# All Questions

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### Complex (topological) K-theory of $S \Sigma$ for a surface $\Sigma$.

In the following question here: What's the K-group of a surface? The $K$-theory of a compact orientable surface is computed. I was curious if it was also possible to compute the "higher" ...
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I'm studying the book 'The Geometry of Syzygies' of David Eisenbud, but I'm having problem with the following step, in page 7 he says the we have a free resolution to the set of ten points in ...
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### An operation with respect to which the set of prime numbers is closed

Like every (semi-)decidable set of natural numbers the set $P$ of prime numbers is diophantine, i.e. there are two polynomials $p(x)$, $q$ with natural coefficients and exponents – the first of ...
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### A simple, yet non-superficial explanation of what “paramodulation” means in the context of automated theorem proving?

Modern automated theorem provers seem to be paramodulation-based. I only have a superficial understanding of what this means: we derive a proposition whose truth is implied from the truth of [two?] ...
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### Which Riemann integrable functions have all lower sums equal?

From Spivak's Calculus, 4th edition, problem 13-11(d): Which (Riemann) integrable functions have the property that all lower sums are equal? (Bear in mind that one such function is $f(x)=0$ for ...
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### Modern notational alternatives for the indefinite integral?

I like the Leibniz notation, and I think the reason it's survived for over 300 years and continued to be almost the only game in town is that in many respects it's a miracle of design. Nevertheless ...
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### What is the probability density function and cumulative distribution function of x

What is the probaiblity density function and cumulative distribution function of $x$ (where $x\in [\dfrac{-\pi}{2},\dfrac{\pi}{2}]$) such that both $y_1=\sin x$ and $y_2=\cos x$ are uniformly ...
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### I need help finding the critical values of this function.

So $h(t)=t^{\frac{3}{4}}-7t^{\frac{1}{4}}$. So I need to set $h'(t)=0$. So for $h'(t)$ the fattest I've gotten to simplifying os $h'(t)=\frac{3}{4 \sqrt[4]{t}}-\frac{7}{4\sqrt[4]{t^3}}$ and that is as ...
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### Anti derivative notation [duplicate]

$F$ is an anti derivative of $f$. $$\int f(x) dx = F(x)+C$$ Can you tell me why there is '$dx$' in the LHS?
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Consider the pair of topological space $(\mathbb{D}^n,X)$, where $X \subset \mathbb{D}^n$ is a subspace. We know that there is a long exact sequence of reduced homology groups, $$\cdots \to ... 1answer 43 views ### I need help proving this theorem (composition of functions) This is the statement: If f and g are functions, the composition g\circ f is a function with$$D(g\circ f)=\{x\in D(f):f(x)\in D(g)\}R(g\circ f)=\{g(f(x)):x\in D(g\circ f)\}$$The ... 1answer 94 views ### A question on smooth morphisms and 'pointwise' smooth morphisms Let X be a scheme, x\in X a point and f\colon \operatorname{Spec}(k(x))\to X the canonical morphism. Is f always a smooth morphism? Now suppose g\colon X\to Y is a scheme over some ... 1answer 230 views ### Union of Increasing Sequences of Monotone Classes is not a Monotone Class. In my text, we define a Monotone Class \mathcal{M} of a non-empty set X to be a collection of subsets of X that is closed under monotone limits: that is, (1) if A_{i} \uparrow A with A_{i} ... 1answer 93 views ### Given A, find invertible B such that B^{-1}AB is positive Given A \in Mat(n,n,\mathbb R), is there always an invertible matrix B, such that B^{-1}AB is positive, assuming all eigenvalues of A are positive and simple ? If yes, is it possible to classify ... 1answer 140 views ### An example in Spivak's Calculus on Manifolds (chain rule). Spivak gives an example which has step that is giving me some problems to get it, even if it's supposed to be trivial. Spivak says: Let f:\mathbb{R}^2\to\mathbb{R} given by f(x,y)=\sin(xy^2). ... 1answer 82 views ### “Visual” interpretation of the Bott Periodicity for complex vector bundles I've just read the Bott Periodicity proof (complex case) in Hatcher book about K-theory. At first it seems that using this theorem I could conclude that every complex vector bundles over S^{2n+1} ... 1answer 137 views ### Does calculus of variations have a close connection to Feynman's ''differentiation under the integral sign''? Most of the calculus I've studied seems separate math problems in to "derivative" or differential applications and integral applications. The one exception seems to be "calculus of variations," which ... 0answers 57 views ### Proof that a set is open. Let (\Lambda_i)_{i\in I} a collection of linear operators from X (Banach space) to Y (Normed space). Let \alpha : X \rightarrow [0,\infty] be the function \alpha(x):=\sup_{i \in I} ... 6answers 152 views ### Solving a trigonometric limit First off, please excuse my n00bishness I have only just begun learning about algebraic manipulation of limits so this is probably a really dumb or obvious question. I'm trying to solve the following ... 1answer 35 views ### k[x_1,\dots,x_n]/\frak{a} is an k-algebra of finite type? Let k be a field and \frak{a}$$\subset k[x_1,\dots,a_n]$be an ideal. Can someone explain to me why$k[x_1,\dots,x_n]/\frak{a}$is an$k$-algebra of finite type? 2answers 55 views ### Lattice Points in x-y plane What are Lattice Points? Which points in x-y planes are Lattice Points? Is (m,n) a lattice point where m,n are any integers? 2answers 591 views ### Function has vertical tangent or vertical cusp? Determine whether or not the graph of the function has a vertical tangent or vertical cusp at the indicated point c.$f(x) = (x+2)^7/3c=-2$I took the first derivative and chain rule and that got ... 2answers 620 views ### Accuracy of football prediction Say you predicted that Brazil would have beaten Germany 2-1, by how large a percentage were you wrong? As Brazil scored 1 out of 2 goals, you could say that part of your prediction was 50% right. ... 1answer 36 views ### Factoring for simplification? I need to show that $$\dfrac{\Gamma\left(\alpha_1 + \alpha_2\right)}{\Gamma\left(\alpha_1\right)\Gamma\left(\alpha_2\right)}\left[\dfrac{\tau y^{\tau \alpha_1}\delta^{\tau \alpha_2}}{y\left(y^{\tau} ... 2answers 98 views ### The closure and the boundary of \mathbb{R}^{\infty} in \mathbb{R}^{\mathbb{N}}. I think that in: Product topology: \overline{\mathbb{R}^{\infty}}=\mathbb{R}^{\mathbb{N}} and \partial\mathbb{R}^{\infty}=\mathbb{R}^{\mathbb{N}}. Box topology: ... 2answers 318 views ### spherical triangle: law of sines Given plane triangle ABC it is well known that the common value of the ratios appearing in the law of sines is equal to the diameter of the circle which passes through the three vertices. ... 1answer 88 views ### Is \mathbb{N}\rightarrow \mathbb{N\times N},f(x)=(x,x) onto? Is \mathbb{N}\rightarrow \mathbb{N\times N},f(x)=(x,x) onto? I am not sure how to tell. Say b\in N\times N this means the codomain is all the different combinations of the natural numbers. But ... 2answers 182 views ### Proving double derivatives with the chain rule (I think?) Hey StackExchange I'm having trouble understating where to start with this problem, I'm supposed to prove something about double derivatives and the chain rule but I'm having trouble understanding ... 1answer 62 views ### Finding a matrix projecting vectors onto column space I can't find P, for vectors you can do P = A(A^{T}A)^{-1}A^T. But here its not working because matrices have dimensions that can't multiply or divide. help 0answers 43 views ### How to solve the following an ODE? Let x,y,z be a given point in \mathbb{R}^3. How to solve (x'(t),y'(t),z'(t))=(x(x+y+z), y(x+y+z),z(x+y+z))? 2answers 231 views ### Describing all points 4000 miles from the north pole I'd like to describe all of the points on the Earth's surface that are exactly 4000 miles from the North Pole. I know that this will eventually give me an equation for a circle; I want to find that ... 2answers 239 views ### Prove that \displaystyle{\sum_{n=1}^{\infty}}(-1)^{n-1} \dfrac{H_n}{n} = \dfrac{\pi^2}{12} - \dfrac{1}{2}\ln^2 2 We know that H_n = \sum_{j=1}^{n}{1 \over j}. Article in The Sum of Certain Series Related To Harmonic Numbers of Omran Kolba, we have proof of this identity which involves some advanced concepts. ... 0answers 156 views ### Eigenvalues problem for generalized Kuramoto-Sivashinsky equation I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem$$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$where \lambda is a ... 1answer 46 views ### How to calculate this expression?(multiplication) How do I show that for any starting n? (I am not really sure it is 0, but I think it is).$$\prod_{i=n}^\infty \left[1-\frac 1 i\right]=\prod_{i=n}^\infty \left[\frac{i-1}{i}\right]=0$$I tried ... 2answers 173 views ### Simple extension fields If I am correct simple extension fields are extensions generated by one element. I have learned that this means that elements of a simple extension can be written as powers of that element as long as ... 1answer 101 views ### The number of partitions of n and the nth Fibonacci number. I'm very sorry if this is a duplicate in any way. There's a lot of material out there on connections between these sequences so it's a possibility . . . Let P_n be the number of partitions of n ... 1answer 128 views ### How to identify continuity or discontinuity of an [Definite] integral? How can I figure out whether an improper integral converges based on the discontinuities in the integrand? For instance, these two both have discontinuities within the intervals of integration, and ... 3answers 537 views ### Find expression for sum of series I want to find a formula for the sum of this series using its general term. How to do it? Series$$ S_n = \underbrace{1/3 + 2/21 + 3/91 + 4/273 + \cdots}_{n \text{ terms}} $$General Term$$ S_n = ... 1answer 52 views ### What are some other applied advanced probability sub-field relevant to finance? What are some other applied advanced probability sub-field relevant to finance? I have heard Martingale, stochastic process, stochastic calculus, monte-carlo statistics I've been searching other ... 2answers 54 views ### A question on distributions Let$\Delta$be a smooth distribution on a smooth manifold$M$and let$X,Y$be 2 vector fields on$M$which are tangent to$\Delta$(namely$X(q),Y(q)\in \Delta_q\leq T_qM$for every$q\in M$. I ... 1answer 72 views ### How to show this is the minimal polynomial I'm trying to the following problem. But I can't show some irreducibility of the polynomials. Put$L=\mathbb{C}(X,Y,Z)$,$\omega=\frac{-1+\sqrt{-3}}{2}$. Define two automorphism$\sigma, \tau$of ... 3answers 332 views ### Basic Group Theory question This is not so much a plea of ignorance, but rather me trying to see whether intuitively I actually understand what is going on in group theory. The question asks What group is ... 1answer 42 views ### Probability of their's constituting a triangle? [duplicate] We have a line segments with length$l$then you choose two random points and cut it from these points so that we have three piece of line segments. What is the probability that these piece ... 2answers 63 views ### Can any measure be made into a bounded measure? Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure? 1answer 104 views ### Faithful Representations of C*-algebras Can anyone give me an example of a represetation of the algebra$M_n(\mathbb{C})$that is not faithul? If it's not possible, could you explain me why it is not? 1answer 32 views ### Discovering the derivatives of functions combined with trig values. Hey StackExchange I have a problem that I don't really understand and I could use some hints for starting it. Suppose$m(\frac{\pi}{3}) = 4$and$ m'(\frac{\pi}{3}) = -2$, and let$g(x) = m(x)\sin x$... 1answer 36 views ### How rigorous is multiplying both sides of an eqaution for the differential of a function? I have to solve this equation: $$-C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A$$ where$C_0$and$A$are two real nonzero constant;$f:\mathcal{R}\to \mathcal{R}$I have seen that the person who ... 2answers 60 views ### How to express a trigonometic equation in$\sin 2\theta $and$\cos 2\theta $? How do I express the given equation in$\sin 2\theta $and$\cos 2\theta $in terms of x?$x + 3 = 7\sin \theta $with$\frac{\pi }{2}{\text{ < }}\theta {\text{ < }}\pi $for$\sin 2\theta ...
what is common surface between: $(x+5)^2+z^2=y$ and $z^2+y^2=25$ ? I have found that at the XY plane the common surface is hiperbola, but it cannot be right because at the paraboloid there aren't any ...