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Ulam Spiral: Discovered by Stanislaw Ulam, the Ulam Spiral or the Prime Spiral depicts the certain quadratic polynomial's tendency to generate large number of primes.Ulam constructed the spiral by arranging the numbers in a rectangular grid . When he marked the prime numbers along this grid, he observed that the prime numbers thus circled show a tendency ...

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In the figure below : To find the remainder on dividing a number by 7, start at node 0, for each digit D of the number, move along D black arrows (for digit 0 do not move at all), and as you pass from one digit to the next, move along a single white arrow. For example, let n = 325. Start at node 0, move along 3 black arrows (to node 3), then 1 white ...

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Here are some random ones that have a narrower scope: they concern more about matrices than about linear algebra in general. If a system of homogeneous linear equations is overdetermined (i.e. it has more equations than variables), it must have only the trivial solution. (False. Consider $Ax=0$, where $A$ has linearly dependent columns.) If $A$ and $B$ are ...

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If $T: \mathbb{R}^3 \to \mathbb{R}$ is a linear transformation, then $T(x,y,z) = ax + by + cz$ for some $a,b,c \in \mathbb{R}$. If $S,T: V \to V$ are linear transformations and $v \in \ker T$, then $v \in \ker S \circ T$. If two non-zero vectors are linearly independent, then one must be a scalar multiple of the other. If $T$ is invertible, then $\ker T = ... 2 Don't know how difficult these need to be. Here are some rather simple ones: For every linear map$f : \mathbb R^m \to \mathbb R^n$there is a unique$m\times n$matrix with$f(v) = Av$for all$v\in \mathbb R^m$. Answer: No, its actually an$n\times m$matrix. The expression$Av$would not even make sense. If$AB = AC$and$A\neq 0$, then$B = C$for all$...

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A simple but not trivial question: (T/F): If $n\times n$ matrix $A$ is of rank $n$, then $A$ is diagonalizable by a similarity transformation $$D = P^{-1}AP$$ Another, slightly harder one: If $P(x)$ is a polynomial and $A$ is a matrix with eigenvalues $\lambda_1 \ldots \lambda_n$, then $P(A)$ has eigenvalues $P(\lambda_1) \ldots P(\lambda_n)$

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A question I am fond of asking my students: Does the vector space of polynomials of degree $\leq n$ have a basis consisting of polynomials with same degree? Answer: yes.

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When I first learned that equations of degree five or higher may not be solved using an equation similar to the infamous quadratic formula, I felt a great amount of despair.

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$1-2\left|x-\frac12\right|$ has the three-cycle $2/7,4/7,6/7$. This example comes from the paper "The Sharkovsky Theorem: A Natural Direct Proof", by Keith Burns and Boris Hasselblatt. A preprint is freely available online, and it's a great read.

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Just pull some function values out of a hat -- for example $$f(0) = 42 \qquad f(42)=117 \qquad f(117)=0$$ and then do Lagrange interpolation (or for that matter linear interpolation, whatever floats your boat) between those points. Your attempt with a first-degree polynomial failed because a first-degree polynomial iterated three times is still a first-...

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Spaces in which countable intersections of open sets are open are called $P$-spaces. (Warning: the same term is also used with a completely different meaning.) The co-countable topology on an uncountable set is an example of a non-discrete $T_1$ $P$-space. In general we can start with any space $\langle X,\tau\rangle$ and let $\tau'$ be the collection of $G_\... 1 For a series of positive terms of the form $$\sum_{n=1}^{\infty}\frac{1}{n^p}$$ a. it converges if$p>1$b. it diverges if$p\le 1$(p-series test) For eg.-$$\sum_{n=1}^{\infty}\frac{2n+3}{n^2+5}$$ diverges as for$n\rightarrow\infty$,$u_n=\frac{2n+3}{n^2+5}\approx\frac{2n}{n^2}\approx\frac{1}{n}$. 0 This is along the lines of a nonmeasurable set, and Feynman may have rejected it on physical grounds, but theorem: There exists a nowhere dense set with positive measure, a fat cantor set 1 When trying to prove that $$\sum^{j}_{k = i} (-1)^{j + k} \binom{j}{k} \binom{k}{i} = \delta_{i j}$$ (which is related to this porblem ) one can start with$xand expand the expression by adding 0, with the goal of getting two binomial coefficents. \begin{align*} x^{j} &= (x + 0)^j = (x +1 -1)^j =((x +1 ) -1 )^{j} \\ &= \sum^{j}_{k = 0} \... 1 Here is a neat test that is relatively unknown. In any case I can think of where it would be practical to apply, I prefer a direct comparison or limit comparison, but it's certainly still interesting and useful. Consider the series\sum_{n=0}^\infty a_n$. Suppose you have a sequence$\{b_n\}_{n=0}^\infty$such that$\sum_{n=0}^\infty 1/b_n$diverges. Then ... 0 If the series is of the form$\sum_{k=1}^\infty a_k^k$, the root test might be useful. It states that in case$\lim_{k\to\infty} |a_k| < 1$the series converges and in the case$\lim_{k\to\infty} |a_k| > 1$the series diverges. Example: Take the series$\sum_{k=1}^\infty \left(\frac{4k+5}{2k+3}\right)^k. For the root test we compute \begin{align} \... 0 As a start one should check the term test. When you investigate a series \sum_{k=1}^\infty a_k and (a_k)_{k\in\mathbb N} does not converge to zero, then \sum_{k=1}^\infty a_k diverges. 0 When I took the math GRE the only trig identities I had memorized were \sin^2\theta + \cos^2\theta = 1 and \tan^2\theta + 1 = \sec^2\theta. I don't recall whether I actually made any use of them. 1 I saw this proof in an extract of the College Mathematics Journal. Consider the Integeral : I = \int_0^{\pi/2}ln(2cosx)dx From 2\cos(x) = e^{ix} + e^{-ix} , we have: \int_0^{\pi/2}In(e^{ix} + e^{-ix})dx = \int_0^{\pi/2}In(e^{ix}(1 + e^{-2ix}))dx =\int_0^{\pi/2}ixdx + \int_0^{\pi/2}In(1 + e^{-2ix})dx The Taylor series expansion of ... 1 Trolling Euclid by Tom Wright is my favorite book about unsolved problems. Very nice read. 0 Topology plays a huge role in contemporary condensed matter physics, and many top-notch researchers have made a career of it---e.g. Xiao-Liang Qi at Stanford. There are more applications of topology to condensed matter physics than I can begin to enumerate (the above linked page names several), but I'll highlight two: Dislocations in crystals. These ... 2 I think Lectures and Exercises on Functional Analysis by A. Y. Helemskii might be exactly what you're looking for. Quoting from the introduction: Perhaps the main idea is that our book is written from the categorical point of view. Everywhere we stress and comment on the categorical nature of the fundamental constructions and results (like the ... 0 The Erdos-Ko-Rado theorem is a result about intersecting set families. Suppose A is a set of r subsets on the set \{1,2,\dots,n\} such that any two sets in A have a non empty subset and n/2\geq r, the maximal size of A is \binom{n-1}{r-1} i.e.,|A| \leq \binom{n-1}{r-1}$$with equality holding if and only if all the sets share a common ... 3 Bruce Sagan's "The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions" is probably exactly what you are looking for. It covers basic representation theory but quickly moves into the representation theory of the symmetric group. 0 Alan Tucker's book is rather unreadable. I'd avoid it. Nick Loehr's Bijective Combinatorics text is much more thorough, and it reads like someone is explaining mathematics to you. It mixes rigor and approachability quite well. 2 The proof of the Littlewood-Offord lemma for sums of real numbers. Erdos noticed that under the correspondence between sequences of \pm signs and finite sets, Sperner's theorem applies and gives the optimal bound for the Littlewood-Offord lemma in dimension 1 (on how many signed sums of n given numbers of absolute value at least 1, can have absolute ... 1 The product of consecutive integers is never a power This was proved by Erdos and Selfridge. http://www.renyi.hu/~p_erdos/1975-46.pdf 2 Erdos - Mordell Inequality : For a point O inside a given triangle ABC, the perpendiculars OP, OQ and OR are drawn to the side$$OA + OB + OC \geq 2(OP + OQ + OR)$$Here's a proof from Donat K. Kazarinoff http://projecteuclid.org/download/pdf_1/euclid.mmj/1028988998 A simple visual proof is provided by Claudi Alsina and Roger Nelson in this ... 9 Erdos' proof of Infinite primes The following proof is taken from the book - "Proofs from THE BOOK" by Martin Aigner and Gunter Ziegler. This proof is attributed to Erdos. This proves that there are infinitely many primes and that the series of the sum of prime reciprocal steps diverges. Let us assume that the infinite series \sum\frac{1}{p}, where p ... 8 Erdos' favourite questions. The following are not research papers but popular questions Erdos used to ask children. If n+1 integers are chosen from the first 2n integers, there will always be two that are co prime. There will be two numbers that are consecutive. These two numbers will be relatively prime. To see that this is not true when n ... 18 I think this is worth posting here, mostly because I really enjoy the simplicity of this proof but also because I have no idea how well it is known. The result is not deep or important, so the main interest is in the simplicity of the argument. Erdős proved a lower bound on the number of primes before an integer n. Wacław Sierpiński, in his Elementary ... 10 One very simple, and yet one of my favorites is the Erdős-Anning theorem: Let A \subseteq \mathbb C be an infinite set of points, such that$$ \forall x, y\in A \quad |x-y| \in \mathbb N $$then there exists some c,k \in \mathbb C , such that all a \in A is of the form a = cx + k for some x \in \mathbb R . It was proved in 1945 in the ... 5 Erdos answered the following question in the affirmative - Are there infinitely many odd numbers that are not expressible as the sum of a prime number and a power of 2. The proof is explained in this paper : http://www.maa.org/sites/default/files/3004416309960.pdf.bannered.pdf The essence of the proof is in showing that for every integral value of k, ... 14 Here is an exposition of the proof that made Erdos famous by David Galvin. An elementary proof of Bertrand's postulate, which states that there is a prime number in between every n and 2n. The essence of this proof is in noticing that the lower bound of$$\binom{2n}{n} \geq \frac{4^n}{2n + 1}$$The binomial expression is the middle term (and the largest)... 4 One example is what is called a fusion system. A fusion system is a category where the objects are the subgroups of some fixed p-group S and where the morphisms is a subset of the set of injective homomorphisms between the subgroups which contains all those induced by conjugation by some element from S. Further, it is required that any morphism \... 2 I like the Mathieu Groupoid M_{13}. Also, for any group G acting on a set X there is the action groupoid X/\!/G with objects the elements of X and with morphisms x_1 \to x_2 given by the elements of G such that g \cdot x_1 =x_2. (Of course, if G and X are finite, so is X/\!/G.) (Maybe these don't quite count, as any groupoid is ... 3 A good candidate would be elliptic curve cryptography. This is a direct practical application of finite fields, number theory, and other arithmetic geometry, that you would otherwise think have no purpose outside of pure mathematics. 1 Perhaps a bit obscure -- finite topological spaces applied to digital analysis Perhaps a bit advanced -- spectral sequences applied to physics 2 I would actually go with the quaternions \mathbb{H}. They form a 4 dimensional, associative division algebra. With the basis i, j , k, 1 which satisfies$$ i^2 = j^2 = k^2 = ijk = -1\$ They first might seem not useful at all, until you notice they are easily created with matrices, what means they are easily computable. Quaternions are used to compute ...

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