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# Johann Carl Friedrich Gauss

### Quick Info

Born
30 April 1777
Brunswick, Duchy of Brunswick (now Germany)
Died
23 February 1855
Göttingen, Hanover (now Germany)

Summary
Carl Friedrich Gauss worked in a wide variety of fields in both mathematics and physics incuding number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. His work has had an immense influence in many areas.

### Biography

At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately. His teacher, Büttner, and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 101.

In 1788 Gauss began his education at the Gymnasium with the help of Büttner and Bartels, where he learnt High German and Latin. After receiving a stipend from the Duke of Brunswick- Wolfenbüttel, Gauss entered Brunswick Collegium Carolinum in 1792. At the academy Gauss independently discovered Bode's law, the binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the prime number theorem.

In 1795 Gauss left Brunswick to study at Göttingen University. Gauss's teacher there was Kästner, whom Gauss often ridiculed. His only known friend amongst the students was Farkas Bolyai. They met in 1799 and corresponded with each other for many years.

Gauss left Göttingen in 1798 without a diploma, but by this time he had made one of his most important discoveries - the construction of a regular 17-gon by ruler and compasses This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae .

Gauss returned to Brunswick where he received a degree in 1799. After the Duke of Brunswick had agreed to continue Gauss's stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. He already knew Pfaff, who was chosen to be his advisor. Gauss's dissertation was a discussion of the fundamental theorem of algebra.

With his stipend to support him, Gauss did not need to find a job so devoted himself to research. He published the book Disquisitiones Arithmeticae in the summer of 1801. There were seven sections, all but the last section, referred to above, being devoted to number theory.

In June 1801, Zach, an astronomer whom Gauss had come to know two or three years previously, published the orbital positions of Ceres, a new "small planet" which was discovered by G Piazzi, an Italian astronomer on 1 January, 1801. Unfortunately, Piazzi had only been able to observe 9 degrees of its orbit before it disappeared behind the Sun. Zach published several predictions of its position, including one by Gauss which differed greatly from the others. When Ceres was rediscovered by Zach on 7 December 1801 it was almost exactly where Gauss had predicted. Although he did not disclose his methods at the time, Gauss had used his least squares approximation method.

In June 1802 Gauss visited Olbers who had discovered Pallas in March of that year and Gauss investigated its orbit. Olbers requested that Gauss be made director of the proposed new observatory in Göttingen, but no action was taken. Gauss began corresponding with Bessel, whom he did not meet until 1825, and with Sophie Germain.

Gauss married Johanna Ostoff on 9 October, 1805. Despite having a happy personal life for the first time, his benefactor, the Duke of Brunswick, was killed fighting for the Prussian army. In 1807 Gauss left Brunswick to take up the position of director of the Göttingen observatory.

Gauss arrived in Göttingen in late 1807. In 1808 his father died, and a year later Gauss's wife Johanna died after giving birth to their second son, who was to die soon after her. Gauss was shattered and wrote to Olbers asking him to give him a home for a few weeks,
to gather new strength in the arms of your friendship - strength for a life which is only valuable because it belongs to my three small children.
Gauss was married for a second time the next year, to Minna the best friend of Johanna, and although they had three children, this marriage seemed to be one of convenience for Gauss.

Gauss's work never seemed to suffer from his personal tragedy. He published his second book, Theoria motus corporum coelestium in sectionibus conicis Solem ambientium , in 1809, a major two volume treatise on the motion of celestial bodies. In the first volume he discussed differential equations, conic sections and elliptic orbits, while in the second volume, the main part of the work, he showed how to estimate and then to refine the estimation of a planet's orbit. Gauss's contributions to theoretical astronomy stopped after 1817, although he went on making observations until the age of 70.

Much of Gauss's time was spent on a new observatory, completed in 1816, but he still found the time to work on other subjects. His publications during this time include Disquisitiones generales circa seriem infinitam , a rigorous treatment of series and an introduction of the hypergeometric function, Methodus nova integralium valores per approximationem inveniendi , a practical essay on approximate integration, Bestimmung der Genauigkeit der Beobachtungen , a discussion of statistical estimators, and Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata . The latter work was inspired by geodesic problems and was principally concerned with potential theory. In fact, Gauss found himself more and more interested in geodesy in the 1820s.

Gauss had been asked in 1818 to carry out a geodesic survey of the state of Hanover to link up with the existing Danish grid. Gauss was pleased to accept and took personal charge of the survey, making measurements during the day and reducing them at night, using his extraordinary mental capacity for calculations. He regularly wrote to Schumacher, Olbers and Bessel, reporting on his progress and discussing problems.

Because of the survey, Gauss invented the heliotrope which worked by reflecting the Sun's rays using a design of mirrors and a small telescope. However, inaccurate base lines were used for the survey and an unsatisfactory network of triangles. Gauss often wondered if he would have been better advised to have pursued some other occupation but he published over 70 papers between 1820 and 1830.

In 1822 Gauss won the Copenhagen University Prize with Theoria attractionis ... together with the idea of mapping one surface onto another so that the two are similar in their smallest parts. This paper was published in 1825 and led to the much later publication of Untersuchungen über Gegenstände der Höheren Geodäsie (1843 and 1846). The paper Theoria combinationis observationum erroribus minimis obnoxiae (1823), with its supplement (1828), was devoted to mathematical statistics, in particular to the least squares method.

From the early 1800s Gauss had an interest in the question of the possible existence of a non-Euclidean geometry. He discussed this topic at length with Farkas Bolyai and in his correspondence with Gerling and Schumacher. In a book review in 1816 he discussed proofs which deduced the axiom of parallels from the other Euclidean axioms, suggesting that he believed in the existence of non-Euclidean geometry, although he was rather vague.
... the vain effort to conceal with an untenable tissue of pseudo proofs the gap which one cannot fill out.
Gauss confided in Schumacher, telling him that he believed his reputation would suffer if he admitted in public that he believed in the existence of such a geometry.

In 1831 Farkas Bolyai sent to Gauss his son János Bolyai's work on the subject. Gauss replied
to praise it would mean to praise myself .
Again, a decade later, when he was informed of Lobachevsky's work on the subject, he praised its "genuinely geometric" character, while in a letter to Schumacher in 1846, states that he
had the same convictions for 54 years
indicating that he had known of the existence of a non-Euclidean geometry since he was 15 years of age (this seems unlikely).

Gauss had a major interest in differential geometry, and published many papers on the subject. Disquisitiones generales circa superficies curva (1828) was his most renowned work in this field. In fact, this paper rose from his geodesic interests, but it contained such geometrical ideas as Gaussian curvature. The paper also includes Gauss's famous theorema egregium :
If an area in $\mathbb{R}^{3}$ can be developed (i.e. mapped isometrically) into another area of $\mathbb{R}^{3}$, the values of the Gaussian curvatures are identical in corresponding points.
The period 1817-1832 was a particularly distressing time for Gauss. He took in his sick mother in 1817, who stayed until her death in 1839, while he was arguing with his wife and her family about whether they should go to Berlin. He had been offered a position at Berlin University and Minna and her family were keen to move there. Gauss, however, never liked change and decided to stay in Göttingen. In 1831 Gauss's second wife died after a long illness.

In 1831, Wilhelm Weber arrived in Göttingen as physics professor filling Tobias Mayer's chair. Gauss had known Weber since 1828 and supported his appointment. Gauss had worked on physics before 1831, publishing Über ein neues allgemeines Grundgesetz der Mechanik , which contained the principle of least constraint, and Principia generalia theoriae figurae fluidorum in statu aequilibrii which discussed forces of attraction. These papers were based on Gauss's potential theory, which proved of great importance in his work on physics. He later came to believe his potential theory and his method of least squares provided vital links between science and nature.

In 1832, Gauss and Weber began investigating the theory of terrestrial magnetism after Alexander von Humboldt attempted to obtain Gauss's assistance in making a grid of magnetic observation points around the Earth. Gauss was excited by this prospect and by 1840 he had written three important papers on the subject: Intensitas vis magneticae terrestris ad mensuram absolutam revocata (1832), Allgemeine Theorie des Erdmagnetismus (1839) and Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskräfte (1840). These papers all dealt with the current theories on terrestrial magnetism, including Poisson's ideas, absolute measure for magnetic force and an empirical definition of terrestrial magnetism. Dirichlet's principle was mentioned without proof.

Allgemeine Theorie ... showed that there can only be two poles in the globe and went on to prove an important theorem, which concerned the determination of the intensity of the horizontal component of the magnetic force along with the angle of inclination. Gauss used the Laplace equation to aid him with his calculations, and ended up specifying a location for the magnetic South pole.

Humboldt had devised a calendar for observations of magnetic declination. However, once Gauss's new magnetic observatory (completed in 1833 - free of all magnetic metals) had been built, he proceeded to alter many of Humboldt's procedures, not pleasing Humboldt greatly. However, Gauss's changes obtained more accurate results with less effort.

Gauss and Weber achieved much in their six years together. They discovered Kirchhoff's laws, as well as building a primitive telegraph device which could send messages over a distance of 5000 ft. However, this was just an enjoyable pastime for Gauss. He was more interested in the task of establishing a world-wide net of magnetic observation points. This occupation produced many concrete results. The Magnetischer Verein and its journal were founded, and the atlas of geomagnetism was published, while Gauss and Weber's own journal in which their results were published ran from 1836 to 1841.

In 1837, Weber was forced to leave Göttingen when he became involved in a political dispute and, from this time, Gauss's activity gradually decreased. He still produced letters in response to fellow scientists' discoveries usually remarking that he had known the methods for years but had never felt the need to publish. Sometimes he seemed extremely pleased with advances made by other mathematicians, particularly that of Eisenstein and of Lobachevsky.

Gauss spent the years from 1845 to 1851 updating the Göttingen University widow's fund. This work gave him practical experience in financial matters, and he went on to make his fortune through shrewd investments in bonds issued by private companies.

Two of Gauss's last doctoral students were Moritz Cantor and Dedekind. Dedekind wrote a fine description of his supervisor
... usually he sat in a comfortable attitude, looking down, slightly stooped, with hands folded above his lap. He spoke quite freely, very clearly, simply and plainly: but when he wanted to emphasise a new viewpoint ... then he lifted his head, turned to one of those sitting next to him, and gazed at him with his beautiful, penetrating blue eyes during the emphatic speech. ... If he proceeded from an explanation of principles to the development of mathematical formulas, then he got up, and in a stately very upright posture he wrote on a blackboard beside him in his peculiarly beautiful handwriting: he always succeeded through economy and deliberate arrangement in making do with a rather small space. For numerical examples, on whose careful completion he placed special value, he brought along the requisite data on little slips of paper.
Gauss presented his golden jubilee lecture in 1849, fifty years after his diploma had been granted by Helmstedt University. It was appropriately a variation on his dissertation of 1799. From the mathematical community only Jacobi and Dirichlet were present, but Gauss received many messages and honours.

From 1850 onwards Gauss's work was again nearly all of a practical nature although he did approve Riemann's doctoral thesis and heard his probationary lecture. His last known scientific exchange was with Gerling. He discussed a modified Foucault pendulum in 1854. He was also able to attend the opening of the new railway link between Hanover and Göttingen, but this proved to be his last outing. His health deteriorated slowly, and Gauss died in his sleep early in the morning of 23 February, 1855.

### References (show)

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Written by J J O'Connor and E F Robertson
Last Update December 1996


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# Number theory derivation from infinity; speculations on equations that are derived in terms of the Field

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Summary: Idea of deriving down to i''' from infinity, potentially establishing number theory from set of sq.rt. of primes, i'. Then, (*highly speculative*) i'' as Kinetic Energy, i'" as Potential Energy from electrostatics for new Hamiltonian in quantum mechanics? In need of critique on mathematics/calculus, in particular. Derivations done with respect to Field "shape", encompassing decreasing dimensions with each derivative for conceptualization, see discussion.

Hello,

Although this hinges on mathematics I will make this only just a discussion with cursory equations. I would encourage criticism, as I suspect my math is completely off-base. That said, I think the idea might have some merit, and could perhaps be improved upon by the forum discussion.

I propose to take the first derivative from infinity (i) with respect to the field shape or conformer that exists. I call this first derivative i' (di/dAc [Ac, Actual conformer]) , and define that the set of departure, or domain, or number set, that emerges is: the set of square roots of prime numbers. I think of this as logically coherent as the set i' could potentially also be infinite, and I do not think any operations can be performed on these numbers, save for further derivation.

IMO, the field shape or conformer here is of the nature of anti-DeSitter saddle conformer (negative curvature), but that is the physics and not mathemathics. That said, to discuss the physics, I think there is a need for 4D here a la Quaternion mathematics. See the Euler identity [wiki: ht tps://en.wikipedia.org/wiki/Euler's_identity]: under Generalizations section: it seems Quaternion exponentiation transforms i (imaginary number, i = sq. rt. (-1)...) found within e^i*pi, into i= (1/sq.rt.(3))* ( i+/-j+/-k)...

Back to maths, the second derivative from the set i' yields i'' as of type (di''/dAc) = (1/2)*(1/ i' ^(1/2)) ; this "function" (I do not know if it is a function) appears to me to have a limit approaching 0. I think of it as this i'' derivation has now opened "top-down" number theory by opening 1 & 2 for operations as 1/2, alongside the set of i' which are now inhabiting the denominator. This 1/2 coefficient could in some way relate to Kinetic Energy, as I'm hopeful this theory is addressing only just Field shape & Energy / information.

To continue with Field discussion, I theorize about the field here as being mappable by hyperbolic geometry on the Poincare disk. There again is the constant negative curvature, and the apparent 2D shape map has perhaps an infinitesimal extension to 3D being understood from the nature of what's going on. Infinitesimal sections here integrate back to more complex (and I do mean perhaps real + imaginary by complex) sections of field math above.

To third derivative: di'''/dAc = [- (1/4)*(1/ (i')^(3/2)]. So, that there is a -1/4 coefficient, which looks to me to dovetail with the Potential Energy derivations from electrostatics that give rise to Bohr's frequency condition. Which is arrived at by applying Coulomb's law I think, and eventually first complexifies to delta E = [{mass(electron)*e^4 / 8e.(permittivity)^2*h^2} {(1/n1^2)-(1/n2^2)}] where n2>n1, which then simplifies to delta E = hv = KE + PE. That, of course, E = (n)hv, is our same result from Planck on blackbody radiation and Einstein on the photoelectric effect. That is then going in to the Hamiltonian's of quantum mechanics. However, I'm hoping to reset the basis of i, the imaginary unit, as a derivative from infinity encompassing all sq. rt. of primes in stead of just i = (-1)^(1/2).

Last paragraph largely follows from initial review of Donald A. McQuarrie's, Quantum Chemistry, 2nd Ed., early chapters. Now, to touch base with field again, here at this i''' derivative level, the field again is simplified and I think can be descibed from a Polar graphing plot only really using theta (angle) and r (radian length) to map vectors of forces [or, Quaternions applied throughout has explanatory potential; I know no Quaternion math], which I think of as electrogravitational force from a circle-dot centralization [see the visualization of a wave function under Preliminaries on the wiki for Schrödinger equation: ht tps://en.wikipedia.org/wiki/Schrödinger_equation]. Apparent 2D vector plot is perhaps really 3D, with infinitesimal extension in 3rd because as soon as a movement or extension takes place, a distortion takes place, and a conformer or shape to space-time begins to take shape.

Some derivations teased out for assessment:

i' = number set of all sq. rt.s of prime no.s

i'' = (1/2)*i(-1/2)

i''' = -(1/4)*i(-3/2)

i'v = (3/8)*i(-5/2)

iv = -(15/16)*i(-7/2)

ivi = (105/32)*i(-9/2)

ivii = -(945/65)*i(-11/2) ...

So, please, please, fire away at the bad math (or physics or chemistry) here -- I know it is in need of more work. There is poorly formalized notation here, and I am not sure I am deriving a continuous function, or that one can derive a number set in relation to a field shape or conformer. But, to me, this mathematics related back to the physics/chemistry makes some logical sense, which I think and hope is providential to better understandings.

Thank you,
NTuft

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37 minutes ago, NTuft said:

I wanted to fire away real bad, but I didn't spot any physics. No chemistry either.

And the only maths is wrong. You can't consider increments of a constant.

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4 minutes ago, joigus said:

I wanted to fire away real bad, but I didn't spot any physics. No chemistry either.

And the only maths is wrong. You can't consider increments of a constant.

I take it you consider infinity to be the constant? Do you suppose that that may be a way of working with a concept that facilitates maths operations built up on a number theory from below it?

I want to consider chemistry as built out of only positive and negative charges, a la electrostatics. I want to consider physics as having gravitation equivalent to magnetism (ergo electrogravitational force) and explained by central points of gravitation/positive charge at nucleus, with negative charges in orbit; again, really just "electrostatics", or maybe that is not the right term.

More targets now available.

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9 minutes ago, NTuft said:

I take it you consider infinity to be the constant?

Who would consider such a stupid thing?

I mean you do, are doing,

$\frac{d}{di}\left(\frac{1}{2}i^{-1/2}\right)=-\frac{1}{4}i^{-3/2}$

So you seem to be considering the imaginary unit, $$i$$, as a constant. variable.

That shows that you don't understand what differentiation is about.

9 minutes ago, NTuft said:

More targets now available.

No target in sight.

Edited by joigus
correction
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Thank you, joigus. I know I needed help on the calculus involved here, and I doubted most of all that that part was feasible or correctly formulated.

I understand a derivative to be a measure of infinitesimal change at a point.

It is mentioned in the discourse that I had an idea that the imaginary unit i could be redefined to encompass all sq. rt.s of primes, instead of just i = (-1)^(1/2). An interjection here -- if you could point me somewhere to learn how to insert the MatLab looking equations I'd appreciate it. Now, getting back to the first, if i is not only i=(-1)^(1/2) , but is instead defined as I posit, could that make it a "variable", or moving target; a differentiable quantity?

I accept that the basis may be off and this all may be "math crankery".

Furthermore, I would add that subsets of the number set of sq. rt.s of primes could be aggregated into the "integer" multipliers that are used to obtain allowable energy states in the Bohr model of allowed energies.

Because this is following from the simplest elucidation, that of the spectral lines of Hydrogen, the more complicated sets of numbers could be conceptualized as "variable", because the size of the central, positive charge in the nucleus, and therefore by extension the base electron cloud (whatever the combination may be) would comprise a differently constituted subset of primes as encompassing the allowed energy excitation states, and hopefully also, somehow, describing the kinetic energy component in motion from the higher-order derivative.

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6 hours ago, NTuft said:

Summary: Idea of deriving down to i''' from infinity, potentially establishing number theory from set of sq.rt. of primes, i'. Then, (*highly speculative*) i'' as Kinetic Energy, i'" as Potential Energy from electrostatics for new Hamiltonian in quantum mechanics? In need of critique on mathematics/calculus, in particular. Derivations done with respect to Field "shape", encompassing decreasing dimensions with each derivative for conceptualization, see discussion.

My problem with what you have written is simple.

As Joigus patiently noted,the quantity commonly denoted by the symbol 'i'  is defined as the square root of minus 1.

5 hours ago, NTuft said:

It is mentioned in the discourse that I had an idea that the imaginary unit i could be redefined

If you really want a quantity that is not the square root of -1 then you should start by saying so, defining what you want your quantity to be, and choosing a suitable symbol for it.

Otherwise you will simply confuse your readers, and lead them to ask what other quantities and terms is the person playing fast and loose with ?

Joigus also noted

6 hours ago, joigus said:

So you seem to be considering the imaginary unit, i , as a constant. variable.

That shows that you don't understand what differentiation is about.

Note he did not say that you cannot differentiate a constant (which of course you can if you know what you are doing)

Reading your response I agree with him that you are attempting to misuse the process of differentiation.

7 hours ago, NTuft said:

Idea of deriving down to i''' from infinity

For your information, you are starting at 'the wrong end'.

Number theory starts at the beginning and uses 'successor theory'.

It does not (can not) start at infinity for the very good reason that infinity is not an end or a number, it is a shorthand way of saying there is no end.

But there is a beginning.

You start by asserting there is a beginning (ie asserting that there is a number called zero, or a number called 1 depending upon your preference)

And then asserting that every number has a successor.

So there must be a successor to your given first number, say 1.

You call this number 2.

This in turn must have a successor, you call 3.

And so on.

Since this process never ends you have defined every number.

But you have not defined or used 'infinity'.

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I should alsao like to point out that what Mathematicians mean by the word 'Field', particularly in relation to number theory,

and what Physicists mean are quite different.

You should disentangle the Physics and the Mathematics please.

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3 hours ago, studiot said:

You should disentangle the Physics and the Mathematics please.

I agree. This is not to say that physics and mathematics don't have interesting connections. They do, and they surface from time to time. But in the way the OP is dealing with it, I find it almost impossible to fathom what the proposed connection actually is. Again, I'm missing a clear statement of what is supposed to be the conjecture. It's more like a purée of mathematical, physical, and chemical terms.

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Studiot, thanks for the assessment. Yes, including or redifining i (i2=(-1)) is like an ad hoc hitchhiker, I don't think it makes sense. Need something other than "i". I also appreciate the number theory run up, I think you get what I was trying. Then from last, & from joigus, I agree, this is tangled up puree, as I can't now see any physical correlates with which this helps. I suppose the hope was the math would have physical correlates, but of course that should show itself.

That said, just what Studiot also mentioned,

17 hours ago, studiot said:

For your information, you are starting at 'the wrong end'.

Number theory starts at the beginning and uses 'successor theory'.

is what I had in mind. To start it from "the wrong end". I don't take it as granted that that is wrong -- that I would say is the math conjecture here.

Edited by NTuft
grammar
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25 minutes ago, NTuft said:

Studiot, thanks for the assessment. Yes, including or redifining i (i2=(-1)) is like an ad hoc hitchhiker, I don't think it makes sense. Need something other than "i". I also appreciate the number theory run up, I think you get what I was trying. Then from last, & from joigus, I agree, this is tangled up puree, as I can't now see any physical correlates with which this helps. I suppose the hope was the math would have physical correlates, but of course that should show itself.

That said, just what Studiot also mentioned,

is what I had in mind. To start it from "the wrong end". I don't take it as granted that that is wrong -- that I would say is the math conjecture here.

I am going to say +1 for your listening attitude here. This is so much better than many who come to lay out their pet cogitations.

I see you are not fuly convinced about the number theory. Nor have you mentioned infinity again.

That is possibly because there are some interesting twists that appear when you study number theory a bit more.

Do you wish to discuss this aspect some more ?

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14 hours ago, studiot said:

I am going to say +1 for your listening attitude here. This is so much better than many who come to lay out their pet cogitations.

Agreed.

I suppose what you wanna do here, @NTuft, is something like considering a complex variable, $$z$$, differentiating by it,

$\frac{d}{dz}\left(\frac{1}{2}z^{-1/2}\right)=-\frac{1}{4}z^{-3/2}$

and then substituting,

$\left.\frac{d}{dz}\left(\frac{1}{2}z^{-1/2}\right)\right|_{z=i}=\left.-\frac{1}{4}z^{-3/2}\right|_{z=i}=-\frac{1}{4}i^{-3/2}$

But be careful; you may be re-discovering complex calculus.

The connection to number theory is something I'm missing.

I'm very skeptic to there being a connection between physics/chemistry and number theory. There is a connection between physics and complex calculus, provided by harmonic functions in two-dimensional problems.

Edited by joigus
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Quote

"...
That is possibly because there are some interesting twists that appear when you study number theory a bit more.

Do you wish to discuss this aspect some more ?"

@studiot
I do need to study number theory, and I woud like to hear exposition from you on what you find interesting, or if you had a resource to recommend to, "get up to speed", I would read into it.

I was using it only in the sense that I could arbitrarily set rules for a number theory on my own terms, and I doubt now anything I've posited provides value other than what is enumerated or established by number theory already developed.

To change it somewhat, I'd say that the first derivative from infinity should include the set of all square roots of numbers. Then, integers can be teased out from the sq. rts of their squares, and the leftover inoperable numbers are the primes and semi-primes that have prime factors still needing the ^(1/2) exponentiation.
More or less, I had the hope that this idea of deriving from infinity was a "bolt from a dark cloud", but it is little else than a fever dream or pipe dream that now I can see has little applicability to any actual problems.

Quote

"I suppose what you wanna do here,..."

Yes, I suppose so -- I had chosen 'i' as my term to define, even though it is already defined as a constant, which is confusing. Furthermore, I was actually ignorant that in complex numbers the term i is actually operated on! From some wiki reading:
cyclic values:
i-3 = i
i-2 = -1
i-1 = -i
i0 = 1
i1=i
i2= -1
i3=-i
i4=1
i5=i
i6=-1

Now, what you're showing I think is a partial differential equation? My math is limited -- hence why in my discussion I was talking about trying to derive "z" (formerly "i") in terms of something else, "the field". This might not make any sense whatsoever. So, again, I'm in need of education.

That said, from above cyclic results, it looks to me that the values that come from i when it is operated on are quite limited, and maybe this set of results could be expanded? The cyclic values of 1,-1,i,-i are what graph onto the complex plane as a magnitude(length?), or argument (direction)? If the magnitudes/arguments were expanded to include the set of larger square roots of primes would this be impossible or without basis? To my mind, this would simply expand the idea of degree of magnitude in complex number graphing. However, my understanding of complex numbers and complex graphing is admittedly lacking.

I am quite ignorant as I keep repeating and likely in need of a Physics 201 or 301 course. I'm going to continue my study of Quantum Chemistry as I think it incorporates a lot of what I want to learn. I'm also looking at some books on basics covering General Relativity and more modern theories (e.g. Hyperspace by Kaku) from a reading list I saw on another board.

I have enough insight I think to see I'm low man on the totem pole in this discussion, so I appreciate your fellows' input, and if you had some reading material on number theory or physics that you'd recommend I would be glad to have those suggestions. I can see you're willing to engage and that alone I can appreciate.

Thanks

On 8/15/2021 at 12:52 AM, joigus said:

The connection to number theory is something I'm missing.

I'm very skeptic to there being a connection between physics/chemistry and number theory. There is a connection between physics and complex calculus, provided by harmonic functions in two-dimensional problems.

noted^

Edited by NTuft
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• 7 months later...
Posted (edited)

Does this serve to "Disproving" Cantor's hypothesis?

So, if we take the set of square roots of prime numbers, each number in that set can match the set of "Natural" numbers by extending the decimal points. As the "Natural" numbers get large a further decimal 0-9 matches each n, n+1, ...; 1:1. The ordinality of i' is the same as the ordinality of "infinity", and there is a degree of aleph=2 to the subset i' before it reaches anything including all the "Reals". So, between the sets "infinity", i', various Real sets, and the Natural set, there are at least two degrees, hence the aleph=2, hence Cantor's conjecture is towards the negative.

Does that make sense?

Thanks

Edited by NTuft
re-writes, Clarity
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5 hours ago, NTuft said:

Does this serve to "Disproving" Cantor's hypothesis?

So, if we take the set of square roots of prime numbers, each number in that set can match the set of "Natural" numbers by extending the decimal points. As the "Natural" numbers get large a further decimal 0-9 matches each n, n+1, ...; 1:1. The ordinality of i' is the same as the ordinality of "infinity", and there is a degree of aleph=2 to the subset i' before it reaches anything including all the "Reals". So, between the sets "infinity", i', various Real sets, and the Natural set, there are at least two degrees, hence the aleph=2, hence Cantor's conjecture is towards the negative.

Does that make sense?

Thanks

Hi. I'm not an expert, but I can tell you what the present status of Cantor's continuum hypothesis is:

Quote

The continuum hypothesis was advanced by Georg Cantor in 1878,[1] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940.[2]

(My emphasis.)

So, within the Zermelo-Frenkel set-theory axioms, plus the axiom of choice, you can neither prove nor disprove the continuum hypothesis.

And that's what I know. The possibility of a cardinality sandwiched between aleph_0 and aleph_1 is still conjectural.

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On top of what @joigus said so clearly (+ 1),

You have to delve much deeper into the philosophy of Mathematics beofre you can start using symbols such a = ;  + ;  n+1 and so on.

Consider.

Until you have defined what a number is how can you define addition ?

So how can you give meaning to n + 1 in the definition of a number ?

Note in Mathematics defining means loosely 'give meaning to' .

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10 hours ago, joigus said:

The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent.

10 hours ago, joigus said:

So, within the Zermelo-Frenkel set-theory axioms, plus the axiom of choice, you can neither prove nor disprove the continuum hypothesis.

Hello again.
I'm no expert either. I don't read what you'd quoted to lead to that following statement.

From: https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gillman544-553.pdf Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis by Leonard Gillman

Quote

Kurt Godel proved in 1938 that the General Continuum Hypothesis and the Axiom of Choice are consistent with the usual (Zermelo-Fraenkel) axioms of set theory [4]. Twenty-five years later, Paul Cohen established that the negations of the Continuum Hypothesis and the Axiom of Choice are also consistent with these axioms [3]. Taken together, these results tell us that the Continuum Hypothesis and the Axiom of Choice are independent of the Zermelo-Fraenkel axioms.

So it may be independent, but, as I think @studiot was alluding to more generally, we have to work from an accepted basis to get anywhere. So, @joigus, bringing in ZF axioms is most pertinent. Since the discussion is under set theory,  here's an overview which I don't think is too tedious:

Quote

ZFC axioms of set theory
(the axioms of Zermelo, Fraenkel, plus the axiom of Choice)

For details see Wikipedia ”Zermelo-Fraenkel set theory”. Note that the descriptions there are quite formal (They need to be, because the goal is to reduce the rest of math to these axioms. So to avoid circular reasoning, you have to state the axioms without using anything you know from the rest of math! That can be done but it does make the precise statements of the axioms too technical for this class).

A1 Axiom of Extensionality.
This Axiom says that two sets are the same if their elements are the same. You can think of this axiom as defining what a set is.

A2 Axiom of Regularity.
I have never seen a set A for which A is an element of A, but that does not imply that such sets do not exist. Do they exist? How would you construct such a set without using a circular construction? Circular definitions are forbidden in math because if you allow them, then it is very easy to run into contradictions. Now about this axiom. This axiom looks quite technical. It implies that a set A cannot cannot be an element of A. Moreover, it also implies that A cannot be an element of an element of A. Also: A cannot be an element of an element of an element of A, etc.

A3 Axiom schema of specification.
If P(x) is a statement (if you plug in a value for x then P(x) becomes either true or false) and if A is a set, then this axiom says that the elements of A that satisfy P also form a set. We denote that set as {x ∈ A | P(x)}. Read that notation as: The set of all x ∈ A for which P(x) is true. The symbol | means: “for which” (some authors use : instead of |).

A4 Axiom of Pairing .
Says that if x, y are sets then {x, y} is also a set. You may wonder: why do we need this axiom? Isn’t this obviously true? Well, the issue is, it doesn’t matter if people feel that this is obviously true, either way, you can not actually prove this statement. The point of the axioms is to explicitly write down all statements we accept without proof, including those that some might feel are “obvious”. Think of it this way, supposed you’ve played chess for many years. Then you might feel that the rules of chess are obviously true. But you can’t mathematically prove that those rules are true. So then why are the rules of chess true? Because chess organizations have decided that those are the rules they want. Likewise, in the game of math, we have settled on certain rules, and these rules are the axioms and the rules of logic.

A5 Axiom of Union.

If A is a set whose elements are again sets, then the union of those sets is again a set.

A6 Axiom of replacement.

Suppose φ(x, y) is a statement such that for every set x there is precisely one set y for which φ(x, y) is true. Denote that y as f(x). Remark: f looks a lot like a function, but we may not yet call f a function. In math you may only call f a function after specifying the domain AND the co-domain. 1 The axiom says that if we apply f to all elements of some set A, so take f(x) for every x ∈ A, then the axiom says that all those f(x) form a set. We denote that set as {f(x) | x ∈ A}.

A7 Axiom of infinity. Says that there exists an infinite set. Note: It does not matter which infinite set, as long as we have at least one infinite set then the other axioms allow us to construct our favorite infinite set, which is N = {0, 1, 2, 3, . . .}. You may wonder, why do we need an axiom that tells us that an infinite set exists, when we’ve already known about {0, 1, 2, 3, ....} for a long time? The issue here is: How do you define the . . . in the notation {0, 1, 2, 3, . . .} without using circular reasoning? The only thing we can do here is to accept the unprovable statement that there exists at least one infinite set, and then work from there. So wait, any time we need an unproven statement, we can simply call it an axiom? No, because other mathematicians won’t accept additional axioms. Although the current axioms are not proven, they are supported by good evidence. They have been thoroughly put to the test for at least a century. It is outside of the scope of this course to detail the many ways in which the axioms have been investigated, but the upshot is that mathematicians are very confident that the standard axioms (called ZFC), combined with the rules of logic, do not lead to errors. Mathematicians are unlikely to accept more axioms; we do not need more axioms, and we are confident about the ones we have.

A8 Axiom of the Power set.

If A is a set then P(A) is a set.

A9 Axiom of Choice. (AC) Chapter 3 [?] proves that this axiom is equivalent to several other statements:

1) Zorn’s lemma

2) The well-ordering theorem

3) For any cardinals d, e we have d ≤ e or e ≤ d.

4) For any sets A, B, if there is a surjective function from A to B then there is an injective function from B to A.

If you replaced AC by one of these four statements, then ZFC set theory stays the same. The axiom of choice, says that if A is a set whose elements are non-empty sets, then one can pick an element from each of these non-empty sets. This sounds harmless, however, if A is an infinite set, then we have to choose one element from infinitely many sets. How do you make infinitely many choices? Even though we may not be able to do this ourselves, AC says that there does exist a function that makes all these choices. In case you think that this is fishy, that is not unreasonable, but consider this: One can prove (using only A1-A8) that if A1-A9 is contradictory, then A1-A8 must also be contradictory. So if you want to prove that something in A1-A9 is wrong, then you also have to prove that something in A1-A8 is wrong (remember from the comments under A7 that the axioms have been put to the test in many ways).

There you have it, a full list of all statements that mathematicians accept without proof. Any statement other than A1-A9 will only be accepted by other mathematicians if it has a proof that only uses the rules of logic and the axioms A1-A9. (The use of other statements is allowed provided that they can be proved from the rules of logic and the axioms.) Proved statements are called theorems, and mathematicians trust them even when they are counter-intuitive (like the existence of infinite sets with different cardinalities!). Conversely, other than A1-A9, mathematicians don’t accept statements that don’t have a proof, no matter how plausible they may sound.

So I think it needs to be formulated under this rubric (A1-A8 or A1-A9) and with understanding of cardinality/ordinality (i.e. using what's agreed upon and the relevent well-defined terms).

Thanks

9 hours ago, studiot said:

On top of what @joigus said so clearly (+ 1),

You have to delve much deeper into the philosophy of Mathematics beofre you can start using symbols such a = ;  + ;  n+1 and so on.

Consider.

Until you have defined what a number is how can you define addition ?

So how can you give meaning to n + 1 in the definition of a number ?

Note in Mathematics defining means loosely 'give meaning to' .

I think that this is the purpose of set theory: numbers are what we commonly refer to as the collection of objects in the Natural/Real set. Once the objects (numbers) are defined, operations can be defined to act upon them, certain inputs giving certain outputs.

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1 hour ago, NTuft said:

I think that this is the purpose of set theory: numbers are what we commonly refer to as the collection of objects in the Natural/Real set.

You are going to be a lot more precise to make any headway.

What is "the Natural/Real set" ?

And what properties do the members of 'the set of square roots of prime numbers enjoy' that have already been proven ?

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Posted (edited)
24 minutes ago, studiot said:

You are going to be a lot more precise to make any headway.

What is "the Natural/Real set" ?

ℕ , the set of all natural numbers: https://infogalactic.com/info/Natural_number#Notation

ℝ , the set of all real numbers: https://infogalactic.com/info/Real_number

Quote

And what properties do the members of 'the set of square roots of prime numbers enjoy' that have already been proven ?

They're irrational? I do not know if they enjoy such being ascribed as such...

Perhaps, also, it's been proven that i' = aleph_1, while the = aleph_0, and the ℝ ≥ aleph_2

Edited by NTuft
tooting my own horn
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On 4/19/2022 at 1:54 PM, studiot said:

And what properties do the members of 'the set of square roots of prime numbers enjoy' that have already been proven ?

They are algebraic numbers?

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• 2 weeks later...

Physical Meaning of Imaginary Unit i

I think this paper is worth the read (9 pgs.), and has some bearing on the relation here between imaginary unit in math, physics of QM, and chemistry.

Thanks

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46 minutes ago, NTuft said:

Physical Meaning of Imaginary Unit i

I think this paper is worth the read (9 pgs.), and has some bearing on the relation here between imaginary unit in math, physics of QM, and chemistry.

Thanks

Successfully solving a problem that does not exist to start with. The solution is beautifully clear:

"the mathematical reflection of the dialectical law of affirmation-negation for qualitatively opposite binary judgments about the nature of any object or process."

"the "imaginary" unit i is a symbol for a qualitatively polar opposite essence (property, number, parameter) obeying to the polar opposite (“negative”) algebra of signs relative to the conventional (“positive”) algebra."

In conclusion,

"we believe that the word "imaginary" as not reflecting reality will eventually be removed from mathematics and physics, harmonizing the mathematical structures with the laws of the Universe, and thus expanding the horizons of knowledge."

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Posted (edited)

Special thanks also to Donald A. McQuarrie, George P. Shpenkov, et al.

Edited by NTuft
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Posted (edited)

2x e-mails:

Re: H2+ and the breakdown of the virial theorem <V>/<T> = -2

Hello,

So the average kinetic and potential energy, <T> and <K>, resp., give us <T> + <V> = <E> , and for the 1s hydrogen atom we have <V>/<T>=-2 properly. (pg. 3321)

Then, even at our simplest nucleation, H2+, we have a discrepancy:

[...] paraphrasing pg.5181, the virial theorem starts to move away from -2 with the H2+ molecule, and already needs "optimization" to fit [...]

even though, "...[the virial theorem] is generally true for any system in which the potential energy is coulombic. Equation 7.25 is an example of the virial theorem and is valid for all atoms and molecules." (pg. 3321)[bold added for emphasis]

[...] to the issue: could it be that we are miscalculating the Potential Energy because we have reduced what should be a set of values to a constant? It works assuming the 1s electron energy, but something has been lost somewhere where we can't scale up, in "normalization", or in an assumption implicit in Ψ*Ψ to Ψ^2 ; not accounting for what could be hidden in the imaginary unit?

What do you think about why the math breaks down at the simplest nucleation, or getting into a 2 electron system, even? I don't know.
--------------------------

Re: Addendum on last, cite source. Plus: are we cheating the harmonic oscillator's power series somehow with the "operator method"?

We have :
"The Schrödinger equation for a harmonic operator [oscillator? it says operator] is

-(hbar^2/2mu)*d''psi/dx'' + (1/2)kx^2psi = E*psi

As we mentioned in Section 5.5, this is a linear differential equation in psi(x), but it does not have constant coefficients. A standard method to solve such equations is to assume that psi(x) can be expressed as a power series in x:

psi(x) =n=0a(sub(n))xn

We substitute this expression into the differential equation and obtain a set of algebraic equations for the a(sub(n)). This method is called the power series method, and is at best tedious. There is an alternative method of solving equation 1 that is based on operator methods, which is not only much easier than the power series method, but is also much more elegant. In this appendix, we shall use operator methods to determine the eigenvalues and eigenfunctions of a harmonic oscillator. Operator methods are used frequently in quantum mechanics and they are well worth assimilating."

/\ = "hat"

"The Hamiltonian operator of a harmonic oscillator is given by

/\H = - (hbar^2)/2mu * d''/dx'' + (1/2)kx^2 =/\P^2/2mu + (k/2/\X^2)

where k=mu*omega^2 is the force constant and /\P and /\X are the momentum and position operators, respectively. If we introduce new operators

/\p = (mu*hbar*omega)^(1/2)/\P    and   /\x = (mu*omega/hbar)^(1/2)/\X

then the Hamiltonian operator can be written as

/\H = [(/\P^2)/2*mu] + (k/2)/\X^2 = [(hbar*omega)/2]*(/\p^2 + /\x^2)

Using the fact that [/\P, /\X]= -i * hbar , we have

[/\p, /\x]=/\p/\x - /\x/\p = (1/hbar)*(/\P/\X-/\X/\P = (1/hbar)(-i*hbar) = -1

We now define the (not necessarily Hermitian) operators /\a+ and /\a_ by

/\a+ = (1/2^(1/2))*(/\x-i/\p)   and /\a_ = (1/2^(1/2))*(/\x+i/\p)

Using these definitions, we have

/\a_/\a+ = (1/2)(/\x+i/\p)(/\x-i/\p) = (1/2)[/\x^2 + i(/\p/\x - /\x/\p) + /\p^2]

= (1/2)(/\x^2+/\p^2 +1) [6]

Similarly,

/\a+/\a_=(1/2)(/\p^2+/\x^2 - 1) [7]

Equations [6] and [7] give us

/\a+/\a_ - /\a_/\a+ = [/\a+, /\a_] = -1 [?...]

Note that the Hamiltonian operator, equation 3, can be written as

/\H=(hbar*omega)*(/\a+/\a_+(1/2)) [9]

To make equation 9 more transparent, we denote the operator /\a+/\a_ by /\v and write equation 9 as

/\H = hbar*omega (/\v + (1/2))" "(pgs. 239-2411)

...
This harmonic oscillator WORKS , but if something got left out in simplification or normalization could that explain why when we try to scale up (continuing to use the simplified Hamiltonian operators methods results) we get inaccurate theoretical vs. experimental energy values?

@joigus I think there is a "physics" target here, at last.

1McQuarrie, Donald A. (Donald Allan). Quantum Chemisty. 2nd ed., University Science Books, 2008.

"Summary: Idea of deriving down to i''' from infinity, potentially establishing number theory from set of sq.rt. of primes, i'. Then, (*highly speculative*) i'' as Kinetic Energy, i'" as Potential Energy from electrostatics for new Hamiltonian in quantum mechanics" QED unless someone can put a hole in it.

Edited by NTuft
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Posted (edited)

@uncool sez it's a pop-Ψ magazine! I read it to say the physiks is (mostly) still intact.

Edited by NTuft
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Posted (edited)
11 hours ago, NTuft said:

even though, "...[the virial theorem] is generally true for any system in which the potential energy is coulombic. Equation 7.25 is an example of the virial theorem and is valid for all atoms and molecules." (pg. 3321)[bold added for emphasis]

I'm not sure what relevance this has on Cantor's hypothesis, but the virial theorem is actually more general: True for any potential that is homogeneous in the spatial coordinates:

$V\left(\lambda\boldsymbol{x}\right)=\lambda^{n}V\left(\boldsymbol{x}\right)$

So ideal gases, harmonic oscillators, and inverse-square systems all have a version of it.

Sorry. I mistook this thread for another one. Oops!

Edited by joigus

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# 3 dimension distance

1. Oct 1, 2005

Hi,
Can someone help me with this:

What is the shortest distance between the two lines:

L1: r= (1,0,0) +t( 2,3,4)
L2 u= (2,1,0) +s(1,2,0)

Thank you very much for your help

B

2. Oct 1, 2005

### quasar987

There is probably a shorter way in terms of vectors and stuff, but here's a way:

Build the function "distance squared btw the lines", defined as the norm squared of the distance vector function

$$\vec{d}(s,t) = \vec{r}(t) - \vec{u}(s)$$

The reason we work with the function distance squared instead of the function distance itself is that the distance is minimum when the distance squared is minimum, and as you'll see, it is much easier to find when d² is minimum then when d is minimum.

This function is a function of the two variables t and s. Use the usual method to find the extrema of the function. (no need to use the second derivative test to determine the nature of each extrema; yuo know the function distance quared as no maxima and only one minimum .

3. Oct 2, 2005


text:
Publications: Abstract

# Asynchronous Multi-Party Computation With Quadratic Communication

## Martin Hirt and Jesper Buus Nielsen and Bartosz Przydatek

 We present an efficient protocol for secure multi-party computation in the asynchronous model with optimal resilience. For $n$ parties, up to $t<n/3$ of them being corrupted, and security parameter $\kappa$, a circuit with $c$ gates can be securely computed with communication complexity $\O(c n^2 \kappa)$ bits, which improves on the previously known solutions by a factor of $\Omega(n)$. The construction of the protocol follows the approach introduced by Franklin and Haber (Crypto'93), based on a public-key encryption scheme with threshold decryption. To achieve the quadratic complexity, we employ several techniques, including circuit randomization due to Beaver (Crypto'91), and an abstraction of certificates, which can be of independent interest.


text:
Hardcover | $16.75 Short | £12.95 | ISBN: 9780262134378 | 360 pp. | 7 x 10 in | February 2004 Paperback |$35.00 Short | £24.95 | ISBN: 9780262632867 | 360 pp. | 7 x 10 in | February 2004

## Overview

This textbook for instruction in biomedical research ethics can also serve as a valuable reference for professionals in the field of bioethics. The 149 cases included in the book are grouped in nine chapters, each of which covers a key area of debate in the field. Some of the case studies are classics, including the famous cases of the Tuskegee Syphilis Study (in which subjects with syphilis were not given treatment) and the Willowbrook hepatitis studies (in which institutionalized subjects were intentionally exposed to hepatitis). Others focus on such current issues as human embryonic stem cell research, cloning by somatic nuclear transfer, and the design and function of institutional review boards. Each chapter begins with a brief introduction that places the issues raised in context; this is followed by a number of cases (each of which is no more than a few pages). Study questions meant to encourage further discussion follow each case.After an introductory discussion of the history and tenets of ethics in medical research, the book's chapters cover the topics of oversight and study design; informed consent; the selection of subjects; conflicts of interest; the social effects of research; embryos, fetuses, and children; genetic research; the use of animals; and authorship and publication. Following these chapters are appendixes with the texts of the Nuremburg Code and the World Declaration of Geneva, two key documents in the establishment of bioethical standards for research. Also included are a glossary, a table of cases by general category, and an alphabetical listing of cases.


text:
A Rate-Optimizing Compiler for Non-malleable Codes Against Bit-wise Tampering and Permutations

Shashank Agrawal, Divya Gupta, Hemanta K. Maji, Omkant Pandey, and Manoj Prabhakaran

Abstract

A non-malleable code protects messages against a class of tampering functions. Informally, a code is non-malleable if the effect of applying any tampering function on an encoded message is to either retain the message or to replace it with an unrelated message. Two main challenges in this area -- apart from establishing the feasibility against different families of tampering -- are to obtain explicit constructions and to obtain high-rates for such constructions. In this work, we present a compiler to transform low-rate (in fact, zero rate) non-malleable codes against certain class of tampering into an optimal-rate -- i.e., rate 1 -- non-malleable codes against the same class. If the original code is explicit, so is the new one. When applied to the family of bit-wise tampering functions, this subsumes (and greatly simplifies) a recent result of Cheraghchi and Guruswami (TCC 2014). Further, our compiler can be applied to non-malleable codes against the class of bit-wise tampering and bit-level permutations. Combined with the rate-0 construction in a companion work, this yields the first explicit rate-1 non-malleable code for this family of tampering functions. Our compiler uses a new technique for boot-strapping non-malleability by introducing errors, that may be of independent interest.

Note: Greatly expanded proof; cleaner notation.

Available format(s)
Category
Foundations
Publication info
A minor revision of an IACR publication in TCC 2015
Keywords
Non-malleable CodesExplicit ConstructionInformation TheoreticRate 1.
Contact author(s)
hemanta maji @ gmail com
History
2015-01-15: last of 5 revisions
See all versions
Short URL
https://ia.cr/2014/842

CC BY

BibTeX

@misc{cryptoeprint:2014/842,
author = {Shashank Agrawal and Divya Gupta and Hemanta K.  Maji and Omkant Pandey and Manoj Prabhakaran},
title = {A Rate-Optimizing Compiler for Non-malleable Codes Against Bit-wise Tampering and Permutations},
howpublished = {Cryptology ePrint Archive, Paper 2014/842},
year = {2014},
note = {\url{https://eprint.iacr.org/2014/842}},
url = {https://eprint.iacr.org/2014/842}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
# VC5.48

Here the given cone $z=2-\sqrt{(x^2-y^2)}\,$ meets the xy plane in a circle C given by $x^2+y^2=4,z=0\,$.Let R be the plane region of the circular disc bounded by C.Let S1 denote the entire closed surface made up of S and R and Let V1 be the volume enclosed by S1.

Then,we have $\iint_{S_1}(\nabla\times F)\cdot n dS=\iint_{S_1}\mathrm{curl}F\cdot n dS\,$

=$\mathrm{div}\mathrm{curl}F dV\,$ by divergence theorm.

=$0\,$ [As $\mathrm{div}\mathrm{curl}F=0\,$]

Therefore,$\iint_S(\nabla\times F)\cdot n dS+\iint_R(\nabla\times F)\cdot n dS=0\,$

<$\iint_S(\nabla\times F)\cdot n dS=-\iint_R \mathrm{curl}F\cdot n dS\,$ --(1)

Here,$\mathrm{curl}F=\begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ (x-z) & (x^3+yz) & -3xy^2\end{vmatrix}\,$

=$(-6xy-y)i+(-1+3y^2)j+(3x^2-0)k\,$ --(2)

Since on R,n=-k,therefore,we have

$\mathrm{curl}F\cdot n=\mathrm{curl}F\cdot(-k)=-3x^2\,$,using (2)

Therefore,from (1),$\iint_S(\nabla\times f)\cdot n dS\,$

=$\iint_R 3x^2 dS=3\int_{r=0}^{2}\int_{\theta=0}^{2\pi}(r\cos\theta)^2\times r d\theta dr\,$,changing to polars

=$3\int_{r=0}^{2}\int_{\theta=0}^{2\pi} r^3[\frac{1+\cos 2\theta}{2}]dr d\theta\,$

=$3\int_0^2 r^3[\frac{\theta}{2}+\frac{\sin 2\theta}{4}]_{0}^{2\pi} dr\,$

=$3\pi\int_0^2 r^3 dr=3\pi[\frac{r^4}{4}]_0^2=12\pi\,$

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text:
Inter‘ means ‘between’ as happening between two things, showing that the term ‘intermolecular forces’ refers to the electrostatic forces of attraction or repulsion between molecules. Another similar term, ‘intramolecular forces’ means the forces present between atoms to constitute molecules. . An example is the word ‘international’ meaning, between two or more nations, and ‘intranational’ meaning, within a nation.

Intermolecular forces (IMF) also known as secondary forces are the forces of attraction that exist between molecules. These forces include dipole-dipole interactions, ion-dipole interactions, ion-induced dipole interactions, van der Waals forces, and hydrogen bonding. These forces are required to determine the physical properties of compounds such as melting point, boiling point, density, vaporization, and enthalpies, etc. Intermolecular forces are much weaker than intramolecular forces (forces of attraction that hold atoms into a molecule) like ionic bonds or covalent bonds, etc.

## Types of intermolecular forces

Mainly, intermolecular forces are divided into three types; dipole to dipole interactions, van der Waals forces (Londen Dispersion forces), and Hydrogen bonding. But the term induced dipoles, referring to Londen dispersion forces (LDF) is divided into further types. This makes five types of intermolecular forces.

• Dipole-dipole forces
• Ion-dipole interaction
• Ion-induced dipole interaction
• Van der Waal forces (Keesom forces, Debye forces, London dispersion)
• Hydrogen bonding

Information on intermolecular forces can be obtained by the study of macroscopic properties like pressure, volume, temperature (PVT) data using instrumental techniques. Such macroscopic properties are quantified by virial coefficients and Lennard-Jones potential.

## 1. Dipole-dipole interaction

Dipole to dipole interaction is due to the partially positive and negative poles across molecules. The negative poles of molecules attract the positive poles of other molecules and result in the establishment of electrostatic forces of attraction between molecules known as dipole-dipole interaction.

This type of electrostatic force of attraction is weaker than ion to ion interaction. This is because the interaction between dipoles is based on partial charges while that of ionic interactions is based on permanent positive and negative charges called cations and anions.

An example of dipole-dipole forces is the interaction between the HCl molecules. There is a partial positive charge on the hydrogen atom and a partial negative charge on the chlorine atom. The partially negative chlorine attracts partially positive hydrogen from other molecules giving rise to a huge network of such interactions.

Dipole to dipole interactions are only possible in polar molecules like water (H2O), ammonia (NH3), sulfur dioxide (SO2), hydrogen disulfide (H2S), nitrogen dioxide (NO2), EDTA (Ethylene diamine tetraacetic acid), etc. The properties of polar compounds are factors of these interactions.

## 2. Ion-dipole interaction

Ion to dipole interactions occur when ions and polar molecules interact and an interactive force is established between them. The polar molecules contain partial positive and negative charges, and as a consequence, when an ion with a positive or negative charge comes closer to such molecules, an electrostatic force of attraction is set between them. As similar charges repel and opposite ones attract, an ion always faces the oppositely charged end of a polar molecule.

When sodium chloride (NaCl) is added to water, the water molecules make interact with the sodium ions and chloride ions.

This property of water to separate ionic compounds into individual ions by making ion-dipole interactions is sometimes referred to as solvation or hydration.

## 3. Ion-induced dipole interaction

Ions are responsible for ion-induced interactions. When an ion comes near a nonpolar molecule, it induces partial positive and negative poles generation on that molecule, converting it into a dipolar structure. In other words, an ion induces a dipole moment on nonpolar molecules and converts them into polar molecules. Such interactions are named ion-induced dipole interactions.

Examples of ion-induced dipole interactions include metal ions approaching oxygen and other molecules and creating a net dipole moment in them, etc.

If this induction of dipoles is done by ions, the resultant forces are called ion-induced dipole interaction. But, if these interactions result from already polar molecules to nonpolar molecules, these interactions are called dipole-induced dipole interactions, also known as Debye forces as explained below.

## 4. Van der Waals forces

Named after Dutch physicist, Johannes Diderik van der Waals, these interactions are the attractive forces between all atoms and molecules. These forces quickly vanish upon the increase in the distance between individual atoms or molecules and the threshold distance for such forces is called van der Waals contact distance.  Unlike other types of bonding interactions, van der Waals interactions do not have an electrostatic or chemical base.

These are the weakest force of attraction between the molecules and are usually seen as Keesom forces, Debye forces, and London dispersion forces (LDF).

### Keesom forces

Keesom forces are the temperature dependent and very weak van der Waals forces that originate between permanent dipoles. As, molecules rotate constantly and it is generally assumed that they cannot be locked in one place, but at some points, they do get locked. These interactions arise at those moments and are seen in all quadrupoles and multipoles.

Keesom forces are named after William Hendrik Keesom and arise due to interactions between permanent polar structures. An aqueous solution that contains electrolytes, does not have Keesom interaction.

Keesom interaction energy depends upon the inverse sixth power of distance between individual polar structures. As the permanent spatially fixed dipole structures depend on inverse third power of distance, Keesom forces are the interactions among multiples.

$\fn_phv&space;V=\frac{-(m_{1}m_{2})^{2}}{24k_{B}(\pi\varepsilon&space;_{0}\varepsilon&space;_{r})^{2}T&space;r^{6}&space;}$

where,

• m1 and m2 = dipole moment of individual molecules
• ε0 = permittivity of free space
• εr = dielectric constant
• T = temperature
• kB = Boltzmann constant (1.38064852 × 10-23 m2 kg s-2 K-1)
• r = distance between polar molecules

### Debye forces

When permanent dipoles rotate and move, non-polar molecules in the vicinity become polarized. This polarization is induced especially by the electronic repulsion of individual molecules. The result of such disturbance of electronic clouds is a molecular attraction called Debye forces.

Debye forces are also termed permanent dipole-induced dipoles interactions. These forces do not depend on temperature as much as that of Keesom forces. These forces are molecular interactions and are not seen between atoms.

Polarization induced by polar molecules is the main cause behind the existence of such forces. These attractive forces are stronger than the London dispersion forces but are weaker than dipole-dipole interactions.

For example,

• Argon molecule experiences dipolar characteristics when attracted by the hydrogen atom of HCl or repelled by the chlorine atom of HCl.
• Polar molecules such as water, create dipoles in neutral molecules like oxygen, etc.

### London dispersion forces (LDF)

London dispersion forces are named after a German-American physicist, Fritz London. They are also known as London forces, dispersion forces, and Instantaneous dipole-induced dipole forces.

These forces are the result of random fluctuation in electronic density. Atom-atom interaction also contains London dispersion forces. These forces depend upon the number of electrons in an atom. The greater the number of electrons, the stronger, the London dispersion forces are there.

Hydrocarbons show weak van der Waals forces, although, this effect can be increased by the introduction of a heteroatom. For example, alkyl halides have considerably more van der Waals interactions than the parent hydrocarbons. It is actually the polarizability of atoms, which determines how easy it is to disturb the electronic cloud of atoms, to introduce LD forces.

## 5. Hydrogen bonding

Hydrogen bonding is a type of dipole-dipole interaction but it is placed separately as another type of intermolecular force due to its extremely strong nature. The hydrogen bond is the only intermolecular force to have the word “bond” in its name because it resembles intramolecular forces’ strength.

The attractive forces between a hydrogen atom from a molecule with a highly electronegative atom like nitrogen (N), oxygen (O), and fluorine (F). Hydrogen bonds are the strongest of the intermolecular forces with bond energies of 4 kJ to 50 kJ per mole.

Hydrogen atom has an electronegativity value of (2.2) which when bound with highly electronegative atoms generate these type of bonds. Sometimes, it is referred to as a covalent bond but it is not entirely covalent, it just resembles covalent bond energy.

Hydrogen bonding is responsible for the higher boiling and melting point of different compounds like water (H2O), alcohols (R-OH), and hydrogen fluoride (HF), etc.

The more electronegative atoms attract the electronic cloud of the hydrogen atoms and a partial positive charge is formed there as (-Hδ+). Electronegative atoms such as oxygen, nitrogen, and fluorine atoms are electron lone pair donors, which donate to this partial positive hydrogen. This way, a partial negative charge is generated on the more electronegative atoms.

Examples of hydrogen bonds in molecules are;

Attractive forces among the following molecules are termed hydrogen bonding.

• Water
• Ammonia
• Alcohols
• Some alkyl halides
• Acetylacetone
• Esters
• Amides
• DNA
• RNA
• Protiens
• Sugars, etc

Some polymers also have hydrogen bonding, such as,

• Nylon
• Kevlar, etc

Hydrogen bonding is the strongest of intermolecular forces. This comparison of strength of intermoleculatr forces can well be understood at; Polar vs Nonpolar bonds.

## Effects of intermolecular forces on chemical compounds

Intermolecular forces deal with the physical effects in chemical substances like boiling and melting point, etc. The stronger these forces are, The more energy is required there, to break the bonds. They considerably increase the melting and boiling point of the compounds.

Parameters affected by intermolecular forces are;

• Melting points
• Boiling points
• Freezing points
• Vapour pressure
• Surface tension, etc

## Strength of intermolecular forces

 Intermolecular forces Strength Dipole-dipole interaction Strong Ion-dipole interaction Strongest Ion-induced dipole force Considerably weak Van der Waals force Weakest Hydrogen bond Very strong

## Applications of intermolecular forces of attraction

Some practical applications of intermolecular forces are as:

• Adhesives are extensively used binding chemicals which use intermolecular forces for binding purpose.
• Construction materials
• Contact cement, tile grouts, and  paint industry is full with intermolecular forces applications.
• Epoxy resins
• Resins use intermolecular forces to become so dense and sometimes decreased water content epoxy resins freeze at room temperatures.
• Electrical devices
• Bakelite, a common electrical switch making material comprises of Novolac and formaldehyde chains bound by hydrogen bonding. which gives characteristic strongness to this material, etc.

Intermolecular forces of attraction can be used statistically to determine melting point, boiling point, vaporization, density, enthalpies of fusion. Furthermore, hydrogen bonds are responsible to hold the nucleotide bases in DNA and RNA, etc.

## Concepts Berg

Are ionic bonds intermolecular forces?

Sometimes, ionic bonds are referred as intermolecular forces because of the ability of ionic compounds to create forces with outside molecules similar to inside ones. In other words, ionic compounds are very polar, which makes them able to attract other atoms, ions or molecules.

How do forces of attraction affect the properties of compounds?

Forces of attraction affect the properties of compounds by increasing the boiling points, melting points, enthalpies of vaporization and fusion, etc.

What are the 5 types of intermolecular forces?

• Dipole-dipole forces
• Ion-dipole interaction
• Ion-induced dipole interaction
• Van der Waals forces
• Hydrogen bonding

What is the meaning of intermolecular forces?

Intermolecular forces means the forces between the molecules like HCl, HI, and water, etc.

What is the strongest intermolecular force?

Among five types of intermolecular forces,

• Dipole-dipole interaction
• Ion-dipole force
• Ion-induced dipole force
• van der Waals interaction
• Hydrogen bonding

The dipole-dipole interaction is the strongest of all these forces and van der Waals is the weakest one.

What type of intermolecular forces does HBr have?

HBr contains dipole-dipole interaction and it can also make hydrogen bonding with more electronegative atom containing chemicals.

What type of intermolecular force holds ethanol molecules together?

Ethanol contains a hydroxyl group that has hydrogen atoms, making it able to make hydrogen bonds. So, hydrogen bonding is the force that holds the ethanol molecules together.

How do intermolecular forces work?

Intermolecular forces are the forces of attraction among molecules. As same charges repel and opposite ones attrcat, the oppositely charged ions or molecules create attractions called intermolecular forces.

Why are intermolecular forces so important?

To study the physical properties of chemical compounds like their melting and boiling points, enthalpies of fusion and vaporization, etc.

What are the intermolecular forces of HF?

Hydrogen fluoride has dipole-dipole interaction. These dipole dipole intractions in this case are internal hydrogen bonding.

Do ionic compounds have intermolecular forces?

Ionic compounds have dipole dipole interactions.

Can intermolecular forces be repulsive?

If the same poles come to contact with each other, repulsive occurs.

What is the difference between intermolecular and electrostatic forces?

Intermolecular forces are discussed in chemistry. These are the forces that hold the molecules together whereas electrostatic forces are dependent on the charges. The greater the charge, the higher the electrostatic forces.

What intermolecular forces are present between two molecules of CF3CF3?

Ethane has weak London dispersion forces (van der Waals forces).

What is the strongest intermolecular force in ammonia?

NH3 has hydrogen and highly electronegative atom nitrogen, so hydrogen bonding.

Do intermolecular forces influence viscosity?

Intermolecular forces are directly proportional to viscosity. Greater the intermolecular forces, greater is the viscosity.

Reference books

• Organic Chemistry (4th edition), Francis A. Carey, University of Virginia


text:
## A - Find Multiple 解説 by en_translator

If you are new to learning programming and do not know where to start, please try Problem A “Welcome to AtCoder” from practice contest. There you can find a sample code for each language.

Let $$X$$ be the quotient of $$B$$ divided by $$C$$, with the remainder rounded down, that is, $$X=\lfloor \frac{B}{C} \rfloor$$.

Let $$Y=X \times C$$.
Then $$Y$$ is the minimum multiple of $$C$$ not exceeding $$B$$.

Therefore, if $$Y \geq A$$, then $$Y$$ is an answer.
If $$Y \lt A$$, then the answer is $$-1$$.

A sample code in C++ follows.

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;

signed main(){
ll a,b,c;
cin>>a>>b>>c;
ll Y = b / c * c; // Y is a multiplt of C not exceeding B
if(a <= Y){
cout<<Y<<endl;
}else{
cout<<-1<<endl;
}
return 0;
}


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# Which of the following is the area of the region that

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Which of the following is the area of the region that  [#permalink]

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13 Aug 2019, 03:02
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Which of the following is the area of the region that includes all the points that satisfy the inequalities : X+Y-3<=0, X>=0, Y>=0?

1) 1.5
2) 2.5
3) 3
4) 4.5
5) 5

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Re: Which of the following is the area of the region that  [#permalink]

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13 Aug 2019, 03:35
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X+Y-3>=0 can be written as X+Y>= 3,

To find the area we can take 2 points that lie on line X+Y>=3
(0,3) and (3,0)
as we already know x>=0 and y>=0.

The line X+Y>=3 forms a triangle with x and y axis
area of triangle = $$\frac{1}{2}$$*b*h
Here B = 3 taking distance from (0,0) to (3,0)
H = 3, taking distance from (0,0) to (0,3)
=$$\frac{1}{2}$$*3*3 = 4.5

So, option D.
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Re: Which of the following is the area of the region that  [#permalink]

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14 Aug 2019, 07:46
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Basically it is asking to find the area of the triangle enclosed between the two axis & the st line - Y=-X+3

Point of intersection with X axis of this line is (3,0) & with Y axis (0,3)

So the Right triangle having base is 3 units & Height is 3 Units.

So area (1/2 * Base * Height) = 4.5

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Re: Which of the following is the area of the region that  [#permalink]

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14 Aug 2019, 08:54
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arunabh12345 wrote:
Which of the following is the area of the region that includes all the points that satisfy the inequalities : X+Y-3<=0, X>=0, Y>=0?

1) 1.5
2) 2.5
3) 3
4) 4.5
5) 5

Posted from my mobile device

These equations form a triangle with base 3 and height 3.
Area = 3*3/2 = 4.5

IMO D
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Email: kinshook.chaturvedi@gmail.com
Re: Which of the following is the area of the region that   [#permalink] 14 Aug 2019, 08:54


text:
# Polynomial Differential Equation

Calculus Level 3

If $f(x)$ is a polynomial such that $f(x) = f'(x) + \left(x+2\right)^2 \; ,$ what is $f(1)?$

Clarification: $f'(x)$ denotes the first derivative of the function $f(x)$.

×