
Me
F
Et
H
H
H
H
3,4-Cl<sub>2</sub>-Ph
H
H
O
0
1A-95
Me
F
Me
H
H
H
H
Ph
H
H
O
0
1A-96
Me
F
Et
H
H
H
H
Ph
H
H
O
0
1A-97
Me
F
Propargyl
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-98
Me
F
<i>i</i>-Pr
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-99
F
Me
H
H
H
H
H
4-OCF<sub>3</sub>-Ph
H
H
O
0
1A-100
F
Me
H
H
H
H
H
4-CF<sub>3</sub>-Ph
H
H
O
0
1A-101
Me
F
Heptafluoro-<i>i</i>-Pr
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-102
Me
F
H
H
H
H
H
4-F-Ph
H
H
O
0
1A-103
Me
F
H
H
H
H
H
4-OMe-Ph
H
H
O
0
1A-104
Me
F
H
H
H
H
H
3-OCF<sub>3</sub>-Ph
H
H
O
0
1A-105
Me
F
H
H
H
H
H
3-CI, 5-CF<sub>3</sub>-Ph
H
H
O
0
1A-106
Me
F
H
H
H
H
H
2-OCF<sub>3</sub>-Ph
H
H
O
0
1A-107
Me
F
H
H
H
H
H
3-CF<sub>3</sub>-Ph
H
H
O
0
1A-108
Me
F
H
H
H
H
H
3-F-Ph
H
H
O
0
1A-109
Me
F
H
H
H
H
H
2-F-Ph
H
H
O
0
1A-110
Me
F
Me
H
H
H
H
4-Cl, 3-CF<sub>3</sub>-Ph
H
H
O
0
1A-111
Me
F
Me
H
H
H
H
2-Cl, 4-F-Ph
H
H
O
0
1A-112
Me
F
4,4,4-trifluoro-<i>n</i>-Bu
H
H
H
H
CF<sub>3</sub>
H
H
O
0
1A-113
Me
F
H
H
H
H
H
2-Cl, 4-F-Ph
H
H
O
0
1A-114
Me
F
H
H
H
H
H
4-Cl, 3-CF<sub>3</sub>-Ph
H
H
O
0
1A-115
Me
F
H
H
H
H
H
5-Pyrimidyl
H
H
O
0
1A-116
Me
F
H
H
H
H
H
CF<sub>3</sub>
H
H
S
0
1A-117
Me
F
Me
H
H
H
H
CF<sub>3</sub>
H
H
S
0
1A-118
Me
F
Me
H
H
H
H
3-F-Ph
H
H
O
0
1A-119
Me
F
Me
H
H
H
H
3-CF<sub>3</sub>-Ph
H
H
O
0
1A-120
Me
F
Me
H
H
H
H
3-Cl, 5-CF<sub>3</sub>-Ph
H
H
O
0
1A-121
Me
F
Me
H
H
H
H
3-OCF<sub>3</sub>-Ph
H
H
O
0
1A-122
Me
F
Me
H
H
H
H
4-OMe-Ph
H
H
O
0
1A-123
Me
F
Me
H
H
H
H
4-F-Ph
H
H
O
0
1A-124
Me
F
Me
H
H
H
H
2-F-Ph
H
H
O
0
1A-125
Me
F
Me
H
H
H
H
2-OCF<sub>3</sub>-Ph
H
H
O
0
1A-126
Me
F
H
F
F
H
H
Cl
H
H
O
0
1A-127
Me
F
Me
H
H
H
H
4-Cl-Ph
H
H
O
0
1A-128
Me
F
Me
H
H
H
H
2-Cl-Ph
H
H
O
0
1A-129
Me
F
Me
F
F
H
H
Cl
H
H
O
0
1A-130
Me
F
Me
H
H
H
H
<i>n</i>-Pr
H
H
O
0
1A-131∗
Me
F
Me
H
H
H
Cl
Cl
H
H
O
0
1A-132*
Me
F
Me
H
H
F
H
H
H
F
O
0
1A-133
Me
F
Me
H
H
H
H
Et
H
H
O
0
1A-134
Me
F
Et
H
H
H
H
SO<sub>2</sub>Me
H
H
O
0
1A-135
Me
F
Me
H
H
H
H
SO<sub>2</sub>Me
H
H
O
0
1A-136
Me
F
H
H
H
H
H
SO<sub>2</sub>Me
H
H
O
0
1A-137
Me
F
H
F
F
H
H
CF3
H
H
O
0
1A-138
Me
F
Me
F
F
H
H
CF3
H
H
O
0
1A-139
Me
F
Me
H
H
H
H
CN
H
H
O
0
1A-140
Me
F
H
H
H
H
H
2,4-Cl<sub>2</sub>-Ph
H
H
O
0
1A-141
Me
F
H
H
H
H
H
3,4,5-F3-Ph
H
H
O
0
1A-142
Me
F
H
H
H
H
H
4-Br-Ph
H
H
O
0
1A-143
Me
F
H
H
H
H
H
3,5-Cl2-Ph
H
H
O
0
1A-144
Me
F
H
H
H
H
H
4-SMe-Ph
H
H
O
0
1A-145
Me
F
H
H
H
H
H
4-Me-Ph
H
H
O
0
1A-146
Me
F
Me
H
H
H
H
4-SMe-Ph
H
H
O
0
1A-147
Me
F
Me
H
H
H
H
3,5-Cl<sub>2</sub>-Ph
H
H
O
0
1A-148
Me
F
Me
H
H
H
H
4-Br-Ph
H
H
O
0
1A-149
Me
F
Me
H
H
H
H
4-Me-Ph
H
H
O
0
Table 2
<b>S. No.</b>
<b><sup>1</sup>H NMR</b>
1A-1
CDCl<sub>3</sub>: <i>δ</i> 8.50 (d, J = 7.6 Hz, 1H), 7.43-7.40 (m, 2H), 7.37-7.33 (m, 3H), 7.27-7.24 (bs, 1H), 6.90 (d, J = 11.2 Hz, 1H), 3.77 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H).
1A-2
CDCl<sub>3</sub>: <i>δ</i> 8.47 (d, J = 8.0 Hz, 1H), 7.39-7.37 (m, 2H), 7.29-7.27 (m, 2H), 7.24 (bs, 1H), 6.93 (d, J = 11.6 Hz, 1H), 3.73 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-3
CDCl<sub>3</sub>: <i>δ</i> 8.59 (d, J = 8.0 Hz, 1H), 7.47-7.41 (m, 3H), 7.33-7.28 (m, 2H), 6.92 (d, J = 11.6 Hz, 1H), 3.89 (s, 2H), 3.37 (q, J = 9.7 Hz, 2H), 2.40 (s, 3H).
1A-4
CDCl<sub>3</sub>: <i>δ</i> 8.47 (d, J = 8.0 Hz, 1H), 7.42-6.42 (m, 2H), 7.28-7.26 (m, 2H), 6.95 (d, J = 11.6 Hz, 1H), 3.84 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-5
DMSO-d<sub>6</sub>: <i>δ</i> 10.02 (s, 1H), 8.06 (d, J = 7.6 Hz, 1H), 7.60 (d, J = 4.0 Hz, 2H), 7.32 (d, J = 8.0 Hz, 1H), 7.25 (d, J = 11.6 Hz, 1H), 3.82-3.76 (m, 4H), 2.37 (s, 3H).
1A-6
CDCl<sub>3</sub>: <i>δ</i> 8.47 (d, J = 8.0 Hz, 1H), 7.34 (s, 1H), 7.26-7.25 (m, 3H), 7.96 (d, J = 11.6 Hz, 1H), 3.70 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.42 (s, 3H).
1A-7
CDCl<sub>3</sub>: <i>δ</i> 8.47 (d, J = 7.6 Hz, 1H), 7.46 (d, J = 9.2 Hz, 1H), 7.39 (bs, 1H), 7.33 (d, J = 6.8 Hz, 1H), 7.27-7.23 (m, 1H), 6.95 (d, J = 11.2 Hz, 1H), 3.93 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-8
CDCl<sub>3</sub>: <i>δ</i> 8.47 (d, <i>J</i>= 8.0 Hz, 1H), 7.47 (d, J = 2.0 Hz, 1H), 7.38 (bs, 1H), 7.36 (d, J = 8.6 Hz, 1H), 7.29 (d, J = 8.4 Hz, 1H), 6.95 (d, J = 11.6 Hz, 1H), 3.85 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-9
CDCl<sub>3</sub>: <i>δ</i> 8.48 (d, J = 8.0 Hz, 1H), 7.32 (m, 3H), 7.26-7.23 (m, 2H), 6.93 (d, J = 11.6 Hz, 1H), 3.73 (s, 2H), 3.37 (q, J = 9.7 Hz, 2H), 2.41 (s, 3H).
1A-10
CDCl<sub>3</sub>: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 7.42 (d, J = 8.4 Hz, 2H), 7.28-7.25 (m, 3H), 6.90 (d, J = 11.2 Hz, 1H), 3.73 (s, 2H), 3.37 (q, J = 9.7 Hz, 2H), 2.39 (s, 3H), 1.33 (s, 9H).
1A-11
CDCl<sub>3</sub>: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 7.40 (d, J = 8.0 Hz, 2H), 7.37 (bs, 1H), 7.22 (d, J = 8.4 Hz, 1H), 6.94 (d, J = 11.6 Hz, 1H), 4.14 (s, 2H), 3.37 (q, J = 9.7 Hz, 2H), 2.41 (s, 3H).
1A-12
CDCl<sub>3</sub>: <i>δ</i> 8.47 (d, J = 7.6 Hz, 1H), 7.66 (d, J = 8.0 Hz, 2H), 7.48 (d, J = 8.0 Hz, 2H), 7.27 (bs, 1H), 6.94 (d, J = 11.2 Hz, 1H), 3.82 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-13
CDCl<sub>3</sub>: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.33-7.29 (m, 2H), 7.23 (bs, 1H), 7.10 (t, J = 8.6 Hz, 2H), 6.92 (d, J = 11.6 Hz, 1H), 3.74 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-14
CDCl<sub>3</sub>: <i>δ</i> 8.48 (d, J = 8.0 Hz, 1H), 7.38 (d, J = 8.4 Hz, 2H), 7.27-7.24 (m, 3H), 6.94 (d, J = 11.6 Hz, 1H), 3.76 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-15
CDCl<sub>3</sub>: <i>δ</i> 8.49 (d, <i>J</i>= 7.6 Hz, 1H), 7.40 (bs, 1H), 6.97 (d, J = 11.6 Hz, 1H), 6.75 (t, J = 7.8 Hz, 2H), 3.77 (s, 2H), 3.36 (q, J = 9.6 Hz, 2H), 2.42 (s, 3H).
1A-16
CDCl<sub>3</sub>: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.38 (bs, 1H), 7.32 (d, J = 8.4 Hz, 1H), 7.18 (d, J = 2.8 Hz, 1H), 6.94-6.89 (m, 2H), 3.83 (s, 2H), 3.81 (s, 3H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-17
CDCl<sub>3</sub>: <i>δ</i> 8.48 (d, J = 7.6 Hz, 1H), 7.38 (q, J = 7.0 Hz, 1H), 7.12 (d, J = 7.6 Hz, 2H), 7.07-7.03 (m, 2H), 6.92 (d, J = 11.6 Hz, 1H), 3.76 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-18
CDCl<sub>3</sub>: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.40-7.31 (m, 3H), 7.20-7.11 (m, 2H), 6.93 (d, J = 11.6 Hz, 1H), 3.78 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-19
CDCl<sub>3</sub>: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 7.43 (bs, 1H), 7.32-7.28 (m, 2H), 6.99-6.94 (m, 3H), 3.83 (s, 2H), 3.36 (q, <i>J</i>= 9.6 Hz, 2H), 2.41 (s, 3H).
1A-20
CDCl<sub>3</sub>: <i>δ</i> 8.48 (d, J = 8.0 Hz, 1H), 7.44 (t, J = 8.4 Hz, 1H), 7.29 (d, J = 8.0 Hz, 1H), 7.22-7.20 (m, 2H), 6.93 (d, J = 11.6 Hz, 1H), 3.78 (s, 2H), 3.37 (q, J = 9.7 Hz, 2H), 2.41 (s, 3H).
1A-21
CDCl<sub>3</sub>: <i>δ</i> 8.47 (d, J = 8.0 Hz, 1H), 7.28-7.26 (m, 4H), 7.18 (bs, 1H). 6.88 (d, J = 11.2 Hz, 1H), 3.78 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H), 2.35 (s, 3H).
1A-22
CDCl<sub>3</sub>: <i>δ</i> 8.43 (d, J = 7.6 Hz, 1H), 7.18 (bs, 1H), 6.95 (s, 2H), 6.87 (d, J = 11.2 Hz, 1H), 3.77 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H), 2.31-2.30 (m, 9H).
1A-23
CDCl<sub>3</sub>: <i>δ</i> 8.47 (d, J = 8.0 Hz, 1H), 7.53 (d, J = 8.0 Hz, 2H), 7.26-7.21 (m, 3H), 6.93 (d, J = 11.6 Hz, 1H), 3.71 (s, 2H), 3.37 (q, J = 9.7 Hz, 2H), 2.40 (s, 3H).
1A-24
CDCl<sub>3</sub>: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.27-7.24 (m, 3H), 6.94 (d, J = 8.4 Hz, 2H), 6.90 (d, J = 11.6 Hz, 1H), 3.83 (s, 3H), 3.70 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H).
1A-25
CDCl<sub>3</sub>: <i>δ</i> 8.49 (d, J = 8.0 Hz, 1H), 7.28-7.22 (m, 3H), 6.94-6.88 (m, 3H), 4.07-4.02 (m, 2H), 3.70 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H), 1.43 (t, J = 8.0 Hz, 3H).
1A-26
CDCl<sub>3</sub>: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.28-7.22 (m, 3H), 6.94-6.88 (m, 3H), 3.93 (t, J = 6.6 Hz, 2H), 3.70 (s, 2H), 3.38 (q, J = 8.8 Hz, 2H), 2.39 (s, 3H), 1.86-1.17 (m, 2H), 1.04 (s, 3H).
1A-27
CDCl<sub>3</sub>: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 7.68 (d, J = 1.6 Hz, 1H), 7.58 (d, J = 8.4 Hz, 2H), 7.51 (d, J = 8.4 Hz, 1H), 7.44-7.41 (m, 3H), 7.29 (bs, 1H), 6.92 (d, J = 11.2 Hz, 1H), 3.81 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-28
CDCl<sub>3</sub>: <i>δ</i> 8.51 (d, J = 7.6 Hz, 1H), 7.62-7.59 (m, 4H), 7.42 (d, J = 8.4 Hz, 2H), 7.30 (d, J = 8.0 Hz, 3H), 6.92 (d, J = 11.6 Hz, 1H), 3.81 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-29
CDCl<sub>3</sub>: <i>δ</i> 8.51 (d, J = 7.6 Hz, 1H), 7.70 (bs, 4H), 7.65 (d, J = 8.0 Hz, 2H), 7.45 (d, J = 8.0 Hz, 2H), 7.32 (bs, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-30
CDCl<sub>3</sub>: <i>δ</i> 8.48 (d, J = 8.0 Hz, 1H), 7.00-6.73 (m, 5H), 6.89 (d, J = 7.6 Hz, 1H), 3.73 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.96-2.89 (m, 1H), 2.39 (s, 3H), 1.26 (d, J = 6.8 Hz, 6H).
1A-31
CDCl<sub>3</sub>: <i>δ</i> 8.49 (d, J = 8.0 Hz, 1H), 7.31 (bs, 1H), 7.10 (d, J = 8.4 Hz, 2H), 6.89 (d, J = 11.6 Hz, 1H), 6.72 (d, J = 8.4 Hz, 2H), 3.72 (bs, 2H), 3.65 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H).
1A-32
CDCl<sub>3</sub>: <i>δ</i> 8.46 (d, J = 8.0 Hz, 1H), 8.26 (d, J = 8.0 Hz, 2H), 7.54 (d, J = 8.4 Hz, 2H), 7.31 (bs, 1H), 6.96 (d, J = 11.6 Hz, 1H), 3.86 (s, 2H), 3.36 (q, J = 9.6 Hz, 2H), 2.42 (s, 3H).
1A-33
CDCl<sub>3</sub>: <i>δ</i> 8.49 (d, J = 8.0 Hz, 1H), 7.35-7.28 (m, 5H), 6.93 (d, J = 11.2 Hz, 1H), 3.37 (q, J = 9.6 Hz, 2H), 2.99 (d, J = 10.0 Hz, 1H), 2.48-2.40 (m, 4H), 1.10 (d, J = 6.8 Hz, 3H), 0.75 (d, J = 6.8 Hz, 3H).
1A-34
CDCl<sub>3</sub>: <i>δ</i> 7.38 (s, 1H), 7.32 (d, J = 3.6 Hz, 2H), 7.22 (d, J = 8.8 Hz, 2H), 7.02 (d, J = 8.8 Hz, 2H), 3.42-2.25 (m, 2H), 2.09 (s, 3H), 1.59 (s, 3H), 1.44 (s, 3H).
1A-35
CDCl<sub>3</sub>: <i>δ</i> 8.77 (d, J = 7.6 Hz, 1H), 7.38 (d, J = 8.0 Hz, 2H), 7.32-7.21 (m, 3H), 6.98 (d, J = 11.2 Hz, 1H), 3.75 (s, 2H), 3.61-3.50 (m, 1H), 3.44-3.33 (m, 1H), 2.37 (s, 3H).
1A-36
CDCl<sub>3</sub>: <i>δ</i> 8.96 (d, J = 7.6 Hz, 1H), 7.39 (d, J = 8.4 Hz, 2H), 7.29-7.26 (m, 3H), 7.07 (d, J = 11.2 Hz, 1H), 3.91 (q, J = 8.8 Hz, 2H), 3.76 (s, 2H), 2.64 (s, 3H).
1A-37
CDCl<sub>3</sub>: δ 8.79 (d, J = 7.6 Hz, 1H), 7.38 (d, J = 8.8 Hz, 2H), 7.32 (bs, 1H), 7.26-7.24 (m, 2H), 6.98 (d, J = 10.8 Hz, 1H), 3.78 (s, 2H), 3.58-3.50 (m, 1H), 3.44-3.33 (m, 1H), 2.38 (s, 3H).
1A-38
CDCl<sub>3</sub>: δ 8.97 (d, J = 7.6 Hz, 1H), 7.38 (d, J = 8.4 Hz, 2H), 7.29-7.25 (m, 3H), 7.07 (d, J = 10.8 Hz, 1H), 3.90 (q, J = 8.9 Hz, 2H), 3.79 (s, 2H), 2.65 (s, 3H).
1A-39
CDCl<sub>3</sub>: δ 8.78 (d, J = 7.6 Hz, 1H), 7.66 (d, J = 8.0 Hz, 2H), 7.48 (d, J = 8.0 Hz, 2H), 7.35 (bs, 1H), 6.99 (d, J = 11.2 Hz, 1H), 3.84 (s, 2H), 3.60-3.49 (m, 1H), 3.44-3.32 (m, 1H), 2.38 (s, 3H).
1A-40
CDCl<sub>3</sub>: δ 8.96 (d, J = 9.5 Hz, 1H), 7.67 (d, J = 8.0 Hz, 2H), 7.48 (d, J = 7.6 Hz, 2H), 7.31 (bs, 1H), 7.08 (d, J = 11.2 Hz, 1H), 3.90 (q, J = 8.9 Hz, 2H), 3.85 (s, 2H), 2.65 (s, 3H).
1A-41
CDCl<sub>3</sub>: δ 7.26 (s, 1H), 7.21 (d, J = 8.4 Hz, 2H), 7.07 (d, J = 10.0 Hz, 1H), 6.96 (d, J = 8.4 Hz, 2H), 3.45-3.34 (m, 2H), 3.28-3.21 (m, 5H), 2.51 (s, 3H).
1A-42
CDCl<sub>3</sub>: δ 7.30 (d, J = 7.6 Hz, 1H), 7.10-7.04 (m, 5H), 3.48-3.38 (m, 2H), 3.29-3.17 (m, 5H), 2.51 (s, 3H).
1A-43
CDCl<sub>3</sub>: δ 7.53 (d, J = 8.0 Hz, 2H), 7.32 (d, J = 7.6 Hz, 1H), 7.19 (d, J = 7.6 Hz, 2H), 7.11 (d, J = 10.4 Hz, 1H), 3.49 (q, J = 13.2 Hz, 2H), 3.30-3.23 (m, 5H), 2.54 (s, 3H).
1A-44
CDCl<sub>3</sub>: δ 8.49 (s, 1H), 7.53 (s, 1H), 7.39 (d, J = 8.4 Hz, 2H), 7.31 (d, J = 11.2 Hz, 2H), 7.26 (s, 1H), 3.76 (s, 2H), 3.44 (q, J = 9.6 Hz, 2H), 2.35 (s, 3H).
1A-45
CDCl<sub>3</sub>: δ 7.22-7.19 (m, 3H), 7.07 (d, J = 10.0 Hz, 1H), 6.95 (d, J = 8.4 Hz, 2H), 3.74-3.66 (m, 2H), 3.42 (s, 2H), 3.26-3.19 (m, 2H), 2.51 (s, 3H), 0.84 (t, J = 8.8 Hz, 3H).
1A-46
CDCl<sub>3</sub>: δ 8.50 (d, J = 8.0 Hz, 1H), 7.20-7.26 (m, 5H), 6.88 (d, J = 11.6 Hz, 1H), 3.73 (s, 2H), 3.41-3.34 (m, 2H), 2.58 (t, J = 7.6 Hz, 2H), 2.39 (s, 3H), 1.68-1.60 (m, 2H), 0.94 (t, J = 7.2 Hz, 3H).
1A-47
CDCl<sub>3</sub>: δ 8.47 (d, J = 8.0 Hz, 1H), 7.68 (d, J = 8.0 Hz, 2H), 7.40 (d, J = 8.0 Hz, 2H), 7.29 (bs, 1H), 6.93 (d, J = 11.2 Hz, 1H), 3.79 (s, 2H), 3.33 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-48
CDCl<sub>3</sub>: δ 7.26 (d, J = 6.4 Hz, 1H), 7.09-7.04 (m, 5H), 3.77-3.63 (m, 2H), 3.44-3.34 (m, 2H), 3.29-3.20 (m, 2H), 2.48 (s, 3H), 0.86 (t, J = 9.6 Hz, 3H).
1A-49
CDCl3: δ 7.51 (d, J = 8.0 Hz, 2H), 7.31 (d, J = 7.6 Hz, 1H), 7.08 (d, J = 8.0 Hz, 2H), 7.04 (d, J = 14.4 Hz, 1H), 3.48 (s, 2H), 3.29-3.22 (s, 5H), 2.51 (s, 3H).
1A-50
CDCl3: δ 7.58 (s, 1H), 7.51 (d, J = 8.0 Hz, 2H), 7.29 (s, 1H), 7.19-7.18 (m, 2H), 3.46 (d, J = 14.8 Hz, 1H), 3.37 (d, J = 14.8 Hz, 1H), 3.29-3.22 (m, 2H), 3.19 (s, 3H), 2.48 (s, 3H).
1A-51
CDCl<sub>3</sub>: δ 7.57 (s, 1H), 7.21 (s, 1H), 7.09 (s, 4H), 3.40 (d, J = 15.2 Hz, 1H), 3.32-3.25 (m, 3H), 3.19 (s, 3H), 2.48 (s, 3H).
1A-52
CDCl<sub>3</sub>: δ 7.57 (s, 1H), 7.21 (d, J = 8.4 Hz, 2H), 7.16 (s, 1H), 6.98 (d, J = 8.4 Hz, 2H), 4.77 (d, J = 15.2 Hz, 1H), 3.30-3.24 (m, 3H), 3.18 (s, 3H), 2.47 (s, 3H).
1A-53
CDCl<sub>3</sub>: δ 8.48 (s, 1H), 7.68 (d, J = 8.0 Hz, 2H), 7.50 (d, J = 8.0 Hz, 3H), 7.33 (s, 1H), 3.84 (d, J = 8.0 Hz, 2H), 3.45 (q, J = 9.6 Hz, 2H), 2.36 (s, 3H).
1A-54
CDCl<sub>3</sub>: δ 8.50 (s, 1H), 7.52 (bs, 1H), 7.40 (d, J = 8.4 Hz, 2H), 7.32-7.26 (m, 3H), 3.80 (s, 2H), 3.45 (q, J = 9.4 Hz, 2H), 2.35 (s, 3H).
1A-55
CDCl<sub>3</sub>: δ 7.40 (s, 1H), 7.21 (d, J = 8.4 Hz, 2H), 7.11 (s, 1H), 6.96 (d, J = 8.0 Hz, 2H), 4.04-3.95 (m, 1H), 3.43-3.35 (m, 2H), 3.28-3.18 (m, 3H), 2.48 (s, 3H), 1.09 (t, J = 7.2 Hz, 3H).
1A-56
CDCl<sub>3</sub>: δ 7.39 (s, 1H), 7.30(s, 1H), 7.11-7.05 (m, 4H), 3.42-3.25 (m, 4H), 3.19 (s, 3H), 2.48 (s, 3H).
1A-57
CDCl<sub>3</sub>: δ 7.39 (s, 1H), 7.21 (d, J = 8.4 Hz, 2H), 7.17 (s, 1H), 6.97 (d, J = 8.4 Hz, 2H), 3.38 (d, J = 15.2 Hz, 1H), 3.27 (q, J = 9.6 Hz, 3H), 3.18 (s, 3H), 2.48 (s, 3H).
1A-58
CDCl<sub>3</sub>: δ 7.40 (s, 1H), 7.17 (s, 1H), 7.10-7.05 (m, 4H), 4.01-3.99 (m, 1H), 3.42-3.37 (m, 2H), 3.30-3.22 (m, 3H), 2.49 (s, 3H), 1.10 (t, J = 7.2 Hz, 3H).
1A-59
CDCl3: δ 8.53 (s, 1H), 7.54 (bs, 1H), 7.40 (d, J = 8.4 Hz, 2H), 7.28-7.26 (m, 2H), 7.16 (s, 1H), 3.80 (s, 2H), 3.44 (q, J = 9.6 Hz, 2H), 2.36 (s, 3H).
1A-60
CDCl<sub>3</sub>: δ 8.51 (s, 1H), 7.68 (d, J = 8.0 Hz, 2H), 7.54 (bs, 1H), 7.49 (d, J = 8.4 Hz, 2H), 7.18 (s, 1H), 3.85 (s, 2H), 3.44 (q, J = 9.6 Hz, 2H), 2.36 (s, 3H).
1A-61
CDCl<sub>3</sub>: δ 8.52 (s, 1H), 7.55 (bs, 1H), 7.39 (d, J = 8.4 Hz, 2H), 7.29 (d, J = 8.4 Hz, 2H), 7.17 (s, 1H), 3.76 (s, 2H), 3.44 (q, J = 9.6 Hz, 2H), 2.36 (s, 3H).
1A-62
CDCl<sub>3</sub>: δ 8.53 (s, 1H), 7.71 (d, J = 8.0 Hz, 2H), 7.52 (bs, 1H), 7.43 (d, J = 8.0 Hz, 2H), 7.16 (s, 1H), 3.82 (s,
2H), 3.44 (q, J = 9.6 Hz, 2H), 2.36 (s, 3H).
1A-63
CDCl<sub>3</sub>: δ 8.47 (d, J = 7.6 Hz, 1H), 7.60-7.52 (m, 3H), 7.41 (d. J = 8.4 Hz, 2H), 7.26-7.21 (m, 4H), 6.95-6.90 (m, 1H), 3.71 (s, 2H), 3.37 (q, J = 9.8 Hz, 2H), 2.40 (s, 3H).
1A-64
CDCl3: δ 8.50 (d, J = 8.0 Hz, 1H), 8.44 (s, 1H), 7.92 (d, J = 8.4 Hz, 1H), 7.41 (d, J = 8.4 Hz, 2H), 7.31 (bs, 1H), 7.20 (d, J = 8.4 Hz, 2H), 7.04 (d, J = 8.8 Hz, 1H), 6.94 (d, J = 11.6 Hz, 1H), 3.79 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-65
CDCl<sub>3</sub>: δ 8.51 (d, J = 8.0 Hz, 1H), 7.65-7.60 (m, 3H), 7.48-7.35 (m, 5H), 7.32 (bs, 1H), 6.91 (d, J = 11.6 Hz, 2H), 3.81 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-66
CDCl<sub>3</sub>: δ 8.50 (d, J = 8.0 Hz, 1H), 7.59 (d, J = 8.4 Hz, 2H), 7.44 (d, J = 8.0 Hz, 2H), 7.30 (bs, 1H), 7.11 (d, J = 8.8 Hz, 2H), 6.93 (d, J = 11.2 Hz, 1H), 6.82-6.78 (m, 1H), 3.81 (s, 2H), 3.38 (q, J = 9.7 Hz, 2H), 2.40 (s, 3H).
1A-67
CDCl<sub>3</sub>: δ 8.52 (d, J = 8.0 Hz, 1H), 7.47 (d, J = 8.0 Hz, 2H), 7.40 (d, J = 8.0 Hz, 2H), 7.41-7.21 (m, 5H), 6.92 (d, J = 11.2 Hz, 1H), 3.82 (s, 2H), 3.39 (q, J = 9.7 Hz, 2H), 2.41 (s, 3H), 2.37 (s, 3H).
1A-68
CDCl<sub>3</sub>: δ 8.52 (d, J = 8.0 Hz, 1H), 7.50-7.47 (m, 3H), 7.41 (d, J = 8.0 Hz, 2H), 7.35-7.29 (m, 4H), 6.92 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-69
CDCl<sub>3</sub>: δ 8.49 (d, J = 7.6 Hz, 1H), 7.26 (bs, 1H), 7.22-7.51 (m, 4H), 6.89 (d, J = 11.6 Hz, 1H), 3.72 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H), 2.38 (s, 3H).
1A-70
CDCl<sub>3</sub>: δ 8.49 (d, J = 8.0 Hz, 1H), 7.28-7.12 (m, 5H), 6.99 (d, J = 11.2 Hz, 1H), 3.73 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.67 (q, J = 7.6 Hz, 2H), 2.39 (s, 3H), 1.25 (m, 3H).
1A-71
CDCl<sub>3</sub>: δ 8.50 (d, J = 8.0 Hz, 1H), 7.28-7.22 (m, 5H), 6.89 (d, J = 11.6 Hz, 1H), 3.73 (s, 2H), 3.38 (q, J = 9.7 Hz, 2H), 2.62 (t, J = 8.0 Hz, 2H), 2.39 (s, 3H), 1.66-1.58 (m, 2H), 1.28-1.36 (m, 4H), 0.91-0.80 (m, 3H).
1A-72
CDCl<sub>3</sub>: δ 7.23 (d, J = 7.6 Hz, 1H), 7.13 (d, J = 8.0 Hz, 2H), 7.07 (d, J = 10.4 Hz, 1H), 6.94 (d, J = 8.4 Hz, 2H), 3.46-3.33 (m, 2H), 3.26-3.19 (m, 5H), 2.50 (s, 3H), 2.45 (s, 3H).
1A-73
CDCl<sub>3</sub>: δ 7.57 (s, 1H), 7.53 (d, J = 8.0 Hz, 2H), 7.22 (s, 1H), 7.12 (d, J = 8.4 Hz, 2H), 3.45-3.26 (m, 4H), 3.19 (s, 3H), 2.48 (s, 3H).
1A-74
CDCl<sub>3</sub>: δ 7.56 (s, 1H), 7.13 (d, J = 8.4 Hz, 2H), 7.08 (s, 1H), 6.95 (d, J = 8.4 Hz, 2H), 3.42-3.38 (m, 1H), 3.27-3.19 (m, 3H), 3.17 (s, 3H), 2.46 (s, 3H), 2.45 (s, 3H).
1A-75
CDCl<sub>3</sub>: δ 8.50 (s, 1H), 7.72 (d, J = 8.0 Hz, 2H), 7.50 (bs, 1H), 7.44 (d, J = 8.0 Hz, 2H), 7.32 (s, 1H), 3.83 (s, 2H), 3.45 (q, J = 9.4 Hz, 2H), 2.35 (s, 3H).
1A-76
CDCl3: δ 8.478 (d, J = 7.6 Hz, 1H), 7.242-7.298 (m, 5H), 6.911 (d, J = 11.6 Hz, 1H), 3.719 (s, 2H), 3.373 (q, J = 9.6 Hz, 2H), 2.497 (s, 3H), 2.398 (s, 3H).
1A-77
CDCl<sub>3</sub>: δ 8.50 (s, 1H), 7.58 (bs, 1H), 7.32-7.29 (m, 5H), 3.74 (s, 2H), 3.46 (q, J = 9.6 Hz, 2H), 2.50 (s, 3H), 2.35 (s, 3H).
1A-78
CDCl<sub>3</sub>: δ 7.16 (d, J = 7.6 Hz, 1H), 7.13 (d, J = 8.0 Hz, 2H), 7.08 (d, J = 10.0 Hz, 1H), 6.93 (d, J = 8.0 Hz, 2H), 3.76-3.62 (m, 2H), 3.41 (d, J = 14.8 Hz, 1H), 3.30 (d, J = 14.8 Hz, 1H), 3.20 (q, J = 9.6 Hz, 2H), 2.51 (s, 3H), 2.45 (s, 3H), 1.08 (t, J = 7.2 Hz, 3H).
1A-79
CDCl<sub>3</sub>: δ 7.95 (d, J = 7.6 Hz, 1H), 7.40 (d, J = 8.4 Hz, 2H), 7.30 (d, J = 8.4 Hz, 2H), 8.90 (d, J = 9.2 Hz, 1H), 8.75 (bs, 1H), 3.75 (s, 2H), 3.50 (q, J = 9.6 Hz, 2H), 1.93 (s, 3H).
1A-80
CDCl<sub>3</sub>: δ 7.98 (d, J = 7.6 Hz, 1H), 7.40 (d, J = 8.4 Hz, 2H), 7.29-7.27 (m, 2H), 8.90 (d, J = 9.2 Hz, 1H), 8.75 (bs, 1H), 3.78 (s, 2H), 3.39 (q, J = 9.6 Hz, 2H), 1.98 (s, 3H).
1A-81
CDCl<sub>3</sub>: δ 7.92 (d, J = 6.8 Hz, 1H), 7.69 (d, J = 8.4 Hz, 2H), 7.50 (d, J = 8.0 Hz, 2H), 6.91 (d, J = 9.6 Hz, 1H), 6.75 (bs, 1H), 3.83 (s, 2H), 3.39 (q, J = 9.6 Hz, 2H), 2.02 (s, 3H).
1A-82
CDCl<sub>3</sub>: δ 7.51 (d, J = 8.4 Hz, 2H), 7.40 (s, 1H), 7.21 (s, 1H), 7.17 (d, J = 8.4 Hz, 2H), 3.43 (d, J = 16.0 Hz, 1H), 3.36 (d, J = 15.6 Hz, 1H), 3.30-3.20 (m, 2H), 3.17 (s, 3H), 2.48 (s, 3H).
1A-83
CDCl<sub>3</sub>: δ 7.50 (d, J = 8.0 Hz, 2H), 7.42 (s, 1H), 7.16 (d, J = 8.4 Hz, 2H), 7.14 (s, 1H), 4.05-3.96 (m, 1H), 3.47-3.31 (m, 3H), 3.23 (q, J = 9.4 Hz, 2H), 2.49 (s, 3H), 1.24 (t, J = 7.2 Hz, 3H).
1A-84
CDCl<sub>3</sub>: δ 8.00 (d, J = 8.4 Hz, 1H), 7.72 (d, J = 7.6 Hz, 2H), 7.44 (d, J = 8.0 Hz, 2H), 6.88 (d, J = 9.6 Hz, 1H), 6.74 (bs, 1H), 3.81 (s, 2H), 3.39 (q, J = 9.4 Hz, 2H), 1.96 (s, 3H).
1A-85
CDCl<sub>3</sub>: δ 7.56 (s, 1H), 7.20 (d, J = 8.4 Hz, 2H), 6.89 (d, J = 8.4 Hz, 2H), 6.56 (s, 1H), 4.07-3.96 (m, 1H), 3.25-3.37 (m, 2H), 3.08-2.97 (m, 2H), 2.46 (s, 3H), 1.37 (t, J = 6.8 Hz, 3H), 1.06 (t, J = 7.2 Hz, 3H).
1A-86
CDCl<sub>3</sub>: δ 7.83 (d, J = 8.0 Hz, 1H), 7.27-7.24 (m, 3H), 6.98 (d, J = 8.4 Hz, 1H), 6.86 (d, J = 8.0 Hz, 1H), 4.14-3.97 (m, 1H), 3.92-3.49 (m, 2H), 3.05 (s, 3H), 2.38 (s, 3H), 1.25 (d, J = 6.8 Hz, 3H).
1A-87
CDCl<sub>3</sub>: δ 7.20-7.09 (m, 3H), 6.94-6.75 (m, 3H), 3.91-3.30 (m, 4H), 3.08-3.04 (m, 1H), 3.52-3.49 (m, 3H), 1.37-1.35 (m, 3H), 1.05 (t, J = 6.8 Hz, 3H).
1A-88
CDCl<sub>3</sub>: δ 7.56 (s, 1H), 7.20 (d, J = 8.4 Hz, 2H), 6.91 (d, J = 8.4 Hz, 2H), 6.68 (s, 1H), 3.34-3.29 (m, 1H), 3.15 (s, 3H), 3.05 (q, J = 9.4 Hz, 2H), 2.44 (s, 3H), 1.38 (d, J = 6.8 Hz, 3H).
1A-89
CDCl<sub>3</sub>: 8.48 (s, 1H), 7.50 (bs, 1H), 7.38 (d, J = 8.4 Hz, 2H), 7.32 (d, J = 8.8 Hz, 3H), 3.71 (q, J = 7.2 Hz, 1H), 3.46 (q, J = 9.6 Hz, 2H), 2.35 (s, 3H), 1.62 (d, J = 6.8 Hz, 3H).
1A-90
CDCl<sub>3</sub>: δ 8.48 (d, J = 8.0 Hz, 1H), 7.36 (d, J = 8.4 Hz, 2H), 7.31 (d, J = 8.4 Hz, 2H), 7.18 (bs, 1H), 6.91 (d, J = 11.6 Hz, 1H), 3.71 (q, J = 7.2 Hz, 1H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H), 1.59 (d, J = 6.8 Hz, 3H).
1A-91
CDCl<sub>3</sub>: δ 8.50 (d, J = 8.0 Hz, 2H), 7.23 (d, J = 8.4 Hz, 1H), 7.15 (d, J = 8.0 Hz, 2H), 7.08 (d, J = 10.0 Hz, 1H), 3.68-3.40 (m, 4H), 3.25-3.17 (m, 2H), 2.51 (s, 2H), 1.50-1.41 (m, 3H), 0.90-0.84 (m, 3H).
1A-92
CDCl<sub>3</sub>: δ 7.49 (d, J = 8.0 Hz, 2H), 7.20 (d, J = 7.6 Hz, 1H), 7.15 (d, J = 8.0 Hz, 2H), 7.09 (d, J = 10.0 Hz, 1H), 3.73-3.66 (m, 2H), 3.62-3.54 (m, 2H), 3.25-3.18 (m, 2H), 2.52 (s, 3H). 1.46-1.41 (m, 2H) 1.32-1.22 (m, 2H), 0.862 (t, J = 7.6 Hz, 3H)
1A-93
CDCl<sub>3</sub>: δ 7.63 (d, J = 3.0 Hz, 1H), 7.48 (d, J = 8.4 Hz, 1H), 7.42 (d, J = 8.0 Hz, 2H), 7.36(s, 1H), 7.30 (d, J = 7.2 Hz, 1H), 7.13-7.08 (m, 3H), 3.53-3.42 (m, 2H), 3.26-3.20 (m, 5H), 2.52(s, 3H).
1A-94
CDCl<sub>3</sub>: δ 7.63 (d, J = 2.4 Hz, 1H), 7.48 (d, J = 8.4 Hz, 1H), 7.40-7.36 (m, 3H), 7.24 (d, J = 8.4 Hz, 1H), 7.12-7.09 (m, 3H), 3.78-3.65 (m, 2H), 3.50-3.37 (m, 2H), 3.25-3.17 (m, 2H), 2.52 (s, 3H). 1.10 (t, J = 7.2 Hz, 3H).
1A-95
CDCl<sub>3</sub>: δ 7.55 (d, J = 7.6 Hz, 2H), 7.47 (d, J = 8.0 Hz, 2H), 7.42(t, J = 7.6 Hz, 2H), 7.33 (t, J = 7.2 Hz, 1H),7.28(s,1 H),7.14-7.06 (m, 3H), 3.55-3.42 (m, 2H), 3.24(s, 3H), 3.16 (s, 2H), 2.51 (s, 3H).
1A-96
CDCl<sub>3</sub>: δ 7.56 (d, J = 7.2 Hz, 2H), 7.46 (d, J = 8.0 Hz, 2H), 7.42 (t, J = 7.2 Hz, 2H), 7.33 (d, J = 7.6 Hz, 1H),7.20(d, J = 7.6 Hz, 1H), 7.09(d, J = 8.4 Hz, 3H),3.80-3.65 (m, 2H), 3.51-3.37 (m, 2H), 3.22-3.14 (m, 2H), 2.51 (s, 3H). 1.10(t, J = 7.6 Hz, 3H).
1A-97
CDCl<sub>3</sub>: δ 7.45-7.53 (m, 2H), 7.29 (s, 1H), 7.18-7.06 (m, 3H), 6.90-6.72 (m, 1H), 5.06-4.84 (m, 1H), 3.59-3.37 (m, 1H), 3.01-2.87 (m, 2H), 2.49 (s, 3H), 2.18-1.90 (m, 2H).
1A-98
CDCl<sub>3</sub>: δ 7.50 (d, J = 8.0 Hz, 2H), 7.15-7.10 (m, 4H), 4.97 (m, 1H), 3.93-3.44 (m, 2H), 3.21-3.14 (m, 2H), 2.53 (s, 3H), 1.10 (d, J = 6.8 Hz, 3H), 0.98 (d, J = 6.8 Hz, 3H).
1A-99
CDCl<sub>3</sub>: δ 7.99 (d, J = 7.2 Hz, 1H), 7.62 (t, J = 7.2 Hz, 4H), 7.45 (d, J = 8.0 Hz, 2H), 7.31 (d, J = 8.0 Hz, 2H), 6.89 (d, J = 9.2 Hz, 1H), 6.84 (s, 1H), 3.83 (s, 2H), 3.43-3.36 (m, 2H), 1.98 (s, 3H).
1A-100
CDCl<sub>3</sub>: δ 7.99 (d, J = 7.2 Hz, 1H), 7.73 (s, 4H), 7.67 (d, J = 7.6 Hz, 2H), 7.48 (d, J = 8.0 Hz, 2H), 6.90 (d, J = 9.2 Hz, 1H), 6.84 (bs, 1H), 3.84 (s, 2H), 3.43-3.36 (m, 2H), 1.99 (s, 3H).
1A-101
CDCl<sub>3</sub>: δ 8.51 (d, J = 9.2 Hz, 1H), 7.73 (d, J = 8.0 Hz, 1H), 7.53 (d, J = 7.6 Hz, 1H), 7.32 (s, 1H), 6.98 (d, J = 8.8 Hz, 1H), 6.86(d, J = 12.0 Hz, 1H), 3.64 (s, 2H), 3.36-3.25 (m, 2H), 2.37 (s, 3H).
1A-102
CDCl<sub>3</sub>: δ 8.50 (d, J = 7.6 Hz, 1H), 7.59-7.54 (m, 4H), 7.41 (d, J = 8.0 Hz, 2H), 7.31 (bs, 1H),7.14 (t, J = 8.8 Hz, 2H), 6.92 (d, J = 11.6 Hz, 1H), 3.80 (s, 2H), 3.42-3.34 (m, 2H), 2.40 (s, 3H).
1A-103
CDCl<sub>3</sub>: δ 8.50 (d, J = 8.0 Hz, 1H), 7.59 (d, J = 8.0 Hz, 2H), 7.54 (d, J = 8.8 Hz, 2H), 7.38 (d, J = 8.0 Hz, 2H), 7.32 (bs, 1H), 7.14 (t, J = 8.8 Hz, 2H), 6.92 (d, J = 11.6, 1H), 3.86 (s, 3H), 3.79 (s, 2H), 3.15-3.34 (m, 2H) 2.40 (s, 3H) .
1A-104
CDCl<sub>3</sub>: δ 7.61 (d, J = 8.0 Hz, 2H), 7.53 (d, J = 8.4 Hz, 1H), 7.48 (d, J = 8.0 Hz, 1H), 7.44 (d, J = 7.6, 4H), 7.31 (bs, 1H), 7.22 (d, J = 8.0 Hz, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.34-3.34 (m, 2H), 2.40 (s, 3H).
1A-105
CDCl3: δ 8.50 (d, J = 7.6 Hz, 1H), 7.73 (d, J = 14.4 Hz, 2H), 7.61 (d, J = 8.0 Hz, 3H), 7.46 (d, J = 8.0 Hz, 2H), 7.30 (bs, 1H), 6.93 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.34-3.41 (m, 2H) 2.41 (s, 3H).
1A-106
CDCl<sub>3</sub>: δ 8.53 (d, J = 8.0 Hz, 1H), 7.51 (d, J = 8.0 Hz, 2H), 7.45-7.36 (m, 6H), 7.19-7.12 (m, 1H), 6.92 (d, J = 11.2 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-107
CDCl<sub>3</sub>: δ 8.50 (d, J = 7.6 Hz, 1H), 7.84 (s, 1H), 7.77 (d, J = 7.6 Hz, 1H), 7.63 (t, J = 7.6 Hz, 3H), 7.57 (t, J = 7.6 Hz, 1H),7.45 (d, J = 8.4 Hz, 2H), 7.33 (s, 1H), 6.92 (d, J = 12.8 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.41 (s , 3H).
1A-108
CDCl<sub>3</sub>: δ 8.51 (d, J = 7.6 Hz, 1H), 7.62 (d, J = 8.0 Hz, 2H), 7.44-7.36 (m, 4H), 7.31-7.26 (m, 2H), 7.07-7.04 (m, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.81 (s, 2H), 3.38 (q, J = 8.8 Hz, 2H), 2.40 (s, 3H).
1A-109
CDCl<sub>3</sub>: δ 8.50 (d, J = 8.0 Hz, 1H), 7.60 (d, J = 6.8 Hz, 2H), 7.47-7.41 (m, 3H), 7.34-7.31 (m, 2H), 7.26-7.14 (m, 2H), 6.92 (d, J = 11.2 Hz, 1H), 3.81 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-110
CDCl<sub>3</sub>: δ 7.85 (s, 1H), 7.64 (d, J = 6.4 Hz, 1H), 7.55 (d, J = 8.4 Hz, 1H), 7.44 (d, J = 8.0 Hz, 2H), 7.31 (d, J = 8.0 Hz, 1H), 7.15 (d, J = 8.0 Hz, 2H), 7.10 (d, J = 10.4 Hz, 1H), 3.54-3.43 (m, 2H), 3.28-3.24 (m, 5H), 2.52 (s, 3H).
1A-111
CDCl<sub>3</sub>: δ 7.69-7.65 (m, 2H), 7.57-7.53 (m, 1H), 7.49-7.44 (m, 1H), 7.32 (s, 1H), 7.29 (s, 2H), 7.09 (d, J = 6.8 Hz, 2H), 3.57-3.48 (m, 2H), 3.29-3.22 (m, 5H), 2.51 (s, 3H).
1A-112
CDCl<sub>3</sub>: δ 7.58 (d, J = 8.0 Hz, 2H), 7.19-7.11 (m, 4H), 3.75-3.83 (m, 1H), 3.70-3.66 (m, 1H), 3.52-3.40 (m, 2H), 3.27-3.20 (m, 2H), 2.53 (s, 3H), 2.15-2.09 (m, 2H), 1.79-1.71 (m, 2H).
1A-113
CDCl<sub>3</sub>: δ 8.52 (d, J = 8.0 Hz, 1H), 7.46-7.40 (m, 4H), 7.34-7.30 (m, 2H), 7.25-7.22 (m, 1H), 7.08-7.03 (m, 1H), 6.93 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-114
CDCl<sub>3</sub>: δ 8.50 (d, J = 8.0 Hz, 1H), 7.89 (d, J = 1.6 Hz, 1H), 7.79 (d, J = 8.0 Hz, 1H), 7.57-7.61 (m, 3H), 7.45 (d, J = 8.0 Hz, 2H), 7.31 (bs, 1H), 6.93 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-115
CDCl<sub>3</sub>: δ 9.22 (s, 1H), 8.97 (s, 2H), 8.50 (d, J = 8.0 Hz, 1H), 7.63 (d, J = 8.0 Hz, 2H), 7.51 (d, J = 8.0 Hz, 2H), 7.33 (bs, 1H), 6.94 (d, J = 11.6 Hz, 1H), 3.84 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-116
CDCl<sub>3</sub>: δ 8.69 (d, J = 7.6 Hz, 1H), 8.42 (bs, 1H), 7.68 (d, J = 8.0 Hz, 2H), 7.51 (d, J = 8.0 Hz, 2H), 6.98 (d, J = 11.2 Hz, 1H), 4.31 (s, 2H), 3.39 (q, J = 9.6 Hz, 2H), 2.44 (s, 3H).
1A-117
CDCl<sub>3</sub>: δ 7.54 (d, J = 8.4 Hz, 2H), 7.13-7.10 (m, 3H), 7.03 (d, J = 10.0 Hz, 1H), 4.14-3.95 (m, 2H), 3.66 (s, 3H), 3.19-3.05 (m, 2H), 2.49 (s, 3H).
1A-118
CDCl<sub>3</sub>: δ 7.45 (d, J = 8.0 Hz, 2H), 7.41-7.30 (m, 3H), 7.28 (s, 1H), 7.23 (s, 1H), 7.11 (d, J = 8.0 Hz, 2H), 7.04-7.00 (m, 1H), 3.54-3.42 (m, 2H), 3.29-3.18 (m, 5H), 2.49 (s, 3H).
1A-119
CDCl<sub>3</sub>: δ 7.37-7.32 (m, 5H), 7.19-7.12 (m, 2H), 7.09-7.06 (m, 2H), 7.04 (s, 1H), 3.55-3.44 (m, 2H), 3.30-3.19 (m, 5H), 2.51 (s, 3H).
1A-120
CDCl<sub>3</sub>: δ 7.68 (d, J = 14.0 Hz, 2H), 7.57 (s, 1H), 7.45 (d, J = 8.4 Hz, 2H), 7.32 (d, J = 7.6 Hz, 1H), 7.15 (d, J = 8.4 Hz, 2H), 7.10 (d, J = 10.0 Hz, 1H), 3.49 (m, 2H), 3.33-3.19 (m, 5H), 2.52 (s, 3H).
1A-121
CDCl<sub>3</sub>: δ 7.79 (s, 1H), 7.73 (d, J = 7.6 Hz, 1H), 7.60-7.52 (m, 2H), 7.48 (d, J = 8.0 Hz, 2H), 7.31-7.27 (m, 1H), 7.19-7.10 (m, 3H), 3.55-3.43 (m, 2H), 3.27-3.19 (m, 5H), 2.50 (s, 3H).
1A-122
CDCl<sub>3</sub>: δ 7.48 (d, J = 8.8 Hz, 2H), 7.42 (d, J = 8.4 Hz, 2H), 7.29-7.24 (m, 1H), 7.19-7.14 (m, 1H), 7.09-7.06 (m, 2H), 6.96 (d, J = 8.4 Hz, 2H), 3.86 (s, 3H), 3.53-3.40 (m, 2H), 3.23-3.16 (m, 5H), 2.48 (s, 3H).
1A-123
CDCl<sub>3</sub>: δ 7.52-7.48 (m, 2H), 7.42 (d, J = 8.0 Hz, 2H), 7.29-7.27 (m, 1H), 7.14-7.07 (m, 5H), 3.53-3.41 (m, 2H), 3.23-3.20 (m, 5H), 2.51 (s, 3H).
1A-124
CDCl<sub>3</sub>: δ 7.43-7.38 (m, 3H), 7.33-7.27 (m, 2H), 7.14-7.06 (m, 5H), 3.52-3.44 (m, 2H), 3.24-3.17 (m, 5H), 2.48 (s, 3H).
1A-125
CDCl<sub>3</sub>: δ 7.40-7.32 (m, 5H), 7.19-7.12 (m, 2H), 7.09-7.04 (m, 3H), 3.55-3.44 (m, 2H), 3.28-3.21 (m, 5H), 2.51 (s, 3H).
1A-126
CDCl<sub>3</sub>: δ 8.46 (d, J = 8.0 Hz, 1H), 8.26 (bs, 1H), 7.62 (d, J = 8.8 Hz, 2H), 7.47 (d, J = 8.8 Hz, 2H), 7.03 (d, J = 11.2 Hz, 1H), 3.40-3.33 (m, 2H), 2.44 (s, 3H).
1A-127
CDCl<sub>3</sub>: δ 7.49-7.35 (m, 4H), 7.29-7.28 (m, 2H), 7.11-7.06 (m, 2H), 6.91 (d, J = 8.8 Hz, 2H), 3.50-3.33 (m, 2H), 3.28-3.21 (m, 5H), 2.51 (s, 3H).
1A-128
CDCl<sub>3</sub>: δ 7.45 (d, J = 6.8 Hz, 1H), 7.33-7.27 (m, 6H), 7.10-7.06 (m, 3H), 3.56-3.44 (m, 2H), 3.24-3.20 (m, 5H), 2.51 (s, 3H).
1A-129
CDCl<sub>3</sub>: δ 7.30 (d, J = 8.4 Hz, 2H), 7.18-7.14 (m, 3H), 6.89 (d, J = 10.4 Hz, 1H), 3.28-3.21 (m, 5H), 2.47 (s, 3H).
1A-130
CDCl<sub>3</sub>: δ 7.21 (d, J = 7.6 Hz, 1H), 7.06-7.03 (m, 3H), 6.91 (d, J = 8.0 Hz, 2H), 3.48-3.34 (m, 2H), 3.25-3.14 (m, 5H), 3.54-2.49 (m, 5H), 1.62-1.50 (m, 2H), 0.84 (t, J = 7.2 Hz, 3H).
1A-131
CDCl<sub>3</sub>: δ 7.69-7.67 (m, 1H), 7.35 (d, J = 7.6 Hz, 1H), 7.12 (d, J = 11.2 Hz, 1H), 6.94-6.83 (m, 2H), 3.56-3.30 (m, 2H), 3.35-3.12 (m, 5H), 2.54 (s, 3H).
1A-132
CDCl<sub>3</sub>: δ 7.50 (d, J = 7.6 Hz, 1H), 7.22-7.14 (m, 1H), 7.11 (d, J = 10.0 Hz, 1H), 6.83 (t, J = 9.4 Hz, 2H), 3.45 (s, 2H), 3.35 (q, J = 9.4 Hz, 2H), 3.25 (s, 3H), 2.51 (s, 3H).
1A-133
CDCl<sub>3</sub>: δ 7.19 (d, J = 7.6 Hz, 1H), 7.06 (d, J = 7.6 Hz, 3H), 6.92 (d, J = 8.0 Hz, 2H), 3.48-3.34 (m, 2H), 3.21-3.13 (m, 5H), 2.59 (q, J = 7.6 Hz, 2H), 2.50 (s, 3H), 1.24-1.14 (m, 3H).
1A-134
CDCl<sub>3</sub>: δ 7.86-7.75 (m, 2H), 7.50-7.37 (m, 1H), 7.22 (d, J = 8.0 Hz, 2H), 6.98-6.08 (m, 1H), 3.37-3.25 (m, 2H), 3.21-3.17 (m, 5H), 2.55-2.41 (m, 5H), 1.17-1.13 (m, 3H).
1A-135
CDCl<sub>3</sub>: δ 8.05 (s, 1H), 7.82-7.76 (m, 2H), 7.18-7.16 (m, 1H), 7.16-7.12 (m, 1H), 6.94-6.89 (m, 1H), 3.73-3.58 (m, 2H), 3.40-3.36 (m, 2H), 3.22 (s, 3H), 3.10-2.98 (m, 3H), 2.47 (s, 3H).
1A-136
CDCl<sub>3</sub>: δ 8.46 (d, J = 8.0 Hz, 1H), 7.96 (d, J = 8.4 Hz, 2H), 7.56 (d, J = 8.0 Hz, 2H), 7.32 (s, 1H), 6.95 (d, J = 11.6 Hz, 1H), 3.85 (s, 2H), 3.40-3.32 (m, 2H), 3.06 (s, 3H), 2.42 (s, 3H).
1A-137
CDCl<sub>3</sub>: δ 8.44 (d, J = 7.6 Hz, 1H), 8.31 (s, 1H), 7.82 (d, J = 8.4 Hz, 2H), 7.78 (s, 1H), 7.76 (s, 1H), 7.03 (d, J = 11.2 Hz, 1H), 3.40-3.33 (m, 2H), 2.45 (s, 3H).
1A-138
CDCl<sub>3</sub>: δ 7.58 (d, J = 8.0 Hz, 2H), 7.35 (d, J = 8.0 Hz, 2H), 7.20 (d, J = 7.6 Hz, 1H), 6.84 (d, J = 10.0 Hz, 1H), 3.32-3.20 (m, 5H), 2.47 (s, 3H).
1A-139
CDCl<sub>3</sub>: δ 7.53 (d, J = 12.4 Hz, 1H), 7.49 (d, J = 6.4 Hz, 1H), 7.16-7.12 (m, 2H), 7.07 (d, J = 8.4 Hz, 1H), 6.91 (d, J = 8.4 Hz, 1H), 3.43-3.35 (m, 2H), 3.18 (d, J = 9.6 Hz, 2H), 2.40 (s, 3H), 1.41 (d, J = 6.4 Hz, 3H).
1A-140
CDCl<sub>3</sub>: δ 8.52 (d, J = 8.0 Hz, 1H), 7.50 (d, J = 1.6 Hz, 1H), 7.46-7.40 (m, 4H), 7.30 (d, J = 2.0 Hz, 1H), 7.31-7.26 (m, 2H), 6.93 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 10.0 Hz, 2H), 2.41 (s, 3H).
1A-141
CDCl<sub>3</sub>: δ 8.50 (d, J = 8.0 Hz, 1H), 7.54 (d, J = 8.0 Hz, 2H), 7.43 (d, J = 7.6 Hz, 2H), 7.29 (bs, 1H), 7.21-7.16 (m, 2H), 6.93 (d, J = 11.6 Hz, 1H), 3.81 (d, J = 4.8 Hz, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
1A-142
CDCl<sub>3</sub>: δ 8.49 (q, J = 4.8 Hz, 1H), 7.69-7.40 (m, 5H), 7.30 (bs, 1H), 7.26-7.21 (m, 3H), 6.99-6.90 (m, 1H), 3.81 (d, J = 6.8 Hz, 2H), 3.41-3.27 (m, 2H), 2.40 (s, 3H).
1A-143
CDCl<sub>3</sub>: δ 8.50 (d, J = 7.6 Hz, 1H), 7.58 (d, J = 8.4 Hz, 2H), 7.46 (d, J = 1.6 Hz, 2H), 7.43 (d, J = 8.0 Hz, 2H), 7.35 (t, J = 1.6 Hz, 1H), 7.29 (bs, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.81 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1A-144
CDCl<sub>3</sub>: δ 8.50 (d, J = 7.6 Hz, 1H), 7.61 (d, J = 8.4 Hz, 2H), 7.53 (d, J = 8.4 Hz, 2H), 7.40 (d, J = 7.6 Hz, 2H), 7.34 (d, J = 8.4 Hz, 3H), 6.91 (d, J = 11.6 1H), 3.80 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.53 (s, 3H), 2.40 (s, 3H).
1A-145
CDCl<sub>3</sub>: δ 8.50 (d, J = 7.6 Hz, 1H), 7.62 (d, J = 8.0 Hz, 2H), 7.50 (d, J = 8.0 Hz, 2H), 7.39 (d, J = 8.0 Hz, 2H), 7.32 (bs, 1H), 7.27-7.26 (m, 2H), 6.91 (d, J = 11.6 1H), 3.80 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 6H).
1A-146
CDCl<sub>3</sub>: δ 7.48 (d, J = 8.4 Hz, 2H), 7.44 (d, J = 8.4 Hz, 2H), 7.31 (d, J = 8.4 Hz, 2H), 7.27 (s, 1H), 7.08 (t, J = 5.6 Hz, 3H), 3.50 (s, 1H), 3.45 (s, 1H), 3.23 (s, 3H), 3.20 (d, J = 10.0 Hz, 2H), 2.52 (s, 6H).
1A-147
CDCl<sub>3</sub>: δ 7.42 (q, J = 6.5 Hz, 4H), 7.32 (s, 2H), 7.14 (s, 3H), 3.23 (m, 7H), 2.52 (s, 3H).
1A-148
CDCl<sub>3</sub>: δ 7.56 (d, J = 6.0 Hz, 1H), 7.44-7.40 (m, 2H), 7.37-7.32 (m, 2H), 7.29-7.26 (m, 1H), 7.12-7.06 (m, 2H), 6.91 (d, J = 8.4 Hz, 2H), 3.49-3.28 (m, 2H), 3.28-3.18 (m, 5H), 2.51 (s, 3H).
1A-149
CDCl<sub>3</sub>: δ 7.45 (d, J = 8.0 Hz, 4H), 7.23 (d, J = 8.4 Hz, 3H), 7.09-7.06 (m, 3H), 3.47 (m, 2H), 3.23 (s, 3H), 3.20-3.16 (m, 2H), 2.50 (s, 3H), 2.39 (s, 3H).
Table 3
<chemistry id="chem0019" num="0019"><img id="ib0019" file="IMGB0019.TIF" wi="63" he="26" img-content="chem" img-format="GIF" /></chemistry>
<b>S. No.</b>
<b>R<sup>4</sup></b>
<b>R<sup>5</sup></b>
<b>R<sup>6</sup></b>
<b>R<sup>7</sup></b>
<b>R<sup>8</sup></b>
<b>R<sup>9</sup></b>
<b>R<sup>10</sup></b>
<b>R<sup>11</sup></b>
<b>x</b>
<b>n</b>
1B-1
H
H
H
H
H
S-Et
H
H
O
0
1B-2
H
H
H
H
H
S-<i>n</i>-Pr
H
H
O
0
1B-3
H
H
H
H
H
S-<i>i</i>-Pr
H
H
O
0
1B-4
H
H
H
H
H
(4-phenyl)-phenyl
H
H
O
0
1B-5
H
H
H
H
H
4-cyano-phenyl
H
H
O
0
1B-6
H
H
H
H
H
4-dimethylaminophenyl
H
H
O
0
1B-7
Me
H
H
H
H
S-Et
H
H
O
0
1B-8
Me
H
H
H
H
S-<i>i</i>-Pr
H
H
O
0
1B-9
Me
H
H
H
H
<i>i</i>-Pr
H
H
O
0
1B-10
Me
H
H
H
H
S-<i>n</i>-Pr
H
H
O
0
1B-11
H
H
H
H
H
2,3,4-Cl<sub>3</sub>-phenyl
H
H
O
0
1B-12
H
H
H
H
H
3-Cl-phenyl
H
H
O
0
1B-13
H
H
H
H
H
4-Ac-phenyl
H
H
O
0
1B-14
Me
H
H
H
H
4-phenyl-phenyl
H
H
O
0
1B-15
Me
H
H
H
H
4-cyano-phenyl
H
H
O
0
1B-16
Me
H
H
H
H
2,3-Cl<sub>2</sub>-phenyl
H
H
O
0
1B-17
Me
H
H
H
H
4-S-Et-phenyl
H
H
O
0
1B-18
H
H
H
H
H
2,3-Cl<sub>2</sub>-phenyl
H
H
O
0
1B-19
H
H
H
H
H
3,4-F<sub>2</sub>-phenyl
H
H
O
0
1B-20
H
H
H
H
H
2,3,4-F<sub>3</sub>-phenyl
H
H
O
0
1B-21
H
H
H
H
H
4-S-Et-phenyl
H
H
O
0
1B-22
H
H
H
H
H
2,2,2-trifluoroethylthio
H
H
O
0
1B-23
Me
H
H
H
H
2,2,2-trifluoroethylthio
H
H
O
0
1B-24
H
H
H
H
H
2,2,2-trifluoroethylthio
H
H
S
0
1B-25
H
H
H
H
H
Cl
H
H
S
0
1B-26
H
H
H
H
H
(3,4-Cl<sub>2</sub>)-phenyl
H
H
S
0
1B-27
H
H
H
H
H
phenyl
H
H
S
0
1B-28
H
H
H
H
H
OCF<sub>3</sub>
H
H
S
0
1B-29
Me
H
H
H
H
(3,4-F<sub>2</sub>)-phenyl
H
H
O
0
1B-30
Me
H
H
H
H
(2,3,4-F<sub>3</sub>)-phenyl
H
H
O
0
1B-31
Me
H
H
H
H
4-Ac-phenyl
H
H
O
0
1B-32
Me
H
H
H
H
(2,3,4-Cl<sub>3</sub>)-phenyl
H
H
O
0
1B-33
Me
H
H
H
H
3-Cl-phenyl
H
H
O
0
1B-34
Me
H
H
H
H
3-CF<sub>3</sub>-phenyl
H
H
O
0
1B-35
Me
H
H
H
H
3-OCF<sub>3</sub>-phenyl
H
H
O
0
1B-36
Me
H
H
H
H
2,2,2-trifluoroethylthio
H
H
S
0
1B-37
Me
H
H
H
H
phenyl
H
H
S
0
1B-38
H
H
H
H
H
3-OCF<sub>3</sub>-phenyl
H
H
S
0
1B-39
Me
H
H
H
H
OCF<sub>3</sub>
H
H
S
0
1B-40
H
F
F
H
H
phenyl
H
H
O
0
1B-41
Me
F
F
H
H
phenyl
H
H
O
0
1B-42∗
H
H
H
3-CF<sub>3</sub>-phenyl
H
H
H
H
O
0
1B-43
H
H
H
H
H
3-CF<sub>3</sub>-phenyl
H
H
S
0
1B-44∗
H
H
H
3-OCF<sub>3</sub>-phenyl
H
H
H
H
O
0
1B-45∗
H
H
H
phenyl
H
H
H
H
O
0
1B-46∗
H
H
H
phenyl
H
H
H
H
S
0
1B-47∗
H
H
H
3-OCF<sub>3</sub>-phenyl
H
H
H
H
S
0
1B-48∗
Me
H
H
phenyl
H
H
H
H
O
0
1B-49∗
Me
H
H
3-CF<sub>3</sub>-phenyl
H
H
H
H
O
0
1B-50∗
Me
H
H
3-OCF<sub>3</sub>-phenyl
H
H
H
H
O
0
1B-51∗
Me
H
H
phenyl
H
H
H
H
S
0
1B-52∗
Me
H
H
3-CF<sub>3</sub>-phenyl
H
H
H
H
S
0
1B-53
H
F
F
H
H
SCF<sub>3</sub>
H
H
O
0
1B-54
Me
F
F
H
H
SCF<sub>3</sub>
H
H
O
0
1B-55∗
Me
H
H
3-OCF<sub>3</sub>-phenyl
H
H
H
H
S
0
1B-56
H
F
H
H
H
CF<sub>3</sub>
H
H
O
0
1B-57∗
H
H
H
H
3-OCF<sub>3</sub>-phenyl
H
H
H
O
0
1B-58∗
H
H
H
H
phenyl
H
H
H
O
0
1B-59∗
H
H
H
H
3-CF<sub>3</sub>-phenyl
H
H
H
O
0
1B-60∗
H
H
H
H
4-CF<sub>3</sub>-phenyl
H
H
H
O
0
1B-61∗
Me
H
H
H
3-OCF<sub>3</sub>-phenyl
H
H
H
O
0
1B-62∗
Me
H
H
H
phenyl
H
H
H
O
0
1B-63∗
Me
H
H
H
3-CF<sub>3</sub>-phenyl
H
H
H
O
0
1B-64∗
Me
H
H
H
4-CF<sub>3</sub>-phenyl
H
H
H
O
0
1B-65∗
Me
H
H
H
4-OCF<sub>3</sub>-phenyl
H
H
H
O
0
1B-66∗
H
H
H
H
2,3-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-67∗
H
H
H
H
2,4-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-68∗
H
H
H
H
2,5-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-69∗
H
H
H
H
3,4-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-70∗
H
H
H
H
3,5-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-71∗
H
H
H
H
3-Cl-phenyl
H
H
H
O
0
1B-72∗
Me
H
H
H
3-Cl-phenyl
H
H
H
O
0
1B-73∗
Me
H
H
H
2,5-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-74∗
Me
H
H
H
3,4-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-75
H
F
F
H
H
SCF<sub>3</sub>
H
H
O
2
1B-76∗
Me
H
H
H
2,3-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-77∗
Me
H
H
H
2,4-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-78∗
Me
H
H
H
3,5-Cl<sub>2</sub>-phenyl
H
H
H
O
0
1B-79
H
H
H
H
H
OCHF<sub>2</sub>
H
H
O
0
1B-80
Me
H
H
H
H
OCHF<sub>2</sub>
H
H
O
0
1B-81
H
H
H
H
H
OCHF<sub>2</sub>
H
H
O
2
1B-82
Me
H
H
H
H
OCHF<sub>2</sub>
H
H
O
2
1B-83
Me
H
H
H
H
OCHF<sub>2</sub>
H
H
O
1
1B-84
H
H
H
H
H
SCF<sub>3</sub>
H
H
O
2
1B-85∗
H
H
H
H
3-CF<sub>3</sub>-Phenyl
H
H
H
O
2
1B-86
H
F
F
H
H
OCF<sub>3</sub>
H
H
O
0
1B-87
Me
F
F
H
H
OCF<sub>3</sub>
H
H
O
0
1B-88
H
F
F
H
H
OCF<sub>3</sub>
H
H
O
2
1B-89
Me
F
F
H
H
OCF<sub>3</sub>
H
H
O
2
1B-90∗
H
H
H
H
3-
H
H
H
O
0
(benzo[d][1,3]di oxol-5-yl)phenyl
1B-91∗
H
H
H
H
3-F-Phenyl
H
H
H
O
0
1B-92∗
H
H
H
H
4-S-Et-Phenyl
H
H
H
O
0
1B-93∗
H
H
H
H
2-S-Me-Phenyl
H
H
H
O
0
1B-94∗
H
H
H
H
(4-Phenyl)Phenyl
H
H
H
O
0
1B-95∗
H
H
H
H
4-CyanoPhenyl
H
H
H
O
0
1B-96∗
Me
H
H
H
4-CyanoPhenyl
H
H
H
O
0
1B-97∗
Me
H
H
H
(4-Phenyl)Phenyl
H
H
H
O
0
1B-98∗
Me
H
H
H
2-S-Me-Phenyl
H
H
H
O
0
1B-99∗
Me
H
H
H
4-S-Et-Phenyl
H
H
H
O
0
1B-100*
Me
H
H
H
3-F-Phenyl
H
H
H
O
0
1B-101*
Me
H
H
H
3-(benzo[d][1,3]di oxol-5-yl)phenyl
H
H
H
O
0
1B-102
H
H
H
H
H
cyano
H
H
O
0
Table 4
<b>S. No.</b>
<sup>1</sup>H NMR
1B-1
CDCl3: <i>δ</i> 8.48 (d, J = 8.0 Hz, 1H), 7.35 (d, J = 8.4 Hz, 2H), 7.26-7.24 (m, 3H), 6.90 (d, J = 11.6 Hz, 1H), 3.72 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.99-2.93 (m, 2H), 2.39 (s, 3H), 1.30 (t, J = 7.2 Hz, 3H).
1B-2
CDCl3: <i>δ</i> 8.48 (d, J = 8.0 Hz, 1H), 7.52 (d, J = 8.4 Hz, 1H), 7.47 (d, J = 8.4 Hz, 1H), 7.32-7.23 (m, 3H), 6.93 (d, J = 11.6 Hz, 1H), 3.73 (s, 2H), 3.36 (d, J = 7.2 Hz, 2H), 3.44 (q, J = 9.6 Hz, 2H), 3.39 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H), 0.88 (t, J = 7.2 Hz, 3H).
1B-3
CDCl3: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.52 (d, J = 8.4 Hz, 1H), 7.45 (d, J = 8.4 Hz, 2H), 7.27(d, J = 8.0 Hz, 2H), 6.91 (d, J = 11.6 Hz, 1H), 3.73 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H), 1.42-141 (m, 1H), 1.31 (d, J = 7.6 Hz, 6H).
1B-4
CDCl3: <i>δ</i> 8.51 (d, J = 8.0 Hz, 1H), 7.69-7.64 (m, 8H), 7.48-7.42 (m, 4H), 7.38-7.35 (m, 2H), 6.91 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-5
CDCl3: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 7.75-7.68 (m, 4H), 7.62 (d, J = 8.0 Hz, 2H), 7.46 (d, J = 8.0 Hz, 2H), 7.31 (bs, 1H), 6.92 (d, J = 12.0 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-6
CDCl3: <i>δ</i> 8.50 (d, J = 7.6 Hz, 1H), 7.59 (d, J = 8.0 Hz, 2H), 7.51(d, J = 8.8 Hz, 2H), 7.36-7.32 (m, 3H), 6.90 (d, J = 7.6 Hz, 1H), 6.81 (d, J = 8.0 Hz, 2H), 3.78 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 3.00 (s, 6H), 2.39 (s, 3H).
1B-7
CDCl3: <i>δ</i> 7.25 (d, J = 8.0 Hz, 2H), 7.29 (d, J = 8.0 Hz, 1H), 7.14-7.12 (m, 1H), 7.04 (t, J = 7.6 Hz, 1H), 6.93 (d, J = 8.0 Hz, 1H), 3.41 (q, J = 9.6 Hz, 2H), 3.25 (d, J = 9.6 Hz, 2H), 3.20 (s, 3H), 2.93-2.86 (m, 2H), 2.50 (s, 3H), 1.30 (t, J = 7.2 Hz, 3H).
1B-8
CDCl3: <i>δ</i> 7.31 (s, 1H), 7.27 (s, 2H), 7.04 (d, J = 10.0 Hz, 1H), 6.94 (d, J = 8.4 Hz, 2H), 3.43-3.25 (m, 5H), 3.21(s, 3H), 2.50 (s, 3H), 1.33-1.21 (m, 6H).
1B-9
CDCl3: <i>δ</i> 7.22 (d, J = 8.0 Hz, 1H), 7.09 (d, J = 8.0 Hz, 2H), 7.05 (d, J = 10.0 Hz, 1H), 6.93 (d, J = 8.0 Hz, 2H), 3.47-3.33 (m, 2H), 3.21 (s, 3H), 3.17 (q, J = 9.4 Hz, 2H), 2.88-2.81 (m, 1H), 2.49 (s, 3H), 1.33-1.21 (m,
6H).
1B-10
CDCl3: <i>δ</i> 7.36 (d, J = 8.4 Hz, 1H), 7.33-7.25 (m, 2H), 7.05 (d, J = 10.4 Hz, 1H), 7.00 (d, J = 8.0 Hz, 1H), 6.96 (d, J = 8.0 Hz, 1H), 3.56-3.23 (m, 6H), 3.21 (s, 3H), 2.50 (s, 3H), 1.42-1.36 (m, 2H), 0.92 (t, J = 7.2 Hz, 3H).
1B-11
CDCl3: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.53 (d, J = 8.4 Hz, 1H), 7.45-7.42 (m, 3H), 7.35-7.30 (m, 1H), 7.23-7.18 (m, 2H), 6.93 (d, J = 12.0 Hz, 1H), 3.82-3.71 (m, 2H), 3.39 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-12
CDCl3: <i>δ</i> 8.50 (d, J = 7.6 Hz, 1H), 7.61-7.58 (m, 3H), 7.48-7.31 (m, 6H), 6.93 (d, J = 11.6 Hz, 1H), 3.81(s, 2H), 3.36 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-13
CDCl3: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 8.04 (d, J = 8.8 Hz, 2H), 7.70-7.66 (m, 4H), 7.45 (d, J = 8.4 Hz, 2H), 7.31 (bs, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.64 (s, 3H), 2.40 (s, 3H).
1B-14
CDCl3: <i>δ</i> 7.67-7.62 (m, 6H), 7.52 (d, J = 8.0 Hz, 2H), 7.45 (d, J = 7.6 Hz, 2H), 7.36 (d, J = 7.2 Hz, 1H), 7.28 (d, J = 7.6 Hz, 1H), 7.13-7.07 (m, 3H), 3.56-3.47 (m, 2H), 3.38 (q, J = 9.6 Hz, 2H), 3.24 (s, 3H), 2.51 (s, 3H).
1B-15
CDCl3: <i>δ</i> 7.68 (d, J = 8.4 Hz, 2H), 7.64 (d, J = 8.4 Hz, 2H), 7.47 (d, J = 8.4 Hz, 2H), 7.31 (d, J = 7.6 Hz, 1H), 7.16 (d, J = 8.0 Hz, 2H), 7.09 (d, J = 10.0 Hz, 1H), 3.53-3.43 (m, 2H), 3.26 (s, 3H), 3.24 (q, J = 9.6 Hz, 2H), 2.51 (s, 3H).
1B-16
CDCl3: <i>δ</i> 7.45 (d, J = 7.6 Hz, 1H), 7.37-7.27 (m, 3H), 7.23-7.18 (m, 1H), 7.10-7.02 (m, 3H), 6.90 (d, J = 8.4 Hz, 1H), 3.64-3.51 (m, 2H), 3.36-3.20 (m, 5H), 2.51 (s, 3H).
1B-17
CDCl3: <i>δ</i> 7.48-7.43 (m, 4H), 7.38-7.35 (m, 2H), 7.27 (s, 1H), 7.10-7.06 (m, 3H), 3.53-3.41 (m, 2H), 3.23 (s, 3H), 3.20 (q, J = 9.6 Hz, 2H), 2.98 (q, J = 7.6 Hz, 2H), 3.50 (s, 3H), 1.29 (t, J = 7.2 Hz, 3H).
1B-18
CDCl3: <i>δ</i> 8.53-8.46 (m, 1H), 7.54-7.40 (m, 4H), 7.32 (s, 1H), 7.24-7.21 (m, 3H), 6.92 (d, J = 11.6 Hz, 1H), 3.82-3.71 (m, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-19
CDCl3: <i>δ</i> 8.50 (d, J = 7.6 Hz, 1H), 7.55 (d, J = 8.4 Hz, 2H), 7.42-7.36 (m, 3H), 7.30 (bs, 1H), 7.28 (s, 1H), 7.23 (d, J = 10.0 Hz, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.80 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-20
CDCl3: <i>δ</i> 8.50 (d, J = 7.6 Hz, 1H), 7.53 (d, J = 7.2 Hz, 2H), 7.43 (d, J = 8.0 Hz, 2H), 7.30 (bs, 1H), 7.14 (s, 1H), 7.02 (d, J = 8.8 Hz, 1H), 6.92 (d, J = 12.0 Hz, 1H), 3.81 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-21
CDCl3: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 7.61 (d, J = 8.4 Hz, 2H), 7.52 (d, J = 8.4 Hz, 2H), 7.41-7.38 (m, 4H), 7.31 (bs, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.80 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.99 (q, J = 7.2 Hz, 2H), 2.40 (s, 3H), 1.35 (t, J = 7.2 Hz, 3H).
1B-22
CDCl3: <i>δ</i> 8.48 (d, J = 8.0 Hz, 1H), 7.52 (d, J = 8.0 Hz, 1H), 7.46(s, 1H), 7.31 (d, J = 8.0 Hz, 2H), 7.24 (s, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.74 (s, 2H), 3.47 (q, J = 9.6 Hz, 2H), 3.32 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-23
CDCl3: <i>δ</i> 7.36 (d, J = 8.0 Hz, 1H), 7.30 (d, J = 9.2 Hz, 1H), 7.07-7.03 (m, 2H), 6.99 (d, J = 8.0 Hz, 2H), 3.44-3.36 (m, 4H), 3.25 (q, J = 9.6 Hz, 2H), 3.21 (s, 3H), 2.50 (s, 3H).
1B-24
CDCl3: <i>δ</i> 8.48 (d, J = 8.0 Hz, 1H), 7.52 (d, J = 8.0 Hz, 2H), 7.46 (s, 1H), 7.31 (d, J = 8.0 Hz, 2H), 6.92 (d, J = 12.4 Hz, 1H), 3.74 (s, 2H), 3.45-3.35 (m, 4H), 2.40 (s, 3H).
1B-25
CDCl3: <i>δ</i> 8.48 (d, J = 8.0 Hz, 1H), 7.37 (d, J = 8.4 Hz, 1H), 7.30 (s, 1H), 7.17 (d, J = 8.0 Hz, 1H), 7.12 (d, J = 8.4 Hz, 1H), 7.02-6.94 (m, 1H), 6.90 (d, J = 8.4 Hz, 1H), 3.68 (s, 2H), 3.27 (q, J = 9.6 Hz, 2H), 2.45 (s, 3H).
1B-26
CDCl3: <i>δ</i> 8.75 (d, J = 7.6 Hz, 1H), 8.48 (s, 1H), 7.68 (s, 1H), 7.60 (d, J = 8.4 Hz, 2H), 7.51 (s, 1H), 7.46-7.41 (m, 3H), 6.98 (d, J = 11.6 Hz, 1H), 4.31 (m, 2H), 3.39 (q, J = 9.6 Hz, 2H), 2.42 (s, 3H).
1B-27
CDCl3: <i>δ</i> 8.75 (d, J = 7.6 Hz, 1H), 8.51 (s, 1H), 7.66 (d, J = 8.0 Hz, 2H), 7.60 (d, J = 7.2 Hz, 2H), 7.47-4.48 (m, 3H), 7.39-7.35 (m, 2H), 6.97 (d, J = 11.6 Hz, 1H), 4.33 (s, 2H), 3.40 (q, J = 9.6 Hz, 2H), 2.50 (s, 3H).
1B-28
CDCl3: <i>δ</i> 8.73 (d, J = 7.6 Hz, 1H), 8.41 (bs, 1H), 7.41 (d, J = 8.4 Hz, 2H), 7.28 (d, J = 8.8 Hz, 2H), 6.98 (d, J = 7.2 Hz, 1H), 4.26 (s, 2H), 3.39 (q, J = 9.6 Hz, 2H), 2.43 (s, 3H).
1B-29
CDCl3: <i>δ</i> 7.39 (d, J = 8.4 Hz, 2H), 7.36-7.30 (m, 2H), 7.23-7.18 (m, 2H), 7.12-7.07 (m, 3H), 3.52-3.41 (m, 2H), 3.26-3.18 (m, 5H), 2.51 (s, 3H).
1B-30
CDCl3: <i>δ</i> 7.37-7.34 (m, 2H), 7.29 (d, J = 8.0 Hz, 1H), 7.13-7.07 (m, 4H), 7.02 (d, J = 7.2 Hz, 1H), 3.50-3.46 (m, 2H), 3.28-3.19 (m, 5H), 2.51 (s, 3H).
1B-31
CDCl3: <i>δ</i> 8.02 (d, J = 8.4 Hz, 2H), 7.65 (d, J = 8.4 Hz, 2H), 7.51 (d, J = 8.4 Hz, 2H), 7.29 (d, J = 7.6 Hz, 1H), 7.14 (d, J = 8.0 Hz, 2H), 7.09 (d, J = 10.4 Hz, 1H), 3.54-3.41 (m, 2H), 3.29-3.18 (m, 5H), 2.64 (s, 3H), 2.51 (s, 3H).
1B-32
CDCl3: <i>δ</i> 7.41 (d, J = 8.4 Hz, 1H), 7.37-7.31 (m, 1H), 7.27 (s, 1H), 7.25 (s, 1H), 7.16-7.06 (m, 3H), 6.90 (d, J = 8.4 Hz, 1H), 3.50-3.46 (m, 2H), 3.28-3.20 (m, 5H), 2.51 (s, 3H).
1B-33
CDCl3: <i>δ</i> 7.53 (s, 1H), 7.45-7.41 (m, 3H), 7.36-7.28 (m, 3H), 7.12-7.07 (m, 3H), 3.48 (q, J = 15.2 Hz, 2H), 3.25-3.18 (m, 5H), 2.51 (s, 3H).
1B-34
CDCl3: <i>δ</i> 7.68-7.63 (m, 4H), 7.48 (d, J = 8.0 Hz, 2H), 7.30 (d, J = 8.0 Hz, 1H), 7.14 (d, J = 8.4 Hz, 2H), 7.09 (d, J = 10.0 Hz, 1H), 3.48 (q, J = 16.0 Hz, 2H), 3.26-3.19 (m, 5H), 2.51 (s, 3H).
1B-35
CDCl3: <i>δ</i> 7.55 (d, J = 8.8 Hz, 2H), 7.43 (d, J = 8.0 Hz, 2H), 7.30-7.25 (m, 3H), 7.12-7.07 (m, 3H), 3.48 (q, J = 16.8 Hz, 2H), 3.25-3.18 (m, 5H), 2.51 (s, 3H).
1B-36
CDCl3: <i>δ</i> 7.30 (d, J = 8.4 Hz, 1H), 7.17-7.10 (m, 2H), 7.00-6.94 (m, 3H), 4.08-3.87 (m, 2H), 3.64 (s, 2H), 3.43-3.29 (m, 3H), 3.21-3.14 (m, 2H), 2.50 (s, 3H).
1B-37
CDCl3: <i>δ</i> 7.54 (d, J = 8.8 Hz, 2H), 7.46-7.40 (m, 5H), 7.34 (d, J = 7.2 Hz, 2H), 7.08-7.01 (m, 2H), 4.19-3.94 (m, 2H), 3.67 (s, 3H), 3.04 (q, J = 9.6 Hz, 2H), 2.50 (s, 3H).
1B-38
CDCl3: <i>δ</i> 8.75 (d, J = 7.6 Hz, 1H), 7.63 (d, J = 8.4 Hz, 1H), 7.53 (d, J = 7.6 Hz, 1H), 7.49-7.44 (m, 4H), 7.43-7.28 (m, 1H), 7.25-7.19 (m, 2H), 6.97 (d, J = 11.2 Hz, 1H), 4.33 (s, 2H), 3.40 (q, J = 9.6 Hz, 2H), 2.42 (s, 3H).
1B-39
CDCl3: <i>δ</i> 7.14 (d, J = 7.2 Hz, 1H), 7.05-6.98 (m, 5H), 4.07-3.93 (m, 2H), 3.65 (s, 3H), 3.17 (q, J = 9.6 Hz, 2H), 2.48 (s, 3H).
1B-40
CDCl3: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 8.30 (bs, 1H), 7.75 (d, J = 8.4 Hz, 2H), 7.70 (d, J = 8.4 Hz, 2H), 7.59 (d, J = 7.6 Hz, 2H), 7.48-7.44 (m, 2H), 7.40 (d, J = 7.2 Hz, 1H), 7.03 (d, J = 11.6 Hz, 1H), 3.39 (q, J = 9.6 Hz, 2H), 2.44 (s, 3H).
1B-41
CDCl3: <i>δ</i> 7.56 (d, J = 7.2 Hz, 2H), 7.53 (d, J = 8.0 Hz, 2H), 7.48-7.44 (m, 2H), 7.40 (d, J = 7.6 Hz, 1H), 7.29 (d, J = 7.6 Hz, 2H), 7.15 (d, J = 7.2 Hz, 1H), 6.85 (d, J = 10.4 Hz, 1H), 3.39 (q, J = 9.6 Hz, 2H), 3.65 (s, 3H), 2.44 (s, 3H).
1B-42
CDCl3: <i>δ</i> 8.40 (d, J = 7.6 Hz, 1H), 7.63 (d, J = 7.2 Hz, 1H), 7.57-7.41 (m, 6H), 7.32 (d, J = 6.8 Hz, 1H), 7.05 (s, 1H), 6.90 (d, J = 9.6 Hz, 1H), 3.68 (s, 2H), 3.36 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H).
1B-43
CDCl3: <i>δ</i> 8.75 (d, J = 7.6 Hz, 1H), 8.49 (bs, 1H), 7.84 (s, 1H), 7.78 (d, J = 8.0 Hz, 1H), 7.66 (d, J = 8.4 Hz, 2H), 7.63-7.55 (m, 2H), 7.47 (d, J = 8.0 Hz, 2H), 6.97 (d, J = 10.8 Hz, 1H), 4.33 (s, 2H), 3.39 (q, J = 9.6 Hz, 2H), 2.42 (s, 3H).
1B-44
CDCl3: <i>δ</i> 8.40 (d, J = 7.6 Hz, 1H), 7.45-7.40 (m, 4H), 7.32 (d, J = 6.8 Hz, 1H), 7.23-7.18 (m, 3H), 7.05 (s, 1H), 6.91 (d, J = 9.6 Hz, 1H), 3.70 (s, 2H), 3.35 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H).
1B-45
CDCl3: <i>δ</i> 8.40 (d, J = 7.6 Hz, 1H), 7.46-7.34 (m, 7H), 7.32-7.29 (m, 2H), 7.04 (s, 1H), 6.89 (d, J = 9.6 Hz, 1H), 3.71 (s, 2H), 3.35 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H).
1B-46
CDCl3: <i>δ</i> 8.65 (d, J = 8.0 Hz, 1H), 8.29 (s, 1H), 7.64 (d, J = 7.6 Hz, 1H), 7.54 (d, J = 8.8 Hz, 2H), 7.52-7.43 (m, 5H), 7.35 (d, J = 6.0 Hz, 1H), 6.96 (d, J = 9.6 Hz, 1H), 4.20 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.42 (s, 3H).
1B-47
CDCl3: <i>δ</i> 8.66 (d, J = 7.6 Hz, 1H), 8.28 (bs, 1H), 7.48-7.42 (m, 4H), 7.35 (d, J = 6.0 Hz, 2H), 7.23 (d, J = 8.8 Hz, 2H), 6.95 (d, J = 11.2 Hz, 1H), 4.22 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.42 (s, 3H).
1B-48
CDCl3: <i>δ</i> 7.60 (d, J = 8.0 Hz, 1H), 7.48 (t, J = 7.6 Hz, 1H), 7.38-7.28 (m, 6H), 7.15 (d, J = 7.6 Hz, 1H), 6.99 (d, J = 7.6 Hz, 1H), 6.92 (d, J = 10.4 Hz, 1H), 3.31 (s, 2H), 3.16 (s, 3H), 3.12 (q, J = 9.6 Hz, 2H), 2.43 (s, 3H).
1B-49
CDCl3: <i>δ</i> 7.33-7.27 (m, 5H), 7.16 (d, J = 7.2 Hz, 1H), 7.07 (d, J = 7.6 Hz, 2H), 6.93 (d, J = 8.0 Hz, 1H), 6.90 (d, J = 10.4 Hz, 1H), 3.38 (s, 2H), 3.16 (s, 3H), 3.09 (q, J = 9.6 Hz, 2H), 2.43 (s, 3H).
1B-50
CDCl3: <i>δ</i> 7.39-7.29 (m, 5H), 7.18 (d, J = 8.4 Hz, 1H), 7.15 (d, J = 7.6 Hz, 1H), 7.05 (d, J = 7.6 Hz, 1H), 7.01 (d, J = 7.6 Hz, 1H), 6.93 (d, J = 10.0 Hz, 1H), 3.33 (s, 2H), 3.16 (s, 3H), 3.09 (q, J = 9.6 Hz, 2H), 2.44 (s, 3H).
1B-51
CDCl3: <i>δ</i> 7.57 (d, J = 7.6 Hz, 1H), 7.47 (t, J = 7.6 Hz, 2H), 7.43 (d, J = 7.6 Hz, 1H), 7.40-7.36 (m, 1H), 7.31 (d, J = 8.4 Hz, 1H), 7.22 (d, J = 7.6 Hz, 1H), 7.16 (d, J = 7.2 Hz, 1H), 7.09 (d, J = 7.6 Hz, 1H), 6.89 (d, J = 6.0 Hz, 1H), 6.82 (d, J = 12.4 Hz, 1H), 3.39-3.67 (m, 2H), 3.62 (s, 3H), 3.08-2.86 (m, 2H), 2.43 (s, 3H).
1B-52
CDCl3: <i>δ</i> 7.49 (d, J = 7.6 Hz, 1H), 7.38-7.28 (m, 3H), 7.16 (d, J = 8.0 Hz, 1H), 7.09 (d, J = 7.2 Hz, 1H), 6.94 (d, J = 7.6 Hz, 1H), 6.89 (d, J = 10.0 Hz, 1H), 6.82 (d, J = 7.6 Hz, 1H), 6.74 (s, 1H), 3.94-3.72 (m, 2H), 3.62 (s, 3H), 3.07-2.91 (m, 2H), 2.44 (s, 3H).
1B-53
CDCl3: <i>δ</i> 10.89 (bs, 1H), 7.93 (d, J = 8.0 Hz, 2H), 7.82 (d, J = 8.4 Hz, 2H), 7.62 (d, J = 7.6 Hz, 1H), 7.29 (d, J = 11.2 Hz, 1H), 3.87 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-54
CDCl3: <i>δ</i> 7.82 (d, J = 7.2 Hz, 1H), 7.73 (d, J = 8.0 Hz, 1H), 7.51 (d, J = 7.6 Hz, 1H), 7.32 (d, J = 7.6 Hz, 1H), 7.27 (d, J = 7.6 Hz, 1H), 7.11 (d, J = 10.8 Hz, 1H), 3.89 (q, J = 9.6 Hz, 2H), 3.17 (s, 3H), 2.34 (s, 3H).
1B-55
CDCl3: <i>δ</i> 7.51 (d, J = 7.2 Hz, 1H), 7.34-7.28 (m, 4H), 7.09 (d, J = 7.6 Hz, 1H), 6.92-6.90 (m, 2H), 6.83 (d, J = 10.0 Hz, 1H), 6.73 (d, J = 7.6 Hz, 1H), 3.99-3.82 (m, 2H), 3.61 (s, 3H), 3.03-2.89 (m, 2H), 2.43 (s, 3H).
1B-56
CDCl3: <i>δ</i> 10.02 (bs, 1H), 7.56-7.50 (m, 3H), 7.08-6.96 (m, 3H), 5.82 (d, J = 12.4 Hz, 1H), 3.30 (q, J = 9.6 Hz, 2H), 2.45 (s, 3H).
1B-57
CDCl3: <i>δ</i> 8.49 (d, J = 8.0 Hz, 1H), 7.59-7.31 (m, 8H), 7.22 (d, J = 8.0 Hz, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.83 (s, 2H), 3.35 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-58
CDCl3: <i>δ</i> 8.50 (d, J = 7.6 Hz, 1H), 7.61-7.55 (m, 4H), 7.51-7.43 (m, 3H), 7.38-7.30 (m, 3H), 6.90 (d, J = 11.2 Hz, 1H), 3.83 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H).
1B-59
CDCl3: <i>δ</i> 8.50 (d, J = 7.6 Hz, 1H), 7.83 (s, 1H), 7.77 (d, J = 7.6 Hz, 1H), 7.63-7.50 (m, 5H), 7.38 (d, J = 7.6 Hz, 1H), 7.37 (s, 1H), 6.92 (d, J = 11.2 Hz, 1H), 3.84 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-60
CDCl3: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 7.70 (m, 4H), 7.57 (d, J = 8.0 Hz, 2H), 7.51 (t, J = 7.6 Hz, 1H), 7.38 (d, J = 7.6 Hz, 1H), 7.33 (bs, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.84 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-61
CDCl3: <i>δ</i> 7.44-7.40 (m, 3H), 7.34-7.28 (m, 3H), 7.19 (s, 1H), 7.11 (bs, 1H), 7.07 (d, J = 7.6 Hz, 1H), 7.28 (d, J = 10.0 Hz, 1H), 3.52 (q, J = 9.6 Hz, 2H), 3.24-3.17 (m, 5H), 2.41 (s, 3H).
1B-62
CDCl3: <i>δ</i> 7.50 (d, J = 7.6 Hz, 1H), 7.41 (t, J = 7.6 Hz, 3H), 7.35-7.30 (m, 3H), 7.13 (bs, 1H), 6.90 (d, J = 10.0 Hz, 1H), 3.52 (q, J = 9.6 Hz, 2H), 3.53 (q, J = 9.6 Hz, 2H), 3.22-3.13 (m, 5H), 2.41 (s, 3H).
1B-63
CDCl3: <i>δ</i> 7.73 (s, 1H), 7.69 (d, J = 7.2 Hz, 1H), 7.59 (d, J = 8.4 Hz, 1H), 7.53 (t, J = 6.8 Hz, 1H), 7.43 (d, J = 7.6 Hz, 1H), 7.34 (t, J = 7.6 Hz, 1H), 7.28 (d, J = 7.6 Hz, 1H), 7.15 (s, 1H), 7.08 (d, J = 8.0 Hz, 1H), 7.02 (d, J = 10.0 Hz, 1H), 3.53 (q, J = 9.6 Hz, 2H), 3.25-3.17 (m, 5H), 2.43 (s, 3H).
1B-64
CDCl3: <i>δ</i> 7.77 (d, J = 8.4 Hz, 2H), 7.61 (d, J = 8.4 Hz, 2H), 7.43 (d, J = 8.0 Hz, 1H), 7.34 (t, J = 8.0 Hz, 1H), 7.29 (d, J = 8.0 Hz, 1H), 7.19 (s, 1H), 7.07 (d, J = 7.6 Hz, 1H), 7.03 (d, J = 10.4 Hz, 1H), 3.53 (q, J = 9.6 Hz, 2H), 3.25-3.18 (m, 5H), 2.44 (s, 3H).
1B-65
CDCl3: <i>δ</i> 7.51 (d, J = 8.4 Hz, 2H), 7.39 (d, J = 8.4 Hz, 1H), 7.32 (d, J = 7.6 Hz, 1H), 7.27 (t, J = 8.8 Hz, 2H), 7.24 (s, 1H), 7.13 (s, 1H), 7.03 (d, J = 7.2 Hz, 2H), 3.53 (q, J = 9.6 Hz, 2H), 3.23-3.16 (m, 5H), 2.43 (s, 3H).
1B-66
CDCl3: <i>δ</i> 8.50 (d, J = 7.6 Hz, 1H), 7.50-7.46 (m, 2H), 7.39-7.37 (m, 3H), 7.32 (s, 1H), 7.24 (d, J = 7.6 Hz, 2H), 6.92 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-67
CDCl3: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 7.50-7.46 (m, 2H), 7.39-7.36 (m, 3H), 7.32-7.27 (m, 3H), 6.91 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-68
CDCl3: <i>δ</i> 10.00 (bs, 1H), 7.61 (d, J = 8.8 Hz, 1H), 7.54 (s, 1H), 7.47-7.13 (m, 7H), 3.81-3.73 (m, 4H), 2.36 (s, 3H).
1B-69
CDCl3: <i>δ</i> 8.49 (d, J = 7.2 Hz, 1H), 7.68 (s, 1H), 7.52-7.47 (m, 4H), 7.42 (d, J = 8.4 Hz, 1H), 7.36 (d, J = 6.8 Hz, 1H), 7.31 (s, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-70
CDCl3: <i>δ</i> 8.49 (d, J = 8.0 Hz, 1H), 7.50-7.48 (m, 3H), 7.46 (s, 2H), 7.39 (d, J = 8.0 Hz, 1H), 7.35 (t, J = 7.6 Hz, 1H), 7.31 (bs, 1H), 6.92 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-71
CDCl3: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.58 (s, 1H), 7.55-7.46 (m, 4H), 7.39-7.32 (m, 4H), 6.92 (d, J = 11.6 Hz, 1H), 3.83 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-72
CDCl3: <i>δ</i> 7.46 (s, 1H), 7.40-7.38 (m, 2H), 7.34 (s, 1H), 7.33-7.27 (m, 3H), 7.09 (s, 1H), 7.05 (d, J = 7.6 Hz, 1H), 7.02 (d, J = 10.4 Hz, 1H), 3.52 (q, J = 9.6 Hz, 2H), 3.24-3.17 (m, 5H), 2.44 (s, 3H).
1B-73
CDCl3: <i>δ</i> 7.37 (d, J = 8.4 Hz, 1H), 7.32-7.27 (m, 4H), 7.23 (s, 1H), 7.07-7.00 (m, 3H), 3.52 (q, J = 9.6 Hz, 2H), 3.24-3.16 (m, 5H), 2.46 (s, 3H).
1B-74
CDCl3: <i>δ</i> 7.58 (s, 1H), 7.48 (d, J = 8.4 Hz, 1H), 7.36-7.27 (m, 4H), 7.11 (s, 1H), 7.06-7.02 (m, 2H), 3.53 (q, J = 9.6 Hz, 2H), 3.26-3.19 (m, 5H), 2.46 (s, 3H).
1B-75
CDCl3: <i>δ</i> 8.95 (d, J = 7.6 Hz, 1H), 8.33 (bs, 1H), 7.82-7.73 (m, 4H), 7.19 (d, J = 10.8 Hz, 1H), 3.90 (q, J = 9.6 Hz, 2H), 2.70 (s, 3H).
1B-76
CDCl3: <i>δ</i> 7.45 (d, J = 6.8 Hz, 1H), 7.30-7.26 (m, 3H), 7.23-7.17 (m, 2H), 7.07-7.03 (m, 3H), 3.49 (q, J = 9.6 Hz, 2H), 3.41-3.18 (m, 5H), 2.45 (s, 3H).
1B-77
CDCl3: <i>δ</i> 7.46 (s, 1H), 7.29-7.26 (m, 3H), 7.22 (d, J = 8.0 Hz, 2H), 7.04-7.02 (m, 3H), 3.50 (q, J = 9.6 Hz, 2H), 3.41-3.18 (m, 5H), 2.46 (s, 3H).
1B-78
CDCl3: <i>δ</i> 7.51-7.47 (m, 1H), 7.39-7.34 (m, 3H), 7.32-7.29 (m, 2H), 7.08 (d, J = 8.0 Hz, 2H), 7.03 (d, J = 10.0 Hz, 1H), 3.50 (q, J = 9.6 Hz, 2H), 3.27-3.20 (m, 5H), 2.47 (s, 3H).
1B-79
CDCl3: <i>δ</i> 8.48 (d, J = 7.6 Hz, 1H), 7.34 (d, J = 8.4 Hz, 2H), 7.26 (s, 1H), 7.15 (d, J = 8.4 Hz, 2H), 6.93 (d, J = 7.6 Hz, 1H), 6.70-6.33 (m, 1H), 3.74 (s, 2H), 3.38 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-80
CDCl3: <i>δ</i> 7.29 (d, J = 7.6 Hz, 1H), 7.07 (d, J = 10.4 Hz, 1H), 7.04-6.97 (m, 4H), 6.64-6.27 (m, 1H), 3.41 (q, J = 9.6 Hz, 2H), 3.28-3.21 (m, 5H), 2.50 (s, 3H).
1B-81
CDCl3: <i>δ</i> 8.95 (d, J = 7.6 Hz, 1H), 7.34 (d, J = 8.4 Hz, 2H), 7.30 (bs, 1H), 7.16 (d, J = 8.4 Hz, 2H), 7.06 (d, J = 10.8 Hz, 1H), 6.71-6.34 (m, 1H), 3.39 (q, J = 9.6 Hz, 2H), 3.77 (s, 2H), 2.64 (s, 3H).
1B-82
CDCl3: <i>δ</i> 8.88 (d, J = 7.6 Hz, 1H), 7.19 (d, J = 9.6 Hz, 1H), 7.10-6.99 (m, 4H), 6.66-6.29 (m, 1H), 3.93 (q, J = 9.6 Hz, 2H), 3.49-3.41 (m, 2H), 3.22 (s, 3H), 2.72 (s, 3H).
1B-83
CDCl3: <i>δ</i> 7.75 (s, 1H), 7.17-6.99 (m, 5H), 6.65-6.28 (m, 1H), 3.93 (q, J = 9.6 Hz, 2H), 3.49-3.41 (m, 2H), 3.24 (s, 3H), 2.42 (s, 3H).
1B-84
CDCl3: <i>δ</i> 8.97 (s, 1H), 7.69 (d, J = 8.0 Hz, 2H), 7.41 (d, J = 8.0 Hz, 2H), 7.30 (bs, 1H), 7.07 (d, J = 11.2 Hz, 1H), 3.89 (q, J = 9.6 Hz, 2H), 3.81 (s, 2H), 2.64 (s, 3H).
1B-85
CDCl3: <i>δ</i> 8.97 (d, J = 7.6 Hz, 1H), 7.83 (s, 1H), 7.77 (d, J = 7.2 Hz, 1H), 7.64-7.50 (m, 5H), 7.38 (d, J = 7.2 Hz, 2H), 7.05 (d, J = 10.8 Hz, 1H), 3.93-3.86 (m, 4H), 2.63 (s, 3H).
1B-86
CDCl3: <i>δ</i> 8.46 (d, J = 7.6 Hz, 1H), 8.29 (bs, 1H), 7.74 (d, J = 8.8 Hz, 2H), 7.33 (d, J = 8.4 Hz, 2H), 7.03 (d, J = 11.6 Hz, 1H), 3.37 (q, J = 9.6 Hz, 2H), 2.44 (s, 3H).
1B-87
CDCl3: <i>δ</i> 7.26-7.23 (m, 3H), 7.15 (d, J = 8.4 Hz, 2H), 6.82 (d, J = 10.4 Hz, 1H), 3.31-3.25 (m, 5H), 2.45 (s, 3H).
1B-88
CDCl3: <i>δ</i> 8.94 (d, J = 7.6 Hz, 1H), 8.33 (bs, 1H), 7.74 (d, J = 8.8 Hz, 2H), 7.34 (d, J = 8.0 Hz, 2H), 7.18 (d, J = 11.2 Hz, 1H), 3.91 (q, J = 9.6 Hz, 2H), 2.68 (s, 3H).
1B-89
CDCl3: <i>δ</i> 7.88 (d, J = 7.2 Hz, 1H), 7.32 (d, J = 8.4 Hz, 2H), 7.20 (d, J = 8.0 Hz, 2H), 7.05 (d, J = 9.6 Hz, 1H), 3.92 (q, J = 9.6 Hz, 2H), 3.27 (s, 3H), 2.71 (s, 3H).
1B-90
CDCl3: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.50-7.41 (m, 3H), 7.32 (bs, 1H), 7.28 (d, J = 7.6 Hz, 1H), 7.06 (d, J = 6.4 Hz, 2H), 6.92 (s, 1H), 6.89 (d, J = 10.8 Hz, 1H), 6.00 (s, 2H), 3.81 (s, 2H), 3.37(q, J = 9.6 Hz, 2H), 3.39 (s, 3H).
1B-91
CDCl3: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.55 (d, J = 9.2 Hz, 2H), 7.49 (t, J = 7.6 Hz, 1H), 7.43-7.28 (m, 5H), 7.05 (t, J = 9.2 Hz, 1H), 6.91 (d, J = 11.2 Hz, 1H), 3.83 (s, 2H), 3.37(q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-92
CDCl3: <i>δ</i> 8.49 (d, J = 8.0 Hz, 1H), 7.56-7.45 (m, 5H), 7.39 (d, J = 8.4 Hz, 2H), 7.31 (d, J = 7.6 Hz, 2H), 6.91 (d, J = 11.6 Hz, 1H), 3.82 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.99 (q, J = 7.2 Hz, 2H), 2.39 (s, 3H), 1.35 (t, J = 7.2 Hz, 3H).
1B-93
CDCl3: <i>δ</i> 8.49 (d, J = 8.0 Hz, 1H), 7.47 (t, J = 8.0 Hz, 1H), 7.39 (d, J = 6.8 Hz, 2H), 7.37-7.33 (m, 3H), 7.33 (s, 1H), 7.29 (s, 1H), 7.22 (t, J = 6.4 Hz, 1H), 6.91 (d, J = 11.6 Hz, 1H), 3.81 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H), 2.35 (s, 3H).
1B-94
CDCl3: <i>δ</i> 8.50 (d, J = 8.0 Hz, 1H), 7.68 (s, 4H), 7.63 (t, J = 7.2 Hz, 4H), 7.52-7.44 (m, 3H), 7.38-7.33 (m, 3H), 6.91 (d, J = 11.6 Hz, 1H), 3.85 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.39 (s, 3H).
1B-95
CDCl3: <i>δ</i> 8.49 (d, J = 7.6 Hz, 1H), 7.47 (d, J = 8.4 Hz, 2H), 7.69 (d, J = 8.0 Hz, 2H), 7.57-7.50 (m, 3H), 7.40 (d, J = 7.2 Hz, 1H), 7.32 (bs, 1H), 6.93 (d, J = 11.6 Hz, 1H), 3.83 (s, 2H), 3.37 (q, J = 9.6 Hz, 2H), 2.40 (s, 3H).
1B-96
CDCl3: <i>δ</i> 7.70 (d, J = 8.4 Hz, 2H), 7.62 (d, J = 8.4 Hz, 2H), 7.43 (d, J = 7.6 Hz, 1H), 7.36 (d, J = 7.6 Hz, 1H), 7.30 (d, J = 7.6 Hz, 1H), 7.22 (s, 1H), 7.07 (d, J = 7.2 Hz, 1H), 7.04 (d, J = 10.4 Hz, 1H), 3.52 (q, J = 9.6 Hz, 2H), 3.27-3.18 (m, 5H), 2.46 (s, 3H).
1B-97
CDCl3: <i>δ</i> 7.66-7.63 (m, 4H), 7.59 (d, J = 8.0 Hz, 2H), 7.48-7.44 (m, 3H), 7.38-7.30 (m, 3H), 7.18 (s, 1H), 7.02-7.00 (m, 2H), 3.54 (q, J = 9.6 Hz, 2H), 3.22-3.15 (m, 5H), 2.39 (s, 3H).
1B-98
CDCl3: <i>δ</i> 7.34-7.27 (m, 4H), 7.20-7.14 (m, 3H), 7.03-6.99 (m, 3H), 3.52 (q, J = 9.6 Hz, 2H), 3.22 (s, 3H), 3.14 (q, J = 9.6 Hz, 2H), 2.44 (s, 3H), 2.36 (s, 3H).
1B-99
CDCl3: <i>δ</i> 7.44-7.35 (m, 5H), 7.29 (d, J = 7.6 Hz, 1H), 7.24 (d, J = 7.6 Hz, 2H), 7.12 (s, 1H) 7.02-6.98 (m, 1H), 3.53 (q, J = 9.6 Hz, 2H), 3.21-3.14 (m, 5H), 2.98 (q, J = 7.2 Hz, 2H), 2.42 (s, 3H), 1.05 (t, J = 7.2 Hz, 3H).
1B-100
CDCl3: <i>δ</i> 7.41-7.34 (m, 2H), 7.33-7.27 (m, 3H), 7.18 (d, J = 10.4 Hz, 1H), 7.10 (s, 1H), 7.06-7.00 (m, 3H), 3.53 (q, J = 9.6 Hz, 2H), 3.23-3.16 (m, 5H), 2.44 (s, 3H).
1B-101
CDCl3: <i>δ</i> 7.34 (d, J = 7.6 Hz, 1H), 7.28 (s, 1H), 7.23 (d, J = 7.6 Hz, 1H), 7.05 (s, 1H), 7.01 (d, J = 10.4 Hz, 1H), 6.98-6.99 (m, 3H), 6.85 (d, J = 8.4 Hz, 1H), 5.99 (s, 2H), 3.52 (q, J = 9.6 Hz, 2H), 3.21-3.14 (m, 5H), 2.44 (s, 3H).
1B-102
CDCl3: <i>δ</i> 8.45 (d, J = 7.6 Hz, 1H), 7.70-7.68 (m, 2H), 7.48-7.46 (m, 2H), 7.23 (bs, 1H), 6.96 (d, J = 11.6 Hz, 1H), 3.81 (s, 2H), 3.36 (q, J = 9.6 Hz, 2H), 2.41 (s, 3H).
The abbreviations in Tables 1 to 4 are as indicated below.
F: fluoro, Cl: chloro, Br: bromo, Me: methyl, Et: ethyl, n-Pr: normal-propyl, i-Pr: isopropyl, n-Bu: normal-butyl, t-Bu: tert-butyl, n-Pent: normal-pentyl, CF: trifluoromethyl, OMe: methoxy, OEt: ethoxy, OCF: trifluoromethoxy, SCF: trifluoromethylthio, SMe: methylthio, NH: amino, NO: nitro, Ph: phenyl, S: sulfur atom, O: oxygen atom, Ac: acetyl, CHF: difluoromethyl.
Below are Preparation Examples in which the "parts" refers to "parts by weight."
10 parts of each compound of the invention was dissolved in 45 parts of Solvesso 150 and 35 parts of N-methylpyrrolidone. 10 parts of an emulsifier (trade name: Sorpol 3005X, produced by Toho Chemical Industry Co., Ltd.) was added thereto. The mixtures were mixed by stirring to give 10% emulsions.
20 parts of each compound of the invention was added to a mixture of 2 parts of sodium lauryl sulfate, 4 parts of sodiumlignin sulfonate, 20 parts of fine powder of synthetic hydrated silicon dioxide, and 54 parts of clay. The mixtures were mixed by stirring with a juice mixer to give 20% wettable powders.
2 parts of sodium dodecylbenzenesulfonate, 10 parts of bentonite, and 83 parts of clay were added to 5 parts of each compound of the invention, and each mixture was sufficiently mixed by stirring. An appropriate amount of water was added thereto. The resulting mixtures were further stirred and granulated with a granulator. The granules were air-dried to give 5% granules.
1 part of each compound of the invention was dissolved in an appropriate amount of acetone. 5 parts of fine powder of synthetic hydrated silicon dioxide, 0.3 parts of acidic isopropyl phosphate (PAP), and 93.7 parts of clay were added thereto. The mixtures were mixed by stirring with a juice mixer, and acetone was removed by evaporation to give 1% dust.
20 parts of each compound of the invention was mixed with 20 parts of water containing 3 parts of polyoxyethylene tristyrylphenyl ether phosphoric acid ester triethanolamine and 0.2 parts of Rhodorsil 426R. The mixtures were subjected to wet pulverization with a DYNO-Mill, and mixed with 60 parts of water containing 8 parts of propylene glycol and 0.32 parts of xanthan gum to give 20% suspensions in water.
Test Examples are given below to demonstrate that the compounds of the invention are useful as an active ingredient for miticides.
A piece of non-woven fabric (4.5×5.5cm) was suspended inside a plastic cup through an incision made in the lid of the plastic cup. After water was poured into the cup, the cup was covered with the lid. A kidney bean leaf (about 3.5×4.5 cm) was then placed on the sufficiently soaked, non-woven fabric. Another kidney bean leaf with two-spotted spider mites (about 30 mite samples) was placed on top of the first leaf, and the fabric and leaves were placed in a thermostatic chamber having a temperature of 25±2° C and a humidity of 40%.
Miticidal formulations containing the compound of the invention (200 ppm) were prepared by adding an aqueous solution (100 ppm) of Sorpol 355 (manufactured by Tobo Kagaku Co. Ltd.) to a methanol solution of the compound of the invention.
These miticidal formulations were sprayed onto the leaves, and the leaves were air-dried and placed in a thermostatic chamber (25±2° C and a humidity of 50%). The mortality rate of the two-spotted spider mites was calculated after 2 days.
The compounds that exhibited the mortality rate of 50% or more are as follows:
Compound Nos.: 1A-2, 1A-5, 1A-8, 1A-12, 1A-13, 1A-14, 1A-15, 1A-20, 1A-23, 1A-24, 1A-27, 1A-28, 1A-30, 1A-33, 1A-42, 1A-43, 1A-45, 1A-46, 1A-47, 1A-48, 1A-49, 1A-50, 1A-51, 1A-52, 1A-53, 1A-54, 1A-55, 1A-56, 1A-57, 1A-58, 1A-59, 1A-60, 1A-62, 1A-63, 1A-65, 1A-67, 1A-68, 1A-72, 1A-73, 1A-74, 1A-75, 1A-76, 1A-77, 1A-78, 1A-82, 1A-83, 1A-85, 1A-86, 1A-87, 1A-88, 1A-90, 1A-91, 1A-92, 1A-93, 1A-94, 1A-95, 1A-96, 1A-97, 1A-103, 1A-104, 1A-107, 1A-108, 1A-109, 1A-111, 1A - 112 , 1A-113, 1A-114, 1A - 11 6, 1A-117, 1A-118, 1A-119, 1A-120, 1A-121, 1A-122, 1A-123, 1A-126, 1A-127, 1A-128, 1B-1, 1B-2, 1B-3, 1B-5, 1B-7, 1B-8, 1B-9, 1B-10, 1B-11, 1B-12, 1B-15, 1B-16, 1B-17, 1B-18, 1B-19, 1B-20, 1B-22, 1B-23, 1B-24, 1B-25, 1B-26, 1B-27, 1B-28, 1B-29, 1B-30, 1B-32, 1B-33, 1B-34, 1B-35, 1B-36, 1B-37, 1B-38, 1B-39, 1B-41, 1B-43, 1B-48, 1B-49, 1B-50, 1B-54, 1B-55, 1B-56, 1B-57, 1B-58, 1B-59, 1B-61, 1B-62, 1B-63, 1B-64, 1B-65, 1B-66, 1B-67, 1B-68, 1B-69, 1B-70, 1B-71, 1B-72, 1B-73, 1B-74, 1B-76, 1B-77, 1B-78, 1B-79, 1B-80, 1B-82, 1B-83, 1B-87, 1B-90, 1B-96, 1B-97, 1B-98, 1B-99, 1B-100, 1B-101, 1B-102.
A piece of non-woven fabric (4.5×5.5cm) was suspended inside a plastic cup through an incision made in the lid of the plastic cup. After water was poured into the cup, the cup was covered with the lid. A kidney bean leaf (about 3.5×4.5 cm) was then placed on the sufficiently soaked, non-woven fabric. Twenty female adults of two-spotted spider mite were placed on the top of the leaf, and the fabric and leaf were placed in a thermostatic chamber having a temperature of 25±2° C and a humidity of 40% and 16L8D.
The next day, after the number of the female adults was adjusted once more to 20, 2 ml of a miticidal formulation containing the compound of the invention (200 ppm) prepared in the same manner as in test example 1 was sprayed onto the leaf, and the leaf was air-dried and placed in a thermostatic chamber (25±2° C and a humidity of 50%). The ovicidal rate of the two-spotted spider mites was calculated 6 days after the spraying of the miticidal formulation. The compounds that exhibited a mortality of 50% or more at 500 ppm are as follows:
Compound Nos.: 1A-2, 1A-8, 1A-12, 1A-13, 1A-14, 1A-20, 1A-23, 1A-27, 1A-33, 1A-42, 1A-43, 1A-47, 1A-48, 1A-49, 1A-50, 1A-51, 1A-52, 1A-53, 1A-54, 1A-55, 1A-56, 1A-57, 1A-58, 1A-59, 1A-60, 1A-61, 1A-63, 1A-65, 1A-67, 1A-68, 1A-69, 1A-70, 1A-71, 1A-72, 1A-73, 1A-74, 1A-76, 1A-77, 1A-78, 1A-82, 1A-83, 1A-85, 1A-86, 1A-87, 1A-88, 1A-90, 1A-91, 1A-93, 1A-94, 1A-95, 1A-96, 1B-1, 1B-2, 1B-3, 1B-5, 1B-7, 1B-8, 1B-9, 1B-10, 1B-11, 1B-12, 1B-15, 1B-16, 1B-17, 1B-18, 1B-19, 1B-20, 1B-22, 1B-23, 1B-24, 1B-25, 1B-26, 1B-27, 1B-28, 1B-29, 1B-30, 1B-32, 1B-33, 1B-34, 1B-35, 1B-36, 1B-37, 1B-38, 1B-39, 1B-41, 1B-43, 1B-48, 1B-49, 1B-50, 1B-54, 1B-55, 1B-56, 1B-57, 1B-58, 1B-59, 1B-61, 1B-62, 1B-63, 1B-64, 1B-65, 1B-66, 1B-67, 1B-68, 1B-69, 1B-70, 1B-71, 1B-72, 1B-73, 1B-74, 1B-76, 1B-77, 1B-78, 1B-79, 1B-80, 1B-82, 1B-83, 1B-86, 1B-87, 1B-88, 1B-90, 1B-96, 1B-97, 1B-98, 1B-99, 1B-100, 1B-101.
[Industrial Applicability]
The present invention provides novel benzylamide compounds, methods for producing the same, and miticides and thus the present inventions are particularly useful in the agricultural industry. | |
TO celebrate the Queen’s Platinum Jubilee, PM Boris Johnson plans to allow UK shops to once again trade in pounds and ounces – imperial measurements.
It would be the first time since an EU directive in 2000 ordered us to march to their metric mantra.Imperial measurements are being brought back for British consumers[/caption]
But are we inching back to the Middle Ages? Will we be rolling out the firkin of ale?
Nipping down to Tesco for a bushel of apples? Does Boris want to make Britain groat again? To put us on the rood to recovery?
To help you prepare, Amy Jones has ten fun questions about our odd imperial measures of old. Answers below.
1) How many groats are there in one old pound?
- A. 30
- B. 60
- C. 90
2) How many ells does a king-size duvet measure lengthways?
- A. 2
- B. 50
- C. 100
READ MORE QUEEN'S JUBILEE
3) The area of a standard football pitch should be how many roods?
- A. 5
- B. 6
- C. 7
4) How many barleycorns would a six-inch Subway sandwich measure?
- A. 18
- B. 12
- C. 9
5) One bushel of apples weighs how many pounds?
- A. 38lb
- B. 48lb
- C. 58lb
6) How many traditional servings of mead would fit into a pint glass?
- A. Half
- B. 2
- C. 4
7) How many leagues is it from Land’s End to John o’Groats?
- A. 291
- B. 591
- C. 791
8) How long is the Grand National in chains?
- A. 2
- B. 345
- C. 568
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9) How many pounds in a hundredweight?
- A. 100lb
- B. 112lb
- C. 150lb
10) How many firkins in a barrel of beer?
- A. 4
- B. 8
- C. 16
Answers
1) B – a groat was a coin worth four old pence, there were 240 pence in a pound
2) A – the ell, around 45in, mostly used for measuring cloth, was assumed to be the average length of a person’s arm
3) C – a rood was equal to a quarter acre or 10,890 sq ft
4) A – a barleycorn was a third of an inch, and is still used as the basis of shoe sizes
5) B – a bushel was a unit of volume for apples, typically about 125 of a medium size and enough to make 15 pies
6) C – a traditional serving of mead was about a quarter-pint, called a gill
7) A – a league is generally three miles
8) B – a chain is 66ft or 22 yards, there are ten in a furlong and 80 per mile. It was also a measuring device made up of metal links
9) B – a hundredweight is 112lb
10) A – a firkin is a quarter barrel. | https://www.brexitnews.tv/2022/06/test-your-knowledge-of-imperial.html |
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If you are still interested in making a Disclosure Request, please contact us at [email protected]. In the Disclosure Request, specify that you seek your “California Customer Choice Privacy Notice.” Please allow thirty (30) days for a response, but note that Other is entitled by law to respond to your Disclosure Request by notifying you of your right to prevent the disclosure of personal information pursuant to our Opt-Out Policy (which is cost free to you).
QUESTIONS? If you have any questions or concerns regarding this Privacy Statement, please contact us at [email protected]. Please note that email communications may not be secure. Accordingly, you should not include personal or other sensitive information in your email correspondence to the Sites or Other. | https://pacetulsa.com/pace-tulsa-privacy-policy/ |
Attending school when you recently arrived in The Netherlands. Your child recently came to the Netherlands, but does not speak (or hardly speaks) any Dutch (yet). Despite this, your child must attend school.
That is not a problem: in Rotterdam, a few classes at ‘regular’ schools are equipped to take in children who have not been in the Netherlands for long, like your child. These classes are called International Transition Classes (Internationale schakelklassen, ISK) in secondary education. In primary education these transition classes are called First Intake Classes (Eerste Opvang). Your child will follow an intense language programme in the ISK class. When your child speaks and writes Dutch sufficiently, he or she will be transferred to a ‘regular’ class in primary or secondary education.
Primary education (children from the age of 6 to 12)
Rotterdam has 13 primary schools that offer a First Intake transition class. You can register your child with the primary school with a First Intake transition class closest to you. They will enter your child’s details and will check to see if they have an opening.
|Name of school||Address||Telephone number||Area|
|Notenkraker||Othelloweg 2 (3190 AE)||010 - 295 72 73||Hoogvliet|
|Over de Slinge||Krabbendijksestraat 243-254 (3086 LR)||010 - 480 35 52||Charlois|
|De Kameleon||Carnissedreef 2-4 (3083 ZG)||010 - 480 85 68||Charlois|
|Pniëlschool||Sandelingplein 25 (3075 AG)||010 - 484 46 88||Feijenoord|
|De Sleutel||Asterstraat 26 (3073 ED)||010 - 484 46 91||Feijenoord|
|Joh. Brandstraat 3-7 (3072 BD)||010 - 484 35 35||Feijenoord|
|De Catamaran||Catullusweg 298 (3076 KH)||010 - 291 67 70||IJsselmonde|
|Kasteel Spangen||Bilderdijkstraat 322 (3027 SN)||010 - 415 49 09||Delfshaven|
|Emmausschool||Tidemanstraat 59 (3022 SE)||010 - 425 83 12||Delfshaven|
|De Rozenhorst||Jan Tooropstraat 1-3-5 (3181 HE)||0181 - 212 565||Rozenburg|
|OBS Dakpark||Catharina Beersmanstraat 80 (3025 EJ)||
|
010 - 476 01 14
|Delfshaven|
|
|
Talmaschool
|
|
Vaandrigstraat 15 (3034 PX)
|
|
010 - 413 66 76
|
|
Kralingen
|Bertrand Russellplaats 7 (3069 CA)||010 - 420 56 29||Prins Alexander|
|Stephanus||Asserweg 360 (3052 AJ)||010 - 418 47 27||Hillegersberg-Schiebroek|
Secondary education
Secondary education in the Netherlands differentiates between various levels of education. Your child will be tested to establish the level at which he or she can start. Employees of Koers-VO will test your child, in collaboration with the secondary education system in Rotterdam. The test will include an intelligence test and a match test. You will then be issued with a recommendation for your child. After testing you will be referred to one of the five schools with an International Transition Class. You can approach Koers-VO for information on secondary education and regional training centres (ROCs) and to register your child
Koers-VO
34 Schiekade
3032 AJ Rotterdam
Telephone 010 - 484 25 76
email [email protected]
koers-vo.nl
The compulsory education department can be reached for questions on telephone number 010 - 498 42 58 or by email: [email protected].
Young people who are 18+
For young people between the age of 18 and 23 who don’t have a Dutch diploma, we would like to know what their level of education is and what they will potentially study. You can provide us with this information by sending an email to [email protected].
If you need help you can contact the Youth Support Desk, located at 122 Westblaak. They can be reached on the Municipality of Rotterdam’s general phone number 010 - 267 67 88. | https://www.rotterdam.nl/english/recently-arrived-and-school/ |
NIHL National Division: Telford Tigers 4 – 3 Leeds Knights
Hexagon Telford Tigers returned to home ice after the Christmas break to take on Leeds Knights.
Telford Tigers v Leeds Knights
The Yorkshire side had recently lost in the final of the Autumn Cup but had found some form in the league – winning their last three games which had seen them climb to fifth in the league standings. League leaders Swindon Wildcats had surprisingly lost to struggling Basingstoke Bison at home on Boxing Day, giving Tigers a chance to close the gap between the two teams.
Tigers were almost back to full strength but saw Nick Oliver and Jack Hopkins join Ricky Plant, Andy McKinney and Jack Watkins as unavailable for the game.
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Tigers made a disastrous start to the game and conceded after just thirty four seconds. Tigers’ defence failed to clear the puck from their defensive zone and gifted possession to Brandon Whistle, the former Tigers player scrambled the puck past Brad Day to open the scoring. The poor start seem to affect Tigers as they struggled to get going and saw Day called upon to prevent a second goal. Midway through the period, Tigers were level. Jason Silverthorn led a 2-on-1 breakaway and his shot was saved by Sam Gospel in the Leeds’ goal but the puck fell perfectly for Austin Mitchell-King to finish into the open net.
Tigers had several power play chances as Leeds committed numerous infractions and the Leeds’ bench were getting upset with every call made, which saw referees Wells and Williams in lengthy discussions with various Leeds’ players. Tigers’ power play unit had little success and barely troubled Gospel.
With Leeds’ players in the ear of the officials with every call and their coach’s histrionics on the bench, it was no surprise when Tigers were called for two innocuous penalties early in the second period. Thomas McKinnon was called for roughing after he reacted to Kieron Brown’s glove to the face but the initial offence from Brown was ignored which was followed by a slashing call on Corey Goodison, giving Leeds an extended 5-on-3 power play. Tigers defended resolutely and killed off both penalties.
Tigers were then given another power play chance when Jordan Griffin was called for hooking. Finally, at the fourth attempt, the power play unit scored. Vladimir Luka picked out the unmarked Scott McKenzie with a cross ice pass and McKenzie hammered home past Gospel.
Another power play chance was converted shortly after. Robert Streetly was serving a hooking penalty and after good play by Goodison, Fin Howells reacted first to scramble the puck past Gospel to increase the lead.
However, Leeds fought back with two quick goals of their own. Brown scored for Leeds on the power play after Silverthorn was given a tripping penalty and then Whistle scored with a shot over Day from close range.
As the buzzer sounded for the end of the period, the referees had finally had enough of the constant complaining from Leeds and awarded Matty Davies a ten minute misconduct penalty.
Tigers went close early in the third period with Brodie Jesson hitting the post on a breakaway and Day had to react sharply at the other end, saving well from Adam Barnes. Both teams looked dangerous on the break and Leeds were cursing the officials again midway through the period. Brown was called for tripping and such were his protestations that he was awarded a game misconduct penalty for unsporting behaviour and ejected from the game. The sides remained level and with no further goals, overtime was required.
Goodison would prove to be the Tigers’ hero, breaking into the Leeds’ offensive zone, skating in on goal and shooting low towards Gospel with the puck deflecting in off a Leeds’ defender to give Tigers a vital win and deserved two points.
With leaders Swindon losing again to Basingstoke, Tigers closed the gap at the top to 4 points with a game in hand.
After the game Head Coach Tom Watkins, commented, “It was a good win tonight in a very close game. It certainly could have gone either way and I thought both teams worked hard with a number of players missing for both clubs. It was great to get the win in front of a packed house tonight.”
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USDE & USDOJ Jointly Release Guidance on Discrimination & Suspensions
The U.S. departments of Education and Justice jointly issued guidance earlier this week on how school leaders can ensure that discipline policies are drafted and applied in a manner that does not discriminate against racial or ethnic groups. The joint departments also urged districts to seek alternatives to "exclusionary" penalties like suspension and expulsion that rob students of valuable classroom time, particularly for non-violent offenses.
Details
The new guidance clarifies how districts can meet their obligations under Title IV and Title VI of the federal Civil Rights Act of 1964, which relate to fair and nondiscriminatory treatment among schools and recipients of federal aid. Schools can end up in violation of the law if they draft policies that unfairly target specific student groups in word or in application if a legitimate educational justification does not exist or is not articulated.
Disciplinary policies, even those drafted without discriminatory intent, may also violate the federal laws if students from certain racial groups are disproportionately affected by them, an effect commonly known as "disparate impact," the guidance says. If students of one race are sanctioned at disproportionately higher rates under a given policy, educators should be prepared to demonstrate that the disciplinary measure is "necessary to meet an important educational goal" and that they have considered alternatives, the document says.
The guidance results from the work of the Supportive School Discipline Initiative, a collaboration that the two federal agencies launched in 2011 to address what's known as the "school-to-prison pipeline," the term critics use for policies that they say result in unnecessary and inappropriate referrals from schools to the criminal justice system. Advocates for school discipline reform have argued that such policies disproportionately impact minority racial and ethnic groups.
The "Dear Colleague letter” outlines schools' obligations to fair and nondiscriminatory discipline under the Civil Rights Act and details what investigators would use to determine if a complaint of discriminatory discipline practice is valid. The agencies also released a directory of federal school climate and discipline resources, an online catalog of state-level school discipline laws and regulations, and a guide of "best practices" for policymakers and district leaders who seek to improve their policies.
The materials also include snapshots of data related to how disciplinary measures affect certain racial groups. Specifically, it indicated that students of certain racial or ethnic groups tend to be disciplined more than their peers. Although African-American students represent 15 percent of students, they make up 35 percent of students suspended once, 44 percent of those suspended more than once, and 36 percent of students expelled. Further, over 50 percent of students who were involved in school-related arrests or referred to law enforcement were Hispanic or African-American.
While recognizing that disparities in student discipline rates in a school or district may be caused by a range of factors the joint group argues that the “substantial racial disparities of the kind reflected in the data are not explained by more frequent or more serious misbehavior by students of color."
The guidance recommends a focus on positive environments and prevention efforts; clear, appropriate, and consistent expectations and consequences; and continuous efforts to ensure equity.
Investigations
The agencies also assured educators that they would continue to investigate allegations of Title IV and Title VI violations triggered by complaints from parents, students, and community members, and that they may also initiate investigations as part of regular compliance monitoring activities. | https://njpsa.org/usde-usdoj-jointly-release-guidance-discrimination-suspensions/ |
CO2CEPTS: An Innovative cloud-based tool for evaluation of new technologies in SAGD facilities
New technologies are continuously being developed to improve the economic and environmental performance of Steam Assisted Gravity Drainage (SAGD) Central Processing Facilities (CPF) in Oil Sands extraction. Some have promise; many are too expensive or impractical. The important question facing producers and technology providers alike is: how to evaluate these new technologies in an efficient manner to identify the best ones and screen the “not so good” options?
Traditionally, process simulation tools are used to evaluate the effect of introducing new technologies on the economics and environmental footprint of the process. However, these process simulation tools are not the best for this purpose given that:
- No universal model is accepted by all producers and technology vendors.
- Models are too detailed for screening technologies.
- TBP of the bitumen, valves, pumps, coolers and heaters are not necessarily required for high-level evaluation of technologies.
- Economic and environmental metrics of the process are not integrated into the simulator and should be calculated in separate spreadsheets.
- The assumptions and the parameters for evaluating such metrics are not clearly stated and are different across different organizations.
- All the SAGD process units cannot be modeled in the process simulation tools and would require custom process units.
- Models would require expert users who need to ensure the adjusts and the recycles typically found in SAGD simulations are properly converged.
- All the potential users for the tool would require a valid license, which can be expensive to obtain.
In order to address these issues and establish a common ground for evaluation of new technologies in a SAGD CPF, Canada's Oil Sands Innovation Alliance (COSIA) developed a series of static excel templates which provided the heat and material balance for the most common configurations of a SAGD CPF. The developed static templates provided a common ground for the evaluation of technologies, however it had many shortcomings especially the fact that the templates were not interactive and users were not able to modify the predefined specifications or configurations.
To extend the functionality of the static templates, COSIA contracted Process Ecology to develop CO2CEPTS. CO2CEPTS is a web-based process simulator for performing mass and energy balances and perform economic evaluations for SAGD plants at the block-flow diagram level. CO2CEPTS has the following features which make it superior to process simulation tools for evaluation of technologies:
- Delivered via web-browser; therefore no installation required.
- Fast and accurate. All the parameters in CO2CEPTS are initialized to default values which are based on feedback from Joint Industry Partner (JIP) Industry members. Users can override the default numbers.
- Supports the efficient evaluation of greenhouse gas emissions reduction technologies for CPF.
- Based on high-level engineering calculations, users can quickly evaluate the environmental and economic impacts of technology changes.
- Integrates cost correlations and GHG calculations.
- Includes detailed economic analysis of the CPF to calculate important financial metrics such as Cash Flow, Net Present Value (NPV), and Internal Rate of Return (IRR).
- Includes validation to guide the user in building the CPF flowsheet.
The process blocks in CO2CEPTS are divided into the following categories:
- Sources: Source blocks represent material sources into the flowsheet (Wellpad, Diluent, Make-up Water, Combustion Air, and Fuel Gas).
- Sinks: There is a set of blocks that represent material “sinks” (Landfill, Injection Well, Product Storage, Stack (ambient), and Utility Steam).
- Process Blocks: These include the set of process blocks usually found in SAGD CPFs (Oil Treating, Evaporation, Blowdown Flashing, Deoiling, Lime Softening, and Glycol System).
- Steam Generation: Steam generation options are separately displayed in the user interface. These include options for OTSG/Drum Boiler or cogeneration as Gas Turbine/HRSG.
- GHG Control: These blocks involve the technologies of interest to reduce GHG emissions from in situ oil sands production. In the current version these blocks include Flue Gas Condensation, CO2 capture, CO2 Compression, CO2 sequestration, and Air Separation Unit (to model oxyfuel options).
The software tool is available through the portal www.co2cepts.com and is accessible for members and associate members of COSIA.
Do you have questions or comments regarding this article? Click here to contact us. | https://processecology.com/articles/co2cepts-an-innovative-cloud-based-tool-for-evaluation-of-new-technologies-in-sagd-facilities |
Self-efficacy and quality of life after low-intensity neuropsychological rehabilitation: A pre-post intervention study.
Being highly self-efficacious is a key factor in successful chronic disease self-management. It is unknown whether neuropsychological rehabilitation improves self-efficacy in managing the consequences of brain injury. To investigate whether levels of general and brain injury specific self-efficacy and quality of life (QoL) increased after neuropsychological rehabilitation and whether cognitive performance was associated with self-efficacy. We conducted a retrospective clinical cohort study of 62 patients with acquired brain injury and cognitive complaints with measurements before start and after completion of treatment. QoL was measured with the visual analogue scale (EQ VAS) of the EuroQol (EQ-5D); self-efficacy with the TBI Self-efficacy Questionnaire (SEsx) and the General Self-efficacy Scale (GSES). Cognitive performance was measured as a compound score of tests for memory, attention and information processing speed. Self-efficacy for managing brain injury-specific symptoms and QoL increased significantly after neuropsychological rehabilitation. Both general and brain injury-specific self-efficacy were positively associated with QoL after completion of the programme. Cognitive performance was not associated with self-efficacy for managing brain injury-specific symptoms nor with general self-efficacy. Self-efficacy and QoL improve after treatment. Further research is needed to identify the specific ingredients responsible for improvement of self-efficacy in patients with cognitive complaints.
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As a senior member of our Product team, I sit at the intersection of many decisions between teams such as Engineering, Design and Product Marketing. These teams may have differing perspectives day-to-day, so it is one of my responsibilities to help communicate and unify the teams towards our shared vision at key moments in the product creation process.
Throughout my six years at Narrative Science, I’ve learned that there are very different types of meetings, and each type benefits from unique organization and processes. In this post, let’s look at “buy-in” meetings and techniques to help them be successful.
Tip 1) Understand when it’s a buy-in meeting
A “buy-in” meeting is one where you need a group of people to agree and commit to a particular direction. The goal is that everyone leaves the meeting dedicated to the agreed-upon direction and (hopefully) becomes mini-advocates for the decision. This way, they can understand the reason for the decision and confidently defend and articulate it to other folks not directly involved in the process.
You’ll quickly realize that you haven’t reached this goal if you see a bunch of vague head nods during the meeting and the conclusions are promptly forgotten or ignored after the meeting is over.
Tip 2) Socialize your proposal ahead of time
Ahead of the meeting, circulate two very important documents with enough lead time so participants can actually read and consider them.
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The first document is the full proposal, which includes whatever detail necessary for people to fully grasp and respond to your proposal.
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The second document is a subset of your full proposal—it is the TL;DR (Too Long; Didn't Read) list of bullet points that captures the most important concepts from the full proposal.
Many of the folks in your meeting (particularly the more senior ones) may lack the time to read through the entire proposal, but there should be no excuses for not reading the TL;DR list. This is why it's super critical that the TL;DR list captures the most controversial or contentious points from the full proposal. Essentially, you're shooting for "If you agree with everything in the TL;DR list then you will also agree with everything in the full proposal."
Make sure to send an explicit email that makes it clear that at least the TL;DR list needs to be read and considered in advance of the meeting (people tend to ignore documents if they are just attached to a meeting invite or shared via Google Docs.)
Tip 3) Use the TL;DR list as a meeting agenda
If the TL;DR list accurately reflects the full proposal, and you can get the entire room to formally agree to each point in the list, then the entire room has also agreed to the full proposal. The actual meeting itself can stay focused by essentially going through each point in the list and marking them as "Agreed-to" or "Needs further discussion".
There's also psychological benefit to tagging each point as Agreed-to as the meeting progresses: it makes each participant a positive supporter of the idea. Harvard psychologist Dan Gilbert believes, “once we’ve committed to a course of action, we stop thinking about alternatives. Or, if we do bother to think about them, we think about how lousy they are compared to our clearly superior and awesome choice."
In other words, positively supporting an idea is a much stronger commitment than, "I didn't comment negatively on the idea", and it will help ensure that the participants of the meeting are active supporters and champions of the ideas and decision to others not involved in the process.
Tip 4) Take notes and incorporate comments
Take notes on the discussion during the meeting in as public a way as possible. One easy way to do this is commenting directly on the TL;DR list on a shared screen as the meeting progresses. Taking notes publicly shows participants that their opinions are important to the process, emphasizing the valuable nature of their feedback by accurately incorporating their ideas into the meeting.
After the meeting, take time to add the participants’ thoughts into the main body of the proposal. As much as possible, be explicit in responding to each comment, including the ones that won’t further the proposal or make their way into the document. This is perfectly fine, but it’s not okay for someone to feel that their contribution was just set aside or ignored.
Tip 5) Recirculate the modified and agreed-to proposal
Finally, recirculate the reconciled proposal. If there was significant disagreement during the first meeting, you may need to repeat the process (hopefully resulting in big chunks of the proposal agreed-to and serving as "ground truth" during further discussion).
If most accept the bulk of the proposal during the first meeting, the reconciled proposal can likely just recirculate along with a helpful reminder that everything in the document was already agreed-to.
Building and delivering an innovative technology like Quill has taught me a lot about team dynamics. It’s also taught me that technologies like Quill wouldn’t exist without each team member contributing their perspective and help. Overall, buy-in meetings are essential to a company’s continued growth and ability to get to market quickly, so hopefully these tips can be a helpful guide next time you must align multiple perspectives around a shared vision. | http://resources.narrativescience.com/h/i/331086225-5-tips-for-running-buy-in-meetings-product-team-spotlight |
It is usually impossible to determine the cause of congenital deafness unless a clear problem has been observed in a breed or carefully planned breedings are performed. In affected breeds, deafness has often been long-established but kept hidden from outsiders to protect reputations. Hereditary deafness can potentially result from any of several mechanisms: autosomal dominant or recessive, X-linked, mitochondrial, or polygenic; in most instances the mechanism is unknown. Incomplete penetrance, where not all aspects of a deafness syndrome are expressed in an affected individual, frequently complicates an understanding of the mode of inheritance. No known X-linked or mitochondrial deafness has been reported in dogs or cats. With a few known exceptions, hereditary deafness is usually associated with pigmentation patterns, where increasing amounts of white in the hair coat increase the likelihood of deafness.
Two pigmentation genes are often associated with deafness in dogs: the merle gene (seen in the Collie, Shetland Sheepdog, Dappled Dachshund, Harlequin Great Dane, American Foxhound, Old English Sheepdog, and Norwegian Dunkerhound among others) and the piebald or extreme piebald gene (Bull Terrier, Samoyed, Greyhound, Great Pyrenees, Sealyham Terrier, Beagle, Bulldog, Dalmatian, English Setter). Not all breeds with these genes have been reported to be affected with deafness.
The merle (dapple) gene (M) produces a mingled or patchwork combination of dark and light areas (Little, 1957; Searle, 1968). This gene is dominant so that heterozygous dogs (Mm) show the pattern, which is considered desirable in many breeds. However, when two dogs with merle are bred, 25% on average will end up with the MM genotype. These dogs usually have a solid white coat and blue irises, are often deaf and/or blind, and are sterile. Experienced breeders of these dogs know not to breed merle to merle. Heterozygous merles can also be deaf, with the likelihood of deafness increasing with increasing amounts of white in the hair coat. In this case the deafness is neither dominant nor recessive, but is linked to a dominant gene that disrupts pigmentation and secondarily produces deaf dogs.
Genetic transmission of deafness in dogs with the piebald (sp) and extreme piebald (sw) pigment genes, such as the Dalmatian, is less clear. These genes affect the amount and distribution of white areas on the body (Little, 1957; Searle, 1968). The canine piebald genes are recessive, but individuals in breeds such as the Dalmatian are homozygous, so all dogs within the breed express the pigment pattern. Deafness in Dalmatians does not appear to be dominant since deaf puppies result from hearing parents. It does not appear to be a simple recessive disorder: we have repeatedly bred pairs of deaf Dalmatians from our research colony and obtained many bilaterally hearing puppies, when all should have been deaf if the disorder was recessive. These findings might be explained by a polygenic cause, the presence of two different autosomal recessive deafness genes, or a syndrome with incomplete penetrance. Suggestions have been made for two different recessive genes, either of which can cause deafness, or two recessive genes where both are required to cause deafness (Hewson-Fruend, 1990), or a recessive multifactorial gene with incomplete penetrance (Greibrokk, 1994). Deafness is still clearly linked to the extreme piebald gene in Dalmatians. In this breed, the underlying coat color is black (B) or liver (b, simple recessive). The extreme piebald gene (sw) covers the color with white, and the dominant ticking gene (T) opens the spots through the white. In Dalmatians with a patch, the sw gene does not completely suppress the underlying coat color; the sw gene is only weakly expressed. Patched Dalmatians have been shown to have significantly lower deafness rates (Strain et al., 1992), but a patch is not allowed in the breed standard. Conversely, blue-eyed Dalmatians, where the normal brown iris pigment is suppressed, are significantly more likely to be deaf (Strain et al., 1992; Greibrokk, 1994). Blue eyes are allowed in the breed standard of the United States, but not in Canada or Europe. Dalmatians that are the offspring of one bilaterally hearing parent and one unilaterally deaf parent are twice as likely to be deaf (unilaterally or bilaterally) as dogs that are the offspring of two bilaterally hearing parents (Strain, 1992b). Efforts through breedings to reduce blue eyes in Norwegian Dalmatians reduced the prevalence of deafness (Greibrokk, 1994).
Recent studies have shown that deafness in Dobermans, which do not carry the merle or piebald genes, results from direct loss of cochlear hair cells without any effects on the stria vascularis (Wilkes & Palmer, 1992). Vestibular system signs, including head tilt and circling, are seen, and the deafness is transmitted by a simple autosomal recessive mechanism. A similar pathology has been described for the Shropshire Terrier (Igarashi et al., 1972).
Numerous references report that most congenital deafness in dogs is autosomal recessive. However, the available data suggest that this is not true for most breeds.
Dr. George M. Strain
Professor of Neuroscience
School of Veterinary Medicine
Associate Vice Chancellor of Research
Office of Research & Graduate Studies
Louisiana State University
Baton Rouge, Louisiana 70803
Phone: 225-388-5833
Visit Dr. Strain's website for a complete discussion of Deafness in Dogs & Cats. We are grateful to Dr. Strain for allowing us to present some of his research and valuable material here.
We are grateful to Dr. Strain for allowing us to present some of his research and valuable material here. | http://fallingbranch.com/library/deafgenetics.htm |
Bill Would Impose Mandatory Penalties for Breaches, Require Cybersecurity Inspections, and Compensate Consumers for Stolen Data
Under This Legislation, Equifax Would Have Paid At Least $1.5 Billion in Penalties for 2017 Data Breach
Lawmakers Unveil New Report Showing Equifax Still Failing Consumers Long After Data Breach and Write to Regulators Demanding Action
Bill Text (PDF) | Fact Sheet (PDF)
New Report on Equifax Complaints (PDF)
Letter to the Federal Trade Commission (PDF) | Letter to the Consumer Financial Protection Bureau (PDF)
Washington, DC - United States Senators Elizabeth Warren (D-Mass.) and Mark Warner (D-Va.), along with Representatives Elijah E. Cummings (D-Md.), Chairman of the House Committee on Oversight and Reform, and Raja Krishnamoorthi (D-Ill.), today reintroduced the Data Breach Prevention and Compensation Act to hold large credit reporting agencies (CRAs)-including Equifax-accountable for data breaches involving consumer data. The bill would give the Federal Trade Commission (FTC) more direct supervisory authority over data security at CRAs, impose mandatory penalties on CRAs for data breaches to incentivize adequate protection of consumer data, and provide robust compensation to consumers for stolen data.
Senators Warren, Warner and Representative Krishnamoorthi, along with Senator Brian Schatz (D-Hawaii), also issued a new analysis of consumer complaints to the Consumer Financial Protection Bureau (CFPB), which revealed that in the 18 months after the Equifax breach was announced, consumers filed over 52,000 complaints related to Equifax, nearly double the number from the same period before the breach was announced. The report shows how the company is still failing consumers by providing inadequate responses to consumer complaints over the course of several months and refusing to remove incorrect information from credit reports despite consumers contacting Equifax multiple times, among other concerns. The senators and Representative Krishnamoorthi also wrote to the FTC and CFPB attaching their new report and asking the agencies to take action.
"It's been nearly two years since Equifax put more than half of the adults in this country at risk by opening the doors to hackers, and this new report shows that this problem is far from fixed," said Senator Warren. "Our bill would hold companies like Equifax accountable for failing to protect consumer data, compensate consumers injured by these breaches, and help ensure that these breaches never happen again."
"It's been nearly two years since hackers accessed the personal information of more than 143 million Americans, yet thousands of individuals continue to grapple with the effects of this massive breach," said Senator Warner. "As personal data becomes more and more valuable in today's information economy, and the scale and impact to consumers of mega-breaches increase, there need to be increased consequences for companies like Equifax that mishandle or neglect to properly safeguard consumer data. By imposing strict penalties for data breaches and facilitating compensations for affected Americans, this legislation will increase accountability and help ensure that credit reporting agencies actively prioritize the security of sensitive consumer information."
"The Equifax data breach was one of the largest and most consequential in United States history," said Congressman Cummings. "It was a wake-up call that credit reporting agencies are not adequately protecting the American public's personal data. Last year, I released a staff report with a number of specific recommendations Congress could take to protect consumers from future cyber attacks, and I am happy that many of those recommendations are now included in the bill we are introducing today. These companies must be held accountable when they fail to protect the personal data entrusted to them by American consumers."
"Working for the people means protecting the personal data of consumers and holding companies accountable for data breaches that compromise consumer health and safety," said Congressman Raja Krishnamoorthi. "As the Chair of the Oversight Subcommittee on Economic and Consumer Policy, I am proud to co-lead this bicameral legislation to prevent the negligence and abuses which could lead to the next consumer data breach."
In September 2017, Equifax announced that hackers stole sensitive personal information -- including Social Security Numbers, birth dates, credit card numbers, driver's license numbers, and passport numbers -- of over 143 million Americans, a number later revised up to 145.5 million people. The attack highlighted that CRAs hold vast amounts of data on millions of Americans but lack adequate safeguards against hackers. Since 2013, Equifax reported at least four separate hacks in which sensitive personal data were compromised.
The Data Breach Prevention and Compensation Act would:
Establish an Office of Cybersecurity at the FTC tasked with annual inspections and supervision of cybersecurity at CRAs.
Impose mandatory, strict liability penalties for breaches involving consumer data, beginning with a base penalty of $100 for each consumer who had one piece of personal identifying information (PII) compromised and another $50 for each additional PII compromised. Under this bill, Equifax would have paid at least a $1.5 billion penalty for their failure to protect Americans' personal information.
beginning with a base penalty of $100 for each consumer who had one piece of personal identifying information (PII) compromised and another $50 for each additional PII compromised. Under this bill, Equifax would have paid at least a $1.5 billion penalty for their failure to protect Americans' personal information. Ensure a robust recovery for affected consumers by requiring the FTC to use 50% of its penalty to compensate consumers.
by requiring the FTC to use 50% of its penalty to compensate consumers. Increase penalties in cases of woefully inadequate cybersecurity or if a CRA fails to provide timely notification to the FTC of a breach.
Enhance FTC enforcement by giving the FTC civil penalty authority under the Gramm-Leach-Bliley Act, as recommended by a Government Accountability Office report requested by Senator Warren and Representative Cummings.
The analysis of Equifax complaint data prepared by the offices of the senators and Representative Krishnamoorthi found that consumers continue to file complaints against Equifax at a higher rate than before the breach. Specific findings of this new analysis include:
In 18 months between September 7, 2017, when Equifax announced the breach of sensitive consumer information, and March 6, 2019, consumers filed 52,031 complaints related to Equifax.
The majority of these complaints-30,372-were filed in one year between March 8, 2018, and March 7, 2019 - revealing that Equifax was still failing to address customer concerns long after the breach was revealed.
Overall, complaints stemming from Equifax's failure to respond effectively to consumer problems make up at least 82% of the complaints about the company in the last year.
The report also found a shift in the type of CFPB complaints filed against Equifax in recent months, indicating that consumers have encountered more and more difficulties with Equifax's response to the breach, and that the problems it has caused millions of Americans do not appear to be fully resolved.
The lawmakers' full report, titled Breach of Trust: CFPB's Complaint Database Shows Failure to Protect Consumers after Equifax Breach, can be read here.
The Data Breach Prevention and Compensation Act is supported by cybersecurity experts and consumer groups:
"This bill requires the FTC to provide much-needed oversight of the credit bureaus for data security. It also imposes real and meaningful penalties when the credit bureaus, who hold our most sensitive financial information, fail to adequately protect that information. I commend Senator Warren, Senator Warner, and Congressmen Cummings and Krishnamoorthi for their continuing efforts to prevent another massive security failure like the Equifax data breach," said National Consumer Law Center Staff Attorney, Chi Chi Wu.
"A concrete response to a serious problem facing American consumers. The ongoing risk of data breach and identity theft have reached epidemic proportions. We clearly need more expertise in the federal government to address this challenge. We hope the Senate will more forward this important and timely effort to safeguard American consumers and Internet users," said Electronic Privacy Information Center President and Executive Director, Marc Rotenberg
"Equifax still hasn't paid a price two years after losing the financial DNA of 150 million Americans. That's why U.S. PIRG commends Senator Warner, Senator Warren, and Congressmen Cummings and Krishnamoorthi for reintroducing the Data Breach Prevention and Compensation Act. The bill provides strong oversight and meaningful financial penalties to incentivize the credit bureaus to protect our data," said U.S. PIRG Consumer Campaign Director, Mike Litt.
"Making the companies that collect and sell consumers' personal information liable when they fail to secure it is a necessary step in ensuring our privacy rights," said Former Chief Technologist at the FTC, Ashkan Soltani.
Read more statements of support here. View a fact sheet about the legislation here. View the bill text here.
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Position Description:
The objective of this individual will be to uphold the integrity of the manufacturing and facility equipment along with maintaining a professional environment within the department. The Maintenance department goals are to improve product output and quality to maintain a competitive edge in the paperboard packaging industry with a safe and skillful approach.
Why Work for Us?
- Medical, Dental, Vision, and 401k with company match
- Culture and engagement committee
- Unique family-owned culture
- Wellness program
- Room for growth and development
Primary Responsibilities:
- Installation and repair of all machinery and building equipment
- Acknowledge and enforce department/company policies
- Work closely wand communicate professionally with all company classifications to assure good effective team concept
- Readily available for technical support to all in-house customers
- Willingness to assist any associate no matter the job classification
- Familiarization and understanding of the company Hazardous Waste Emergency Contingency Plan
Troubleshooting and technical skill requirements:
- Machinist
- Millwright
- Plumbing
- Carpentry
- Welding
- Electrical (light)
- Pneumatic
- Hydraulic
- Comprehension of multiple types of prints and schematics
Work with Maintenance Leadman:
- Preventative maintenance activities
- Equipment modifications and upgrades
- Spare part and supply inventories
Mathematical Ability:
Able to use basic math to Add / Subtract / Multiply / Divide
Communication Skills/Requirements:
Must be able to effectively communicate verbally and in writing with management, peers, and other employees. Have active listening skills, attention to details, and follow-through. Have a strong customer focus. Required to identify areas of need, while defining problems, and identifying potential options and solutions to meet and exceed customer expectations. Must demonstrate tact, professionalism, teamwork, flexibility, attention to detail, an organized approach to work with a positive attitude.
Reasoning Ability:
Majority of tasks require some judgment. Needs to define problems, draw valid conclusions. Must exhibit solid attention to detail. Must be able to perform multiple tasks and work under pressure in a fast-paced manufacturing environment. Normal level of concentration is required for accuracy.
Safety:
All duties to be performed in compliance with the company guidelines, and safety policy.
MUST WEAR Ear plug hearing protection, safety glasses, and face mask in Designated Areas.
MUST WEAR Hair Net when handling direct food contact packaging.
Reports to:
Maintenance Supervisor
Experience:
No experience necessary. Manufacturing experience is a plus.
Education:
High School Diploma or GED required
Company Overview:
Zumbiel Packaging is a fourth-generation family-owned business that has been in the printing business for over 170 years. We are a printing company offering a wide range of products from beverage to consumer, and one of the leading in the industry. This is a company that offers the opportunity for a career and a unique culture with our family-owned atmosphere.
Must be authorized to work in the United States
Zumbiel Packaging provides equal employment opportunities to all employees and applicants for employment and prohibits discrimination and harassment of any type without regard to race, color, religion, age, sex, national origin, disability status, genetics, protected veteran status, sexual orientation, gender identity or expression, or any other characteristic protected by federal, state or local laws. | https://www.zumbiel.com/careers/mechanic-trainee/ |
•Warns illegal sand miners
The Lagos State Government has restated commitment to ensuring that miners comply with internationally acceptable standards in the sand mining industry in the state.
The Director, Human Resources and Administration, Mr. Fashola Adeyemi Taofik, said this during an on-the-spot- assessment of some sand mining sites in the Ajah/Lekki axis of the state.
He said the Ministry is poised to ensure that miners operate according to regulations to ensure sustainable development in the state.
In a statement, the Public Relations Officer of the Ministry, Olaoye Olusegun, Taofik noted that to obtain the mining standard, the Ministry has to continually monitor the activities of sand miners and ensure that rules are followed strictly.
He said: “In furtherance of the Ministry’s mandate to ensuring a sustainable mining operation, constant monitoring and clinical review of mining methods and its impacts, taking stock of degraded area as well as putting in place a restoration plan is being done by the Ministry. | https://thenationonlineng.net/lagos-restates-commitment-to-mining-standards/ |
As Lord Krishna is the center attraction in the spiritual realm, similarly in the material world Lord Krishna manifested his expansion in the form of Shiva to Lord over the decentralized world called the material world, where everyone misuses their freewill and make themselves the center. Lord Shiva is neither Jiva Tattva nor Visnu Tattva, Krishna has made Shiva as his own Tattva (Shiva-Tattva). Shiva is the husband of Durga and Lord of the material world but he has his own original position in the spiritual world serving as the gatekeeper of Vrindavan in the form known as Gopisvara Mahadev. One can see Lord Shiva as the protector of the Holy Dham who can allow us to enter there.
Lord Shiva is commonly known as the destroyer of the world. When there is an imbalance and disturbance in the universe and when the sustenance of life becomes impossible, Lord Shiva destroys the universe so that the creation of the next cycle can begin. One analogy is given that just as a goldsmith does not destroy the gold whenhe melts old unusable golden jewelry to create new ornaments, in the same way Lord Shiva annihilates the universe so that souls that were not liberated will have another chance to liberate themselves from the bondage of birth and death. In this way, Lord Shiva is known as the destroyer of the world.
Generally we see the form of Lord Shiva known as Tat Purusha. This form of Lord Shiva is his meditating form. He is sitting cross-legged in a yogic position and his eyes are half closed and in deep meditation. When He opens his eyes a new cycle of creation begins and when they are closed the universe is annihilated for creation of the next cycle. Moreover, his half opened eyes mean that creation is going through the cycle process.
Lord Shiva’s bodily luster is depicted as bluish or white in color. His neck is dark blue in color due to his drinking the poison during the churning of the milk ocean. He did this because he didn’t want to disturb his Istadev (Lord) so he is known as Nila-Kanta. Lord Shiva’s body is smeared with ashes. These ashes symbolize that there is nothing in this world for us to enjoy but that Krsna is the supreme enjoyer. Everything in this world will in the end turn into ashes so we should not think we are the enjoyers. It is explained that Lord Shiva has a third eye which rests between his eyebrows. This third eye is known as the eye of wisdom, knowledge, and power. When Lord Shiva opens his third eye he destroys illusion, false ego, anger, lust, and greed. His third eye also eminates blazing fire that burns everything. The tiger skin that Shiva wears is a symbol of his ability to control and change animal nature. His matted hair represents his magnificent powers and his spiritual life. In addition, the moon that ornaments his head signifies the waxing and waning process in comparison to the evolution of the time
cycle from the beginning to the end. Lord Shiva is shown with a snake around his neck and arms. These snakes stand for His yogic power and His control over desire and sensuality and the snake on his head represents his worshipable lord Sankarshan who is representative of Krishna. Furthermore, Lord Shiva wears a Rudraksha necklace which has 108 beads that symbolize the elements used in creating the world. Lastly, the Trisula or three- pronged trident carried by Lord Shiva represents the three modes of nature namely, tamas (ignorance), rajas (passion), and sattva (goodness). The trident is also a symbol of His desire to destroy ignorance and evil.
It appears that Lord Shiva gives benedictions to demonic persons who apparently fight in battle with Lord Krishna as his enemy, but in reality this is Lord Shiva’s blessings to those demons to receive the mercy of Lord Krishna. Yogisvar the master of all the yoga, krsna is known as yogesvaw or the object of all the yoga processes.
Shiva Blue
19 inches tall... | http://radhakrishnadollsoflove.com/ganesh/blueshiva/blueshiva.html |
Operational risk refers to the risk of loss resulting from inadequate or failed processes or systems, from personnel or from external events. This definition includes compliance risk but excludes risks resulting from strategic decisions. The risks may realize for instance as a consequence of:
- Internal misconduct
- External misconduct
- Insufficient human resources management
- Insufficiencies in operating policies as far as customers, products or business activities are concerned
- Damage to physical property
- Interruption of activities and system failures
- Defects in the operating process.
Materialized operational risks can cause an immediate negative impact on the financial results due to additional costs or loss of earnings. In the longer term, materialized operational risks can lead to a loss of reputation and, eventually, a loss of customers which endangers the company’s ability to conduct business activities in accordance with the strategy.
Compliance risk is the risk of legal or regulatory sanctions, material financial losses or loss of reputation resulting from a company’s failure to comply with laws, regulations and administrative orders as applicable to its activities. A compliance risk is usually the consequence of internal misconduct and hence it can be seen as a part of operational risk. | https://ar2015.sampo.com/en/risk-management/appendix-2-risk-definitions/operational-risks/ |
Get Un-Stuck! Tips to Unlocking Symbolic Meanings for Guidance in Life
When it comes to unlocking symbolic meanings, it’s best to start with basics and work our way through a narrative to gain insights.
When signs and symbols come into my awareness, sometimes I’m unsure about their meaning. It happens to all of us!
I believe, if a sign or symbol surfaces, then the symbolic meanings are inherent within our understanding as well. It just makes sense.
Why be given a sign if we don’t have the capacity or faculties to interpret? Remember, my sweet peeps, like attracts like.
That means if something is floating in your awareness…like an odd bird, repetitive numbers showing up, a funky encounter with rose petals…whatever! That has your attention.
If it has your attention, that means your consciousness has the solution to solve it.
We are never given a stimulus that is disconnected from us. That’s a weird way of saying:
If something is poking at you, you absolutely have the tools and talent to poke back!
So, whether geometric, totemic or otherwise I know I have the right keys for unlocking symbolic meanings.
Armed with this knowledge, when unclear about a symbolic meaning, these are some ideas about what we can do to get clarity…
Tips for Interpreting Symbolic Meanings for Guidance
♦ Meditate:
Sometimes lightly, sometimes deeply, sometimes while doing the dishes or while walking. My energy rolls all over the symbol in question – thoughtfully, lovingly. Inspiration always comes from this. Always. Once I get an inspirational hit, I either know the message that symbol has for me, or I am directed to other resources for more information.
♦ Research:
If the message of the symbol doesn’t come instantly, I’ve got to research it. If the symbol is geometric, I begin to contemplate its basic features. For example, if the symbol is comprised of a triangle – I’d start there, exploring all the symbolic implications (personal, historical, cultural, mythological, etc) of the triangle.
♦ Play:
I can’t allow symbols to control me. They are oracles, designed to help gain guidance and clarity – if they consume or obsess me – then I am a slave. So, I incorporate a lot of play into finding symbolic meanings. I paint them, write about them, lucid dream them, meditate with them (meditation is a form of play, for me). I draw them in the snow or sand or the earth, I make up silly songs about them. You get the idea.
Sometimes symbol meanings won’t give themselves easily. When this happens, I save all my observations about the symbol (from my journals, paintings, songs, meditation results, etc) and put them away.
The answer always comes, but not always in my perceived timing. I’ve waited as long as 3 years to unlock some symbolic meanings.
These are just some tricks of my own that may help you on your own personalized journey to unlocking symbolic meanings.
I hope this article on interpreting and tips to unlocking symbolic meanings proves helpful on your path.
As always, thank you for reading.
Brightest always,
Avia
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Six Simple Steps to Interpreting Dreams
Interpreting dreams is so important. Dreams are a reflection of who we are. When interpreting dreams, we uncover rich, rewarding insight. This article gives you insight about dream meanings and six simple steps to interpreting dreams. | https://www.whats-your-sign.com/tips-to-unlocking-symbolic-meanings.html |
async Music By Ryuichi Sakamoto
async is Ryuichi Sakamoto’s first solo album in 8 years. Taking inspiration from everyday objects, sculpture, and nature,Sakamoto composed and arranged the sounds/music that he most wanted to listen to. Paying close attention to the essence of each track and carefully balancing the sounds with a less-is-more perspective, what remains are singular expressions of Sakamoto’s current mindset, and one of his most personal albums.
The album was primarily recorded and conceived in NYC, with some elements drawn from field or location recordings and museums around the world. During the production process Sakamoto came upon the concept of creating a soundtrack for an Andrei Tarkovsky film that does not exist. Needless to add, Tarkovsky is one of his most favorite film directors.
The pallet includes conventional instrumentation such as piano and orchestra, but also a deep selection of unique acoustic and electric sounds both programmed and organic. The album plays with ideas of a-synchronism, prime numbers, chaos, quantum physics and the blurred lines of life and artificiality/noise and music. | https://www.milanrecords.com/release/async/ |
Hate Speech Detection In Twitter: A Selectively Trained Ensemble Method
Houston, Jackson (2020)
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Title
Hate Speech Detection In Twitter: A Selectively Trained Ensemble Method
Authors
Houston, Jackson
Issue Date
2020-05
Type
Thesis or Dissertation
Abstract
This thesis tests classification models from Natural Language Processing and Machine learning in the task of identifying hate speech. We tested on multiple annotated data sets (Davidson et al. 2017) of tweet data labeled as hate speech, offensive speech, both, or neither. Hate speech has become an unavoidable topic in the current social media environment due to poorly monitored comment sections and news feeds. With that, studies showing the negative affects that it brings to people’s well-being have also begun to surface (Gelber and McNamara 2015). Therefore, being able to identify hate speech accurately and precisely has grown in importance. Hate speech is often contextual, subjective, and a matter of opinion which makes creating an accurate model of such speech all the more difficult. We have found that using an ensemble method of a classic Naive Bayes classifier (Pedregosa et al. 2019c), Random Forest (Pedregosa et al. 2019b), K-Means (Pedregosa et al. 2019d), and Bernoulli (Pedregosa et al. 2019a) performed better than similar studies in precision, accuracy, recall, and f-score (Malmasi and Zampieri 2018). The ensemble performed better than using the strongest of the individual models, Random Forest, by a small but useful margin. We believe this to be due to the nuanced nature and context behind hate speech being more than one model can fully encompass. In addition to the ensemble strategy, training on data which was labeled as ‘clean’ (not hate speech or offensive) or labeled ‘dirty’ (hate speech) with higher confidence ratings increased the precision of our model by around 10% in some cases when compared to training on the complete data set including the tweets which have a blurred sentiment such as offensive but not hate speech tweets. Having an accurate and precise model such as this will allow organizations to protect their users from such language to prevent the negative effects of hate speech. Additionally, it will allow us to identify more hate speech tweets or statements to have more data to research in the future and find deeper trends than simply the tweet text, such as replies, retweets, and user biographies.
Keywords
Ensemble
Hate Speech
Machine Learning
Natural Language Processing
Selective Training
Twitter
Appears in collections
Master's Theses (Plan A and Professional Engineering Design Projects)
Description
University of Minnesota M.S. thesis. May 2020. Major: Computer Science. Advisor: Richard Maclin. 1 computer file (PDF); viii, 35 pages.
Suggested Citation
Houston, Jackson
.
(2020).
Hate Speech Detection In Twitter: A Selectively Trained Ensemble Method.
Retrieved from the University of Minnesota Digital Conservancy, https://hdl.handle.net/11299/216080.
Content distributed via the University of Minnesota's Digital Conservancy may be subject to additional license and use restrictions applied by the depositor. | https://conservancy.umn.edu/handle/11299/216080 |
1. Technical Field
This invention relates generally to a method for improving the efficiency obtained in the burning of a heavy fuel oil in a boiler or other combustion device and to methods and means for preparing the fuel oil for combustion.
More specifically, this invention relates to a method for improving the combustion efficiency of a heavy fuel oil by mixing an effective amount of a combustion enhancing material comprising a high molecular weight, viscoelastic polymer to a heated fuel oil just prior to its combustion.
This invention further relates to a means for producing a homogeneous mixture of a heavy fuel oil and a viscoelastic polymer without degradation of the polymer or interfering with normal operations of the combustion device
2. Description of Related Art
It is well known that the addition of a low concentration of a long chain, high molecular weight, hydrocarbon polymer to a liquid hydrocarbon can result in significant changes to the flow behavior of the resulting solution. If the hydrocarbon polymer has a sufficiently high molecular weight, of at least about 2.5 million daltons, then the resulting solution will exhibit viscoelasticity, displaying both viscous and elastic characteristics when undergoing rapid flow or other deformation. Low concentrations of relatively high molecular weight polymers, particularly such polymers as polyisobutylene, are known to reduce turbulence and have been used as drag reducing agents in pipelines, see for example, U.S. Pat. No. 4,837,249. These same polymers are also known to impart anti-misting properties to jet fuels to reduce flammability of fuel sprays occurring during aircraft crashes, see for example, U.S. Pat. No. 4,789,383.
High molecular weight hydrocarbon polymers, especially those substantially linear polymers such as polyisobutylene, have also found use as a fuel modifying agent for gasoline and diesel engines. As is described in U.S. Pat. No. 5,906,665, a high molecular weight polyisobutylene is first dissolved by extended gentle stirring in a petroleum solvent such as isooctane to obtain a stock solution containing about 2% polyisobutylene. That stock solution can then added directly at the time of refueling to the fuel tank of a vehicle where it readily dissolves. Substantial improvements in fuel consumption and acceleration were observed.
The effect of adding a high molecular weight hydrocarbon polymer, in this case polyisobutylene, on diesel engine performance was also extensively tested and the results reported in SAE Publication No. 2007-01-3981. As reported in that paper, the addition of about 5 ppm of the polyisobutylene agent to the fuel burned in several different commercial (Cummins, Caterpillar and Detroit) diesel engines resulted in a reduction in exhaust particulate matter on the order of 20% to 50% while also reducing NOx on the order of 5% to 25%. In some cases, fuel economy was also improved upon use of the polymer additive.
Attempts have also been made to improve the combustion efficiency of stationary industrial boilers burning heavy fuel oils by adding polymeric fuel modifying agents to the heavy fuel oil prior to combustion. Those attempts have met with failure in that no measurable increase in combustion efficiency could be observed.
This invention provides a technique for successfully employing a polymer additive to increase the combustion efficiency of a boiler or other combustion device burning heavy fuel oils and provides means for practicing that technique.
Other features and advantages of this invention are set out in the following description of the presently preferred embodiments of the invention.
| |
How do I know if I need professional help with a behavioral, emotional, or life problem?
When deciding if your or a loved one needs professional help, consider the following questions:
- How much time is consumed by the problem
- How much does the problem interfere with normal life activities such as school, work, or relationships?
- How much distress is the problem causing, either to the person him or herself or to those who are involved in that individual’s life?
Technicalities aside, sometimes the best answer may be found by honest and accurate self-examination or by heeding the concerns of loved ones. Sometimes just speaking with a professional can give you the answers you need. For mental health issues, the earlier that intervention occurs, the greater the chances of success.
How do I know if my child’s condition warrants professional evaluation and/or intervention?
Essentially, the answer here is about the same as to the question above. However, children, particularly very young children, often lack the self-awareness needed to tell when they are in real psychological trouble. They may not have the communication skills to convey the extent of their difficulties effectively to those in a position to help them, like their parents or teachers. It is important not to explain away or otherwise minimize the need to seek professional assistance if your child asks for help or is having a difficult time in social settings.
What steps do I need to take if a member of my family or I need psychological care?
In many cases, you can ask for a referral to a reputable professional from your family medical practitioner (It’s always a good idea to have a physical check-up to rule out a specific physiological cause.) You may also want to check which providers are covered by your insurance plan. Advice from friends and family who have had similar experiences can also be helpful.
Whatever your path, it is important to make that first appointment. Procrastination can result in the exacerbation of psychological issues. Sometimes it is necessary to speak to a number of clinicians before finding the best approach for yourself or a family member. Whether to consult a psychiatrist, psychologist, or social worker can be a difficult decision and may require some trial and error.
What do I do if I think that another adult in my family needs assistance but other people in the family don’t agree?
Differences of opinion regarding whether a member of a family has clinically significant mental or behavioral health issues is actually quite common. Even within the same family, there can be major differences in levels of education and awareness of psychological problems. Stigma is another issue. For some, having a mental health issue in a family member is a cause for embarrassment and shame.
It takes a lot of conviction and courage to do what is needed in the face of opposition from loved ones but sometimes there is no other choice. As they say, “you can’t go wrong doing the right thing.” If you are convinced that your family member needs help, do your research then speak to your relatives personally in a calm, non-judgmental way. It helps to have some written material or websites to discuss with them. Then you can give them the contact information for the resources you have found. Of course, if they are incapable or refuse to participate in this process you will probably have to do what is needed without their consent. In the end, the most important thing is to do what's right for the person in trouble.
What do I do if I think my child, teenager or an adult in my immediate family needs treatment but they are resistant?
Unfortunately, there are times when intervention is necessary, and action must be taken even against a person's will. Although some would argue that there are no circumstances that warrant such kinds of action, it is unlikely that they have ever encountered a family member that is acutely psychotic, dangerous to him or herself or others, has become unable to function, or has life-threatening health problems (e.g. they are not eating). Usually, the best course of action is to call for emergency help from the local health/legal authorities.
Don’t expect gratitude, to say the least, from your family member if you have to have them receive intervention to which they are opposed. Try not to take any hostility that they show at the time personally, this is to be expected given the circumstances. However, in the long-run, most will understand your actions.
For less urgent, but still problematic situations, you cannot have the authorities intervene against an adult’s will. There are many instances when an adult has severe depression, anxiety, obsessions and compulsions, eating issues, substance abuse problems or other difficulties, but is against getting help. Reasons for this include:
- Fear of losing control over their lives
- Distrust of health professionals
- Lack of insight
- Rationalizing their own behaviors as "no big deal"
This, of course, can be very frustrating. It must be recognized that, even for someone you love, there are limitations to what one can do. It is important to stay patient, try not to be an enabler, and not neglect your own life and health. Sometimes it is a good idea to talk about the situation with a mental health professional on your own. They may be able to generate ideas and assist in mediating the situation.
Teenagers, not surprisingly, present a great dilemma when it comes to how to deal with resistance. Your leverage to get them into treatment is higher if they are under eighteen. Calmly withholding privileges until they are compliant can be a good strategy. Yelling, threatening, and guilt-tripping are usually counter-productive. Strategizing with a clinician who is experienced in these matters can often help. You may also want to model the behaviors that you want to see in your child by going to see the therapist when he/she refuses to go. It will help your child see how much you believe that getting help is essential.
In regard to resistant children, the key is to think of their resistance the same way you would if they did not want to go to school, the doctor, or the dentist. What would you do in these instances? Like most parents, you simply would have to get the child to where they had to be, regardless of the fuss or comments they make. Sometimes a child will stonewall or complain in the initial stages of treatment in order to escape. For your child’s sake, as long as you are confident in the professionals with whom you are working, it will better to hang in there and not let this happen. Many times the resistant child becomes the most successful and engaged patient over time.
How do I know if the person who is evaluating a member of my family or me is qualified?
Mental health professionals are required to display their licenses to practice. You can check the practitioner’s status with the appropriate state licensing board (usually this is in the Department of Health or the Department of Education). However, licensure alone does not ensure that the practitioner is qualified. You can ask about the professional’s participation in subspecialty organizations and ask them directly about their experience level in a given field.
One idea is to attend a support group and ask the members about practitioners they have found helpful. Sometimes the true test of qualifications occurs only in the privacy of your sessions with the practitioner. This does not mean that you will necessarily like what you are hearing or even like a particular clinician at first, just that they have an evidence-based approach and a high level of scientific knowledge in your area of concern.
Is it O.K. to ask questions?
Not only is it appropriate to ask questions and clarify your concerns with your behavioral and mental health practitioners, it is also advisable. Having numerous questions is the norm rather than the exception when it comes to the field of behavioral and mental health. In fact, it is typical to have a chance for “questions and answers” as part of the initial consultation. There are no restrictions on the kind of questions you may ask. Behavioral "is a place that anything may be discussed, no matter how unusual or potentially awkward”.
By definition, sessions are totally confidential (with the major exception being if there is imminent and specific threat to self or others). Remember, you have sought out your practitioner because you are looking for answers. So ask your questions without hesitation.
What happens if I want a second opinion?
Given the complexity of many cases, it is perfectly appropriate to seek a variety of perspectives, just as you might if you needed surgery. Also, different clinicians may be able to offer alternative treatment options. It is not a good sign if a practitioner does not respond to your concerns about their diagnosis or recommendations in a straightforward manner, even if it concerns seeking other opinions. Ethical clinicians are not threatened by other opinions- to the contrary they are welcomed. Responsible practitioners will even offer to assist you in getting the best second or third opinions, if that is your wish. There are many times when the clinician him or her self suggests that getting another opinion is indicated. After all, nobody, however well trained or experienced, has all of the answers all of the time.
For information on any of the services we offer or to schedule an appointment: | https://www.nbiweston.com/resources/faqs/ |
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Development of a context-sensitive physical activity intervention for persons living with HIV and AIDS of low socioeconomic status using the behaviour change wheel.
Mabweazara, S Z
;
Leach, L L
;
Ley, C
.
BMC Public Health
; 19(1): 774, 2019 Jun 17.
Artigo em Inglês | MEDLINE | ID: mdl-31208375
BACKGROUND:
Regular
physical activity
(PA) has been recommended for the management of
HIV
and
AIDS
. The purpose of this study was to develop a contextualised intervention for promoting PA among
women
living with
HIV
and
AIDS
(WLWHA) of low
socioeconomic status
(SES). A secondary aim of the study was to optimise the PA intervention using behavioural theory/ frameworks derived from preliminary studies and the
literature
.
METHODS:
The Behaviour Change Wheel (BCW) for designing behaviour change interventions was used. This
method
was further supplemented by evidence from the
literature
, systematic
literature
review
(SLR), a concurrent mixed
methods
study and two
cross-sectional studies
. The SLR aided in determining the theoretical frameworks to inform the intervention, the specific PA behaviours to be targeted by the intervention, the intervention functions, the intervention
policy
category and the mode of delivery of the intervention. The concurrent mixed
methods
study was used to identify key factors that needed to change in order for participants to engage in regular PA. The first
cross-sectional study
was used to determine the
gender
to be targeted by the study. The second
cross-sectional study
was used to determine the domain and intensity of PA to target in the intervention.
RESULTS:
A
face
-to-
face
context-sensitive PA intervention employing 14 behavioural change
techniques
was designed. The PA intervention (a) utilised the Transtheoretical model of behaviour change and the Social Cognitive theory as the underpinning theoretical frameworks (b) included convenient PAs, such as
walking
, doing simple home-based
exercises
, engaging in
activities of daily living
or doing simple
exercises
at the
community
centre (c) used
education
,
reward
,
training
in PA, modelling
exercise
activities and enablement to increase the opportunity to engage in PA as intervention functions (d) used service
provision
as
policy
priorities, and (e) used a direct
face
-to-
face
mode of delivery.
CONCLUSIONS:
The PA intervention emphasises behavioural
techniques
for increasing PA participation, such as
goal
-setting,
self
-
monitoring
,
strategies
for overcoming PA barriers,
social support
and
rewards
. The intervention employs
strategies
that highlight low-
cost
local PA
resources
and opportunities to help
HIV
infected
women
of low SES to participate in PA. The BCW provides a useful and comprehensive framework for the development of evidence and theory-based PA interventions for PLWHA of low SES. The BCW can thus be used in the development of interventions that 'talk' to
policy
by bridging the
health
inequality
gap. | https://pesquisa.bvsalud.org/sms/resource/pt/mdl-31208375 |
Email or call for price.
Email or call for price.
Description
Traditionally attributed to Chinese philosopher Lao Tzu, the true authorship of the Tao Te Ching, as well as the date around which it was written (usually said to be 6th c. BCE to 4th century BCE), is often debated. The Tao Te Ching is one of the most famous Chinese classic texts and one of the founding texts of Taoism, an ancient Chinese philosophical and religious tradition. The Tao Te Ching includes short verses regarding a number of central aspects of Taoism, such as action, the duality of nature, knowledge, and virtue. However, the true basis of the Tao Te Ching, as well as of Taoism overall, is the "Tao"--an abstract concept most commonly translated as the "Way." The Tao refers to, in rough terms, the natural order and progression of the universe. While Taoism describes nature as the interaction of two opposite but complementary forces, the Tao itself is unified, eternal and indescribable, and such aspects of its nature are emphasized throughout the Tao Te Ching. The goal of adherence to Taoism is to harmonize oneself with the Tao, and therefore with nature and with the universe. | https://www.explorebooksellers.com/book/9798701965186 |
This week, from the 2nd November 2021, a group of ASOS.com Garment Technologists came in for an introductory masterclass on Optitex pattern design software. Tutor Claire Solley, explained the many benefits of the package, which is rapidly being adopted by the fashion industry, including how to prepare patterns for 3D simulation, garment fit, applying colour, pattern and texture and much more.
The insightful session included:
-Working with the Avatar editor.
-How it is possible to adjust the digital mannequin size to be as per your customer.
-PDS 2D how to prepare patterns for 3D simulation.
-Pattern placement, applying location and shape for accurate simulation adding stitches to the garment, flipping stitches that are incorrect applying fabric parameters so simulation knows how much the fabric will stretch/drape, applying stitch properties, lock stitch for woven garments, 4 thread for stretch, 3D placement on avatar so pattern simulates well.
-Garment fit.
-Checking the fit of the garment using mesh, tension maps, and stretch map.
-PDS applying colour and print and fabric texture.
-Creating multiple colourways for the garment, creating printed versions, scaling print up and down, what happens if the print is not repeated correctly.
-Applying fabric textures to give realistic effect in rendering.
Feedback from the ASOS team included:
“I love Optitex. I learnt how I will use Optitex in my day job.” – Milli
“I now have a clear idea of how to use the basics of Optitex software. I would be confident using the areas we have been shown. The seminar was interesting and interactive, two days is needed for the amount of information. I would like a top up in a few weeks to run through any problems, overall I am really pleased.” – Jessica
“Teaching was adapted to our questions and how we work in physical form – relating our needs to the Optitex simulation.” – Gaia
Feedback included: | https://www.fashioncapital.co.uk/insights/asos-garment-techs-gain-optitex-insight/ |
In an age of increasing consumer awareness and connectedness, demand for company level innovation that reduces harmful environmental effects has morphed into a baseline expectation. However, an attitude behavior gap is present between consumers’ stated preferences for sustainable innovation in the products they purchase and their follow through purchase behavior. Research presents conflicting evidence concerning the primary motivation for purchasing with the environment in mind, is it concern for the planet, are consumers just following the way of the crowd, or do they not even care at all? Companies often fail to address the sustainable attributes of products due, in part, to the liability that accompanies mentioning attributes focused on sustainability innovations. While eco-innovations have become far more common in all industries, the athletic and outdoor industry has consumers whom are particularly connected to the environment and companies still struggle to tell sustainability stories. This research contributes findings to consumers’ preferences for specific attributes of sustainability, between material, supply chain, and ethical innovations. A qualitative industry survey established baselines for these innovations which were tested in two iterations of consumer facing surveys (n=23, 103). Emergent findings presented consumer preference for ethical innovation over innovation in material or supply chain and conflicting preference for material durability and material environmental friendliness and conflicting preference between material and supply chain environmental friendliness which may be moderated by product function or measured by physical proximity. These emergent findings are being tested in a national sample (n=200) with intent to contribute to academic and industry knowledge about consumer preferences of different aspects of sustainability innovations.
Rights
In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
Persistent Identifier
http://archives.pdx.edu/ds/psu/25398
Recommended Citation
Cotton, Ethan, "Consumer Acceptance of Eco-Innovations in the Athletic & Outdoor Industry" (2018). University Honors Theses. Paper 592. | https://pdxscholar.library.pdx.edu/honorstheses/592/ |
About Open Book Therapy
Kyla Dannelke, MA, LCPC.
Licensed Clinical Professional Counselor
Originally from the Lehigh Valley in Pennsylvania, Kyla earned her undergraduate degree in Counseling Psychology at Delaware Valley University in Doylestown, PA.
While completing her undergraduate program, she worked providing services at an adult group home specializing in working with individuals with trauma, individuals experiencing homelessness, and individuals seeking recovery support for substance abuse. During this period she worked to refine her skills at running groups, working with individuals to create treatment goals, and taking the time to assist residents in major life transitions.
In 2014 she completed her Master’s Degree in Counseling Psychology, with a special focus in Community Mental Health from Adler University in Chicago, IL.
Over the last several years, Kyla has had the opportunity to work with some of Chicagoland's mostly reputable mental health facilities and programs. During her time in these various positions and programs she has been allowed to grow her knowledge and understanding of how mental health impacts day to day functioning along with how it informs interpersonal relationships. Kyla has also made it a priority to spend time on specialized training in CBT, DBT, EMDR and substance abuse. Her work has a speciality focus of working with those who have attained long-term recovery and sobriety and are looking to continue to gain further interpersonal growth. Most importantly, she has taken the time to hone her skills to be able to help tailor each therapeutic experience so that each individual truly feels allowed to have the space they needs to be able to tell their story in a way that feels authentic to them.
Awards and Achievements
Academic Awards and Achievements
-
Psi Beta Honors Society, Inducted Spring 2008.
-
1st place winner of the American Psychological Association Electronic Project Contest, "Diffusion of Responsibility". Spring 2008.
-
Presenter at International Society for Exploring Teaching and Learning Conference, co-presenting with Dr. Allison Buskirk-Cohen on, "The Impact of Technology Use on Emotional Intelligence: A Student’s First Hand Experience of Designing a Research Project". Presented in October 2011.
Licenses
Licensed Clinical Professional Counselor
- Issued by Illinois Department of Financial and Professional Regulation.
Certifications
-
Completed two part EMDRIA approved training for completion of EMDR training competency, working towards EMDR certification through EMDRIA.
-
Trained and certified in S.O.A.R. application process for the Social Security Administration.
-
Completed 18 hours of Certified DASA DUI Service Provider Orientation Training. | https://www.openbooktherapychi.com/about |
Lebanon’s anti-corruption protesters were on Sunday flocking to Beirut's Riad al-Solh and Martyrs squares to take part in a central demo dubbed "Sunday of Unity" and "Sunday of Pressure".
As protesters from across Lebanon joined demonstrators in downtown Beirut, the anti-corruption rallies continued in the northern city of Tripoli, the southern city of Tyre, the southern city of Sidon and the Mount Lebanon city of Aley.
Activists had called for a million-strong protest in downtown Beirut.
Other protesters also marched from the southern town of Kfar Rumman towards the city of Nabatiyeh.
Earlier in the day, thousands of Free Patriotic Movement supporters had rallied near the Baabda Palace in support of President Michel Aoun.
Addressing the Baabda demo, Aoun called on citizens to unite behind reforms, after more than two weeks of nationwide anti-graft protests that brought down the government.
Unprecedented cross-sectarian demonstrations have gripped Lebanon since October 17, demanding a complete overhaul of a political system deemed inefficient and corrupt.
The cabinet stepped down on Tuesday, but protesters have said this was not enough.
Along with its allies including powerful Iran-backed party Hizbullah, Aoun's political party holds the majority in parliament.
The FPM is now headed by his son-in-law Jebran Bassil, who has emerged as one of the most reviled figures in the protests.
Before the cabinet resigned on Tuesday, Bassil was foreign minister.
A proposed tax on calls via free phone applications such as WhatsApp triggered protests last month.
But they soon morphed into a huge nationwide movement to denounce a raft of woes including a lack of basic services, a failing economy, and rampant sectarianism.
On Tuesday, prime minister Saad Hariri announced his government would be stepping down.
But it is still unclear what a new cabinet will look like, and if it will include independent technocrats as demanded by demonstrators.
After around two weeks of closure, banks and some schools re-opened this week.
But protesters have vowed to press ahead with their demands.
On Saturday night, thousands of anti-government protesters had flocked together in the impoverished northern city of Tripoli to keep the popular movement alive.
Several said they had traveled to the Sunni-majority city from other parts of the country, inspired by the after-dark street parties that earned it the title "bride of the revolution".
More than 25 percent of Lebanese citizens live in poverty, the World Bank says.
The country's economic growth has stalled in recent years in the wake of repeated political crises, compounded by an eight-year civil war in neighboring Syria.
In 2006 after Cedar Revolution was guiding Lebanon to prosperity, freedom and independence. When there was consensus to remove Hizb arms serving Iran and Syria, Aoun and FPM stabbed our freedom and independence in the back and sold their soul to Hizballah for political gain. Today, Aoun and Bassil are again stabbing Lebanon in the back under Hizbollah's orders. They have driven Lebanon to bankrupcy and sold our freedom and democracy. Hariri should refuse any future role. Salameh should resign. Let them destroy Lebanon so the people will march towards the presidency and free this country from this nightmare protected by foreign arms.
Demonstrators should be careful. The same Basij that oppressed the 2010 Green Revolution in Iran, are running Lebanon using FPM as marionettes. Ultimately they will create division and use force. May God save the army and demonstrators from these criminal Basij.
Howdy thepatriot, I got it on WhatsApp in the afternoon. I don’t think the very serious and trustworthy annahar would make such things up. I know it does sound like a lot, but given theFatmagül sultan and Orhan Bey scams.... I doesn’t surprise me. Plus the humongous kickbacks from the oil exploration file with Ned Hariri.... | |
How to Calculate Steel Quantity for Slab, Footing and Column?, Estimation of steel reinforcement quantity for concrete slab, footing and column, beams etc is crucial for the cost evaluation for the construction Design drawings are used as a base for computing rebar quantity in different structural elements This article presents steel quantity computation process for slabs, columns, and footings Contents:Calculate Steel Quantity for SlabCalculate Steel ,How much quantity of Water is added in M20 grade concrete ,, May 02, 2020· Here you can learn about the quantity of water is added in M20 grade concrete (all grades) We know that Concrete is a mixture of Cement, Sand, Aggregate and water Water-Cement ratio in concrete possesses a great rank in acquiring desired strength of concrete Different grades of concrete have different proportionsCONNECTIONS BETWEEN STEEL AND OTHER MATERIALS, Concrete Beam Steel Prop to Existing Concrete Floor 15 20 25 30 35 40 45 50 55 59 64 69 73 78 APPENDIX A - Fixings in Concrete and Masonry 83 APPENDIX B - Grouts and Resins 86 vi SUMMARY Many building and refurbishment projects require structural connections between steelwork and other materials, such as concrete or masonry ,Thumb Rules used in the Construction by Civil Engineering, Dec 14, 2018· Step 2: Calculate the steel quantity using formula As per the above table, the steel quantity of slab is 1% of the total volume of concrete utilized Thumb rule to calculate Steel quantity of above slab = Volume of Concrete x Density of Steel x % of Steel of Member Steel quantity required for above slab = 3 x 7850 x 001 = 235Kgshow many kg steel for 1cu meter rcc slab, How much steel in one cubic meter of concrete for a RCC slab , Someone said: as far as i can remember, 50 kg of cement = 1 cu ft in volume per bag cement specific gravity of cement = 315 or equal to 3150 kg/ cu mtr without , »More detailed.
Civil Notes (Quantity surveying, Concrete, Steel), Sep 22, 2018· All important Quantity Surveying, Concrete and Steel notes are available in this Civil Notes appCivil engineering is one of the best and oldest discipline in engineering work In this civil engineering app we cover up to 150+ topics related to quantity, steel and concretehow much deduct steel quantity in concrete quantity | Civil4M, Feb 29, 2020· NO deduction will be done due to reason that concrete after compaction it releases air entrapped in the concrete 2% of air will be removed from concrete after compaction This removed air quantity and occupied by reinforcement is equal in quantity Due to above reason, deduction is ,Reinforced Concrete Design, transformation coefficient for steel to concrete na = shorthand for neutral axis (NA) pH = chemical alkalinity P = name for load or axial force vector A sc f cc f sc A b A c l dh ARCH 631 Note Set 101 F2013abn 2 P o = maximum axial force with no concurrent bending moment in aMaterial consumption norms for various civil works | Civil4M, Mar 09, 2020· Quantity Of Steel Required :- per cum of concrete For Normal Slab, Chajja, Lintel = 785kg or 0785 Ton , Of Mild Steel = 7850kg Weight Of Concrete 1 cum Of Plain Cement Concrete = 2400kg Weight Of Concrete 1 cum Of Reinforced Cement Concrete = ,How to Calculate Cement, Sand and Aggregate required for 1 ,, Total weight of concrete ingredients = 50+115+209+275 = 4015 say 400 kg Density of concrete = 2400 kg/cum So, 1 bag of cement produces = 400/2400 = 0167 cum No of bags required for 01 cum of concrete = 1/0167 = 598 bags ~ 6 bags From above, if the concrete mix is 1:2:4 , to get a cubic meter of concrete we require 1Cement = 6 bags ,.
Quantity analysis for Materials In 1 cum Cement Concrete ,, May 26, 2020· Concrete is basically a heterogeneous mixture, which consists of cement, sand, stone aggregates and water For a big project, where the higher grade of concrete is used, mix design is must to determine the quantities of the materials required to maintain the grade But when a lesser grade of concrete is used, quantity analysis for Cement, Sand And Aggregate is done by some basic calculationWhat Is In 1 Cubic Metre Mix of Concrete? [Infographic ,, Aggregates directly influence the compressive strength of concrete In a standard cubic metre of concrete, this amount of crushed rock forms a sturdy framework for the mix, while the size and shape of the aggregates will influence workability 1 Cubic Meter of Concrete 1m 3 of concrete = 150l water + 250kg cement + 700kg sand + 1200kg aggregatFor one meter cube of concrete how much kg steel is ,, For example, according to SP (Russian norms for Nonprestressed Concrete Structures) minimum reinforcement area should be determined Asmin = mus * b * h0, where Asmin - is the minimum area of reinforcing steel within the tensile zone, mus - is the minimum reinforcement ratio, for bending mus =0001, for bending with axial force mus =00025, b ,How To Calculate Steel Quantity For Slab, As we think that calculating steel quantity for the slab is a too difficult task, it required good skills in civil engineering but i am sharing with you an excel sheet, which will make slab steel quantity calculation too easy for anyone who wants to know the exact quantity of steel required for his/her slabHow to Calculate Steel Quantity for Slab?, In this post, we are going to explain how to calculate steel quantity for the slab? For b oth one way and two way slab with an example Note: For Better View, Please read this post in landscape mode if you are on the mobile device.
Chapter 5 Quantities Calculations, project has two structures, both of them have removal of concrete, class B concrete, and new strip seal expansion devic The quantities for Removal of Concrete and Class B Concrete are 215 CY for Site A and 212 CY for Site B when taken to the tenth of a cubic yard (01 CY) but the Removal of Concrete quantity is rounded up Introduction GeneralHow much quantity of steel required for 1m3 concrete?, Quantity of steel in concrete depending on type of structure and load of structure For Foundation :- 05 to 08 % steel are required; Let we take minimum 05 % of steel is required on 1 m3 concrete Then quantity of steel = (05/100)×1× 7850 = 3925 kgQuantity Of Steel Required For 1 M3 Concrete, Contents1 Beam:2 Column:3 Slab/Lintel:4 Foundation: First of all, the quantity of steel does not depend on the concrete quantity instead, it depends on the type of structure and its dimension, loading condition, etc However, if you need an approximate quantity you can use thumb rul These thumb rules have been adopted based on the practice [,]How to Calculate Quantities of Cement, Sand and Aggregate ,, One bag of cement and other ingredients can produce = 400/2400 = 0167 Cum of concrete (1:2:4) 01 bag cement yield = 0167 cum concrete with a proportion of 1:2:4 01 cum of concrete will require Cement required = 1/0167 = 598 Bags ~ 6 Bags Sand required = 115/0167 = 688 Kgs or 1498 cft Aggregate required = 209/0167 = 1251 kgs or 2996 cftHow To Calculate Of Cement, Sand And Aggregate For M10 ,, Grade of concrete M20 (1:15:3) Find-out the Quantity of concrete (Wet volume and Dry volume) Wet volume of concrete = 1 m³ Dry volume = Wet volume x 154 (You can also take it as 150 to 155, But it became standard practice to considering 154 as multiplication factor) Dry volume of concrete = 1 x 154 =154 m³ 1- Find-out quantity of Cement.
How to Calculate the Steel Quantity in Column Footing ,, Oct 15, 2017· The steel quantity in column footing can be calculated very easily Before calculating the steel read carefully given footing drawing and note all the important points like Footing (Length, Width, Thickness) The diameter of Footing reinforcement A grade of reinforcement is going to use The spacing of the reinforcement (c/c)How To Calculate Steel Quantity For RCC Beam, Column And Slab, Aug 04, 2017· Following are the steps to calculate the quantity of steel for RCC slab 1 Prepare a bar bending schedule in order to classify different shapes of bars (bent up bar, straight anchor bar, eos bar, curtail bar etc) and diameters 2 List down all the shapes ,How is the steel quantity estimated for a structure?, Enter "Materials Quantity manager" to view the real steel quantity Or: Select "Weight review" A document will be generated In this case, the steel quantity from the column is only 007% of the total quantity estimated and 004% of the estimation for the first level This property is ,Concrete Calculator, Example calculation Estimate the quantity of cement, sand and stone aggregate required for 1 cubic meter of 1:2:4 concrete mix Ans Materials required are 7 nos of 50 kg bag of cement, 042 m 3 of sand and 083 m 3 of stone aggregateALL IN ONE: STEEL QUANTITY OF BEAM EXCEL SHEET Click Here ,, May 16, 2018 - STEEL QUANTITY OF BEAM EXCEL SHEET Click Here To Download.
How much quantity of steel required for 1m3 concrete ,, Quantity of steel required for 1m3 concrete beam Thumb rule for steel in beam = 1% to 2% Minimum quantity of steel required for 1m3 concrete beam is 1%, now 1% of 1m3 = 001 m3, and we know that 1m3 steel weight is 7850 Kg, so weight of 001m3 steel = 001 × 7850 = 7850 kg, ,Analysis & Rates, Analysis and rates for construction work ,, Brick ballast for lime concrete 100cuft for 100 cuft 100cum for 100 cum Sl# Particulars Quantity 8 Dry mortar for lime concrete I foundation and floor 35% 35cuft for 100cuft 35cum for 100 cum 9 Dry mortar for lime concrete in roof terracing 45% 45cuft for 100cuft 45cum for 100 cum 10 Materials for cement concrete 1:2:4 Ballast ,Quantity of Steel and Concrete in RCC Beam, This video shows how to calculate quantity of steel and concrete in RCC beam If you have known dimensions of concrete beam you can easily find out the concr,How much steel is required for 1 cubic meter concrete?, You may require the following quanties of steel reinforcements per one cum of cement concrete which is expressed in kg:- 1Column footings 75 kg use 10 or 12 mm dia rods 2Grade beams 100kg use 12,16 mm dia rods -85%; 8 and 6mmdia rods -15% 3Plin,Concrete Material Calculation / Concrete Quantity, Concrete material calculation first proportions of matrial such as 1 : 2 : 4 (M15), 1 : 15 : 3 (M20), 1 : 1 : 2 (M25), 1 : 05 : 1 (M30), Above M 25 Grade concrete need mix design Here 1 : 2 : 4 is Material how to define which material first number 1 is a Cement , Secon number material is a Fine Aggregate (Sand), here the third number is ,. | https://www.labergerie1700.fr/plan/4116/steel-quantity-in-1cum-of-concrete.html |
Filling a crucial gap in the clinical practice revolution
Cutting-edge technologies, like next-generation-sequencing (NGS), combined with the development of targeted therapies are now revolutionizing clinical practice, bringing new complexity to the field of oncology. These high-throughput technologies produce ever-larger amounts of data, thus providing an unprecedented level of molecular information, but also raising new challenges for clinicians and clinical laboratory professionals.
In oncology and hemato-oncology, genomic profiles are now increasingly being used to prescribe the right drug for the right patient. The process required to generate, analyze and interpret genomic data is however complex and calls upon a plurality of skills.
There is therefore a need for continuing education in this rapidly evolving field, to ensure that professionals with various backgrounds can communicate and collaborate efficiently to optimize the personalized oncology process, for the benefit of patient care.
About the CAS in personalized molecular oncology
Coordinated by SIB Clinical Bioinformatics, the CAS aims at providing a comprehensive and integrative view of the field, by covering:
- tumour biology and genetics;
- molecular pathology;
- clinical bioinformatics;
- clinical oncology.
The course will focus on the methodologies used to generate, analyze and interpret patients’ molecular profiles. Ultimately benefiting the patients, the CAS will address both the potential and limitations of these cutting-edge technologies for personalized oncology, also touching upon the associated technical, regulatory and ethical challenges.
As an important outcome, it will contribute to establish a common language between the wide range of professionals involved in the personalized oncology process, from biologists and bioinformaticians to pathologists and clinicians. It will thus enable an efficient and better-informed use of genomic data for both routine clinical practice and clinical research.
Overall, it should therefore empower professionals to develop a vision in their own institution, by critically evaluating the potential benefits and limitations of current and future developments in personalized oncology.
Practical information
- Who is it for? The CAS targets a multidisciplinary audience of professionals involved in personalized molecular oncology and hemato-oncology, including laboratory managers, biologists, bioinformaticians, pathologists, geneticists, clinicians and pharmaceutical company employees
- Applications: starting early 2018
- Start: autumn of 2018
- CAS delivered by: University of Basel
- Partners: | https://www.sib.swiss/about/news/news-2017/10228-first-certificate-of-advanced-studies-cas-in-personalized-molecular-oncology |
Every so often, someone finds a new way to do an old thing. Mind you, that's not necessarily a prescription for success in life because a lot of bad guys put their minds toward finding ways over, under, and around the law but, regardless, it's nice to know that the entrepreneurial spirit is still alive. Consider a recent regulatory settlement involving an enterprising fellow who found a way to cash the same paycheck twice!
Case In Point
For the purpose of proposing a settlement of rule violations alleged by the Financial Industry Regulatory Authority ("FINRA"), without admitting or denying the findings, prior to a regulatory hearing, and without an adjudication of any issue, Thomas J. Hind, Respondent, Respondent submitted a Letter of Acceptance, Waiver and Consent ("AWC"), which FINRA accepted. In the Matter of Thomas J. Hind, Respondent (AWC 2015045049101, November 30, 2015).
Hind was first registered in 2006 and by November 2015 was associated with FINRA Member Firm MBSC Securities Corp., where he remained until his April 7, 2015 discharge.
The Ol' Double Dip With A New Digital Twist
As set forth in the AWC:
In January and February 2015, Hind misappropriated funds by depositing three paychecks via a mobile application and also cashing the physical copies of those same checks. Hind previously received paychecks from MBSC via direct deposit to his checking account. In January 2015, Hind notified MBSC that he had closed his checking account and wished to receive physical paychecks going forward. The firm issued Hind physical paychecks on January 16, January 30, and February 27, totaling $5,109.15. For each paycheck, Hind first deposited the check into his checking account via a mobile application on his cellphone and then cashed the same check at a check cashing center. Through this strategy, Hind received double payment from each of the three paychecks. Because each paycheck had already been deposited and paid, the paychecks were returned unpaid to the check cashing centers, resulting in losses to the check cashing centers. FINRA Rule 2010 provides that"[a] member, in the conduct of its business, shall observe high standards of commercial honor and just and equitable principles of trade." By reason of the conduct described above, Hind misappropriated funds in violation of FINRA Rule 2010.
SIDE BAR: According to online FINRA BrokerCheck records as of December 11, 2015, MBSC "Discharged" Hind on March 9, 2015, based upon allegations that:
MR. HIND MANIPULATED PAYROLL COMPENSATION. THERE WERE TWO SEPARATE INSTANCES OF THIS ACTION. THIS ACTION IS NOT CONSISTENT WITH THE FIRM'S POLICY. IT WAS NOT SECURITIES RELATED.
In accordance with the terms of the AWC, FINRA imposed upon Hind a Bar in all capacities from associating with any member firm.
Bill Singer's Commentary
Who a thunk it? Hind's employer directly deposited his paycheck into his bank account. Ah yes, how convenient that is. On the other hand, Eureka!, Hind apparently was hit by the realization that there now existed a mobile app that would allow him to take a snapshot of his paycheck and digitally deposit his salary into his bank account. What if . . . yes, our young FINRA respondent may have had ia "what if" moment . . . Hind could get his hands on a physical, hard-copy, old-fashioned, analog paycheck? Hmmmm, if he could take the photo of that check and digitally deposit it into his personal bank account via his cellphone and, quickly, run like a jack-rabbit to one of those check-cashing places and cash that paycheck, well, yeah, maybe, you know, yeah, maybe it would work: the Digitized Double Dip!!!
Alas, not all that bad an idea but, perhaps, Hind could have -- should have -- thought things through a bit more? At some point, his employer would be presented with two demands for payment of the same paycheck. Then, of course, there could be that whole other problem involving that guy who is standing outside your apartment building, the guy with no neck, the guy wearing a long, black raincoat on a hot, sunny August day, the guy that's normally sent to get the Vig you owe and didn't pay on time, the guy who was paid by some guy who knows some guy who knows the guy who owns the check-cashing place?
For now on, when you take a paycheck, just take one dip and end it! | https://www.brokeandbroker.com/2981/double-dip-paychecks/ |
Fuel efficiency of vehicles has been increased by reducing the rolling resistance of tires to suppress heat build-up. In recent years, such contributions of tires to an increase in fuel efficiency have been increasingly demanded. There has previously been a great need to improve the fuel efficiency of treads, among other tire components, which constitute a large portion of a tire. More recently, there has been a need not only for treads but also for sidewalls, insulation components, breaker cushions, and the like to have higher fuel efficiency.
Known approaches to improve the fuel efficiency of rubber compositions include the use of low reinforcing filler, or the use of a smaller amount of reinforcing filler. Another known approach for improving fuel efficiency is to use silica filler to reduce rolling resistance. However, these approaches reduce the reinforcing properties of rubber compositions, and therefore disadvantageously reduce breaking performance, such as flex crack growth resistance, and abrasion resistance.
In particular, tires for trucks and buses are used under very severe conditions, and thus treads of tires for trucks and buses require high abrasion resistance as well as high breaking performance sufficient to prevent chipping of treads, etc. To satisfy this requirement, the compositions for treads of tires for trucks and buses typically contain natural rubber and/or polybutadiene rubber. There have been attempts in recent years to further improve abrasion resistance by increasing the cis content or the molecular weight of polybutadiene rubber, or by subjecting polybutadiene rubber to modification for carbon black. In contrast, unlike polybutadiene rubber, there is at present very little development of techniques for natural rubber, which is a main component of the composition for treads of tires for trucks and buses, because it is a natural product.
Accordingly, the development of a natural rubber having high abrasion resistance and high breaking performance has been required.
Natural rubber is mainly formed of polyisoprene. Unlike synthetic polyisoprenes, it has a high gel fraction. The term “gel fraction” refers to a fraction poorly soluble in a solvent. It is believed that the gel fraction is derived from branched structures formed by large amounts of protein, lipids and the like contained as impurities in natural rubber. In fact, it is known that the gel component is reduced to some extent by deproteinization to remove proteins as allergenic substances.
For example, Patent Literatures 1 and 2 disclose methods of reducing proteins and the like contained in natural rubber by adding a proteolytic enzyme and a surfactant to natural rubber latex, and aging the mixture. Besides, in order to reduce the gel content in natural rubber, Patent Literature 3 discloses a method of immersing a solid natural rubber swollen with a solvent in an alkali hydroxide; Patent Literature 4 discloses a method of adding a phosphate to natural rubber latex and then removing magnesium phosphate; and Patent Literature 5 discloses a method of adding a surfactant to natural rubber latex followed by washing. Moreover, Patent Literature 6 discloses a method of preparing a natural rubber having a low Mooney viscosity by treating natural rubber latex with a phospholipase or lipase. Furthermore, Patent Literature 7 discloses a method for coagulation of natural rubber using a polymer flocculant.
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Brown Publishing Co. and its biggest bank lender Wednesday accused unsecured creditors of making last-minute, unsubstantiated claims in seeking to prevent the bank from using its debt to bid on the bankrupt Cincinnati-based newspaper publisher.
In separate papers filed in U.S. Bankruptcy Court in Central Islip, N.Y., Brown Publishing Co. (BPC) and PNC Bank said there was no merit to claims by the official committee of unsecured creditors that the debt taken on by entity owned by Brown Publishing CEO Roy Brown and other company insiders made the deal a “fraudulent conveyance,” that is, the entity, Brown Media Holdings Co., was insolvent from day one, PNC Bank knew it, and therefore it shouldn’t be allowed to use its debt to make a “credit bid” for Brown Publishing 15 dailies and numerous other papers and business publications.
“The committee has not provided any evidence to substantiate its claims,” the PNC filing said. “Rather, the conclusions reached by the committee in the motion rely solely on the unfounded hope that the lenders’ liens and perfected security interests in BMHC’s assets constitute fraudulent transfers.” In fact, the loans BMHC accepted responsibility for, along with other Brown companies, did provide value to BMHC and therefore cannot be used to prevent PNC from making a credit bid.
PNC is owed more than $74 million by Brown Publishing, and has said it will make a $20 million credit bid for the company by the bid deadline of this Friday.
Roy Brown’s three-year-old separate company, Brown Media, has made an opening bid of $15.9 million for BPC, which claimed a book value of $94.1 million when it filed for bankruptcy May 1. BPC listed $104.6 million in debts.
Judge Dorothy Eisenberg has previously approved an auction sales plan that would allow PNC to make a credit bid, and without putting up a 5% cash deposit required of other bidders, including Roy Brown’s group. But after the unsecured creditors filed on Monday asking for a hearing on prohibiting a credit bid, the judge quickly agreed. The hearing is set for Thursday July 15.
The unsecured creditors say allowing PNC to bid up to $74 million in credit alone has a chilling effect on other potential bidders for the newspapers.
Brown Publishing said in its filing that the unsecured creditors, who stand to get nothing from a sale of the company, had not presented any evidence that a credit bid was chilling interest in the company. The unsecured creditors have argued in past court documents that the pre-bankruptcy sale was structured to discourage bidding interest because the company with many widely separated clusters of publications was offered for sale as a whole rather than piecemeal. They noted, too, that a separate Brown family company owns the actual real estate and office and production facilities for many of its publications and leases them to the Brown Publishing publications.
But Brown Publishing argued that time was of the essence in proceeding with the auction. | https://www.editorandpublisher.com/news/brown-publishing-bank-cry-foul-on-attempt-to-stop-credit-bid-in-bankruptcy-auction/ |
I’m not imagining it when I say that Emma Stone has been compared to Diane Keaton, right? I know that she herself has talked openly many times about her admiration for Diane. But, like, people have called Emma an actress in the mold of Diane Keaton, right? Yes, yes, let her be the first Emma Stone and not the next Diane Keaton. Of course. But what I’m asking is that they’ve been connected, that they remind people of each other? No? Maybe I was just influenced a few years ago after reading Diane interviewing Emma in Interview. Have you read it before? If not it’s worth a revisit, right from the beginning. Diane opens with a question about ambition, revealing that she used to hide it, hide her desire for things. Out of fear that they wouldn’t happen, that she had not right to want. And Emma, she doesn’t challenge her, but here she is, talking to her “hero”, and she’s like…really? Because I only feel “safe” when I know what I want. It’s a fascinating contrast. Is it generational? Or is it simply their personalities?
Then there’s the part when Diane just blurts out, with no warning, “So, Emma, look. The way I see it is, you're unconventionally beautiful”.
And even on paper, if you’re familiar with Diane’s communication style, is just so Diane Keaton, non? And I feel like I know where that question comes from – which is that this is how Diane herself has been described. It’s a compliment to her, and she means it as a compliment. If you’ve read any of her books though, you might know that it took her a while to get there.
Emma was at the AFI event honouring Diane Keaton last night. And she was also photographed with Patty Jenkins. Patty’s wearing Wonder Woman inspiration on her arm. And both she and Emma are giving us the Wonder Woman arms. I love this photo: | https://www.laineygossip.com/emma-stone-honours-diane-keaton-afi-lifetime-achievement-event/47209 |
What would you do if someone handed you a quilt pattern with one of Moda Fabrics’ fat quarter bundles and just said, “Here—go make this quilt and have fun?” That is, after you stopped pinching yourself to make sure you weren’t dreaming?
That’s the situation Eileen Fowler found herself in a few months ago when we asked her to remake Sherri’s Scrappy Stars by Sherri Bain Driver as part of McCall’s Quilting’s 30th anniversary celebration this year for the May/June 2018 issue. Originally published in 2010, Sherri’s Scrappy Stars was made with a traditional selection of fabrics in a variety of colors. The Moda bundle Eileen got to take home and play with is in a more controlled palette of yellows, blues and grays. So how did she set about making the most of it?
“I first sorted the lights and the darks and the yellows,” she says. “There were some that didn’t fit any of those categories; I set them aside along with the prints that were also used in the borders. That gave me enough fat quarters to work with to make the blocks.”
Eileen didn’t just jump into cutting up the fat quarters. First, she took a little time to think about her options and a plan of attack.
She decided to use as many yellow prints as possible in place of the light prints originally used in the star block backgrounds and filled in the rest with low volume prints. Once she’d designated where those fabrics would go, she cut the all dark and light patches for the star points and block backgrounds.
Once that was done, she cut and assembled the small squares for the nine-patches that go in the centers of the blocks from the scraps. “I tried to mix up a variety of fabrics for the nine patches,” she says. “They could be really scrappy as long as there was some amount of value contrast.”
With all the nine-patches made, she returned to the star points and background patches she’d already cut and started auditioning combinations to find a good variety of fabrics within each block. She had two goals: to maintain as much contrast as possible in each block, and to make no identical blocks, which is no small task when you have 63 blocks to make from a limited number of prints.
Eileen is a really fast piecer thanks to experience with quick techniques like chain piecing, but with this quilt she worked on only a few blocks at a time. “I was trying to keep all the patches straight and I think it would have been too confusing to work on all the blocks at once.”
Did she achieve her two goals? Mostly, she says. “Obviously some of the stars really pop out because of the strong contrast and others are more low volume.” As for no duplicates? “I thought I succeeded until I got it all sewn together and realized, ‘Hey, those two blocks are the same.’ But I placed them far enough away from each other that you can’t see them.” (If you spot the matching blocks, let me know–she wouldn’t tell me which ones they are.)
All of Eileen’s careful attention to placement and contrast yielded a sparkling, scrappy look from a controlled selection of prints. She added what was originally a scrap quilt design to McCall’s library of fat quarter patterns and provided a lesson on how to use color and value to maximize your fabrics.
Comment (1)
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Ellen B
Anything with yellow is the best and this one is a beauty! | https://www.quiltingcompany.com/sparkling-anniversary-stars/ |
|Credit: NASA / WMAP Science Team (usage does not imply endorsement of site contents)|
One modification for the Big Bang is the "inflation theory", which looks good on computer screens but has no real observational evidence. The universe is expanding? Probably. Such a concept is well within biblical creation science views. Obviously, scientists on either side disagree on the details. It is quite clear, however, that the Big Bang is irreconcilable with the days of Creation a few thousand years ago.
Three main arguments are commonly used to support the Big Bang model of the universe’s origin:To read the rest, click on "Does the Cosmic Microwave Background Confirm the Big Bang?"
Although an expanding universe is consistent with the Big Bang, it doesn’t necessarily demand a Big Bang as its cause. One could imagine that for some reason God imposed an expansion on His created universe, perhaps to keep the universe from collapsing under its own gravity. Of course, this assumes that secular scientists’ interpretation of the redshift data is correct, which some creation scientists are starting to question.
- The apparent expansion of the universe, inferred from redshifted spectra of distant galaxies;
- The fact that the Big Bang can account for the observed relative abundances of hydrogen and helium;
- The observed cosmic microwave background (CMB) radiation, thought to be an “afterglow” from a time about 400,000 years after the supposed Big Bang.
Looking for a comment area?
You can start your own conversation by using the buttons below! | https://www.piltdownsuperman.com/2018/06/the-big-bang-and-cmb-radiation.html |
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