# All Questions

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### nonlinear first ODE : Solve $\displaystyle\frac{dy}{dx}=(x^2+y^2)^2$

Solve $\displaystyle\frac{dy}{dx}=(x^2+y^2)^2$ Any hints for me the solve the problem??
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### Why was the term “integral” used to represent the area under a curve?

I have a colleague in the English dept. who is wondering the reason why the word "integral" came to be used to represent the process by which the area under a curve can be found.
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### Reversing an Arithmetic Sequence

So, it's been a long time since I've studied math, so I'm having more trouble with this problem than I thought I would as for some help. I have an arithmetic sequence $0,...,99$ with the difference ...
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### Lines on a cubic surface in $\mathbb{P}^3$ using Chern classes

I'm trying to figure out why there are 27 lines on a smooth cubic hypersurface in $\mathbb{P}^3$ using Chern classes, without looking up the proof in 3264 and All That. One thing confusing me is the ...
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### How do you solve the system given the contraint?

Using lagrange i got something like $$3x = 4z = 6y$$ And the constraint is $$z^2 = x^2 + y^2$$ Where do you get from here? I usually get $x=y=z$, but here i got $3$ variables with different values. ...
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### Weak Convergence of Positive Part

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain and $p\in (1,\infty)$. Suppose $u_n\in L^p(\Omega)$ is such that $u_n\rightharpoonup u$ in $L^p(\Omega)$. Define the positive part of $u$ by ...
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### Existence of roots of a polynomial equation when coefficients have varying weights

I have two $n-$degree polynomials $f_{1}(p)$ and $f_{2}(p)$, where the domain of $p\in[0,1]$. I know that $\exists$ $0 < p_{1} < 1$ such that: $f_{1}(p_{1}) = f_{2}(p_{1})$. Let ...
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### Strong mathematical induction to prove $n=4x+5y$

Use the principle of strong mathematical induction to prove that if $n\in\mathbb N, n\geq12$ , then there exist non-negative integers $x$ and $y$ such that $n=4x+5y$.
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### Proof of a complex identity

This might be a very obvious one, but I am stuck on this from a long time. If $F(s) = M(s) + N(s)$ where $M(s)$ is even polynomial function and $N(s)$ is odd polynomial function (where $s$ is a ...
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### Explanation of easy statement regarding derivative and Jacobian needed

Let $\Phi:S \to T$ be a map between surfaces in $\mathbb{R}^n$. What precisely does this mean: Let $\text{det}(\mathbf{D}_S \Phi(.))$ denote the Jacobian determinant of the matrix representation ...
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### How many numbers exists that are smaller than $p$ and prime with $p$?

I have a homework to hand in and they asked this question. I don't know if I'm supposed to count 1 as a prime to that number or not. In my case $p=3947$, so I count 3945 numbers fitting that criteria ...
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### $\int_{a}^{b}f(x)\cos(kx)dx\rightarrow 0 (k\rightarrow \infty)$ only need $f\in L([a,b])$?

This is strange result $$\int_{a}^{b}f(x)\cos(kx)dx\rightarrow 0$$ when $k\rightarrow \infty$. Similarly under the same condition,$\int_{a}^{b}f(x)\sin(kx)dx\rightarrow 0 (k\rightarrow \infty)$ ...
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### How to express inclusion with arrows?

Does$$\forall x \left(1 \stackrel{x}\longrightarrow X\right) \Rightarrow 1 \stackrel{x} \longrightarrow A$$ means $A \subset B$? Is there any better way to express this with arrows?
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### Tossing a fair coin [duplicate]

Possible Duplicate: Probability for the length of the longest run in $n$ Bernoulli trials Suppose a fair coin is tossed n times. Determine the probability that exactly r consecutive heads ...
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I need help with the following trigonometric development: $x = r(\theta)\cos\theta$ $y = r(\theta)\sin\theta$ this gives: $x' = r'(\theta)\cos\theta - r(\theta)\sin\theta$ $y' = ... 1answer 138 views ### Separable form by substitution Please suggest appropriate substitution to reduce it to separable form $$\frac{dy}{dx} = \frac{4x+7y+2}{4x+7y+3}$$ let $$z=4x+7y$$ then $$\frac{dz}{dx}=4+7\frac{dy}{dx}$$ $$\frac {dy}{dx}= ... 3answers 431 views ### Leading Digit of 2^{4242} How could I solve this problem? Find the first digit of 2^{4242} without using a calculator. I know how to find the last digit with modular arithmetic, but I can't use that here. 2answers 107 views ### Mathematics Induction on Inequality I want to prove 2^n \ge 3n^2 +5--call this statement S(n)--for n\ge8 Basis step with n = 8, which \text{LHS} \ge \text{RHS}, and S(8) is true. Then I proceed to inductive step by ... 3answers 96 views ### Finding the ones digit for 2^{98} How can i find the ones digit for the number$$2^{98}$$1answer 259 views ### Showing boundedness and a coercivity condition for a bilinear form Suppose \Omega \subset \mathbb{R}^n is a compact domain. Let f and J (and also \frac 1J) be C^1 functions on \Omega. Consider the bilinear form a:H^1(\Omega) \times H^1(\Omega) \to ... 1answer 83 views ### Constant f:[\mathbb{N}]^2\to \{1,2\} (part 2) Edit 2: solved! In this post, "we" proved that exist infinite R_1, such that f is constant on elements of the form \{n_1,r\} where r\in R_1. By the same considerations we can show that if we ... 1answer 84 views ### x^y \bmod n seems to repeat itself after some steps (when iterating over n) Given 1516^{2627} \bmod 13 I tried several things to find the solution without a calculator, such as examining some powers like 1516^{1} \bmod 13, 1516^{2} \text{mod} 13, 1516^{3} \bmod 13 and ... 0answers 75 views ### Ergodic/stochastic convergence I do have a problem with my homework, and to be honest I am simply lacking any idea on how to begin- maybe someone could give me a tip. First off, here is the assignment: The whole assignment deals ... 3answers 2k views ### Cardioid: converting parametric form into polar coordinates I am interested in converting parametric equations:$$\varphi=\left(\varphi_{1},\varphi_{2}\right)=\left(2\cos{t}-\cos{2t},2\sin{t}-\sin{2t}\right)$$which describe a cardioid, into polar ... 1answer 91 views ### Probability conditioned on disjunction of events This seems elementary, but I cannot see a quick proof: For events A, B, C we have$$ P(A \mid B \cup C) \leq \max(P(A \mid B), P(A \mid C)) $$Is this right? 1answer 236 views ### Regular Language : \{a^m b^n \mid mn \ge 10\} I am little bit confuse here about below language is it regular language$$ L= \{a^m b^n \mid mn \ge 10 \}. $$1answer 95 views ### Transformation of coordinates Given a point P with spherical coordinates (r_p, \phi_p, \theta_p) on the sphere:$$(x-a)^2 +(y-b)^2 +(z-c)^2 = R^2$$and a line through the center of the sphere with equation :$x=a+\alpha\$ , ...
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How to learn about interesting topics in a small group of people?It seems very useful to broaden your mathematical background and get to know topics that are away from your field of specialization. ...

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