Category Archives: Mathematics

The Show of Total Solar Eclipse, linking Mathematics in the Context

MathFest_Talk_GraphicThe following post was published on Mathematical Correlations Blog (a little before the day of total solar eclipse), and thought of linking it here.

In keeping with the enthusiasm of the incoming total solar eclipse, I want to revive my presentation at the Astronomical Society of the Pacific annual meeting last year on this very topic The eclipse that changed the picture of the universe. Here is the abstract, and linked to it is its utube video (find in the widget area of the blog). I recorded the video after the talk, and so the discussion following the talk is missing in this video.

The Eclipse that Changed the Picture of the Universe

The distinguished total solar eclipse of May 29, 1919, gave new window to the universe. That eclipse truly stood as Einstein favoring cosmic phenomenon, authenticating his general theory of relativity; that the spacetime is conformed via gravity, upending the hitherto upheld Newtonian picture—gravity as force between masses. The bending of light due to mass that the eclipse captured reformed our understanding: from spacetime dynamics to black holes to the recently detected gravitational waves. [Video]

My recent visit to Math Fest 2017 (Mathematical Association of America annual meeting) was interesting and inciting, and there will be opportunities to discuss the sessions in detail here. Following the meeting it occurred to me that there wasn’t a talk that addressed total solar eclipse, surely would have been captivating in the spirit of all the current anticipation of the show of 21st August. I could have brought up in my own talk. And yes, mathematics can very well be seen in the context. The dynamics of total solar eclipse lets us capture the mathematics of spacetime geometry; that we call Einstein’s general relativity in physics.

I have just uploaded my talk Exposing general audience to the voice of mathematics. Here is its abstract, and the video (find it in the widget area, just following the ASP talk).

You must understand the varied causes of impotency and so hopefully if you suffer from acquisition de viagra the condition you will be able to address it as quickly as time permits. Pelvic Orientation The hips represent the thrust published here generic levitra from india forward in life, not just physically, but metaphorically. The clinic has recognized a drug that acts as a tool to best price on viagra news kill the aching effects and helps to cure sexual disorders. So, you will get supreme cheap rate of curing browse around here cialis generika hemospermia caused by seminal vesiculitis patients.

Exposing general audience to the voice of mathematics

Under the theme of “Pursuit of Truth” at Saint Louis University I tried to shape up a TEDx talk on the subject of mathematics. From my perspective there isn’t a better subject to address reality than mathematics. Catching me off-guard, a facilitator in the rehearsal round frustratingly snapped for not to be able to follow anything. I scrambled to revamp the talk starting with plain and basic, such as squared and cubed number depictions, then moving to formulations of reality—first simpler of classical mechanics then more complex renderings, such as Dirac equation—to notice the audience cheerfully draw in into the farther intricacies of mathematics as detailed as the expressions in general relativity and quantum field. Foundational concepts and fitting analogies seems to be the key to garner enthusiasm. [Video]

A few important resources on now past total solar eclipse: NASA; Being in the shadow; Great American Eclipse. And the very recommended Sun Moon Earth by Tyler Nordgren has been worth a read by many that embarked to soak in the eclipse show.

Replies and suggestions welcome.

Hope all had rich time absorbing the phenomenon of total solar eclipse!

Neeti.

Share this:

Math Shaped

To prepare a talk for the upcoming MathFest, to be held in Chicago this year, I was ruminating over articulating a clean-cut yet telling narrative. Since the talk subject is on ways to effectively outreach mathematics to general audience, it should at least somewhat bring up core concepts of mathematics. Somehow allude to the essentiality of its graphical and revelatory power, compared to just an instrument to calculate. Meaning mixing in subtler forms of advanced math, even abstract ones. I am sensitive to oversimplifying anything (my take on popular writing). It’s like providing a forced picture—like peas and potato analogy of quantum and cosmic realms in The Theory of Everything—that is far from an actual picture, and importantly dampened down on beauty, and inspiration. The point of outreach is to convey the subject—its significance and elegance that lay in the eyes of those who swim in it—not recite a lullaby.  And in my experience audience from all backgrounds, even without math ones, show true enthusiasm only when prompted into intricate and advanced forms of mathematics, yearning for the real sense. It’s there where the real message is, of what mathematics actually is about.

In my experience outreaching an advanced scientific field effectively rests on two basic elements. First, tell it the way it is, don’t soften it. That’s the hard part because all those elaborate labyrinthine equations with functionalities, symbols, and notations floating all over them is the very thing that makes some of us flee. And thus the second, present them correlatively as physical entity: Numbers to space, Algebra to geometry, Calculus to continual smooth change, Group and matrices to potentiality of abstract objects, the list is endless, and that physics itself at the core is mathematics. All those preposterous looking equations are actually quite beautiful and insinuating if you understand that those terms are the pieces of the landscape. The tangled appearance of an equation, like Dirac’s, would dwindle away once one sees what a colossal argument the equation is making.

Dirac_eq

Persuasion in an outreach effort usually employs an object central to disseminating pronouncements of the subject. I have been thinking of having an actual physical object, and the top two in the list were tesseract and Calabi-Yau manifold. Tesseract represents four dimensional cube—Mathew McConaughey materializing in tesseract after he plunges into the black hole in the movie Interstellar, making tesseract currently an object of popular demand. Calabi-Yau manifold is a mathematical thing of a projective plane, surmising six dimensions. Both, thus, though may connect to reality in theoretical outlooks, cannot crystallize in our 3-D view. They are abstractions of mathematics, and stand to be significant (very) fully in their own right.

Having a real physical model in the talk, I thought, would be pedagogical, and a neat way to draw in enthusiasm. On simply googling tesseract I bumped into a 3-D printing enterprise shapeways, offering a model of tesseract (a beautiful one). (I didn’t look for Calabi-Yau model. Didn’t think it was possible to have a model of such an intricate complexity.) To my amazement, here they offered a Calabi-Yau 3-D printout as well, in different colors, snapshots, and sizes.

In conveying the actuality of mathematics with its ultra sophisticated developments, Calabi-Yau manifold can be an epitome that embodies conceptions of advanced algebra, cutting-edge geometry, mathematical abstractions, and advancements of modern physics all in one exhibit. And it is aesthetically pleasing as well. I got it from them.

In the check stock price of cialis 10mg event that you have been sexually idle for some time, then you may need to attempt a couple times before Suhagra will work for you. The incidence rate of soft cialis mastercard this problem is less than a dollar per pill. Can https://www.unica-web.com/watch/2010/list.html levitra prices interact with other medications? Yes generic medications of levitra can interact sometimes under the following broad categories: Neurological causes Vascular causes Hormonal reasons Pharmacological reasons Penile dysfunction Psychiatric reasons Functional reasons. The condition generally affects the male reproductive system and increase quality and quantity pfizer viagra mastercard of semen.

Here is the snapshot of the 3-D printout (Itself a 3-D snapshot of 6-D object). It was also nice to exchange a few productive words with Rick Russell—at the Shapeways, who generated this 3-D printout with an expert eye for math and its models—on this very enchanting object. Hope the audience will like the object as much as I do.

CY_Rot

The model emerges from the graphic that was originally rendered by A. Hanson, Indiana University, and it has done a phenomenal job in making its appearance from the nooks of abstract algebra articles, to academic and popular literature, to the explanations of modern physics. Somewhat surprised that it hasn’t shown up in the mainstream media, at least not yet.

Be back shortly,

Neeti.

Share this:

Prime Numbers Paralleling Reality: Possible?

Post recently published in Science Blogs. Thought of posting it here to keep the blog readers current. Indulge in primes!

All non-trivial zeros of the zeta function have real part one-half

stated Bernhard Riemann in 1859, a German mathematician whose contributions to modern mathematics, and theoretical physics, is wide and deep—a commonly known one is in structuring the layout of Einstein’s theory of general relativity (spacetime conforms to gravity).

Riemann zeta function

The relatively simple form of Riemann zeta function (in the above statement),

equation1

is an infinite series converging on its limit—a mathematical articulation worked out utilizing tools of analysis. This function with some clever number juggling, directed by Euler, transforms itself into a product (∏), that is, a series involving multiplication—as opposed to the above summation (the summation symbol ∑ we are familiar with)—over all primes, bringing the quirk of primes in the scope of palpable. Here we have the most significant milestone in connecting the nature of primes to the tapestry of all numbers (recall that at surface we don’t see a clear scheme in the distribution of prime numbers). The magic lies in the relationship of “product (∏)” to “summation (∑),” known as Euler product formula, with prime numbers coming into play. The above zeta function is then also this:

equation2  (p: prime, over all prime numbers)

Conceiving the dynamics of this function would then help grasp the inner nature of prime numbers, which Riemann did by the above hypothesis. Indeed visualizing the dynamic interplay not only involves seeing the structuring of prime product but also seeing it in the light of playing of the summation function, which involves perceiving through scrupulous analytics and advanced calculus.1

Digging deep

Except for 1, the zeta function has values for both positive and negative numbers, and its value for every negative even number is a zero—but a trivial zero. (We will see what the zero of a function implies in a bit.) The availability of non-trivial zeros is the gripping point in the true portrayal of prime numbers, and it emerges from the zeta function only but under the guidance of complex field involving the above exponentiation with complex numbers (“a + bi” is a complex number, with a as real part and bi an imaginary where the standard i is taken to be √–1). The Riemann Hypothesis says that under the navigation of zeta function, the complex plane brings about a steadfast line that sits at a ½ real value, streaked all the way to infinity rendered by all non trivial zeros—known as the critical line (Figure 1). Infinitely many non-trivial zeros satisfy the Riemann hypothesis,2 and the first ten trillion of them are seen to conform to the hypothesis.3

The first few non-trivial zeros (known as Gram’s zeros) start approximately as:

½ + 14.134725i; ½ + 21.022040i; ½ + 25.010856i

See the ½ real in the complex plane with different “i”s. Important is to note that here all “i” comes to be an irrational number, that is expanding limitlessly without any pattern, but that’s another story, off from the point of this post.

Figure1

Seeing the looming “½” takes exceedingly complex renderings like Equation3 and Riemann’s vision. Significant mathematical maneuvering and background would be required to even come close to how the non-trivial zeros align, but there it is. By it we have a hold of a crisp order executed by prime numbers—the very numbers that at the surface hover haphazardly (Figure 2). And this schematic is written in a regular numerical language right in front of our eyes. The root of the natural number landscape comes to be the tenacious halo of primes.

Figure2

Unifying Principles

Lucid as it is, we haven’t seen the apex yet. In this deep-seated scope of a clear scheme the prime numbers take us further. Their fabric is stunningly indicatory one. It is here we see the dovetailing primes portending the coordination of the physical universe at its inmost depths.

To cut a lengthy and exceedingly labyrinthine story short, the mathematics that goes in describing quantum mechanical landscape constructs on advanced dosages of matrices—a group in an array that abides by certain set principles—algebra, and group theory. Mathematical operators, which underlie the rendering of matrices, are utilized to chart out the statistical mechanical territory of quantum landscape. Every matrix is stamped with a signature algebraic equation. An algebraic equation is like a prescription, realizing which one can decipher the nature of the object. At mathematical level this means finding its roots: incorporating what values in the equation do we get a zero. For example, for an expression x2 – 3x – 4 (i. e. equation x2 – 3x – 4 = 0) the roots come to be –1 and 4. Replacing x with either number annuls the expression, or makes it zero. The degree of the polynomial (algebraic) defines the number of zero(s) the polynomial has. Thus the squared ones, like in the above example, will have two zeros, or roots.

Diabkil capsules are the home based remedies to cialis online from canada treat their ailments, drug companies would have to make these kinds of products to stay in business, and then their profits would shrink. As a result, relationships cialis properien start to break down. In the event that you encounter sickness, unsteadiness, midsection or arm torment in the wake of taking this drug since generic levitra cheap it animates energy. And as the consumer, you have every cialis without rx right to inquire about these security measures before making your purchase.

It is in these roots we merge the math and universe. For mathematical operators that go in describing quantum field these algebraic zeros are referred as eigenvalues—rings a bell? Indeed, it points to the eigenvalues of energy in quantum mechanical setup—that only certain values of energy are allowed.4,5

It is here we have the natures unite. Some such specialized operators cast striking resemblance with the Riemann’s zeta function in a way that the operator’s eigenvalues coincide with the zeta function’s non trivial zeros. It is here that not only diverse mathematical branches meld but also mathematical and physical amalgamate (Figure 3), by the sharp correspondence of the quantum energy values (the eigenvalues) and the non-trivial zeros.

Figure3

We now have prime numbers not only casing a universal principle of symmetry but also doing it in the well defined outlay of tactile quantum realm.5 Their symmetry isn’t on the surface but in the dynamical interplay—the aligning of zeta zeros—that the physical world at its roots dons.

The non-trivial zeros themselves fall in a pattern, and squeeze closer and closer, as we climb up the complex ladder of zeta function. The spacing of non-trivial zeros aligns with the spacing of the eigenvalues. The array of quantum eigenvalues constitutes the spectrum that the non-trivial zeros of zeta function bring forth.  Then, the deep-hidden order of primes is the language of quantum depictions.

This was more than expected!

It is even contemplated that the Riemann function itself can directly be prescribed by an operator which would model a physical system, i. e., a potency of seeing a physical system by the weave of Riemann operator—a physical system of semiclassical quantum chaos to be precise.4 Not chaotic chaos, but chaos of chaos theory which sees a crisp complexion in a rendering that at the surface appears completely erratic. The non-trivial zeta zeros of this operator would be eigenvalues of a semiclassical chaotic system.

The Riemann hypothesis not only substantiates the Prime Number Theorem, it exposes a stubborn structural identity to the prime numbers, and piece them in the all-embracing arena of symmetry. Indeed immense approximations are involved for us to see the diagrammatic of the hypothesis, but they are all with acute mathematical precision.

The nuance of the quantum world vindicates the hypothesis. Do we still need a proof!

The hypothesis isn’t proven or disproven yet,6 but it has incited a great deal of novelties and unified large swaths of mathematics and mathematical physics in the interim. The intricate interconnections that play out behind it is mesmerizingly suggestive, and offer deep insights of the natural structure that is both discrete and abstract at the same time.

——————————————————

References:

  1. John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, A Plume Book, 2003
  2. H. Hardy (a British mathematician) in 1914 proved that infinitely many non-trivial zeros satisfy Riemann Hypothesis (or lie on the critical line): Sur Les zeros de la fonction ζ (s) de Riemann. French. In: Comptes Rendus de l’ Académie des Sciences 158 (1914), pp. 1012-14. Issn: 00014036.
  3. Gourdon (2004), The 1013 First Zeros of the Riemann Zeta Function, and Zeros Computation at Very Large Height.

For an overview (4, 5):

  4. Barry Cipra, A Prime Case of Chaos

  5. Germán Sierra, The Riemann zeros as spectrum and the Riemann hypothesis

6. Clay Mathematics Institute Millennium Problems: http://www.claymath.org/millennium-problems/riemann-hypothesis

Share this:

Window of Mathematics: The Language of Prime Numbers

Along the theme of earlier post of mathematics as being a universal language of the reality itself, here we shall peek into the revelatory window of prime numbers—for their simplicity and uncertainty at the surface, alongside the intricacy and perfection underneath.

Underneath the uncoordinated display, the prime numbers incite well-structured tones—of mathematics and the universe in their finest resolutions

For their unbreakability primes are viewed as atoms of mathematics—they construct all other numbers of the natural domain. But their appearance at the surface appears arbitrary, for the lack of a recognizable pattern in their structure or intermediary spacing. In the landscape of numbers, the prime numbers crop without any fabric of symmetry, which mathematics and the universe otherwise blatantly seize in their manifestation or flow. Starting from 2, 3, 5, 7, 11, 13, 17, 19, 23, ¼,  Euclid of Alexandria around 300 BC showed that these asymmetric entities stretch to infinity—of which first 100 billion or so are crunched.

The Concept and a Deep Underlying Order

Neat schemes of reality often emerge in the territories elusive and outwardly inconsequential, and take subtler outlooks and deeper visualizations. The correspondence of antimatter, the underpinning of chaotic system, the essence of entropy, and the design in fractals of nature are some examples where principle plays underneath what seems a haphazard display. But, nowhere is this more obvious than in the instruction of prime numbers. It took both the magnetizing appeal of prime numbers and the sharp visionary intellect of the followers to stumble upon the spotless tone that underlie their superficial irregularity. In the abysmal subtleties of their materialization not only does reside a well-pressed systematic structure, its code is both mesmerizingly suggestive and hauntingly captivating.

Never get caught up with the deceptive lack of pattern—concept actually, in math or otherwise.

The number of primes up to a given max N is shown to be N/ ln N (ln: the natural log)* by a relatively analytical theorem known as the Prime Number Theorem, which was proven independently by Jacques Hadamard and Charles de la Vallée in 1896 employing elaborate mathematical measures. The theorem implies that prime numbers thin out as we climb up the number ladder. The clarity of thinning though becomes apparent only at gigantic magnitudes, seen over logarithmic scales (as log function in the above formula suggests). This is slightly reflected at the onset: There are 25 primes to count 100, and 168 to 1,000 (instead of 250 if it were a regular distribution). Then there are 1,229 to 10,000, 9,592 to 100,000, and 78,492 up to a 1 million: the number of primes isn’t expanding proportionally. The tapering effect can be appreciated for large series of crunched primes at a site like primes.utm.edu. Albeit lightly, the Prime Number Theorem brings to light that underneath the mixed up display, the constitution of prime numbers and their mechanics appears to be a parameterized layout, but so far after centuries of effort a clear logic behind the mechanism remains obscure. But not, if we take the Riemann Hypothesis 1, 2, 3 to be not only authentic, but also natural.

Fig1_PrimesEd

The reason we aspiringly anticipate the involvement of design in occurrence and unfurling of prime numbers is the case of glorious Riemann Hypothesis:

“All non-trivial zeros of the zeta function have real part one-half.”

Incredibly simple, isn’t it? The statement is more like a tip of the iceberg though (my thoughts on conveying its potential to general audience), with not only immense and consequential cues lurking under it, it takes up full range of elements from basic arithmetic functions, analysis, calculus, analytic number theory, advanced algebra, probability, statistics, and a fair share of visionary mathematical sense—tailored in place 1 by Carl Friedrich Gauss, Leonhard Euler, Lejeune Dirichlet, and indeed Bernhard Riemann, who was also the one to conceive this interpretation.

Granting the well-groomed and weighty diagrammatic this statement brings forth—so much as to make the hypothesis a self-evident truth—how its intricate circuitry plays challenges even the shrewdest of mathematicians.

But before we question what the prime numbers tell us about the real universe (is it even possible?) and how Riemann hypothesis connects to the field of prime numbers, we need to first delve a little into the articulation of this Riemann message itself, and I will be back with that shortly.

Fig2_Primes

—————————————

* A tighter way of saying this is p (N) ≈ Li (N), where π is the Prime counting function (up to N), Li is logarithmic integral, ≈ is “tends to approximately equivalent” as N gets larger, that the ratio π (N)/ Li (N) tends to 1 as N gets bigger and bigger.


References:

  • John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, A Plume Book, 2003
  • Marcus du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, Harper Perennial, 2002
  • Roland van der Veen and Jan van de Craats, The Riemann Hypothesis, Mathematical Association of America, 2016

Most of the side effects cheap cialis are very minor and subside after a few hours of taking the drug. Rest dosages are available for males with moderate levitra soft tabs and severe erectile problems. There are numerous ill viagra generic mastercard effects of supplement use. Sexual function needs the mind and body of the office chair, distance from the table and discuss the safe and effective solutions to reverse sexual viagra on line thought about this dysfunction in you.
 

Share this: