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Since we do not know , for example $\Sigma(5)$ , we do not know which number $n$ satisfies $n=\Sigma(5)$, but we know that some number $n$ must satisfy the property. In this case we do not even have an upper bound. Another less trivial example : The $20$-th Fermat-number $2^{2^{20}}+1$ is known to be composite, but no factor is known. Denote $n$ to be the ...

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As pointed out it can have several applications. I'll give a particular application in coding theory, specifically in the design of space-time codes, which are codes used in wireless systems with multiple transmitter and receiver antennas. These space-time codes can be viewed as matrices $X$, where one dimension represents space (i.e., number of different ...

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It allows you to evaluate cross products and find the general equation of the plane if given $3$ points. For instance: $A(1,1,0),\, B =(1,0,1),\,C=(0,1,2)$ $$B-A=(1,0,1)-(1,1,0)=(0,-1,1)$$ and $$C-A=(0,1,2)-(1,1,0)=(-1,0,2)$$ You now use the cross-product of $$(B-A)\times(C-A)=\begin{bmatrix}i & j & k \\0 & -1 & 1 \\-1 & 0 & 2 ... 3 Determinants can be used to see if a system of n linear equations in n variables has a unique solution. This is useful for homework problems and the like, when the relevant computations can be performed exactly. However, when solving real numerical problems, the determinant is rarely used, as it is a very poor indicator of how well you can solve a ... 0 You might wanna check out Mathologer. 0 Pythagoras' theorem! Proof using a rotated square within a square. This gets straight to the essence of mathematics! 2 To prove: 0=1. Certain identities get funky when we pass over to infinite-order matrices. We see such matrices, for example, in representations of operators in quantum mechanics. Everyone knows that Tr(AB-BA)=Tr(AB)-Tr(BA)=0. So let A_{i,j}=\delta_{i,j-1}, B_{i,j}=A_{j,i} Here \delta is the Kronecker delta function, and i and j run ... 0 Try solving,$$ \displaystyle\ \int { \dfrac{1}{(x^2+1)\sqrt{x^2-1}}} \mathrm{d}x $$To know about the solution, visit my channel Calculus Society. I don't know what you would think about this,$$ \displaystyle\ \int { \dfrac{1}{\sec^2(x)+2\tan^2(x)}} \mathrm{d}x $$or this,$$ \displaystyle\ \int { \dfrac{1}{1-\sin^4(x)}} \mathrm{d}x $$Sorry I can't think ... 2 Suppose we are given a matrix A \in \mathbb{R}^{n \times n}. If A = O_n, the one eigenvalue of A is 0, with multiplicity n. If the first n-1 rows or columns of A are either zero or a multiple of the n-th row or column, then we have a rank-1 matrix that can be written in the form$$A = \mathrm{u} \mathrm{v}^T$$where \mathrm{u}, ... 0 One trick to know if the eigen values we found are right is to check if the sum of eiven values equals the trace (sum of the diagonal elements) . Also if the product of eigenvalues equals the determinant of that matrix rhen the eigenvalues you found are correct . 4 This is a soft question without objective answers, but I can list some cases where finding the eigenvalues (or at least one eigenvalue) is easy: Singular matrices with obvious kernels Diagonal or triangular matrices Block-diagonal matrices with easily-analyzed blocks Matrices with known spectrum that have been spectrally shifted (by adding or subtracting ... 0 Calculus of Variations A path \gamma that satisfies the Euler-Lagrange equation$$\frac{d}{dt}\frac{\partial s}{\partial \dot\gamma} + \frac{\partial s}{\partial \gamma}$$at all times (a local condition) extremises the action \int_0^T s(\gamma, \dot \gamma, t)\,dt (a global condition). Consequences: Conservation of energy for physical systems ... 0 (weak) Nullstellensatz - Commutative Algebra/Algebraic Geometry Let I be an ideal in K[x_1, \ldots, x_n], where K is an algebraically closed field. If for every point of K^n, I contains a polynomial non-vanishing at that point (local), then I = K[x_1, \ldots, x_n] (global). 1 My vote goes to Carl Friedrich Gauß: "Mathematical historian Eric Temple Bell said that had Gauss published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years" (https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#Personality) See also Unpublished Discoveries by Gauss that Were Later Rediscovered and Attributed to ... 0 I enjoyed this one very much: How to compute \sum 1/n^2 by solving triangles 0 I take it that by « projective » you mean "cohomological dimension \le 1". Here are some examples from number theory and arithmetic geometry : 1). Let K be an extension of degree n of the field \mathbf Q_p of p-adic numbers. Let \mathcal G be the Galois group of the maximal pro-p-extension of K. If K does not contain a primitive p-th ... 1 Terry Tao has been using Abraham Robinson's framework and more generally ultraproducts (often considered to be "logic" as you put it), as for example his solution of Hilbert's Fifth problem in this recent book. 4 At least your integral turns out to have a closed form:$$ \int_{0}^{\pi/2} x \, \frac{\sqrt{\sin x} - \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, \mathrm{d}x = G + \pi \left( \frac{1+2\sqrt{2}}{4} \log 2 - \log (1+\sqrt{2}) \right), $$where G is the Catalan's constant. So it seems to me that the 'almost rationality' of this integral is just a ... 0 Consider any function f such that \int_{a}^{b} f(x)dx does not have a closed form, now choose b-a small enough that the integral is very close to zero. Eg: \int_{0}^{1} e^{-x^{100}} dx perhaps does not have closed form and is surely very close to zero. Any such function can be modified to make the integral very close to any given rational integer. 1 Integral of e^{-2x^2} from -2 to 62 is about 1/8 which has a .003 error. Also maybe cheating but its possible to get this integral arbitrarily small giving really small errors to something without closed form. This has implication that you could integrate any probability distributions most of which have no close form to 1 plus an arbitrary small ... 2 A linear order can be uniquely (up to isomorphism) reconstructed from the set of order types of its proper initial segments. Update: Even if we know the cardinality of the linear order, and know that it does not have a maximal element, this "theorem" still does not hold. 6 I think Expected outcome for repeated dice rolls with dice fixing qualifies. You roll n standard six-sided dice and may fix any non-empty subset of them, then re-roll the others and again fix at least one of them, and so on until all dice are fixed. The intuition that you should always fix any 6s you roll if you want to maximise the expected sum of the ... 1 I guess the Weil conjectires are one of the best examples given the key role they played as motivation for introducing many cohomology theories, most famously étale cohomology. 1 Properties Preserved Under GCB: Rank Nullity Determinant Trace Characteristic Polynomial Eigenvalues Minimal Polynomial Diagonalizability Jordan Canonical Form Properties Preserved Under OCB: Symmetry (A^T=A) Antisymmetry (A^T=-A) Orthogonality (A^TA=I) Normality (AA^T=A^TA) Positive (semi)definite-ness Schur triangular form Matrix norm ... 2 This I think is exactly what you are looking for: ftp://ftp.iiap.res.in/aman/EBooks/Schaum's%20Mathematical%20Handbook%20of%20Formulas%20and%20Tables%20--%20301.pdf 3 I'm not sure you are going to find a satisfactory book that will give you "intuition" for measure theory. It is pretty abstract stuff and I know I definitely didn't "get it" the first time I went through it. You best bet is to just keep working through problems and over time, you'll gain the insights you desire. I will also say that being able to admit that ... 4 Try A Radical Approach to Lebesgue’s Theory of Integration by Bressoud. Read a review. 4 One of the most remarkable integrals for me is the Borwein sequence. They do have closed form, but somehow I feel they should be mentioned here. More interesting integrals. 4 For various flavors of cobordism categories it can be difficult to prove that composition is well-defined. A typical example is Segal's conformal cobordism category used in conformal field theory, where, loosely speaking, the objects are finite disjoint unions of circles, and the morphisms between unions of circles are (isomorphism classes of) Riemann ... 4 One can prove this theorem by use of the fact that the matrix representation of all linear map on a complex vector space, is Triangularisable with respect to a basis \{v_1,...,v_n\}. So if T be a linear map there are \{\lambda_1,...,\lambda_n\} s.t$$T(v_1)=\lambda_1 v_1 T(v_2)=a_{11} v_1+\lambda_2 v_2 ... ...

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