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The aim of this study was to evaluate the effect of oregano essential oil, carvacrol and thymol on biofilm-grown and Staphylococcus epidermidis strains, as well as the effects of the oils on biofilm formation. For most of the S. aureus (n=6) and S. epidermidis (n=6) strains tested, the biofilm inhibitory concentration (0.125–0.500 %, v/v, for oregano, and 0.031–0.125 %, v/v, for carvacrol and thymol) and biofilm eradication concentration (0.25–1.0 %, v/v, for oregano and 0.125–0.500 %, v/v, for carvacrol and thymol) values were twofold or fourfold greater than the concentration required to inhibit planktonic growth. Subinhibitory concentrations of the oils attenuated biofilm formation of S. aureus and S. epidermidis strains on polystyrene microtitre plates. | http://www.siroe.it/servizi/106-journal-of-medical-microbiology.html |
Can you use Medifast pudding instead of shakes in muffins??
I made muffins out of all the shakes I got in the initial variety pack - except the chocolate ones, which I couldn't do even as muffins. I gave those chocolate shakes and puddings to my MIL, and she gave me vanilla pudding in return..
The puddings make excellent shakes! thick and creamy and smooth!..
No, not great for muffins, but they are awesome for cookies!.
CLICK HERE for butter cookie recipe!..
I used them in muffins and they were always good. My standard recipe weas.
1 Medifast pudding or shake or cocoa.
Favorite was egg, choc pudding and brownie..
1. do you bake that into 3 muffins?.
2. and then it's 3 Medifast meals?.
3. how long to do they stay fresh?.
4. fridge or countertop storage?.
5. baking in ...... a ramikin? a muffin tin with paper liner?..
Could someone post the shake muffin recipe please? I ordered a zillion shakes. Thank you very much..
Muffin make 3 big or 6 regular. 3=3mf meals 6=3 (2 muffins each) meals..
I baked in regular size muffin tins sprayed with Pam or in silicone muffin cups..
They last a couple days out, longer in fridge or freeze. Store in airtight container like lock and lock..
Use the shake in above recipe with Medifast eggs and your choice of other, ( oatmeal, pudding, cocoa or brownie) Find your favorite combos and enjoy..
Greenteadrinker, I use essentially that recipe for the muffins I make now - just with the shakes, obviously. I also add cinnamon to mine, and depending on what I'm making, some flavoring. Current favorite is lemon-blueberry (made with vanilla shake, scrambled eggs, and blueberry oatmeal... and adding in lemon flavoring). It's really good!.
I never buy the eggs what happens if you don't use the eggs? I have some peach oatmeal I'm trying to use up...... yucky as oatmeal! Anyone have a non-egg recipe to make muffins or a cookie.......anything except just eating it as oatmeal. thanks.. | http://snubbr.com/can-you-use-medifast-pudding-instead-of-shakes-in-muffins-40313/ |
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臘月 the twelfth moon of the lunar year.
年貨 food and other articles for use during the New Year season（Nienhuo）.
除舊佈新to remove the old and introduce the new.
除夕 Lunar New Year's Eve.
年糕 New Year pudding made of glutinous rice flour
春聯 couplets written on strips of red paper and pasted around door panels during the Lunar New Year （usually containing words of luck）.
年畫 drawings or picture sold in lunar new year for good luck.
平安順利 peace and smoothly, easy, having no trouble.
財神爺 the God of Wealth; Mammon.
財源滾滾 source of finance and no end.
祭祀 to honor by a service or rite; to offer sacrifices to .
祖先 ancestors; forebears; forefathers.
神靈 gods; spirits.
慎終思遠 thoroughgoing about the funeral rites for parents and the worship of ancestors.
飲水思源 When one drinks water, one thinks of its source-grateful for favors received;not to forget one's origin.
團圓 to reunite; reunion （esp. of a family）.
年夜飯 New Year's Eve dinner.
紅包 red package（also called壓歲錢）.
拜年 to call on others and offer New Year's greetings.
恭喜 Congratulations!
國曆 national calendar （of the Republic of China） （also called 陽曆 solar calendar） .
農曆 farmer's calendar（also called 陰曆 lunar calendar）.
麻將 mahjong pieces.
沖天炮 to shoot up to the sky.
舞龍舞獅 dragon dance（a team of men dancing with a cloth-made dragon on Chinese festivals） and lion daance（a two-man team dancing inside a paper-made lion on Chinese festivals）.
吉利 be lucky; be propitious.
元宵節 The story of the Lantern Fesfival.
source from:
http://www.mdnkids.com.tw/learn/learn_c/c2_5_3.html
Posted on: 2007/2/6 10:59
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Lunar New Year in Taiwan
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Home away from home
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(Lunar New Year in Taiwan)
The Lunar New Year is the most significant festival for ethnic Chinese around the world, wherever they come from. It is a very jubilant occasion mainly because it is the time when people take a break from work to get together with family and friends.
The origin of the Lunar New Year Festival can be traced back thousands of years, involving a series of colorful legends and traditions. One of the most famous legends is Nian 年, an extremely cruel and ferocious beast that the ancients believed would devour people on New Year's Eve. To keep Nian away, red-paper couplets are pasted on doors, torches are lit, and firecrackers are set off throughout the night, because Nian is said to fear the color red, the light of fire, and loud noises. Early the next morning, as feelings of triumph and renewal fill the air at successfully keeping Nian away for another year, the most popular greeting heard is gong si 恭喜,or "congratulations."
Even though Lunar New Year celebrations generally last for only several days, starting on New Year's Eve, the festival itself is actually about three weeks long. It begins on the twenty-fourth day of the twelfth lunar month, the day, it is believed, when various gods ascend to heaven to pay their respects and report on household affairs to the Jade Emperor, the supreme Taoist deity. According to tradition, households busily honor these gods by burning ritualistic paper money to provide for their traveling expenses. Another ritual is to smear malt sugar on the lips of the Kitchen God, one of the traveling deities, to ensure that he either submits a favorable report to the Jade Emperor or keeps silent.
Next, "spring couplets" are hung up around the house. Spring couplets are paper scrolls and squares inscribed with blessings and auspicious words, such as "good fortune," "wealth," "longevity," and "springtime." The paper squares are usually pasted upside down, because the Mandarin word for "upside down," dao, is a homonym of the word "arrival." Thus, the paper squares represent the "arrival" of spring and the "coming" of prosperous times
On Lunar New Year's Eve, family members who are no longer living at home make a special effort to return home for reunion and share in a sumptuous meal. At that time, family members hand out hong bao, or "lucky money" in red envelopes, to elders and children. They also try to stay up all night to welcome the New Year, as it was long believed that by doing so on New Year's Eve, their parents would live a longer life. Thus, lights are kept on the entire night--not just to drive away Nian, as in ancient times, but also as an excuse to make the most of the family get-together. In addition, some families even hold religious ceremonies after midnight to welcome the God of the New Year into their homes, a ritual that is often concluded with a huge barrage of firecrackers.
The first order of business on Lunar New Year's Day is offering ritual homage to one's ancestors. Reverence is then paid to the gods, followed by younger family members paying their respects to their living elders. New clothes are worn, and visits are made to friends, neighbors, and relatives to exchange good wishes of gong si fa cai 恭喜發財, which means "congratulations and prosperity." As an occasion for reconciliation, it's a time when old grudges are cast aside amidst an atmosphere of warmth and friendliness.
One of the most spectacular sights during the Lunar New Year Festival is the dragon and lion dance. The heads of these fearsome beasts are supposed to ward off evil, and the nimble movements of the dancers provide a grand spectacle enjoyable to everyone.
The second day of the Lunar New Year Festival is the day that married daughters return to their parents' home. If she is a newlywed, her husband must accompany her and bring gifts for her family. According to a charming legend, the third day of the Lunar New Year is the day when mice marry off their daughters. Thus, on that night, people are supposed to go to bed early so that the mice can perform their wedding ceremonies.
On the fourth day, the fervor begins to ebb. In the afternoon, people prepare offerings of food to welcome the return of the Kitchen God and his retinue from their trip to the Jade Emperor's court. The Kitchen God's return signifies the end of freedom from spiritual surveillance, hence the popular saying: "It's never too early to send off the gods, and never too late to invite them back."
Day five almost brings the Lunar New Year festivities to a close. All offerings are removed from the altars and life returns to normal. Finally, on the ninth day, numerous offerings are set out in the forecourt or central courtyard of temples to celebrate the birthday of the Jade Emperor, who was believed to have been born immediately after midnight on the ninth day.
As in all such festivals, food plays an important role throughout the Lunar New Year Festival, and dinners tend to be especially lavish. Many of the dishes made at this time are served because they are regarded as symbols of good luck. For instance, fish (yu) represent "having enough to spare," garlic chives (jiou cai) stand for "everlasting," turnips (cai tou) mean "good omens," and fish balls (yu wan) and meat balls (rou wan) represent "reunion." Auspicious refreshments are also prepared at this time, such as glutinous rice flour pudding (nian-gao), which is said to make people "advance toward higher positions and prosperity step by step." People usually have dumplings (shuei jiao) too, which look like shoe-shaped gold and are supposed to help those who eat them to amass fortunes and wealth
The Lunar New Year Festival is not all freewheeling fun, however, and certain negative superstitions and taboos at this time have never quite lost their pervasive force. For example, people believe it is unlucky to sweep the floor during the first five days of the Lunar New Year, because one might accidentally sweep one's good luck and wealth out of the house. Bad language and talk of death are severely frowned upon. If a dish is broken, it is vital to say suei suei ping an 歲歲平安, which means "peace throughout the year," as quickly as possible. Joss sticks and altar candles must be kept burning day and night to encourage longevity; and in some households, knives and scissors are put away so that no one will accidentally cut the "thread of good luck" in the year to come.
A few of these superstitions and rituals have a spiritual aspect to them, and all of Taiwan's temples are usually very busy during this time of year as large numbers of people crowd into them with elevated incense sticks to pray for good luck. Indeed, some of the major temples close their main gates before midnight on Lunar New Year's Eve as noisy and expectant crowds gather outside. At the stroke of midnight, the doors are thrown wide open and people surge forward in an attempt to be the first to place their incense sticks into the censer, as another long-standing tradition states that the first person to do so will be blessed with good luck throughout the coming year.
Although some of the Lunar New Year's magic has worn off in Taiwan because of the island's steady march toward industrialization, the festival and accompanying celebration are still unrivaled in importance. Long before the holiday, street vendors have already begun to seek out the best sites to display their "spring couplets." Shopping for Lunar New Year's fare also begins early and is still one of the holiday's most characteristic activities in Taiwan. For instance, Taipei's Dihua Street in an older part of town, which is renowned for its foodstuffs, often bustles with people buying groceries for the Lunar New Year Festival. Familiar songs and traditional music associated with Lunar New Year are broadcast through loudspeakers in department stores, many of which hold year-end sales to attract wage earners, whose pockets are weighted down by the traditional annual bonus that is always paid at this time of year.
Thus, several days before Lunar New Year's Eve, people living far away from their families begin to prepare for their journey home. In an attempt to beat the traffic jams, many hit the road on the previous day. Those relying on public transportation will often camp out in sleeping bags at airports, train stations, and bus terminals to ensure that they get reservations for the dates they want, as tickets are usually snapped up the second they go on sale. Trains, buses, and planes are always packed.
No matter how grueling the journey may turn out to be, though, all of the inconveniences are considered to be worth it once the family has gathered around the table to eat their Lunar New Year's Eve dinner, the most important meal of the year. Indeed, no matter what changes may occur over time, the notion of getting together with the family will always lie at the heart of Lunar New Year celebrations.
source from:
http://www.gio.gov.tw/taiwan-website/5-gp/culture/lunar-NY/
Posted on: 2007/2/6 11:22
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Re: Some terms about Chinese new year in English
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Just popping in
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So the New Year in taiwan is similarto this in the mainland.
Posted on: 2007/2/24 19:51
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Re: Some terms about Chinese new year in English
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Just popping in
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how great it is !let's start it ,and look forward to it !
Posted on: 2007/2/24 23:07
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You cannot post without approval. | http://okenglish.tw/modules/newbb/viewtopic.php?topic_id=1599&forum=12 |
IT BEGAN when they found a box containing a bloody hand in a stolen car. Later, the killer would taunt them — sending them clues and the victim’s eyeball.
Then, the investigators unearthed the scene of what might have been one of the most gruesome crimes the city has seen.
If it were real, that is.
The body parts are fake. The investigators are students at James Logan High School. And the crime? A lab exercise in the forensic science class being taught at the school.
The class, offered by the Mission Valley Regional Occupation Program, teaches nearly every aspect of crime-solving science, from evidence collection procedures to DNA analysis. While forensic science also is taught at a few other area high schools, biology teacher Emily Panico said she developed the Logan class on her own, and it is her touches that make the Logan class unique.
For one thing, Panico estimates she has given fewer than five lectures during the yearlong course. Students learned ballistics by shooting paint balls. They yanked a plastic knife out of fake blood to see the “cast off” — the blood trail that flies from the knife. They found out how to identify fellow class members from a solitary fingerprint or strand of hair.
“It’s all about getting into the crime scene and getting dirty,” senior Gerardo Paz said.
The crime scene investigation is part of the students’ final exam. But just getting to the scene was a challenge in itself.
“They’ve had to follow clues to get there. They are doing everything on their own. They have to find out who did it,
et cetera,” Panico said.Ever since the severed hand was discovered in April — while following a trail of blood — her students have been receiving clues in the form of letters signed “The Killer.”
By analyzing the DNA in hair samples plucked from the letters, they determined that both the victim and killer are female. Just last week, the students received a transparency with an “X” marked on it. When a few thought to hold the transparency over a map of the school and investigate the areas that were pointed to, they discovered a patch of loose dirt in one of those areas.
Early Friday morning, students strung up yellow tape to mark the scene’s perimeter. With one student logging who went to and from the scene, junior Brian Ross shoveled the dirt, unearthing potential evidence — a sealed box, a compact disc, a candy wrapper and empty soda cans. And several bones.
Before Noemi Hernandez carefully picked up and bagged the numbered items, Nichole Griffin “triangulated” the location of each one — marking it on a diagram so the scene could be reconstructed if necessary.
“It’s broken down like a real unit. … It’s a process they have mastered well — going in and out of a crime scene,” Panico said.
The students still need to find the killer. All Panico is saying is that it is one of the 4,600 faculty, staff members and students at Logan.
“It’s still a mystery we’re solving,” senior Eric Westcott said.
Students had only praise for the hands-on approach Panico has developed, citing one team from the class that took first place in a forensic science competition at American High
School in Fremont earlier this spring.
“We use all the equipment in the crime scenes,” said junior Veronica Schlachter.
With this being the first year the class was taught at Logan, Panico has even bigger plans for future ones.
She has applied to the University of California for her students to receive college lab credit.
And she has spoken with the Logan student who made the life-like hand.
“Next year, we might do a whole corpse,” she said.
Barry Shatzman covers Union City for The Argus. He can be reached at (510) 353-7003, or bshatzman@angnewspapers
.com.
Join the Conversation
We invite you to use our commenting platform to engage in insightful conversations about issues in our community. We reserve the right at all times to remove any information or materials that are unlawful, threatening, abusive, libelous, defamatory, obscene, vulgar, pornographic, profane, indecent or otherwise objectionable to us, and to disclose any information necessary to satisfy the law, regulation, or government request. We might permanently block any user who abuses these conditions. | https://www.eastbaytimes.com/2005/06/01/logan-students-learn-forensics-hands-on/ |
North America's Premier International Resort Community
On May 21, the Vail Police Department responded to the Northside Grab and Go, located at 2271 N. Frontage Rd. W on a report that the business had been burglarized the previous night. A black safe containing employee paychecks and business documentation was reported stolen. Investigators could not find any signs of forced entry.
According to police, the suspects used stolen W2 tax paperwork, as well as their own true identities to create fraudulent checking accounts. Two suspects were identified when they allegedly attempted to deposit the checks into the fictitious accounts using a web-based banking app. This particular app requires an in-person identity verification at the bank. Video security footage of the suspects was captured when they went into a local bank to attempt to confirm the validity of the accounts. The bank denied their applications to create the new accounts as neither suspect could produce valid identification.
Vail detectives were also able to subpoena the IP addresses used and identify the suspects. The owners of the IP addresses used for the fraudulent banking were a match to the security images captured at the bank. Felony arrest warrants were issued for both suspects.
On June 29, Vail police arrested Cheyenne Carter, 22, of Edwards on identity theft, forgery and criminal possession of a financial device. On June 30, Bryan Sohn, 26, of Edwards, was arrested and charged with second-degree burglary, identification theft, forgery and criminal possession of a financial device. Both suspects were transported to the Eagle County Detention Facility where they were held on bond.
Both suspects are presumed innocent until proven guilty in the court of law.
Further inquires can be directed to Detective Sgt. Luke Causey at 970-479-2346 or [email protected]. | https://www.vailgov.com/announcements/vail-police-arrest-burglary-and-identity-theft-suspects |
At Mrs. Miller's house, Mrs. Honour laments losing Sophia. Jones, thinking that Sophia must have died, frantically begs Honour to tell him what has happened. When Jones finally extracts the news that Western has locked up Sophia and dismissed Honour, Tom is thankful that Sophia is alive. Honour chides Jones for not having compassion for her misfortune, since she says that she has always taken his part against Blifil. Honour is scared that Western will hurt Sophia. She says she wishes Sophia had some of her courage—if her father withheld her from the man she loved, she would tear out his eyes. Partridge runs into the room to inform Jones that Lady Bellaston has arrived. Jones hides Honour behind the bed. Lady Bellaston plops herself on the bed and scolds Jones for not contacting her. Then she flirts with him. Lady Bellaston waits in surprise as Jones stands awkwardly, not knowing what to do. A very drunk Nightingale suddenly bursts into Tom's room, mistaking it for his own. Partridge manages to lead Nightingale away. While Tom was occupied with Nightingale, Lady Bellaston tried to hide herself behind the bed, coming face to face with Honour. The ladies are horrified. Lady Bellaston implies that she will bribe Mrs. Honour, after which Honour calms down. Lady Bellaston leaves, shunning Tom's attempts to hold her hand. Honour is upset about Tom's infidelity to Sophia, but Tom "at last found means to reconcile her."
Mrs. Miller gently scolds Tom for the upheaval in his room the previous night. Nightingale and Nancy are married that day, with Tom acting as father to Nancy. Before the wedding, Nightingale's uncle tries to intoxicate him and dissuade him from marrying Nancy. News arrives during this meeting that Harriet, the daughter of Nightingale's uncle, has run away with a neighboring clergyman. This destroys his case with Nightingale.
Tom receives three letters from Lady Bellaston summoning him immediately. Nightingale enters the room while Tom is reading and reveals that he knows about Tom's affair with Lady Bellaston. Tom asks for more details on the affairs of Lady Bellaston, but the narrator refuses to repeat Nightingale's words for fear of being accused of spreading scandals.
Nightingale's stories greatly reduce Tom's gratitude to Lady Bellaston and he realizes that he has been in "commerce" with this lady rather than in "love." Nightingale advises Jones that the easiest way for him to rid himself of Lady Bellaston is by proposing marriage. Together they compose a letter of proposal, to which Lady Bellaston replies that she is offended that Tom is so covetous of her fortune. Tom responds that he is insulted by her suspicion and will return her gifts to him. At the wedding dinner that night, Mrs. Miller devotes more attention to Tom than to Nightingale and Nancy.
Mrs. Miller has received a letter from Allworthy informing her that he and Blifil are coming immediately to London. He wishes to reserve the first and second floors of her house. The truth is that when Allworthy started paying Mrs. Miller an annuity of fifty pounds, it was on condition that he could occupy the first floor of her house whenever he came to town. Mrs. Miller thus has to comply with Allworthy's wishes, but she is distressed that Jones and Nightingale have to leave. Jones says that he does not mind at all. Honour sends Jones a letter saying that she is sure he will attain Sophia in the end, but she can no longer be of service to him. Lady Bellaston has hired her.
Mrs. Arabella Hunt, a friend of Mrs. Miller's, sends Tom a marriage proposal. She is twenty-six and a little plump, but otherwise attractive. She has recently been widowed by a turkey merchant who left her a rich woman. Tom is at first excited by the prospect of having so much money, but—thinking of Sophia—writes a courteous refusal. | https://www.sparknotes.com/lit/tomjones/section15/page/2/ |
---
abstract: 'We calculate the vacuum polarization functions on the lattice using the overlap fermion formulation. By matching the lattice data at large momentum scales with the perturbative expansion supplemented by Operator Product Expansion (OPE), we extract the strong coupling constant $\alpha_s(\mu)$ in two-flavor QCD as $\Lambda^{(2)}_{\overline{MS}}$ = $0.234(9)(^{+16}_{-\ 0})$ GeV, where the errors are statistical and systematic, respectively. In addition, from the analysis of the difference between the vector and axial-vector channels, we obtain some of the four-quark condensates.'
author:
- 'E. Shintani'
- 'S. Aoki'
- 'T. W. Chiu'
- 'S. Hashimoto'
- 'T. H. Hsieh'
- 'T. Kaneko'
- 'H. Matsufuru'
- 'J. Noaki'
- 'T. Onogi'
- 'N. Yamada'
title: Lattice study of vacuum polarization function and determination of strong coupling constant
---
Introduction
============
In Quantum Chromodynamics (QCD) the vacuum polarization, defined through the (axial-)vector current correlator, contains rich information of its perturbative and non-perturbative dynamics. In the long distance regime it is sensitive to the low-lying particle spectrum. The short distance regime, on the other hand, can be analyzed using perturbation theory supplemented by the Operator Product Expansion (OPE). The current correlator can be expressed as an expansion in terms of the strong coupling constant $\alpha_s$ together with power corrections of the form $\langle\mathcal{O}^{(n)}\rangle/Q^n$. Here, the local operator $\mathcal{O}^{(n)}$ has a mass dimension $n$ and $Q$ is the momentum scale flowing into the correlator. Determination of $\alpha_s$ (and of the vacuum expectation values $\langle\mathcal{O}^{(n)}\rangle$, in principle) can be performed by applying the formulae for experimental results of $e^+e^-$ cross section or $\tau$ decay distributions [@Davier:2005xq], for instance. On the other hand, if one can [*calculate*]{} the correlators non-perturbatively, theoretical determination of those fundamental parameters is made possible.
Lattice QCD calculation offers such a non-perturbative technique. Two-point correlators can be calculated for space-like separations. In this work we investigate the use of the perturbative formulae of the correlators for the lattice data obtained in the high $Q^2$ regime. The strong coupling constant $\alpha_s$ may then be extracted. In such an analysis, it is essential to find the region of $Q^2$ where the perturbative expression can be applied and at the same time the discretization error is under control. By inspecting the numerical data, we find that this is indeed possible at a lattice spacing $a\simeq$ 0.12 fm if we subtract the bulk of the discretization effects non-perturbatively. The remaining effect can be estimated using the perturbation theory.
The idea of analyzing the short distance regime is not new: in fact, the analysis of hadron correlators in the whole length-scales was proposed 15 year ago [@Shuryak:1993kg], but to our knowledge quantitative analysis including the determination of $\alpha_s$ and $\langle\mathcal{O}^{(n)}\rangle$ has been missing until recently. (Calculation of the vacuum polarization from the vector current correlator in lattice QCD may be found in [@Blum:2002ii; @Gockeler:2003cw]. More recently, an analysis of charmonium correlator has been published [@Allison:2008xk].)
While the vacuum polarizations $\Pi_J(Q^2)$ ($J$ denotes vector or axial-vector channel) are ultraviolet divergent and their precise value depends on the renormalization scheme, their derivative $D_J(Q^2)=-Q^2d\Pi_J(Q^2)/dQ^2$, called the Adler function [@Adler:1974gd], is finite and renormalization scheme independent. Therefore, the continuum perturbative expansion of $D_J(Q^2)$ to order $\alpha_s^3$ [@Surguladze:1990tg; @Gorishnii:1990vf], can be directly applied to the lattice data. At relatively low $Q^2$ region, higher order terms of OPE become relevant. They include the parameters describing the gluon condensate $\langle\alpha_s G^2\rangle$ and the quark condensate $\langle m\bar{q}q\rangle$ (we suppress quark flavor index assuming degenerate up and down quark masses) at $O(1/Q^4)$, and four-quark condensates $\langle O_8\rangle$ and $\langle O_1\rangle$ at $O(1/Q^6)$ [@Donoghue:1999ku; @Cirigliano:2003kc]. (The explicit form of $O_8$ and $O_1$ will be given in Section \[sec:V-A\].)
We use the lattice QCD data containing two dynamical flavors described by the overlap fermions [@Aoki:2008tq]. The simulations are performed at lattice spacing $a$ = 0.118(2) fm on a $16^3\times 32$ lattice. For the details of the simulation including the choice of the lattice actions and parameters, we refer [@Aoki:2008tq]. The physical volume is about (1.9 fm)$^3$, which is relatively small compared to the present large scale QCD simulations. The finite volume effect is, however, not significant for the short distance quantities considered in this work. The quark masses $m_q$ in this analysis are 0.015, 0.025, 0.035 and 0.050 in the lattice unit, that cover the range $[m_s/6,m_s/2]$ with $m_s$ the physical strange quark mass. An analysis of pion mass and decay constant is presented in [@Noaki:2008iy].
The main advantage of this data set is that both the sea and valence quarks preserve exact chiral and flavor symmetries by the use of the overlap fermion formulation [@Neuberger:1997fp; @Neuberger:1998wv]. (Although the fermionic currents used in our calculation are not conserved at finite lattice spacings, it does not change the following argument of the operator mixing.) The perturbative formulae for the vacuum polarizations can therefore be applied without any modification due to explicit violation of the chiral symmetry. For instance, the scalar density operator $\bar{q}q$ to define the quark condensate is free from the leading power divergence which scales as $1/a^3$. This means that a term of the form $m a^{-3}/Q^4$ is forbidden in the OPE formula as in the continuum theory. With the Wilson-type fermion formulation, this term may appear and has to be identified and subtracted non-perturbatively. With the staggered fermion formulation, there is no such problem because of its remnant chiral symmetry, while the effect of taste-breaking may become significant when $(aQ)^2$ becomes $O(1)$.
This paper is organized as follows. In Section \[sec:vacuum\_polarization\] we define the vacuum polarization functions and explain the method to calculate them on the lattice. Subtraction of lattice artifacts is discussed in some detail. Section \[sec:OPE\] summarizes the perturbative formulae of OPE. Then, in Section \[sec:results\] we show the results of fitting of our data with the perturbative formulae. Estimate of the systematic errors is also given. Conclusions are given in Section \[sec:conclusion\].
Vacuum polarization function {#sec:vacuum_polarization}
============================
Definition
----------
In the continuum theory, the vacuum polarization functions $\Pi_J^{(\ell)}(Q^2)$ are defined through two-point correlation functions as $$\begin{aligned}
\label{eq:Jmunu}
\lefteqn{\langle J_\mu J_\nu\rangle(Q)
\equiv \int d^4 x e^{iQ\cdot x}\langle
T\{J_\mu^{ij}(x)J_\nu^{ji}(0)\}\rangle}
\nonumber\\
&=&
(\delta_{\mu\nu}Q^2-Q_\mu Q_\nu)\Pi_J^{(1)}(Q^2)
-Q_\mu Q_\nu \Pi_J^{(0)}(Q^2),
\label{eq:VVcont}\end{aligned}$$ where the current $J_\mu^{ij}$ may either be a vector current $V_\mu^{ij} = \bar q_i \gamma_\mu q_j$ or an axial-vector current $A_\mu^{ij} = \bar q_i\gamma_\mu\gamma_5 q_j$ with flavor indices $i\ne j$. $\Pi_J^{(1)}(Q^2)$ and $\Pi_J^{(0)}(Q^2)$ denote the transverse and longitudinal parts of the vacuum polarization, respectively. For the vector channel ($J=V$), $\Pi_V^{(0)}(Q^2)=0$ is satisfied due to current conservation. For the axial-vector channel ($J=A$), the longitudinal component may appear when the quark mass is finite.
In the lattice calculation we employ the overlap fermion formulation [@Neuberger:1997fp; @Neuberger:1998wv], for which the Dirac operator is given by $$D(m) = \left(m_0+\frac{m}{2}\right)
+ \left(m_0-\frac{m}{2}\right)\gamma_5\mathrm{sgn}\left[H_W(-m_0)\right]$$ for a bare quark mass $m$. The kernel operator $H_W(-m_0)\equiv\gamma_5D_W(-m_0)$ is constructed from the conventional Wilson-Dirac operator $D_W(-m_0)$ at a large negative mass $-m_0$. We set $m_0=1.6$ in the numerical simulation. We use the vector and axial-current operators of the form $$\begin{aligned}
\label{eq:vector}
V_\mu^{ij} &=& Z\bar q_i \gamma_\mu\left(1-\frac{D}{2m_0}\right) q_j,
\\
\label{eq:axial}
A_\mu^{ij} &=& Z\bar q_i \gamma_\mu\gamma_5
\left(1-\frac{D}{2m_0}\right) q_j.\end{aligned}$$ With this choice, the vector and axial charges form a multiplet under the axial transformation $\delta_A^aq_i = \varepsilon \tau^a_{ij} \gamma_5(1-D/m_0)q_j$, $\delta_A^a\bar q_i = \varepsilon\bar q_j \tau^a_{ji} \gamma_5$, where $\varepsilon$ denotes an infinitesimal parameter and $\tau^a$ is a generator of the flavor $SU(2)$ symmetry. The overlap fermion action is invariant under this modified chiral transformation [@Luscher:1998pqa], as it satisfies the Ginsparg-Wilson relation $D\gamma_5 + \gamma_5 D = D\gamma_5 D/m_0$ [@Ginsparg:1981bj]. The common renormalization factor $Z$ has been calculated non-perturbatively as $Z$ = 1.3842(3) [@Noaki:2008iy].
An obvious drawback of the (axial-)vector currents in (\[eq:vector\]) and (\[eq:axial\]) is that the current conservation property $\partial_\mu J_\mu=0$ ($J=V$ or $A$) is not satisfied at finite lattice spacing. It leads to a significant complication in the extraction of the functions $\Pi_J^{(0)}(Q^2)$ and $\Pi_J^{(1)}(Q^2)$, as described in the next subsection. The use of the conserved (axial-)vector current [@Kikukawa:1998py] reduces this complication. Once we have extracted the functions $\Pi_J^{(0)}(Q^2)$ and $\Pi_J^{(1)}(Q^2)$, these two types of currents should give an equally good approximation to the continuum one up to the unphysical constant shift (and the discretization error). Our preliminary study employing the conserved currents shows that this is indeed the case.
Non-perturbative subtraction of lattice artifact
------------------------------------------------
Due to the discretization effects including the current non-conservation effect, the two-point correlation functions (\[eq:Jmunu\]) may have more complicated structures. Taking account of remaining symmetries on the lattice (parity and cubic symmetries) but without the current conservation, the correlators on the lattice $\langle J_\mu J_\nu\rangle^{\mathrm{lat}}(Q)$ can be expressed as an expansion in $Q_\mu$: $$\begin{aligned}
\lefteqn{
\langle J_\mu J_\nu\rangle^{\rm lat}(Q)
= \Pi_J^{(1)}(Q) {Q}^2\delta_{\mu\nu}
- \Pi_J^{(0+1)}(Q) {Q}_\mu{Q}_\nu
}\nonumber\\
&-& \sum_{n=0}^\infty B_n^J(Q) Q_\mu^{2n}\delta_{\mu\nu}
- \sum_{m,n=1}^\infty C_{mn}^J(Q)\big\{Q_\mu^{2m+1}Q_\nu^{2n-1}
+ Q_\nu^{2m+1} Q_\mu^{2n-1}\big\},
\label{eq:JJlat}\end{aligned}$$ in the momentum space. The lattice momentum ${Q}_\mu$ is defined as $Q_\mu = (2/a)\sin(\pi n_\mu/L_\mu)$ with an integer four-vector $n_\mu$ whose components take values in $(-L_\mu/2,L_\mu/2]$ on a lattice of size $L_\mu$ in the $\mu$-th direction ($L_{i=1,2,3}=16$ and $L_t=32$ in our case). The functions corresponding to the continuum counterparts, $\Pi_J^{(1)}(Q)$ and $\Pi_J^{(0+1)}(Q)$ ($\equiv\Pi_J^{(0)}(Q)+\Pi_J^{(1)}(Q)$), may also have Lorentz-violating effects and could be a function of $Q_\mu$ in general rather than a function of just a single argument $Q^2$.
The term $B_{0}^J(Q)\delta_{\mu\nu}$, which has the same Lorentz structure as the term of physical $\Pi_J^{(1)}(Q)$ does, contains a quadratically divergent contact term. Since one cannot disentangle the physical contribution from the unphysical divergence using the Lorentz structure alone, we focus on extracting $\Pi_J^{(0+1)}(Q)$, which is free from the contact term.
The terms including functions $B_{n>0}^J(Q)$ and $C_{mn}^J(Q)$ represent the lattice artifacts that violate the Lorentz symmetry. They are generally written in terms of an expansion in $aQ_\mu$ and $aQ_\nu$. (Physically relevant terms are separately written with a conventional notation $\Pi_J^{(1)}(Q)$ and $\Pi_J^{(0+1)}(Q)$.) The lowest order term $B_1^J(Q)$ remains constant in the continuum limit $aQ\to 0$, while the terms of $B_2^J(Q)$ and $C_{11}^J(Q)$ are relatively suppressed by $O((aQ)^2)$ and vanish in the continuum limit. Higher order terms are suppressed by additional powers of $a$ at a fixed $Q$. Since the momentum scale $Q$ of interest is not much less than the lattice cutoff $1/a$, the convergence of the expansion at our lattice spacing must be carefully investigated for the lattice data. These terms can be identified non-perturbatively, and we found that the lowest non-trivial terms including $B_2^J(Q)$ and $C_{11}^J(Q)$ are already very small as described below. Higher order terms are thus safely neglected.
Extraction of $B_{1,2}^J(Q)$ and $C_{11}^J(Q)$ from the lattice data goes as follows. The off-diagonal components $\langle J_\mu J_\nu\rangle^{\mathrm{lat}}(Q)$ ($\mu\ne\nu$) contain $\Pi_J^{(0+1)}(Q)$ and $C_{11}^J(Q)$, hence by taking the data with two different momentum configurations giving the same ${Q}^2$ one can solve a linear equation to disentangle $\Pi_J^{(0+1)}(Q)$ from the lattice artifact. To be explicit, for two different momentum configurations $aQ^{(1)}$ and $aQ^{(2)}$ giving the same $(aQ^{(1)})^2=(aQ^{(2)})^2=(aQ)^2$, the linear equation is written as $$\begin{aligned}
\langle J_\mu J_\nu\rangle^{\rm lat}|_{\mu\ne\nu}(Q^{(1)})
&=& aQ_\mu^{(1)} aQ_\nu^{(1)}\Pi_J^{(0+1)}(Q^{(1)})
- \left(aQ^{(1)}_\mu(aQ^{(1)}_\nu)^3 +
aQ^{(1)}_\nu(aQ^{(1)}_\mu)^3\right)
C_{11}^J(Q^{(1)}),
\nonumber\\
\langle J_\mu J_\nu\rangle^{\rm lat}|_{\mu\ne\nu}(Q^{(2)})
&=& aQ_\mu^{(2)} aQ_\nu^{(2)}\Pi_J^{(0+1)}(Q^{(2)})
- \left(aQ^{(2)}_\mu(aQ^{(2)}_\nu)^3 + aQ^{(2)}_\nu(aQ^{(2)}_\mu)^3
\right)C_{11}^J(Q^{(2)}).
\nonumber\\
\label{eq:JJ_linear}\end{aligned}$$ We may assume the equalities $\Pi_J^{(0+1)}(Q^{(1)})=\Pi_J^{(0+1)}(Q^{(2)})$ and $C_{11}^J(Q^{(1)})=C_{11}^J(Q^{(2)})$ for small enough $(aQ)^2$, because $aQ^{(1)}$ and $aQ^{(2)}$ are different only by permutations of space-time directions. The linear equation (\[eq:JJ\_linear\]) can be solved when $$\label{eq:cond}
Q_\mu^{(1)}Q_\nu^{(1)}Q_\mu^{(2)}Q_\nu^{(2)}
\left[
(Q_\mu^{(2)})^2 + (Q_\nu^{(2)})^2 - (Q_\mu^{(1)})^2 -(Q_\nu^{(1)})^2
\right] \ne 0.$$ It is easy to see that three different non-zero components must be contained in $aQ^{(1)}$ and $aQ^{(2)}$ to satisfy (\[eq:cond\]). The smallest possible momentum assignment corresponds to the combination $|n_\mu^{(1)}| = (2,1,0,1)$, $|n_\mu^{(2)}| = (1,2,0,1)$ with $(\mu,\nu)=(1,4)$ and its permutations. Since the fourth (temporal) direction is longer for our lattice ($L_{i=1,2,3}=16$ while $L_4=32$), the fourth component of $Q_\mu$ is effectively 1/2 of spatial components when they are the same in $n_\mu$. The corresponding momentum squared for this choice is $(aQ)^2\simeq$ 0.776. For larger lattice momenta, there are many possible choices that this procedure is applied.
The lattice artifact in the diagonal pieces, $B_1^J(Q)$ and $B_2^J(Q)$, can be extracted in a similar manner by solving linear equations for $\mu=\nu$ after subtracting the $C_{11}^J(Q)$ terms. For instance, the leading contribution $(\Pi_J^{(1)}(Q)Q^2-B_0^J(Q))\delta_{\mu\nu}$ is extracted by subtracting the sub-leading contribution $B_1^J(Q)Q_\mu^2\delta_{\mu\nu}$, which can be identified from a difference between $\langle J_1 J_1\rangle^{\mathrm{lat}}(Q)$ and $\langle J_2 J_2\rangle^{\mathrm{lat}}(Q)$ at the same $Q^2$, for instance.
![ Momentum dependence of $B^J_1(Q)$, $B^J_2(Q)/a^2$, and $C^J_{11}(Q)/a^2$ at $m_q=0.015$. Circles (crosses) show the vector (axial-vector) channel. The solid curves represent a polynomial fit and the dashed curves show the one-loop results.[]{data-label="fig:BandC"}](vioABC_qm0.015.eps){width="100mm"}
Figure \[fig:BandC\] shows the numerical results for $B_1^J(Q)$, $B_2^J(Q)/a^2$ and $C_{11}^J(Q)/a^2$ at the smallest quark mass ($m_q=0.015$) as a function of $(aQ)^2$ for both vector and axial channels. In the momentum region $(aQ)^2<$ 2.3 only the $B_1^J(Q)$ term gives sizable contribution, while the others are an order of magnitude smaller even without the suppression due to $(aQ)^2$. Their dependence on $(aQ)^2$ is rather mild, so that it seems reasonable to fit these functions as a polynomial of $(aQ)^2$. We use a third-order polynomial to model these functions. This is used to subtract the artifacts at the momentum points for which the above procedure is not applicable, [*e.g.*]{} below the lowest $(aQ)^2\simeq$ 0.776.
We notice that the difference between $J=V$ and $J=A$ is consistent with zero within statistical errors. This indicates that these lattice artifacts are strongly constrained by the exact chiral symmetry of the overlap fermion, and the effect of the finite quark mass is negligible. It also suggests that such short distance quantities are insensitive to the spontaneous chiral symmetry breaking, as it should be. This property is essential in the calculation of the difference $\Pi^{(\ell)}_V(Q)-\Pi^{(\ell)}_A(Q)$, which is related to the electromagnetic mass difference of pions [@Shintani:2007ub; @Shintani:2008qe].
Perturbative calculation of the lattice artifacts {#sec:pert}
-------------------------------------------------
Since the lattice artifacts are most significant in the high $(aQ)^2$ region, perturbative analysis of the discretization effects is expected to give a reasonable estimate. We calculate the vacuum polarization functions in the lattice perturbation theory at one-loop level, which means that only the zeroth order of $\alpha_s$ is included. We then extract the terms corresponding to $\Pi_J^{(0+1)}(Q)$, $B_{1,2}^J(Q)$ and $C_{11}^J(Q)$.
We calculate the vacuum polarization diagram in which two (axial-)vector currents (\[eq:vector\]) and (\[eq:axial\]) are inserted. The renormalization factor $Z$ is set equal to 1 at this order. In the momentum space, the two-point function is written as $$\begin{aligned}
\langle V_\mu V_\nu\rangle^{\rm lat}(Q)
&=& \int^\pi_{-\pi} \frac{d^4 K}{(2\pi)^4}
\mathrm{Tr}
\left[
\left(1-\frac{1}{2m_0}D_0(K)\right) S_0(K) \gamma_\mu
\right.
\nonumber\\
&&
\left.
\times \left(1-\frac{1}{2m_0}D_0(K-Q)\right) S_0(K-Q) \gamma_\nu
\right],
\label{eq:LatPT}\end{aligned}$$ where the fermion propagator $S_0(K)$ is given by $$S_0(K) =
\frac{1}{2m_0}\left[
\frac{-i\sum_\mu\gamma_\mu\sin(K_\mu)}{\omega(K)+b(K)}+1
\right]$$ with $$\begin{aligned}
\omega(K) & = & \sqrt{\sum_\mu \sin^2(K_\mu) + b(K)^2}, \\
b(K) & = & \sum_\mu\left(1-\cos(K_\mu)\right)-m_0\end{aligned}$$ for the overlap fermion and $D_0(K)^{-1}=S_0(K)$. We set $a=1$ in this subsection. In the perturbative calculation, $m_0$ may be set equal to 1. At the perturbative level, the vector and axial-vector current correlators are equivalent in the massless limit, because of the exact chiral symmetry of the overlap fermion.
![ Momentum dependence of $B^J_1(Q)$, $C^J_{11}(Q)/a^2$ and $B^J_2(Q)/a^2$ calculated in perturbation theory. []{data-label="fig:BandC_PT"}](ABC_latPT.eps){width="100mm"}
After performing the numerical integral in (\[eq:LatPT\]) we extract $B_{1,2}(Q)$, $C_{11}(Q)$ and $\Pi_V(Q)$ in (\[eq:JJlat\]) through the same numerical procedure as we used in the non-perturbative extraction. To be explicit, we take representative values of $(aQ)^2$ between 0.4 and 2.3 and consider two different momentum configurations $aQ^{(1)}$ and $aQ^{(2)}$. The results for $B_1(Q)$, $B_2(Q)/a^2$, and $C_{11}(Q)/a^2$ are shown in Figure \[fig:BandC\_PT\]. As we found in the non-perturbative calculation, the $(aQ)^2$ dependence is rather mild and we may precisely model these functions by quadratic functions: $B_1^{PT}(Q)$ = $0.06930(59) - 0.00332(85) (aQ)^2 + 0.00009(27) (aQ)^4$, $B_2^{PT}(Q)$ = $0.0025(22) + 0.0023(30) (aQ)^2 - 0.0009(9) (aQ)^4$, and $C_{11}^{PT}(Q)$ = $-0.00507(14) + 0.00227(20) (aQ)^2 - 0.00046(6) (aQ)^4$. The fit curves are shown in Figure \[fig:BandC\_PT\].
The same curves are also plotted in Figure \[fig:BandC\] by dashed lines. These perturbative results show reasonable agreement with the lattice data. It indicates that the lattice artifacts are indeed well described by the perturbation theory.
Results for the vacuum polarization functions
---------------------------------------------
![ $\Pi_V^{(0+1)}(Q)$ from off-diagonal $\mu\ne\nu$ and diagonal $\mu=\nu$ correlators with (lower panel) and without (upper panel) the subtraction of $B^J_{1,2}(Q)$ and $C^J_{11}(Q)$. []{data-label="fig:PiV"}](PiV_munu.qm0.015.eps){width="100mm"}
Lattice results for the vacuum polarization function $\Pi_J^{(0+1)}(Q)$ for $J=V$ are shown in Figure \[fig:PiV\]. The vacuum polarization function can be extracted from off-diagonal $\mu\ne\nu$ (triangles) and from diagonal $\mu=\nu$ (circles) components. Upper and lower panels show the data before and after the subtraction of $B_n^J(Q)$ and $C_{mn}(Q)^J(Q)$ terms. Namely, for the upper panel, $\Pi_J^{(0+1)}(Q)$ is identified with the formula (\[eq:JJlat\]) but without the $B_n^J(Q)$ and $C_{mn}^J(Q)$ terms. As discussed above, raw lattice data of the diagonal components receive large contamination from $B_1^J(Q)$ while the artifact for the off-diagonal components is much smaller (below 0.01).
After the non-perturbative subtraction of $B_{1,2}^J(Q)$ and $C_{11}^J(Q)$, we observe that the off-diagonal and diagonal components give consistent results. It strongly indicates that the higher order lattice artifacts are unimportant. We average the diagonal and off-diagonal data in the following analysis.
Operator product expansion {#sec:OPE}
==========================
$V$ and $A$ channels
--------------------
We now discuss the fit of the lattice data to the OPE expression of the form [@Shifman:1978bx] $$\begin{aligned}
\left.\Pi_J^{(0+1)}\right|_{\rm OPE}(Q^2)
&=& c + C_0(Q^2,\mu^2) + \frac{m^2}{Q^2}C_m^J(Q^2,\mu^2)
+ C^J_{\bar qq}(Q^2)\frac{\langle m\bar q q\rangle}{Q^4}
\nonumber\\
& & + C_{GG}(Q^2)\frac{\langle(\alpha_s/\pi) GG\rangle}{Q^4}.
\label{eq:pi_J_OPE}\end{aligned}$$ Instead of directly treating the Adler function, we analyze its indefinite integral $\left.\Pi_J^{(0+1)}\right|_{\rm OPE}(Q^2)$. The coefficient functions $C_0(Q^2,\mu^2)$, $C_m^J(Q^2,\mu^2)$, $C_{\bar qq}^J(Q^2)$ and $C_{GG}(Q^2)$ are analytically calculated in perturbation theory. The terms of order $1/Q^6$ and higher are not included.
A constant $c$ is divergent and thus scheme-dependent, while other terms are finite and well-defined. Although we need to specify the renormalization scheme, the scheme dependence should disappear as the higher order terms are included. The following formulae are consistently given in the $\overline{\mathrm{MS}}$ scheme, so that the strong coupling constant $\alpha_s(\mu)$ is defined in this conventional scheme.
The leading term $C_0(Q^2,\mu^2)$ is known to $\mathcal{O}(\alpha_s^2)$ in the massless limit [@Surguladze:1990tg; @Gorishnii:1990vf] as $$\begin{aligned}
C_0(Q^2,\mu^2) &=&
\frac{1}{16\pi^2}
\left\{
\frac{20}{3} + 4\ln\frac{\mu^2}{Q^2}
+ \frac{\alpha_s(\mu^2)}{\pi}
\left[ \frac{55}{3}-16\zeta(3) + 4\ln\frac{\mu^2}{Q^2}
\right]
\right.
\nonumber\\
& & + \left(\frac{\alpha_s(\mu^2)}{\pi}\right)^2
\left.\left[ \frac{41927}{216} - \frac{3701}{324}N_f
- \left(\frac{1658}{9}-\frac{76}{9}N_f\right)\zeta(3)
+ \frac{100}{3}\zeta(5)
\right.\right.
\nonumber\\
& &
\left.\left.
+ \left\{ \frac{365}{6}-\frac{11}{3}N_f -
\left(44-\frac{8}{3}N_f\right)\zeta(3)
+\left(\frac{11}{2}-\frac{1}{3}N_f\right)\ln\frac{\mu^2}{Q^2}
\right\} \ln\frac{\mu^2}{Q^2}
\right] \right\}, \label{eq:c0}
\nonumber\\\end{aligned}$$ where $N_f$ denotes the number of flavors, and the zeta function is numerically given as $\zeta(3)=1.20205\cdots$, $\zeta(5)=1.03692\cdots$. For a finite quark mass there is a contribution of $\mathcal{O}(m^2/Q^2)$ with running mass $m=m(\mu)$. This term is represented by $C_m^J(Q^2,\mu^2)$, which is also calculated to $\mathcal{O}(\alpha_s^2)$ as $$\begin{aligned}
C_m^V(Q^2,\mu^2) &=& \frac{1}{4\pi^2} \bigg[
-6 + \frac{\alpha_s(\mu^2)}{\pi}\Big(-16-12\ln\frac{\mu^2}{Q^2}\Big) \nonumber\\
&+& \Big(\frac{\alpha_s(\mu^2)}{\pi}\Big)^2 \Big\{
-\frac{19691}{72} + \frac{95}{12}N_f - \frac{124}{9}\zeta(3) + \frac{1045}{9}\zeta(5)\nonumber\\
&-& \Big(55 + 12\ln\frac{\mu^2}{Q^2}\Big)\ln\frac{\mu^2}{Q^2}
- \Big(11-\frac{2}{3}N_f\Big)\Big(\frac{13}{2}+\frac{3}{2}\ln\frac{\mu^2}{Q^2}\Big)
\ln\frac{\mu^2}{Q^2}\Big\} \bigg]\nonumber\\
&+& \frac{N_f}{16\pi^2}\Big(\frac{\alpha_s(\mu^2)}{\pi}\Big)^2
\Big[\frac{128}{3}-32\zeta(3)\Big],
\\
C_m^A(Q^2,\mu^2) &=& \frac{1}{4\pi^2} \bigg[
-6+\frac{\alpha_s(\mu^2)}{\pi}\Big(-12-12\ln\frac{\mu^2}{Q^2}\Big) \nonumber\\
&+& \Big(\frac{\alpha_s(\mu^2)}{\pi}\Big)^2 \Big\{
-\frac{4681}{24} + \frac{55}{12}N_f - \Big(34-\frac{8}{3}N_f\Big)\zeta(3) + 115\zeta(5)
\nonumber\\
&-& \Big( 47+12\ln\frac{\mu^2}{Q^2}\Big)\ln\frac{\mu^2}{Q^2}-\Big(11-\frac{2}{3}N_f\Big)
\Big(\frac{11}{2}+\frac{3}{2}\ln\frac{\mu^2}{Q^2}\Big)\ln\frac{\mu^2}{Q^2}\Big\} \bigg]
\nonumber\\
&+& \frac{N_f}{16\pi^2}\Big(\frac{\alpha_s(\mu^2)}{\pi}\Big)^2\Big[\frac{128}{3}-32\zeta(3)\Big].\end{aligned}$$ We ignore terms of $\mathcal{O}(m^4)$ and higher.
The OPE corrections of the form $\langle O^{(n)}\rangle/Q^n$ start from the dimension-four operators $m\bar{q}q$ and $(\alpha_s/\pi)GG$. Their Wilson coefficients $C^J_{\bar{q}q}(Q^2)$ and $C_{GG}(Q^2)$ are known to $\mathcal{O}(\alpha_s^2)$ and to $\mathcal{O}(\alpha_s)$, respectively, as [@Chetyrkin:1985kn] $$\begin{aligned}
C_{\bar qq}^{V/A}(Q^2) &=& -2\frac{\alpha_s(\mu^2)}{\pi}\Big[ 1+\frac{1}{24}
\frac{\alpha_s(\mu^2)}{\pi}\Big\{(116-4N_f)+(66-4N_f)\ln\frac{\mu^2}{Q^2}\Big\}\Big]\nonumber\\
&+/-& 2\Big[ 1+\frac{4}{3}\frac{\alpha_s(\mu^2)}{\pi}
+ \frac{4}{3}\Big(\frac{\alpha_s(\mu^2)}{\pi}\Big)^2 \Big\{
\Big(\frac{191}{24}-\frac{7}{36}N_f\Big)
+\Big(\frac{11}{4}-\frac{1}{6}N_f\Big)\ln\frac{\mu^2}{Q^2}\Big\}\Big]
\nonumber\\
&+& \frac{N_f}{3}\Big(\frac{\alpha_s(\mu^2)}{\pi}\Big)^2\Big(4\zeta(3)-3+\ln\frac{\mu^2}{Q^2}\Big) + 0/4,\\
C_{GG}(Q^2) &=& \frac{1}{12}\Big[ 1 - \frac{11}{18}\frac{\alpha_s(Q)}{\pi}\Big].\end{aligned}$$ Here we note that the “gluon condensate” $\langle(\alpha_s/\pi)GG\rangle$ is defined only through the perturbative expression like (\[eq:pi\_J\_OPE\]). Due to an operator mixing with the identity operator, the operator $(\alpha_s/\pi)GG$ contains a quartic power divergence that cannot be unambiguously subtracted within perturbation theory, which is known as the renormalon ambiguity [@Martinelli:1996pk]. Therefore, the term $\langle(\alpha_s/\pi)GG\rangle$ in (\[eq:pi\_J\_OPE\]) only has a meaning of a parameter in OPE, that may depend on the order of the perturbative expansion, for instance.
The quark condensate $\langle \bar{q}q\rangle$ is, on the other hand, well-defined in the massless limit, since it does not mix with lower dimensional operators, provided that the chiral symmetry is preserved on the lattice. Power divergence may appear at finite quark mass as $ma^{-2}$. In the OPE formula (\[eq:pi\_J\_OPE\]), it thus leads to a functional dependence $m^2a^{-2}/Q^4$. Since the quark mass in the lattice unit is small (0.015–0.050) and $(aQ)^2$ is of $O(1)$ in our lattice setup, this divergent contribution is tiny ($\sim$ 0.1–0.2%). In fact, we do not find any significant $m^2$ dependence in the lattice data. We therefore neglect this $m^2$ dependence in the numerical analysis.
$V-A$ channel {#sec:V-A}
-------------
In addition to the individual vector and axial-vector correlators, we consider the $V-A$ vacuum polarization function. For the difference $\Pi^{(0+1)}_{V-A}(Q)\equiv\Pi^{(0+1)}_V(Q)-\Pi^{(0+1)}_A(Q)$, the lattice data are more precise than the individual $\Pi^{(0+1)}_J(Q)$, so that the $1/Q^6$ and $1/Q^8$ terms are also necessary: $$\begin{aligned}
\left.\Pi_{V-A}^{(0+1)}\right|_{\rm OPE}(Q^2)
&=& (C_m^V-C_m^A)(Q^2)\frac{1}{Q^2}
+ \left(C^V_{\bar qq}-C^A_{\bar qq}\right)(Q^2)
\frac{\langle m\bar q q\rangle}{Q^4}\nonumber\\
& + & \left(a_6(\mu) + b_6(\mu)\ln\frac{Q^2}{\mu^2} + c_6 m_q\right)\frac{1}{Q^6}
+ \frac{a_8}{Q^8}.
\label{eq:pi_V-A_OPE}\end{aligned}$$ In the $V-A$ combination the coefficients $C_m^V-C_m^A$ and $C^V_{\bar qq}-C^A_{\bar qq}$ start at $\mathcal O(\alpha_s)$. The coefficients $a_6(\mu)$ and $b_6(\mu)$ contain dimension six operators $O_8$ and $O_1$ as [@Donoghue:1999ku; @Cirigliano:2003kc] $$\begin{aligned}
a_6(\mu) &=& 2\pi\langle\alpha_sO_8\rangle(\mu)
+ \frac 25 4\langle\alpha_s^2 O_8\rangle(\mu)
+ 2\langle \alpha_s^2 O_1\rangle(\mu),\\
b_6(\mu) &=& -\langle\alpha_s^2 O_8\rangle(\mu)
+ \frac 8 3\langle \alpha_s^2 O_1\rangle(\mu),\end{aligned}$$ and the definition of these operators is given by $$\begin{aligned}
\langle O_8\rangle &=& \sum_{\mu,i,j} \left\langle
(\bar q_i\gamma_\mu\tau^3_{ij} q_j)(\bar q_i\gamma_\mu\tau^3_{ij} q_j)
-(\bar q_i\gamma_\mu\gamma_5\tau^3_{ij} q_j)
(\bar q_i\gamma_\mu\gamma_5\tau^3_{ij} q_j) \right\rangle,\\
\langle O_1\rangle &=& \sum_{\mu,a,i,j} \left\langle
(\bar q_i\gamma_\mu\lambda^a\tau^3_{ij} q_j)
(\bar q_i\gamma_\mu\lambda^a\tau^3_{ij} q_j)
-(\bar q_i\gamma_\mu\gamma_5\lambda^a\tau^3_{ij} q_j)
(\bar q_i\gamma_\mu\gamma_5\lambda^a\tau^3_{ij} q_j) \right\rangle,\end{aligned}$$ with generator matrices $\tau^3$ and $\lambda^a$ of flavor SU(2) and color SU(3) symmetries, respectively. The numerical coefficients in the definition of $a_6$ and $b_6$ correspond to those of the Naive Dimensional Regularization (NDR) of $\gamma_5$.
Unlike the dimension-four quark condensate $\langle m\bar{q}q\rangle$, $\langle O_8\rangle$ and $\langle O_1\rangle$ remain finite in the massless limit, hence gives leading contribution. The term $c_6$, which has a mass-dimension five, describes their dependence on the quark mass. The term $a_8/Q^8$ represents the contributions from dimension-eight operators.
Fitting results {#sec:results}
===============
Fit parameters
--------------
In the fitting of the lattice data with the functions (\[eq:pi\_J\_OPE\]) and (\[eq:pi\_V-A\_OPE\]), we fix the scale $\mu$ to 2 GeV. We use the value of the quark condensate obtained from a simulation in the $\epsilon$-regime using the same lattice formulation at slightly smaller lattice spacing, $\langle\bar{q}q\rangle(\mathrm{2~GeV})$ = $-$\[0.251(7)(11) GeV\]$^3$ [@Fukaya:2007fb]. (The values quoted in [@Fukaya:2007yv; @Fukaya:2007pn; @Noaki:2008iy] are slightly different from but consistent with this number. The precise value does not affect the fit much, since the contribution of the $C_{\bar qq}$ term is sub-dominant.) The quark mass is renormalized in the $\overline{\mathrm{MS}}$ scheme using the non-perturbative matching factor $Z_m(\mathrm{2~GeV})$ = 0.838(17) [@Noaki:2008iy] as $m(\mu)=Z_m(\mu)m_q$. The coupling constant $\alpha_s(\mu)$ is transformed to the scale of two-flavor QCD, $\Lambda^{(2)}_{\overline{\mathrm{MS}}}$, using the four-loop formula [@van; @Ritbergen:1997va] $$\begin{aligned}
\frac{\alpha_s(\mu^2)}{\pi}
&=& \frac{1}{\beta_0 L}
\left[
1-\frac{\beta_1}{\beta_0^2}\frac{\ln L}{L}
+ \frac{1}{\beta_0^2L^2}
\left\{\frac{\beta_1^2}{\beta_0^2}
\left(\ln^2L-\ln L-1\right)+\frac{\beta_2}{\beta_0}
\right\}
\right.
\nonumber\\
& &
\left. +
\frac{1}{\beta_0^3L^3}\left\{\frac{\beta_1^3}{\beta_0^3}
\left(-\ln^3L+\frac{5}{2}\ln^2L+2\ln L-\frac{1}{2}\right)
-3\frac{\beta_1\beta_2}{\beta_0^2}\ln L+\frac{\beta_3}{2\beta_0}
\right\}
\right]\end{aligned}$$ with $$\begin{aligned}
\beta_0 &=& \frac{1}{4}\left(11-\frac{2}{3}N_f\right),\\
\beta_1 &=& \frac{1}{4^2}\left(102-\frac{38}{3}N_f\right),\\
\beta_2 &=& \frac{1}{4^3}\left(\frac{2857}{2}-\frac{5033}{18}N_f +
\frac{325}{54}N_f^2\right),\\
\beta_3 &=& \frac{1}{4^4}\left[\frac{149753}{6}+3564\zeta(3)-
\left(\frac{1078361}{162}+\frac{6508}{27}\zeta(3)\right)N_f
\right.
\nonumber\\
&& +
\left. \left(\frac{50065}{162}+\frac{6472}{81}\zeta(3)\right)N_f^2+
\frac{1093}{729}N_f^3\right],\end{aligned}$$ and $L=\ln(\mu^2/\Lambda_{\overline{MS}}^{(N_f)2})$.
Then, the free parameters in the fit are the scheme-dependent constant $c$, the gluon condensate paemeter $\langle(\alpha_s/\pi)GG\rangle$, and the QCD scale $\Lambda^{(2)}_{\overline{\mathrm{MS}}}$ for the fit of an average $\Pi_{V+A}^{(0+1)}(Q)\equiv\Pi_V^{(0+1)}(Q)+\Pi_A^{(0+1)}(Q)$. For the difference $\Pi_{V-A}^{(0+1)}(Q)$, $\Lambda^{(2)}_{\overline{\mathrm{MS}}}$ obtained above is used as an input and the dimension-six condensates $a_6$, $b_6$ and $c_6$ are free parameters.
$V+A$ channel
-------------
![ Fit range dependence of $\Lambda^{(2)}_{\overline{MS}}$, $\langle(\alpha_s/\pi)GG\rangle$ and the constant term $c$. The maximum momentum squared $(aQ)^2_{max}$ is 1.324. []{data-label="fig2a"}](fitdep_sim.eps){width="100mm"}
The OPE analysis requires a window in $Q^2$ where the systematic errors are under control. The upper limit $(aQ)_{\mathrm{max}}^2\simeq 1.3238$ is set by taking the points where different definitions of the lattice momentum, [*i.e.*]{} $Q_\mu = (2/a)\sin(\pi n_\mu/L_\mu)$ and $Q_\mu=(2/a)\pi n_\mu/L_\mu$, give consistent results within one standard deviation. In the physical unit, this corresponds to 1.92 GeV. To determine $(aQ)_{\mathrm{min}}^2$, we investigate the dependence of the fit parameters on $(aQ)_{\mathrm{min}}^2$ in Figure \[fig2a\]. We observe that the results for $\Lambda^{(2)}_{\overline {\mathrm{MS}}}$, $\langle(\alpha_s/\pi)GG\rangle$, and $c$ are stable between $(aQ)_{\mathrm{min}}^2\simeq$ 0.48 and 0.65, which correspond to the momentum scale 1.16–1.35 GeV. Above $(aQ)_{\mathrm{min}}^2\simeq 0.65$ the fit becomes unstable; the results are still consistent within one standard deviation.
![ $\Pi^{(0+1)}_{V+A}(Q)$ as a function of $(aQ)^2$. The lattice data at different quark masses are shown by open symbols. Fit curves for each quark mass and in the chiral limit are drawn. The full result in the chiral limit (dashed-dots curves are at the finite masses, and solid curve is in the chiral limit), as well as that without $\langle\alpha_s G^2\rangle/Q^4$ term (dashed curve), are shown. []{data-label="fig2b"}](PiV+A_allav_high.allqm.eps){width="100mm"}
Figure \[fig2b\] shows the lattice data for $\Pi_{V+A}^{(0+1)}(Q)$ at each quark mass and corresponding fit curves. It is clear that the $Q^2$ dependence of the lattice data is well reproduced by the analytic formula. The quark mass dependence of $\Pi_{V+A}^{(0+1)}(Q)$ is, on the other hand, not substantial as expected from the fit function (\[eq:pi\_J\_OPE\]). Our fit with the known value of $\langle\bar{q}q\rangle$ reproduces the data well. In the chiral limit, (\[eq:pi\_J\_OPE\]) is controlled by two parameters: $\Lambda^{(2)}_{\overline{MS}}$ and $\langle(\alpha_s/\pi)GG\rangle$ (apart from the unphysical constant term $c$). The fit result in the chiral limit is drawn by a solid curve. The dashed curve drifting upwards towards low $Q^2$ region shows the result when the contribution from the $\langle(\alpha_s/\pi)GG\rangle$ term is omitted by hand. It indicates that the $Q^2$ dependence is mainly controlled by the perturbative piece while the dimension-four term gives a minor contribution, which becomes slightly more important in the low $Q^2$ regime. Numerically, we obtain $\Lambda^{(2)}_{\overline{MS}}$ = $0.234(9)$ GeV and $\langle(\alpha_s/\pi)GG\rangle$ = $-0.058(7)$ GeV$^4$ from a global fit of the lattice data at four different quark masses. The fit range of $(aQ)^2$ is \[0.65, 1.3238\].
![ $\Lambda^{(2)}_{\overline{MS}}$ from the data at each quark mass. []{data-label="fig3"}](lam_mdep_sim.eps){width="100mm"}
Figure \[fig3\] shows $\Lambda^{(2)}_{\overline{MS}}$ extracted from the lattice data at each quark mass. The flat behavior provides another evidence that the lattice data are consistent with the perturbative prediction (\[eq:pi\_J\_OPE\]).
Systematic errors
-----------------
In this subsection we discuss on possible systematic errors in this determination of $\Lambda^{(2)}_{\overline{MS}}$. That includes an estimate of the discretization effects and that of the truncation of perturbative and operator product expansions.
As indicated from the perturbative analysis presented in Section \[sec:pert\], the discretization effects are estimated reasonably well using the perturbation theory. Here, we discuss on the one-loop results for $\Pi_J(Q)$ on the lattice for our choice of the fermion action and the current operators. This aims at estimating the remaining systematic errors due to the discretization effects after explicitly subtracting the $B_n^J(Q)$ and $C_{mn}^J(Q)$ terms.
We again calculate the same one-loop vacuum polarization diagram at representative values of $(aQ)^2$ between 0.1 and 2.0. After subtracting the $B_{1,2}^J(Q)$ and $C_{11}^J(Q)$ terms determined perturbatively in Section \[sec:pert\] we numerically obtain the piece corresponding to $\Pi_J(Q)$, which contains the physical logarithmic dependence $-1/(4\pi)^2\ln((aQ^2))$ as well as the lattice artifacts. In the continuum theory (or the perturbative calculation with the dimensional regularization, to be specific) only this logarithmic term appears, hence we may identify the remaining terms as the lattice artifacts. They are parametrized by a polynomial of $(aQ)^2$.
![ One-loop calculation of $\Pi_V(Q)$ (upper panel) at representative values of $(aQ)^2$ (circles) and a fit with the log-plus-polynomial form. The dashed curve shows the purely logarithmic contribution. The term that represents the lattice artifact $\Delta\Pi_V(Q)$ is shown in the lower panel. []{data-label="fig4"}](VP.eps){width="100mm"}
The result of the one-loop calculation is shown in Figure \[fig4\] (upper panel). We fit the data with a function including the known logarithmic term plus a quadratic function of $(aQ)^2$ and obtain the numerical result $\Pi_V^{\rm LatPT}(Q^2) = -\frac{1}{4\pi^2}\ln((aQ)^2)
+ 0.03085(9) + 0.00952(30)(aQ)^2 -0.00132(20)(aQ)^4$.
In order to estimate the impact of this size of the discretization effect, we add this term to the fit function (\[eq:pi\_J\_OPE\]) and repeat the whole analysis. The result is $\Lambda^{(2)}_{\overline{MS}}$ = $0.249(37)$ GeV and $\langle(\alpha_s/\pi)GG\rangle$ = $+0.11(15)$ GeV$^4$. We find that $\Lambda^{(2)}_{\overline{MS}}$ is not largely affected, while $\langle(\alpha_s/\pi)GG\rangle$ is very sensitive to the lattice artifact and in fact changes its sign.
Other (Lorentz-violating) discretization effects due to $B_n^J(Q)$ and $C_{mn}^J(Q)$ are subtracted non-perturbatively so that the associated error should be negligible. With our preliminary calculation of the above mentioned conserved vector and axial-vector currents for the overlap fermion, we confirmed that the results are consistent with the calculation presented in this paper obtained with the non-conserved currents (\[eq:vector\]) and (\[eq:axial\]) up to the unphysical constant term $c$. This observation confirms that our procedure to subtract the $B_n^J(Q)$ and $C_{mn}^J(Q)$ terms is working as expected.
The truncation of the perturbative and operator product expansions is also a possible source of the systematic error. In order to estimate the size of the former, we repeat the analysis using the fit formulae truncated at a lower order (two-loop level), and find that the change of $\Lambda^{(2)}_{\overline{MS}}$ is much less than one standard deviation. It indicates that the higher order effects are negligible. The error from the truncation of OPE is estimated by dropping the terms of ${\cal O}(1/Q^4)$ from (\[eq:pi\_J\_OPE\]). From fits with higher $(aQ)^2_{\mathrm{min}}$ (between 0.79 and 0.89) to avoid contamination from the $1/Q^4$ effects, we obtain $\Lambda^{(2)}_{\overline{MS}}$ = 0.247(3) GeV. The deviation of $\Lambda^{(2)}_{\overline{MS}}$ is about the same size as that due to the discretization effect.
The errors due to finite physical volume and the fixed topological charge in our simulation [@Aoki:2007ka] are unimportant for the short-distance quantities considered in this work. A simple order counting gives an error of order $1/(QL)^2\lesssim$ 0.4% or smaller.
To quote the final result, we take the central value from the fit without the discretization effect $$\Lambda^{(2)}_{\overline{MS}} = 0.234(9)(^{+16}_{-\ 0}) \mathrm{~GeV},$$ where the first error is statistical and the second is systematic due to the discretization and truncation errors. The result is compatible with previous calculations of $\alpha_s$ in two-flavor QCD: $\Lambda^{(2)}_{\overline{MS}}$ = 0.250(16)(16) GeV [@DellaMorte:2004bc] and 0.249(16)(25) GeV [@Gockeler:2005rv]. (The physical scale is normalized with an input $r_0$ = 0.49 fm.)
$V-A$ channel {#v-a-channel}
-------------
![ Fit range dependence of $a_6(\mu)$, $b_6(\mu)$ and $a_8$. The horizontal axis denotes the minimum momentum squared $(aQ)_{\rm min}^2$. []{data-label="fig5a"}](a6b6a8_fitdep.eps){width="100mm"}
For the fit of the $V-A$ vacuum polarization $\Pi_{V-A}^{(0+1)}(Q)$, we also examine the fit range dependence. In Figure \[fig5a\] the fit parameters $a_6$, $b_6$ and $a_8$ are shown as a function of $(aQ)^2_{\mathrm{min}}$ while fixing $(aQ)^2_{\mathrm{max}}$ at the same value 1.3238. We attempt to fit with (filled symbols) and without (open symbols) the $a_8/Q^8$ term in order to investigate how stable the results are against the change of the order of the $1/Q^2$ expansion. We find that the fit with $a_8/Q^8$ is stable down to $(aQ)_{\mathrm{min}}^2\simeq 0.46$, while the other could not be extended below $(aQ)_{\mathrm{min}}^2\simeq 0.58$. The difference between filled and open symbols is marginal for $a_6$ (circles), but too large to make a reliable prediction for $b_6$ (squares). To quote the results we set $(aQ)_{\mathrm{min}}^2=0.586$ for both $\Pi^{(0+1)}_{V+A}(Q)$ and $\Pi^{(0+1)}_{V-A}(Q)$.
![ $Q^6\Pi_{V-A}^{(0+1)}(Q)$ as a function of $(aQ)^2$. The lattice data at different quark masses are shown by open symbols. Fit curves for each quark mass and in the chiral limit are drawn. []{data-label="fig5b"}](PiV-Ap6_allav_high.allqm.eps){width="100mm"}
In Figure \[fig5b\], we plot $Q^6\Pi_{V-A}^{(0+1)}(Q)$ as a function of $(aQ)^2$ for four different values of the quark mass $m_q$. The quark mass dependence is clearly observed. The main contribution comes from a dimension-six term $c_6 m_q/Q^6$, while the dimension-four term $\langle m\bar qq\rangle/Q^4$ is sub-dominant ($\sim 20\%$), as its coefficient starts at $\mathcal O(\alpha_s)$. In the chiral limit, there is a small but non-zero value remaining in $Q^6\Pi_{V-A}^{(0+1)}|_{\rm OPE}(Q^2)$ as shown by a dashed curve in the plot. This is due to the four-quark condensates $a_6$ and $b_6$.
The four-quark condensate $a_6$ obtained from $\Pi_{V-A}^{(0+1)}(Q)$ is $$a_6(2\mbox{~GeV}) = -0.0038(3)(^{+16}_{-\ 0})\,{\rm GeV^6},$$ where the first error is statistical. The second error represents an uncertainty due to the truncation of the $1/Q^2$ expansion. The central value is taken from the fit with $a_8/Q^8$ in (\[eq:pi\_V-A\_OPE\]) and the error reflects the shift when this term is discarded. The result agrees with the previous phenomenological estimates $-(0.003\sim 0.009)$ GeV$^6$ [@Almasy:2008al]. The other condensate is less stable; we obtain $b_6(2~GeV)$ = $+$0.0017(7) GeV$^6$ or $-$0.0008(2) GeV$^6$ with or without the $\mathcal{O}(1/Q^8)$ term, respectively.
Conclusion {#sec:conclusion}
==========
Many of the lattice calculations to date have analyzed the two-point correlation functions to extract physical quantities such as the hadron mass spectra and decay constants. Usually the exponential fall-off of the correlator at large Euclidean time separation is used to isolate the ground state contribution. In this way, however, many interesting pieces of information are lost. They are in the short and middle distance regime where the perturbative analysis is also applicable. We use the two-point current correlators calculated on the lattice to extract the strong coupling constant with the help of the continuum perturbation theory and the operator product expansion. The recent work by Allison [*et al.*]{} has exploited [@Allison:2008xk] the similar idea and applied it to the charmonium correlator to extract the charm quark mass and the strong coupling constant.
With the exact chiral symmetry realized by the overlap fermion formulation, the analysis of the lattice data is simplified. For the case of the vacuum polarizations, the continuum form of OPE may be applied without suffering from additional operator mixings, such as the additive renormalization of the operator $\bar{q}q$, which appears in the Wilson-type fermion formulations. We also obtain the four-quark condensates $a_6$ and $b_6$, which are relevant to the analysis of kaon decays [@Donoghue:1999ku].
In principle, our analysis does not require lattice perturbation theory, which is too complicated to carry out to the loop orders available in the continuum theory. But the perturbative calculation is still useful to estimate the discretization effects, which is well-described by perturbation theory in the asymptotic free theories.
The result for the strong coupling constant is compatible with previous lattice calculations. The size of statistical and systematic errors is also comparable with them. An obvious extension of this work is the calculation in 2+1-flavor QCD, which is underway [@Hashimoto:2007vv]. We also study the improvement of the analysis by using the conserved current for the overlap fermion formulation.
Numerical calculations are performed on IBM System Blue Gene Solution and Hitachi SR11000 at High Energy Accelerator Research Organization (KEK) under a support of its Large Scale Simulation Program (No. 07-16). This work is supported by the Grant-in-Aid of the Japanese Ministry of Education (No. 18034011, 18340075, 18740167, 19540286, 19740121, 19740160, 20025010, 20340047, 20740156), and National Science Council of Taiwan (No. NSC96-2112-M-002-020-MY3, NSC96-2112-M-001-017-MY3).
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How yummy do these sound? AND I can make them in advance, which always is a bonus for a tailgate party. They are probably best when still warm, so you could reheat them a bit for added yumminess!
Preheat oven to 350 degrees F. Butter and flour 24 standard muffin cups (or use muffin liners).
Make batter: In a medium bowl, whisk together flour, baking powder, baking soda, salt, nutmeg, and allspice.
In a small bowl, whisk together buttermilk and pumpkin puree.
In a large bowl, using an electric mixer, beat butter and brown sugar until light and fluffy.
Beat in eggs, one at a time, scraping down bowl as needed.
With mixer on low, add flour mixture in three additions, alternating with two additions pumpkin mixture, and beat to combine.
Spoon 1/3 cup batter into each muffin cup (I always use an ice cream scoop – it’s the perfect size for cupcake and muffin batter).
Bake eat 350 degrees F until a toothpick inserted in center of a muffin comes out clean, about 20-25 minutes.
Meanwhile, in a medium bowl, combine granulated sugar and cinnamon.
Let muffins cool 10 minutes in pan on a wire rack. Working with one at a time, remove muffins from pan, dunk all tops in the melted butter, then toss to coat in sugar mixture.
Serve immediately or place in an airtight container to serve later.
Let muffins cool completely on a wire rack. Serve immediately or place in an airtight container to serve later. | http://tailgateguru.net/recipe/pumpkin-doughnut-muffins/ |
Hi, beaders! It’s time for our monthly challenge color palette.
This month’s challenge piece is “Nighthawks” by Edward Hopper, and is a true masterpiece of American art. When I studied it in college, there was never a final assessment about what the meaning of it was, since it’s considered one of Hopper’s most ambiguous paintings. Several of my fellow students said it felt lonely, but I always felt like “Nighthawks” was more of a statement of sanctuary – four night owls in a restaurant in the middle of the night.
I think that initial feeling I had back then has influenced the way I view and relate to the colors in the painting; my eye is drawn first to the saturated warm colors of red, reddish browns, oranges, and yellow, colors that I personally associate with creativity. So, color-wise, my first instinct is to find the happier, warmer colors in the painting.
But the longer I looked, the more colors I saw, and I have to say that I was really surprised by all of the colors I found. Cool blues, slightly desaturated greens, and warm browns started appearing. I wouldn’t have thought it at first, but “Nighthawks” has a ton of colors in it.
So, with that in mind, and because each palette has a slightly different feel, I’m sharing all four of the color palettes I made for this month’s challenge.
What colors do you see in “Nighthawks”? Which palette do you gravitate to?
The third palette really calls to me. I don't know if it is the topaz & turquoise or the two greens with the ecru, but it really set my Muse to wondering. I think I see something in my future with this color palette.
That is one of my all time favorite paintings. And I've never before seen so many colors in it. Thanks Brandi for letting me see more into this wonderful work of art!
As an unrepentant lover of GREEN (must be the distant relatives from County Kerry on the Emerald Isle), I fall hard for the last palette… just gorgeous. As for the painting, my eyes immediately seek out the gentleman at the center: he is obviously lost in thought, unlike the three other figures who seem to be engaged in light conversation. I'm assuming this is intentioned by the artist? I don't have any "art training" so I've never heard of Hopper or this work. I just know I like the colour choices in the last set the best, Brandi.
The color palette is so helpful to me in getting started on these challenges. And having three to consider is absolutely wonderful! My eye keeps coming back to the third one, but they are all so appealing.
I've never thought to use a painting for a color palette before and I must admit, I am NOT a lover of green as someone else mentioned…BUT something about this painting and these palettes speaks to me. I'll do this (even though I'm a bit terrified of the green)! Thanks for the inspiration. | https://www.artbeadscenestudio.com/june-2012-challenge-color-palette/ |
Long-term therapy and retreatment of hepatorenal syndrome type 1 with ornipressin and dopamine.
Peripheral vasodilation is considered an important factor in the pathophysiology of the hepatorenal syndrome (HRS). Therefore, the aim of this study was to evaluate the therapeutic potential of the vasoconstrictor ornipressin plus dopamine in the treatment of the most severe form of HRS, namely HRS type 1. Seven cirrhotic patients (creatinine clearance 15 +/- 1 mL/min, UNaV 7 +/- 2 mmol/24 h) with HRS type 1 were included in the study after normalization of central venous pressure with intravenous albumin and low-dose dopamine had failed to prevent further deterioration of renal function. Ornipressin was given continuously (intravenous 6 IU/h) in combination with dopamine (2-3 microgram/kg/min) until creatinine clearance had increased to above 40 mL/min or adverse events prevented further treatment. HRS was reverted in 4 of 7 patients after 5 to 27 days (creatinine clearance 51 +/- 4 mL/min, UNaV 47 +/- 11 mmol/24 h) of treatment. Withdrawal was necessary in 1 patient after 15 days because of intestinal ischemia. Treatment failure was observed in 2 of 7 patients (creatinine clearance 19 +/- 10 mL/min, UNaV 8 +/- 3 mmol/24 h). Two of 4 responders had recidivant HRS 2 and 8 months after initial therapy, respectively. HRS in 1 of these patients was reverted with 18 days of ornipressin retreatment. The other patient had to be withdrawn from ornipressin after 2 hours because of ventricular tachyarrhythmia. Altogether, 3 of 7 patients survived HRS type 1, 1 after successful ornipressin therapy and liver transplantation, 1 with 2 successful courses of ornipressin, and 1 with liver transplantation after ornipressin treatment had failed. Thus, ornipressin plus dopamine can be a useful therapeutic option in patients with HRS type 1, especially as bridge to liver transplantation.
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---
abstract: 'This work study the finite temperature effects of a mass dimension one fermionic field, sometimes called Elko field. The equilibrium partition function was calculated by means of the imaginary time formalism and the result obtained was the same for a Dirac fermionic field, even though the Elko field does not satisfy a Dirac like equation. The high and low temperature limits were obtained, and for the last case the degeneracy pressure due to Pauli exclusion principle can be responsible for the dark matter halos around galaxies to be greater than or of the same order of the galaxy radius. Also, for a light particle of about $0.1$eV and a density of just 0.1 particle per cubic centimeter, the value of the total dark matter mass due to Elko particles is of the same order of a typical galaxy. Such a result satisfactorily explains the dark matter as being formed just by Elko fermionic particles and also the existence of galactic halos that go beyond the observable limit.'
author:
- 'S. H. Pereira'
- 'Richard S. Costa'
title: Partition function for a mass dimension one fermionic field and the dark matter halo of galaxies
---
Introduction
============
Finite temperature effects in quantum field theory, or thermal field theories, are studied long time ago and the main results concerning relativistic degenerate gas, super-dense nuclear matter, quark-gluon phase transition, spontaneous symmetry breaking of electroweak unification and Higgs model at finite temperature have been discussed exhaustively in several textbooks [@bellac; @kapusta; @das; @love], including astrophysical and cosmological applications in white dwarf stars, neutron stars and baryogenesis after big bang.
Most of works in thermal field theory concerns the study of equilibrium thermodynamic properties of systems formed by spin-0 real or complex scalar fields (bosonic particles), spin-${1\over 2}$ fields (fermionic Dirac like particles) and spin-1 fields (gauge vector bosons). In order to extract physical thermodynamic properties of relativistic fields at equilibrium, the standard method is by means of the study of the partition function from the statistical mechanics, constructed through a path integral over the fields. Once the partition function has been evaluated, the canonical ensemble assures that the Helmholtz free energy can be obtained, from which follows the total energy density, pressure and entropy density of the system.
As already mentioned at the beginning, finite temperature effects for scalar fields, fermionic fields and gauge vector bosons are well known in literature. Nevertheless, in which concerns the fermionic fields, the calculations are done just for fermionic Dirac like fields, that is, fermionic fields with spin-${1\over 2}$ and mass dimension-${3\over 2}$ that satisfies a Dirac like equation. Recently, a new class of mass dimension one fermionic fields has been discovered [@AHL1; @AHL2; @ahl2011a; @AHL4; @WT], sometimes called Elko[^1]. The new fermionic fields constructed from such new spinors are natural candidates to dark matter particles in the universe, once they are eigenstate of the charge conjugation operator, being neutral and coupling very weakly or even not coupling to other particles of the standard model. The signature of such new fermionic mass dimension one particle at LHC has been proposed in [@dias1; @alves1; @clee; @julio2], cosmological applications have also been recently studied in [@FABBRI; @BOE4; @BOE6; @GREDAT; @BASAK; @sadja; @kouwn; @saj; @js; @asf; @sajf; @st; @sra], a sigma model has been obtained in [@elkosigma], its Casimir effect has been calculated in [@elkocasimir] and an one loop effective Lagrangian has been obtained in [@elkoeffective].
In this work we aims to study the finite temperature effects due to Elko fermionic fields and its contribution to high and low temperature systems formed by such fields. While Elko fields satisfy just a Klein-Gordon like equation, as a scalar field, it must satisfies anti-periodic boundary conditions, as Dirac like fields. Thus the calculation of its partition function must take into account a mixture of both properties of bosonic and fermionic fields. In Section II we present a brief review of the results of finite temperature systems, in Section III we makes a explicit calculation of the partition function for Elko fields and present the main results at high temperature limit, in Section IV we calculated the degeneracy pressure in low temperature limit and its relation to dark matter halos of galaxies. We conclude in Section V.
Finite temperature field theory
===============================
The study of different quantum fields at finite temperature is done by means of the partition function $Z$ of the system [@bellac; @kapusta; @das; @love], which is related to the Helmholtz free energy $F$ by $Z=\exp(-\beta F)$, where[^2] $\beta \equiv 1/T$, from which follows the thermodynamic quantities as total energy $E$, pressure $P$ and entropy $S$: $$E=F+TS,\hspace{0.5cm} P=-\Bigg(\frac{\partial F}{\partial V}\Bigg)_T,\hspace{0.5cm} S=-\Bigg(\frac{\partial F}{\partial T}\Bigg)_V\,.\label{EPS}$$
The temperature is introduced by means of the imaginary time formalism, making a rotation of the real time axis to the complex one, $x_0=t\to -i\tau$, similar to go from Minkowski to Euclidean space-time metric, with new variables $\bar{x}^\mu=(\bar{x}_0,\bar{\bf{x}})=(ix_0,\bf{x})=(\tau,\bf{x})$. Once the system is at equilibrium (not evolving in time), the equilibrium thermodynamic temperature is introduced by means of $\tau = \beta =1/T$.
The partition function for a standard real scalar field $\phi$ is [@bellac; @kapusta; @das; @love]: $$Z=N(\beta)\int_{periodic}D\phi\exp\int_0^\beta d\tau \int d^3x\mathcal{L}(\phi,\bar{\partial}_\mu\phi)\,,\label{Z1}$$ where $N(\beta)$ is just a normalisation constant ($\beta$ dependent) and $\mathcal{L}$ is the Lagrangian density of the field $\phi$ and its derivative, with $\bar{\partial}_\mu \equiv (i\partial_\tau, \partial_i)$. The functional integral over the field must be understood to be periodic in the interval $0<\tau<\beta$, $$\phi(\tau=0,\bf{x})=\phi(\tau=\beta,\bf{x})\,.$$
For a general fermionic field $\psi$ the partition function is: $$Z=N(\beta)\int_{antiperiodic}D\bar{\psi}D\psi\exp\int_0^\beta d\tau \int d^3x\,\mathcal{L}(\psi,\bar{\psi})\,,\label{Z2}$$ where $N(\beta)$ is also a normalisation constant and $\mathcal{L}$ is the Lagrangian density of the field $\psi(\bar{x})$ (and its dual $\bar{\psi}(\bar{x})$) and the functional integral over the fields must be anti-periodic in the interval $0<\tau<\beta$, $$\psi(\tau=0,\bf{x})=-\psi(\tau=\beta,\bf{x})\,.$$
A standard real scalar field $\phi$ satisfies a Klein-Gordon like Lagrangian: $$\mathcal{L}(\phi,\bar{\partial}_\mu\phi)={1\over 2}\partial^\mu\phi\partial_\mu\phi-{1\over 2}m^2\phi^2\,,$$ while for a standard Dirac free-field, the Lagrangian is: $$\mathcal{L}(\psi,\bar{\psi})=\bar{\psi}(x)(i\gamma^\mu\partial_\mu-m)\psi(x)\,.\label{lagranD}$$ However, for a mass dimension one fermionic field $\lambda(x)$ (and its dual $\stackrel{\neg}{\lambda}$) the Lagrangian is of Klein-Gordon type: $$\mathcal{L}(\lambda,\stackrel{\neg}{\lambda})={1\over 2}\partial^\mu\stackrel{\neg}{\lambda}\partial_\mu\lambda-{1\over 2}m^2\stackrel{\neg}{\lambda}\lambda\,.\label{lagran}$$ Here is the main difference concerning the calculation of the partition function involving different kind of fields. While the calculation of (\[Z1\]) for bosonic fields (satisfying a Klein-Gordon like equation) must be periodic in $\tau$, the calculation of (\[Z2\]) for Dirac fields must be anti-periodic. In the case of a mass dimension one fermionic field as Elko, although satisfying a Klein-Gordon like equation, the partition function must be done with anti-periodic boundary condition, as in (\[Z2\]). As far as we know, this calculation has not been done so far.
Partition function for a mass dimension one fermionic field
===========================================================
[In order to calculate the partition function for a mass dimension one fermionic field satisfying (\[lagran\]) we must be careful. We start with the functional integral in Minkowski space-time: $$Z=N\int_{antiperiodic}D\stackrel{\neg}{\lambda}D\lambda\exp\Bigg(\frac{i}{2}\int d^4x'\int d^4x\,\stackrel{\neg}{\lambda}({x}')\mathbf{A}({x}',{x})\lambda({x})\Bigg)\,,\label{Z3}$$ where $$\mathbf{A}({x}',{x})=-({\partial}'_\mu{\partial}^\mu+m^2) \delta({x}'-{x})\mathbf{I}\,,$$ is the Klein-Gordon like operator that comes from (\[lagran\]) after a partial integration and $\mathbf{I}$ stands for a $4\times 4$ identity matrix. The functional integral over Grassmannian like functions in Minkowski space-time is well established]{}[^3], [and the path integral over $\stackrel{\neg}{\lambda}$ and $\lambda$ can be done, giving: $$Z=N\exp\big(\mathrm{Tr}\ln {i\over 2}\mathbf{A}\big)\,.$$ Once the functional integral has been done in Minkowski space, we make a rotation to Euclidean space-time $\bar{x}^\mu$ in order to establish contact with thermodynamic by means of $\tau$. The trace must be evaluated in the Euclidean space by introducing the Fourier transform of the field as:]{} $$\lambda(\bar{x})={1\over \beta}\sum_n\int\frac{d^3p}{(2\pi)^3}\mathrm{e}^{-i\omega_n\tau}\mathrm{e}^{i\bf{p}\cdot \bf{x}}\tilde{\lambda}(\omega_n,\bf{p})\,,$$ with a similar one for the dual $\stackrel{\neg}{\lambda}(\bar{x})$, and where the sum is done over the Matsubara frequencies for fermion $$\omega_n=\frac{(2n+1)\pi}{\beta}\,,\hspace{1cm} n \textrm{ integer}.$$ We obtain: $$\mathrm{Tr}\ln \mathbf{A}=4\int_0^\beta d\tau \int d^3x{1\over \beta}\sum_n\int\frac{d^3p}{(2\pi)^3}\ln(\omega_n^2+{\bf p}^2+m^2)\,,$$ and performing the $\tau$ integration and the fermionic Matsubara frequency sum[^4], we obtain finally the partition function as: $$Z=N(\beta)\exp\Bigg\{2\int d^3x \int\frac{d^3p}{(2\pi)^3}\Bigg[\beta \sqrt{{\bf p}^2+m^2}+ 2 \ln\bigg(1+\exp(-\beta \sqrt{{\bf p}^2+m^2})\bigg) + C\Bigg]\Bigg\}\,,$$ where $C$ is a $\bf{p}$ independent constant. The Helmholtz free energy is: $$F=-{1\over \beta}\ln Z = -2\int d^3x \int\frac{d^3p}{(2\pi)^3}\Bigg[\sqrt{{\bf p}^2+m^2}+ {2\over \beta} \ln\bigg(1+\exp(-\beta \sqrt{{\bf p}^2+m^2})\bigg)\Bigg]\,,\label{F1}$$ where the constant $C$ has been cancelled against $N(\beta)$. The first term inside integral represents the divergent zero temperature contribution that must be removed by regularisation and the second term the temperature dependent one. It is interesting to notice that such result is exactly the same that one obtained by standard Dirac fermion, even though the Lagrangian for the mass dimension one Elko field is very different from the Dirac case.
Taking just the temperature dependent term of (\[F1\]) and making an integration by parts we are left with: $${F\over V}=-\frac{2}{3\pi^2}\int_0^\infty\frac{p^4}{\sqrt{p^2+m^2}}\frac{1}{\textrm{e}^{\beta\sqrt{p^2+m^2}}+1}dp\,,\label{F2}$$ where $V=\int d^3x$. In the last term we recognise the occupation number that characterises the Fermi-Dirac distribution function, $n_\varepsilon\equiv [\textrm{e}^{\beta(\varepsilon-\mu)}+1]^{-1}$, with relativistic energy $\varepsilon=\sqrt{p^2+m^2}$ and null chemical potential $\mu$.
In the high temperature limit $(T>>m)$ the integral in (\[F2\]) is easily calculated [@bellac; @kapusta; @das; @love] and the results for the Helmholtz free energy density, energy density, pressure and entropy density from (\[EPS\]) are: $$\frac{F}{V}=-\frac{7\pi^2T^4}{180}, \hspace{0.5cm}\frac{E}{V}=\frac{7\pi^2T^4}{60}, \hspace{0.5cm} P =\frac{7\pi^2T^4}{180}, \hspace{0.5cm}\frac{S}{V}=\frac{7\pi^2T^3}{45}.$$
We can summarise the results including the well known results for bosonic fields, Dirac fermions and now the contribution from Elko fields, generalising the results of [@love]: $$\begin{aligned}
\frac{F}{V}&=&-\frac{\pi^2T^4}{90}(N_B+{7\over 8}N_D + {7\over 8}N_E),\\
\frac{E}{V}&=&\frac{\pi^2T^4}{30}(N_B+{7\over 8}N_D + {7\over 8}N_E),\\
P&=&\frac{\pi^2T^4}{90}(N_B+{7\over 8}N_D + {7\over 8}N_E),\\
\frac{S}{V}&=&\frac{2\pi^2T^3}{45}(N_B+{7\over 8}N_D + {7\over 8}N_E),\end{aligned}$$ where $N_B=1$ for neutral scalar field, $N_B=2$ for neutral gauge field, $N_D=4$ for a Dirac field, $N_D=2$ for a Weyl field and $N_E=4$ for a Elko field.
Degeneracy pressure and dark matter halo of galaxies
====================================================
The low temperature limit is much more interesting to study and is related to the degeneracy pressure of a fermionic system. In the presence of a non-null chemical potential $\mu$, the limit $T\to 0$ makes the occupation number $n_\varepsilon$ to behave as a step function which stays constant at the value 1 for $\varepsilon<\mu_0$ and at the value 0 for $\varepsilon>\mu_0$. Thus, at $T=0$ all single-particle states are completely filled with one particle per state up to $\varepsilon=\mu_0$, following the Pauli exclusion principle, while all single-particle states with $\varepsilon>\mu_0$ are empty. This makes the integral on (\[F2\]) easy to be calculated and the Fermi energy is defined as $\varepsilon_F = \mu_0$, which also defines the Fermi momentum $p_F$ at which the integral must be done. For the degeneracy pressure $P_0$ at $T\to 0$ we have: $$P_0=-\frac{\partial F}{\partial V}=\frac{2}{3\pi^2}\int_0^{p_F}\frac{p^4}{\sqrt{p^2+m^2}}dp\,.$$ Apart from constant terms, the integration is exactly that one for a relativistic particle in classical statistical systems [@pathria]. The Fermi momentum can be written as $p_F=(3n/8\pi)^{1/3}$, where $n\equiv N/V$ is the particle number density[^5]. Defining $x\equiv p_F/m$, the integral can be done [@pathria] and in the limits $x<<1$ and $x>>1$ it is given by, respectively: $$P_0\simeq\frac{8\pi m^4}{15}x^5\,,\hspace{2cm}P_0\simeq\frac{2\pi m^4}{3}x^4\,.\label{P0}$$ Given the values of $n$ and $m$ the degeneracy pressure can be calculated.
Suppose that the whole universe was filled with a gas of free Elko particles that were at thermal equilibrium with all matter in the past[^6]. Along the universe evolution, the temperature decreases and today it must be very small, satisfying the condition $T\to 0$, similar to a cosmic microwave background temperature of about $2.75$K. As good candidates to dark matter, the Elko particles do not interact electromagnetically and falls within the gravitational potential wells around the galactic nuclei. But the gravitational attraction must be counterbalanced by the degeneracy pressure due to Pauli exclusion principle, reaching equilibrium within a ray $R$, similar to what happens in a white dwarf star or a neutron star before collapsing. The equilibrium equation can be written as [@pathria]: $$P_0(R)=\frac{\alpha}{4\pi}\frac{GM^2}{R^4}\,,$$ where $M=nmV$ is the total mass within a volume $V$, $G$ is the gravitational constant and $\alpha\simeq 1$ is a constant whose value depends upon the nature of the variation of the number density inside the gas. By supposing a density number $n\simeq 0.1$cm$^{-3}$ and a mass $m\simeq 0.1$eV, we have $x\simeq 0.00028$ and the first approximation in (\[P0\]) is valid. In this case, the equilibrium radius $R=R_0$ can be written as: $$R_0=\frac{3^{4/3}}{20\pi^{5/6}}\sqrt{\frac{5}{\alpha G}}\Bigg(\frac{1}{n^{1/6}m^{3/2}}\Bigg)\,.\label{R}$$ It can be seen that greater the values of $m$ or $n$ lower the value of the radius, as expected.
For the above values of $m$ and $n$ we found $R_0\simeq 4.7\times 10^{26}$cm, four orders greater than the radius of our galaxy[^7]. Such very rough estimate shows that dark matter formed by Elko particles may be uniformly distributed beyond the radius of the galaxy, as indicated by several observations, maintained at equilibrium due to the degeneracy pressure of its fermionic particles. Moreover, for these values of mass $m$, density number $n$ and radius $R_0$, we found the total dark matter mass inside a sphere of radius $R_0$ as $M=nmV = 4.3\times 10^{78}$eV $=7.7\times 10^{45}$g, which is of the same order of the total mass $M$ of a typical galaxy like ours$^7$. In the limit of very low density, $n\simeq 10^{-10}$cm$^{-3}$, and high mass, $m\simeq 1.0$GeV, we find very low values of radius and total mass, namely $R_0\simeq 10^{13}$cm and $M\simeq 10^6$g, incompatible with observations.
[An interesting and complete analysis considering the possibility that dark matter halos are described by the Fermi-Dirac distribution at finite temperature was done in [@chavanis], where dark matter could be a self-gravitating quantum gas made of massive neutrinos at statistical equilibrium or if dark matter can be treated as a self-gravitating collisionless gas.]{}
Conclusion
==========
The partition function for a mass dimension one fermionic field was calculated and the result is the same as for Dirac like fermions. Although these new kind of fields not obey a Dirac like equation, the finite temperature effects and thermodynamic properties obtained by the standard method of Matsubara frequencies sum for fermions are exactly the same. This opens the possibility to use very known results from thermofield dynamics to systems of Elko particles. In particular, results for energy density, entropy density and pressure at high temperatures are the same of standard Dirac particles.
The low temperature limit exhibits the phenomenon of degeneracy pressure, which could be responsible for maintain a dark matter halo around galaxies nuclei. For a low mass Elko particle of about $m\simeq 0.1$eV and a density number of about $n\simeq 0.1$cm$^{-3}$ the equilibrium radius found is greater than the observable radius of a typical galaxy as ours, explaining the existence of a large halo of dark matter around galactic nucleus. Moreover, the total Elko mass attributed to dark matter is of the same order of the galactic masses, in good agreement to observations. Additionally, due to Elko particles do not interact electromagnetically, its distribution around the galactic nucleus is nearly spherical and uniform, modelled just by equilibrium between gravitational pressure and degeneracy pressure, while baryonic matter condensates at the centre of the galaxies, forming heavier particles that does not suffer from degeneracy pressure and interact electromagnetically, which make it to loses energy and condensate. This shows correctly that most of the mass of the galaxy may be in the form of dark matter representing about 25% of dark matter against less than 5% of baryonic matter of the total content of the universe, in good agreement to the $\Lambda$CDM model data.
[It is important to notice that while some estimates at LHC searches indicate a very huge mass to Elko (about $100$GeV in [@clee] and 1TeV in [@julio2]), cosmological estimates based on some observational constraints point to very small masses (about $10^{-32}$eV in [@sajf; @st; @sra]). The value estimated here is of same order of neutrino masses, also considered a good dark matter candidate. More realistic models of varying mass distribution around the galactic nuclei need to be analysed in order to confirm Elko particles as good candidate to dark matter in the universe. Also, a more complete discussion about the thermal decoupling of Elko dark matter particles and its present temperature is an interesting subject to be studied, as it is done for WIMPS [@ultimo].]{}
SHP acknowledges CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazilian research agency, for financial support, No. 303583/2018-5 and 400924/2016-1. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. RSC acknowledges support from CAPES. We thank J. M. Hoff da Silva for helpful discussions.
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[^1]: From German, [*Eigenspinoren des ladungskonjugationsoperators*]{}, or eigenspinor of charge conjugation
[^2]: We are using units where $c=\hbar=k_B=1$
[^3]: [According to [@love], for $n$ Grassmann variables $\theta_1, ..., \theta_n$ satisfying anti commuting relations as $\{\theta_i, \theta_j\}=0$ we have $\int d\theta_i = 0$ and $\int d\theta_i \theta_i=1$. For a Gaussian integral over $n$ Grassmann variables, $I_n\equiv \int d\theta_1\,...d\theta_n\exp(-{1\over 2}\Theta^T \mathbf{A} \Theta)$, where $\mathbf{A}$ is a real anti-symmetric matrix and $\Theta$ a column vector with components $(\theta_1, ..., \theta_n)$. For $n$ odd we have $I_n=0$ and for $n$ even $I_n=(\det \mathbf{A})^{1/2}=\exp({1\over 2}Tr\ln \mathbf{A})$. For complex Grassmann variables the generalisation is $\int d\theta^{*}_1 d\theta_1\,...\int d\theta^{*}_n d\theta_n\exp(-{1\over 2}\Theta^\dagger \mathbf{B} \Theta)=\det({1\over 2}\mathbf{B})$, where $\mathbf{B}$ is a skew Hermitean matrix. For a complex Gaussian, as occurs in a Minkowski space, the generalisation is $\int D\varphi^{*} D\varphi\exp(i\varphi^{*}\mathbf{C}\varphi)=\det(i\mathbf{C})= \exp Tr \ln(i\mathbf{C})$, where $\mathbf{C}$ is Hermitean.]{}
[^4]: See [@love] for further details.
[^5]: For photons in the whole universe, $n_\gamma \simeq 422$cm$^{-3}$ and for neutrinos $n_\nu \simeq 115$cm$^{-3}$ at present time [@kolb].
[^6]: Although Elko does not couple to electromagnetic fields, at very high energies it could couple to Higgs fields, which would be responsible to thermalize the Elko particles with other matter fields.
[^7]: For the Milk-Way or Andromedae we have $R\simeq 50,000$l.y. $\simeq 4.7\times 10^{22}$cm and $M\simeq 10^{12}M_\circ \simeq 2\times 10^{45}$g, where $M_\odot$ is the solar mass.
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---
abstract: 'We develop the technique of weight truncation in the context of wall-crossings in birational cobordisms, parallel to that in [@HL15; @BFK19]. More precisely, for each such wall-crossing, we embed the bounded above derived category of coherent sheaves of the semistable part as a semi-orthogonal summand of that of the stack in question. Our construction does not require any smoothness assumptions, and exhibits a strong symmetry across the two sides of the wall-crossing. As an application, we show that for wall-crossings satisfying suitable regularity conditions, a certain duality of local cohomology complexes implies the existence of a fully faithful functor/equivalence between the derived categories under wall-crossings.'
address: 'Department of Mathematics, Indiana University, Bloomington, IN 47405, USA'
author:
- 'Wai-Kit Yeung'
title: 'Weight truncation for wall-crossings in birational cobordisms'
---
Introduction
============
A central problem in the geometry of derived categories is to study the change in the derived categories under birational transforms. In recent years, significant progress on this problem have been obtained in the setting of variations of GIT quotients, or more broadly wall-crossings in moduli problems. This is a convenient setting because, on the one hand, the spaces of semistable objects are often birational to each other under wall-crossings; on the other hand, the fact that they arise as different open subspaces of the same space $\mathfrak{X}$ allows one to relate their derived categories via their mutual relations to the derived category of the space $\mathfrak{X}$.
For example, if $G$ is a reductive group acting on a quasi-projective variety $X$, then each choice of $L \in {{\rm NS}}^G(X)_{\bQ}$ determines an open substack $\mathfrak{X}^{ss} = \mathfrak{X}^{ss}(L)$ of the stack $\mathfrak{X} := [X / G]$. By a standard abuse of language, we will refer to $\mathfrak{X}^{ss}$ as the GIT quotient, the usual GIT quotient being its scheme-theoretic categorical quotient. As an open substack, the derived category of $\mathfrak{X}^{ss}$ is a localization of that of $\mathfrak{X}$. As $L$ changes, say from $L^-$ to $L^+$, then the derived categories of $\mathfrak{X}^{\pm} := \mathfrak{X}^{ss}(L^{\pm})$ are thus two different localizations of the same derived category. This setting is a baseline for the comparison of the derived categories of $\mathfrak{X}^-$ and $\mathfrak{X}^+$.
Specifically, one may compare the derived categories of $\mathfrak{X}^-$ and $\mathfrak{X}^+$ by finding suitable full triangulated subcategories $\cE^{\pm} \subset {\cD^b_{{\rm coh}}}(\mathfrak{X})$ such that the restriction functors $(j^{\pm})^* : \cE^{\pm} {\rightarrow}{\cD^b_{{\rm coh}}}(\mathfrak{X}^{\pm})$ are equivalences. There is often a direct comparison between $\cE^-$ and $\cE^+$, which in turn allows one to compare ${\cD^b_{{\rm coh}}}(\mathfrak{X}^-)$ and ${\cD^b_{{\rm coh}}}(\mathfrak{X}^+)$.
Weight truncation, also known as grade restriction rule or GIT window (see, [[*e.g.*]{}]{}, [@HL15; @BFK19]), is a particular way to realize this idea. In this paper, we develop the technique of weight truncation, focusing on the abelian case $G = \bG_m$, or more precisely the case of birational cobordism, where the element $L \in {{\rm NS}}^{\bG_m}(X)_{\bQ}$ changes via twisting by a (fractional) character of $\bG_m$. Indeed, there are several techniques to reduce the more general case to this abelian case. For example, the “master space construction” of Thaddeus [@Tha96] realizes every variation of GIT quotients as a birational cobordism. On the other hand, the structure of HKKN stratification essentially allows for a strata-by-strata reduction to the abelian case. This last picture also extends beyond GIT quotients (see, [[*e.g.*]{}]{}, [@HL]). As such, the abelian case has a special significance.
Compared with [@HL15; @BFK19], our approach has several advantages. Firstly, our construction of weight truncation does not require any smoothness assumption, although some regularity condition is still required in order to guarantee certain desired properties. Secondly, our construction exhibits a manifest symmetry across the wall. Thirdly, our construction carries over to the noncommutative case, in the spririt of noncommutative projective geometry (see, [[*e.g.*]{}]{}, [@AZ94; @Orl09]), although this will not be made explicit in this paper.
Let us now explain our construction of weight truncation. Given a $\bG_m$-space $X$, consider a family $L(t) \in {{\rm NS}}^{\bG_m}(X)_{\bQ}$ obtained by twisted a given $L$ by fractional characters $t \in \bQ$. Under a wall-crossing from $t^-$ to $t^+$, passing through the wall $t_0$, the semistable loci satisfy $X^{ss}(L(t^-)) \subset X^{ss}(L(t_0)) \supset X^{ss}(L(t^+))$. Denote by $W := X^{ss}(L(t_0))$, and $Y := W // \bG_m$ the scheme-theoretic categorical quotient. Then we have $W = {{\rm Spec}}_Y(\cA)$ for a sheaf of $\bZ$-graded rings on $Y$, and the (stacky) GIT quotients across the wall are $$\label{birat_cobord_log_flip_diag_intro}
\left(
\begin{tikzcd} [row sep = 0, column sep = 0]
{\rm [} X^{ss}(L(t^-)) \, / \, \bG_m {\rm ]} \ar[rd]
& & {\rm [} X^{ss}(L(t^+)) \, / \, \bG_m {\rm ]} \ar[ld] \\
& Y &
\end{tikzcd}
\right)
\, \cong \,
\left(
\begin{tikzcd} [row sep = 0, column sep = 5]
{\mathpzc{Proj}}^-_Y(\cA) \ar[rd]
& & {\mathpzc{Proj}}^+_Y(\cA) \ar[ld] \\
& Y &
\end{tikzcd}
\right)$$ where ${\mathpzc{Proj}}^{+}_Y(\cA)$ is a stacky version of the scheme-theoretic GIT quotient ${{\rm Proj}}^+_Y(\cA) := {{\rm Proj}}_Y(\cA_{\geq 0})$, and similarly for ${\mathpzc{Proj}}^{-}_Y(\cA)$. We will write $\mathfrak{X}^{\pm} := {\mathpzc{Proj}}^{\pm}_Y(\cA)$.
Thus, the study of wall-crossing in this abelian case is reduced to the study of a sheaf of $\bZ$-graded rings. Since all our constructions are local on $Y$, we may assume that $Y$ is affine, so that we may focus on a Noetherian $\bZ$-graded ring $A$ ([*cf.*]{} Proposition \[Noeth\_gr\_ring\] below for the Noetherian property). The category of quasi-coherent sheaves on $\mathfrak{X} := [{{\rm Spec}}\, A / \bG_m]$ in this case is simply the category ${{\rm Gr}(A)}$ of graded $A$-modules. Thus, we are reduced to studying the derived category $\cD({{\rm Gr}(A)})$ of graded $A$-modules.
Let $I^+ := A_{>0} \cdot A$ and $I^- := A_{<0} \cdot A$. Then the semistable loci $X^{ss}(L^{\pm})$ are the complement of the $\bG_m$-invariant closed subsets defined by the graded ideals $I^{\pm}$. Thus, if we denote by $j^{\pm} : \mathfrak{X}^{\pm} \subset \mathfrak{X}$ the open inclusions, then for any $M \in \cD({{\rm Gr}(A)}) \simeq \cD({{\rm QCoh}}(\mathfrak{X}))$, the derived pushforward ${\bm R}j^{+}_*(j^{+})^*(M)$ is computed by a certain Čech complex ${\check{\cC}}_{I^+}(M)$ (see Definition \[Cech\_cplx\_def\] below), which sits in an exact triangle $$\label{RGI_CI_seq_intro}
\ldots {\,\rightarrow \,}{{\bm R}\Gamma}_{I^+}(M) {\, \xrightarrow[]{\epsilon} \,} M {\, \xrightarrow[]{\eta} \,} {\check{\cC}}_{I^+}(M) {\, \xrightarrow[]{\delta} \,} {{\bm R}\Gamma}_{I^+}(M)[1] {\,\rightarrow \,}\ldots$$
In fact, this triangle is the decomposition triangle for the semi-orthogonal decomposition $$\label{torsion_SOD_intro}
\cD({{\rm Gr}(A)}) \, = \, \langle \, \cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}) \, , \, \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)}) \, \rangle$$ where $\cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$ are objects of $\cD({{\rm QCoh}}(\mathfrak{X}))$ supported on the unstable locus, and $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$ is equivalent to $\cD({{\rm QCoh}}(\mathfrak{X}^+))$ via the functor ${\bm R}j^{\pm}_*$. While this gives an embedding of $\cD({{\rm QCoh}}(\mathfrak{X}^+))$ as a semi-orthogonal summand of $\cD({{\rm Gr}(A)})$, this embedding is usually very far[^1] from preserving ${\cD^b_{{\rm coh}}}$: $$\label{Cech_coherent_intro_1}
\parbox{40em}{For $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, the \v{C}ech complex ${\check{\cC}}_{I^+}(M)$ is almost never in ${\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$.}$$
The Čech complex ${\check{\cC}}_{I^+}(M)$ also plays a role for the classical GIT quotient. Indeed, let $X^+ := {{\rm Proj}}(A_{\geq 0})$, and $\pi^+ : X^+ {\rightarrow}Y = {{\rm Spec}}\, A_0$ the projection, then since the cohomology of quasi-coherent sheaves can be computed by the Čech complex, we have $$\label{CI_Rpi_intro}
{\bm R}\pi^+_* (\widetilde{M(i)}) \, \cong \, {\check{\cC}}_{I^+}(M) _i$$
It is a classical fact that the derived pushforward of the projective morphism $\pi^+ : X^+ {\rightarrow}Y$ sends ${\cD^b_{{\rm coh}}}(X^+)$ to ${\cD^b_{{\rm coh}}}(Y)$. Thus, by , each weight component ${\check{\cC}}_{I^+}(M)_i$ of the Čech complex is in ${\cD^b_{{\rm coh}}}(A_0)$. In fact, more is true (see the proof of Proposition \[weight\_Cech\_Dbcoh\] below): $$\label{Cech_coherent_intro_2}
\parbox{40em}{If $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, then for each $w \in \bZ$, take the restriction ${\check{\cC}}_{I^+}(M)_{\geq w}$ of the \v{C}ech complex to weights $i \geq w$, and consider it as an object in $\cD({{\rm Gr}}(A_{\geq 0}))$. Then we have ${\check{\cC}}_{I^+}(M)_{\geq w} \in {\cD^b_{{\rm coh}}}({{\rm Gr}}(A_{\geq 0}))$.}$$
A comparison of and suggests that, by considering the restriction ${\check{\cC}}_{I^+}(M)_{\geq w}$ of the Čech complex, one might be able to obtain an embedding of $\cD({{\rm QCoh}}(\mathfrak{X}^+))$ into $\cD({{\rm Gr}(A)})$ that has more chance of preserving ${\cD^b_{{\rm coh}}}(-)$. This is achieved by the technique of weight truncation. For this, we will reformulate the observation .
In , we considered the restriction ${\check{\cC}}_{I^+}(M)_{\geq w}$ as an object in $\cD({{\rm Gr}}(A_{\geq 0}))$. However, the passage from $A$ to $A_{\geq 0}$ is somewhat unnatural, and it is difficult to compare their derived categories. In fact, there is a better way to organize the data of the complexes ${\check{\cC}}_{I^+}(M)_{i}$ for $i \geq w$. For this, we use the language of a small pre-additive category.
By definition, a pre-additive category is a category whose Hom sets have structures of abelian groups, and whose composition maps are bilinear. As in [@Mit72], one may view a small pre-additive category as an associative algebra with many objects, and as such, the usual notions about associative algebras, such as left/right modules, left/right Noetherian properites, tensor product of modules, derived categories of modules, etc, carries over to the case of small pre-additive categories. See Appendix \[app\_mod\_preadd\] for a summary of all the results needed for our discussion.
Let $\cF$ be the pre-additive category with objects set ${{\rm Ob}}(\cF) = \bZ$, and with Hom sets $\cF(i,j) := A_{i-j}$. Compositions are defined by the multiplication in $A$. It is clear that a (right) module of $\cF$ is the same as a graded module over $A$. Let ${\cF_{\geq w}}\subset \cF$ be the full subcategory with object set $\bZ_{\geq w}$. Then for any graded module $M$, the data $M_{\geq w}$ is naturally organized into a right module over ${\cF_{\geq w}}$. The observation can then be rewritten in the following form: $$\label{Cech_coherent_intro_3}
\parbox{40em}{If $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, then for each $w \in \bZ$, the restriction ${\check{\cC}}_{I^+}(M)_{\geq w}$ is in ${\cD^b_{{\rm coh}}}({\cF_{\geq w}})$.}$$
This observation is the crucial entry into the proof of the following result, which asserts that the “weight truncated” analogue of always preserve ${\cD^b_{{\rm coh}}}$. \[Dbcoh\_Fgw\_SOD\_intro\_thm\] There is a semi-orthogonal decomposition $$\label{Dbcoh_Fgw_SOD_intro}
{\cD^b_{{\rm coh}}}({\cF_{\geq w}}) = \langle \, \cD^b_{{\rm coh}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) \, , \, \cD^b_{{\rm coh}, \, {{\rm Tor}^+}}({\cF_{\geq w}}) \, \rangle$$ Moreover, there is an exact equivalence $\cD^b_{{\rm coh}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) \simeq {\cD^b_{{\rm coh}}}(\mathfrak{X}^+)$.
One can then use Theorem \[Dbcoh\_Fgw\_SOD\_intro\_thm\] to obtain results on $\cD({{\rm Gr}(A)})$. Indeed, since ${\cF_{\geq w}}$ is a full subcategory of $\cF$, one may use (left) Kan extensions to obtain a fully faithful functor $\scL_{[\geq w]} : \cD({\cF_{\geq w}}) {\hookrightarrow}\cD(\cF) \simeq \cD({{\rm Gr}(A)})$. By proving a certain Noetherian property of ${\cF_{\geq w}}$ (see Corollary \[Fgw\_Noeth\] below), one can show that this embedding always preserves ${\cD^{-}_{{\rm coh}}}$. Hence, a version of Theorem \[Dbcoh\_Fgw\_SOD\_intro\_thm\] with ${\cD^b_{{\rm coh}}}$ replaced by ${\cD^{-}_{{\rm coh}}}$ then implies that there is a three-term semi-orthogonal decomposition (see Theorem \[three\_term\_SOD\_Dmcoh\]) $${\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)}) \, = \, \langle \, \cD^-_{{{\rm coh}},<w}({{\rm Gr}(A)}) \, , \, \mathscr{L}_{[\geq w]} ( \cD^-_{{{\rm coh}}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) ) \, , \, \mathscr{L}_{[\geq w]} ( \cD^-_{{{\rm coh}}, \, {{\rm Tor}^+}}({\cF_{\geq w}}) ) \, \rangle$$ with the middle component equivalent to ${\cD^{-}_{{\rm coh}}}(\mathfrak{X}^+)$. Similar result was obtained in [@HL15b] using different methods. We believe that these give rise to the same semi-orthogonal decomposition. We also give a sufficient condition for this semi-orthogonal decomposition to restrict to ${\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$ (see Theorem \[cond\_then\_reg\]). In particular, this weakens the “Assumption (A)” in [@HL15]. When these conditions are satisfied, we also show that the semi-orthogonal decomposition thus obtained coincide with the one in [@HL15].
As we mentioned above, the technique of weight truncation is designed to faciliate the comparison of derived categories under wall-crossings. Theorem \[Dbcoh\_Fgw\_SOD\_intro\_thm\] is particularly convenient for that purpose, as it exhibits a strong symmetry between the two sides of the wall-crossing. Indeed, the analogue of Theorem \[Dbcoh\_Fgw\_SOD\_intro\_thm\] for the negative direction concerns the category ${\cF_{\leq -w}}$, which is in fact isomorphic to the opposite category of ${\cF_{\geq w}}$. In other words, ${\cD^b_{{\rm coh}}}(\mathfrak{X}^+)$ is a semi-orthogonal summand of the derived category ${\cD^b_{{\rm coh}}}({\cF_{\geq w}})$ of *right* ${\cF_{\geq w}}$-modules; while ${\cD^b_{{\rm coh}}}(\mathfrak{X}^-)$ is a semi-orthogonal summand of the derived category ${\cD^b_{{\rm coh}}}(({\cF_{\geq w}})^{{{\rm op}}})$ of *left* ${\cF_{\geq w}}$-modules. This suggests one to relate the two via a duality functor $$\label{bD_Fgw_intro}
\bD_{{\cF_{\geq w}}} : \cD({\cF_{\geq w}})^{{{\rm op}}} {\rightarrow}\cD(({\cF_{\geq w}})^{{{\rm op}}}) \simeq \cD({\cF_{\leq -w}}) \, , \qquad \quad
\cM \, \mapsto \, {{\bm R}{\rm Hom}}_{{\cF_{\geq w}}}(\cM, {\cF_{\geq w}})$$
We will show below that this works well when there is a certain duality between the local cohomology complexes ${{\bm R}\Gamma}_{I^+}(A)$ and ${{\bm R}\Gamma}_{I^-}(A)$. Let $\omega_Y^{\bullet} \in {\cD^b_{{\rm coh}}}(A_0)$ be a dualizing complex, and take the weight degreewise dualizing functor $$\bD_Y : \cD({{\rm Gr}(A)})^{{{\rm op}}} {\rightarrow}\cD({{\rm Gr}(A)}) \, , \qquad \quad \bD_Y(M)_i \simeq {{\bm R}{\rm Hom}}_{A_0}(M_{-i}, \omega_Y^{\bullet})$$ Then we assume that $$\label{local_cohom_duality_intro}
\parbox{40em}{There is an isomorphism ${{\bm R}\Gamma}_{I^+}(A)(a)[1] {\xrightarrow[]{\cong }}\bD_Y({{\bm R}\Gamma}_{I^-}(A))$ in $\cD({{\rm Gr}(A)})$, where $a \geq 0$.}$$
Indeed, one of the main results of [@Yeu20a] is that, for a large class of flips and flops, the assumption is satisfied for the sheaf of graded rings that controls the flip/flop, where $a=0$ for flops and $a=1$ for flips. We have the following (see Proposition \[D\_Fgw\_ITR\] below)
If holds for $a \geq 0$, then $\bD_{{\cF_{\geq w}}}$ sends $\cD^-_{{\rm coh}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$ to $\cD^+_{{\rm coh}, \, {I^-\text{-}{\rm triv}}}({\cF_{\leq -w}})$.
If holds for $a = 0$, then $\bD_{{\cF_{\leq -w}}}$ also sends $\cD^-_{{\rm coh}, \, {I^-\text{-}{\rm triv}}}({\cF_{\leq -w}})$ to $\cD^+_{{\rm coh}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$.
Thus, if the duality functor is suitably involutive, say if ${\cF_{\geq w}}$ is “Gorenstein” in a suitable sense, then this result says that the derived categories behave in the expected way: it shrinks under flips and remain equivalent under flops. In any case, the duality functor is always involutive on ${\cD_{{\rm perf}}}({\cF_{\geq w}})$, and we have the following (see Corollary \[flip\_flop\_Dperf\])
\[Dperf\_X\_pm\_intro\] Assume that the semi-orthogonal decomopsition and its negative version restrict to ${\cD_{{\rm perf}}}({\cF_{\geq w}})$ and ${\cD_{{\rm perf}}}({\cF_{\leq -w}})$ respectively. If holds for $a \geq 0$, then there is a fully faithful exact functor ${\cD_{{\rm perf}}}(\mathfrak{X}^+) {\hookrightarrow}{\cD_{{\rm perf}}}(\mathfrak{X}^-)$; if holds for $a = 0$, then there is an exact equivalence ${\cD_{{\rm perf}}}(\mathfrak{X}^+) \simeq {\cD_{{\rm perf}}}(\mathfrak{X}^-)$.
In fact, if $A$ is smooth, then we have ${\cD^b_{{\rm coh}}}({\cF_{\geq w}}) = {\cD_{{\rm perf}}}({\cF_{\geq w}})$, so that the assumption of this Corollary is indeed satisfied in good cases. In general, a further sharpening of Corollary \[Dperf\_X\_pm\_intro\] shows that the analogous conclusions at the level of Ind-completions hold provided that certain spectral sequences converge (see Remark \[pairing\_remark\]). Such a convergence issue seems to be similar to those arising in Koszul duality. It seems possible that a formal modification of our arguments might lead to a more satisfactory statement. This might open up a way to tackle a conjecture of Bondal and Orlov (see also [@Yeu20a]).
Derived categories of graded modules {#DGrA_sec}
====================================
In this section, we recall some basic results about graded rings, in order to establish notations and conventions. A more detailed discussion may be found in [@Yeu20c].
A *$\bZ$-graded ring* is a commutative ring $A$ with a $\bZ$-grading $A = \bigoplus_{n \in \bZ} A_n$. Here, by commutative we mean $xy = yx$, not $xy = (-1)^{|x||y|} yx$.
A *graded module* over $A$ will always mean a $\bZ$-graded module $M = \bigoplus_{n \in \bZ} M_n$.
We first recall the following result (see, e.g., [@BH93 Theorem 1.5.5]): \[Noeth\_gr\_ring\] Let $A$ be a $\bZ$-graded ring. Then the followings are equivalent:
1. $A$ is a Noetherian ring;
2. every graded ideal of $A$ is finitely generated;
3. $A_0$ is Noetherian, and both $A_{\geq 0}$ and $A_{\leq 0}$ are finitely generated over $A_0$;
4. $A_0$ is Noetherian, and $A$ is finitely generated over $A_0$.
Denote by ${{\rm Gr}}(A)$ the category of graded modules over $A$, whose morphisms are maps of graded modules of degree $0$. Given two graded modules $M,N \in {{\rm Gr}}(A)$, then the $A$-module $M \otimes_A N$ has a natural grading where $\deg(x\otimes y) = \deg(x) + \deg(y)$ for homogeneous $x,y \in A$. Moreover, one can define a graded $A$-module ${\underline{{\rm Hom}}}_A(M,N)$ whose degree $i$ part is the set of $A$-linear homomorphism from $M$ to $N$ of homogeneous degree $i$. Thus, in particular, we have ${{\rm Hom}}_A(M,N) := {{\rm Hom}}_{{{\rm Gr}}(A)}(M,N) = {\underline{{\rm Hom}}}_A(M,N)_0$. These form the internal Hom objects with respect to the graded tensor product. The abelian category ${{\rm Gr}(A)}$ is a Grothendieck category, with a set $\{A(-i)\}_{i \in \bZ}$ of generators. The same set is also a set of compact generators in the derived category $\cD({{\rm Gr}(A)})$. Since ${{\rm Gr}(A)}$ is a Grothendieck category, the category of complexes has enough K-injectives (see, e.g., [@Sta Tag 079P]). Moreover, as in the ungraded case, it also has enough K-projectives (see, e.g., [@Sta Tag 06XX]). As a result, the bifunctors $-\otimes_A - $ and ${\underline{{\rm Hom}}}_A(-,-)$ admit derived functors $$\begin{split}
- \otimes_{A}^{{\bm L}} - \, &: \, \cD({{\rm Gr}(A)}) \, \times \, \cD( {{\rm Gr}(A)}) {\,\rightarrow \,}\cD({{\rm Gr}(A)}) \\
{{\bm R} \underline{{\rm Hom}}}_{A}(-,-) \, &: \, \cD({{\rm Gr}(A)})^{{{\rm op}}} \, \times \, \cD( {{\rm Gr}(A)}) {\,\rightarrow \,}\cD({{\rm Gr}(A)})
\end{split}$$ which can in turn be used to define ${\underline{{\rm Ext}}}^{\bullet}_A(M,N)$ and ${{\rm Tor}}^A_{\bullet}(M,N)$.
We now extend some standard results on the derived categories of modules to the graded case. We start with the following
An object $M \in \cD({{\rm Gr}(A)})$ is said to be *pseudo-coherent* if it can be represented by a bounded above complex of finitely generated projective graded $A$-modules. Denote by ${\cD_{{\rm pc}}}({{\rm Gr}(A)}) \subset \cD({{\rm Gr}(A)})$ the full subcategory consisting of pseudo-coherent objects.
Given a Noetherian $\bZ$-graded ring $A$, then for $\spadesuit \in \{\, \, \, ,+,-,b\}$, define ${\cD^{\tiny \mbox{$\spadesuit $}}}_{{{\rm coh}}}({{\rm Gr}(A)})$ the full subcategory of ${\cD^{\tiny \mbox{$\spadesuit $}}}({{\rm Gr}(A)})$ consisting of complexes $M \in {\cD^{\tiny \mbox{$\spadesuit $}}}({{\rm Gr}(A)})$ such that $H^p(M)$ is finitely generated for all $p \in \bZ$.
Then we have the following standard result (see Lemma \[Dbcoh\_pc\_lem\]):
Given a Noetherian $\bZ$-graded ring $A$, then for any $M \in \cD(\cA)$, the followings are equivalent:
1. $M \in {\cD_{{\rm pc}}}({{\rm Gr}(A)})$;
2. $M \in {\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)})$;
3. $M$ is quasi-isomorphic to a bounded above complex of free graded modules of finite rank.
Let $\cD_{{{\rm perf}}}({{\rm Gr}(A)})$ be the smallest split-closed triangulated subcategories containing the set $\{A(-i)\}_{i \in \bZ}$ of objects. By [@Rou08 Theorem 4.22] (see also [@Nee92 Lemma 2.2]), $\cD_{{{\rm perf}}}({{\rm Gr}(A)})$ is precisely the full subcategory of compact objects in $\cD({{\rm Gr}(A)})$. We mention the following graded analogue of [@Sta Tag 0ATK], whose proof is completely parallel to the ungraded case:
\[tensor\_in\_Hom\_target\] For any $N \in {\cD_{{\rm pc}}}({{\rm Gr}(A)})$, $L \in \cD^+({{\rm Gr}(A)})$, and $M \in \cD({{\rm Gr}(A)})$ of finite Tor dimension, the canonical map $$M \otimes_A^{{\bm L}} {{\bm R} \underline{{\rm Hom}}}_A(N,L) {\,\rightarrow \,}{{\bm R} \underline{{\rm Hom}}}_A(N, M\otimes_A^{{\bm L}} L)$$ is an isomorphism in $\cD({{\rm Gr}(A)})$.
Now we briefly discuss local cohomology on a graded ring.
\[I\_infty\_torsion\_def\] Let $I$ be a graded ideal in a $\bZ$-graded ring $A$. Given any graded module $M$ over $A$, an element $x\in M$ is said to be *$I^{\infty}$-torsion* if for every $f \in I$ there exists some $n > 0$ such that $f^n x = 0$. If $I$ is finitely generated, this is equivalent to $I^n x = 0$ for some $n > 0$. The graded module $M$ is said to be *$I^{\infty}$-torsion* if every element in it is $I^{\infty}$-torsion. Denote by ${I^{\infty}\text{-}{\rm Tor}}\subset {{\rm Gr}}(A)$ the full subcategory consisting of $I^{\infty}$-torsion modules.
It is clear that ${I^{\infty}\text{-}{\rm Tor}}\subset {{\rm Gr}(A)}$ is a Serre subcategory. Thus the full subcategory $\cD_{{I^{\infty}\text{-}{\rm Tor}}}({{\rm Gr}(A)}) \subset \cD({{\rm Gr}(A)})$ is a triangulated subcategory. If $I$ is finitely generated, then this inclusion has a right adjoint, which has a simple and useful description. To this end, we recall the following
\[RGam\_f\_def\] Let $f_1,\ldots,f_r$ be homogeneous elements in $A$, of degrees $d_1,\ldots,d_r$ respectively. For any graded module $M \in {{\rm Gr}}(A)$, we define the *local cohomology complex* (or *extended Čech complex*) of $M$ with respect to the tuple $(f_1,\ldots,f_r)$ to be the cochain complex of graded modules $$\label{RGam_f}
{{\bm R}\Gamma}_{(f_1,\ldots,f_r)}(M) \, := \, \bigl[ \, M {\xrightarrow[]{d^0}} \prod_{1 \leq i_0 \leq r} M_{f_{i_0}} {\xrightarrow[]{d^1}}
\prod_{1 \leq i_0 < i_1 \leq r} M_{f_{i_0}f_{i_1}} {\xrightarrow[]{d^2}} \ldots {\xrightarrow[]{d^{r-1}}}
M_{f_1\ldots f_r} \, \bigr]$$ whose differentials are defined by $d^m := \sum_{j=0}^m (-1)^j d^m_j$, where $d^m_j$ is the direct product of the canonical maps $d^m_j : A_{i_0\ldots \hat{i_j} \ldots i_m} {\rightarrow}A_{i_0\ldots i_m}$. Here, the first term $M$ is put in cohomological degree $0$.
For a cochain complex $M \in {{\rm Ch}}({{\rm Gr}}(A))$ of graded modules, we define ${{\bm R}\Gamma}_{(f_1,\ldots,f_r)}(M)$ to be the total complex of the double complex $C^{p,q} = {{\bm R}\Gamma}_{(f_1,\ldots,f_r)}(M^p)^q$.
The functor $M \mapsto {{\bm R}\Gamma}_{(f_1,\ldots,f_r)}(M)$ on ${{\rm Ch}}({{\rm Gr}}(A))$ is exact, and hence descend to a functor at the level of derived categories. Moreover, if we let $I := (f_1,\ldots,f_r)$ be the graded ideal generated by the elements $f_1$, then this functor has image inside the full subcategory $\cD_{{I^{\infty}\text{-}{\rm Tor}}}({{\rm Gr}}(A))$. Thus, this gives a functor $$\label{RGI_def}
{{\bm R}\Gamma_{I}}:= {{\bm R}\Gamma}_{(f_1,\ldots,f_r)} \, : \, \cD({{\rm Gr}}(A)) {\,\rightarrow \,}\cD_{{I^{\infty}\text{-}{\rm Tor}}}({{\rm Gr}}(A))$$
Moreover, the map $\epsilon_M : {{\bm R}\Gamma}_{(f_1,\ldots,f_r)}(M) {\rightarrow}M$ defined by projecting to the first component of gives rise to a natural transformation $$\label{RGam_counit}
\epsilon \, : \, \iota \circ {{\bm R}\Gamma_{I}}\, \Rightarrow \, {{\rm id}}$$ where $\iota : \cD_{{I^{\infty}\text{-}{\rm Tor}}}({{\rm Gr}}(A)) {\rightarrow}\cD({{\rm Gr}}(A))$ is the inclusion functor. Then we have
\[RGam\_right\_adj\] The functor is a right adjoint to the inclusion $\iota : \cD_{{I^{\infty}\text{-}{\rm Tor}}}({{\rm Gr}}(A)) {\rightarrow}\cD({{\rm Gr}}(A))$, with counit given by . In particular the functor depends only on the graded ideal $I$.
The cone of $\epsilon_M$ is homotopic to the kernel of $\epsilon_M$ shifted by $1$, which is given by the following \[Cech\_cplx\_def\] The *Čech complex* of a graded module $M$ with respect to a tuple $(f_1,\ldots,f_r)$ of homogeneous elements is the cochain complex of graded modules $$\label{Ce_f}
{\check{\cC}_I}(M) \, = \, {\check{\cC}}_{(f_1,\ldots,f_r)}(M) \, := \, \bigl[ \, \prod_{1 \leq i_0 \leq r} M_{f_{i_0}} {\, \xrightarrow[]{-d^1} \,}
\prod_{1 \leq i_0 < i_1 \leq r} M_{f_{i_0}f_{i_1}} {\, \xrightarrow[]{-d^2} \,} \ldots {\, \xrightarrow[]{-d^{r-1}} \,}
M_{f_1\ldots f_r} \, \bigr]$$ given as a subcomplex of , shifted by one. As in Definition \[RGam\_f\_def\], this definition can be extended to cochain complexes $M \in {{\rm Ch}}({{\rm Gr}}(A))$ by taking the total complex.
Thus, for each $M \in \cD({{\rm Gr}(A)})$, there is an exact triangle $$\label{RGam_Ce_seq}
\ldots {\,\rightarrow \,}{{\bm R}\Gamma_{I}}(M) {\, \xrightarrow[]{\epsilon_M} \,} M {\, \xrightarrow[]{\eta_M} \,} {\check{\cC}_I}(M) {\, \xrightarrow[]{\delta_M} \,} {{\bm R}\Gamma_{I}}(M)[1] {\,\rightarrow \,}\ldots$$ where $\eta_M = -d^0$, the negative of the first differential in , and $\delta_M$ is the inclusion.
The exact triangle turns out to be the decomposition triangle associated to a semi-orthogonal decomposition. To this end, we recall the following \[I\_triv\_equiv\] For any $M \in \cD({{\rm Gr}(A)})$, we have ${{\bm R}\Gamma_{I}}(M) \simeq 0$ if and only if ${{\bm R} \underline{{\rm Hom}}}_A({{\bm R}\Gamma_{I}}(A),M) \simeq 0$.
We say that $M \in \cD({{\rm Gr}(A)})$ is *$I$-trivial* if we have ${{\bm R}\Gamma_{I}}(M) \simeq 0$, or equivalently ${{\bm R} \underline{{\rm Hom}}}_A({{\bm R}\Gamma_{I}}(A),M) \simeq 0$ by Lemma \[I\_triv\_equiv\]. Denote by $\cD_{{I\text{-}{\rm triv}}}({{\rm Gr}(A)}) \subset \cD({{\rm Gr}(A)})$ the full subcategory consisting of $I$-trivial objects.
There is a semi-orthogonal decomposition $$\label{local_cohom_SOD}
\cD({{\rm Gr}(A)}) \, = \, \langle \, \cD_{{I\text{-}{\rm triv}}}({{\rm Gr}(A)}) \, , \, \cD_{{I^{\infty}\text{-}{\rm Tor}}}({{\rm Gr}(A)}) \, \rangle$$ whose associated decomposition triangle is given by .
We are mostly interested in the special case $I = I^+ := A_{>0} \cdot A$, where the corresponding semi-orthogonal decomposition is closely related to the projective space $X^+ = {{\rm Proj}}^+(A) := {{\rm Proj}}(A_{\geq 0})$. Similarly, the semi-orthogonal decomposition for $I = I^- := A_{<0} \cdot A$ is closely related to the projective space $X^- = {{\rm Proj}}^-(A) := {{\rm Proj}}(A_{\leq 0})$.
A first instance of this relation is the interpretation of ${\check{\cC}}_{I^+}(M)$ in terms of a derived pushforward functor. Namely, consider the map $\pi^+ : {{\rm Proj}}^+(A) {\rightarrow}{{\rm Spec}}(A_0) =: Y$, then for each $M \in \cD({{\rm Gr}(A)})$ and for each $i \in \bZ$, there is a canonical isomorphism in $\cD(A_0)$: $$\label{Ce_i_RGam}
{\check{\cC}}_{I^+}(M)_i \, \cong \, {\bm R}\pi^+_*\, \widetilde{M(i)}$$ where we denote by $\widetilde{M} \in {{\rm QCoh}}(X^+)$ the quasi-coherent sheaf on $X^+$ associated to $M \in {{\rm Gr}(A)}$. Indeed, follows from the usual way of computing cohomology of quasi-coherent sheaves from the Čech complex (see [@Yeu20c (4.10)]). This gives a proof of the following two results (see [@Yeu20c Lemma 4.12, 4.13] for details): \[RGam\_Dbcoh\] If $A$ is Noetherian, then for any $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, we have ${\check{\cC}}_{I^+}(M)_i \in {\cD^b_{{\rm coh}}}(A_0)$ and ${{\bm R}\Gamma}_{I^+}(M)_i \in {\cD^b_{{\rm coh}}}(A_0)$ for each weight $i \in \bZ$.
\[local\_cohom\_weight\_bounded\] If $A$ is Noetherian, then for any $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, there exists $c^+, c^- \in \bZ$ such that ${{\bm R}\Gamma}_{I^+}(M)_i \simeq 0$ for all $i \geq c^+$ and ${{\bm R}\Gamma}_{I^-}(M)_i \simeq 0$ for all $i \leq c^-$.
Serre’s equivalence gives another relation. We will use the version of Serre’s equivalence proved in [@Yeu20c], where the usual condition that $A$ be generated by $A_1$ over $A_0$ is replaced by the following condition: \[frac\_Cartier\] A $\bZ$-graded ring $A$ is said to be *positively $\tfrac{1}{d}$-Cartier*, for an integer $d > 0$, if the canonical map $\widetilde{A(di)} \otimes_{\cO_{X^+}} \widetilde{A(dj)} {\rightarrow}\widetilde{A(di+dj)}$ in ${{\rm QCoh}}(X^+)$ is an isomorphism for all $i,j \in \bZ$. In the case $d = 1$, we simply say that $A$ is *positively Cartier*.
For example, if $A_{\geq 0}$ is generated over $A_0$ by homogeneous elements $f_1,\ldots,f_p$ of positive degrees $d_i := \deg(f_i) > 0$, then it can be shown that $A$ is positively $\tfrac{1}{d}$-Cartier for any $d > 0$ that is divisible by each of $d_i$ (see [@Yeu20c Lemma 3.5]).
Denote by ${{\rm Tor}}^+ := {(I^+)^{\infty}\text{-}{\rm Tor}}\subset {{\rm Gr}(A)}$ the full subcategory consisting of $(I^+)^{\infty}$-torsion modules in the sense of Definition \[I\_infty\_torsion\_def\], and denote by ${{\rm Q}^+{\rm Gr}}(A)$ the Serre quotient ${{\rm Q}^+{\rm Gr}}(A) := {{\rm Gr}(A)}/ {{\rm Tor}^+}$. Then we have (see [@Yeu20c Theorem 3.15]) \[Serre\_equiv\] Suppose that $A$ is Noetherian and positively Cartier, then there is an equivalence of categories $$\begin{tikzcd}
\!\,^0 \! \cL^{+} \,:\, {{\rm QCoh}}(X^+) \ar[r, shift left]
& {{\rm Q}^+{\rm Gr}}(A) \, : \, (-)^{\sim} \ar[l, shift left]
\end{tikzcd}$$
\[stacky\_Proj\] When $A$ is not positively Cartier, the category ${{\rm Q}^+{\rm Gr}}(A)$ also admit a description in terms of quasi-coherent sheaves on a stacky projective space. Namely, one can show that the map of $\bG_m$-equivariant schemes ${{\rm Spec}}(A) {\rightarrow}{{\rm Spec}}(A_{\geq 0})$ induces an isomorphism on the $\bG_m$-invariant open subschemes $${{\rm Spec}}(A) \setminus {{\rm Spec}}(A/I^+) {\, \xrightarrow[]{\cong} \,} {{\rm Spec}}(A_{\geq 0}) \setminus {{\rm Spec}}(A_0)$$
If we denote this $\bG_m$-equivariant scheme by $W^{ss}(+)$, and let $\, {\mathpzc{Proj}}^+(A)$ be the quotient stack $[W^{ss}(+)/\bG_m]$, then we have an equivalence $
{{\rm Q}^+{\rm Gr}}(A) \simeq {{\rm QCoh}}({\mathpzc{Proj}}^+(A))
$. See, e.g., [@AKO08 Proposition 2.3] or [@HL15 Example 2.15].
Given a Serre quotient $\phi^* : \cC {\rightarrow}\cC/\cT$, there are some general criteria laid out in [@Yeu20c Appendix A] which guarantees that the derievd category of the quotient $\cD(\cC/\cT)$ coincides with the Verdier quotient $\cD(\cC)/\cD_{\cT}(\cC)$ of the derived categories. If $A$ is Noetherian and positively Cartier, then Theorem \[Serre\_equiv\] asserts that ${{\rm QCoh}}(X^+)$ is a Serre quotient. The corresponding criteria can be easily verified, and we have
The functor $\phi^* : \cD({{\rm Gr}(A)}) {\rightarrow}\cD({{\rm Q}^+{\rm Gr}}(A))$ has a fully faithful right adjoint ${\bm R}\phi_* : \cD({{\rm Q}^+{\rm Gr}}(A)) {\rightarrow}\cD({{\rm Gr}(A)})$, which induces a semi-orthogonal decomposition $$\cD({{\rm Gr}(A)}) \, = \, \langle \, {\bm R}\phi_* ( \cD({{\rm Q}^+{\rm Gr}}(A))) \, , \, \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)}) \, \rangle$$ Comparing with , we see that the following is a pair of inverse exact equivalences: $$\label{QpGrA_IpTR_equiv}
\begin{tikzcd}
\phi^* \, : \, {\cD^{\tiny \mbox{$\spadesuit $}}}_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}) \ar[r, shift left]
& {\cD^{\tiny \mbox{$\spadesuit $}}}({{\rm Q}^+{\rm Gr}}(A)) \, : \, {\bm R}\phi_* \ar[l, shift left]
\end{tikzcd}$$
The exact equivalence restricts to certain bounded coherent subcategories. One has to be careful that the relevant subcategory of $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$ is not given by $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}) \cap {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$. Instead, one considers the following \[IpTR\_coh\_def\] Denote by ${{\rm gr}}(A) \subset {{\rm Gr}(A)}$ the full subcategory of finitely generated graded modules. Let ${{\rm q}^+{\rm gr}}(A) \subset {{\rm Q}^+{\rm Gr}}(A)$ be the essentially image of ${{\rm gr}}(A)$ under $\phi^* : {{\rm Gr}(A)}{\rightarrow}{{\rm Q}^+{\rm Gr}}(A)$. For each $\spadesuit \in \{\, \, \, ,+,-,b\}$,
1. let ${\cD^{\tiny \mbox{$\spadesuit $}}}_{{{\rm coh}}}({{\rm Q}^+{\rm Gr}}(A)) \subset {\cD^{\tiny \mbox{$\spadesuit $}}}({{\rm Q}^+{\rm Gr}}(A))$ be the full subcategory consisting of complexes whose cohomology lies in ${{\rm q}^+{\rm gr}}(A)$; and
2. let ${\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}({I^+\text{-}{\rm triv}})}({{\rm Gr}(A)}) \subset {\cD^{\tiny \mbox{$\spadesuit $}}}_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$ be the essential image of ${\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}}({{\rm Gr}(A)})$ under the functor ${\check{\cC}}_{I^+} : {\cD^{\tiny \mbox{$\spadesuit $}}}({{\rm Gr}(A)}) {\rightarrow}{\cD^{\tiny \mbox{$\spadesuit $}}}_{{I^+\text{-}{\rm triv}}}(GrA)$.
\[coh\_IpTR\_QpGr\] For $\spadesuit \in \{-,b\}$, the equivalence restricts to give an exact equivalence $$\begin{tikzcd}
\phi^* \, : \, {\cD^{\tiny \mbox{$\spadesuit $}}}_{{{\rm coh}}({I^+\text{-}{\rm triv}})}({{\rm Gr}(A)}) \ar[r, shift left]
& {\cD^{\tiny \mbox{$\spadesuit $}}}_{{{\rm coh}}}({{\rm Q}^+{\rm Gr}}(A)) \, : \, {\bm R}\phi_* \ar[l, shift left]
\end{tikzcd}$$
Thus, if $A$ is positively Cartier, then the subcategory $\cD^b_{{{\rm coh}}({I^+\text{-}{\rm triv}})}({{\rm Gr}(A)})$ is equivalent to the usual bounded derived category of coherent sheaves on $X^+ = {{\rm Proj}}^+(A)$. In general, it is equivalent to that of the stacky projective space ${\mathpzc{Proj}}^+(A)$ by Remark \[stacky\_Proj\].
Weight truncation
=================
In this section, we use freely the language of modules over small pre-additive categories. The reader is referred to Appendix \[app\_mod\_preadd\] for details.
Given any $\bZ$-graded ring $A$, let $\cF = \cF_A$ be the pre-additive category with object set ${{\rm Ob}}(\cF) = \bZ$, and Hom spaces $\cF(i,j) := A_{i-j}$. Compositions in $\cF$ are defined by multiplication in $A$ in the obvious way. A right $\cF$-module is nothing but a graded (right) $A$-module. More precisely, there is an equivalence of abelian categories $$\label{Fmod_GrA_1}
(-)^{\sharp} \, : \, {{\rm Gr}(A)}{\, \xrightarrow[]{\simeq} \,} {{\rm Mod}}(\cF) \, , \qquad \quad (M^{\sharp})_i := M_i$$ whose inverse will be denoted as $(-)^{\flat} : {{\rm Mod}}(\cF) {\xrightarrow[]{\simeq}} {{\rm Gr}(A)}$.
Since $A$ is assumed to be commutative, any graded module $M \in {{\rm Gr}(A)}$ in fact induces an $\cF$-bimodule. In other words, there is an additive functor $$\label{tilde_F_bimod}
{{\rm Gr}}(A) {\,\rightarrow \,}{{\rm Mod}}(\cF^e) \, , \quad M \mapsto \widetilde{M} \, , \quad \text{where} \quad \,_j\widetilde{M}_i := M_{i-j}$$ which recovers the functor by restricting to $\,_0\widetilde{M}_{*}$.
The graded tensor products and graded Hom spaces between graded modules can be expressed naturally in terms of $\cF$-bimodules. Namely, for any $M,N \in {{\rm Gr}(A)}$, there are natural isomorphisms of $\cF$-bimodules $$\label{cF_A_tensor_Hom}
\widetilde{M} \otimes_{\cF} \widetilde{N} \, \cong \, \widetilde{M \otimes_A N}
\qquad \text{ and } \qquad
{{\rm Hom}}_{\cF}(\widetilde{M} , \widetilde{N}) \, \cong \, \widetilde{{\underline{{\rm Hom}}}_A(M,N)}$$ In particular, if we only remember the right $\cF$-module structure of $\widetilde{M}$, then we have $$\label{sharp_tensor_tilde}
M^{\sharp} \otimes_{\cF} \widetilde{N} \, \cong \, (M \otimes_A N)^{\sharp}$$
Since $A$ is assumed to be commutative, the pre-additive category $\cF = \cF_A$ admits an involution, [[*i.e.*]{}]{}, it comes equipped with an isomorphism $\cF \cong \cF^{{{\rm op}}}$ of pre-additive categories, defined by $i \mapsto -i$ on objects, and $\cF(i,j) = A_{i-j} = \cF^{{{\rm op}}}(-i,-j)$ on Hom sets. In fact, the commutativity of $A$ is equivalent to the fact that this assignment $\cF {\rightarrow}\cF^{{{\rm op}}}$ is a functor. This involution induces an isomorphism of categories $$(-)^{\tau} \, : \, {{\rm Mod}}(\cF) {\, \xrightarrow[]{\cong} \,} {{\rm Mod}}(\cF^{{{\rm op}}}) \, , \qquad \,_i(M^{\tau}) := M_{-i}$$ whose (strict) inverse will also be denoted as $(-)^{\tau}$.
For any integer $a \in \bZ$, let ${\cF_{\geq a}}$ be the full subcategory of $\cF$ on the subset ${{\rm Ob}}({\cF_{\geq a}}) = \bZ_{\geq a} \subset \bZ = {{\rm Ob}}(\cF)$. Define ${\cF_{\leq a}}\subset \cF$ in the similar way. We will also write $\cF_{\leq \infty} = \cF = \cF_{\geq -\infty}$. For any $-\infty \leq a' \leq a$, denote by $(-)_{\geq a} : {{\rm Mod}}(\cF_{\geq a'}) {\rightarrow}{{\rm Mod}}(\cF_{\geq a})$ the restriction functor.
Notice that the involution on $\cF$ restricts to an isomorphism $({\cF_{\geq a}})^{{{\rm op}}} \cong {\cF_{\leq -a}}$. As a result, there is again an isomorphism of categories $$\label{transpose_Fga}
(-)^{\tau} \, : \, {{\rm Mod}}({\cF_{\geq a}}) {\, \xrightarrow[]{\cong} \,} {{\rm Mod}}(({\cF_{\leq -a}})^{{{\rm op}}}) \, , \qquad \,_i(M^{\tau}) := M_{-i}$$ whose inverse will also be denoted as $(-)^{\tau}$. Similarly, the functor $(M^{\tau})_i := \!\,_{-i}M$ gives an isomorphism of categories $(-)^{\tau} : {{\rm Mod}}(({\cF_{\geq a}})^{{{\rm op}}}) {\xrightarrow[]{\cong}} {{\rm Mod}}({\cF_{\leq -a}})$ whose inverse will also be denoted as $(-)^{\tau}$.
From now on, we fix an integer $w \in \bZ$. Notice that the inclusion functor ${\cF_{\geq w}}{\hookrightarrow}\cF$ induces a three-way adjunction $$\label{Fga_F_adj}
\begin{tikzcd}
{{\rm Mod}}({\cF_{\geq w}}) \ar[rr, bend left, "-\otimes_{{\cF_{\geq w}}} \cF"] \ar[rr, bend right, "{{\rm Hom}}_{{\cF_{\geq w}}}(\cF\text{,}-)"']
& & {{\rm Mod}}(\cF) \ar[ll, "(-)_{\geq w}"']
\end{tikzcd}$$
In fact, under the equivalence ${{\rm Mod}}(\cF) \simeq {{\rm Gr}(A)}$, the right-pointing functor $- \otimes_{{\cF_{\geq w}}} \cF$ on the top may be characterized as the unqiue cocontinuous functor satisfying $$\label{weight_tensor_free}
\parbox{40em}{For each $i \geq w$, the functor $(- \otimes_{{\cF_{\geq w}}} \cF)^{\flat} : {{\rm Mod}}({\cF_{\geq w}}) {\rightarrow}{{\rm Gr}}(A)$ sends the free module $\!\,_{i}{\cF_{\geq w}}$ to the free graded module $A(-i)$}$$
Since the inclusion functor ${\cF_{\geq w}}{\rightarrow}\cF$ is fully faithful, we have $$\label{tensor_restrict_Fgw}
\begin{split}
(M \otimes_{{\cF_{\geq w}}} \cF)_{\geq w} \, &\cong \, M \otimes_{{\cF_{\geq w}}} {\cF_{\geq w}}\, \cong \, M \\
{{\rm Hom}}_{{\cF_{\geq w}}}(\cF, M)_{\geq w} \, &\cong \, {{\rm Hom}}_{{\cF_{\geq w}}}({\cF_{\geq w}}, M) \, \cong \, M
\end{split}$$
One can also relate ${\cF_{\geq w}}$-modules to graded $A_{\geq 0}$-modules. Indeed, for any $\cM \in {{\rm Mod}}({\cF_{\geq w}})$, the assignment $M_i := \cM_i$ defines a graded $A_{\geq 0}$-module, which will be denoted as $\cM|_{A_{\geq 0}}$. This can be used to characterize finite generated ${\cF_{\geq w}}$-modules:
\[Fga\_fg\_module\] Suppose that $A$ is Noetherian, then for any $\cM \in {{\rm Mod}}({\cF_{\geq w}})$, the followings are equivalent:
1. $\cM$ is a finitely generated ${\cF_{\geq w}}$-module in the sense of Definition \[fin\_gen\_mod\];
2. there exists a finitely generated graded $A$-module $M$ such that $\cM \cong M^{\sharp}_{\geq w} := (M^{\sharp})_{\geq w}$;
3. $\cM|_{A_{\geq 0}}$ is a finitely generaeted graded $A_{\geq 0}$-module.
For $(1) \Rightarrow (2)$, simply notice that if there is an epimorphism $\oplus_{j = 1}^m \, \,_{i_j}{\cF_{\geq w}}{\twoheadrightarrow}\cM$, then applying $-\otimes_{{\cF_{\geq w}}} \cF$, we have by an epimorphism $\oplus_{j = 1}^m \, A(-i_j) {\twoheadrightarrow}(\cM \otimes_{{\cF_{\geq w}}} \cF)^{\flat}$, which shows that $M := (\cM \otimes_{{\cF_{\geq w}}} \cF)^{\flat}$ is a finitely generated graded $A$-module. Moreover, it satisfies $\cM \cong M^{\sharp}_{[\geq w]}$ by .
For $(2) \Rightarrow (3)$, we use the Noetherian condition. By Proposition \[Noeth\_gr\_ring\], $A_0$ is Noetherian, and each $A_i$ is finitely generated as a module over $A_0$. Moreover, there exists integers $d>0$ and $N_0 > 0$ such that whenever $N \geq N_0$, we have $A_{N} = A_{d} \cdot A_{N-d}$ (see, for example, [@Yeu20c Lemma 3.2]). Now if $M$ is generated by $\xi_1,\ldots,\xi_r$ over $A$, then for any $j \geq N_0 + \max_i\{ \deg(\xi_i) \}$, we have $M_j = A_d \cdot M_{j-d}$. Since each $M_i$ is clearly finitely generated over $A_0$, the same is true for $\oplus_{w \leq j \leq N_0 + \max_i\{ \deg(\xi_i) \}} M_i$, which then generates $M^{\#}_{[\geq w]}|_{A_{\geq 0}}$ as an $A_{\geq 0}$-module.
The implication $(3) \Rightarrow (1)$ follows directly from the definitions.
\[Fgw\_Noeth\] If the $\bZ$-graded ring $A$ is Noetherian, then the small pre-additive category ${\cF_{\geq w}}$ is Noetherian.
By the characterization of finitely generated right ${\cF_{\geq w}}$-modules in Lemma \[Fga\_fg\_module\](3), we see that ${\cF_{\geq w}}$ is right Noetherian. The same argument also shows that $\cF_{\leq -w}$ is right Noetherian. By the isomorphism of categories $(-)^{\tau} : {{\rm Mod}}(({\cF_{\geq w}})^{{{\rm op}}}) {\xrightarrow[]{\cong}} {{\rm Mod}}(\cF_{\leq -w})$, we see that ${\cF_{\geq w}}$ is left Noetherian as well.
Now we take the derived functors of the functors appearing in . Under the identification $\cD(\cF) \simeq \cD({{\rm Gr}(A)})$, we denote these derived functors as $$\begin{split}
\mathscr{L}_{[\geq w]} \, &: \, \cD({\cF_{\geq w}}) {\,\rightarrow \,}\cD({{\rm Gr}(A)}) \, , \qquad \mathscr{L}_{[\geq w]}(\cM) \, := \, (\cM \otimes^{{\bm L}}_{{\cF_{\geq w}}} \cF)^{\flat} \\
\mathscr{R}_{\{\geq w\}} \, &: \, \cD({\cF_{\geq w}}) {\,\rightarrow \,}\cD({{\rm Gr}(A)}) \, , \qquad \mathscr{R}_{\{\geq w\}}(\cM) \, := \,
({{\bm R}{\rm Hom}}_{{\cF_{\geq w}}}(\cF , \cM) ) ^{\flat}
\end{split}$$
Let $\cD_{<w}({{\rm Gr}(A)}) \subset \cD({{\rm Gr}(A)})$ be the full subcategory consisting of $M \in \cD({{\rm Gr}(A)})$ such that $M_i \simeq 0$ for all $i \geq w$.
Then we have a recollement $$\label{weight_recollement}
\begin{tikzcd}
\cD_{<w}({{\rm Gr}(A)}) \ar[rr, "\iota"]
&& \cD({{\rm Gr}(A)}) \ar[rr, "(-)^{\sharp}_{\geq w}"] \ar[ll, bend right, "\cL_{<w}"'] \ar[ll, bend left, "\cR_{<w}"']
&& \cD(\cF_{\geq w}) \ar[ll, bend right, "\mathscr{L}_{[\geq w]}"'] \ar[ll, bend left, "\mathscr{R}_{\{\geq w\}}"']
\end{tikzcd}$$ where we have written $M^{\sharp}_{\geq w} := (M^{\sharp})_{\geq w}$.
Indeed, if we denote by $\cL_{[\geq w]}$ and $\cR_{\{\geq w\}}$ the endofunctors on $\cD({{\rm Gr}(A)})$ given by the compositions $$\cL_{[\geq w]}(M) \, := \, \mathscr{L}_{[\geq w]}(M^{\#}_{\geq w}) \qquad \text{and} \qquad
\cR_{\{\geq w\}}(M) \, := \, \mathscr{R}_{\{\geq w\}}(M^{\#}_{\geq w})$$ then the functors $\cL_{< w}$ and $\cR_{<w}$ are defined by the exact triangles $$\label{weight_trunc_exact_tri}
\begin{split}
\ldots {\,\rightarrow \,}\cL_{[\geq w]}(M) {\,\rightarrow \,}&M {\,\rightarrow \,}\cL_{< w}(M) {\,\rightarrow \,}\cL_{[\geq w]}(M)[1] {\,\rightarrow \,}\ldots \\
\ldots {\,\rightarrow \,}\cR_{< w}(M) {\,\rightarrow \,}&M {\,\rightarrow \,}\cR_{\{\geq w\}}(M) {\,\rightarrow \,}\cR_{<w}(M)[1] {\,\rightarrow \,}\ldots
\end{split}$$
We are mostly interested in the functors $\mathscr{L}_{[\geq w]}$, $\cL_{[\geq w]}$ and $\cL_{<w}$ instead of their counterpart $\mathscr{R}_{\{\geq w\}}$, $\cR_{\{\geq w\}}$ and $\cR_{<w}$. In particular, we emphasize the following
Let $\cD_{[\geq w]}({{\rm Gr}(A)}) \subset \cD({{\rm Gr}(A)})$ be the essential image of the fully faithful functor $\mathscr{L}_{[\geq w]} : \cD({\cF_{\geq w}}) {\rightarrow}\cD({{\rm Gr}(A)})$.
Alternatively, the subcategory $\cD_{[\geq w]}({{\rm Gr}(A)}) \subset \cD({{\rm Gr}(A)})$ may be characterized as follows: \[D\_geq\_w\_gen\] $\cD_{[\geq w]}({{\rm Gr}(A)})$ is the smallest strictly full triangulated subcategory containing the objects $A(-i)$ for $i \geq w$ and is closed under small coproducts. Therefore, we have $$\cD_{[\geq w]}({{\rm Gr}(A)}) \cap \cD({{\rm Gr}(A)})_c \, = \, \cD_{[\geq w]}({{\rm Gr}(A)})_c \, = \, {\rm EssIm}(\,\scL_{[\geq w]} : \cD_{{{\rm perf}}}({\cF_{\geq w}}) {\rightarrow}\cD({{\rm Gr}(A)}) \,)$$ where the subscript $(-)_c$ denotes the full subcategory of compact objects.
The first statement follows from and the fact that $\scL_{[\geq w]}$ preserves small coproducts. For the second statement, the first equality is standard (see, e.g., [@Nee92 Lemma 2.2] or [@Rou08 Theorem 5.3]). The second equality follows from the standard fact (see, e.g., ) that $\cD_{{{\rm perf}}}({\cF_{\geq w}}) = \cD({\cF_{\geq w}})_c$.
It follows immediately from the recollement that there is a semi-orthogonal decomposition $$\label{weight_SOD}
\cD({{\rm Gr}(A)}) = \langle \, \cD_{< w}({{\rm Gr}(A)}) \, , \, \cD_{[\geq w]}({{\rm Gr}(A)}) \, \rangle$$
The notation $\cD_{[\geq w]}$ is meant to convey the idea that these are objects “generated in weight $\geq w$"; while the notation $\cD_{< w}$ means that these are objects “concentrated in weight $<w$".
Local cohomology and weight truncation
======================================
Now we study local cohomology under weight truncation. For any finitely generated graded ideal $I \subset A$, the objects ${{\bm R}\Gamma_{I}}(A)$ and ${\check{\cC}_I}(A)$ in $\cD({{\rm Gr}(A)})$ give rise to the objects $\!\,_{\geq w}\widetilde{{{\bm R}\Gamma_{I}}(A)}_{\geq w}$ and $\!\,_{\geq w}\widetilde{{\check{\cC}_I}(A)}_{\geq w}$ in the derived category $\cD(({\cF_{\geq w}})^e)$ of ${\cF_{\geq w}}$-bimodules. Tensoring over these give rise to functors $$\label{RGam_Ce_weight}
\begin{split}
{{\bm R}\Gamma}_{I,\geq w} &: \cD({\cF_{\geq w}}) {\rightarrow}\cD({\cF_{\geq w}}) , \quad \cM \mapsto \cM \otimes^{{\bm L}}_{{\cF_{\geq w}}} \widetilde{{{\bm R}\Gamma_{I}}(A)}_{\geq w} \cong
((\mathscr{L}_{[\geq w]}\cM) \otimes_A^{{\bm L}} {{\bm R}\Gamma_{I}}(A) )^{\sharp}_{\geq w} \\
{\check{\cC}}_{I,\geq w} &: \cD({\cF_{\geq w}}) {\rightarrow}\cD({\cF_{\geq w}}) , \quad \cM \mapsto \cM \otimes^{{\bm L}}_{{\cF_{\geq w}}} \widetilde{{\check{\cC}_I}(A)}_{\geq w} \cong
((\mathscr{L}_{[\geq w]}\cM) \otimes_A^{{\bm L}} {\check{\cC}_I}(A) )^{\sharp}_{\geq w}
\end{split}$$ where the last isomorphisms on each line is obtained by applying and .
We are mostly interested in the case $I = I^+$, where these functors behave very similarly to the corresponding ones on $\cD({{\rm Gr}(A)})$ (see Proposition \[RGam\_Ce\_weight\_descend\] and Theorem \[RGam\_Ce\_weight\_ortho\] below). In fact, these properties are formal consequences of the following easy \[weight\_subset\_tor\] We have $\cD_{<w}({{\rm Gr}(A)}) \subset \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$.
which can be used to prove the following two results:
\[Torp\_weight\_lem\] If $M \in \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$ then $\cL_{[\geq w]}(M) \in \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$.
If ${{\bm R}\Gamma}_{I^+}(M) \in \cD_{<w}({{\rm Gr}(A)})$ then $\cL_{[\geq w]}(M) \otimes_A^{{\bm L}} {{\bm R}\Gamma}_{I^+}(A) \in \cD_{< w}({{\rm Gr}(A)})$.
For any $M \in \cD({{\rm Gr}(A)})$, we have $\cL_{<w}(M) \in \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$ by Lemma \[weight\_subset\_tor\], so that the first statement follows directly from the exact triangle in the first row of . For the second statement, apply ${{\bm R}\Gamma}_{I^+}(-)$ to the same exact triangle, and notice that $\cL_{<w}(M) \otimes_A^{{\bm L}} {{\bm R}\Gamma}_{I^+}(A) \simeq \cL_{<w}(M) \in \cD_{<w}({{\rm Gr}(A)})$ because $\cL_{<w}(M) \in \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$.
\[RGam\_Ce\_weight\_descend\] The following two functors commute up to isomorphism of functors: $$\begin{tikzcd}
\cD({{\rm Gr}(A)}) \ar[r, "{{\bm R}\Gamma}_{I^+}"] \ar[d, "(-)^{\sharp}_{\geq w}"'] & \cD({{\rm Gr}(A)}) \ar[d, "(-)^{\sharp}_{\geq w}"]
& & \cD({{\rm Gr}(A)}) \ar[r, "{\check{\cC}}_{I^+}"] \ar[d, "(-)^{\sharp}_{\geq w}"'] & \cD({{\rm Gr}(A)}) \ar[d, "(-)^{\sharp}_{\geq w}"] \\
\cD({\cF_{\geq w}}) \ar[r, "{{\bm R}\Gamma}_{I^+,\geq w}"] & \cD({\cF_{\geq w}})
& & \cD({\cF_{\geq w}}) \ar[r, "{\check{\cC}}_{I^+,\geq w}"] & \cD({\cF_{\geq w}})
\end{tikzcd}$$
Given any $M \in \cD({{\rm Gr}(A)})$, take the exact triangle in the first row of . By Lemma \[weight\_subset\_tor\], we have $\cL_{<w}(M) \in \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$. Applying ${\check{\cC}}_{I^+}$ to this exact triangle, we have ${\check{\cC}}_{I^+}(\cL_{[\geq w]} M) \cong {\check{\cC}}_{I^+}(M)$ in $\cD({{\rm Gr}(A)})$. Applying $(-)^{\sharp}_{\geq w}$ to this isomorphism gives the commutativity of the second square. Similarly, applying ${{\bm R}\Gamma}_{I^+}$ to the same exact triangle, we have an exact triangle $$\ldots {\,\rightarrow \,}{{\bm R}\Gamma}_{I^+}(\cL_{[\geq w]}(M)) {\,\rightarrow \,}{{\bm R}\Gamma}_{I^+}(M) {\,\rightarrow \,}\cL_{< w}(M) {\,\rightarrow \,}{{\bm R}\Gamma}_{I^+}(\cL_{[\geq w]}(M))[1] {\,\rightarrow \,}\ldots$$ Applying $(-)^{\#}_{\geq w}$ therefore gives an isomorphism $({{\bm R}\Gamma}_{I^+}(\cL_{[\geq w]}(M)))^{\sharp}_{\geq w} \cong ({{\bm R}\Gamma}_{I^+}(M))^{\sharp}_{\geq w}$, proving the commutativity of the first square.
Let $\cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$ and $\cD_{{{\rm Tor}^+}}({\cF_{\geq w}})$ be the full subcategories of $\cD({\cF_{\geq w}})$ defined by $$\begin{split}
\cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) \,&:= \, \{ \, \cM \in \cD({\cF_{\geq w}}) \, | \, {{\bm R}\Gamma}_{I^+,\geq w}(\cM) \simeq 0 \, \} \\
\cD_{{{\rm Tor}^+}}({\cF_{\geq w}}) \,&:= \, \{ \, \cM \in \cD({\cF_{\geq w}}) \, | \, {\check{\cC}}_{I^+,\geq w}(\cM) \simeq 0 \, \}
\end{split}$$
Then we have the following
\[RGam\_Ce\_weight\_ortho\] For $I = I^+$, the functors form a semi-orthogonal pair of idempotents, in the sense that the followings hold:
1. for any $\cM \in \cD({\cF_{\geq w}})$, we have ${{\bm R}\Gamma}_{I^+,\geq w}(\cM) \in \cD_{{{\rm Tor}^+}}({\cF_{\geq w}})$ and ${\check{\cC}}_{I^+,\geq w}(\cM) \in \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$;
2. if $\cM \in \cD_{{{\rm Tor}^+}}({\cF_{\geq w}})$ then ${{\bm R}\Gamma}_{I^+,\geq w}(\cM) \cong \cM$;
3. if $\cM \in \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$ then ${\check{\cC}}_{I^+,\geq w}(\cM) \cong \cM$;
4. there is a semi-orthogonal decomposition $\cD({\cF_{\geq w}}) = \langle \, \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) \, , \, \cD_{{{\rm Tor}^+}}({\cF_{\geq w}}) \, \rangle$.
Statement (1) follows immediately from Proposition \[RGam\_Ce\_weight\_descend\]. For statements (2) and (3), notice that every $\cM \in \cD({\cF_{\geq w}})$ can be written as $\cM \cong M^{\sharp}_{\geq w}$ for some $M \in \cD({{\rm Gr}(A)})$. For example one can take $M = \scL_{[\geq w]}(\cM)$. For this $M$, take the exact triangle for $I = I^+$, and apply $(-)^{\sharp}_{\geq w}$ to the triangle. In view of Proposition \[RGam\_Ce\_weight\_descend\], we have an exact triangle $$\ldots {\,\rightarrow \,}{{\bm R}\Gamma}_{I^+,\geq w}(\cM) {\, \xrightarrow[]{\epsilon_{\cM}} \,} \cM {\, \xrightarrow[]{\eta_{\cM}} \,} {\check{\cC}}_{I^+,\geq w}(\cM) {\, \xrightarrow[]{\delta_{\cM}} \,} {{\bm R}\Gamma}_{I^+,\geq w}(\cM)[1] {\,\rightarrow \,}\ldots$$ which immediately shows (2) and (3). In fact, because of (1), this exact triangle also establishes the decomposition for (4), so that it suffices to show the semi-orthogonality $\cD_{{{\rm Tor}^+}}({\cF_{\geq w}}) \perp \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$. Thus, let $\cM \in \cD_{{{\rm Tor}^+}}({\cF_{\geq w}})$ and $\cN \in \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$, then by (2) and (3), we may write $\cM \cong M^{\sharp}_{\geq w}$ and $\cN \cong N^{\sharp}_{\geq w}$ for some $M \in \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$ and $N \in \cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$. By the adjunction , we have $${{\rm Hom}}_{\cD({\cF_{\geq w}})}( \cM , \cN ) \, \cong \, {{\rm Hom}}_{\cD({{\rm Gr}(A)})} ( \cL_{[\geq w]}(M), N)$$ which is zero because of the first statement of Lemma \[Torp\_weight\_lem\].
Combined with , this gives the first statement of the following
\[three\_term\_SOD\] There is a semi-orthogonal decomposition $$\cD({{\rm Gr}(A)}) \, = \, \langle \, \cD_{<w}({{\rm Gr}(A)}) \, , \, \mathscr{L}_{[\geq w]} ( \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) ) \, , \, \mathscr{L}_{[\geq w]} ( \cD_{{{\rm Tor}^+}}({\cF_{\geq w}}) ) \, \rangle$$ where the functor $\mathscr{L}_{[\geq w]} : \cD({\cF_{\geq w}}) {\rightarrow}\cD({{\rm Gr}(A)})$ is fully faithful. Moreover, the latter two semi-orthogonal components can be identified as $$\label{SOD_comp_descr_1}
\begin{split}
\mathscr{L}_{[\geq w]} ( \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) ) \, &= \, \{ \, M \in \cD_{[\geq w]}({{\rm Gr}(A)}) \, | \, {{\bm R}\Gamma}_{I^+}(M) \in \cD_{<w}({{\rm Gr}(A)}) \, \} \\
\mathscr{L}_{[\geq w]} ( \cD_{ {{\rm Tor}^+}}({\cF_{\geq w}}) ) \, &= \,
\cD_{{{\rm Tor}^+},[\geq w]}({{\rm Gr}(A)}) \, := \, \cD_{[\geq w]}({{\rm Gr}(A)}) \cap \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})
\end{split}$$
Only the identifications of the semi-orthogonal components needs proof. It is clear that all the subcategories in lie in $\cD_{[\geq w]}({{\rm Gr}(A)})$. Moreover, given $M \in \cD_{[\geq w]}({{\rm Gr}(A)})$, then by Proposition \[RGam\_Ce\_weight\_descend\], we have $M^{\sharp}_{\geq w} \in \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$ if and only if $({{\bm R}\Gamma}_{I^+}(M))^{\sharp}_{\geq w} = 0$. The latter condition is precisely ${{\bm R}\Gamma}_{I^+}(M) \in \cD_{<w}({{\rm Gr}(A)})$, hence proving the first row of . For the second row, we similarly observe that, given any $M \in \cD_{[\geq w]}({{\rm Gr}(A)})$, then by Proposition \[RGam\_Ce\_weight\_descend\], we have $M^{\sharp}_{\geq w} \in \cD_{{{\rm Tor}^+}}({\cF_{\geq w}})$ if and only if ${\check{\cC}}_{I^+}(M) \in \cD_{<w}({{\rm Gr}(A)})$. But ${\check{\cC}}_{I^+}(M)$ is always in $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$, so that the latter is true if and only if ${\check{\cC}}_{I^+}(M) = 0$.
Unravelling the definitions, we see that this semi-orthogonal decomposition decomposes every $M \in \cD({{\rm Gr}(A)})$ into the diagram $$\label{triple_SOD_terms}
\begin{tikzcd} [row sep = 12, column sep = 15]
\cL_{[\geq w]} {{\bm R}\Gamma}_{I^+}(M) \ar[rr, "\cL_{[\geq w]} (\epsilon_M)"]
& & \cL_{[\geq w]} M \ar[rr, "\text{counit}"]
\ar[ld, " \cL_{[\geq w]}(\eta_M)"]
& & M \ar[ld, "\text{unit}"] \\
& \cL_{[\geq w]} {\check{\cC}}_{I^+}(M) \ar[ul, "\text{[1]}" description]
& & \cL_{<w}(M) \ar[ul, "\text{[1]}" description]
\end{tikzcd}$$ with the decomposition terms
1. $\cL_{<w}(M) \in \cD_{<w}({{\rm Gr}(A)})$,
2. $\cL_{[\geq w]} {\check{\cC}}_{I^+}(M) \in \mathscr{L}_{[\geq w]} ( \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) )$, and
3. $\cL_{[\geq w]} {{\bm R}\Gamma}_{I^+}(M) \in \mathscr{L}_{[\geq w]} ( \cD_{{{\rm Tor}^+}}({\cF_{\geq w}}) ) = \cD_{{{\rm Tor}^+},[\geq w]}({{\rm Gr}(A)})$
We also have the following two characterizations of semi-orthogonal components \[Torp\_SOD\] There is a semi-orthogonal decomposition $$\cD_{{{\rm Tor}^+}}({{\rm Gr}(A)}) \, = \, \langle \, \cD_{<w}({{\rm Gr}(A)}) \, , \, \cD_{{{\rm Tor}^+},[\geq w]}({{\rm Gr}(A)}) ) \, \rangle$$
For any $M \in \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$, take the exact sequence in the first row of . Since $\cL_{<w}M \in \cD_{<w}({{\rm Gr}(A)}) \subset \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$, we have $\cL_{[\geq w]}(M) \in \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$ as well. This proves the claimed decomposition. Orthogonality was already shown in Theorem \[three\_term\_SOD\].
\[first\_two\_terms\_SOD\] The composite of the first two components in Theorem \[three\_term\_SOD\] can be identified as $$\langle \, \cD_{<w}({{\rm Gr}(A)}) \, , \, \mathscr{L}_{[\geq w]} ( \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) ) \, \rangle \,
= \, \{ \, M \in \cD({{\rm Gr}(A)}) \, | \, {{\bm R}\Gamma}_{I^+}(M) \in \cD_{<w}({{\rm Gr}(A)}) \, \}$$
An object $M \in \cD({{\rm Gr}(A)})$ lies in $\langle \, \cD_{<w}({{\rm Gr}(A)}) \, , \, \mathscr{L}_{[\geq w]} ( \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) ) \, \rangle$ if and only if the component $\cL_{[\geq w]} {{\bm R}\Gamma}_{I^+}(M)$ in vanishes. This is true if and only if ${{\bm R}\Gamma}_{I^+}(M) \in \cD_{<w}({{\rm Gr}(A)})$.
Our next goal is to show that the triangulated category $\cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$ is in fact equivalent to $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$ (see Theorem \[I\_triv\_weight\_equiv\] below). First, notice that, by the second statement of Lemma \[Torp\_weight\_lem\], the functor $(-)^{\sharp}_{\geq w} : \cD({{\rm Gr}(A)}) {\rightarrow}\cD({\cF_{\geq w}})$ sends the subcategory $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}) \subset \cD({{\rm Gr}(A)})$ to the subcategory $\cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) \subset \cD({\cF_{\geq w}})$, so that we have an exact functor $$\label{I_triv_weight_functor_1}
(-)^{\sharp}_{\geq w} \, : \, \cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}) {\,\rightarrow \,}\cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$$
In the other direction, there is the exact functor $$\label{I_triv_weight_functor_2}
{\check{\cC}}_{I^+} \circ \mathscr{L}_{[\geq w]} \, : \, \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) {\,\rightarrow \,}\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$$
\[I\_triv\_weight\_equiv\] The functors and are inverse equivalences.
First, notice that the functors and are restrictions of the composite adjunctions $$\begin{tikzcd}
\cD({\cF_{\geq w}}) \ar[r, shift left, "\mathscr{L}_{[\geq w]}"]
& \cD({{\rm Gr}(A)}) \ar[r, shift left, "{\check{\cC}}_{I^+}"] \ar[l, shift left, "(-)^{\sharp}_{\geq w}"]
& \cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}) \ar[l, shift left, "\iota"]
\end{tikzcd}$$ and are therefore adjoints to each other.
The fact that the adjunction unit ${{\rm id}}\Rightarrow (-)^{\sharp}_{\geq w} \circ {\check{\cC}}_{I^+} \circ \mathscr{L}_{[\geq w]}$ is an isomorphism on $\cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$ is precisely statement (3) of Theorem \[RGam\_Ce\_weight\_ortho\]. This shows that, for any $M \in \cD({{\rm Gr}(A)})$, the adjunction counit ${\check{\cC}}_{I^+} ( \mathscr{L}_{[\geq w]} (M^{\sharp}_{\geq w})) {\rightarrow}M$ becomes an isomorphism after applying $(-)^{\sharp}_{\geq w}$. In other words, its cone lies in $\cD_{<w}({{\rm Gr}(A)}) \subset \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})$. If $M \in \cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$, then this cone also lies in $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$, which means it must be zero.
The semi-orthogonal decomposition in Theorem \[three\_term\_SOD\] can be rewritten in the form $$\label{SOD_no_F}
\cD({{\rm Gr}(A)}) \, = \, \langle \, \cD_{<w}({{\rm Gr}(A)}) \, , \, \cL_{[\geq w]} (\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})) \, , \, \cD_{{{\rm Tor}^+},[\geq w]}({{\rm Gr}(A)}) \, \rangle$$ where $\cL_{[\geq w]} (\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}))$ is the essential image of the functor $\cL_{[\geq w]} : \cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}) {\rightarrow}\cD({{\rm Gr}(A)})$, which is fully faithful by Theorem \[I\_triv\_weight\_equiv\].
Our discussion so far about local cohomology on weight truncation have a completely parallel version for local homology ([[*i.e.*]{}]{}, derived completion)[^2]. Indeed, one can show that, in place of Lemma \[weight\_subset\_tor\], we have $\cD_{<w}({{\rm Gr}(A)}) \subset \cD_{{I^-\text{-}{\rm comp}}}({{\rm Gr}(A)})$ (see [@Yeu20c Proposition 2.42]). This allows one to develop formal analogues of Proposition \[RGam\_Ce\_weight\_descend\], Theorem \[RGam\_Ce\_weight\_ortho\], Theorem \[three\_term\_SOD\] and Theorem \[I\_triv\_weight\_equiv\], where the functors $({{\bm R}\Gamma}_{I^+}, {\check{\cC}}_{I^+}, \mathscr{L}_{[\geq w]}, \cL_{[\geq w]})$ are replaced by $({{\bm L}\Lambda}_{I^-}, \cE_{I^-}, \mathscr{R}_{\{\geq w\}}, \cR_{\{\geq w\}})$. However, this seems to be less useful for us because the results in the next section seems to have no analogue for this dual version.
For later use, we also show that the equivalence in Theorem \[I\_triv\_weight\_equiv\] have finite cohomological dimension. This is clear for the functor since it descends from an exact functor on the abelian categories. For the functor , we have the following \[I\_triv\_weight\_cohom\_dim\] The functor ${\check{\cC}}_{I^+} \circ \mathscr{L}_{[\geq w]} : \cD({\cF_{\geq w}}) {\rightarrow}\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$ has finite cohomological dimension.
Since the exact functor $(-)^{\sharp}_{\geq w} : {{\rm Gr}(A)}{\rightarrow}{{\rm Mod}}({\cF_{\geq w}})$ is essentially surjective, the statement in the Lemma is equivalent to the statement that the functor ${\check{\cC}}_{I^+} \circ \cL_{[\geq w]} : \cD({{\rm Gr}(A)}) {\rightarrow}\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$ has finite cohomological dimension. To show this, simply apply ${\check{\cC}}_{I^+}$ to the exact triangle in the first row of . By Lemma \[weight\_subset\_tor\], we therefore have $ {\check{\cC}}_{I^+}(\cL_{[\geq w]}(M)) \cong {\check{\cC}}_{I^+}(M)$, which has cohomology in degrees $[p,q+r]$ if $M$ has cohomology in degrees $[p,q]$ (here $r$ is the number of generators of $I^+$).
Coherent subcategories
======================
The goal of this section is two-fold. We first show that the semi-orthogonal decomposition in Theorem \[three\_term\_SOD\] restricts to a semi-orthogonal decomposition on the subcategory ${\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)})$ (see Theorem \[three\_term\_SOD\_Dmcoh\] below). Then we show that the equivalence in Theorem \[I\_triv\_weight\_equiv\] restricts to an equivalence between suitable bounded coherent subcategories (see Theorem \[I\_triv\_weight\_equiv\_Dsuitcoh\]).
From now on, assume that the $\bZ$-graded ring $A$ is Noetherian. Then by Corollary \[Fgw\_Noeth\], ${\cF_{\geq w}}$ is also Noetherian, a fact that we will use without any more explicit mention. We start with the following \[weight\_Cech\_Dbcoh\] If $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, then we have ${\check{\cC}}_{I^+}(M)^{\sharp}_{\geq w} \in {\cD^b_{{\rm coh}}}({\cF_{\geq w}})$.
It suffices to assume that $M$ is a finitely generated graded $A$-module concentrated in cohomological degree $0$. Recall from that $H^p({\check{\cC}}_{I^+}(M))_i \cong H^p(X^+ , \widetilde{M(i)})$. Thus, by Lemma \[Fga\_fg\_module\], it suffices to show that $\bigoplus_{i \geq w} H^p(X^+ , \widetilde{M(i)})$ is finitely generated over $A_{\geq 0}$.
Recall (see, e.g., [@Sta Tag 0B5T]) that if $\cF$ is a coherent sheaf on $X^+$ and $\cL$ is an ample invertible sheaf on $X^+$, then $\bigoplus_{i \geq w} H^p(X^+ , \cF \otimes \cL^{\otimes i})$ is finitely generated over $B_{\geq 0} := \bigoplus_{i \geq 0} H^0(X^+ , \cL^{\otimes i})$. In the present case, suppose that $A$ is positively $\tfrac{1}{d}$-Cartier, then our statement follows by applying this to $\cF = \widetilde{M}, \widetilde{M(1)},\ldots, \widetilde{M(d-1)}$ and $\cL = \widetilde{A(d)}$, because the $\bN$-graded algebra $B_{\geq 0} := \bigoplus_{i \geq 0} H^0(X^+ , \widetilde{A(di)}$ is itself finite over $A^{(d)}$.
\[weight\_Cech\_Dbcoh\_remark\] Proposition \[weight\_Cech\_Dbcoh\] (and its proof) is one of the major advantages of imposing the weight truncation. Namely, while ${\check{\cC}}_{I^+}(M)$ has bounded cohomology, its cohomology groups $H^p({\check{\cC}}_{I^+}(M)) = \bigoplus_{i \in \bZ} H^p(X^+ , \tilde{M}(i))$ are *not* finitely generated over $A$. This is because, while the sheaf $\bigoplus_{i \geq w} \widetilde{M(i)}$ is finitely generated over the sheaf $\pi^*(A_{\geq 0})$ of algebras, the sheaf $\bigoplus_{i \in \bZ} \widetilde{M(i)}$ is *not* finitely generated over the sheaf $\pi^*(A)$ of algebras ([*cf*]{}. [@Sta Tag 0897]).
\[weight\_Cech\_Dsuitcoh\] For each $\spadesuit \in \{ \, \, , +,-,b\}$, if $M \in {\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}}({{\rm Gr}(A)})$, then we have ${\check{\cC}}_{I^+}(M)^{\#}_{\geq w} \in {\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}}({\cF_{\geq w}})$.
Since ${\check{\cC}}_{I^+}$ has bounded cohomological dimension, say ${\check{\cC}}_{I^+}(\cD^{\leq p}({{\rm Gr}(A)})) \subset \cD^{\leq p+m}({{\rm Gr}(A)})$ and ${\check{\cC}}_{I^+}(\cD^{\geq q}({{\rm Gr}(A)})) \subset \cD^{\geq q}({{\rm Gr}(A)})$, we have $H^p( {\check{\cC}}_{I^+}(M)^{\sharp}_{\geq w} ) \cong H^p( {\check{\cC}}_{I^+}(\tau_{\geq p-m}\tau_{\leq p}M)^{\sharp}_{\geq w} )$. Apply Proposition \[weight\_Cech\_Dbcoh\] to $\tau_{\geq p-m}\tau_{\leq p}M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$ to conclude that $H^p( {\check{\cC}}_{I^+}(M)^{\sharp}_{\geq w} ) \in {{\rm Mod}}({\cF_{\geq w}})$ is finitely generated.
For each $\spadesuit \in \{ \, \, , +,-,b\}$, define the following subcategories of $\cD({\cF_{\geq w}})$: $$\begin{split}
{\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) \, &:= \, {\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}}({\cF_{\geq w}}) \, \cap \, \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) \\
{\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}, \, {{\rm Tor}^+}}({\cF_{\geq w}}) \, &:= \, {\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}}({\cF_{\geq w}}) \, \cap \, \cD_{{{\rm Tor}^+}}({\cF_{\geq w}})
\end{split}$$ Then Corollary \[weight\_Cech\_Dsuitcoh\] gives the following \[weight\_Cech\_Dsuitcoh\_SOD\] For each $\spadesuit \in \{ \, \, , +,-,b\}$, the semi-orthogonal decomposition in Theorem \[RGam\_Ce\_weight\_ortho\](4) restricts to a semi-orthogonal decomposition $$\label{Dbcoh_Fgw_SOD}
{\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}}({\cF_{\geq w}}) = \langle \, {\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) \, , \, {\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}, \, {{\rm Tor}^+}}({\cF_{\geq w}}) \, \rangle$$
We wish to combine this with the semi-orthogonal decomposition to obtain a three-term semi-orthogonal decomposition on the full subcategory ${\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)})$. To this end, we have to show that the weight truncation functor $\cL_{[\geq w]}$, and hence $\cL_{<w}$, preserve the full subcategory ${\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)}) \subset \cD({{\rm Gr}(A)})$. This follows from the following two lemmae:
\[preserve\_Dmcoh\_1\] If $\cM \in {\cD^{-}_{{\rm coh}}}({\cF_{\geq w}})$, then $\mathscr{L}_{[\geq w]}(\cM) \in {\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)})$.
Recall from Proposition \[Dmcoh\_Dpc\_right\_Noeth\] that ${\cD^b_{{\rm coh}}}({\cF_{\geq w}}) = {\cD_{{\rm pc}}}({\cF_{\geq w}})$, so that $\cM$ may be represented by a bounded above complex $\cP^{\bullet}$ of free modules of finite rank. By , the complex $(\cP^{\bullet} \otimes_{{\cF_{\geq w}}} \cF)^{\flat}$ is therefore in ${\cD_{{\rm pc}}}({{\rm Gr}(A)}) = {\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)})$.
\[preserve\_Dmcoh\_2\] For each $\spadesuit \in \{ \, \, , +,-,b\}$, if $M \in {\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}}({{\rm Gr}(A)})$, then $M^{\sharp}_{\geq w} \in {\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}}({\cF_{\geq w}})$.
By Lemma \[Fga\_fg\_module\], the exact functor $(-)^{\sharp}_{\geq w} : {{\rm Gr}}(A) {\rightarrow}{{\rm Mod}}({\cF_{\geq w}})$ sends finitely generated graded modules to finitely generated modules.
As a result, if we let $$\begin{split}
\cD^-_{{{{\rm coh}}}, [\geq w]}({{\rm Gr}(A)}) \, &:= \, {\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)}) \, \cap \, \cD_{[\geq w]}({{\rm Gr}(A)}) \\
\cD^-_{{{{\rm coh}}}, <w}({{\rm Gr}(A)}) \, &:= \, {\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)}) \, \cap \, \cD_{<w}({{\rm Gr}(A)})
\end{split}$$ then we have the following \[weight\_SOD\_Dmcoh\] The full subcategory $\cD^-_{{{{\rm coh}}}, [\geq w]}({{\rm Gr}(A)}) \subset \cD({{\rm Gr}(A)})$ is the essential image of ${\cD^{-}_{{\rm coh}}}({\cF_{\geq w}})$ under the fully faithful functor $\mathscr{L}_{[\geq w]} : \cD({\cF_{\geq w}}) {\rightarrow}\cD({{\rm Gr}(A)})$. Moreover, the semi-orthogonal decomposition restricts to a semi-orthogonal decomposition $${\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)}) = \langle \, \cD^-_{{{\rm coh}},< w}({{\rm Gr}(A)}) \, , \, \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}(A)}) \, \rangle$$
Combining Corollary \[weight\_Cech\_Dsuitcoh\_SOD\] and Proposition \[weight\_SOD\_Dmcoh\], we see that all the decomposition terms in lie in ${\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)})$. As a result, we have the following \[three\_term\_SOD\_Dmcoh\] The semi-orthogonal decomposition in Theorem \[three\_term\_SOD\] restricts to a semi-orthogonal decomposition $${\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)}) \, = \, \langle \, \cD^-_{{{\rm coh}},<w}({{\rm Gr}(A)}) \, , \, \mathscr{L}_{[\geq w]} ( \cD^-_{{{\rm coh}}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) ) \, , \, \mathscr{L}_{[\geq w]} ( \cD^-_{{{\rm coh}}, \, {{\rm Tor}^+}}({\cF_{\geq w}}) ) \, \rangle$$ where the latter two semi-orthogonal components can be identified as $$\begin{split}
\mathscr{L}_{[\geq w]} ( \cD^-_{{{\rm coh}}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) ) \, &= \, \{ \, M \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}(A)}) \, | \, {{\bm R}\Gamma}_{I^+}(M) \in \cD_{<w}({{\rm Gr}(A)}) \, \} \\
\mathscr{L}_{[\geq w]} ( \cD^-_{{{\rm coh}}, \, {{\rm Tor}^+}}({\cF_{\geq w}}) )
\, &= \, \cD^-_{{{\rm coh}}, {{\rm Tor}^+}, [\geq w]}({{\rm Gr}(A)}) \, := \, \cD^-_{{{\rm coh}}, [\geq w]}({{\rm Gr}(A)}) \cap \cD_{{{\rm Tor}^+}}({{\rm Gr}(A)})
\end{split}$$
Next we show that the equivalence in Theorem \[I\_triv\_weight\_equiv\] restricts to an equivalence on coherent subcategories (see Theorem \[I\_triv\_weight\_equiv\_Dsuitcoh\] below). Recall from Definition \[IpTR\_coh\_def\] that, for each $\spadesuit \in \{ \, \, , +,-,b\}$, the full subcategory ${\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}({I^+\text{-}{\rm triv}})}({{\rm Gr}(A)}) \subset {\cD^{\tiny \mbox{$\spadesuit $}}}_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$ is defined to be the essential image of ${\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}}({{\rm Gr}(A)})$ under the functor ${\check{\cC}}_{I^+} : {\cD^{\tiny \mbox{$\spadesuit $}}}({{\rm Gr}(A)}) {\rightarrow}{\cD^{\tiny \mbox{$\spadesuit $}}}_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$. In view of Proposition \[coh\_IpTR\_QpGr\], this is the “correct” coherent subcategory to consider.
\[I\_triv\_weight\_equiv\_Dsuitcoh\] For each $\spadesuit \in \{ \, \, , +,-,b\}$, the equivalences in Theorem \[I\_triv\_weight\_equiv\] restricts to equivalences $$\begin{tikzcd}
{\check{\cC}}_{I^+} \circ \mathscr{L}_{[\geq w]} \, : \, {\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) \ar[r, shift left = 2] \ar[r, phantom, "\simeq" description]
& {\cD^{\tiny \mbox{$\spadesuit $}}}_{{{\rm coh}}({I^+\text{-}{\rm triv}})}({{\rm Gr}(A)}) \, : \, (-)^{\#}_{\geq w} \ar[l, shift left = 2]
\end{tikzcd}$$
The fact that the functor sends ${\cD^{\tiny \mbox{$\spadesuit $}}}_{{{\rm coh}}({I^+\text{-}{\rm triv}})}({{\rm Gr}(A)})$ to ${\cD^{\tiny \mbox{$\spadesuit $}}}_{{\rm coh}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$ is precisely the content of Corollary \[weight\_Cech\_Dsuitcoh\]. For the other direction, notice that Lemma \[preserve\_Dmcoh\_1\] establishes the statement for $\spadesuit = -$. The general case then follows from Lemma \[I\_triv\_weight\_cohom\_dim\] by using a standard truncation argument as in the proof of Corollary \[weight\_Cech\_Dsuitcoh\].
Boundedness
===========
In this section, we first give an alternative characterization of the full subcategory $\cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}(A)})$ (see, e.g., Proposition \[Dmcoh\_geq\_w\_equiv\] below). This allows us to show that our three-term semi-orthogonal decomposition in Theorem \[three\_term\_SOD\_Dmcoh\] coincides with that of [@HL15] when the regularity assumptions in [@HL15] are satisfied (see Remark \[SOD\_coincide\_remark\]). We also follow the arguments of [@HL15] to give a sufficient condition for the semi-orthogonal decomposition in Theorem \[three\_term\_SOD\_Dmcoh\] to restrict to one on ${\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$ (see Theorem \[cond\_then\_reg\] below). By using Lemma \[local\_cohom\_weight\_bounded\], we are able to circumvent [@HL15 Proposition 3.31] in the proof of an analogue of [@HL15 Lemma 3.36]. This in turn allows us to weaken the “Assumption (A)” in [@HL15].
For any Noetherian $\bZ$-graded ring $A$, let ${{\rm Gr}}_{<w}(A) \subset {{\rm Gr}}(A)$ be the Serre subcategory consisting of graded modules $M \in {{\rm Gr}}(A)$ such that $M_i = 0$ for all $i \geq w$. Let ${{\rm gr}}(A) \subset {{\rm Gr}}(A)$ be the full subcategory of finitely generated graded modules, and let ${{\rm gr}}_{<w}(A) := {{\rm gr}}(A) \cap {{\rm Gr}}_{<w}(A)$.
Throughout this section, we fix a graded ideal $I^{\prime +} \subset A$ such that $$\label{I_prime_plus_cond}
I^+ \subset I^{\prime +} \subset \sqrt{I^+}$$ Since the notion of $I^{\infty}$-torsion modules, $I$-trivial complexes, etc, depend only on $\sqrt{I}$, all our previous discussions remain valid if we replace $I^+$ by $I^{\prime +}$ everywhere.
We start with the following simple
The Serre subcategory ${{\rm gr}}_{<w}(A) \subset {{\rm gr}}(A)$ is the smallest Serre subcategory containing the essential image of ${{\rm gr}}_{<w}(A/I^{\prime +})$ under the (fully faithful) functor ${{\rm gr}}(A/I^{\prime +}) {\rightarrow}{{\rm gr}}(A)$.
Any module in ${{\rm Gr}}_{<w}(A)$ is clearly $(I^+)^{\infty}$-torsion. Thus if $M \in {{\rm gr}}_{<w}$ then let $M[(I^{\prime +})^m] := \{ x \in M \, | \, (I^{\prime +})^m \cdot x = 0\}$, we have an increasing filtration $0 \subset M[I^{\prime +}] \subset M[(I^{\prime +})^2] \subset \ldots$ whose union is $M$. Since $M$ is Noetherian, this must stabilize after finitely many terms. Since all the successive quotients in this filtration lies in ${{\rm gr}}_{<w}(A/I^{\prime +})$, we have the desired result.
\[Dbcoh\_a\_ess\_im\] The triangulated subcategory $\cD^b_{{{\rm coh}},<w}({{\rm Gr}(A)}) \subset {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$ is the smallest triangulated subcategory of ${\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$ that contains the essential image of $\cD^b_{{{\rm coh}},<w}({{\rm Gr}}(A/I^{\prime +}))$ under the functor ${\cD^b_{{\rm coh}}}({{\rm Gr}}(A/I^{\prime +})) {\rightarrow}{\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$.
A completely parallel proof also shows the following \[Torp\_ess\_im\] The Serre subcategory ${{\rm gr}}(A) \cap {{\rm Tor}^+}(A) \subset {{\rm gr}}(A)$ is the smallest Serre subcategory containing the essential image of ${{\rm gr}}(A/I^{\prime +}) {\rightarrow}{{\rm gr}}(A)$. Therefore, the triangulated subcategory $\cD^b_{{{\rm coh}},{{\rm Tor}^+}}({{\rm Gr}(A)}) \subset {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$ is the smallest triangulated subcategory that contains the essential image of the functor ${\cD^b_{{\rm coh}}}({{\rm Gr}}(A/I^{\prime +})) {\rightarrow}{\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$.
\[D\_geq\_a\_coh\_prop1\] For any $M \in \cD^-_{{{\rm coh}}}({{\rm Gr}(A)})$, we have $M \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}(A)})$ if and only if $M \otimes_A^{{\bm L}} (A/I^{\prime +}) \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}}(A/I^{\prime +}))$.
The implication “$\Rightarrow$” is clear. For the converse, suppose that $M \otimes_A^{{\bm L}} (A/I^{\prime +}) \in \cD^-_{[\geq w],{{\rm coh}}}({{\rm Gr}}(A/I^{\prime +}))$. Since the functor $-\otimes_A^{{\bm L}} (A/I^{\prime +}) $ is left adjoint to the restriction of scalar functor $(-)_A$, we have ${{\bm R}{\rm Hom}}_A(M,Q_A) \simeq 0$ for each $Q \in \cD^-_{{{\rm coh}},<w}({{\rm Gr}}(A/I^{\prime +}))$. In view of Corollary \[Dbcoh\_a\_ess\_im\], we have in particular ${{\bm R}{\rm Hom}}_A(M,K) \simeq 0$ for each $K \in \cD^b_{{{\rm coh}},<w}({{\rm Gr}(A)})$. If $M$ is not in $\cD^-_{[\geq w],{{\rm coh}}}({{\rm Gr}(A)})$, then by Proposition \[weight\_SOD\_Dmcoh\], there exists a nonzero map to some $N \in \cD^-_{{{\rm coh}},<w}({{\rm Gr}(A)})$, which must remain nonzero after passing to the truncation $N {\rightarrow}\tau_{\geq m}(N)$ for some $m \in \bZ$. Since $\tau_{\geq m}(N) \in \cD^b_{{{\rm coh}},<w}({{\rm Gr}(A)})$, this gives a contradiction.
Now suppose that $B$ is a Noetherian ($-\bN$)-graded ring. [[*i.e.*]{}]{}, $B$ is a $\bZ$-graded ring such that $B_i = 0$ for all $i >0$. Then we have $I^-(B) = B_{<0}$, and hence $B/I^-(B) = B_0$. In this case, weight truncation can often be performed inductively: \[B\_weight\_trunc\] Suppose that $M \in \cD_{<w}({{\rm Gr}}(B))$, then we have $\cL_{[\geq w-1]}(M) \cong M_{w-1}(-w+1) \otimes_{B_0}^{{\bm L}} B$.
Take the adjunction counit $M_{w-1}(-w+1) \otimes_{B_0}^{{\bm L}} B {\rightarrow}M$, which is clearly a quasi-isomorphism in weight $a-1$, so that its cone lies in $\cD_{<w-1}({{\rm Gr}}(B))$, and hence is equal to $\cL_{<w-1}(M)$, since we have $M_{w-1}(-w+1) \otimes_{B_0}^{{\bm L}} B \in \cD_{[\geq w-1]}({{\rm Gr}}(B))$.
The functor $-\otimes^{\bm L}_{B} B_0$ preserves both the subcategories $\cD_{[\geq w]}$ and $\cD_{<w}$:
\[D\_geq\_a\_lem1\] If $M \in \cD_{[\geq w]}({{\rm Gr}}(B))$ then $M \otimes^{{\bm L}}_{B} B_0 \in \cD_{[\geq w]}({{\rm Gr}}(B_0))$.
If $M \in \cD_{<w}({{\rm Gr}}(B))$ then $M \otimes^{{\bm L}}_{B} B_0 \in \cD_{<w}({{\rm Gr}}(B_0))$.
The first statement is obvious. For the second statement, we apply Lemma \[B\_weight\_trunc\]. It is clear that $(M_{w-1}(-w+1) \otimes_{B_0}^{{\bm L}} B) \otimes^{{\bm L}}_{B} B_0$ is concentrated in weight $w-1$. Thus, a repeated application of Lemma \[B\_weight\_trunc\] gives a sequence of maps $$M = \cL_{<w}(M) {\,\rightarrow \,}\cL_{<w-1}(M) {\,\rightarrow \,}\cL_{<w-2}(M) {\,\rightarrow \,}\ldots$$ such that ${{\rm cone}}[\, \cL_{<i+1}(M) {\rightarrow}\cL_{<i}(M) \,] \otimes^{{\bm L}}_{B} B_0$ is concentrated in weight $i$. Thus, for any $i < w$, the weight truncation $\cL_{[\geq i]}(M) = {{\rm cone}}(M {\rightarrow}\cL_{<i}(M))[-1]$ satisfies $$\parbox{40em}{$\cL_{[\geq i]}(M) \otimes_B^{{\bm L}} B_0$ is concentrated in weight $[i,w-1]$. }$$ Since the sequence of maps $$\cL_{[\geq w-1]}(M) {\rightarrow}\cL_{[\geq w-2]}(M) {\rightarrow}\ldots {\rightarrow}M$$ exhibits $M$ as a homotopy colimit in $\cD({{\rm Gr}}(B))$, and since homotopy colimit commutes with the functor $-\otimes_B^{{\bm L}} B_0$, we have $M \otimes^{{\bm L}}_{B} B_0 \in \cD_{<w}({{\rm Gr}}(B_0))$.
We also have the following
\[D\_geq\_a\_lem2\] If $M \in {\cD^{-}_{{\rm coh}}}({{\rm Gr}}(B))$ is a nonzero object, then $M \otimes^{{\bm L}}_{B} B_0 \in {\cD^{-}_{{\rm coh}}}({{\rm Gr}}(B_0))$ is also nonzero.
Take the highest nonvanishing cohomology degree $H^p(M) \neq 0$. Then by the Nakayama lemma for $\bN$-graded rings, we have $0 \neq H^p(M) \otimes_B B_0 = H^p( M \otimes^{{\bm L}}_{B} B_0)$.
Combining these two lemmae, we have
\[B\_geq\_w\_B\_zero\] For any $N \in {\cD^{-}_{{\rm coh}}}({{\rm Gr}}(B))$, we have $N \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}}(B))$ if and only if $N \otimes^{{\bm L}}_B B_0 \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}}(B_0))$.
The direction “$\Rightarrow$” is obvious. For the direction “$\Leftarrow$”, suppose that $N \notin \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}}(B))$ so that $\cL_{<w}(N) \neq 0$. Then by Lemma \[D\_geq\_a\_lem1\], the exact triangle $$\ldots {\,\rightarrow \,}\cL_{[\geq w]}(N) \otimes^{{\bm L}}_B B_0 {\,\rightarrow \,}N \otimes^{{\bm L}}_B B_0 {\,\rightarrow \,}\cL_{[< w]}(N) \otimes^{{\bm L}}_B B_0 {\, \xrightarrow[]{[1]} \,} \ldots$$ is precisely the weight truncation sequence for $N \otimes^{{\bm L}}_B B_0$ in $\cD({{\rm Gr}}(B_0))$. By Lemma \[D\_geq\_a\_lem2\], we have $ \cL_{<w}(N) \otimes^{{\bm L}}_B B_0 \neq 0$, which therefore shows that $N \otimes^{{\bm L}}_B B_0 \notin \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}}(B_0))$.
\[Dmcoh\_geq\_w\_equiv\] For any $M \in {\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)})$, the followings are equivalent:
1. $M \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}(A)})$
2. $M \otimes^{{\bm L}}_A (A/(I^- + I^+)) \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}}(A/(I^- + I^+))$.
3. $M \otimes^{{\bm L}}_A (A/\sqrt{I^- + I^+}) \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}}(A/\sqrt{I^- + I^+})$.
Take $B = A/I^{\prime +}$ for $I^{\prime +} = I^+$ or $I^{\prime +} = \sqrt{I^+}$. In the former case, we have $B_0 = B/I^-(B) = A/(I^- + I^+)$. In the latter case, $B_0$ is a subring of the reduced ring $B$, and hence is reduced. In other words, $I^- + \sqrt{I^+} \subset A $ is equal to its radical, and must therefore be equal to $\sqrt{I^- + I^+}$. Thus, we have $B_0 = A/\sqrt{I^- + I^+}$, and it suffices to show that $$M \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}(A)}) \quad \Leftrightarrow \quad
M \otimes^{{\bm L}}_A B_0 \in \cD^-_{{{\rm coh}},[\geq w]}({{\rm Gr}}(B_0))$$ for $B = A/I^{\prime +}$, where $I^{\prime +}$ is any graded ideal satisfying . Take $N := M \otimes_A^{{\bm L}} B \in {\cD^{-}_{{\rm coh}}}({{\rm Gr}}(B))$. The result then follows from Propositions \[D\_geq\_a\_coh\_prop1\] and \[B\_geq\_w\_B\_zero\].
Now we give a sufficient condition for weight truncation to preserve ${\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$ (see Theorem \[cond\_then\_reg\]). The arguments for Lemma \[B\_flat\_then\_reg\], Propositon \[Torp\_reg\] and Theorem \[cond\_then\_reg\] below are adapted from those in [@HL15]. However, we weaken the assumption $(A)$ in [*loc.cit.*]{}.
Take a graded ideal $I^{\prime +} \subset A$ satisfying . Consider the conditions $$\label{two_conds}
\parbox{40em}{(a) The graded ring $B := A/I^{\prime +}$ has finite Tor-dimension over the subring $B_0 \subset B$. \\
(b) As a quotient, the ring $B_0 = B/I^-(B)$ has finite Tor-dimension over $B$.\\
(c) $A/I'^+ \in \cD_{[\geq 0]}({{\rm Gr}(A)})$.}
$$ Notice that by cocontinuity, condition (c) is equivalent to the condition that the restriction of scalar functor sends $\cD_{[\geq w]}({{\rm Gr}}(A/I'^+))$ to $\cD_{[\geq w]}({{\rm Gr}(A)})$.
Under the first two conditions, we have the following \[B\_flat\_then\_reg\] Suppose that (a) holds, then weight truncation for $B$ preserves ${\cD^b_{{\rm coh}}}({{\rm Gr}}(B))$. [[*i.e.*]{}]{}, for all $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}}(B))$, we have $\cL_{< w}(M) \in {\cD^b_{{\rm coh}}}({{\rm Gr}}(B))$.
Suppose that (b) holds, then for every $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}}(B))$ there exists $i \in \bZ$ such that $M \in \cD^b_{{{\rm coh}},[\geq i]}({{\rm Gr}}(B))$.
Since $B$ is ($-\bN$)-graded, there exists some $w' \in \bZ$ such that $M \in \cD_{<w'}({{\rm Gr}}(B))$. Apply Lemma \[B\_weight\_trunc\], we see that $\cL_{[\geq w'-1]}(M) \cong M_{w-1}(-w'+1) \otimes_{B_0}^{{\bm L}} B$, which is in ${\cD^b_{{\rm coh}}}({{\rm Gr}}(B))$ by the assumption (a). Thus, $\cL_{< w'-1}(M) \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$. A repeated application of the argument then shows that $\cL_{<w}(M) \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$ for all $w \in \bZ$.
For the second statement, the assumption (b) guarantees that $M \otimes_B^{{\bm L}} B_0 \in {\cD^b_{{\rm coh}}}({{\rm Gr}}(B_0))$. Since $B_0$ is concentrated in weight $0$, any finitely generated graded $B_0$-module must be concentrated in finitely many weight components. Hence, $M \otimes_B^{{\bm L}} B_0 \in \cD_{[\geq i]}({{\rm Gr}}(B_0))$ for some $i \in \bZ$. By Proposition \[B\_geq\_w\_B\_zero\], this is precisely the sought for statement.
\[Torp\_reg\] Suppose that conditions (a) and (c) hold, then
1. $\cD^b_{{{\rm coh}},{{\rm Tor}^+},[\geq w]}({{\rm Gr}(A)})$ is the smallest triangulated subcategory containing the essential image of the functor $\cD^b_{{{\rm coh}},[\geq w]}({{\rm Gr}}(A/I^{\prime +})) {\rightarrow}{\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$.
2. For any $M \in \cD^b_{{{\rm coh}}, {{\rm Tor}^+}}({{\rm Gr}(A)})$, we have $\cL_{[\geq w]}(M) \in {\cD^b_{{\rm coh}}}(M)$.
If (b) also hold, then we also have
1. for any $M \in \cD^b_{{{\rm coh}},{{\rm Tor}^+}}({{\rm Gr}(A)})$ there exists $i \in \bZ$ such that $M \in \cD^b_{{{\rm coh}},[\geq i]}({{\rm Gr}}(A))$.
Consider the following full subcategories of ${\cD^b_{{\rm coh}}}({{\rm Gr}}(B))$: $$\begin{split}
\cE_1 \,& := \, {\rm EssIm}( \, \cD^b_{{{\rm coh}},<w}({{\rm Gr}}(B)) {\,\rightarrow \,}{\cD^b_{{\rm coh}}}({{\rm Gr}(A)}) \, ) \\
\cE_2 \,& := \, {\rm EssIm}( \, \cD^b_{{{\rm coh}},[\geq w]}({{\rm Gr}}(B)) {\,\rightarrow \,}{\cD^b_{{\rm coh}}}({{\rm Gr}(A)}) \, )
\end{split}$$
We shall adopt the notation from [@Yeu20c Appendix A]. In particular, for any full subcategory $\cE \subset {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, we denote by $\langle \cE \rangle$ the smallest triangulated subcategory containing $\cE$. In this notation, Corollary \[Dbcoh\_a\_ess\_im\] asserts that $\langle \cE_1 \rangle = \cD^b_{{{\rm coh}},<w}({{\rm Gr}(A)})$. Condition (c) implies that $\cE_2 \subset \cD^b_{{{\rm coh}}, {{\rm Tor}^+}, [\geq w]}({{\rm Gr}(A)})$, so that $\cE_1$ and $\cE_2$ are strongly orthogonal, [[*i.e.*]{}]{}, ${{\rm Hom}}_{\cD({{\rm Gr}(A)})}(E_2,E_1[i]) = 0$ for all $i \in \bZ$, $E_1 \in \cE_1$ and $E_2 \in \cE_2$. Recall from [@Yeu20c Corollary A.9] that this implies that $\langle \cE_1 \rangle \vec{*} \langle \cE_2 \rangle = \langle \cE_1 \vec{*} \, \cE_2 \rangle$. By Lemma \[B\_flat\_then\_reg\], we have ${\cD^b_{{\rm coh}}}({{\rm Gr}}(B)) = \langle \cD^b_{{{\rm coh}},<w}({{\rm Gr}}(B)), \cD^b_{{{\rm coh}},[\geq w]}({{\rm Gr}}(B)) \rangle$ under condition (a), so that $$\cE \, := \, {\rm EssIm}( \, \cD^b_{{{\rm coh}}}({{\rm Gr}}(B)) {\,\rightarrow \,}{\cD^b_{{\rm coh}}}({{\rm Gr}(A)}) \, ) \, \subset \, \cE_1 * \cE_2$$ By Lemma \[Torp\_ess\_im\], we have $\langle \cE \rangle = \cD^b_{{{\rm coh}},{{\rm Tor}^+}}({{\rm Gr}(A)})$. Combining these facts, we have $$\begin{split}
\langle \cE \rangle \, &\subset \, \langle \cE_1 \vec{*} \, \cE_2 \rangle \, = \, \langle \cE_1 \rangle \vec{*} \langle \cE_2 \rangle \\
&\subset \, (\cD^b_{{{\rm coh}},<w}({{\rm Gr}(A)})) \, \vec{*} \, (\cD^b_{{{\rm coh}}, {{\rm Tor}^+}, [\geq w]}({{\rm Gr}(A)})) \\
&\subset \, \cD^b_{{{\rm coh}},{{\rm Tor}^+}}({{\rm Gr}(A)}) \, = \, \langle \cE \rangle
\end{split}$$ Since the first and last term are the same, we must have equalities. In particular, the equality for the second inclusion implies that $\langle \cE_2 \rangle = \cD^b_{{{\rm coh}}, {{\rm Tor}^+}, [\geq w]}({{\rm Gr}(A)})$, which is the first sought for statement. The equality for the third inclusion is precisely the second sought for statement.
If (b) holds, then applying the second statement of Lemma \[B\_flat\_then\_reg\], together with (c), we see that for every object $N \in \cE$ there is some $i \in \bZ$ such that $N \in \cD^b_{{{\rm coh}},{{\rm Tor}^+},[\geq i]}({{\rm Gr}(A)})$. Since $\cD^b_{{{\rm coh}},{{\rm Tor}^+}}({{\rm Gr}(A)}) \, = \, \langle \cE \rangle$, every $M \in \cD^b_{{{\rm coh}},{{\rm Tor}^+}}({{\rm Gr}(A)})$ also has this property.
\[cond\_then\_reg\] Suppose that the conditions (a),(b),(c) hold. Then
1. For every object $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, there exists some $i \in \bZ$ such that $M \in \cD^b_{{{\rm coh}},[\geq i]}({{\rm Gr}(A)})$.
2. Weight truncation for $A$ preserves ${\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$. [[*i.e.*]{}]{}, for any $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, we have $\cL_{<w}(M) \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$.
Let $f_1,\ldots,f_r \in A$ be a set of elements of positive degrees $\deg(f_i) = d_i > 0$ that generate $I^+$, and let $
K^{\bullet}(A,f_1,\ldots,f_r) \, = \, \bigwedge_A (A \theta_1 \oplus \ldots \oplus A \theta_r)
$ be the (cohomological) Koszul complex, which is a finite complex of free graded $A$-modules with a set $\{\wedge_{s \in S} \, \theta_s\}_{S \subset \{1,\ldots,r\}}$ of $2^r$ generators of weight $-\sum_{s \in S} d_s$ and cohomological degree $|S|$. Moreover, the differentials of the Koszul complex satisfies $$\label{Koszul_diff_in_I}
d(K^{\bullet}(A,f_1,\ldots,f_r)) \, \subset \, I^+ \cdot K^{\bullet}(A,f_1,\ldots,f_r)$$ For any $M \in {\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, let $K^{\bullet}(M,f_1,\ldots,f_r) := K^{\bullet}(A,f_1,\ldots,f_r) \otimes_A M$. Then implies that $$K^{\bullet}(M,f_1,\ldots,f_r) \otimes_{A}^{{\bm L}} B \, \, \, \cong \, \bigoplus_{S \subset \{1,\ldots,r\}} (M \otimes_A^{{\bm L}} B) \,( \, \textstyle \sum_{s \in S} d_s \,)[ \,- |S| \, ]$$ where $B := A/I^{\prime +}$. By Proposition \[D\_geq\_a\_coh\_prop1\], we therefore see that $K^{\bullet}(M,f_1,\ldots,f_r) \in \cD^b_{{{\rm coh}},[\geq i]}({{\rm Gr}(A)})$ if and only if $M \in \cD^b_{{{\rm coh}},[\geq i]}({{\rm Gr}(A)})$. Since we always have $K^{\bullet}(M,f_1,\ldots,f_r) \in \cD^b_{{{\rm coh}},{{\rm Tor}^+}}({{\rm Gr}(A)})$, the first statement of the present Theorem follows from Proposition \[Torp\_reg\](3).
Let $K_{\bullet}(A,f_1^j,\ldots,f_r^j)$ be the homological Koszul complex, [[*i.e.*]{}]{}, it is the $A$-linear dual ${\underline{{\rm Hom}}}_A(-,A)$ of $K^{\bullet}(A,f_1^j,\ldots,f_r^j)$. Thus, it is a finite complex of free graded $A$-modules with a set $\{\wedge_{s \in S} \, \theta_s^{\vee}\}_{S \subset \{1,\ldots,r\}}$ of $2^r$ generators of weight $(\sum_{s \in S} d_s)j$ and cohomological degree $-|S|$. Let $K_{\bullet}(M,f_1^j,\ldots,f_r^j) := M \otimes_A K_{\bullet}(A,f_1^j,\ldots,f_r^j)$, then by statement (1) we have just proved, we have $${{\rm cone}}\, [ \, M {\,\rightarrow \,}K_{\bullet}(M,f_1^j,\ldots,f_r^j) \, ] \, \in \, \cD_{[\geq w]}({{\rm Gr}(A)})
\qquad \text{for } \, j \gg 0$$ As a result, we have $\cL_{<w}(M) \cong \cL_{<w} ( K_{\bullet}(M,f_1^j,\ldots,f_r^j))$ for $j \gg 0$. Since $K_{\bullet}(M,f_1^j,\ldots,f_r^j) \in \cD^b_{{{\rm coh}},{{\rm Tor}^+}}({{\rm Gr}(A)})$, the second statement follows from Proposition \[Torp\_reg\](2).
Notice that if weight truncation preserves ${\cD^b_{{\rm coh}}}({{\rm Gr}(A)})$, then the semi-orthogonal decomposition in Theorem \[three\_term\_SOD\_Dmcoh\] restricts to a semi-orthogonal decomposition $$\label{three_term_SOD_Dbcoh}
{\cD^b_{{\rm coh}}}({{\rm Gr}(A)}) \, = \, \langle \, \cD^b_{{{\rm coh}},<w}({{\rm Gr}(A)}) \, , \, \mathscr{L}_{[\geq w]} ( \cD^b_{{{\rm coh}}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) ) \, , \, \cD^b_{{{\rm coh}}, {{\rm Tor}^+}, [\geq w]}({{\rm Gr}(A)}) \, \rangle$$ Moreover, the latter two semi-orthogonal components can be described as in Theorem \[three\_term\_SOD\_Dmcoh\], with $\cD^-$ replaced by $\cD^b$ everywhere.
The following is the main class of examples of Noetherian $\bZ$-graded rings that satisfies the conditions (a),(b),(c):
\[smooth\_then\_cond\] If $A$ is a $\bZ$-graded ring finitely generated over a field $k$ of characteristic zero, and if the underlying ungraded algebra $A$ is smooth over $k$, then we have
1. The algebras $B^+ = A/I^+$, $B^- = A / I^-$ and $B_0 = A/(I^- + I^+)$ are smooth over $k$.
2. The projections $\rho^+ : {{\rm Spec}}\, B^+ {\rightarrow}{{\rm Spec}}\, B_0$ and $\rho^- : {{\rm Spec}}\, B^- {\rightarrow}{{\rm Spec}}\, B_0$ are locally trivial bundle of weighted affine spaces.
3. Along each connected component $Z_i \subset {{\rm Spec}}\, B_0$, we have $\dim((\rho^+)^{-1}(Z_i)) + \dim((\rho^-)^{-1}(Z_i)) = \dim(Z_i) + \dim(A)$.
Therefore, the conditions (a),(b),(c) are satisfied.
Notice that $B_0 = A/(I^- + I^+)$ is the weight zero part of both $B^+ = A/I^+$ and $B^- = A/I^-$, so that the second statement make sense. These statements are then a special case of a result of Białynicki-Birula [@BB73]. The conditions (a),(b),(c) then follows (see, [[*e.g.*]{}]{}, [@HL15 Lemma 2.7]).
As a consequence, we have the following \[smooth\_then\_reg\_and\_perf\] If $A$ is a $\bZ$-graded ring finitely generated over a field $k$ of characteristic zero, and if the underlying ungraded algebra $A$ is smooth over $k$, then the weight truncation functor $\cL_{[\geq w]}$ preserves ${\cD^b_{{\rm coh}}}({{\rm Gr}(A)}) = {\cD_{{\rm perf}}}({{\rm Gr}(A)})$.
For applications in Section \[Dcat\_flip\_flop\_sec\] below, we will be mostly interested in the case when the semi-orthogonal decomposition restricts to one on ${\cD_{{\rm perf}}}({\cF_{\geq w}})$. If the weight truncation functor $\cL_{[\geq w]}$ preserves ${\cD_{{\rm perf}}}({{\rm Gr}(A)})$, then the semi-orthogonal decomposition restricts to one on ${\cD_{{\rm perf}}}({\cF_{\geq w}})$.
Recall that there exists some $n_0 \geq 0$ such that any sequence ${\check{\cC}}_{I^+}(A)(i), \ldots, {\check{\cC}}_{I^+}(A)(i+n_0)$ of length $n_0 + 1$ is a set of compact generators for $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$. Take some $c \in \bZ$ such that ${{\bm R}\Gamma}_{I^+}(A) \in \cD_{<c}({{\rm Gr}(A)})$ (see Lemma \[local\_cohom\_weight\_bounded\]), so that ${\check{\cC}}_{I^+}(A)(i)^{\sharp}_{\geq w} = A(i)^{\sharp}_{\geq w}$ for any $i \geq c - w$. Thus, for these values of $i$, we have $\cL_{[\geq w]}({\check{\cC}}_{I^+}(A)(i)) = \cL_{[\geq w]}( A(i) )$, which is in ${\cD_{{\rm perf}}}({{\rm Gr}(A)})$ by assumption. Since $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})_c$ is split generated by these objects, we see that $\cL_{[\geq w]}(C) \in {\cD_{{\rm perf}}}({{\rm Gr}(A)})$ for all $C \in \cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})_c$. In view of Lemma \[D\_geq\_w\_gen\], this shows that the functor ${\check{\cC}}_{I^+,\geq w}$ in preserves ${\cD_{{\rm perf}}}({\cF_{\geq w}})$.
\[SOD\_coincide\_remark\] Proposition \[Dmcoh\_geq\_w\_equiv\] allows us to compare our construction of weight truncation with the ones in [@HL15] and [@BFK19]. Namely, if the assumptions (L+) and (A) in [@HL15] are satisfied, then so does our assumptions . By comparing Proposition \[Dmcoh\_geq\_w\_equiv\] with [@HL15 Definition 2.8] for $\mathfrak{X} := [{{\rm Spec}}\, A / \bG_m]$, we see that $$\cD^b_{{{\rm coh}},{{\rm Tor}^+},[\geq w]}({{\rm Gr}(A)}) ) \, = \, \cD^b_{\mathfrak{X}^u}(\mathfrak{X})_{\geq w}$$ Then, notice that Lemma \[Torp\_SOD\] restricts to $\cD^b_{{{\rm coh}},{{\rm Tor}^+}}({{\rm Gr}(A)})$. Comparing this with [@HL15 Theorem 2.10(5)], we see that $$\cD^b_{{{\rm coh}},<w}({{\rm Gr}(A)}) \, = \, \cD^b_{\mathfrak{X}^u}(\mathfrak{X})_{< w}$$ Finally, if we compare the three term semi-orthogonal decomposition in with the corresponding one in [@HL15 Theorem 2.10(6)], then we see that $$\mathscr{L}_{[\geq w]} ( \cD^b_{{{\rm coh}}, \, {I^+\text{-}{\rm triv}}}({\cF_{\geq w}}) ) \, = \, \mathbf{G}_w$$ This shows that the three-term semi-orthogonal decomposition of [@HL15], and hence of [@BFK19], coincides with the one in \[three\_term\_SOD\_Dbcoh\] in this abelian case.
The case of non-affine base
===========================
In this section, we provide the formal arguments to extend our previous discussion to the case of non-affine base. More precisely, we work in the following setting: $$\label{sheaf_A_setting}
\parbox{40em}{$Y$ is a Noetherian separated scheme, and $\cA$ is a quasi-coherent sheaf of Noetherian $\bZ$-graded rings on $Y$, such that $\cA_0$ (and hence every $\cA_i$) is coherent over $\cO_X$.}$$
Denote by ${{\rm Gr}}(\cA)$ the category of quasi-coherent graded $\cA$-modules. Then for any quasi-coherent sheaf of graded ideal $\scI \subset \cA$, there is a semi-orthogonal decomposition $$\cD({{\rm Gr}}(\cA)) \, = \, \langle \, \cD_{{\scI\text{-}{\rm triv}}}({{\rm Gr}}(\cA)) \, , \, \cD_{{\scI^{\infty}\text{-}{\rm Tor}}}({{\rm Gr}}(\cA)) \, \rangle$$ whose restriction to each open affine subscheme is precisely . More precisely, if we take the decomposition triangle $$\label{RGam_Ce_seq_sheaf}
\ldots {\,\rightarrow \,}{{\bm R}\Gamma}_{\scI}(\cM) {\, \xrightarrow[]{\epsilon_M} \,} \cM {\, \xrightarrow[]{\eta_{\cM}} \,} {\check{\cC}}_{\scI}(\cM) {\, \xrightarrow[]{\delta_{\cM}} \,} {{\bm R}\Gamma}_{\scI}(\cM)[1] {\,\rightarrow \,}\ldots$$ for $\cM\in \cD({{\rm Gr}}(\cA))$, then the value on $U = {{\rm Spec}}\, R \subset Y$ is precisely the for $M = \cM(U)$ (see, [[*e.g.*]{}]{}, [@Yeu20c Section 5] for details). Once we have defined these functors, their properties can then be checked affine-locally.
One can also consider weight truncation for pairs $(Y,\cA)$ in . As in the above discussion of local cohomology, it suffices to construct the relevant weight truncation functors that reduce to the ones above over any open affine subscheme ${{\rm Spec}}\, R \subset Y$. Then the properties of such functors can be checked locally. We start with the following Let $\cD_{[\geq w]}({{\rm Gr}}(\cA)) \subset \cD({{\rm Gr}}(\cA))$ be the smallest strictly full triangulated subcategory closed under small coproducts, and containing the object of the form $$\label{gen_obj_Dgw}
\cF \otimes_{\cO_Y}^{{\bm L}} \cA(-i) \, , \qquad \text{where} \quad \cF \in \cD_{{{\rm perf}}}({{\rm QCoh}}(Y)) \quad \text{and} \quad i \geq w$$
Clearly, each of the objects of the form is compact in $\cD({{\rm Gr}}(\cA))$, and hence also in $\cD_{[\geq w]}({{\rm Gr}}(\cA))$. Thus, $\cD_{[\geq w]}({{\rm Gr}}(\cA))$ is compactly generated, and the inclusion functor $\cD_{[\geq w]}({{\rm Gr}}(\cA)) {\hookrightarrow}\cD({{\rm Gr}}(\cA))$ preserves small coproducts. By the Brown-Neeman representability theorem [@Nee96 Theorem 4.1], this inclusion therefore has a right adjoint, which will be denoted as $$\cL_{[\geq w]} \, : \, \cD({{\rm Gr}}(\cA)) {\,\rightarrow \,}\cD_{[\geq w]}({{\rm Gr}}(\cA))$$
Since the full triangulated subcategory $\cD_{[\geq w]}({{\rm Gr}}(\cA)) \subset \cD({{\rm Gr}}(\cA))$ is right admissible, there exists a semi-orthogonal decomposition of the form $$\label{weight_SOD_nonaffine}
\cD({{\rm Gr}}(\cA)) \, = \, \langle \, \cD_{<w}({{\rm Gr}}(\cA)) \, , \, \cD_{[\geq w]}({{\rm Gr}}(\cA)) \, \rangle$$ where $\cD_{<w}({{\rm Gr}}(\cA)) := \cD_{[\geq w]}({{\rm Gr}}(\cA))^{\perp}$. Alternatively, it can be characterized as follows: \[Dlw\_nonaffine\] An object $\cM \in \cD({{\rm Gr}}(\cA))$ is in $\cD_{<w}({{\rm Gr}}(\cA))$ if and only if its $i$-th weight component $\cM_i \in \cD({{\rm QCoh}}(Y))$ is zero for all $i \geq w$.
For any $\cM \in \cD({{\rm Gr}}(\cA))$, we have $\cM \in \cD_{[\geq w]}({{\rm Gr}}(\cA))^{\perp} $ if and only if $$\label{left_orth_gen}
{{\rm Hom}}_{\cD({{\rm Gr}}(\cA))}( \cF \otimes_{\cO_Y}^{{\bm L}} \cA(-i) , \cM[j]) = 0 \quad \text{for all} \quad \cF \in \cD_{{{\rm perf}}}({{\rm Gr}}(\cA)) , \, i \geq w \, \, \text{and} \, \, j \in \bZ$$ Indeed, by the simple fact below, applied to $\cD := \cD({{\rm Gr}}(\cA))$ and $X := \cM$, we see that the objects are in $\,^{\perp}({\bm \Sigma} \cM)$ if and only if $\cD_{[\geq w]}({{\rm Gr}}(\cA)) \subset \,^{\perp}({\bm \Sigma} \cM)$. $$\label{left_orth_coprod}
\parbox{40em}{Suppose $\cD$ is a triangulated category that admits small coproducts. Then for any $X \in \cD$, the full subcategory $\! \,^{\perp}({\bm \Sigma} X) := \{ \, Y \in \cD \, | \, {{\rm Hom}}_{\cD}(Y,X[i]) = 0 \text{ for all } i \in \bZ \, \}$ is a triangulated subcategory that is closed under small coproducts.}$$
Notice that we have $${{\rm Hom}}_{\cD({{\rm Gr}}(\cA))}( \cF \otimes_{\cO_Y}^{{\bm L}} \cA(-i) , \cM[j]) \, \cong \, {{\rm Hom}}_{\cD({{\rm QCoh}}(Y))} ( \cF , \cM_i )$$ Recall that $\cD({{\rm QCoh}}(Y))$ is compactly generated by $\cD_{{{\rm perf}}}({{\rm QCoh}}(Y))$ (see, e.g., [@Nee96 Corollary 2.3, Proposition 2.5] and [@BVdB03 Theorem 3.1.1]). The result therefore follows from the characterization of $\cD_{[\geq w]}({{\rm Gr}}(\cA))^{\perp}$.
By the semi-orthogonal decomposition , we see that the inclusion $\cD_{<w}({{\rm Gr}}(\cA)) {\hookrightarrow}\cD({{\rm Gr}}(\cA))$ has a left adjoint, which we denote as $$\cL_{< w} \, : \, \cD({{\rm Gr}}(\cA)) {\,\rightarrow \,}\cD_{< w}({{\rm Gr}}(\cA))$$
For any $\cM \in \cD({{\rm Gr}}(\cA))$, the semi-orthogonal decomposition then gives us a decomposition sequence $$\label{weight_decomp_seq_nonaffine}
\ldots {\,\rightarrow \,}\cL_{[\geq w]}(\cM) {\,\rightarrow \,}\cM {\,\rightarrow \,}\cL_{< w}(\cM) {\,\rightarrow \,}\cL_{[\geq w]}(\cM)[1] {\,\rightarrow \,}\ldots$$
Given any open affine subscheme $U = {{\rm Spec}}\, R \subset Y$, let $A := \cA(U)$. If $\cM$ is of the form , then its restriction to $U$ has the form $K \otimes_R^{{\bm L}} A (-i)$, where $\cF \in \cD_{{{\rm perf}}}(R)$ and $i \geq w$. Since $R$ split generates $\cD_{{{\rm perf}}}(R)$, we see that these are all contained in $\cD_{[\geq w]}({{\rm Gr}(A)})$. Thus, we have $$\cD_{[\geq w]}({{\rm Gr}}(\cA))|_U \, \subset \, \cD_{[\geq w]}({{\rm Gr}(A)})$$
By Lemma \[Dlw\_nonaffine\], we also have $$\cD_{< w]}({{\rm Gr}}(\cA))|_U \, \subset \, \cD_{< w}({{\rm Gr}(A)})$$ Therefore the restriction of to $U$ becomes precisely the first row of . This allows us to verify properties of weight truncations locally.
Combining the weight truncation sequence with the local cohomology sequence , this allows us to extend to the case of non-affine base: $$\label{triple_SOD_terms_nonaffine}
\begin{tikzcd} [row sep = 12, column sep = 15]
\cL_{[\geq w]} {{\bm R}\Gamma}_{\scI^+}(\cM) \ar[rr, "\cL_{[\geq w]} (\epsilon_{\cM})"]
& & \cL_{[\geq w]} \cM \ar[rr, "\text{counit}"]
\ar[ld, " \cL_{[\geq w]}(\eta_\cM)"]
& & \cM \ar[ld, "\text{unit}"] \\
& \cL_{[\geq w]} {\check{\cC}}_{\scI^+}(\cM) \ar[ul, "\text{[1]}" description]
& & \cL_{<w}(\cM) \ar[ul, "\text{[1]}" description]
\end{tikzcd}$$ which gives the following generalization of to the non-affine case: $$\cD({{\rm Gr}}(\cA)) \, = \, \langle \, \cD_{<w}({{\rm Gr}}(\cA)) \, , \, \cL_{[\geq w]} (\cD_{{\scI^+\text{-}{\rm triv}}}({{\rm Gr}}(\cA))) \, , \, \cD_{{{\rm Tor}^+},[\geq w]}({{\rm Gr}}(\cA)) \, \rangle$$ where the component in the middle is the essential image of the functor $\cL_{[\geq w]} : \cD_{{\scI^+\text{-}{\rm triv}}}({{\rm Gr}}(\cA)) {\rightarrow}\cD({{\rm Gr}}(\cA))$, which is fully faithful with left quasi-inverse ${\check{\cC}}_{\scI^+}$. Alternatively, it may be described as $$\cL_{[\geq w]} (\cD_{{\scI^+\text{-}{\rm triv}}}({{\rm Gr}}(\cA))) \, = \,
\{ \, \cM \in \cD_{[\geq w]}({{\rm Gr}}(\cA)) \, | \, {{\bm R}\Gamma}_{\scI^+}(\cM) \in \cD_{<w}({{\rm Gr}}(\cA)) \, \}$$
Again, once we have formally defined the relevant functors, their properties can be checked affine-locally. For example, the analogues of Theorem \[three\_term\_SOD\_Dmcoh\], \[I\_triv\_weight\_equiv\_Dsuitcoh\], \[cond\_then\_reg\], and \[smooth\_then\_reg\_and\_perf\] hold without change in the setting .
As we explained in the introduction (see the paragraph of ), every wall-crossing in birational cobordism can be reduced to the setting of a sheaf of $\bZ$-graded ring. Moreover, the “master space construction” of Thaddeus [@Tha96] realizes every variation of GIT quotient as a birational cobordism.
Derived categories under flips and flops {#Dcat_flip_flop_sec}
========================================
Let $A$ be a Noetherian $\bZ$-graded ring. Recall from Proposition \[coh\_IpTR\_QpGr\] and Remark \[stacky\_Proj\] that the derived category ${\cD^b_{{\rm coh}}}({\mathpzc{Proj}}^+(A))$ of the stacky projective space is equivalent to $\cD^b_{{{\rm coh}}({I^+\text{-}{\rm triv}})}({{\rm Gr}(A)})$ defined in Definition \[IpTR\_coh\_def\]. Thus, Corollary \[weight\_Cech\_Dsuitcoh\_SOD\] and Theorem \[I\_triv\_weight\_equiv\_Dsuitcoh\] combine to show that ${\cD^b_{{\rm coh}}}({\mathpzc{Proj}}^+(A))$ is a semi-orthogonal summand of ${\cD^b_{{\rm coh}}}({\cF_{\geq w}})$. By symmetry, all these results hold for the negative direction as well, where the weight truncation is controlled by the full subcategory ${\cF_{\leq -w}}\subset \cF$. Thus, we see likewise that ${\cD^b_{{\rm coh}}}({\mathpzc{Proj}}^-(A))$ is a semi-orthogonal summand of ${\cD^b_{{\rm coh}}}({\cF_{\leq -w}})$.
As we observed in , the pre-additive category ${\cF_{\leq -w}}$ is isomorphic to the opposite of ${\cF_{\geq w}}$. Thus, ${\cD^b_{{\rm coh}}}({\mathpzc{Proj}}^+(A))$ is a semi-orthogonal summand of the derived category ${\cD^b_{{\rm coh}}}({\cF_{\geq w}})$ of *right* modules; while ${\cD^b_{{\rm coh}}}({\mathpzc{Proj}}^-(A))$ is a semi-orthogonal summand of the derived category ${\cD^b_{{\rm coh}}}(({\cF_{\geq w}})^{{{\rm op}}})$ of *left* modules. This suggests one to relate the derived categories by taking a duality functor (see and ) $$\label{D_Fgw_functor}
\bD_{{\cF_{\geq w}}} \, : \, \cD({\cF_{\geq w}})^{{{\rm op}}} {\,\rightarrow \,}\cD({\cF_{\leq -w}})\, , \qquad \cM \, \mapsto \, \bD_{{\cF_{\geq w}}}(\cM) := {{\bm R}{\rm Hom}}_{{\cF_{\geq w}}}(\cM, {\cF_{\geq w}})^{\tau}$$
We will see that this tends to works well when there is a certain duality between the local cohomology complexes ${{\bm R}\Gamma}_{I^+}(A)$ and ${{\bm R}\Gamma}_{I^-}(A)$. Let $\omega_Y^{\bullet} \in {\cD^b_{{\rm coh}}}(A_0)$ be a dualizing complex, and take the weight degreewise dualizing functor $$\bD_Y : \cD({{\rm Gr}(A)})^{{{\rm op}}} {\rightarrow}\cD({{\rm Gr}(A)}) \, , \qquad \quad \bD_Y(M)_i \simeq {{\bm R}{\rm Hom}}_{A_0}(M_{-i}, \omega_Y^{\bullet})$$ Then we will consider the assumptions $$\label{Gor_and_local_cohom_dual_2}
\parbox{40em}{(i) $A$ is Gorenstein. \\
(ii) There is an isomorphism $\Psi : {{\bm R}\Gamma}_{I^+}(A)(a)[1] {\xrightarrow[]{\cong }}\bD_Y({{\bm R}\Gamma}_{I^-}(A))$ in $\cD({{\rm Gr}(A)})$.}$$ In fact, one of the main results in [@Yeu20a] is that a large class of flips and flops are controlled by sheaves of rings that satisfy this assumption, where $a = 0$ for a flop, and $a = 1$ for a flip. We will see below that $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)})$ tends to be smaller than $\cD_{{I^-\text{-}{\rm triv}}}({{\rm Gr}(A)})$ if $a > 0$, and they tend to be equivalent if $a = 0$. For some other results along this line, see [@Yeu20a Section 6].
First, we observe that the duality functor can be expressed alternatively in terms of a duality on $\cD({{\rm Gr}(A)})$. Indeed, consider the functor $${\bD_A}\, : \, \cD({{\rm Gr}(A)})^{{{\rm op}}} {\,\rightarrow \,}\cD({{\rm Gr}(A)}) \, , \qquad {\bD_A}(M) := {{\bm R} \underline{{\rm Hom}}}_A(M,A)$$ Then the following diagram commutes up to isomorphism of functors $$\label{DAO_DFgw_diag}
\begin{tikzcd}[column sep = 80]
\cD_{[\geq w]}({{\rm Gr}(A)})^{
{{\rm op}}} \ar[r, " \cL_{[\leq -w]} \circ {\bD_A}"] \ar[d, "(-)^{\sharp}_{\geq w}"', "\simeq"] & \cD_{[\leq -w]}({{\rm Gr}(A)}) \ar[d, "(-)^{\sharp}_{\leq -w}", "\simeq"']\\
\cD({\cF_{\geq w}})^{{{\rm op}}} \ar[r, "\bD_{{\cF_{\geq w}}}"] & \cD(\cF_{[\leq -w]})
\end{tikzcd}$$ Indeed, it suffices to prove the commutativity after replacing the vertical arrow on the left by its inverse $\scL_{[\geq w]} : \cD({\cF_{\geq w}}) {\xrightarrow[]{\simeq}} \cD_{[\geq w]}({{\rm Gr}(A)})$, so that the commutativity follows (see ) from taking the isomorphism $${{\bm R}{\rm Hom}}_{\cF}(\cM \otimes_{{\cF_{\geq w}}}^{{\bm L}} \cF ,\cF) \, \cong \, {{\bm R}{\rm Hom}}_{{\cF_{\geq w}}}(\cM, \cF)$$ in $\cD((\cF)^{{{\rm op}}})$, restrict to $\cD(({\cF_{\geq w}})^{{{\rm op}}})$, and take the transpose .
\[D\_Fgw\_ITR\] If (ii) holds for $a \geq 0$, then $\bD_{{\cF_{\geq w}}}$ sends $\cD^-_{{{\rm coh}},{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$ to $\cD^+_{{{\rm coh}},{I^-\text{-}{\rm triv}}}({\cF_{\leq -w}})$.
If (ii) holds for $a = 0$, then $\bD_{{\cF_{\leq -w}}}$ also sends $\cD^-_{{{\rm coh}},{I^-\text{-}{\rm triv}}}({\cF_{\leq -w}})$ to $\cD^+_{{{\rm coh}},{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$.
It suffices to prove the first statement as the second statement follows by symmetry. Recall that the equivalence $(-)^{\sharp}_{\geq w} : \cD_{[\geq w]}({{\rm Gr}(A)}) {\xrightarrow[]{\simeq}} \cD({\cF_{\geq w}})$ restricts to functors ${\cD^{\tiny \mbox{$\spadesuit $}}}_{{{\rm coh}},[\geq w]}({{\rm Gr}(A)}) {\rightarrow}{\cD^{\tiny \mbox{$\spadesuit $}}}_{{{\rm coh}}}({\cF_{\geq w}})$ for each $\spadesuit \in \{ \, \, , +,-,b\}$, which is moreover an equivalence if $\spadesuit = -$. Recall also that $M^{\sharp}_{\geq w} \cong (\cL_{[\geq w]} M)^{\sharp}_{\geq w}$ for all $M \in \cD({{\rm Gr}(A)})$. Moreover, we have $M^{\sharp}_{\geq w} \in \cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})$ if and only if ${{\bm R}\Gamma}_{I^+}(M) \in \cD_{<w}({{\rm Gr}(A)})$. Thus, in view of , the result follows from the following Lemma.
\[RGam\_w\_DAO\] Suppose that (ii) holds for $a \geq 0$. If $M \in {\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)})$ satisfies ${{\bm R}\Gamma}_{I^+}(M) \in \cD_{<w}({{\rm Gr}(A)})$, then we have ${{\bm R}\Gamma}_{I^-}({\bD_A}(M)) \in \cD_{>-w+a}({{\rm Gr}(A)})$.
Since $\bD_Y$ is involutive on complexes with locally coherent cohomology (see Lemma \[RGam\_Dbcoh\]), condition (ii) can be rewritten as an isomorphism ${{\bm R}\Gamma}_{I^-}(A) \cong \bD_Y( {{\bm R}\Gamma}_{I^+}(A) )(-a)[-1]$. As a result, we have the following isomorphism in $\cD({{\rm Gr}(A)})$: $$\begin{split}
{{\bm R} \underline{{\rm Hom}}}_A( M , A ) \otimes_A^{{\bm L}} {{\bm R}\Gamma}_{I^-}(A)
\, &\cong \, {{\bm R} \underline{{\rm Hom}}}_A( M , {{\bm R}\Gamma}_{I^-}(A) ) \\
&\cong \, {{\bm R} \underline{{\rm Hom}}}_A( M , \bD_Y( {{\bm R}\Gamma}_{I^+}(A) ))(-a)[-1] \\
& = \, \bD_Y( \, {{\bm R}\Gamma}_{I^+}(M) \,)(-a)[-1]
\end{split}$$ where the first isomorphism uses Proposition \[tensor\_in\_Hom\_target\].
In Proposition \[D\_Fgw\_ITR\], if the small pre-additive category ${\cF_{\geq w}}$ is “Gorenstein” in a suitable sense, then the functor $\bD_{{\cF_{\geq w}}}$ gives a contravariant equivalence between ${\cD^b_{{\rm coh}}}({\cF_{\geq w}})$ and ${\cD^b_{{\rm coh}}}({\cF_{\leq -w}})$. Proposition \[D\_Fgw\_ITR\] then shows that $\cD^b_{{{\rm coh}}({I^+\text{-}{\rm triv}})}({{\rm Gr}(A)})$ and $\cD^b_{{{\rm coh}}({I^-\text{-}{\rm triv}})}({{\rm Gr}(A)})$ are related in the expected way: [[*i.e.*]{}]{}, the former is smaller for flips, and the two are equivalent for flops. While ${\cF_{\geq w}}$ may not be “Goresntein” in general, the duality functor $\bD_{{\cF_{\geq w}}}$ is always well-behaved on ${\cD_{{\rm perf}}}({\cF_{\geq w}})$. As a result, we have the following
\[flip\_flop\_Dperf\] Denote by $\cD_c \subset \cD$ the subcategory of compact objects.
Suppose that (ii) holds for $a \geq 0$. If the semi-orthogonal decomposition on ${\cD^b_{{\rm coh}}}({\cF_{\geq w}})$ restricts to one on ${\cD_{{\rm perf}}}({\cF_{\geq w}})$, then the functor restricts to a fully faithful functor $\bD_{{\cF_{\geq w}}} : (\cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})_c)^{{{\rm op}}} {\hookrightarrow}\cD_{{I^-\text{-}{\rm triv}}}({\cF_{\leq -w}})_c$.
Suppose that (ii) holds for $a = 0$. If the semi-orthogonal decomposition , and its negative version, restrict to ${\cD_{{\rm perf}}}({\cF_{\geq w}})$ and ${\cD_{{\rm perf}}}({\cF_{\leq -w}})$ respectively, then the functor restricts to an equivalence $\bD_{{\cF_{\geq w}}} : (\cD_{{I^+\text{-}{\rm triv}}}({\cF_{\geq w}})_c)^{{{\rm op}}} {\xrightarrow[]{\simeq}} \cD_{{I^-\text{-}{\rm triv}}}({\cF_{\leq -w}})_c$.
\[pairing\_remark\] In Corollary \[flip\_flop\_Dperf\], we required to restrict to ${\cD_{{\rm perf}}}({\cF_{\geq w}})$ because the duality functor $\bD_{{\cF_{\geq w}}}$ is well-behaved there. There is an alternative formulation of duality in terms of pairings of DG categories, which is well-behaved on the entire category. Namely, let $\Omega : \cA \times \cB {\rightarrow}\cD(k)$ be a bi-cocontinuous exact functors between compactly generated triangulated category, with given DG enhancements. Then $\Omega$ induces co-continuous functors $\Omega^L : \cA {\rightarrow}\cB^{\vee}$ and $\Omega^R : \cB {\rightarrow}\cA^{\vee}$, where $\cB^{\vee}$ is the Ind-completion of $(\cB_c)^{{{\rm op}}}$, carried out at the DG level, and similarly for $\cA$. One can show that, if the essential image of $\Omega^L$ (resp. $\Omega^R$) contains all the compact objects, then $\Omega^R$ (resp. $\Omega^L$) is fully faithful. To apply this general formalism to our situation, consider $C^{(i)}_{\pm} \in {\cD^{-}_{{\rm coh}}}({{\rm Gr}(A)})$ defined by $$C^{(i)}_+ := \cL_{[\geq w]}({\check{\cC}}_{I^+}(A)(-i))
\qquad \text{and} \qquad
C^{(i)}_- := \cL_{[\leq -w]}({\check{\cC}}_{I^-}(A)(-i))$$ One can show that, if the canonical evaluation maps $$\label{evaluation_map_weight_trunc}
\begin{split}
{{\bm R} \underline{{\rm Hom}}}_A( C^{(i)}_+ , A) \otimes_A^{{\bm L}} C^{(j)}_+ &{\,\rightarrow \,}{{\bm R} \underline{{\rm Hom}}}_A( C^{(i)}_+ , C^{(j)}_+ ) \\
{{\bm R} \underline{{\rm Hom}}}_A( C^{(i)}_- , A) \otimes_A^{{\bm L}} C^{(j)}_- &{\,\rightarrow \,}{{\bm R} \underline{{\rm Hom}}}_A( C^{(i)}_- , C^{(j)}_- )
\end{split}$$ are quasi-isomorphism in weight $0$, then we have $$\parbox{42em}{(1) If \eqref{Gor_and_local_cohom_dual_2}(ii) holds for $a \geq 0$, there is a fully faithful $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}) {\hookrightarrow}\cD_{{I^-\text{-}{\rm triv}}}({{\rm Gr}(A)})$. \\
(2) If \eqref{Gor_and_local_cohom_dual_2}(ii) holds for $a = 0$, there is an equivalence $\cD_{{I^+\text{-}{\rm triv}}}({{\rm Gr}(A)}) \simeq \cD_{{I^-\text{-}{\rm triv}}}({{\rm Gr}(A)})$.}$$ If the semi-orthogonal decomposition , and its negative version, restrict to ${\cD_{{\rm perf}}}({\cF_{\geq w}})$ and ${\cD_{{\rm perf}}}({\cF_{\leq -w}})$ respectively, then $C^{(i)}_{\pm}$ are in ${\cD_{{\rm perf}}}({{\rm Gr}(A)})$, so that are quasi-isomorphisms. In general, the evaluation map being a quasi-isomorphism is essentially an issue about convergence of spectral sequences, and seems to be similar to the issues one encounters in Koszul duality. It seems possible that a formal modification of our arguments might lead to a more satisfactory statement. This might open up a way to tackle a conjecture of Bondal and Orlov.
Modules over pre-additive categories {#app_mod_preadd}
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A *pre-additive category* is a category $\cA$ enriched over the monoidal category $({{\rm Ab}}, \otimes)$ of abelian groups. It is said to be *small* if the objects of $\cA$ form a set ${{\rm Ob}}(\cA)$. It is helpful to think of a small pre-additive category as an “associative ring with many objects”, as in [@Mit72]. This allows us to define the notions of left/right modules, tensor products, Hom spaces, etc, which we recall now.
Given a small pre-additive category $\cA$, a *left $\cA$-module* is an additive functor $\cA {\rightarrow}{{\rm Ab}}$, while a *right $\cA$-module* is an additive functor $\cA^{{{\rm op}}} {\rightarrow}{{\rm Ab}}$. Maps between left or right modules are simply natural transformations. We will mostly work with right modules, and we denote the category of right $\cA$-modules by ${{\rm Mod}}(\cA)$. In more concrete terms, a right $\cA$-module associates an abelian group $M_a$ to each $a \in {{\rm Ob}}(\cA)$, together with maps $M_a \otimes \cA(a',a) {\rightarrow}M_{a'}$, satisfying the obvious associativity and unitality conditions.
Given small pre-additive categories $\cA$ and $\cB$, an *$(\cA,\cB)$-bimodule* consists of a collection $M(b,a) = \!\,_a M_b$ of abelian groups, one for each pair $a \in {{\rm Ob}}(\cA)$ and $b \in {{\rm Ob}}(\cB)$, together with maps $\cA(a,a') \otimes \!\,_a M_b \otimes \cB(b',b) {\rightarrow}\!\,_{a'} M_{b'}$, satisfying the obvious associativity and unitality conditions. For example, $\cA$ is canonically a bimodule over itself. Denote by $\!\,_{\cA}{{\rm Mod}}_{\cB}$ the category of $(\cA,\cB)$-bimodules. If $M \in \!\,_{\cA}{{\rm Mod}}_{\cB}$ and $N \in \!\,_{\cB}{{\rm Mod}}_{\cC}$, then define $M \otimes_{\cB} N \in \!\,_{\cA}{{\rm Mod}}_{\cC}$ by $$\label{bimod_tensor}
\! \,_a(M \otimes_{\cB} N)_{c} \, := \bigl( \, \bigoplus_{b \in {{\rm Ob}}(\cB)} \! \,_aM_b \otimes \! \,_b N_c \bigr) \big/ ( \, \xi f \otimes \eta - \xi \otimes f \eta \,)$$ where we mod out the abelian subgroup generated by the displayed relations, for $\xi \in \,_aM_{b'}$, $f \in \cB(b,b')$, and $\eta \in \,_bM_{c}$. In particular, if $\cA = \cC = \ast$ is the pre-additive category with one object, with endomorphism algebra $\bZ$, then this gives the notion of a tensor product $M \otimes_{\cB} N \in {{\rm Ab}}$ between a right $\cB$-module $M$ and a left $\cB$-module $N$.
Similarly, if $M \in \!\,_{\cA}{{\rm Mod}}_{\cB}$ and $N \in \!\,_{\cC}{{\rm Mod}}_{\cB}$, then we define ${{\rm Hom}}_{\cB}(M,N) \in \!\,_{\cC}{{\rm Mod}}_{\cA}$ by $$\label{bimod_Hom}
\! \,_c {{\rm Hom}}_{\cB}(M,N)_{a} \, := \, \bigl \{ (\varphi_b) \in \prod_{b \in {{\rm Ob}}(\cB)} {{\rm Hom}}_{{{\rm Ab}}}( \! \,_a M_b, \! \,_c N_b) \, \big| \, \varphi_{b}(\xi f) = \varphi_{b'} (\xi) f \, \, \, \forall \,
\xi \in \! \,_a M_{b'} , \, f \in \cB(b,b') \,\bigr \}$$ In other words, $\! \,_c {{\rm Hom}}_{\cB}(M,N)_{a}$ is the Hom-space ${{\rm Hom}}_{\cB}(\!\,_a M , \!\,_c N)$ in the (big) additive category ${{\rm Mod}}(\cB)$.
As for usual associative algebras, there are canonical isomorphisms $$\label{triv_Hom_ten}
M \otimes_{\cB} \cB \cong M \cong \cA \otimes_{\cA} M
\qquad \text{ and } \qquad
{{\rm Hom}}_{\cB}(\cB,M) \cong M$$
For any $M \in \!\,_{\cA}{{\rm Mod}}_{\cB}$, $N \in \!\,_{\cB}{{\rm Mod}}_{\cC}$, and $L \in \!\,_{\cE}{{\rm Mod}}_{\cC}$, there is a usual Hom-tensor adjunction, given by the canonical isomorphism of $(\cE,\cA)$-bimodules $$\label{bimod_Hom_ten_adj}
{{\rm Hom}}_{\cC}(M \otimes_{\cB} N , L) \, \cong \, {{\rm Hom}}_{\cB}(M , {{\rm Hom}}_{\cC}(N,L))$$
For each $a \in {{\rm Ob}}(\cA)$, denote by $\!\,_a\cA$ the right $\cA$-module represented by $a$. In other words, $\!\,_a\cA_{a'} := \cA(a',a)$. A right module $M$ is said to be *free* if there is an indexed set of objects $\varphi : S {\rightarrow}{{\rm Ob}}(\cA)$, together with an isomorphism $M \cong \oplus_{s \in S} \, (\!\,_{\varphi(s)}\cA)$. In more concrete terms, this means that there is a set $S$ of elements $\xi_s \in M_{\varphi(s)}$ such that, for any $a \in {{\rm Ob}}(\cA)$, any element $\xi \in M_a$ can be written uniquely as a finite sum $\xi = \sum \xi_s f_s$, for $f_s \in \cA(a,\varphi(s))$. The cardinality of $S$ is said to be the *rank* of the free module $M$.
Clearly the category ${{\rm Mod}}(\cA)$ of right modules is an abelian category, where limits and colimits are determined objectwise. Thus, it also satisfies the usual Ab5 and Ab3\* axioms of an abelian category. Moreover, the set $\{ \,_a\cA \, \}_{a \in {{\rm Ob}}(\cA)}$ of right modules forms a set of generators for ${{\rm Mod}}(\cA)$, so that ${{\rm Mod}}(\cA)$ is a Grothendieck category (see, e.g., [@Sta Tag 079B]). Projective objects in ${{\rm Mod}}(\cA)$ are precisely retracts of free modules. A projective right module is said to be of *finite rank* if it is a retract of a free module of finite rank.
\[fin\_gen\_mod\] We say that a right module $M \in {{\rm Mod}}(\cA)$ is *finitely generated* if there is an epimorphism $\oplus_{s \in S} \, (\!\,_{\varphi(s)}\cA) {\twoheadrightarrow}M$ for a finite indexed set of objects $\varphi : S {\rightarrow}{{\rm Ob}}(\cA)$.
In more concrete terms, this means that there is a finite set $S$ of elements $\xi_s \in M_{\varphi(s)}$ such that, for any $a \in {{\rm Ob}}(\cA)$, any element $\xi \in M_a$ can be written as a finite sum $\xi = \sum \xi_s f_s$, for $f_s \in \cA(a,\varphi(s))$.
A small pre-additive category $\cA$ is said to be *right Noetherian* (resp. *left Noetherian*) if every submodule of a finitely generated right (resp. left) $\cA$-module is finitely generated. It is said to be *Noetherian* if it is both left and right Noetherian.
Since ${{\rm Mod}}(\cA)$ is a Grothendieck category, it has enough injectives (see, e.g., [@Sta Tag 079H]). Moreover, complexes in ${{\rm Mod}}(\cA)$ admit K-injective resolutions (see, e.g., [@Sta Tag 079P]). The category ${{\rm Mod}}(\cA)$ clearly has enough projectives. Thus, by [@Spa88 Theorem 3.4] (see also [@Sta Tag 06XX]), complexes in ${{\rm Mod}}(\cA)$ admit K-projective resolutions. This allows us to take derived functors of the above Hom functors and tensor functors. However, a subtlety arises when one wants to take the derived tensor product or the derived Hom bimodule between bimodules. For example, even if $M \in \!\,_{\cA}{{\rm Mod}}_{\cB}$ is projective in the category $\!\,_{\cA}{{\rm Mod}}_{\cB}$, it may not be true that each $\!\,_a M \in {{\rm Mod}}(\cB)$ is projective, or even flat, so that it might be problematic if one wants to derive and naively. However, for our purposes, we will only need to consider the derived tensor product (or derived Hom) between a module and a bimodule. For these, there are no problems, and we may define $$\label{derived_Hom_ten_1}
\begin{split}
- \otimes_{\cA}^{{\bm L}} - \, &: \, \cD(\cA) \, \times \, \cD( \!\,_{\cA}{{\rm Mod}}_{\cB} ) {\,\rightarrow \,}\cD(\cB) \\
{{\bm R}{\rm Hom}}_{\cA}(-,-) \, &: \, \cD(\cA)^{{{\rm op}}} \, \times \, \cD( \!\,_{\cB}{{\rm Mod}}_{\cA} ) {\,\rightarrow \,}\cD(\cB^{{{\rm op}}})
\end{split}$$
In particular, given an additive functor $F : \cA {\rightarrow}\cB$, then $\cB$ may be viewed as an $(\cA,\cB)$-bimodule in the obvious way, so that the extension functor $$- \otimes_{\cA}^{{\bm L}} \cB \, : \, \cD(\cA) {\,\rightarrow \,}\cD(\cB)$$ is well-defined. Moreover, it satisfies the usual “cancellation rule” for tensor products $$\label{tensor_two_step}
(M \otimes^{{\bm L}}_{\cA} \cB) \otimes^{{\bm L}}_{\cB} N \, \cong \, M \otimes^{{\bm L}}_{\cA} N$$ for any $M \in \cD(\cA)$ and $N \in \cD(\!\,_{\cB}{{\rm Mod}}_{\cC})$.
An object $M \in \cD(\cA)$ is said to be *pseudo-coherent* if it is quasi-isomorphic to a bounded above complex $P^{\bullet}$ of projective modules of finite rank. Denote by ${\cD_{{\rm pc}}}(\cA) \subset \cD(\cA)$ the full subcategory consisting of pseudo-coherent objects.
Suppose $\cA$ is right Noetherian, then denote by ${\cD^{-}_{{\rm coh}}}(\cA) \subset \cD(\cA)$ the full subcategory consisting of objects $M \in \cD(\cA)$ such that each $H^p(M)$ is finitely generated, and $H^p(M) = 0$ for $p \gg 0$.
\[Dmcoh\_Dpc\_right\_Noeth\] Suppose $\cA$ is right Noetherian, then for any $M \in \cD(\cA)$, the followings are equivalent:
1. $M \in {\cD_{{\rm pc}}}(\cA)$;
2. $M \in {\cD^{-}_{{\rm coh}}}(\cA)$;
3. $M$ is quasi-isomorphic to a bounded above complex of free modules of finite rank.
Clearly only the implication $(2) \Rightarrow (3)$ needs proof. It follows from the well-known Lemma \[Dbcoh\_pc\_lem\] below (for a proof, see, [[*e.g.*]{}]{}, [@Yeu20c Lemma A.40]).
\[Dbcoh\_pc\_lem\] Let $\cC$ be an abelian category, and let $\cP \subset {{\rm Ob}}(\cC)$ be a collection of projective objects closed under finite direct sum. Denote by $Q(\cP) \subset {{\rm Ob}}(\cC)$ the collection of objects $M$ such that there exists an epimorphism $P {\twoheadrightarrow}M$ from some $P \in \cP$. Suppose $Q(\cP)$ is closed under taking subobjects, then for any bounded above complex $M^{\bullet}$ in $\cC$ whose cohomology objects lie in $Q(\cP)$, there exists a bounded above complex $P^{\bullet}$ of objects in $\cP$, together with a quasi-isomorphism $\varphi: P^{\bullet} {\stackrel{\sim}{\rightarrow}}M^{\bullet}$.
Clearly, the set $\{ \!\,_a \cA \}_{a \in {{\rm Ob}}(\cA)}$ forms a set of compact generators of $\cD(\cA)$. Denote by ${\cD_{{\rm perf}}}(\cA)$ the smallest split-closed triangulated subcategory of $\cD(\cA)$ containing the set $\{ \!\,_a \cA \}_{a \in {{\rm Ob}}(\cA)}$ of objects, then it is a standard fact (see, e.g., [@Rou08 Theorem 4.22] or [@Nee92 Lemma 2.2]) that $$\label{cpt_equal_perf}
\cD(\cA)_c \, = \, {\cD_{{\rm perf}}}(\cA)$$ where the subscript $(-)_c$ denotes the subcategory of compact objects.
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[^1]: By the exact triangle , this is the same as asking whether the local cohomology modules $H^j({{\bm R}\Gamma}_{I^+}(M))$ are finitely generated. For a result along this line, see, [[*e.g.*]{}]{}, [@Sta Tag 0BJV].
[^2]: For details about local homology, see [@Yeu20c], whose notation is adopted here.
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The German shepherd named Rex was found at the top of a remote hill, abandoned and chained to a tree.
When a dog is abandoned, it's usually left during a remote location in the hopes that it'll not return home. Such animals become strays, living their lives on the road, occasionally being rescued and brought to the local shelter.
In once case in Greece, a German shepherd was left chained to a tree at the highest of a hill. Fortunately for him, he was rescued by some kind people and was ready to live a cheerful life. Tied up and abandoned, The German shepherd, named Rex, was found at the highest of a foreign hill, abandoned and chained to a tree. The police believe that he was left there to stay away from the sheep. The police soon called Takis Proestakis, an area man who had made it his life mission to save lots of and supply a home for the strays on his island home of Crete. Exposed to the weather
As soon as Takis arrived, he approached Rex. The poor dog had been left, exposed to the cold and rain, for who knows how long. Rex has no food or water, so people who leave him probably won't care about his welfare. Rex flinched in fear as Takis, so the police decided what to do next. Takis brought some food for Rex before approaching him. Confident that he wouldn’t bite, Takis got closer. surely, asOf course, as Takis approached, Rex seemed to feel the kindness of the men. Fortunately, Rex was ready to muster his courage and eventually allowed the Takis to the touch him. Rex’s friendly nature shines through
As he grew familiar with Takis’ touch, Rex’s tail began to wag, a sure sign that he was friendly. Soon, Takis brought Rex some water for a drink before he led him to the car for a visit back to Takis’ shelter, where he houses and cares for many stray dogs from the world. Rex fit right in
Takis took Rex to the shelter, where he was verified, and Rex went straight to the bucket outside. He then drank for a minimum of three minutes, getting his fill. Additionally to food, water, and shelter, Rex was also ready to make some new friends among the opposite dogs at the shelter. Thanks to Takis, Rex now has a new opportunity to live an honest life.
Takis runs a sanctuary on the island of Crete, fearing that most of the animals he rescues come from nearby garbage dumps.dump. He eventually built an outsized shelter for all of them to measure at, called Takis Shelter, where he cares for the dogs, cats, and goats that he has rescued over the years. For more on Takis and his shelter, visit his Website. You can watch the video below of Rex’s rescue. | https://www.androdass.com/2020/08/neglected-german-shepherd-found.html |
- This is the number of instructions a single processor core can carry out per second (Hz). For most desktop computers, this will be somewhere around 3.5 GHz (3.5 billion instructions per second).
- The higher the clock speed, the greater the number of instructions that can be carriedd out per second.
- Some CPUs can be overclocked to make then run at a higher clock speed than the factory-set rate. But it's risky if not done properly - it can make CPUs overheat, causing crashes or permanent damage to the system. High performance cooling systems are usually needed.
1 of 5
Number of Cores
- Each core in a CPU can process data independently of rest.
- The more cores a CPU has, the more instructions it can carry out at once, so the faster it can process a batch of data.
2 of 5
Cache Size
- The Cache is data stored inside the CPU that's much faster than RAM.
- A larger CPU cache gives the CPU faster access to more data it needs to process.
- Generally speaking, CPUs with higher clock speeds, more cores or larger caches will have better performance, but will also be more expensive.
3 of 5
More RAM
- If a computer has too little RAM it may run slowly due to the use of virtual memory.
- The more RAM, the more applications or more memory intensive applications it can smoothly run, making it faster overall.
- It's easy to upgrade RAM on a PC or laptop - it's just a matter of replacing the RAM stick with higher capacity (or higher speed) ones.
- If the computer already has plenty of RAM to run everything the user wants, increasing RAM may make no difference to performance.
4 of 5
GPUs
- GPUs (graphics processing units) are specialised circuits for handling graphics and image proccessing. They relieve the processing load on the CPU, freeing it to do other things.
- Computers have basic GPUs integrated onto the motherboard or the CPU. For better graphics performance, a dedicated GPU is often used.
- Using high-end graphics cards can greatly improve performance in graphics-intensive applications. | https://getrevising.co.uk/revision-cards/cpu-and-system-performance |
When I was about 5 years old my brother, Gordon came home on leave and he brought his new wife Lily and her 2 children with him. Lily was born and raised in France. She had met Gordon while he was stationed there, and it didn’t take long before they were married. When they showed up at the door, we were all so surprised and thrilled to learn that they were expecting a baby. It was odd for me to have a niece and a nephew that were only a couple of years younger than I but I was also excited about having a baby niece or nephew.
Lily was a fun person to be around. She had learned to speak English pretty well, although she still had a little trouble making us understand some things she said. One day Lily and I were out in the yard walking around and she was looking very nervous. I asked her what was wrong, and she said, “I like spiders and I am looking for one.” I thought she was nuts! I hate spiders; they are probably the only creepy crawler that scares me. The next day my brother took me with him to the store and he gave me some money to spend. I bought a rubber spider. When I got home. I put it in a box and wrapped it up really nice. I was so excited; I ran to the dinner table and gave it to Lily. She was so touched that I thought of her she eagerly tore the wrapping off and opened the box. The next thing I knew chaos broke out. Lily threw the box into the air and was screaming hysterically! My brother was trying to calm her down, my Dad found the box which had bounced off the wall and had flown over the table and he tossed it outside, my sister was laughing so hard she had tears flowing down her face, the two kids were crying, and my mother fainted.
When everything calmed down my mother asked me “How could you do such a horrible thing to Lily?” I told her what she had told me the day before and then Lily started laughing. She realized she had said that she liked spiders instead of disliked them. Everyone was fine with it except my mother. I got in a lot of trouble for it and almost every day until my Nephew Earl was born I had to hear my mother tell me that the poor little baby was now going to have a spider-shaped birthmark on it and that it will be all my fault. After Earl was born Gordon called to tell us he was fine, no birthmarks of any kind. My mother never believed them; she thought they were just saying that to make me feel better.
I am a professional genealogist, writer, photographer, wife, mother, and grandma. I have two books available on Amazon.com: Your Family History: Doing It Right the First Time and Planning Your Genealogy Research Trip. You can also connect with me via Facebook or Twitter. | https://genealogywithvalerie.com/2020/03/02/mondays-for-me-a-lesson-learned/ |
After moving to Dubai in the United Arab Emirates in December 2015, Lisa O’Donohue immersed herself in the Dubai-Irish community. Although she is now a member of the Dubai Celts GAA Club, she never played Gaelic football until she moved to the desert, 8,000km away from her hometown of Galway. The 25-year-old, who works in marketing, recently represented Dubai in the Rose of Tralee.
Where is the first place you bring people when they’re visiting?
Anyone who visits me will be ushered to Old Dubai within the first few days of their trip. It is a rich cultural oasis in a city that is more often known for its glamorous, cosmopolitan way of life. Winding streets filled with the aromas of fresh spices, traditional abras to carry you across the creek, and bustling market stalls each promising a better bargain than the last. This part of Dubai is immersed in traditional Arabic charm.
The top three things to do in Dubai that don’t cost money are . . .
Although Dubai can be an expensive place to visit, there are plenty of things to do and see here that won’t make your eyes pop and your purse flop. I found it difficult to narrow it down to just three.
1. Wander around Al Bastakiya’s enchanting streets that date back to the 1800s
2. Watch the world-famous dancing fountain show that takes place in front of the world’s tallest building, the Burj Khalifa
3. Embrace the local landscape and go camping in the desert
Where do you recommend for a great meal that gives a flavour of Dubai?
Living in a city where there are more expats than locals, we are exposed to so many cultures and their traditional foods. Fresh falafel and Arabic bread baked on the walls of hot stone ovens are foods that you indulge in when experiencing a desert safari (or if you want a flavour of India and Pakistan, the unassuming Ravi’s is your place to go).
Where is the best place to get a sense of Dubai’s place in history?
The Dubai Museum is located in Al Fahidi Fort, an old defence fort that was renovated to showcase Dubai’s history. Transitioning from a small fishing village into one of the world’s largest cities, the history of the Emirate of Dubai is a fascinating one.
What should visitors save room for in their suitcase after a visit to Dubai?
There are beautiful little gold or silver necklaces on which your name, or a word of your choice, is displayed in Arabic. I still haven’t bought one yet, but it’s definitely on my list before I eventually move home to Ireland.
If you’d like to share your little black book of places to visit where you live, please email your answers to the five questions above to [email protected], including a brief description of what you do there and a photograph of yourself. We’d love to hear from you. | https://www.irishtimes.com/life-and-style/abroad/welcome-to-my-place-dubai-1.3360384 |
Q:
Python import dependencies
I am dynamically creating some python code that I want to run inside a wrapper. Here is an overly simplified example.
[wrapper.py]
import cv2
img = cv2.imread('pic.png',0)
__import__("fragment")
cv2.imshow('pic',img)
[fragment.py]
img = cv2.cvtColor(img, cv2.COLOR_GRAY2BGR)
I want the wrapper to set up any imports and variables, then import the fragment which will do stuff (i.e. make the image grayscale) and then do some standardized stuff afterwards (i.e. display image).
The fragments will be changing (genetic algorithm) so I would prefer to keep them separate from the setup which will be constant and will just get make manipulating the fragments more complicated.
When I run the program I get dependency errors on the fragment because cv2 and img are not defined (scope errors). Is there a way to achieve this either with a correction to the method I have used above or with another method?
I expect I might be able to also create the composite of the files in ram and then exec it or write over the fragment with a version of itself that contains all of the needed wrapping, but I wanted to see if there was something cleaner first.
Sincerely, Paul.
A:
The fragments will be changing (genetic algorithm) so I would prefer
to keep them separate from the setup which will be constant and will
just get make manipulating the fragments more complicated.
Whatever the complexity of the genetic algorithms you implemented in fragment.py is, I do not see how importing cv2 (and eventually more modules) will impact it in a way or an other.
However, I agree with the first part of your statement in that you want to respect the principle of separation of concerns and make your code cleaner.
The solution I see for your problem is to set a configuration file config.py in which you set all your imports. But importing config.py into other files is useless unless you succeed to make modules such as cv2 available elsewhere once for all. You can achieve that by dynamically importing them within config.py file:
cv2=__import__('cv2')
in your main program, fragment.py file or whatever module, you can make use of cv2 by simply running this:
import config
config.cv2.imread('pic.png')
import config ↔ you do not need anymore to run: import cv2. This is because this trick renders cv2 as a global variable available across multiple modules.
The same idea is valid for your other variables such as img that you need to declare in your config.py file too.
Given these facts, here is my solution for your problem. Note that I am not using classes and functions: I prefer to address your problem straightforwardly and keep things too simple and clear instead.
Organization of the code:
The config.py file corresponds to your wrapper.py:
solution/
├── application.py
├── cfg
│ ├── config.py
│ └── __init__.pyc
├── gallery
│ └── pic.png
└── genalgos
├── fragment.py
└── __init__.py
config.py:
# This will make cv2 global and thus you won't need to import it in ./genalgos/fragment.py
# You can use the same idea for all your other imports
cv2=__import__('cv2')
imgc=cv2.imread('./gallery/pic.png') # imgc is global
fragment.py:
# The only import you can not avoid is this one
import cfg.config
# imgs is global
# By importing cfg.config you do not need to import cv2 here
imgf=cfg.config.cv2.cvtColor(cfg.config.imgc,cfg.config.cv2.COLOR_BGR2GRAY)
application.py:
import cfg.config
import genalgos.fragment
if __name__=="__main__":
"""
Display the image 'imgc' as it is in 'cfg/config' file
"""
cfg.config.cv2.imshow('Pic in BGR',cfg.config.imgc)
cfg.config.cv2.waitKey(0)
cfg.config.cv2.destroyAllWindows()
"""
Display the grascaled image 'imgf' as it is in 'genalgos/fragment' file which
itself is obtained after transforming imgc of 'cfg/config' file.
"""
cfg.config.cv2.imshow('PIC Grayscaled',genalgos.fragment.imgf)
cfg.config.cv2.waitKey(0) # Press any key to exit
cfg.config.cv2.destroyAllWindows() # Unpaint windows and leave
| |
The objective of this project is to investigate and determine farming practices that allow Relay Cropping Wheat and Soybeans to have a consistent net economic return higher than either crop grown as a monocrop.
Methods
Three replicate field scale trials were conducted at 3 sites in 2018, and 3 sites in 2019. There were 3 main treatments in this trial which include:
- Wheat planted at 7.5” spacing
- Twin row wheat (two paired 7.5” wheat rows on 30” centres)
- Twin row wheat with relay soybeans (soybeans planted in the midrow gaps the following spring)
All wheat was planted using a 1560 John Deere no-till drill. To plant the twin row wheat several rows of the drill were blocked off as shown in table 1. Starter fertilizer was still applied to the rows with the blocked seed tubes.
Table 1: Twin Row Wheat Drill Setup
Soybeans were inter-seeded into the wheat using a 3pt hitch no-till drill. The soybeans were planted in 30” rows in between the twin rows of wheat (Image 1). The extra row units on the drill were removed so they wouldn’t damage the wheat. To harvest the twin row wheat with soybeans interseeded pieces of 6” drainage tile were cut to slide over the cutter bar on the header to prevent the soybeans from getting clipped (image 2).
All treatments received 120 lbs of nitrogen and 12 lbs of sulphur in the spring. All sites were sprayed with herbicide before the soybeans emerged. All sites received a T3 fungicide.
Figure 1: Soybeans Being Inter-seeded into Twin Row Wheat
Figure 2: Header with Tile Covers for Wheat Harvest
Results
The planting and harvest dates for the wheat and beans are shown Table 2. In 2018 the beans were all planted on May 11th, while in 2019 planting was delayed until June due to wet field conditions.
Yield results are shown in Table 3. Overall there was a 10% (8.6 bu/acre) reduction in wheat yield going from 7.5” inch rows to twin row wheat. An additional 8% (6.5 bu/acre) reduction in yield was experienced with the inter-seeded soybean treatment. A large portion of this additional yield loss likely occurred during the harvest process. Some wheat heads get pushed under the knife by the pieces of tile that are protecting the soybean plants (image 3). This could be corrected by use of a row crop header, but we have been unable to find one that would work on the research combine. It is also possible that some wheat yield loss is due to the soybean planting process, either due to tramping of wheat or the process of planting the soybeans cutting off wheat roots. Tramping of wheat was minimized and appeared insignificant, but root pruning of the wheat crop was inevitable in order to plant the soybean crop.
The summer of 2018 was very dry in many areas, which had a major impact on the relay soybeans. At the Lucan site in 2018 the soybeans suffered severely from a lack of moisture as the early planted, high yield wheat crop used most of the soil moisture. By wheat harvest nearly 50% of the soybean plants had died from lack of moisture. The remaining soybean plants aborted most of the flowers, and by the time moisture did come after wheat harvest they did not re-flower. Since most plants did not have any pods the soybeans were not harvested. The Woodstock location also had very few pods and what pods were there had only tiny seeds inside. The soybeans were not mechanically harvested at the Woodstock location but the yield was estimated to be 6 to 7 bu/acre by plant and pod counts.
The spring of 2019 was extremely wet which resulted in poor and variable wheat stands at both the Bornholm and Belmont locations. At the Bornholm location there were areas within the plots where the wheat was completely killed out. Overall soybean seed quality was good with moisture’s ranging from 15% to 17%. However, at the Bornholm site in 2019 the soybeans were still green, and at harvest had a bad odour. The soybeans planted at Bornholm were the same maturity as the field soybeans at that site, so the difference in seed quality was surprising and showed the impact of the wheat competition on the relay soybeans. At both Belmont and Lucan in 2019, longer season soybeans were planted to try to maintain flowering beyond the timing of wheat harvest. At both of these sites soybean seed quality at harvest was fine, although seed moisture was high, and soybean maturity was extremely late.
The total revenue from each treatment is shown in Table 4. At the time this report was written wheat was valued at $7.53/bushel while soybeans were at $11.17/bushel and these are the values used to calculate total revenue. This does not account for any additional cost or savings a treatment may provide. On average the relay bean treatment provided gross extra revenue of $28.50 an acre but this does not account for the cost of 2 additional passes over the field to plant and harvest the soybeans, or the cost of the soybean seed. Returns were larger at the sites with lower wheat yields (Bornholm 2018, 2019, and Belmont 2019) where returns ranged from $65/acre to $142/acre. These gross economic values also ignore the value of straw, which cannot be harvested from the relay soybean plots, or the value of cover crops and a place to spread manure, both of which are not possible with relay soybeans.
A treatment including an increased seeding rate was also included at several locations. The high seeding rate was typically about 50% above the normal seeding rate (drill set wide open). These results are shown in table 5. Increased seeding rate rarely had an impact on final wheat or soybean yield in any of the treatments.
It is important to note that the twin row wheat contained many more weeds then the 7.5” wheat (image 4). The value of crop canopy cannot be overlooked when evaluating cropping systems. As well, planting relay soybeans into the wheat crop drastically limits the herbicide options that can be used in the wheat crop. While all treatments had a herbicide applied prior to wheat planting, the herbicide options that are registered both in winter wheat and pre-plant in soybeans is extremely limited. In some cases there may be problem weeds that are not controlled by these herbicides. At the Belmont location in 2019, grass weeds were particularly problematic in the twin row and relay soybean treatments. In the solid stand of wheat, crop competition provided essentially 100% control of grass weeds. In this case, an additional herbicide cost would need to be included with the relay soybean economics.
Summary
Overall there was a 10% (8.6 bu/acre) reduction in wheat yield going from 7.5” inch rows to twin row wheat. An additional 8% (6.5 bu/acre) reduction in yield was found with the inter-seeded soybean treatment. A large portion of this additional yield loss likely occurred during the harvest process. The relay soybeans yields averaged 12.7 bu/acre with a range from 0 to 18 bu/acre. Economic analysis showed a net increase in gross return of $28.50/ac with the relay soybeans system before any additional costs of establishment were included, and without considering the value of the straw. Once these values are included, it will be difficult to find any site that had an economic benefit to the relay intercropping system versus growing wheat as a monoculture.
Next Steps
This trial was established at 2 more locations in the fall of 2019.
Acknowledgements
This project would not be possible without the leadership and funding of the Ontario Soil and Crop Improvement Association, as well as the Quinte Region SCIA and regional director Mark Burnham and the Middlesex SCIA. A big thank-you to co-operators Kevin Rivers, Christine and Andrew Brown, and the Perth SCIA (Bornholm location). Thanks to Marcel Meyer for harvesting the Belmont location soybeans, and to Horst Bohner for always looking after soybean harvest at Bornholm. Thanks to Pioneer and Maizex for supplying the soybean seed for these plots. And finally, thanks to Marian Desjardine from MSCIA for her administrative skill, and a HUGE thank-you to Shane McClure, who makes sure all this work actually happens. | https://www.osciaresearch.org/current-projects/relay-beans/ |
When citing a play with numbered lines, the MLA parenthetical citation need to incorporate the writer name and the act, scene and also line number(s). If the lines are not numbered, incorporate the web page number rather.
You are watching: How to cite medea in text mla
When quoting dialogue, incorporate the character names in all resources followed by a duration, and also pay attention to indentation.Quoting and citing a play
ROSS. I’ll view it done.DUNCAN. What he hath shed noble Macbeth hath won. (Shakespeare 1.2.94–95)
Citing plays in MLA
An MLA in-message citation consists of the author’s last name and also a page number:In-message citation for a play(Beckett 8)
If the message of the play has act, scene, and also line numbers, replace the web page number with the act, scene, and line numbers, separated by periods:In-text citation through act, scene, and line numbers(Shakespeare 1.3.188–90)
If the text provides lines only, clarify what the numbers mean by writing “line(s)” beforehand also in the initially citation of that play, separated from the author name or title with a comma. Subsequent citations of the same play can omit “line(s).”In-text citation with line numbers(Aeschylus, lines 15–26)(Aeschylus 35–40)
Citing multiple plays by one playwright
In documents focusing on multiple works by one playwappropriate (for example, the functions of Shakespeare), usage the italicized play name rather of the author name in each citation:In-text citation through play title(Macbeth 1.3.188–90)
To avoid repeating play names throughout your dissertation, the MLA style guide recommends composing the complete name in the initially citation, then using abbreviations for subsequent mentions.
If your research study is focused on Shakespeare, tright here are universally accepted play name abbreviations you can usage. Do not devise your very own, as your supervisor will be expecting these typical abbreviations:In-message citation via abbreviated play title(Mac. 2.1.25)
How to quote dialogue from a play
When quoting multiple lines of dialogue from a play or screenplay:Set the quote on a brand-new line, indented fifty percent an inch from the left margin.Start the dialogue with the character’s name in resources letters, followed by a period.If a character’s dialogue runs over one line, indent subsequent lines a better half inch.Add the citation at the end, after the punctuation mark.Quoting dialogue from a play
Throughout the play, memory is associated through both religion and fantasy:
VLADIMIR. Do you remember the gospels?ESTRAGON. I remember the maps of the Divine Land. Coloured they were. Very pretty. The Dead Sea was pale blue. The incredibly look of it made me thirsty. That’s where we’ll go, I used to say, that’s wright here we’ll go for our honeymoon. We’ll swim. We’ll be happy.VLADIMIR. You must have actually been a poet. (Beckett 5)
Plays in an MLA Works Cited list
The Works Cited section is wbelow you list the complete recommendations for sources cited in the message. The recommendation for a play looks different depending upon whether it was published overall book, accumulated in an anthology, or performed live.
Book
If the play is published as a stand-alone book, it looks the same as a conventional MLA book citation.
|Format||Author last name, First name. Play Title. Publisher, Year.|
|Works Cited entry||Friel, Brian. Translations. Faber and also Faber, 1981.|
|In-text citation||(Friel 57)|
Collection or anthology
If the play is publimelted in an anthology or arsenal, location a period after the play’s title, complied with by full details of the book in which it appears.
|Format||Author last name, First name. Play Title. Collection/Anthology Title, edited by Editor first name Last name, Publisher, Year, Page selection.|
|Works Cited entry||Shakespeare, William. The Tragedy of Macbeth. The Oxford Shakespeare: The Complete Works, edited by John Jowett et al., 2nd ed., Oxford UP, 1998, pp. 2501–2565.|
|In-message citation||(Shakespeare 1.2.20)|
If there is no called editor, simply omit this component and also continue straight from the anthology name to the publisher information.
Live performance
|Format||Author last name, First name. Play Title. Directed by Director initially name Last name, Publisher, Day Month Year, Theater Name, City. Performance.|
|Works Cited entry||Parker, Trey, et al. The Publication of Mormon.|
See more: Which Is True Of A Collective Biography : An Introduction, Collective Biography In The 1980S
Directed by Casey Nicholegislation and Trey Parker, 20 Feb. 2019, Prince of Wales Theatre, London. | https://muzic-ivan.info/how-to-cite-medea-in-text-mla/ |
The frilled shark is a fascinating creature that not many people know about. Swimming in some deep watery depths, these beings don’t see humans too often. Here are some facts about the animals that many people don’t know. Stay tuned for the second part of the article, which is coming soon!
Number Fifteen: It Was Named by a German Ichthyologist
Ludwig H.P. Döderlein was the person who described the frilled shark. In short, the animal has been described as “snaky”.
Number Fourteen: The Frilled Shark Has Amazing Teeth
The frilled shark is no joke. The mouth of this shark has backward-facing teeth that are 300 in total. Looking into the jaws of the shark. the teeth almost look like white planets, but deadly ones at that.
Number Thirteen: It’s Amazingly Buoyant
The frilled shark has a huge liver which is filled with hydrocarbons and oils, which allow it to be quite buoyant in the water. It gives it the impression that it is floating. This seems like it would be quite the eerie thing to witness.
Number Twelve: It Has a Strange Diet
The frilled shark certainly has a weird diet. For the most part, it eats cephalopods and teleost fishes primarily, but sometimes, it feels like it needs to be cannibalistic.
Number Eleven: It’s Everywhere, and Nowhere
These beasts can be found everywhere in the world, but because they sleep in deep waters, they usually won’t be seen. They swim as deep as 4200 feet, which is quite profound.
Number Ten: The Females Are Big
Like many other species on this planet, the frilled shark females are bigger than the males. They can even reach six and a half feet, which is, of course, longer than most people are tall!
Number Nine: It Takes a While for Them to Give Birth
While most creatures take a relatively short time to give birth or lay eggs, the frilled shark does not. In fact, it takes sometimes three and a half years for these creatures to have children. There’s lots more to learn about the frilled shark than meets the eye. Stay tuned for part two, coming soon! | http://ppcorn.com/us/frilled-shark-15-things-you-didnt-know-part-1/ |
MiniZoo exotic terrarium
The MiniZoo is located in the centre of Pärnu where visitors can see many interesting animals living in its exotic terrarium. The zoo’s collection consists of snakes from smaller species to large pythons and boa constrictors. The venomous snakes found in the zoo include the common adder, cobras, rattlesnakes, etc. Many of the snakes on display at MiniZoo are unique in European zoos. We also have some geckos, iguanas, monitor lizards, and crocodiles. Braver visitors can also pet and hold the animals! For groups, please register in advance.
Features and amenities
Amenities
Paid parking
Wheelchair accessible
Activities for children
Information boards
Suitable for children
Estonian
English
Russian
Finnish
Additional services
Souvenirs
Payment
Cash only
Getting there
The entrance is on Vana-Tallinna Street. | https://visitparnu.com/en/objekt/minizoo-exotic-terrarium/ |
Devta followers beg for money on Shimla’s Mall
Despite Shimla’s Mall street is out of bounds for religious processions.Deities are often circulated in palanquins by greedy followers to raise quick money.
In recent years the deity(devta) is carried during rush hour. Locals and curious tourists stop to donate money.
Often these deities are placed on the road on the mall and more people drop by to hand over money.
Police quietly pass by and never stop them. So that the devta caretakers merrily go home with loads of money.
Deities are mostly confined to a set of villages in the mid and higher hills of the state.They are rarely moved out of their territories.
But now a new practice of bringing devtas to the heart of Shimla and other towns has started.This needs to stop. Will the devta followers stop using the deities to beg for money! | https://sunpost.in/devta-followers-beg-for-money-on-shimlas-mall/ |
This Gummy Packer is used to fully automate the packaging process for all HaHa Gummies.
We begin by loading large bags of gummies into the machine, then loading empty, child proof mylar bags into the system. The machine begins to vibrate, forcing the gummies through. The gummy packer sorts the edible pieces into a single file line, which then go through a counting system. Every time a gummy passes through, the system registers and counts each unit. Once ten pieces are counted, the machine opens a mylar bag and releases the gummies into it. The system signals that ten pieces have been loaded, which allows the machine to move the bag to the next station, heat seal the unit, and discharge a finished ten pack of HaHa Gummies.
Planet 13 carries 9 flavors of HaHa Gummies ranging from Blue Raspberry to Sour Mystery Berry.
Have you tried them all? | https://www.planet13lasvegas.com/gummy-packager/ |
Are you weary of waiting for him and not knowing whether or not he will return? It’s time to use the Is He Coming Back Tarot Spread to find out.
In my Tarot practice, one of the questions I get asked most frequently is, “Is he coming back? The one that breaks my heart the most is also this one.
It breaks my heart to watch so many people waiting for an ex who doesn’t intend to come back. Or even worse, he can return with the wrong intentions and wind up shattering her heart once more.
I designed the Is He Coming Back Tarot Spread to help you find out immediately, which is why:
- if he is coming back or not
- what his real objectives are
- What can you do to improve the situation?
There will be no more hesitance, waiting, or disappointment. Try this Tarot spread right away if you’re prepared to learn the truth about whether or not he’s returning.
How do you recognize a reversed tarot card?
It’s important to pay attention to how you’re flipping a card over when you’re purposefully reading the significance of one, adds Walsh. “If you place a card face down on a table, Walsh explains, you can flip it up or sideways. The direction of the card will change if you flip it up. You might interpret a card that has been flipped up to be right side up because it was previously upside down.
What does a reversed tarot card represent?
Every card in a tarot deck has two meanings: the upright meaning and the inverted meaning. With the average deck having 78 cards, there are 156 meanings to become familiar with.
Concerned that you’ll spend all of your reading time engrossed in a book? Fear not, you will be able to set the book down and play with reversals on your own once you get familiar with the upright major and minor arcana meanings. Naturally, as you develop and broaden your tarot practice, you can go back to the written interpretations.
There are several ways to interpret reversals, many of which are dependent on intuition or the upright meaning of the card. That is to say, there is no one right way to read reversals, and mastering the skill doesn’t require you to commit a term to memory.
You come closer to mastery and figuring out your own unique style the more you read, practice, and try different ways.
Here are some ways to explore reversals as you get started:
Look at the position while keeping the significance of the card that you have learnt so far in mind. Consider the energy entering your life or the querent’s life if you are upright. Consider it to be reversing if its effect is waning.
Consider upright cards as characteristics, individuals, or elements that have a significant influence on the current situation. The card’s reversed meaning denotes anything with a weak influence.
Think of upright cards as fully realized facets of a circumstance, subject, or character. Cards that are reversed point to a personality trait or aspect of life that needs work.
Cards that are upright often refer to aspects of life that are going well and don’t require much effort. Reversals may be a sign of problematic or difficult characteristics.
The Point: Reversals frequently serve to make the current situation more clear. Consider viewing reversals as areas you need to improve upon or qualities you need to foster, for instance, if you are conducting a reading for self-improvement.
The Situation: Take a look at the nearby cards. What enhances and what diminishes the card in question?
The Position: Does the reversed card represent the past or the future? Or a trait to value? Make use of this knowledge to direct your interpretation.
Does the reading have a dominant suit? What could that possibly indicate about the reversed card(s)?
How is a Tarot deck updated?
While rearranging the cards in the tarot deck is a good approach to purify and clear their energy, there are some circumstances in which you might wish to perform a more specialized ritual. If you’re just getting started with tarot, cleaning your deck can be an excellent place to start.
You might want to clean your tarot deck for a variety of reasons, including:
- beginning with a fresh deck
- readings for other people
- You think you need to recharge.
- Your card readings seem a touch “odd” or “disconnected”
- Your deck hasn’t been used recently.
- Your deck has been handled by others
- You think you’ve been utilizing your deck a lot. A LOT, especially for books with strong emotional content
Why should you cleanse or clear your tarot deck?
Tarot deck cleansing helps keep the energy flowing between you and your deck. Consider it as a little spiritual hygiene to maintain a strong and clear connection. It’s not necessary, but if you have any of the aforementioned symptoms, try a few of the energetic cleansing techniques listed below and note which ones seem to work the best for you.
How often should you cleanse your tarot deck?
This is another way of stating USE YOUR INTUITION: there are no hard and fast laws. Don’t stress if you don’t believe it is necessary for your deck. Alternately, if you like to cleanse them once per week or once per month, that’s great. If it feels appropriate to you, you can even place your favorite crystal on the balcony each night.
If you frequently place crystals on your deck and store it on an altar while not in use, you might not feel the need to cleanse it frequently because this quick ritual will likely be sufficient to keep your deck feeling nice.
There are numerous ways to cleanse your cards, just as there are numerous reasons why you might desire to do so.
Different ways to cleanse your tarot deck
Use holy smoke. Light a dried rosemary, lavender, cedar, sage, or palo santo cleansing wand until it begins to smoke. Hold the smoke a safe distance below the deck while holding the burning herbs in one hand and the deck in the other so that the smoke drifts upward onto the cards. Turn the deck so that the smoke covers it from all angles. Next, safely put your deck to the ground and put out the fire.
On the deck, set a selenite stone (or a black tourmaline or a transparent quartz). It works well to leave it like way for an hour, but I prefer to leave it overnight.
Set them on display during a new moon. The New Moon is energy of a blank slate; you can purify the deck by setting it on a window sill on a new moon night. At this moment, you can also make a brand-new intention for your deck.
Place the cards in a salty dish. A strong and stabilizing cleaner is salt. My preferred choice for a thorough cleansing is this. Allow it to sit anywhere from one to eight hours in a dry area.
Unorderly shuffle. Spread the cards out on the ground, then shuffle them around like a child playing in dirt. This method’s freedom and randomization serve as an excellent reset.
the shuffle and sort. Set up the deck in rows of seven cards across, commencing with the Major Arcana numbers 0 to 22. (see photo above). Next, arrange the cards, Ace through King, one for each suit, as follows: Swords, Pentacles, Cups, and Wands. View the deck in this configuration, then mix everything up (like the chaotic!) and shuffle it thoroughly.
Does the tarot card order matter?
Because so many subsequent cards drew inspiration from its iconic artwork, the Rider-Waite Tarot Deck is an excellent starting deck. The deck is referenced in practical tarot publications like 78 Degrees of Wisdom as well. Start with the Rider-Waite deck and then add more decks that speak to you to your collection. Although it is available online, you are welcome to visit your neighborhood occult shop to see what appeals to you there.
Tarot cards fly out for what reason?
I adore proverbs with a witchy theme. They are a part of an oral tradition that most likely began when illiteracy rates among rural residents were high. Witches created rhymes and other catchy words to help people remember their rituals before they could record their spells in intricate grimoires.
I’ve never been able to determine where the adage first appeared “What hits the ground makes its way to the door, but I believe it’s a keeper. The statement is applied by tarot readers to cards that fly out of the deck during the shuffle, whether they “either touch the table or the floor. Jumping cards is most definitely a message to pay attention to if, like me, you see the tarot as an oracle and a doorway to a higher plane of awareness.
Why Do Tarot Cards Jump Out of the Deck?
Cards may jump as a result of luck, inexperienced handling, or subconscious energy transference from the reader.
When seasoned tarot readers manipulate their decks, they infuse the cards with energy and intention. Empaths are particularly adept at transferring energy, so if you belong to this mystical group, you should be aware of any strange occurrences when you shuffle the cards.
amateur tarot readers
Additionally, anxious clients who shuffle the deck before a reading are more likely to make poor shuffles that cause cards to fall to the table or floor. In spite of this, their jumpers shouldn’t be dismissed as “accidents.” Regardless of the shuffler’s skill, every card that leaves the deck needs to be recorded.
How Do Cards Jump?
A card can emerge from the deck in a number of ways. Jumper cards are ranked in the following order, from least to most significant:
- Several cards from the deck drop to the ground or the table. This mishap was probably just the result of a careless shuffle.
- Without any ceremony or drama, one card is dealt face-down to the tabletop.
- One card is dealt face-up and is placed on the table.
- From the deck, one card flips enthusiastically and lands face-up on the surface of the table. Please read this carefully, dear reader. Hey, says the greeting card. Observe me! I want to share something with you.
Methods to Deal with a Jumping Tarot Card
It takes a lot of honesty and trust to read the tarot, especially for someone else. Even if you’ve only recently met and even if you’re reading for yourself, take a moment to pause and focus into the vibes surrounding your relationship with the querent whenever a card jumps out of the deck during a shuffle.
From the most cautious to the most important, here are the six ways to deal with an escaped card:
- Reshuffle the deck after placing the card back in it as if nothing had happened.
- Make a mental note of the jumper, reshuffle it, and only pay attention to it if it reappears in the spread you laid.
- Lay your spread separately as usual, with the jumper face up on the table to the side. After that, assess whether the jumper has any bearing on the cards you laid. Only incorporate it into your reading if it “you and makes sense in the given situation.
- The jumper should serve as the signifier. Particularly in spreads that feature a card meant to represent the inquirer, such as Card 1 in the Celtic Cross spread, treat this card as the beginning point for the remainder of your reading by placing it in the first place.
- Think of the jumper as resetting the reading. The true question is frequently avoided by respondents out of fear. They are hesitant to discover their murkier, more hidden sides. Even though you are the one asking the question, there could be an opportunity to do so “Maybe the question you asked wasn’t quite the correct one. What exactly do you want to know?
- Give the jumper a reading of its own. Because they lack the context that comes from reading cards in connection to other cards, one-card readings are probably the most challenging. However, there are instances when the most challenging tasks are also the ones that are most important. Examine the sweater thoroughly and attentively. Really go to it! Take into account all the information you have available about this card, including conventional keywords, your own interpretation of the symbolism, color, and numerology. Ask yourself if the jumper card might be a communication from the afterlife if your belief system includes communicating with the spirit realm.
Tarot card reading is a practice rather than a craft that can be mastered. There are numerous factors that effect every reading, making them unique. Avoid putting too much restriction on your practice. To make every reading the most meaningful and pertinent experience possible, open your heart, intellect, and sixth sense. This includes paying attention to feisty cards that demand your attention.
Should your tarot deck be reset?
Every time you take a reading from it. That’s right, Magdaleno advises cleaning your deck before each reading to ensure the most accurate results. She advises starting with a cleansing to make the cards clear for your reading, whether it’s for yourself or someone else.
Can Tarot cards be washed?
Therefore, it would appear that mastering the art of cleansing tarot cards is something you might wish to perform before purchasing a new deck, if you already own one, or if you’re giving a new or used deck to someone else. Actually, if the cards are a gift for someone else, it’s a good idea to cleanse, bless, or activate them. Tarot cards are typically offered as gifts, according to Tarra. Before giving the cards to a new owner, perform a protection ritual on them and clean them to remove any strange energies they may have picked up along the route.
There is a legend.
Myth: Purchasing your own deck of tarot cards brings bad luck. And I suppose giving my mother a cursed one would be extremely impolite of me. So Tarra gave me a royal blue mesh bag of goodies after dinner and wisely explained how to clean them, and I promptly forgot all about it until we spoke again this morning.
Can you perform a tarot reading on yourself?
Is it feasible for a beginner to perform Tarot spreads on themselves? Yes! It most certainly is. Tarot is a technique that aids in deepening our understanding of the present moment, honoring our intuition, and predicting future possibilities.
Are tarot cards supposed to be shuffled?
Because the left side is connected to intuitive and receptive energy, traditional tarot readers used their left hand to shuffle the cards. For similar reasons, some people think you should deal and shuffle tarot cards with your nondominant hand. I’ve never done this, though, because I can’t shuffle with one hand!
Apparently, it takes seven precise shuffles to get a completely random tarot deck, although in all honesty, you don’t have to follow that rule. As they focus on their query, some people prefer to shuffle their tarot deck a few times, while others prefer to shuffle for at least a minute. Use a non-traditional shuffling technique if it works for you. The most crucial step is to simply shuffle the cards.
Questions you don’t really want answered
Even though it might seem apparent, it’s advisable to refrain from asking the tarot cards questions that you aren’t prepared to hear the answers to. That’s because answers to these questions can reveal information you’re just not quite ready to hear.
“Tarot can definitely come off as offensive if you’re not willing to hear the truth or consider an opposing opinion. Tarot reading Nicole Fortunaso
According to tarot reader and life coach Nicole Fortunaso, “tarot may truly come out as offensive if you are not willing to hear the truth of the problem or look at an alternate viewpoint.” She advises analyzing why you’re reacting the way you are in order to reflect on how to effectively address the underlying problem if you ask the question and aren’t satisfied with the response. | https://www.hiszodiac.com/tarot/when-is-he-coming-back-tarot/ |
Halloween fun!
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Halloween fun!
We used tools to explore the inside of a Pumpkin! We looked at, felt and smelt the seeds which we had scraped out of the middle.
We have been developing our gross and fine motor skills and hand/eye co-ordination by using the wooden tools to hammer the tees into the pumpkins
*** Attendance for w/c 6th February 2020... RW 94%, RF 90%,1W 96%, 1F 94%, 2W 98%, 2F 95%, 3W 98%, 3F 96%, 4WF 96%, 4W 97%, 4F 96%, 5W 97%, 5F 97%, 6W 97%, 6F 95% ** ** Congratulations to 2W and 3W for achieving the highest attendance. Well Done!**
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MEMBERS of the San community in Tsholotsho will receive 100 loaves of bread three times a week from Bakers Inn in Bulawayo as the corporate world begins to respond to pleas to help alleviate starvation in the area.
LINDA CHINOBVA
OWN CORRESPONDENT
Bakers Inn’s gesture was in response to a plea by Jabulani Nkomo, son of the late Vice-President John Nkomo who with his father have been supporting the marginised community with food.
The Nkomo family also built a school for the community to help intergrate them into mainstream society.
Bakers Inn general manager Felix Vhazhure said the donation was meant to honour the late VP, who had adopted seven San children and paid for their education.
“We have stepped in to assist them by donating 100 loaves of bread three times a week for as long as it is necessary,” he said.
Jabulani said he approached Bakers Inn after Tsholotsho authorties informed him the San community was seriously affected by hunger.
“The manager did not hesitate to assist them and that is why today he is presenting bread to them,” he added.
He said he -was looking for more donations like mealie meal to help the community.
A representative of the San, Watch Moyo, said the donation would go a long way in alleviating starvation.
“We really appreciate what you have done for us. It has really come as a surprise to us,” he said.
“The insufficient rainfall has affected crop production leaving us hunger-stricken.”
The San have struggled to transform from a life of hunter-gatherers and do little subsistence farming. | http://www.southerneye.co.zw/2013/09/20/bread-galore-san-community/ |
Police have released the sketch of a suspect sought in the murder of a 32-year-old transgender victim in Hollywood on Nov. 17, and have been canvassing the area near Lexington Avenue and Gower Street searching for information that may help them find the killer.
The same suspect is also wanted for an attempted murder and robbery in West Hollywood’s Plummer Park that occurred shortly after the murder in Hollywood. The suspect is considered armed and dangerous.
The deceased victim, who was identified by police as Nathan Henry Vickers, and who goes by the name Cassidy Vickers, was shot multiple times around 10 p.m. in the 6100 block of Lexington Avenue. Lt. Michael Oreb, with the Los Angeles Police Department’s Hollywood Division, said police received a 911 call about the shooting, and arriving officers found the victim lying unconscious in the street. Oreb said the paramedics pronounced Vickers dead at the scene. Witnesses reportedly told police that she had been standing next to a vehicle and talking to the driver when he fired several shots and drove away. Oreb said the street where Vickers was killed is known for prostitution.
“It’s hard to tell a motive. We don’t have a lot to go on, and we are still going around the neighborhood trying to find out exactly what happened,” Oreb added. “That street is known for transgender prostitution, so she might have been involved in prostitution, but we are still looking in to that.”
The incident in Plummer Park occurred around 10:35 p.m. Authorities from the West Hollywood Sheriff’s Station received a 911 call about a gunshot being fired in the park, and located the victim, a transgender woman, a short time later. The unidentified victim told police she had been walking in the park when an African American man approached on foot and demanded her purse. The victim refused to give the suspect her purse, and he pulled out a handgun and fired. The shot missed the victim and she ran away. The suspect also fled, and no property was taken.
Sgt. Michael Caprioli, of the West Hollywood Sheriff’s Station, said he could not release any further details because of the ongoing investigation, but added that investigators are searching for witnesses. Sheriff’s department detectives are working with investigators from Hollywood, and patrol deputies have been advised to be on the lookout for the suspect.
Oreb said officers from the Hollywood Division’s vice and gang units have been assigned to the case, and patrols have been increased in the neighborhoods around the murder scene. The suspect is described as African American, in his mid 20s to 30s, five-feet-nine-inches tall and 150 pounds, with a medium complexion. He had light facial hair, and was wearing black clothing and a black hat.
In response to the murder in Hollywood and the attempted murder and robbery in Plummer Park, West Hollywood Mayor John Duran warned people to be cautious when out at night.
“West Hollywood is home to a very large transgender community, much more so than the surrounding cities, and any time there is an attack on one of our people, we are concerned about it,” Duran added. “We have to remind people to be wary and watch out for one another. Walking in Plummer Park alone in the evening can be risky, so people are encouraged to walk in lighted areas.”
Anyone with information about the incidents is asked to call homicide detectives with the Hollywood Division at (213)972-2910, or the West Hollywood Sheriff’s Station at (310)855-8850.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | https://beverlypress.com/2011/11/suspect-sought-in-murder-of-transgender-victim/ |
Q:
Differences of $V_1 \cup V_2$ and $V_1 +V_2$?
Let $V_1,V_2$ are subspaces of vector space
$V$ . Differences of $V_1 \cup V_2$ and $V_1 +V_2$ ?
A:
Let $V = \Bbb R^2,V_1 = \{(x,0)\mid x \in \Bbb R\}$ and $V_2 = \{(0,y)\mid y \in \Bbb R\}$. Then $$V_1 + V_2 = \big\{(x,y)=(x,0)+(0,y)\in \Bbb R^2 \mid (x,0) \in V_1 \text{ and }(0,y)\in V_2\big\}=\Bbb R^2$$ but $$V_1 \cup V_2 =\{(a,b)\in \Bbb R^2\mid a=0 \text{ or } b = 0\} \neq \Bbb R^2$$
A:
The first is in general not a subspace and is the set of vectors that are in $V_1$ or in $V_2$. The second is the set of sums $v_1 + v_2$ where $v_i \ in V_i$ and is a subspace. It is by the way generated as vector space by the first one.
| |
Heyshott Down 233m (764ft)
Classification: Hill Trigpoint, Notable.
OS Grid Ref: SU899166 (10 figure Grid Ref: SU8999316628).
The original GR data for this hill was only accurate to a 6 figure grid reference (c.a 100m).
We have used additional user / trigpoint data to bring it up to 10 figures.
UK Area: 42 - South-East England & the Isle of Wight (OS Maps 1:50k 197. 1:25k 120.)
Latitude: 50.94208, Longitude: -0.72046, Height: 233 metres (764 feet)
Unique Waypoint POI Name: 0233.- (Hill ID: 301236)
Associated Trigpoint:
TP3080 'Heyshott Down' Grid Ref: SU8999316628 Separation: 29m from Hill grid ref.
External Links for Heyshott Down:
-GeoHack link... -OS Map Bing link... -Google Map link... -Geograph link... -Heyshott Down on trigpointinguk.com...
Dewey Notables data courtesy of Nick Wakelam
Selected mapping: Google or Bing / Ordnance Survey:All | :Nearby Hill | :Nearby Trigpoint | : mouseover
Heyshott Down List & GPS Waypoints:
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- '0233.-' SU899166 Heyshott Down (233m)
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Heyshott Down's Ten Nearest Neighbouring Hills...
Heyshott Down East Top (1.0km)Upper Polecats Copse (2.4km)Pendean (3.6km)Hat Hill (4.4km)Selhurstpark Hill (5.3km)Heathbarn Down (6.5km)Treyford Hill (7.3km)Goldrings Warren (7.9km)Chilgrove Hill (8.3km)Welchs Common (8.4km)
Heyshott Down has been climbed by...
adoling Adrian arrans BadNewsWalking captainslow Carole carolml Chris Pearson daveyf2001 Griefmiester Griggsy Ian Baines jonglew Kevgbl Nick Wakelam rhw Rhys Thomas slateloose TimJB1 wrose
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* The waypoint data in italics is only accurate to a 6 fig grid reference. Click here to learn how to help improve it. | https://www.haroldstreet.org.uk/waypoints/download/?hillnumber=301236&/Heyshott+Down |
People often like to quote from the stories of Sherlock Holmes and it was possibly the most famous line of all, “…when you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth”, that made me feel that the attitude displayed by Conan Doyle’s detective was that required by a good Software Maintenance Engineer.
From my experience one of the hardest and most undervalued skills to really master is that of analysing and resolving customer issues. Even the cleverest, most practised team of developers have issues, though the tolerance level of the domain may affect at what stage of the development lifecycle these become apparent. It is fairly clear that the systems on an aeroplane need to be somewhat more robust say than your average computer game. However the principles of resolving an issue apply at whatever stage the problem is identified. In fact, if you are adept at addressing these problems with the more limited resources concomitant with customer issues, failures identified during the development phases should be a much easier prospect; so it is never a redundant skill.
Being a skilled maintenance engineer starts well before the issue arrives at your desk. Initial development planning must consider how you will maintain your code. The chances are some other poor developer is going to have to fix it! Plan to provide as much evidence as possible to deduce the cause of a problem.
The quality of your logging is important, whether it be product specific log files, Windows Event Log, syslog, an audit table in your database, or some equivalent. You cannot assume you will be able to reproduce an issue, so you need to be able to plan to provide rely on historical evidence.
Make it simple, make it succinct and make it clear. I have seen code with extremely complex logging mechanisms allowing logging levels to be controlled at an almost infinitesimally precise level. From my experience, if you think carefully about what you are logging the generally common model of four levels; information, warning, error and debug, will suffice in most circumstances.
Look at the log files whilst you are developing your code.
Can it be more succinct without losing clarity?
Can you tell the user how to resolve the issue, and if so do you?
If you have multiple data sources are they structured consistently?
These may seem petty considerations, but when you are trawling through thousands of lines in a log file they become highly relevant. If you need to cross-reference data from two files a consistent structure simplifies this. If you can avoid customers having to raise an issue in the first place, why not?
You need the evidence to identify suspects. Your Support Team should have a standard set of items they automatically collect when a customer raises an issue. By the time you get your hands on the problem you should have details of the customer environment, including any bespoke tweaks that they have. You should have log files and dump files if appropriate, clear steps to reproduce, actual outcome and expected outcomes; you may even want database backups. You can never have too much information, because generally you find that fairly quickly you can filter out the irrelevant and focus on the valuable.
You cannot come to an issue with a conclusion already made. You must analyse all the data; which means you need to have an excellent Support Team that will get you this. Take time to look at the errors, the logs, the databases, the environment, that the software is running in.
Reproduce the error. The Support Team need to have the skills and information to be able to at least attempt to reproduce the issue in a simplified environment. The means the Development Team must have taken time to give them training on the products. Clearly they must be able to install, configure and run your products, but are there other technologies they need to be familiar with; SQL, AD, IIS? They need at least a basic understanding of the architecture and what each component is responsible for. If you have a third line support team that can do some basic debugging and database analysis all the better, but you must give them the skills. Time taken to give this additional training will pay dividends by freeing up the development team; reducing the impact caused by context switching, changing development environments and diverting development away from newer projects.
You have to be willing and able to dive wholesale into a new domain, unknown code and study. I have always enjoyed the challenge of this part of the job. Whether it be working on a wholly new product or in the realm of maintenance, this is a job where you can never rest on your laurels.
One of the first issues I addressed when I joined 1E over five years ago was a customer issue with IIS. A technology that as a developer I had not had to touch in my previous roles. Within two weeks I had gone from complete novice to resolving an issue with configuration inheritance causing application pools to be recycled when they shouldn’t have and hence causing the user session state to be lost. Probably not the most complex of IIS issues for an expert, but I wasn’t.
The challenge needs to push you on, direct your study and inform your investigation.
Very few of us are geniuses, but as a developer trying to resolve a customer issue you cannot take short-cuts. As with Holmes’s magnifying glass you need to be able to get into the minutiae of the crime scene. This may be the analysis of hundreds of lines of code or investigating the contents of a database table with a million rows of data; you just have to do it. What you do need to do is make sure you know the basic techniques that can reduce this pain.
Basic tools and techniques, like setting breakpoints, viewing the content of the registers or other such things; are covered in a million articles and books. Any experienced developer should already have a toolbox full of these techniques, but what about the junior members of your team? Make sure your senior engineers work with them on real issues. Just knowing the different kinds of breakpoints and how to use them may not have been something they have been exposed to. Many products contain lines of code might be iterated over thousands of times within seconds, try using a standard positional breakpoint to step through that and you will never find anything. Learn how to debug multithreaded applications, know how to look at the call stack or check the contents of the registers.
Sometimes you may have work with optimised code, so you make sure you understand how to do this. The disassembly window shows what is really being executed, the code may not be an accurate description of the actual behaviour. Play with the various tools around that can help you, above and beyond the standard debuggers for your environment, things like Fiddler for examining HTTP traffic, or HexEdit for examining binary files. Learn some simple techniques for reducing the amount of data you have to deal with, like binary splitting (bisection algorithm).
Make sure you are teaching the novices in your team. Like any knowledge silo, having a limited set of developers with these skills is only going to cause you problems down the line.
I myself have spent many days investigating an issue which finally was identified as being caused by a particular string value of one column in a single row of data being imported with thousands of other rows.
Other examples of unexpected causes of issues include a bug in a third party library, where you had to call one particular method twice, with no other call in between with the same parameters for it to commit the change you requested.
Let the information lead you to the answer, however obscure. | https://www.1e.com/news-insights/blogs/sherlock-holmes-was-a-software-maintenance-engineer/ |
The available transitions are Fade and Cut, by screeplay default (the Cut setting effectively deactivates transition effects by just making a quick cut screeplay transitions between scenes). 1: Use scene transitions to shift between time periods. Location Headings — Should Only Have Three Pieces of Information. If you ever read an old screenplay, you’ll notice how densely written they are. and FADE TO BLACK.
Well, they’re somewhat self-explanatory. Jump Cut: These transitions are interesting because they fracture time in a very noticeable way. The transition property is a shorthand property for the four transition properties: transitionProperty, transitionDuration, transitionTimingFunction, and transitionDelay. There are many ways to use scene transitions and breaks in your book. MATCH CUT: PULL BACK TO REVEAL: WIPE TO: DISSOLVE TO: There are dozens more.
Whenever there is a jump in time and/or space, we may bridge the gap by means of a transitional instruction. . INT/EXT LOCATION – DAY/NIGHT. Transitions you may be familiar with are: CUT TO: DISSOLVE TO: SMASH CUT: QUICK CUT: FADE TO: FADE OUT (never at the end screeplay transitions screeplay transitions of the script) Writing Tip: The only time to use a Transition in a spec script is if it&39;s integral to telling the story.
Simultaneous entering and leaving transitions aren’t always desirable screeplay though, so Vue offers some alternative transition modes: in-out: screeplay transitions New element screeplay transitions transitions in first, then when complete, the current element transitions out. To pull it off, just take one of your longer clips, cut bits and pieces out of it, and put the fragments you want to include in your timeline back together. I never know if I should actually use one, or just move to the next scene.
Transitions are an element which are added to Shooting Scripts with a few notable exceptions. Subheaders screeplay transitions are used to time jump or move in time within the same location. But in this day and age, the transition is implied by a change of scene. Transitions within paragraphs: As with transitions between sections and paragraphs, transitions within paragraphs act as cues by helping readers to screeplay transitions screeplay anticipate screeplay transitions what is coming before they read it. Search only for screeplay transitions. Transitional instructions should always be used to indicate these changes. The wife ends the row by slamming the bathroom door shut.
0" Right: 0. 0" Width: 2. Start early and make transitions part of the scene.
Here are five examples of scene transitions to help your story flow through your screenplay. DISSOLVE TO is the proper transition to use within the script, if needed. By Elaine Radford Scene transitions in a screenplay indicate changes from one setting to a new setting, or from one time screeplay transitions frame to a different time frame. It&39;ll look like the clip is "jumping" around through time. WAREHOUSE DISTRICT - DAY “EXT” stands for “Exterior.
More about that soon. The transition-timing-function property can have the following values: screeplay transitions ease - specifies a transition effect with a slow start, then fast, then end slowly (this is default) linear - specifies a transition effect with the same speed from start to end. The script won’t work right now because SceneTransitioner doesn’t exist yet. A very typical transition is “DISSOLVE TO:” DISSOLVES evoke a passage of time, or a “slowing down” of tempo as two images overlap. Transitions should be at a tab stop 6-1/2 inches from the left side (the right margin should be set at 1 inch. Most of the time you don&39;t need Transitions.
Transition Definition. The combination of CSS3, along with the right JS library can produce compelling visuals. Transitions are appearing less and less in modern screenplays.
Use one to indicate when you’re initiating a dream sequence, triggering a flashback sequence, or going inside of a character’s head. A properly formatted screenplay serves two purposes. However, if you click the plus (+) button under the transition select dropdown it will give you the additional options: Swipe, Slide, Fade to Color and Luma Wipe. Together, Mazin and August outline five different types of scene transitions that work well in screenplays, including: Size: screeplay going from a very tight shot to a super wide shot or vice versa (this transition helps readers see the movie in their heads) Music or sound: audio, usually in a prelap (meaning introduced over the visuals at the end of one scene and continuing over the cut) gives the reader and.
A written work by screenwriters for a film, video game, or television program. Note: Always specify the transitionDuration property, otherwise the duration is 0, and the transition will have no effect. This training video screeplay transitions will show you how to create stinger transitions in vMix! It’s a common mistake, but a very annoying one. Transitions are formatted in all caps and almost always follow an Action and precede Scene Headings. They can sometimes be used, but you should do so sparingly. Transitions can be at the end of the first paragraph, at the beginning of the second paragraph, or in both places.
An example would be these screeplay transitions two scenarios: a) A husband and wife argue at home late at night. screeplay transitions Screenplay transitions were cues to the editing team that communicated how transitions between shots were to be handled. A transition tells the editing crew how quickly screeplay transitions they should move to the next scene. ” and “DISSOLVE TO:” A common way to lead into a dream sequence, for example, is with the transitional instruction “RIPPLE DISSOLVE TO:” Some other transitions include “WIPE TO:,” “IRIS IN:,” and “IRIS OUT. Transition verbiage includes: CUT TO: DISSOLVE TO: SMASH CUT: QUICK CUT: FADE TO: As a spec script writer, you should avoid using a transition unless there is no other way to indicate a story element. FADE TO BLACK screeplay transitions is just what it sounds like: the image fades to total black and the screen goes dark. The transition-timing-function property specifies the speed curve of the transition effect.
Can you specify how to use transitions while writing a screenplay? One way to use scene screeplay transitions transitions is to switch between present experiences and backstory. When delivering presentations it&39;s important for screeplay transitions your words and ideas to flow so your audience can screeplay transitions understand how everything links together and why it&39;s all relevant. See more videos for Screenplay Transitions.
Not only is the description more detailed (and, at times, borders on prose), you’ll find camera directions and all sorts of transitions. This can be done using speech transitions because these act as signposts to the audience - signalling screeplay transitions the relationship between points and ideas. Examples are CUT TO, SMASH TO, DISSOLVE TO, etc. The scene headings are written in all caps, as screeplay transitions well as INT or EXT for Interior or exterior. "CUT TO:" is the simplest screeplay transitions form of transition.
Specific examples include: INT. 0" screeplay Transitions are film editing instructions, and generally only appear in a shooting script. This is the truly final draft used on set by the production people, actors, and director to make the movie from the screenplay. Transitions are elements that can help a Writer move from one scene to the next. A subheader is usually after an action line and is capitalized. screeplay I’m referring to all those "cut to"-s and other transitions.
Slug Lines — a. Spice up your video productions with animated screeplay transitions transitions. It was very prevalent in the screenplays of early cinema because, at that time, screenplays were more of a technical document. vMix Video Tools is i. Your Scene 1 should look something like this. A unique sound is located in each transition-layer What resolution projects are supported.
Get up to 91% screeplay transitions OFF yearly plans using the code "BRACKEYS": Transition Indent: Left: 4. The first screenplay Element screeplay transitions type is screeplay transitions a Scene Heading, also known as a Slugline. By contrast, action sequences, arguably the fastest-paced sequences written for the screen, can appear in a script like dull blocks of words crowding the page. A fplay written for the screen.
out-in: Current element transitions out first, then when complete, the new element transitions in. The transitions used most often are “FADE IN:,” “ FADE OUT. While most of a script is "left-justified" screeplay transitions (all text lined up against the left margin,) there are "tab stops" for dialog, parentheticals, character names and transitions. The exceptions are, FADE IN: FADE OUT. Writing screeplay transitions an effective scene to scene transition can help crank up the screeplay transitions pace or even provide a subtle subtext through screeplay transitions sound and visuals that enhance a piece of dramatic script writing. If there&39;s a cut, you&39;ve changed shots.
. 861 Best Transitions Free Video Clip Downloads from the Videezy community. ,” but these are considered screeplay transitions old-fashioned and rarely used.
Those are the general screeplay choices. Example of introducing backstory in a transition. Screenplays written in the master scene format are broken into scenes not cuts. Here are five tips screeplay transitions for transitions – how to add them into your show and what things you need to consider when working on screeplay them. Script for quickly change the resolution of the transitions for your project (required screeplay only for manual method) Sound effects pack. Free Transitions Stock Video Footage licensed under creative commons, open source, and more! The following is a guest post by Zach Saucier. If you write your screenplay well, your description of a great battle will explode in the reader’s ears, your dialogue between two lovers will cause the reader’s eyes to tear up, and that emotional speech you write from a great leader will leave a lump in your reader’s throat.
But first, what are they? Transitions in a Properly Formatted Screenplay Transitions are always aligned to the right side of screeplay transitions the page and are used to move from screeplay transitions one scene to the next. Zadie Smith uses this type of screeplay transitions scene transition effectively in her novel White. Transition to a surreal scene. When used in a responsible manner, transitions can a fun way to enhance UX. Transitions are right justified on a script. This way, transitions are taken care screeplay transitions of right screeplay transitions away and can be practiced frequently.
A Transition indicates that we are, in some way, moving to a different scene or shot. Transitions The first formatting element is the scene heading – also dubbed the slug line. Block and/or choreograph transitions as soon as you block a scene. The bonus Split package, which includes some transitions from the package Split Handy Transitions. Note on the image that we’ve connected the button to the script using the SerializeField attribute. Every scene begins with one. | http://bflr.specmontag37.ru/98210 |
Home/Workpackages/ Analytical methods for performing hydrogen purity testing to enable the full implementation of the revised ISO 14687‐2 standard (WP2)
Analytical methods for performing hydrogen purity testing to enable the full implementation of the revised ISO 14687‐2 standard (WP2)
The aim of this work package is to propose optimised analytical protocols (including fit‐for‐purpose analytical methods) that enable the implementation of ISO 14687‐2.
This will be done by evaluating the performance of analytical methods, existing and under development, for performing hydrogen purity testing (number of parameters covered, uncertainties, risk for interferences, robustness…) and by developing new methods when needed. The requirements specified by the ISO 14687‐2 standard present a series of analytical challenges, mostly due to the very low detection limit (e.g. carbon monoxide, total sulphur, formaldehyde) or total quantification of one element (e.g. sulphur) or family of compounds (e.g. total halogenated, total hydrocarbons). As a large number of components need to be quantified, it is not possible to analyse all the components using a single method; on the contrary, many methods are required. Moreover, a number of these components are unstable and/or reactive which implies that sampling vessels need to be passivated and extreme care has to be taken during analysis to ensure that these components do not adsorb to sample lines and other analytical equipment.
By 2016, hydrogen purity should comply with the tolerance limits set in ISO 14687‐2. For routine laboratory/analysis, performing the whole set of analysis is currently extremely challenging. Traceability of measurements, trueness and detection limits are some of the critical points. By defining relevant performance characteristics for methods (selectivity, measurement uncertainties, detection limits, working range, robustness, trueness and precision), laboratories would be able to choose adequate analytical methods for each parameter.
This work package also aims to discuss the presence of other potentially harmful impurities not yet specified in ISO 14687‐2 by using the analyses performed in WP1.
The outcomes of WP2 will be directly used to identify the challenge in implementing ISO 14687‐2 in routine laboratory/analysis and for a revision of the standard.
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Happy Sunday!
Figures that vacation week (March Break) goes by super quickly. We did not do too much this week. We did the exciting weeks in life: clean up the cluttered basement, clean the floors, go to the accountants office. Exciting stuff! We are going to the hockey game on Sunday. I’m not really a Senators Fan (don’t judge) but hockey games are fun! I really don’t want to go back to work on Monday and I’m totally jealous of anyone who went on a sunny vacation!
One fun thing that we all did together was take Max to see his very first movie. We took him to Frozen at the South Key’s cinemas. We were a bit worried that his attention span would be short, but he did a pretty good job. He sat there for the first few minutes clutching his nalgene and chocolate bar (don’t judge) with this “what is going on?” face:
Max liked the movie. He liked Sven and Olaf the best – but had a big smile whenever Elsa came on (because she looks like a Barbie, I think). He danced and sang – and laughed quite a bit. At the very end of the movie (which was long for kids), he ended up getting up and walking on the stairs, all while trying to eat popcorn off the ground. He had never had popcorn before and he loved that too – kept grabbing for the bag!
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This is the first week that I’ve really started to get into half marathon training. Since I’ve been running at least once a week (mostly) all winter, I already have a bit of a base – so I’m not too concerned about being 11 weeks til the half and just really getting started. I am following a modified version of a few different training plans, including one from our ORW running coach, the Running Room 1:55 Half Marathon plan and the philosophy of the Run Less, Run Faster training plan with three solid runs per week: long run, tempo and a speed workout. I love my speed workouts and since they are the one thing that I am almost always able to do – they are in my weekly plan (except this week because we are on break).
This week’s runs:
Sunday: 4KM Tempo
This was the run I did when I fell. My finger is not black and blue anymore, but is still a bit swollen – but just a bit. This run was fairly steady, but anytime I had open, clear sidewalk I went faster.
Tuesday: 8KM slow, long run
Done in 53:00 minutes – this 8 KM run was fairly steady, holding at a 6:40 pace which is bang on for my long run pace. The different I am feeling between this year’s training and the last half marathon I ran is that I felt very comfortable the whole time and did not need to talk a walking break (I did stop for traffic lights and slowed right down during the ice bits). Before I was always staring at my watching thinking “how much longer til 10 minutes” with my long run pace more like 7:00. Feeling much better this year. Considering this was my longest run in a very long time – this was a good one! If only the route was better – I was stuck doing loops around my neighbourhood – BORING.
Saturday: 3KM steady-fast run (subbing for speed)
I had a longer run scheduled, but because of our movie adventure, I just went out quickly at 6pm for a quick 3K run. It was getting rather chilly out, and I could see the water freezing in front of me – so 3K was all it was. Just a quick run out the back of the condo complex to the bus station and back. I was running faster than two people on a bike at one point. I completed this run in 16:40 at an average pace of 5:34. I could have gone faster if I wasn’t dodging ice puddles.
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This Week’s plan is to run a 10K before track practice on Friday, get a 5K in on Thursday and do another run at some point. P is working all next weekend, and the next, and the next (and away for a whole pile of days)- so I really have to squeeze my runs in when I get home early from work.
Alright – time to get ready to go to the hockey game! | http://runningfoodbaby.com/2014/03/first-movie-half-marathon-training/ |
Every year I try and take a picture of each child on the first day as it’s amazing to look at at the end of the year and see the change. I’ve seen a lot of first day frames on Pinterest and decided this may be the way to go this year. So that became today’s little project.
I drove to the Reject Shop (discount store) and picked up a few supplies. The wooden frame cost $8, the alphabet stickers were $2 and the jungle stickers were $2. I took the glass and board out of the frame and will find another use for them. I pushed the brackets back on the back so they wouldn’t poke into anyone and then stuck the stickers around the outside.
I’ll get the children to hold the frame for me and take their picture on the first day of school. Sometimes I get a child who is too timid to have a photo taken but that’s okay. We leave it for another day if we have to but generally after they’ve seen all their classmates have a picture most are happy to have oblige. | http://lessonsfromateacher.com/2013/01/18/first-day-frame/ |
You may have noticed that your vehicle has a strange engine problem that is causing it misfire. You might think this is a minor problem and that you can take it to the mechanic for them to look at. You might discover that your vehicle’s manual, or manufacturer, did not mention the problem. Here are some of the most common engine misfire causes that you should be aware of.
Common Engine Misfires
If the exhaust pipe is damaged, it can cause misfires that sound like the engine is having problems. The pipes are responsible for carrying toxic gasses away from the vehicle in the form of exhaust. They can become blocked and stop the vehicle from releasing toxic gases into the atmosphere. The muffler on the vehicle must also be clear and free of obstructions so that the exhaust gases can pass through smoothly.
Another reason for engine misfires is if the ignition system’s electrical components stop working. These include the ignition coils and the injectors. If these fail to work, the injectors will not work properly and will cause the fuel to sputter out as it is being burned. This will result in the emission of noxious fumes into the environment.
When it comes to determining the cause of misfires, you must remember that spark plugs are essential to the engine’s operation. Without the proper plugs, the engine can not function. Spark plugs are made out of metal alloys and are very precious to mechanics. They must be connected properly in order to function properly. If you notice any problems with your spark plugs, you should immediately inspect them and replace them if necessary.
The opening and closing the exhaust ports can also cause engine misfires. If you find that the spark plug wires are losing their fire or they are not closing properly, it could mean that either the exhaust port clamps have become loose, the intake port has become clogged with grease or debris, or that both the exhaust port and the intake port have seized together. This can have a negative impact on the vehicle’s performance. You should diagnose the problem and make sure that the port is open and clear before you do anything else. This will prevent the misfires from happening again. Also, ensure that the idle speed is set correctly and that there is no maintenance required for the engine.
A common mistake that many people make is checking the fuel pump by looking at the check engine light. The fuel pump is not the problem with the misfire of the vehicle. When the check engine light comes on, you should check the battery for leaks. If the battery isn’t leaking, it is most likely that there is a blown gasket or bad fuel line.
There are many possible causes for a misfire, but these are the most common. A compression problem could mean that the spark plug, or the entire fuel delivery system, is not working properly. The vehicle may not be able to work, but the owner can save it by simply taking it to a mechanic and having the fuel delivery system repaired. The fuel delivery switch is another possible reason why the engine may not start. If you find that the vehicle produces very little power and even loses power rapidly, then you may want to have the fuel delivery switch checked out.
A bad control module or a bad switch in the ignition system could also be a cause of misfires. The switch is usually located next to the ignition coils on the left side of the vehicle. It is possible to check this part by removing the spark plugs, finding the control module and then checking its springiness. You should replace the switch if it is extremely stiff or poor in performance.
Common Causes Of A Vehicle’s Engine Mismatch
When the check engine light shines, your vehicle’s primary computer, which is called the powertrain control Module (PCM) will first store a single diagnostic trouble code (Dtc) in its data memory. The code will then be determined by all other components of the vehicle’s computer and, if applicable, the appropriate action to take. Dtc codes are basically strings of letters or numbers, representing a problem. A letter or number combination is usually interpreted by the computer hardware device and interpreted as a possible problem code. When multiple DTC codes are in a row, that means there is a possible problem with one or more of your vehicle’s engine components. Your car’s PCM then performs some diagnostics on that problem code to try to find out what the problem is.
One of the possible reasons why your engine misfire occurs is because of faulty exhaust system. It could be caused by clogging up of the exhaust system, causing too much pressure and heat to build up in the engine compartment. Overheating the engine parts can lead to engine failure. To avoid this kind of malfunctions, you should always keep your exhaust system clean. Clean it thoroughly with a high-pressure air cleaner, before and after every use of your vehicle.
Another possible reason why your engine misfire occurs is because of wrong or failing exhaust pipe. The hot combustion end of the engine’s exhaust will cause too much heat to build up in the piston area. If the pipe is blocked, or covered, the heat generated by the hot exhaust will escape to the exterior of the cylinder. The cylinder will eventually burn itself out because it can no longer supply enough fuel to the engine. To fix this problem, you need to open and clean the exhaust pipe to remove any excess metal build-up or dust.
Inadequate oxygen supply can also cause engine misfires. An engine’s proper functioning depends on oxygen. The oxygen is needed by the different engine parts to get fuel or energy to function. The engine cannot function at its full potential if it doesn’t have enough oxygen. For instance, when you put fuel in the engine, the oxygen is first passed through the combustion chamber.
Then, the oxygen is pushed to the cylinders where it is mixed with water to make the mixture of gas. The mixture is then ignited by the electric spark and burned to make the desired power. A typical gas engine has an enclosed cylinder. This cylinder is filled with compressed oxygen to prevent the mixture spilling over into other parts such as the crankcase. An OBD test can quickly diagnose any problems with your vehicle’s idle mix. But if you notice that the engine misfire continues even after the complete diagnosis, it is best to have it checked by a mechanic.
One cause of a vehicle’s misfiring could be faulty exhaust system. The defective exhaust pipes can cause the engine misfires. The pipes of the exhaust system often accumulate dust and debris, making it difficult for the fuel to flow. The mechanic should clean and replace the exhaust pipes to fix this problem.
Another common causes of a misfire includes the monitor itself. If the monitor is not correctly installed, the electricity-to-gas ratio is improperly set, thereby causing the misfire. Dry-fire is the most common problem with ignition. This occurs when the metal prongs of your monitor don’t fit properly and overlap. This is when the mechanic will need to properly install the prongs to ensure proper operation of the ignitor.
There are many possible causes for a vehicle to stop working when it comes to compression testing. The first is when the compression test is not done correctly or is too low. The second is when the compression gauge has not been calibrated correctly. Also, excessive compression can be caused by the presence of a coolant leak or when the fuel injector is clogged. Last but not the least; poor air pressure can also be one of the common causes of engine misfiring. | https://fleetserviceshocrv.com/common-engine-misfires-causes-of-a-vehicles-engine-mismatch/ |
PROBLEM TO BE SOLVED: To expand an adjusting range of a natural period of a mass damper, and simplify a structure by fixing a pendulum to a low speed side rotary shaft of a speed reducer, fixing a pinion to a high speed side rotary shaft, and meshing the pinion with a rack installed on a horizontally moving mass arranged on a structure.
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SOLUTION: A pendulum 5 is fixed to a low speed side rotary shaft of a speed reducer 3, and a pinion is fixed to a high speed side rotary shaft. A horizontally moving mass 8 is arranged on a structure 50 so as to reciprocate in the same direction as the pendulum 5, and the pinion is meshed with a rack installed on the horizontally moving mass 8. When swinging in the arrow X direction is caused in the structure 50, a reciprocally rotating motor rotates in the clockwise direction, and the pendulum 5 swings in the arrow (a) direction, and the horizontally moving mass 8 moves in the same direction. When swinging in the arrow X direction is caused in the structure 50, the reciprocally rotating motor rotates in the anticlockwise direction, and the pendulum 5 swings in the arrow (b) direction, and the horizontally moving mass 8 moves in the same direction, and damps vibration of the structure 50.
COPYRIGHT: (C)1998,JPO | |
LESSONS FROM MY SPRING GARDEN, 2019
It is important to get an early start on spring gardening here along the South Texas Coast because the hot humid weather often descends in May. I started under the Grow Light my Celebrity, Caspian Pink, Pink Girl, and Sun Sugar tomatoes; 12 varieties of peppers; Cutting Celery; 2 varieties of broccoli, and Caraflex Cabbage which has a pointed head with sweet tender leaves. My San Antonio friend gave me a Harris Moran Tomato plant, which replaces Tycoon. Another friend gave a Mortgage Lifter and Brad’s Atomic Grape tomato plants. I repotted all of these plants to larger pots.
After pulling up all the thick tall weeds in my garden caused by the heavy rains of the fall and winter, I used my four cycle small tiller to prepare the seven raised rows for planting. On top of the raised rows I spread cotton seed meal, bone meal, dry sorghum meal, composted chicken manure, and 2 cubic yards of compost. I buy the compost from a friend who makes it out of oak leaves, pre-garbage from a hotel, plus shrimp and crab shell. I tilled all these fertilizers in and then pulled the tiller backwards to till the top of each row deeper. I used the till pulled backwards and turned at a slight angle to throw dirt up on the sides of the rows. I finished by using a rake to level the tops of the raised rows.
After using the hose to water each row, I planted the plants and seeds so that each different variety was in a different place than last year if possible. The seeds I planted were watermelon, cantaloupe, Maxabell bush beans, Fortex and Algarve pole beans, Purple Hull peas, Ambrosia and Temptress bicolor corn, straight neck and zucchini squash, 3 varieties of cucumbers, and Cowhorn, Star of David, and Bull Dog okra. The okra, which needs the warm sun, takes a while to germinate and may have to be replanted. The Purple Hull peas had to be replanted because the seeds were pre-Harvey old.
The short day onions planted in December, the bunching onions planted in January, the late winter lettuce, and the one winter tomato plant which is still producing ripe tomatoes, are all growing and used in salads. The new tomato plants have thick root stems and the pepper plants are starting to add many leaves as they grow. I have one hill of double Caraflex cabbage plants. I will harvest one of them for making a salad soon and then cut it off so the other one can make a head. I will need to thin the beans and corn soon. I hope the peas and okra come up thick enough so I have to thin them.
One of the draw backs of planting so soon after putting down the organic fertilizers is the composting process uses a lot of nitrogen before it returns it to be used by the plants. Some of the plants can even have a yellow green color in the process. I will compensate by sprinkling a fish emulsion, kelp liquid, molasses, and soluble organic nitrogen mixture. | http://gardening.spray-n-growgardening.com/lessons-from-my-spring-garden-2019 |
Is Zar pegged to USD?
Is Zar pegged to USD?
Key Takeaways. The South African Rand (ZAR) was introduced in February 1961 and mostly held a steady peg against the US dollar until the end of apartheid. Since then, its value has depreciated as the South African economy has become increasingly linked to the rest of the world.
Why is the ZAR so weak?
Since South Africa relies more on mineral exports, low commodity prices have also led to a weakening of the Rand. ... The reserve bank is expected to adopt an aggressive monetary policy and come up with bigger rate hikes to prevent the Rand from declining further.
Is the rand going to get weaker?
So, while the rand is currently enjoying the benefit of global tailwinds, it is likely to weaken during the course of the year. “However, the extent of this weakening will ultimately depend on the government's progress on fiscal reforms, without which we could see the local currency head north of R18/dollar.
What SA coins are valuable?
Most valuable South African coins value list
- Single 9 Pond (1898) Single 9 Pond. ...
- Kruger Double Nine Ponds (1899) ...
- Burgers Pond Coarse Beard (1874) ...
- Sammy Marks Tickey (1898) ...
- Burgers Pond Fine Beard (1874) ...
- VeldPond (1902) ...
- Mandela 90th Birthday Coin (2008) ...
- Mandela's 100th Birthday R5 coin (2018)
How much does a 5 rand Mandela coin worth?
As mentioned above, the Mandela R5 Coin's value soared to new heights after the President's death. Today, this coin is considered to be the most valuable Mandela coin of all. Several years ago, three of these coins sold for R100,000 or $10,000 each. Today, the coins may be worth even more.
What is a 1977 1 rand coin worth?
Mintage, Worth:
|Year||Mintage||Value, USD|
|Unc|
|1980||2.
|
What is the value of 1 rand coin?
0.
How much is a 1990 1 rand coin worth?
Information:
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- Why is there a shortage of clay pots? | https://archinfos.com/library/lecture/read/17146-is-zar-pegged-to-usd |
In Minecraft, you will inevitably run out of storage room for all of the items you will gather up during your playthrough, luckily you can solve this problem by building a few barrels.
How to craft Barrels in Java Minecraft
To make a barrel in Minecraft, you will need wood planks and two wood slabs placed in a crafting table. You will need to fill both the far left and far right slots going down with the six wood planks. After the wood planks, you will need to place the two wood slabs on the top middle and the bottom space. If you need an example of how to arrange the pattern, check our image below.
Before you can make barrels in Minecraft, you will need to gather the materials required to craft them. First, let's start with the wood planks, as they are probably the most straightforward block to make. To make wood planks, all you need to do is find a tree and chop it down with an ax tool. Afterward, you will get wood logs, take these back to a crafting table, and set the wood logs in any slot in the table, and you will craft wood planks.
Once you get some wood planks, place three in a row, like how we did in the image above, this, in turn, will create the wood slabs you need for the barrel. After this, you can then create the barrel using the pattern we showed earlier in this guide, so go back now if you are ready to make some barrels in Minecraft.
Related: How to make a Cartography Table in Minecraft
How to craft Barrels in Bedrock Minecraft
To make barrels in the bedrock edition is pretty much the same deal as the Java version, except you replace the wood planks with sticks. If you didn't know already, you can make sticks out of wood planks using the pattern shown below.
Once you craft the sticks, place them in the same spot as the wood planks, as shown in the image at the start of this guide.
For more on Minecraft, PGG has you covered with guides like how to find a village and how to change villager jobs. | https://progameguides.com/minecraft/how-to-make-barrels-in-minecraft/ |
a very important criterion for|criterion that is extremely important the caliber of dissertation work is the criterion of this effectiveness of dissertation research. We suggest dissertation outcomes.
Do you know the details in dissertation outcomes?
The effectiveness regarding the outcomes of your dissertation is fundamentally Justified and established. Why don’t we cite frequently employed arguments in justifying the effectiveness of dissertation research. It offers the existence of:
¦ positive results of this utilization of the thesis in culture, industry, technology, any training;
¦ good outcomes of the employment of inventions and energy models;
¦ practical strategies for developing a specific system, a situation for attaining the outcome;
¦ recommendations meant for the style and technical divisions and bureaus associated with the industry;
¦ proposals to enhance the research methodology, manufacturing technology, dimension precision;
¦ knowledge useful for use into the academic means of additional or advanced schooling.
Credibility of research
Evidently, no sense is made by it to persuade opponents for the relevance, novelty and effectiveness regarding the link between dissertation research, if the answers are maybe not dependable. The substantiation of clinical knowledge and bringing it in to a harmonious unified system have actually for ages been the many critical indicators when you look at the development of technology.
Whenever justifying theoretical outcomes, the following requirements are mandatory:
¦ consistency;
¦ consistency with empirical information;
¦ consistency in explaining understood phenomena;
¦ ability to anticipate phenomena that are new.
It is crucial to strictly observe regulations of logic. It’s the legislation of enough explanation: every idea, to become authentic, should be substantiated by other ideas, the reality of which can be shown or self-evident.
The legitimacy associated with outcomes of the dissertation scientific studies are accomplished
¦ predicated on strictly proven and properly utilized conclusions regarding the Applied and fundamental sciences, the conditions of that have discovered application into the work;
¦ verification of theoretical jobs and brand brand brand brand brand brand new solutions, tips, experimental studies;
¦ metrological help of experimental studies;
¦ complex use of well-known theoretical and research that is empirical;
¦ theoretical conditions produced by the writer because of this task that is particular
¦ coordination of the latest conditions with currently understood theoretical jobs of technology;
¦ coordination of this brand new theories using the training and data that are experimental of this writer and other writers;
¦ eradication of contradictions involving the positions that are theoretical by the writer while the known regulations for the development of technology, technology, and knowledge; reason of this outcomes making use of design that is well-known procedures, means of finding solutions, along with real and mathematical modeling;
¦ comparing the outcomes of the test and tests conducted because of the applicant with known data that are experimental other scientists on a single dilemmas;
write essays for money ¦ magazines for the primary outcomes of peer-reviewed main magazines;
¦ speaking about the outcomes of the dissertation at seminars and symposia, getting reviews from leading professionals on problems of work;
¦ with the total outcomes in training with all the assessment of outcomes.
The necessary completeness of this solution associated with the dilemma of dependability is accomplished through experimental verification for the theoretical roles of the dissertation, plus the persistence of its own experimental information because of the experimental information of other scientists.
The adequacy associated with the solution is based on the persistence regarding the experimental information acquired by the applicant with known theoretical jobs of other writers in accordance with noise and constant solutions that are theoretical really by the applicant.
Nota bene! Just expert composing service provides a dissertation that is decent result help! | http://tazaindianrestaurant.co.uk/2019/04/10/dissertation-outcomes-assist-for-post-graduate-46/ |
The utility model discloses a car adjustable internal and external full-view mirror comprising a car internal main view mirror, an inner front view mirror, an inner rear view mirror, a car external left outer main view mirror, a car external right outer main view mirror and secondary mirrors hidden in the concave surface of a front left side car door and the concave surface of a front right side car door. A driver seat is arranged in a cab. The car top or the middle portion corresponding to the middle of the cab is provided with the inner front view mirror which is a convex mirror. The inner front view mirror and a perpendicular line L of a car body form an included angle. The car internal main view mirror is arranged right in the front of the cab. An included angle is also formed between the car internal main view mirror and the perpendicular line L of the car body. The reflection surface of the car internal main view mirror is arranged opposite to the reflection surface of the inner front view mirror. As the middle of the cab is provided with the car internal front view mirror and the car internal main view mirror, drivers can see view angle of the middle of the cab from the driver seat, can accurately see side situations of the left main view mirror and the right main view mirror on two sides of the car, can easily observe the situations in front of the car, so view of the drivers is enlarged, and easy, accurate and safe driving is ensured for the drivers. | |
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Adjustments to the recipe and predicting/explaining the impact it has on the finished product:
When adjusting the recipe, peanut butter was both substituted and omitted. This will impact the final product in different ways. The batch with almond butter will marginally shift from the batch with peanut butter, because they have similar properties, ingredients, and effects when cooked. Properties of peanut butter, like its fat content for example, are similar in almond butter. However, while both are a good source of nutrients, their differences cause the muffins to have an altered texture and taste. In the batch with the peanut butter omitted, the muffins will have a more liquidy texture, because it does not have the nut butter’s properties to bind it together. One can predict that the omission of peanut butter will cause both the batter, baking, and final results to have an altered result, and they will not hold the way that original recipe muffins do.
After baking trials, the above theories were confirmed as the almond butter muffins crisp seemed marginally greater than the crisp of the peanut butter. Though this was not a major change, it shows that substitutions inevitably cause differences in final results because of the chemical properties the substituted ingredients possess. The muffins that lacked peanut or almond butter did not cook as well as previous batches, as shown by the results, liquidy and breakable muffin. Again, this can be a result of the lack of nutrients within peanut butter, like fat and water, which help the muffins bake the way that they are supposed to.
The adjusted muffins also differ in taste. Because the muffins have very few ingredients, it is easy to taste the peanut butter and almond butter when it is present. In the case of omission, the muffins seem to be lacking an ingredient, though anyone who does not know the original recipe may not be able to tell what said ingredient is. Through the trials it was demonstrated that while omission of peanut butter does solve the allergen issue, adequate substitution, like almond butter must be included in the recipe and omission is not possible when trying to achieve a tasty and enjoyable muffin. | https://blogs.dickinson.edu/chemistryinthekitchensp22/2022/05/02/chocolate-chip-oatmeal-muffin-adjustments/ |
Plant: Annual or short-lived perennial herb; to 1.5 m tall; glabrate to pubescent, with a short, slightly swollen tap root Leaves: widely ovate, 5-18.5 cm long, 3-16 cm wide, both surfaces variously pubescent, acute, entire to few lobed or dentate, truncate to subtruncate Flowers: with calyx 4-8 cm long, the teeth ovate, acute or acuminate, 5-15 mm long, the base persistent, rotate or reflexed; corolla white with purple throat, the tube (8-)10(-15) cm long, the limb 4-8 cm wide, the acumens acute to acuminate, 1-10 mm long Fruit: FRUITS pendent, dehiscing along 4 sutures at least half the length, green to purple, subglobose, 2.5-4 cm in diameter, with 200-300 spines, these 1-3.2 cm long; pericarp glandular-puberulent; persistent calyx base rotate or reflexed; SEEDS black, reniform, 3-4.5 mm long, 2.4-3.5 mm wide, rugose; caruncle white, present in fresh seeds Misc: Roadsides and waste grounds; usually below 600 m (2000 ft), although up to 1750 m (5500 ft); Mar-Oct REFERENCES: Bye, Robert. 2001. Solanaceae. JJ. Ariz. - Nev. Acad. Sci. Volume 33(1). | http://swbiodiversity.org/seinet/taxa/index.php?taxauthid=1&taxon=3869&clid=4133 |
Wong, Nguok Ling (2017) Keberkesanan pembelajaran koperatif (STAD) ke atas pemahaman, komunikasi, pencapaian dan sikap matematik bagi topik pecahan murid sekolah rendah daerah Sarikei, Sarawak. PhD. thesis, Universiti Utara Malaysia.
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Abstract
This study aims to investigate the comprehension, communication, attitude and effects of Student Teams-Achievement Division (STAD) cooperative learning on Mathematics achievement in Sarikei District, Sarawak. This study also explore the students’ and teachers’ perceptions of STAD cooperative learning. The number of subjects involved this study is seventy students from Year Five in Sarikei District, Sarawak. A total of 35 students were involved in the experimental group (20 males & 15 females) while another 35 students were involved in the control group (19 males & 16 females). Data collection was done twice which were the pretest and the post test. The gap between the exam was four weeks. The Mathematics test has 20 items which consists of 10 comprehension items and another 10 communication items. The questions were adapted from the Primary School Achievement Test (Ujian Pencapaian Sekolah Rendah). Mathematics attitude was measured using 19 items, while the 20 items used to measure students’ attitude on STAD. Semi-structured interviews and observations were used to obtain students’ and teachers’ views on STAD cooperative activities and learning. Data were analyzed using SPSS version 19 technique ANCOVA. The findings shown that the use of STAD techniques in Mathematics learning towards comprehension, communication, and Mathematics attitude can improves Mathematics achievement. Additionally, this study has also shown main effect and direct interaction in comprehension, communication, and students’ Mathematics achievement and attitude in post test between the experimental and control groups. This suggests that STAD cooperative learning plays an active pedagogical role to improve comprehension, communication, as well as Mathematics achievement and attitude. Interviews show that STAD cooperative learning increases interest, motivation and Mathematics knowledge sharing among students as compared with conventional learning. Observation of teaching and learning using STAD in the classroom shows that student-student and student-teacher interaction is strengthened. STAD encourages the students and teachers to be innovative and creative in improve teaching and learning of Mathematics in the classroom. These benefit the students in Sarikei District and enable them to compete healthily with the other students from urban areas in Mathematics. | http://etd.uum.edu.my/7074/ |
Just a 2-minute walk from Wiesbaden Spa Park, these apartments are situated within a senior citizens residence. They include a 24-hour reception, and a kitchen and flat-screen TV. The bright apartments at GDA Hildastift am Kurpark come with a minibar, a balcony and a bathroom with a hairdryer. Guests can prepare their own meals in the fully equipped kitchen, or dine in the Hildastift am Kurpark’s restaurant. A buffet breakfast is served each morning, and special diabetic menus are available. The property features an indoor pool, and massages can be booked. The Hessisches Staatstheater Theatre is a 10-minute walk away. The A643 motorway is 4 km away.
Wiesbaden
Travel Guide
Wiesbaden› Hotel
GDA Hildastift am Kurpark
Hildastraße 2, Wiesbaden, 65189, DE
Our Rating Neighborhood around town Prices 49 EUR to 125 EUR Amenities Grounds; Library; Breakfast in the Room; Restaurant With Dining Menu; Room Service; Special Diet Meals (upon request); 24-Hour Front Desk; Free Parking; Misc Parking; On-site Parking; Private Parking; Internet; Hair/Beauty Salon; Shops (on site)Book a Room Now
MapHildastraße 2 Wiesbaden 65189 DE Wiesbaden
Note: This information was accurate when it was published, but can change without notice. Please be sure to confirm all rates and details directly with the companies in question before planning your trip. | https://www.frommers.com/destinations/wiesbaden/hotels/gda-hildastift-am-kurpark |
Nov 12, 2018 · What Does Mbps Really Mean? We have been speaking in terms of “higher” Mbps and “lower” Mbps. Let’s bring numbers into the mix. Internet service providers usually attach figures to their data plans. One service provider might speak of speeds of up to 7Mbps and another will promise 10Mbps (or 50Mbps). What do these figures mean?
Nov 14, 2019 · The data rate of a network connection is normally measured in units of bits per second, generally abbreviated as bps instead of b/s. Network equipment manufacturers rate the maximum network bandwidth level their products support using the standard units of Kbps, Mbps, and Gbps. Apr 05, 2019 · If you stream a lot of movies and TV shows, or play online games, you might be wondering if your internet is fast enough. Consumer Reports helps you evaluate and improve your broadband speed. 10Mbps is astonishingly fast, I have it. I can download 4 anime episodes while playing Modern Warfare 3 with no problems what so ever. I have at times downloaded 10 episodes while playing Battlefield 3 with no problems, but that is the highest and 10 episodes are smaller than seasons worth of downloading so you have to do upload and download math to get a feel for what your bandwidth can handle… May 30, 2018 · The definition of “broadband” has changed over the years to reflect the way we use the Internet. From a mere 200 kilobits per second (Kbps) in download speed in 1996 to 25 megabits per second (Mbps) in 2015, it’s also important to remember that download speed is only part of the definition. While lobbyists for big ISPs argue the 25/3 standard is too high, 25 Mbps (download) and 3 Mbps How Fast is 10 Megabits Per Second? It is a hard question to answer, because every people uses internet in their own way. If you just surf the web, read and write blogs, check your e-mails, stream standard definition contents and download small files, then yes, 10Mbps internet would be fine for you.
10 MB mean 10 Megabytes. It is an amount of data, there are 1024 Megabytes in a Gigabyte(GB), there are 1024 Kilobytes(KB) in a Megabyte, there are 1024 Bytes in a Kilobyte. A song file might be 5
Dec 29, 2016 · What does 10Base-T Mean? Dec 29, 2016. This is an Ethernet standard that is designed to transmit data at the rate of 10Mbps over a twisted pair cable. It is supposed What does mbps mean? Putting it simply, mbps is the unit of measurement used to measure how fast an internet connection is. The higher the mbps, the faster the internet and vice versa. For example, you can get broadband plans with speeds of up to 100mbps. You can also get broadband plans with faster speeds than that, but they generally cost a
The answer is, it depends. It depends on how high a quality you want, and how tolerant you are of either delays or reduced video quality. First, the bad news: Standard definition YouTube videos have a target bandwidth of 8 Mbps. | https://superbvpnzsfm.web.app/huntress76433jywo/what-does-10mbps-mean-365.html |
DAY 1: Prepare Poolish:
Weigh water, flour and yeast into a clean planetary mixer bowl, and mix on 1st speed with dough hook for 3 minutes, or until amalgamated, scraping down bowl where required.
Place into a clean, lightly oiled, appropriately sized, plastic rectangular container.
Affix lid, and leave in ambient temperature of 25°C for 1 hour, before placing in a retarder set at 10°C for 17 hours.
DAY 2: Create Dough:
Using correctly tempered water is essential to achieve desired dough temperature of 25°C.
Weigh Ingredients:
Weigh tempered water into Poolish, and pour into spiral mixer, then add oil, flour and dried yeast. Delay the addition of salt until final minutes of mixing.
Mixing Time Guide:
Mix for 5 minutes on low speed. Scrape down the bowl and reconfigure dough.
Mix for another 5 minutes high speed.
Add salt, and turn by hand into dough.
Mix for a further 4 minutes on high speed.
Dough will be sticky, extensible and glossy.
Bulk Proof:
Place dough into a clean, lightly oiled, appropriately sized, plastic rectangular container, affix lid, and bulk proof for 45 minutes in ambient temperature of 25°C.
Remove lid, and give the dough 3 folds, left - ⅔ over to right, right - over to left and finally, top - down to bottom.
Affix lid and proof for a further 45 minutes.
Divide, Final Shape:
Divide dough into required amounts, and place onto a light covering of flour.
Immediately shape Turkish loaves by gently lengthening out to approximately 20 cm, and folding in half from top - up, around and down evenly to the bottom line to create a straight seam.
Gently roll loaf into cradling hands, and carefully place seam down onto boards or trays heavily dusted with fine semolina, allowing enough room for growth.
Place in a rack with a cover.
If shaping Pide rolls, lift out adjacent sides with fingertips and stretching up to meet above the centre.
Continue around sides to form a 'money bag' shape.
Place seam down onto boards or trays heavily dusted with fine semolina leaving approximately 3 cm between each.
Place into a rack with a cover.
Final Proof:
Cover rack and dry proof in bakery ambience (26°C ± 2°C) for 1 hour, or until fully proofed.N.B. Loaf is fully proofed when touched gently with finger on the end of the loaf, it is slow to spring back, and does not spring back the whole way.
Load and Bake:
Brush Turkish loaves and Pide rolls liberally with egg (whisked with a pinch of salt).
Dip fingers in egg, and firmly dock loaves/rolls evenly in centre leaving a 1 cm border undocked.
Sprinkle loaves with Zatar Mix and rolls with Nigella and/or Sesame seeds.
Walk fingers under loaves and stretch lengthways to 35 cm, stretch rolls to form an oval shape and place onto a peel, or back onto trays if required.
Load loaves immediately into preheated oven, and bake directly on stone or ceramic tile floor where possible, and steam for 3 -5 seconds.
Bake at 250°C for 15 minutes (Turkish) or 12 minutes (Pide). N.B, Loaves and rolls should have a light golden crust, allowing bread to be toasted in a sandwich press without burning.
Loaves and rolls may be left to cool on trays to soften base crust. | https://www.laucke.com.au/recipies/turkish-bread-5-pre-ferment |
Week 8: Sub-50 10-K (and Half-Marathon) Training | Fit Girl. Happy Girl.
Monday– I did a yoga workout for about 30 minutes to cross-train. I didn’t have much time because it was Christmas Eve and my family and I planned to spend the day in Boston before I had to go back to Pennsylvania. Sometimes you have to sacrifice a workout for family time but I was totally fine with that because I knew I wouldn’t get to see them for awhile.
Tuesday– Christmas! No workout for me today because I spent the morning with my family, squeezing every last minute out of my visit home for the holidays. I then spent the rest of the afternoon/evening driving back to Pennsylvania for work the next day.
Friday– 4.2 miles- I decided to do speed work on Friday on the treadmill because it was FREEZING outside! I did a five minute warm up and then 200m at easy pace followed by 4x400m at race pace (8:06/mile) with 200m at easy pace in between. I finished the run with about a mile cool down at easy pace.
Saturday– ran 4.2 miles in the snow! I ran 9:17/mile pace and it was such a relaxing run because the snow was just starting to fall but hadn’t stuck to the ground yet.
Sunday– I was supposed to do my long run on Sunday but when I woke up to 30mph winds with gusts of 50mph and below freezing temps, I figured I could push it off to Monday since I didn’t have work.
Monday– 12 miles at 9:20/mile pace – I was completely shocked at how well this run went! I felt really strong and comfortable the entire time. I didn’t have to stop or walk at all and I finished feeling like I could keep running. This run was an amazing indicator of how the half-marathon will go (knock on wood!) and I can’t wait to see the results of this training!
Congrats on your good long run.
Thank you so much!! I wish you happy runs in 2013! | https://fitgirlhappygirl.com/2013/01/03/week-8-sub-50-10-k-and-half-marathon-training/ |
GATE | GATE-CS-2016 (Set 1) | Question 23
The worst case running times of Insertion sort, Merge sort and Quick sort, respectively, are:
(A) Θ(n log n), Θ(n log n) and Θ(n2)
(B) Θ(n2), Θ(n2) and Θ(n Log n)
(C) Θ(n2), Θ(n log n) and Θ(n log n)
(D) Θ(n2), Θ(n log n) and Θ(n2)
Answer: (D)
Explanation:
- Insertion Sort takes Θ(n2) in worst case as we need to run two loops. The outer loop is needed to one by one pick an element to be inserted at right position. Inner loop is used for two things, to find position of the element to be inserted and moving all sorted greater elements one position ahead. Therefore the worst case recursive formula is T(n) = T(n-1) + Θ(n).
- Merge Sort takes Θ(n Log n) time in all cases. We always divide array in two halves, sort the two halves and merge them. The recursive formula is T(n) = 2T(n/2) + Θ(n).
- QuickSort takes Θ(n2) in worst case. In QuickSort, we take an element as pivot and partition the array around it. In worst case, the picked element is always a corner element and recursive formula becomes T(n) = T(n-1) + Θ(n). An example scenario when worst case happens is, arrays is sorted and our code always picks a corner element as pivot.
Attention reader! Don’t stop learning now. Practice GATE exam well before the actual exam with the subject-wise and overall quizzes available in GATE Test Series Course.
Learn all GATE CS concepts with Free Live Classes on our youtube channel. | https://www.geeksforgeeks.org/gate-gate-cs-2016-set-1-question-23/?ref=lbp |
While Nobel Peace Prize recipient Elie Wiesel penned several autobiographies and works of fiction, his best-known work is Night—a memoir based on his experiences as a prisoner in WWII concentration camps, specifically Auschwitz and Buchenwald. A harrowing read offering a firsthand account of one of our world’s most tragic time periods, Night helped shine a light on the Holocaust and to this day is regarded as one of the world’s most important works of literature.
A fact that is not as widely known as the work itself is that Night is part one of a trilogy: Night, Dawn, and Day. Each book focuses on specific parts of Wiesel’s transformative renaissance—darkness to light, horror to healing. With Night, we know of Wiesel’s intent:
“I wanted to show the end, the finality of the event. Everything came to an end—man, history, literature, religion, God. There was nothing left. And yet we begin again with night.”
For the trilogy’s subsequent works, Wiesel took a different approach, saying, “In Night it is the ‘I’ who speaks. In the other two, it is the ‘I’ who listens and questions.” The final book, Day (not a memoir but a work of fiction), completes the transformation arc: an injured man reflects on his relationships and experiences during WWII and comes to grips with his survival and the deaths of loved ones.
Memoirs, like Night, offer a clear window into the thoughts and experiences of others, especially those who write them. They are also a subgenre of autobiography—though the exact categorizations of “memoir” and “autobiography” are a bit fuzzy and at times almost entirely overlapping. Essentially, a memoir is autobiographical, while not all autobiographies meet the criteria for a memoir. Loosely, autobiographies will encompass the subject’s entire lifespan, whereas memoirs—depending on the work—tend to be more flexible and focused on a specific point in time or subject matter, like WWII. Though there has been some debate over the years about Night‘s designation as a memoir, most publishers agree that the story speaks to Wiesel’s personal experiences—something we can all learn from.
Night will surely live on as part of the historical canon and as a must-read memoir for generations to come. Keep reading for ten more memorable, must-read memoirs handpicked by our staff. Some are new, some are old, and many you may not have heard of just yet (but should definitely check out now!). | https://blog.enotes.com/2017/06/29/night-and-10-other-must-read-memoirs/ |
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Is quadratic probing open addressing?
Quadratic probing is an open addressing scheme in computer programming for resolving hash collisions in hash tables. Quadratic probing operates by taking the original hash index and adding successive values of an arbitrary quadratic polynomial until an open slot is found.
How do you implement quadratic probing in hashing?
Let hash(x) be the slot index computed using the hash function.
- If the slot hash(x) % S is full, then we try (hash(x) + 1*1) % S.
- If (hash(x) + 1*1) % S is also full, then we try (hash(x) + 2*2) % S.
- If (hash(x) + 2*2) % S is also full, then we try (hash(x) + 3*3) % S.
What is open addressing in hash table?
The open addressing is another technique for collision resolution. Unlike chaining, it does not insert elements to some other data-structures. It inserts the data into the hash table itself. The size of the hash table should be larger than the number of keys.
Why do we use quadratic probing?
Why would someone use quadratic probing? Assuming we need some collision resolution algorithm, Quadratic probing can be a more efficient algorithm in a closed hash table, since it better avoids the clustering problem that can occur with linear probing, although it is not immune.
What is linear probing and quadratic probing?
Linear Probing has the best cache performance but suffers from clustering. Quadratic probing lies between the two in terms of cache performance and clustering. Double caching has poor cache performance but no clustering.
What is open addressing and give example?
Some open addressing methods, such as Hopscotch hashing, Robin Hood hashing, last-come-first-served hashing and cuckoo hashing move existing keys around in the array to make room for the new key. This gives better maximum search times than the methods based on probing.
What is open addressing and linear probing?
Along with quadratic probing and double hashing, linear probing is a form of open addressing. In these schemes, each cell of a hash table stores a single key–value pair.
What is the difference between linear and quadratic probing?
What is the main problem of quadratic probing?
There is one problem with quadratic probing: Its probe sequence typically will not visit all slots in the hash table. Unfortunately, quadratic probing has the disadvantage that typically not all hash table slots will be on the probe sequence.
How linear probing differ from quadratic probing with example?
Is linear probing and open addressing same?
How many probing techniques are there in open addressing?
Three techniques are commonly used to compute the probe sequence required for open addressing: Linear Probing. Quadratic Probing. Double Hashing. | https://bigsurspiritgarden.com/2022/09/26/is-quadratic-probing-open-addressing/ |
Artist Paolo Cirio’s introductory video to Loophole4All.com teaches everyday people how to pursue tax loopholes by becoming a pirate and hijacking an offshore company.
With leaders from eight of the world’s wealthiest countries discussing economic policies that will affect citizens worldwide, the G8, always a symbol of undemocratic governance, is particularly contradictory this year. British Prime Minister David Cameron, host of the summit in Northern Ireland, is calling for a crackdown on widespread global tax evasion. But he might as well be called prime minister of major tax havens for his role overseeing London and the Crown Dependencies.
As usual, G8 members will advance measures aimed at maintaining, rather than resolving, these contradictions. Their proposals will not contain any changes that might distress corporate interests.
A mere 100 miles southeast of the summit, Dublin serves as a tax haven and a center of massive tax evasion for many of the wealthiest corporations. American companies like Cisco and Apple set up subsidiaries in Dublin to evade U.S. taxes, since corporate tax rates are roughly three times lower in Ireland than in the United States. Apple’s Irish affiliate actually paid no taxes on $30 billion in profit over the last four years. Corporate CEOs defend such tax evasions by presenting themselves as job creators acting for the benefit of the economy as a whole, but they leave out the data showing the decline of the middle class and consequent increases in poverty and homelessness. Ireland is not such a great place for normal people, with its severe austerity and outrageously high unemployment.
If corporations don’t pay taxes, then it follows that when people buy iPhones, search on Google or order items on Amazon, everyone loses hospitals, schools, road maintenance and eventually pensions. Meanwhile, the lucky employees of untaxed companies get higher wages that directly produce unaffordable living costs for people employed by local and public businesses. Only those who work for the regime of major firms can survive.
We need to supplement public demonstrations with creative strategies of subversion.
People who raise their voices against the injustice of this situation by taking to the streets outside G8 summits have met escalating violence from security forces. In my twenties, I joined several anti-G8 protests across Europe, facing riot police that regularly employed tear gas, water cannons and clubs against peaceful demonstrators. At the Genoa summit in 2001, I dodged the massacre at the Diaz school out of pure luck. I have not always had the same fortune, and violent repression and mass arrests have become ordinary in the decade since.
Although street protests are crucial in manifesting dissent, we need to supplement public demonstrations with new, creative strategies of subversion. For instance, with the project Loophole4All.com, I managed to unsettle corrupt Cayman Islands authorities and international accounting firms by creating a caricature of the Certificate of Incorporation used by shell companies set up in the Caymans. At the same time, I drew attention to thousands of fraudulent companies, engaging the public in an unusual form of civil disobedience that threatens the offshore financial system.
Political innovation should be considered an art form that challenges brutal repression and creates solutions for global governance. I believe that artists can create legislative and financial models for the complex needs of the 21st century, incorporating humor, beauty and interactivity into new forms of social organization. Just as creativity and concrete social goals come together in architecture, contemporary artists should intervene in proposing policies that work for our times, while guiding us in interpreting and unveiling the invisible truths of our world.
The absurdity of the unsolved legality of offshore business helps to expose to everyone the disorder of our times and the need for radical change. The vast exploitation of discrepancies among legal jurisdictions undermines the notions of law and national borders that are central to contemporary civilization. Globalization has outstripped the power of governments, businesses and citizens; each is left powerless against the other.
Paolo Cirio interviews John Christensen, executive director of the Tax Justice Network; William Brittain-Catlin, author of The Dark Side of the Global Economy; and Jack Blum, chair of Tax Justice Network USA; and Chris Taggart, co-founder of OpenCorporates.
The leaders at the G8 summit may recommend some weak international regulations. However, they won’t resolve the implicit conflict of global economic competition. Take the cases of Russia and Cyprus, China and Macao, the United States and Delaware or the United Kingdom and its Crown Dependencies. Every economic power has its own offshore center as a structural financial instrument that cannot be dismantled without major consequences. The use of offshore finance is too big to fail. The financial centers of London, New York, Frankfurt and Hong Kong are today’s toxic factories, and they exploit offshore jurisdictions like the Caymans, Jersey, Zurich and Singapore as noxious, yet legal resources.
When impunity and injustice are the new normal, transparency becomes an empty word.
As the American Senate and the British House of Commons interrogated the CEOs of Apple, Amazon, Starbucks and Google about their massive tax evasions, it was evident that these companies would get away with the biggest robberies in recent history with nothing more than a slap on the wrist. The public, otherwise powerless, could only laugh at this nonsense.
When impunity and injustice are the new normal, transparency becomes an empty word. Corruption is no secret in Ireland, for example, which never needed to hide the fact that corporations pretend to be based there in order to evade taxes everywhere else.
Embedded in digital technologies, transparency is unavoidable, but it isn’t enough to tackle present and future abuses of power. Leaks of unclassified information are important; however, information doesn’t make any sense by itself. The huge quantity of data published by WikiLeaks and the recent Offshore Leaks can only generate political change if mainstream media filter the leaks sensitively and honestly. (Hence, whistleblower Edward Snowden turned to Glenn Greenwald, because he trusted the journalist, and his outlet The Guardian, to tell the story of the U.S. National Security Agency’s surveillance.) Real change can only come about when people incisively interpret the political and ethical value of information.
We should all be involved in designing alternative tax structures in a process similar to the participatory budgeting initiatives that have spread from Brazil to Mexico to the United States. For example, people and businesses could be empowered with tools that let them determine which area of society needs their funds. People should be able to enact change in a more participatory and fluid manner, rather than waiting on a slow and corrupt legislative system to deliver tax reforms.
We already have the tools for a direct and open democracy. What we need is a cultural and educational revolution that can bring it into being. Designing new ideas for governance is the real creative challenge of today. Faced with the austerity recommended by politicians and economists, artists can activate the utopian imagination, fostering innovative forms of participation and shared cultural values in social structures.
This piece, commissioned by Creative Time Reports, has also been published by Alternet. | https://creativetimereports.org/2013/06/16/tax-loopholes-for-all/ |
Julia Gillard leadership Essay Example
Julia Gillard was Australia’s 27th primeminsiter and she was the first woman to assume such a high leadership role in Australia. Gillard was born in Wales in 1961 before her family migrated into Australia in 1966 after which she became an Australian citizen in 1974. Her leadership traits started manifesting in her early age when she was elected president of the Australian Union of Students in 1983. After her studies, she worked as a solicitor in Melbourne with the law firm Slater and Gordon becoming a partner in 1990 where she focused on securing fairer treatment for workers. She first contested for a federal seat of Lolor for Australian labor party in 1998 and was elected. She served as the MP for the 1998, 2001, 2004 and 2007 terms. She has also held various portfolios in the Australian government including minister for various ministries including education, employment and workplace relations, minister for social inclusion, deputy prime minister and finally the first female prime minister in Australia before quitting politics. Julian is of interest to me as a leader owing to the fact that she has been able to demonstrate her leadership skills throughout her life becoming the first female prime minister in Australia despite politics being male dominated. She also interests me for her leadership traits including being calm, polite and being rarely irritated even when things seems to be tough. In fact, she has been described as having a tough, steely personality in how she was able to confront the pressures of being the first female prime minister.
Literature review
A great deal of literature exists on leadership, leadership traits, teamwork, ethics, diversity and change. Ololube (2013) for instance sees leadership as the ability of an individual to influence others to achieve a common goal. This according to Jenkins (2013) calls for strong character and selfless devotion to the course. Many theories of leadership also exist that try to explain leadership and how leaders behave in certain circumstances. Such theories include the great man theory, the trait theory, contingency theories and situational theory among others. According to Charry (2012), the contingency theory postulates that no single leadership style is appropriate in all situations and that it will be contingent with the situation. On the other hand, situational theory postulates that a leader should choose the best course of action based on the situational conditions. Lamb on the other hand states that a leader should possess certain traits including enthusiasm, boldness, self-assurance, competitiveness and conscientiousness. According to Wolinski, a good leader ought to also be a good team player as organizations are almost entirely ran through teamwork. In addition, leaders ought to exhibit ethical leadership as a leader without integrity cannot be termed as a leader (Mulford, 2003). According to Hargreaves and Fink, a good leader is also able to deal with diversity since organizations do not exist in vacuums while being able to cope with change since organizations are evolving on a daily basis (2004). It should thus be concluded that a good leader ought to be all round being able to deal with different situations as they arise while keeping everyone focused to the ultimate goal.
Using the theories to analyze leaders
Julia can be described as possessing leadership traits since her young age in line with the greatman and traits theory as she is seen leading from the time she was in school as a president of the students union. It can be said that she applies many of the leadership theories studied in the module. For instance, she is also able to apply contingency and situational leadership theories successfully especially when faced with difficult political decisions. This was seen in her courage to ask for a leadership vote when her credibility as a leader faced so many questions. Even when she wins with a minority, she is able to form government by working with other minor parties to ensure that she runs the government in line with the current situation. She can also be hailed for being a team player as is seen in her ability to work with various leaders both in the opposition and in the government especially under Rudd to ensure the success of her party. Her ethical standing can also not be questioned since it is through her integrity that she gets the mandate of the people as she goes up the ladder until she becomes the prime minister. Being able to work with various people up the ladder shows her ability to cope with diversity. However, her ability to cope with change can be questioned as she quits politics after she is defeated in an election instead of holding on.
Conclusion
This essay has looked at Julia Gillard as a leader. There are many character traits that depict her as a leader that are worth noting. Julia is calm, polite and rarely irritated even when faced with difficult situations especially politically. These have been part of her character since her early age in that she is always herself and is not that easily pushed or pulled by others views. She has a tough personality that enabled her confront the pressures of being Australia’s first female prime minister. She was able to endure constant disapproval of many political issues. Her top leadership traits include intelligent, hardworking and aggressive. However, to become a better leader, she needs to improve on her relations with people as she has been accused of being out of touch with ordinary people, being arrogant, superficial, erratic l and narrow minded. She also needs to improve on her honesty since she has been accused of doing things that she promised not to do during her campaigns such as introduction of carbon tax. She also needs to be a better team player since her defeat result from her poor working relationship with her labor party. If she can improve on the traits below, then she is able to become even a better leader.
References: | https://freestudyhelp.net/critical-review |
Implementing technology, large-scale, can be a herculean task for a district. Making sure tools are actually being used to support learning and determining the effect on student outcomes requires thoughtful planning, committed leadership, and engaging professional development.
In Learning Transformed: 8 Keys to Designing Tomorrow's Schools, Today, advocates and former educators Eric Sheninger and Tom Murray attempt to provide a roadmap for that process. The book outlines research-backed strategies for integrating technology and implementing personalized learning.
"It's really important for school leaders to know what the research actually shows does work," said Murray, the director for innovation at Future Ready Schools, a project of the Alliance for Excellent Education, and a former district tech director and teacher.
The simple fact that schools have devices for students "doesn't tell us anything about learning," he said. "It's how they're being used."
Learning Transformed combines best practices from educational research with Murray's and Sheninger's experiences working in schools and, in their current positions, with educators across the country. Sheninger, a senior fellow at the International Center for Leadership in Education, is a former principal and teacher. He is also the author of several books on leadership and technology in education.
Murray and Sheninger especially wanted to highlight districts innovating in instruction with tech and bridging the digital divide, despite facing financial constraints. It was important to present district leaders' solutions that were not only evidence-based, but that also seemed accessible and attainable, said Sheninger.
"There is no more powerful way to convey a message to motivate and inspire people to change," he said, "than by exposing them to those people that are doing it."
Education Week spoke with Murray and Sheninger about how districts can better evaluate instructional technology, personalize professional learning, and make smarter purchasing decisions.
The Q&A has been edited for brevity and clarity.
Before starting the purchasing process, what kind of evidence should districts require from vendors that the tool or device in question actually improves teaching and learning? And once technology is integrated, how should school and district leaders determine whether tools are being used effectively?
Murray: We cite, a couple of times [in the book], the report that the Alliance for Excellent Education did with Linda Darling-Hammond and her team at Stanford. It was a meta-analysis of ed-tech studies, on what actually works—what instructional practices. It's important for school leaders to know, number one, that learning with ed tech needs to be interactive. Number two, [what works is] the use of technology to explore, to design, to create. It's not the digital drill-and-kill, or the electronic worksheet—which is very prevalent.
School leaders really need to be asking and pushing vendors, "What evidence and research do you have—particularly independent evidence and research that you have—that your product can help support the work that we're trying to do?"
Sheninger: We talk about the concept of "return on instruction." We spend all this money on technology—how do we know it's having an impact? We present multiple concrete areas to show that efficacy, such as looking at qualitative and quantitative data, using portfolios for students and for educators to show growth over time, to show change over time.
The book provides a 10-step strategy for districts making purchasing decisions, as well as tips for developing a "refresh cycle": a check-in evaluation to make sure that, after a few years of use, tools are still functional and doing what they were purchased to do. What factors are most important for districts to keep in mind when selecting and evaluating technology?
Murray: That very first step of that 10-step process for selecting devices is your vision for teaching and learning. Quite often, what happens is, people focus [on the] device first, and then figure out how the learning can fit into the device.
Those who are in the classroom often have the least amount of voice in the purchasing process, and that's just a massive problem. Teachers need to be a vital part of that decision making, because they're where the rubber hits the road. Purchasing is often left in the hands of a handful of high-level administrators, and it really needs to be a much more thorough process. Not just that we're getting the best price—we're purchasing things that are reliable, sustainable, and solid investments in the longer haul.
The refresh piece was an important topic to cover, again from my own personal experience as a tech director. Eric and I will hear [from administrators], "We're 1:1 in our district, this is wonderful!" And then a teacher will whisper, "Yeah, but my laptops are 8 years old and they take 26 minutes to boot." If we don't have a solid refresh plan in place, sustainability becomes a massive issue, and technology can start to impede learning when the devices start to get in the way.
You cite research that professional development, in its current form, doesn't necessarily improve teacher performance. A more effective approach, you argue, would embed PD regularly within the school day, give teachers a voice in deciding what learning opportunities would be most helpful for them, and focus on improving pedagogy—not simply on how to use tech tools. How can districts implement these strategies, especially when it comes to PD around instructional technology?
Scheninger: When you look at professional development, a lot of it is structured how traditional education is. We get everyone together; districts have [educators] sit through PD sessions. It's not really authentically engaging and applicable to the needs of everyone in attendance.
When it comes to technology, we look at [professional development] this way. [The first step] is exposure to the tool. Number two is giving educators time to tinker, play around, learn how to use the tool. The third step is the most important—ensuring that the tool is being integrated with purpose, that there is accountability for sound instructional design, and that it leads to evidence of improvement in professional practice.
By [exposing educators] to different pathways to learn—utilizing webinars that are available, virtual PLCs, and the creation of personal learning networks—everyone can follow their own topic and work at their own time, path, and pace, to focus on areas that really have an impact on their prospective role.
Photos courtesy of Tom Murray and Eric Sheninger.
See more: | http://blogs.edweek.org/edweek/DigitalEducation/2017/06/Q&A_Tom_Murray_Eric_Scheninger.html |
IntroductionIn vitro gas production technique is one of the methods used for evaluating ruminal fermentation kinetic of feedstuffs. In this method, the volume of gas produced during the incubation is presented as a curve. The mathematical description of gas production profile is performed by fitting data set to a nonlinear model. Recently, several non-linear models have been developed to estimate gas production profile however, some of these models are not accurate enough. Therefore, the aim of this study was to investigate the accuracy of some nonlinear models for predicting ruminal fermentation kinetic of a forage feed.
Materials and methods In this experiment, corn silages (samples on 0, 30 and 60 days after ensiling) were used as fermentation substrates. Dry matter and chemical composition (organic matter, crude protein, NDF and ADF) of the samples were determined using standard methods. The rumen fluid was obtained from three fistulated rams before the morning feeding. The collected ruminal fluids were pooled and transferred into a flask to the laboratory. The rumen fluid was filtered through four layers cheesecloth, flushed continuously with CO2 and maintained at 39oC before incubation. The rumen fluid was then mixed with buffered mineral solution at the ratio of 1:2 (V/V). Gas production technique was completed in three separate runs on three different days (each run lasted 6 days). In each run, the samples were incubated in triplicate and two vials (without the substrate) were considered as the blanks. The volume of gas produced was measured at 0, 2, 4, 6, 8, 12, 16, 20, 24, 48, 72, 96,120 and 144 hours of incubation. The prediction of the gas volume at different times of incubation was compared by four nonlinear models and results were expressed in ml per 200 mg of DM incubated. The selected models (experimental treatments) included Exponential (EXP), Fitzhugh (FZH), logistic (LOG) and Gompertz (GOM). The goodness of fit of the models were evaluated using mean square error (MSE), coefficient of determination (R2), residual mean absolute deviation (RMAD) and mean percentage error (MPE). In addition, Durbin-Watson test (DW), run test and linear regression analysis (between observed and predicted values of the gas volume at different incubation times) were used to assess the accuracy of the models in fitting the data. The estimated ruminal fermentation parameters (the asymptotic gas volume and gas production rate) and goodness of fit parameters obtained from each model (MSE, R2, RMAD and MPE statistics) were analyzed using completely randomized design.
Results The studied models had no difference in terms of predicting asymptotic gas volume (A) on 0, 30 and 60 days after ensiling and the value of parameter A predicted by the models were in the range of 98.37 (for GOM model on day 0) to 76.09 (for LOG model on day 60) ml per 200 mg DM. The EXP and LOG models had the highest and lowest MSE and R2 values, respectively, indicating their lower accuracy compared with GOM and FZH models. The RMAD value was lowest in GOM and FZH models (2.591 and 2.879, respectively) and was highest in EXP model (3.807). The RMAD value is used as an indicator for evaluating the goodness of fit of models. the lower values of RMAD (closer to zero), represents a better ability of the model in fitting data. Based on these results, GOM and FZH models had a higher accuracy than EXP model in fitting data. The MPE value in the EXP model (5.527) was significantly higher than the other models (p < 0.05). In other words, the predicted values (the volume of gas produced at different times of incubation) by the EXP model were lower than the observed values (it was underestimated). Based on Durbin Watson (DW) test results, the DW statistics in the EXP, FZH, GOM and LOG models were 0.392, 0.691, 0.705 and 0.675, respectively, indicating that EXP and GOM models had the lowest and highest accuracy, respectively, in predicting the rumen fermentation kinetic of corn silage. According to the run test, all the curves in EXP model had the lowest run (3 ≥) implying a poor performance of EXP model in predicting the results. The linear regression between the observed versus predicted values (regression parameters) showed a significant difference between intercept with 0; and slope with 1 in all the studied models (p < 0.05). However, based on the goodness of fit parameters obtained from the linear regression, FZH and GOM models had a better prediction of the gas production profile.
Conclusion The EXP model had lower accuracy in predicting the rumen fermentation kinetic of corn silage compared with the other studied models. It is recommended that other nonlinear models be used in addition to the EXP model for investigating the ruminal fermentation kinetics of corn silage. | https://ijasr.um.ac.ir/article_37010.html |
Friends and supporters of Greenwich Hospital gathered at the Belle Haven Club on April 13 for an evening reception and viewing of the short film, Restoration: Thread and Gold, created in gratitude to the community.
The short documentary, which was produced by the Greenwich Hospital Office of Development, features Gen Saratani, a Japanese laquer artist, and Dr. Sarah Lambert, a Greenwich-based Yale Medicine pediatric urologist and associate professor of urology with the Yale School of Medicine. The two film subjects, who attended the event, each shined light on the art of healing through their respective disciplines, helping the audience discover how mending someone or something restores to us their essence and humanity.
Greenwich Hospital President Diane P. Kelly welcomed guests and reflected on the core meaning of the film: “Our focus this evening surrounds the heart of what Greenwich Hospital stands for. Our hospital represents a space in people’s lives that centers around healing — for ourselves and for our loved ones. It’s a treasured responsibility for each and every one of us at Greenwich Hospital that has and will continue to guide us.”
Greenwich Hospital Board Chairman W. Robert Berkley, Jr. introduced the film and thanked guests, many of whom are long-time donors, hospital friends, volunteers, and dedicated supporters. | https://fairfieldcountylook.com/parties/greenwich-hospital-celebrates-supporters-with-reception-and-film-premiere/ |
Under Esslingen Coimbatore Association (Esscom), Municipal Council of Esslingen am Neckar, Germany and Municipal Corporation of Coimbatore, India signed a treaty for friendship, rich cultural and technical exchange for mutual cooperation and upliftment. The areas of upliftment would be in education, culture, sports, waste management, environment, tourism etc.
As a team we were excited for such a reputed project. Lot of research went into it with cultural enlightenment, learning and focused research.
Our contribution was brand identity creation. A simple approach was to merge the cultures of these countries with easy recall ability.
The brand identity shows the national bird of Germany (Golden Eagle) and India (Peacock). The logo embarked global simplicity and significance.
Result : The significant was identified for its genuine representation and showing cultural values through icons and colors. | http://signatures1.com/works-esslingen.html |
Stellar Wind, 2015 Virginia-bred Horse of the Year, will face powerful Beholder Saturday in Del Mar’s 8th race, scheduled to go off at 8:33 PM EDT. The 4 year old daughter of Curlin will try to become the latest Virginia-bred million dollar earning horse. She enters the stakes with a lifetime bankroll of $993,200 from just 9 starts. In her only ’16 outing, she was runner-up to Beholder in Santa Anita’s Vanity Stakes. Stellar Wind, who was bred by Keswick Stables and Stonestreet Thoroughbred Holdings LLC, is out of the Malibu Moon mare, Evening Star. Victor Espinoza will ride for trainer John Sadler.
Virginia-bred Rapid Rhythm will compete in Sunday’s $150,000 Royal North Stakes (Gr. 3) at Woodbine while Virginia-owned Disco Barbie has entered the $100,000 Caress Stakes at Saratoga the same day. The former is a 4 year old by Successful Appeal out of Patriot Miss by Quiet American and was bred by the Lazy Lane Farms. The latter is a Kentucky bred that is trained by Gary Capuano and is owned by Virginian Diane Manning. The Woodbine stakes is the 8th race on that card and the Caress is the 10th at Saratoga.
Sunday’s highly anticipated $1,000,000 Haskell Invitational (Gr. I) at Monmouth pits Kentucky Derby winner Nyquist against Preakness winner Exaggerated. Four other graded stakes are on tap at Monmouth that day in their biggest program of the year. The Haskell is the 12th race and goes off at 5:47 PM.
Saratoga has a powerhouse card Saturday that includes the $500,000 Alfred Vanderbilt Handicap (Gr. I), $600,000 Jim Dandy Stakes (Gr. 2), $250,000 Bowling Green Handicap (Gr. 2) and $200,000 Amsterdam Stakes (Gr. 2). The Vanderbilt has been carded as Race 9 at 5:40 PM and the Jim Dandy follows as Race 10 at 6:18 PM.
For harness fans, Saturday’s focus is at The Meadows just outside of Pittsburgh, where a 17 race card is highlighted by the $500,000 Adios, a 3 Year Old Open Pace which is scheduled as Race 12 (3:40 PM). Post time is 12 Noon.
Watch. Wager. Win! | https://www.virginiahorseracing.com/2016/07/30/july-30-31-weekend-racing-preview-stakes-galore-with-derby-preakness-winner-15-va-bred-horse-of-the-year/ |
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Taking pictures of the community at special events. www.Unitedwestandforpeace.org
29 More opportunities with Peace Institute1 Review
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To facilitate and coordinate outreach efforts in the community by promoting understanding and acceptance among people of all faiths and all ethnicities. With that in mind, we believe that education builds acceptance and understanding. The Peace Institute continues to bring people for dialogue and peace by providing accurate information about diverse topics, developing a platform for critical thinking and promoting community engagement through service.
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The Peace Institute was founded to respond to the need for accurate information about diverse groups and interfaith awareness after the tragedies of 9/11, and works to promote understanding and acceptance amongst people of all faiths and backgrounds.
Currently, the Institute focuses on bringing awareness about the social issues that face the general community. | https://www.volunteermatch.org/search/opp456332.jsp |
What your current position?
PhD student, Imperial College London
The A-Levels (or equivalent) that you did?
Physics, Chemistry, Mathematics, Economics, General Paper and Mathematics Special Paper (Singapore A-levels)
Why did you choose a career in Materials Science and Engineering (MSE)?
I chose to do a Materials Science and Engineering (MEng) undergraduate degree because I couldn’t decide between engineering, physics and chemistry, and a Materials degree is all three of these combined.
In my third year, I got involved in research projects and realised I enjoyed research a lot -this inspires me to continue with a PhD in the same field.
What did you enjoy most about MSE course?
I really enjoyed how my course allowed me to specialise in a particular field of materials science.
I chose to specialise in Aerospace Materials which allowed me to use my materials knowledge in aerospace applications. I got to do research for Rolls-Royce plc and one of my modules in my final year gave me the chance to design an actual passenger aeroplane!
What is your research about?
I research 3D printed titanium alloys for biomedical applications. Specifically, I work with a company who manufactures 3D printed titanium implants and bone replacements. My project in particular aims to determine the corrosion mechanisms of these implants when exposed to fluid in our body.
What is the coolest thing you have done in your career so far?
Doing outreach! I never knew it is so fun to share cool science with people of all ages! I have played with cornstarch and water (a non-Newtonian fluid) with five year olds, performed toughness experiments on different chocolate bars with teenagers and demonstrated shape-memory paperclips to adults, among others.
It’s such great fun to talk about science to curious children, teenagers and adults alike. I even have ongoing outreach projects including one where we are planning to make a spacecraft (model, of course)!
What do you see yourself doing in the future?
At the moment, I don’t know (yet). However, I currently enjoy helping with teaching undergraduate students, so teaching is something I am considering. If not, I can also see myself continuing with research in the future, perhaps working in the research and development (R&D) section of a company.
What is your favourite material (and why)?
Since I am currently working with titanium, I don’t know if I am obliged to say titanium -but definitely metals!
Metals are used in so many different things, from cans, batteries and electrical wires to steel bridges, blood vessel stents and magnets. They have very unique properties such as shape-memory capabilities (think planes that can change wing shapes to optimise aerodynamics) and chromicity (think colour-changing dyes).
What advice would you give your 16-year old self?
Take it easy, and stay curious! | https://discovermaterials.co.uk/discover-materials-ambassadors/jessica-tjandra/ |
Falling business confidence, record high petrol prices and stabilising house prices in larger cities are just some of the headlines dominating national media, and while there are concerns the national economy is cooling off, it is still growing strongly. GDP rose by 1 per cent in the three months to June 2018, the strongest quarterly growth in two years and quarterly retail card spending in the September 2018 quarter rose at its fastest pace in seven and a half years, Stats NZ said.
Palmerston North and Manawatu’s regional economy is strong and growing, and business and consumer confidence is also on the rise. This sentiment is driven by strong export trade conditions, record high house prices supporting rising consumer spending and a great deal of construction activity that’s showing no signs of slowing. But that’s not to say Manawatū is immune from the weakening national confidence and barriers to growth like labour skills and house shortages, the pain of which are already being felt.
To get insights into what’s currently happening and factors that affect the region’s economy, we asked Palmerston North City Council’s Economic Policy Advisor Peter Crawford (PC) and Manawatū District Council’s Economic Development Advisor Stacey Bell (SB) to share their insights.
What’s driving the region’s economic growth?
SB: The regions are the powerhouse of the New Zealand economy. The city’s economic growth increased by 3.4 per cent alongside the district’s growth of 4.1 per cent, in the year ended June 2018. This compares with the national economic growth of 2.7 per cent. While Infometrics indicates strong growth across rural New Zealand due to export trade, levels of growth in Palmerston North now exceed both national growth and growth of our largest urban centres. This is good news for the Manawatū Region!
PC: Record levels of building activity in Palmerston North is really driving growth in the city. $293 million worth of building consents (for new housing and non-residential) were approved in the year to August 2018, well above the previous peak of $221 million the year to April 2008.
SB: Construction investment peaked in the district in the year ended January 2018 at $97.7m. This included $14.5 million planned investment in the New World complex in Feilding, which is now complete. Coming off this high base, construction investment remains strong in historical terms with a total of $80.9 million in building consents issued to the year ended August 2018.
How does this peak compare to previous years and what trends are you seeing as a result?
PC: Consent values in the city were just $106 million in the year to August 2014 alongside $41 million in the district to the year ended April 2014. This rapid increase in building activity over the past four years has led to demand for more labour. To meet construction demand, building businesses from outside the region are coming here to take up work opportunities. This growth in employment and workers is adding pressure to an already tight housing market in the wider region, which is driving the city and district’s house prices up, while national growth in house prices is slowing.
The region’s population is growing, what insights can you share?
SB: The Manawatū region’s population has grown by an estimated 1,900 people, or 1.6% in the year to June 2018, according to Statistics New Zealand. The district has seen an increase in its population of 600 people. This is a 1.9% increase in population in the year to June 2018. The strong rate of population growth across the district is driven by employment growth, increasing employment opportunities and household incomes, high quality schools and social infrastructure alongside proximity to the labour markets of Palmerston North, and specialist health services of MidCentral District Health Board.
PC: For the city, this is the strongest growth in population since the early 1990’s, with an extra 1400 people now living here, in the year to June 2018.
View the 2018 annual population growth report here.
Housing affordability is a drawing card for the region – are rising house and rent prices putting the city and district at risk of losing that?
PC: QV data shows that average house prices in Palmerston North in the three months ended August 2018 were 35 per cent higher than three years ago, while national house prices increased by 26 per cent over this period. However, the dollar value increase in the city was smaller than the national increase, due to a lower starting point in 2015. Over the three years, average house values in Palmerston North increased by $104,000 while national house values increased by $138,000.
SB: The figures are similar for the district, with house prices rising 41.8 per cent to the three years ended August 2018. The average price of houses in the district are now $102,000 higher than they were three years ago. The increase in house prices reflects the current strength of economic conditions in the region. However, it also signals an under supply of housing relative to strong demand. Palmerston North City Council is aware there is a shortage of land for housing development and is working with developers to release more land to the market. The Manawatū District Council has a lead programme in place to roll out key infrastructure over the next 20 years. This initiative will support the staged development of key residential and industrial growth areas in and around Feilding.
Manawatū district’s economy is underpinned by farming, primary production and manufacturing. Is the district’s economy relatively stable given strong dairy and meat prices and the low NZ dollar?
SB: Growth in the district’s economy remains strong due to the above factors. Supply conditions have further supported export incomes with volumes up to the year ended August 2018. As a result, we are seeing strong labour market conditions, the continuation of strong housing demand and house price growth, and strong growth in consumer spending. Concerns include the impact of petrol prices, future trade and global demand conditions, and changing central government policy. In the current climate, the risk of rising international interest rates is also raising concerns for heavily indebted households and farming operations.
Domestic tourism spending has softened in the city, why?
PC: Growth in domestic tourism spending has been weak in Palmerston North in the first half of 2018, but that followed very strong growth in late 2016 and early 2017 due to the impact of the Kaikoura earthquake on retailing in Wellington’s central city and Lower Hutt. There was a 16.5 per cent increase in Wellington region visitor spending in the year ended August 2017, but a 4 per cent decline in the year to August 2018.
Strong growth in international visitor spending – where are these visitors from?
PC: Both the city and the district have seen strong growth in international spending from the rest of Asia (other than China, Japan and Korea), the rest of Europe (other than the UK and Germany), the rest of Americas (other than Canada and the USA) and the rest of Oceania (other than Australia).
Growth from ‘Rest of Europe’ has slowed but China and ‘Rest of Asia’ have grown strongly over the past 4 years, particularly China. The UK and USA used to be the biggest markets after Australia but have been overtaken by China and ‘Rest of Asia’.
For more information:
[email protected]
06 350 1830
Or, | https://ceda.nz/latest-news/manawatu-economic-update-september-october/ |
Oklahoma State University Center for Health Sciences, which is located in Tulsa, OK is a public university. The university covers an area of 16 acres. It was in 1972 when the Oklahoma State University Center for Health Sciences was founded. Notably, the official colors of the university are orange and black. The school has a total of 95 (full time) 611 (part time) academic staff. These faculty teach different courses to the students. Several notable personalities have been produced by the school in its 48 years of existence. OSUCHS has produced some well-known personalities over the years. The school believes in the all-round growth of the students and therefore, encourages students to take part in several sporting activities, conducted inside the campus. A total of 22 programs are currently offered by the varsity to the students. It includes both the residential as well as online programs. The university also has a long list of accreditations. It has been accredited by Higher Learning Commission.
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Address1111 W 17th St
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Phone: 9185821972
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It becomes easier for the aspirants to apply to a university and have their doubts regarding the same cleared if the university has a ranking. Usually, the rankings of colleges/universities by Forbes and US News are considered trustworthy by applicants and experts. However, Oklahoma State University Center for Health Sciences doesn't feature in the list of top colleges, according to the latest rankings of these two publications. Generally, the ranking of a university is a great way to gauge its performance on several factors like research excellence, the performance of its students after graduation, student experience, and academic success.
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Accept Common App
FALSE
The students can make the most out of the admission process as the admission at the university is fairly easy. Along with the admission form, the students have to deposit a non-refundable fee to complete the application process. As a general rule, securing a good score in a standardised test like SAT or ACT improves an applicant’s chances of securing admission. However, at Oklahoma State University Center for Health Sciences, the average SAT or ACT scores of students securing admission is not available. As expected, the candidates can complete the process either online or by visiting the school's office at 1111 W 17th St, Tulsa, Oklahoma 74107-1898. To gather more information, they can call them at (918) 582-1972.
Annual Cost to Attend
The next two things to keep in mind before applying to OSUCHS are the tuition fees and the cost of living during the course tenure. If it sounds unaffordable to the applicant, there are numerous financial aid options available that they can explore.
Academics
Highest Degree Offered
Doctor's degree
Student:Faculty Ratio
NA
Full-Time Retention Rate
N/A
Part-Time Retention Rate
N/A
Academic Calendar
Semester
Research Funding per Student
N/A
Faculty Overview
Male Professors
Female Professors
Non-traditional Learning
Evening Degree Programs
Teacher Certification
Distance Education
Study Abroad
Student Body
Undergraduate Students
NA
Socio-Economic Diversity
N/A
Full-time
N/A
Part-time
N/A
Students Gender
Male
N/A
US National: 44%
Female
N/A
US National: 56%
LGBTQIA STUDENT RESOURCE GROUPS
ON CAMPUS WOMEN'S CENTER
Race/Ethnicity
0% White
0% Unknown
0% Black
FALSE% Hispanic
0% Two or more races
0% Non Resident alien
0% Asian
0% American Indian/Alaska native
0% Native Hawaiians/Pacific islander
Geographic Density
Geographic Density
High
Unknown Location
Out Of State
In State
Foreign
Economic Diversity
36% percentage of students who recieved an income-based Federal Pell Grant intended for low-income students.
Median Household Income
$56,744 per year
Campus Life
Housing
The institution does not provide on-campus housing facilities for students.
On Campus Housing Available
No
Freshmen Required to Live on Campus
Not applicable
Freshmen Live on Campus
N/A
Undergrads in College Housing
N/A
Averege Housing Cost
N/A
Campus Food
The institution does not provide any meal plan options.
Meal Plan Available
No
Average Meal Plan Available
N/A
Athletes
Division Sports Nickname
N/A
School Colors
N/A
Varsity Athletics Association
N/A
Varsity Athletics Conference Primary
Not applicable
Total Male Athletes
N/A
Total Female Athletes
N/A
Intramural Sports
N/A
Sports Club
N/A
Campus Safety
24-Hour Security Patrol
No
Campus Emergency Phones
No
24-Hour Escort Safety rides
No
Mobile Campus Emergency Alert
No
Since the campus life of the students assumes an imperative part of their development and advancement, students at OSUCHS invest a lot of their time learning from the faculty members. Additionally, the students have an incredible breadth to learn and turn into a change-maker. The university has enrolled around 49% male and 51% female students.
After Graduation
Median Earnings 6 Years After Graduation
PrivacySuppressed
Median Earnings 10 Years After Graduation
$241,900/year
Typical Monthly Loan Payment
N/A - N/A
Financial Aid
The financial aid department of Oklahoma State University Center for Health Sciences helps students by assisting them with scholarships, grants, and student loans in the event that they find the fees and the cost of living at the campus to be expensive. | https://www.thecollegemonk.com/colleges/oklahoma-state-university-center-for-health-sciences |
It is no surprise that funding for research—and for health research in particular—is declining. Whether lower funding is the result of the health care crisis, sequestration implications, or both, federal agencies, research institutions, and foundations increasingly have to prioritize projects and make difficult fiscal decisions. Despite the challenges that are associated with more limited funding, Russell Glasgow of the National Cancer Institute believes that one silver lining is that the tough times are forcing people to think more in terms of a “systems approach” to health care. Across government agencies, non-profit organizations, academic institutions and foundations, groups are attempting to harmonize the measures and data they have and making those data sources publicly available. In the Sunday “Funding Priorities” meeting track, funders explained how their respective agencies are prioritizing projects, while attempting to lower costs:
- National Institutes of Health - Robert Kaplan, in "The Health Services Research Agenda and Funding Opportunities at the National Institutes of Health" session said that, in the social sciences portfolio of NIH, there are five subcategories that have been chosen for closer focus: prevention, social epidemiology, measurement development, decision science, and mHealth. When asked how NIH is prioritizing activities after the sequester, Kaplan said there was an intention to focus more on patients with multiple chronic conditions, and he's hopeful that NIH will see more of this work in the years to come. At the National Heart, Lung, and Blood Institute (NHLBI), the director has made clear that the number one priority is traditional R01s, or investigator-initiated grants. Steven Clauser of the National Cancer Institute (NCI) also echoed the importance of investigator-initiated research. At NCI, discretionary spending, such as travel, is being cut drastically to honor the commitment to research. Institution grants will be scaled down. In addition, NHLBI special advisor George Mensah is attempting increase the Institute’s work in implementation research, while still keeping the overall budget down. As for current funding opportunities, NIH recently announced a trans-NIH Dissemination and Implementation Research in Health grant. Participating institutes include NIMH, NIDA, NCI, NIAID, NHLBI, and NNR. Russell Glasgow urged all attendees interested in dissemination and implementation work to look into this opportunity.
- Foundations - Several of the panelists in “Foundations’ Research and Policy Agendas” articulated that their respective organizations were working in the health reform. Kimberly VanPelt of St. Luke’s Health Initiatives said a big focus of the organization was the implementation of the Affordable Care Act (ACA) and moving Arizona forward with Medicaid expansion. Lisa Sugarman of The SCAN Foundation also indicated her organization was working on the integration of Medicare and Medicaid. Debra Perez of the Robert Wood Johnson Foundation, noted that although RWJF’s portfolio is diverse, there is currently a focus on quality/equality, big data, and utilization. Finally, Judith Meyers, Child Health & Development Institute, said a focus area for her organization has been on integrated tools, specifically in mental health. The overarching mission of the organization - communicating research to policymakers - is still the priority, and the Child Health & Development Institute is currently working with academic foundations and other institutions to continue that mission, moving research from “the bedside to the community.”
- Center for Medicare and Medicaid Innovation - CMMI, according to Gerald Riley, Office of Research, Development, and Information at CMS, in a session “Center for Medicare and Medicaid Innovation Research Agenda,” highlighted the many programs in which the center is currently involved. Current projects include Partnership for Patients and models that target the Medicaid programs, such as the Strong Start for Mothers and Newborns Initiative. One current funding opportunity is the second round of Health Care Innovation Awards, which fund “projects from across the country that test new payment and service delivery models that will deliver better care and lower costs for Medicare, Medicaid, and Children’s Health Insurance Program (CHIP) enrollees.” Letters of intent for the project are due on June 28 by 3:00 p.m. | https://academyhealth.org/blog/2013-06/arm-highlights-funding-priorities-sessions |
Siletz Bay's coastal habitats include salt marsh, mudflats, sloughs and conifer-hardwood forests, all of which are essential to shorebirds, waterfowl, wading birds, and Coho and Chinook Salmon.
Salmon migrations constitute a wave of nutrients that ebbs and flows like a piscine tide, bringing nourishment to animal and plant alike.
Forested wetlands at Siletz Bay NWR are host to an evolutionary arms race between two common but uniquely adapted critters.
The dam-building Beaver is perhaps the most famous aquatic rodent, but just as industrious (and fascinating) is the homely Muskrat.
Images of life found in the marshes, tidal sloughs, mudflats, and coniferous and deciduous forestland of Siletz Bay NWR.
In many ways, salmon are the lifeblood of the Pacific coast. Their annual migrations inland and seaward amount to a wave of nutrients that ebbs and flows like a piscine tide, bringing nourishment to animal and plant alike. For the former, it is the flesh of these vimful fish that sustains them—even enriches them, as is the case with humans. For the latter, it is rather the elemental constituents of that flesh, liberated by decay and repurposed as fertilizer for lush forests.
The largest of North America's four elk subspecies, Roosevelt Elk are an impressive and conspicuous resident in Oregon. Huge herds migrate from forested slopes to coastal flats in winter. Watch for them at lower elevations in fields and clearings, browsing on vegetation or simply lolling about.
Eelgrass beds form one of the many estuarine habitats at Siletz Bay, sustaining all manner of life from the grassroots on up. Often mistaken for "seaweed", eelgrass is not algae but a true flowering plant, or angiosperm.
Siletz Bay National Wildlife Refuge is managed as part of the Oregon Coastal Refuge Complex.
Page Photo Credits – Snake and newt - ©Richard Greene, Roosevelt Elk on beach - ©Terry W. Smith, Purple Martin - ©Tom Benson, All photos courtesy of USFWS unless otherwise noted. | https://www.fws.gov/refuge/Siletz_Bay/ |
Proposal for Export Marketing Group
By Ray Kronquist, President of Virtual Classrooms.
While I was recently in Russia, in Akademgorodok, the academic community where most of my Virtual Classrooms online tutors live, I came across what I feel is a great business opportunity. Here is a summary:
Proposal Summary:
I am proposing an alliance between my company, Virtual Classroom, the institutes and companies of Akademgorodok, the Economics Department of Novosibirsk State University (NSU) and U.S. business schools. Students at these U.S. business schools will take on the task of researching and developing the U.S. market for these products and services. This will provide American business school students with entrepreneurial experience that incorporates international business and cutting-edge technology and could generate income for the students before and after graduation.
Akademgorodok, established by the Russian Academy of Sciences in 1958, is located in Novosibirsk, the third largest city in Russia. A great many technically advanced products and services have been developed in the research institutes and small companies of Akademgorodok. The leaders of the institutes and companies would like very much to market these products and services in the U.S., but in most cases, they lack the marketing capability to be successful in this effort.
Proposal Details, Roles of the Participants:
I propose that each of the participants be responsible for the following:
Benefits to the U.S. Business Schools:
I see the following benefits:
Keys to Success:
There are two major reasons why this venture is likely to succeed:
Investment Required:
No investment on the part of the U.S. business school would be required to start this program, other than the time of a professor to negotiate the terms of the project and to supervise students working on it. Students would need Internet access and the capability to make phone calls from an office at the school. In the future, it may be attractive to arrange financing for one or more of the Akademgorodok ventures, but this will be down the road when much more information about the products and their sales potential will be known.
Background Information:
Here is some additional information that should help you to understand community of Akademgorodok and the opportunity it offers.
Akademgorodok was set up with a university, Novosibirsk State University (NSU) and a number of research institutes, all of which have close relationships with one or more of the academic departments in NSU. Currently there are a total of 44 institutes, some of which concentrate on pure research, and others which concentrate on applied work in science and technology. Most of the land and buildings in Akademgorodok are the property of the Siberian Branch of the Russian Academy of Science (SBRAS).
NSU is one of the most prestigious universities in Russia. Normally only 10% to 30% of the students who apply are admitted. Faculty members normally work also at one of the institutes. Students usually do their graduate theses at one of the institutes. Thus, the university and the institutes constitute a very large academic and research complex covering most fields of science and technology.
SBRAS has an exhibit displaying about 300 of the innovations, products and services that have been developed at its various institutes. Sections of this exhibit are taken to trade show from time to time. The institutes that have developed these innovations are seeking to market them within Russia and abroad. Some are products ready to be sold or licensed. Some need further development, and the institutes are seeking investors or joint venture partners to fund the development.
In addition to the innovations developed by the institutes, there are a number of software development companies in Akademgorodok who are seeking to sell their services abroad. I visited seven of these companies during my visit, and they range from start-ups with a half dozen people to the largest with 400 employees. The areas of expertise cover game developers, E-commerce, web site development, PDA software and more.
All the software companies I visited were very interested in my help in expanding their sales in the U.S. The people at the institutes are also very interested in sales of their products in the U.S.
Some of the institutes have been very successful in selling their products, such as the Institute for Catalysis. However, many institutes have not done an effective job in marketing their products.
During my visit, I met with the Chair of the Economics Department at NSU and proposed that they set up a marketing group to work with Virtual Classrooms on marketing in the U.S. the products developed by the institutes and the services of some of the local software companies. He was very favorable to the idea, and he arranged a meeting for me with three of the professors in his department that afternoon. They also liked the idea and expressed a willingness to put four graduate business students to work on the project.
I am now seeking business schools in the U.S. who are interested in collaborating with Virtual Classrooms and with the Economics Dept. at NSU in this marketing program.
I feel that business students who work on this project would learn a great deal in a real entrepreneurial environment. Also, there would be an opportunity for the students and the business school to benefit financially due to commissions on successful sales.
Details of U.S. Students Responsibilities:
If this project is of interest to you, please give me a call, 408-929-9066, or send me an e-mail, [email protected] with your thoughts. | http://virclass.com/marketing.html |
Why Go On A Retreat?
Stillness is an essential part of developing a relationship with God. He is always present in us, but many times we’re simply too busy and distracted to sense His presence and hear His voice. When we slow down and eliminate distractions, we can enter into a deeper relationship with Him that renews and strengthens us.
Stillness and peace are not a function of separation. Retreats help us focus our attention on God and build our spiritual capacity to hear His voice in the midst of our day-to-day lives. This capacity allows us to cope with our daily challenges from a position of strength, confidence, and surrender to our Holy Lord. Spiritual renewal and maturity are not ends unto themselves. They are the fuel that we need to minister and serve without exhaustion, in God’s name.
Writings of great Christians throughout history suggest a consistent practice of getting away from the busyness of life to reflect on Scripture and talk to God. Jesus himself (Matthew 26:36-44; Mark 1:35; Luke 5:16, 6:12, 9:18) we are told often separated himself from the crowds that followed him and even from his disciples to talk to His Father. At the beginning of his Christian ministry Paul went to Arabia for three years (Galatians 1:17-18) and some say during this time he was reflecting on his conversion experience and building his relationship with the Lord.
In the early Church it was not uncommon for the spiritual leaders and guides to spend years in the desert. These Desert Fathers and Mothers sought God’s voice and teachings and were sought out by others seeking to grow in the new movement known as Christianity.
Fourteen and fifteenth century Western Europe was alive with Christian experience. Many of those who would eventually become Saints of the Catholic church sought to separate themselves, temporarily or permanently from the pressures of the world so that they could hear the voice of God. St. Teresa of Avila, Julian of Norwich, St. John of the Cross and others lived a life of solitude and devotion. Their experiences and writings have, over the centuries, inspired others to grow in their faith through stillness and solitude.
In more modern time, Thomas Merton, Evelyn Underhill, Marjorie Thompson, Richard Foster, Ruth Haley Barton and many others have written about and led others into stillness and silence as a devotion practice. The history and practice of Christian stillness is rich and deep, but in today’s world where busyness, efficiency, and accomplishment have become idols, it is not often comfortable or sought out.
We have become a society of doers. Our worth is often determined by how much we get done, how much we produce, how much we contribute. Even in the church idleness is often equated with laziness or lack of worth. Even pastors and others in religious life struggle to take time to pray and build their relationship with the Lord.
The experience of being still can be difficult at first because we are addicted to activity and outcomes. Sitting still, listening for God’s voice, enjoying God’s creation, and even taking a nap or a walk feel like frivolous luxuries that we can’t afford. We tell ourselves that we are wasting time doing these things because we can’t measure the value of the time spent. It is only when God changes our perspective and we see that who we are in Him is more important than what we do (even for Him), that we begin to appreciate and crave Him in stillness.
God values the time we spend with Him. He grieves time not spent in relationship with us and He seeks to draw us to him. In Psalm 139: 23-24, David writes: “Search me, God, and know my heart; test me and know my anxious thoughts. See if there is any offensive way in me, and lead me in the way everlasting.” This is the cry of a man who is seeking God’s direction. Many pray prayers like this, but it is only when they slow down, to listen to God’s voice, that they sense his direction. Stillness slows us down, quiets our internal voices and distractions, and allows us to be with God.
About Seeking Stillness Retreats
Each Seeking Stillness Ministries retreat is unique. From pre-retreat discussions to considerable times of prayer, the retreat staff discerns God’s will for that particular retreat and designs it accordingly.
While times of teaching, discussion, and fellowship are part of each retreat, the majority of your time at a Seeking Stillness retreat is spent in prayer and discussion with God. This is your time with the Lord. The retreat staff is available throughout to provide guidance and support, but participants are free to spend the retreat time in whatever way they feel will help them draw closer to God.
The retreats are held in places that provide the space and environment for stillness. Participants are encouraged to spend time outdoors, in quiet spaces, and in the Word.
All meals and accommodations are provided through the retreat registration fee.
Partial scholarships are available upon request. | https://seekingstillness.org/retreats/?shared=email&msg=fail |
What does acorn taste like
Can acorns be eaten by humans?
Acorns can be used in a variety of ways. They can be eaten whole, ground up into acorn meal or flour, or made into mush to have their oil extracted. Once you’ve safely leached the tannins from your raw acorns, you can roast them for 15 to 20 minutes and sprinkle them with salt for a snack.
What does acorn meal taste like?
Pre-leaching, the acorn flour initially tastes sweet, almost like maple sugar until the tannins flood in. The aftertaste is disconcertingly bitter, like a coated pill held too long on the tongue. Post-leaching, both the sweet and bitter tastes fade away.
Can you die from eating a acorn?
Untreated raw acorns contain high concentrations of tannic acid, causing their taste to be bitter and them to be toxic to humans if eaten in large quantities. It looks like eating raw acorns won’t kill you but its not likely to be pleasant and can harm you.
What does an oak tree taste like?
The high tannin amount in Southern Live Oak acorns makes them bitter, but the bitterness can be removed by leaching the tannins in either a cold or hot water process. Acorn meat tastes somewhat like a chestnut, with a nutty sweetness.
What can I do with fallen acorns?
5 Creative Uses for Acorns
- Make a rustic wreath. Get a simple foam wreath form and gather dozens of acorns.
- Use as a vase filler. Buy clear vases in assorted sizes and fill them with acorns.
- Feed your feathered friends.
- Donate them!
- Start seedlings.
Are oak trees poisonous to humans?
Oaks at any stage of growth are poisonous, but are particularly toxic when the leaf and flower buds are just opening in the spring. As the leaves mature they become less toxic. Ripe acorns are less toxic than when green.
Is Oak Tree sap poisonous?
Oaks are very tannin-heavy, and while I can’t find anything that actually says this, I would not be the least bit surprised to find out that oak sap has lots of tannins in it, which would make it at best very bitter and at worst toxic, especially once boiled down to a syrup. They do produce sap.
Is Oak toxic?
Most species of oak (Quercus spp) found in Europe and North America are considered toxic. Clinical signs occur 3–7 days after consumption of large quantities of young oak leaves in the spring or green acorns in the fall. Fallen trees associated with a recent storm are often reported with outbreaks.
Is oak tree fruit edible?
The acorns of the Oak Tree are the only edible part. They contain tannin which is a bitter chemical that prevents animals from eating them to abundantly.
How can you tell if an acorn is edible?
Roasted Acorns
Place the damp nut chunks on a baking sheet and sprinkle with fine salt. Toast them for 15-20 minutes at 375 degrees in a pre-heated oven, or roll them around in a dry frying pan over the camp fire. You can tell they’re done when the color has changed a little, and the nut pieces smell like roasted nuts.
Who eats oak trees?
What Eats Oak Trees?
- Birds. The acorn woodpecker relies on oak woodlands on the West Coast and in the Southwest.
- Mammals. White-tailed deer forage heavily on acorns in the autumn.
- Grizzly Bears. Grizzly bears historically indulged in acorns on the West Coast and in the Southwest.
- Feral Hogs.
Are oak leaves good for you?
One of the best things organic mulches such as oak leaves do is recycle nutrients and organic matter back into the soil. Florida gardens are typically created on sandy soil with low organic matter content. An important part of the soil, organic matter primarily comes from plant and animal residues.
Are oak leaves bad for your lawn?
Burning leaves contributes to our poor air quality and also removes a valuable nutrient resource from your yard. What you have heard about oak leaves is somewhat true in that they have high levels of tannins that will slow the decomposition process, but they can still be composted.
Can you plant under an oak tree?
Some California Native plants thrive under the shade of oaks, including Coffeeberry (Rhamnus californica) and many native ferns, especially Sword Fern (Polystichum munitum) and Giant Chain Fern (Woodwardia fimbriata). Heuchera does beautifully in dry shade and can brighten up your oak understory.
Are oak tree leaves poisonous to dogs?
Not only are they a choking hazard, but oak acorns and young oak leaves also contain a chemical called gallotannin, which can cause severe gastrointestinal distress in dogs, including vomiting, lethargy, diarrhea, and damage to the liver and kidneys.
Will one acorn hurt my dog?
Acorns are poisonous if eaten by dogs. They contain tannins, and possibly other compounds, which can cause stomach upset and in very severe cases, kidney failure and death. They are also hard and sharp and can cause an internal obstruction if ingested.
What happens if a dog eats an acorn?
If a dog has eaten an acorn, symptoms can include stomach discomfort, vomiting and diarrhea. More severe poisoning may occur in smaller dogs or dogs who have eaten a larger quantity of acorns. Because they are hard and sharp, acorns can also cause obstruction and internal damage.
Can one acorn kill a dog?
Firstly, acorns contain a chemical called gallotannin. This can make your dog seriously unwell. Occasionally, it can prove fatal. Thirdly, if your pet manages to eat an excessive amount of acorns, they can cause an obstruction in your dog’s digestive tract. | https://peoplequestions.com/sb/what-does-acorn-taste-like/ |
Magnesium is a naturally occurring mineral that is important for many systems in the body, including energy metabolism, DNA production and bone structure. When magnesium is coupled with other elements, a wide variety of uses results. For example, magnesium oxide (CAS 1309-48-4) and magnesium citrate (CAS 7779-25-1; 3344-18-1 (anhydrous)) are two compounds frequently used in human consumption.
Though both compounds are used as dietary supplements and osmotic laxatives, magnesium oxide is poorly absorbed by the body. Around 4% of its elemental magnesium is absorbed, equivalent to about 9.5 mg out of a 400 mg tablet (with 60% elemental magnesium). On the other hand, magnesium citrate is much better absorbed by the body than magnesium oxide. It appears to have a bioavailability of 25-30%, meaning that between 25 and 30% of the compound reaches the systemic circulation. For this reason, magnesium citrate is the most common type of magnesium supplement and is the recommended choice for most issues and deficiencies.
In addition to dietary supplements and laxatives, magnesium oxide is also used as a heat-resistant material. For example, magnesium oxide wallboards are very appealing in construction due to their ability to withstand fire, termites, moisture, mold and mildew. Another niche use of magnesium oxide is as an antacid to relieve heartburn, a sour stomach or indigestion.
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The school lunch lady (known as just "Lunch Lady") is a secret crime fighter who uncovers evil plots. Hector, Terrence, and Dee are three school children in on her secret, and help her out whenever they can.
2) Ms. Marvel
Author
Series
Language
English
Appears on list
The complete first season of Marvel's international sensation between two covers! Kamala Khan is an ordinary girl from Jersey City - until she suddenly gains extraordinary gifts. But who is she now? Teenager? Muslim? Inhuman? As Kamala explores her newfound powers in the guise of the new Ms. Marvel, she takes on the maniacal Inventor, teams up with Wolverine and Spider-Man, meets the Inhuman royal family, adopts their teleporting dog Lockjaw, faces...
Author
Pub. Date
2016
Language
English
It was the night before finals and the student body is hard at work... and nothing is going right! Wonder Woman, Supergirl, Harley Quinn, Batgirl and their friends are learning to become heroes, but no one knew the trials that awaited them. In the first original graphic novel from the DC Super Hero Girls line, meet the students of Superhero High School as they find out that fun, friendship and hard work are all parts of growing up!
4) Spider-Gwen
Author
Language
English
"The breakout star of Spider-Verse takes the comics world by storm with her own series! Gwen Stacy is the Spider-Woman of her world, but you knew that already. What you don't know is what's waiting for her when she comes to the other side of the Spider-Verse! On the most tragic day of her life, Gwen was convinced that the Lizard died in her arms along with Peter Parker. But when a new reptilian rampage begins, she is left in doubt...not only about...
Author
Series
Unbeatable Squirrel Girl volume 4
Marvel now!
Marvel now!
Pub. Date
2015-
Language
English
"Wolverine, Deadpool, Doctor Doom, Thanos: There's one hero that's beaten them all-and now she's got her own ongoing series! (Not that she's bragging.) That's right, you asked for it, you got it, it's Squirrel Girl! (She's also starting college this semester.) It's the start of a brand-new set of adventures starring the nuttiest and most upbeat super hero in the world!"--provided from Amazon.com.
Author
Series
Moon Girl and Devil Dinosaur volume 8
Pub. Date
-
Language
English
"Lunella Lafayette is an inhuman preteen genius who wants to change the world! THAT JOB would be a lot easier if she wasn't living in mortal fear of her latent inhuman gene. There's no telling what she'll turn into -- but Luna's got a plan. All she needs is an Omni-Wave Projector. Easy, right? That is, until a red-scaled beast is teleported from the prehistoric past to a far-flung future we call ... today! Together they're the most Marvelous Team-Up...
7) Faith
Author
Pub. Date
2016-
Language
English
Orphaned at a young age, Faith Herbert - a psionically gifted "psiot" discovered by the Harbinger Foundation - has always aspired to greatness. But now this once ordinary teenager is taking control of her destiny and becoming the hard-hitting hero she's always known she can be - complete with a mild-mannered secret identity, unsuspecting colleagues, and a day job as a reporter that routinely throws into her harms way! Well, at least she thought it...
8) Thor
Author
Series
Thor (2014) volume 2
Marvel now!
Marvel now!
Language
English
Formats
"Mjolnir lies on the moon, unable to be lifted! Something dark has befallen the God of Thunder, leaving him unworthy for the first time ever! But when Frost Giants invade Earth, the hammer will be lifted - and a mysterious woman will be transformed into an all-new version of the mighty Thor! Who is this new Goddess of Thunder? Not even Odin knows, but she may be Earth's only hope against the Frost Giants! Get ready for a Thor like you've never seen...
Author
Pub. Date
Language
English
Squirrel Girl will encounter her most unbeatable, powerful and dangerous enemy--herself!
10) The mighty Thor
Author
Pub. Date
2016-
Language
English
"When Dr. Jane Foster lifts the mystic hammer Mjolnir, she is transformed into the Goddess of Thunder, the Mighty Thor! Her enemies are many, as Asgard descends further into chaos and unrest threatens to spread throughout the Ten Realms. Yet her greatest battle is against a far more personal foe: the cancer that is killing her mortal form. When Loki steps back into Thor's life, will it ease her troubles or only add to her pain? It's time to find out...
Author
Series
Pub. Date
Language
English
Eleven-year-old Diana, the gangly, sometimes clumsy, only child on the island of Themyscira, struggles to live up to the high Amazonian standards and longs for someone her own age whom she can talk to.
Author
Series
Pub. Date
Language
English
Princess Diana of Themyscira's 16th birthday celebrations are cut short when refugees break through to her island home and she defies her Amazon elders by trying to bring the outsiders to safety, but a stormy sea sweeps her away to where she must learn to survive in a foreign world full of danger and injustice.
Author
Pub. Date
2016-
Language
English
"Marinette is the sweetest girl in Paris. With a big crush on a boy at school, a big dream of becoming a fashion designer, and a big problem with being totally awkward, she's just your average teenage girl, right? Did we mention she's also the crime fighting superhero, Ladybug? Becoming Ladybug is a complicated process! First, Marinette needs a 'Kwami,' which is a tiny magical assistant (hers is named Tikki). She also needs a 'Miraculous,' which is...
14) Silk
Author
Pub. Date
2015-
Language
English
"Cindy Moon exploded into the [Marvel Universe] when we learned she was bitten by the same radioactive spider that empowered Spider-Man! It was good timing, too: she was just in time to save Peter Parker's life (more than once!) and take on the inheritors in Spider-Verse. Now, as Silk, Cindy is on her own in New York City, searching for her past, defining her own future, and webbing up wrongdoers along the way! But she's about to cross the Black Cat's...
15) Captain Marvel
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Carol Danvers has a new name, a new mission - and all the power she needs as the new Captain Marvel. After changing her superhero name, Captain Marvel tries to live up to the legacy of her fallen predecessor, while facing a time-traveling adventure that's intricately linked with the woman who inspired her military career.
16) Anti/Hero
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2020-
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"Piper Pájaro is strong and longs to be a superhero while Sloane MacBrute puts her smarts to use for her villainous grandfather, but when a mission to steal an experimental technological device brings the two 13-year-old girls face to face with each other, the device sparks and the two girls switch bodies."--Provided by publisher.
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Thirteen-year-old Dinah Lance is in a rock band with her two best friends and has a good relationship with her mom, but when a mysterious figure threatens her friends and family, she learns more about herself and her mother's secret past.
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"Even after everything that's happened, the world outside the Horde's walls is still a strange one to Adora. Each day she's learning more about her growing powers as She-Ra, including something new: the ability to heal corrupted runestones. Runestones are the magical source from which princesses like Frosta, Mermista, and Perfuma draw their power, but Glimmer knows of another runestone -- one with a dark past. Long ago a fire princess ruled the lands...
19) Captain Marvel
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2016-
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"Face front, true believers! The Captain has made her return and oh, how mighty it is. It's a bold new frontier for Carol Danvers as she soars to new heights in her greatest mission yet--leader of the all-new Alpha Flight space program. Yup, Alpha Flight. As earth's first line of defense, Carol and her team aim to protect the planet from extraterrestrial threats. But can Carol be a soldier and a diplomat? Especially when an unknown enemy emerges that...
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Teen Titans (Garcia) volume 1
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"When a tragic accident takes the life of the only family she's ever known, 16-year-old Raven is sent to New Orleans to start over. She soon discovers that she can hear the thoughts of others around her ... and another, more disturbing, voice in her head."--
Didn't find it? | https://catalog.haltomcitytx.com/Search/Results?lookfor=%22Women%20superheroes%20Comic%20books%2C%20strips%2C%20etc.%20Juvenile%20fiction.%22&searchIndex=Subject |
Certification & Information security
Increasing digitization brings new challenges in terms of information security.
The importance of certifications has increased.
The increased percentage of digital work is a challenge for IT and information security. – At the same time, more restrictive legal requirements are becoming more important. A functioning Information Security Management System (ISMS) is necessary to safeguard the confidentiality, integrity and availability of data. Certifications such as ISO 27001 or TISAX provide objective information about the respective information security maturity level of organizations.
The implementation and certification of an ISMS is also an opportunity to reduce costs by optimizing process and IT costs, to differentiate performance from suppliers and to minimize risks.
EFS’s modular system for “Information Security & Certification Support” offers different modules according to the respective requirements in order to set up an Information Security Management System (ISMS) with reasonable effort and to achieve certification.
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The EFS Information Security Check is a standardized procedure for quickly obtaining a transparent assessment of an organization’s information security maturity level. It creates the base for developing or improving the Information Security Management System (ISMS).
Modules:
- Analyzing and evaluating information security measures
- Identification of gaps, risks and potential for improvement
- Evaluation of protection requirements
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Together with customers – and from the perspective of the business units – we identify sensible points for setting up the Information Security Management System (ISMS) and optimizing it continuously and according to plan.
Modules:
- Identifying and implementing measures for setting up the Information Security Management System
- Designing and developing strategic and operational processes for the information security management system
- Structuring and detailing guidelines and documents
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For the organizational implementation of the Information Security Management System (ISMS), we determine the right level of control, according to the requirements of the business processes.
Modules:
- Setting up the governance framework (organizational structures, roles, documentation systems)
- Assessment and implementation of variations and tools for the technological realization of the Information Security Management System
- Creation of the communication strategy and resource planning
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We prepare the Information Security Management System (ISMS) and the organization for the audit by sensibly considering risk minimization, benefits, and effort and create the requirements for obtaining certification.
Modules:
- Planning and implementation of an internal assessment
- Identify risks, vulnerabilities and gaps regarding to certification requirements
- Develop and implement a perfect action plan
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For an efficient completion of the audit, we define an optimized preparation together with our customers in advance. – We know how to realistically assess the scope for action and how to use it sensibly in the audit.
Modules:
- Coaching and preparation for management and internal experts before the audit
- Active support during and participation in audit meetings
- Evaluation of audit reports
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Taking into account optimized lead times, we carry out a needs-based definition and implementation of measures for acceptance by the auditor and for successful attainment of certification.
Modules: | https://efs.consulting/en/certification-information-security/ |
- 1Bardez, Joan N.
- 1Bryan, Robert
- 1Dillon, Martin
- 1Fisher, Gerald
- 3Digital humanities
- 11964-2017
- 1Artificial intelligence
- 1Automatic language processing
- 1Blogs
- 1Collaboration
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Sedelow, Sally Y., Sedelow, Walter A., Ruggles, Terry, Gerbner, George, Holsti, Ole R., Krippendorff, Klaus, Paisley, William K., Stone, Philip J., Horowitz, Floyd R., Kachru, Braj B., Stahlke, Herbert, Bryan, Robert, Ford, Frank, Harris, Herbert, Taylor, Scott, Smith, Walter L., Bardez, Joan N., H. Buttelmann, William, Hickok, William G., Peters, Joan, Gerig, Thomas, Rosen, Larry, Smith, John B., Lewis, Peggy, Warfel, Sam, Dillon, Martin, Shaffer, Juliet, Joyce, Frank, Kosakowski, Thomas, Wagner, David, Wright, Harrel, Fisher, Gerald, Sawin, Lewis, Rockwell, Geoffrey, Nyhan, Julianne, Sinclair, Stéfan
This is a collection of items from the personal collection of Sally Yeates Sedelow, sent to Dr. Geoffrey Rockwell at the University of Alberta over the course of several years in the 2010s. The collection contains published journal articles, conference presentations, Sally Sedelow's C.V., working...
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The design of an international social media event: A day in the life of the digital humanitiesDownload
2012
Organisciak, Peter, Meredith-Lobay, Megan, Rockwell, Geoffrey, Ruecker, Stan, Nyhan, Julianne, Ranaweera, Kamal
A Day in the Life of the Digital Humanities (Day of DH) is a community documentation project that brings together digital humanists from around the world to document what they do on one day, typically March 18. The goal of the project, which has been run three times since 2009, is to bring...
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2012
Welsh, Anne, Nyhan, Julianne, Salmon, Jessica, Rockwell, Geoffrey
This extended interview with Geoffrey Rockwell was carried out via Skype on the 28th April 2012. He narrates that he had been aware of computing developments when growing up in Italy but it was in college in the late 1970s that he took formal training in computing. He bought his first computer,... | https://era.library.ualberta.ca/search?facets%5Ball_contributors_sim%5D%5B%5D=Nyhan%2C+Julianne |
On transgender rights, Massachusetts and Trump administration moving in opposite directions
“Transgender could be defined out of existence by Trump administration” read the headline on the New York Times story from Monday, reporting on a leaked memo that showed the Department of Health and Human Services was seeking to define gender on a “biological basis”. On Wednesday, the Justice Department argued to the Supreme Court that discrimination against workers on the basis of their gender identity would not be illegal.
These moves follow a 2017 move banning transgender people from serving in the military, a move opposed by 73 percent of Americans in a CNN poll taken late last year. There is no public polling yet on the idea of undermining the very idea of transgender identity, as the concept was not in the public space until just this week.
Meanwhile, Massachusetts voters seem poised to send the opposite message. In 2016, state lawmakers added “gender identity” to the list of characteristics protected from discrimination in public accommodations. The law paved the way for the protection of transgender individuals in public spaces such as restaurants, parks, beaches, or bathrooms.
Opponents of the law initiated what is now Question 3 on the November ballot. Early polling by WBUR and Suffolk found the “yes” side leading, but only slightly. More recent polling has shown Yes on 3 comfortably ahead, as the campaign has heated up, and more voters have focused on the question. The issue of transgender rights is somewhat newer and less familiar for many voters, meaning more voters were open to persuasion. For similar reasons, it is likely that opinion will remain fluid for the near future.Just 37 percent of Americans say they know someone who is transgender, compared to 87 percent who knew someone gay or lesbian in a 2013 Pew poll. This sort of personal contact matters. A 2017 Pew survey found that Americans who knew someone transgender were 21 points more likely to think society had not gone far enough in accepting transgender people than those who didn’t. Support for same-sex marriage grew over time, significantly through increased personal familiarity.
When the Trump administration takes public positions, it has often had the effect of strengthening and polarizing public opinion. We’ll know in less than two weeks what effect, if any, Trump has on Question 3. | https://commonwealthmagazine.org/news-analysis/on-transgender-rights-massachusetts-and-trump-administration-moving-in-opposite-directions/ |
The word KASUGAI comes from the construction and architecture world of construction and means “a bond, a clamp”. In modern Japan, the term has been given a broader interpretation and a special meaning: kasugai is an invisible connection between people, important things, and significant events. The concept of interconnection has become a defining one for KASUGAI Architects+.
Our mission is to bring and adapt the ideas of Japanese architects to Russia, and, more broadly, to allow our customers to create their own harmonious space, taking into account the world outlook and experience that have been accumulated over the centuries in Japan.
The fact that the Japanese concept of space is not outdated and inherent only in traditional culture is very important for us. In the 21st century, the concept of understanding housing as an extension of a person’s inner world is more relevant than ever. The house became a special spiritual environment that harmonizes people’s minds and nourishes their souls.
An invisible connection between man and nature, man and his environment, is the link, the kasugai that is important for us to implement in our work. A man and his dwelling are part of a vast world. We should not defend ourselves from this world, but start a conscious, respectful, and enriching dialogue with it. | https://kasugai-plus.com/about/ |
Example: Design considerations (checklist) for restoration and rehabilitation of semi-arid lands • Have the effects of climate change and land use change been considered? • Is the use of native species being encouraged? • Are multi-species plantings encouraged? • Has the potential increased risk of invasiveness due to use of any exotics been assessed? • Have tradeoffs between livelihood needs (eg. firewood), biodiversity goals and reduction in land degradation been considered? • Has the potential increase in water demand posed by use of any exotic species been assessed? | https://present5.com/scientific-and-technical-advisory-panel-stap-advice-activities-related/ |
There are a lot of ways to earn a living. In today’s world, it’s easier than ever to make your mark and earn income at the same time. There was a time when earning potential was dictated by factors that included education, location, financial backing, and more. Nowadays, these barriers don’t have to stand between you and all the unlimited opportunities, thanks to entrepreneurialism.
Google defines an entrepreneur as a person who organizes or operates a business or businesses, taking on greater than normal financial risk to do so. While entrepreneurs do take great financial risks, they also have the potential to reap even more than financial rewards.
For only $27 you'll get all this: | https://www.planningaddicts.com/blog/entrepreneurial-education-plr |
By Russel West-Pavlov
Space in concept: Kristeva, Foucault, Deleuze seeks to provide an in depth yet succinct review of the function of spatial mirrored image in 3 of the main influential French serious thinkers of modern a long time. It proposes a step by step research of the altering position of area of their theories, focussing at the universal complex all 3 critics deal with, yet highlighting the numerous adjustments among them. It goals to rectify an unaccountable absence of special research to the importance of area of their paintings up in the past. Space in Theory argues that Kristeva, Foucault and Deleuze handle the query: How are that means and data produced in modern society? What makes it attainable to talk and imagine in methods we take without any consideration? the reply which all 3 thinkers supply is: area. This house takes quite a few varieties: psychic, subjective area in Kristeva, power-knowledge-space in Foucault, and the areas of lifestyles as a number of flows of turning into in Deleuze. This ebook alternates among analyses of those thinkers' theoretical texts, and short digressions into literary texts through Barrico, de Beauvoir, Beckett, Bodrožić or Bonnefoy, through Borges, Forster, Gide, Gilbert, Glissant, corridor, to Kafka, Ondaatje, Perec, Proust, Sartre, Warner and Woolf. those detours via literature goal to render extra concrete and available the hugely advanced conceptulization of up to date spatial thought. This quantity is aimed toward scholars, postgraduates and researchers drawn to the components of French poststructuralist idea, spatial mirrored image, or extra in general modern cultural thought and cultural stories.
Read or Download Space in Theory: Kristeva, Foucault, Deleuze. (Spatial Practices) PDF
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C. S. Lewis on the Final Frontier: Science and the Supernatural in the Space Trilogy
C. S. Lewis's celebrated house Trilogy - Out of the Silent Planet, Perelandra, and That Hideous power - used to be accomplished over sixty years in the past and has remained in print ever considering the fact that. during this groundbreaking examine, Sanford Schwartz bargains a brand new studying that demanding situations the normal view of those novels as portraying a uncomplicated fight among a pre-modern cosmology and the fashionable clinical paradigm that supplanted it.
Particularism and the Space of Moral Reasons
Particularism and the gap of ethical purposes severely assesses the startling concept that our ethical reasoning doesn't have to use ethical ideas. If we do not have rules, how will we determine what to do? This ebook examines 'moral particularism', a debatable notion on the vanguard of latest ethical conception.
Space Systems Failures: Disasters and Rescues of Satellites, Rocket and Space Probes (Springer Praxis Books)
The first actual e-book on area platforms mess ups written from an engineering viewpoint. makes a speciality of the reasons of the mess ups and discusses how the engineering wisdom base has been greater by way of the teachings realized. Discusses non-fatal anomalies which don't impact the final word good fortune of a project, yet that are disasters however.
Territory, Identity and Spatial Planning: Spatial Governance in a Fragmented Nation
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Born: 23 January 1888, London, United Kingdom.
Died: 11 July 1966, Ramsgate, Kent, United Kingdom.
Edward James Wayland, geologist and archaeologist, was the son of Edward Wayland and his wife Emily, born Street. He was educated at the City of London College, the Royal College of Science, and the Royal School of Mines (the latter two forming part of the Imperial College of Science and Technology, London), and also completed an apprenticeship in building and architecture. In 1909 he conducted geological research in Egypt as a Marshall Research Scholar in palaeontology of the Royal College of Science. Two years later he went to Portuguese East Africa (now Mozambique) to conduct geological exploration as part of the Memba Minerals Expedition. He collected some stone implements in the Monapo gravels near the town of Mozambique (the implements are now in the British Museum) and described them in 'Notes on the occurrence of stone implements in the province of Mozambique' (Man, 1915).
In 1912 Wayland was appointed as assistant mineral surveyor in the government service of Ceylon, where he also collected stone artefacts, and from 1916 to 1919 did war service in France during World War I (1914-1918). In 1919 the British government sent him to Uganda as a geological expert, where he remained as government geologist and later as the first director of the Geological Survey of Uganda. During the next 20 years he published numerous reports and papers on the geology and prehistory of Uganda, many of them on the Pleistocene and its pluvial periods and their associated Stone Age artefacts. He recorded hundreds of cave sites with Wilton industries and found evidence of profound climate changes. Among others he collaborated with L.S.B. Leakey and C. van Riet Lowe and with the latter wrote The Pleistocene geology and prehistory of Uganda (1952). He was particularly interested in the earliest prehistoric stone artefacts and believed that he had found these in the simply fractured pebbles and flakes in the earliest terraces of the rivers of northern Uganda, particularly the Kafu River. The existence of this so-called Kafuan culture has not been widely accepted and its artefacts are believed to be natural products rather than being made by humans.
In 1922 Wayland visited South Africa to study South African mining methods and compare its geological problems with those of Uganda. During his visit he found a pebble industry at Belfast which he likened to the Kafuan culture. At another site near Belfast he found artefacts which he thought equivalent to the Sangoan culture on the shore of Lake Victoria, which he had described in 1920. He and M.R. Drennan* reported on some of this work in 'Some account of a pebble industry in the Transvaal' (Transactions of the Royal Society of South Africa, 1929, Vol. 17(4), pp. 333-340). Years later he wrote 'From an archaeological notebook' (South African Archaeological Bulletin, 1950), dealing with the antiquity of humans in southern Africa.
During World War II (1939-1945) Wayland did war duty in Dover and Gibraltar and then joined a bomb disposal unit of the Royal Engineers. In 1943 he was sent to the Bechuanaland Protectorate (now Botswana) to develop water resources and served as the first director of the Geological Survey of the territory. During and after this period he published 'Drodsky's Cave' (Geographical Journal, 1944) on a cave system in Ngamiland, 'More about the Kalahari' (Ibid, 1953), and 'Outlines of prehistory and Stone Age climatology in the Bechuanaland Protectorate' (Academie royale des Sciences coloniales, Section des Sciences naturelles et medicales, 1954).
Wayland was elected a Fellow of the Geological Society of London in 1912, was a member of the Institution of Mining and Metallurgy, a Fellow of the Royal Anthropological Institute, an associate of the Royal College of Science and was honoured as a commander of the Order of the British Empire (CBE) in 1938. He was awarded the Bigsby Medal of the Geological Society of London in 1933 and the Victoria Medal of the Royal Geographical Society in 1935. In 1917 he married Ellen Morrison.
Cohen, A.B. Notes on southern African stone tool collectors represented in the British Museum (Natural History), 21 March 2001. Copy received from author.
Hall, A.L. A bibliography of South African geology. Pretoria: Geological Survey, Memoirs No. 25 (1927), 27 (1931), and 30 (1937).
Transactions of the Royal Society of South Africa, 1929, Vol. 17, paper by Wayland. | http://www.s2a3.org.za/bio/Biograph_final.php?serial=250 |
How to Prepare for Spine Surgery and Recovery
If you have scheduled an upcoming spine surgery, you may be feeling nervous or anxious in the days prior to the procedure. The best way to decrease this stress is to be fully prepared for what lies ahead. Knowing that you have everything ready and taken care of will give you less to worry about and can make the recovery period much easier on you and your family.
The first step in preparation is to obtain all the information you can about what to expect during the entire process. Most of the information will be given to you by your surgeon, but if there are questions that pop into your mind, write them down and ask the doctor. Once you have an idea of how long you will be away from home, how restricted your movement or activities will be during recovery, and how long that recovery period should last, you are ready to start planning.
Depending on how much you will physically be capable of doing after surgery, you will need to plan ahead for the things that will be difficult or impossible for you to do yourself. You may want to arrange for child care, pet care, or maybe even have a close friend or relative come stay with you for a while. Prepare your home beforehand to ensure that you will be able to reach necessary items easily. If you will spend a lot of time in bed, set up a table with reading material, the TV remote, a pitcher of water, etc. beside the bed for easy access.
Run errands that you may not be able to handle for a while. Stock up on groceries and prepare meals that can be frozen and reheated as needed. Make a list of bills that need to be paid during your down time and take care of them before the surgery. Pre-schedule follow-up appointments with your doctor and arrange for any therapy that is recommended. See if you can get any prescriptions filled before surgery so that they will be ready for you to take when you get home. Plan your vacation or sick time with your job, and make sure your health insurance or any supplemental disability insurance is taken care of.
As the day of your surgery nears, plan your transportation to and from the hospital and pack a bag for your stay. Even if you are having outpatient surgery, you may want to bring a change of loose, comfortable clothing, slip on shoes, a book or magazine, etc. Last, but not least, prepare yourself mentally for the surgery. Relaxation techniques like deep breathing, envisioning a positive outcome, and peaceful music can be extremely helpful in calming your nerves and easing your stress. Once you have planned physically and emotionally for surgery, you can become excited about how the procedure will benefit you, ease your pain, or increase your mobility. | https://centerfordiscreplacement.com/how-to-prepare-for-spine-surgery-and-recovery/ |
Housuke Nojiri’s novel Usurper of the Sun is a novel of first contact, of enigmatic artifacts and ineffable extraterrestrials, of cognitive science and astroengineering. When high school astronomer Aki Shiraishi notices a tower on Mercury during its transit of the Sun, she immediately finds herself regarded as the expert on the phenomena—soon to extend to a giant Ring which blocks out the Sun and gradually plunges Earth into an ice age. The UN quickly forms a task force (the UN Space Force, but everyone knows its secret and true name is UN Spacy), and Aki, by now a fully trained astronomer with a specialty in the Ring and its Builders, volunteers herself for the trip to investigate and destroy the Ring. They succeed, but Aki’s work has barely begun, for now she must prepare Earth for the imminent arrival of the Builders, and convince the remnants of humanity that they are not malevolent entities bent on the destruction of Earth.
Usurper is profoundly, thoroughly capital-H hard science fiction: it is foremost a fictive investigation into alien consciousness and cognition, with dalliances on the side with alien nanotechnology, artificial intelligence, and interstellar flight. Debates on the nature of the Builders—are they intelligent? conscious? sapient? benevolent? apathetic? bent on galactic destruction?—dominate the book, particularly between Aki and programmer Raul Sanchez. Raul has developed an AI in a closed system that appears to be conscious and highly intelligent, yet has not developed language or a sense of “other”; he argues that the AI, which he has named Natalia, is closer to how the Builders may be themselves, beings of pure cognition, and therefore poses that Natalia is the best method by which to contact them.
This frequently fascinating debate on alternative forms of consciousness permeates the novel, twining with the time limit until the Builders arrive in the solar system to provide the main narrative thrust. Unfortunately, the novel serves primarily as a way to explain the concepts Nojiri presents, and other aspects of the book suffer as a result. Aki is not unlikable, but she’s rather bland and uninteresting as a protagonist; most of the other characters are flat and tend to represent concepts or viewpoints more than people. This fits well with the naturalist style of the novel, and wouldn’t really be a problem—unfair as it may be, science fiction, and in particular hard science fiction, is not exactly known for rich, complex characters—except that sometimes Nojiri attempts pathos, which comes off as awkward at best, and downright annoying at worst. The most egregious example is Aki’s brief love affair with a fellow member of the UN Space Force: in the span of twenty pages, the two meet, a potential relationship between them is barely hinted at, they kiss, and then he dies. For an event which has a deep effect on Aki, and is fairly important later in the novel, the reader isn’t really given a chance to invest any emotion into the relationship, making later references to the event ring hollow instead of provoking sympathy.
All this detachment could also be the product of the prose style used in this translation. Unlike most contemporary third-person novels I’ve read, where the narration is through a specific character, Usurper’s narrative style sounds as though the narrator is telling a story about Aki. It’s a subtle shift of narration, but it adds a distinct level of detachment from the events and characters. I’m not sure whether this is a common feature in Japanese third-person novels (nearly every Japanese novel I’ve read has been written in the first-person) or a specific, intentional stylistic quirk of Nojiri; if it’s the latter, it’s quite possible to read the style as though the “narrator” is the Builders themselves, which could add an interesting layer on top of the story as presented.
Usurper is not a bad read at all; you could certainly do a lot worse, especially if you’re looking for a good hard SF novel that deals with theories of consciousness and cognition and isn’t written by Robert J. Sawyer. And even if you’re not, the story is quite enjoyable simply as an alien artifact / first contact story: I grew up reading and enjoying Clarke’s Rendezvous with Rama and Sagan’s Contact (and, more recently, Alastair Reynolds’s Pushing Ice), so I quite unabashedly love novels with either of those elements on principle alone. I also admit to being easily entertained, so there’s your grain of salt for that. | http://www.otakuusamagazine.com/usurper-of-the-sun/ |
Last Updated on November 6, 2021
There really is no greater sight than a group of horses roaming across the wilderness! These magnificent creatures look truly mythical, with their flowing manes and elegant movements. But what’s a group of horses called, and how do they interact with each other?
If you’ve ever seen horses in the wild, you’ll have noticed that they stick together in a group. Our domesticated horses mimic this behavior in the paddock and barn, developing close bonds and relationships. Let’s find out everything we need to know about groups of horses!
What’s A Group Of Horses Called?
As with everything horse-related, there isn’t a straightforward answer to this! There are several names for groups of horses, depending on the type of horse and the situation they are kept in. Here are the most common names you will hear to describe a group of horses:
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Herd Of Horses
The most common term you will hear used to describe a group of horses is a herd. This is normally used to refer to a group of horses living in their wild and natural state. A herd of horses will have a unique and fascinating group dynamic, without any outside interference from humans.
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Team Of Horses
Horses are often called a team when they work together as a group in team activities. For example, this could be used to refer to a team of horses pulling a plow or dray. Horses are also called a team when they compete as a group in competitive events.
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Stud Of Horses
A stud of horses is the name for a group of horses who are kept solely for breeding purposes.
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Rag Of Horses
A rag of horses is the name for a group of young male horses, known as colts.
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String Of Horses
A string of horses is a group of horses belonging to or used by one person or organization. For example, a string of horses from a racing yard or showjumping trainer.
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Harras Of Horses
Harras is an old word that was used to describe a group of horses. The term harras is not widely used in modern times, however, you may still hear it in use on ranches in the US.
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Stable Of Horses
A stable of horses is a group of horses kept in the same stableyard or barn. This name may be linked to a certain rider, trainer, or owner.
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Troop Of Horses
A troop of horses is a group that serves in the military or as horse guards. In the UK, the Queen’s Birthday Parade includes a ritual called ‘Trooping the Colour’, where two hundred mounted officers march to Buckingham Palace accompanied by a marching band and hundreds of foot soldiers.
How Big Is A Herd Of Horses?
In a wild or feral state, herds of horses will roam across wide areas of land. It is not uncommon for two or more herds to co-exist in the same area or territory, but they will rarely cross paths or interact with each other.
A herd of wild horses is normally made up of one or two stallions. The herd will also have a group of mares and their young offspring.
Wild Horse Country: The History, Myth, and Future of the Mustang
The size of a herd of horses varies widely, according to their physical terrain and social dynamics. There are normally around 8 mares per stallion, plus their offspring. This means that a herd could be made up of around 20 horses.
Horse Group Dynamics Explained
The way a herd of horses interacts is fascinating! They have a complicated social structure, with each horse playing a different role. Let’s take a look at how the dynamics of a herd of wild horses works:
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The Stallion
The entire male horse, or stallion, is the protector and guardian of the herd. They also have a key role in reproduction, impregnating the mares during the breeding season.
The stallion has to constantly battle to keep his place within the herd. During the breeding season, he will face threats from other stallions wishing to take his place. If he gets ousted from the herd, he will seek to take over a new herd or join a bachelor group.
Most stallions remain part of a herd for about two years. In some cases, they can last for over ten years as herd protectors and guardians.
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The Alpha Mare
The alpha mare is the leader of the herd. It is unusual for the alpha mare to be replaced, and she will remain with the herd throughout her life.
She is normally an older mare, with a lot of experience and dominant nature. The alpha mare is not always the strongest or biggest female in the herd, but she will be the one who makes decisions and keeps the herd in check.
In a domesticated situation, you will find that one mare will quickly become the alpha mare in your paddock. She will dictate when and where the other horses graze, sleep and relax.
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Colts
As there are only one or two stallions in each herd of horses, colt foals are not permitted to stay in the herd once they become old enough to survive on their own. They will leave the herd and form a group of horses made up entirely of males. This is called a bachelor herd.
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Fillies
Young female horses are not always ousted from the herd, and some will stay in the same herd as their dam. This is not always a problem, as it is likely that the stallion will be replaced at some point so she will not be impregnated by her sire.
More commonly, fillies will leave their birth herd and join another herd of horses. They may also join up with an entire male from the bachelor herd, forming an entirely new herd of horses.
Summary
So, as we’ve learned, the name for a group of horses living naturally in the wild is a herd. Groups of domesticated horses can also be referred to as teams, studs, rags, as well as several other names. In the wild, a herd of horses has a complicated and intricate social structure, with herd dynamics changing constantly.
We’d love to hear about your experiences – have you ever seen a herd of horses running free in the wild? Perhaps you’ve got a cute name for the group of horses in your yard? Add a comment below and we’ll get back to you!
Kate Chalmers is a qualified veterinary nurse who has specialized in horse
care for the vast majority of her career. She has been around horses since
she was a child, starting out riding ponies and helping out at the local
stables before going on to college to study Horse Care & Management. She
has backed and trained many horses during her lifetime and competed in
various equestrian sports at different levels.
After Kate qualified as a veterinary nurse, she provided nursing care to the
patients of a large equine veterinary hospital for many years. She then went
on to teach horse care and veterinary nursing at one of the top colleges in
the country. This has led to an in-depth knowledge of the care needs of
horses and their various medical ailments, as well as a life-long passion for
educating horse owners on how to provide the best possible care for their
four-legged friends. | https://www.besthorserider.com/whats-a-group-of-horses-called/ |
Description of property
The listed residential and commercial building was constructed in 1893 by the renowned architect Paul Segesser in the neo-Renaissance style for his client, Wilhelm Weingartner.
The property comprises a ground floor constructed in granite with the original wooden windows, four sandstone-clad upper storeys and a gabled roof with a terrace area. There is a bay window with a balcony on the first two upper storeys, flanked on the first floor by wide balconies.
As part of the block, the property has had a strong influence on the development of Löwenstrasse and is one of the many protected listed buildings in Switzerland.
Renovations (2010 – 2012)
The property was elaborately and expertly restored in consultation with the department for the preservation of historical monuments of the city of Lucerne. The windows were replaced and soundproofed according to the stipulations of the department for the preservation of historical monument and sun protection elements were also installed in accordance with historical specifications.
New balconies have been added on the façade on the inner courtyard, to ensure every apartment has at least one balcony. Two maisonette apartments have been created in the raised roof storey. The technical facilities and risers have been replaced, and the energy supply has been converted to gas heating with a central water treatment system. The groundwater protection in the basement has been renovated with a new well system.
A glass elevator has been installed at ground level to serve all the apartments.
Extensive renovations have been carried out in the existing apartments. All the bathrooms, kitchens and electrical systems have been refurbished, and fire separation areas and soundproofing have been installed between the dwellings. The entrance doors have been doubled and soundproofed.
All the units have their own washing machine and tumble dryer in the apartment and a private space in the basement. | https://loewenstrasse.com/description-of-property.html |
Gold miner Gold Fields has acknowledged the lack of female representation in the workplace, citing it as “a global business issue” across industries and geographies.
Women in the South African mining industry alone face a range of challenges, which often prevent them from either choosing a career in mining or remaining in mining roles.
To put it in perspective, in South Africa, women make up only 12% of the mining industry, compared with 17% in Australia and 16% in Canada.
However, female employees account for 23% of all employees at Gold Fields’ South Deep mine, indicating the progress the mine has made, but also the work that remains to be done, the miner says, adding that it “has a key focus on removing barriers that have traditionally made mining less appealing to women”.
The mine aims to advance a diverse workforce by ensuring that all members of the leadership team are committed to an inclusive and diverse workplace, while also setting very clear targets to increase the level of female representation at all levels of leadership.
Additionally, the mine is aiming to ensure that talent and performance management programmes promote the identification of high-potential women and offer them appropriate development opportunities, while simultaneously addressing systemic and cultural barriers that may impede gender equality throughout the business.
The South Deep mine also fosters a safe space to report and address all forms of harassment, while engaging and collaborating with all stakeholders to promote and support impactful gender equality initiatives, the miner says.
Gold Fields executive VP Martin Preece notes that “women play many critical roles in our lives, our society and in our work environments. This is also true at South Deep, where many women are ambassadors and role models to our people and in particular to other women who want to follow careers in mining”.
He adds that the miner believes that gender equality is key to a fair and trusting work environment that will strengthen organisational culture and ultimately improve business performance. | https://www.miningweekly.com/article/gold-fields-south-deep-mine-addressing-gender-equality-in-mining-2020-08-11 |
1. To sit, lean, or recline into something and rest one's body. After such a long day at work, it feels good to finally crack open a beer and relax into my favorite armchair. I relaxed into my bath to try to ease the tension in my aching muscles.
2. To feel more comfortable, confident, and able in a particular role, job, position, etc. I'm not sued to giving people orders, so it took me a while to relax into this new position of authority. She seemed quite nervous at first, but she's finally relaxing into the role.
See also: relax
Farlex Dictionary of Idioms. © 2022 Farlex, Inc, all rights reserved.
relax into something
1. to sit or lie down in something, relaxing. I want to go home and relax into my easy chair. I relaxed into the reclining chair and was asleep in a few moments.
2. [for something that is tense] to assume a more relaxed shape or condition. His cramped muscle finally relaxed into a soft mass of tissue. As her tight neck relaxed into softness, her face brightened.
See also: relax
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs. © 2002 by The McGraw-Hill Companies, Inc. | https://idioms.thefreedictionary.com/relax+into |
Entanglement is basically an artistic project at the crossroads of scientific research inspired by the concept and properties of entanglement in quantum physics and quantum fields theory. The sources of inspiration of France Jobin and Markus Heckmann are based on the two current dominant theories explaining quantum entanglement: the Copenhagen interpretation and the multiverse (quantum decoherence). In addition to these two theories, there is the fluidity of time and the principle of quantum entanglement. The Entanglement performance is inspired by the same theories; an ambitious immersive experience that will be presented as a world premiere at the 22nd edition of MUTEK.
Salomé Perli FR/QC — TSUNAMI
World Premiere
Immersive performance merging music, dance and visual arts, TSUNAMI plunges deep into the core of our weaknesses. Akin to the devastating phenomenon, TSUNAMI submerges audiences with sound textures from hydrophones manipulated in real time by Salomé Perli and her creatures, starkly contrasted with erratic no-input loops. Jules Roze’s visuals highlight Stefania Skoryna’s flowing choreography, manifesting the quake’s raw power. TSUNAMI invites us to let go by reminding us that this catastrophe may simply be the result of the endless movement of life. | https://montreal.mutek.org/en/shows/2021/play-4-2021 |
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Tuesday, November 23, 2010
We anchored Thursday night on the Tensaw River, near some rough-hewn fish camps, and early Friday morning we set out on the last leg of our journey down the Tombigbee River. We were a mere 12 miles north of Mobile Bay and we marveled at the mostly unspoiled landscape: a bald eagle perched at the top of a cypress snag, a raft of pelicans foraging in a little cove bordered by palmettos.
Mirage? No, Mobile.
We started to see evidence of civilization: dumpsters, one, then another, then another, mangled and mashed and shoved into the mud on shore. After days of anchoring out, we joked about how convenient this was for boaters with a boatload o' trash. But we held on to our Glad bags. (We figure the dumpsters were carried up river during a storm or hurricane and never retrieved.)
Then we spotted an incongruous sight: an elegant spire, rearing above the marsh grass. It was the top of a skyscraper. The city of Mobile was just ahead.Passing under the Route 90 bridge, we also spotted power line transmission towers sporting solar panels. The towers had flashing aircraft lights, and we figured the solar panels were to power the lights in the event of a power failure.
Solar panels in the foreground, solar panels in the background!
A little research revealed a number of other cool applications these panels might be supporting. Some power-line towers have security webcams powered by solar panels--to monitor for vandals stealing copper, steel, or other metals, or to alert the substation operator to downed lines or other problems. Solar panels can power other kinds of monitoring equipment as well: sensors that detect icing on the lines, current leakage, wind speed, and so on.We couldn't help but smile at the thought that these enormous towers--carrying the megawatts of juice that makes the busy city run--stay safe thanks to the power of the sun.Lots more photos showing this stage of the trip on Facebook!
2 comments:
Cynthia, have a nice holiday break. I do think the shrink-wrapped "sausages" really were heat exchangers. The turbine guess wasn't bad either, and maybe 2nd runner up might be engines for the new Aries rocket headed up to Huntsville.
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Most Canadians approaching retirement know that they will have some retirement income through the Canada Pension Plan (CPP) and Old Age Security (OAS) programs. Many, however, are unaware that there is a third federal program, the Guaranteed Income Supplement (GIS), which provides an additional monthly income amount to eligible individuals. While there is no need for an individual to apply to receive an Old Age Security benefit, anyone who wishes to receive the GIS must apply to do so. Automatic enrollment in GIS is something that is planned for future implementation, but is not yet in place. Finally, while the OAS benefit is a standard amount for most recipients, the rules governing eligibility for GIS, and the amount an individual will receive, are more complex.
The first and most basic rule of GIS eligibility is that GIS is paid only to individuals who are already receiving the Old Age Security benefit. Canadians can begin receiving such OAS benefit at age 65, or can defer receipt of that benefit up until the age of 70. However, regardless of the age at which an individual chooses to begin collecting OAS, he or she cannot receive the GIS until that OAS benefit has started.
There is a perception that GIS benefits are available to only the lowest income seniors. While it is true that eligibility for the GIS is tied to income, the current reality is that in the first quarter of 2017, nearly 2 million Canadians, or nearly one-third of those who collect OAS, also received GIS benefits.
The basic rule is that single (or divorced or widowed) individuals who have less than $17,688 in net income for the previous year are eligible to receive at least partial GIS benefits each month. Once net income exceeds the $17,688 threshold, eligibility for GIS is eliminated. That figure is somewhat deceiving, however, as not all income sources are treated the same way when it comes to determining net income for purposes of assessing GIS eligibility. When determining such eligibility, the sources from which income is received is nearly as important as the amount of that income.
Generally, in calculating net income for purposes of determining GIS eligibility, the following income amounts are included:
The following income amounts are not included in net income for purposes of determining GIS eligibility:
Finally, many retirees work part-time, whether out of financial need or for social reasons. In calculating net income to determine GIS eligibility, an exemption is provided for the first $3,500 in employment income earned each year.
In 2017, an individual who is single, divorced, or widowed and is eligible for a full GIS amount will receive $871.86 per month. That amount is reduced as income increases and is eliminated entirely where the individual’s net income exceeds the $17,688 cut-off.
A similar calculation is required for taxpayers who are married. The net income calculation is the same, but the cut-off amount above which GIS eligibility for both spouses is eliminated, where both spouses are receiving OAS, is $23,376. Where one of the spouses does not receive OAS, the combined income threshold for GIS eligibility is $42,384. More information on the benefit and income cut-off amounts for the current quarter (July to September 2017), as well as links to tables which will show the exact amount of GIS payable at different income levels, can be found on the Canada.ca website at https://www.canada.ca/en/services/benefits/publicpensions/cpp/old-age-security/payments.html.
A final note — where individuals receive the Guaranteed Income Supplement, whether the full benefit or partial amounts, all such amounts received are non-taxable. | https://segalllp.com/claiming-the-guaranteed-income-supplement-september-2017/ |
Pollution - Chemical Handling - Climate Resiliency Planning
We have extensive experience in Superfund response and allocation matters. Our work involves not only litigation defense, but multi-party allocation, natural resource damage resolution and defending claims made by State and federal environmental regulators like EPA, the Department of Justice and Natural Resource Trustee Council.
Our attorneys have extensive experience defending clients in some of the largest National Priority List sites including the Portland Harbor site in Oregon, and the Duwamish and Commencement Bay sites in Washington.
We also defend clients on matters related to waste management, groundwater contamination, risk management planning, data quality, and industrial plant incidents. Our attorneys have handled and responded to state and federal environmental claims including RCRA, CAA, CWA, SDWA, EPCRA, FIFRA and CERCLA.
In addition to litigation experience, we have attorneys with substantive environmental backgrounds and experience, and relationships with industry-leading engineers and scientists with significant expertise in a wide variety of environmental, pollution and hazardous materials issues.
Recently, the Firm represented a tank sampling and investigation company in a federal court RCRA citizens suit action alleging the testing company’s activities had resulted in a release of fuel oil into the environment. We also recently represented a company in a claim regarding improper handling and storage of hazardous waste streams, and a trucking company involved in an accidental release of fuel. | https://www.foleymansfield.com/practice-areas/environmental-pollution-law/ |
We have carved a niche in the market as a prime organization engaged in manufacturing and supplying Karl Fischer Titrator. It is microcontroller based titrator, uses peristatlitic pump for delivery of KF reagent. We manufacture this equipment at our sound production unit using high quality materials and components. The result, including drift, is displayed on back lighted liquid crystal display screen and can be printed if required using external dot matrix printer. Karl Fischer Titrator is used for measuring water content in various substances including chemicals, oils, pharmaceuticals and food.
Features:
Specification:
Resolution : 0.01 ml
Mode : Blank, Titer, Sample
Results units : %w/w, % w/v, ppm, mg, mg/ml
Real time clock : For Date & Time of Titration
Report : Result report on display / Document on printer & PC
Address:
W- 446, Rabale MIDC, Rabale, Navi Mumbai - 400701
Ph : + 91 22 2764 4030 to 34,
Fax : + 91 22 2764 4035
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Developed and Managed by Infocom Network Private Limited. | http://www.spectralabinstruments.net/karl-fischer-titrator-815677.html |
Being charged with driving under the influence (DUI) in California can be many things. It can be humiliating, humbling, aggravating, annoying, eye opening, and much more. For those who have never before been in trouble with the law, the experience can be especially harrowing, which is why the help of a professional DUI defense attorney is always recommended. A DUI defense attorney is an attorney who dedicates a majority of his or her time to the practice of defending those charged with driving under the influence. Because it is such a unique area of the law, an attorney with knowledge and experience specific to DUI defense can be an especially valuable ally in the fight against charges filed by a local prosecutor.
Immediately After an Arrest for DUI
Immediately after a person has been arrested on suspicion of driving under the influence, that person will be detained by police, kept in police custody, and brought before a judge (usually the next business day) to be arraigned. An arraignment is a formal reading of the charges which a person has been charged with, and an opportunity for a formal response (guilty or not guilty). In between arrest and arraignment, the police will most likely ask a variety of questions. Instead of answering these questions, anyone who believes that the police may suspect them of DUI, no matter the true circumstances, should exercise his or her right to remain silent.
In order to receive the full protections of the right not to incriminate oneself, the right to remain silent must be “activated” by an affirmative statement or other obvious indication. Sitting in silence is no longer a valid method of exercising the right to remain silent, and may actually be used against a person in court as evidence of a guilty conscience (as of 2013, when the U.S. Supreme Court ruled on Salinas v. Texas). Exercising the right to remain silent is the best way to ensure that nothing you say is taken out of context or misinterpreted by an overzealous police officer / investigator.
Legal Appointment
Since most people don’t have any practical way of securing legal representation after an arrest but before an arraignment, courts will usually appoint a public defender to represent the suspect during the arraignment. If the suspect does not believe that he or she has committed a crime, then the suspect should enter a plea of not guilty during arraignment with the help of the appointed public defender, but the suspect shouldn’t make any long term representation plans with the public defender. After arraignment, the suspect will have plenty of time to secure private legal counsel, which is exactly what should be done. Most public defenders, especially those in heavily populated cities, are too overworked and under motivated to give their appointed clients the representation that they deserve.
Once Secured
As soon as private legal counsel has been secured to handle DUI charges on a suspect’s behalf, that suspect will have far less on his or her mind to worry about. This is because when a private attorney takes on a case, the attorney will immediately get to work crafting a competent defense for the suspect, reviewing the evidence in a case, and making contact with the necessary witnesses for more insight into the charges. | https://www.toddlandgren.com/blog/arrest-to-arraignment-after-dui-in-california/ |
Davis, Louise A (2018) Can a non-phonics based intervention scheme enable children who are falling behind in literacy to make better progress than normal classroom teaching? A pilot/feasibility study. EdD thesis, University of Sheffield.
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LAD Non-phonics Thesis 16Feb18.pdf
Available under License Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 UK: England & Wales.
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Abstract
The aim of this study was to investigate whether a non-phonics-based intervention, Fischer Family Trust (FFT) Wave 3, might help children in Years 1 and 2 who were falling behind in literacy to catch up with their peers. The study consisted of an ‘outer’ and an ‘inner’ study. The outer study was a pilot/feasibility study (PFS) which investigated whether the quantitative approach used in the inner study would be suitable for scaling up to investigate the research questions. The inner study was a quantitative evaluation of FFT Wave 3. It consisted mainly of a randomised control trial (RCT) using standardised literacy tests, supplemented by a number of non-standardised assessments. A small (n=24) two-armed RCT lasting 10 weeks was undertaken in a state primary school in Derbyshire. Standardised tests of reading comprehension (primary outcome), of spelling and of oral word and sentence reading (secondary outcomes), were administered to both groups pre and post, and after the control group had received the intervention. At pre- and post-tests both groups answered an attitudes questionnaire, and the intervention group provided one-sentence writing samples, and their reading ages were estimated (exploratory outcomes). Results for the primary and secondary outcomes fell into a confusing pattern, and were inconclusive, and results for the exploratory outcome of attitudes to reading were null. Results for the exploratory outcomes of reading ages and writing showed statistically significant gains, but could not be considered definitive because no parallel data were gathered from the control group. Thus the inner, quantitative study failed to show conclusively whether the FFT Wave 3 intervention had real impact. On the other hand, the PFS successfully showed that, with adjustments, a quantitative, mainly RCT, approach could be a suitable method for assessing a non-phonics-based intervention.
|Item Type:||Thesis (EdD)|
|Academic Units:||The University of Sheffield > Faculty of Social Sciences (Sheffield) > School of Education (Sheffield)|
|Identification Number/EthosID:||uk.bl.ethos.745669|
|Depositing User:||Mrs Louise A Davis|
|Date Deposited:||12 Jun 2018 08:48|
|Last Modified:||12 Oct 2018 09:54|
|URI:||http://etheses.whiterose.ac.uk/id/eprint/20618|
You do not need to contact us to get a copy of this thesis. Please use the 'Download' link(s) above to get a copy.
You can contact us about this thesis. If you need to make a general enquiry, please see the Contact us page. | http://etheses.whiterose.ac.uk/20618/ |
Extracellular SH3 domain containing proteins--features of a new protein family.
In the year 1994, the protein MIA (melanoma inhibitory activity) was found to be strongly expressed and secreted by malignant melanomas and subsequent studies revealed that MIA has an important function in melanoma development and invasion. Multidimensional NMR-spectroscopy and x-ray crystallography revealed that recombinant human MIA adopts a Src homology 3 (SH3) domain-like fold in solution, a structure with two perpendicular antiparallel three- and five-stranded beta-sheets. SH3 domains are protein modules that are found in many intracellular signalling proteins and mediate protein-protein interactions by binding to proline-rich peptide sequences. Unlike previously described protein structures with SH3 domain folds, MIA is a secreted single-domain protein of 12 kDa that contains an additional antiparallel beta-sheet and two disulfide bonds. Furthermore, the charge surrounding the canonical binding site differs from that of classical SH3 domains. The two disulfide bonds are crucial for correct folding and function as revealed by mutation analysis. Therefore, MIA appears to be the first extracellular protein adopting an SH3 domain-like fold. MIA was shown to interact with fibronectin, and MIA-interacting peptide ligands identified by phage display screening are similar to the consensus sequence of type III human fibronectin repeats, especially FN14. Interestingly, recent data revealed that MIA can also directly bind to integrin alpha 4 beta 1 and alpha 5 beta1 and that it modulates integrin activity negatively. These findings suggest an interesting role of the SH3-domain proteins in the extracellular compartment. Recently, MIA homologous proteins with a sequence identity of 44% and a sequence homology of approximately 80% were determined (TANGO, MIA-2, OTOR). This clearly suggests that this structural device is used more frequently, in processes ranging from developmental changes to the interference of cell attachment in the extracellular matrix. Detailed studies are necessary to determine the exact function of the MIA homologous proteins. It will be interesting to know whether additional protein families can be identified which are secreted and carry SH3 domain-like modules, in addition to elucidate what the specific physiological targets of this protein family are.
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Origami in Nature: Protein Structure Prediction
Predicting how proteins will fold in vivo is a Holy Grail of proteomics and theoretical chemistry. Current hopes are that this can be achieved by designing an in silico platform that can predict protein folding, either de novo (a.k.a. from scratch) or using known proteins as a guide. What would we need to do, why would we want to, why is it so hard, and where are we with this now? Let’s delve into the world of predicting protein folding. Whether you’re a novice or hoping to learn more, this article is for you!
1. What do we mean by protein folding:
Proteins are composed of building blocks called amino acids. Some describe it as being like pearls on a string to make a necklace. However, this is only helpful when you’re thinking of the protein as a long, unfolded strand. In actuality, proteins exist in intricately folded and twisted arrays, often interacting with other proteins in a specific environment. Each amino acid has its own unique chemical properties, which results in a preference or disdain for certain other amino acids and aqueous environments.
2. Reasons we’d want to predict protein structure:
The Holy Grail is automated protein structure prediction. To be able to predict how a string of amino acids will fold, the program will need to know certain information including, but not limited to:
- Amino acid sequence
- Properties of each amino acid in the sequence
- Properties of the environment in which the folding will occur
- Whether the protein will interact with other proteins (called a quaternary structure)
- Whether the sequence has similarities to other sequences for which the folding pattern is well understood.
3. How we try to predict protein folding:
The key for the program is to be able to identify likely patterns in the folding. The program will look at the primary structure of the protein and the extended chain of amino acids, and pick out features that suggest its likelihood to fold in a particular manner. For example, a ubiquitous folding pattern is the alpha (?) helix. For example, regions richer in alanine, glutamic acid, leucine, and methionine and lower in proline, glycine, tyrosine, and serine tend to form an alpha helix. Depending on where the helix will reside on the protein, it will have certain properties. So, helices exposed on the surface of a protein folding in a water-rich solution will have a higher proportion of hydrophilic amino acids than those that form with a protein’s covered core or on the surface of a protein that exists in a lipid-rich solution. By picking out all these features, the program will begin to work out the most energetically favorable way for the protein to fold.
4. Why it’s so hard to predict:
Firstly, two completely different amino acid chains from totally different sources and evolutionary backgrounds that share little sequence similarity may fold into very similar structures. So, sequence similarity may not tell the whole story for predicting protein structure.
Secondly, two proteins that share a statistically significant degree of sequence similarity likely evolved from a common ancestor. However, gene duplication and genetic rearrangements during evolution may give rise to new gene copies, which can then evolve into proteins with new function and structure. This means that, although the two protein sequences may share a similar sequence, they may fold very differently!
Thirdly, it takes extraordinarily powerful computers and highly experienced experts to be able to even attempt protein structure prediction because there are so many variables. So, the high cost is a hindrance in some cases.
Fourthly, there are so many unknowns. It can be hard to know enough about a protein, its particular microenvironment, and the in vivo folding process to predict its structure. The sheer number of variables and presumption on which prediction software is based is also an issue.
Right now, the most advanced software can predict protein folding with about 80% accuracy and weekly tables are available, such as LiveBench and EVA. Some labs have made their software open source to allow for a crowdsourcing approach—even allowing the “common man” to be involved—including Rosetta@home, the Human Proteome Folding Project, Nutritious Rice for the World, and Folding@home. A publicly known project is called FoldIt, and is a very clever and unique online game that teaches you about protein folding as well as providing new solutions to scientists. Players can puzzle away real protein problems like targeting and eradicating diseases and creating biological innovations. A 2010 paper in the journal Nature credited Foldit’s 57,000 players with providing useful results that matched or outperformed algorithmically computed solutions.
5. What’s next?
Personally, I feel that everything goes in cycles. Science started out with the common man and s/he then became more and more educated and specialized until you have the scientists today who are experts in their own little niches. However, to be able to do something as complex as protein folding, you need some many experts. Experts get into the habit of thinking of things a certain way and as you get to the top of the food chain of scientists, people are so specialized they struggle to communicate with each other (generally speaking!). The common man has the ability to add a new perspective, a new way of thinking, and there are far more common men than expert scientists. I feel that huge progress can be made by using the “hive mind” of public involvement!
What do you think about using crowd-based methods to solve complex problems? Tell us more in the comments.
Feature image courtesy of katsuuu 44.
3 Comments
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[…] This is an original post from Bitesizebio […]
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Predicting is good analysis, although wouldn’t the important key be figuring out how to fix the improperly unfolded proteins? Thanks for writing this article. This definitely gives me a better understand of the topic! 🙂
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Hi Aurora! Thanks for your comment, I’m very happy you found my article useful! You’re right, figuring out how and why proteins misfold and how to correct this is extremely important. I think over the next few years, research like into protein folding will help us come up with better treatment options for many protein misfolding diseases. Super exciting :)!
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You must be logged in to post a comment. | https://bitesizebio.com/26539/origami-in-nature-protein-structure-prediction/ |
Why Linking Narcissistic Abuse and Codependency Is Dangerous
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Codependency is real. The link between narcissistic abuse and codependency, however, is a stereotype I’d like to dispel because it can be harmful.
More specifically, it’s a myth that all men or women that get into relationships across the board who have been in relationships with narcissists were codependent before they got there. It’s inaccurate and inappropriate for outsiders to apply that term to survivors simply because they have been abused by narcissists.
In addition, narcissists can specifically target strengths of individuals that happen to overlap with some of the so-called characteristics of codependency if those characteristics are taken to excess or merely worded differently when we talk about them. For example, someone may look for the good in other people, or have an easygoing temperament or be known as a caretaker, and have no issues in other areas of their lives. When taken advantage of by a narcissist, who sees these as opportunities to exploit, the traits that were in other instances seen as positive suddenly become “evidence” of codependency and low self-worth (“5 Questions for Your Narcissistic Abuse Recovery Program“).
Finally, a relationship with a narcissist may erode the boundaries and self-esteem of someone so that they do ultimately become codependent, however, there is no evidence to support the idea that being codependent is why they entered the relationship to begin with– especially when coupled with the idea that they were fooled by the idealization heaped upon them by the narcissist at the beginning of the relationship. And as Shahida Arabi says, once they have become attached, partners who stay in relationships with narcissists and find it difficult to leave are “trauma bonded.”
Yet it’s not just that the term can be inadequate as a whole to describe survivors of narcissistic abuse. What’s more important is that it can also be downright dangerous.
Why the Terms “Narcissist” and “Codependent” Can Be Dangerous When Used Together
1. Narcissists Blameshift as a Form of Psychological Abuse and Pathologizing Survivors Reinforces the Abuse.
Christine Hammond describes how narcissists flip the script in their relationships and convince themselves that they are justified in their abuse because they feel victimized by their partner’s reactions to what they do. They are blind to their own narcissism and fail to see their abusive actions as a catalyst and their partner’s reactions to their behavior as normal.
Narcissists continually use tactics such as gaslighting, blame-shifting, and stonewalling to condition their partners to believe that the relationship issues are their own fault. Thus, many survivors in the relationship are brainwashed by the narcissist to believe that they are to blame for their own abuse and they just need to try harder to get him or her to stop. (“5 Reasons Why Verbal Abuse is Not Your Fault“).
Calling partners “codependent” can trigger the same feelings of shame, helplessness, and defeat that are being or were manufactured in the relationship. It feeds into the narcissist’s narrative that something is wrong with the partner. It ignores the undue influence that the narcissist had on the partner that influenced the partner’s behavior, further lowering his or her self-esteem. The result can be paralyzing — making it difficult to leave or recover.
2. The Label “Codependent” Can Provide Survivors with a False Sense of Security.
The label can be potentially dangerous because a survivor who has only looked inward and hasn’t understood the special nature of narcissistic abuse can still be victimized again because narcissists con victims from the start about who they really are.
Jackson MacKenzie, author of Psychopath Free, writes, “Most of us have no problem spotting nasty people– we avoid them. But psychopaths present themselves as your mirror image. A soulmate. They quickly declare that no one has ever made them so happy in their life– they compare you to past exes, holding you high above everyone else. They sniff out your vulnerabilities, insecurities, and dreams. And based on their findings, they transform their entire personalities to become your perfect match.”
By encouraging survivors to focus on themselves and ignore how narcissists were able to abuse them in the first place, a survivor may not learn the warning signs or understand how the cycle perpetuates a bond that is difficult to break no matter who you are. Survivors may believe that if they only work on themselves, they can never be victimized again.
3. Pathologizing Survivors Perpetuates the Myth that Only Some People Are Susceptible to Narcissistic Abuse.
If survivors who have been victimized believe they need to “fix” themselves and they will be immune to narcissists, the public-at-large may be under the false impression that only certain people may be victimized.
Although it is true that narcissists may target people with certain characteristics that they find appealing, let’s talk about what that really means.
The National Coalition Against Domestic Violence states that “Anyone can be a victim of domestic violence. There is NO “typical victim.” Victims of domestic violence come from all walks of life, varying age groups, all backgrounds, all communities, all education levels, all economic levels, all cultures, all ethnicities, all religions, all abilities, and all lifestyles.”
This would seem especially to be the case with narcissistic abuse, given the deception involved. They can deceive anyone across any situation.
Robert Hare, one of the world’s leading experts on psychopaths says in his book Without Conscience, “Everyone, including the experts, can be taken in, manipulated, conned and left bewildered by them. A good psychopath can play a concerto on anyone’s heartstrings” (p. 207).
Narcissists will always look out for their own best interests, which means even though they may find certain characteristics appealing– characteristics that help them feel glorified– those characteristics fall on a spectrum. Their goal is to ensure that in the long run, they keep people around who will glorify them and discredit those who don’t, and they may constantly shuffle people in and out of their lives using a variety of tactics when people “fall out of favor.”
In short, they use manipulative tactics on everyone.
This extends beyond romantic relationships to friendships and in the workplace, but may be particularly damaging in romantic relationships.
This is because finances may be comingled, legal relationships such as marriages and property contracts may be formed, and children may be born, setting in motion seeds of destruction and abuse long before the true character of the narcissist is revealed to the partner who entered the relationship. The effects of those seeds may last a lifetime or extend into generations beyond the original survivor who entered the relationship.
Focusing on the survivor as pathological denies that this is a societal problem. It doesn’t take into account the various ways that narcissists move within their social circles manipulating and harming people in them to various degrees, such as by stealing from them, assaulting them, causing social drama or malignant harm through abuses of trust and lack of conscience.
Staying focused on the survivors of narcissistic abuse may make life feel more comfortable because it makes it seem as if it could never happen. Yet that mentality makes it easy for narcissists to keep hiding in plain sight, sliding under the radar, while we victim-blame as a society and they move from victim to victim, leaving behind a trail of destruction.
How to Use the Term Codependent Effectively
I’ve made it no secret that I find the term “codependent” problematic, as it is used in general by many in its application to survivors as a whole. I find it unhelpful, even potentially detrimental in a lot of cases.
There are times when it can be useful, however. Given the literature on the topic and the many people who find its affirmations and descriptions beneficial, the circumstances when I think it can be useful are when the label is applied:
individually and not categorically to survivors as a whole;
by an individual to himself or herself and not by an outsider, or if in conjunction with someone else, if the individual is in agreement that the term seems applicable to his or her experiences;
therapeutically when the focus is on healing certain aspects of the self that extend beyond personality characteristics and not merely for explanatory purposes;
in conjunction with understanding the abuser and the dynamics of the abuse as a whole.
Understanding the dynamics of the abuse as a whole can set the stage for understanding the entirety of the damage that was done so that the healing from the trauma can be thorough enough not only to understand the past, but to set the stage for healthy relationships in the future.
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2 Comments
Reet
Thank again Kristin
It makes so much sense to me.
I am both easy going and a caretaker.
He love bombed and lied while putting on a fake smile.
I never identified with the codependent mainly because I have lived my life independently I have had a son with kidney disease since he was 8 and spent yrs in and out of hospital. I worked at a high school while doing a full time degree,I have 3 grown children.
And then I got ME and I have been quiet poorly for a decade,not relying on a soul. He was my 1st relationship in a decade.
I was so far from being able to depend on a single soul
So I don’t see codependent as negativity it’s just doesn’t fit into my story.
Kristen Milstead
Hi Reet: Thank you for leaving a comment. I’m glad you feel confident in defining your own story– I don’t feel there is a need to take on labels that don’t seem to define us. And I agree with you– it doesn’t mean seeing “codependent” as bad, just not necessarily accurate in all situations. I hope you are doing better these days and are on your way to recovery. Stay strong! -Kristen
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ROTTERDAM, Netherlands and SAN DIEGO, June 23, 2020 /PRNewswire/ -- Last week, SkylineDx signed the 10th collaboration agreement with an academic partner for research under the extensive Falcon R&D Program to further validate both melanoma (skin cancer) tests. The 10 clinical centers represent 6 countries on 3 continents with data on over 3,500 cutaneous melanoma patients. The data generated will be used in the validation of the Merlin and Peregrine assay. The Merlin assay has been developed to predict a patient's risk of having metastasis in the sentinel lymph node. If a patient is identified as low-risk, the surgery that removes the sentinel lymph node can be safely avoided. The Merlin assay is developed on a US patient dataset[2] and validated in a European dataset[3]. The group of patients without metastasis in their sentinel lymph nodes, are currently considered low risk, although a significant number of patients will see their melanoma returning within 5 years. The Peregrine assay has been developed to identify patients at high risk of disease recurrence within this group of patients now considered low-risk, so treatment options can be discussed[4-5].
"I am very pleased that we are continuing our research with global partners for retrospective validation studies. Even more clinical groups have expressed interest to collaborate and are likely to be added in the near future to this research initiative. Processing of the biobanked samples is in full swing and we expect to have all the results for analyses by the end of 2020. The peer reviewed publication will follow shortly in 2021," explains Dharminder Chahal, CEO SkylineDx.
About Merlin & Peregrine
Both assays are using the CP-GEP model, a powerful algorithm that calculates the risk of metastasis in a patient's sentinel lymph nodes (predictive use) and the risk of the melanoma returning (prognostic use). The model is able to calculate risk on an individual basis through a combination analysis of 8 genes from the patient's primary tumor, the tumor thickness and the patient's age. The model has been previously published in JCO Precision Oncology[2]. The predictive use of the CP-GEP model is the main focus of the Merlin Study Initiative. The prognostic use of the CP-GEP model is the main focus of the Peregrine Study Initiative. Both are developed under the wings of the Falcon R&D Program. More information on www.falconprogram.com.
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Join us for some straight talk about the economy and why you should be excited about the 2017 forecast. Join us as we welcome back acclaimed economist, Alan Beaulieu, for the seventh time.
With the presidential election fresh in our minds, Alan shared with us the potential impact it has on his forecasts from last year, and going forward. He also discussed what we can expect for GDP and the stock market, the effects of the proposed $15/hr minimum wage legislation, and its potential impact on unemployment. And, as always, he gave an analysis of the implications of Fed policy and potential interest rate changes on your business. There are a lot of myths about the US economy, and Beaulieu looks into the ones that matter most to the manufacturing community in Ohio and the country as a whole. Together, we will see what the future holds for manufacturing and key markets in the year to come.
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TECHNICAL FIELD
BACKGROUND OF THE INVENTION
SUMMARY OF INVENTION
DETAILED DESCRIPTION OF THE INVENTION
The instant invention relates to motor vehicle accessories, and more particularly, to a wind deflector and screened ventilation system with cigarette egress passageway that easily attaches to a vehicle door, which thereby allows fresh air to circulate through the passenger compartment of the vehicle, prevents entry into the passenger compartment of outside debris, and provides a smoker-friendly system that enables passengers who are smoking cigarettes to more safely dispose of their cigarette ash.
Wind deflectors, or vent visors, as they are sometimes referred to, have been a popular motor vehicle accessory for many years. The success of the wind deflector stems from the fact that it provides several benefits to motor vehicle passengers.
One of the benefits achieved by the wind deflector is apparent when the vehicle is operated during a rain shower. In this situation, the deflector operates to prevent rain from entering the passenger compartment of the vehicle. This benefit is especially advantageous for those motor vehicle operators who smoke and prefer to dispose of their cigarette ash outside the vehicle window.
Another advantage of the wind deflector is that it decreases the noise level created by the wind while driving a motor vehicle with the windows slightly open. A further benefit of the wind deflector is heat dissipation. The wind deflector allows a motor vehicle operator to slightly open the windows of a parked vehicle, thereby dissipating heat and preventing the “greenhouse effect” experienced by many motor vehicle operators who enter their vehicle on a hot summer day. In this same capacity, the wind deflector can prevent a human or animal passenger from suffocating by providing adequate ventilation into the passenger compartment.
A final benefit of the wind deflector is that it allows fresh air to circulate within the passenger compartment of a motor vehicle no matter what the weather conditions may be. This can be helpful if a passenger in a motor vehicle is smoking, but the other passengers would prefer not to inhale the second-hand smoke. The fresh air circulating through the passenger compartment will help to displace the smoke contaminated air, thus providing a more enjoyable passenger compartment.
The typical wind deflector is made of acrylic, although deflectors may conceivably be manufactured from various other plastics and lightweight metals. Wind deflectors are designed to easily attach to a vehicle door. Some wind deflectors mount directly to the frame using double-sided acrylic foam tape, while other deflectors are installed in the window channel.
Although the current art wind deflector provides several benefits, it also has several disadvantages. A primary disadvantage is that outside debris can easily enter the passenger compartment of a vehicle by virtue of the space created between the wind deflector and the open window. Another problem encountered with current wind deflectors is that they are not as smoker-friendly as they could be. For example, a safety issue arises from a motor vehicle passenger who is smoking and trying to dispose of their ash between the wind deflector and window. The air circulating into the passenger compartment may dislodge the burning cigarette from the passenger's grip, resulting in damage to the interior of the vehicle, or even worse, burns to a passenger within the vehicle.
There remains an unfilled need to provide a wind deflector that retains the many benefits associated with current wind deflectors, yet is also more smoker-friendly and capable of preventing unwanted outside debris from entering a vehicle's passenger compartment.
In its most general configuration, the present invention advances the state of the art with a variety of new capabilities and overcomes many of the shortcomings of prior devices in new and novel ways. In its most general sense, the present invention overcomes the shortcomings and limitations of the prior art in any of a number of generally effective configurations. The instant invention demonstrates such capabilities and overcomes many of the shortcomings of prior methods in new and novel ways.
The present invention is a wind deflector and screened ventilation system with cigarette egress passageway. The wind deflector and screened ventilation system with cigarette egress passageway is configured to be releasably attached to a vehicle door.
The window deflector and screened ventilation system with cigarette egress passageway includes a deflector, a screen, and a cigarette egress passageway. The deflector has a deflector attachment portion and a wind deflection portion. The deflector attachment portion provides the means for mounting the window deflector and screened ventilation system with cigarette egress passageway to a frame of the vehicle door.
The screen is releasably attached to the deflector and includes a screen attachment portion and a screen ventilation portion. The screen attachment portion provides means for releasably attaching the screen to the deflector.
The screen ventilation portion cooperates with a window to provide an air-permeable barrier to the passenger compartment of the vehicle. The cooperation between the screen ventilation portion and the window allows the window to be in an open position while the screen ventilation portion lets fresh air into the passenger compartment of the vehicle and prevents insects, debris, and the like from entering the passenger compartment of the vehicle.
The present invention further includes a cigarette egress passageway formed in the screen. The cigarette egress passageway has an egress perimeter configured to permit a cigarette to pass from a screen ventilation portion interior surface to a screen ventilation portion exterior surface. Furthermore, the configuration of the cigarette egress passageway in the screen is such that the window need not be lowered below the screen before the cigarette egress passageway may be utilized. This feature allows the window and the screen ventilation portion to remain in cooperation with one another, creating an air-permeable barrier to the vehicle's passenger compartment. Thus, a passenger who is smoking a cigarette and wishes to dispose of their ash outside of the vehicle may do so with a substantially decreased likelihood that outside debris will enter the vehicle's passenger compartment.
The system of the instant invention enables a significant advance in the state of the art. The instant invention is, in addition, widely applicable to a large number of applications. Variations, modifications, alternatives, and alterations of the various preferred embodiments may be used alone or in combination with one another, as will become more readily apparent to those with skill in the art with reference to the following detailed description of the preferred embodiments and the accompanying figures and drawings.
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The wind deflector and screened ventilation system with cigarette egress passageway () of the instant invention enables a significant advance in the state of the art. The preferred embodiments of the device accomplish this by new and novel arrangements of elements and methods that are configured in unique and novel ways and which demonstrate previously unavailable but preferred and desirable capabilities. The detailed description set forth below in connection with the drawings is intended merely as a description of the presently preferred embodiments of the invention, and is not intended to represent the only form in which the present invention may be constructed or utilized. The description sets forth the designs, functions, means, and methods of implementing the invention in connection with the illustrated embodiments. It is to be understood, however, that the same or equivalent functions and features may be accomplished by different embodiments that are also intended to be encompassed within the spirit and scope of the invention.
FIGS. 1 through 20
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Referring now generally to , the present invention is a wind deflector and screened ventilation system with cigarette egress passageway (). The device may generally be described as having a deflector (), a screen (), and a cigarette egress passageway ().
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In one embodiment, as seen in , the wind deflector and screened ventilation system with cigarette egress passageway () is shown attached to a vehicle door (). Although shows the wind deflector and screened ventilation system with cigarette egress passageway () on a front passenger side door, one skilled in the art would recognize that the wind deflector and screened ventilation system with cigarette egress passageway () of the present invention can be attached to the front driver side door, as well as both the driver and passenger side rear doors. One skilled in the art would also recognize that the wind deflector and screened ventilation system with cigarette egress passageway () of the present invention can be manufactured to be compatible with a multitude of different vehicle door and window configurations. Furthermore, the wind deflector and screened ventilation system with cigarette egress passageway () may be available as an aftermarket kit or as a vehicle's original equipment manufacturer (“OEM”) part.
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Referring generally to , the vehicle door () has a frame () and a window (). The frame () has a window opening () and a window receiving channel (). The window () has a top window edge (), a leading window edge (), a trailing window edge (), a bottom window edge (), an interior window surface (), an exterior window surface (), a sealed position (), and a point of exposure (). As best seen in , the window () is in the sealed position (), which refers to the position where the top window edge () is securely seated within the window receiving channel (). The sealed position () is also used to define the point of exposure (). Again referring to , the point of exposure () is defined as the point on the exterior window surface () where the window () is first exposed to the outside elements when the window () is in the sealed position (). The point of exposure () is used as a reference point to measure two distances, which will be described in detail below.
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FIG. 7
FIG. 10
The window deflector and screened ventilation system with cigarette egress passageway () includes a deflector (), a screen (), and a cigarette egress passageway (). As seen in , the deflector () has a deflector attachment portion () and a wind deflection portion (). The deflector () serves as a type of weather guard to prevent rain and other inclement weather from entering the vehicle's passenger compartment when the windows are slightly opened. The deflector () may be formed from a variety of materials, including, but not limited to, aluminum, acrylic, and other types of plastics and metals. The deflector attachment portion () provides the means for mounting the window deflector and screened ventilation system with cigarette egress passageway () to the frame () of a vehicle door (). In one embodiment of the instant invention, the deflector attachment portion () is a window channel cooperation member (), as seen in . The window channel cooperation member () may be designed to fully seat within the window receiving channel (), or alternatively, the window channel cooperation member () may only cooperate with a portion of the window receiving channel ().
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FIG. 9
In another embodiment of the present invention, the deflector attachment portion () is a surface mounting flange (), as seen in . The surface mounting flange () can be releasably attached to the frame () using any type of releasable attaching means including, but not limited to, two-sided acrylic foam tape, hook and loop fasteners, and plastic or metal clips, or combinations thereof.
FIG. 7
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FIGS. 7-11
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Referring again to , the wind deflection portion () includes a wind deflection portion distal edge (), a wind deflection portion proximal edge (), and a wind deflection portion proximal edge projection distance (). As seen in , the wind deflection proximal edge projection distance () is the longitudinal distance measured from the point of exposure () to the orthogonal projection of the wind deflection portion distal edge () on the exterior window surface (). Furthermore, the wind deflection portion () projects outwardly from the door () to deflect wind. By way of example only, set forth a geometry of the wind deflection portion () as a half-semi-oval. However, those having skill in the art would recognize and appreciate that other geometries for the wind deflection portion () may be utilized.
FIG. 8
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With reference now to , the screen () is releasably attached to the deflector () and includes a screen attachment portion () and a screen ventilation portion (). The screen () may be formed from a variety of materials, including, but not limited to, various types of plastic, wire mesh, natural fibers, and other materials suitable for functioning as an air-permeable screen. The screen attachment portion () provides means for releasably attaching the screen () to the deflector (). The screen attachment portion () may be attached to the deflector by any type of releasable attaching means, including, but not limited to, nuts and bolts, hook and loop fasteners, plastic or metal clips, adhesives or bonding agents, and fusion techniques, or combinations thereof.
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With continued reference to , the screen ventilation portion () has a screen ventilation portion distal edge (), a screen ventilation portion proximal edge (), a screen ventilation portion proximal edge projection distance (), seen only in , a screen ventilation portion interior surface (), and a screen ventilation portion exterior surface (). The screen ventilation portion proximal edge () is represented by a dashed line perpendicular to the screen ventilation portion interior () and exterior () surfaces, approximately even with the wind deflection portion proximal edge (). The screen ventilation portion proximal edge projection distance (), as seen in , is the orthogonal distance measured from the point of exposure () to the screen ventilation portion distal edge () along the exterior window surface ().
FIGS. 9-12
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Referring generally to , the screen ventilation portion () cooperates with the window () to provide an air-permeable barrier to the passenger compartment of a vehicle. The cooperation between the screen ventilation portion () and the window () allows the window () to be in an open position while the screen ventilation portion () lets fresh air into the passenger compartment of a vehicle and prevents insects, debris, and the like from entering the passenger compartment of the vehicle.
FIG. 1
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As seen in an embodiment of the instant invention in , the screen ventilation portion proximal edge projection distance () is less than the wind deflection portion proximal edge projection distance (). While in other embodiments of the instant invention the screen ventilation portion proximal edge projection distance () may be a greater distance, the screen ventilation portion proximal edge projection distance () should be no more than the wind deflection portion proximal edge projection distance (). Having a screen ventilation portion proximal edge projection distance () that is longer than the wind deflection portion proximal edge projection distance () would not be desirable. For example, if the screen ventilation portion proximal edge projection distance () was longer than the wind deflection portion proximal edge projection distance (), then the shielding effect of the deflector () would be decreased and the possibility of rain, for example, entering a vehicle's passenger compartment would be greatly increased.
FIGS. 12-17
FIG. 13
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Referring now generally to , a cigarette egress passageway () is formed in the screen (). The cigarette egress passageway () has an egress perimeter () configured to permit a cigarette to pass from the screen ventilation portion interior surface () to the screen ventilation portion exterior surface (), as seen in . Furthermore, the cigarette egress passageway () has a passageway length () and a passageway depth (). In an embodiment of the instant invention, the passageway depth () is configured such that it is less than 75% of the screen ventilation portion proximal edge projection distance (). In another embodiment of the present invention, the passageway length () is designed to be at least 7.62 centimeters (three inches). A passageway length () of at least 7.62 centimeters (three inches) is desirable due to the fact that the length of a typical cigarette measures anywhere from 8 to 10 centimeters (3.15 to 3.94 inches). Thus, a passageway length () of at least 7.62 centimeters (three inches) will allow the cigarette user to more effectively use the cigarette egress passageway () by decreasing the chance that a cigarette will unintentionally come into contact with a point on the egress perimeter (), which can lead to cigarette burns on the cigarette user and cigarette burns to the interior of the vehicle.
FIG. 14
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Referring now to , in one embodiment of the present invention, two points of the egress perimeter () are located on the screen ventilation portion distal edge () to form the cigarette egress passageway (). Thus, in the embodiment seen in , the egress perimeter () is a concave notch in the screen ventilation portion () along the screen ventilation portion distal edge (). As those skilled in the art will observe and appreciate, and by way of example and not limitation, the egress perimeter () may be formed using a variety of different geometries, such as a semi-oval, rectangle, square, or triangle, just to name a few.
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FIGS. 15-17
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In another embodiment of the present invention, the cigarette egress passageway () is disposed between the screen ventilation portion distal edge () and the screen ventilation portion proximal edge (), as seen generally in . Thus, in this embodiment the cigarette egress passageway () is a dedicated penetration through the screen () located between the screen ventilation portion distal () and proximal () edges. In one embodiment of the invention, the cigarette egress passageway () is defined by an oval perimeter (), as seen in . Referring now to , another embodiment of the present invention is shown with the cigarette egress passageway () having a rectangular perimeter (). In still another embodiment of the instant invention, as seen in , the cigarette egress passageway () is defined by a triangular perimeter ().
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The geometries of the egress perimeter () can be chosen based on functional or aesthetic value. For example, the triangular perimeter (), as seen in , allows for the window () to be open a minimal distance while providing a maximum passageway length (). Furthermore, when the window () is below the entire triangular perimeter (), the bottom point of the triangular perimeter () provides a wedge that the cigarette user may rest his or her cigarette in. However, the triangular perimeter () seen in may also be rotated 180° such that the base of the triangular perimeter is closer to the screen ventilation portion distal edge (). This particular geometry (not shown) creates a gradually increasing passageway length () as the window () is further opened. On the other hand, the rectangular perimeter (), as seen in , maintains a constant maximum passageway length (), but the passageway depth () increases as the window () is lowered. The oval perimeter (), shown in , provides a maximum passageway length () when the window () has moved below the top-half of the oval perimeter ().
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FIG. 18
In another embodiment of the present invention, a screen gasket () is releasably attached to the screen ventilation portion distal edge (), as seen in . The screen gasket () provides means for allowing the screen () to form a tighter seal against the exterior window surface (), thus preventing the chance that outside objects might enter into a vehicle's passenger compartment between the screen () and window (). The screen gasket () provides a further benefit in that it allows the screen () to cooperate more smoothly along the exterior window surface () and also prevents the screen () from scratching the exterior window surface (). The screen gasket () may be made from a wide variety of materials, including, but not limited to, rubber, plastic, silicone, PTFE, and various types of elastomers.
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In still another embodiment of the present invention, the egress perimeter () may be configured to have a cigarette retaining recess (), as seen in , which shows a rectangular perimeter (). The cigarette retaining recess () provides passengers with a notch, which may be used to secure their cigarette while they are not smoking. Referring now to , another embodiment of the instant invention is shown with a cigarette holding clip () releasably attached to the screen (). The cigarette holding clip () is configured to releasably secure a cigarette when not being used. The cigarette holding clip () may be formed from a variety of materials, including, but not limited to, various plastics and lightweight metals.
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Although the instant invention is described as having a deflector (), one having skill in the art would immediately recognize and appreciate that the wind deflector and screened ventilation system with cigarette egress () would retain several of its beneficial attributes with the deflector () removed, but overall would be less effective for its intended purpose.
Numerous alterations, modifications, and variations of the preferred embodiments disclosed herein will be apparent to those skilled in the art and they are all anticipated and contemplated to be within the spirit and scope of the instant invention. For example, although specific embodiments have been described in detail, those with skill in the art will understand that the preceding embodiments and variations can be modified to incorporate various types of substitute and or additional or alternative materials, relative arrangement of elements, and dimensional configurations. Accordingly, even though only few variations of the present invention are described herein, it is to be understood that the practice of such additional modifications and variations and the equivalents thereof, are within the spirit and scope of the invention as defined in the following claims. The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or acts for performing the functions in combination with other claimed elements as specifically claimed.
BRIEF DESCRIPTION OF THE DRAWINGS
Without limiting the scope of the present invention as claimed below and referring now to the drawings and figures:
FIG. 1
is an elevation view of an embodiment of the wind deflector and screened ventilation system with cigarette egress passageway attached to a vehicle door, not to scale;
FIG. 2
is an elevation view of a vehicle door, not to scale;
FIG. 3
is an elevation view of a vehicle door window, not to scale;
FIG. 4
is a cross-sectional view of a vehicle door frame and window, showing the window in a slightly open position, not to scale;
FIG. 5
FIG. 2
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is a cross-sectional view of the vehicle door taken along section line - of , showing the window in a sealed position, not to scale;
FIG. 6
is a cross-sectional view of a vehicle door frame and window, showing the window in a position between the sealed position and the open position, not to scale;
FIG. 7
is a cross-sectional view of an embodiment of the deflector of the wind deflector and screened ventilation system with cigarette egress passageway, not to scale;
FIG. 8
is a cross-sectional view of an embodiment of the wind deflector and screened ventilation system with cigarette egress passageway, not to scale;
FIG. 9
is a cross-sectional view of an embodiment of the wind deflector and screened ventilation system with cigarette egress passageway attached to a vehicle door, showing the window in an open position, not to scale;
FIG. 10
is a cross-sectional view of an embodiment of the wind deflector and screened ventilation system with cigarette egress passageway attached to a vehicle door, showing the window in the sealed position, not to scale;
FIG. 11
is a cross-sectional view of an embodiment of the wind deflector and screened ventilation system with cigarette egress passageway attached to a vehicle door, showing the window in the sealed position, not to scale;
FIG. 12
is an elevation view of a vehicle door with an embodiment of the screen of the wind deflector and screened ventilation system with cigarette egress passageway attached to the vehicle door, not to scale;
FIG. 13
FIG. 12
13
13
is a cross-sectional view of the vehicle door with an embodiment of the screen of the wind deflector and screened ventilation system with cigarette egress passageway attached to the vehicle door taken along section line - of , illustrating a cigarette passing through the cigarette egress passageway, not to scale;
FIG. 14
is an elevation view of an embodiment of the screen of the wind deflector and screened ventilation system with cigarette egress passageway, not to scale;
FIG. 15
is an elevation view of an embodiment of the screen of the wind deflector and screened ventilation system with cigarette egress passageway, not to scale;
FIG. 16
is an elevation view of an embodiment of the screen of the wind deflector and screened ventilation system with cigarette egress passageway, not to scale;
FIG. 17
is an elevation view of an embodiment of the screen of the wind deflector and screened ventilation system with cigarette egress passageway, not to scale;
FIG. 18
is a cross-sectional view of an embodiment of the wind deflector and screened ventilation system with cigarette egress passageway attached to a vehicle door, showing the window in an open position, not to scale;
FIG. 19
is an elevation view of an embodiment of the screen of the wind deflector and screened ventilation system with cigarette egress passageway, not to scale; and
FIG. 20
is an elevation view of an embodiment of the screen of the wind deflector and screened ventilation system with cigarette egress passageway, not to scale. |
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