XEVEX Spring 2023
Sarah Silverman ‘24 Grace Kollander ‘25
The Ramaz Mathematics Publication
3.14159265358
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TABLE OF CONTENTS
The Birthday Paradox Julius Zimbler ’24
...3 Fast-Fourier Transform Philip-David Medows ’24 ...10 Probability and The Law of AverMachine Learning: A Brief Introages duction Grace Kollander ’25 Philip-David Medows ’24 ...11 ...4 Fermat’s Enigma Review Sarah Silverman ’24 ...5 Fermat’s Enigma Review Leo Eigen ’25 ...7 Erdos Numbers Brayden Kohler ’23
...8
Things Named After Euler Brayden Kohler ’23
...9
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The Birthday Paradox Julius Zimbler
The Birthday Paradox is a statistical phenomenon that may seem counterintuitive at first glance. The paradox states that in a group of 23 people, there is approximately a 50% that at least two people will have the same birthday. This might seem impossible, but what is even more surprising is that if there are 50 people, the odds jump to 90%, and if there are 75 people, there is a 99.97% chance two people will have the same birthday. To understand why this is true, we have to understand the basic principles of probability. How is it possible that to have two of the same birthdays, we only need a sixth of 365 to be sure there is a match? One might think the probability is 23/365, but this is not true. The easiest way to show the likelihood is not to explain the probability that there is a match but to find the likelihood of when there isn’t a match. We have to assume that every person has the same 1/365 chance of being born on any day of the year. There are no leap years and no twins to make the math easier. When the first baby is born, there is a probability of 365/365 (or 1) that that person shares a birthday. We then multiply that by the second baby’s chance of not sharing that birthday, which is 364/365. We repeat this process for how many people we have in the group. By baby #23, we are multiplying by 343/365. To write this into an equation, we would put 364! on the numerator. It’s 364 and not 365 on the numerator because 365/365 is just 1. We then put 342! on the denominator because 365-23 is 342. We then multiply the denominator by 365 days in the year and raise that two to the 22nd power (which is the first baby
subtracted from our 23 babies). Formally written, the equation would then be 364!342! * 36522. The result of the equation gives us .4927, or 49.3%. But this is the probability that we don’t share a birthday, so we would subtract the answer from 1, and the result would be .5072, or a 51% chance. Now we understand how this is possible mathematically, but it still doesn’t seem to make much sense. To simplify, it’s not as if ten people are trying to match the same birthday with a pre-determined person, but everyone is being evaluated against each other. So we are saying that any two babies out of the 23 shares a birthday. Probability is a fundamental concept that is present everywhere in our world. It is used to model and understand the behavior of random and uncertain phenomena and to make decisions based on incomplete or uncertain information. Even our own actions and choices are influenced by probability. Probability is a powerful tool that can be used in any field to help us understand and navigate the complexity and uncertainty of our world.
Probability and The Law of Averages Grace Kollander Probability is how likely an event might occur. Probability is significant in ways one might not expect. For example, Meteorologists use weather patterns to predict the probability of rain. Probability is measured by the ratio of the favorable cases to the whole number of cases possible. If one wants to know how likely a certain outcome of an event is, one can calculate the probability. In order to calculate the probability you take the number of favor3
able outcomes divided by the total number of outcomes (x) , P(H) = x. One simple example to explain probability is flipping a coin. What is the probability of flipping the coin and getting head? Intuitively you understand that the probability is 50%, but if you had to plug that into the equation to find the probability, it looks like this, P(heads) = ½. There is one favorable outcome and the total number of possible outcomes is 2. Another example is rolling a die. For instance, you are playing a game of monopoly, with only a single die, and you are three spaces away from free parking, and you want to know the probability of you landing on free parking. You would start by noting that there are 6 outcomes when rolling a single die. What is the probability of rolling 1/ 6 of those numbers, in this case, a 3? You would P(3) = 1/3, which would equal approximately 0.1667 if you were to ask what the probability of rolling an even number you would do P(even)= 3/6. The Law of Averages, created by Jacob Bernoulli, is the idea that a certain outcome is bound to happen because of the amount of times it generally happens. There is a misconception that the Law of Averages occurs with a small number of consecutive experiments, and they will eventually average out. For example, if you flip a coin 8 times and all 8 times the coin lands on tails, you are wrong to assume you will roll a heads next because it must average out; this is not how that works. The Law of Averages is based on the Law of Large Numbers. The Law of Large Numbers is if you take an unpredictable experiment and repeat it a lot of times, you can find an average. This is confused with the Law of Averages because people think the same goes for if you experiment only a few times, but that is not how it works. 4
Fermat’s Enigma Review Sarah Silverman
Fermat’s Enigma is not a book I would have picked up of my own accord. To be completely frank, I have seen it on the library shelves and have never had an impetus to read the back cover page. Writing this review has led me to reconsider my previous rejection of math-related books. Perhaps, as is the case with Mathematics in general, proofs and theories seem incredibly intimidating to the average teenager or even adult, to call a spade a spade. There is a larger narrative in our culture of math being too difficult for the average person to understand. Even though I am at the honors level and do well, I still don’t feel comfortable expressing my mathematical abilities to others, fearful that one of the people who just “get everything” will outpace me in answering a question or expressing an idea. A societal myth exists that there are two types of math learners; those who get everything at first glance and the rest of us (who even wonder about the point of studying a subject that they will never master). This is fundamentally untrue, and by requiring students in our class to read Fermat’s Enigma, we are breaking down stereotypes of who Math books are “supposed to be for.” Although Singh, the author of Fermat’s Enigma, tries to portray Wiles as Math’s Clark Kent, who, with his 130-page manuscript filled with the blood, sweat, and tears of eight years of solitude and public pressure due to a faulty gap, conquered all odds and brought number theory into a new era of discovery. The effort is a bit overdone. He even pulls in the sentimental note of how this was Andrew’s lifelong dream, something that most of the readers of this book are unable
to accomplish in their own lives. This makes the Ivy League mathematical genius’s story quasi-relatable to his audience. The narrative is all well and good, but it leaves me with the fundamental question of what did Wiles actually do? How did he solve it? Singh can list the page quantity and a brief, almost matter-of-thefact statement of how the mathematician put the Iwasawa theory and Kolyvagin and Flach principle together, but with nearly 300 pages of buildup, doesn’t this influential step deserve a longer, more descriptive explanation? I, for one, was somewhat dissatisfied with the deflated ending. Now I understand that Wiles’ mindset requires a deep understanding of advanced math, something I, and most readers, do not have. Singh cannot explain years of research in a couple of pages. However, if a person were to ask me, after hearing a synopsis of the book, “How did Wiles finally solve Fermat’s theorem?” I could not even begin to explain Wiles’ thought process in the final stages; I would only know the names of the two concepts of the things he put together. A slimmed-down version of Wiles’ proof would have been helpful for the average reader to understand how it was derived, as well as possibly enabling the reader to learn new mathematical skills while reading the excellent history Singh has laid out for us. Reading Fermat’s Enigma has changed my perception of math. Math theories are not isolated rules that have existed since the beginning of time. Like other branches of knowledge, mathematics constantly develops new ideas. It is a fluid system whose history is complex and still being written. By providing a narrative overview of the cult of Pythagoras and discussing the mathematical climate
around Pierre de Fermat’s life, Singh helps the reader understand the field’s changes throughout the ages. Dr. Tugendhalf’s guest lectures helped me gain a deeper appreciation of this, and I enjoyed the mixing of these two Ramaz departments. The practical math inside Fermat’s Enigma was easy enough to understand. Sometimes I had to reread certain chapter sections, but the conceptual knowledge I have learned will help me throughout my academic career. When my friends in super-honors Math initially told me I would be reading a book in Rabbi Stern’s class this year, I was skeptical. An English book in Math class? What is this? The point of school is to expand students’ comfort/ knowledge zone. Having this book as required reading enabled me to acquire skills I would not have necessarily received in a regular pre-calculus course. Through this experience, I have gained insight into the world of math academia while learning about the process of solving one of the world’s greatest problems. And who knows, next time I see a theoretical math-related book on the shelf, I might just pick it up and read it!
Fermat’s Enigma Review
Leo Eigen a^n + b^n = c^n Though it seems quite simple at first glance, this equation was considered for centuries to be an unsolvable beast. It was proposed by French mathematician Pierre de Fermat in 1637, who famously wrote that “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.” In the centuries that followed, mathematicians grappled with the unsuspecting complexity of Fermat’s Last Theorem— and it came to the point where people were discouraged from even trying to 5
solve it out of fear of wasting a career on an unfruitful mission. The book Fermat’s Enigma by Simon Singh focuses on the centuries-long quest in the mathematical community to find a solution (or lack thereof) to Fermat’s Last Theorem. Singh chronicles the lives and works of the mathematicians who tried and failed to be the one to solve it, and culminates with British mathematician Andrew Wiles solving it in the 1990s along with the help of Richard Taylor. Throughout the book, Singh explains how numerous discoveries and people in the mathematical world contributed directly or indirectly to the eventual solving of the Theorem. Singh’s book is engaging, and it lays out the information in a clear and interesting manner which allows the reader to understand the inner workings of the academic world of mathematics and which presents famous mathematicians not only as lifeless historical figures but as individual personalities each of whose surroundings made an impact on their works. Though I can’t claim to suddenly understand the Taniyama-Shimura Conjecture or Modularity Theorem after having read the book, at least I have gained an appreciation for the immense devotion and years of work that go into making a mathematical breakthrough. Mathematicians, just like everyone else, experience frustration and difficulty when trying to make a name for themselves, as shown by the numerous failed attempts at solving Fermat’s Last Theorem. Perhaps what made the biggest impression on me from the book came in the epilogue. Singh makes an interesting point: considering that Andrew Wiles made use of numerous techniques in his proof which were first hypothesized in the 19th and 20th, is there a chance that even the proof of 6
Pierre de Fermat himself was faulty? Is it possible that the man for whom the theorem was named was also misguided in his logic? There is no way to be certain, but it is definitely an interesting question for Singh to raise at the end of a book which is based on the very premise that Fermat had stumped the mathematical community for years. In conclusion, I highly recommend the book. I would like to end my review by quoting one of the reviews on the back of the book, written by the Library Journal: “Singh captures the joys and frustrations of this quest for an extremely elusive proof...and builds to a truly engrossing climax...”
Erdos Numbers Brayden Kohler
Paul Erdos (1913-1996) was an incredibly influential mathematician. To this date, Erdos has published more papers than any other mathematician in history (at least 1,525) that are mostly co-written. Euler published more pages of mathematics, however, he only published about 800 separate papers. As a tribute to Erdos’s impact, his friends invented the Erdos number. A person’s Erdos number describes your “distance” to Erdos in terms of co-authorship. Paul Erdos has an Erdos number of 0 and all of his co-authors have an Erdos number of 1. 509 people have an Erdos number of 1 and ~12,600 people have an Erdos number of 2. Many historical figures have Erdos numbers as low as 3, such as Srinivasa Ramanujan who died in 1920. The median Erdos number of a Field’s Medalist is 3 and the median of all people with an Erdos number is 5. As time goes on, the median Erdos number will decrease as people with smaller Erdos numbers no longer
live to co-author papers. If one has no route to get to Erdos (Their only collaborator has no other collaborators themself), their Erdos number is infinite. The largest finite Erdos number is 15 which very few people have. Terence Tao, has an Erdos number of 2. John von Neumann has an Erdos number of 3. And Dr. Fabio Nironi has an Erdos number of 4: Nironi → Ginibre → Kasteleyn → Richmond → Erdos In 2004, William Tozier with an Erdos number of 4, auctioned off the possibility of co-authorship. People debated the ethics of selling Erdos numbers and how it affects amateur mathematicians. In the end, William Tozier’s auction was sabotaged by an anonymous bidder. They bid $1,031 but refused to pay it. They said that they had only made that bid “to stop the mockery this person is doing of
the paper/journal system.”
Things Named after Euler Brayden Kohler
Leonard Euler is considered the most prolific mathematician of all time. Pierre-Simon Laplace once said, “Read Euler, read Euler, he is the master of us all” and he couldn’t be more correct. Euler has had his hand in all sorts of studies from mathematics to geography and as a result, many things have been named after
him. To avoid naming too many things after Euler, discoveries are sometimes named after the second person to discover them. This list doesn’t include everything named after him, just selected ones of my interest. Euler’s Number The famous number e is Euler’s number. This number appears in many places, especially ones that relate to exponentials (like compound interest). e can be defined in multiple ways, here are a few:
Euler’s Formula This is a very important equation that establishes the relationship between trigonometry and complex exponentials. e^ix = cos(x) + isin(x) From this equation we see that: e^i* pi = cos(pi) + i(sin(pi)) = -1 e^i*pi+1=0 (Euler’s identity) Euler’s Sum of Powers Conjecture Euler conjectured that if the sum of n positive integers raised to the kth power equals another kth power, provided that k is an integer, then k n. An example where this conjecture holds is as follows: 3^3+ 4^3+5^3=6^3 This conjecture has been recently disproven, however, with the following counterexample: 2755+8455+11055+13355= 14455 Here, k=5 but n=4. אתבוית Euler’s Rotational Equation This equation describes the rotation of an object where the reference is itself rotating with angular velocity (speed of rotation) of w: 7
Ia+w * (iw) = t Where I is the inertia matrix, the vector α is the angular acceleration, and t is the applied torques (the rotational analog to forces). Euler’s Four Square Identity This identity states that if you take two numbers, each of which are the sum of four squares, and multiply them by each other, the product is also a sum of four squares. An example case is as follows: ( 1 ^ 2 + 2 ^ 2 + 2 ^ 2 + 2 ^ 2 ) (1^2+1^2+1^2+2^2) = (1^2+1^1+5^2+8^2) Lucky Numbers of Euler A lucky number of Euler is a positive integer n such that for all integers k where 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. Only 7 lucky numbers of Euler exist, they are: 1, 2, 3, 5, 11, 17, and 41. So, with the number 3: 2^2-2 + 3 = 5 But for the number 7: 5^2-5 + 7 = 27 = 9*3 and therefore 7 is not a lucky number of Euler Euler’s Laws of Motion Euler’s first law of motion states that the net external force on an object is equal to the rate of change of its linear momentum: F = dP/dt Euler’s second law of motion states that the net external torque on an object is equal to the rate of change of its angular momentum 2002 Euler 2002 Euler is an asteroid named after Euler due to his contributions to astronomy. Euler Spiral An Euler spiral is a curve whose curvature changes linearly with the curve’s length. One interesting property of this is that if the globe were to be cut along a spiral of width 1/N, when 8
flattened out it would make an Euler spiral. An Euler spiral is a curve whose curvature changes linearly with the curve’s length. One interesting property of this is that if the globe were to be cut along a spiral of width 1/N, when flattened out it would make an Euler spiral. This map projection will have minimal distortion (and zero distortion as N tends to infinity ).
Fast-Fourier Transform Philip-David Medows
What do the detection of nuclear tests, cell phone signals, and music files all have in common? They all rely on the Fast Fourier Transform (FFT). To understand what the FFT is, and what it does, one needs to take a step back and understand how signals are represented mathematically. Most signals, like the ones broadcasting from a phone to the internet look something like a random squiggly line that appears to have no internal mathematical structure. However, this line can actually be broken up into many different sine and cosine waves that add up, or combine, to form a complex looking signal. To visualize this, consider this photo: An example of this is the equation: 3+ cos(x) +1.5 sin(x)+2cos(2x) +8
sin(2x)+... + a to the nth cos(nx)+ b to the n sin(nx) It is relatively easy to add up these sine and cosine functions, but it is much harder to decompose the resulting signal into the original sine and cosine waves that make it up. However, before we go into how this process works, it is very useful to understand why so many analysts and researchers would want to decompose these signals into sine and cosine functions in the first place. Signals in both a physical and abstract way, convey information. Physically, signals are composed of amplitude and frequency. The amplitude is the strength of the energy of the signal, which, mathematically, is the sum of the area underneath the various sine and cosine waves that make up the signal. The frequency of the signal is how fast the signal is moving, which mathematically, is how fast the sine and cosine waves are repeating, or their periodicity. However, these amplitudes and frequencies are not necessarily obvious in the original physical manifestation of the signal, as the signal is just one line, rather than many that display individual frequencies and amplitudes. In an abstract sense, the various frequencies and amplitudes can communicate bits of information (as in 1’s and 0’s) or other physical properties, like frequencies emitted by musical instruments. Interestingly enough, frequencies and amplitudes are useful in the detection of earthquakes and nuclear tests underground via seismographs. Thus, the importance of translating these signals into frequency waves cannot be overstated. The impetus for the development of the FFT was surprisingly enough the Cold War. The U.S.A. and the U.S.S.R were both working on treaties to severely limit nuclear missile tests both above
ground and underground. However, while the detection of above ground nuclear missile tests is relatively simple to enforce the treaties, the detection of underground nuclear tests is not as straightforward. A common method to detect these tests underground is to use a seismograph to detect unexpected variations in both amplitudes and frequencies that could signal that a test has occurred. Nevertheless, it is difficult to distinguish a test from a minor earthquake, which occurs very often. Thus, the signal has to be transformed in some way to obtain the various amplitudes and frequencies it contains. At the time, the leading method to accomplish this was called the Discrete Fourier Transform (DFT). A big issue with the DFT is its computational complexity: the number of steps needed for the DFT scales with the square of the number of data points. This meant that the computations necessary to detect underground tests would take months, if not years, on computers during the 1960’s. As a result, underground nuclear tests were not banned in resultant treaties since a ban would have been unenforceable.
In an effort to include a ban on underground tests, James Cooley and John Tukey discovered a faster algorithm to replace the DFT. This was dubbed the Fast Fourier Transform (FFT). The algorithm takes the final output signal composed of n data 9
points and takes at most N * logN steps to decompose the signal. This may not seem that important at first, but when compared to its predecessor the benefits were massive. For example, a signal with a million data points would take 10^18 steps to complete its decomposition via the DFT, while the same signal would take approximately 3* 10^10 steps to run via the FFT. The magnitude of this difference is enormous, as the FFT is 100,000 times faster than the DFT in this case. However, the advent of the FFT came too late, as it never allowed a comprehensive ban on underground nuclear tests.
Machine Learning: A Brief Introduction Philip-David Medows
When most people hear the phrase ‘machine learning’, they envision a computer learning like a human: how to draw, how to speak, and so on. While this is partially true, machine learning is in fact somewhat true, as the mathematical structure behind machine learning is modeled after that of neurons in the brain. Machine learning usually makes reference to some form of an artificial neural network (ANN). To understand what an ANN is, it is helpful to first understand what a neural network is. Broadly speaking, a neural network is a biological structure that is composed of neurons, or messenger cells found in the brain. These neurons, often numbering in the millions or billions, often cannot individually compute anything, such as the text of a sign, but collectively can learn to recognize patterns and learn from data over time. Neurons are able to collectively accomplish amazing feats because of their interconnectedness. For example, one neuron 10
can have thousands of connections to other neurons, making the overall learning structure very large, which allows for complex learning. An ANN takes this structure and represents it via layers of artificial neurons. Naturally, such a mathematical structure should have layers that operate on previous layers’ output to mimic real-life neurons. A neat way to mathematically model this is through an artificial neuron. An artificial neuron is a linear function that operates on the outputs of the previous layer. This means that it can be represented via a matrix, which can describe a linear function.
The main benefit of this approach is the fast computability of the output of each layer of neurons by a computer. However, someone may wonder what the original input to this neural network is, since the first neurons need to operate on some form of input. This is typically problem-specif An ellipse is a plane curve surrounding two foci, such that the sum of the two distances from both foci to any point on the ellipse is a constant, which is equal to 2a . The general equation for an ellipse whose major axis is on the x or y axis is x2a2 + y2b2 = 1. This article will discuss the derivation of
a rotated ellipse, whose major axis is oblique, and the equation of the major axis being (y-k)=m(x-h). To start, find the foci of a rotated ellipse (which lie on the major axis) in terms of m (the slope of the major axis), c (the distance from the foci to the center), x and y. Using the pythagorean theorem, we can establish that x2+y2=c2. In addition, the equation of the major axis is established as y=mx. Therefore, we can substitute mx for y in the first equation, giving you x2+m2x2=c2, which factors to x2(1+m2)=c2 . Using some algebraic manipulations, you can get x=c1+m2, which you can plug back into the equation y=mx, giving you y=mc1+m2, which gives you the coordinates for one of the foci of the ellipse, (c1+m2,mc1+m2). Since the ellipse is centered about the origin, the coordinates for the second focus will be the same, except they’re in the negative direction, giving you (-c1+m2,-mc1+m2). Using the distance formula to find the distance from the foci to a generic point on the ellipse (x, y), we can derive the formula of a rotated ellipse that is centered about the origin. The distance formula is d= (x2-x1)2+(y2-y1)2. So, substituting the x coordinate of first focus for x2, and the y coordinate of first focus for y2, we can find the distance of a generic point, (x,y) from the first focus, which will become d1=(c1+m2x)2+(mc1+m2-y)2. We can do the same for the second focus, which will give you d2=(-c1+m2- x)2+(mc1+m2-y)2 . Since we know that the sum of these two distances will always equal a constant, 2a, we can combine two equations, becoming
(c1+m2- x)2+(mc1+m2-y)2+(-c1+m2x)2+(-mc1+m2-y)2=2a. Now, to account for shifts, in terms of h and k, the equation becomes (c1+m2- (x-h))2+(mc1+m2(y-k))2+ (-c1+m2- (x-h))2+(-mc1+m2(y-k))2=2a. Using the calculations attached, we can see that the equation of a rotated ellipse simplifies to c21+m2(x+my)2+(a2-x2-y2)=0 , or, to account for shifts, c21+m2((xh)+m(y-k))2+(a2-(x-h)2-(y-k)2)=0 . Not including shifts, the expanded version of this equation is: c21+m2x2+2mc21+m2xy+m21+m2y2+a2 -x2-y2=0, which, when converted into the general form of an ellipse (Ax2+By2+Cxy+Dx+Ey+F=0), would make A=c2-m2-11+m2, B=-11+m2, C=2mc21+m2, and F=-a2. Because this equation doesn’t account for shifts, there are no D and E terms included yet. But, if we were to account for shifts, the equation would have these D and E terms, which would be D=hc1+m2, and E=mhc1+m2. Therefore, in the given equation, 3x2+4xy+4y2-3x+2y-6=0, Which when you solve for each coefficient, the result gives you that m=58, c=4312, and a=6, thus making the equation of the major axis y=-58 x, the foci (43121+58-1.5,4312 (58)1+581.18), and (-43121+58+3.5,-4312 (58)1+58-0.4), and the vertices (-61+58-2.82,-6 (58)1+58-0.8), and (43121+58+4.77,6 (58)1+58-0.73). ic, but can be thought of abstractly as information. A great example of this is the problem of digit classification, or determining what digit a drawn image represents. For simplicity, assume that the picture
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is a grayscale image, so all of the pixels are simply a number between 0 and 1, with 0 being black and 256 being white. An example of this would be the image below:
. In this case, the inputs would be the value of each of the pixels. The challenge then would be to train a neural network that can ‘read’ an image to determine what digit is being depicted. At first, this task seems very hard, if not downright impossible. However, with a little bit of calculus, this problem can be solved by a computer (not by hand!). For those acquainted with calculus, recall that a derivative describes whether a function is increasing or decreasing. The connection between a derivative and a neural network is that of a loss function.
The trick here is to create a loss function, which measures the error of the model, or the distance from the output of the neural network (in this case a digit from 0 to 9) to the correct output. This is possible because a neural network can 12
be thought of as a giant function with thousands of variables, since the coefficients of the various matrices of the artificial neurons can be tweaked accordingly. So, to improve the model’s accuracy, this loss function should be minimized. To see how much to change each variable in the matrices that make up the neural network to minimize the loss function, all one needs to do is to take the derivative of the loss function with respect to that variable. This can be accomplished with the chain rule for multiple variables (see image). Afterwards, this derivative is multiplied by -1 to lower the function’s value, since we are trying to minimize the error. So, the model’s various variables can be adjusted accordingly to decrease the error rate. A very important factor to consider is that the input should have a pre-labelled output when training the model, as otherwise, the loss function cannot be computed and other methods must be used (see unsupervised learning for more details). This stage of the learning is called the training phase. Since this process is repeated many times for thousands of variables, it is a virtual necessity to have a computer compute these derivatives and adjust the model. After this initial stage of training data is completed, the next step is called the validation phase. The validation phase is almost identical to the first training phase, as the data used is often of the same form of the training set. This is where the model is tested to determine its accuracy and to continue improving it performance as well. This method yields suprisingly accurate results with enough training data and computing power, and it is the reason that Siri can understand your
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44184263129860809988868741326047215695162396586457302163159819319516735381297416772947867 24229246543668009806769282382806899640048243540370141631496589794092432378969070697794223 62508221688957383798623001593776471651228935786015881617557829735233446042815126272037343 14653197777416031990665541876397929334419521541341899485444734567383162499341913181480927 77710386387734317720754565453220777092120190516609628049092636019759882816133231666365286 1932668633606273567630354477w628035045077723554710585954870279081435624014517180624643626 79456127531813407833033625423278394497538243720583531147711992606381334677687969597030983 39130771098704085913374641442822772634659470474587847787201927715280731767907707157213444 73060570073349243693113835049316312840425121925651798069411352801314701304781643788518529 Reach out to: 09285452011658393419656213491434159562586586557055269049652098580338507224264829397285847 Sarah Silverman (silvermans@ramaz.org) 83163057777560688876446248246857926039535277348030480290058760758251047470916439613626760 44925627420420832085661190625454337213153595845068772460290161876679524061634252257719542 Grace Kollander(kollanderg@ramaz.org) 91629919306455377991403734043287526288896399587947572917464263574552540790914513571113694 10911939325191076020825202618798531887705842972591677813149699009019211697173727847684726 86084900337702424291651300500516832336435038951702989392233451722013812806965011784408745 Faculty Advisor: Dr. Fabio Nironi (nironif@ramaz.org) 19601212285993716231301711444846409038906449544400619869075485160263275052983491874078668 08818338510228334508504860825039302133219715518430635455007668282949304137765527939751754 61395398468339363830474611996653858153842056853386218672523340283087112328278921250771262 94632295639898989358211674562701021835646220134967151881909730381198004973407239610368540 66431939509790190699639552453005450580685501956730229219139339185680344903982059551002263 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Thank you to everyone who contributed to this issue of Xevex! Interested in becoming a writer?