Justpassingby
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3 answers
1 votes
166 views
Converting Polar Equation to Cartesian Equation problem
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You could substitute for $r$ and $\theta$ their expressions in terms of $x$ and $y$. But remember that trigonometric functions of $\theta$ have easier expressions than $\theta$ itself.

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2 answers
4 votes
184 views
"Extension" of orthogonal group
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Orthogonal transformations preserve the norm of a vector. You can take the product of that group with the one-dimensional multiplicative group that scales the vectors by nonzero real factors (a copy ...

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1 answers
1 votes
326 views
What is " the two-parameter family of vectors"
Accepted answer
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u is a given function of x and t, so the triplet (x,t,u) is really (x,t,u(x,t)) which leaves two degrees of freedom, namely, x and t. The capital T means "transpose": exchanging rows and columns of a ...

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1 answers
1 votes
46 views
What, in set-theoretical terms, is a satisfactory definition of the complex logarithm?
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For fixed \alpha your function \log^{(\alpha)} is not a bijection onto C. Its range is a horizontal strip of height 2\pi.

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3 answers
2 votes
69 views
Trying to create a inverse quare algorithm for expanding sphere
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You should define what you mean by "inverse square". This is by no means a standard concept. In you formula it means that the radius of the sphere is divided by the square of time, which indeed ...

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1 answers
0 votes
75 views
Reflection about Gaussian measures
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If an arbitrary centered Gaussian measure is given on a product of a Banach space X with itself, then it is not guaranteed to be the tensor product of two centered Gaussian measures on X. As an ...

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3 answers
0 votes
52 views
Normed Linear Space ,$p \neq 2$ is $\left \| f\right \|_{p}= \sqrt{f,f}$ for each $ f \in L^P([0,1])$?
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Hint: if a norm is derived from an inner product in that way, then the inner product is uniquely determined by the norm and there is an explicit algebraic expression for (f,g) in terms of ||f||, ||g||,...

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2 answers
3 votes
88 views
Checking Continuity in General.
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The concept that you are thinking of is sequence continuity (which is equivalent to continuity in metric spaces such as R^n but that requires proof). If you check the condition "for every given open ...

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2 answers
1 votes
333 views
Non negative eigenvalues of a 10 by 10 matrix.
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You can eliminate answers A and D by noting that the zero matrix B always satisfies that identity.

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1 answers
0 votes
476 views
Free commutative ring functor
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Intuitively I understand the first definition but not the second. In fact the standard notation for a polynomial with integer coefficients and variables taken from the set X suggests first taking ...

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1 answers
0 votes
467 views
How to describe the locus of $z$ where $\arg \left(\frac{z-z_1}{z-z_2}\right)=\theta$.
-1 votes

Since $\theta$ is a given real number, the arc is oriented. Only one of two possible circles has the right orientation of the angle from $z_1$ via $z$ to $z_2.$

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5 answers
8 votes
4k views
Is the Cartesian product of sets associative?
-1 votes

There is an abstract notion of products in category theory. A category where every finite collection of objects has a product is called a cartesian category. In those categories (and Set is one of ...

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1 answers
3 votes
332 views
Convergence of Series Whose Terms are Defined Recursively
-1 votes

I would greatly prefer your approach no.3 because such series are actually easy to find. One example is the geometric series with ratio 1/2 (up to an initial "stutter" term which is easy to discard).

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2 answers
1 votes
776 views
Prove composition is a measurable function
-2 votes

On an abstract level, the composition of two measurable maps $f:X\to Y$ and $g:Y\to Z$ is measurable; this is very easy to verify from the definition of a measurable map. The only remaining step is ...

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