Recent Questions - Mathematics Stack Exchange |
- Find all functions $f: \mathbb{N}^*\to \mathbb{Z}$ satisfies $f(x+|f(y)|)=x+f(y), \forall x, y\in\mathbb{N}^*$
- ODE of two-variable function $u_x$=$2u(x,y)$
- How to complete gcd number theory proof with induction?
- How to classify these subsets of order complexity?
- Expected value and variance for a biased 1d random walk with a boundary condition
- When the image of operator be subset of $l_p$
- Suppose that x5 = 3. How many nonnegative integer solutions are there to the equation x1 + x2 + x3 + x4 + 5x5 = 20. How many are there total?
- Clarification on predicate logic question
- On closed subspaces of a Banach space
- Is Z plus Cartesian Product sufficient for the Recursion Theorem?
- Finding local/global minimum points of $f(x_{1},x_{2}) = 8x_{1}x_{2} + 4(x_{1}-x_{2})^4$
- Research work to obtain a bachelor degree
- A version of Hilbert Nullstellensatz for co-ordinate ring of a general irreducible affine variety
- Is every isometric relation derivable and constructible (from/to some set of points) via rigid transformations? Can these be expressed exhaustively?
- Ask for a proof of logarithmically complete monotonicity of a power-exponential function involving the difference of the psi and logarithmic functions
- Proof that cube root of product of 2 prime numbers are irrational
- What statistical test I should use?
- Help me solve this number theory problem concerning odd squares and exponents
- Partition of sets in topological spaces that preserves limit point for each member
- A function with some differential equation properties.
- For a Banach space $E$, is it sufficient to check differentiability of $F: U \to E$ on $\phi \circ F$ for all linear functionals $\phi$?
- How to evaluate $\sum\limits_{x=0}^\infty \text{erfc}(x)= 1.1619990479471263635323…?$
- Is integration unary or binary operator?
- URGENT: Questions and concepts in statistics (in physics context), as I really need to improve my understanding...
- Evaluate the integral $I=\int_{1}^{\infty}[x^{3/2}\psi^{'}(x)]^{'}\log x \ dx$
- Implication Logic Truth Table Explained
- Whether the below matrices are in row-echelon form / reduced row-echelon form
- Why does logarithmic scale give linearization here? [Low-pass-filter]
- Why are the roots of $y^2 - iy + 2 = 0$ not complex conjugates?
- Correct way to calculate a matrix trace with negative values
| Posted: 19 Sep 2021 08:30 PM PDT Find all functions $f: \mathbb{N}^*\to \mathbb{Z}$ satisfies $$f(x+|f(y)|)=x+f(y), \forall x, y\in\mathbb{N}^*$$ My current progress:
... And that's it. I don't know what to do next, because I'm only familiar with $f: \mathbb{R}\to \mathbb{R}$.  |
| ODE of two-variable function $u_x$=$2u(x,y)$ Posted: 19 Sep 2021 08:27 PM PDT Let $u(x,y)$ : $\mathbb{R}^2$ -> $\mathbb{R}$, solve the ODE $u_x$ = $2u$, or in another form, $\frac{\partial u}{\partial x}=2u$. I know how to solve $\frac{du}{dx}=2u$ when $u$ is $\mathbb{R}$ -> $\mathbb{R}$, but the case when $u$ is two variable function makes me confused.  |
| How to complete gcd number theory proof with induction? Posted: 19 Sep 2021 08:25 PM PDT Consider the following proof question: prove for all natural numbers $n$, if $n|a$ and $n|b$ then $n|$gcd($a,b$). Using the prime factorization of $a, b$ and gcd($a,b$), this proof should be straightforward. Namely, since $n|a$ and $n|b$, then the prime factorization of $n$ must contain the same primes in gcd($a,b$), since the prime factorization of gcd($a,b$) will contain the minimal number of primes from both $a$ and $b$. That said, is there a way to do this proof with induction? If so, what would the induction hypothesis ($P(k)$) be and how could I use that to prove the following case ($P(k+1)$) after the inductive hypothesis?  |
| How to classify these subsets of order complexity? Posted: 19 Sep 2021 08:23 PM PDT Let: $$f(n) = n^{2\log_2(n)}$$ $$g(n) = (nln(n))^2$$ $$h(n) = 3n^4$$ $$t(n) = \begin{cases} f(n) &\quad\text{if n is a prime number}\\ g(n) &\quad\text{otherwise} \\ \end{cases}$$ Given that $O(g) \subset O(h)$, I want to find the relationship between the different sets $O$ of each function: Here are my steps: $$ \lim_{n \to \infty} \frac{f}{h} = \lim_{n\to\infty} \frac{n^{2\log_2(n)}} {3n^4} = \lim_{n\to\infty} n^{2\log_2(n)-4} *\frac{1}{3} = \infty \\\Rightarrow O(h) \subset O(f)$$ $$ \text{since}\; O(g)\subset O(h) \land O(h)\subset O(f) \\ \Rightarrow O(g)\subset O(f) $$ I am confused at how to proceed to find the relationship between $O(t)$ with respect to $O(g), O(h) \text{ and } O(f)$. My guess is so following(but I am not sure at all or don't know how to prove it formally): $$O(g) \subseteq O(t)$$ $$O(h) \neq O(t)$$ $$O(t) \subseteq O(f)$$ Can you please help me or at least guide me? I don't know how to prove it Thanks  |
| Expected value and variance for a biased 1d random walk with a boundary condition Posted: 19 Sep 2021 08:19 PM PDT I'm working on a population genetics problem. Let's say that N independent elements start with a value of 1 copy. At each generation they can either:(i) lose a copy (-1) with a probability nu;(ii) suffer a rearrangement with a probability theta that can lead to lose a copy (-1) or gain a copy (+1) with equal chance of either outcome; or (iii) stay the same with probability 1-nu-theta. If an element loses all copies (reaches value 0) then it stays like that for good. Other than that, the outcome of each generation does not affect probabilities down the road. I see this as a biased 1d random walk with an absorbing boundary condition at 0. I need expressions for the expected value and the variance of the sum of all N elements at generation g, as a funtion of N, g, theta and nu. I can work out the model without the boundary condition. From simulations, this seems good enough for small values of nu and theta compared to 1/Ng, but it breaks down for larger values. I guess that this is because as more elements reach 0 the stop contributing to the mean and the variance. I've been reading everything I can about this, starting with Feldman, but i don't know how to use the "gambler's ruin" concepts to refine the expected value and variance. Can anyone give some guidance?  |
| When the image of operator be subset of $l_p$ Posted: 19 Sep 2021 08:10 PM PDT Let $f:l_{\infty} \rightarrow l_{\infty}$ defined by $f(x_1,x_2,...)=(x_1,\frac{1}{2}x_2,...,\frac{1}{2}x_n,...)$. Then we need to (1) prove that $f$ is continuous operator and (2) find $p\geq 1$ such that $Im(f) \subset l_p$. My attempt: (1) I proved that $f$ is bounded, namely, $||f(x)||_{\infty} \leq ||x||_{\infty}$, and so it continuous (2) We need to find all $p\geq 1$ such that: every $x \in l_{\infty}$ we have $f(x) \in l_p$, i.e. $||f(x)||_p < \infty$, i.e. $|x_1|^p +\sum_{n \geq 2} |\frac{1}{2}x_n|^p < \infty$. Taking $x=(1,1,...) \in l_{\infty}$ we have $|1|^p +\sum_{n \geq 2} |\frac{1}{2}|^p < \infty$ which is a contradiction, so can we say there is no such $p$ exists.  |
| Posted: 19 Sep 2021 08:08 PM PDT I am not quite sure how to approach this question. I have seen numerous examples similar in process, but none that have a coefficient >1 for one of the xi variables.  |
| Clarification on predicate logic question Posted: 19 Sep 2021 08:08 PM PDT I am attempting to answer the following question: To this question I got the following solution: However, the solution only gives the following : So I tried to then convert my solution to the one in the answer to prove that it gives the same conclusion: Is my working out correct here?  |
| On closed subspaces of a Banach space Posted: 19 Sep 2021 08:06 PM PDT Let $X$ be a Banach space and $M, N$ be its closed subspaces. Suppose $M+N$ is closed. Prove that there exists a constant $c$ such that we can decompose $x$ into $m+n$ in $M+N$ satisfying $\Vert m\Vert\leq c\Vert x \Vert$ and $\Vert n\Vert\leq c\Vert x \Vert$.  |
| Is Z plus Cartesian Product sufficient for the Recursion Theorem? Posted: 19 Sep 2021 08:04 PM PDT In a response, including a quite long attached proof, by the author of the following question, How to Apply Theorem the Recursion in Practice, it is suggested that Z plus Cartesian Product suffices for the Recursion Theorem. Is this correct?  |
| Finding local/global minimum points of $f(x_{1},x_{2}) = 8x_{1}x_{2} + 4(x_{1}-x_{2})^4$ Posted: 19 Sep 2021 08:09 PM PDT Consider the function $f(x_{1},x_{2}) = 8x_{1}x_{2} + 4(x_{1} - x_{2})^4$.
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| Research work to obtain a bachelor degree Posted: 19 Sep 2021 07:51 PM PDT I am currently looking for a subject(Paper) to mathematical models or artificial intelligence related to the maintenance of machinery in the mining sector in order to obtain my bachelor degree. I would appreciate your support if you can share a mathematical paper with me, so that I can develop my research work that helps me achieve my goal. Thank you  |
| A version of Hilbert Nullstellensatz for co-ordinate ring of a general irreducible affine variety Posted: 19 Sep 2021 08:10 PM PDT Let $\mathbb{K}$ be an algebraically closed field. A version of Hilbert Nullstellensatz for polynomials rings says that,
I was wondering if such a statement is true for a general co-ordinate ring of an irreducible affine variety, i.e.,
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| Posted: 19 Sep 2021 08:01 PM PDT There are three types of rigid transformations that can be combined to create an isometry: reflection, translation, rotation. The former two can be applied to any function to create a new (or trivially, identical) function in isometric correspondence, whereas the latter cannot in general (only to bijections) although can be applied to non-invertible relations. Can these be expressed as a mapping $f$? Let $\{f\}$ denote the set of all mappings $f$ from one set-relation to all its isometries, where $c$ and $d$ represent fixed constants (causing a translation) and a negative sign ($-$ from $±$) before one or more of the variables represents a reflection. $\stackrel{A}{\{(x,\:y)\}_1}\quad\stackrel{\{f\}}{\mapsto}\quad\big\{\{(±x+c,\:±y+d)\}_{\{≅A_1\}}\big\}$ Does this look right (for two variables, without rotation)? If so, does it extend without issues to more dimensions?, and How can rotations be applied (without, and if possible also with, precluding multiple-to-one mappings with or without restrictions)?  |
| Posted: 19 Sep 2021 08:03 PM PDT It is common knowledge that the classical Euler gamma function $\Gamma(z)$ can defined by \begin{equation*} \Gamma(z)=\int^\infty_0t^{z-1} e^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and the psi (digamma) function is defined by $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$. A non-negative function $f$ is said to be completely monotonic on an interval $I$ if $f$ has derivatives of all orders on $I$ and \begin{equation*} 0\le(-1)^{n-1}f^{(n-1)}(x)<\infty \end{equation*} for all $x\in I$ and $n\in\mathbb{N}=\{1,2,3,\dotsc\}$. A positive function $f$ is said to be logarithmically completely monotonic on an interval $I$ if it is infinitely differentiable (smooth) and satisfies \begin{equation*} (-1)^k[\ln f(t)]^{(k)}\ge0 \end{equation*} on $I$ for $k\in\mathbb{N}=\{1,2,\dotsc\}$. A logarithmically completely function on an interval $I$ must be also completely monotonic on $I$, but not conversely. The Bernstein--Widder theorem reads that a function $f$ is completely monotonic on $(0,\infty)$ if and only if it can be represented as a Laplace transform \begin{equation}\label{Laplace-mu(t)-INT}\tag{1} f(x)=\int_0^\infty e^{-xt}\textrm{d}\mu(t), \quad x\in(0,\infty), \end{equation} where $\mu(t)$ is non-decreasing and the above integral converges for $x\in(0,\infty)$. For $\alpha\in\mathbb{R}$, let \begin{equation*} h_\alpha(t)=t^{t[\psi(t)-\ln t]-\alpha}, \quad t>0. \end{equation*} It is easy to prove that the necessary condition for $h_\alpha(t)$ to be logarithmically completely monotonic on $(0,\infty)$ is $\alpha\ge-\frac12$. Is the necessary condition $\alpha\ge-\frac12$ a sufficient condition for $h_\alpha(t)$ to be logarithmically completely monotonic on $(0,\infty)$? A hint: the function $t[\psi(t)-\ln t]+\frac{1}{2}$ is completely monotonic on $(0,\infty)$. References
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| Proof that cube root of product of 2 prime numbers are irrational Posted: 19 Sep 2021 07:58 PM PDT I am stuck with this problem from my son's homework: Given $p$ and $q$ are prime numbers, prove that $\sqrt[3]{pq}$ is irrational Could someone please shed some light? Thanks!  |
| What statistical test I should use? Posted: 19 Sep 2021 07:54 PM PDT I'm an undergrad and my stats knowledge is passable, but the current project we're doing is a bit out of my depth and would appreciate some help in figuring out how to analyse the data. We are trying to see how "personality" very's by size in cat's eyes (an intertidal snail). We recorded weight and size of our test subjects and did two tests on each to see how long they would hide in their shell after 1): being moved 2): being tapped on their shell We timed how long it would take for them to open their operculum - a trap-door like seal they use to cover the entrance to their shell - and how long it would take them to subsequently emerge from their shell. We recorded if they hid in their shell, and if they closed their operculum, and recorded if they opened their operculum and if they emerged from their shell. Recording went for 5 minutes max, and some did not open their operculum or emerge at all in that time. If they didn't retreat or didn't open their operculum for the first test, they were not treated in the second test and were given an N/A for that one. If they didn't emerge or open their operculum, they were given an N, these were therefore the most fearful of the subjects, but I don't know how I should incorporate them in the rest of the continuous data. So just looking for some help on figuring out what statistical tests I should use, any help is appreciated. Also, our professor showed us this site and suggested we use it if we're not confident in our stats, before anyone asks if I'm allowed to ask this here. Any help is appreciated, let me know if clarification is needed, cheers.
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| Help me solve this number theory problem concerning odd squares and exponents Posted: 19 Sep 2021 08:04 PM PDT $$a^{a+b}b^a+1=k^2$$ all integer values, some results for this is: $$a=2, b=3, k=17$$ this isn't the only square I found but it is the only odd square I found, how can I simplify or approach this?  |
| Partition of sets in topological spaces that preserves limit point for each member Posted: 19 Sep 2021 08:01 PM PDT Recall that for a sequence $\{x_{n}\}$ in a metric space, if $\{x_{n}\} \rightarrow x$ while $x_{n} \neq x \ \ \forall n$, then $\{x_{2k}\} \rightarrow x$ and $\{x_{2k+1}\} \rightarrow x$. As a corollary, for arbitrary limit point $a$ of a countable set $A$, there exists an n-partition $\{A_{1}, ... ,A_{n} \} $of $A$ s.t. $t \in \overline{A_{i}},\forall i \in {1,...,n}$. I wish to generalize this to arbitrary compact Hausdorff spaces and arbitrary sets (without assuming countability). More precisely, I wish to prove:
Note that direct application of sequential methods fails since $X$ might not be first-countable.  |
| A function with some differential equation properties. Posted: 19 Sep 2021 07:55 PM PDT I am dealing with a family of functions that take the form $$y(x)=-\frac{f(x)}{(x-1)^{2v+1}}.$$ By taking $y'(x)$, we obtain a new rational function $$y'(x)=\frac{g(x)}{(x-1)^{2v+2}}.$$ What I notice is that although $f(x)\neq g(x)$, it is always the case that $$(2v+1)f(1)=g(1).$$ Here are some examples: Consider \begin{align*} y(x)=-\frac{(3x^2+12x)}{(x-1)^7}.\tag1 \end{align*} Then \begin{align*} y'(x)=\frac{15x^2+78x+12}{(x-1)^8} \end{align*} Although $3x^2+12x\neq 15x^2+78x+12$, we have $$(2*3+1)(3(1)^2+12(1))=15(1)^2+78(1)+12$$ Another example: Let $$y(x)=-\frac{(-5x^3+10x^2+100x)}{(x-1)^9}$$ We have $$y'(x)=\frac{(100 + 820 x + 55 x^2 - 30 x^3)}{(x-1)^{10}}$$ Observe that $$9*(-5+10+100)=-30+55+820+100$$ Is there a general solution to $(2v+1)f(1)=g(1)$?  |
| Posted: 19 Sep 2021 08:05 PM PDT Let $E$ be a complex Banach space and and $U \subset \mathbb{C}$ an open set containing $0$. Consider a function $F: U \to E$. If for every linear functional $\phi \in E^*$, the function $\phi \circ F: U \to \mathbb{C}$ is (complex) differentiable at $0$, does this imply that $F$ is differentiable at $0$? related questions/generalizations: What about the case of real Banach spaces and real differentiability? Also, what happens if $U$ is also taken to be a subset of a complex Banach space and we consider Frechet derivatives? My ideas so far: We are assuming that for every $\phi \in E^*$ there exists a $g(\phi) \in E$ s.t. $$ \phi(F(h)) = \phi(F(0)) + h \cdot g(\phi) + o(h) $$ It's easy to see that $\phi \to g(\phi)$ is linear so $g \in E^{**}$. If we assume reflexivity of $E$ then we can say $g \in E$and obtain that $\phi(F(h) - h\, g) = o(h)$ for any $\phi \in E^*$.  |
| How to evaluate $\sum\limits_{x=0}^\infty \text{erfc}(x)= 1.1619990479471263635323…?$ Posted: 19 Sep 2021 08:27 PM PDT This will be the $5$th in a series of an infinite series of a single function. Here are 2 related sums: and Our goal sum uses the Complementary Error function integrating term by term. Note the Fresnel Integrals. There are also Gamma type functions: $$\sum_{\Bbb N^0}\text{erfc}(x)=\sum_{x=0}^\infty \text{erfc}(x) =\lim_{n\to\infty} \left(n-\sum_0^n \text{erf}(x)\right)= \frac{2}{\sqrt \pi}\sum_{x\in \Bbb N^0}\int_x^\infty e^{-t^2} dt= \sum_{x\in \Bbb N^0}\left(1-\frac 2\pi \int_0^\infty \frac{\sin(2tx)}{te^{t^2}}dt\right)=\sum_{x\in\Bbb N^0}\left(1 - (1 + i) \left(\text C\left(\frac{((1 - i) z)}{\sqrt\pi}\right) - i\,\text S\left(\frac{((1 - i) z)}{\sqrt\pi}\right)\right)\right)=\sum_{\Bbb N^0}\text{erf}(x,\infty)= \frac{1}{\sqrt \pi}\sum_{\Bbb N^0}Γ\left(\frac12,x^2\right)= \sum_{\Bbb N^0}Q\left(\frac12,x^2\right) =\frac{1}{\sqrt \pi} \sum_{\Bbb N^0}x \text E_\frac12 \left(x^2\right)= \frac{1}{\sqrt \pi}\int_1^\infty t^{-\frac12}\sum_{x\in\Bbb N^0}xe^{-tx^2}dt =1.16199904794712636353230832245579717…$$ The final integral-sum representation reminds me of a Differentiated Jacobi Theta function of the Third Kind. Here is a question about it although there are others. How can I evaluate the constant? Please correct me and give me feedback!  |
| Is integration unary or binary operator? Posted: 19 Sep 2021 08:01 PM PDT Is integration unary operation or binary operation? If we think about numbers, then integrals are interated binary operation. But if we think in terms of functions, then integrals seem unary operators. I'm confused.  |
| Posted: 19 Sep 2021 08:14 PM PDT I have never done a statistics course and I am really struggling to understand these concepts. This is in the context of physics but the concepts should be the same... First question is: Suppose we want to fit a model of the form ymodelled(t) = a sin ωt + b cos ωt to a set of magnetotelluric data, where ymodelled(t) is the predicted or modelled data point at time t. The least squares misfit function to observed data yobserved(t) with standard deviation σ(t) is given by chi squared = $\sum_{t=1}^N$ ( yobserved(t) - ymodelled(t) / σ(t) )2. So, how can we find the best set of parameters (a,b) that minimise the chi squared function. Second question is: Suppose we want to fit a model of the form ymodelled(t) = a0 + a1T + a2T2 to a set of magnetotelluric response data, where T is the periodicity of the signal in seconds and ymodelled(t) is the predicted or modelled data point at a periodicity of T. Explain how we can determine the best set of parameters from a Monte Carlo simulation. Why is the Monte Carlo simulation is better then finding the minima of the chi squared function with respect to each of the model parameters. Third question is: Explain why we may need to use robust methods, in some cases, to better determine model parameters from a set of observations. A Jackknife error estimation of the mean of a set of N data systematically removes one data point at a time (N-1 remaining) and calculates the mean; thus we obtain N different estimates of the mean. How can the Jackknife method identify outliers? Last question is: In terms of modelling, what is the essential difference between a linear equation and a non-linear equation? For a linear equation, briefly define the terms: (a) model parameters; (b) basis functions. In terms of modelling, what is the fundamental advantage of linear over non-linear equations? I don't have a strong mathematical background and so I am looking for a more intuitive explanation of these concepts. Thank you.  |
| Evaluate the integral $I=\int_{1}^{\infty}[x^{3/2}\psi^{'}(x)]^{'}\log x \ dx$ Posted: 19 Sep 2021 08:16 PM PDT
Using integration by parts taking $u=\log x$ as first function and $v=[x^{3/2}\psi^{'}(x)]^{'}$ as second function we have $$I= (\log x) x^{3/2}\psi^{'}(x) \Biggr|_{1}^{\infty} -\int_{1}^{\infty}\sqrt{x}\psi^{'}(x)dx $$ Consider $$J= \int_{1}^{\infty}\sqrt{x}\psi^{'}(x)dx $$ Integrating by parts taking $\sqrt{x}$ as first function and $\psi^{'}(x)$ as second function we have $$J=\sqrt{x}\psi(x)\Biggr|_{1}^{\infty}-\frac{1}{2}\int_{1}^{\infty}\frac{\psi(x)}{\sqrt{x}}dx$$ Any help is appreciated. Thanks a lot!  |
| Implication Logic Truth Table Explained Posted: 19 Sep 2021 08:08 PM PDT The question that has bothered me for a while has been answered and closed here (Implication in mathematics - How can A imply B when A is False?) and probably many other posts. Although all the answers are accurate and correct in their own way of explaining (or went above my head), those still didn't "click" for me so I kept trying to find a more specific example. Here's what I came up with and would like folks to review and comment. First the question:
My conclusion and probably the mistake I've been making to understand here is that this truth table is not about when the result will be true if you use the equivalent (~A V B) logic. What this truth table represents is the fact that if you have a data set (or situations) that results in a false value of (~A V B) then your assumption that A implies B is violated (or is not correct). In simpler words, the true values in the truth table are for the statement A implies B. Conversely, if the result is false that means that the statement A implies B is also false. And now as I read it, I guess, I'm stating the obvious but let's try it with an example. Let's say we have an assumption that the State of California is the only State with a city named Los Angeles. So we setup two tests; 'A' (City=Los Angeles) and 'B' (State=CA). Now, based on our assumption, whenever we find an address with city of Los Angeles, we can infer that the State must be California (if A is true it will imply that B is also true). However, if we were not correct in making that earlier assumption, and there is another State which has a city of Los Angeles, in that case the test (~A V B) will result in false thus proving that the assumption "City=Los Angeles implies State=CA" is wrong.  |
| Whether the below matrices are in row-echelon form / reduced row-echelon form Posted: 19 Sep 2021 07:55 PM PDT I would like to ask whether the following matrix is a row echelon matrix (and reduced row-echelon matrix). The below are $n\times1$ matrices, where $n\in \mathbb{R}$ and $n \geq 2$ Here are my opinions. I would like for my opinions be confirmed (or to be countered if mine is wrong): (For (d), it is inspired by " Is a $1 \times n$ matrix already in echelon form? ".)  |
| Why does logarithmic scale give linearization here? [Low-pass-filter] Posted: 19 Sep 2021 08:01 PM PDT I'm given the formula for describing the voltage ratio bewerten in- and output for a low-pass filter (neglecting physical aspects): $\dfrac{U_{out}}{U_{in}} = \dfrac{1}{\sqrt{\left(\frac{f}{f_G}\right)^2+1}}$. Here $f$ is the variable frequency, $f_G$ a constant depending on the assemble. Plotting that "function", gives a graph, decreasing for higher frequency. Don't mind the physics The mathematical thing I do not understand: if I plot in double-logarithmic scale, say $\log\left(\dfrac{U_{out}}{U_{in}}\right) = \log\left(\dfrac{1}{\sqrt{\left(\frac{f}{f_G}\right)^2+1}}\right)$ - why does this lead to a linearization with higher frequencies? I'd just aspect this happening for $\textsf{exponential}$ functions.  |
| Why are the roots of $y^2 - iy + 2 = 0$ not complex conjugates? Posted: 19 Sep 2021 08:17 PM PDT So I have the equation $y^2 - iy + 2 = 0$. I use the quadratic formula to solve it, and the solutions I got are $2i$ and $-i$. Why aren't the solutions here complex conjugates? Why didn't the quadratic formula produce complex conjugates as solutions in this case?  |
| Correct way to calculate a matrix trace with negative values Posted: 19 Sep 2021 08:04 PM PDT I have a 10x10 symmetrical variance-covariance matrix, such that the variances for 10 vectors are on the main diagonal and the covariance between all vectors are on the off-diagonals. I want to quantify the amount of variance in total. I can easily take the matrix trace as the sum of the eigenvalues on the main diagonal. However, the matrix can be split into meaningful (biologically meaningful, in my case) sub-matrices: 4 submatrices, 5x5 each, in each corner of the original matrix. If I then want to quantify the variation within each sub-matrix using the matrix trace, I run into some trouble with the top-right/bottom-left sub-matrices. These are formed of covariance estimates and are therefore not necessarily positive. My question is, what is the correct way to calculate the matrix trace here? If I sum the eigenvalues, I will have some negative values subtracting from the total, so should I use absolute values? Is the matrix trace the best method to use here or is there a more appropriate way of summarising the amount of variance in the sub-matrices? Any guidance would be gratefully received. Thanks, Fiona  |
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