Our first challenge was to create a paper-plate marble maze. We looked at different videos of old-style pinball machines and talked about the different elements we saw, like ramps and pockets. The materials that we used included pipe-cleaners, paper, straws and white glue to create our own marble mazes. The goal of the task was to create a maze track that would guide the marble into the hole in the plate.

The students found through a LOT of trial and error that this was a bit more challenging than they expected! However, they had a **great** time experimenting with different materials and creating their marble tracks. Here are a few finished products! (First picture is mine!)

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As ABC News 7 reported, the Hendel family of Santa Rosa first saw the Tubbs fire flames spread across Sonoma County at 11 p.m. on Sunday, Oct. 8, and by 11:15 p.m. Roland Hendel knew that he and his daughter had to flee their home. They quickly began loading up their cats and dogs, but knew they’d be unable to save their eight bottle-fed rescue goats. However, one furry member of the family, Odin the Great Pyrenees, wasn’t going anywhere.

“Despite the sounds of exploding propane tanks, twisting metal, and the hot swirling winds, Odin refused to leave our family of eight bottle-fed rescue goats,” Hendel wrote on a YouCaring.com fundraising page.

“He was determined to stay with the goats and I had to let him do it,” Hendel told ABC News.

This meant that Odin was even risking separation from his sister, Tessa. Hendel had rescued both dogs as puppies, and they’d never spent more than a day apart in their lives. He was certain they’d lost Odin and the goats to the all-consuming flames.

“I was sure I had sentenced them to a horrific and agonizing death,” he wrote on YouCaring.

**RELATED VIDEO: What Do I Need to Know About Becoming a Pet Parent?**

[bc_video video_id=”5444174709001″ account_id=”416418724″ player_id=”rJSWQ1RE”]

Instead, all the animals shockingly beat the odds. When Hendel checked in with his neighbors, they reported back that they’d spotted Odin and the goats alive and well on Hendel’s property.

By Tuesday, Hendel was determined to return to his animals. He dodged police roadblocks throughout his still-burning neighborhood to get home, where he found his heroic, albeit injured dog, safe and continuing to protect the goats as well as a few baby deer who’d also joined the herd. The steadfast pup’s fur was singed, his whiskers melted and he had a limp — plus, everything the Hendels’ owned was gone — but all things considered, the animals’ survival was a miracle.

“We lost a million dollars worth of property; I don’t really care, we can replace all that. I just want my animals safe,” Hendel told ABC News.

Odin and the goats stayed on the property since Hendel didn’t have anywhere he could take them, but he gave them food and water, and he returned the next day to reunite the hero dog with his sister, Tessa.

Since then, Odin has been getting stronger, but Hendel says they are not out of the woods yet.

“Our pumphouse was destroyed, and we have no fresh water supply for them. All structures on our property were decimated, including the barn we had lovingly rebuilt for them earlier this year. And winter is coming. We have been overwhelmed with offers for help and assistance from our extended family of friends and loved ones. We would like to ask that you all direct this compassion to Odin, Tessa, and our goats, so we can re-establish a fresh water supply and rebuild their barn before the winter. Any and all donations are greatly appreciated,” wrote on YouCaring. “When you lose everything, it is easy to see what is important. I pray we can get our animal family what they need to win their heroic fight for survival, so Odin’s bravery and sacrifice are not in vain.”

Since Hendel first posted the YouCaring page, the family has raised more than $43,000 of a $45,000 goal. The page has been shared more than 5,000 times, but Odin and the Hendels can use all the help they can get. Click here to further help the cause.

As of Monday, Hendel had a happy update to share with the Facebook community.

“Ariel and I saw Odin, Tessa and the goats at the shelter yesterday. They are all doing great, and Odin is his old happy self! He is sure enjoying the attention and spotlight,” Hendel wrote. “I got some steak for Odin and Tessa, but did not have time to cook it yesterday, so I am going to do that now. A local pet groomer has offered to give both Odin and Tessa a full treatment for their coats! What a treat for them both.”

“I am overwhelmed by the support that we have received from all of you. So much kindness and compassion. The human spirit is truly a remarkable and wondrous thing.”

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I love this picture from one of the presentations that Matthew Oldridge and I do on this topic:

What it shows is a continuum of teaching. At times, we may be closer to the fully guided while at times we do some unstructured unguided lessons. However, most of the time we are some where near the middle. For myself I lean more towards the 3/4 mark of the line.

A couple of years ago I wrote a post about a balanced math class but since then I’ve had some small tweaks that I thought would be useful to highlight.

When I first thought of this subject I thought of six things that should be in the program (you can read about each section in my post):

**Guided Mathematics****Shared Mathematics: Students work together to “Mathematize”****Conferencing/ Monitoring****Congress****Reflection****Math Games and Math Facts**

Now my opinion about these things haven’t changed I still think you need to have all of these components but I want to simplify a bit and think more about the practical side. For this reason I want to steal a little line from the Leaf’s Head Coach Mike Babcock, think of a five day block of time.

Now, before I go into detail I want to preface that this is just my opinion and in no way is this the only way. I think as teachers we need to have professional judgement to choose what is best. I also don’t expect to have these ideas prescribed like a five day must follow. I just want you to reflect on these components.

I broke it into five days because I really felt that it was easy to look a five day segment in time. Some times these components may take more time or less but on average I try hard to stick to this.

Day 1: Problem Solving

I am a firm believer that our math program should be predominately a place where students are problem solving and exploring math concepts. During this time, the teachers role is to explore the concepts with the students. It is a fine balance between a guided approach for some to a more let kids explore. As a teacher I am also conferencing, questioning and monitoring students work. I am checking it to landscapes of learning and thinking about how I will debrief the learning. What misconceptions are students having? How are they tackling the problem? What collective conclusions are they making? are all questions that go through my head.

Day 2: Congress

This to me is one of the most important things we can do in a math class and where that shared, guided and explicit instruction is happening. During this time, I am questioning and explicitly linking the math concepts to their problem solving. Where I may allow students to wander a bit in exploration I am tightly keeping the reigns around the big ideas and misconceptions I observed in the problem.

Day 3: Number Talks

These have been one of the best decisions that I have made as a teacher. Number talks allow me to discuss strategies, talk through misconceptions and help students visually see the mathematics that is happening around them. Number talks is also a 15 to 20 minute exercise so they happen frequently and often in the classroom. Another great aspect is that it allows students to communicate and talk about math in a meaningful way.

Day 4: Reflection

The more I read about this topic the more I believe that this needs to be integrated more in the classroom. We need to explicitly show students how to reflect about their learning and how to set goals in order to improve. This year in my class I have purposefully set time aside for students to regularly talk about their math learning.

Day 5: Purposeful Practise (Math games, Centers and regular practise)

Yes I said it Purposeful practise. This may be in a worksheet but if it is I hope it is geared toward each child’s needs. For me purposeful practise is about seeing where a child is developmentally and finding things that may work for them. This year it has been center work, using board games or math games and digital games like knowledgehook and Mpower. The important part is understanding that it is purposeful and meaningful.

Overall, I think we need to think less of this war between concept and procedure and meet in the middle. How can we help our students learn and build bridges mathematically.

I would also love to hear your thoughts. If you have any opinions or questions please feel free to leave a comment.

Here is my slide deck on a balanced math approach.

]]>這個題目有許多不同版本，比如說 Algorithms to Live By: The Computer Science of Human Decisions 就問如果你搬到一個新城市，需要找地方住，找到你最合意的公寓的找房策略是什麼？或者你想找一個新祕書，你要面試多少人才找到合適的祕書呢（這問題有個名字叫「祕書問題」）。

If you want the best odds of getting the best apartment, spend 37% of your apartment hunt (eleven days, if you’ve given yourself a month for the search) noncommittally exploring options. Leave the checkbook at home; you’re just calibrating. But after that point, be prepared to immediately commit—deposit and all—to the very first place you see that beats whatever you’ve already seen. This is not merely an intuitively satisfying compromise between looking and leaping. It is the provably optimal solution.

那麼這個問題的最佳策略是什麼呢？從數學觀點，確實有最佳策略，就是大名鼎鼎的**37%法則**。Hannah Fry 告訴我們，如果你一生要談10次戀愛，找到最佳對象的機率發生在拒絕4個人之後；如果你有無數個伴侶，拒絕前37%的人，成功率最高。如果以時間軸來考量，在你遊戲花叢時間的前37%，千萬不要定下來。

**37** 這個數字，怎麼算出來的？Hannah Fry 在書裡講的很簡略，告訴我們最優停止理論給了我們一個極為簡潔的公式，用這個公式算出來的數字就是 37%。書末的參考文獻指向一篇1997年出版的論文《Searching for the Next Best Mate^{1}》，這篇論文的摘要（abstract）也只是直接說出 37 這個數字，沒有解釋來由。這篇論文的訂價是 24.95 歐元（Oops），直接放棄購買或下載的打算。

…In this paper, we analyze the third approach of mate choice as applicant screening and show through simulation analyses that a traditional optimal solution to this problem-the 37% rude-can be beaten along several dimensions by a class of simple “satisficing” algorithms we call the Take the Next Best mate choice rules. Thus, human mate search behavior should not necessarily be compared to the lofty optimal ideal, but instead may be more usefully studied through the development and analysis of possible “fast and frugal” mental mechanisms.

透過谷歌，在 plus.math.org 找到兩篇文章《Strategic dating: The 37% rule》和《Kissing the frog: A mathematician’s guide to mating》解釋答案為什麼是 37%，看完推導，只能說一個字：**服**！維基百科的解釋看起來不一樣，其實精神和 plus.math.org 提出來的解法是一致的。

簡而言之，這公式其實就是求 1/x 的積分，當 N 接近無窮大，最後我們得到

P 的最佳值在 x=1/e 的時候出現，大約等於 **0.3679**。

- Todd P.M. (1997) Searching for the Next Best Mate. In: Conte R., Hegselmann R., Terna P. (eds) Simulating Social Phenomena. Lecture Notes in Economics and Mathematical Systems, vol 456. Springer, Berlin, Heidelberg ↩

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Hölder and locally Hölder Continuous Functions, and Open Sets of Class C^k, C^{k,lambda} (Frontiers in Mathematics) by Renato Fiorenza

English | 22 Jan. 2017 | ISBN: 3319479393 | 166 Pages | PDF | 2.46 MB

This book offers a systematic treatment of a classic topic in Analysis. It fills a gap in the existing literature by presenting in detail the classic λ-Hölder condition and introducing the notion of locally Hölder-continuous function in an open set Ω in Rn. Further, it provides the essen…

After a drought of a collective **Wu-Tang** project, and the gross misstep of the Clan’s 2014’s opus **“A Better Tomorrow“**, the Clan is back to form like Voltron with Wu disciple **AllahMathematics** at the production helm. **Allah Math** does this project justice by reinvigorating the dirty drums introduced by the **RZA** with **“Enter the Wu-Tang (36 Chambers)“**, but still manages to keep the LP from sounding stale, or outdated. The spirit of the Wu shines through on each track, meticulously crafted by the Queens-born DeeJay/Producer.

The 9-member ensemble hasn’t lost a lyrical step, and most importantly speaks wisdom to the newer generation of rap fans, while not forgetting to satisfy their original fanbase aka the *“Older Gods”.* Lol But unfortunately, one of the original Wu members is missing from the album. Can you guess who?

Guest appearances from the likes of **Sean Price (RIP), Redman**, **Chris Rivers, R-Mean**, and **Mzee Jones** to name a few, help to bridge the gap between the old and the new. And touch boro borders beyond the Island of Shaolin.

Peppered with classic samples of the Kung Fu flicks that the Wu is known for, **“The** **Saga Continues”** is a triumphant return that mimics their meteoric impact in 1993. And with prominent members like **Raekwon **and** Masta Killa**, and their latest projects “**The Wild**” (Rae) and “**Loyalty is Royalty**” (MK) — both released earlier this year, the Wu is poised to shift the current landscape of Hip-Hop and bring lyricism back to the forefront.

Cause, *“when the Wu comes through, the outcome is critical!”*

*Wu-Tang Clan – “People Say” feat. Redman*

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Take two ellipses, one within the other. Take a point on the outer ellipse, and draw one of the two tangents to the inner ellipse, and find its second intersection with the outer ellipse. Use this point to start the process again, and again. You will get a polygonal path in the ellipse that will most likely not close up. But in case you are lucky, something miraculous happens: If you pick any other point and repeat the game, the polygon will again close up.

This is the content of a famous theorem by Jean-Victor Poncelet.

In spirit, it is similar to a theorem of Jakob Steiner that asserts that a chain of circles in an annulus bounded by two circles either always or never closes up. While Steiner’s theorem follows immediately by inverting the circles into a pair of concentric circles, such a simple proof is not available for Poncelet’s theorem. Until recently, all proofs I know of were, let’s say, *advanced*.

At the core of a new proof by Lorenz Halbeisen and Norbert Hungerbühler are some fundamental theorems from projective geometry.

Let’s first recall that *five* points, no three collinear, determine a unique conic.

This is because through four points, you can find two different degenerate conics consisting each of a pair of lines, and by forming linear combinations, accommodate a fifth point. Below we will need the dual theorem: Given five lines, no three concurrent, there is a unique conic tangent to them.

Pascal’s theorem is a condition for *six* points to lie on a conic: They do if and only if opposite sides intersect in collinear points. Above you see this for six points on the two branches of a hyperbola.

Dual to this is Brianchon’s theorem (illustrated above): The sides of a hexagons are tangent to a conic if and only of its diagonals are concurrent.

As an application, Halbeisen and Hungerbühler show: If the six vertices of two triangles a1,a2,a3 and b1,b2,b3 lie on a conic, than there is a conic tangent to the six sides of the triangles. The proof is easy: Applying Pascal to the hexagon a1,b2,a3,b1,a2,b3 gives us three collinear points c12,c13,c23.

Then applying Brianchon to the hexagon a1,c12,b1,b3,c23,a3 shows that it is tangent to a conic. But the sides of this hexagon are the same as the sides of the two triangles, so we are done

From here, we obtain Poncelet’s theorem for triangles: Suppose you have two ellipses inside each other, and a triangle whose vertices lie on the outer ellipse and whose sides are tangent to the inner. Take another point on the outer ellipse, and form a second triangle by drawing the tangents to the inner ellipse. We have to show that the third side of the triangle is also tangent to the inner ellipse.

By the theorem by Halbeisen and Hungerbühler, the two triangles have an inscribed common ellipse. The given inner ellipse touches five of the same six lines by construction. But a conic is uniquely determined by five tangent lines.

The general case follows of n-gons the same idea, but requires more bookkeeping.

]]>The missing links between galaxies have finally been found. This is the first detection of the roughly half of the normal matter in our universe – protons, neutrons and electrons – unaccounted for by previous observations of stars, galaxies and other bright objects in space.

You have probably heard about the hunt for dark matter, a mysterious substance thought to permeate the universe, the effects of which we can see through its gravitational pull. But our models of the universe also say there should be about twice as much ordinary matter out there, compared with what we have observed so far.

Two separate teams found the missing matter – made of particles called baryons rather than dark matter – linking galaxies together through filaments of hot, diffuse gas…

Get galactic at: “Half the universe’s missing matter has just been finally found.”

* meme

###

**As we heed E.M. Forster,** we might recall that it was on this date in 1843 that Sir William Rowan Hamilton conceived the theory of quaternions. A physicist, astronomer, and mathematician who made important contributions to classical mechanics, optics, and algebra, he had been working since the late 1830s on the basic principles of algebra, resulting in a theory of conjugate functions, or algebraic couples, in which complex numbers are expressed as ordered pairs of real numbers. But he hadn’t succeeded in developing a theory of triplets that could be applied to three-dimensional geometric problems. Walking with his wife along the Royal Canal in Dublin, Hamilton realized that the theory should involve quadruplets, not triplets– at which point he stopped to carve carve the underlying equations in a nearby bridge lest he forget them.

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A stochastic process (“a mathematical object usually defined as a collection of random variables”) is said to have the Markov Property if, conditional on the present value, the future is independent on the past.

Let’s firstly introduce some notation: let **S** be a countable set called the *state space* and let **X **= (X_{t}: t ≥ 0) be a sequence of random variables taking values in **S**.

Then, the sequence **X** is called a Markov Chain if it satisfies the **Markov Property**:

for all t ≥ 0 and all x_{0}, x_{1}, …, x_{t} ϵ S.

Notation is simplified in the case where the Markov chain is **homogeneous**. This is when for all i, j ϵ S, the conditional probability P(X_{t+1} = j | X_{t} = i) does not depend on the value of *t*.

**Branching Process**: The branching process is a simple model of the growth of a population; each member of the*n*^{th }generation has a number of offspring that is independent of the past; X_{n}= size of the n^{th}generation.**Random Walk**: A particle performs a random walk on the line: let Z_{1}, Z_{2}, …, be independent with P(Z_{i}= 1) = p and P(Z_{i}= -1) = 1-p, then X_{n}=

Z_{1}+ … + Z_{n}; at each epoch of time, it jumps a random distance that is independent of the previous jumps.**Poisson Process**: the Poisson process satisfies a Markov property in which time is a*continuous*variable rather than a discrete variable, and thus the Poisson process is an example of a*continuous*-time Markov chain; the Markov property still holds as arrivals after time*t*are independent of arrivals before this time.

For simplification (and as this is only an *intro* to Markov chains) we’ll assume that the Markov chains are homogeneous.

Two quantities that are needed in order to calculate the probabilities in a chain are the:

**transition matrix**: P = (p_{i,j}: i,j ϵ S) given by p_{i,j}= P(X_{1}= j | X_{0}= i);**initial distribution**: λ = (λ_{i}: i ϵ S) given by λ_{i}= P(X_{0}= i).

As we have assumed homogeneity we have that

p_{i,j} = P(Xn+1 = j | Xn = i) for n ≥ 0.

These quantities are characterised in the following way:

## Proposition:

a) The vector λ is a

distributionin that λ_{i}≥ 0 for i ϵ S and the sum of λ_{i}over i = 1.b) The matrix P = (p

_{i,j}) is astochastic matrixin that p_{i,j}≥ 0 for i, j ϵ S and the sum of p_{i,j}over j = 1 for i ϵ S (i.e. that P row sums to 1)## Proof:

a) As λ

_{i}is a probability, it is clearly non-negative. Additionally, the sum of λ_{i}over i = the sum of P(X_{0}= i) over i = 1.b) Since p

_{i,j}is a probability, it is non-negative. Finally, the sum of p_{i,j}over j = the sum of P(Xn+1 = j | Xn = i) over j = P(Xn+1 ϵ S | Xn = i) = 1.

Hope you enjoyed this brief introduction to a Markov Chain!

M x

]]>In the Gospel of Luke Jesus told us “One greater than Solomon is here,” and yet we fail to live as though God has given us the mission of demonstrating the wisdom, excellence, and king-level influence implied in Jesus’ statement.

As I dig deeper into something I’m calling the *Solomon Standard*, I’m discovering many examples just like this exist all around us. In other words, I’ve been completing the formula “If X, then Y,” where X = a true statement and Y = the result of that truth. Here’s an example of what I mean:

“If every human being carries a divine spark – is made in Gods image – then even the most disagreeable person on earth has something within himself that I can honor, respect and champion on his behalf.”

Following this same model, what’s your “if, then” truth?

]]>So… Well… Many nice things came out of this – and I wrote about that at length – but last night I was thinking this interpretation may also offer an explanation of relativistic length contraction. Before we get there, let us re-visit our hypothesis.

**The geometry of the wavefunction**

The elementary wavefunction is written as:

ψ = *a·e*^{−i(E·t − p∙x)/ħ} = *a·cos*(**p**∙**x**/ħ – E∙t/ħ) *+** i·a·sin*(**p**∙**x**/ħ – E∙t/ħ)

Nature should not care about our conventions for measuring the phase angle clockwise or counterclockwise and, therefore, the ψ = *a·e*^{i}^{[E·t − p∙x]/ħ} function may also be permitted. We know that *cos*(θ) = *cos**(**–*θ) and *sin*θ = *–**sin**(**–*θ), so we can write: * *

ψ = *a·e*^{i}^{(E·t − p∙x)/ħ} = *a·cos*(E∙t/ħ – **p**∙**x**/ħ) *+** i·a·sin*(E∙t/ħ – **p**∙**x**/ħ)

*= **a·cos*(**p**∙**x**/ħ – E∙t/ħ) *–** i·a·sin*(**p**∙**x**/ħ – E∙t/ħ)

The vectors **p** and **x** are the the momentum and position vector respectively: **p** = (p_{x}, p_{y}, p_{z}) and **x** = (x, y, z). However, if we assume there is no uncertainty about **p** – not about the direction nor the magnitude – then we may choose an x-axis which reflects the direction of **p**. As such, **x** = (x, y, z) reduces to (x, 0, 0), and **p**∙**x**/ħ reduces to p∙x/ħ. This amounts to saying our particle is traveling along the x-axis or, if p = 0, that our particle is located somewhere on the x-axis. Hence, the analysis is one-dimensional only.

The geometry of the elementary wavefunction is illustrated below. The x-axis is the direction of propagation, and the y- and z-axes represent the real and imaginary part of the wavefunction respectively.

Note that, when applying the right-hand rule for the axes, the vertical axis is the y-axis, not the z-axis. Hence, we may associate the vertical axis with the cosine component, and the horizontal axis with the sine component. You can check this as follows: if the origin is the (x, t) = (0, 0) point, then cos(θ) = cos(0) = 1 and sin(θ) = sin(0) = 0. This is reflected in both illustrations, which show a left- and a right-handed wave respectively. We speculated this should correspond to the two possible values for the quantum-mechanical spin of the wave: +ħ/2 or −ħ/2. The cosine and sine components for the left-handed wave are shown below. Needless to say, the cosine and sine function are the same, except for a phase difference of π/2: sin(θ) = cos(θ − π/2).

As for the wave velocity, and its direction of propagation, we know that the (phase) velocity of any wave F(kx – ωt) is given by *v*_{p} = ω/k = (E/ħ)/(p/ħ) = E/p. Of course, the momentum might also be in the negative x-direction, in which case k would be equal to -p and, therefore, we would get a negative phase velocity: *v*_{p} = ω/k = *–*E/p.

**The de Broglie relations**

E/ħ = ω gives the frequency in time (expressed in radians per second), while p/ħ = k gives us the wavenumber, or the frequency in space (expressed in radians per meter). Of course, we may write: f = ω/2π and λ = 2π/k, which gives us the two de Broglie relations:

- E = ħ∙ω = h∙f
- p = ħ∙k = h/λ

The frequency in time is easy to interpret. The wavefunction of a particle with more energy, or more mass, will have a *higher density in time* than a particle with less energy.

In contrast, the second *de Broglie *relation is somewhat harder to interpret. According to the p = h/λ relation, the wavelength is *inversely *proportional to the momentum: λ = h/p. The velocity of a photon, or a (theoretical) particle with zero rest mass (m_{0} = 0), is *c* and, therefore, we find that p = m* _{v}*∙

λ = h/p = hc/E = h/mc

However, this is a limiting situation – applicable to photons only. Real-life *matter*-particles should have *some *mass[1] and, therefore, their velocity will never be *c*.[2]

Hence, if p goes to zero, then the wavelength becomes infinitely long: if p → 0 then λ* → ∞*. How should we interpret this inverse proportionality between λ and p? To answer this question, let us first see what this wavelength λ actually represents.

If we look at the ψ = *a*·cos(p∙x/ħ – E∙t/ħ) – *i*·*a*·sin(p∙x/ħ – E∙t/ħ) once more, and if we write p∙x/ħ as Δ, then we can look at p∙x/ħ as a phase factor, and so we will be interested to know for what x this phase factor Δ = p∙x/ħ will be equal to 2π. So we write:

Δ =p∙x/ħ = 2π ⇔ x = 2π∙ħ/p = h/p = λ

So now we get a meaningful interpretation for that wavelength. It is the distance between the crests (or the troughs) of the wave, so to speak, as illustrated below. Of course, this two-dimensional wave has no real crests or troughs: we measure crests and troughs against the y-axis here. Hence, our definition depend on the frame of reference.

Now we know what λ actually represents for our one-dimensional elementary wavefunction. Now, the time that is needed for one cycle is equal to T = 1/*f *= 2π·(ħ/E). Hence, we can now calculate the wave velocity:

v = λ/T = (h/p)/[2π·(ħ/E)] = E/p

Unsurprisingly, we just get the phase velocity that we had calculated already: *v *= *v*_{p} = E/p. The question remains: what if p is zero? What if we are looking at some particle at rest? It is an intriguing question: we get an infinitely long wavelength, and an infinite wave velocity.

Now, re-writing the *v *= E/p as *v *= m∙*c*^{2}/m∙*v*_{g }* *= *c*/β_{g}, in which β_{g} is the relative *classical *velocity[3] of our particle β_{g} = *v*_{g}/*c*) tells us that the *phase *velocities will effectively be superluminal (β_{g} < 1 so 1/ β_{g} > 1), but what if β_{g} approaches zero? The conclusion seems unavoidable: for a particle at rest, we only have a frequency *in time*, as the wavefunction reduces to:

ψ = a·e^{−i·E·t/ħ} = a·cos(E∙t/ħ) – i·a·sin(E∙t/ħ)

How should we interpret this?

**A physical interpretation of relativistic length contraction?**

In my previous posts, we argued that the oscillations of the wavefunction pack energy. Because the energy of our particle is finite, the wave train cannot be infinitely long. If we assume some *definite* number of oscillations, then the string of oscillations will be shorter as λ decreases. Hence, the physical interpretation of the wavefunction that is offered here may explain relativistic length contraction.

:-)

Yep. Think about it. :-)

[1] Even neutrinos have some (rest) mass. This was first confirmed by the US-Japan Super-Kamiokande collaboration in 1998. Neutrinos oscillate between three so-called flavors: electron neutrinos, muon neutrinos and *tau *neutrinos. Recent data suggests that the *sum *of their masses is less than a millionth of the rest mass of an electron. Hence, they propagate at speeds that are very near to the speed of light.

[2] Using the Lorentz factor (γ), we can write the relativistically correct formula for the kinetic energy as KE = E − E_{0} = m_{v}*c*^{2} − m_{0}*c*^{2} = m_{0}γ*c*^{2} − m_{0}*c*^{2} = m_{0}*c*^{2}(γ − 1). As *v *approaches *c*, γ approaches infinity and, therefore, the kinetic energy would become infinite as well.

[3] Because our particle will be represented by a wave *packet*, i.e. a superimposition of elementary waves with different E and p, the classical velocity of the particle becomes the *group *velocity of the wave, which is why we denote it by *v*_{g}.

DILEMMA

Faith is often blind, and that seems tragic.

It drops to its knees, humble and devout.

Science’s problem is lack of magic.

It can’t accept the mystical throughout.

Each sees light stream

through a prism of glass.

The pious think of stained glass

and God’s bliss,

and all but simplicity they let pass.

They have no need for a hypothesis.

The logical need to know how light’s bent,

and measure photon wavelength to decide

if particle-waves end the argument

or there are more dimensions to divide.

The first has all the answers that it needs.

The other must seek before it accedes.

]]>

*“The only legitimate child of Lord Byron, the most brilliant, revered, and scandalous of the Romantic poets, Ada was destined for fame long before her birth. Estranged from Ada’s father, who was infamously “mad, bad, and dangerous to know,” Ada’s mathematician mother is determined to save her only child from her perilous Byron heritage. Banishing fairy tales and make-believe from the nursery, Ada’s mother provides her daughter with a rigorous education grounded in mathematics and science. Any troubling spark of imagination—or worse yet, passion or poetry—is promptly extinguished. Or so her mother believes.*

*When Ada is introduced into London society as a highly eligible young heiress, she at last discovers the intellectual and social circles she has craved all her life. Little does she realize that her delightful new friendship with inventor Charles Babbage—brilliant, charming, and occasionally curmudgeonly—will shape her destiny. Intrigued by the prototype of his first calculating machine, the Difference Engine, and enthralled by the plans for his even more advanced Analytical Engine, Ada resolves to help Babbage realize his extraordinary vision, unique in her understanding of how his invention could transform the world. All the while, she passionately studies mathematics—ignoring skeptics who consider it an unusual, even unhealthy pursuit for a woman—falls in love, discovers the shocking secrets behind her parents’ estrangement, and comes to terms with the unquenchable fire of her imagination.” *(Book Description via Good Reads)

Published: 2017 (The edition I have is an advance copy, this book doesn’t come out until December 5th).

Read and Reviewed: September 2017

I won this book in a Good Reads giveaway. I’m happy to say these are my thoughts on it.

My goodness! What a life Ada Lovelace led!

I have to say that this author succeeded wonderfully in getting me to sympathize with Ada. Her parents were not easy to deal with at all. Much of her fame in her lifetime was attributed to her infamous poetic father, Lord Byron. He didn’t even know much of his daughter, having separated from her mother when Ada was two. Her mother, Lady Byron, demonizes him and in many cases takes out her anger on Ada because of what Lord Byron did to her. He is shown as being both neglectful and extremely promiscuous (having and flaunting his extramarital affairs with other lovers openly). While I don’t agree with how Lord Byron treated his wife at all, I thought it was pretty cruel at points how Ada’s mother treats her. She definitely takes out a lot of her hurt on poor Ada. There may have been some exaggerations, for we mostly get Ada’s viewpoint. Generally, following the shift from her mother as narrator to Ada, her mother does not remain a very sympathetic character for me.

Despite reading a lot of literature, I’m not overly familiar with Lord Byron as a writer though I’ve probably read him before in one or two of my poetry classes in school at one point or another. So I’m not particularly on his side over his wife’s side as far as their fighting goes. It’s a pretty central part of Ada’s life and in the novel, which is why I mention it at all. Ada basically isn’t allowed to know anything about her father until after she is married and separated from her mother. Even then her mother is very reluctant to tell her anything about him because he broke her heart terribly.

Ada herself is a remarkable person despite how she grew up. She becomes one of the most accomplished female mathematicians of her era. She is mainly credited with translating and expanding upon a document which explains Mr. Babbage’s Differential Machine, an early model of a computer. Babbage himself admitted that aside from himself, no one knew the machine better than Ada. Unfortunately, Ada dies tragically young from illnesses, so she was not able to produce any other major scientific or mathematic works. Ada is buried in her father’s family tomb so that she can be with him and the rest of her family in death because she was denied them for most of her life.

I would rate this book 4.5 stars. I could sympathize with Ada more so than most people I’ve read about. Her story made me very sad with how she grew up and wasn’t able to live how she wanted to for most of her short life. It misses the other half a star rating because there were points in the middle of the book which were a little slow but other than that I would suggest this book. Though if you’re a die-hard Lord Byron fan, you probably won’t like this book because it doesn’t favor him very well (nor his wife, Ada’s mother, either, in all honesty). There is a mention of the supposed affair with his half-sister which is a controversy surrounding Byron’s life. Ada’s mother leads Ada to believe this is true and I’m not sure if this was true but that’s pretty sad if that’s the case.

https://www.goodreads.com/book/show/34627441-enchantress-of-numbers

]]>Once we get the period, we compute for , the inverse of e modulo r:

The inverse can then be used to decrypt the ciphertext:

In our previous example, we encrypted the message

THIS IS A SECRET MESSAGE

using public key p=53, q=59, N=pq=3127 and e=7 and private key d=431. The “plain text” is

1907 0818 2608 1826 0026 1804 0217 0419 2612 0418 1800 0604

and the ciphertext is:

0794 1832 1403 2474 1231 1453 0268 2223 0678 0540 0773 1095

Let’s compute the period of the first block of our ciphertext:

Using the python script below, we can compute the period

for r in range(1,100): p=pow(794,r,N) if p == 1: print "%d %d" % (r,p)

The result of running the above program gives r=58. We can then compute using the following equation:

The above equation is satisfied when m=3 and . Using this value of , we can compute for

which gives us the original message!

However, unlike using the private key, you need to compute the period r and for every block of the ciphertext (unless the ciphertext is composed of only one block). However, that should not stop a cracker from deciphering all the blocks.

]]>Bob should be able to send the key to Alice encrypted so that Eve will not be able to read it. In order to do this, Bob will have to create a second key to encrypt the key he wants to send to Alice. The problem now is how will Alice decrypt the message (which is the encrypted key) if she does not have the second key?

This is where RSA encryption is used. Suppose we want to encrypt a message using RSA, what we’ll do is find 2 large prime numbers p and q and get their product N = pq. We will need another number e, which we will use to encode the message into a ciphertext. The set of numbers N and e is called the **public key** which Alice can send to Bob via email. Bob will use these numbers to encrypt the secret key before sending to Alice.

We will represent a textual message like “THIS IS A SECRET MESSAGE” into numbers. To accomplish this, we need to map letters into numbers like the following:

Using the above mapping, we can write ‘THIS IS A SECRET MESSAGE’ as:

19 7 8 18 26 8 18 26 0 26 18 4 2 17 4 19 26 12 4 18 18 0 6 4

If a number is less than 10, we pad it with a zero to the left. The message then becomes:

19 07 08 18 26 08 18 26 00 26 18 04 02 17 04 19 26 12 04 18 18 00 06 04

To conserve some space, we can group the numbers into groups of 4:

1907 0818 2608 1826 0026 1804 0217 0419 2612 0418 1800 0604

Now for each number above, we will encode it using the formula:

Let’s say we choose p = 53, q=59 and e = 7. This gives us . To encode 1907, we do

The number 0794 is now the ciphertext. It is the number we give to the recipient of the message. We can use python to generate the ciphertext above.

for n in ("1907","0818","2608","1826","0026","1804","0217","0419","2612","0418","1800","0604"): print pow(int(n),7,3127)

Doing this for all numbers we get:

0794 1832 1403 2474 1231 1453 0268 2223 0678 0540 0773 1095

When the recipient gets this message, she can decipher it using a key which she keeps private to herself. The key d is the inverse of e modulo . The key d is called the **private key**. We can retrieve the original message using the formula:

The inverse of e=7 modulo can be calculated using the equation

for some integer m. Computing for d, we find that when m=1, the equation is satisfied and the value of d = 431.

Using d = 431 and applying the decipher formula to the first block, we get

which is our original message!

We now apply this to the entire ciphertext

0794 1832 1403 2474 1231 1453 0268 2223 0678 0540 0773 1095

using the python program below:

for n in ("0794","1832","1403","2474","1231","1453","0268","2223","0678","0540","0773","1095"): print pow(int(n),431,N)

we get

1907 0818 2608 1826 0026 1804 0217 0419 2612 0418 1800 0604

which is our original full message!

Using this mechanism, Alice will send 2 numbers N and e to Bob which he will use to encrypt the secret key and send to Alice. When Alice receives the encrypted secret key, she will use her private key d to decrypt it and get the secret key. After that, they can now start using the secret key to encrypt messages between them.

]]>https://www.asme.org/engineering-topics/articles/robotics/a-realistic-robotic-hand

**Can you tell us about your work?**

Vikash Kumar, a University of Washington doctoral student in computer science and engineering, is part of a team of researchers making great strides in developing a robot for a human body part that is one of the most difficult areas to replicate with the same functionality: the hand.

**What is your educational background? How did you end up in such an offbeat, unconventional and interesting career?**

Kumar grew interested in robotics after becoming disillusioned with the study of mathematics because it was only about numbers and equations on the board and on paper. “The only way I could appreciate those things was if I could use them in a real-world application. That’s when I saw robotics as a means to channel all my learning into something tangible,” Kumar says.

After earning his bachelor’s and master’s degrees at the Indian Institute of Technology Kharagpur (Mathematics and Computing), he enrolled at UW to study for his doctorate (Computer Science & Engineering) working in the university’s Movement Control Lab under the supervision of Professor Emanuel Todorov.

**How does your research benefit the industry?**

The five-fingered hand he and Todorov have built is a breakthrough because it can perform tasks that require highly dexterous manipulation, which traditionally has been very difficult for robots, plus it also can learn from its own experience and build upon that learning without ongoing human direction.

Most work with commercial robots has involved software engineers imagining every movement to perform a specific task and then programming those movements. Because of machine-learning algorithms that help the robot model both the basic physics involved and plan which actions it should take to achieve the desired result, the robot gets progressively more adept each time it performs a specific task, eventually learning by experience to respond as quickly as, or even faster than, a human hand.

Starting in 2011, Kumar spent two and a half years designing the hand, which has 40 tendons, 24 joints, and more than 130 sensors. Then he turned to working with different methods of controlling the hand. “I eventually ended up with combining both optimal control and the learning-based method because [combining them] has produced a much better result,” he explains. The benefit is that the kind of task that used to take thousands and thousands of trials of iterations now takes orders of magnitude less trial and less data, and this combination gives us an efficient way of learning, he adds.

The eventual goal is to have a functional equivalent of the human hand. The grand vision is to have robots mimic more closely what nature has done in creating what an animal can do, he said. “We are on the right track. It will be just a matter of time before we get there,” says Kumar.

“Now we have tools good at solving one particular job: a copy machine, a dishwasher, a washing machine. Soon you realize that every task you have to solve is asking for another piece of hardware. That’s why a robotic butler is something in the grand vision of robotics because that is one machine that does all the household chores for you,” Kumar says. “That’s why it’s important to get a functional equivalent of a human hand because the human hand is capable of solving various different tasks; it’s capable of peeling a label, capable of loading a dishwasher, capable of assembling IKEA furniture. But any one robot right now is good at only one of those.”

To date, the team has demonstrated local learning with the robot, meaning that the hand learns from and improves by performing a task repeatedly. The next step is to tackle global learning, in other words, have the robot figure out how to perform a task with an unfamiliar object or in a new situation it hasn’t faced before.

**What is your vision for the future?**

Kumar envisions endless possible end uses eventually for the basic technology, everything from defusing explosive devices to prosthetics to space or marine exploration to disaster recovery to health care. But his team’s focus will continue to be on developing the fundamental building blocks that underlie all the possible uses, making sure that all the steps involved are in the right sequence.

As excited as he is about the breakthrough work his team is doing, he is more excited to see where other scientists around the world may take the basic capability and begin to collaborate and combine their individual expertise for specific uses. He acknowledges that it’s not something that is going to happen overnight. “It’s not going to be one day or one month where we say it doesn’t work until it works. It’s a gradual process, probably five to ten years,” Kumar says.

But he does expect to see more results starting to come pretty soon as “more people jump in now that the problem seems more tractable. We will start to have small but gradual progress from now onwards.”

]]>If we take a look at the general definition of the term *Regression* we will find something like: “transition to a simpler or less perfect state”. *Perfection* is quite a subjective category, and, depending on the context and point of view, the same phenomena, by the same person, may be viewed as more or less perfect for one or another purpose. For example a “more simple” state or model may be viewed as “less perfect” for purposes of the simulation accuracy, or “more perfect” for easiness and clarity of understanding. So, let’s rather stick to a more clear and distinct “simplicity” aspect of the definition.

In application to the Data Science’s meaning of “simplicity”, and especially in the context of the space mappings, it would, obviously, mean reduction of the dimensionality and/or number and complexity of *Relations* between the *Space* objects. Which, actually, means *Projection* of our Data from a Super- to a Subspace. We won’t usually know beforehand which Subspace is more suitable for our purposes, but we may have an idea about possible variants from which we may chose, applying particular criteria, the best one. As it was already mentioned, any objects may be members of a Space, including other Spaces, or Subspaces. There, concept of Quotient (or Factor) Spaces may be useful. In such a Space its members are its disjoint (not having common elements) Subspaces.

Let’s imagine a Crape Cake, which, as a whole, is a 3D Space, but also it can be thought of as a 1D Quotient Space of the 2D Crapes. Also, let’s imagine we have Blueberries somehow stuffed in between our Crapes. And then, we somehow want to associate (via so called Equivalence Relation) all these Blueberries with only one Crape, for example by protruding (*Projecting*) those Blueberries through other Crapes by our fingers and smashing them into One Chosen Crape (or, making sure they somehow squeezed and moved through the holes made by toothpicks). All these Blue Spots on the One Chosen Crape we may call Equivalence Class. And we may want to minimize the ruin we have just done to our Cake by choosing the One Crape that would ensure that, and that will be condition for our Equivalence Relation.

Of course, there may be other criteria chosen, for example a Crape with biggest holes in it, or something else. Also we may want to choose boring holes in the Cake not with the straight, but crooked fingers (that won’t be a Linear Transformation), or put the Cake on the edge of table, and let it bend like Clocks on Salvador Dali paintings (that won’t be Linear/Vector Subspaces), and then bore it with straight fingers. We may decide those non-linear Blueberry Transformations and Subspaces are even cooler than the linear ones (for example the crooked holes in the Cake would make it a Piece of Art), but for the Linear, One Output Parameter Regression from 3D Space into 2D Subspace we will stick to the algorithm (Linear Projections from Vector Superspace into Vector Subspaces) described in the former paragraph.

Technically, we may use Linear Transformations (but not Projections (which immediately eliminate dimension(s) of the original Vector Space)) that vary from one Data element to another, and actually may be a way to linearize non-linear transformations (not your usual Linear Regression), but that will call for a bit different mathematical treatment (adding one more transformation in the target subspace) of the Transformation presented in the next chapter.

…

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Einstein was brilliant, we all can agree.

He opened up new possibility –

the Theory of Relativity

he worked out, questing for simplicity.

His equation, E=mc2,

is a well-accepted reality;

there’s no doubt about how well it has fared

in destruction and practicality.

Though people know this, most don’t understand

the mathematics and don’t even try

to comprehend a mind that was so grand.

They claim their brains simply don’t qualify.

Yet these same folks don’t even think it odd

to claim they understand the mind of God.

]]>

Okay here we are, I have presented a presentation in my mathematics club on the fibonacci numbers and the consequences of fibonacci numbers.This is about the history of fibonacci numbers in India. So enjoy it, if you found any thing which should be not there, please let me know.

There is the link and do not forget to run it in slide show mode.

If you have any query, I have provided the contact form, feel free to contact.