Recent Questions - Mathematics Stack Exchange |
- How can you determine coefficient polarity of cubic polynomial regression
- Show whether ⟨(e, 3)⟩ is normal in the subgroup K4 × ℤ6?
- Convex Set with Empty Interior Lies in an Affine Set Intuition
- A TEXT BOOK OF CONVERGANCE
- Question about normal operator, skew-hermitian operator and unitary operator
- What is the tightest bound for a given converge series?
- What does $\sum_{i/k_i = k}$ mean in the equation below?
- How to prove $\pi_1(\mathbb{A}_k^n) = 0$ when $k$ is an algebraic closed field of char 0?
- Linear Algebra - SPAN
- Does a wrong/false proposition should be p or `p?
- Show that $B_f(x, y)$ (Bergman Distance) is convex in $x$ but not necessarily in $y$.
- Is the union of open intervals always open?
- Convexity of next state with respect to control input when executing zero-order hold (ZOH) control input
- If a univariate function has directional derivatives on the whole domain, does this imply that it is differentiable except on a countable set?
- Deriving a bell curve
- Sequence of functions is uniformly convergent to a function (f). Is sequence of their derivatives converging to a derivative (f') at only one point?
- Are absolutely continuous functions analytic?
- Factor $f(x)= 6x^3-14x^2-44x+16$. Find all zeros of the function.
- confidence interval explanation
- Proving that $[\mathcal F, \mathcal F]$ is an ideal.
- Schwarz-Christoffel formula for a half-plane
- Solving $v'(x) - ie^{x/2} v(x) = e^{2ie^{x/2}}$
- Formal or informal definability of the standard model of natural numbers
- Solving 2D Poisson equation using complex analysis
- Is it so that $\sigma(\cup_i \sigma(A_i)) = \sigma(\cup_i A_i)$?
- How to solve an equation where $x$ is exponent of $2$ different bases?
- Handling elementary but messy computations in proofs
- Solve $\frac{A^2 + B^2}{AB + 1} = K$ for integral values of $A$, $B$, & $K$. Prove that $K$ is a perfect square
- Real numbers $x,y$ satisfies $x^2+y^2=1.$If the minimum and maximum value of the expression $z=\frac{4-y}{7-x}$ are $m$ and $M$
- How to estimate the number of decimal places required for a division?
| How can you determine coefficient polarity of cubic polynomial regression Posted: 12 Apr 2021 07:39 PM PDT I understand that the polarity (positive/negative value) of the coefficients for a cubic polynomial alters its form, so if you were given a dataset of four coordinates could you determine the polarity of the coefficients and thus the general form without using software or reduced row echelon to determine the cubic regression?  |
| Show whether ⟨(e, 3)⟩ is normal in the subgroup K4 × ℤ6? Posted: 12 Apr 2021 07:36 PM PDT ⟨(a, 0)⟩ = {(a, 0), (e, 0)} I need to show whether or not a series of subgroups are normal and wanted to use this subgroup as an example of how they are done.  |
| Convex Set with Empty Interior Lies in an Affine Set Intuition Posted: 12 Apr 2021 07:30 PM PDT In Section 2.5.2 of the book Convex Optimization by Boyd and Vandenberghe, the authors claim that "a convex set in $\mathbb{R}^n$ with empty interior must lie in an affine set of dimension less than $n$." Can someone provide some intuitive explanations of what this means?  |
| Posted: 12 Apr 2021 07:33 PM PDT after the sequence A, in the seconde line, the writter say we can make............they are not multiples of 3 just I dont understand this paragraphes ,can you help me?
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| Question about normal operator, skew-hermitian operator and unitary operator Posted: 12 Apr 2021 07:18 PM PDT This problem is about the normal operator. In finite dimensional complex inner product space Cn, a operator N is normal if N†N=NN†. The self-adjoint operator T, the skew-Hermitian operator S, and the unitary operator U are all normal operators. Let N be a normal operator in Cn (a)An operator C is skew-unitary if C†C=CC†=−I. Obviously, C is normal. Show that the skew-unitary operator does not exist.  |
| What is the tightest bound for a given converge series? Posted: 12 Apr 2021 07:17 PM PDT I currently consider the following question. Given a converging series $$ \sum_{i=1}^{\infty} (\frac{x_i}{i^{\alpha}})^2 < \infty \ \textit{for} \ \alpha>0 $$ what is the tightest upper bound for each $x_i$? My current idea is assume $x_i \asymp i^{-\beta}$. Then it leads to $$ \sum_{i=1}^{\infty} i^{-2(\alpha+\beta)} < \infty. $$ By the fact that $\sum_{i=1}^{\infty} i^{-\alpha}<\infty$ for $\alpha>1$, we obtain $\beta>\frac{1}{2}-\alpha$ and therefore a possible upper bound for $x_i$ is $i^{\alpha-\frac{1}{2}}$. But my question is whether it is possible to prove this is the tightest bound for $x_i$, if not, is there any possible method to get the tightest bound?  |
| What does $\sum_{i/k_i = k}$ mean in the equation below? Posted: 12 Apr 2021 07:18 PM PDT I was reading chapter 5 of Dynamical Processes on Complex Networks, which discusses the Ising model, where I encountered the following equation for the average magnetization of the class of nodes with degree k: $$\langle \sigma \rangle_k = \frac{1}{N_k}\sum_{i/k_i = k} \langle \sigma_i \rangle$$ I'm particularly confused by the notation $$\sum_{i/k_i = k}$$ Can anyone explain what this means?  |
| How to prove $\pi_1(\mathbb{A}_k^n) = 0$ when $k$ is an algebraic closed field of char 0? Posted: 12 Apr 2021 07:42 PM PDT I know that to prove $\pi_1(\mathbb{P}_k^n) = 0$, one can just use the curve case and proceed by induction. For example, if $n=2$ and there is any connected etale covering, one can choose any line in $\mathbb{P}_k^2$. The inverse image of the line is the support of an ample divisor hence connected by Hartshorne III.7.9. Then on can conclude the covering is of degree 1 and finish the proof. But I don't know how to do this in the affine case and I don't want to use comparison theorem to the topological fundamental group. Can someone tell me how to do this? Thank you very much!  |
| Posted: 12 Apr 2021 07:13 PM PDT How can I know if this is is span? I am studying linear algebra and I'm having problems with this type of given questions  |
| Does a wrong/false proposition should be p or `p? Posted: 12 Apr 2021 07:32 PM PDT What should I use between (p or ~p) if the given proposition is a wrong example of proposition like  |
| Show that $B_f(x, y)$ (Bergman Distance) is convex in $x$ but not necessarily in $y$. Posted: 12 Apr 2021 07:02 PM PDT Here is a related post: Why does Bregman's function require strict convexity? Any guidance/answer is deeply appreciated.  |
| Is the union of open intervals always open? Posted: 12 Apr 2021 06:59 PM PDT Is the union $A:=\bigcup_{x\in \mathbb{R}, x \neq0}(a_x,b_x) \cup \{x\}$, where $0\in (a_x,b_x) \: \forall x$, always open in $\mathbb{R}$? The answer should be no, but I need help coming up with a counterexample. Any ideas would be appreciated.  |
| Posted: 12 Apr 2021 06:55 PM PDT I'd like to know whether the resulting next state of a continuous-time dynamical system is convex with respect to control input when executing zero-order hold (ZOH) input. For simplicity, suppose that the given dynamics $\dot{x} = f(x) + g(x)u$ is input-affine system and consider time span $[t, t+\Delta t)$. The next state $x(t+\Delta t)$ can be written as $x(t+\Delta t) = x(t) + \int_{t}^{t+\Delta t} f(x(\tau)) + g(x(\tau)) u d\tau $ where the control input $u(\tau) \equiv u$ is constant for all $\tau \in [t, t+\Delta t)$. The difficulty is that the state variable at intermediate time, namely, $x(\tau)$ for all $\tau \in (t, t+\Delta t)$, is (probably) a function of $u$. Therefore, I'm not sure that we can conclude that $x(t+\Delta t)$ is convex in u (or, other conclusions).
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| Posted: 12 Apr 2021 07:05 PM PDT Suppose we have a univariate function $f:\mathbb{R} \rightarrow \mathbb{R}$. It is well-known that if both directional derivatives of the function exist at a point then the function is differentiable at that point. Of course, even if both directional derivatives exist, it is possible that they are different values at a point, in which case the function is not differentiable at that point. My Question: If both directional derivatives exist over the whole domain (i.e., all real numbers), does this imply that $f$ is differentiable except on a countable set (i.e., that the directional derivatives match except on a countable set)? Intuitively that seems right to me, but I'm not sure how to prove it.  |
| Posted: 12 Apr 2021 07:14 PM PDT I am trying to see if it possible to derive a bell curve for a profession's annual salary. If I know how many people are part of the profession (such as 30,000 persons) and I know the mean annual salary (such as \$234,000) and that I know that the lowest 10% earn \$143,000, how can I derive a bell curve to find out what the highest 10% earn?  |
| Posted: 12 Apr 2021 07:21 PM PDT For each $n \in N$ let the function $f_n : \mathbb{R} \to \mathbb{R}$ be differentiable, let the function $f : \mathbb{R} \to \mathbb{R}$ be differentiable and let $f_n \rightrightarrows f$. Is it possible that the sequence $f'_{n}(x)$ converges to $f'(x)$ for exactly one $x ∈ R$?  |
| Are absolutely continuous functions analytic? Posted: 12 Apr 2021 07:18 PM PDT I am asking for a proof that any absolutely continuous function with absolutely continuous derivatives is analytic, once I am studying a function with the first property and 'd like to obtain the second one. I know that exist smooth functions that are not analytical, but and if the derivatives are absolutely continuous? If I am wrong, a counterexample is really welcome. Thank you.  |
| Factor $f(x)= 6x^3-14x^2-44x+16$. Find all zeros of the function. Posted: 12 Apr 2021 07:39 PM PDT I need help with the factoring of the polynomial $f(x)= 6x^3-14x^2-44x+16$. Thank you.  |
| confidence interval explanation Posted: 12 Apr 2021 07:16 PM PDT Can someone explain what is the confidence interval? Why it should be 95%? When it is used and what is it measuring? I understand it's some kind of evaluation metric, but I can't seem to find a decent explanation by connecting it to real-world examples. Any help would be greatly appreciated. thanks  |
| Proving that $[\mathcal F, \mathcal F]$ is an ideal. Posted: 12 Apr 2021 07:28 PM PDT Let $\mathcal{F}$ be a Lie algebra over a field $k.$ A lie subalgebra of $\mathcal{F}$ is a vector subspace $S \subset \mathcal{F}$ such that $[S,S] \subset S.$ An ideal of $\mathcal{F}$ is a vector subspace $I \subset \mathcal{F}$ such that $[I, \mathcal{F}] \subset I$. Now, I want to show that Show that $[\mathcal F, \mathcal F] = \{\sum_{i}^n[x_i,y_i]| x_i, y_i \in \mathcal{F}$} is an ideal of $\mathcal{F}.$ My questions are: -1- (closure under Lie bracket)My intuition says that $[\mathcal F, \mathcal F] \in \mathcal F,$ but I am unable to give an appropriate wording because I do not understand what is [,] exactly. (any help with that wording will be appreciated!) 0- I have a question, does this [,] here always means the commutator or just the Lie bracket? 1-Also, I have a question, can I say $[0,0] = 0 \in F$? my intuition say no (can the commutator by any means be zero?), but how can I show that the additive identity is in $\mathcal F$? Also, do I have to show that 2- Showing that it is closed under addition (I know that the given definition is giving it to me as the set of finite linear combinations of the commutator. ): so assume $a,b \in [\mathcal F, \mathcal F],$ we want to show that $a+b \in [\mathcal F, \mathcal F],$ i.e., it is a set of finite linear combinations of the commutator. Since $a \in [\mathcal F, \mathcal F],$ then $a = \{\sum_{i}^n[x_i,y_i]| x_i, y_i \in \mathcal{L}\},$ Since $b \in [\mathcal F, \mathcal F],$ then $b = \{\sum_{j}^m[x_j,y_j]| x_j, y_j \in \mathcal{L}\},$ But then how can I add 2 sets? Should I use induction here? if so, how? 3- Showing that it is closed under scalar multiplication with elements from the field $k.$ Let $c \in k, a \in [\mathcal F, \mathcal F],$ we want to show that $ca \in [\mathcal F, \mathcal F].$ How multiplying a lie bracket (or a commutator ) by a scalar look like ? do I have to multiply the first coordinate only or the two coordinates inside [,]? 4- Then showing $[[\mathcal F, \mathcal F], \mathcal F] \in [\mathcal F, \mathcal F]$ is extremely hard to me, Any help in that part will be greatly appreciated! Can anyone help me answer all the above questions so that I can have a peaceful, alert mind instead of the confusion I am having now? thanks in advance!  |
| Schwarz-Christoffel formula for a half-plane Posted: 12 Apr 2021 07:33 PM PDT I can't understand the example that was given in the book Schwarz-Christoffel Mapping by Tobin Driscoll and Lloyd Trefethen. It's formula 2.5 at page 12. By using Schwarz–Christoffel formula for a half-plane: $$ f(z) = A + C \int^{z} \prod_{k=1}^{n-1} (\zeta-z_{k})^{\alpha_{k}-1} d\zeta $$ he concludes that for $n=1$, with $w_{1}=\infty$ and $\alpha_{1}=-1$, we have a line (that I can understand), and gets to: $$ f(z) = A + Cz $$ I know that he can absorb constants into $C$, nevertheless I can't understand why we have anything in this integrand because the product is empty for $n=1$, therefore it should be zero, and we would have $f(z) = A$.  |
| Solving $v'(x) - ie^{x/2} v(x) = e^{2ie^{x/2}}$ Posted: 12 Apr 2021 07:11 PM PDT While considering the first order inhomogenous ODE $$ v'(x) - ie^{x/2} v(x) = e^{2ie^{x/2}}, $$ I became curious about the most concise way to produce a solution. For the homogenous equation $$v_h'(x)-ie^{x/2}v_h(x) = 0$$ that I will use to satisfy the boundary conditions, I employed a separation of variables to determine $v_h(x) = Ce^{2ie^{x/2}}$. For a particular solution $v_p(x)$ of the inhomogeneous equation, I had thought about employing a method of integrating factors, though I noticed such a method worked well only for particular coefficients. I feel as though a variation of parameters or Laplace transform would be far too much for the problem. Does anyone have simple method for solving the above equation? Thank you all.  |
| Formal or informal definability of the standard model of natural numbers Posted: 12 Apr 2021 06:52 PM PDT I started a discussion in comments to this answer, but it grew beyond what fits comments, so I'm promoting it to this separate question. Here is a recap:
I cannot say for sure whether (or to what degree) my last question belongs to mathematics proper or to philosophy of mathematics. To clarify, I am not asking whether the standard model of natural numbers exists in some platonic sense. I hold a view that mathematics studies formal and formalizable reasoning, and ways to put our ideas into a precise enough form that they can be communicated to other people and understood by them, so that when we discuss them we can be confident we are on the same page. I have a feeling that when I think about various models of artithmetic, I can recognize or pinpoint the standard model among them. Perhaps, this is because my informal reasoning resembles the second-order logic with full semantics, where Dedekind's proof of categoricity of arithmetic holds. I also have a feeling that when I want to discuss the standard model with another person, I can reliably communicate, using a mixture of formal and informal approaches, which model I have in mind. So, when I say something like "consider a formula of a finite length" or "the length of a proof is a standard natural number", I am fairly confident that the other person understands exactly what I have in mind. But because the second-order logic does not have a complete proof system, and first-order theories (that are complex enough to be capable to represent arithmetic) are inherently non-categorical, I have a lingering doubt that my confidence here might be an illusion. So, my question is whether there is a way to reliably pinpoint the standard model of natural numbers and communicate it to other mathematicians (assuming they cooperate in good faith, and do not just try to troll me). Does Dedekind's proof of categoricity play any role in it? Does it make sense to say our informal reasoning corresponds to second-order logic with full semantics? Here are some references to provide more context to this discussion:
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| Solving 2D Poisson equation using complex analysis Posted: 12 Apr 2021 06:56 PM PDT In 2D, the Poisson equation $$ \nabla^2 \Phi(\vec{x}) = \delta^{(2)}(\vec{x}) $$ admits the solution $$ \Phi(\vec{x}) = \frac{1}{2\pi}\log(|\vec{x}|) + \text{constant}\,. \tag{1} $$ By using Fourier transforms, we know that the integral $$ \Phi(\vec{x}) = \int \frac{d^2 \vec{k}}{(2\pi)^2} \frac{-e^{i \vec{k}\cdot \vec{x}}}{\vec{k}^2} \tag{2} $$ is a formal solution but it is divergent. What is the consistent way to give a meaning to Eq.(2) such that we can use complex analysis to compute the solution Eq.(1)? I don't want to solve this equation by using polar coordinates. I want to use the analytic structure of the integrand in Eq.(2), together will complex analysis theorems, e.g. Cauchy theorems. So my question would be really: what is the path in the complex $\vec{k}$-plane that allows me to integrate Eq.(2)? Denoting $\vec{k}=(k_1,k_2)$, we see that in the complex $k_1$-plane there is a pole at $k_1=\pm i k_2$. Is there a prescription to reduce the computation of Eq.(1) to computation of residues, for example?  |
| Is it so that $\sigma(\cup_i \sigma(A_i)) = \sigma(\cup_i A_i)$? Posted: 12 Apr 2021 07:33 PM PDT You are given a set $\Omega$ with some subsets $A_i \subset \Omega$, for some indices $i\in I$. We define a generated $\sigma$-algebra $\sigma(A_i)$ to be the smallest $\sigma$-algebra that contains $A_i$. Is it always the case that $ \sigma\left( \bigcup_{i\in I} \sigma(A_i) \right) = \sigma\left( \bigcup_{i\in I} A_i \right) $? My intuition says yes, and it seems clear in the case where $A_i=\{a_i\},\,a_i\in\Omega$, i.e. singleton sets. Please provide a proof or a counterexample for the general case.  |
| How to solve an equation where $x$ is exponent of $2$ different bases? Posted: 12 Apr 2021 07:28 PM PDT I've got an equation where variable I'm looking for is an exponent of $2$ different bases: $$a^x+b^x=c$$ If I know $a, b$ and $c$, how can I calculate $x$? I was thinking of taking logarithm of both sides to somehow use the rule that $\log(a^x) = x\cdot\log(a)$, but this doesn't help me much as I have the logarithm of sum on the left side: $$\log(a^x+b^x)=\log(c)$$ Now I'm a bit stuck at this point, any hints as to how to approach this?  |
| Handling elementary but messy computations in proofs Posted: 12 Apr 2021 07:44 PM PDT Often when working on a proof, I get to a computation which appears to be elementary (e.g. requiring only standard algebra and perhaps calculus) but messy. Solving this via pen and paper is tedious and error prone, yet the path to a solution is not always elegant (or, at the least, an elegant path is not always apparent to this amateur mathematician). How would you advise a beginning mathmeatician to handle these cases? How do seasoned mathematicians handle this? Often I proceed forwards with pen and paper, but more often than not this results in elementary errors (sometimes simply because the amount of writing gets huge, my penmanship gets sloppy, and I misread my writing). Should I simply be more patient and explicit, and learn to do these computations by hand, accurately if laboriously? Should I learn to use software such as Sage to do them? Or should I take the computational ugliness as a sign that a more elegant proof should be approached? UpdateA good example of the algebraic manipulations I'm talking about are those referenced (but not spelled out) in this answer https://math.stackexchange.com/a/246288/  |
| Posted: 12 Apr 2021 07:07 PM PDT This question was originally asked in the 1988 IMO. It has been answered many times and in almost as many ways. I have not been able to find a complete and logical set of solutions this problem. Two solution sets are known. 1) K=B^2,A=B^3.; & 2) K=M^2/4,B=M^3/8 . I know how Set 1) was derived. How was Set 2) derived?  |
| Posted: 12 Apr 2021 07:26 PM PDT Real numbers $x,y$ satisfies $x^2+y^2=1.$If the minimum and maximum value of the expression $z=\frac{4-y}{7-x}$ are $m$ and $M$ respectively,then find $2M+6m.$ Let $x=\cos\theta$ and $y=\sin\theta$,because $\sin^2\theta+\cos^2\theta=1$.  |
| How to estimate the number of decimal places required for a division? Posted: 12 Apr 2021 07:03 PM PDT Given two decimal numbers, is it possible to estimate the number of decimal places required to fit the result of their division? Provided that the division yields a finite number of decimals, of course. For example:
By estimate, I don't necessarily need the exact number of decimals in the result, but a number of decimals at least equal to the required amount, so that it is guaranteed that the result fits. What I've triedI was able to roughly solve my problem using rational numbers & prime factorization, but this is very compute expensive. Here are the steps:
How I came to the conclusion aboveOut of all the prime numbers, only dividing by Each division by 10 extends the scale of the decimal number by 1 digit. Each combination of In 2x * 5y, there are Now each division by Maximum required digits = Which simplifies to: Which further simplifies to I feel like my approach, although it works, is overly complex. The direct consequence on my software is the slowness of the algorithm. Is there a more straightforward approach to estimating the number of decimal places required for the result of the division to fit? Note that I've read this question: Number of decimal places to be considered in division but it didn't help.  |
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