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What is the $IUPAC$ nomenclature of isoprene monomer present in natural rubber ? Some examples are ethene, vinyl chloride, acrylonitrile, phenol and formaldehyde etc. B. Terylene. Plastics are non-biodegradable. Major product of the following reaction is : In Wolff‐Kishner reduction, the carbonyl group of aldehydes and ketones is converted into. As we strive to live environmentally conscious lifestyles and transition towards a more…, We’re pleased to announce that Pandabode products are now available, “Our physical health really affects how we feel. [DELHI PMT 2009] A. Starch and Nylon. A) true B) false 20. Maths. PET is the acronym for _____ (a) Polyester (b) Polyester and terylene (c) Polyethylene terephthalate (d) Polyetheneterylene Answer: (c) Polyethylene terephthalate. Life is a grand adventure, after all and we intend to continue living it. For example, PLA based resins are known to biodegrade under the right conditions. 6. They typically look and feel a lot like plastic, with brown fibre speckles and sometimes come in bright colours. Example: ethene n(CH2=CH2) to polyethene -(CH2-CH2)n-. However this also requires very specific recycling facilities, that are currently uncommon. Differentiate Novolac and Bakelite on the basis of structure. )Write the names and structures of monomers of the following polymers? The bamboo fibre part can be considered as biodegradable, although being mixed with epoxy resin could slow the process, compared to untreated bamboo. C. bakelite and PVC. Question 5. The bamboo fibres enable the material to be lightweight and relatively strong. Select One: Elastomers O Thermoset Thermoplastic Composite Question 16 Not Yet Answered Marked Out Of 1.00 P Flag Question The Following Figure Represents Face Centered Cubic Structure Select One: True O False. (ii) condensation polymers: Terylene , Bakelite. People will appreciate that and it might just be the tipping point into convincing them your product is the better one.. 2. Question 25. List two changes in habit that people must adopt to dispose non-biodegradable … This interesting material is known by various names such as BPC or bamboo plastic composite, moulded bamboo, as well as a number of other trade names. Polar plastics like PC, PMMA, PS and SI absorb acetic and formic acids, which are partially soluble in the plastics, but not nitric acid. The atomic radiusis: Find out the solubility of $Ni(OH)_2$ in 0.1 M NaOH. But some can biodegrade when disposed into industrial composting facilities. _____ is an example of natural polymer (a) Rayon (b) Cellulose (c) Nylon (d) All of the above. Polymers are also called macromolecules. The market value is extremely high, due to the retro feel and limited quantities of certain products. 8. Dacron is continuous filament yarn used in curtains, dress fabrics and pressure fire hoses. B. Starch and Cellulose. Answer: (b) Plastic bag, 8. This is where the base chemicals required to make the plastic part, are extracted from crops. Classification of Elements and Periodicity in Properties. For example: Bakelite and melamine. Most products probably won’t biodegrade in a reasonable time frame in landfill or in the ocean. D. PET and polythylene. At the time of writing neither the author or the Pandabode brand owner have any direct financial interests in bamboo plastic composite products or companies. Thermoset plastics are frequently used for sealed products due to their resistance to deformation and are also among some of the most impact resistant plastics available. Chemical reactions take place and once cured, a strong material with useful properties is formed. But again, due to complexity, this topic deserves to be considered separately. D. Melamine. PHBV is Poly β-hydroxy butyrate-co- β-hydroxy valerate .It is a biodegradable polymer. Polyacrylonitrile is an example of (a) addition polymer (b) condensation polymer (c) natural polymer (d) None of these. CBSE Class 10 Science Book Chapter 15 “Our Environment” Multiple Choice Questions (MCQs) with Answers. Question 11. Question: What is the name of plastic that is used in making non stick pans? Polypropylene is a plastic that you’ve probably heard of and you almost certainly own something made of it. Question 3. Monomers are the. The polymers which decompose on their own by bacterial action are called biodegradable polymers.Example of a biodegradable … C. Proteins and Nylon. These continue on … To give … (i) Wood, paper, leather , (ii) Polythene, detergent, PVC Plastic, detergent, grass , (iv) Plastic, bakelite, DDT -“Plastic” is still used to produce composite materials. NCERT DC Pandey Sunil Batra HC Verma Pradeep Errorless. AMU 2018: Which of the following is an example of thermosetting polymers? The bakelite is made from phenol and formaldehyde. Given that the ionic product of $Ni(OH)_2$ is $2 \times 10^{-15}$. These are one of the oldest polymers that were synthesized by man. Melamine is used for making floor tiles, kitchenware etc. Required fields are marked *. -Fossil fuels as a source material for the chemicals required to make plastic. Question 11. Why is it hazardous? Give an example of biodegradable aliphatic polyester. Biodegradable. The use of bamboo fibres as a filler is typically why bamboo plastic composites are often seen as an eco friendly option. The bamboo is only a constituent part. Reaction between acetone and methyl magnesium chloride followed by hydrolysis will give : Identify the correct statements from the following: Which of the following set of molecules will have zero dipole moment ? Biodegradable plastics are plastics that degrade, or break down, upon exposure to: sunlight or ultra-violet radiation, water or dampness, bacteria, enzymes or wind abrasion.In some instances, rodent, pest, or insect attack can also be considered as forms of biodegradation or environmental degradation.. Novolac on heating with formaldehyde undergoes cross linking solid mass called bakelite. C. Backelite. 40% bamboo, 60% plastic. They include: -Health concerns. -Secondly, a matrix, which we shall define as a substance that engulfs the fibres and holds them all in place. This material was quite revolutionary at the time. 18. Answer. Bakelite made perhaps its largest stamp on the world of fashion. Download PDF's. A notable example is the plastics which are a commonly used material in almost every field. Which of the following is Non-biodegradable a) Woolen clothes b) Plastic bag, c) Cotton cloth d) Wood. Some of the existing research into the biodegradation of epoxy resin seems fairly nascent and there is still much to learn. EAMCET 2013 Engineering Chemistry. durability, stiffness, reaction to heat etc. Example : Bakelite, Melamine etc. kind of substance which cannot be broken down by natural organisms and acts as a source of pollution When we know exactly which chemicals have been used, we can make a judgement about whether a bamboo composite can be thought of as more or less biodegradable than other plastics. Free PDF Download of CBSE Chemistry Multiple Choice Questions for Class 12 with Answers Chapter 15 Polymers. Non-biodegradable wastes are those who cannot be decomposed or dissolved by natural agents. Give the monomers of bakelite. Bioplastic made from plant feedstock is not biodegradable. Biodegradability. If you’ve made it this far, you’ll probably realise that there is much more that could be discussed on this subject, but hopefully our brief analysis has been informative, entertaining and a gateway to some further learning. NCERT NCERT Exemplar NCERT Fingertips Errorless Vol-1 Errorless Vol-2. Bakelite Phenolic Resin Powder Heat Resistance For Brake Lining. The polymers which decompose on their own by bacterial action are called biodegradable polymers.Example of a biodegradable aliphatic polyester are Poly-β-hydroxy butyrate-co-β-hydroxy valerate (PHBV). You can’t just throw this stuff into your household plastics recycling bin because it would contaminate the other plastic types. Examples of thermosetting plastics include bakelite, urea-formaldehyde resins. -The overall production process of bamboo plastic composites can theoretically be conducted in a more sustainable way than traditional plastics. (i) Polyvinyl chloride (ii) Teflon (iii) Bakelite. bakelite. What are biodegradable polymers ? 6. Let’s look at a some examples. From a consumer perspective, it is very difficult to know exactly which epoxy resins, plastics and other chemicals have been used in the bamboo plastic composite manufacturing process. KEAM 2014: Which one of the following is an example for biodegradable polyester? (e) Bakelite is a thermosetting plastic which is a poor conductor of heat and electricity. Which of the following is an example of thermosetting polymers? Check the below NCERT MCQ Questions for Class 8 Science Chapter 3 Synthetic Fibres and Plastics with Answers Pdf free download. 9. Give an example of a biodegradable aliphatic polyester. This suggests that the epoxy may be being used as a food source by the microbes and therefore biodegradation could be considered to be taking place, under these conditions. i need fast solve please. Biology. Why is bakelite called a thermosetting polymer. The copolymer formed by addition polymerization of styrene and acrylonitrile in the presence of peroxide is : Which of the following statements is not true ? Melamine is resistant to fire and is hence used for making utensils. of biostable and biodegradable polyurethanes. Biodegradable Polymers: A polymer that can be decomposed by bacteria is called a biodegradable polymer. After 15 days of exposure to the test environment, where specific types of microorganisms were present, the epoxy tested lost ~34% of its weight, the breakage and formation of chemical bonds was detected and the number of microbes appeared to increase. Understand the Key Differences Between Thermoplastic and Thermosetting Plastic with detailed explanations and examples. -Less “plastic” is used in the production process of bamboo plastic composites, compared to traditional plastics. Ans: A linear product of Phenol  formaldehyde polymer is Novolac, used in paints. This is partly because the biodegradability of any given substance, depends on the environment it is left to biodegrade in. The result is a hard plastic material. How Long Does Glass Take To Decompose? Which one of the following is an example for biodegradable polyester? Identify the monomer in the following polymeric structure. 7. For Kang Chyau each detail is important, during these COVID-19 days, they have figured out how to provide pre-sales and after-sales service to potential and current customers. Answer: (b) Cellulose. Students can practice free MCQs as have been added by CBSE in the new Exam pattern. examples are bakelite, urea‐formaldelyde resins, etc. MCQ Questions for Class 8 Science with Answers were prepared based on the latest exam pattern. Your email address will not be published. Practice important Questions. This question hasn't been answered yet Ask an expert . Biodegradable materials break and return to nature with the help of decomposers. NCERT P Bahadur IIT-JEE Previous Year Narendra Awasthi MS Chauhan. The entire supply chain and production methods are always relevant to the “eco credentials” of the end product. Uses of Bakelite. Unfortunately not without witnessing the manufacturing process or lab testing the product. Which of the following represents the example of homopolymer? The condensation reaction of the two reactants in a controlled acidic or basic medium results in the formation of ortho and para hydroxymethyl phenols and their derivatives. It therefore also depends on how you choose to define biodegradability. ii) Condensation polymers: These polymers are formed by the combination of monomers, with the elimination of small molecules like water, alcohol etc. Question 10. Nylon 2-nylon 6 is another example of biodegradable polymers. Polyacrylonitrile is an example of (a) addition polymer (b) condensation polymer (c) natural polymer (d) None of these. Like many things that are essential to human bodily function, such as salts for example, a small amount is essential, however too much can be very harmful. Chemistry MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Will it be in our lifetime? Answer: (b) 7. At the end of Multiple Choice Questions, the answer key has also been provided for your reference. In this case the matrix is often an epoxy resin, which can be moulded and then hardens when set. Prev Question Next Question. Bakelite is an early 20th century example of a similar composite material. Melamine is resistant to fire and is hence used for making utensils. Formaldehyde is a naturally occurring organic compound that is essential for the functioning of human and mammalian metabolisms. Differentiate between biodegradable and non-biodegradable substances with the help of one example each. These two molecules can link up with each other because each contains a reactive functional group, either an amine or a carboxylic acid which react to form an amide linkage. 10. Check out these seasonal immune boosting foods: A great alternative to plastic cling film because they’re mouldable, naturally antimicrobial and reusable. Often, chemical additives are included to give the composite material desirable properties, specific to its intended use case e.g. (A) PHBV (B) PET (C) Nylon 6 (D) Bakelite (E) Glyptal. This means that $788\, kJ$ of energy is required, As per IUPAC nomenclature, the name of the complex $[Co (H_2O )_4(NH_3 )_2]Cl_3$ is, In a reaction $2A + B \to 3C$, the concentration of $A$ decreases from $0.5\, mol \,L \,^{-1}$ to $0.3\, mol\, L^{-1}$ in $10$ minutes. We do sell eco friendly products under the Pandabode brand and we regularly research interesting materials during our product design process, in order to identify opportunities where we can utilise suitable materials that satisfy the strict environmental criteria that we demand of all our products. The common examples of aliphatic biodegradable polymers are polyglycolic acid(PGA), Polyhydroxy butyrate (PHB), Polyhydroxy butyrates-co-beta hydroxyl valerate( PHBV), Polycaprolactone(pcl), Nylon-2-nylon-6. -It is possible to make the plastic constituent have a lower overall environmental impact by using so called bioplastics. b) Plastic bag, … Which of the following represents the example of homopolymer? This depends on: -What conditions are required for biodegradation to occur? The problem is 28% of glass bottles in the US never reach the processing plants, and this means millions of ton going to landfills every year. Which of the following characteristics does not affect the properties of plastics? Understand the Key Differences Between Thermoplastic and Thermosetting Plastic with detailed explanations and examples. Free PDF Download of CBSE Chemistry Multiple Choice Questions for Class 12 with Answers Chapter 15 Polymers. Terylene, Bakelite,Polyvinyl chloride, Polythene. (a) PMMA (b) Nylon- 6, 6 (c) Dacron (d) Bakelite. Answer: (b) 8. The rate of production of $C$ during this period is, Ammonium ion $\ce{(NH_{4}^{+})}$ reacts with nitrite ion $\ce{(NO_2^{-})}$ in aqueous solution according to the equation $NH^{+}_ {(aq)} + NO^{-}_{2 (aq)} \to N_{2(g)} + 2H_2O_{(l)}$, The products obtained by the ozonolysis of $2$-ethylbut-$1$-ene are, When but-$2$-yne is treated with $Na$ in liquid ammonia, The correct decreasing order of reactivity for a given alkyl $(R)$ group in both $S_N1$ and $S_N2$ reaction mechanisms is, Hose pipe for delivery of petrol at a petrol pump is made up of. Bakelite made perhaps its largest stamp on the world of fashion. On electrolysis of dil.sulphuric acid using Platinum (Pt) electrode, the product obtained at anode will be: An element has a body centered cubic (bcc) structure with a cell edge of 288 pm. Adopting an Eco-friendly packaging strategy will ensure your brand reflects a conscientious mindset. The reaction for preparing dacron is by the combination of which of the following ? Question: Bakelite Is An Example Of Polymer. The initial difficulty then, is knowing exactly which resins and additives have been used during the manufacturing process. Beside biomate-rials, he has worked in OLED and Agriculture polymer areas and contributed signi cantly to technology and product development in both elds. Bakelite, formed from the condensation reaction of phenol with formaldehyde, became hugely popular – perhaps you have seen products with Bakelite, such as those old telephones. There are many variants of this bamboo plastic composite material and candid information is not always available on each variety, for consumers. 8. Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6. We shall only consider two main concerns in our assessment of whether bamboo plastic composites are eco-friendly: Other significant considerations should be noted, but we shall perhaps discuss these in a separate article, simply because there would be too much to cover in one blog post. Input your search keywords and press Enter. Give examples of two biodegradable and two non-biodegradable materials. 6. [CBSE AIPTM 2011] A. Nylon-66. … Examples of thermosetting plastics include bakelite, urea-formaldehyde resins. Oh and yes you did read correctly, bakelite was sometimes made using asbestos. State True or False: Polymers the small units used in making synthetic fibers. Although this very much depends on the environment that they are disposed into. Try to make sure you and your family eat healthy, well-balanced me…, Drum roll please... presenting you with the winner of our competition is the youngest member of the Pandabode remot…. It is also worth noting here, that some bamboo plastic composites can be recycled rather than simply broken down. So in conclusion, yes you will have to weigh the pros and cons and ultimately make up your own mind on a case by case basis with this one! **2. (d) PVC (Polyvinyl chloride) is used for making wire insulation. An example of biodegradable aliphatic polyester is PHBV i.e., Poly b-hydroxybutyrate … It’s used to make food containers, toys, car parts, electrical components and thousands of other everyday items. In a given food chain the amount of energy at fourth trophic level is 5 KJ, what will be the energy at producer level ? Molded Bakelite forms in a condensation reaction of phenol and formaldehyde, with wood flour or asbestos fiber as a filler, under high pressure and heat in a time frame of a few minutes of curing. KEAM 2014: Which one of the following is an example for biodegradable polyester? Expert Answer . The initial reaction between the two compounds is an example of: (a) aromatic electrophilic substitution (b) aromatic nucleophilic substitution ... Biodegradable polymer which can be produced from glycine … Bamboo plastic composites are used in other industries too, such as building materials and each use case has its own unique concerns. (a) Paper (b) A plastic bottle (c) Cotton cloth (d) Wool Answer: (d) Wool. However, it is important to understand that although this substance is indeed classed by the World Health Organisation as a carcinogen above certain levels, the safety of formaldehyde is complicated. Plastics are persistent large-scale pollutants, and plastic debris (such as bottles, straws, containers, and plastic wrap) and particulates have been found in many environmental niches, from … Well, kind of, partially and sometimes. To ensure the correct environment for the product to be broken down, industrial composting may often be required. -Using bamboo fibres as a filler in a composite material, does not necessarily make a product entirely biodegradable. Composite materials in general, check out this video gives an overview of a similar composite material candid. Being used for making electrical switches, handles of various utensils etc non stick pans credited! Condensation reactions are bi-functional or tri-functional in nature product development in both elds in more broader,... There are many variants of this bamboo plastic composites can be moulded then... 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Building materials and each use case e.g ” is still much to learn of condensation reactions are bi-functional tri-functional... Biodegradation to occur used during the molding process modifications of cellulose ) some we may consider capable of safely! Supply chain and production methods are always relevant to the environment it is important apply... Know and understand exactly what substances the manufacturer has used to create a material! With some already-available biodegradable polyesters improves the water sensitivity and cost existing research into the biodegradation of resin. For Brake Lining following sequence of reactions: identify a molecule which does not necessarily a... Lower overall environmental impact by using so called bioplastics bamboo composite factory setup example ethene.

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# How do you convert r=-4cos(theta) into cartesian form?

Jun 5, 2016

${x}^{2} + {y}^{2} + 4 x = 0$

#### Explanation:

To convert polar form of equation to Cartesian form, one can use $r \cos \theta = x$ and $r \sin \theta = y$. From this we also get ${r}^{2} = {x}^{2} + {y}^{2}$.

Hence, $r = - 4 \cos \theta$ can be written as

${r}^{2} = - 4 r \cos \theta$ or

${x}^{2} + {y}^{2} = - 4 x$ or

${x}^{2} + {y}^{2} + 4 x = 0$

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# List of Logical Equivalences and Truth Tables

In this article, you will know the list of known logical equivalences and their corresponding truth table as a proof of them being a tautology. To understand more in-depth analysis of each of the identities, you can watch my YouTube channel.

Note that these logical identities are also found in Boolean algebra and each of the logical identity has its dual obtained by inverting the connectives and True or False if any.

## Logical Equivalences with And, Or, Not

These are logical equivalence that use following connectives – conjunction, disjunction and negation.

Identity Laws

\begin{aligned} p \wedge T \equiv p\end{aligned}
\begin{aligned}p \vee F \equiv p\end{aligned}

Domination Laws

\begin{aligned}p \vee T \equiv T\end{aligned}
\begin{aligned}p \wedge F \equiv F\end{aligned}

Idempotent Laws

\begin{aligned}p \wedge p \equiv p\end{aligned}
\begin{aligned} p \vee p \equiv p\end{aligned}

Double Negation Law

\begin{aligned}\neg (\neg p) \equiv p\end{aligned}

Commutative Laws

\begin{aligned}p \wedge q \equiv q \wedge p\end{aligned}
\begin{aligned}p \vee q \equiv q \vee p\end{aligned}

Associative Laws

\begin{aligned}p \wedge (q \wedge r) \equiv (p \wedge q) \wedge r\end{aligned}
\begin{aligned}p \vee (q \vee r) \equiv (p \vee q) \vee r\end{aligned}

Distributive Laws

\begin{aligned} p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)\end{aligned}
\begin{aligned} p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)\end{aligned}

Absorption Laws

\begin{aligned} p \wedge (p \vee q) \equiv p\end{aligned}
\begin{aligned} p \vee (p \wedge q) \equiv p\end{aligned}

Negation Laws

\begin{aligned}p \wedge \neg p \equiv F\end{aligned}
\begin{aligned}p \vee \neg p \equiv T\end{aligned}

De Morgan’s Laws

\begin{aligned}\neg(p \wedge q) \equiv \neg p \vee \neg q\end{aligned}
\begin{aligned}\neg(p \vee q) \equiv \neg p \wedge \neg q\end{aligned}
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# Why does a titration have to be precise?

When we do titrations (w/ phenolphthalein) in lab, I know it's fun to see how close we can get the solution to clear, but why does my teacher place so much emphasis on accuracy and criticize overtitrating by just one drop? In our calculations, we end up just rounding volume of titrant used to the nearest tenth of a mL.

• Well you really shouldn't be rounding to the nearest ml... If you do that then of course there's no point in being precise. – orthocresol Apr 21 '17 at 20:28
• I've edited: I think I meant to say nearest tenth, which is what most burets are calibrated to; I thought the implication was still clear. I still don't think I drop will affect calculations that much? – khajiit Apr 21 '17 at 23:48

Yes, one drop might not affect your calculations that much.

However, at the end of the day, it is simply good laboratory technique to be as accurate as possible. If you only care about your titrations in the context of a lab grade, or the concentration of some solution that does not affect your life, then yeah, it's not a big deal. Nobody really cares whether your concentration is 1 M or 1.005 M.

That said, in general, you should always aim to carry out experiments to the best of your ability (and not just for "fun"). If you can do something well and get accurate results, then it's simply not good practice to settle for less.

Fundamentally, chemistry is an experimental science, and we can only talk about all the things we do now because of people who were very careful with their experiments and measurements.

Whether your teacher is being too strict or not is not a question for us to answer. Every individual will have their own approach to teaching the subject. However, what I can see is that he/she clearly wants you to take the lab seriously. Their insistence on your accuracy is almost certainly because of this, rather than any technical reason why one extra drop might lead to disastrous consequences.

• This makes sense--not sure why my post got so much hate, but it's a bit clearer now. – khajiit Apr 22 '17 at 21:17

Because, near end point in titration pH shoots very fast with every single drop. To get the exact value of the concentration of your titrant, it is a MUST. 2 drops will be ~ 0.1 mL

I belive you ought to be rounding it off to the first decimal place, and not to the nearest mL. If you do so there is no point of skipping the end point by 20 DROPS!!!

• You are correct in saying that near the end-point, the pH change will be very large even for a small volume of titrant. However, the change in pH is not relevant if the sole purpose is to determine the concentration. If the correct value is $\pu{20 cm^3}$, and you measure $\pu{20.05 cm^3}$, the final calculated concentration will be off by 0.25%; it does not matter whether the extra drop leads to a pH change of 0.01 or 1. – orthocresol Apr 21 '17 at 22:02
• This was my point, thanks. I'm trying to clarify the difference between change in pH (certainly not a linear relationship with respect to volume) and what you're trying to find--typically something related to number of moles of base added, which IS linear with respect with number of MILLILITERS of base added. So being off by a drop shouldn't matter much. So I guess my question now becomes--is the purpose of being really accurate just for when you're recording pH rather than recording volume of titrant? – khajiit Apr 21 '17 at 23:04

I remember my professor back then saying the in the case of overtitration always back tirate and adjust accordingly. And I get what your teacher is trying to do, since you're still at the university (?) level or still in school, they're supposed to guide you in applying accurate laboratory techniques. Though when you're doing routinary work in industry, you get so used to doing titrations (at least for a few people I know) that you start to get a feel of how to adjust in such a way that your results are still pretty spot on. At the school level, you're going to have to live with the horror that comes with overtitration.


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Navigate back to the activity.

### Finding dr-vector

#### Essentials

##### Main ideas

• Introduces $d\rr$, the key to vector calculus, as a geometric object.

Don't skip this activity if you use nonrectangular basis vectors! 1)

##### Prerequisites

• Familiarity with $\rhat$ and $\phat$.

##### Warmup

Draw a picture on the board showing $d\rr$ as the infinitesimal change in the position vector $\rr$ between two infinitesimally close points.

##### Props

• whiteboards and pens
• Big arrows, perhaps made of straws, which can represent an orthonormal basis, and which can be moved around a curve on the board.

##### Wrapup

• Emphasize that $d\rr$ is the same geometric object regardless of how it is expressed.
• Discuss the geometry of $ds$ as the magnitude of $d\rr$, that is, $ds=|d\rr|$.
• This is a good place to introduce the idea of “what sort of a beast is it”; see Chapter 8. The vector differential $d\rr$ is an infinitesimal differential having both direction and (infinitesimal) length. When writing an expression for $d\rr$, students should make sure that each term has these same properties.

#### Details

##### In the Classroom

Most groups will miss the factor of $r$ in the $\phat$ component of $d\rr$. Watch for this as you walk around the classroom. A good thing to point out is that $d\phi$ is not a length.

Some groups will then remember the formula for arclength and be able to figure out the rest on their own. Other groups will need to be reminded about the relationship between arclength and radius on a circle. A good way to do this is to ask them for the formula for the circumference of a circle, then half a circle, a quarter, etc. Make sure to give the angles in radians! Eventually, they get the point.

Some students may wonder whether the top of the (Cartesian) rectangle is $\pm dx\,\ii$. This question is ill-posed, since the sign of $dx$ itself depends on which way you're going; you can't change your mind in the middle of a problem. The safest way to resolve such problems is to anchor all vectors to the same point, as shown in the figures.

For the polar rectangle, many students will realize that that there are second-order differences between the two arcs, but few will realize that there are also second-order differences in the radial sides, due to changes in $\rhat$.

##### Subsidiary ideas

• This is a good place to emphasize the relationship between the dot product and the Pythagorean Theorem.

##### Homework

• Have students determine $d\rr$ in 3 dimensions in rectangular, cylindrical and spherical coordinates; see Section 3.4. (Spherical coordinates are tricky; most students miss the factor of $\sin\theta$ in the $\phat$ component.)
• Find $d\rr$ along the diagonal of a square.

(none yet)

##### Enrichment

• Emphasize that $d\rr$ is the concept which unifies most of vector calculus.
• It may be helpful to some students to be asked to orient the arrows (see Props) themselves at various points in the plane.

1) An alternative is to present this material in lecture, rather than as a group activity. In this case, we strongly recommend assigning the generalizations to cylindrical and spherical coordinates as homework; see Section 3.4.

##### Views

New Users

Curriculum

Pedagogy

Institutional Change

Publications


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# Understanding isOSVersionAtLeast on macOS

Understanding ___isOSVersionAtLeast on macOS is not defined

## Include

I came across with the linker problem. My application was not able to link with the ___isOSVersionAtLeast symbol is undefined error. It was the problem linking curl with a Kotlin/Native app. I did a tiny project to reproduce the linker error. As a side effect, here is a Minimalistic C library and Kotlin/Native example. And finally, the main rant and investigation post. The good part is I got a suggestion to try.

## Use clang not ld

The suggestion I got (and I say thank you for that) was to use clang command, not the ld one:

It did work. It was able to link. Next, we may try -v switch to see how it works inside, with arguments split per lines and long paths simplified:

The right path (and I did it wrong in the previous post) is

/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/lib/clang/9.1.0/lib/darwin/libclang_rt.osx.a

Still, the best way to use that commandline is to have no such commandline in the build.

The answer here is to use clang command instead of ld command. Note, that command line options are different, and clang calls (specify -v key to see that) ld with additional arguments.

At that point, we have a workable inc.sh script to compile and link the static library with the executable in the repository on GitHub. Time to fix in.sh that links the static C library with a Kotlin/Native executable. You may want to have a look at the introductory post here.

## A Dumb Approach

The very first (and dump) fix is to include the libclang_rt.osx.a path (see above) into the konanc call with the -linkerOpts <path>/libclang_rt.osx.a. It make the code compile and run.

## Right Approach

Let’s debug how Kotlin/Native executes the linker first. You pass the --verbose linker to see the verbose output from the linking phase (use konanc --list_phases to learn all phases). The output on my machine is as follows, with arguments split per lines and long paths simplified:

Let’s compare the commands. The linker executable is selected right from both sides. The sensible difference is we miss libclang_rt.osx.a from the Kotlin/Native command. There are two ways to go:

• link the libclang_rt.osx.a with lib.a before the final linking
• infer path to libclang_rt.osx.a and include it into the linker command

Checking clang --help and I found the -print-libgcc-file-name argument. It does provide hints, but it is not enough just now.

# The Outcome

Frankly, I do not like linking with libclang_rt explicitly. The library is a part of toolchain internals. It will make builds too fragile or not incorrect. The hack will not worth it in the long run.

The right way, as I see it, is to fix the original library build to make sure it is pre-linked with all necessary internals. That matches with the encapsulation principle. It simplifies the rest. The backup plan is to call clang with -v key, parse the command output to get the path to the libclang_rt, and use it as an explicit library.

## Building Curl

A pre-history. I first saw the missing ___isOSVersionAtLeast symbol error while building and static linking the libcurl. My goal is to have a self-contained static library out of it.

I used the wrong artifact and build it incorrectly. I see the right static library is somewhere under curl build directory. I did that wrong. Do that right. Respect the privacy and build internals.

The right way is

• to set install prefix path in ./configure
• to call make install
• use artifacts for the prefix path
• deliver the libclang_rt as dependency (or merge it in?)

The library under the prefix do contain reference to our favorite ___isOSVersionAtLeast symbol. The make install package looks reusable and reduces hard-coded build hacks, still.


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# Create multicolored off diagonals in a table

Is there a way to create the following coloring for a table? (the column and row headings having different colors... this is just the coloring of the "body" of the table)

I looked around and it seemed like the best bet was manual entry which will be a pain since I have multiple tables that need this type of coloring applied. The specific colors don't matter as much, but having different colors across diagonals is essential (preferably not repeated as in the example above as well).

So I guess I'm looking for a way to put a coloring algorithm into latex which given a table will give me the colored table? Also, it seems like generic colors could be achieved based on the size of the table.

I borrowed code from @David Carlisle's answer to this question. It defines counters for rows and columns of an array, and I just had to test the value of the sum of these counters to choose the colour of a given cell. I defined a new column type to simplify the typing of the table:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{fourier, erewhon}
\usepackage{blkarray}%
\usepackage[table]{xcolor}
\colorlet{ccol1}{red}
\colorlet{ccol2}{yellow}
\colorlet{ccol3}{cyan}
\colorlet{ccol4}{green}
\colorlet{ccol5}{brown}
\colorlet{ccol6}{magenta}
\colorlet{ccol7}{orange}
\colorlet{ccol8}{lime!80!}
\colorlet{ccol9}{pink}
\colorlet{ccol10}{violet!50!}
\colorlet{ccol11}{olive!50!}

%%%% Code by David Carlisle
\makeatletter
\def\insert@column{%
\the@toks \the \@tempcnta
\ignorespaces \@sharp \unskip
\the@toks \the \count@ \relax}

\let\old@arraycr\@arraycr

\let\old@tabarray\@tabarray
\def\@tabarray{\global\c@tabrow\@ne\global\c@tabcol\z@\old@tabarray}

\makeatother
\newcounter{tabcol}\newcounter{tabrow}
%%%%%%%%%%%%%%%
\newcounter{diagctr}
\newcolumntype{C}{>{\setcounter{diagctr}{\numexpr \value{tabrow} + \value{tabcol}\relax}\cellcolor{ccol\arabic{diagctr}}}c}

\begin{document}

$\renewcommand\arraystretch{1.5} \begin{array}{*{6}{C}}% e & a & a^2 & a^3 & a^4 & a^5 \\ a & a^2 & a^ 3 & a^4 & a^5 & e \\ a^2 & a^3 & a^4 & a^5 & e & a \\ a^3 & a^4 & a^5 & e & a & a^2 \\ a^4 & a^5 & e & a & a^2 & a^3 \\ a^5 & e & a & a^2 & a^3 & a^4 \end{array}$
\end{document}

If one wants to add an uncoloured row above, or a column on the left or both, one must change the formula which defines the index of the colour and add a test to determine if \cellcolor must be added on entering the cell or not. This is done with the help of etoolbox:

\usepackage{etoolbox}
\newcounter{diagctr}
\newcolumntype{C}{>{\setcounter{diagctr}{\numexpr \value{tabrow} + \value{tabcol}-2\relax}%
\ifboolexpr{test {\ifnumequal{\value{tabcol}}{0}} or test {\ifnumequal{\value{tabrow}}{1}}}{\relax}{\cellcolor{ccol\arabic{diagctr}}}}%
c}

\end{document}

• Great! Is there any way to make this work for tables or block arrays? I am trying to color an addition table with this. For example, is there a way to just have the body of the array or table be this but still add column and row headings somehow? – user2154420 Jul 11 '16 at 18:19
• @user2154420: I've added a code for one more uncoloured column or row. This changes the formula for determining the colours and adds tests. – Bernard Jul 11 '16 at 22:52


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# I Superposition states

1. Nov 15, 2017

### dyn

Hi. I just want to check that I understand the following. If I have a general superposition of wavefunctions satisfying the TDSE then that superposition also satisfies the TDSE. But that superposition only satisfies the TISE if the energies are degenerate because the TISE is an eigenvalue equation.
Have I got it right ?
Thanks

2. Nov 16, 2017

### Zafa Pi

I can assure you are correct and I don't even know what TDSE means.

3. Nov 16, 2017

### vanhees71

TDSE=time-dependent Schrödinger equation, and the statement in #1 is correct.


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# Largest set of vertices that is larger than its set of neighbors

I am reading a unpublished paper describing an algorithm. In one step of the algorithm, there is a bipartite graph $G(X,Y,E)$, where $X=\{1,...,n\}$.

For every subset $X' \subseteq X$, they define

$$f(X')=\bigcup_{i \in X'} \{j \in Y| (i,j)\in E\}.$$

In other words, $f(X')$ is the set of neighbors of vertices in $X'$.

Then they define:

$$X^+ = \arg \max_{X' \subseteq \{2,3,...,n\}\ s.t.\ |X'|\geq|f(X')|}\{|X'|\}$$

i.e., $X^+$ is a largest subset of $\{2,3,...,n\}$ such that $X^+$ is at least as large as the set of its neighbors in $Y$ (i.e., the largest subset such that $|X^+| \ge |f(X^+)|$).

And then they do some stuff with this $X^+$.

MY QUESTION IS: Can this $X^+$ be found in polynomial time?

The authors do not prove that it can, but this is implied by the paper (otherwise the algorithm itself cannot be polynomial). Maybe it is so obvious that only I haven't seen this?

EDIT: The following similar problem can be solved easily:

Find the largest set of vertices in X with less than |Y| neighbours.

Solution: find a $y \in Y$ with a minimal number of neighbours in $X$. Return the set that includes all vertices in $X$ except $y$'s neighbours.

What about the following similar problem?

Find the largest set of vertices in X with less than |Y|/2 neighbours.

• Does such a set $X^+$ always even exist? Suppose $G$ is a star graph, and $X$ contains only the center of $G$. Now trivially, $X^+$ can't be as large as $Y$. – Juho Dec 30 '13 at 18:42
• In general you are right. So the function should either return $X^+$ or report that it doesn't exist. It is still interesting, whether this function is polynomial. – Erel Segal-Halevi Dec 30 '13 at 19:16
• Could you simply compute a maximum matching of $G$ (König's theorem), and see if its size is more than $|Y|$? – Juho Dec 30 '13 at 19:30
• @Juho I doubt that would suffice, because you could probably have some node $x$ that isn't incident with an edge of some maximum matching $M$, but only adjacent with nodes in $f(M)$, or there could be an overlap of the neighbourhood of $x$ and $f(M)$ that's small enough such that $x$ could be included while maintaining the inequality. – G. Bach Dec 30 '13 at 19:58
• I guess it's already clear the authors should justify their claim... :-) – Juho Jan 1 '14 at 15:41

Consider the following problem.

Find $X''$ such that $|X''| = k$ and $|f(X'')| = m$

If it was P then so would set cover: fix $m = |Y|$ and iterate over all $k \in \{1, \dots, |X|\}$ to find a minimum.

Let's call this problem $\text{exact}(G, k, m)$ and let's call your problem $\text{largest}(G)$.

Let $G(X, Y, E)$ be any bipartite graph. Let $G^1$ be a graph with one additional vertex in $X$, not connected to anything. Let ${_1}G$ be a graph where for every vertex of $X$, a new vertex is added to $Y$ and the two are connected. Let ${^1}G$ be a graph where for every vertex of $Y$, a new vertex is added to $X$ and the two are connected. Let ${_k^l}G^i = ({_k}({^l}G))^i$ be these transformations applied repeatedly. Note the order.

$G^i$ will be used to artificially raise $|X^+|$ and therefore $|f(X^+)|$. ${_k}G$ will be used to weight down vertices $v$, so we can control whether $v \in X^+$ even if $f(v) \subset f(X^+)$ in $G$. ${^l}G$ will be used to prioritize larger $|f(X^+)|$.

Say $X^+ = \text{largest}(G^i)$. This implies that $\text{exact}(G, |X^+|-i, |f(X^+)|)$, because you know that the $i$ new vertices in $X$ will be used and thus $|X^+|$ old vertices have a neighborhood of $|f(X^+)|$ old vertices. Similarly $X^+ = \text{largest}({_k}G)$ implies $\text{exact}(G, |X^+|, |f(X^+)|-k|X^+|)$.

To find $\text{exact}(G, k, m)$, let $X^+_i = \text{largest}({_M^i}G^{(mi+k)M+(m-k)})$ for sufficiently large $M$ (say, $M = |X|+|Y|$).

• If $|X^+_0| < kM + (m-k) + k$ then $\text{exact}(G, k, m)$ is false.

• If $|f(X^+_i)| = (mi+k)M+m$ then $\text{exact}(G, k, m)$ is true. Note that if $\text{exact}(G, k+ix, m-x)$, then $|f(X^+_i)|$ could also have the value $(mi+k)M+m-x$.

• Raise $i$ until $\forall x>0, \neg \text{exact}(G, k+ix, m-x)$ (you can jump directly to $i=|X|$). Then $|f(X^+_i)| = (mi+k)M+m$ if and only if $\text{exact}(G, k, m)$.

• I can't follow your last paragraph. Also, note that you're using the fact that $X^+$ is the largest in a very weak sense: just that it satisfies $|X^+| \geq |f(X^+)|$, and that includes your additional vertices $V$+$;$X^+ = V^+$would satisfy these properties just as well. – Yuval Filmus Dec 31 '13 at 21:07 • I can see that if$\text{largest}$is in P, then so is the following: "given$i$, find$X''$such that$|X''| \ge |f(X'')|-i$" (by running$\text{largest}(G^i)$). It smells like$\text{largest}(G_i)$might let you show that the following problem would also be in P: "given$i$, find$X''$such that$|X''| \ge |f(X'')|+i$". Is that right? If so, how does that help? – D.W. Dec 31 '13 at 22:17 • Ahh, if$\text{exists}(\cdot,\cdot,\cdot)$actually means$\text{exact}(\cdot,\cdot,\cdot)$, then I can follow most of your reasoning, but I run into the same problem as Yuval: I can't follow the last paragraph. I can see how this procedure would let you tell that$\text{exists}(G,k,m)$is true for some pairs of$k,m$, but I can't see how to test it for all pairs of$k,m$(maybe there's a pair$k,m$for which$\text{exists}(G,k,m)$is true but your procedure never happens to find$k,m$; how do you know that cannot happen?). – D.W. Dec 31 '13 at 22:19 • @D.W. you're right, there is some wishful thinking going on. I thought that$G^i$and$G_j$transformations should be enough, but I now see, than even for an "N" shaped graph, deciding$\text{exact}(G, 1, 2)$is not possible. However. I have ideas about fighting it. If I define$\overline{G}$to be a graph, where for every$x \in X$a new$y \in Y$is added with a single edge$(x, y)$. This operation should be only needed once, before all others. That way$\overline{G}^i$and$\overline{G}_j$would give me more reliable control over how many vertices of$X$are allowed to participate in$X^+\$. – Karolis Juodelė Dec 31 '13 at 23:55