-2
votes
1answer
111 views

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??
3
votes
2answers
212 views

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.
0
votes
3answers
199 views

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 ...
2
votes
1answer
110 views

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 ...
0
votes
1answer
39 views

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. ...
0
votes
1answer
45 views

Is a statement concerning the future part of a decidable problem?

Let Did I ever get 100% in an exam? be a problem and the corresponding (characteristic) function $$\chi(x)=\begin{cases}1,& \text{if the statement can be answered with ...
0
votes
2answers
50 views

Doubt on Logarithms multiplication

today I'm in doubt on calculating the follow expression $\log_4 3 * \log_9 32$ Changing all to base 4: Working on: $\log_4 3 * \dfrac{\log_4 32}{\log_4 9}$ Ending with: $\log_4 3 * \dfrac{2 + \log_4 ...
1
vote
2answers
193 views

Finding the degree and coefficients of the Polynomial

A polynomial is denoted by $f(x)$. The coefficients of the polynomial are positive integers. $$f(1) =17$$ $$f(20)=421350$$ Could you tell if such a polynomial is possible? If ye, find the degree of ...
1
vote
1answer
237 views

Question on mixed nash equilibrium!

The question is as follows: Think of the Golden Ball game. Now player 1 is money-minded and jealous, and player 2 is very good-hearted, so the payoff matrix is follows: ...
6
votes
3answers
325 views

Application of Central Limit Theorem

In the book Probability Essentials, by Jacod and Protter, the following question has bugged me for a long while and I'm wondering if it is bugged. The question is an application of Central Limit ...
1
vote
0answers
217 views

Integral with respect to Wiener process.

Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process. My First Question What is ...
1
vote
2answers
68 views

Help solving $\frac{1}{{2\pi}}\int_{-\infty}^{+\infty}{{e^{-{{\left({\frac{t}{2}} \right)}^2}}}{e^{-i\omega t}}dt}$

I need help with what seems like a pretty simple integral for a Fourier Transformation. I need to transform $\psi \left( {0,t} \right) = {\exp^{ - {{\left( {\frac{t}{2}} \right)}^2}}}$ into ...
4
votes
4answers
272 views

Show $\sqrt{2 - \sqrt{2}} \notin \mathbb{Q}(\sqrt{2 + \sqrt{2}})$

I'm in the middle of a proof where I'd like to show that $\sqrt{2 - \sqrt{2}} \notin \mathbb{Q}(\sqrt{2 + \sqrt{2}})$ The only way I can think of involves finding an explicit set representation for ...
-4
votes
1answer
109 views

strong equation!

Find $(x;y)$ sastisfied that: ...
0
votes
1answer
185 views

Maximum of a harmonic function

Let $D$ denote the open unit disk around the origin in the complex plane. Let $f:D\rightarrow D$ holomorphic and $f$, $f^\prime$ extend continuously to $\overline{D}$. Let $u$ be the real part of $f$. ...
2
votes
1answer
472 views

A path has only two vertices which are not cut-vertices

Prove that a simple undirected graph $G$ is a path if and only if $G$ has exactly two vertices which are not cut-vertices. If $G$ is a path then it is obvious that there only two vertices which ...
2
votes
2answers
31 views

Describing a sequence of terms

I'm currently in a Discrete Mathematics class in college, and my professor is giving a quiz soon and told us what will be on it. Problem is, I missed a day of class and I have no idea to figure out ...
3
votes
1answer
531 views

Incomparable Elements In A Poset

The problem I am working on is, Find two incomparable elements in these posets. a) $(P(\{0,1,2\}),⊆)$ b) $(\{1,2,4,6,8\},|)$ For a, I said that $R \subseteq p(\{0,1,2,3\}) \times p(\{0,1,2,3\})$, ...
1
vote
0answers
16 views

On the computational complexity of plugging in numbers into general expressions to obtain special ones

There are many expressions, which can be considered straight generalizations of others. I'm motivated by values of integral expressions specifically, for example there is $$\int_0^\infty e^{-a ...
1
vote
0answers
54 views

Valid Distribution Function?

Given a Probability Density Function ${f_X}(x)$, how can I prove that ${f_X}(x)$ is a valid distribution function? I already known that ${f_X}(x) \ge 0$ and $\int\limits_{ - \infty }^\infty ...
0
votes
1answer
97 views

Absolute continuity and integration formula (explain a statement please)

I read this: For $v$, $w$ in $L^2(0,T;H^1(S))$ (with weak derivatives in $H^{-1}(S)$ for each time), the product $(v(t), w(t))_{L^2(S)}$ is absolutely continuous wrt. $t \in [0,T]$ and ...
0
votes
1answer
132 views

An infinite group is cyclic if and only if it is isomorphic to each of its non-trivial subgroups

The question has been described in the title. How to prove it?
1
vote
2answers
143 views

Vakil's Foundations of Algebraic Geometry, Exercise 5.5.E

Exercise 5.5.E Let $A = \sum_{n\in\mathbb{Z}} A_n$ be a graded commutative ring with a $\mathbb{Z}$ type grading. Let $f \in A_d, d > 0$. Suppose $f$ is invertible. Let h-$Spec(A)$ be the set of ...
2
votes
0answers
136 views

using stokes theorem to calculate a line integral [duplicate]

Use stokes theorem to show that: $$\int_c ydx + zdy +xdz = -\sqrt{3} \pi a^2$$ Where c is the suitably oriented intersection of the surfaces $x^2 + y^2 +z^2=a^2$ and the plane $x+y+z=0$.
2
votes
1answer
353 views

Pólya’s Enumeration Theorem and chemical compounds

The hydrocarbon naphthalene has ten carbon atoms arranged in a double hexagon, and eight hydrogen atoms attached at each of the corners of the hexagons. Naphthol is obtained by replacing one of ...
0
votes
0answers
114 views

repeated Galois conjugate .

Thanks for your answer Ram. But I will change my question , because I want a more directly answer. And I'll write exactly what I want. Edited question: Let's consider a finite Galois extension ...
0
votes
1answer
471 views

Contour Integral of $f(z) \; \cot(\pi z)$

In lecture, my professor stated that $$ \lim_{N \to \infty} \int_{\gamma_N} f(z) \cot(\pi z) \; dz = 0 $$ where $\gamma_N$ is the square contour with vertices at $\pm (N + \frac12) \pm i(N + ...
2
votes
2answers
115 views

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 ...
3
votes
0answers
65 views

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 ...
1
vote
2answers
254 views

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 $n=4x+5y$.
1
vote
3answers
102 views

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 ...
1
vote
1answer
106 views

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 ...
1
vote
1answer
83 views

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 ...
2
votes
3answers
105 views

$\int_{a}^{b}f(x)\cos(kx)dx\rightarrow 0 (k\rightarrow \infty)$ 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,$\int_{a}^{b}f(x)\sin(kx)dx\rightarrow 0 (k\rightarrow \infty)$ ...
0
votes
1answer
103 views

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?
1
vote
0answers
33 views

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 ...
0
votes
2answers
24 views

Trigonometric development help.

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' = ...
1
vote
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}= ...
11
votes
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.
1
vote
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 ...
0
votes
3answers
96 views

Finding the ones digit for $2^{98}$

How can i find the ones digit for the number $$2^{98}$$
2
votes
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 ...
0
votes
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 ...
2
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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?
0
votes
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 $$ L= \{a^m b^n \mid mn \ge 10 \}. $$
0
votes
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 : $x=a+\alpha$ , ...
4
votes
1answer
203 views

Reading circle in mathematics?

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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