text:
## "Biotechnological Applications to Produce Animal-free Meat and Milk: Ethical Considerations"

John D. Loike*

Touro College and University Systems, New York, USA

*Corresponding author: John D. Loike, Touro College and University Systems, New York, USA. Tel: +12123051510; Email: jdl5@cumc.columbia.edu

Received Date: 17 May, 2018; Accepted Date: 06 June, 2018; Published Date: 14 June, 2018

1.       Introduction

Biotechnologies have consistently improved the production of many types of foods [1], including cheeses [2], wine [3], fruits [4], and crops [5]. However, new stem cell and gene editing biotechnologies may cause a paradigm shift in these industries. In this article, I explore the potential scientific, health, cultural/religious, ethical, and economic ramifications of using these innovative biotechnologies to make milk, meat and poultry products from non-traditional food sources.

1.1.  Milk

Cow’s milk is considered a staple in many people’s diets. It is consumed as a beverage, poured on cereal and added to smoothies, tea or coffee. Cow’s milk boasts an impressive nutrient profile. It’s rich in high-quality protein and important vitamins and minerals, including calcium, phosphorus and B vitamins. The history of including milk in the human diet is rich and fascinating [6]. Milk consumption has been part of the human diet for thousands of years. In ancient Egypt milk and other dairy products were generally reserved for Royalty or the wealthy. There are many reasons and theories why milk became such a popular food in western countries. Some believe that as infant mortality rose in American cities in the 1800’s milk emerged as a healthy food for children.  Clearly US government sponsored ads over the past 40 years has  influenced the acceptance of milk as a staple of the western diet.

However, there are many health-related concerns associated with the consumption of milk. For example, there is a large market for non-dairy milk targeted to individuals who present with milk (e.g., lactose) allergies, people to subscribe to vegan diets, and individuals who are concerned with the presence of antibiotics, pesticides and hormones commonly found in commercial milk. It is only recently that companies are developing synthetic milk that has the same composition as cow’s milk without antibiotics, pesticides and hormones. Perfect Day is a California start-up company that is developing milk proteins derived from genetically engineered yeast [7]. They expect to be on the market sometime in 2020.

1.2.  Meat

Red meat is long been established as an important dietary source of protein and essential nutrients including iron, zinc and vitamin B12. According to some scholars (e.g., Leslie Aiello) human beings starting to eat calorie-dense meat and marrow instead of the low-quality plant diet of apes, to provide enough extra energy at each meal to help fuel a bigger brain. Digesting a higher quality diet and less bulky plant fiber  have allowed these humans to have much smaller guts. The energy freed up as a result of smaller guts could be used by the greedy brain [8]. Today, the rise in “fast foods” that rely on meat consumption has perpetuated throughout our culture. Yet, while meat is an important staple of western diets, there are many reports that describe health risks associated with meat consumption including increased risk of cancer and heart disease [9]. Moreover, the huge demand for land required to grow animals for meat has triggered an examination whether alternative methods to produce healthier meat products without animals has resulted in the development of synthetic or cloned meat and poultry.

2.       Applying Synthetic Biotechnologies to Make Animal-Free Milk

The food tech company Perfect Day is creating an "animal-free milk" product made from proteins derived from yeast fermentation [10]. Using a Genetically Modified (GM) strain of yeast (nicknamed Buttercup) as the biological machinery, this company will produce a milk product that retains the taste and texture of cow’s milk. While the proteins in this milk (such as casein and whey) product will be synthesized from yeast, the fats in this product will be obtained from vegetable sources. Moreover, sugars and minerals will be added to the product to improve its texture and taste. Although Perfect Day’s milk will be lactose-free and vegan, the casein and whey contained in this product are still potentially antigenic and would restrict its consumption by people with milk allergies. One of the main motivations to produce milk in yeast is that the taste of plant derived non-dairy milk is not the same as animal generated milk [11].

3.       Applying New Biotechnologies to Make Animal-Free Meats

The basic composition of meat includes water, protein and amino acids, minerals, fats and fatty acids, vitamins and other bioactive components, and small quantities of carbohydrates. While there are many companies focusing on making tasty plant-derived meats and poultry, there are various considerations that have stimulated companies to create animal-free meats that have the same composition and cooking qualities (e.g., the ability to barbecue) as animal derived meats. Stem cell biotechnologies are being employed to create animal-free meats that are referred to as “clean meat”, “cloned meat” or “synthetic meat” [12]. In creating animal-free meats, a small sample of muscle is carefully removed, via a syringe, from a living cow’s neck muscle without harming the animal. Muscle precursor cells, called myosatellite cells, are then separated from other cells in the extracted tissue and are grown in vitro in a bioreactor. These myosatellite cells proliferate and eventually differentiate to generate muscle fibers that form the essential muscle components of animal-derived meat. A similar stem cell process is being applied to generate cloned poultry meats from chickens [13].

The possibility of growing animal-free meat within an industrial setting has long captured the imagination of many people. Winston Churchill suggested in 1931: “We shall escape the absurdity of growing a whole chicken in order to eat the breast or wing, by growing these parts separately under a suitable medium” [14]. In 2002, scientists from NASA reported the need to develop viable means of supplying safe, healthy, nutritious food to Space voyagers on long journeys [15]. They developed a method to culture in the laboratory adult dorsal abdominal skeletal muscle from goldfish as a source to generate crude “meat” explants that resembled fresh fish filets. They hoped the method of growing fish filets in the laboratory could serve as a renewable food source for human space travel. Unfortunately, cultured goldfish fillets never caught on [16].

In 2008, People for the Ethical Treatment of Animals (PETA) offered a $1 million prize to the first company to bring lab-grown chicken meat to consumers by 2012 [17]. Even a 2-year extension from the original deadline of 2012 wasn’t enough to lure applicants for PETA’s prize. Even though their$1 million prize expired in 2014, PETA continued to fund research at the University of Missouri to develop methods to produce cloned meat [18]. In November 2009, scientists from the Netherlands announced they had managed to grow meat in the laboratory using the cells from a live pig. The project, was backed by a sausage manufacturer and a 2-million-pound grant from the Dutch government [19]. Time magazine declared cultured meat production to be one of the 50 breakthrough ideas of 2009. Since then, over 30 laboratories from around the world have announced that they are working on cultured meat research. The commercial potential of such meats is evident as Tyson Ventures, the venture capital arm of Tyson Foods, Inc., invested an undisclosed amount in Memphis Meats, a developer of cultured meat processed through animal cells [20]. In addition, the cost of producing one hamburger has dropped significantly. The price of the first lab-grown beef burger, created by Mosa Meats, was equivalent to about $1.2 million per pound, retail. Now, lab-grown hamburger runs for about$11.36 per pound [21].

4.       Advantages in The Production of Synthetic Milk, Cloned Meat and Poultry

The potential advantages to cultured meat and milk include environmental benefits [22], health benefits [23], and marketing benefits (such as animal welfare by killing fewer animals to produce meat) [24].

4.1.              Environmental Advantages: Environmentally, the production of animal-free milk requires 98% less water, 91% less land, and 65% less energy than milk produced in a typical dairy farm. The production of animal-free milk will also significantly reduce greenhouse gas emissions,  as the cows in dairy farms are responsible for roughly 3 percent of global greenhouse gas emissions each year [25].

Traditional methods to produce meat from livestock also requires large amounts of cattle, pigs, and chickens that take up about 30 percent of the planet’s ice-free land and can wreak incredible environmental damage. Cattle and pigs generate large amounts of methane and waste that pollute the atmosphere, rivers, and streams. Moving meat production from the farm to industrial bioreactors could mean a significantly greener industry by lowering methane production and animal waste as well as utilizing significantly less land, water, and energy [26].

4.2.              Health Benefits [27]: It is reported that organic milk contains 50% more omega-3 fatty acids than conventionally produced products and also contains less saturated fat than non-organic.  One advantage of producing cultured milk is the capacity to manipulate the product’s flavor, fatty acid composition, fat content and ratio of saturated to poly-unsaturated fatty acids through the composition of the culture medium or co-culturing with other cell types.  Furthermore, health aspects of the milk can be enhanced by adding factors like certain types of vitamins to the culture medium which might have an advantageous effect on the health [28].

The dairy industry has long advertised its products as an excellent source of nutrition. Dairy products provide a rich source of many nutrients including protein, calcium, magnesium, and vitamins A and B. Calcium is one of the key nutrients believed to be responsible for the positive findings between dairy and prevention of weight gain.

During the last decade, more than a dozen studies found that higher intakes of milk or dairy calcium may help in preventing weight gain and/or changes in body composition among adolescents [29]. However, other studies show no relationship between dairy consumption and body weight or body fat composition. This has raised a considerable controversy about whether or not the amount or level of dairy intake is associated with loss or gain of body weight or fat. One study [8] reported that dairy intake to be significantly associated with body growth in an adolescent population. In addition, this study found total dairy intake to be positively associated with fat mass, and overall BMI, but only in boys. Many more studies are needed to scientifically assess the benefits and risks of milk free diets.

There are many reports that emphasize the health benefits of diets that exclude meat or poultry consumption. However, very few studies rigorously evaluate and compare the health outcomes of people on omnivorous, vegetarian, and vegan diets as distinct experimental groups. In general, people on vegan diets have lower blood pressure, and lower fasting triacylglycerol and glucose concentrations than omnivorous subjects, as well as a biochemical profile found to be cardio-protective and diabetes protective. Vegetarian diets are also reported to improve insulin resistance, lower diabetes risk, and lower cardiometabolic disease risk [30]. Some research suggests that vegetarian diets, especially vegan diets, are associated with lower Bone Mineral Density (BMD), but this does not appear to be clinically significant [23]. The problem with many of these studies is that they only compared the health outcomes of vegans to omnivores, and not to individuals who are vegetarian. While there are reports of the health risks associated with meat [31], other reports [32] suggest that diets which allow moderate amounts of animal products may be as protective against disease as vegan diets. For example, a pooled analysis of five prospective cohort studies, involving 76,000 subjects, found that both vegetarians and those who followed a “prudent” diet allowing small amounts of red meat benefited from a reduced risk of coronary heart disease and type 2 diabetes [33]. Another health advantage to producing cloned poultry in bioreactors instead of using live birds is that the reduction in chicken farms would lower the risk of viral epidemics (avian flu) that can spread via chickens [34].

Natural meat is reported to contain some cancer-causing agents. In muscle tissue, heme iron is found primarily as the oxygen-binding protein, myoglobin. Heme iron can cause DNA damage and induce the formation of N-nitroso compounds, some of which are potent carcinogens. A study that followed nearly 200,000 post-menopausal women found that the amount of heme iron in their diet was positively associated with an increased risk of breast cancer [35]. Other studies show connections between heme iron intake and various forms of cancer [36]. According to its producers, lab-cultured beef or pork can be made completely free of heme iron. But because lab-grown meat lacks a circulatory system, it’s kept in a very high-oxygen environment, which has the unintended consequence of reducing cellular myoglobin expression resulting is less iron.

4.3.              Marketing Advantages: There are many individuals across the globe who are concerned with animal cruelty issues associated with farm-based productions of meat and poultry [37]. Cloning meat in bioreactors would reduce the number of animals that must be sacrificed and reduce the potential for animal cruelty. In addition, there may be a significant marketing advantage that the public would find cloned meat tastier and  a better “meat’ to barbecue over plant-derived meats. Finally, this technology can be applied to create a new market of exotic cultured meats or even meat from rare or endangered animals.

Americans love hamburgers and chicken wings. 50 billion hamburgers are sold in America each year and one billion chicken wings are consumed on Super Bowl Sunday alone. Recent polls report [38]  that 96% of Americans eat fast food meats. Moreover, Americans love the taste of real meat and believe plant made meat does not adequately mimic the taste, convenience, and barbecuing potential of animal-based meats. Surveys about meat consumption report that:

1.                   Eight in ten Americans report eating at fast-food restaurants at least monthly, with almost half saying they eat fast food at least weekly [39]. About two-thirds of Americans report that their doctors share the benefits of a healthy diet with them, yet, they prefer to eat fast food regularly.

2.                   Americans tend to ignore nutritional information while eating out: Less than half of Americans  say that they pay a "great deal" or a "fair amount" of attention to nutritional information on menus.

One outcome from surveys is that not only do Americans enjoy eating out at fast food establishments but the health of foods offered in these establishments is not a major concern. Thus, one might conclude that given an equal pricing, the majority of Americans would favor animal-free meat over plant-derived meats. A 2017 online survey of US participants [40] concluded that most respondents were willing to try cloned meat, but only one third were definitely or probably willing to eat it regularly or as a replacement for farm-produced meat. This survey also showed that vegetarians and vegans were more likely to perceive potential benefits of cloned meat compared to farmed meat but were less likely to try it than meat eaters. The main concerns of those who did not want to try cloned meat were an anticipated high price, limited taste, and a concern that the product was unnatural and therefore unhealthy. The results of the survey showed that people in the USA are likely to try cloned meat, but few believed that it would replace farmed meat in their diet.

5.       Challenges in the Production of Cloned Meat and Poultry

There are various scientific challenges in culturing myosatellite cells from animal muscle to produced cloned meat. First, myosatellite cells are a rare cell within muscles and display limited regenerative potential. Second, these cells are prone to malignant transformation in long term culture. While the health risks of consuming such malignant cells have not been extensively studied [41],  there is concern whether the consumer might want to avoid consuming “bovine cancer cells” that may be present in cloned meat. To minimize malignant transformation by these cells, companies producing cloned meat will use stem cells that are constantly re-harvested from new cows in the production system. Third, at this time, maintaining these cells in culture  requires the use of animal (bovine) serum, in the media that nourishes these cells. Bovine serum is a costly component that may not be readily acceptable to potential consumers because obtaining animal serum requires the sacrifice of many fetal calves. The use of animal serum in culturing meat cells also questions whether this process will really reduce animal use and suffering. However, scientists are rapidly trying to develop serum-free media to avoid the use of animal serum in producing cloned meat. Finally, producers of cloned meats must demonstrate that these meats have the same taste, texture, juiciness, and barbecue potential as animal-derived meat.

6.       Companies Gearing Up to Produce Animal-Free Meat and Poultry

The technology to produced laboratory-grown meat is moving along at a rapid pace. Mosa Meats [42], based on the research and technology developed by Mark Post. produced the first commercial cloned hamburger for culinary experts to taste. In 2013, Dr. Post made headline news around the world for producing the world’s first lab-grown burger that was cooked and eaten at a news conference in London. The burger was made from real meat grown in a lab (using 20,000 strips of muscle tissue) at a cost of $325,000. As stated above, a recent ABC News interview with Post reported that the cost of a cloned burger has dramatically dropped to just over$11 for a burger (\$80 per kilogram of meat) [43].

The first cloned burger was cooked by Chef Richard McGeown of Couch’s Great House Restaurant in Cornwall and tasted by Hanni Ruetzler, a food critic and food researcher at the Future Food Studio in London. Ruetzler described the experience as follows: “There is really a bite to it, there is quite some flavor with the browning. I know there is no fat in it so I didn’t really know how juicy it would be, but there is quite some intense taste; it’s close to meat, it’s not that juicy, but the consistency is perfect. This is meat to me… It’s really something to bite on and I think the look is quite similar.”

Since the first public trial, several other startups have made advances in the field. Memphis Meats [44], a Silicon Valley startup founded by a cardiologist, launched a video in February 2016 showcasing its cultured beef meatball. In March 2017, it showcased chicken tenders and duck a-l’orange, the first cultured poultry-based foods shown to the public [45]. At a tasting event in March of 2017, people who tasted lab-grown chicken strips swear it tastes just like chicken [46].

Other major corporate players in this market include:

1.                   An Israeli company, SuperMeat [47], ran a viral crowdfunding campaign in 2016 to produce cloned chicken meat.

2.                   Perfect Day Foods: a startup focused on animal-free dairy products.

3.                   Shojinmeat: a Japanese biohacker community that is developing cloned meat.

4.                   Finless Foods, a California-based biotech startup, takes a small sample of cells from a living marine animal and cultured them in a brewery-like environment in the shape of a fish fillet. The process takes several months to complete, with a product that features the same texture and appearance of real fish meat. Unlike commercial fish farming, Finless Foods’ process requires no antibiotics, hormones or harmful chemicals and delivers only the part of the fish that consumers want to eat [48].

5.                   New Harvest, Good Food Institute, and the Modern Ag Foundation: Non-profit research institutes dedicated to the field of cellular agriculture, focused on the development of animal-free eggs, milk, meat, and more non-animal products.

7.       Ethical, Cultural/Religious Issues with Animal-Free Meat and Milk

As mentioned above, animal cruelty is one of the most discussed ethical issues in meat production. Despite many governmental safeguards, there is still evidence of animal cruelty in generating meat from cows, pigs and chickens [49]. Many articles [50] have been written that discusses the issue of animal cruelty in the meat industry. In principle, the production of cloned meat should help resolve this ethical concern.

Aside from animal cruelty, there are several other ethical concerns that emerge from producing cloned meat. First, potential consumers may be repulsed by cloned meat on grounds that it is unnatural or some perversion of biology. Second, the public may be concerned whether cloned meat is harmful to human health. Although there is much confusion in the USA about what is considered a healthy diet [51], the publication of cookbooks on the use of cloned meat is a first step in marketing a product to the public. Two recent books [The “In Vitro Meat Cookbook”, by Koert Mensvoort, and Hendrik-Jen-Grievink, and “In vitro meat: protein for twelve billion?” by Adam May, and Jean Fleming], may alleviate some of the health concerns and fears of consuming cloned meat. Another final issue is how vegetarians and vegans will view cloned meat compared to plant-derived? The answer to this question will depend on the reasons why people choose to eliminate meat from their diets. If the reasons are to avoid cruelty to animals, then cloned meat would be a valid alternative to animal-derived meat. However, if the concerns are related to the perceived negative health effects of consuming animal products, then cloned meat will still require the use of animal cells and may not be a solution for vegetarians.

8.       Religious Concerns Related to Cloned Meat

Two major religions of the world - Islam and Judaism - have restrictions on the consumption of meat and dairy products. In Islam, animal meat cannot be porcine in origin and must be obtained from appropriate animals (“Halal”- which means lawful or permissible). In Judaism, animal meat must be obtained from “Kosher animals” (primarily domesticated animals, such as cow, sheep, goat, deer, and buffalo). In Halal- and Kosher-certified facilities, meat must be obtained from animals that are sacrificed using a slaughtering procedure designed to reduce animal suffering.

There are fundamental religious principles regarding Halal meats. For example, the majority of Islamic scholar’s view that it is permissible to eat horse meat.  It was narrated from Jaabir ibn ‘Abd-Allaah that he said: “On the day of Khaybar, the Messenger of Allaah forbade eating the flesh of domestic donkeys, but he granted a concession with regard to horses.” Narrated by al-Bukhaari, 3982; Muslim, 1941.  In contrast, the following living organisms are not Halal-acceptable [52]:

1.                   Insects considered ugly or filthy such as worms, lice, flies, etc.

2.                   Animals with fangs such as tigers, lions, cats etc,

3.                   Birds that have talons with which they catch their prey such as owls, eagles, etc.

4.                   Animals which Islam encourages to kill such as scorpions, centipedes, rats etc.

5.                   Dogs

6.                   Animals which Islam forbids to kill such as bees etc.

7.                   Animals which have toxins, poisons or produce ill effects when eaten such as some fish etc.

8.                   Amphibian animals such as crocodiles, turtles, frogs etc.

9.                   Lawful animals not slaughtered according to Islamic rites. (Fish is exempt from slaughtering).

10.                Carrion or dead animals [53].

There also are at least two differences regarding the views of Muslim and Jewish scholars regarding cloned meat. First, as stated above there are many “non-Kosher” animals that are acceptable as Halal-certified. Second, observant Jews will not consume or cook milk and meat products together. Both Jewish and Islamic legal scholars have stated that cloned meat would only be permissible for consumption if the cloned meat were produced from muscle-derived stem cells that were obtained from either a Kosher or Halal-accepted animal and certainly not porcine-derived. Second, the muscle sample must be obtained from the animal after slaughter because according to both Jewish and Islamic law one is not allowed to extract a tissue from a live animal for food consumption. Third, the maintenance of cultured meat in the bioreactors currently requires the cells to be grown in media containing bovine serum. Bovine serum used in traditional bioreactors are not typically obtained from Kosher or Halal certified companies and might render the meat as non-Kosher or non-Halal certified. In the Jewish tradition, for example, the biblical verse that forbids the consumption of blood might apply to the consumption of animal serum as well. In the Islamic tradition (according to Sharee’ah), the presence of blood in meat is one reason why it is forbidden to eat meat that has not been slaughtered. Sharee’ah directs that the slaughtered animal should be drained of blood as completely as possible, and that is because of the extreme spiritual harm that would result from eating its blood. Currently, there are no publications that address the specific religious issue of using bovine serum to produce cloned meat. However, in the future, the religious concerns of using bovine serum in cloning meat may not be germane for either religious Jews or Muslims because commercial companies are trying to eliminate the use of bovine serum for maintaining stem cells in bioreactors. For example, SuperMeat has developed a technology to grow and maintain poultry-derived stem cells in the absence of animal serum in order to generate kosher cloned chicken meat.

Regarding Jewish law, there are some additional complications regarding the use of kosher cloned meat. As mentioned above Jewish law prohibits the consumption or cooking of meat and milk products together. Thus, the question is whether cloned meat is viewed as animal meat or another byproduct of the animal (similar to milk) that does not carry a meat status? I propose that since the biological origin and source of animal meat is myosatellite cells then any product subsequently generated from these stem cells will retain their original status as essential components of muscle or meat. Thus, many Jewish scholars would classify cloned meat as being technically considered equal to animal meat and would prohibit its consumption with any dairy products [54]. This ruling differs from the general Jewish legal/religious ruling of gelatin. Gelatin is not considered to possess the status of meat because it is derived from bones or skin and not flesh. Thus, many Jewish authorities permit the consumption of gelatin with dairy products.

Cloned poultry (obtained from stem cells of a chicken and turkey) will also require that the bird source be a Kosher species and that the cells be obtained after the bird is slaughtered. However, cloned chicken or turkey meat differs from cloned beef meat in one important aspect. From a Jewish perspective, meat obtained from poultry is biblically classified as a non-meat product. It is only through a rabbinical decree that meat obtained from poultry is not cooked or consumed with any dairy products. Because this is a rabbinical decree, some current Jewish authorities would not institute another rabbinical prohibition above and beyond an existing rabbinical decree. Thus, using myosatellite cells to generate chicken meat may render this product as a non-meat product or parve [40,55]. Such a rabbinical ruling would allow the production of cheeseburgers using cloned chicken meat and real cheese.

Currently, there may consenting views from either Jewish or Islamic scholars regarding the religious issues of cloned meat discussed in the article. However, history has taught us that only when a product actually hits the public market, will religious scholars carefully review how the product is made and re-assess the religious status of such a product. Thus, we expect that until cloned meat or yeast-derived milk products appears on the market shelf, there will be various opinions issued by Jewish and Islamic scholars. If history is a guide, I would predict that many Jewish and Islamic scholars might readily accept cloned meat derived from cows or poultry, rather than pigs, that has been properly slaughtered as being a religiously permitted food.  However, it is much more difficult to predict how the vegan communities will accept cloned meat as compared to plant-derived meat. I imagine that marketing, texture, and taste - along with the health risks, health benefits, and affordability - will be important considerations in how the vegan community will accept or reject the incorporation of cloned meat and poultry into their diets.

Regarding milk-derived from yeast, the major religious issue will be from the Jewish religious community. I predict that such products will be classified within Jewish law as a non-dairy product because it is not obtained from a mammalian source. Thus, milk-derived from yeast will be no different than soymilk, rice milk or almond milk and will be considered non-dairy (or Parve) in Jewish Law - allowing it to be consumed with any animal-derived meats or poultry.

9.       Conclusions

There is no doubt that the use of gene editing and stem cell biotechnologies will continue to have a huge cultural, environmental, and financial impact on the food industry. Besides using these technologies to generate meat and milk, many companies are using genetic methodologies to produce wine products [56], opioids [57], cheeses [58], and gelatin [59]. The practical challenges for this industry are four-fold. Will their products be healthier than animal-derived products, less expensive, mimic the taste of the animal-derived counterparts, and will the production of these products have a positive impact on resources and the environment?  From an ethical perspective, vegan, Jewish, and Muslim communities will have to decide under what conditions and provisions will cloned meat be a viable food entity.

Citation: Loike JD (2018) Biotechnological Applications to Produce Animal-free Meat and Milk: Ethical Considerations. Food Nutr J 3: 181. DOI: 10.29011/2575-7091.100081


text:
# Matchings in Graphs Variations of the Problem

Ontology type: schema:Chapter

### Chapter Info

DATE

2007-01-01

AUTHORS ABSTRACT

Many real-life optimization problems are naturally formulated as questions about matchings in (bipartite) graphs.We have a bipartite graph. The edge set is partitioned into classes E1, E2, . . . , Er. For a matching M, let si be the number of edges in M ∩ Ei. A rank − maximal matching maximizes the vector (s1, s2, . . . , sr). We show how to compute a rank-maximal matching in time O(r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sqrt{nm}$\end{document}) [IKM + 06].We have a bipartite graph. The vertices on one side of the graph rank the vertices on the other side; there are no ties. We call a matching M more popular than a matching N if the number of nodes preferring M over N is larger than the number of nodes preferring N over M. We call a matching popular, if there is no matching which is more popular. We characterize the instances with a popular matching, decide the existence of a popular matching, and compute a popular matching (if one exists) in time O(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sqrt{nm}$\end{document}) [AIKM05].We have a bipartite graph. The vertices on both sides rank the edges incident to them with ties allowed. A matching M is stable if there is no pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(a, b) \in E{\backslash}M$\end{document} such that a prefers b over her mate in M and b prefers a over his mate in M or is indifferent between a and his mate. We show how to compute stable matchings in time O(nm) [KMMP04].In a random graph, edges are present with probability p independent of other edges. We show that for p ≥ c0/n and c0 a suitable constant, every non-maximal matching has a logarithmic length augmenting path. As a consequence the average running time of matching algorithms on random graphs is O(m log n) [BMSH05]. More... »

PAGES

1-2

### Book

TITLE

Combinatorial Optimization and Applications

ISBN

978-3-540-73555-7
978-3-540-73556-4

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-540-73556-4_1

DOI

http://dx.doi.org/10.1007/978-3-540-73556-4_1

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https://app.dimensions.ai/details/publication/pub.1044019902

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"description": "Many real-life optimization problems are naturally formulated as questions about matchings in (bipartite) graphs.We have a bipartite graph. The edge set is partitioned into classes E1, E2, . . . , Er. For a matching M, let si be the number of edges in M\u2009\u2229\u2009Ei. A rank\u2009\u2212\u2009maximal matching maximizes the vector (s1, s2, . . . , sr). We show how to compute a rank-maximal matching in time O(r\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\sqrt{nm}$\\end{document}) [IKM\u2009+\u200906].We have a bipartite graph. The vertices on one side of the graph rank the vertices on the other side; there are no ties. We call a matching M more popular than a matching N if the number of nodes preferring M over N is larger than the number of nodes preferring N over M. We call a matching popular, if there is no matching which is more popular. We characterize the instances with a popular matching, decide the existence of a popular matching, and compute a popular matching (if one exists) in time O(\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\sqrt{nm}$\\end{document}) [AIKM05].We have a bipartite graph. The vertices on both sides rank the edges incident to them with ties allowed. A matching M is stable if there is no pair \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$(a, b) \\in E{\\backslash}M$\\end{document} such that a prefers b over her mate in M and b prefers a over his mate in M or is indifferent between a and his mate. We show how to compute stable matchings in time O(nm) [KMMP04].In a random graph, edges are present with probability p independent of other edges. We show that for p \u2265 c0/n and c0 a suitable constant, every non-maximal matching has a logarithmic length augmenting path. As a consequence the average running time of matching algorithms on random graphs is O(m log n) [BMSH05].",
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text:
# Python line condenser function

This is a function I just wrote that tries to condense a set of strings into grouped lines. It actually works, but looks ugly. Is there a better way to achieve the same thing?

Take 4 filtering empty strings first and dropping the if line on the final yield

def mergeToLines(strings, length=40, sep="    "):
strs = (st for st in sorted(strings, key=len, reverse=True) if st)
line = strs.next()
for s in strs:
if (len(line) + len(s) + len(sep)) >= length:
yield line
line = s
else:
line += sep + s
yield line

The idea is that it takes a bunch of strings, and combines them into lines about as long as lineLength. Each line has one or more of the original strings in it, but how many isn't clear until I actually get to slotting them in. Also, note that if a single entry in the list is longer than lineLength, it's still added as its own line (dropping elements is not an option here).

• In take 3, if line: is redundant unless you have empty strings in strings. If you do have empty strings, you probably should filter out all of them earlier on. – Janne Karila Apr 26 '13 at 6:36
• @JanneKarila - good point. – Inaimathi Apr 26 '13 at 13:46

RECURSIVE SOLUTION: NOTE: it assumes that the input strings is already sorted. You could sort this at every call of the function, but this seems like unnecessary overhead.

def recursiveMerge(strings=[], current_line='', line_length=40, result=[]):
""" Gather groups of strings into a line length of approx. 40 """

# base case
if not strings:
result.append(current_line)
return result

else:
next_line = strings.pop(0)

# if current_line + next_line < line_length, add them
# otherwise append current_line to result
if len(current_line + next_line) < line_length:
current_line = '    '.join([current_line, next_line]).lstrip(' ')
else:
result.append(current_line)
current_line = next_line

# recursively call function
return recursiveMerge(strings, current_line = current_line, result=result)

ITERATIVE SOLUTION:

def myMerge(strings, line_length=40, res=[]):
""" Gather groups of strings into a line length of approx. 40 """

# sort the list of string by len
strings = sorted(strings, key=len, reverse=True)

# loop through the group of strings, until all have been used
current_line = strings.pop(0)
while strings:
next_line = strings.pop(0)

# if current_line + next_line shorter than line_length, add them
# otherwise, append current_line to the result

if len(current_line + next_line) < line_length:
current_line = '    '.join([current_line, next_line])
else:
res.append(current_line)
current_line = next_line

# add any remaining lines to result
res.append(current_line)

return res

• l_len should be line_length, no? – William Morris Apr 26 '13 at 0:59

Your code isn't so ugly, guys :) But I can write it better

def cool_merger(strings, line_len=40, sep="    "):
ss = list(sorted(strings, key=len, reverse=True))
return [line for line in [(lambda n, s: s if n+1 < len(ss) and (len(s) + len(ss[n+1]) >= line_len) else (ss.__setitem__(n+1, sep.join((str(ss[n]), str(ss[n+1]))))) if len(ss) > n+1 else ss[n] )(n, s) if len(s) < line_len else s for n, s in enumerate(ss)] if line != None]

(joke)

By the way, works fast.

• +1 for it being crazy short! Very nice. Wouldn't want to be your team-mate who had to maintain it though :) – Nick Burns Apr 26 '13 at 0:19
• @NickBurns Thanks :) Surely it's written not for production but just for fun. Maintainability should be on the first place. – gaart Apr 26 '13 at 6:53


text:
### Investigating Ulam sequences题号：167 难度： 75 中英对照

For two positive integers a and b, the Ulam sequence U(a,b) is defined by U(a,b)1 = a, U(a,b)2 = b and for k > 2, U(a,b)k is the smallest integer greater than U(a,b)(k-1) which can be written in exactly one way as the sum of two distinct previous members of U(a,b).

For example, the sequence U(1,2) begins with
1, 2, 3 = 1 + 2, 4 = 1 + 3, 6 = 2 + 4, 8 = 2 + 6, 11 = 3 + 8;
5 does not belong to it because 5 = 1 + 4 = 2 + 3 has two representations as the sum of two previous members, likewise 7 = 1 + 6 = 3 + 4.

Find ∑U(2,2n+1)k for 2 ≤ n ≤10, where k = 1011.

### Code

import java.util.*;

public final class p167 {
public static void main(String[] args) {
long start=System.nanoTime();
String result = run();
long end=System.nanoTime();
System.out.println(result);
System.out.println( (end-start)/1000000 + "ms" );
}

static public String run() {
final long k=100000000000L;
long ans=0;
for(int n=2;n<=10;n++)
ans+=solve(2*n+1,k);
return ans+"";
}

final static long N[]=new long[]{0,0,32,26,444,1628,5906,80,126960,380882,2097152};
final static long D[]=new long[]{0,0,126,126,1778,6510,23622,510,507842,1523526,8388606};
static long solve(int v,long k){
long n=N[v>>1],d=D[v>>1];
long even=(v<<1)+2;
long r=k%n+n;
long p=k/n-1;
long i=0;
long now=0;
boolean flag=true;
while(i<r){
if(i==0){
now=2;i++;
}else if(i==1){
now=v;i++;
}else if(flag){
now+=2;
if(now<even) i++;
else{
i+=2;
flag=false;
}
}else{
boolean galf=true;//标记now是否在序列中
while(true){
now+=2;
boolean con=set.contains(now-even);
if(galf && con)
galf=false;
else if(galf || con)
break;
}


text:
## Stream: Lean Together 2021

### Topic: Tactics

#### Lukas Stevens (Jan 06 2021 at 18:44):

In @Jannis Limperg 's talk, @Patrick Massot asked whether the new tactic framework would support simprocs and it was not clear what those exactly are. I'll try to clarify them to the best of my understanding: to define a simproc, you give a term pattern for which the simplifier should look for in the goal. Additionally you give a conversion, that is, a function that for a given term $t$ gives you a theorem $t \equiv t'$ where $t'$ is a simplified version of $t$.

I will explain why you would do this instead of using the simplifier by giving an example: suppose we register a simproc that looks for occurences of $\text{is\_prime}\ c$ where $c$ is a literal number. Then, our function looks at the number and efficiently (without going through the inference kernel) computes a certificate that proves whether the number is prime or not. We then use the simplifier to turn the certificiate into a proof of primality. With this approach we avoid using the simplifier to compute the certificate which is slow (at least in Isabelle).

The architecture that @Jannis Limperg talked about where you can register tactics as unsafe resolution rules roughly corresponds to what are called solvers/loopers in Isabelle. Those are tactics which are called when auto cannot make progress any more using the simplifier or resolution rules. I am not exactly sure about the order of those procedures in Isabelle but simprocs are also tried at that level, probably before loopers and solvers. Of course, the architecture could just make the pattern optional and thus unify the concept of a simproc and a solver. However, there should be some smart way to handle precedence between those procedures then. Maybe try from most specific to most general pattern first and then try those procedures without patterns or even with integer preferences?

Anyway, I hope you will find this (hopefully mostly correct) explanation useful.

#### Jannis Limperg (Jan 06 2021 at 21:36):

Thank you very much! This matches what I thought simprocs were (which I did not convey very well during the Q&A). I think of them as 'custom simp rules' in analogy with the 'custom resolution rules' I described: both are arbitrary tactics which effectively construct a rule on the fly.

The exact order of things in Isabelle is something I should definitely look at again. My initial impression was that Isabelle's architecture is a bit cumbersome; it seemed like it may have developed as piecemeal extensions of the simplifier more so than a bigger search procedure which incorporates the simplifier. But it is equally possible that I just don't understand the reasons why the architecture is what it is.

#### Patrick Massot (Jan 06 2021 at 21:42):

Can't you at least find out who wrote the Isabelle version? Or at least the name of someone knowing a lot about it. Talking to such people would certainly be much more productive than reading the source code, at least in the beginning.

#### Jacques Carette (Jan 06 2021 at 21:47):

I saw Makarius was attending multiple of the talks today. He'd be my first lead into the Isabelle community.

#### Sebastien Gouezel (Jan 06 2021 at 21:54):

I guess blast and auto (at least the first versions) were probably written by Larry Paulson.

#### Sebastien Gouezel (Jan 06 2021 at 21:56):

A keyword is tableau-prover.

#### Sebastien Gouezel (Jan 06 2021 at 21:56):

https://www.cl.cam.ac.uk/~lp15/papers/Isabelle/blast.pdf

#### Jeremy Avigad (Jan 06 2021 at 22:05):

Yes, the credit for auto goes to Larry. It might also be worth taking a look at Bohua Zhan's auto2 . There are some good ideas there, though to some extent he was re-discovering existing methods, so his terminology is idiosyncratic.

I think a contemporary update to auto should make use of key ideas from SMT solvers, most importantly, e-matching and the use of triggers to instantiate quantifiers.

#### Yury G. Kudryashov (Jan 06 2021 at 23:30):

BTW, one more related feature I missed while porting some proof from Isabelle: ability to say that lemma

constant f : nat → nat
lemma foo : f n = f 0 + n * 3 := sorry

applies only if n is not exactly 0. I had to turn this lemma into ∃ c, ∀ n, f n = c + n * 3.

#### Eric Wieser (Jan 06 2021 at 23:34):

That would be useful for a lemma like foo [subsingleton A] [has_one A] (a : A) : a = 1 too, which today can't be simp as it loops

#### Yakov Pechersky (Jan 06 2021 at 23:36):

What about bar [subsingleton A] [has_zero A] (a : A) : a = 0

#### Yakov Pechersky (Jan 06 2021 at 23:37):

Neither can really be simp

#### Eric Wieser (Jan 06 2021 at 23:45):

Yeah, I guess you can't really have both, but if you picked just one and used a "don't apply if syntactically equal" rule you might be ok

#### Yakov Pechersky (Jan 06 2021 at 23:46):

What do you do about fin 1?

#### Eric Wieser (Jan 07 2021 at 00:05):

Perhaps then a canonical_subsingleton which specifies a canonical syntactic value

#### Eric Wieser (Jan 07 2021 at 00:05):

Which would obviously be 0 for fin 1

#### Mario Carneiro (Jan 07 2021 at 06:25):

Eric Wieser said:

That would be useful for a lemma like foo [subsingleton A] [has_one A] (a : A) : a = 1 too, which today can't be simp as it loops

Actually it can't be a simp lemma because it has a variable on the lhs, and even if it wasn't syntactically disallowed it would trigger on literally everything

#### Gabriel Ebner (Jan 07 2021 at 09:31):

Eric Wieser said:

That would be useful for a lemma like foo [subsingleton A] [has_one A] (a : A) : a = 1 too, which today can't be simp as it loops

As Mario has already said, the reason this doesn't work is because a is a variable and Lean ignores such lemmas. However, it wouldn't loop:

variables {α : Type*} [has_zero α]
def a : α := 0
lemma foo : (a : α) = @a _ ⟨has_zero.zero⟩ := rfl
set_option trace.simplify.rewrite true
example : a = a + 0 := by simp [foo] -- rewrites using foo, but doesn't loop

(This is known as Floris' trick and we also use it in mathlib.)

#### Gabriel Ebner (Jan 07 2021 at 09:32):

The eta-expansion of the instance is only necessary so that the simplifier doesn't reject the lemma outright.

#### Eric Wieser (Jan 07 2021 at 11:48):

Do you have an example of where this is used in mathlib in mind?

#### Gabriel Ebner (Jan 07 2021 at 12:49):

/- We can simplify an application of filter where the decidability is inferred in "the wrong way" -/
@[simp] lemma filter_congr_decidable {α} (s : finset α) (p : α → Prop) (h : decidable_pred p)
[decidable_pred p] : @filter α p h s = s.filter p :=
by congr

Thanks!

#### Jannis Limperg (Jan 07 2021 at 13:46):

Thanks you for all the references. To the degree that auto is similar to blast, I understand what's going on, but it seems like auto is significantly more popular, so I'd like to know where that comes from. I'll track down Makarius or Larry. E-matching and triggers are on the reading queue, and auto2 on the re-reading queue, for when my CPP talk is done.

The discussion about simp lemmas that require some syntactic condition on the term is very interesting. Seems like there are some compelling use cases for 'simprocs', so I'll look into how hard it would be to add something like this.

#### Lukas Stevens (Jan 07 2021 at 13:50):

In auto2, it is quite easy to make rules conditional to avoid loops.

Last updated: May 08 2021 at 21:09 UTC


text:
# Math Help - did i do this correctly?

1. ## did i do this correctly?

find $f'(\frac{\pi}{4})$
if $f(x)=\frac{2\sqrt{x}}{tan x}$

my answer: $f'(\frac{\pi}{4})=\frac{2-4\pi}{\sqrt{\pi}}$

2. Originally Posted by yoman360
find $f'(\frac{\pi}{4})$
if $f(x)=\frac{2\sqrt{x}}{tan x}$

my answer: $f'(\frac{\pi}{4})=\frac{2-4\pi}{\sqrt{\pi}}$
Close. The derivative is of $f$ is:

$\frac{\tan x\cdot\frac{1}{\sqrt{x}}-2\sqrt{x}\cdot\sec^2x}{\tan^2x}$

Plugging in $\frac{\pi}{4}$ gives us

$\frac{\tan(\pi/4)\cdot\frac{1}{\sqrt{\pi/4}}-2\sqrt{\pi/4}\cdot\sec^2(\pi/4)}{\tan^2(\pi/4)}=\frac{\frac{2}{\sqrt{\pi}}-\sqrt{\pi}\cdot2}{1} = \boxed{\frac{2-2\pi}{\sqrt{\pi}}}$

3. Originally Posted by redsoxfan325
Close. The derivative is of $f$ is:

$\frac{\tan x\cdot\frac{1}{\sqrt{x}}-2\sqrt{x}\cdot\sec^2x}{\tan^2x}$

Plugging in $\frac{\pi}{4}$ gives us

$\frac{\tan(\pi/4)\cdot\frac{1}{\sqrt{\pi/4}}-2\sqrt{\pi/4}\cdot\sec^2(\pi/4)}{\tan^2(\pi/4)}=\frac{\frac{2}{\sqrt{\pi}}-\sqrt{\pi}\cdot2}{1} = \boxed{\frac{2-2\pi}{\sqrt{\pi}}}$
Ok thanks 1 little error messed me up. Thanks


text:
Science and technology

# Using Python and GNU Octave to plot knowledge

Data science is a website of data that spans programming languages. Some are well-known for fixing issues on this area, whereas others are lesser-known. This article will allow you to change into acquainted with doing knowledge science with some widespread languages.

## Choosing Python and GNU Octave for knowledge science

Every so usually, I attempt to be taught a brand new programming language. Why? It is generally a mix of boredom with the outdated methods and curiosity in regards to the new methods. When I began programming, the one language I knew was C. Life was exhausting and harmful in these years, as I needed to manually allocate reminiscence, handle pointers, and bear in mind to free reminiscence.

Then a pal urged I attempt Python, and life grew to become a lot simpler. Programs grew to become a lot slower, however I didn’t should endure by means of writing evaluation software program. However, I quickly realized that every language was extra appropriate than others for some functions. I later studied another languages, and every one introduced some new little bit of enlightenment. Discovering new programming kinds let me backport some options to different languages, and all the things grew to become way more attention-grabbing.

To get a sense for a brand new programming language (and its documentation), I all the time begin by writing some instance applications that carry out a process I do know effectively. To that ends, I’ll clarify write a program in Python and GNU Octave for a specific process you would classify as knowledge science. If you’re already accustomed to one of many languages, begin with that one and undergo the others to look for similarities and variations. It will not be supposed to be an exhaustive comparability of the languages, just a bit showcase.

All of the applications are supposed to be run on the command line, not with a graphical user interface (GUI). The full examples can be found within the polyglot_fit repository.

## The programming process

The program you’ll write on this collection:

• Reads knowledge from a CSV file
• Interpolates the info with a straight line (i.e., f(x)=m ⋅ x + q)
• Plots the consequence to a picture file

This is a standard scenario that many knowledge scientists have encountered. The instance knowledge is the primary set of Anscombe’s quartet, proven within the desk under. This is a set of artificially constructed knowledge that offers the identical outcomes when fitted with a straight line, however their plots are very completely different. The knowledge file is a textual content file with tabs as column separators and some strains as a header. This process will use solely the primary set (i.e., the primary two columns).

Anscombe’s quartet

I II III IV
x y x y x y x y
10.zero eight.04 10.zero 9.14 10.zero 7.46 eight.zero 6.58
eight.zero 6.95 eight.zero eight.14 eight.zero 6.77 eight.zero 5.76
13.zero 7.58 13.zero eight.74 13.zero 12.74 eight.zero 7.71
9.zero eight.81 9.zero eight.77 9.zero 7.11 eight.zero eight.84
11.zero eight.33 11.zero 9.26 11.zero 7.81 eight.zero eight.47
14.zero 9.96 14.zero eight.10 14.zero eight.84 eight.zero 7.04
6.zero 7.24 6.zero 6.13 6.zero 6.08 eight.zero 5.25
four.zero four.26 four.zero three.10 four.zero 5.39 19.zero 12.50
12.zero 10.84 12.zero 9.13 12.zero eight.15 eight.zero 5.56
7.zero four.82 7.zero 7.26 7.zero 6.42 eight.zero 7.91
5.zero 5.68 5.zero four.74 5.zero 5.73 eight.zero 6.89

## The Python method

Python is a general-purpose programming language that’s among the many hottest languages in use as we speak (as evidenced by findings from the TIOBE index, RedMonk Programming Language Rankings, Popularity of Programming Language Index, State of the Octoverse of GitHub, and different sources). It is an interpreted language; subsequently, the supply code is learn and evaluated by a program that executes the directions. It has a complete standard library and is usually very nice to make use of (I’ve no reference for this final assertion; it’s simply my humble opinion).

### Installation

To develop with Python, you want the interpreter and some libraries. The minimal necessities are:

• NumPy for handy array and matrices manipulation
• SciPy for scientific calculations
• Matplotlib for plotting

Installing them in Fedora is straightforward:

sudo dnf set up python3 python3-numpy python3-scipy python3-matplotlib

In Python, comments are achieved by placing a # in the beginning of the road, and the remainder of the road might be discarded by the interpreter:

# This is a remark ignored by the interpreter.

The fitting_python.py instance makes use of feedback to insert licensing info within the supply code, and the primary line is a special comment that permits the script to be executed on the command line:

#! /usr/bin/env python3

This line informs the command-line interpreter that the script must be executed by this system python3.

### Required libraries

Libraries and modules might be imported in Python as an object (as within the first line within the instance) with all of the capabilities and members of the library. There is a handy choice to rename them with a customized label through the use of the as specification:

import numpy as npfrom scipy import statsimport matplotlib.pyplot as plt

You might also resolve to import solely a submodule (as within the second and third strains). The syntax has two (roughly) equal choices: import module.submodule and from module import submodule.

### Defining variables

Python’s variables are declared the primary time a worth is assigned to them:

input_file_name = "anscombe.csv"delimiter = "t"skip_header = threecolumn_x = zerocolumn_y = 1

The variable sorts are inferred by the worth that’s assigned to the variable. There aren’t any variables with fixed values except they’re declared in a module and might solely be learn. Idiomatically, variables that shouldn’t be modified must be named in uppercase.

### Printing output

Running the applications by means of the command line implies that the output is simply printed on the terminal. Python has the print() perform that, by default, prints its argument and provides a newline on the finish of the output:

print("#### Anscombe's first set with Python ####")

It is feasible to mix the print() perform with the formatting power of the string class in Python. Strings have the format methodology that can be utilized so as to add some formatted textual content to the string itself. For occasion, it’s doable so as to add a formatted float quantity, e.g.:

print("Slope: ".format(slope))

Reading CSV information could be very straightforward with NumPy and the perform genfromtxt(), which generates a NumPy array:

knowledge = np.genfromtxt(input_file_name, delimiter = delimiter, skip_header = skip_header)

In Python, a perform can have a variable variety of arguments, and you’ll have it move a subset by specifying the specified ones. Arrays are very highly effective matrix-like objects that may be simply sliced into smaller arrays:

x = knowledge[:, column_x]y = knowledge[:, column_y]

The colons choose the entire vary, and so they may also be used to pick a subrange. For occasion, to pick the primary two rows of the array, you’d use:

first_two_rows = knowledge[zero:1, :]

### Fitting knowledge

SciPy gives handy capabilities for knowledge becoming, such because the linregress() perform. This perform gives some important values associated to the match, such because the slope, intercept, and the correlation coefficient of the 2 datasets:

slope, intercept, r_value, p_value, std_err = stats.linregress(x, y)print("Slope: ".format(slope))print("Intercept: ".format(intercept))print("Correlation coefficient: ".format(r_value))

Since linregress() gives a number of items of data, the consequence might be saved to a number of variables on the identical time.

### Plotting

The Matplotlib library plots solely knowledge factors; subsequently, you need to outline the factors you wish to plot. The x and y arrays had been already outlined, so you’ll be able to instantly plot them, however you additionally want knowledge factors that may symbolize the straight line.

fit_x = np.linspace(x.min() - 1, x.max() + 1, 100)

The linspace() perform conveniently generates a set of equally spaced values between two values. The ordinates might be simply calculated by exploiting the highly effective NumPy arrays, which can be utilized in a method as in the event that they had been unusual numeric variables:

fit_y = slope * fit_x + intercept

The method is utilized element-by-element on the array; subsequently, the consequence has the identical variety of entries in the preliminary array.

To create the plot, first, outline a figure object that may include all of the graphics:

fig_width = 7 #inch
fig_height = fig_width / 16 * 9 #inch
fig_dpi = 100fig = plt.determine(figsize = (fig_width, fig_height), dpi = fig_dpi)

Several plots might be drawn on a determine; in Matplotlib, the plots are referred to as axes. This instance defines a single axis object to plot the info factors:

ax = fig.add_subplot(111)ax.plot(fit_x, fit_y, label = "Fit", linestyle = '-')ax.plot(x, y, label = "Data", marker = '.', linestyle = '')ax.legend()ax.set_xlim(min(x) - 1, max(x) + 1)ax.set_ylim(min(y) - 1, max(y) + 1)ax.set_xlabel('x')ax.set_ylabel('y')

Save the determine to a PNG image file with:

fig.savefig('fit_python.png')

If you wish to show (as an alternative of saving) the plot, name:

plt.present()

This instance references all of the objects used within the plotting part: it defines the article fig and the article ax. This technicality will not be obligatory, because the plt object can be used on to plot the datasets. The Matplotlib tutorial reveals an interface comparable to:

plt.plot(fit_x, fit_y)

Frankly, I don’t like this strategy as a result of it hides the non-trivial interactions that occur between the assorted objects. Unfortunately, generally the official examples are a bit complicated as a result of they have a tendency to make use of completely different approaches. Referencing graphical objects will not be obligatory on this easy instance, but it surely turns into vital in additional complicated ones (comparable to when embedding plots in GUIs).

### Results

The output on the command line is:

#### Anscombe's first set with Python ####Slope: zero.500091Intercept: three.000091Correlation coefficient: zero.816421

Here is the picture Matplotlib generates.

## The GNU Octave method

The GNU Octave language is primarily supposed for numerical computations. It affords a easy syntax for manipulating vectors and matrices and has some highly effective plotting amenities. It is an interpreted language like Python. Since Octave’s syntax is mostly compatible with MATLAB, it’s usually described as a free various to MATLAB. Octave will not be listed among the many hottest programming languages, however MATLAB is, so Octave is somewhat widespread in a way. MATLAB predates NumPy, and I’ve the sensation that it was impressed by the previous. While you undergo the instance, you will note the analogies.

### Installation

The fitting_octave.m instance solely wants the essential Octave bundle, making the set up in Fedora somewhat easy:

sudo dnf set up octave

In Octave, you’ll be able to add feedback to code with the p.c image (%), and it’s also possible to use # if MATLAB compatibility will not be wanted. The choice to make use of # permits you to write the identical particular remark line from the Python instance to execute the script instantly on the command line.

### Necessary libraries

Everything used on this instance is contained within the fundamental bundle, so you do not want to load any new libraries. If you want a library, the syntax is pkg load module. This command provides the module’s capabilities to the listing of accessible capabilities. In this regard, Python has extra flexibility.

### Defining variables

Variables are outlined with just about the identical syntax as Python:

input_file_name = "anscombe.csv";delimiter = "t";skip_header = three;column_x = 1;column_y = 2;

Note that the tip of the road has a semicolon; this isn’t obligatory, but it surely suppresses the output of the outcomes of the road. Without a semicolon, the interpreter would print the results of the expression:

octave:1> input_file_name = "anscombe.csv"input_file_name = anscombe.csvoctave:2> sqrt(2)ans =  1.4142

### Printing output

The highly effective perform printf() is used to print on the terminal. Unlike in Python, the printf() perform doesn’t mechanically add a newline on the finish of the printed string, so you need to add it. The first argument is a string that may include format info for the opposite arguments to be handed to the perform, comparable to:

printf("Slope: %fn", slope);

In Python, the formatting is constructed into the string itself, however in Octave, it’s particular to the printf() perform.

The dlmread() perform can learn textual content information structured like CSV information:

knowledge = dlmread(input_file_name, delimiter, skip_header, zero);

The result’s a matrix object, which is likely one of the elementary knowledge sorts in Octave. Matrices could also be sliced with a syntax much like Python:

x = knowledge(:, column_x);y = knowledge(:, column_y);

The elementary distinction is that the indexes begin at one as an alternative of zero. Therefore, within the instance, the

x column is column primary.

### Fitting knowledge

To match the info with a straight line, you need to use the polyfit() perform. It matches the enter knowledge with a polynomial, so that you simply want to make use of a polynomial of order one:

p = polyfit(x, y, 1);slope = p(1);intercept = p(2);

The result’s a matrix with the polynomial coefficients; subsequently, it selects the primary two indexes. To decide the correlation coefficient, use the corr() perform:

r_value = corr(x, y);

Finally, print the outcomes with the printf() perform:

printf("Slope: %fn", slope);printf("Intercept: %fn", intercept);printf("Correlation coefficient: %fn", r_value);

### Plotting

As within the Matplotlib instance, you first have to create a dataset that represents the fitted line:

fit_x = linspace(min(x) - 1, max(x) + 1, 100);fit_y = slope * fit_x + intercept;

The analogy with NumPy can be evident right here, because it makes use of the linspace() perform that behaves identical to the Python’s equal model.

Again, as with Matplotlib, create a figure object first, then create an axes object to carry the plots:

fig_width = 7; %inch
fig_height = fig_width / 16 * 9; %inch
fig_dpi = 100;fig = determine("units", "inches",             "position", [1, 1, fig_width, fig_height]);ax = axes("parent", fig);set(ax, "fontsize", 14);set(ax, "linewidth", 2);

To set properties of the axes object, use the set() perform. The interface is somewhat complicated, although, because the perform expects a comma-separated listing of property and worth pairs. These pairs are only a succession of a string representing the property title and a second object representing the worth for that property. There are additionally different capabilities to set numerous properties:

xlim(ax, [min(x) - 1, max(x) + 1]);ylim(ax, [min(y) - 1, max(y) + 1]);xlabel(ax, 'x');ylabel(ax, 'y');

Plotting is achieved with the plot() perform. The default habits is that every name resets the axes, so you might want to use the perform hold().

maintain(ax, "on");plot(ax, fit_x, fit_y,     "marker", "none",     "linestyle", "-",     "linewidth", 2);plot(ax, x, y,     "marker", ".",     "markersize", 20,     "linestyle", "none");maintain(ax, "off");

Also, it’s doable within the plot() perform so as to add the property and worth pairs. The legend should be created individually, and the labels must be acknowledged manually:

lg = legend(ax, "Fit", "Data");set(lg, "location", "northwest");

Finally, save the output to a PNG picture:

image_size = sprintf("-S%f,%f", fig_width * fig_dpi, fig_height * fig_dpi);
image_resolution = sprintf("-r%f,%f", fig_dpi);print(fig, 'fit_octave.png',      '-dpng',      image_size,      image_resolution);

Confusingly, on this case, the choices are handed as a single string with the property title and the worth. Since in Octave strings shouldn’t have the formatting amenities of Python, you need to use the sprintf() perform. It behaves identical to the printf() perform, however its consequence will not be printed, somewhat it’s returned as a string.

In this instance, as within the Python one, the graphical objects are referenced to maintain their interactions evident. If Python’s documentation on this regard is a bit bit complicated, Octave’s documentation is even worse. Most of the examples I discovered didn’t care about referencing the objects; as an alternative, they depend on the truth that the plotting instructions act on the at present energetic determine. A world root graphics object retains monitor of the prevailing figures and axes.

### Results

The ensuing output on the command line is:

#### Anscombe's first set with Octave ####Slope: zero.500091Intercept: three.000091Correlation coefficient: zero.816421

And this reveals the ensuing picture generated with Octave.

## Next up

Both Python and GNU Octave can plot the identical info, although they differ in how they get there. If you are trying to discover different languages to finish related duties, I extremely advocate  Rosetta Code. It’s a wonderful useful resource to see resolve the identical issues in lots of languages.

What language do you wish to plot knowledge in? Share your ideas within the feedback.

### Most Popular

breakingExpress.com features the latest multimedia technologies, from live video streaming to audio packages to searchable archives of news features and background information. The site is updated continuously throughout the day.


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# What is the value of x^5?

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What is the value of x^5? [#permalink]

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23 Nov 2017, 05:51
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What is the value of x^5?

(1) x^2 = 9
(2) x^3 > 9
[Reveal] Spoiler: OA

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Re: What is the value of x^5? [#permalink]

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23 Nov 2017, 05:56
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KUDOS
Bunuel wrote:
What is the value of x^5?

(1) x^2 = 9
(2) x^3 > 9

S1) x = -3,3
Insufficient.

S2) $$x^3$$ > 9
x = 3,4,5,....
Insufficient.

Quote:
combining both statements

x = -3,3
$$x^3$$ > 9 => x > 0
=> x = 3
We can find the value of $$x^5$$

Sufficient.

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Re: What is the value of x^5? [#permalink]

### Show Tags

23 Nov 2017, 13:00
Bunuel wrote:
What is the value of x^5?

(1) x^2 = 9
(2) x^3 > 9

(1) x^2 = 9 : x = + or - \sqrt{9} we need to find out if the root is positive or negative, insuf.
(2) x^3 > 9 : since x^3 is > 9, then x must be a positive. But we don't know the value of x, insuf.
(C) from (2) we know that 'x' is a positive and from (1) we know the value of 'x', so sufficient.
Re: What is the value of x^5?   [#permalink] 23 Nov 2017, 13:00
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### অসীম গুণোত্তর ধারার বর্গ এবং ঘন

##### Score: 2 Points

$a_1+a_2+a_3+ ...$ is an infinite geometric series whose sum is $3$. Replacing each of the terms of the series by their squares results in a series whose sum is the same. Replacing each of the terms of the series by their cubes results in a series whose sum can be expressed by $\frac ab$ where $a$ and $b$ are co-pime positive integers. What is the value of $a+b$?

Source: BdMO

Algebra

Tried 83

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Let $P(t)$ represent the price of a share of stock of a corporation at time $t .$ What does each of the following statements tell us about the signs of the first and second derivatives of $P(t) ?$(a) "The price of the stock is rising faster and faster."(b) "The price of the stock is close to bottoming out."
Let $P(t)$ represent the price of a share of stock of a corporation at time $t .$ What does each of the following statements tell us about the signs of the first and second derivatives of $P(t) ?$ (a) "The price of the stock is rising faster and faster." (b) "The price of the stock is...


text:
queryparser-0.1.0.1: Analysis and parsing library for SQL queries.

Database.Sql.Util.Lineage.ColumnPlus

Synopsis

# Documentation

ColumnLineagePlus is a set of descendants, each with an associated set of ancestors. Descendents may be a column (representing values in that column) or a table (representing row-count).

Ancestors are the same, but that ancestor columns may be further specialized by field path.

Tracking impacts on row-count is necessary because row-count can have impacts on data without any columns being involved. For a clear example, consider CREATE TABLE foo AS SELECT COUNT(1) FROM BAR;

It also gives us something coherent to speak about with respect to EXISTS - the value depends on the row count of the subquery.

N.b. While it looks like we're talking about "tables", this is *not* the same thing as table-level lineage. Changes to values in existing rows does not impact row count. Following an UPDATE, if we ask "has this table changed, such that we need to rerun things downstream?", the answer is clearly yes. If we ask "has the row-count of this table changed, such that we need to rerun things that depend only on row-count?" the answer is clearly not.

class HasColumnLineage q where Source #

Minimal complete definition

getColumnLineage

Methods

Instances

 Source # Methods

Constructors

 ColumnPlusSet FieldscolumnPlusColumns :: Map FQCN (Map FieldChain (Set Range)) columnPlusTables :: Map FQTN (Set Range)

Instances

 Source # Methods Source # MethodsshowList :: [ColumnPlusSet] -> ShowS # Source # Methodsstimes :: Integral b => b -> ColumnPlusSet -> ColumnPlusSet # Source # Methods


text:
# $\text{GL}_n:\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{GL}_n(R)$ is a representable functor

Following the definition of $\text{GL}_n:\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{GL}_n(R)$ found here: functor from $\mathbf{Alg}$ to $\mathbf{Set}$

I would like to show that it is a representable functor.

EDIT: wrong!

For ease of notation, $F:=\text{GL}_n$ and $G:= \text{Hom}(k,-)$

Where $\text{Hom}(k,-):\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{Hom}(k,R):f\mapsto (\sigma\mapsto f\circ \sigma)$

I'll define the natural transformation as:

$$\alpha_R:\text{GL}_n(R)\rightarrow \text{Hom}(k,R):(a_{i,j})\mapsto (k\mapsto \sum a_{i,j}k)$$

this defines a natural transformation. But how natural is it? I.e. are there any more intuitive ways of doing this?

For ease of notation, $F:=\text{GL}_n$ and $G:= \text{Hom}(A_n,-)$

where $$A_n=k[X_{11},\ldots,X_{nn},Y]/\left<1-Y\det(X_{ij})\right>.$$

Where $\text{Hom}(A_n,-):\mathbf{Alg}_k\rightarrow\mathbf{Set}:R\mapsto \text{Hom}(A_n,R):f\mapsto (\sigma\mapsto f\circ \sigma)$

I'll define the natural transformation as:

$$\alpha_R:\text{GL}_n(R)\rightarrow \text{Hom}(A_n,R)\\(a_{i,j})\mapsto (P(X_{11},\ldots,X_{nn},Y)+ (1-Y\det(X_{ij}))\mapsto P(a_{1,1},\ldots,a_{n,n},\text{det}(a_{i,j})))$$

• It will certaibly not be represented by $k$. Do you think $GL_n(k) \simeq Hom(k,k)$ ? – Max Dec 30 '17 at 11:52
• Oups, I forgot to check that it is a bijection. What I gave is a natural transformation but not necessarily a natural isomorphism. – tomak Dec 30 '17 at 12:00
• Only other idea is to represent it by $k^n$ but not sure that works – tomak Dec 30 '17 at 12:01
• There's the same problem with $Hom(k^n, k) \simeq GL_n(k)$ – Max Dec 30 '17 at 12:40

Yes, $\text{GL}_n$ is an affine group scheme. It is represented by the $k$-algebra $$A_n=k[X_{11},X_{12},\ldots,X_{21},\ldots,X_{nn},Y]/\left<1-Y\det(X_{ij})\right>.$$ An algebra homomorphism $\phi:A_n\to R$ is given by choosing the images of $X_{ij}$ and $Y$, which must go to the inverse of $\det(\phi(X_{ij}))$ which ensures that $(\phi(X_{ij}))\in\text{GL}_n(R)$.


text:
Last edited by Taurg
Wednesday, August 5, 2020 | History

2 edition of Nucleophilic substitution reactions in cobalt (III) metal complexes. found in the catalog.

Nucleophilic substitution reactions in cobalt (III) metal complexes.

Moustafa Hassan Mohamed Abou El-Wafa

# Nucleophilic substitution reactions in cobalt (III) metal complexes.

## by Moustafa Hassan Mohamed Abou El-Wafa

Published .
Written in English

Edition Notes

Thesis (Ph. D.)--The Queen"s University of Belfast, 1984.

The Physical Object
Pagination1 v
ID Numbers
Open LibraryOL19188013M

We have developed an environmentally benign synthetic approach to nucleophilic substitution reactions of alcohols that minimizes or eliminates the formation of byproducts, resulting in a highly atom-efficient chemical process. Proton- and metal-exchanged montmorillonites (H- and Mn+-mont) were prepared easily by treating Na+-mont with an aqueous solution of hydrogen chloride or metal salt Cited by: They are equilibrium reactions. The equilibrium concentrations mainly depend on the characters of the nucleophile, the leaving group and the solvent. S N reactions are classified by chemical kinetics: Bimolecular nucleophilic substitution, or S N 2 reaction, and unimolecular nucleophilic substitution, or S .

Nucleophilic Substitution Reactions. STUDY. Flashcards. Learn. Write. Spell. Test. PLAY. Match. Gravity. Created by. Jasmyn Terms in this set (33) objective. to synthesize substitution products from an alkyl halide and alcohol. substitution. nucleophiles will attack at the electrophilic carbon halide bond replacing the halide. In organohalogen compound: Nucleophilic substitution. Nucleophilic substitution, which can be represented by the following general equation, permits the halogen to be replaced by oxygen, sulfur, nitrogen, or another carbon.. Read More; substitution reaction. In substitution reaction of reaction is known as nucleophilic substitution. Examples of nucleophilic reagents are the halogen anions.

Other articles where Nucleophilic aromatic substitution is discussed: organohalogen compound: Reactions: Nucleophilic aromatic substitution is a practical synthetic reaction only when the aryl halide bears a strongly electron-attracting substituent, such as a nitro group NO2, at a position ortho or para to the halogen, as in 1-chloronitrobenzene. The side of attack of the nucleophile for an SN1 reaction will be the side opposite to where the leaving group is attached BECAUSE even if the breaking of the bond between the leaving group and the carbon is complete, it is not uncommon for the leaving group to remain associated for a short time after that.

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The Petition of divers of His Majesties faithfull subjects, of the true Protestant religion, in the county Palatine of Lancaster

The Petition of divers of His Majesties faithfull subjects, of the true Protestant religion, in the county Palatine of Lancaster

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Appealing And Preventing Denied Claims

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### Nucleophilic substitution reactions in cobalt (III) metal complexes by Moustafa Hassan Mohamed Abou El-Wafa Download PDF EPUB FB2

Nucleophilic substitution reactions are commonplace in organic chemistry, and they can be broadly categorised as taking place at a saturated aliphatic carbon or at (less often) an aromatic or other unsaturated carbon centre.

1 Saturated carbon centres. SN1 and SN2 reactions. Borderline mechanism. Other mechanisms. 8 Nucleophilic Substitution and Elimination Reactions pair.

Reagents that acquire an electron pair in chemical reactions are said to be electrophilic ("electron-loving"). We can picture this in a general way as a heterolytic bond breaking of compound X:Y by an electrophile E such that E becomes bonded to Y by the electron pair of the XY bond. ThusFile Size: 1MB.

Nucleophilic substitution. In organic (and inorganic) chemistry, nucleophilic substitution is a fundamental class of reactions in which a nucleophile selectively bonds with or attacks the positive or partially positive charge on an atom or a group of atoms.

As it does so, it replaces a weaker nucleophile which then becomes a leaving group; The remaining positive or partially positive atom. Nucleophilic Substitution Reactions. Nucleophilic reaction on the 1,2,5-thiadiazole ring at the carbon and sulfur atoms is quite facile. However, nucleophilic functional group displacement reactions also occur smoothly as the thiadiazole ring itself acts as an electron-withdrawing substituent and activates the leaving group for substitution.

Substitution Reaction Example. These type of reactions are said to possess primary importance in the field of organic chemistry. For example, when CH 3 Cl is reacted with the hydroxyl ion (OH-), it will lead to the formation of the original molecule called methanol with that hydroxyl ion.

The following reaction is. This is a general image, and says nothing yet about mechanism. The arrangement of R groups is entirely irrelevant at this point. It simply represents the Substitution of a Nucleophile for the Leaving Group. There are 2 types of Nucleophilic Substitution Reactions: SN1 & SN2, each having its own mechanism.

The SN2 reaction is usually learned first. Experimental data from nucleophilic substitution reactions on substrates that have optical activity (the ability to rotate plane‐polarized light) shows that two general mechanisms exist for these types of reactions.

The first type is called an S N 2 mechanism. This mechanism follows second‐order kinetics (the reaction rate depends on the concentrations of two reactants), and its. What is nucleophilic substitution.

Includes background material on the bonding in halogenoalkanes, and general mechanisms for their nucleophilic substitution reactions. Substitution reactions involving hydroxide ions The mechanisms for the formation of alcohols from.

Elimination Reactions: Nucleophiles are Lewis bases. In the reaction with alkyl halides, they can also promote elimination reactions rather than substitution. RCC: _ R X 1° alkyl halide SN2 RCCCH2R RCC: RX HR _ Helimination H RH R 2° alkyl halide Elimination is a.

The difference in nucleophilic acyl substitution is that when the nucleophile adds to the electrophilic C, it becomes tetrahedral and an intermediate forms, then the leaving group departs as shown below. QUESTION. In nucleophilic substitution (e.g.

for alkyl halides) the nucleophile cannot attack until the leaving group leaves, why is there a difference. Many substitution reactions involve a kind of reacting group called a nucleophile. A nucleophile contains an unshared pair of electrons that reacts with a site in an organic molecule that has a deficiency of electrons.

Nucleophilic substitution reactions share several characteristics with acid-base reactions, as shown in Figure Size: KB. Nucleophilic Aromatic Substitution for Hydrogen Reduction in chemical waste generation elimination of 74% of organic waste 99% of inorganic waste Eliminates use of chlorine Reduction in waste water more than 97% savings Eliminates use of xylene a SARA chemical Improves process.

Lab Report: Alkyl Halide Nucleophilic Substitution Experiment. Total word count: Time spent researching and writing: 4 hrs Lab Conclusion: Alkyl Halide Nucleophilic Substitution Experiment.

The actual results correlated somewhat to the predicted reactivity of the alkyl halides. In the S N 1 reactions, all products were formed. It was. THE FORMATION OF COBALT(III)-ALKYL COMPOUNDS BY CARBANION NUCLEOPHILIC SUBSTITUTION REACTIONS OF ACTIVATED METI{YI~E GROUPS N.A.

Bailey, B.M. '{igson, and E.D. McKenzie Chemistry Department, The University, Sheffield, \$3 by: Summary. What does the term "nucleophilic substitution" imply?A nucleophile is an the electron rich species that will react with an electron poor species; A substitution implies that one group replaces another.; Nucleophilic substitution reactions occur when an electron rich species, the nucleophile, reacts at an electrophilic saturated C atom attached to an electronegative group (important.

A review has been made of the kinetic data for the substitution reactions of the Co(CN) 5 X-3 ions, where X refers to any one of various ligands. The evidence suggests that the penta-coordinate Co(CN) is generated as a reactive intermediate in the substitution of water in Co(CN) 5 OH by various nucleophiles and in the aquation of the various Co(CN) 5 X-3 ions.

Mechanism 3. An alternative mechanism, usually labeled the $${\mathrm{S}}_{\mathrm{N}}\mathrm{1CB}$$ (Substitution, Nucleophilic, first-order in the Conjugate Base mechanism) mechanism, is also consistent with the second-order rate this mechanism, hydroxide removes a proton from one of the ammine ligands, to give a six-coordinate intermediate, containing an.

Modern Nucleophilic Aromatic Substitution. Francois Terrier. ISBN: pages. August Description. This book provides a comprehensive overview of nucleophilic aromatic substitutions, focusing on the mechanistic and synthetic features that govern these reactions.

Last time we saw an overview of the nucleophilic substitution mechanisms of alkyl halides. We examined one of these, the S N 2 mechanism in detail. Today we'll examine the other, the S N 1 mechanism, and then go on to look at elimination reactions, the major competition for substitutions.

Here's the outline of the S N 1 mechanism. Recalling what the "2" in S N 2 meant -- that the reaction. PDF | A thorough mechanistic study on cobalt-catalysed direct methoxycarbonylation reactions of chlorobenzenes in the presence of methyl oxirane on a | Find, read and cite all the research you.

Nucleophilic Substitution Reactions of Haloalkanes Nucleophilic substitution reactions are ionic reactions that break and make chemical bonds by transfers of pairs of electrons.

We illustrate this using a general representation of a nucleophilic substitution reaction in File Size: 4MB.The book began with chemistry that is by no means uncommon, but the bulk of organic chemistry has and continues to be based on two electron processes.

Substitution reactions comprise one class of reactions involving such chemistry. Addition reactions comprise the other. You should have been introduced to both by now.Nucleophilic substitution is one of the most fundamental reactions used in organic synthesis.

A "nucleophile" is an electron-rich species. In a nucleophilic substitution, a nucleophile reacts with an alkyl halide to form a product with a new functional group. This reaction is the starting point for a vast array of organic syntheses.


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# Numerical Approach to find the Many-Body Ground State Wavefunction of Tight-Binding model

I encounter a problem in finding the many-body ground state wave-function of the tight-binding model in python. The Hamiltonian of tight-binding model is follow: $$$$H = -t \sum_{} c^{\dagger}_{i} c_{j} + \mu \sum_{i}c^{\dagger}_{i}c_{i}$$$$ Suppose I want to solve the 1D Tight-binding model, I can re-write the Hamiltonian in the following matrix form: $$$$H = \begin{bmatrix} \mu & -t & 0 & 0 & \dots & -t \\ -t & \mu & -t & 0 &\dots & 0 \\ 0 & -t & \mu & -t &\dots & 0 \\ 0 & 0 & -t & \mu &\dots & 0 \\ \vdots & \vdots & \vdots &\ddots & \ddots & \ddots \\ -t & 0 & 0& \dots & -t & \mu \end{bmatrix}$$$$

Therefore, I can directly solve the ground-state eigenvalue and the eigenvector. However, the ground state that I solve is the ground state correspond to single particle case. In tight-binding model, we know that the many-body ground state should be the Fermi sea: $$$$| \Psi_{GS} \rangle = \prod_{k \leq k_{F}} c^{\dagger}_{k} | 0 \rangle$$$$ Therefore, my problem is that how can I calculate the many-body ground state wave function once I solve all the eigenvectors and eigenvalues of the matrix $$H$$? Besides, how can I relate real-space eigenvectors to momentum space eigenvector like the $$|\Psi_{GS} \rangle$$?

• Do you have periodic boundary conditions? From your matrix it looks like the answer is maybe no? – jacob1729 Apr 22 at 15:15
• I forgot to impose the period boundary condition for the Hamiltonian. Thanks for reminding. – Ricky Pang Apr 22 at 16:13