Comment by user458276 on Find all the ring morphisms from $ℤ_{15}$ to $ℤ_3$.
As other users have mentioned, it's sufficient to consider what happens when one applies an ring homomorphism $f$ to $[1]$.
View ArticleComment by user458276 on Left multiplication is homeomorphism of topological...
@tomasz I elaborated a bit. Hopefully this makes more sense.
View ArticleComment by user458276 on Translation Request from French - Comparison of the...
@SylvainJulien No deadline for the translation, mainly for enrichment. Thank you so much.
View ArticleComment by user458276 on What is the cardinality of the set of all roots of...
Both proofs seem correct to me.
View ArticleComment by user458276 on Convergence using the ratio test
Have you tried Stirling's approximation? It may simplify things a little.
View ArticleComment by user458276 on Solving trigonometric equations: do I need to...
Oops. Fixed it. Cheers!
View ArticleComment by user458276 on Solving system of ODE with elimination method for...
You should first solve the system by eliminating either x' or y' from the equation. If you do that you get a trivial equation with the solution you mentioned. Working with the differential operators...
View ArticleAnswer by user458276 for Integrate the improper integral (need help with...
Hint:The integral should be:$$\int_{0}^{5} \frac{x}{x-2}dx = \lim_{\epsilon \rightarrow 2^{-}} \int_{0}^{\epsilon} \frac{x}{x-2}dx +\lim_{\delta \rightarrow 2^{+}}\int_{\delta}^{5}...
View ArticleAnswer by user458276 for Element is not in intersection of A and B
An element $x$ can be in one of the sets, but not the other:If $A =\{a,b,c\}, B = \{a,b\}$, $c \notin (A\cap B)$ and $c \notin B$, but $c \in A$.
View ArticleAnswer by user458276 for Why is an adjunction a “weak form of equivalence”?
As the comments have mentioned, there are a few ways to think of this:Equivalences are Adjunctions, but not conversely. The unit/counit are natural isomorphisms if the adjunction is an equivalence.An...
View ArticleAnswer by user458276 for Trig Differentaition
If $y = 27\sec^{3}(x)$, then you can find $y'$ in a couple of ways.Using the method you suggested:$$y = 27\sec^{3}(x) = 27\sec^{2}(x)\cdot\sec(x)$$Applying the product rule, we have:$$y' =...
View ArticleAnswer by user458276 for Problem about properties of poset category
As others have stated, $Q$ has an intial object if $P$ has a least element $s$ - meaning no other element is less than $s$. In a poset, if $a\leq b$ and $b\leq a$, then $a=b$ as stated. This implies...
View ArticleAnswer by user458276 for Definition of triangulated functor
Both questions you ask are true - you need the triangulated functor to preserve distinguished triangles, and to do so would need a natural isomorphism $\eta:F\circ T_{D}\rightarrow T_{D'}\circ F$.Note...
View ArticleAnswer by user458276 for Formalize "naturality" of a functor?
A "Natural" functor is typically a functor that arises from a natural mathematical situation,ie. $U:\textbf{Ring} \rightarrow \textbf{Grp}$ described as earlier, $F:\textbf{Ring}\rightarrow...
View ArticleAnswer by user458276 for Composition of relation and transitive closure?
There are a couple of things you should check:If $R\subset X\times Y$ and $S \subset Y\times Z$ are two binary relations, their composition $S\circ R$ is defined as follows:$$S\circ R = \{(x,z) \in...
View ArticleFunctorial comparison of Different Models of Set Theory
I'm very much a novice to Model Theory/Categorical Model Theory (though I very much would like to learn). I apologize if my question is improperly stated.Fix a small (or locally small if desired)...
View ArticleAnswer by user458276 for Finding density of a thin plate
The only thing off about your solution is the bounds.You'l need to integrate over the region: $0\leq x \leq 2$, and $0\leq y \leq 4-x^2$.In your work you integrate over $[0,2]\times [0,4]$, which is...
View ArticleAnswer by user458276 for Prove that if $f(x)$ divides $g(x)$ and $f(x)$...
If $f(x)$ divides $g(x)$, then $g(x) = f(x)q(x)$ and likewise $h(x) = f(x)r(x)$.Also note that $f(x)$ divides $s(x)g(x)$, since $s(x)g(x) = s(x) f(x)q(x)$.
View ArticleWhy are (Pre)sheaves more important than Co(pre)sheaves?
I'm learning Sheaf Theory, and this is an issue that's been bothering me.Fix a small category $\mathcal{C}$.A $\mathcal{V}$-valued presheaf on the small category $\mathcal{C}$ is a functor...
View ArticleAnswer by user458276 for If partial derivatives of a harmonic function are...
No.If $u(x,y) = k_{1}x+k_{2}y+C$, then:$$\dfrac{\partial u(x,y)}{\partial x} = k_{1}$$$$\dfrac{\partial u(x,y)}{\partial y} = k_{2}$$But $u(x,y)$ is not linear - this kind of function is an affine...
View ArticleAnswer by user458276 for Hint for solving this problem about constant...
Hint:Consider the sets where the function is not equal to the certain constants from the two described functions. What is it’s measure?Now consider the subset of $\mathbb{R}^2$ which is not equal to...
View ArticleAnswer by user458276 for Find $\lim\limits_{x\to0}\frac{\sqrt{1+\tan...
This can be (almost) completely solved using algebraic manipulation:$$\frac{\sqrt{\tan x+1}-\sqrt{x+1}}{\sin^2(x)} = \frac{\tan x -x}{\sin^2 x\cdot(\sqrt{\tan x+1} +\sqrt{x+1})}$$Now,...
View ArticleAnswer by user458276 for See if $\sum_{n=2}^{\infty}(\ln n)^{-20}$ converge
Let's see if we can use the comparison test with $\sum_{n=2}^{\infty}(\ln n)^{-20}$ and $\sum_{n=2}^{\infty}n^{-20}$.We know that $$\sum_{n=2}^{\infty}n^{-20}=\sum_{n=2}^{\infty}\frac{1}{n^{20}}$$...
View ArticleComment by user458276 on A result concerning local rings
If the ring $R$ is not-necessarily local, you need to consider $\{R_{\frak p}\}$ where $\frak p$ is a prime ideal to calculate dimension.
View ArticleComment by user458276 on Expressing $F(x,y,z)$ in terms of $F(x,\alpha(x))$.
Your second approach is fine, but you need to show that your choice of $y=y(x)$ exists on an open interval which includes zero, and is differentiable at x=0.
View ArticleAnswer by user458276 for Expressing $F(x,y,z)$ in terms of $F(x,\alpha(x))$.
Let $\alpha:U\rightarrow\mathbb{R}^2$:We can write the components as follows:$$ x\mapsto(\alpha_1(x),\alpha_2(x))$$By the definition of...
View ArticleAnswer by user458276 for Finding Taylor Series from existing Series
Notice that:$$\int f(x)dx = \int \dfrac{x}{1-x^2}dx=-\frac{1}{2}\int \frac{du}{u}=-\frac{1}{2}\ln|1-x^2|+C$$(via the $u$-substitution $u=1-x^2$)From your...
View ArticleAnswer by user458276 for Integration Question - MGF
Note that the integral is of the form:$$\int_{-\infty}^{\infty}e^{-a\phi^2}d\phi=\sqrt{\frac{\pi}{a}}$$And if you substitute again with $$\phi=z-\frac{t}{1-2s}$$You'll get the integral into the...
View ArticleComment by user458276 on Prove difference of infinite series of decreasing...
What is your question?
View ArticleComment by user458276 on Uniqueness in analysing coordinate geometry situations
What is $a$? Is it the value of $x$ on $x,f(x)$?
View ArticleAnswer by user458276 for N variables Quadratic Form matrix operations proof
Welcome to Math.Stackexchamge!We can work out $\bf{x}^{\it{T}}\cdot A\cdot x$:For Matrices $\bf C,D$:$$\left(\bf C\cdot D\right)_{i}=\sum_{l=1}^{n}C_{\it jl}\cdot D_{li}$$Then:$$\left(\bf x^{T}\cdot...
View ArticleAnswer by user458276 for Solve system of equations $9^y=3^x$ and...
Hint:Rewrite $$2\log_{2}x-\frac{1}{2}\log_{2}3\rightarrow\log_{2}x^2-\log_{2}3^{1/2}=\log_{2}\frac{x^{2}}{3^{1/2}}$$
View ArticleComment by Thomas Davis on Shouldn't that be $g(x)$ instead of $f(g)$ in my...
I'm assuming that $f(g)$ would refer to the composite function $f(g(x))$ when suppressing the $x$ value, similar to how one might write $f$ instead of $f(x)$.
View ArticleComment by Thomas Davis on For how many values of $x$ does $f(x)=\cos...
Wolfram Alpha thinks that the absolute minimum is -2. I'm curious to see if that's correct.
View ArticleComment by Thomas Davis on Example of a homeomorphism that's not diffeomorphism
Do you know what a local homeomorphism is? Any continuous function $f:\mathbb{R}\to\mathbb{R}$ is a local homeomorphism, and there are many non-differentiable continuous functions.
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Answer by Thomas Davis for Graphical interpretation of discontinuity of...
It doesn't really appear until you graph the first derivative.Here the left and right hand side (respectively) are the left and right hand derivatives. There you can see that the left side is...
View ArticleAnswer by Thomas Davis for What allows a function to remain equivalent to...
There is!As functions, they have different expressions and exist at different points, so they are NOT equivalent of course.However, it is true that:$$\lim_{x\to c}\dfrac{x^2+2x-15}{x-3}=\lim_{x\to...
View ArticleComment by Thomas Davis on How do I get the time it takes to travel between...
@Andrei Good suggestions! I edited my response.
View ArticleComment by Thomas Davis on Integral of the usual mollifier function: finding...
I'm finding that $c_3\approx 2.267, c_4 \approx 2.611, c_5\approx 3.2305, c_6\approx 4.262$. Julia ran out of memory on my computer to calculate $c_7$.
View ArticleAnswer by Thomas Davis for How does this differentiation map work?
Since differentiation is a linear map from polynomials of degree 3 to polynomials of degree 2, we can express it as matrix multiplication:$$p_3(x)=a_0+a_1x+a_2x^2+a_3x^3\to...
View ArticleDo Monoid Homomorphisms preserve the identity?
In both my textbook (Hungerford's Algebra), and in class, it is claimed that Monoid Homomorphisms are not required to preserve the identity. Interestingly enough, the Wikipedia page for Monoids...
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