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## What is Loop Quantum Gravity?

15 Mar

Loop Quantum Gravity (also known as Canonical Quantum General Relativity) is a quantization of General Relativity (GR) including its conventional matter coupling. It merges General Relativity and Quantum Mechanics without extra speculative assumptions (e.g., no extra-dimensions, just 4 dimensions; no strings; not assuming that space is formed by individual discrete points). LQG has no ambition to do unification of forces or to add more than 4 spacetime dimension, nor supersymmetry [1]. In this sense, LQG has a less ambitious research program than String Theory and is its biggest competitor.

General Relativity envisages spacetime and the gravitational field as the same entity, “spacetime” itself, that, in many ways, can be seen as a physical object analog to the electromagnetic field. Quantum Mechanics (QM) was formulated by means of an external time variable t like it appears in Schrödinger equation

$i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$

Fig. 1 below shows the solutions of SE when applied to the simplest atom in Nature, the hydrogen atom, where the potential is $V=\frac{1}{r}$. Electrons that rotate around the positive nucleus have the energy is quantified according to the law $\latex E_n=\frac{13.6}{n^2}$ and the wave function $\latex \Psi$ are given by the mathematical functions given in Figure 1.

However, in General Relativity (GR) this external time (represented above by the letter t) is incompatible because the role of time becomes dynamical in the framework of Minkowski spacetime. Time is no longer absolute (as Sir Isaac Newton once stated) but is relative to a frame of measurement. In addition, GR was formulated formerly by Albert Einstein in the framework of Riemannian geometry, where it is assumed that the metric is a smooth and deterministic dynamical field (Fig.2).

Fig.2 – Example of Riemann surface. Image courtesy: http://virtualmathmuseum.org (See for details about this specific surface here: http://virtualmathmuseum.org/Surface/riemann/riemann.html

This raises an immediate problem, since QM requires that any dynamical field be quantized, that is, be made of discrete quanta that follows probabilistic laws… This would mean that we should treat quanta of space and quanta of time…

All the known forces in the universe have been quantized, except gravity. The first approach to quantization of gravity consists of writing the gravitational field as composed of the sum of two terms, a background field $g_{background}$ and a perturbation $h(x)$. So, its full metric $g_{\mu \nu}$:

$g(x)=\eta_{background} (x)+ h(x)$

where $\eta_{\mu \nu}$ represents the background spacetime, normally Minkowski) and $h_{\mu \nu}$ a perturbation of the field (representing the graviton). The Minkowski space united space and time as a single entity introducing the new concept of space-time manifold where two points are distant by

$ds^2= c^2 dt^2-d \bf{x}^2$.

The problem resides in the intrinsic difficulty that this approach face when describing extreme astrophysical (near a black hole) or cosmological scenarios (Big Bang singularity). The inconsistency between GR and QM becomes more clear when looking at Einstein equation of GR:

$R_{\mu \nu} - \frac{1}{2}gR=\kappa T_{\mu \nu}(g)$

$R_{\mu \nu}$ is the Ricci curvature tensor, $R$ is the curvature, and $T_{\mu \nu}$ is the energy-momentum tensor. $\kappa \equiv \frac{8 \pi G}{c^4}$. While the left-hand side is described by a classical theory of fields, the right-hand side is described by the quantum theory of fields…

LQG avoids any background metric structure (described by the metric g), choosing a background independent approach, along the suggestion of Roger Penrose on the spin-networks where a system is supposed to be built of discrete “units” (anything from the system can be known on purely combinatorial principles) and all is purely relational (avoiding the use of space and time…) In GR spacetime is represented as a well-defined grid of lines, even if curved in the presence of a massive body, such as

In LQG, spacetime is represented rather as background-independent, the geometry is not fixed, is a spin network of points defined by field quantities and angular momentum, more like a mesh of polygons; spacetime is more a derived concept rather than a pre-structure, pre-concept on which events take place, as shown here

Image credit: http://www.timeone.ca/glossary/spin-network/

This new representation of fields has the advantage of representing both their intrinsic attributes but also their induction attributes. That is, the field quantities depend not only of the point where it exist but also on the neighboring points connected by a line. That’s why the mathematical idea that best express this representation is the holonomy of the gauge potential $A$ along the loop (line) $\alpha$, $U(A,\alpha)$, which is given by the integral

$U(A,\alpha)=exp{\int_0^{2\pi} ds A_a(\alpha(s))\frac{d\alpha^a(s)}{ds}}.$

REFERENCES:

[1] Carlo Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, 2004)

## Tribalistic Science – what it is?…

25 Jun

It is hard to believe, but Science is becoming a tribalistic activity, with hub inside groups of people with similar interests and that defend themselves and their publication like we were living in a doomsday era. Yes, we have a new type of science: the TRIBALISTIC SCIENCE. The index that institutions and governments use to measure the level of interestingness in tribalistic science are pseudo-indexes that measure:

• the number of publications, disregarding the number of authors (those working at CERN can publish 100 or more papers in one year, among hundreds of co-authors, a must for scientific career)
• the number of papers on the same subject, even if continued during a time life guarantee to the authors a place in the podium and possibly the Nobel Prize (this level is much harder to attain due to the related politics and involvement of governments and institutions, eager of prestige)
• the way authors in the same exact field cite each others, increasing exponentially the number of citations, disregarding the real usefulness of the work (some of them explicitly ask you to cite them for better impact on their institutions)
• …really don’t measure the real future impact of new ideas, their potential, or if the author is alone striving to invent or develop new concepts that may, in the future, or right now, contribute to the welfare of humanity.
• possibly, more significant items could be added, it’s up to you, reader.

This is my definition of the new type of science that is being done worldwide with huge success, even if destroying the public fate in science…

Since the Bomb exploded over Hiroshima, the prestige of science in the United States has mushroomed like na atomic cloud. In schools and colleges, more students than ever before are choosing some branch of science for their careers. Military budgets earmarked for scientific research have never been so fantastically huge. Books and magazines devoted to science are coming off the presses in greater numbers than at any previous time in history. Even in the realm of escape literature, science fiction threatens seriously to replace the detective story. – Martin Gardner, in Fads and Fallacies in the name of science.

References:

[2] – Mario J. Pinheiro, The Art of Academia Guerrilla – how Academia can help society to progress (ISBN-13: 978-1514370612 (CreateSpace-Assigned)

## The Laws of Causality and… Synchronicity

21 Jun

Human opinions are children’s toysHeraclitus

All matter originates and exists only by virtue of a force which brings the particle of an atom to vibration and hold this must minute solar system of the atom together. We must assume behind this force the existence of a consciousness and intelligent Mind. This Mind is the matrix of all matter. – Max Planck, Nobel Prize speech in 1944.

Oswald Spengler (1880-1936) gave a vivid account of the causal thinking that we, humans, are proud of.

With his hand, his weapon, and his personal thinking man became creative. All that animals do remains inside the limits of their genus-activity   and does not enrich life at all. Man, however, the creative animal, has spread   such a wealth of inventive thought There is already activity in the existence of the animals, but deeds begin only with Man.   Nothing is more enlightening in this connexion than the story of fire. Man   sees (cause and effect) how a fire starts, and so also do many of the beasts. But Man alone (end and means) thinks out a process   of starting it. No other act so impresses us with the sense of creation as this one. One of the most uncanny, violent, enigmatic phenomena   of Nature — lightning, forest fire, volcano — is henceforth called into   life by Man and action all over the world that he seems perfectly   entitled to call his brief history “world-history” and to regard his entourage as “humanity,”   with all the rest of Nature as a background, an object, and a means.The act of   the thinking hand we call a deed. himself, against Nature. What it must have been to man’s soul, that first   sight of a fire evoked by himself! – Oswald Spengler [1]

The discovery of fire is the first instance when humankind understood the cause and effect kind of law, and from that moment this law serve as a guide to us. Quite interestingly, fire was discovered by human species well before what previously was thought. Researchers recently discovered evidence of human use of fire dating back about 1 million years ago, in the Wonderwerk Cave, in South Africa, a massive cavern located near the edge of the Kalahari Desert.

Every problem contains its own solution, and it is part of human life to be repeatedly confronted with sources of difficulties that requires a solution. In fact, these situations are the ones that keep us alert, sharpens our senses and challenges our rational mind.

We live in a materialistic world bound by a narrow logic, and constrained “physical laws” that restrain the appealing of the mysterious in us.

The usual causal thinking follows a linear trail, where events A,B,C follow one after the other, C takes place because of B, and B is due to A. Jung hypothesized that causal effects have place with transmission of energy from cause to effect. However, there are some effects that apparently occur without Exchange of energy, and the event was called acausal by Carl Jung. A well-known physical effect representing an acausal effect is the Einstein-Podolsky-Rosen effect, which is also a non-local effect.

I am therefore using the general concept of synchronicity in the special sense of a coincidence in time of two or more causally unrelated events which have the same or similar meaning…Synchronicity therefore means the simultaneous occurrence of a certain psychic state with one or more external events which appears as meaningful parallels to the momentary subjective state-and, in certain cases, vice-versa. – Carl Jung, p. 251 Ref. [3].

Chinese philosophy thought occurrence of events differently from us. We usually ask: what causes this? Classic chinese texts ask: what likes to occur with what?…

But there is mysterious and inexplicable coincidences in our lives that we feel are full of meaning, although we don’t understand if we follow the stringent logic of reason, as teached by Aristotle and others in the West. These coincidences are what Carl Jung called “meaningful coincidences”, James Joyce’s “epiphanies”, and those that experienced them (and we all did) feel as they are occasions when a bridge are formed in order to connect the inner and outer worlds. We quote now the very funny text wrote by James Joyce and entitled “Stephen Hero” where he gave his definition of epiphany {FN1}:

He [Stephen Hero] was passing through Eccles’ St one evening, one misty evening, with all these thoughts dancing the dance of unrest in his brain when a trivial incident set him composing some ardent verses which he entitled a “villanelle of the Temptress.” A young lady was standing on the steps of one of those brown brick houses which seem the very incarnation of Irish paralysis. A young gentleman was leaning on the rusty railings of the area. Stephen as he passed on his quest heard the following fragment of colloquy out of which he received an impression keen enough to afflict his sensitiveness very severely.

The Young Lady-(drawling discreetly) … 0, yes … I was … at the … cha … pel …

The Young Gentleman- (inaudibly) … I … (again inaudibly) … I …

The Young Lady-(softly) … 0 … but you’re … ve … ry … wick … ed .

This trivialit*y made him think of collecting many such moments together in a book of epiphanies. By an epiphany he meant ‘ a sudden spiritual manifestation, whether in the vulgarity of speech or of gesture or in a memorable phase of the mind itself. He believed that it was for the man of letters to record these epiphanies with extreme care, seeing that they themselves are the most delicate and evanescent of moments. He told Cranly that the clock of the Ballast Office was capable of an epiphany. Cranly questioned the inscrutable dial of the Ballast Office with his no less inscrutable countenance:

-Yes, said Stephen. I will pass it time after time, allude to it, refer to it, catch a glimpse of it. It is only an item in the catalogue of Dublin’s street furniture. Then all at once I see it and I know at once what it is: epiphany.

-What?

-Imagine my glimpses at that clock as the gropings of a spiritual eye which seeks to adjust its vision to an exact focus. The moment the focus is reached the object is epiphanised. It is just in this epiphany that I find the third, the supreme quality of beauty.

-Yes? said Cranly absently.

What we see here, is the apparent lack of rational explanation in terms of the usual methods that are currently teached in school, in terms of causal links and connections that hide a fabric of underlying patterns where the mental and the material coincide in what seems to be connectors with potential to transform our lives. The transformative power of synchronicities led the outstanding physicist Wolfgang Pauli to explore the deep connections between the psyche and matter in the book “Resurrection of Spirit within the World of Matter” [2]{3}.

Why a great physicist, as Wolfgang Pauli was, become interested in Synchronicity and the role of the psychic life, may be found in the tragedy that attained him when his mother committed suicide, after knowing about his father’s infidelity. This tragedy accompanied Pauli all his entire life, and the series of dreams that so much have perturbed him, lead him to Carl Jung, and from the therapeutics sessions he had with Jung, it followed an intense collaboration and sharing of thoughts about these matters, that culminated in Jung’s book “Psychology and Alchemy”. One recurrent dream that Pauli had, called “The World Clock”. Pauli recurrently saw two clocks with a common axis, but one clock was at 90 degrees with other, connecting two different patterns of movement, and apparently showing a link between the real time and the complex time, since complex numbers can be represented since Euler, Wessel and D’Argand in a kind of Cartesian frame where the Ox axis represents real numbers and the Oy axis the complex numbers (see Complex Plane).

Wolfgang Pauli 45’s birthday.

Pauli’s mind was full of strange dreams that reflect his deep interest about all this world we are living in. He was deeply interest in a dimensionless constant of physics that is near the cabalistic number “137” (the fine-structure constant representing the coupling of light to matter). We may say here that Pauli was na outstanding physicist, maybe greater than Albert Einstein, but instead to be willing at the front with other succesful physicists, he choose to stay behind, thinking and giving ideas to others. He died in a hospital room number 137…

Another strange dream that was reported about Pauli involves a Persian figure. Pauli asks: “Are you my shadow?” The Persian replies: “I am between you and the light, hence you are my shadow. Not the other way around.” Pauli: “Are you studying physics?” The Persian: “Your language thereof is too difficult for me, but in my language you would not understand physics.”{4}

As David Peat wrote in his inspiring book “Synchronicity: the Bridge between Matter and Mind and the Resurrection of Spirit in the World” [4]:

The universe was perceived not so much in terms of separately distinct objects connected by forces but through sympathies, influences, humors, resonances and patterns that belong together. It was not that movements of the planets causally influenced events on earth, but that an essential harmony was maintained between the patterns of heaven and earth. Within such a world-view, synchronicity is perfectly natural.

The concept of Causal relationships implies a linear concept of time, introduced with the rise of banking and commerce, when lending money implies accumulating interest. With the Renaissance, all other concepts of time (including the idea of a circular time, the eternal return) were definitively banned, and replaced with a “time” used for prediction, control, accumulation rates, and wealth. With this time it followed the concept of biological evolution (Krishnamurty and David Bohm they don’t believe on this concept, see important interview), and the general idea that pervades our societies that linear time lead us all to progress.

It is progressively clear that Science is not serving humanity the way it should. Since the Atomic Bombs and the horror provoked in Hiroshima and Nagasaki, the Cold War and its sequels, the use of science (including fractal geometries) to finance that ultimately led us to the bankruptcy, and the end of the dream, the use of technology to submit man to powerful people who the only good they seek is money and power, is not a good asset for our future. Scientists must depart in a new journey, developing a new Science where the soul and qualities are treated  equally, finishing with the general mechanistic view, and thinking about what kind of Science is good for all of us. Otherwise, Science risks to become a real danger, or at minimum no more an interesting subject with important outcomes for Society (for example, David Bohm and even Einstein, they wouldn’t choose the same profession, if reborn). In a way, I believe that a new Era for humanity is already beginning, scientists (including me) are committed to explore new paths, aiming to serve humanity, helping us to more deeply what means to be a human being, and for what purpose we are here on this little planet.

We recall here Pauli, again, as reported in his autobiography by the brilliant Dutch theoretician H.B.G. Casimir who discover an important proof of the existence of the vacuum as having an internal structure: “We are living in curious times. Christianity has lost its grip on humanity. Other times should come. I think that I know, what will come. I know it quite certain. But I won’t tell anyone, because otherwise they will think that I am crazy.”

REFERENCES:

[5] H.B.G. Casimir, Haphazard Reality: Half a Century of Science, 1984, Harper Collins.

SITES:

## Leo Szilard and the Foundation Mark Gable

16 Sep

“Gloire et louange à toi, Satan, dans les hauteurs

Du Ciel, où tu régnas, et dans les profondeurs

De l’Enfer, où, vaincu, tu rêves en silence!

Fais que mon âme un jour, sous l’Arbre de Science,

Près de toi se repose, à l’heure où sur ton front

Comme un Temple nouveau ses rameaux s’épandront!

Charles Baudelaire, in Les Fleurs du Mal [1]

Science is in danger of survival. Technology and Burocracy took its place. How to create anew the conditions that give birth to the wonderful discoveries of the last century that endure until the 1950’s with the discovery of the laser? This is an actual and fundamental problem that strikes our civilization.

“Only daring have made the main contributions  to Science. Notwithstanding this, the contemporary heavy administration in the
system of modern scientific education and scientific research still suppresses  everyone who wishes to develop new and productive ideas. That is why the  freedom  of scientific work and the free initiative of original studies should  undeniably be defended, because only these factors were and will be the most  productive sources of great progress of Science”—Louis de Broglie, Necessity of  Freedom of Scientific Work, Annales de la Fondation Louis de Broglie, vol. 4, nº  1, p.62 (1979)

But here we are talking about a singular man of science. Leo Szilard, the great physicist of Hungarian ascent after Hiroshima left physics to dedicate his interests to biology and science-fiction…Szilard was born in Hungary in 1898 and come to Berlin in 1919 to study with Albert Einstein, Max von Laue and become a friend with Eugen Wigner. In 1938 he went to the United States, rapidly recognizing the evil nature of nazism. In Chicago, he worked with Enrico Fermi at the Metallurgical Laboratory in the framework of the Manhattan Project. At this laboratory, Leo Szilard developed the concept of nuclear chain reaction into a nuclear  reactor, and he patented the idea together with another great physicist Enrico Fermi (this kind of people are disappearing in our societies unfortunately, there is no place for them), see their patent [1a].

At a certain moment, Szilard felt the need to eliminate the sinister side of his nuclear reactor and atomic bomb, not being pleased with the political turn of the Manhattan Project.

What was his idea about Big Science? «It was not the kind of physics I like, and I even wonder if it is physics at all». In a  series of tales, Szilard, describes with humor a meeting with a millionaire, Mark Gable, that enriched with a bank of sperm. We refer now to a part of the tale appearing in Leo Szilard book “The voice of the dolphins and other stories” [2].

“Would you intend to do anything for the advancement of science?” I asked.

“No”, Mark Gable said. “I believe scientific progress is too fast as it is.”

“That I would very much like to do,” Mark Gable said, “but how do I go about it?”

“Well,” I said, “I think that shouldn’t be very difficult. As a matter of fact, I think it would be quite easy. You could set up a foundation, with an annual endowment of thirty million dollars. Research workers in need of funds could apply for grants, if they could mail out a convincing case. Have ten committees, each committee, each composed of twelve scientists, appointed to pass on these applications. Take the most active scientists out of the laboratory and make them members of these committees. And the very best men in the field should be appointed as chairman at salaries of fifty thousand dollars each. Also have about twenty prizes of one hundred thousand dollars each for the best scientific papers of the year. This is just about all you would have to do. Your lawyers could easily prepare a charter for the foundation. As a matter of fact, any of the National Science Foundation bills which were introduced in the Seventy-ninth and Eightieth Congress could perfectly well serve as a model.”

This question is of huge importance nowadays and must be taken seriously by all of us. What kind of civilization are we leaving to our sons? [3]

REFERENCES:

[2] Leo Szilard, The voice of Dolphins and Other Stories

## The Art of Scientific Illustration

21 Apr

One Picture is Worth Ten Thousand Words – Confucius.

Create, as if your life depends upon it – Jessie Shaw

There is hardly a more familiar artifact of modern life than the so-called scientific illustrations. That is, the diagram or picture in isometric or linear perspective with notations for scale and measurement which show how machines or houses or even human beings are put together and taken apart and how they work. Who, indeed, has never depended on such an illustration for assembling a Christmas bicycle or a Sears & Roebuch porch swing (not to mention for constructing an atomic reactor or preparing for open heart surgery)? So taken for granted is the ubiquitous scientific illustration that few scholars have ever sensed that it has any historical interest. – Samuel Y. Edgerton, Jr. [1]

Leonardo’s drawing of an ornithopter. He was inspired by the observation of birds flying, and this drawing is considered the first scientific illustration. Image credit: NASA/Photo Researchers, Inc.

The art of scientific drawing is an irreplaceable method for the better apprehension of ideas and a way to prepare the ground for new discoveries. All along the history of science, we know of great genius that relied on drawing to discover or better express his/her ideas.

The first men of science illustrating their writings were Leonardo da Vinci (see the ornithopter), Francis Bacon, Galileo, William Harvey, Descartes.

Leonardo was very eager to keep his secrets, not wanting they fall into the wrong hands. For this motive, he had a preferred left-hand writing in his notebooks (called ‘mirror writing’), while employing his hand-writing in conventional communication (in his letters). Movement specialist Grant Ramey [2] sustains that Da Vinci uses ‘mirror writing’ because he was passionate by symmetry and the human form in art and science. Apparently, Da Vinci wrote in his notebooks from right to left, with his left hand, in order to keep thinking (instead of to remain focused on his own writing), see here.

A scientific illustration is an important form of art, intending at the same time accurately transmit scientific knowledge. Interestingly, Goethe is quoted to have said that you really do not see a plant until you actually draw it…

Why we should start to draw figures in our intellectual and aesthetic activities? In order to understand the power of drawing, let us start to quote here another great man, Thomas H. Huxley, since it is with them that we learn: «[…] I should, in the first place, secure that training of the young in reading and writing, and in the habit of attention and observation, both to that which is told them, and that which they see, which everybody agrees to. But in addition to that, I should make it absolutely necessary for everybody, for a longer or shorter period, to learn to draw.»

In Meno, famous dialogue between Socrates and one of Meno’s slave (a boy), see Ref.[3] to read the complete dialogue, it is clear that by drawing and with the right questions, we may “recall” the knowledge we have in our minds. This famous dialogue depicts the problem of teaching science, in fact, a very old topic in philosophy of science, the problem of the “tacit knowledge” that we all may eventually possess.

We quote next to a short part of this important dialogue, led by Socrates while drawing on the ground.

«Meno: Yes, Socrates. But what do you mean by this, that we do not learn and what is called learning is recollection? Can you teach me that this is so? […]

Meno: Certainly. Step forward here.

Socrates: Now, is he Greek and speaks Greek?

Meno: Absolutely. He was born in the house.

Socrates: Then pay close attention to see whether he seems to recollect or to be learning from me.

M: I certainly will.

So: Tell me, boy, do you know that a square is like this? [Socrates draw a square on the ground, see 1]

Slave: I do.

So: And so a square has these lines, four of them, all equal? [see 2]

Slave: Of course.

So: And these ones going through the center are also equal? [see 3]

Slave: Yes.

So: And so there would be larger and smaller versions of this area? [see 4]

Slave: Certainly.

So: Now, if this side were two feet and this side two feet also, how many feet would the whole be? Look at it like this: if this one were two feet

but this one only one foot, wouldn’t the area have to be two feet taken once? [see 5]

Slave: Yes.

So: When this one is also two feet, there would be twice two?

Slave: There would.

So: An area of twice two feet?

Slave: Yes.

So: How much are twice two feet? Calculate and tell me. [see 6]

Slave: Four, Socrates. [see 7]

So: Couldn’t there be one different from this, doubled, but of the same kind, with all the lines equal, as in that one? [see 8]

Slave: Yes.

So: And how many feet in the area?

Slave: Eight.

So: Now could one draw another figure double the size of this, but similar, that is with all its sides equal like this one? [see 9]

Slave: Yes.

So: How many feet will its area be?

Slave: Eight.

So: Now then, try to tell me how long each of its sides will be. The present figure has a side of two feet. What will be the side

of the double-sized one?

Slave: It will be double, Socrates, obviously.

So: You see, Meno, that I am not teaching him anything, only asking. Now he thinks he knows the length of the side of the

eight-feet square.

MENO: Yes.

So: But does he?

Meno: Certainly not.

So: He thinks it is twice the length of the other.

MENO: Yes.

So: Now watch how he recollects things in order — the proper way to recollect.

Archimedes was also known to write in whatever surfaces he had at hand, on the sawdust-covered floors, on the sand, drawing geometric shapes on the extinguished fires. That ‘s why the majority of Archimedes drawings are forever lost. He spent hours, sited on the floor, like most geometers at his time did, since it was too expensive to scribbled on a papyrus and then thrown it away.

Galileo Galilei with the help of his telescope (invented by him) also made the first drawing of the moon. The consequences were controversial since the Catholic Church saw in his drawings, his sketches of the moon, irregular surface, full of craters, a proof that the heavenly bodies were not perfect, as supposed before. But the dialectical fight between science and religion was just beginning {5}. the old war between the Catholic Church and science made the popes suspicious of the scientific findings and induce them to create the Vatican Observatory with headquarters at the papal summer residence in Castel Gandolfo, Italy, outside Rome. Quite surprisingly, they also have a research center, the Vatican Observatory Research Group, hosted by Steward Observatory at the University of Arizona, Tucson, USA. Located at the Mount Graham International Observatory, in southeastern Arizona, the Vatican possesses the 1.8m Alice P. Lennon Telescope with its Thomas J. Bannan Astrophysics Facility, known together as the Vatican Advanced Technology Telescope. Vatican astronomers said recently that it is okay that people believe in ET’s [5].

Hiero II calling Archimedes to fortify Syracuse. Archimedes was considered at the time a great mind in matters of military strategy. Painting by Sebastiano Ricci. Image credit: http://www.artfund.org

Galileo first drawing of the Moon.

Remark that by drawing you may understand the Pythagoras theorem (see also here).

We must not lose sight that analytical equations represent spatial structures. Our mind must deal with this “hidden” aspect of the mathematical formalism. This is most important for people working in visual science, like computers programming {2}.

Researches were done by Professor Shaaron Ainsworth of the University of Nottingham’s School of Psychology, and colleagues from La Trobe and Deakin Universities in Australia, have shown that students learn better when they are endowed to draw, a method which helps students in visualizing abstract concepts, to recall and to more easily engage in communicating with each other. Teachers at school should endeavor to teach and encourage students to draw what they have learned since this is a powerful method to apprehend any subject, and a powerful process when aiming to transmit ideas to other people {4}. According to Horst Bredekamp [6], it was the ability to draw shown by Galileo that allowed him to better understand Nature; due to their artistic abilities, he could see better than others not gifted in the arts of illustration.

REFERENCES:

SITES:

{1} The Craft of Scientific Illustrations [contains important advice on how to draw a scientific illustration] []

{2} The importance of drawing

{3} Drawing pictures key to learn science

{5} Catholic Church and Science

## Superconductivity and its applications

15 Mar

“If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is that…all things are made of atoms.”
Richard P. Feynman, winner of the 1965 Nobel Prize in Physics

PART 1

History

The idea of the absolute zero of temperature was advanced by J. A. C. Charles and J. L. Gay-Lussac (Fig.1). In fact, they have shown that by extrapolating their measurements of the volume of any gas as a function of the temperature (at constant pressure), their volume would tends toward a zero value at a temperature near -273 degree Celsius (see fig.2).

Gay-Lussac. Image credits: chemistryland.com

By extrapolation of the curve of volume versus temperature, Charles and Gay-Lussac predicted the existence of the absolute zero. Im. credit: chempaths.chemeddl.org

In 1911, Kamerlingh Onnes discovered that mercury suddenly lose its resisitivity  when the temperature dropped to -260 degree Celsius. Omnes started investigating the field of low-temperature physics in 1908, when he succeeded in liquifying helium. Whith these findings we may say that the field of superconductivity is open, bringing to us the compreension of astounding phenomena, characterized by the manifestation of the wave nature of particles at a macroscopic scale.

It is not always fundamental to reduce friction to put the body at zero degrees Kelvin. In fact, russian scientists succeeded that achievement with a simple experiment. They launched a sphere of steel on a surface made of molybdenium and sulphur, while at the same time thay beam a cloud of electrons on the surface. They observe a strong reduction of friction from a coefficient 0.9 to 0.0015!

Fundamental Properties of Superconductors
superconducting or superfluid phase is considered a different thermodynamical phase when compared to the normal state that exists when the temperature T>T_c, where T_c means the critical temperature below which the superconducting or superfuid state appears. For example, liquid 4He under its vapor pressure becomes superfluid at T_c=2.17 K. So, the passage from one state to the other is called a phase transition, and it is accompanied by a strong increase of the specifi heat when the temperature is near T_c, with a strong release of entropy. This phenomena, experimentally seen, leads to the formation of a more ordered state of matter (see also precedent video). Fundamentally, a superconductor is a conductor that has undergone a phase transition to a lower energy state below the critical temperature T_c, characterized by the appearance of groups of paired electrons, the so called Cooper pairs, carrying electrical current without any resistance and responsible, among other properties, of perfect diamagnetism.

We may distinguish two types of superconductors. A Type I supercondcutor, is characterized by the following properties [1]:

• zero electrical resistance and perfect diamagnetism at a temperature below T_c. Normally, at temperatures above T_c this material is a normal metal, although not a very good conductor;
• perfect diamagnetism, also called Meissner effect, the magnetic field stays outside the material, cann’t penetrate the material. Curiosly enough, if you appply an external magnetic field  B_app above a given critial magnetic field, B_c, the material suffers a transition from superconductor to normal state. An approximate functional dependence on temperature for this critical magnetic field is given by

And what is the diamagnetic property of matter? This property  is a kind of negative magnetism. This effect was studied in the framework of  classical mechanical by Paul Langevin in 1905 [2], using previous and revolutionary ideas proposed formerly by André-Marie Ampère[3]  and Wilhelm Weber [4] (nowadays we barely talk about these two great men of science that really use their minds for the advancement of science and the progress of mankind…). Langevin found, in the classic framework provided by Ampère and Weber, that N electrons moving in orbits around the nucleus at an average distance <r²>, such that the (constant and negative) magnetic susceptibility χ is given by

The magnetic susceptibility is the ratio of M/H, where M is the magnetization field and H the magnetic field. The susceptibility χ is slightly negative for diamagnets, but acquires small positive values for paramagnetic substances (e.g., ), and is strongly positive for ferromagnetics substances (e.g., Fe). It can be shown that a material constitute of paramagnetic ions with magnetic moment μ, obeys the Curie-Weiss law:

where n means the concentration of paramagnetic ions, Θ is the Curie-Weiss constant () . when perfect diamagnetism is achieved, χ=-1, that is, the magnetization M is directed opposite to the H field, cancelling it, M=-H. For example, when a superconductor with spherical form is placed nearby the poles of a magnet, it results a superposition of the applied magnetic field B_app, and the resulting dipole field (Fig.1a) , giving a curvature of the magnetic field lines of the form as shown in Fig.1b.

The dipole result when a uniform permanent magnetization M is parallel to the axis Oz (see Section 5.10 in the textbook of J. D. Jackson [4]).

Besides perfect diamagnetism, the other important property of the superconducting state is its zero resistance. In ideal conditions, an electric current established in a loop of supercuncting wire will last indefenitely. The surface resistance of the material with a current flowing along a film of thickness d, must satisfy the condition

where ρ is the electric resistivity, h is the Planck constant and e is the absolute charge of the electron.

Methods of Quantum field Theory

In quantum field theory (QFT), the Lagrangian plays a central role in working out mathematically the physical problems and from which the dynamic field equations are generated. In the Hamiltonian formulation of classical mechanics, the equations of motion of a system of particles can be obtained from a special function called the Action. Usually, the Lagrangian is dependent on the positions and velocities of the particles:

Hamilton’s minimum action principle states that Nature prefer to follow movements that  extremize this quantity called action.

In QFT, this formulation can be translated, replacing que classical quantities by a “field”, in the manner of the Table 1 below:

In field theory, the fields φ(x,t) play the role of the generalized coordinates, {qi(t)}, where the discrete index i represents the number of discrete coordinates of the system. In local field theories the Lagrangian may be written as an integral over another function called the Lagrangian density, L, such as

depending on the set of a possible fields present on the given point of space and their first derivatives:

When seeking to extremize the functional called Action, it is obtained the so called Euler-Lagrange equations for any field:The Fig. below represents several possible “trajectories”, or “stories” associated to a given dynamical field. But Nature prefer only one of them, the one “story” which is represented by the above Euler-Lagrange equations…

As written by Pierre Fermat in a letter of 1662 to M. de la Chambre:

Natura operatur per modos faciliores et expeditiones (Nature works by the easiest and readiest means).

Various possible "stories" that a physical system may undergo. However, Nature prefer the one that extremize the Action.

Maupertius argued that the principle of least action (was the first to enunciate it), showed the wisdom of the Creator. Maupertuis believed that the vis viva (which today is twice the kinetic energy) should be minimal.}: “ The action is proportional to the product of mass and velocity and space. Now here this principle, so wise, so worthy of the Supreme Being: once a change occurs in nature, the amount of action employed for this variation is always as small as possible. ” Pierre Louis Moreau de Maupertuis was a French mathematician and astronomer. Born July 7, 1698 in Saint-Malo and died in the Bale July 27, 1759. Interestingly, he was the son of a pirate.

Let us take as an example (although for non-mathematicians, a little bit too complex problem that can however be grasped as an example of the methods in QFT), the Lagrangian of a physical system composed by matter + electromagnetic field, such as described below, and we will recover essential properties of the superconductor state.
matter field                                                                             EM field                   Current

PART 2.

To follow

REFERENCES:

[1] Handbook of Superconductivity, Charles P. Poole, Jr. (Academic Press, San Diego, 2000)

[2] Paul Langevin

[FN1] Annales de Chimie et Physique, 5 (1905); see paper by Paul Langevin at p.70-127

________________________________________________________________________________

PART 2 – To follow soon…

## Pataphysics: the beginning of a new science…

13 Feb

All mankind have an instinctive desire of knowledge. This illustrated by our enjoyment of our sense-perceptions. – Aristotles, in his first book on Methaphysics.

“Blind and unwavering undisciplined at all times constitutes the real strength of all free men.” – Alfred Jarry (1873 to 1907)

“Duration is the transformation of a succession into a reversion. In other words: THE BECOMING OF A MEMORY.” – Alfred Jarry (1873 to 1907) from How to Construct a Time Machine

“God is the tangential point between zero and infinity.” – Alfred Jarry (1873 to 1907)

Aristotles wrote fourteen books on the ultimate conceptions of philosophy. In fact, he entitled the books with the word “the lectures that come after the lectures on Physics”  (the lectures on Physics are composed by eight books). He dedicated his thoughts about being, what is a being in regard to its existence, independently of any secondary quality that may qualify it. The name “Metaphysics” was given by the editor of his works on first philosophy, because they appear after the researches done on the physical world.

Aristotle school at Athens, in a fresco painted by Rafael, located at the Apostolic Palace, Vatican city. Image credit: wikipedia

Aristotles understood that just searching around the physical world would not give to him any clue about the universe and the being, about the essential attributes of the existence. Mathematics and Physics, only treat the laws of the universe based on ideal assumptions: line, points, plane, space, time. But the meaning of them are not enough to approach a functional understanding of life, its purpose, and the marvellous meaning of Being and Time. One wonderful example of thinking can be witnessed in this video with Martin Heidegger talking about his understanding of what is a being and our relationship with it.

The problem of being is not only philosophical, it should be also a concern to physicists (how matter is created locally, how the universe appeared apparently from nothing, from a Big Bang?…). This is well stated by William James: “How comes the world to be here at all instead of the nonentity which might be imagined in its place? … from nothing to being there is no logical bridge.”

But nobody could imagined that, later, Alfred Jarry will invent a new area of though called Pataphysics (a contraction of the pseudo-Greek term τὰ ἐπὶ τὰ μετὰφυσικά (ta epi ta metaphusika – “that which is above metaphysics“). Jarry intent with this denomination to suggest a humorous variation of Aristotle’s work “Metaphysics”, and so Pataphysics means “that which is above that which is after physics”.

Despite the disaster that represented the opening of the piece Ubo Roi [2], and the shocking first sentence uttered , “merdre” (watch the video, in French), together with the scene of boxing that followed in the audience, there is an interesting tentative of conceptualization of a phenomenon rarely spoken and even less studied. The Pataphysics is actually a critique of science and an attempt “to subvert the procrusteans constraints of science” [3]. It is a critique of the society as a all, turned to useless purposes, plain of vanity, and with grandiloquent speeches, but with its “scientific effort” bringing no results with interest to the humankind. It is a critique of “science” as is done nowadays, searching for “reports” that end on the managers archives, useless, a “science” that is not concerned with  understanding Nature, nor the meaning of being (and its enormous importance to our achievement as human beings), but a “science” that serves a nomenclature that uses it for self-promotion, as a living commerce (science sells).

In a dellusional work, criticising the scientific establishment, tempted by the big shows and impressing the public, fainting to be an astrophysicist or some specialist alike, he wrote “How to build a time machine”. Another work full of genial veine is “Exploits and Opinions of Dr. Faustroll, Pataphysician” [5].

Pataphysics will examine the laws governing exceptions, and will explain the universe supplementary to this one; or, less ambitiously, will describe a universe which can be – and perhaps should be – envisaged in the place of the traditional one,  since the laws that are supposed to have been discovered in the traditional universe are also correlations of exceptions, albeit more frequent ones, but in any case accidental data which, rediced to the status of unexcpetional exceptions, possess no longer even the virtue of originality. – Alfred Jarry, in Ref.[5], p. 854.

It is not a surprise to see that, in this era of fake reality, where things appear to be serious due to marketing and the fragility of people minds, there is a growing “[…]body of art that adopts the language and trappings of officialdom. Examples include Los Angeles’s Center for Land Use Interpretation, the Center for Tactical Magic from Oakland, Calif., and, locally, the Institute for Infinitely Small Things, the National Bitter Melon Council, and the Institute for Applied Autonomy.”[Blog1].

These social/cultural phenomena are a reaction from the society to the bad use of science, or better, of the despise with which real science is treated by governments and “managers” looking for numbers, statistics, statistics of things they don’t fully grasp, statistics that only aim for the immediate use of technology and huge profits for corporations, instead to be concerned with the preparation of minds (particularly the young generation), and the quest of the fundamental laws of the universe, that might help humanity escape from war, poverty and delusion. That’s why the great French philosopher, Jean Braudillard, called Pataphysics as the imaginary science of excess of emptiness and insignificance [4]. Clearly this new kind of non-philosophy, is anti-establishment without being truly political.

Two asteroids crossed the space between earth and moon recently. It could be too late to avoid disaster...Im. credit:http://www.space.com

It is but due to the benevolence of gods that we are still here, writing on the Web, reading, thinking, and fighting for a better and a more dignified world, since otherwise we are not really protected from cosmic disasters, as it remains clear from the recent event of two asteroids that crossed the space between earth and the moon. It could be too late to avoid disaster. Because science is not being taken seriously, as it should be.

References:

[5] Exploits and Opinions od Dr. Faustroll, Pathaphysician, in Poems for the Millenium: the University of California of Romantic and Post-Romantic Poetry, Vol. III

Other Blogs:

[Blog1] Cabinet of Wonders

Museums:

Institutes:


text:
# Ideal gas law problem with two cylinders

• Chemistry
DottZakapa
Homework Statement:
There are two cylinders A and B with the same capacity and at the same temperature. If the mass of hydrogen contained in cylinder A is double the mass of helium contained in cylinder B. How does the pressure vary? (Consider the ideal gases)
Relevant Equations:
ideal gas law
my answer will be ##P_1=2 P_2## but I have some doubts, if that is correct or not

Mentor
That would be correct if both cylinders were filled with the same gas.

DottZakapa
ok so the reasoning will be:
Concerning H, in the first container, the moles will be given by ## \frac {2 *He (g)} {\text{molar mass of} H_2 \text{(because comes as diatomic molecule} } ##
that is :

## \frac {2*He (g)}{2\frac {g} {mol} } = He\text{ mol } \rightarrow P_1=\frac {He \text{ mol} * R*T}{V} ##

Concerning the second, which contains He :

## \frac {He (g)}{4\frac {g} {mol} } = \frac {He}{4} mol \rightarrow P_2=\frac {He \text{ mol} * R*T}{4V} \rightarrow 4P_2=\frac {He \text{ mol} * R*T}{V}##

concluding, merging everything :

## P_1=4*P_2##

is it how it should be managed ?

Homework Helper
The answer is correct, but your working is a bit confusing. Is He a variable equal to the number of grams of helium in cylinder 2? Then it is correct that there are He moles of hydrogen in cylinder 1; but the notation is confusing. "He mol" could easily be understood as "the number of moles of He" rather than, as I think is meant, " a number of moles equal to the number of grams of He". Then the pressures would be out by a factor of 4, but the ratio would still be correct. It would be better to use e.g. mHe for the mass of helium - or a variable like x; mass of helium = x g. Then P1 = xRT/V and P2 = xRT/4V.

DottZakapa
The answer is correct, but your working is a bit confusing. Is He a variable equal to the number of grams of helium in cylinder 2? Then it is correct that there are He moles of hydrogen in cylinder 1; but the notation is confusing. "He mol" could easily be understood as "the number of moles of He" rather than, as I think is meant, " a number of moles equal to the number of grams of He". Then the pressures would be out by a factor of 4, but the ratio would still be correct. It would be better to use e.g. mHe for the mass of helium - or a variable like x; mass of helium = x g. Then P1 = xRT/V and P2 = xRT/4V.
The problem says:
"the mass of hydrogen contained in cylinder A is double the mass of helium contained in cylinder B"
so, if :
PV=nRT
where n are the moles of gas.
If n is equal to the mass of the gas divided by the Molecular Mass of the gas, then, knowing that in cylinder A the mass of Hydrogen is twice the mass of Helium, in the above mentioned mass of the gas I should write twice the mass of Helium (g)divided by the Molecular mass of Hydrogen (## \frac g {mol}##).
that is :
##n_a= \frac {H (g)}{H \frac g {mol}}= \frac {2He (g)}{H\frac {g} {mol} } = \frac {2He(g)}{2 \frac g {mol}} ##
This is for cylinder A.
For cylinder B always mass of gas (Helium) divided by molecular mass of Helium.
##n_b= \frac {He (g)}{He \frac g {mol}}= \frac {He (g)}{4\frac {g} {mol} } = \frac {He(g)}{4\frac g {mol}} ##
correct?

Homework Helper
Your method is correct. It is your notation that I'm concerned about. I don't like "He (g)" for "the mass of helium in grams", because it's potentially confusing, especially when you go on to write "He mol", not meaning "the number of moles of helium". Further confusion is caused by writing H when you should write H2. The molar mass of H is 1 g/mol, not 2. But you don't have H, you have H2. So your calculation is right, but the way you write it is confusing.

DottZakapa
Your method is correct. It is your notation that I'm concerned about. I don't like "He (g)" for "the mass of helium in grams", because it's potentially confusing, especially when you go on to write "He mol", not meaning "the number of moles of helium". Further confusion is caused by writing H when you should write H2. The molar mass of H is 1 g/mol, not 2. But you don't have H, you have H2. So your calculation is right, but the way you write it is confusing.
it was an MCQ, but, just in case I will have it in a written test, would you Kindly explain how should I write? Yes in H it was a typo.

Homework Helper
Let mass of helium be x g; MW of He = 4. No. of moles helium = x/4. PB = nBRT/V = xRT/4V.
Mass of hydrogen is twice mass of helium = 2x g. MW of H2 = 2. No. of moles hydrogen = 2x/2 =x.
PA = nART/V = xRT/V = 4PB.

DottZakapa
DottZakapa
Let mass of helium be x g; MW of He = 4. No. of moles helium = x/4. PB = nBRT/V = xRT/4V.
Mass of hydrogen is twice mass of helium = 2x g. MW of H2 = 2. No. of moles hydrogen = 2x/2 =x.
PA = nART/V = xRT/V = 4PB.
thank you :)

format: plain


text:
# Volume of Void

Written by Jerry Ratzlaff on . Posted in Geotechnical Engineering

Volume of void, abbreviated as V_v, is the ratio of total volume of soil minus volume of solid soil.

## Volume of Void formulas

 $$\large{ V_v = V_t - V_s }$$ $$\large{ V_v = V_w + V_a }$$ $$\large{ V_v = \frac{ e \; V_t }{ 1 \;+\; e } }$$

### Where:

 Units English Metric $$\large{ V_v }$$ = volume of void $$\large{ft^3}$$ $$\large{m^3}$$ $$\large{ V_t }$$ = total volume of soil $$\large{ft^3}$$ $$\large{m^3}$$ $$\large{ e }$$ = void ratio $$\large{dimensionless}$$ $$\large{ V_a }$$ = volume of air $$\large{ft^3}$$ $$\large{m^3}$$ $$\large{ V_s }$$ = volume of solid soil $$\large{ft^3}$$ $$\large{m^3}$$ $$\large{ V_w }$$ = volume of water $$\large{ft^3}$$ $$\large{m^3}$$

## Related Volume of Void formulas

 $$\large{ V_v = n \; V_t }$$ (porosity) $$\large{ V_v = e \; V_s }$$ (void ratio)

### Where:

$$\large{ V_v }$$ = volume of void

$$\large{ n }$$ = porosity

$$\large{ V_t }$$ = total volume of soil

$$\large{ e }$$ = void ratio

$$\large{ V_s }$$ = volume of solid soil


text:
# Symbol: “maps isomorphically” [duplicate]

I want to make a symbol for "maps isomorphically" which consists of a relatively large tilde above a right arrow. Overset doesn't work (besides, the tilde would be too small). MWE:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
Thus $X$ is isomorphic to $Y$ via $f$:
$f : X \overset{\tilde}{\to} Y.$
\end{document}

I also tried using array but that didn't work either. MWE:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
Thus $X$ is isomorphic to $Y$ via $f$:
$f: X \begin{array}{c}\tilde \\ \to \end{array} Y$

If I could get this to work, the spacing would need to be adjusted significantly.

The \xrightarrow should work as pointed out by Alex. Here is how to implement this:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
{\tiny
Thus $X$ is isomorphic to $Y$ via $f$:
$f : X \xrightarrow{\sim} Y.$
}
{\Large
Thus $X$ is isomorphic to $Y$ via $f$:
$f : X \xrightarrow{\sim} Y.$
}
\end{document}

The output is shown for two different font sizes:

• It's not perfect though. There is a small visual "glitch" which is also visible in the screenshot. – paldepind Nov 7 '18 at 11:26


text:
## Taiwanese Journal of Mathematics

### MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS IN UNBOUNDED CYLINDER DOMAINS

#### Abstract

In this paper, we show that if $Q(x)$ satisfies some suitable conditions, then the quasilinear elliptic Dirichlet problem $-\Delta_p u + |u|^{p-2} u = Q(x)|u|^{q-2}u$ in an unbounded cylinder domain $\Omega$ has at least two solutions in which one is a positive ground state solution and the other is a nodal solution.

#### Article information

Source
Taiwanese J. Math., Volume 16, Number 2 (2012), 409-428.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406593

Digital Object Identifier
doi:10.11650/twjm/1500406593

Mathematical Reviews number (MathSciNet)
MR2892890

Zentralblatt MATH identifier
1247.35022

#### Citation

Hsu, Tsing-San; Lin, Huei-Li. MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS IN UNBOUNDED CYLINDER DOMAINS. Taiwanese J. Math. 16 (2012), no. 2, 409--428. doi:10.11650/twjm/1500406593. https://projecteuclid.org/euclid.twjm/1500406593

#### References

• C. O. Alves, P. C. Carri$\rm\tilde{a}$o and E. S. Medeiros, Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions, Abstr. Appl. Anal., 3 (2004), 251-268.
• C. O. Alves, Existence of positive solutions for a problem with lack of compactness involving the $p$-Laplacian, Nonlinear Analysis T.M.A., 51 (2002), 1187-1206.
• \label1 A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. I. H. P. Analyse Non Lineaire, 14 (1997), 365-413.
• \label2 A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\Bbb R^N$, Revi. Mate. Iber., 6 (1990), 1-15.
• V. Benci and G. Cerami, Positive solution of semilinear elliptic equation in exterior domains, Arch. Rat. Mech. Anal., 99 (1987), 283-300.
• \label3 H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I, II, Arch. Rati. Mech. Anal., 82 (1983), 313-376.
• \label4 W. Ding and W. M. Ni, On the existence of positive entire solutions of semilinear elliptic equations, Arch. Rati. Mech. Anal., 91 (1986), 288-308.
• I. Ekeland, Non-convex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474.
• \label9 D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.
• T. S. Hsu, Multiple solutions for semilinear elliptic equations in unbounded cylinder domains, Proc. Roy. Soc. Edin., 134A (2004), 719-731.
• T. S. Hsu and H. L. Lin, Multiple solutions for some Neumann problems in exterior domains, Bull. Austral. Math. Soc., 73 (2006), 353-364.
• D. Huang and Y. Li, A concentration-compactness principle at infinity and positive solutions of some quasilinear elliptic equations in unbounded domains, J. Math. Anal. Appl., 304 (2005), 58-73.
• Y. Jianfu, Positive solutions of quasilinear elliptic obstacle problems with critical exponents, Nonlinear Analysis T.M.A., 25 (1995), 1283-1306.
• G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on $\mathbb{R}^N$, Comm. Partial Differential Equations, 14 (1989), 1291-1314.
• \label5 P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case I, Ann. Inst. H. Poincaré, Analyse Nonlinéaire, 1 (1984), 109-145.
• \label6 P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case II, Ann. Inst. H. Poincaré, Analyse nonlinéaire, 1 (1984), 223-283.
• \label7 P. L. Lions, On positive solutions of semilinear elliptic equation in unbounded domains, in: Nonlinear Diffusion Equations and their Equilibrium States, (Ni, Peletier and Serrin, eds.), Springer-Verlarg, Berlin, 1988.
• \label10 M. $\rm \hat{O}$tani, Existence and nonexistence of nontrivial solutions for some nonlinear degenerate elliptic equations, J. Functional Anal., 76 (1988), 140-159.
• P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, in: Eigenvalues of nonlinear problems, Rome: Ediz. Cremonese, 1972.
• \label11 J. Serrin, Local behavior of solutions of quasilinear elliptic equations, Acta. Math., 111 (1964), 247-302.
• J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana University Mathematics Journal, 49 (2000), 897-923.
• \label12 P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Equ., 51 (1984), 126-150.
• N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., XX (1967), 721-747.
• \label13 J. L. V\vaa zquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
• \label8 X. P. Zhu, Multiple entire solutions of a semilinear elliptic equation, Nonlinear Analysis T.M.A., 12 (1988), 1297-1316.


text:
# What options are there for programming MCU's in production?

For my Super OSD project I'm going to need to program large batches of MCU's, probably about 10-15 boards an hour. I could just use my trusty PICkit 2 to program them (there are two - maybe three - on each board) but it would be very laborious to load each file (up to 128KB and 256KB in size), hit program, and repeat for each MCU, and probably wouldn't allow me to meet my target speed. Also, the MCU's are surface mount, so I have to have headers on the board to allow this. Ideally, I'd like to remove these headers, but it may not be an option.

So what options are there for mass programming of MCU's - are there faster ways to do it? I'm kind of on a budget of less than $100 USD, so things like Microchip's MPLAB PM3 at$895 USD, while nice, wouldn't be an option.

I'm dealing with PIC microcontrollers of various types. There's a PIC16F887, PIC24FJ64GA002 and dsPIC33FJ128GP802. However, I am also interested in options for AT32 and STM32 processors, as I may also be using them.

Please note, I'm not currently considering ordering chips from the manufacturers pre-programmed because the software may change by the time the chips arrive, and because of the additional cost involved.

• 15 boards per hour is four minutes per board. How long does the PICkit 2 take to program your part? – markrages Nov 27 '10 at 23:36
• @markrages: It's about 20 seconds with a small code base (~6KB). I imagine with more code, it could take up to a minute and a half for each board, which doesn't give me much breathing room. – Thomas O Nov 28 '10 at 0:17
• @Markrages, I can vouch, I wrote a very large project with the ICD2 LE and it would take 2 or 3 minutes to program. Quite annoying. With the pickit 3 however, it is screaming fast. – Kortuk Nov 28 '10 at 0:20

What we do for production, is to first load a program into the PIC that tests out the board (using a small test board that independently verifies the 3.3v rail is within spec using a couple of comparators, and then we use the ADC on the PIC to check everything else out. We had enough pins left over to allow this (it required some extra resistors to act as voltage dividers for the voltages over 3v).

After the tests pass, the real production code is flashed into the micro. Some additional tests are run, and the PCB is ready for assembly into a case.

This is all done via a program on the PC that only requires an operator to connect the board, click one button, and wait for the result PASS/FAIL. All test results (including ADC readings) are logged. The entire process (including the programming of the PICs via an ICD 3) is controlled via the PC program, which runs batch scripts to do the actual programming. Communication to the PIC to control the tests is done via one of the UARTs, whose pins are brought out to the test board (so in addition to the pins required for programming, we also have TX/RX as a minimum).

We set up several stations like this at our contract manufacturer.

BTW the ICD 3 is much faster than the ICD 2 (USB 2.0 vs 1.1).

• The pickit 3 also runs at USB 2.0 which is why I love it. ICD3 is a little expensive for me. My company uses MSP430s so it is quite different. – Kortuk Nov 29 '10 at 1:18
• Can you clarify how you program the IC? Your own brewed up ICSP or what? – Erik Friesen Jan 15 '16 at 13:13
• @ErikFriesen I use either an ICD 3 or REAL Ice for Microchip PICs. Both are probably too expensive for most hobbyists ($200 and$500 respectively). Other chips families (Freescale, Silicon Labs, etc.) have their own line of programmers. I include headers on the PCB's to connect with the programmers. After a device is out in the field, if it has wireless connectivity (Cellular or WiFi) it can be updated over the air. – tcrosley Jan 15 '16 at 18:42
• I have both and the pm3. The pm3 won't do all parts without connecting to the pc on > X2.15, it splits on the pic32 line. – Erik Friesen Jan 15 '16 at 19:10

The PICkit 2 isn't a production programmer. You really need to get an ICD 3, which is classed as a production programmer, if you want to be sure that the devices are being programmed properly. Alternatively, you could buy your chips pre-programmed from Microchip or a distributor, if you have the code finalised.

• This would be the correct way to go about this. – Kortuk Nov 27 '10 at 22:28
• Please provide a citation for the "production programmer" comment. Clearly, the Pickit isn't physically robust enough for an industrial application. But the signals it presents to the target are the same as any other programmer. (The original meaning of "production programmer" is that certain programmers were able to vary the supply voltage and read back the program contents to verify correct programming. Microchip hasn't mentioned this in their datasheets for about a decade, but the folklore lives on. Ironically, the PICkit 2 can vary the supply voltage...) – markrages Nov 27 '10 at 23:32
• This cropped up on the Microchip forum recently. The ICD 3 documentation states that it is a production programmer, unlike the PICkit 2/3 which someone from Microchip stated were not production programmers. – Leon Heller Nov 27 '10 at 23:45

Microcontroller programming is usually a small part of a larger functional testing process. What is the rest of your test plan like?

For any non-trivial product, you will need to design another board to test the board under production. Generally, the test board will have a computer interface and will connect the target to test equipment with RS-232 or GPIB capabilities. Then a computer can run through a script, programming the board, then running whatever tests are appropriate.

It is helpful to have this test board running for firmware development as well. Implement enough tests to catch regressions in the firmware.

The start of your test board is probably the PICkit 2 circuit. The firmware is open-source, it will take care of the programming part, it has a handy bootloader, and you can extend it to do anything else your functional test will need.

I like to write my production test scripts in Python, but I think the industry standard is Labview.

• I use Python lots, but never for general scripting - usually I leave that up to bash/shell or Perl. Thanks for the tips. – Thomas O Nov 28 '10 at 0:19

Get a pickit 3 and put it in programmer to go mode.

You tell it you want programmer to go mode and you load your rom, then you no longer need a usb port for more than power(yes, you still need it for power).

As you plug into devices you hit the program button and it loads the program and lets you know when it is done, takes less then 20 seconds a board. Costs no more than 50\$

One method you could use is to make a clamp or something akin to a test bed (example video) with pogo pins, so that you don't need to solder pins for the programming connectors. A PCB edge connector, such as an old style floppy cable, might be a cheaper (but more board area demanding) option.

You'll have to come up with a way to program each microcontroller, probably using either multiple programmers or a batch file like David suggested. You could combine it with a large switch (perhaps one of the rotary switches from an old printer sharing switchbox, or a few relays), to reconnect the programmer from one PIC to another without moving the connector.

The techniques also apply for other in-system programmable controllers, although the JTAG capable ones may make it yet easier because JTAG TAP is designed to be chained, so there'd be no moving the programmer from chip to chip. Also, for chips like AVRs that can use ordinary logic levels for programming, the switching could become easier.

• You don't even need pogo pins: I put a header pin in the PICkit 2 and I can program by inserting it into the holes in the board and pressing sideways a little bit, then maintaining this pressure until programming is complete. – markrages Nov 27 '10 at 23:37
• @Markrages, that is how I do it at work. – Kortuk Nov 28 '10 at 6:00

There is a command-line utility called PK2CMD for Windows and Linux that lets you program your PICs with the PicKit (so you don't have to use MPLAB or some other GUI). You could use your favorite scripting language to make a little program that runs the correct PK2CMD command when you press a key. This would make the computer side of your production less laborious (just 1 key press per MCU) but you would still need a way to connect the PicKit to each of your 3 MCUs.

Microchip's programming service is very cheap last time I used it, Once you cover the setup cost it's very cheap - pennies per chip for 12F's - not looked at higher end ones. They can also do marking/labelling etc. Even if you intend to change the FW, having a bootloader preprogrammed may make this easier.

For a production programmer I like the Asix Presto - tons of options for stuff like serialising, and very fast


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# Torsion for Lie algebras and Lie groups

This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for associated Lie groups.

Let $\mathfrak{g}$ be a complex simple Lie algebra. Then we can choose a basis of Chevalley generators for $\mathfrak{g}$. The structure constants describing the Lie brackets among the Chevalley generators are all integers, so the Chevalley basis spans a Lie algebra over the integers, which I shall denote by $\mathfrak{g}_\mathbb{Z}$. We can compute the (Chevalley-Eilenberg) Lie algebra cohomology of $\mathfrak{g}_\mathbb{Z}$ with coefficients in $\mathbb{Z}$ using the usual Koszul complex, and thus get for each $1 \leq n \leq \dim \mathfrak{g}$ a finitely-generated abelian group $H^n(\mathfrak{g}_\mathbb{Z},\mathbb{Z})$.

On the other hand, (as I understand it) the Lie algebra cohomology of $\mathfrak{g}$ with coefficients in $\mathbb{C}$, denoted $H^*(\mathfrak{g},\mathbb{C})$, is the same as the cohomology of some associated (compact?) connected Lie group $G$. This philosophy is described in the section "Motivation" of the Wikipedia article on Lie algebra cohomology. (I'm not an expert on this matter, so I invite someone more knowledgable to correct me if I've got something incorrect. I think that the associated Lie group $G$ will have $\mathfrak{g}$ as its Lie algebra, or perhaps $\mathfrak{g}$ will be the complexification of the Lie algebra of $G$.) We can also compute $H^n(G,\mathbb{Z})$, the cohomology of $G$ with coefficients in $\mathbb{Z}$.

Is there, or ought there to be, a connection between the torsion of $H^n(G,\mathbb{Z})$ and the torsion of $H^n(\mathfrak{g}_\mathbb{Z},\mathbb{Z})$? Should the torsion primes of the two abelian groups be the same? Should the torsion primes of one be a subset of the torsion primes for the other?

To the best of my knowledge, the torsion primes of the compact connected simple Lie groups were worked out in the 1950s and/or 1960s, especially by Armand Borel, but at present no such list seems readily available for the Lie algebra cohomology side of the picture.

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A related question: mathoverflow.net/questions/11703/… –  algori Jun 4 '11 at 4:12

I don't know the answer to the actual question but here is a situation which should be similar but simpler. Consider an integral polynomial group law $G$, i.e., a group scheme structure on the affine space over $\mathbb Z$. We then have a torsion free finitely generated nilpotent group $G(\mathbb Z)$ and an integral Lie algebra $\mathfrak g_{\mathbb Z}$. We also have a compact manifold $X:=G(\mathbb R)/\mathbb(\mathbb Z)$ and as $G(\mathbb R)$ is contractible we have that the cohomology of $X$ is equal to the cohomology of $G(\mathbb Z)$ (as $X$ is a $K(G(\mathbb Z),1)$). We also have that $G(\mathbb Q)$ is the Malcev completion of $G(\mathbb Z)$ and (consequently) we have that the inclusion $G(\mathbb Z)\hookrightarrow G(\mathbb Q)$ induces an isomorphism on cohomology with rational coefficients. For more obvious reasons (the Chevalley-Eilenberg complexes are isomorphic) $\mathfrak g_{\mathbb Z}\hookrightarrow \mathfrak g_{\mathbb Q}$ induces an isomorphism in cohomology.

Furthermore, $\mathfrak g_{\mathbb Q}$ is the Lie algebra that correspond to the torsion free, divisible nilpotent group $G(\mathbb Q)$ under the Malcev correspondence (i.e., the Campbell-Baker-Hausdorff formula) which results in an isomorphism between the Lie algebra cohomology (with rational coefficients) of $\mathfrak g_{\mathbb Q}$ and the group cohomology (again with rational coefficients) of $G(\mathbb Q)$. Combining everything we get an isomorphism between the cohomology of $\mathfrak g_{\mathbb Z}$ and of $G(\mathbb Z)$ (equal to that of $X$) both with rational coefficients. If I remember correctly Larry Lambe computed examples where the torsion of the integral cohomology was different (perhaps even for strictly upper triangular $4\times4$-matrices).

We can approach the comparison somewhat differently: To begin with, there is a third cohomology involved, the cohomology of the group scheme (over $\mathbb Z$) $G$. We can put all three of these cohomologies on an equal footing. We may consider the subcomplex of the standard complex for computing group cohomology consisting of polynomial cochains $G(\mathbb Z)\hookrightarrow \mathbb Z$, i.e., as $G$ as a scheme is just affine space $G(\mathbb Z)=\mathbb Z^m$ for some $m$ and we demand that the cochain $\mathbb Z^{mn}\hookrightarrow \mathbb Z$ be given by a polynomial. This is clearly a subcomplex and computes the group scheme cohomology of $G$. We may consider another subcomplex consisting of the functions given by polynomials with rational coefficients (still inducing a map $G(\mathbb Z)\hookrightarrow \mathbb Z$). It again is a subcomplex which has been shown to compute the cohomology of the group $G(\mathbb Z)$ (i.e., the inclusion into all cochains is a quasi-isomorphism). Note that such functions are sums of products of binomal polynomials in the projection functions (of $\mathbb Z^{mn}$ to its factors). The third complex is not quite a subcomplex consisting it does of rational valued functions that can be written as sums of products of divided powers of the projection functions. I have never completely checked all the details but am pretty certain that this complex computes the Lie algebra cohomology of $\mathfrak g_{\mathbb Z}$.

As polynomials with integer coefficients are both polynomials with rational coefficients and divided power polynomials we get an explicit map from group scheme cohomology to both group cohomology of integral points and to the integral Lie algebra cohomology. Tensoring with $\mathbb Q$ induces isomorphisms of complexes giving an explicit realisation of the rational isomorphisms. Note however that the group scheme cohomology is definitely distinct from the Lie algebra cohomology and the group cohomology already in the case of the additive group scheme: The $2$-cocycle $((x+y)^p-x^p-y^p)/p$, $p$ a prime, is polynomial which is not a polynomial coboundary. It is however the coboundary of the cochain $(x^p-x)/p$ which is integer-valued as well as of the divided power cochain $x^p/p$.

-

There is a simple observation about some odd behaviour of the prime $2$. The Lie group $SL_2(\mathbb{C})$ has torsion-free cohomology, because as a space it is weakly equivalent to $S^3$.

Consider the Lie algebra $\mathfrak{sl}_2(\mathbb{Z})$ of $2\times 2$-matrices with trace zero, and the usual commutator. This is a $\mathbb{Z}$-form of $\mathfrak{sl}_2$. Since the Cartan acts with weight $2$, the reduction of the above $\mathbb{Z}$-form to $\mathbb{F}_2$ is in fact not simple but solvable - the Cartan is an ideal and the quotient is abelian. This is still visible for the $\mathbb{Z}$-form, which has abelianization isomorphic to $(\mathbb{Z}/2)^{\oplus 2}$, generated by the matrices $$\left(\begin{array}{cc} 0&1\\0&0\end{array}\right) \qquad\textrm{ and }\qquad \left(\begin{array}{cc} 0&0\\1&0\end{array}\right).$$ The universal coefficient formula then implies $H^2(\mathfrak{sl}_2(\mathbb{Z}),\mathbb{Z})\cong(\mathbb{Z}/2)^{\oplus 2}$.

In this case, $2$ would be a torsion prime for the Lie algebra, but not for the Lie group. For all other primes, the cohomology of the Lie algebra and of the Lie group coincide. At least the reason for this behaviour (the weight $2$ action of the Cartan on the weight spaces) persists for all the $\mathfrak{sl}_n$, but I have not made the calculations to see if this really implies the existence of non-trivial $2$-torsion in the homology of $\mathfrak{sl}_n$, $n\geq 3$.

Whether or not a relation should exist between torsion in the homology of the Lie algebra and the Lie group seems to be related to the mod $p$ splitting behaviour of the Lie group. In some cases, the mod $p$ homotopy type of the Lie group splits as a product of spheres. In these cases, I would expect stronger relationships (maybe excluding the odd behaviour of the prime $2$). However, I do not really know how to possibly relate these things - the $\mathbb{Z}$-form of the Lie algebra is related to differential forms, the splitting of the Lie groups is serious algebraic topology, and these only seem to interact nicely with rational coefficients via rational homotopy theory.

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# Why is angular momentum of the Earth/Moon system conserved?

Why is the angular momentum of the Earth/Moon system conserved, apparently unaffected by external forces such as force of the Sun?

• Welcome to SE.Physics! Since this question has attracted 3 answers already, I'm guessing that other folks are getting the gist of what's being asked, however I'm a bit unclear about it personally. Are you basically asking how the conservation of angular momentum can apply to partial subsets of the universe? – Nat Sep 12 '18 at 15:13
• But momentum is conserved about centre of rotation – Ashutosh Sep 12 '18 at 19:02

Angular momentum is conserved when the net external torque is zero (you're thinking about linear momentum) via $\tau = \frac{dL}{dt}$. Anyway, as long as we assume that the earth, moon, and planets are all orbiting around in a disk (which is a decent approximation), then the position vector to the Earth/Moon system from any other of the planets is in the same plane as their perturbing gravitational forces, and furthermore these other planets are sufficiently far away from the Earth/Moon system so that the position vector and force vector are essentially parallel, meaning that the torque, $\tau = \vec{r} \times \vec{F}_{Grav}$, is zero, and thus angular momentum is conserved.


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# Is there an “inverse” of the dot-product?

The dot product of $$[a_1,a_2]$$ and $$[b_1, b_2]$$ is $$a_1b_1 + a_2b_2$$ (and so on for bigger vectors). What I'm wondering is if there's any definition of a function s.t. the "invdot" product of $$[a_1,a_2]$$ and $$[b_1, b_2]$$ is $$\frac{a_1}{b_1} + \frac{a_2}{b_2}$$, and if so, are there any other properties/derivations of this function from the dot product? I can see it's the same as taking the reciprocal of all the components of $$b$$ and then doing the dot product on that, but I can't understand how I'd do that in my specific use case.

For context, I'm trying to solve a problem where I want to algebraically manipulate $$a_1b_1 + a_2b_2$$ to $$\frac{a_1}{b_1} + \frac{a_2}{b_2}$$, and think that this might help if it's a well known operation.

• Note that the dot product sends two vectors to a scalar, the inverse of that won't make much sense (there are infinitely many such mappings). – lightxbulb Jan 23 '19 at 22:54
• The dot product is not injective, so there cannot be an inverse. However, maybe you used the term "inverse" by mistake, since you are talking about reciprocals. – Ben W Jan 23 '19 at 22:59
• It's "inverse" as in "opposite", not "reverse". It's not the common meaning in mathematics, but it's pedantic to require everyone here to adhere to such standards. Particularly when they're experimenting. – Arthur Jan 23 '19 at 23:03
• Yep, +Arthur's got it - though I see where you guys are coming too. – Robin Aldabanx Jan 23 '19 at 23:07

Your operation (it is not an "inverse" of the dot product, so let's not call it that) takes two vectors and produces a number, but the number is not invariant under coordinate transformations (e.g. it becomes infinite when you take a coordinate system where $$\vec{b}$$ is along the x-axis): there is no geometric meaning that can be assigned to it, because it is not the same in every coordinate system.


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1. ## projectile

a baseball fielder can throw a ball faster at low angles than high angles. this is modelled by assuming that at an angle a he can throw the baseball with a speed of k√(cosa) . where k is a constant. show the horizontal distance he can throw the ball is given by 2k^2/g *sina*√(cosa)

so i am using the equation ut-.5at^2=0 to find this. so i find what the time is when the vertical distance is zero. k√(cosa)sinat-.5gt^2
so time=2
k√(cosa)sinat /g . i the sub this into the equation distance in the x direction=ut so

k√(cosa)cosa ( 2k√(cosa)sinat /g) is what i get but i dont think this is right

2. ## Re: projectile

My first question is, what, exactly, do you mean by "horizontal distance"? Is that when the ball comes back to the same level as when it was thrown (the height of the thrower's arm) or to the ground? Given that he throws at angle a, with initial speed s, the height, above his arm, after time t, is given by $\displaystyle h= -\frac{g}{2}t^2+ s sin(a)t$ and the horizontal distance by $\displaystyle d= s cos(a) t$. Assuming you only asking for the distance when the ball comes back to the same height (the other option requires that we know the initial height) then we need to find $\displaystyle s cos(a) t$ with t such that $\displaystyle -\frac{g}{2}t^2+ s sin(a) t= 0$. Solve that quadratic equation for t, then use that value of t to find $\displaystyle s cos(a) t$.

3. ## Re: projectile

i think they mean when the ball comes back to the same level.

so when i solve that i get t=2ssina/g and then when i sub this in i get 2s^2cosasina/g sadly this is not right

4. ## Re: projectile

sorry in the opening post you have to prove the distance is 2k^2/g *sinacos^2a and not what i said it was

5. ## Re: projectile

range equation ...

$R = \dfrac{v_0^2 \sin(2\alpha)}{g}$

substitute for $v_0 = k\sqrt{\cos{\alpha}}$

$R = \dfrac{ (k\sqrt{\cos{\alpha}})^2 \sin(2\alpha)}{g}$

$R = \dfrac{ k^2\cos{\alpha} \cdot 2\sin{\alpha}\cos{\alpha}}{g}$

$R = \dfrac{ 2k^2\cos^2{\alpha} \cdot \sin{\alpha}}{g}$

6. ## Re: projectile

Thanks. The next part of the question says by writing it in the form of 2k^2 sina cos^2 a find the max distance he can throw the ball and the angle he throws it at to get the max distance.

I know the max angle is usually when sina=1 or do you solve this using calculus

7. ## Re: projectile

Under normal conditions, max range occurs when $\sin(2\alpha)=1$, but not in this case since the launch angle affects the initial speed.

Assuming $k$ is a constant, find $\dfrac{dR}{d\alpha}$ and maximize.

8. ## Re: projectile

So when I'm differentiating it am i supposed to leave k out of it all together?

9. ## Re: projectile

Originally Posted by markosheehan
So when I'm differentiating it am i supposed to leave k out of it all together?
you have the function ...

$R = \dfrac{2k^2}{g} \left(\cos^2{\alpha} \cdot \sin{\alpha} \right)$

you use the constant multiple rule ... the derivative of a constant times a function = the constant times the function's derivative, i.e. ...

$\dfrac{d}{dx} \left[c \cdot f(x)\right] = c \cdot f'(x)$

so ...

$\dfrac{dR}{d\alpha} = \dfrac{2k^2}{g} \cdot \dfrac{d}{d\alpha}\left(\cos^2{\alpha} \cdot \sin{\alpha} \right)$

10. ## Re: projectile

so i get 2k^2/g (cos^2 a -2sin^2a cosa). how do i maximise this? i cant just let it equal to zero like i would normally do because it has a k in it.

11. ## Re: projectile

Originally Posted by markosheehan
so i get 2k^2/g (cos^2 a -2sin^2a cosa). how do i maximise this? i cant just let it equal to zero like i would normally do because it has a k in it.
$R = \dfrac{2k^2}{g}(\cos^2{\alpha} \cdot \sin{\alpha})$

product rule ...

$\dfrac{dR}{d\alpha} = \dfrac{2k^2}{g}\bigg[\cos^2{\alpha} \cdot \cos{\alpha} + \sin{\alpha} \cdot 2\cos{\alpha} \cdot (-\sin{\alpha})\bigg]$

$\dfrac{dR}{d\alpha} = \dfrac{2k^2}{g}\bigg[\cos^3{\alpha} - 2\sin^2{\alpha} \cdot \cos{\alpha}\bigg]$

$\dfrac{dR}{d\alpha} = \dfrac{2k^2}{g}\bigg[\cos{\alpha}\left(\cos^2{\alpha} - 2\sin^2{\alpha}\right)\bigg]$

$\dfrac{dR}{d\alpha} = \dfrac{2k^2}{g}\bigg[\cos{\alpha}\left(1 - 3\sin^2{\alpha}\right)\bigg] = 0$

you only need to set the trig factors equal to zero because $\dfrac{2k^2}{g} \ne 0$ ...

$\cos{\alpha}=0 \implies \alpha = 90^\circ$ ... that would be a minimum, correct?

$1 - 3\sin^2{\alpha} = 0 \implies \sin{\alpha} = \dfrac{1}{\sqrt{3}} \implies \alpha = \arcsin\left(\dfrac{1}{\sqrt{3}} \right) \approx 35^\circ$

12. ## Re: projectile

thanks. just wondering why do you only set the trig factors equal to zero and not 2k^2/g

13. ## Re: projectile

$g \ne 0$. Do you agree with that?

... at an angle a he can throw the baseball with a speed of k√(cosa) .
If $k^2 = 0 \implies k = 0 \implies v_0 = k\sqrt{\cos{\alpha}} = 0$, which means the initial speed of the ball would be zero. Is that what happens?

So, what can you say about the value of $\dfrac{2k^2}{g}$ ?

14. ## Re: projectile

its not zero. but why does this mean i cant multiply it but the trig function and let it equal to zero.?

15. ## Re: projectile

Originally Posted by markosheehan
its not zero. but why does this mean i cant multiply it but the trig function and let it equal to zero.?
huh? multiply what?


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# Graphical representation of limit superior and limit inferior of a sequence

Consider this image on Wikipedia that attempts to graphically represent the concepts of $$\underset{n \rightarrow \infty}{\text{lim sup}} \space a_n$$ and $$\underset{n \rightarrow \infty}{\text{lim inf}} \space a_n$$.

Now, I realize that there is some sequence $$(x_n)_{n \in \mathbb{N}}$$, whose graph for $$n \leq 600$$ is identical to the one shown on the photo, such that the dashed black lines represent the limit superior and limit inferior of that sequence.

But what if I define $$x_n := 0$$ for every $$n > 600$$? Wouldn't the sequence, then, be convergent and make the current depiction of limit superior and limit inferior (as shown on the photo) wrong?

In other words, it seems that when one only has this photo, then one cannot determine precisely what the limit superior and limit inferior of the sequence are (but only an upper bound and a lower bound for those limits), because one wouldn't know how the sequence is defined for large values of $$n$$.

Am I getting something wrong?

• You aren't wrong. The picture comes with the implicit assumption that the sequence continues to "look like that" for larger values of $n$. We cannot plot infinitely many values after all. Nov 17, 2019 at 14:28

If $$(x_n)$$ is a sequence and only the values of $$x_n$$ for $$n\leq m$$ are known where $$m$$ is some fixed positive integer then nothing is known about $$\limsup x_n$$ and $$\liminf x_n$$.
This is illustrated by the fact that for the sequence $$(y_n)_n$$ prescribed by $$y_n=x_{n+m}$$ we have:$$\limsup y_n=\limsup x_n\text{ and }\liminf y_n=\liminf x_n$$
We could say that $$\limsup$$ and $$\liminf$$ are determined by the "tail" of the sequence.
• But if we do additionally know $\underset{n \geq m}{\text{sup}} \space x_n$ and $\underset{n \geq m}{\text{inf}} \space x_n$, then it seems we would have an upper bound and a lower bound, respectively, for the limit superior and limit inferior of the sequence. Is that correct? Nov 17, 2019 at 14:47


text:
39 views
25, 100, ?, 1600, 6400
A.  400
B.  300
C.  360
D.  420
E.  None of these

Here,the number is multiplied by 4 successively to get the next result.

$25 \times 4 =100$

$100 \times 4 =400$

$400 \times 4 =1600$

$1600 \times 4 =6400$

Hence,Option(A)$400$ is the correct choice.
The given series is completely divisible series each number is div by 4 hence

the options A) 400 is correct


text:
Prerequisites

### The Dirac Delta Function in Three Dimensions

The three-dimensional delta function must satisfy: $$\int\limits_{\hbox{\scriptstyle all space}} \delta^3(\rr-\rr_0)\,d\tau=1$$ where $\rr=x\,\xhat +y \,\yhat +z\,\zhat$ is the position vector and $\rr_0=x_0\,\xhat +y_0 \,\yhat +z_0\,\zhat$ is the position at which the “peak” of the delta function occurs. In rectangular coordinates, it is just the product of three one-dimensional delta functions: $$\delta^3(\rr-\rr_0)=\delta(x-x_0)\,\delta(y-y_0)\,\delta(z-z_0)$$ so that: $$\int\limits_{\hbox{\scriptstyle all space}} \delta^3(\rr)\,d\tau = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \delta(x)\,\delta(y)\,\delta(z)\, dx\, dy\, dz = 1$$

But in curvilinear coordinates, with $d\tau=h_u h_v h_w\,du\,dv\,dw$, it has a Jacobian in it. Thus, $$\delta^3(\rr-\rr_0)\ne\delta(u-u_0)\,\delta(v-v_0)\,\delta(w-w_0)$$ but instead: $$\delta^3(\rr-\rr_0) = {1\over h_u h_v h_w} \delta(u-u_0)\,\delta(v-v_0)\,\delta(w-w_0)$$

In particular, in cylindrical coordinates: $$\delta^3(\rr-\rr_0) = {1\over r} \delta(r-r_0)\,\delta(\theta-\theta_0)\,\delta(z-z_0)$$ and in spherical coordinates: $$\delta^3(\rr-\rr_0) = {1\over r^2\sin(\theta)} \delta(r-r_0)\,\delta(\theta-\theta_0)\,\delta(\phi-\phi_0)$$

Just as with the delta function in one dimension, when the three-dimensional delta function is part of an integrand, the integral just picks out the value of the rest of the integrand at the point where the delta function has its peak. $$\int\limits_{\hbox{\scriptstyle all space}} f(\rr)\,\delta^3(\rr-\rr_0) \,d\tau = f(\rr_0)$$


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# INEQUALITIES FOR CHORD POWER INTEGRALS

• Xiong, Ge (DEPARTMENT OF MATHEMATICS SHANGHAI UNIVERSITY) ;
• Song, Xiaogang (DEPARTMENT OF MATHEMATICS SHANGHAI UNIVERSITY)
• Published : 2008.03.31

#### Abstract

For convex bodies, chord power integrals were introduced and studied in several papers (see [3], [6], [14], [15], etc.). The aim of this article is to study them further, that is, we establish the Brunn-Minkowski-type inequalities and get the upper bound for chord power integrals of convex bodies. Finally, we get the famous Zhang projection inequality as a corollary. Here, it is deserved to mention that we make use of a completely distinct method, that is using the theory of inclusion measure, to establish the inequality.

#### References

1. A. D. Alexandrov, On the theory of mixed volumes of convex bodies, Mat. Sb. 3 (1938), no. 45, 28-46
2. Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer, Berlin, 1980
3. G. D. Chakerian, Inequalities for the difference body of a convex body, Proc. Amer. Math. Soc. 18 (1967), 879-884 https://doi.org/10.2307/2035131
4. H. G. Eggleston, B. Grunbaum, and V. Klee, Some semicontinuity theorems for convex polytopes and cell-complexes, Comment. Math. Helv. 39 (1964), 165-188 https://doi.org/10.1007/BF02566949
5. R. J. Gardner and G. Zhang, Affine inequalities and radial mean bodies, Amer. J. Math. 120 (1998), no. 3, 505-528 https://doi.org/10.1353/ajm.1998.0021
6. D. Ren, An Introduction to Integral Geometry, Science and Technology Press, Shanghai, 1988
7. L. A. Santalo, Integral Geometry and Geometric Probability, With a foreword by Mark Kac. Encyclopedia of Mathematics and its Applications, Vol. 1. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976
8. G. Xiong, Wing-Sum Cheung, and De-Yi Li, Bounds for inclusion measures of convex bodies, Adv. Appl. Math., in press, 2008 https://doi.org/10.1016/j.aam.2007.09.004
9. G. Xiong, D. Li, and L. Zheng, Intersection bodies and inclusion measures of convex bodies, J. Math. (Wuhan) 23 (2003), no. 1, 6-10
10. G. Xiong, J. Ni, and Z. Ruan, Inclusion measures of convex bodies, J. Math. Res. Exp. 27 (2007), no. 1, 81-86
11. G. Zhang, Integral geometric inequalities, Acta Math. Sinica 34 (1991), no. 1, 72-90
12. G. Zhang, Restricted chord projection and affine inequalities, Geom. Dedicata 39 (1991), no. 2, 213-222
13. G. Zhang, Geometric inequalities and inclusion measures of convex bodies, Mathematika 41 (1994), no. 1, 95-116 https://doi.org/10.1112/S0025579300007208
14. G. Zhang, Dual kinematic formulas, Trans. Amer. Math. Soc. 351 (1999), no. 3, 985-995 https://doi.org/10.1090/S0002-9947-99-02053-X
15. G. Zhang and J. Zhou, Containment Measures in Integral Geometry, Integral geometry and convexity, 153-168, World Sci. Publ., Hackensack, NJ, 2006

#### Cited by

1. CHORD POWER INTEGRALS OF SIMPLICES vol.02, pp.04, 2009, https://doi.org/10.1142/S1793557109000479
2. Random chord distributions and containment functions vol.58, 2014, https://doi.org/10.1016/j.aam.2014.05.003
3. Some inequalities for chord power integrals of parallelotopes vol.181, pp.4, 2016, https://doi.org/10.1007/s00605-016-0888-y


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# Gauss Theorem and Weil Conjecuters for elliptic curves

Hi there!

It is known (by Gauss) that for a prime $p \equiv 1 \ (mod \ 3)$ there is a "unique" writing of $4p=A^2+27B^2$ where $A=1+p-M_p$ and $M_p$ is the number of solutions of $X^3+Y^3+Z^3=0$ in the projective plane $\mathbb{P^2(\mathbb F_p)}$.

It's also known that the points on the cubic $y^3=x^3+1$ are in 1-1 correspondance with the points on the elliptic curve $y^2=x^3-432$.

I noted that $A$ is the same (up to its sign) of the Frobenius trace in the Weil conjectures for elliptic curves; and the number $A^2-4p$ is actually the discriminant of the caracteristic polinomyal of the map the Frobenius automorphism induces on the Tate module.

So, I've the following question to you:

Is there a proof of $p \equiv 1 \ (mod \ 3) \Rightarrow 4p=A^2+27B^2, \ A=1+p-M_p$ using Weil conjectures?

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What does "using Weil conjectures" mean? That statement follows from identifying Frobenius in the ring of endomorphisms of the elliptic curve, which is an order in the field of cube roots of one. –  Felipe Voloch Sep 23 '11 at 11:49
The problem is that I do not know how to show that Frobenius is in $\mathbb Z[\zeta_3]$. –  user18019 Sep 23 '11 at 12:00
The elliptic curve has an automorphism of order 3, $x \mapsto \zeta_3 x$, so the ring of endomorphisms contains $\zeta_3$. The condition $p \equiv 1 \mod 3$ implies that the curve is ordinary so the endomorphism ring is an order in a quadratic field. So the quadratic fiels has to be generated by $\zeta_3$ and you are done. But the "Weil conjectures" are not used anywhere. –  Felipe Voloch Sep 23 '11 at 12:32
I've read of this. I found this proof in many books, but the point it that I would like to know if or not a prove by WCs exists. Thank you anyway!! –  user18019 Sep 23 '11 at 13:03
Just becouse I would like -if possible of course!- to put in my thesis this fact as an application of the Weil conjectures. –  user18019 Sep 23 '11 at 14:00

I am going to interpret "the Weil conjectures" as a synechdoche for "thinking about counting points in terms of eigenvalues of Forbenius operators." In this case, the answer is a resounding yes! However, the specific facts about eigenvalues of Frobenius which which are being used here are easier than the ones discussed in the Weil conjectures. Silverman's The Arithmetic of Elliptic Curves might be a good reference for you.

Let $E$ be the genus one curve $X^3+Y^3+Z^3$ over $\mathbb{F}_p$. This curve has an automorphism $\Phi: (x:y:z) \mapsto (z^p:y^p:z^p)$. The map $\Phi$ sends every point with values in $\mathbb{F}_p$ to itself. However, points with values in $\mathbb{F}_{p^n}$ for higher values of $n$ are permuted. Our goal is to find the number of fixed points of $\Phi$.

The Weil conjecture way to do this is to associate to $E$ certain vector spaces $H^0(E)$, $H^1(E)$ and $H^2(E)$. The map $\Phi$ acts functorially on cohomology, and we have $$\#(\mbox{Fixed points of }\Phi) = Tr(\Phi^*:H^0 \to H^0) - Tr(\Phi^*:H^1 \to H^1) + Tr(\Phi^*:H^2 \to H^2).$$ Now, in this case, you can really get quite explicit about what these vector spaces are. $H^0$ and $H^2$ are both $1$-dimensional, and $\Phi$ acts by $1$ and $p$ respectfully. $H^1$ the Tate module of $E$ (Silverman III.7); a two dimensional space.

So the number of fixed points is $p+1-a$ where $a$ is the trace of $\Phi^*$ acting on the Tate module.

One can show that the endomorphism ring of $E$ is $\mathbb{Z}[\omega]$ where $\omega$ is a primitive cube root of unity (see Silverman V.3). So we must have $\Phi = c+d \omega$ for some integers $c$ and $d$. The fact that $\Phi$ has degree $p$ (see Silverman II.2) means that $N(c \omega +d) = c^2 - cd + d^2 =p$. And one computes that $a = Tr(\Phi) = 2c - d$.

So, in short, the number of points on $E$ is $p+1-a$, where $a=2c-d$ with $(c,d)$ such that $N(c+d \omega)=p$. Converting that into your statement, and getting the signs right, is left to you.

Focusing on the more specific question of using the Weil conjectures. The fact that $N(c+d \omega)$ is $p$ comes from the Riemann hypothesis part of the Weil conjectures. But you also need to know that $\mathbb{Z}[\omega]$ is the full ring of automorphisms, which is not particularly related to the Weil conjectures. And you need to figure out which of the $6$ elements of $\mathbb{Z}[\omega]$ which have norm $p$ is the correct one, which is also not a Weil computation. So I would say that this is not specifically a Weil conjecture result, although definitely can be proven using the same sort of ideas as in the Weil conjectures.

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I found it! One can write the number of solution of $y^2=x^3-432$ in term of a Jacobi sum $S \in \mathbb Z[\zeta_3]$, namenly $\# E(\mathbb F_p)=1+p+S+\overline S$. It has absolute value $p$. Furthermore, from what I know from Weil Conjectures, $S+\overline S$ is equal to minus the norm of Frobenius. Then Frobenius is $-S$ and hence it is in the form $c+d\zeta_3$. The rest comes from simple aritmetics calculations and the fact that $3$ (and $9$ too) divides $\# E(\mathbb F_p)$. Thak you very much David, you've been very usefull to me! –  user18019 Sep 25 '11 at 7:25
Isn't $H^1$ dual (rather than equal) to the tate module? –  Daniel Loughran Oct 3 '11 at 21:54


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# Obscure determinant question

1. Aug 14, 2009

### grawil

I'm working on some math that falls out of using a Kalman filter for estimation. I'd like to show that $$|G^T C_1^{-1} G + C_2^{-1}| \leq |G^T [diag(C_1)]^{-1} G + C_2^{-1}|$$ where $$C_1$$ and $$C_2$$ are covariance matrices and $$diag(C_1)$$ denotes the diagonal of matrix A.

I've been able to show this is true when $$G=G^T=I$$ and $$C_2$$ is also diagonal. Numerical simulations suggest it is more generally true but I've been unable to convince myself of it on paper.