diff --git "a/science/test/questions.forum.tsv" "b/science/test/questions.forum.tsv" new file mode 100644--- /dev/null +++ "b/science/test/questions.forum.tsv" @@ -0,0 +1,2017 @@ +0 Cooling a cup of coffee with help of a spoon +1 Can I compute the mass of a coin based on the sound of its fall? +2 How does light bend around my finger tip? +3 How does gravity escape a black hole? +4 Did the Big Bang happen at a point? +5 Why are mirror images flipped horizontally but not vertically? +6 Do we know why there is a speed limit in our universe? +7 Book recommendations +8 What experiment would disprove string theory? +9 Why dont metals bond when touched together? +10 Why do ballpoint pens write better on pages that have pages below them? +11 What is Chirped Pulse Amplification, and why is it important enough to warrant a Nobel Prize? +12 Why does kinetic energy increase quadratically, not linearly, with speed? +13 What really allows airplanes to fly? +14 Why are four-legged chairs so common? +15 If I sliced the universe in half, would the slice go through a star? +16 Dont heavier objects actually fall faster because they exert their own gravity? +17 How do towels stay on hooks? +18 What exactly is a photon? +19 Does Earth really have two high-tide bulges on opposite sides? +20 Strange ice found in my garden +21 Why is the detection of gravitational waves so significant? +22 Surviving under water in air bubble +23 Why do we bend a book to keep it straight? +24 If photons have no mass, how can they have momentum? +25 Why doesnt matter pass through other matter if atoms are 99.999% empty space? +26 Why is Googles quantum supremacy experiment impressive? +27 Are units of angle really dimensionless? +28 Why dont electrons crash into the nuclei they orbit? +29 Why does space expansion not expand matter? +30 When separating an Oreo cookie, why does the cream stick to just one side only? +31 Why does NASA use gold foil on equipment and gold-coated visors? +32 How do moving charges produce magnetic fields? +33 How does a knife cut things at the atomic level? +34 Why do we not have spin greater than 2? +35 Why do shadows from the sun join each other when near enough? +36 Gauge symmetry is not a symmetry? +37 Why do we actually see the sun? +38 Whats the point of Hamiltonian mechanics? +39 Could Legolas actually see that far? +40 How do you make more precise instruments while only using less precise instruments? +41 Why does holding something up cost energy while no work is being done? +42 What makes a theory Quantum? +43 Given Newtons third law, why are things capable of moving? +44 Why does Stephen Hawking say black holes dont exist? +45 Why do sunbeams diverge even though the sun is much more than a few kilometers away? +46 How do I explain to a six year old why people on the other side of the Earth dont fall off? +47 Simple check for the global shape of the Earth +48 Why are the harmonics of a piano tone not multiples of the base frequency? +49 What is a field, really? +50 Why are the wet patches on these floor tiles circular? +51 Why is nuclear waste more dangerous than the original nuclear fuel? +52 Is $\pi^2 \approx g$ a coincidence? +53 Why does a mirror split my laser beam? +54 Why is my hand not burned by the air in an oven at 200 °C? +55 Why would spacetime curvature cause gravity? +56 Does someone falling into a black hole see the end of the universe? +57 Why are the windows of bridges of ships always inclined? +58 Does the Planck scale imply that spacetime is discrete? +59 Is temperature a Lorentz invariant in relativity? +60 Why do people categorically dismiss some simple quantum models? +61 If you view the Earth from far enough away can you observe its past? +62 Is time continuous or discrete? +63 How can you weigh your own head in an accurate way? +64 Why does my tea periodically alternate its rotational speed after stirring? (Link to video below) +65 How fast does gravity propagate? +66 How do I experimentally measure the surface area of a rock? +67 Is the butterfly effect real? +68 Why doesnt water boil in the oven? +69 A list of inconveniences between quantum mechanics and (general) relativity? +70 Why are there only derivatives to the first order in the Lagrangian? +71 Reading the Feynman lectures in 2012 +72 Why is the vibration in my wire acting so oddly? +73 Why can I touch aluminum foil in the oven and not get burned? +74 Why does paper cut through things so well? +75 What does it mean for two objects to touch? +76 Why do spaceships heat up when entering earth but not when exiting? +77 Does a particle exert force on itself? +78 Why does Newtons Third Law actually work? +79 Why does ice cream get harder when colder? +80 Why does the humidifier make a stoves flame orange? +81 How does mass leave the body when you lose weight? +82 Is anti-matter matter going backwards in time? +83 Calculus of variations -- how does it make sense to vary the position and the velocity independently? +84 Why dont we use weights to store energy? +85 Why do wet objects become darker? +86 Why am I not burned by a strong wind? +87 The Role of Rigor +88 On this infinite grid of resistors, whats the equivalent resistance? +89 If all motion is relative, how does light have a finite speed? +90 Could we send a man safely to the Moon in a rocket without knowledge of general relativity? +91 Superfields and the Inconsistency of regularization by dimensional reduction +92 Is it necessary to consume energy to perform computation? +93 Toilet paper dilemma +94 Is it possible for information to be transmitted faster than light by using a rigid pole? +95 Why do electrons, according to my textbook, exist forever? +96 What happens to the energy when waves perfectly cancel each other? +97 Why does a helium filled balloon move forward in a car when the car is accelerating? +98 What is the actual significance of the amplituhedron? +99 Can the Heisenberg Uncertainty Principle be explained intuitively? +100 What, in simplest terms, is gauge invariance? +101 Can Maxwells equations be derived from Coulombs Law and Special Relativity? +102 Is the universe fundamentally deterministic? +103 Why are most metals gray/silver? +104 Is it possible to start fire using moonlight? +105 What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis? +106 Books for general relativity +107 Why is glass transparent? +108 How can anything ever fall into a black hole as seen from an outside observer? +109 Whats inside a proton? +110 Why is oil a better lubricant than water? +111 Why do tuning forks have two prongs? +112 Why do fusion and fission both release energy? +113 Why is quantum entanglement considered to be an active link between particles? +114 Why are some people are claiming that the Big Bang never happened? +115 How do we know that radioactive decay rates are constant over billions of years? +116 How can I stand on the ground? EM or/and Pauli? +117 Is Angular Momentum truly fundamental? +118 What is $\Delta t$ in the time-energy uncertainty principle? +119 Why the Principle of Least Action? +120 Why does the Suns (or other stars) nuclear reaction not use up all its fuel immediately? +121 Intuitively, why are bundles so important in Physics? +122 Why does a sticker slowly peel off, but if it is pulled quickly it tears? +123 What is known about the topological structure of spacetime? +124 Does the $\frac{4}{3}$ problem of classical electromagnetism remain in quantum mechanics? +125 How can we see an atom now? What was the scale of this equipment? +126 Is there such thing as imaginary time dilation? +127 Thought experiment - would you notice if you fell into a black hole? +128 Explain it to me like Im a physics grad: Greenhouse Effect +129 QM without complex numbers +130 What Is Energy? Where did it come from? +131 Why is filling a balloon from your mouth much harder initially? +132 If gravity isnt a force, then why do we learn in school that it is? +133 If I pull a metal bar for long enough with a constant small force, will it eventually break? +134 Which is stronger, a rope without knots or a rope with knots? +135 Would a pin head heated to 15 million degrees Celsius kill everyone in a 1000 mile radius? +136 Why is there a scarcity of lithium? +137 Why do most formulas in physics have integer and rational exponents? +138 Is there something similar to Noethers theorem for discrete symmetries? +139 What is a good introductory book on quantum mechanics? +140 Best books for mathematical background? +141 What is the physical meaning of commutators in quantum mechanics? +142 Differentiating Propagator, Greens function, Correlation function, etc +143 What justifies dimensional analysis? +144 How is a quantum superposition different from a mixed state? +145 Why dont miners get boiled to death at $4$ km deep? +146 Trace of a commutator is zero - but what about the commutator of $x$ and $p$? +147 About the complex nature of the wave function? +148 Why was carbon-12 chosen for the atomic mass unit? +149 Why does the shower curtain move towards me when I am taking a hot shower? +150 Where is the flaw in this machine that decreases the entropy of a closed system? +151 Why does the atmosphere rotate along with the earth? +152 Why does hot water clean better than cold water? +153 What is the speed of sound in space? +154 How can a black hole produce sound? +155 What is the proper way to explain the twin paradox? +156 Why are radiators always placed under windows? +157 Can we theoretically balance a perfectly symmetrical pencil on its one-atom tip? +158 If Earth had rings? +159 How can time dilation be symmetric? +160 Quantum Entanglement - Whats the big deal? +161 If we had a perfectly efficient computer and all the energy in the Milky-way available, what number could it count to? +162 Why does the LIGO observation disprove higher dimensions? +163 Why does nature favour the Laplacian? +164 What software programs are used to draw physics diagrams, and what are their relative merits? +165 Why do same/opposite electric charges repel/attract each other, respectively? +166 What conservation law corresponds to Lorentz boosts? +167 Why quantum mechanics? +168 Does a gun exert enough gravity on the bullet it fired to stop it? +169 Why does my wooden door disperse light into a rainbow color spectrum? +170 What is the relation between electromagnetic wave and photon? +171 Why is the $S_{z} =0$ state forbidden for photons? +172 Seeing something from only one angle means you have only seen (what?)% of its surface area at most? +173 Are Newtons laws of motion laws or definitions of force and mass? +174 Does juggling balls reduce the total weight of the juggler and balls? +175 Why does a remote car key work when held to your head/body? +176 Why does matter exist in 3 states (liquids, solid, gas)? +177 How did Einstein know the speed of light was constant? +178 Why does a billiard ball stop when it hits another billiard ball head on? +179 Am I attracting Pluto? +180 Why does a yellow object turn white under a yellow light? Shouldnt it turn yellow instead? +181 Why is a laserpointer able to erase a glow-in-the-dark sticker? +182 Can tin foil hats block anything? +183 Why is a $5-60 mph$ time slower than a $0-60 mph$ time for some automobiles? +184 Why do scientists think that all the laws of physics that apply in our galaxy apply in other galaxies? +185 Why do phones land face down? +186 What is a manifold? +187 Why cant $ i\hbar\frac{\partial}{\partial t}$ be considered the Hamiltonian operator? +188 How does this baby rattle work? +189 Superluminal neutrinos +190 Why do metals only glow red, yellow and white and not through the full range of the spectrum? +191 What is time, does it flow, and if so what defines its direction? +192 How exactly do you avoid fooling yourself? +193 Why are differential equations for fields in physics of order two? +194 Quantum Field Theory from a mathematical point of view +195 Why is light bent but not accelerated? +196 Why can Hiroshima be inhabited when Chernobyl cannot? +197 What makes running so much less energy-efficient than bicycling? +198 Why is it bad taste to have a dimensional quantity in the argument of a logarithm or exponential function? +199 What is spontaneous symmetry breaking in quantum systems? +200 Why does a full moon seem uniformly bright from earth, shouldnt it be dimmer at the border? +201 Why is information indestructible? +202 Why dont fluorescent lights produce shadows? +203 If you are vacuuming your carpet and you wrap the cord around your body do you become a magnet? +204 When I walk down the stairs where does my potential energy go? +205 What is more fundamental, fields or particles? +206 Why does rainwater form moving waves on the ground? Is there a name for this effect? +207 List of freely available physics books +208 In the earths crust, why is there far more uranium than gold? +209 Could a living planet alter its own trajectory only by changing its shape? +210 Classical and quantum anomalies +211 What causes the water in this fountain to reverse direction? +212 What is spin as it relates to subatomic particles? +213 If dark matter only interacts with gravity, why doesnt it all clump together in a single point? +214 Why doesnt water actually perfectly wet glass? +215 Can photons be detected without being absorbed? +216 Is there a symmetry associated to the conservation of information? +217 What does one second after big bang mean? +218 What happened to David John Candlin? +219 Why do travelling waves continue after amplitude sum = 0? +220 Will a hole cut into a metal disk expand or shrink when the disc is heated? +221 Why does public mains power use 50-60 Hz and 100-240 V? +222 How can magnets be used to pick up pieces of metal when the force from a magnetic field does no work? +223 How do laser tape measures work? +224 Why must a physical theory be mathematically self-consistent? +225 Why does a rubber band become a lighter color when stretched? +226 Does centrifugal force exist? +227 Why does water stop boiling immediately after turning off the heat? +228 Why does fire make very little sound? +229 Where does the extra kinetic energy come from in a gravitational slingshot? +230 What was the major discovery on gravitational waves made March 17th, 2014, in the BICEP2 experiment? +231 Why doesnt the Moon fall onto the Earth? +232 How and why do accelerating charges radiate electromagnetic radiation? +233 Visually stunning math concepts which are easy to explain +234 Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? +235 How long will it take Marie to saw another board into 3 pieces? +236 Can I use my powers for good? +237 The staircase paradox, or why $\pi\ne4$ +238 How to study math to really understand it and have a healthy lifestyle with free time? +239 Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem) +240 Whats an intuitive way to think about the determinant? +241 Does $\pi$ contain all possible number combinations? +242 Splitting a sandwich and not feeling deceived +243 What was the first bit of mathematics that made you realize that math is beautiful? (For childrens book) +244 Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real? +245 Why can you turn clothing right-side-out? +246 Examples of patterns that eventually fail +247 Mathematical difference between white and black notes in a piano +248 Do complex numbers really exist? +249 What are imaginary numbers? +250 How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? +251 Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$ +252 The Egg: Bizarre behavior of the roots of a family of polynomials. +253 Is this Batman equation for real? +254 Best Sets of Lecture Notes and Articles +255 My sons Sum of Some is beautiful! But what is the proof or explanation? +256 Proofs that every mathematician should know. +257 Why does $1+2+3+\cdots = -\frac{1}{12}$? +258 What is the intuitive relationship between SVD and PCA? +259 How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? +260 On familiarity (or How to avoid going down the Math Rabbit Hole?) +261 Fourier transform for dummies +262 The Ring Game on $K[x,y,z]$ +263 Pedagogy: How to cure students of the law of universal linearity? +264 Find five positive integers whose reciprocals sum to $1$ +265 Can every proof by contradiction also be shown without contradiction? +266 Obvious theorems that are actually false +267 Zero to the zero power – is $0^0=1$? +268 If $AB = I$ then $BA = I$ +269 Calculating the length of the paper on a toilet paper roll +270 How can you prove that a function has no closed form integral? +271 A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language +272 Surprising identities / equations +273 Is it true that $0.999999999\ldots=1$? +274 Why dont we define imaginary numbers for every impossibility? +275 What is the importance of eigenvalues/eigenvectors? +276 Nice examples of groups which are not obviously groups +277 Multiple-choice question about the probability of a random answer to itself being correct +278 In Russian roulette, is it best to go first? +279 One question to know if the number is 1, 2 or 3 +280 Why does this innovative method of subtraction from a third grader always work? +281 Intuition for the definition of the Gamma function? +282 Why can ALL quadratic equations be solved by the quadratic formula? +283 Math without pencil and paper +284 V.I. Arnold says Russian students cant solve this problem, but American students can -- why? +285 Really advanced techniques of integration (definite or indefinite) +286 Is mathematics one big tautology? +287 Help with a prime number spiral which turns 90 degrees at each prime +288 Is $10$ a magical number or I am missing something? +289 Funny identities +290 Cant argue with success? Looking for bad math that gets away with it +291 Is $7$ the only prime followed by a cube? +292 In the history of mathematics, has there ever been a mistake? +293 The Mathematics of Tetris +294 Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet? +295 Why do mathematicians use single-letter variables? +296 How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$? +297 Is a matrix multiplied with its transpose something special? +298 How discontinuous can a derivative be? +299 Too old to start math +300 Evaluate $ \int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\:\mathrm dx$ +301 What is the maximum volume that can be contained by a sheet of paper? +302 Please explain the intuition behind the dual problem in optimization. +303 Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$? +304 Norms Induced by Inner Products and the Parallelogram Law +305 Fun but serious mathematics books to gift advanced undergraduates. +306 Which answer in this list is the correct answer to this question? +307 What is the practical difference between a differential and a derivative? +308 Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ +309 Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers +310 Can you answer my sons fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place? +311 The Integral that Stumped Feynman? +312 In (relatively) simple words: What is an inverse limit? +313 What is the importance of the Collatz conjecture? +314 What are some examples of when Mathematics accidentally discovered something about the world? +315 Derivative of sigmoid function $\sigma (x) = \frac{1}{1+e^{-x}}$ +316 Conjectures that have been disproved with extremely large counterexamples? +317 Best Fake Proofs? (A M.SE April Fools Day collection) +318 Any open subset of $\Bbb R$ is a countable union of disjoint open intervals +319 Why study Algebraic Geometry? +320 Why are rings called rings? +321 What is the result of $\infty - \infty$? +322 Integral Milking +323 A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I +324 How to read a book in mathematics? +325 Why do we care about dual spaces? +326 What are some counter-intuitive results in mathematics that involve only finite objects? +327 How many fours are needed to represent numbers up to $N$? +328 Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer? +329 Your favourite application of the Baire Category Theorem +330 What books must every math undergraduate read? +331 Proving you *cant* make $2011$ out of $1,2,3,4$: nice twist on the usual +332 Does the open mapping theorem imply the Baire category theorem? +333 Optimizing response times of an ambulance corp: short-term versus average +334 Is there a 0-1 law for the theory of groups? +335 How can a piece of A4 paper be folded in exactly three equal parts? +336 Eigenvectors of real symmetric matrices are orthogonal +337 Do men or women have more brothers? +338 In simple English, what does it mean to be transcendental? +339 How to check if a point is inside a rectangle? +340 Good books and lecture notes about category theory. +341 Why does this matrix give the derivative of a function? +342 Is it faster to count to the infinite going one by one or two by two? +343 What are the Differences Between a Matrix and a Tensor? +344 How do I convince someone that $1+1=2$ may not necessarily be true? +345 How do I sell out with abstract algebra? +346 Why can a Venn diagram for $4+$ sets not be constructed using circles? +347 Identification of a quadrilateral as a trapezoid, rectangle, or square +348 Meaning of Rays in Polar Plot of Prime Numbers +349 What does $2^x$ really mean when $x$ is not an integer? +350 What do modern-day analysts actually do? +351 Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$ +352 Is computer science a branch of mathematics? +353 What is the difference between linear and affine function +354 Which one result in mathematics has surprised you the most? +355 Books on Number Theory for Layman +356 Teaching myself differential topology and differential geometry +357 List of interesting math videos / documentaries +358 Exterior Derivative vs. Covariant Derivative vs. Lie Derivative +359 How to show $e^{e^{e^{79}}}$ is not an integer +360 Are $14$ and $21$ the only interesting numbers? +361 How could we define the factorial of a matrix? +362 When can you switch the order of limits? +363 List of Interesting Math Blogs +364 Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$ +365 How do we prove that something is unprovable? +366 How many sides does a circle have? +367 Whats the intuition behind Pythagoras theorem? +368 Why is compactness so important? +369 Can a coin with an unknown bias be treated as fair? +370 A Topology such that the continuous functions are exactly the polynomials +371 What is the geometric interpretation of the transpose? +372 Counterintuitive examples in probability +373 What is a good complex analysis textbook, barring Ahlforss? +374 What were some major mathematical breakthroughs in 2016? +375 How to intuitively understand eigenvalue and eigenvector? +376 Importance of Representation Theory +377 Advice to young mathematicians +378 Why cant differentiability be generalized as nicely as continuity? +379 How to define a bijection between $(0,1)$ and $(0,1]$? +380 Derivative of Softmax loss function +381 Some users are mind bogglingly skilled at integration. How did they get there? +382 How do people perform mental arithmetic for complicated expressions? +383 Can we ascertain that there exists an epimorphism $G\rightarrow H$? +384 What Does it Really Mean to Have Different Kinds of Infinities? +385 How do you revise material that you already half-know, without getting bored and demotivated? +386 Are there any series whose convergence is unknown? +387 When can a sum and integral be interchanged? +388 A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$ +389 Symmetry of function defined by integral +390 Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$? +391 Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$ +392 There are apparently $3072$ ways to draw this flower. But why? +393 Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction +394 Proof that the trace of a matrix is the sum of its eigenvalues +395 Limit of $L^p$ norm +396 Is there a categorical definition of submetry? +397 Deleting any digit yields a prime... is there a name for this? +398 Inverse of the sum of matrices +399 What is $dx$ in integration? +400 Is $2048$ the highest power of $2$ with all even digits (base ten)? +401 Why is Eulers Gamma function the best extension of the factorial function to the reals? +402 $L^p$ and $L^q$ space inclusion +403 Do we have negative prime numbers? +404 Why do people use it is easy to prove? +405 Taking Seats on a Plane +406 Why is $1^{\infty}$ considered to be an indeterminate form +407 Mathematical ideas that took long to define rigorously +408 Open problems in General Relativity +409 How far can one get in analysis without leaving $\mathbb{Q}$? +410 How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? +411 Rational roots of polynomials +412 Lesser-known integration tricks +413 A variation of Fermats little theorem in the form $a^{n-d}\equiv a$ (mod $p$). +414 Is there a homology theory that counts connected components of a space? +415 Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ +416 An Introduction to Tensors +417 Transpose of inverse vs inverse of transpose +418 How to prove Eulers formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$? +419 Intuition behind Matrix Multiplication +420 Striking applications of integration by parts +421 Why is the Penrose triangle impossible? +422 Software for drawing geometry diagrams +423 How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number? +424 Examples of mathematical discoveries which were kept as a secret +425 What properties of busy beaver numbers are computable? +426 What happens when we (incorrectly) make improper fractions proper again? +427 Can you provide me historical examples of pure mathematics becoming useful? +428 What are the most overpowered theorems in mathematics? +429 Do we know if there exist true mathematical statements that can not be proven? +430 Is there any integral for the Golden Ratio? +431 Using proof by contradiction vs proof of the contrapositive +432 Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$ +433 Induction on Real Numbers +434 Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger. +435 The Hole in One Pizza +436 Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$ +437 What functions can be made continuous by mixing up their domain? +438 Is there a general formula for solving 4th degree equations (quartic)? +439 Discontinuous derivative. +440 Monty hall problem extended. +441 The sum of an uncountable number of positive numbers +442 Is Apple ipad / tablet good for mathematics students? +443 Stopping the Will I need this for the test question +444 Are there any open mathematical puzzles? +445 Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$? +446 How to distinguish between walking on a sphere and walking on a torus? +447 Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? +448 Is there another simpler method to solve this elementary school math problem? +449 Most ambiguous and inconsistent phrases and notations in maths +450 Why does factoring eliminate a hole in the limit? +451 Whats the point in being a skeptical learner +452 Good book for self study of a First Course in Real Analysis +453 Overview of basic results about images and preimages +454 How to determine with certainty that a function has no elementary antiderivative? +455 What actually is a polynomial? +456 Sum of random decreasing numbers between 0 and 1: does it converge?? +457 Are there real-life relations which are symmetric and reflexive but not transitive? +458 Calculate Rotation Matrix to align Vector A to Vector B in 3d? +459 Whats new in higher dimensions? +460 How do you describe your mathematical research in laymans terms? +461 Pythagorean triples that survive Eulers totient function +462 Whats the significance of Tates thesis? +463 What is the difference between singular value and eigenvalue? +464 Intuition of the meaning of homology groups +465 Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? +466 Why does an argument similiar to 0.999...=1 show 999...=-1? +467 Does a cubic matrix exist? +468 Is the following matrix invertible? +469 Do most mathematicians know most topics in mathematics? +470 Slice of pizza with no crust +471 Has lack of mathematical rigour killed anybody before? +472 Alternative notation for exponents, logs and roots? +473 Math and mental fatigue +474 What is the Riemann-Zeta function? +475 Why do both sine and cosine exist? +476 What is the best book to learn probability? +477 The square roots of different primes are linearly independent over the field of rationals +478 Can someone explain the math behind tessellation? +479 Where to start learning Linear Algebra? +480 Intuitively, what is the difference between Eigendecomposition and Singular Value Decomposition? +481 Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix +482 Example of infinite field of characteristic $p\neq 0$ +483 Is non-standard analysis worth learning? +484 Connection between Fourier transform and Taylor series +485 Are we allowed to compare infinities? +486 Online tool for making graphs (vertices and edges)? +487 Whats 4 times more likely than 80%? +488 Simple theorems that are instances of deep mathematics +489 Do Arithmetic Mean and Geometric Mean of Prime Numbers converge? +490 Partial derivative in gradient descent for two variables +491 Apparently sometimes $1/2 < 1/4$? +492 Is the product of two Gaussian random variables also a Gaussian? +493 Intuitive explanation of entropy +494 The Best of Dover Books (a.k.a the best cheap mathematical texts) +495 Are mathematical articles on Wikipedia reliable? +496 What are the differences between rings, groups, and fields? +497 Why do units (from physics) behave like numbers? +498 Studying Euclidean geometry using hyperbolic criteria +499 Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ +500 Identification of a curious function +501 What is the difference between regression and classification? +502 How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen? +503 How do you respond to I was always bad at math? +504 What are the numbers before and after the decimal point referred to in mathematics? +505 Can you explain the Axiom of choice in simple terms? +506 Why do we use the word scalar and not number in Linear Algebra? +507 What is the difference between Fourier series and Fourier transformation? +508 What is the difference between a point and a vector? +509 The direct sum $\oplus$ versus the cartesian product $\times$ +510 Is the blue area greater than the red area? +511 Does $R[x] \cong S[x]$ imply $R \cong S$? +512 Why does mathematical convention deal so ineptly with multisets? +513 List of interesting math podcasts? +514 Can you raise a number to an irrational exponent? +515 Examples of mathematical results discovered late +516 $\pi$ in arbitrary metric spaces +517 Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$ +518 Is there any mathematical reason for this digit-repetition-show? +519 Visually deceptive proofs which are mathematically wrong +520 Best book of topology for beginner? +521 Can someone explain Gödels incompleteness theorems in layman terms? +522 Why is $1$ not a prime number? +523 Studying for the Putnam Exam +524 Intuitive interpretation of the Laplacian +525 what is expected from a PhD student? +526 Why is the eigenvector of a covariance matrix equal to a principal component? +527 Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$ +528 Is it possible for a function to be in $L^p$ for only one $p$? +529 What are some examples of notation that really improved mathematics? +530 Classification of prime ideals of $\mathbb{Z}[X]$ +531 Division by $0$ +532 Application of Hilberts basis theorem in representation theory +533 Is $0$ a natural number? +534 Why is negative times negative = positive? +535 On Ph.D. Qualifying Exams +536 Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ +537 What are some examples of a mathematical result being counterintuitive? +538 Can you be 1/12th Cherokee? +539 Why did mathematicians take Russells paradox seriously? +540 Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$ +541 How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$ +542 What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges? +543 Why is learning modern algebraic geometry so complicated? +544 What should be the intuition when working with compactness? +545 Motivation of irrationality measure +546 Show that the determinant of $A$ is equal to the product of its eigenvalues +547 The median minimizes the sum of absolute deviations (the $ {\ell}_{1} $ norm) +548 Why is gradient the direction of steepest ascent? +549 Learning Lambda Calculus +550 Is there a function that grows faster than exponentially but slower than a factorial? +551 How to find solutions of linear Diophantine ax + by = c? +552 What is category theory useful for? +553 Are half of all numbers odd? +554 Can $x^{x^{x^x}}$ be a rational number? +555 Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f(x)$ +556 Are if and iff interchangeable in definitions? +557 Is there an integral that proves $\pi > 333/106$? +558 Past open problems with sudden and easy-to-understand solutions +559 How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$ +560 What remains in a students mind +561 Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$ +562 Why is $\infty \cdot 0$ not clearly equal to $0$? +563 What are good books to learn graph theory? +564 Can an irrational number raised to an irrational power be rational? +565 Advantages of Mathematics competition/olympiad students in Mathematical Research +566 Whats the difference between predicate and propositional logic? +567 How can I find the surface area of a normal chicken egg? +568 Why does LHopitals rule fail in calculating $\lim_{x \to \infty} \frac{x}{x+\sin(x)}$? +569 Why study linear algebra? +570 Why is integration so much harder than differentiation? +571 Why is $\cos (90)=-0.4$ in WebGL? +572 Why “characteristic zero” and not “infinite characteristic”? +573 Why are the solutions of polynomial equations so unconstrained over the quaternions? +574 Making Friends around a Circular Table +575 Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$? +576 Real life applications of Topology +577 Whats your favorite proof accessible to a general audience? +578 Pullback and Pushforward Isomorphism of Sheaves +579 Physical meaning of the null space of a matrix +580 How do you explain to a 5th grader why division by zero is meaningless? +581 Overview of basic facts about Cauchy functional equation +582 Is 10 closer to infinity than 1? +583 Examples of problems that are easier in the infinite case than in the finite case. +584 Help find hard integrals that evaluate to $59$? +585 Is the inverse of a symmetric matrix also symmetric? +586 When to learn category theory? +587 In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False? +588 Modular exponentiation by hand ($a^b\bmod c$) +589 How were old-school mathematics graphics created? +590 Infiniteness of non-twin primes. +591 probability $2/4$ vs $3/6$ +592 Continuous projections on $\ell_1$ with norm $>1$ +593 Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$? +594 All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$ +595 Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$ +596 What is the difference between independent and mutually exclusive events? +597 What is the Jacobian matrix? +598 If squaring a number means multiplying that number with itself then shouldnt taking square root of a number mean to divide a number by itself? +599 Is there a known well ordering of the reals? +600 Intuition behind Conditional Expectation +601 Is there an inverted dot product? +602 Find the average of $\sin^{100} (x)$ in 5 minutes? +603 How to find the Galois group of a polynomial? +604 Strategies for Effective Self-Study +605 What did Alan Turing mean when he said he didnt fully understand dy/dx? +606 Is there a domain larger than (i.e., a supserset of) the complex number domain? +607 Motivation for the rigour of real analysis +608 Probability that a stick randomly broken in five places can form a tetrahedron +609 The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$) +610 Expected time to roll all 1 through 6 on a die +611 Proofs of AM-GM inequality +612 The math behind Warren Buffetts famous rule – never lose money +613 Compute $\int_0^{\pi/4}\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)} x\exp(\frac{x^2-1}{x^2+1}) dx$ +614 derivative of cost function for Logistic Regression +615 Construction of a Borel set with positive but not full measure in each interval +616 Are the proofs by contradiction weaker than other proofs? +617 Why are There No Triernions (3-dimensional analogue of complex numbers / quaternions)? +618 What parts of a pure mathematics undergraduate curriculum have been discovered since $1964?$ +619 Is black hole pattern possible in Conways Game of Life that eats/clears everything? +620 A multiplication algorithm found in a book by Paul Erdős: how does it work? +621 Are all limits solvable without LHôpital Rule or Series Expansion +622 $n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesnt use Chebyshevs theorem? +623 Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$ +624 When is matrix multiplication commutative? +625 What is the difference and relationship between the binomial and Bernoulli distributions? +626 Is infinity a number? +627 Am I just not smart enough? +628 What is the smallest unknown natural number? +629 Is there a positive definition for irrational numbers? +630 Solving Special Function Equations Using Lie Symmetries +631 Gross-Zagier formulae outside of number theory +632 Why is the derivative of a circles area its perimeter (and similarly for spheres)? +633 Getting better at proofs +634 Mathematical equivalent of Feynmans Lectures on Physics? +635 How often does it happen that the oldest person alive dies? +636 Do mathematicians, in the end, always agree? +637 Can an infinite sum of irrational numbers be rational? +638 Unexpected examples of natural logarithm +639 What are the Axiom of Choice and Axiom of Determinacy? +640 Least prime of the form $38^n+31$ +641 Different ways to prove there are infinitely many primes? +642 Can you give an example of a complex math problem that is easy to solve? +643 Proof of Frullanis theorem +644 More than 99% of groups of order less than 2000 are of order 1024? +645 Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square +646 Does a four-variable analog of the Hall-Witt identity exist? +647 How do I get the square root of a complex number? +648 Prove that $||x|-|y||\le |x-y|$ +649 Prove that simultaneously diagonalizable matrices commute +650 Could someone explain conditional independence? +651 Good Book On Combinatorics +652 A good way to retain mathematical understanding? +653 Why is it important for a matrix to be square? +654 Are there real world applications of finite group theory? +655 Why cant calculus be done on the rational numbers? +656 Why is there no remainder in multiplication +657 What are the issues in modern set theory? +658 What makes a theorem fundamental? +659 Why is an average of an average usually incorrect? +660 Finding a primitive root of a prime number +661 Do factorials really grow faster than exponential functions? +662 What exactly is the difference between a derivative and a total derivative? +663 Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$? +664 Why, historically, do we multiply matrices as we do? +665 Why do we still do symbolic math? +666 Fastest way to meet, without communication, on a sphere? +667 Grothendieck s question - any update? +668 How to sum this series for $\pi/2$ directly? +669 In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? +670 Can someone clearly explain about the lim sup and lim inf? +671 How can I understand and prove the sum and difference formulas in trigonometry? +672 Is zero odd or even? +673 Continuity of the roots of a polynomial in terms of its coefficients +674 In calculus, which questions can the naive ask that the learned cannot answer? +675 Why are mathematical proofs that rely on computers controversial? +676 Getting Students to Not Fear Confusion +677 Topology: The Board Game +678 Fibonacci number that ends with 2014 zeros? +679 What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? +680 Generalization of Liouvilles theorem +681 Complete course of self-study +682 Motivation for Ramanujans mysterious $\pi$ formula +683 What is the most unusual proof you know that $\sqrt{2}$ is irrational? +684 How do we know an $ \aleph_1 $ exists at all? +685 Connected metric spaces with disjoint open balls +686 Why is the volume of a sphere $\frac{4}{3}\pi r^3$? +687 Incremental averageing +688 How to show that a set of discontinuous points of an increasing function is at most countable +689 If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that its an unfair coin? +690 What made you choose your research field? +691 How Do You Actually Do Your Mathematics? +692 Is this continuous analogue to the AM–GM inequality true? +693 Open mathematical questions for which we really, really have no idea what the answer is +694 Is the derivative the natural logarithm of the left-shift? +695 Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? +696 lim sup and lim inf of sequence of sets. +697 Why is it hard to prove whether $\pi+e$ is an irrational number? +698 How do you go about learning mathematics? +699 Is $[0,1]$ a countable disjoint union of closed sets? +700 Chatting about mathematics (with real-time LaTeX rendering) +701 Calculating the integral $\int_0^\infty \frac{\cos x}{1+x^2}\, \mathrm{d}x$ without using complex analysis +702 Finding the limit of $\frac {n}{\sqrt[n]{n!}}$ +703 What is the single most influential book every mathematician should read? +704 Logic puzzle: Which octopus is telling the truth? +705 Why is a geometric progression called so? +706 Why, intuitively, is the order reversed when taking the transpose of the product? +707 Prove the theorem on analytic geometry in the picture. +708 Good Physical Demonstrations of Abstract Mathematics +709 Can I think of Algebra like this? +710 Ways to evaluate $\int \sec \theta \, \mathrm d \theta$ +711 Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ +712 Does this property characterize a space as Hausdorff? +713 All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$ +714 How to prove that eigenvectors from different eigenvalues are linearly independent +715 Lebesgue integral basics +716 Why rationalize the denominator? +717 Mathematicians ahead of their time? +718 Are all algebraic integers with absolute value 1 roots of unity? +719 Can we remove any prime number with this strange process? +720 Produce an explicit bijection between rationals and naturals? +721 Is math built on assumptions? +722 Riddles that can be solved by meta-assumptions +723 Why are vector spaces not isomorphic to their duals? +724 What are some examples of mathematics that had unintended useful applications much later? +725 Division by $0$ and its restrictions +726 Theorems with an extraordinary exception or a small number of sporadic exceptions +727 True or false? $x^2\ne x\implies x\ne 1$ +728 $\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis +729 An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$ +730 Does notation ever become easier? +731 Mathematician vs. Computer: A Game +732 Non-textbook Math book recommendation to read to my kids +733 Cardioid in coffee mug? +734 Direct proof that the wedge product preserves integral cohomology classes? +735 How did Hermite calculate $e^{\pi\sqrt{163}}$ in 1859? +736 What does it mean to have a determinant equal to zero? +737 Why is the volume of a cone one third of the volume of a cylinder? +738 Dont see the point of the Fundamental Theorem of Calculus. +739 Comparing $\pi^e$ and $e^\pi$ without calculating them +740 Is Bayes Theorem really that interesting? +741 Unconventional mathematics books +742 Does LHôpitals work the other way? +743 Geometric interpretation of the Riemann-Roch for curves +744 Whats the difference between theorem, lemma and corollary? +745 Unusual mathematical terms +746 Is infinity an odd or even number? +747 How do you find the center of a circle with a pencil and a book? +748 Using we have in maths papers +749 How to learn from proofs? +750 What are Some Tricks to Remember Fatous Lemma? +751 Why dont analysts do category theory? +752 Defining a manifold without reference to the reals +753 Does a non-trivial solution exist for $f(x)=f(f(x))$? +754 A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent +755 A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational +756 Expectation of the maximum of gaussian random variables +757 Simplest or nicest proof that $1+x \le e^x$ +758 What would base $1$ be? +759 Should I put number combinations like 1111111 onto my lottery ticket? +760 Continuous bijection from $(0,1)$ to $[0,1]$ +761 How is a system of axioms different from a system of beliefs? +762 Why cant you add apples and oranges, but you can multiply and divide them? +763 How can adding an infinite number of rationals yield an irrational number? +764 Example of a very simple math statement in old literature which is (verbatim) a pain to understand +765 Whats the largest possible volume of a taco, and how do I make one that big? +766 Limit of sequence in which each term is defined by the average of preceding two terms +767 Why are all the interesting constants so small? +768 Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on +769 $4494410$ and friends +770 A semigroup $X$ is a group iff for every $g\in X$, $\exists! x\in X$ such that $gxg = g$ +771 Arithmetic-geometric mean of 3 numbers +772 Geometric way to view the truncated braid groups? +773 Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$ +774 How to find a general sum formula for the series: 5+55+555+5555+.....? +775 What is the difference between a class and a set? +776 Why is $\frac{987654321}{123456789} = 8.0000000729?!$ +777 The Langlands program for beginners +778 What makes elementary functions elementary? +779 Why would I want to multiply two polynomials? +780 If $x$, $y$, $x+y$, and $x-y$ are prime numbers, what is their sum? +781 Mathematicians Tensors vs. Physicists Tensors +782 Is the determinant that shows up accidental? +783 Given a die, what is the probability that the second roll of a die will be less than the first roll? +784 Why is the set of all sets a paradox, in Laymans terms? +785 Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$ +786 What seemingly innocuous results in mathematics require advanced proofs? +787 Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$ +788 How to convince a math teacher of this simple and obvious fact? +789 What kind of symmetry is the symmetric group about? +790 What are good math habits that have improved your mathematical practice?$ $ +791 math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$? +792 Cover of Gödel, Escher, Bach +793 Problems that become easier in a more general form +794 How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$ +795 When does a sequence of rotated-and-circumscribed rectangles converge to a square? +796 Why is the area under a curve the integral? +797 Good 1st PDE book for self study +798 Importance of matrix rank +799 Much less than, what does that mean? +800 Simple beautiful math proof +801 Anecdotes about famous mathematicians or physicists +802 Why do bell curves appear everywhere? +803 What are the differences between class, set, family, and collection? +804 What is the solution to Nashs problem presented in A Beautiful Mind? +805 Factorial and exponential dual identities +806 The pepperoni pizza problem +807 How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis? +808 A Case Against the Math Gene +809 Complexity class of comparison of power towers +810 Does $\lfloor \sqrt{p} \rfloor$ generate all natural numbers? +811 Pointwise vs. Uniform Convergence +812 What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6? +813 GRE Subject Test - Past Papers, Books, Advice +814 Can the golden ratio accurately be expressed in terms of $e$ and $\pi$ +815 How do you explain the concept of logarithm to a five year old? +816 If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$ +817 What is a real-world metaphor for irrational numbers? +818 Intuitive explanation of Cauchys Integral Formula in Complex Analysis +819 Why should I believe in weak solutions to PDEs? +820 Explain homotopy to me +821 Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing +822 Root Calculation by Hand +823 What does the dot product of two vectors represent? +824 Whats the difference between simple induction and strong induction? +825 What is the most elegant proof of the Pythagorean theorem? +826 Is the vector cross product only defined for 3D? +827 Best Maths Books for Non-Mathematicians +828 Self-studying real analysis — Tao or Rudin? +829 What are differences between affine space and vector space? +830 If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum a_n<\infty$, then $\lim{n a_n} = 0$ +831 Is arrow notation for vectors not mathematically mature? +832 I roll a die repeatedly until I get 6, and then count the number of 3s I got. Whats my expected number of 3s? +833 What is the difference between a Hamel basis and a Schauder basis? +834 Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known? +835 Results that came out of nowhere. +836 Can the product of infinitely many elements from $\mathbb Q$ be irrational? +837 Divisibility by 7 rule, and Congruence Arithmetic Laws +838 Can you make a sphere out of a plane? +839 Cutting sticks puzzle +840 The Right Triangle Game +841 Geometric interpretation of $\det(A^T) = \det(A)$ +842 How to prove that exponential grows faster than polynomial? +843 Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when its an integer +844 Good book for self study of functional analysis +845 If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable +846 $x^y = y^x$ for integers $x$ and $y$ +847 100 blue-eyed islanders puzzle: 3 questions +848 Surprise exam paradox? +849 Demystify integration of $\int \frac{1}{x} \mathrm dx$ +850 Your favourite maths puzzles +851 Number of simple edge-disjoint paths needed to cover a planar graph +852 Good abstract algebra books for self study +853 Does convergence in $L^p$ imply convergence almost everywhere? +854 Is there any difference between mapping and function? +855 Normal subgroup of prime index +856 Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$) +857 Learning mathematics as if an absolute beginner? +858 Why did no student correctly find a pair of $2\times 2$ matrices with the same determinant and trace that are not similar? +859 Will it become impossible to learn math? +860 Should an undergrad accept that some things dont make sense, or study the foundation of mathematics to resolve this? +861 Is $0! = 1$ because there is only one way to do nothing? +862 Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$ +863 Why are groups more important than semigroups? +864 Can someone explain these strange properties of $10, 11, 12$ and $13$? +865 What is the best book for studying discrete mathematics? +866 Why cant you square both sides of an equation? +867 For any prime $p > 3$, why is $p^2-1$ always divisible by 24? +868 Is the notorious $n^2 + n + 41$ prime generator the last of its type? +869 Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$ +870 Easy math proofs or visual examples to make high school students enthusiastic about math +871 Why do we define quotient groups for normal subgroups only? +872 Formal proof for $(-1) \times (-1) = 1$ +873 Mathematically, why was the Enigma machine so hard to crack? +874 Is it possible to represent every huge number in abbreviated form? +875 $1=2$ | Continued fraction fallacy +876 How is a group made up of simple groups? +877 Cute Determinant Question +878 Modelling the Moving Sofa +879 Graph theoretic proof: For six irrational numbers, there are three among them such that the sum of any two of them is irrational. +880 Contest problem: Show $\sum_{n = 1}^\infty \frac{n^2a_n}{(a_1+\cdots+a_n)^2}0$, $\sum_{n = 1}^\infty \frac{1}{a_n}<\infty$ +881 Proof of every convex function is continuous +882 Are calculus and real analysis the same thing? +883 A goat tied to a corner of a rectangle +884 Predicting Real Numbers +885 Why is abuse of notation tolerated? +886 Why are gauge integrals not more popular? +887 Do numbers get worse than transcendental? +888 Whats so special about standard deviation? +889 What concept does an open set axiomatise? +890 Would you ever stop rolling the die? +891 Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc... & Representation Theory of Special Functions +892 References for multivariable calculus +893 When is the closure of an open ball equal to the closed ball? +894 Can manholes be made in other shapes than circles, that prevent the cover from being able to fall down its own hole? +895 In what sense are math axioms true? +896 Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$ +897 Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$ +898 What are some conceptualizations that work in mathematics but are not strictly true? +899 Why do mathematicians sometimes assume famous conjectures in their research? +900 Where are the axioms? +901 How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$? +902 How do we know that Cantors diagonalization isnt creating a different decimal of the same number? +903 In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual? +904 Determining information in minimum trials (combinatorics problem) +905 What is the term for a factorial type operation, but with summation instead of products? +906 Difference between gradient and Jacobian +907 Why does Turn! Turn! Turn! equal 241217.524881? +908 Is there a size of rectangle that retains its ratio when its folded in half? +909 What is ultrafinitism and why do people believe it? +910 Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture +911 Intuition in algebra? +912 A continuous, nowhere differentiable but invertible function? +913 Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$ +914 Can someone explain this integration trick for log-sine integrals? +915 Polynomials such that roots=coefficients +916 Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$? +917 Do groups, rings and fields have practical applications in CS? If so, what are some? +918 What is special about the numbers 9801, 998001, 99980001 ..? +919 How does Cantors diagonal argument work? +920 Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$ +921 Why is a circle in a plane surrounded by 6 other circles? +922 Expected number of unpecked chicks - NYT article +923 Whats the difference between $\mathbb{R}^2$ and the complex plane? +924 What is your favorite application of the Pigeonhole Principle? +925 Is mathematics just a bunch of nested empty sets? +926 Easy example why complex numbers are cool +927 Why is the construction of the real numbers important? +928 Any rectangular shape on a calculator numpad when divided by 11 gives an integer. Why? +929 Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top +930 Paul Erdoss Two-Line Functional Analysis Proof +931 Numbers $n$ such that the digit sum of $n^2$ is a square +932 How to solve these two simultaneous divisibilities : $n+1\mid m^2+1$ and $m+1\mid n^2+1$ +933 Do most numbers have exactly $3$ prime factors? +934 Can a row of five equilateral triangles tile a big equilateral triangle? +935 Variance of sample variance? +936 Union of two vector subspaces not a subspace? +937 Probability density function vs. probability mass function +938 Determinant of a non-square matrix +939 What is a simple example of an unprovable statement? +940 Mathematical precise definition of a PDE being elliptic, parabolic or hyperbolic +941 Does commutativity imply Associativity? +942 What is the Tor functor? +943 100 Soldiers riddle +944 Why does Friedberg say that the role of the determinant is less central than in former times? +945 Is Lagranges theorem the most basic result in finite group theory? +946 Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental? +947 $6!\cdot 7!=10!$. Is there a natural bijection between $S_6\times S_7$ and $S_{10}$? +948 Show that the set of all finite subsets of $\mathbb{N}$ is countable. +949 $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed +950 Why are rotational matrices not commutative? +951 How to use the Extended Euclidean Algorithm manually? +952 How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? +953 What are some interpretations of Von Neumanns quote? +954 Why does the Mandelbrot set contain (slightly deformed) copies of itself? +955 If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$ +956 General request for a book on mathematical history, for a VERY advanced reader. +957 Terence Tao–type books in other fields? +958 Alice and Bob play the determinant game +959 Is there a definitive guide to speaking mathematics? +960 $\frac{1}{n}$ as a difference of Egyptian fractions with all denominators $5$, topology = algebra +997 Conjectured formula for the Fabius function +998 Element-wise (or pointwise) operations notation? +999 Where exactly are complex numbers used in the real world? +1000 Is zero positive or negative? +1001 Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$. +1002 Finding the Transform matrix from 4 projected points (with Javascript) +1003 Reference book on measure theory +1004 How to effectively and efficiently learn mathematics +1005 Logic problem: Identifying poisoned wines out of a sample, minimizing test subjects with constraints +1006 What is a universal property? +1007 Why are differentiable complex functions infinitely differentiable? +1008 Sheaf cohomology: what is it and where can I learn it? +1009 Koch snowflake paradox: finite area, but infinite perimeter +1010 I lost my love of math; Im getting it back. How can I determine if math is actually right for me? +1011 Is there a function with a removable discontinuity at every point? +1012 Direct proof that $\pi$ is not constructible +1013 Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$ +1014 Rigorous nature of combinatorics +1015 Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions +1016 Escaping infinitely many pursuers +1017 Relations between p norms +1018 Projection map being a closed map +1019 Math behind rotation in MS Paint +1020 The Monty Hall problem +1021 Why is the Möbius strip not orientable? +1022 Help me solve my fathers riddle and get my book back +1023 Why did my friend lose all his money? +1024 What is lost when we move from reals to complex numbers? +1025 Is linear algebra more “fully understood” than other maths disciplines? +1026 Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct? +1027 How deep is the liquid in a half-full hemisphere? +1028 Why do books titled Abstract Algebra mostly deal with groups/rings/fields? +1029 What exactly are eigen-things? +1030 What is an example of a sequence which thins out and is finite? +1031 Why is $\omega$ the smallest $\infty$? +1032 Besides proving new theorems, how can a person contribute to mathematics? +1033 The Intuition behind lHopitals Rule +1034 Motivation for spectral graph theory. +1035 Mathematicians shocked(?) to find pattern in prime numbers +1036 Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$ +1037 Why cant the Polynomial Ring be a Field? +1038 A path to truly understanding probability and statistics +1039 Is there a shape with infinite area but finite perimeter? +1040 Infinite sets dont exist!? +1041 Fastest way to check if $x^y > y^x$? +1042 What is integration by parts, really? +1043 Conjectures (or intuitions) that turned out wrong in an interesting or useful way +1044 What is the intuition behind uniform continuity? +1045 Dividing 100% by 3 without any left +1046 Inscribing square in circle in just seven compass-and-straightedge steps +1047 List of interesting integrals for early calculus students +1048 The Duals of $l^\infty$ and $L^{\infty}$ +1049 Finite subgroups of the multiplicative group of a field are cyclic +1050 Whats a proof that the angles of a triangle add up to 180°? +1051 Stirlings formula: proof? +1052 Is learning (theoretical) physics useful/important for a mathematician? +1053 An example of a problem which is difficult but is made easier when a diagram is drawn +1054 Multiple-choice: sum of primes below $1000$ +1055 Fake induction proofs +1056 Theorem that von Neumann proved in five minutes. +1057 Why did mathematicians introduce the concept of uniform continuity? +1058 Why is $i! = 0.498015668 - 0.154949828i$? +1059 Gerrymandering on a high-genus surface/can I use my powers for evil? +1060 A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$ +1061 Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$ +1062 How are mathematicians taught to write with such an expository style? +1063 Number of ways to write n as a sum of k nonnegative integers +1064 What is $\sqrt{i}$? +1065 Teenager solves Newton dynamics problem - where is the paper? +1066 Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$ +1067 What is the meaning of the third derivative of a function at a point +1068 Continuous mapping on a compact metric space is uniformly continuous +1069 Prove that the set of all algebraic numbers is countable +1070 Difference between metric and norm made concrete: The case of Euclid +1071 (Theoretical) Multivariable Calculus Textbooks +1072 Quotient ring of Gaussian integers +1073 String Theory: What to do? +1074 What is the chance to get a parking ticket in half an hour if the chance to get a ticket is 80% in 1 hour? +1075 Why there is no sign of logic symbols in mathematical texts? +1076 Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces) +1077 Interesting and unexpected applications of $\pi$ +1078 How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$? +1079 Thurstons 37th way of thinking about the derivative +1080 Volumes of n-balls: what is so special about n=5? +1081 Ramanujan log-trigonometric integrals +1082 Finding out the area of a triangle if the coordinates of the three vertices are given +1083 Factorial, but with addition +1084 A simple explanation of eigenvectors and eigenvalues with big picture ideas of why on earth they matter +1085 Zero probability and impossibility +1086 Whats the difference between stochastic and random? +1087 How many connected components does $\mathrm{GL}_n(\mathbb R)$ have? +1088 Mathematical subjects you wish you learned earlier +1089 Closed form for $\sum \frac{1}{n^n}$ +1090 Why cant you pick socks using coin flips? +1091 Is it bad form to write mysterious proofs without explaining what one intends to do? +1092 Does $G\cong G/H$ imply that $H$ is trivial? +1093 What does the mysterious constant marked by C on a slide rule indicate? +1094 Two curious identities on $x^x$,$e$,and $\pi$ +1095 How to find perpendicular vector to another vector? +1096 How to calculate $\,(a-b)\bmod n\,$ and $ {-}b \bmod n$ +1097 How was the normal distribution derived? +1098 What is the proper way to study (more advanced) math? +1099 What are the applications of functional analysis? +1100 Very *mathematical* general physics book +1101 Why not include as a requirement that all functions must be continuous to be differentiable? +1102 Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis +1103 What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle. +1104 Does associativity imply commutativity? +1105 Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra +1106 A new imaginary number? $x^c = -x$ +1107 About Euclids Elements and modern video games +1108 A matrix and its transpose have the same set of eigenvalues/other version: $A$ and $A^T$ have the same spectrum +1109 What is the math behind the game Spot It? +1110 Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!) +1111 Why determinant of a 2 by 2 matrix is the area of a parallelogram? +1112 Intuition behind using complementary CDF to compute expectation for nonnegative random variables +1113 Proof that a Combination is an integer +1114 A map is continuous if and only if for every set, the image of closure is contained in the closure of image +1115 Examples of finite nonabelian groups. +1116 Whats going on with compact implies sequentially compact? +1117 Best Algebraic Geometry text book? (other than Hartshorne) +1118 Why does the symbol for the multiplication operation change shape? +1119 Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down +1120 Why does the google calculator give $\tan 90^{\circ} = 1.6331779e^{+16}$? +1121 Reference request: introduction to commutative algebra +1122 When writing in math, do you use a comma or colon preceding an equation? +1123 Is linear algebra laying the foundation for something important? +1124 How come $32.5 = 31.5$? (The Missing Square puzzle.) +1125 Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods? +1126 Is there an easy way to show which spheres can be Lie groups? +1127 How to solve fifth-degree equations by elliptic functions? +1128 Intuition behind conjugation in group theory +1129 How to stop forgetting proofs - for a first course in Real Analysis? +1130 Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$ +1131 How to find the inverse modulo $m$? +1132 Finding a point along a line a certain distance away from another point! +1133 Graph theory: adjacency vs incident +1134 Space of bounded continuous functions is complete +1135 Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$? +1136 Simplest proof of Taylors theorem +1137 What algorithm is used by computers to calculate logarithms? +1138 Why do we need to learn integration techniques? +1139 Solving $DEF+FEF=GHH$, $KLM+KLM=NKL$, $ABC+ABC+ABC=BBB$ +1140 Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers? +1141 Mathematical research of Pokémon +1142 Why Zariski topology? +1143 Why should we prove obvious things? +1144 Drunk man with a set of keys. +1145 Will assuming the existence of a solution ever lead to a contradiction? +1146 What is $x^y$? How to understand it? +1147 Lebesgue measure theory vs differential forms? +1148 Naturally occurring non-Hausdorff spaces? +1149 What are some mathematical topics that involve adding and multiplying pictures? +1150 How to cut a cube out of a tree stump, such that a pair of opposing vertices are in the center? +1151 Why are the last two numbers of this sequence never prime? +1152 Is $ 0.112123123412345123456\dots $ algebraic or transcendental? +1153 Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx$ +1154 Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$ +1155 Difference between Fourier transform and Wavelets +1156 Is there a known mathematical equation to find the nth prime? +1157 Understanding Borel sets +1158 $\infty = -1 $ paradox +1159 What do $\pi$ and $e$ stand for in the normal distribution formula? +1160 Completion of rational numbers via Cauchy sequences +1161 Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$ +1162 Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even +1163 Why does being holomorphic imply so much about a function? +1164 Why should I care about adjoint functors +1165 How did Euler prove the Mersenne number $2^{31}-1$ is a prime so early in history? +1166 Pseudo Proofs that are intuitively reasonable +1167 Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer? +1168 Is this determinant identity known? +1169 How was the Monsters existence originally suspected? +1170 How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$ +1171 Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$ +1172 What exactly is Laplace transform? +1173 $\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this? +1174 Can a limit of an integral be moved inside the integral? +1175 If $G/Z(G)$ is cyclic, then $G$ is abelian +1176 Prove if $n^2$ is even, then $n$ is even. +1177 An Explanation of the Kalman Filter +1178 Geometric interpretation for complex eigenvectors of a 2×2 rotation matrix +1179 Proving that $1$- and $2D$ simple symmetric random walks return to the origin with probability $1$ +1180 incremental computation of standard deviation +1181 Intuitive explanation of a definition of the Fisher information +1182 Not every metric is induced from a norm +1183 Why does multiplying a number on a clock face by 10 and then halving, give the minutes? ${}{}$ +1184 Equivalent Definitions of the Operator Norm +1185 Is it morally right and pedagogically right to google answers to homework? +1186 Why is the complex plane shaped like it is? +1187 What are some applications of elementary linear algebra outside of math? +1188 How can one prove that $e<\pi$? +1189 Why does this miracle method for matrix inversion work? +1190 Is there an intuitive reason for a certain operation to be associative? +1191 Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z...: unified treatment of transforms? +1192 Are there infinitely many super-palindromes? +1193 Let, $A\subset\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$. +1194 Numerical phenomenon. Who can explain? +1195 Probability of 3 people in a room of 30 having the same birthday +1196 Difference between continuity and uniform continuity +1197 Compact sets are closed? +1198 What is mathematical research like? +1199 Difference between complete and closed set +1200 When does L Hopitals rule fail? +1201 In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True? +1202 Whats a good place to learn Lie groups? +1203 Why do we use a Least Squares fit? +1204 Chance of meeting in a bar +1205 What is the most expensive item I could buy with £50? +1206 Can someone please explain the Riemann Hypothesis to me... in English? +1207 Prove $0! = 1$ from first principles +1208 Why cant the second fundamental theorem of calculus be proved in just two lines? +1209 What is so interesting about the zeroes of the Riemann $\zeta$ function? +1210 Does every Abelian group admit a ring structure? +1211 Are there an infinite number of prime numbers where removing any number of digits leaves a prime? +1212 Japanese Temple Problem From 1844 +1213 The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$ +1214 Understandable questions which are hard for non-mathematicians but easy for mathematicians +1215 Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers? +1216 Fractal behavior along the boundary of convergence? +1217 Are there any valid continuous Sudoku grids? +1218 A nasty integral of a rational function +1219 Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$ +1220 How much does symbolic integration mean to mathematics? +1221 Evaluating sums and integrals using Taylors Theorem +1222 How to show that $\det(AB) =\det(A) \det(B)$? +1223 What is the equation for a 3D line? +1224 Density of sum of two independent uniform random variables on $[0,1]$ +1225 Use of without loss of generality +1226 Why is $\pi $ equal to $3.14159...$? +1227 $\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation +1228 Limit $\frac{x^2y}{x^4+y^2}$ is found using polar coordinates but it is not supposed to exist. +1229 What is the largest eigenvalue of the following matrix? +1230 Good books on Math History +1231 Why is a circle 1-dimensional? +1232 Overview of basic results on cardinal arithmetic +1233 Is The empty set is a subset of any set a convention? +1234 How do people apply the Lebesgue integration theory? +1235 Why is radian so common in maths? +1236 Why do engineers use derivatives in discontinuous functions? Is it correct? +1237 Why are addition and multiplication commutative, but not exponentiation? +1238 $100$-th derivative of the function $f(x)=e^{x}\cos(x)$ +1239 Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number? +1240 Is memory unimportant in doing mathematics? +1241 Theorems names that dont credit the right people +1242 What is the oldest open problem in geometry? +1243 Why the emphasis on Projective Space in Algebraic Geometry? +1244 Is there a great mathematical example for a 12-year-old? +1245 Am I too young to learn more advanced math and get a teacher? +1246 How to straighten a parabola? +1247 Geometric intuition for the tensor product of vector spaces +1248 How likely is it not to be anyones best friend? +1249 What is a real world application of polynomial factoring? +1250 What is the proof that the total number of subsets of a set is $2^n$? +1251 How to explain for my daughter that $\frac {2}{3}$ is greater than $\frac {3}{5}$? +1252 Where is the flaw in this proof that 1=2? (Derivative of repeated addition) +1253 How to write a good mathematical paper? +1254 What are the names of numbers in the binary system? +1255 Why $9$ & $11$ are special in divisibility tests using decimal digit sums? (casting out nines & elevens) +1256 Is it technically incorrect to write proofs forward? +1257 What Is Exponentiation? +1258 Why do I get one extra wrong solution when solving $2-x=-\sqrt{x}$? +1259 Mathematicians dont quit, they fade away +1260 If I know the order of every element in a group, do I know the group? +1261 Is it possible to find an infinite set of points in the plane where the distance between any pair is rational? +1262 Have there been efforts to introduce non Greek or Latin alphabets into mathematics? +1263 Algebra: Best mental images +1264 What is the average rational number? +1265 Is this similarity to the Fourier transform of the von Mangoldt function real? +1266 Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$ +1267 Topological spaces admitting an averaging function +1268 How do I tell if matrices are similar? +1269 What isnt a vector space? +1270 Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ +1271 Why is Euclids proof on the infinitude of primes considered a proof? +1272 When not to treat dy/dx as a fraction in single-variable calculus? +1273 how to read a mathematical paper? +1274 Why do people lose in chess? +1275 Is it mathematically valid to separate variables in a differential equation? +1276 How much Math do you REALLY do in your job? +1277 What is the importance of the infinitesimal generator of Brownian motion? +1278 Do Parabolic Trigonometric Functions exist? +1279 Why is the tensor product important when we already have direct and semidirect products? +1280 Nuking the Mosquito — ridiculously complicated ways to achieve very simple results +1281 Are proofs by contradiction really logical? +1282 Is it possible to formulate category theory without set theory? +1283 What is the logic/rationale behind the vector cross product? +1284 Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$ +1285 Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$ +1286 Can $18$ consecutive integers be separated into two groups,such that their product is equal? +1287 Integration of forms and integration on a measure space +1288 Simplicial homology of real projective space by Mayer-Vietoris +1289 Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function +1290 Does the string of prime numbers contain all natural numbers? +1291 Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even +1292 Differential forms on fuzzy manifolds +1293 Difference between NFA and DFA +1294 Given this transformation matrix, how do I decompose it into translation, rotation and scale matrices? +1295 How can a probability density be greater than one and integrate to one +1296 Good books on mathematical logic? +1297 There are 4 cups of liquid. Three are water and one is poison. If you were to drink 3 of the 4 cups, what is the probability of being poisoned? +1298 How to convince a layperson that the $\pi = 4$ proof is wrong? +1299 Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$. +1300 Intuitive explanation of a positive semidefinite matrix +1301 Proving that the tensor product is right exact +1302 Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$ +1303 Express 99 2/3% as a fraction? No calculator +1304 What exactly is calculus? +1305 Does it ever make sense NOT to go to the most prestigious graduate school you can get into? +1306 Set of continuity points of a real function +1307 Does every prime divide some Fibonacci number? +1308 Does $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ converge? +1309 Understanding the Laplace operator conceptually +1310 What is Cauchy Schwarz in 8th grade terms? +1311 Proving the existence of a proof without actually giving a proof +1312 How and why does Grothendiecks work provide tools to attack problems in number theory? +1313 Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$ +1314 Does multiplying all a numbers roots together give a product of infinity? +1315 What are some good ways to get children excited about math? +1316 Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse? +1317 When is an infinite product of natural numbers regularizable? +1318 Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$? +1319 Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$ +1320 Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental? +1321 Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure? +1322 Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$. +1323 What is a good book for learning math, from middle school level? +1324 difference between dot product and inner product +1325 Check if a point is within an ellipse +1326 What does proving the Riemann Hypothesis accomplish? +1327 Is Serge Langs Algebra still worth reading? +1328 Highest power of a prime $p$ dividing $N!$ +1329 Proof that Pi is constant (the same for all circles), without using limits +1330 Interesting real life applications of serious theorems +1331 How do I motivate myself to do math again? +1332 Under what conditions the quotient space of a manifold is a manifold? +1333 How to effectively study math? +1334 Is there any branch of Mathematics which has no applications in any other field or in real world? +1335 Nice expression for minimum of three variables? +1336 What exactly is a number? +1337 Dominoes and induction, or how does induction work? +1338 Riemann hypothesis: is Bender-Brody-Müller Hamiltonian a new line of attack? +1339 Why do we not have to prove definitions? +1340 Optimal strategy for cutting a sausage? +1341 Symbol for probably equal to (barring pathology)? +1342 Is it not effective to learn math top-down? +1343 If we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect? +1344 A new general formula for the quadratic equation? +1345 Conways Murder Weapon +1346 How to explain to the layperson what mathematics is, why its important, and why its interesting +1347 Is the box topology good for anything? +1348 Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge? +1349 Does Fermats Last Theorem hold for cyclotomic integers in $\mathbb{Q(\zeta_{37})}$? +1350 In combinatorics, how can one verify that one has counted correctly? +1351 Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres +1352 Fourier Transform of Derivative +1353 What is the relation between rank of a matrix, its eigenvalues and eigenvectors +1354 What is the standard basis for fields of complex numbers? +1355 Good books for a high schooler self-studying Abstract Algebra? +1356 Why do the French count so strangely? +1357 A comprehensive list of binomial identities? +1358 What is a covector and what is it used for? +1359 First-Order Logic vs. Second-Order Logic +1360 Why is the determinant the volume of a parallelepiped in any dimensions? +1361 Introductory texts on manifolds +1362 Functions that are their own inverse. +1363 How do I convince my students that the choice of variable of integration is irrelevant? +1364 If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss? +1365 A way to find this shaded area without calculus? +1366 Learning Roadmap for Algebraic Topology +1367 What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity? +1368 How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical communitys standards? +1369 When to give up on a hard math problem? +1370 Simplicial Complex vs Delta Complex vs CW Complex +1371 Why are primes considered to be the building blocks of the integers? +1372 What are some mathematically interesting computations involving matrices? +1373 Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power +1374 Given real numbers: define integers? +1375 Arc length contest! Minimize the arc length of $f(x)$ when given three conditions. +1376 Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|^n}n$ convergent？ +1377 What does strength refer to in mathematics? +1378 Is there a reason it is so rare we can solve differential equations? +1379 Why it is important to write a function as sum of even and odd functions? +1380 What is a proof? +1381 Area covered by a constant length segment rotating around the center of a square. +1382 Is it normal to treat Math Theorems as Black Boxes +1383 Good history of mathematical ideas book? +1384 Abstract nonsense proof of snake lemma +1385 How to find a total order with constrained comparisons +1386 Integers $n$ such that $i(i+1)(i+2) \cdots (i+n)$ is real or pure imaginary +1387 Which Algebraic Properties Distinguish Lie Groups from Abstract Groups? +1388 I have learned that 1/0 is infinity, why isnt it minus infinity? +1389 Difference between axioms, theorems, postulates, corollaries, and hypotheses +1390 Is the sum and difference of two irrationals always irrational? +1391 What is the difference between the Frobenius norm and the 2-norm of a matrix? +1392 How does a calculator calculate the sine, cosine, tangent using just a number? +1393 How do you compute negative numbers to fractional powers? +1394 Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction +1395 Probability that random moves in the game 2048 will win +1396 Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$? +1397 Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$ +1398 How can you prove that the square root of two is irrational? +1399 Is the Law of Large Numbers empirically proven? +1400 Cardinality of Borel sigma algebra +1401 Motivation and methods for self-study +1402 Intuition on group homomorphisms +1403 Can someone explain the Yoneda Lemma to an applied mathematician? +1404 Polynomial division: an obvious trick? [reducing mod $\textit{simpler}$ multiples] +1405 How can I introduce complex numbers to precalculus students? +1406 $n$th derivative of $e^{1/x}$ +1407 What should an amateur do with a proof of an open problem? +1408 Why does an exponential function eventually get bigger than a quadratic +1409 Big List of Erdős elementary proofs +1410 Continuity and the Axiom of Choice +1411 Graphs for which a calculus student can reasonably compute the arclength +1412 Conjugate subgroup strictly contained in the initial subgroup? +1413 Rigour in mathematics +1414 Where does the word torsion in algebra come from? +1415 Computation with a memory wiped computer +1416 Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers. +1417 Unexpected approximations which have led to important mathematical discoveries +1418 On Ramanujans curious equality for $\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots} $ +1419 Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$? +1420 How do I prove that a function is well defined? +1421 Is the rank of a matrix the same of its transpose? If yes, how can I prove it? +1422 Evaluating the indefinite integral $ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $ +1423 Proof that the irrational numbers are uncountable +1424 Can a complex number ever be considered bigger or smaller than a real number, or vice versa? +1425 Prove every odd integer is the difference of two squares +1426 how to be good at proving? +1427 Are there any examples of non-computable real numbers? +1428 What is a good book to study linear algebra? +1429 Why is empty set an open set? +1430 How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$? +1431 Probability for the length of the longest run in $n$ Bernoulli trials +1432 Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ +1433 What do prime ideals in $k[x,y]$ look like? +1434 Characterizing units in polynomial rings +1435 Entire one-to-one functions are linear +1436 Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable. +1437 How do you show monotonicity of the $\ell^p$ norms? +1438 Strategies to denest nested radicals $\sqrt{a+b\sqrt{c}}$ +1439 How would you explain to a 9th grader the negative exponent rule? +1440 Why learn to solve differential equations when computers can do it? +1441 Big List of Fun Math Books +1442 What is the definition of a set? +1443 Is there such a thing as proof by example (not counter example) +1444 Can there be two distinct, continuous functions that are equal at all rationals? +1445 Whats the goal of mathematics? +1446 How to debug math? +1447 What does communicated by mean in math papers? +1448 Using Gröbner bases for solving polynomial equations +1449 Which is larger? $20!$ or $2^{40}$? +1450 Unsolved Problems due to Lack of Computational Power +1451 Whats wrong with this reasoning that $\frac{\infty}{\infty}=0$? +1452 Are there theoretical applications of trigonometry? +1453 The last digit of $2^{2006}$ +1454 Intuition for the Importance of Modular Forms +1455 Linear Algebra Versus Functional Analysis +1456 Whats the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds? +1457 Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$... +1458 Surprisingly elementary and direct proofs +1459 Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$ +1460 Is $n \sin n$ dense on the real line? +1461 When are nonintersecting finite degree field extensions linearly disjoint? +1462 The relation between trace and determinant of a matrix +1463 How to prove every closed interval in R is compact? +1464 Is there a step by step checklist to check if a multivariable limit exists and find its value? +1465 Is an automorphism of the field of real numbers the identity map? +1466 What do eigenvalues have to do with pictures? +1467 Why cant you flatten a sphere? +1468 List of problem books in undergraduate and graduate mathematics +1469 What is the importance of Calculus in todays Mathematics? +1470 Why does $\frac{1}{x} < 4$ have two answers? +1471 Explain this mathematical meme (Geometers bird interrupting Topologists bird) +1472 Coin flipping probability game ; 7 flips vs 8 flips +1473 Finite Groups with exactly $n$ conjugacy classes $(n=2,3,...)$ +1474 Help understanding Algebraic Geometry +1475 7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish. +1476 Can a number have infinitely many digits before the decimal point? +1477 How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists? +1478 Whats the intuition with partitions of unity? +1479 Center-commutator duality +1480 In how many different ways can a 9-panel comic grid be used? +1481 I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 +1482 I almost quit self-studying mathematics, but should I continue? +1483 What is Trinity Hall Prime number? +1484 Examples of morphisms of schemes to keep in mind? +1485 Review of my T-shirt design +1486 Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix +1487 Is A New Kind of Science a new kind of science? +1488 What am I doing when I separate the variables of a differential equation? +1489 Does every set have a group structure? +1490 Why do we use trig functions in Fourier transforms, and not other periodic functions? +1491 How to evaluate $I=\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$ +1492 Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$? +1493 Can squares of infinite area always cover a unit square? +1494 Whats the probability that a sequence of coin flips never has twice as many heads as tails? +1495 $5^n+n$ is never prime? +1496 What does it take to divide by $2$? +1497 If a two variable smooth function has two global minima, will it necessarily have a third critical point? +1498 Subgroups as isotropy subgroups and regular orbits on tuples +1499 A question about Sylow subgroups and $C_G(x)$ +1500 How to know if a point is inside a circle? +1501 Poisson Distribution of sum of two random independent variables $X$, $Y$ +1502 Explanation on arg min +1503 Product of two Gaussian PDFs is a Gaussian PDF, but Product of two Gaussian Variables is not Gaussian +1504 Is the product of symmetric positive semidefinite matrices positive definite? +1505 Matrices commute if and only if they share a common basis of eigenvectors? +1506 What are the formal names of operands and results for basic operations? +1507 Differentiating an Inner Product +1508 How to show that the commutator subgroup is a normal subgroup +1509 Nobody told me that self teaching could be so damaging... +1510 (undergraduate) Algebraic Geometry Textbook Recommendations +1511 Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD? +1512 Dot Product Intuition +1513 Divisor -- line bundle correspondence in algebraic geometry +1514 How to tell if Im good enough for graduate school? +1515 How is the derivative truly, literally the best linear approximation near a point? +1516 Why is the Continuum Hypothesis (not) true? +1517 Good examples of double induction +1518 A very general method for proving inequalities. Too good to be true? +1519 Is it possible to have three real numbers that have both their sum and product equal to $1$? +1520 What was the book that opened your mind to the beauty of mathematics? +1521 Research done by high-school students +1522 Why does the discriminant in the Quadratic Formula reveal the number of real solutions? +1523 Understanding the intuition behind math +1524 A circle rolls along a parabola +1525 Why is $\varphi$ called the most irrational number? +1526 How can I intuitively understand complex exponents? +1527 It looks straightforward, but actually it isnt +1528 Refuting the Anti-Cantor Cranks +1529 Could I be using proof by contradiction too much? +1530 Is there any geometric intuition for the factorials in Taylor expansions? +1531 Why do differential forms have a much richer structure than vector fields? +1532 A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$ +1533 Quadratic reciprocity via generalized Fibonacci numbers? +1534 Penroses remark on impossible figures +1535 Why are asymptotically one half of the integer compositions gap-free? +1536 Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them? +1537 Difference between ≈, ≃, and ≅ +1538 What is the difference between only if and iff? +1539 Gradient of squared $2$-norm +1540 Distance/Similarity between two matrices +1541 Are all eigenvectors, of any matrix, always orthogonal? +1542 Area of a square inside a square created by connecting point-opposite midpoint +1543 How to generate random points on a sphere? +1544 Number of monic irreducible polynomials of prime degree $p$ over finite fields +1545 Should I be worried that I am doing well in analysis and not well in algebra? +1546 Negative versus Minus +1547 Discontinuous linear functional +1548 Geometric interpretation for sum of fourth powers +1549 The two-daughter-problem +1550 Has there ever been an application of dividing by $0$? +1551 Why do we miss 8 in the decimal expansion of 1/81, and 98 in the decimal expansion of 1/9801? +1552 Why does the Cauchy-Schwarz Inequality even have a name? +1553 What use is the Yoneda lemma? +1554 Is there a function whose antiderivative can be found but whose derivative cannot? +1555 Similar matrices and field extensions +1556 Are there mathematical concepts that exist in dimension $4$, but not in dimension $3$? +1557 Is there a definition of determinants that does not rely on how they are calculated? +1558 Unexpected use of topology in proofs +1559 Does this Fractal Have a Name? +1560 How is the Gödels Completeness Theorem not a tautology? +1561 How many 7-note musical scales are possible within the 12-note system? +1562 Approximating a $\sigma$-algebra by a generating algebra +1563 How do different definitions of degree coincide? +1564 What is the size of each side of the square? +1565 A non-losing strategy for tic-tac-toe $\times$ tic-tac-toe +1566 Can there be an injective function whose derivative is equivalent to its inverse function? +1567 Evaluating $\int_0^\infty \frac{dx}{\sqrt{x}[x^2+(1+2\sqrt{2})x+1][1-x+x^2-x^3+...+x^{50}]}$ +1568 Trigonometric sums related to the Verlinde formula +1569 A bestiary about adjunctions +1570 Is OEIS A248049 an integer sequence? +1571 Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $ +1572 Golden Number Theory +1573 Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$ +1574 An illusionist and their assistant are about to perform the following magic trick +1575 The most effective windshield-wiper setup. (Packing a square with sectors) +1576 How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$ +1577 Period of the sum/product of two functions +1578 Example of Partial Order thats not a Total Order and why? +1579 What does curly (curved) less than sign $\succcurlyeq$ mean? +1580 Why we consider log likelihood instead of Likelihood in Gaussian Distribution +1581 Is there an equation to describe regular polygons? +1582 Eigenvalues of the rank one matrix $uv^T$ +1583 Under what condition we can interchange order of a limit and a summation? +1584 Calculating pi manually +1585 Difference between basis and subbasis in a topology? +1586 Visualizing the 4th dimension. +1587 Why does the derivative of sine only work for radians? +1588 Origin of the dot and cross product? +1589 What is the arithmetic mean of no numbers? +1590 The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$ +1591 Would a proof to the Riemann Hypothesis affect security? +1592 What lies beyond the Sedenions +1593 Is it possible to determine if you were on a Möbius strip? +1594 Big O Notation is element of or is equal +1595 Is there a slowest rate of divergence of a series? +1596 Any smart ideas on finding the area of this shaded region? +1597 Can we calculate $ i\sqrt { i\sqrt { i\sqrt { \cdots } } }$? +1598 Is it possible that A counter-example exists but it cannot be found +1599 Applications of complex numbers to solve non-complex problems +1600 Is it an abuse of language to say *the* integers, *the* rational numbers, or *the* real numbers, etc.? +1601 Paradox: increasing sequence that goes to $0$? +1602 Is it possible to prove a mathematical statement by proving that a proof exists? +1603 Is $6.12345678910111213141516171819202122\ldots$ transcendental? +1604 Big list of guided discovery books +1605 Are there any Turing-undecidable problems whose undecidability is independent of the Halting problem? +1606 Funny double infinite sum +1607 Geometry problem involving infinite number of circles +1608 What proportion of positive integers have two factors that differ by 1? +1609 What are Different Approaches to Introduce the Elementary Functions? +1610 Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$ +1611 How to think of the group ring as a Hopf algebra? +1612 Whats the value of this Viète-style product involving the golden ratio? +1613 Is $1+x+\frac{x^2}2+\dots+\frac{x^n}{n!}$ irreducible? +1614 How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis? +1615 Looking for a definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel +1616 Whats the sign of $\det\left(\sqrt{i^2+j^2}\right)_{1\le i,j\le n}$? +1617 Is there a good reason why $\left\lfloor \frac{n!}{11e}\right\rfloor$ is always even? +1618 What does E mean in 9.0122222900391E-5? +1619 What’s the difference between analytical and numerical approaches to problems? +1620 How to find the multiplicity of eigenvalues? +1621 Is the power set of the natural numbers countable? +1622 Finding $\int x^xdx$ +1623 Calculate on which side of a straight line is a given point located? +1624 Generating correlated random numbers: Why does Cholesky decomposition work? +1625 product distribution of two uniform distribution, what about 3 or more +1626 Dantzigs unsolved homework problems +1627 Interesting math-facts that are visually attractive +1628 How does TREE(3) grow to get so big? (Laymen explanation) +1629 Order of general- and special linear groups over finite fields. +1630 Why are two permutations conjugate iff they have the same cycle structure? +1631 What is an operator in mathematics? +1632 Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral? +1633 Are there any differences between tensors and multidimensional arrays? +1634 Is it possible to write a sum as an integral to solve it? +1635 Why divide by $2m$ +1636 Can math be subjective? +1637 Is there an easy way to see associativity or non-associativity from an operations table? +1638 Matrix is conjugate to its own transpose +1639 Is there a third dimension of numbers? +1640 Proving that $m+n\sqrt{2}$ is dense in $\mathbb R$ +1641 What is the antiderivative of $e^{-x^2}$ +1642 Is there a math expression equivalent to the conditional ternary operator? +1643 Understanding Gödels Incompleteness Theorem +1644 Do harmonic numbers have a “closed-form” expression? +1645 Is $\{0\}$ a field? +1646 How hard is the proof of $\pi$ or $e$ being transcendental? +1647 Interesting results easily achieved using complex numbers +1648 All natural numbers are equal. +1649 Geometrical interpretation of Ricci curvature +1650 Why are real numbers useful? +1651 $x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible? +1652 What kind of work do modern day algebraists do? +1653 Choice of $q$ in Baby Rudins Example 1.1 +1654 Why is there never a proof that extending the reals to the complex numbers will not cause contradictions? +1655 Is there any conjecture that we know is provable/disprovable but we havent found a proof of yet? +1656 Gaps or holes in rational number system +1657 What is $-i$ exactly? +1658 What knot is this? +1659 Doesnt the unprovability of the continuum hypothesis prove the continuum hypothesis? +1660 What is the fastest/most efficient algorithm for estimating Eulers Constant $\gamma$? +1661 What is the crime of lèse-Bourbaki? +1662 Natural example of cosets +1663 Is Complex Analysis equivalent Real Analysis with $f:\mathbb R^2 \to \mathbb R^2$? +1664 If $e^A$ and $e^B$ commute, do $A$ and $B$ commute? +1665 A discrete math riddle +1666 How does one give a mathematical talk? +1667 Probability that a quadratic equation has real roots +1668 How to identify surfaces of revolution +1669 Is there an abstract definition of a matrix being upper triangular? +1670 Self-learning mathematics - help needed! +1671 What is this pattern called? +1672 The $100$th derivative of $(x^2 + 1)/(x^3 - x)$ +1673 Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$ +1674 Infinite Series $\sum\limits_{n=1}^\infty\frac{(H_n)^2}{n^3}$ +1675 In which ordered fields does absolute convergence imply convergence? +1676 Crazy pattern in the simple continued fraction for $\sum_{k=1}^\infty \frac{1}{(2^k)!}$ +1677 A sequence that avoids both arithmetic and geometric progressions +1678 Proof of triangle inequality +1679 How to find the factorial of a fraction? +1680 What is the general equation of the ellipse that is not in the origin and rotated by an angle? +1681 Prove that the union of countably many countable sets is countable. +1682 Symbol for elementwise multiplication of vectors +1683 Difference between supremum and maximum +1684 What is difference between a ring and a field? +1685 Picking random points in the volume of sphere with uniform probability +1686 Necessity/Advantage of LU Decomposition over Gaussian Elimination +1687 Introductory Group theory textbook +1688 Does non-symmetric positive definite matrix have positive eigenvalues? +1689 Difference between Analytic and Holomorphic function +1690 Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $. +1691 Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials? +1692 What does it take to get a job at a top 50 math program in the U.S.? +1693 Is an irrational number odd or even? +1694 Definitions of Hessian in Riemannian Geometry +1695 How to calculate the pullback of a $k$-form explicitly +1696 How to prove that $\log(x)1$? +1697 Exam with $12$ yes/no questions (half yes, half no) and $8$ correct needed to pass, is it better to answer randomly or answer exactly 6 times yes? +1698 Distinguishing probability measure, function and distribution +1699 False proof: $\pi = 4$, but why? +1700 Books about the Riemann Hypothesis +1701 Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? +1702 What is exactly the difference between a definition and an axiom? +1703 Non-associative operations +1704 Is $\sqrt{64}$ considered $8$? or is it $8,-8$? +1705 What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$? +1706 Mathematical explanation behind a picture posted (lifted from facebook) +1707 Path-connected and locally connected space that is not locally path-connected +1708 Can I search for factors of $\ (11!)!+11!+1\ $ efficiently? +1709 Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields +1710 Square root confusion: Why am I getting an answer if it doesnt work? +1711 Geometric understanding of differential forms. +1712 Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions? +1713 What is the formula for pi used in the Python decimal library? +1714 Intuition for why the difference between $\frac{2x^2-x}{x^2-x+1}$ and $\frac{x-2}{x^2-x+1}$ is a constant? +1715 Axiom of choice and automorphisms of vector spaces +1716 Show that floating point $\sqrt{x \cdot x} \geq x$ for all long $x$. +1717 How to entertain a crowd with mathematics? +1718 Colliding Bullets +1719 False beliefs about Lebesgue measure on $\mathbb{R}$ +1720 Evaluating $\int P(\sin x, \cos x) \text{d}x$ +1721 Is there a geometric idea behind Sylows theorems? +1722 Is $29$ the only prime of the form $p^p+2$? +1723 Is Stokes Theorem natural in the sense of category theory? +1724 Trying to define $\mathbb{R}^{0.5}$ topologically +1725 Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$ +1726 Elementary proof that the derivative of a real function is continuous somewhere +1727 Can $\sqrt{p}^{\sqrt{p}^{\sqrt{p}}}$ be an integer, if $p$ is a non-square positive integer? +1728 How to tell if a set of vectors spans a space? +1729 Easy way of memorizing values of sine, cosine, and tangent +1730 What does ∈ mean? +1731 Recognizable vs Decidable +1732 Why have we chosen our number system to be decimal (base 10)? +1733 How the product of two integrals is iterated integral? $\int\cdot \int = \iint$ +1734 How to take the gradient of the quadratic form? +1735 Why cant a set have two elements of the same value? +1736 Order of finite fields is $p^n$ +1737 Why is $A^TA$ invertible if $A$ has independent columns? +1738 Proof that the largest eigenvalue of a stochastic matrix is $1$ +1739 how does expectation maximization work? +1740 Are functions of independent variables also independent? +1741 What is an intuitive explanation for $\operatorname{div} \operatorname{curl} F = 0$? +1742 What is the difference between projected gradient descent and ordinary gradient descent? +1743 Functions which are Continuous, but not Bicontinuous +1744 Choose a random number between $0$ and $1$ and record its value. Keep doing it until the sum of the numbers exceeds $1$. How many tries do we need? +1745 How I can prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2? +1746 Difference between bijection and isomorphism? +1747 $\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite +1748 An easy example of a non-constructive proof without an obvious fix? +1749 Why doesnt induction extend to infinity? (re: Fourier series) +1750 A really complicated calculus book +1751 Why isnt several complex variables as fundamental as multivariable calculus? +1752 Parabola is an ellipse, but with one focal point at infinity +1753 Formal definition of conditional probability +1754 Why is it worth spending time on type theory? +1755 Why is cross product defined in the way that it is? +1756 What are the rules for equals signs with big-O and little-o? +1757 What book can bridge high school math and the more advanced topics? +1758 Foundation for analysis without axiom of choice? +1759 Intuition behind ideal +1760 How far is the list of known primes known to be complete? +1761 Why the real and imaginary parts of a complex analytic function are not independent? +1762 Do circles divide the plane into more regions than lines? +1763 On the functional square root of $x^2+1$ +1764 $n!+1$ being a perfect square +1765 Does $\zeta(3)$ have a connection with $\pi$? +1766 Why do we need so many trigonometric definitions? +1767 Prove that the sum of pythagorean triples is always even +1768 What is the explanation for this visual proof of the sum of squares? +1769 Why are topological spaces interesting to study? +1770 Good math books to discover stuff by yourself +1771 Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$ +1772 Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space? +1773 Why is the Laplacian important in Riemannian geometry? +1774 MIT 2015 Integration Question +1775 Is there a name for this type of polygon? +1776 Intuition behind Snake Lemma +1777 Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius? +1778 Proof that a trigonometric function of a rational angle must be non-transcendental +1779 Does $|n^2 \cos n|$ diverge to $+\infty$? +1780 What are exact sequences, metaphysically speaking? +1781 What structure does the alternating group preserve? +1782 About the integral $\int_{0}^{+\infty}\sin(x\,\log x)\,dx$ +1783 Effective Research Notes +1784 A new continued fraction for Apérys constant, $\zeta(3)$? +1785 Two questions about weakly convergent series related to $\sin(n^2)$ and Weyls inequality +1786 What is the integral of 1/x? +1787 How does one denote the set of all positive real numbers? +1788 Prove: If a sequence converges, then every subsequence converges to the same limit. +1789 How many prime numbers are known? +1790 What books are prerequisites for Spivaks Calculus? +1791 Create unique number from 2 numbers +1792 What is the main difference between a vector space and a field? +1793 If eigenvalues are positive, is the matrix positive definite? +1794 $\epsilon$-$\delta$ proof that $\lim\limits_{x \to 1} \frac{1}{x} = 1$. +1795 Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$ +1796 Pen, pencils and paper to write math +1797 Video lectures on Real Analysis? +1798 If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative. +1799 Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator? +1800 Zero divided by zero must be equal to zero +1801 Proof that every number ≥ $8$ can be represented by a sum of fives and threes. +1802 What is the difference between a Ring and an Algebra? +1803 Intuitive understanding of the derivatives of $\sin x$ and $\cos x$ +1804 A linear operator commuting with all such operators is a scalar multiple of the identity. +1805 Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian +1806 Is $\pi$ equal to $180^\circ$? +1807 Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions? +1808 Seeking a laymans guide to Measure Theory +1809 What are the Laws of Rational Exponents? +1810 An intuitive approach to the Jordan Normal form +1811 Intuitive Understanding of the constant $e$ +1812 How far can one see over the ocean? +1813 Every power series is the Taylor series of some $C^{\infty}$ function +1814 Is the axiom of choice really all that important? +1815 On average, how many friends would I need to have to have at least one friends birthday every day? +1816 Is there a purely algebraic proof of the Fundamental Theorem of Algebra? +1817 What books should I get to self study beyond Calculus for someone about to start undergrad mathematics? +1818 Are there any objects which arent sets? +1819 What is the theme of analysis? +1820 Is there any geometric way to characterize $e$? +1821 Why and How do certain manipulations in indefinite integrations just work? +1822 History of the Concept of a Ring +1823 Simplest way to get the lower bound $\pi > 3.14$ +1824 What to answer when people ask what I do in mathematics +1825 How many super imaginary numbers are there? +1826 Fibonacci infinite sum resulting in $\pi$ +1827 Is $i = \sqrt{e^{\pi\sqrt{e^{\pi\sqrt\ldots}}}}$? +1828 Example of a compact set that isnt the spectrum of an operator +1829 Why is one $\infty$ number enough for complex numbers? +1830 How do the Catalan numbers turn up here? +1831 What is so special about $\alpha=-1$ in the integral of $x^\alpha$? +1832 To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$ +1833 On a long proof +1834 Why do all the Platonic Solids exist? +1835 Bijection $f\colon\mathbb{N}\to\mathbb{N}$ with $f(0)=0$ and $|f(n)-f(n-1)|=n$ +1836 How to solve an nth degree polynomial equation +1837 Number of onto functions +1838 What does := mean? +1839 Adding two polar vectors +1840 Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $ +1841 Average Distance Between Random Points on a Line Segment +1842 Understanding Dot and Cross Product +1843 What are good books/other readings for elementary set theory? +1844 Best Algebraic Topology book/Alternative to Allen Hatcher free book? +1845 What is the implicit function theorem? +1846 What does it mean to integrate with respect to the distribution function? +1847 Very good linear algebra book. +1848 Can a piece of A4 paper be folded so that its thick enough to reach the moon? +1849 Every nonzero element in a finite ring is either a unit or a zero divisor +1850 Projection is an open map +1851 Can someone explain Cayleys Theorem step by step? +1852 Lemma/Proposition/Theorem, which one should we pick? +1853 Game theory - self study +1854 What are the required backgrounds of Robin Hartshornes Algebraic Geometry book? +1855 Semi-direct v.s. Direct products +1856 Why is it that if I count years from 2011 to 2014 as intervals I get 3 years, but if I count each year separately I get 4 years? +1857 What exactly is infinity? +1858 How do we know the ratio between circumference and diameter is the same for all circles? +1859 Good introductory book on geometric algebra +1860 Jacobi identity - intuitive explanation +1861 Why does the volume of the unit sphere go to zero? +1862 Is it possible to place 26 points inside a rectangle that is 20 cm by 15 cm so that the distance between every pair of points is greater than 5 cm? +1863 What does it mean to solve an equation? +1864 Isomorphic quotients by isomorphic normal subgroups +1865 How can this function have two different antiderivatives? +1866 Why arent integration and differentiation inverses of each other? +1867 Is 128 the only multi-digit power of 2 such that each of its digits is also a power of 2? +1868 Do we really need reals? +1869 How can someone reject a math result if everything has to be proved? +1870 What are some things we can prove they must exist, but have no idea what they are? +1871 I dont understand Gödels incompleteness theorem anymore +1872 Where to begin with foundations of mathematics +1873 Development of the Idea of the Determinant +1874 Do we need to formally teach the Greek Alphabet? +1875 The Meaning of the Fundamental Theorem of Calculus +1876 A conjecture involving prime numbers and circles +1877 Is the percentage symbol a constant? +1878 Is a proof still valid if only the writer understands it? +1879 Set Theoretic Definition of Numbers +1880 Why cant Antoines necklace fall apart? +1881 Areas versus volumes of revolution: why does the area require approximation by a cone? +1882 Designing an Irrational Numbers Wall Clock +1883 Must eigenvalues be numbers? +1884 How does the divisibility graphs work? +1885 Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem. +1886 The identity cannot be a commutator in a Banach algebra? +1887 If the field of a vector space werent characteristic zero, then what would change in the theory? +1888 What are some math books written in dialogue or story form, e.g., a teacher explaining to a student? +1889 A zero sum subset of a sum-full set +1890 Why cant we define more elementary functions? +1891 A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$ +1892 Sheldon Cooper Primes +1893 Notations that are mnemonic outside of English +1894 Hidden patterns in $\sin(a x^2)$ +1895 AM-GM-HM Triplets +1896 Why is the Galois Correspondence intuitively plausible? +1897 Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers +1898 A curious equality of integrals involving the prime counting function? +1899 Xmas Maths 2015 +1900 How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$? +1901 Time to reach a final state in a random dynamical system (answer known, proof unknown) +1902 What numbers can be created by $1-x^2$ and $x/2$? +1903 Is there a way to get trig functions without a calculator? +1904 Sample Standard Deviation vs. Population Standard Deviation +1905 Why does an integral change signs when flipping the boundaries? +1906 Why is the absolute value function not differentiable at $x=0$? +1907 Effect of elementary row operations on determinant? +1908 Do all square matrices have eigenvectors? +1909 Is zero a prime number? +1910 How to find a basis for the intersection of two vector spaces in $\mathbb{R}^n$? +1911 Subgroup of index $2$ is Normal +1912 The generating function for the Fibonacci numbers +1913 Examples of infinite groups such that all their respective elements are of finite order. +1914 Why are $3D$ transformation matrices $4 \times 4$ instead of $3 \times 3$? +1915 Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$ +1916 Every linear mapping on a finite dimensional space is continuous +1917 difference between maximal element and greatest element +1918 How can a set contain itself? +1919 Sum of two closed sets in $\mathbb R$ is closed? +1920 How unique are $U$ and $V$ in the Singular Value Decomposition? +1921 Are normal subgroups transitive? +1922 Why is the ring of matrices over a field simple? +1923 In set theory, how are real numbers represented as sets? +1924 Is it possible to plot a graph of any shape? +1925 Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$ +1926 Subgroups of finitely generated groups are not necessarily finitely generated +1927 $|G|>2$ implies $G$ has non trivial automorphism +1928 There exists a power of 2 such that the last five digits are all 3s or 6s. Find the last 5 digits of this number +1929 Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense? +1930 $\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is ..... +1931 How many fair dice exist? +1932 How to start a math blog? +1933 Modus moron rule of inference? +1934 precise official definition of a cell complex and CW-complex +1935 Alternative proofs that $A_5$ is simple +1936 Can you give me some concrete examples of magmas? +1937 Does Monty Hall logic apply to this real world situation? +1938 Are there dictionaries in math? +1939 How did early mathematicians make it without Set theory? +1940 Sharing a pepperoni pizza with your worst enemy +1941 Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $∞$? +1942 Why is the Daniell integral not so popular? +1943 Say $a=b$. Is Do the same thing to both sides of an equation, and it still holds an axiom? +1944 Why is there antagonism towards extended real numbers? +1945 What is a topological space good for? +1946 Are rational points dense on every circle in the coordinate plane? +1947 What do mathematicians mean by equipped? +1948 Category-theoretic limit related to topological limit? +1949 Notation for an interval when you dont know which bound is greater +1950 Why can we use induction when studying metamathematics? +1951 Finding the value of $\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}$ +1952 What if $\pi$ was an algebraic number? (significance of algebraic numbers) +1953 Solution to the equation of a polynomial raised to the power of a polynomial. +1954 Are commutative C*-algebras really dual to locally compact Hausdorff spaces? +1955 Is $dx\,dy$ really a multiplication of $dx$ and $dy$? +1956 Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$ +1957 How do you prove Gautschis inequality for the gamma function? +1958 Function that is the sum of all of its derivatives +1959 Identity for simple 1D random walk +1960 Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$ +1961 How to maintain enthusiasm and joy in teaching when the material grows stale +1962 An exotic sequence +1963 Making trigonometric substitutions rigorous +1964 A new kind of fractal? +1965 How do you go about doing mathematics on a day to day basis? +1966 $C^{k}$-manifolds: how and why? +1967 Why do I only breathe out of one nostril? +1968 Why are so few foods blue? +1969 Does DNA have the equivalent of IF-statements, WHILE loops, or function calls? How about GOTO? +1970 How many times did terrestrial life emerge from the ocean? +1971 Do bacteria die of old age? +1972 Is there a reason why human eyesight and plants make use of the same wavelength of light? +1973 Why is thymine rather than uracil used in DNA? +1974 Are male and female brains physically different from birth? +1975 What is the effect of non-vaccinated people on vaccinated people? +1976 Why do plants have green leaves and not red? +1977 Can HIV be transmitted via mosquitos? +1978 Whats the evidence against SARS-CoV-2 being engineered by humans? +1979 Do beneficial viruses exist? If so, what examples are there? +1980 Does the string ...CATCAT... appear in the DNA of Felis catus? +1981 Why did the process of sleep evolve in many animals? What is its evolutionary advantage? +1982 Are humans the only species who drink milk as adults? +1983 How could humans have interbred with Neanderthals if were a different species? +1984 Why do some bad traits evolve, and good ones dont? +1985 Death because of distilled water consumption +1986 Why is the heart not in the middle of the body? +1987 Why 20 amino acids instead of 64? +1988 Is there an RGB equivalent for smells? +1989 Whats up with this leaf? +1990 How does the brains energy consumption depend on mental activity? +1991 Do animals exhibit handedness (paw-ness?) preference? +1992 Why do Humans not produce Vitamin C like other mammals? +1993 Why shouldnt dogs eat chocolate? +1994 Why are there no wheeled animals? +1995 What is the evolutionary advantage of red-green color blindness? +1996 Do ants or other insects sleep, and if so why? +1997 What could cause a forest of bent trees? +1998 Is there any evidence that sexual selection may lead to extinction of species? +1999 Why does evolution not make our life longer? +2000 Why does it hurt the next day after doing significant exercise? +2001 Can a woman give birth to twins with different fathers? +2002 Are there organisms with fewer than 1000 neurons? +2003 How come large herbivores have such thin legs? +2004 A new species of small bird? +2005 Why dont mammals have more than 4 limbs? +2006 Why do smaller mammals move intermittently? +2007 Are the social-distancing measures implemented against SARS-CoV-2 also suppressing the spread of other viruses? +2008 Why do the fastest runners tend to be black? +2009 Life without DNA? +2010 What is the benefit of fever during infections? +2011 Why isnt a virus alive? +2012 Why do men have nipples? +2013 Why are there no organisms with metal body parts, like weapons, bones, and armour? (Or are there?) +2014 Human perception of time depending on age +2015 Do large animals experience a meaningful delay when moving their most distant appendages? +2016 Why does cracking a joint make noise?