Recent Questions - Mathematics Stack Exchange |
- How to proof this conclusion? I don't know if it is related to ring homomorphism.
- Shortest path in conformal maps of a surface
- Countable sets problem
- Linear independence of permutations of a set
- Where i am wrong? A question on uniformly continuous function in functional analysis.
- Probability : The emperor's death sentence
- question about decomposition of positive terms series $\sum_{n=1}^\infty a_k$ where $a_k=b_k + c_k$
- Regularity of a language checker
- $\lim_{n\to\infty}\frac{n+\sqrt{n}+\sqrt[3]{n}+\cdots+\sqrt[n]{n}}{n}$
- Is Heine's theorem realted to Bolzano-Weierstrass theorem?
- How to tackle the integral $\int_{0}^{\infty} \frac{\ln x}{x^{n}-1} d x$?
- Why is modular arithmetic called "modular"?
- show $X^{(0)}$ is a discrete finite subset of $X$
- I don't know how to prove some class of models is whether an elementary class or an elementary class in the wider sense. How can I prove what it is?
- what is q & n in Primality Testing using Elliptic Curves
- Proof that a nonzero point is continuous for g(x)
- Independence of long times coin toss model
- Product, inverse, determinant, exponential of $n \times n \times n$ and $n \times n \times n \times n$ matrices
- Projecting a band of a sphere onto a 2D surface
- Why is $\text{maximize} \frac{1}{\lVert x \rVert}$ equivalent to $\text{minimize}\ \lVert x \rVert^2$?
- Notation of derivative w.r.t. the multi-variable function argument of an univariate function
- Calculate the limit $\lim\limits_{n\to\infty }(1+\frac{1}{a_n})^{a_n}$ Given $a_n$ is an increasing monotone sequence of integers
- What does mean of Topological distortion in perspective projection?
- All pairs of letters (de Bruijn sequence with $n=2,k=52$)
- Minimal spanning set ("conical basis") for 2x2 Hermitian PSD (positive semi-definite) cone?
- Finding all polynomials $P(x) \in \mathbb R[x]$ such that $P(x)^2=4P\left(x^2-5x+1\right)+2$
- $S^1$-valued function on $T^n$
- Need Recommendation for High Level proof book
- How to take integral of absolute value(x) on a Casio fx-991ms
| How to proof this conclusion? I don't know if it is related to ring homomorphism. Posted: 28 Nov 2021 11:04 PM PST Assuming $F$ is a number field, and $M_n(F)$ represents the set of all $n\times n$ matrixes on the number field $F$, $M_m(F)$ is defined similarly. Map $f:M_n(F)\to M_m(F)$ meets conditions below: 1) $f$ is injection, 2) $f(A+B)=f(A)+f(B)$, 3) $f(AB)=f(A)f(B)$, 4)$f(I_n)=I_m,f(0_n)=0_m$.Then how do we proof that $n| m$? |
| Shortest path in conformal maps of a surface Posted: 28 Nov 2021 11:04 PM PST My intuition tells me that the shortest distance between two points on the surface corresponds to a line segment joining the two points on the map of a surface. Since the path on the surface is same as the shortest path in the map. However, this turns out to be wrong. Take for instance, the Beltrami-Poincare half-plane model of $\mathbb{H}^2$, the shortest path between two points seems to be an arc of a semi circle centered at somewhere on the horizon. Picture:
Why is the shortest distance not a straight line in the map here? Probably I am missing something quite basic, but I just can't seem to figure it out. |
| Posted: 28 Nov 2021 11:04 PM PST I'm stuck on this problem, and I'm afraid I don't have no idea how I would start it: If S is a subset of the set of real numbers and if |x_1 + x_2 + ... + x_n|<=1 for all finite sets {x_1, x_2, ..., x_n} subset of S, then S is countable. Any ideas are welcome, thank you! |
| Linear independence of permutations of a set Posted: 28 Nov 2021 11:04 PM PST Let $(a_1,a_2,..a_n)$ be some tuple of reals. I'm wondering if the set of permutations of the tuples always generates at least $n$ linearly independent vectors. Trivially, I have found that the all zeros tuple generates no L.I, vectors and all equal tuple generates only one L.I. vector. Further, experimentation with some small examples leads me to believe that even if one $a_i$ is distinct then, $n$ L.I. vectors are generated. Is there a way to rigorously show this? |
| Where i am wrong? A question on uniformly continuous function in functional analysis. Posted: 28 Nov 2021 11:01 PM PST Given metric spaces $(X, d) $ and $(Y, d') $ , a mapping $T:X \to Y$ is uniformly continuous on $X$ iff for every Cauchy sequence $(x_n) $ in $(X, d) $, the sequence $ (Tx_n) $ is Cauchy in $(Y, d') $. One sided implication (uniformly continuous function map Cauchy sequence to Cauchy sequence) is easy to prove. But what about the converse. I think it is false. $f:\mathbb{R} \to \mathbb{R}$ defined by $$f(x) =x^2$$ maps Cauchy sequence to Cauchy sequence but this function is not uniformly continuous. Please verify the proof that I upload as an image. Is the proof correct?
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| Probability : The emperor's death sentence Posted: 28 Nov 2021 10:58 PM PST please help, I don't understand how to do this. Click link You are a prisoner sentenced to death. The emperor (who has sentenced you) offers you a chance to live by playing a simple game. He gives you 10 black marbles, 10 white marbles, and 2 empty bowls. He then says, "Divide these 20 marbles into these 2 bowls. You can divide them any way you like as long as you place all the marbles in the bowls. Then I will blindfold you, and ask you to choose a bowl, and remove ONE marble from that bowl you have chosen. You must leave the blindfold on the entire time, and must remove the first marble you touch. If the marble is white you will live, but if the marble is black, you will die." 1.If you put 5 black and 5 white marbles in each bowl, then complete the tree diagram (make sure you have written a probability on every branch!) and use it to determine your chance of living. |
| question about decomposition of positive terms series $\sum_{n=1}^\infty a_k$ where $a_k=b_k + c_k$ Posted: 28 Nov 2021 11:03 PM PST let $\sum_{n=1}^\infty a_k$ be a series at positive terms. and with some positive free parameter x. now suppose $a_k=b_k + c_k$ for all k. what can we say in general about the series by looking at $\sum_{n=1}^\infty b_k(x)$ and $\sum_{n=1}^\infty c_k$. does it converge if and only if both of them converge? or in general we can't really say anything about such a series? to give context if I may wasn't clear here's an example of what i mean: $$\sum_{n=1}^\infty \frac{n^x+1}{\log(n)}, \quad x>0\tag{i}$$ notice: $$\frac{n^x+1}{\log(n)}=\frac{n^x}{\log(n)} +\frac{1}{\log(n)}$$ then we can assure that the terms where (i) converges are the same where both of the series $\sum_{n=1}^\infty \frac{n^x}{\log(n)}$ and $\sum_{n=1}^\infty \frac{1}{\log(n)}$ converge? and those where (i) diverges are the same where at least 1 of the former 2 diverge? thanks in advance |
| Regularity of a language checker Posted: 28 Nov 2021 10:54 PM PST I have to check if this language is regular or not:$$L = \{w(bb)^nw^R:w\in\{a,b\}^* \land n \in \mathbb{N}\}$$ My thoughts are if this language is regular so the RE for this is: $(11)^*$ where $w$ and $w^R$ are empty strings. But if this language is not regular, the pumping lemma doesn't work on this language since there are $2$ different exponents. What do you guys think? Is this language regular or not? |
| $\lim_{n\to\infty}\frac{n+\sqrt{n}+\sqrt[3]{n}+\cdots+\sqrt[n]{n}}{n}$ Posted: 28 Nov 2021 10:54 PM PST How to calculate the limit of this sequence $$\lim_{n\to\infty}\frac{n+\sqrt{n}+\sqrt[3]{n}\cdots+\sqrt[n]{n}}{n}$$ is it convergent or divergent? |
| Is Heine's theorem realted to Bolzano-Weierstrass theorem? Posted: 28 Nov 2021 10:44 PM PST The Bolzano-Weierstrass theorem states that: Any Bounded sequence has subsequences that converge to a finite limit. Heine's theorem states that: $\lim\limits_{n \to a} f(x)=L \ \iff \ \forall \ subseqeuences \ x_n \ s.t \ \lim\limits_{n \to \infty} x_n=a \ and \ x_n \ne a \lim\limits_{n \to \infty} f(x_n)=L$ For example, if we use $f(x)=\frac{sinx}{x}$ if we use the two subsequences one with all zeroes that indicate all the crossings two and one with all the peaks with decreasing amplitudes as $x \to \infty$ both tend to zero. I know this is just one example. But I want to know if this is just coincidence, or are the two theorems related on anyway? |
| How to tackle the integral $\int_{0}^{\infty} \frac{\ln x}{x^{n}-1} d x$? Posted: 28 Nov 2021 10:56 PM PST In my post, I started to investigate the integral $\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^{2}-1} d x$. Fortunately, $$\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^{2}-1} d x =2 \int_{0}^{1} \frac{\ln x}{x^{2}-1} d x.$$ So we need only to evaluate the integral $J$ using series and integration by part. $\displaystyle \begin{aligned} J\displaystyle & = \int_{0}^{1} \frac{\ln x}{1-x^{2}} d x =\sum_{k=0}^{\infty} \int_{0}^{1} x^{2 k} \ln x d x=\sum_{k=0}^{\infty}\left(\left[\frac{x^{2 k+1} \ln x}{2 k+1}\right]_{0}^{1}-\frac{1}{2 k+1} \int_{0}^{1} x^{2 k+1} \cdot \frac{1}{x} d x\right) \\\displaystyle &=-\sum_{k=0}^{\infty}\frac{1}{(2 k+1)^{2}}=-\frac{\pi^{2}}{8} \end{aligned} \tag*{} $ $$\therefore \displaystyle \int_{0}^{\infty} \frac{\ln x}{x^{2}-1} d x =-2J=\frac{\pi^{2}}{4} $$ However, when I began to increase the power $n$, I found, in Wolframalpha, that there is a pattern for the integral$$ I_{n}=\int_{0}^{\infty} \frac{\ln x}{x^{n}-1} d x $$ $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline I_{n} & \text { Diverges } & \frac{\pi^{2}}{4} & \frac{4 \pi^{2}}{27} & \frac{\pi 2}{8} & \frac{8 \pi^{2}}{25(5-\sqrt{5})} & \frac{\pi^{2}}{9} & \frac{\pi^{2}}{49} \csc ^{2}\left(\frac{\pi}{7}\right) & \frac{\pi^{2}}{64} \csc ^{2}\left(\frac{\pi}{8}\right) \\ \hline \end{array} $$ By the pattern, let's guess the formula for $I_n$ as $$ I_{n}=\left(\frac{\pi}{n}\right)^{2}\csc ^{2}\left(\frac{\pi}{n}\right). $$ How to prove it? Is it ifficult or interesting? Looking forward to your suggestions and proofs. |
| Why is modular arithmetic called "modular"? Posted: 28 Nov 2021 10:39 PM PST I think of modular in the context of "modular design" which is basically subdivided a system into self-contained chunks. modular is based on the word modules which are like self contained lessons or sections. There is almost a sense of functional programming here where you can think of "modular" as meaning "atomic" in the way that a pure function is the most distilled version of a black box yielding output from a given output. Based on the above concept, here is my hypothesis regarding why modular arithmetic is called modular. If you do modular arithmetic on a given modulus, say $n$, then you are dividing the set of integers into $n-1$ discrete chunks, reminiscent of how a system is broken down in modular design. |
| show $X^{(0)}$ is a discrete finite subset of $X$ Posted: 28 Nov 2021 10:37 PM PST If $(X,\mathcal{C})$ is a CW complex, then $X^{(0)}$ is a discrete finite subset of $X$. Let $A\subset X^{0}$. For any cell $c\in\mathcal{C}$, let $X_c$ be a finite subcomplex that contains $\bar{c}$. Note that $A\cap X_c=X_c\setminus(X_c\cap A)$. As $X_c$ is finite, we have $X_c\setminus(X_c\cap A)$ is finite. Additionally, $X$ is Hausdorff, $A\cap X_c=X_c\setminus(X_c\cap A)$ is closed because finite set is closed in Hausdorff space. Thus, $A$ is closed by the weak topology. Moreover, $A\cap\bar{c}=A\cap(X_c\cap\bar{c})$ which is finite. this is what I have so far, and I don't know how to move forward. |
| Posted: 28 Nov 2021 10:28 PM PST First some definitions: For a set Σ of L-sentences, Mod(Σ) denotes the class of all models that satisfy Σ. For a class M of models, we say it is EC if M=Mod(σ) for some sentence σ and ECΔ if M=Mod(Σ). Problem. Let T be a theory having arbitrary large finite models. (For example, T can be axioms for groups, or fields, or linear orderings.) Let Kinf={M : M⊨T, Card(|M|) is infinite.} and Kfin={M : M⊨T, Card(|M|) is finite}. Question. (a) Kinf is ECΔ (b) Kfin is not ECΔ (c) Kinf is not EC |
| what is q & n in Primality Testing using Elliptic Curves Posted: 28 Nov 2021 10:25 PM PST Shafi Goldwasser and Joe Kilian's paper on Primality Testing using Elliptic Curves under the header "A PRIMALITY CRITERION USING ELLIPTIC CURVES" on page 9 contains a formula $$\ q > n^\frac12+n^\frac14 + 1$$ and don't understand where the values of q and n are coming from. |
| Proof that a nonzero point is continuous for g(x) Posted: 28 Nov 2021 10:23 PM PST I have come across this problem in my math book for my numerical analysis class: Let $g(x)=\sqrt[3]{x}$. Prove that $g$ is continuous at a point $c \neq 0$. I start my proof off the typical way for proving a continuous function: Given some arbitrary $\epsilon > 0$, let $\delta=$ ... and then I get lost. I cannot find what my delta should be for the life of me. I am primarily struggling with the triangle inequalities for this problem, in order to find what our delta should be. Guidance is much appreciated! |
| Independence of long times coin toss model Posted: 28 Nov 2021 10:35 PM PST A coin is tossed independently $n$ times. The probability of heads at each toss is $p$. At each time $k (k = 2,3,\cdots,n)$ we get a reward at time $k+1$ if $k^{th}$ toss was a head and the previous toss was a tail. Let $A_k$ be the event that a reward is obtained at time $k$. a.Are events $A_k$ and $A_{k+1}$ independent? b. Are events $A_k$ and $A_{k+2} independent? In this problem as the event $A_k$ depends on the previous out comes so I think bit a will be dependant however bit b is independent. Is it the correct way to show. |
| Posted: 28 Nov 2021 10:40 PM PST There are many MO or MSE questions, some very popular, about the product of 3-D matrices (cubes) and 4-D matrices, and more. Of course they are considered as tensor operations, but so far, I haven't seen a non-trivial product of "cubes" ($n \times n \times n$ matrices) resulting in a cube. I propose examples here, indeed in any dimension, generalizing the standard 2-D case, and I am wondering if these examples exhaust all the nice possibilities. The objects investigated here have $n^d$ elements, all real numbers, and $d$ is called the dimension ($n$ is not called the dimension). To cut it short, it seems to work quite nicely only if $n=2$ or $n=4$. First, define a product $A\cdot B = C$ of cubes (there are many possible definitions) by reducing it to 2-D, as follows:
The choice for the $\sigma_k$'s and $\epsilon_{k,j}$'s is very important and discussed below. Note that the generic definition of $A\cdot B$ generalizes easily to any dimension $d>2$, that is to hypercubes and so on. In that case $A_k$ is an hyper-slice of dimension $d-1$, of $A$, and $A$ has $n$ such hyper-slices. The product is computed iteratively by going to lower and lower dimensions until we reach standard $n\times n$ matrices (dimension $=$ 2). It is a combinatorial problem. Examples I see only two cases that are interesting and non-trivial: the case $n=2$ and $n=4$, regardless of the dimension (cube, hypercube and so on). Are there any other? The case $n=2$ is related to complex numbers, and $n=4$ to quaternions. For $n=2$, the $2\times 2 \times 2$ cube $A$ is equivalent to $A_1 + A_2i$, where $i$ is the imaginary unit. Since $$(A_1 + A_2i)\cdot(B_1+B_2i)=A_1 B_1 - A_2 B_2 + (A_1 B_2 + A_2 B_1) i,$$ we have
that is: $\sigma_1(1,2)=(1,2)$, $\sigma_2(1,2)=(2,1)$, $\epsilon_{1,1}=1,\epsilon_{1,2} = -1, \epsilon_{2,1}=1, \epsilon_{2,2}=1$. So the case $n=2$ (for cubes) essentially corresponds to complex matrices, despite the elements being real numbers. The inverse, exponential function, or determinant of a $2\times 2\times 2$ cube can be computed easily. In particular, the identity cube has $A_1 = I$ and $A_2=0$ (respectively, the identity and the zero $2\times 2$ matrices). The case $n=4$ can be handled in the same way if we replace complex numbers by quaternions. For other values of $n$ there is no great complex-like extension of real numbers in $n$ dimension, thus my chagrin. See however an attempt with $n=3$, here. Do you know of other products, possibly even better than those mentioned here, for cubes, hypercubes and so on, especially if $n=3$ or $n>4$? |
| Projecting a band of a sphere onto a 2D surface Posted: 28 Nov 2021 10:26 PM PST For a craft project, I want to take a "band" of a sphere (i.e. the area between two latitudes) and project it onto a plane, so that I can fold the 2d shape onto the sphere and recreate the band in 3 dimensions. The sphere I am working with has a radius of 13cm. The band I wish to project is the area between a great circle and a small circle with radius 11.2cm. If my trig is correct, the distance between these two circles, following the surface of the sphere, is about 6.76cm.
I do not know how to perform a projection given this setup. From similar crafts I can see that one strategy is to create a sort of arced rectangle, which looks as though it could be constructed by aligning a circle and an offset of the same circle.
However I am unsure how the arcs could be calculated. |
| Posted: 28 Nov 2021 11:06 PM PST I know it is possible to solve $\text{minimize}\ \lVert x \rVert^2$ instead of $\text{maximize} \frac{1}{\lVert x \rVert}$, since the former behaves better around zero. However, I am looking for a theorem or a rule that tells me this is allowed. Edit: This is the objective of a constrained optimization problem. I just omit the constraints here. |
| Notation of derivative w.r.t. the multi-variable function argument of an univariate function Posted: 28 Nov 2021 10:55 PM PST Suppose I have a univariate function, whose variable is a multi-variable function, i.e. $$ f=f(g)\,,\quad g=g(x,y)\,.$$ Should I write the derivative of $f$ as $\frac{df}{dg}$ or $\frac{\partial f}{\partial g}$? When using chain rule, which nontation is correct or better? $$\frac{\partial f}{\partial x}=\frac{d f}{d g}\frac{\partial g}{\partial x}$$ or $$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial g}\frac{\partial g}{\partial x}$$ I saw the second notation (all partial derivative) on a book about solid mechanics, but $\frac{df}{dg}$ makes more sense to me. |
| Posted: 28 Nov 2021 10:47 PM PST
NOTE - I noticed these 2 questions here (1) and here (2) but I believe that my question is a bit different because these questions answer only the case of $a_1 \geq 0$ according to the information we can understand that if $a_1 \geq 0$ then it is immediately solved as $\lim\limits_{n\to\infty }(1+\frac{1}{a_n})^{a_n} =e$ since then we can look at $a_n$ as in $a_n=n$ and then the limit would be just an identity $\lim\limits_{n\to\infty }(1+\frac{1}{n})^{n}=e$ otherwise what if $a_1 <0$? we will need to show that there exists an $N \in \Bbb N$ such that $a_N \geq 0$ we can also understand that $a_{n+1}-a_n \geq 1$ then for ever $n$ we get $a_{n+1}-a_n = (a_{n+1}-a_n)+(a_n-a_{n+1})...+(a_2 - a_1) \geq 1+1+1...+1 =n$ I do not know how to continue from here.. I usually have more ideas and stuff I tried on my posts but I really cannot figure out what to do here. thanks for any help and tips! |
| What does mean of Topological distortion in perspective projection? Posted: 28 Nov 2021 10:42 PM PST I asked this question and I read from this websites that The points of the plane that is parallel to the view plane & also passes through the Centre of projection are projected to infinity by the perspective projection. When we join the point which is back of the viewer to the point which is front of the viewer then the line will be projected as a broken line of infinite extent. This is called the topological distortion.
I am not understanding what does mean of topological distortion!! And how the point behind the viewer projected to front of the viewer along the infinite line? |
| All pairs of letters (de Bruijn sequence with $n=2,k=52$) Posted: 28 Nov 2021 10:26 PM PST Suppose I'm designing a font and want to verify its kerning. Ignoring punctuation, I want to generate a string made of $52$ symbols (uppercase I looked up similar problems, and it seems that what I want is a de Bruijn sequence. However, I couldn't find anything online relating to an alphabet size $k>9$. How long would my de Bruijn sequence be, and how would I go about generating said sequence? |
| Minimal spanning set ("conical basis") for 2x2 Hermitian PSD (positive semi-definite) cone? Posted: 28 Nov 2021 10:28 PM PST A linear combination $ax + by$ is called conic(al) if $a, b \ge 0$ (cf. section 2.1.5 of Boyd, Vandenberghe). I.e. conic(al) combinations are just linear combinations where the coefficients are restricted to be non-negative. Here I am treating the space of $2 \times 2$ Hermitian (complex) PSD matrices as a real vector space (i.e. scalars are real numbers, so saying coefficients are $\ge 0$ makes sense).
Attempt: Under that assumption, and via trial and error, I think the following might be a minimal spanning set for the $2 \times 2$ complex Hermitian PSD cone. It seems like none of them can be written as conic combinations of each other (even though they're linearly dependent), so all that would need to be shown is that every PSD matrix can be written as a conic combination of these six matrices. $$ (A) \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$$ $$ (B) \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$$ $$ (C) \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$$ $$ (D) \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$$ $$ (E) \begin{bmatrix} 1 & i \\ -i & 1 \end{bmatrix}$$ $$ (F) \begin{bmatrix} 1 & -i \\ i & 1 \end{bmatrix}$$ Note that a linear basis for Hermitian matrices should have $n^2 = 4$ elements, whereas the above has $2n^2 - n$ elements. This is reflected in how the above set is linearly dependent, e.g. (C) = -(D) + 2(A) + 2(B), and (E) = -(F) + 2(A) + 2(B). But clearly neither of these are conic(al) combinations. Obviously being linearly independent implies also being conically independent. So showing that the above set is minimal (assuming that every PSD matrix can be writen as a conical combination of them) requires showing that (D) and (F) cannot be written as conic(al) combinations of (A), (B), (C), (E), and/or that (C) and (E) cannot be written as conic(al) combinations of (A), (B), (D), (F). This might follow from how (A), (B), (C), (E) form a basis for Hermitian matrices, as do (A), (B), (D), (F), and so by uniqueness of linear combinations with respect to a basis, e.g. (C) = -(D) + 2(A) + 2(B) is the unique way to write (C) as a linear combination of (A), (B), (D), (E) or (A), (B), (D), (F), it is not a conic(al) combination, so no such conic(al) combination exists? Anyway as hinted above, is this isn't difficult to generalize to the $n \times n$ case, I would love to hear your thoughts about that too. Right now I am trying to understand "the qubit", so $2 \times 2$ case suffices for me. Related question about minimal spanning sets for conic(al) combinations |
| Finding all polynomials $P(x) \in \mathbb R[x]$ such that $P(x)^2=4P\left(x^2-5x+1\right)+2$ Posted: 28 Nov 2021 10:53 PM PST
This comes from a no-solution class problem so it should have a definitive solution, unless my teacher wrongly wrote it as it is now. Approach This is like solving a functional equation, but only that it's a polynomial. I first thought I would solve for $x=x^2-5x+1$ or $x^2-6x+1=0$ which yields $x_1=3+2\sqrt2$ or $x_2=3-2\sqrt2$. Which yields the value for $P(x_1)$ and $P(x_2)$. But then I couldn't proceed. If $d>0$ and $ax^d$ is the leading term of $P(x)$, then $a^2x^{2d}=4ax^{2d}$, and thus $a=4$. Another way I considered is using sequence to prove that there's infinitely many values of $P(x)=P(y)$ but that doesn't work either. Any help is appreciated! |
| $S^1$-valued function on $T^n$ Posted: 28 Nov 2021 10:46 PM PST Let $f:T^n\to S^1$ be a smooth function on the $n$-torus $T^n=S^1\times \cdots \times S^1$. The differential $df$ can be viewed as a closed 1-form on $T^n$ (not exact). Moreover, it should give a nonzero cohomology class $[df]$ in $H^1(T^n;\mathbb Z)$ in $\mathbb Z$-coefficients. Concerning the natural isomorphism $H^1(T^n;\mathbb Z)\cong \mathrm{Hom}(\pi_1(T^n),\mathbb Z)$, this should mean the map sending a loop $\sigma$ in $T^n$ to the integer $\mathrm{deg}(f\circ \sigma)$. (Note that $f\circ \sigma: S^1\to S^1$) I believe all these are standard, but I fail to find a reference or write down enough details to convince myself. Could you help me to make this clear? Or, maybe I was mistaken somewhere? |
| Need Recommendation for High Level proof book Posted: 28 Nov 2021 10:29 PM PST I know how solve geometry, combinatorics, algebra/Precalc, and number theory non-proof problems pretty well. However, I lack the ability to prove theorems, certain parts of recursive functions (ex prove ,a2020 smaller than a2019-7)(first time using mathstackexchange srry bad formatting), certain geometric ideas, etc. I'm also bad at using AM-GM and other tools to prove inequalities. |
| How to take integral of absolute value(x) on a Casio fx-991ms Posted: 28 Nov 2021 11:03 PM PST I don't know if this is the correct place for this question, if there is a more appropriate forum please let me know. Trying to solve the probability density function which has an absolute value in it, how can I integrate an absolute value on a Casio fx-991ms calculator? In order to enter absolute value (x) I need to switch to Mode 2, but once I do that everything I entered for the integral disappears. If I try to do this in Mode 2 to begin with than the integral symbol simply does not work in Mode 2. |
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