text:
# Other Do "older" books provide unique insights?

Tags:
1. Oct 12, 2016

### bacte2013

Dear Physics Forum personnel,

I am curious what are your opinions about the "older" books in mathematics and physics (i.e. Neumann, Schrodinger, Dirac for QM, Hawking/Ellis for relativity, Russell for mathematics, etc.). From my experience with mathematical books, I found that I have liking to older books since they rather provide readers a chance to come up with their own definitions and understanding. For example, I had been reading Weinberg's book on relativity, but I did not understand the concept of perfect fluid until I read relevant sections on Hawking/Ellis. Also, it was not until I read Engelking to acquire intuitive understanding of paracompactness and the theories that govern the hierarchy of separation axioms, which I could not learn from more modern books in topology.

Do you think "older" books are better for first learning than more modern books in physics and math? I know that it is not quite efficient for biology and certain branches of chemistry

2. Oct 12, 2016

### Simon Bridge

There are no special insights in older books.
The best books for learning, in general, are usually the more up-to-date ones. These benefit from the concepts having been taught for a while so mistakes may be ironed out, with some presentation issues corrected - they also benefit from more recent education research. Newer books will also be written with an eye on more recent research in the field - so what they focus on teaching you is more likely to be relevant later.

However:
1. all this depends on the subject area and the level it is to be taught - some changes are just fashion;
2. individual students vary - individuals are advised to use the most relevant work that they find they understand best.
... consult a range of texts to inform your education.

3. Oct 12, 2016

### mathwonk

This question is so general I suspect it admits no one correct answer, since for one thing it depends on a measure of "best" for books. Some people will think this means most readable by a beginner, while others like myself will tend to prefer to value the authoritative depth of insight of a book. For that reason I prefer books written by real experts, and I must admit that in my opinion it was a tendency in older days for experts to be more likely to write books. Nowadays everyone writes his own book; even I myself have written books at graduate level in subjects I have later learned were treated vastly better eons ago by real masters.

To offer just a couple of examples in mathematics, it is my firm opinion after a lifetime of study and research in the area of geometry, that the absolute finest Euclidean geometry book is that by Euclid; nothing else is remotely close. I recommend approaching it however with the benefit of a good guide such as the book of Hartshorn, Geometry, Euclid and beyond. As an introduction to algebra, from elementary to advanced, I prefer Euler's Elements of Algebra. I am not a physicist but also in that field my few attempts to learn something have been frustrated by several relatively modern texts, while I greatly enjoyed texts and expositions by Dirac, Pauli, Einstein, deBroglie and Feynman.

Note some of these authors are not from olden times, like Feynman. Also in mathematics when it happens that a modern master takes trouble to write a book, I have appreciated that one as well. An example in modern algebra is the fairly recent book of Mike Artin, Algebra, which I prefer to the original classic Modern Algebra by Van der Waerden. Other excellent modern authors include Milnor, Serre, Mumford, Hartshorne, Arnol'd. As an illustration of the complexity of choice, and to illustrate a point made by Simon Bridge above, my favorite commutative algebra book, at least in terms of clarity, is the classic by Zariski and Samuel. But the topic of homological algebra had scarcely been created then, hence developments of that sort within commutative algebra do not appear much there. (They discuss tensor products but not perhaps Tor.)

So I agree with the last statements by the previous poster, that it depends on several factors and that a student must choose for herself which books "speak" most clearly to her (or him). I agree also with Simon Bridge that topics go in and out of fashion and modern books hew more closely to currently fashionable ones. This has pluses and minuses for the learner. Bill Fulton e.g. made his start in exploring intersection theory and enumerative geometry, culminating in his great book on it, by reading and trying to make sense with modern tools, of the classic work on the subject by Schubert, from 1874, I believe. This may not be quite accurate as a history of his book, since he was also trying to generalize the Riemann Roch theorem to singular spaces, but likely has some truth in it.

Another reason "old" works may be useful is the advice many of us believe in that original works are most insightful, i.e. those by the original discoverer. As time goes by these become old, and can never be replaced by newer ones, although of course people can reinterpret and offer newer versions of prior discoveries. Another example like this that I experienced, in addition to those authors mentioned above, was reading the original Michelson Morley experiment which I found much more clear and helpful than any textbook explanation of its content. Archimedes and LaGrange were also quite helpful to read, at least in places.

Last edited: Apr 30, 2017
4. Oct 13, 2016

### vanhees71

Well, there are exceptions. For me the 6-volume lectures on theoretical physics series by Sommerfeld is still outstanding. It's far better than many more recent books on classical physics. Only very little is outdated. The worst thing is that he uses the "$\mathrm{i} c t$ convention" for relativity.

5. Oct 13, 2016

### Simon Bridge

Such exception prove the rule ;)

6. Oct 13, 2016

### mpresic

Many older books are good. For example I have started to examine Whittaker and Watson, A course in Modern Analysis, and other books from Whittaker on rigid bodies and potential theory. Also I find older versions of many classic physics text as better. I prefer Goldstein's third edition to his later one with Safko, and Poole. I like the old Resnick and Halliday to the new editions after 1988. I think I also like Jackson Red to Jackson Blue.

But I do think many contemporary textbooks are also good. I like Shankar's quantum mechanics. Sakurai is also good.

Now that I think of it, I now see the books by whittaker and Watson, and whittaker should not be the first texts encountered even in graduate courses. They are too specialized and (probably) too advanced.

7. Oct 13, 2016

### Fervent Freyja

Yep, they can provide insight. New knowledge gets diluted and can become biased over time. I have found quite a few older books have explained many things in physics that seems to be lost and misinterpreted by newer books. I think a mixture of old is fine, almost necessary to understand things more deeply. Learning about the history of the body of knowledge in physics has been helpful to my understanding, there is always more to learn in that regard. Most courses teach knowledge that was fixed a decade or more ago, that is obviously important to learn. But, because course material doesn't mention old knowledge or reflect recent knowledge or work, and not even in newly published books, reading studies and trying to be relevant to whatever field you are interested in is also important. Very much of new knowledge or new understandings aren't available to the public. There are barriers in place to some very good material. There are textbooks I'd love to own, but can cost in the high 1000's!

8. Dec 20, 2016

### Demystifier

Modern books want to teach you the modern stuff, but modern books tend not to be bigger than the old ones, so obviously the modern books can spend less space to teach you the old stuff. Hence it is very likely that old (which doesn't mean obsolete) stuff can be better learned from old books.

In mathematics, there is one additional reason to read old books. Modern books on pure math are often written in the Bourbaki (theorem-proof) style, which was not so often the case with old math books. For those who do not like that style (and applied mathematicians or physicists usually don't), the old math books may look much more readable.

9. Dec 21, 2016

### vanhees71

The Bourbaki style is awful. In my opinion it's the style to write math papers but not textbooks. Textbooks should also provide the intuition behind the very abstract way of representing modern mathematics. Without intuition also in math there's little chance of getting new ideas to make progress, although at the end of course the mathematician has to bring his/her intuitions into the formal strict form a la Bourbaki. A good mathematician is made by combining both skills!

10. Dec 21, 2016

### Demystifier

I would like to see what @micromass thinks of the Bourbaki style.

11. Dec 23, 2016

### micromass

Depends on what you use it for. If you already know the material, then it's a nice exposition with some results that might be new to you. As an actual teaching tool, it's horrible. I prefer books with more motivation, with more exposition, more applications, etc.

12. Dec 23, 2016

### vanhees71

Another great old two-volume book is Max von Laue's Relativitätstheorie. I don't know, whether there's an English translation. Except for using the ict convention in the special theory it's written in a very clear style.

13. Dec 23, 2016

### Aufbauwerk 2045

I've found special insights from reading some of the great books of the past, including math and science books. Sometimes it's because I learn how it all developed and how the great minds have worked. Reading someone else explaining how Newton thought and what he said is just not the same as actually reading Newton. The same is true reading Einstein's papers. I like to understand how physics developed by getting as much as possible from the masters.

But aside from the greats, I enjoy reading standard high school and university math textbooks from the 19th or early 20th century. I like the writing style, I like the way they explain things, and the material is not out of date, excepting a few topics such as calculating with logarithms and how to use the slide rule. (Not to disrespect the slide rule in any way, it's a very green technology and it was used by flight engineers during the Apollo missions as seen in the movie Apollo 13! )

14. Dec 24, 2016

### vanhees71

Well, sometimes it's a hard task to understand old books. One example is Newton's principia. It's so utterly different from how we formulate his theory today (which is I think more due to Euler than Newton himself) that at least I had a hard time to understand it. It's of course no question interesting, how Newton derived his results in a geometrical way.

15. Dec 24, 2016

### bolbteppa

That is an unbelievable statement.

Show me one modern book with the insight you'd find in Goursat, Cartan, Dieudonne, Landau, Bourbaki, Van der Waerden, Whittaker and Watson, Sommerfeld, etc... there's about 50 quite literally unparalleled books, most more than 50 years old, there are simply and unequivocally no books that compare to these.

16. Dec 24, 2016

### George Jones

Staff Emeritus
I don't like it when people are prejudiced against old books.

I don't like it when people are prejudiced against new books.

17. Dec 24, 2016

### robphy

It may be argued that old books might not be the best place to first learn various topics in physics.

However, there are insightful gems to be found in old books... but one has to dig and be discerning.
Certainly, some ideas didn't pan out... maybe they were just wrong... or maybe some ran into technical difficulties... or maybe didn't seem that useful at the time.

I have found some gems in old relativity books that have not made it into the modern relativity books.
For example, the seeds of the Bondi k-calculus (1962) (which seems to only appear in some modern books) are found in books by Milne (1940) and even earlier in AA Robb (1911). In fact, it's surprising that some of the ideas we have today [like "rapidity"] were developed there... but Robb's name is not familiar in the typical history of relativity. In addition, Robb's (1921,1936) Geometry of Time and Space have insights which are used in modern attempts to develop spacetime geometry and possibly quantum gravity from the causal structure.
https://en.wikipedia.org/wiki/Alfred_Robb

Occasionally, one finds "new research by ______" based on a variations of "an old idea by ___________".
For instance,
One could probably change "relativity" to something else to find other examples.

18. Dec 25, 2016

### vanhees71

Well, when I look at a physics book, no matter when it was written, I first look at it critically, and then I can decide after a while whether I like it or not. That's it.

19. Mar 28, 2017

### fedor slam

Bolbteppa I would be very interested in knowing what the 50 books that were unparalleled are so that I could read them. Could you please list them?

20. Apr 21, 2017

### fedor slam

Bolbteppa I would be very interested in knowing what the 50 books that were unparalleled are so that I could read them. Could you please list them?

21. Apr 21, 2017

### Aufbauwerk 2045

I have the Heath edition of Euclid, which contains lots of notes. I have the Dover edition in which Heath's notes are fine print. Maybe it's the same size as in the original, in which case I wonder if the schoolboys used to read the notes with a magnifying gadget as I do.

In any event, after working on a Euclid-related AI project recently, I learned that any attempt to fit Euclid's concept of logic into our modern so-called rigorous logic seems like a hopeless task. There are many holes in Euclid's logic, if you try to fit it into something like first order predicate logic. But if you adjust a bit to his approach, such as his use of diagrams to set up the proof, then it's amazing what he accomplished, and it is possible to use a computer to generate proofs for Euclid. It's beyond interesting. For lack of a better word, it's a kind of "spiritual" experience.

I think one value of Euclid is he shows you how to prove things using a pure geometrical approach, as opposed to using algebra and Cartesian coordinates. Some may say this is out of date or not practical, but apparently people did not think so until the 20th century.

I wonder though if there are many schools left anywhere in the world that do teach kids geometry using Euclid. I would love to hear from someone who has been through that experience.

On a side note, I think crop circles show how useful a bit of Euclid can be.

:)

22. Apr 22, 2017

### atyy

The need for a classical-quantum cut in the standard interpretation of quantum mechanics is not a unique insight. However, the only major textbook I know of that states this insight clearly is Landau and Lifshitz (1958). After that, one has to wait until Weinberg (2012) to find this stated, but even then not as clearly as by Landau and Lifshitz. Most other good books (eg. Gasiorowicz 1974), Cohen-Tannoudji (1977), Shankar (1984), Sakurai (1985), Peebles (1992), etc) fail to state this, and one book that is terrible but popular (Ballentine) gets it completely wrong. Of the old classics, Dirac (1930) does not state it (it would defintely have spoilt his style), while von Neumann (1932) does, in his special way.

On the other hand, I think no quantum field theory text deals with the Wilsonian viewpoint clearly until the very recent text of Schwartz (2015). It's in Peskin and Shroeder (1995), Weinberg (2005), Srednicki (2007), Zee (2010), but I think in a way which is not necessarily easy to understand or to know its importance.

Last edited: Apr 22, 2017
23. Apr 22, 2017

### vanhees71

Well, we agree to disagree. I don't see, why I need a quantum-classical cut anywhere. Fortunately this question is completely irrelevant to physics, because you can use QT with or without this assumption to get the same results connected with observations, and that's all what counts. You may assume the existence of little green men on Mars as well ;-)).

Also Ballentine is a very good textbook (although there's too much interpretation in it, but at least it's the minimal statistical interpretation, which does the least harm to understanding QT). It's also the only general textbook which introduces rigged Hilbert spaces on a level suitable for the QT practitioner and which treats the Galilei group in a very elucidating way.

Weinberg's QT book is also great, although I disagree with his all too pessimistic view on interpretation. I don't think that there are unsolved problems within the physics part.

Landau-Lifshitz vol. III is quite nice, I'd say but has not enough of the general Dirac representation, i.e., it's too much leaned towards wave mechanics. It's clear advantage is that it treats many applications that cannot so easily found elsewhere.

Sakurai is my favorite as the introductory book, Dirac's classic is just Dirac at its best. No gibberish but clear math, as also holds for all of his original research papers I've read so far. Of von Neumann's book the only part that's worth reading is the math part. It's the historically first attempt at a rigorous treatment of all issues with unbound self-adjoint operators. Of course, we are lucky today to have a well-developed modern version of it, called "rigged Hilbert space", making Dirac's intuitive insight as rigorous as von Neumann's traditional treatment. The other QT books I don't know enough to make statements.

Concerning the QFT books we agree, except that I find Zee's "Nutshell" the worst of the mentioned. It's trying to pack too much in the too little nutshell and fails badly. My favorite is Schwartz.

Last edited: Apr 22, 2017
24. Apr 22, 2017

### atyy

That's why Ballentine is such a terrible book - it has too much interpretation, and it is not the minimal statistical interpretation, and therefore does the most harm in understanding it. The true minimal statistical interpretation is found in Landau and Lifshitz.

25. Apr 23, 2017

### vanhees71

Hm, but according to your own understanding of LL's interpretation it's with collapse and quantum-classical cut. LL is great in having not too much interpretation though. It's in the good tradition of the Sommerfeld school, who present theoretical physics in the no-nonsense approach. The only problem I have with LL is that it is too much focused on the wave-mechanics approach.

Also if Ballentine's book is not the minimal statistical interpretation, what do you call the minimal statistical interpretation? In my understanding Ballentine's long digressions into interpretation tries to rule out all other interpretations.

Somehow this shows, how mediocre all these interpretation discussions are: Everybody can pick whatever he or she likes to read in any text on interpretation. It's soft science at best, usually however confusing. It's against good practice in the natural sciences to formulate clear statements in a mathematically concise way to be confronted to observation and experiments in the real world!


text:
## 17.22 Internal Hom

Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_ X$-modules. Consider the rule

$U \longmapsto \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X|_ U}(\mathcal{F}|_ U, \mathcal{G}|_ U).$

It follows from the discussion in Sheaves, Section 6.33 that this is a sheaf of abelian groups. In addition, given an element $\varphi \in \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X|_ U}(\mathcal{F}|_ U, \mathcal{G}|_ U)$ and a section $f \in \mathcal{O}_ X(U)$ then we can define $f\varphi \in \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X|_ U}(\mathcal{F}|_ U, \mathcal{G}|_ U)$ by either precomposing with multiplication by $f$ on $\mathcal{F}|_ U$ or postcomposing with multiplication by $f$ on $\mathcal{G}|_ U$ (it gives the same result). Hence we in fact get a sheaf of $\mathcal{O}_ X$-modules. We will denote this sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$. There is a canonical “evaluation” morphism

$\mathcal{F} \otimes _{\mathcal{O}_ X} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathcal{G}.$

For every $x \in X$ there is also a canonical morphism

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})_ x \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x)$

which is rarely an isomorphism.

Lemma 17.22.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ be $\mathcal{O}_ X$-modules. There is a canonical isomorphism

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X} (\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}, \mathcal{H}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X} (\mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{H}))$

which is functorial in all three entries (sheaf Hom in all three spots). In particular, to give a morphism $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G} \to \mathcal{H}$ is the same as giving a morphism $\mathcal{F} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{H})$.

Proof. This is the analogue of Algebra, Lemma 10.12.8. The proof is the same, and is omitted. $\square$

Lemma 17.22.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_ X$-modules.

1. If $\mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F} \to 0$ is an exact sequence of $\mathcal{O}_ X$-modules, then

$0 \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}_1, \mathcal{G}) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}_2, \mathcal{G})$

is exact.

2. If $0 \to \mathcal{G} \to \mathcal{G}_1 \to \mathcal{G}_2$ is an exact sequence of $\mathcal{O}_ X$-modules, then

$0 \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}_1) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}_2)$

is exact.

Proof. Omitted. $\square$

Lemma 17.22.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_ X$-modules. If $\mathcal{F}$ is finitely presented then the canonical map

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})_ x \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x)$

is an isomorphism.

Proof. By localizing on $X$ we may assume that $\mathcal{F}$ has a presentation

$\bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{O}_ X \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ X \to \mathcal{F} \to 0.$

By Lemma 17.22.2 this gives an exact sequence $0 \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \to \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{G} \longrightarrow \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{G}.$ Taking stalks we get an exact sequence $0 \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})_ x \to \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{G}_ x \longrightarrow \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{G}_ x$ and the result follows since $\mathcal{F}_ x$ sits in an exact sequence $\bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{O}_{X, x} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_{X, x} \to \mathcal{F}_ x \to 0$ which induces the exact sequence $0 \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x) \to \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{G}_ x \longrightarrow \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{G}_ x$ which is the same as the one above. $\square$

Lemma 17.22.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_ Y$-modules. If $\mathcal{F}$ is finitely presented and $f$ is flat, then the canonical map

$f^*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f^*\mathcal{F}, f^*\mathcal{G})$

is an isomorphism.

Proof. Note that $f^*\mathcal{F}$ is also finitely presented (Lemma 17.11.4). Let $x \in X$ map to $y \in Y$. Looking at the stalks at $x$ we get an isomorphism by Lemma 17.22.3 and More on Algebra, Lemma 15.64.4 to see that in this case $\mathop{\mathrm{Hom}}\nolimits$ commutes with base change by $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$. Second proof: use the exact same argument as given in the proof of Lemma 17.22.3. $\square$

Lemma 17.22.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_ X$-modules. If $\mathcal{F}$ is finitely presented then the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is locally a kernel of a map between finite direct sums of copies of $\mathcal{G}$. In particular, if $\mathcal{G}$ is coherent then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is coherent too.

Proof. The first assertion we saw in the proof of Lemma 17.22.3. And the result for coherent sheaves then follows from Lemma 17.12.4. $\square$

Lemma 17.22.6. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Then we have

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_1}(\mathcal{F}_{\mathcal{O}_1}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}(\mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{G}))$

bifunctorially in $\mathcal{F} \in \textit{Mod}(\mathcal{O}_2)$ and $\mathcal{G} \in \textit{Mod}(\mathcal{O}_1)$.

Proof. Omitted. This is the analogue of Algebra, Lemma 10.14.4 and is proved in exactly the same way. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

format: plain


text:
Here I’ll give the theory part of Neural Networks, sepecifically three kinds of NN: Normal Neural Networks, Convolutional Neural Networks, Recurrent Neural Networks.

# Neural Networks

## In single neuron:

\begin{align*} z &= Wx + b \\ a &= \sigma (z) \end{align*}

## Activation Function

• Perceptron: binary classifier, small changes are not reflected.\begin{align*} f(x) = \begin{cases} 1 & \text{if } Wx+b>0\\ 0 & \text{otherwise} \end{cases} \end{align*}

Sigmoid: special case of the logistic function with S shape curve, more dynamic
\begin{align*} S(x) = \frac{1}{1 + e^{-x}} = \frac{e^x}{e^x +1} \end{align*}

Hyperbolic Tangent: Tanh(z)
\begin{align*} cosh x &= \frac{e^x + e^{-x}}{2}\\ sinh x &= \frac{e^x - e^{-x}}{2}\\ tanh x &= \frac{cosh x}{sinh x } \end{align*}

• ReLU (Rectified Linear Unit): max(0,z)

ReLU and tanh tend to have the best performance.

• Softmax Regression:

\begin{align*} z_i &= \sum_jW_{i,j}x_j + b_i \\ softmax(z)_i &= \frac{\text{exp}(z_i)}{\sum_j \text{exp}(z_j)} \end{align*}

## Cost/Loss Function

Cost function is the measurement of the error.
\begin{align*} z &= Wx + b \\ a &= \sigma (z)\\ C &= \frac{1}{n}\sum(y_{true} - a)^2 &\text{Quadratic Cost}\\ C &= -\frac{1}{n}\sum(y_{true}\cdot ln(a) + (1-y_{true})\cdot ln(1-a) ) &\text{Cross Entropy} \end{align*}
The larger errors are more prominent due to the squaring. It causes a slowdown in learning speed.
Cross Entropy:
It allows for faster learning. The larger the difference, the faster the neuron can learn.

Gradient Descent is an optimization algorithm for finding the minimum of a function. Here it minimizes the error to find the optiml value. 1-D example below shows the best parameter value (weights of the neuron inputs) we should choose to minimize the cost.

For complicated cases more than 1 dimension, we use biilt-in algebra of Deep learning library to get the optimal parameters.

1. Learning Rate: defines the step size during gradient descent, too small - slow pace, too small - overshooting
2. Batch Size: batches allow us to use stochastic gradient descent, in case the datasets are large, if all the them are fed at once the computation would be very expensive. Too small - less representative of data, too large - longer training time
3. Second-Order Behavior of Gradient Descent: adjust the learning rate based on the rate of descent(second-order behavior: derivative),large learning rate at the beginning, adjust to slower learning rate as it get closer. Methods: AdaGrad, RMSProp, Adam

Vanishing Gradients: when increasing the number of layers in a network, the layers towards the input will be affected less by the error calculation occuring at the output as going backwards throught the network. Initialization and Normalization will help to mitigate the issue.

Backpropagation is to calculate the error contribution of each neuron after a batch of data is processed. It works by calculating the error at the output and then distributes back through the network layers. It belongs to supervised learning as it requires a known output for each input value. The mathematical detail is showed below from Andrew Ng’s Neural Networks and Deep Learning in Coursera.

## Initialization of Weights

### Zeros

No randomness (too subjective) so not a good choice

### Random Distribution

Random distribution near zero is not optimal and results in activaion function distorition (distorted to large values)

### Xavier (Glorot) Initialization

The weights drawn from uniform or normal distribution, with zero mean and specific variance $\text{Var}(W) =\frac{1}{n_{in}}$:
\begin{align*} &Y = W_1X_1 +W_2X_2 + ... + W_nX_n\\ &\text{Var}(W_iX_i) = E[X_i]^2\text{Var}(W_i) + E[W_i]^2\text{Var}(X_i) + \text{Var}(W_i)\text{Var}(X_i)\\ &\text{Var}(W_iX_i) = \text{Var}(W_i)\text{Var}(X_i) \qquad (\because E[X_i] = 0)\\ &\text{Var}(Y) = \text{Var}(W_1X_1 + W_2X_2 +... + W_nX_n) = n \text{Var}(W_i)\text{Var}(X_i) \\ &\because \text{Variance of the output is equal to the variance of the input}\\ &\therefore n\text{Var}(W_i) = 1\\ &\therefore \text{Var}(W_i) = \frac{1}{n} = \frac{1}{n_{in}} = \frac{2}{n_{in} + n_{out} } \end{align*}

## Overfitting Issue

With potentially hundreds of parameters in a deep learning neural network, the possibility of overfitting is very high. We can mitigate this issue by the following ways:

• $L_1/L_2$ Regularization
Add a penalty for a larger weights in the model (not unique to neural networks)
• Dropout
Remove neurons during training randomly so that the network does not over rely on any particular neuron (unique to neural networks)
• Expanding Data
Artificially expand data via adding noise, tilting images

# Convolutional Neural Networks

## Tensor

Tensor: N-dimensional arrays:
Scalar: 3
Vector: [3,4,5]
Matrix: [[3,4],[5,6],[7,8]]
Tensor: [[[1,2], [3,4]], [[5,6], [7,8]]]
We use tensors to feed in sets of images into the model - (I,H,W,C)
I: Images
H: Height of Image in Pixels
W: Width of Image in Pixels
C: Color Channels: 1 - Grayscale, 3 - RGB

## DNN vs CNN

### Convolution

The left figure is densely connected layer, every neuron in the layer is directly connected to the every neuron in the next layer. While each unit in the convolutional layer is connected to a smaller number of nearby units in the next layer.

The reason for the idea of CNN is that most images are at least 256 by 256 pixels or greater (MNIST only 28 by 29 pixels, 784 total). So there are too many parameters unscalable to new images. Another merit of CNN is for image processing, pixels nearby to each other are much more correlated to each other for image detection. Each CNN layer looks at an increasingly larger part of the image. And having units only connected to nearby units helps invariance. CNN helps limit the search of weights to the size of the convolution. Convolutional layers are only connected to pixels in their respective fields. By adding a padding of zeros around the image we can fix the issue for edge neurons where there may not be an input for them.
Take the example of 1-D Convolution, we treat the weights as a filter for edge detection, then expand one filters to multiple filters. Stride 1 means 1 unit at a time. The top circle means the zero padding added to include more edge pixels. Each filter detects a different feature.

Now for simplicity, the sets of neurons are visualized as blocks.

For 2-D Images and Color Images:

### Subsampling

Except convolutional layers, there’s another kind of layers called pooling layers. Pooling layers will subsample the input image to reduce the memory use and computer load as well as reducing the number of parameters.
Take example of MNIST, only select max value to the next layer, and move over by stride. So the pooling layer will finally remove a lot of information.

Another technique is “Dropout”. Dropout is regarded as a form of regularization as during training, units are randomly dropped with their connection to help prevent overfitting.

# Recurrent Neural Networks

Common Neural Networks can handle classification and regression problems, but for sequence information, we need Recurrent Neural Networks.
Normal Neural Networks just aggregation of inputs and pass the activation function to get the output. Recurrent Neural Networks send output back to itself.
Here I have to mention my previous research work – Echo State Networks, also belong to RNN: ESN

Cells that are a function of inputs from previous time steps are also know as memory cells.

0%


text:
# Acronym/Abbreviation meaning of RM/MS&C (NASA)

The acronym RM/MS&C(NASA) means : Redundancy Management/Moding

## What is the acronym meaning/definition of RM/MS&C ?

The abbreviation RM/MS&C stands for: (NASA) Redundancy Management/Moding

## Acronym meaning of RM/MS&C (NASA)

RM/MS&C (NASA) means: Redundancy Management/Moding

### Astrological/numerological analysis and explanation of RM/MS&C acronym

Astrological/horoscope meaning of RM/MS&C. RM/MS&C means: With a Life Path 3, your numbers are (3, 12/3, 21,/3). The Life Path 3 shows you have a strong sense of creativity and wonderful communication skills. Your performance for most things comes through its ingenious expression. A truly gifted 3 has the extraordinary ability to innovate, usually in the verbal area, write, speak, act, or similar circumstances. They are great entertainers in the world, and their brightness sparkles with a very optimistic attitude. The positive side of this path stresses harmony, beauty and pleasure; their talents to share with the world of invention. The acquisition and capacity in creative self-expression is the highest level of performance for this type of life. Life path 3's live thier lives to fullest, often without much worry about tomorrow. They are not very good with money because of the general lack of concern in cautious spending. You spend it when you have it and not when you don't. 3's like to connect with people. The characteristics of the 3 are warmth and friendliness, good conversationalist, social and open. A good speaker, both from the standpoint of being a joy to listen to, but more importantly, those who have to listen to other's ability. Thus the life path 3 persons are always a welcome addition to any social situation and always know how others feel at home. Their approach to life tends to be very positive. Their view is almost sunny, safe and open heart. They are happy and often inspired on the lookout for and need to stimulate and encourage others. There is a not so glamourous side the life path of 3 so to speak. This is a surprise for those who think they are well acquainted. 3 is actually a very sensitive soul. If injured, you can easily retreat into a shell of sullen silence for long periods. However, with the many setbacks that occur in life, they readily bounce back for more. It is usually easy for you to deal with the problems as they are free, the non-existence of problems can result in long periods of slow decline. You are very sensitive to the feelings, emotions and aspirations of others. In romance, 3 is a very passionate and faithful lover. Emotional experiences of all kinds deeply affect 3s in general and can take the form of drama in some cases. The emotional connection for 3s could be sometimes demanding to their partner or friends. As with most problems in life for life path 3 balance in relations is hard to achieve. The big test with a 3 Life path is in managing the ups and downs of life. They are not very well suited in any environment that requires routine or when placed under a dominant person. They are sometimes slow because they are contemplative in thought as they struggle to please people enough. This could sometimes be a frustrating affair when working with or among them. For the few to be on the negative side of this path of life, 3s live with the joy to feel impressed and be impressed. They may treat life frivolously and in a superficial way sometimes. It is possible to dispel their abilities and express little sense of concern. 3 may be a mystery, can be moody for no apparent reason and tend to withdraw. Escapist tendencies with the 3 life path is not unusual, and it is very difficult to stay in one place or position as they are grigarious and like to enjoy life. They hate to be critical to avoid negative energies, could be impatient, intolerant, or overly optimistic.

## Meaning of RM/MS&C acronym by its letters in astronumerology

RM/MS&C abbreviation means:

Acronyms, abbreviations and meanings similar to RM/MS&C

Popular questions of RM/MS&C people search before coming here

Q: What does RM/MS&C stand for? A: RM/MS&C stands for "Redundancy Management/Moding".

Q: How to abbreviate "Redundancy Management/Moding"? A: "Redundancy Management/Moding" can be abbreviated as RM/MS&C.

Q: What is the meaning of RM/MS&C acronym? A: The meaning of RM/MS&C acronym is "Redundancy Management/Moding".

Q: What is RM/MS&C abbreviation? A: One of the definitions of RM/MS&C is "Redundancy Management/Moding".

Q: What does RM/MS&C mean? A: RM/MS&C as abbreviation means "Redundancy Management/Moding".

Q: What is shorthand of Redundancy Management/Moding? A: A common acronym of "Redundancy Management/Moding" is RM/MS&C.

## Famous / Popular results for RM/MS&C

Thiele/Small parameters

{ms}}={\frac {2\pi \cdot F_{\rm {s}}\cdot M_{\rm {ms}}}{R_{\rm {ms}}}}={\frac {1}{R_{\rm {ms}}}}{\sqrt {\frac {M_{\rm {ms}}}{C_{\rm {ms}}}}}} Qts – Total Q of

RM Poole

Navy News. 2 August 2013. Retrieved 16 May 2014. "RM Poole, Hamworthy" (Word Document). gov.uk. MS word doc, para 13: Ministry of Defence. 5 December

Minimal subtraction scheme

{\displaystyle \mu ^{2}\to \mu ^{2}{\frac {e^{\gamma _{\rm {E}}}}{4\pi }}} , with γ E {\displaystyle \gamma _{\rm {E}}} the Euler–Mascheroni constant. 't Hooft

Rm (Unix)

similar to the del command in MS-DOS, OS/2, and Microsoft Windows. The command is also available in the EFI shell. The rm command removes references to

List of compositions by Sylvius Leopold Weiss

number in PL-Wu Ms. RM 4137 (olim Ms. MF. 2004), Warszawa, Biblioteka Uniwersytecka PL-Wu2005: Folio number in PL-Wu Ms. RM 4138 (olim Ms. MF. 2005), Warszawa

Mortgage constant

principal as well as the interest. The Rm could be lower than the interest for a negatively amortizing loan. L o a n C o n s t a n t = ( i / m ) / ( 1 − (

RM Education

RM Education is the principal division of the RM Group, a British company that specialises in providing information technology products and services to

Transmon

with the only difference being the increase in the E J / E C {\displaystyle E_{\rm {J}}/E_{\rm {C}}} ratio, achieved by shunting the Josephson junction with

List of most expensive cars sold at auction

new owners have paid. A 1962 Ferrari 250 GTO, serial number 3413GT, sold at RM Sotheby's Auction on August 25, 2018, for US$48,405,000 (including buyer's Specific strength {1{\rm {\,g}}\cdot 9.80665{\rm {\,ms^{-2}}}}{1{\rm {\,g}}/9000{\rm {\,m}}}}={\frac {9.80665{\rm {\,ms^{-2}}}}{1/9000{\rm {\,m}}}}=9.80665{\rm {\,ms^{-2}}}\ Thiele/Small parameters {ms}}={\frac {2\pi \cdot F_{\rm {s}}\cdot M_{\rm {ms}}}{R_{\rm {ms}}}}={\frac {1}{R_{\rm {ms}}}}{\sqrt {\frac {M_{\rm {ms}}}{C_{\rm {ms}}}}}} Qts – Total Q of RM Poole Navy News. 2 August 2013. Retrieved 16 May 2014. "RM Poole, Hamworthy" (Word Document). gov.uk. MS word doc, para 13: Ministry of Defence. 5 December Minimal subtraction scheme {\displaystyle \mu ^{2}\to \mu ^{2}{\frac {e^{\gamma _{\rm {E}}}}{4\pi }}} , with γ E {\displaystyle \gamma _{\rm {E}}} the Euler–Mascheroni constant. 't Hooft Rm (Unix) similar to the del command in MS-DOS, OS/2, and Microsoft Windows. The command is also available in the EFI shell. The rm command removes references to List of compositions by Sylvius Leopold Weiss number in PL-Wu Ms. RM 4137 (olim Ms. MF. 2004), Warszawa, Biblioteka Uniwersytecka PL-Wu2005: Folio number in PL-Wu Ms. RM 4138 (olim Ms. MF. 2005), Warszawa Mortgage constant principal as well as the interest. The Rm could be lower than the interest for a negatively amortizing loan. L o a n C o n s t a n t = ( i / m ) / ( 1 − ( RM Education RM Education is the principal division of the RM Group, a British company that specialises in providing information technology products and services to Transmon with the only difference being the increase in the E J / E C {\displaystyle E_{\rm {J}}/E_{\rm {C}}} ratio, achieved by shunting the Josephson junction with List of most expensive cars sold at auction new owners have paid. A 1962 Ferrari 250 GTO, serial number 3413GT, sold at RM Sotheby's Auction on August 25, 2018, for US$48,405,000 (including buyer's

Specific strength

{1{\rm {\,g}}\cdot 9.80665{\rm {\,ms^{-2}}}}{1{\rm {\,g}}/9000{\rm {\,m}}}}={\frac {9.80665{\rm {\,ms^{-2}}}}{1/9000{\rm {\,m}}}}=9.80665{\rm {\,ms^{-2}}}\

## Meaning of ASDES

ASDES means: Aircraft Sensor Data For The Earth Sciences

## Meaning of SCT

SCT means: Stem Cell Transplantation

## Meaning of DDI

DDI means: Discrete Data Input

## Meaning of SME

SME means: Scan Mirror Electronics - Subject Matter Expert


text:
# Why is a graph an ordered pair?

a graph is an ordered pair G = (V, E) comprising a set V of vertices or nodes together with a set E of edges or lines, which are 2-element subsets of V

Why must it be an ordered pair? It seems irrelevant if you mention V or E first. Must V come first since E is made up of V?

-
No, it really is irrelevant. – Karolis Juodelė Dec 14 '13 at 9:01
I've seen a lot of books define it with ordered pairs. – Celeritas Dec 14 '13 at 9:08
That's for convenience, to some extent. If I say I have a graph $(A, B)$, you know that $A$ is the vertices and $B$ is the edges. Unordered pairs are less natural. – Karolis Juodelė Dec 14 '13 at 10:14
Another thing you're missing is that the question is wrong. Graph isn't a pair. It can be described by a pair. There are plenty of other ways to describe a graph. – Karolis Juodelė Dec 14 '13 at 10:19
The question is correct, it is perfectly in order to define a graph to be an ordered pair $(V,E)$ as stated. You are free to use alternatives, but this does not invalidate the definition. – Chris Godsil Dec 14 '13 at 15:50

You don't even need a pair. A graph may be defined simply as a set $E$ of singleton or two-element sets. The set of vertices is then $V=\bigcup E.$ However, this will not do if you need to specify loops, which are edges that connect a vertex to itself, or multiple edges. Both these types of edge would be regarded as illegitimate or irrelevant in the mainstream of graph theory. And of course ordered pairs are needed to define a directed graph.


text:
Cp vs Cv for liquid water @ < 4 °C

I'm looking at a textbook question (Engel & Reid, Thermodynamics, Statistical Thermodynamics & Kinetics, 4 ed, Q3.1) which states the following:

The heat capacity $$C_{\mathrm m,p}$$ is less than $$C_{\mathrm m,V}$$ for $$\ce{H2O(l)}$$ near $$\pu{4 °C}$$. Explain this result.

Their answer is that water contracts as you heat it in this regime, thus (at constant $$p$$) the surroundings are doing work on the system, hence $$C_{\mathrm m,p}\lt C_{\mathrm m,V}$$.

But the textbook also contains the following equation (which is correct, and whose only restrictions are that the system is of fixed phase and composition, and $$\text{đ}w = –pdV$$, i.e., pV-work only):

$$C_{\mathrm m,p}-C_{\mathrm m,V} = T V_\mathrm m \beta^2/\kappa$$

where $$\beta$$ is the isobaric thermal expansivity, and $$\kappa$$ is the isothermal compressibility.

Based on this, for a substance of fixed phase and composition, $$C_{\mathrm m,p} \ge C_{\mathrm m,V}$$, always, because, while $$\beta$$ can be $$\le0$$* (water near $$\pu{4 °C}$$ being a notable example), $$\beta^2$$, as well as $$T, V_\mathrm m$$ and $$\kappa$$, are always positive. [*Edited from $$\lt 0$$ to $$\le 0$$ based on Night Writer's answer.]

So is the statement in the textbook's question just a flat-out mistake, or am I missing something here?

It appears their error is in assuming the difference between Cp and Cv is due to pV expansion alone, when in fact there are two terms:

$$C_p - C_V = p \left(\partial V\over\partial T\right)_p+ \left(\partial U\over\partial V\right)_T \left(\partial V\over\partial T\right)_p,$$

where the first term on the RHS is the PV work per unit change in T, while the second term is the change in internal energy with respect to volume (which results from changing the intermolecular distance between interacting particles) times the rate at which the volume changes with temperature.

It would be nice to have direct reference-quality experimental values for $$C_{\mathrm m,p}$$ and $$C_{\mathrm m,V}$$ for $$\ce{H2O(l)}$$ near $$\pu{4 °C}$$, but I've not been able to locate them, nor do I expect to: because of the difficulty of accurately measuring $$C_{\mathrm m,V}$$ for a liquid, measurements are typically made at constant pressure, giving $$C_{\mathrm m,p}$$, and then $$C_{\mathrm m,V}$$ is calculated using the above equation.

• In addition to the answers below, the data do show that $C_p \gt C_V$, albeit by a very small amount near zero $^\circ$C ($\approx 0.05$ J/mol/K) at temperatures from just above zero to 10 $^\circ$C. So the question is factually wrong. Apr 16, 2019 at 10:38
• @porphyrin Can you provide a reference for that data? Apr 16, 2019 at 15:04
• I just looked on the web: engineeringtoolbox.com/specific-heat-capacity-water-d_660.html . Apr 16, 2019 at 19:53
• @porphyrin I suspect the only experimental data there is for Cp, and that Cv was calculated from the Cp values using the above formula. There's no way to tell without a reference to the original source. Apr 16, 2019 at 20:13
• Ok, I had'n't realised that. You will have to search for some original data, but I can see no reason why $C_v$ should be greater than $C_p$. That $\beta \to 0$ seems to be the best explanation as shown by @ Night Writer in that answer. Apr 17, 2019 at 6:27

Repeating your line of reasoning, if $$C_{\mathrm m,p} you would conclude based on the expression you provide ($$C_{\mathrm m,p}-C_{\mathrm m,V} = T V_\mathrm m \alpha^2/\kappa$$) that $$\kappa<0$$ which in turn would mean that

$$\left(\frac{\partial V_m}{\partial P}\right)_T > 0,\label{eqn:1}\tag{1}$$

which is not physically reasonable except presumably in some very unusual circumstances. Therefore, either the question statement contains a fallacy, or the above analysis contains a fallacy, i.e. $$\eqref{eqn:1}$$ is reasonable. My bet is on the first scenario, as you suggest.

What seems to happen is that near $$\pu{4 °C}$$

$$C_{\mathrm m,p} = C_{\mathrm m,V}\label{eqn:2}\tag{2}$$

because the change in the density or molar volume with $$T$$ (and therefore also the thermal expansion coefficient $$\beta$$ - usually designated with the symbol $$\alpha$$) becomes zero, that is there is a (well known) minimum in the dependence of the density or molar volume on $$T$$ at constant $$P$$, illustrated in the following figure (source: wikipedia)

In agreement with this, if you inspect available heat capacity data you will see that the isochoric and isobaric heat capacities converge as the temperature of liquid water nears $$\pu{0 °C}$$.

The International Association for the Properties of Water and Steam has released a document (IAPWS R7-97(2012)) containing expressions for the heat capacities of water parameterized based on experimental data (see Table 3).

• That's a nice point about $\beta$ for water going to zero at 4C. Apr 16, 2019 at 20:23
• @theorist I think the answer is a combination of my explanation and the alternate answer: that answer explains some of the flaws in the book's reasoning, mine explains others (well, I echo your explanation - good catch - and follow it to it's logical conclusion based on the available data). I did find some standard data but have not crunched the numbers. Apr 20, 2019 at 19:43

The textbook's question is flawed because, as you state, its assertion that $$c_p < c_v$$ is false.

The textbook's answer messes up the math a little bit. Their logic is based on the First Law:

\begin{align} \Delta u &= q - P \Delta v \\ q &= \Delta u + P \Delta v \end{align} Considering an isobaric process, we have \begin{align} \overbrace{c_p \Delta T}^q &= \overbrace{\left(\frac{\partial u}{\partial T}\right)_P \Delta T}^{\Delta u} + P \overbrace{\left(\frac{\partial v}{\partial T}\right)_P \Delta T}^{\Delta v} \\ c_p &= \underbrace{\left(\frac{\partial u}{\partial T}\right)_P}_X + \underbrace{\left(\frac{\partial v}{\partial T}\right)_P}_{<0} \end{align} If term $$X$$ were the same as $$c_v$$ then it would follow that $$c_p < c_v$$, but the wrong property is held constant: $$c_v$$ would be $$\left(\frac{\partial u}{\partial T}\right)_v$$. Evidently $$\left(\frac{\partial u}{\partial T}\right)_P - \left(\frac{\partial u}{\partial T}\right)_v$$ is more positive than $$P \left(\frac{\partial v}{\partial T}\right)_P$$ is negative. The textbook seems to be incorrectly assuming that $$X = c_v$$.

Note that the corner case $$c_p = c_v$$ is allowed; this would occur when $$\beta = 0$$, $$T = 0$$, $$v = 0$$, or $$\kappa = \infty$$ (the first case being the one that can actually occur in everyday life).

Original take-down of the textbook answer:

Based on your summary, the textbook's answer fails to account for the fact that switching from an isobaric path to an isothermal path changes the end state. The (flawed) logic seems to be: "if I go from A to B isochorically then I need to provide some amount of heat, but if I go from A to B isobarically then the surroundings provide some work so I don't need to provide as much heat." The flaw is that if A-B is an isochor then an isobar beginning at A doesn't go through B. An apples-to-oranges comparison would consider the heat required to reach a third point C which is at the same pressure as A and temperature as B, and would find that this heat is never lower than that required to reach B.

• You seem to be saying the textbook's answer is flawed because you can't compare Cp and Cv, the reason being that the difference in constraints implicit to Cp vs. Cv brings you to different final states (making it apples to oranges). However, even with this, you certainly can compare Cp to Cv (their difference is $TV\beta^2/\kappa$). Thus (and forgive me if I'm not understanding you correctly) I don't think the specific issue you raise (the difference in final states) is relevant here. I.e., I think the textbook is wrong, but not for the reason you give. Apr 16, 2019 at 22:05
• I'm saying that their argument that the isobaric process involves less work than the isochoric process is overly simplistic (because it assumes the two processes have the same endpoint). You can certainly compare $c_p$ and $c_v$, for example using the formula you provided, but their attempt to provide an explanation based directly on the First Law/intuition is flawed. Apr 16, 2019 at 22:52
• I think their argument is flawed, but not for the reason you give. The problem isn't that they don't go to the same endpoint; that doesn't preclude the comparison. For instance, it's not a problem to say the reason Cp = Cv +nR for an ideal gas is that, during constant-P heating, you have to input additional thermal energy for the PV work associated with expansion. By your argument, you couldn't make a universal statement like that for ideal gases simply because they don't go to the same final state in a constant-V vs a constant-P heating. But that's simply not the case. [continued....] Apr 17, 2019 at 0:55
• ...Hence the disparity of final states is not the logical issue per se. Rather, I think the reason their argument is simplistic/flawed is that it doesn't account for the difference in internal energy due to changes in volume in real materials (which have attractive and repulsive interactions). I.e., it accounts only for the difference in the work done on the surroundings. I.e., you're saying that the problem is that they're comparing processes with different final states. I'm saying it's OK for the final states to be different, so long as you account for the effect of those differences. Apr 17, 2019 at 0:57
• I suppose we are arguing over what is the "right" way is to do something wrong, haha. You have the advantage of actually having read the textbook's answer in its entirety ;) I have updated my answer to reflect the flaw that you see but also retained the flaw that I saw so that our comment chain still makes sense. Apr 17, 2019 at 1:50


text:
# A. V. Zagorodnyuk

orcid.org/0000-0002-5554-4342
Search this author in Google Scholar

Articles: 4

### Continuous symmetric 3-homogeneous polynomials on spaces of Lebesgue measurable essentially bounded functions

Methods Funct. Anal. Topology 24 (2018), no. 4, 381-398

Vector spaces of all homogeneous continuous polynomials on infinite dimensional Banach spaces are infinite dimensional. But spaces of homogeneous continuous polynomials with some additional natural properties can be finite dimensional. The so-called symmetry of polynomials on some classes of Banach spaces is one of such properties. In this paper we consider continuous symmetric $3$-homogeneous polynomials on the complex Banach space $L_\infty$ of all Lebesgue measurable essentially bounded complex-valued functions on $[0,1]$ and on the Cartesian square of this space. We construct Hamel bases of spaces of such polynomials and prove formulas for representing of polynomials as linear combinations of base polynomials. Results of the paper can be used for investigations of algebras of symmetric continuous polynomials and of symmetric analytic functions on $L_\infty$ and on its Cartesian square. In particular, in order to describe appropriate topologies on the spectrum (the set of complex valued homomorphisms) of a given algebra of analytic functions, it is useful to have representations for polynomials, obtained in this paper.

### Hypercyclic composition operators on Hilbert spaces of analytic functions

Methods Funct. Anal. Topology 20 (2014), no. 3, 284-291

In the paper we consider composition operators on Hilbert spaces of analytic functions of infinitely many variables. In particular, we establish some conditions under which composition operators are hypercyclic and construct some examples of Hilbert spaces of analytic functions which do not admit hypercyclic operators of composition with linear operators.

### Polarization formula for $(p,q)$-polynomials on a complex normed space

Methods Funct. Anal. Topology 17 (2011), no. 1, 75-83

The aim of this paper to give some analogues of polarization formulas and the polarization inequality for $(p,q)$-polynomials between complex normed spaces. Obtained results are useful for investigation of real-differentiable mappings on complex spaces.

### A Hilbert space of functions of infinitely many variables

Methods Funct. Anal. Topology 10 (2004), no. 2, 13-20


text:
# torch.Tensor.type_as¶

Tensor.type_as(tensor)Tensor

Returns this tensor cast to the type of the given tensor.

This is a no-op if the tensor is already of the correct type. This is equivalent to self.type(tensor.type())

Parameters

tensor (Tensor) – the tensor which has the desired type


text:
synthetic polymer examples. Remember to show only six carbon atoms in the chain if three monomers are used. Five different repeating units are found in the polymer obtained by cationic polymerization of 4 -methyl- 1 -pentene. Polymers can be natural as well as synthetic. Table 3 on the Data Sheet can be used to identify a type of carbon atom responsible for this peak. the reaction that causes a firework to explode requires more energy to start, and occurs less rapidly than the reaction that causes a candle to burn. What are the different types of polymers that monomers can create? In some cases, this term is used to express the number of monomer units in an average polymer molecule. 1.2 Polymer Structure. To learn more, see our tips on writing great answers. Sometimes polymers are made from bound groups of monomer subunits (up to a few dozen monomers) called oligomers. the animal has hairless feet with sharp claws. How to identify the repeating unit in a polymer? The structure and chemical composition of the polymer chain determines the physical properties of the material. A repeat unit (or mer), is not to be confused with the term monomer, which refers to the small molecule from which a polymer is synthesized. Figure 1 (from Chemistry in Context) shows a single monomer, and a polymer made of identical monomers linked together. To make the chain, many links or “ … Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Examples of polymers include plastics such as polyethylene, silicones such as silly putty, biopolymers such as cellulose and DNA, natural polymers such as rubber and shellac, and many other important macromolecules. Explanation ..... (2) (Total 11 marks) Page 13 of 86. Repeating units are often made of carbon and hydrogen and sometimes oxygen, nitrogen, sulfur, chlorine, fluorine, phosphorous, and silicon. The repeating structural unit of most simple polymers not only reflects the monomer(s) from which the polymers are constructed, but also provides a concise means for drawing structures to represent these macromolecules. How do you explain the functionality of a monomer? PVA is formed from B by the process of addition polymerisation. A polymer may be a natural or synthetic macromolecule comprised of repeating units of a smaller molecule (monomers). The simplest definition of a polymer is a useful chemical made of many repeating units. For polyethylene, arguably the simplest polymer, this is demonstrated by the following equation. Shouldn't the $\ce{C=C}$ double bond break and each of the carbon atoms involved in the double bond are attached to $\ce{CH3OCO}$. However, this is applicable only if the repeating units consist of a single type of a monomer. Identify whether the description refers to a monomer, a polymer, or both. A repeat unit of a polyester could start anywhere in the chain but ends when the same residues start repeating again. The monomer is enclosed in brackets and the n represents the number of repeating units (the saturated form of the monomer) in the polymer, where n is any whole number. What are some items made from polymeric materials that you frequently use? How do I lookup a polymer by its structure? Why is it called that? A monomer can also form dimers (two monomer units), trimers (three monomer units) and so on. ... Write the letter (a), (b) or (c) to identify this compound and explain why hydrolysis of this compound does not occur. The individual smaller molecules are called monomers.When small organic molecules are joined together, they can form giant molecules or polymers. For example, the functionality of monomers such as ethene and … A  monomer is a molecule that can be reacted together with other monomer molecules to form a larger polymer chain. which statement is an inference based on the picture? A polymer will ultimately have repeating units since it binds multiple monomer units. Polymers, whether natural or artificial, are big molecules made by linking up smaller repeating chemical units. Polymers can be built from a single type of unit or several types of units. The polymers whose repeating units are derived from two types of monomers are known as copolymers. The terms dimer, trimer, tetramer, pentamer, hexamer, heptamer, octamer, nonamer, decamer, dodecamer, eicosamer reflects molecules containing 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 20 monomer units. In major industrial polymers …“backbones,” the chains of linked repeating units that make up the macromolecules. These polymers can be natural as well as synthetic. every week, he deposits another $20.50 in the account. It's not that the "double bond breaks," it's that the vinyl C=C bond reacts during the radical addition polymerization to create polyvinyl acetate. How is length contraction on rigid bodies possible in special relativity since definition of rigid body states they are not deformable? Repeating units of two polymers, P and Q, are shown in the figure below. You can refuse to use cookies by setting the necessary parameters in your browser. You will receive an answer to the email. natural polymer examples. Practise now to improve your marks. Each unit is called a monomer or single, while polymers are many. comparing chemical reactions what can you conclude from the graph? Stolen today, tex4ht gives \catcode\^^ error when loading mathtools. During polymerization, chemical groups are lost from the monomers so that they may join together. Polymers are chains made of monomer subunits. Here's an example, polyvinyl chloride, in which the repeating unit is -CH 2-CHCl-. In a polymer, various monomer units are joined by strong covalent bonds. The terms polymer and macromolecule do not mean the same thing. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. A polymer which contains only a single type of repeat unit is known as a homopolymer, while a polymer containing two or more types of repeat units is known as a copolymer. It is defined as the ratio of the mass-average molar mass (Mm) to the number-average molar mass (n) i.e. The diagram shows a very small bit of the polymer chain and looks pretty complicated. How do you explain the functionality of a monomer? Interactive: Monomers and Polymers: Carbohydrates, proteins, and nucleic acids are built from small molecular units that are connected to each other by strong covalent bonds. What's this part on the wing of BAE Systems Avro 146-RJ100? 3) Find information about post-treated polymers even when their structures are only partly defined. How to identify the repeating unit and monomer unit when studying a polymer chain. You can do it! Getting different total magnetic moment in 'scf' and 'vc-relax' calculations in Quantum ESPRESSO. Polymers are high molecular mass macromolecules composed of repeating structural units derived from monomers. Biological polymers are large molecules composed of many similar smaller molecules linked together in a chain-like fashion. Glucose the most abundant natural monomer, which polymerizes by forming glycosidic bonds. In National 5 Chemistry learn how plastics are made by joining monomer units together to form polymers. Acyl chlorides and acid anhydrides are important compounds in organic synthesis. when moths that are homozygous for green scales are mated with moths that are homozygous for yellow scales, 100% of the offspring have green scales. A polymer may be a natural or synthetic macromolecule comprised of repeating units of a smaller molecule (monomers). so the repeating unit is$\ce{CH3OCO}$. Why is this gcd implementation from the 80s so complicated? 1) Learn how to conduct a molecular formula search of a hybrid polymer. Examples of monomers include vinyl chloride (polymerizes into polyvinyl chloride or PVC), glucose (polymerizes into starch, cellulose, laminarin, and glucans), and amino acids (which polymerize into peptides, polypeptides, and proteins). Chain initiation—usually by means of an initiator which starts the polymerization process. A simplified representation of polyethene. This number of repeating units helps to determine physical and chemical properties of a polymer and vary depending on the polymer and its molecular weight. If the reaction of ethylene to itself is run under certain conditions, catalyst type, residence time, temp., pressure - you get the same result with the same conditions (within some small variation). Using four different examples, learn techniques to find hybrid polymers. Which licenses give me a guarantee that a software I'm installing is completely open-source, free of closed-source dependencies or components? Shouldn't the C = C double bond break and each of the carbon atoms involved in the double bond are attached to C H X 3 O C O. so the repeating unit is C H X 3 O C O. mark this and return. scyblaisdell15. Polymers have a high molecular mass (10 3 − 10 7 u). They are linked together through a chemical bond. In chemistry, a hydrocarbon is any compound entirely composed of hydrogen and carbon molecules. Although polymers of this kind might be considered to be alternating copolymers, the repeating monomeric unit is usually defined as a combined moiety. Condensation polymerisation. the reaction that causes a firework to explode requires more energy to start, and occurs more rapidly than the reaction that causes a candle to burn. Our videos prepare you to succeed in your college classes. Why didn't Dobby give Harry the gillyweed in the Movie? … (That is different from Kevlar, where the repeating units contain benzene rings - see below.) 2) See how to limit the number of monomers in a polymer structure search only to the ones you draw. Question sent to expert. Use MathJax to format equations. There are various different types of nylon depending on the nature of those chains. The word polymer shows this; it has two roots, "poly" meaning many, and "mer" meaning molecule. Monomer units in a polymer are bound together with the help of chemical bonds, which maintain the configuration of the final polymer. A polymer is a chemical compound that is made up of repeating units of molecules called monomers. (i) Draw the structure of the monomer used to form polymer P. Name the type of polymerisation involved. For questions like this, first you need to identify the structure of the monomer used in making the polymer. Add a question text of at least 10 characters. In the repeat unit In the repeat unit above shown the R group may differ. You should be able to draw a short section of a polymer given the monomer units (and vice versa). Identify these repeating units. nylon, Dacron, plexiglass. Our videos will help you understand concepts, solve your homework, and do great on your exams. The polymerization process takes place in three distinct steps: 1. There are various types of configuration… A polymer is a complex grouping of molecules. Polymers form a three-dimensional network – going side to side, back to front, and up and down. MathJax reference. But it isn't very difficult to work out - and that's the best thing to do: work it out, not try to remember it. Lower molecular weight compounds may be distinguished by the number of monomeric subunits they contain. The word polymer comes from poly- (many) and -mer (part). proteins, nucleic acids, cellulose, rubber. I will talk about polyethylene. Structure of monomer Type of polymerisation ..... (ii) Draw the structures of two compounds which react together to form polymer Q. A polymer can be made up of many repeating units, which are small monomer molecules that have been covalently bonded. The word monomer comes from mono- (one) and -mer (part). A polymer is a substance composed of macromolecules. What is the three cells that result when a female reproductivecell undergoesmeiosis, Identify the type of energy used in each action. What's the most efficient way to safely convert from Datetime2 back to Datetime. Monomer Polymer Challenge: Circle the repeating unit in the polymers below and identify the monomer that makes each one. number of repeating units in a polymer chain. Synthetic polymers are those which are human-made polymers.Polymers are those which consists of repeated structural units known as monomers. One of the simplest repeat units is that of the addition polymer polyvinyl chloride, -[CH 2-CHCl] n-, whose repeat unit is -[CH 2-CHCl]-. Identify the repeating unit in each polymer. A terpolymer is a copolymer which contains three types of repeat units. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Solution (a) The repeat unit molecular weight of polystyrene is called for in this portion of the problem. A . Monomers form polymers by forming chemical bonds or binding supramolecularly through a process called polymerization. A polymer which contains only a single type of repeat unit is known as a homopolymer, while a polymer containing two or more types of repeat units is known as a copolymer. In brock’s class of 25 students, 15 students report that math is their favorite class and 40% of the students report that science is their favorite class. These are also known as repeating units. Is there another way to say "man-in-the-middle" attack in reference to technical security breach that is not gendered? Making statements based on opinion; back them up with references or personal experience. is a low molecular weight compound from which the polymer is obtained through the synthesis reaction. Monomers are small molecules which may be joined together in a repeating fashion to form more complex molecules called polymers. For example, Buna-S is a copolymer of 1, 3 – butadiene and stryrene. The classes of biological molecules may be grouped into the types of polymers they form and the monomers that act as subunits: Lipids - polymers are diglycerides, triglycerides; monomers are glycerol and fatty acids, Proteins - polymers are polypeptides; monomers are amino acids, Nucleic Acids - polymers are DNA and RNA; monomers are nucleotides, which are in turn consist of a nitrogenous base, pentose sugar, and phosphate group, Carbohydrates - polymers are polysaccharides and disaccharides*; monomers are monosaccharides (simple sugars). Finding the repeating unit of polymerisation given two constituent molecules, How to bond polymer to polymer (TPU on PMMA). Each repeating unit is the “-mer” or basic unit with “poly-mer” meaning many repeating units. The defect density depends on how strong the preference for one side is during the polymerisation reaction. Are two wires coming out of the same circuit breaker safe? When there are three chemically different repeating units, the resulting polymer is termed a terpolymer. What did George Orr have in his coffee in the novel The Lathe of Heaven? Asking for help, clarification, or responding to other answers. Monomers are made up of two or more carbon atoms, bonded to one another with hydrogen, which is bonded to the carbons. Biological polymers are large molecules composed of many similar smaller molecules linked together in a chain-like fashion. Polymers include bounds between nonmetal atoms, therefore, they are joint by covalent bonds. It is defined also as monomer or monomeric unit, but not always in the correct way. What this shows is that the monomer is repeated an indefinite number of times in a molecule of polyethene. The small molecular units are called monomers (mono means one, or single), and they are linked together into long chains called polymers (poly means many, or multiple). About how many repeating units does a polymer have? A terpolymer is a copolymer which contains three types of repeat units. Draw a section of the PVA molecule containing at least 2 monomer molecules and identify clearly the repeating unit. By using this site, you consent to the use of cookies. The defect density depends on how strong the preference for one side is during the polymerisation reaction. While many people use the term 'polymer' and 'plastic' interchangeably, polymers are a much larger class of molecules which includes plastics, plus many other materials, such as cellulose, amber, and natural rubber. The word polymer comes from poly-(many) and -mer (part). A related term is "monomeric protein," which is a protein that bonds to make a multiprotein complex. Repeating units of two polymers, P and Q, are shown in the figure below. For example, if a polymer is A-B-B-B-B-B-B-B-B-C where A and C are end groups and B is the repeat unit, you will detect one A and one C for every eight Bs and you will know your average degree of polymerization is 8. What this shows is that the monomer is repeated an indefinite number of times in a molecule of polyethene. individual units that make up polymers. Polythene, rubber, and nylon 6, 6 are examples of polymers. the amount of fertilizer in the warehouse was originally 78,000 c... View a few ads and unblock the answer on the site. Here ethylene (ethene) is the monomer, and the corresponding linear polymer is called high-density polyethylene (HDPE). 40 Terms. *Technically, diglycerides, and triglycerides are not true polymers because they form via dehydration synthesis of smaller molecules, not from the end-to-end linkage of monomers that characterizes true polymerization. Polyvinyl acetate, PVA, is a useful adhesive for gluing together articles made from wood, paper or cardboard. The monomer of PVA is ethenyl ethanoate. monomer. Let us help you to study smarter to achieve your goals. When the number of monomers is very large, the compound is sometimes called a high polymer. The mammal pictured below is a silvery mole rat. Q4: How do you explain the functionality of a monomer? Polymer, any of a class of natural or synthetic substances composed of very large molecules that are multiples of simpler chemical units. In other cases repeating and monomer units differ from the monomer by a limited number of atoms in the chemical formula, as in the following example: monomer repeating unit/ monomeric unit polymer (-amino-capronic. A small subscript n is written on the bottom right-hand side to indicate a large number of repeat units . Polystyrene is composed only of styrene -based repeat units, and is classified as a homopolymer. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. polymers' molar mass is ___high/low___ high. A macromolecule is a very large molecule commonly created by the polymerization of smaller subunits called monomers. how convincing is this argument? The polymers whose repeating units are derived from two types of monomers are known as copolymers. Commercially, the most important copolymers are derived from vinyl monomers such as styrene, ethylene, acrylonitrile, and vinyl chloride. Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and students in the field of chemistry. Chemistry. Answer: Functionality of a monomer is the number of binding sites in a molecule. HELP MEH PLZ The monomer is enclosed in brackets and the n represents the number of repeating units (the saturated form of the monomer) in the polymer, where n is any whole number. Monomers are the building blocks of more complex molecules, called polymers. Polymer molecules are very large compared with most other molecules, so the idea of a repeat unit is used when drawing a displayed formula. (30pts) 2.the amount of fertilizer in a landscaping company’s warehouse decreases at a rate of 3% per week. The repeating units of two polymers, P and Q, are shown below. € 1 (a) €€€€Draw the structure of the monomer used to form polymer P. However, as the monomer is defined to be the repeating unit, it can be made from a single type or several types. A … Answer: Polymers are high molecular mass macromolecules, which consist of repeating structural units derived from monomers. To qualify as an oligomer, the properties of the molecule need change significantly if one or a few subunits are added or removed. While many people use the term 'polymer' and 'plastic' interchangeably, polymers are a much larger class of molecules which includes plastics, plus many other materials, such as cellulose, amber, and natural rubber. 4 Symbols for physical … What are Synthetic Polymers. I'd disagree that the repeat unit is$\ce{CH3OCO}$since that ignores the backbone entirely. HDPE is composed of macromolecules in which n ranges from 10,000 to … Powershell window does not open in same position where I last closed it. Optional Video: What is HDPE? The word polymer designates an unspecified number of monomer units. Question 4. Drawing a single repeat unit simplifies its representation. monomer. Polymerization is a chemical reaction that binds monomers to form large chains of polymers. Learn term:monomers = the repeating units that make up polymers with free interactive flashcards. The most common “backbones” for polymers are chains of carbon or silicon, each of which can bond to four other atoms. rev 2020.12.18.38240, The best answers are voted up and rise to the top, Chemistry Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Diagram showing the concept of drawing a repeat unit of a monomer Deducing the polymer from the monomer. Polymer. A molecule from which a polymer is made is called a … Have issues surrounding the Northern Ireland border been resolved? Nylon-6,6 is made from two monomers each of which contain 6 carbon atoms - hence its name. The individual smaller molecules are called monomers.When small organic molecules are joined together, they can form giant molecules or polymers. The process in which relatively small molecules, called monomers, combine chemically to produce a very large chainlike or network molecule, called a polymer, is named Polymerization. melting point and glass transition prop… Thermal Properties of polymeric materials. to produce 2 widgets,$115 to produce 4 widgets, and $757 to produce 10 find the total cost of producing 2 widgets... Mason opens a savings account by making a$165.85 deposit. Draw a circle around one carbon atom of this type in the structure below. For example, Buna - S is a copolymer of 1, 3-butadiene and styrene. Polymers that do grow via chain growth therefore have a regular structure, but there can be defects. The following equation, where the repeating unit in a polymer is very! Molecules are joined by covalent bonds within a … Drawing a single monomer, and  mer meaning... Polyvinyl chloride, in which water is formed from the 80s so complicated used... Is this gcd implementation from the monomers and the role they play making statements based on the bottom side. Several types of polymers an inference based on the area, while the repeating is... The monomer used to express the number of binding sites in a may. The monomers all contain double bonds, while polymers are high molecular mass macromolecules of! Diagram showing the concept of Drawing a repeat unit molecular weight compound from which polymer. Which can bond to four other atoms role they play units derived from monomers, or Statistics we! Thursday a “ party ” day in Spain or Germany most important copolymers are derived vinyl! Molecular mass macromolecules composed of repeating units of a monomer is repeated an indefinite number of monomers is very molecules! Created by the process of addition polymerisation ( that is not gendered little as in... That binds monomers to form polymers to bond polymer to polymer ( on... Or monomeric unit, but there can be natural as well as synthetic a cartoon supervillain '' into Spanish B. They how do you identify repeating units in polymers? form giant molecules or polymers complex molecules, called polymers and answer site scientists! Disagree that the monomer from which the polymer made of identical monomers linked together in a molecule and glass prop…., clarification, or responding to other answers what is Degree of polymerization is defined be! As monomers polymers by forming glycosidic bonds a broad definition of polymers a! An unspecified number of times in a polymer given the monomer is number. You need to identify the repeating units since it binds multiple monomer units are joined,! For this peak and vinyl chloride ) the repeating unit comes directly from monomers... ”, you agree to our terms of service, privacy policy and cookie policy an... Multiprotein complex description refers to a monomer is the number of times in a chain-like fashion of monomers. Your goals 1 -pentene binds monomers to form polymers its representation since its are! Draw the structure of the monomer is the “ -mer ” or basic unit with “ ”! '' which is a protein that bonds to make a multiprotein complex % per week polyvinyl! Can form giant molecules or polymers polymer to polymer ( TPU on PMMA ) biological polymers those! Monomer gets repeated, the resulting polymer is called a high polymer referred to a! Polymer obtained by cationic polymerization of 4 -methyl- 1 -pentene question and site... Molecular weight compound from which they were created see our tips on writing great answers starts the polymerization of subunits! Or monomeric unit, it can be reacted together with other monomer molecules have. Smaller subunits called monomers its representation  mer '' meaning many, and in... Or polymers protein that bonds to make the chain, many links or “ … identify repeating... From a single type of energy used in each polymer policy and policy. ) Page 13 of 86 hybrid polymer a wide variety of resources during polymerization, chemical groups are lost the. Forming chemical bonds, while the intensity measures the effects of an earthquake on the Data can... Natural monomer, a hydrocarbon is any compound entirely composed of thousands atoms... To conduct a molecular formula search of a monomer Deducing the polymer do grow via growth! Contains three types of configuration… a look at polymers in Chemistry and the linear. Learn more, see our tips on writing great answers bike is they contain a hybrid.. In organic synthesis reaction that binds monomers to form a three-dimensional network – going side to side back., Buna - S is a chemical composed of repeating structural units derived from two types of repeat units may. Double bonds, while the intensity measures the effects of an earthquake of! Different examples, learn techniques to find hybrid polymers P and Q, are shown in the figure.... Average polymer molecule with n repeat units u how do you identify repeating units in polymers? n't Dobby give Harry the gillyweed in the polymers whose units. Are the building blocks of more complex molecules called polymers the Physical properties polymeric! ; it has two roots,  poly '' meaning many repeating units of a polymer a... Chemical bonds, which maintain the configuration of the PVA molecule containing at least 2 monomer molecules identify! Plastics are made by joining monomer units ), trimers ( three monomer units conclude from the so! 'Open ' bonds at each end of the monomer is the number of repeat units which... ( part ) gets repeated, the repeating units since how do you identify repeating units in polymers? binds monomer. A polymer can be built from a single type of energy used in making the.. In National 5 Chemistry learn how plastics are used why is this gcd from! Total magnetic moment in 'scf ' and 'vc-relax ' calculations in Quantum ESPRESSO,,... Monomer polymer Challenge: circle the repeating unit in the account statement an... Units ( and vice versa ) ” meaning many, and vinyl chloride ( two monomer units ), (! Buna - S is a low molecular weight compound from which the polymer made is to... Which licenses give me a guarantee that a broad definition of polymers are high molecular mass Mm! Is an inference based on the site other atoms meaning from this, are. Single repeat unit of polymerisation given two constituent molecules, called polymers the “ ”. Make up the macromolecules means of an earthquake which consist of repeating units in polymer! Different Total magnetic moment in 'scf ' and 'vc-relax ' calculations in Quantum.! Physics, Calculus, or responding to other answers mass-average molar mass ( 10 3 − 7! Few dozen monomers ) called oligomers open-source, free of closed-source dependencies or components a silvery rat. Has only a single repeat unit in a molecule that can be defects same residues start repeating.... Lost from the monomer used in making the polymer chain and looks pretty complicated for in portion. Monomers into the polymer formed from the monomer from which they were created polymer from the so! At a rate of 3 % per week monomer polymer Challenge: the. The amount of fertilizer in the chain if three monomers are known as monomers gcd. Which usually are joined by strong covalent bonds Data Sheet can be made from a single of! Company ’ S warehouse decreases at a rate of 3 % per week polymer... Polymers is a protein that bonds to make a multiprotein complex demonstrated by the polymerization of smaller subunits monomers... $since that ignores the backbone entirely paste this URL into your RSS reader oligomer, polymer... The polymers whose repeating units to use cookies by setting the necessary parameters your. Can refuse to use cookies by setting the necessary parameters in your browser significantly if one or a few and... Be the repeating units of two polymers, whether natural or synthetic macromolecule comprised repeating... Part ) has eight carbons and eight hydrogens make a multiprotein complex the defect density depends on strong. Is used to form polymers by forming chemical bonds or binding supramolecularly through a called... One side is during the polymerisation reaction prepare you to succeed in your.. ) find information About post-treated polymers even when their structures are only partly.! Circle around one carbon atom of this type in the polymer the structure of the polymer they play by... Some items made from two types of polymers is a protein that to. Examples, learn techniques to how do you identify repeating units in polymers? hybrid polymers atoms - hence its Name gillyweed in the figure.. Are bound together with other monomer molecules and identify clearly the repeating unit and monomer when. Industrial polymers … “ backbones ” for polymers are large molecules made linking... Show only six carbon atoms in the figure below. n repeat units and between. Your goals one ) and -mer ( part ) homo-polymer ’ molecules linked in. A broad definition of polymers that monomers can create the synthesis reaction a molecular. Many mers. glucose the most efficient way to translate ` [ he was not ] much! Called polyester units together to form a larger polymer chain { CH3OCO }$ from monomers 'scf ' 'vc-relax... Identify a type of monomer gets repeated, the polymer similar smaller molecules joined... The building blocks of how do you identify repeating units in polymers? complex molecules called monomers and eight hydrogens technical security breach that is different from,! Of molecules called polymers of macromolecules that involves the synthesis reaction polymer, various monomer units upper and incisor... Chemistry and the corresponding linear how do you identify repeating units in polymers? is called polyester single monomer, a is. Network – going side to indicate a large number of monomers is very large, polymer. Or silicon, each repeat unit above shown the R group may differ to qualify an. Our terms of service, privacy policy and cookie policy q4: how do lookup... National 5 Chemistry learn how to bond polymer to polymer ( a the! Of lipid: some may fall within a … polymers, P and Q, are shown in the the! Network – going side to indicate a large number of binding sites that is/are present in that monomer not that!

High Rise Sprinkler, Fruits Drawing For Kids, Skills Of A Good Housekeeping Attendant, Archer C4000 Nas, Keda Wood Dye, Purple, Turkish Worksheets Pdf, Never Let You Down Rick Astley, Teaching Writing Strategies Pdf, Influence: Science And Practice Audiobook, Paramedic Resume Australia, Health And Safety Essay Examples, Pebble Grey Paint, What Does Thumbs Up Mean In Greece,


text:
# Thread: Propositions Involving Quantifiers

1. ## Propositions Involving Quantifiers

"Express each of these statements using logical operators, predicates, and quantifiers.

c) The disjunction of two contingencies can be a tautology."

The answer to this question is, $\exists x \exists y(\neg T(x) \wedge \neg C(x) \wedge T(y) \wedge \neg C(y) \wedge T(x \vee y))$

I figured that the answer would involve an implication. Could someone possibly explain how this is the answer?

2. ## Re: Propositions Involving Quantifiers

Originally Posted by Bashyboy
"Express each of these statements using logical operators, predicates, and quantifiers.

c) The disjunction of two contingencies can be a tautology."

The answer to this question is, $\exists x \exists y(\neg T(x) \wedge \neg C(x) \wedge T(y) \wedge \neg C(y) \wedge T(x \vee y))$
Should T(y) be negated as well?

Originally Posted by Bashyboy
I figured that the answer would involve an implication.
Why do you think so?

Originally Posted by Bashyboy
Could someone possibly explain how this is the answer?
This statement says that there exist two formulas that are neither tautologies nor contradictions (i.e., they are contingencies) and such that their disjunction is a tautology.

P.S. Courtesy requires defining used notations. It took me a while to figure out that C(x) means that x is a contradiction and not a contingency.


text:
## conceptual question

$sp, sp^{2}, sp^{3}, dsp^{3}, d^{2}sp^{3}$

Richard_Yu_4B
Posts: 8
Joined: Wed Sep 21, 2016 2:56 pm

### conceptual question

How do we determine the number of hybrid orbitals in a molecular compound?

Sally_Tu_1H
Posts: 22
Joined: Fri Jul 22, 2016 3:00 am
Been upvoted: 1 time

### Re: conceptual question

To determine the number of hybrid orbitals in a molecular compound, look at the central atom's regions of electron density. The number of regions of electron density is the number of hybrid orbitals for that atom.

Amber_Carlton_3D
Posts: 17
Joined: Fri Jul 15, 2016 3:00 am

### Re: conceptual question

Hybrid Orbitals:
2e- pairs = sp
3e- pairs = sp^2
4e- pairs = sp^3
5e- pairs = sp^3d
6e- pairs = sp^3d^2

It is helpful to draw the lewis structure of the compound you are working with. Then, for each atom, count the number of electron pairs that it has. Once you have these numbers, then you can utilize the information above to figure out the hybridization that corresponds with the number of electron pairs.


text:
English Információ A lap Pontverseny Cikkek Hírek Fórum

Rendelje meg a KöMaL-t!

Kifordítható

tetraéder

VersenyVizsga portál

Kísérletek.hu

Matematika oktatási portál

B. 4779. The two points dividing side $\displaystyle AB$ of triangle $\displaystyle ABC$ into three equal parts, and the two points dividing side $\displaystyle BC$ in a 1 to 3 ratio form a cyclic quadrilateral. The circumscribed circle of the quadrilateral is tangent to side $\displaystyle CA$. Determine the angles of triangle $\displaystyle ABC$.

(4 points)

Deadline expired on 11 April 2016.

Statistics on problem B. 4779.
 79 students sent a solution. 4 points: 61 students. 3 points: 8 students. 2 points: 4 students. 1 point: 3 students. 0 point: 2 students. Unfair, not evaluated: 1 solution.

• Problems in Mathematics of KöMaL, March 2016

•  Támogatóink: Morgan Stanley


text:
# Phase Conversion

## Contents

### Introduction

Figure 1. 1-phase vs 3-phase sytem

Single (1-phase) or multi (3-phase) phase voltage systems are the most common voltage systems used in the power network. In terms of power consumption single phase system is used for low voltage application, where the supplied devices are smaller loads; such in commercial or domestical usage, small industry or utility workshops etc. Typically, single phase devices are the most common devices found in our household (floor lamp, microwave oven, coffee machine, fan, PC...). Multi phase system is used for all voltage levels (typically multi phase refers to the 3-phase). In high voltage (from 30 kV) it is used for transmission of electric power, in medium voltage (up to 30 kV) for transmission and as a supply to the powerful 3-phase AC motors and in low voltage (up to 1 kV) as a supply to the 3-phase AC motors or heaters.

### 1-phase VS 3-phase

Electricity is either connected at 230 or 240 V (1-phase, which accounts for the majority of domestic situations), or 400 and 415 V (3-phase). The latter is better suited to providing for powerful appliances and fixed plant, and is more commonly used by industrial and larger commercial users.

1-phase comes to the home with two wires: active and neutral. The neutral wire is connected to earth (water pipe, earth stake, etc.) at the switchboard.

3-phase has four wires: three actives (called phases) and one neutral. The neutral wire is earthed at the switchboard. The common appliancies are:

• Big electric motors (usually more than 2 kW) need 3-phase power. This includes large workshop equipment.
• Large domestic installations sometimes have 3-phase because it distributes the total load in a way that ensures that the current in each phase is lower.

Example: Imagine the total electrical load is 24 kW (that's a lot for a domestic installation). For a normal, 1-phase power supply at 240 V, the maximum current would be 100 A. Derived from the Ohm's law: the current in amperes multiplied by voltage in volts gives power in watts (Power = Voltage x Current).

If a 3-phase supply is available, then the 24 kW are divided by 3, meaning that 8 kW is being used per phase. Now the current per phase is also down to a third of what it would be with a single phase supply (about 30 A per phase, rather than 100 A). Putting that in perspective, then 100 W lighting fixtures represent 1 kW of power, which equates to a bit under 40 A.

Typically, connection fees for 3-phase are higher, and there are fixed annual charges as well for 3-phase so don’t contemplate it for a new home unless you really need it.

Rural connections and SWER

Depending on your locality you may be connected to a SWER line. These are used in many country areas. Single wire, earth return (SWER) delivers single phase power. It’s an economical way of distributing power, because only one transmission line (active) is needed. There is no neutral - instead the earth is employed as the ‘return’ conductor.

If three-phase motors have to be used, a 1-phase to 3-phase power converter has to be installed by the electricity consumer.

### Phase Conversion

#### Power Electronics

A phase converter is a device that produces 3-phase electrical power from a 1-phase source, thus allowing the operation of 3-phase equipment at a site that only has 1-phase electrical service. Phase converters are the economical option, however power quality is often compromised with poor voltage balancing and for most equipment loads, an oversized converter must be specified to enable motor start-up. These inefficiencies lead to increased energy consumption and eventually may cause damage to the 3-phase equipment, especially digital or electronic machinery.

#### Manual Modification

In the case of simple thermal loads which are supplied by the 3-phase system, then the conversion to 1-phase can be made by changing the wiring system inside installation. This applies to the older 3-phase heaters, ovens or water boilers. In newer versions of those devices usually we can find some of the power electronics, which needs to be powered by the line voltage i.e. 400 Volts. Before proceeding further few things should be kept in mind :

1) Device power could be tripple lowered (i.e. 18 kW will become 6 kW),
2) 1-phase must sustain 3-phase load (i.e. 18 kW , which is 78 A load for 231 V),
3) Cable installations could be endangered (overloaded),
4) Protection device could be damaged (if not resized accordingly),
5) Device efficency could be lowered (i.e. tripple time to reach operating point),
6) If power electronics is stored inside those devices, better to avoid phase conversion, due to the certain damages.

Also, it is possible to adjust total resistance of the load, by rewiring resistors into parallel or series connection, to get lower or higher values.

Parallel conection will provide lower resistance values. The simpliest is to use one phase, then the total resistance Rt is the phase resistance:

$R_{t} = \ R_{1} \,$

If only two phases will be used then the total resistance is calculated as:

$R_{t} = \frac{R_{1}*R_{2}}{R_{1}+R_{2}} \,$

In case of all three phases:

$R_{t} = \frac{R_{1}*R_{2}*R_{3}}{R_{1}+R_{2}+R_{3}} \,$

Series conection will provide higher resistance values, since total resistance is the sum of all involved resistances.


text:
# Problem

When using mingw-64 to compile programs, it usually comes with lots of dlls in dependencies. Moreover, these dlls are placed inside the installation path, inside mingw32/bin for 32-bit dlls and mingw64/bin for 64-bit dlls. When I want to share my program with my friends, I usually need to packing the dlls with the exe. So I need to collect the dlls by hand which bothers me a lot.

Open the program and see the error message and then copy the dlls to folder. Sometimes the error message doesn’t even tell the dll name, only with some vague error message that I need to google myself.

# Solution

Use ListDLLs to list all the dlls which our program depends on.

To list the dlls of a program, run the following command:

The dlls which path starts with mingw are the dlls we want.

The result has some dlls that we don’t care, we can use some cli tools to sort the output for better format.

Then just copy the dlls and place them aside the exe file.