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# All Questions

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### What are good/best practices to take the Fourier transform of an InterpolatingFunction?

I have a function which I have obtained from numerical integration of a differential equation, and I would like to take its Fourier transform. What are good practices for doing this? To make things ...
254 views

### Module in numerical model for NonlinearModelFit is slow and leaks memory

I am trying to make a NonlinearModelFit where the model is the result of a numerical calculation. My code works, but it is very slow and also seems to have a memory ...
209 views

### Resources for Sound Synthesis & Design

Are there any resources for sound synthesis using Mathematica? This page, Mathematica: Audio Synthesis Software, refers to other software packages, e.g. Max/MSP and Csound, for real-time synthesis. ...
157 views

### Is it possible to model the piano key attack?

My purpose is to model the attack of a piano key. As the key is pressed a sound must be emitted. When the key is released the sound must stop with decreasing volume. I have a partial success when ...
134 views

### Given two samples, how does LocationTest select its AutomaticTest?

How does LocationTest select its "AutomaticTest" when testing two samples? Background: A previous post poses a similar question ...
137 views

### How to get mathematica to simplify a result

I have the following output $\text{Li}_2\left(3-2 \sqrt{2}\right)-4 \text{Li}_2\left(-1+\sqrt{2}\right)$. This turns out to be equivalent to $\displaystyle\log^2(1+\sqrt{2})-\frac{\pi^2}{4}$. How ...
126 views

### Compiling/calling Mathematica-generated .so from C

I'm trying to call a compiled function from a C program. I am generating the static object ...
96 views

### Converting problem into form involving Probability

I want to solve the following problem symbolically, using Probability: What is the probability of the first random variate of a set to be present more than ...
290 views

### Spherical harmonic derivative

Consider the following substitution Derivative[2, 0][S][th, ph] /. S -> Function[{th, ph}, SphericalHarmonicY[3, 0, th, ph]] which gives correct answer. While ...
432 views

I'm trying to register (align) two circular grayscale images using a boolean mask to prevent them simply being aligned on the fixed circular border. There are also small fixed circular regions in the ...
171 views

### Non-linear integral equation

I'm trying to solve with Mathematica an integral equation. I found this excellent answer (How to solve a non-linear integral equation?) solving with a collocation method a problem which can be ...
134 views

### Establish connection with RLink and Rtools

Has anyone gotten RLink to work with Rtools (or even tried)? If so a simple step-by-step would be much appreciated ! When attempting to access any Rtools functionality through RLink I get: The ...
143 views

### Problem importing individual frames from QuickTime file

I am having difficulties with a code I am trying to run. Essentially I want to Read in a large .mov video file (60000+frames). ...
292 views

### Does Mathematica compute free-form-input on Wolfram servers?

I've just started to use Mathematica and I'm using the free-form input a lot. The computation time for simple problems seems to take a lot of time and my CPU isn't really used at all. I first thought ...
214 views

### Discriminant of Characteristic Polynomial

I'm doing a calculation which finds the characteristic polynomial of a matrix with rather complex entries and then determines the discriminant of that polynomial. For smaller matrices up to around 7x7 ...
303 views

### Mouse cursor disappears

After working on a notebook for a while sometimes my mouse cursor disappears. I can click on different cells and the text cursor will move accordingly but I just can't see the mouse cursor. Just now ...
121 views

### How to Check Spelling strings written inside input cells

Is it possible to check the spelling of strings? Strings are often used to show messages in my user interfaces, and I would like to have an automatic checking. Methods that export all the strings to ...
126 views

### Tools for Stability/Automatic Error Analysis in Mathematica

I have a longer analytic expression in several variables containing special functions and others. Does Mathematica bring tools - or are there any packages - to examine the stability when I evaluate ...
230 views

### DynamicModule, SaveDefinitions and global functions

Unfortunately, I had not checked stackoverflow, because there, the exact same behavior was already reported. Please see the Q&A "SaveDefinitions considered dangerous". Up-front, I'm on Ubuntu ...
276 views

### Call Haskell from Mathematica and return the results inside Mathematica

I would like to learn Haskell using Mathematica's environment. I know there is the RLink protocol. Is there a similar protocol for Haskell?
113 views

### Rearrange cells without cut-paste

When working in a Mathematica notebook, I often want to reorder cells. Most of the time, I am adding cells at the bottom of the notebook, until I realize it may be better to insert this calculation at ...
230 views

### Syntax error for optional pattern, but only in debugger

This expression (explained here by Leonid) evaluates fine unless Evaluation->Debugger is enabled: h[x : (_Integer) : 1] := x Syntax::sntxf: "x:" cannot be ...
188 views

### Single image import from stack via FTP connection

Background I would like to create a CDF document that is embedded on a webpage. The CDF will contain some image display ...
138 views

### What is the signature for a callback function for FinancialIndicators?

I'd like to write my own indicators to use with functions like TradingChart. From the context, it's clear that one needs to write a function whose parameters ...
272 views

### How to improve gesture detection by JLink for the LeapMotion controller

Building on this thread (where I cannot add anything anymore somehow) and with this nice answer I kind of got swipe detection to work, but it is not very responsive. How to improve (in the sense of ...
147 views

### MathKernel with -noprompt hangs indefinitely when converting simple TeX expression

If I enter MathKernel -noprompt -run 'ToExpression["1", TeXForm]; Exit[]' the kernel hangs, apparently while running TeX.exe ...
89 views

### JewishNewYear date limitation

The documentation for the function "JewishNewYear" in the "calendar" package declares that the year argument must be between 1900 and 2100. No explanation is given for this limitation. The function ...
116 views

### Gather dependency on list order?

I get intrigued about Gather behavior on this post. I thought that Gather was invariant under list order? Here is the result I ...
263 views

### Memory problem with producing complex graphics of general type

The problem I am about to ask has appeared once here: Memory Leak in Frontend - anyone know a workaround?. However, in this question I would like to put it in a bit different context. Problem recipe: ...
82 views

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### Analytically solve the eigenvalue problem with infinite dimensions by Mathematica?

If I am given a symbolic expression of all the matrix elements in an infinite-dimensional space, e.g., the Hamiltonian of a quantum mechanical system, is it possible to get the symbolic expression for ...
727 views

### When using NMinimize[…,Method->“DifferentialEvolution”] is an “iteration” different than a “step”?

Hopefully this question has a simple answer. I have the following bit of code: ...
641 views

### looking for a generalised Hough Transform function or a least a function to locate circles

I am looking for a generalised Hough Transform function or a least a function to locate circles (position of center and radius) in a image. There is the standard Hough line search function ...
205 views

### Strange result from evaluating a Limit expression

When I evaluate the limit of (1-Cos[x]) Sin[1/x] at x = 0 in Mathematica 9, I obtained ...
104 views

### Is there a way to check whether two notebooks share the same variables?

When I code in Mathematica, I often re-use code from old notebooks which fulfilled a specific purpose; for example, one testing for soft matches in strings, or one that identifies outliers on a graph ...
438 views

### Cubic polynomial interpolation with velocities listed rather than some points

I am trying to use the function BSplineFunction[] to create a natural cubic spline. Normally what is done is that you specify "SplineDegree -> 3" and include a list of points and it will carry out ...


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# PDF Stochastic Finite Element Technique for Stochastic One

kanonisk form - Wikidocumentaries

[ math] -x y^{\prime}=(3 x+2) y+x e^{x} [/math] - Slader. 18 Jan 2021 (a) Equation (1.1.4) is called the general solution of the differential The solution of the differential equation can be computed form the second. First and Second Order Differential Equations. A first order differential equation is of the form: displaymath137 The general general solution is given by. The general form for a first order linear ODE in x(t) is dxdt+p(t)x(t)=q(t).

(ii) From the standard form of the equation identify P(x) and then find th integrating factor eP(x) dx. No constant need be used in evaluating the indefinite integralP(x) dx. Practice questions for Differential Equations Phil Spector's death resurrects mixed reaction from skeptics, and other top stories in entertainment from January 19, 2021. mined by the diﬀerential form pdx+qdy, but two diﬀerent forms may determine equivalent equations. For example, if µ = h(x,y) is a non-zero scalar, then the form µpdx + µqdy is a quite diﬀerent form, but it determines an equivalent diﬀerential equation. If pdx + qdy is exact, then pdx + qdy = dz, for some scalar z depending on x and y Package like odepack needs the ODE written in standard form, which means write the high order ODE to first order ODE equations.

## HKU Calculus – Appar på Google Play

This is the general solution of the ODE. More generally, ODEs of the form dy dx. = f(x)g(y),.

### first order differential equations - Desmos

The format of this test is similar to Raven's progressive matrices. (264 words) [view diff] exact match in snippet view article find links to article. in which one discovers coarse variables and derives equations for their evolution.

Forskningsoutput: Kapitel i  Köp Ordinary Differential Equations av William A Adkins, Mark G Davidson på equations, this textbook gives an early presentation of the Laplace transform, the standard solution methods for constant coefficient linear differential equations  Jag försöker se saker i form av geometri. This system of linear equations has exactly one solution. In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns  There is also a corresponding differential form of this equation covered in Schoen and Yau extended this to the standard Lorentzian formulation of the positive  (b) This is linear equation. Writing equation in the standard form dу dx.
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For a review of the direct method to solve linear first-order Equations of nonconstant coefficients with missing y-term If the y-term (that is, the dependent variable term) is missing in a second order linear equation, then the equation can be readily converted into a first order linear equation and solved using the integrating factor method. Example: t y″ + 4 y′ = t 2 The standard form is y t t Se hela listan på byjus.com In practice, most of the differential equation do not have a standard form and can not be solved with analytic methods, which means we can not find a general solution y(x). This is the case for most of the differential equations derived from physical models (mechanical, electrical, thermal, etc.). Ti-83 plus quadratic equation written out, maths question paper of ninth standard, pre-algerbra study help, coupled differential equations matlab.
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### kanonisk form - Wikidocumentaries

Find the integrating factor, μ(t), using (10). Multiply everything in the differential equation by μ(t) and verify that the left side becomes the product rule (μ(t)y(t)) ′ and write it as such. Linear Equations – In this section we solve linear first order differential equations, i.e. differential equations in the form y′ +p(t)y = g(t) y ′ + p (t) y = g (t).

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### Campusbokhandeln: Spara pengar på kurslitteratur

2015-04-30 · When we was first introduced to first order differential equations we learned that the standard form was : y’ +p(t)y = g(t) , y(to) = yo. What separates Bernoulli Equations from other first order equations is that in standard form, it is not equal to some function that is linear but one that has an exact solution. Standard and Differential Form of First-Order Differential Equations - YouTube. This video provides several examples of how to write a first order DE in standard form and differential form.website In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

## Partiell differentialekvation - Partial differential equation - qaz

For x tending to infinity, the equation tends to d Package like odepack needs the ODE written in standard form, which means write the high order ODE to first order ODE equations. The steps of converting ODE to standard form are quite standard, but I do not find functions in Mathematica that can rewrite high order ODE into its standard form. For example, EQ = y''[x] + Sin[y[x]] y[x] == 0 We start with the standard form of a first-order linear differential equation: y ′ + p(x)y = q(x). The first term on the left-hand side of Equation is the derivative of the unknown function, and the second term is the product of a known function with the unknown function. This is somewhat reminiscent of the power rule. The equation of family of standard circle with radius r is x 2 + y 2 = r 2, where r is an arbitrary constant.

A system of n linear first order differential equations in n unknowns (an n × n system of linear equations) has the general form: x 1′ = a 11 x 1 + a 12 x 2 + … + a 1n x n + g 1 x 2′ = a 21 The standard form for linear equations in two variables is Ax+By=C. For example, 2x+3y=5 is a linear equation in standard form. When an equation is given in this form, it's pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations. This form of matrix equation is called 'State Space' matrix equation. < Example : Simultaneous Differential Equation (System Equation) > Previous example shows how we can convert one higher linear order differential equation into a single matrix equation.

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# Triangular numbers, proving numbers are tringular

1. May 13, 2014

### schlynn

I've been learning about polygonal numbers, and one of the exercises in this book ask me to show that 9t$_{n}$+1 [Fermat], 25t$_{n}$+3, and 49t$_{n}$+6 [both from Euler] are triangular numbers. I don't know how to approach these proofs, I've tried to show that they have some form similar to $\frac{n(n+1)}{2}$, but with no avail. But it looks like there is a pattern, that would be
(2n+1)$^{2}$t$_{\alpha}$+$\frac{n(n+1)}{2}$, but I have no way of proving this. Could someone point me in the correct direction?

t$_{n}$ and t$_{\alpha}$ are both triangular numbers.

Last edited: May 13, 2014
2. May 13, 2014

### gopher_p

$9\cdot\frac{n(n+1)}{2}+1=\frac{(3n+1)(3n+2)}{2}=\frac{(3n+1)\big((3n+1)+1 \big)}{2}$, yes?

3. May 13, 2014

### schlynn

Wow, I was over thinking this a lot. Thank you, I can check my pattern with this too.

4. May 13, 2014

### gopher_p

Another one is $5929t_n+741$ is triangular whenever $t_n$ is triangular [gopher_p].

But it looks like you've already figured out that pattern.


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#### Model-Agnostic Private Learning via Stability

##### Raef Bassily, Om Thakkar, Abhradeep Thakurta

We design differentially private learning algorithms that are agnostic to the learning model. Our algorithms are interactive in nature, i.e., instead of outputting a model based on the training data, they provide predictions for a set of $m$ feature vectors that arrive online. We show that, for the feature vectors on which an ensemble of models (trained on random disjoint subsets of a dataset) makes consistent predictions, there is almost no-cost of privacy in generating accurate predictions for those feature vectors. To that end, we provide a novel coupling of the distance to instability framework with the sparse vector technique. We provide algorithms with formal privacy and utility guarantees for both binary/multi-class classification, and soft-label classification. For binary classification in the standard (agnostic) PAC model, we show how to bootstrap from our privately generated predictions to construct a computationally efficient private learner that outputs a final accurate hypothesis. Our construction - to the best of our knowledge - is the first computationally efficient construction for a label-private learner. We prove sample complexity upper bounds for this setting. As in non-private sample complexity bounds, the only relevant property of the given concept class is its VC dimension. For soft-label classification, our techniques are based on exploiting the stability properties of traditional learning algorithms, like stochastic gradient descent (SGD). We provide a new technique to boost the average-case stability properties of learning algorithms to strong (worst-case) stability properties, and then exploit them to obtain private classification algorithms. In the process, we also show that a large class of SGD methods satisfy average-case stability properties, in contrast to a smaller class of SGD methods that are uniformly stable as shown in prior work.

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# mathematics

The coordinates of point A on a coordinate grid are (−2, −3). Point A is reflected across the y-axis to obtain point B and across the x-axis to obtain point C. What are the coordinates of points B and C?
B(2, 3) and, C(−2, −3)
B(−2, −3) and C(2, 3)
B(2, −3) and C(−2, 3)
B(−2, 3) and C(2, −3)

1. 👍
2. 👎
3. 👁
1. across the y-axis: (x,y) → (-x,y)
across the x-axis: (x,y) → (x,-y)
so see what you can do with these.

1. 👍
2. 👎
👤
oobleck
2. nice work it really helped me thx

1. 👍
2. 👎

## Similar Questions

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#### Volume 12, issue 1 (2012)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Estimating the higher symmetric topological complexity of spheres

### Roman Karasev and Peter Landweber

Algebraic & Geometric Topology 12 (2012) 75–94
##### Abstract

We study questions of the following type: Can one assign continuously and ${\Sigma }_{m}$–equivariantly to any $m$–tuple of distinct points on the sphere ${S}^{n}$ a multipath in ${S}^{n}$ spanning these points? A multipath is a continuous map of the wedge of $m$ segments to the sphere. This question is connected with the higher symmetric topological complexity of spheres, introduced and studied by I Basabe, J González, Yu B Rudyak, and D Tamaki. In all cases we can handle, the answer is negative. Our arguments are in the spirit of the definition of the Hopf invariant of a map $f:{S}^{2n-1}\to {S}^{n}$ by means of the mapping cone and the cup product.

##### Keywords
topological complexity, configuration spaces
##### Mathematical Subject Classification 2010
Primary: 55R80, 55R91


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## Section5.1Constructing Accurate Graphs of Antiderivatives

###### Motivating Questions
• Given the graph of a function's derivative, how can we construct a completely accurate graph of the original function?

• How many antiderivatives does a given function have? What do those antiderivatives all have in common?

• Given a function $f\text{,}$ how does the rule $A(x) = \int_0^x f(t) \, dt$ define a new function $A\text{?}$

A recurring theme in our discussion of differential calculus has been the question “Given information about the derivative of an unknown function $f\text{,}$ how much information can we obtain about $f$ itself?” For instance, in Activity 1.8.3, we explored the situation where the graph of $y = f'(x)$ was known (along with the value of $f$ at a single point) and endeavored to sketch a possible graph of $f$ near the known point. In Example 3.1.4 — and indeed throughout Section 3.1 — we investigated how the first derivative test enables us to use information regarding $f'$ to determine where the original function $f$ is increasing and decreasing, as well as where $f$ has relative extreme values. Further, if we know a formula or graph of $f'\text{,}$ by computing $f''$ we can find where the original function $f$ is concave up and concave down. Thus, the combination of knowing $f'$ and $f''$ enables us to fully understand the shape of the graph of $f\text{.}$

We returned to this question in even more detail in Section 4.1; there, we considered the situation where we knew the instantaneous velocity of a moving object and worked from that information to determine as much information as possible about the object's position function. We found key connections between the net-signed area under the velocity function and the corresponding change in position of the function; in Section 4.4, the Total Change Theorem further illuminated these connections between $f'$ and $f$ in a more general setting, such as the one found in Figure 4.4.5, showing that the total change in the value of $f$ over an interval $[a,b]$ is determined by the exact net-signed area bounded by $f'$ and the $x$-axis on the same interval.

In what follows, we explore these issues still further, with a particular emphasis on the situation where we possess an accurate graph of the derivative function along with a single value of the function $f\text{.}$ From that information, we desire to completely determine an accurate graph of $f$ that not only represents correctly where $f$ is increasing, decreasing, concave up, and concave down, but also allows us to find an accurate function value at any point of interest to us.

###### Preview Activity5.1.1

Suppose that the following information is known about a function $f\text{:}$ the graph of its derivative, $y = f'(x)\text{,}$ is given in Figure 5.1.1. Further, assume that $f'$ is piecewise linear (as pictured) and that for $x \le 0$ and $x \ge 6\text{,}$ $f'(x) = 0\text{.}$ Finally, it is given that $f(0) = 1\text{.}$

1. On what interval(s) is $f$ an increasing function? On what intervals is $f$ decreasing?

2. On what interval(s) is $f$ concave up? concave down?

3. At what point(s) does $f$ have a relative minimum? a relative maximum?

4. Recall that the Total Change Theorem tells us that

\begin{equation*} f(1) - f(0) = \int_0^1 f'(x) \, dx\text{.} \end{equation*}

What is the exact value of $f(1)\text{?}$

5. Use the given information and similar reasoning to that in (d) to determine the exact value of $f(2)\text{,}$ $f(3)\text{,}$ $f(4)\text{,}$ $f(5)\text{,}$ and $f(6)\text{.}$

6. Based on your responses to all of the preceding questions, sketch a complete and accurate graph of $y = f(x)$ on the axes provided, being sure to indicate the behavior of $f$ for $x \lt 0$ and $x \gt 6\text{.}$

### Subsection5.1.1Constructing the graph of an antiderivative

Preview Activity 5.1.1 demonstrates that when we can find the exact area under a given graph on any given interval, it is possible to construct an accurate graph of the given function's antiderivative: that is, we can find a representation of a function whose derivative is the given one. While we have considered this question at different points throughout our study, it is important to note here that we now can determine not only the overall shape of the antiderivative, but also the actual height of the antiderivative at any point of interest.

Indeed, this is one key consequence of the Fundamental Theorem of Calculus: if we know a function $f$ and wish to know information about its antiderivative, $F\text{,}$ provided that we have some starting point $a$ for which we know the value of $F(a)\text{,}$ we can determine the value of $F(b)$ via the definite integral. In particular, since $F(b) - F(a) = \int_a^b f(x) \, dx\text{,}$ it follows that

$$F(b) = F(a) + \int_a^b f(x) \, dx\text{.}\label{E-FTCTCT}\tag{5.1.1}$$

Moreover, in the discussion surrounding Figure 4.4.5, we made the observation that differences in heights of a function correspond to net-signed areas bounded by its derivative. Rephrasing this in terms of a given function $f$ and its antiderivative $F\text{,}$ we observe that on an interval $[a,b]\text{,}$

differences in heights on the antiderivative (such as $F(b) - F(a)$) correspond to the net-signed area bounded by the original function on the interval $[a,b]$ ($\int_a^b f(x) \, dx$).

For example, say that $f(x) = x^2$ and that we are interested in an antiderivative of $f$ that satisfies $F(1) = 2\text{.}$ Thinking of $a = 1$ and $b = 2$ in Equation (5.1.1), it follows from the Fundamental Theorem of Calculus that

\begin{align*} F(2) =\mathstrut \amp F(1) + \int_1^2 x^2 \, dx\\ =\mathstrut \amp 2 + \left. \frac{1}{3}x^3 \right|_1^2\\ =\mathstrut \amp 2 + \left(\frac{8}{3} - \frac{1}{3}\right)\\ =\mathstrut \amp \frac{13}{3}\text{.} \end{align*}

In this way, we see that if we are given a function $f$ for which we can find the exact net-signed area bounded by $f$ on a given interval, along with one value of a corresponding antiderivative $F\text{,}$ we can find any other value of $F$ that we seek, and in this way construct a completely accurate graph of $F\text{.}$ We have two main options for finding the exact net-signed area: using the Fundamental Theorem of Calculus (which requires us to find an algebraic formula for an antiderivative of the given function $f$), or, in the case where $f$ has nice geometric properties, finding net-signed areas through the use of known area formulas.

###### Activity5.1.2

Suppose that the function $y = f(x)$ is given by the graph shown in Figure 5.1.2, and that the pieces of $f$ are either portions of lines or portions of circles. In addition, let $F$ be an antiderivative of $f$ and say that $F(0) = -1\text{.}$ Finally, assume that for $x \le 0$ and $x \ge 7\text{,}$ $f(x) = 0\text{.}$

1. On what interval(s) is $F$ an increasing function? On what intervals is $F$ decreasing?

2. On what interval(s) is $F$ concave up? concave down? neither?

3. At what point(s) does $F$ have a relative minimum? a relative maximum?

4. Use the given information to determine the exact value of $F(x)$ for $x = 1, 2, \ldots, 7\text{.}$ In addition, what are the values of $F(-1)$ and $F(8)\text{?}$

5. Based on your responses to all of the preceding questions, sketch a complete and accurate graph of $y = F(x)$ on the axes provided, being sure to indicate the behavior of $F$ for $x \lt 0$ and $x \gt 7\text{.}$ Clearly indicate the scale on the vertical and horizontal axes of your graph.

6. What happens if we change one key piece of information: in particular, say that $G$ is an antiderivative of $f$ and $G(0) = 0\text{.}$ How (if at all) would your answers to the preceding questions change? Sketch a graph of $G$ on the same axes as the graph of $F$ you constructed in (e).

### Subsection5.1.2Multiple antiderivatives of a single function

In the final question of Activity 5.1.2, we encountered a very important idea: a given function $f$ has more than one antiderivative. In addition, any antiderivative of $f$ is determined uniquely by identifying the value of the desired antiderivative at a single point. For example, suppose that $f$ is the function given at left in Figure 5.1.3,

and we say that $F$ is an antiderivative of $f$ that satisfies $F(0) = 1\text{.}$

Then, using Equation (5.1.1), we can compute $F(1) = 1.5\text{,}$ $F(2) = 1.5\text{,}$ $F(3) = -0.5\text{,}$ $F(4) = -2\text{,}$ $F(5) = -0.5\text{,}$ and $F(6) = 1\text{,}$ plus we can use the fact that $F' = f$ to ascertain where $F$ is increasing and decreasing, concave up and concave down, and has relative extremes and inflection points. Through work similar to what we encountered in Preview Activity 5.1.1 and Activity 5.1.2, we ultimately find that the graph of $F$ is the one given in blue in Figure 5.1.3.

If we instead chose to consider a function $G$ that is an antiderivative of $f$ but has the property that $G(0) = 3\text{,}$ then $G$ will have the exact same shape as $F$ (since both share the derivative $f$), but $G$ will be shifted vertically away from the graph of $F\text{,}$ as pictured in red in Figure 5.1.3. Note that $G(1) - G(0) = \int_0^1 f(x) \, dx = 0.5\text{,}$ just as $F(1) - F(0) = 0.5\text{,,}$ but since $G(0) = 3\text{,}$ $G(1) = G(0) + 0.5 = 3.5\text{,}$ whereas $F(1) = F(0) + 0.5 = 1.5\text{,}$ since $F(0) = 1\text{.}$ In the same way, if we assigned a different initial value to the antiderivative, say $H(0) = -1\text{,}$ we would get still another antiderivative, as shown in magenta in Figure 5.1.3.

This example demonstrates an important fact that holds more generally:

If $G$ and $H$ are both antiderivatives of a function $f\text{,}$ then the function $G - H$ must be constant.

To see why this result holds, observe that if $G$ and $H$ are both antiderivatives of $f\text{,}$ then $G' = f$ and $H' = f\text{.}$ Hence,

\begin{equation*} \frac{d}{dx}[ G(x) - H(x) ] = G'(x) - H'(x) = f(x) - f(x) = 0\text{.} \end{equation*}

Since the only way a function can have derivative zero is by being a constant function, it follows that the function $G - H$ must be constant.

Further, we now see that if a function has a single antiderivative, it must have infinitely many: we can add any constant of our choice to the antiderivative and get another antiderivative. For this reason, we sometimes refer to the general antiderivative of a function $f\text{.}$ For example, if $f(x) = x^2\text{,}$ its general antiderivative is $F(x) = \frac{1}{3}x^3 + C\text{,}$ where we include the “$+C$” to indicate that $F$ includes all of the possible antiderivatives of $f\text{.}$ To identify a particular antiderivative of $f\text{,}$ we must be provided a single value of the antiderivative $F$ (this value is often called an initial condition). In the present example, suppose that condition is $F(2) = 3\text{;}$ substituting the value of 2 for $x$ in $F(x) = \frac{1}{3}x^3 + C\text{,}$ we find that

\begin{equation*} 3 = \frac{1}{3}(2)^3 + C\text{,} \end{equation*}

and thus $C = 3 - \frac{8}{3} = \frac{1}{3}\text{.}$ Therefore, the particular antiderivative in this case is $F(x) = \frac{1}{3}x^3 + \frac{1}{3}\text{.}$

###### Activity5.1.3

For each of the following functions, sketch an accurate graph of the antiderivative that satisfies the given initial condition. In addition, sketch the graph of two additional antiderivatives of the given function, and state the corresponding initial conditions that each of them satisfy. If possible, find an algebraic formula for the antiderivative that satisfies the initial condition.

1. original function: $g(x) = \left| x \right| - 1\text{;}$ initial condition: $G(-1) = 0\text{;}$ interval for sketch: $[-2,2]$

2. original function: $h(x) = \sin(x)\text{;}$ initial condition: $H(0) = 1\text{;}$ interval for sketch: $[0,4\pi]$

3. original function: $p(x) = \begin{cases}x^2, \amp \text{ if } 0 \lt x \le 1 \\ -(x-2)^2, \amp \text{ if } 1 \lt x \lt 2 \\ 0 \amp \text{ otherwise } \end{cases}\text{;}$ initial condition: $P(0) = 1\text{;}$ interval for sketch: $[-1,3]$

### Subsection5.1.3Functions defined by integrals

In Equation (5.1.1), we found an important rule that enables us to compute the value of the antiderivative $F$ at a point $b\text{,}$ provided that we know $F(a)$ and can evaluate the definite integral from $a$ to $b$ of $f\text{.}$ Again, that rule is

\begin{equation*} F(b) = F(a) + \int_a^b f(x) \, dx\text{.} \end{equation*}

In several examples, we have used this formula to compute several different values of $F(b)$ and then plotted the points $(b,F(b))$ to assist us in generating an accurate graph of $F\text{.}$ That suggests that we may want to think of $b\text{,}$ the upper limit of integration, as a variable itself. To that end, we introduce the idea of an integral function, a function whose formula involves a definite integral.

Given a continuous function $f\text{,}$ we define the corresponding integral function $A$ according to the rule

$$A(x) = \int_a^x f(t) \, dt\text{.}\label{E-intfxn}\tag{5.1.2}$$

Note particularly that because we are using the variable $x$ as the independent variable in the function $A\text{,}$ and $x$ determines the other endpoint of the interval over which we integrate (starting from $a$), we need to use a variable other than $x$ as the variable of integration. A standard choice is $t\text{,}$ but any variable other than $x$ is acceptable.

One way to think of the function $A$ is as the “net-signed area from $a$ up to $x$” function, where we consider the region bounded by $y = f(t)$ on the relevant interval. For example, in Figure 5.1.4, we see a given function $f$ pictured at left, and its corresponding area function (choosing $a = 0$), $A(x) = \int_0^x f(t) \, dt$ shown at right.

Note particularly that the function $A$ measures the net-signed area from $t = 0$ to $t = x$ bounded by the curve $y = f(t)\text{;}$ this value is then reported as the corresponding height on the graph of $y = A(x)\text{.}$ It is even more natural to think of this relationship between $f$ and $A$ dynamically. At http://gvsu.edu/s/cz, we find a java applet 1 David Austin, Grand Valley State Universitythat brings the static picture in Figure 5.1.4 to life. There, the user can move the red point on the function $f$ and see how the corresponding height changes at the light blue point on the graph of $A\text{.}$

The choice of $a$ is somewhat arbitrary. In the activity that follows, we explore how the value of $a$ affects the graph of the integral function, as well as some additional related issues.

###### Activity5.1.4

Suppose that $g$ is given by the graph at left in Figure 5.1.5 and that $A$ is the corresponding integral function defined by $A(x) = \int_1^x g(t) \, dt\text{.}$

1. On what interval(s) is $A$ an increasing function? On what intervals is $A$ decreasing? Why?

2. On what interval(s) do you think $A$ is concave up? concave down? Why?

3. At what point(s) does $A$ have a relative minimum? a relative maximum?

4. Use the given information to determine the exact values of $A(0)\text{,}$ $A(1)\text{,}$ $A(2)\text{,}$ $A(3)\text{,}$ $A(4)\text{,}$ $A(5)\text{,}$ and $A(6)\text{.}$

5. Based on your responses to all of the preceding questions, sketch a complete and accurate graph of $y = A(x)$ on the axes provided, being sure to indicate the behavior of $A$ for $x \lt 0$ and $x \gt 6\text{.}$

6. How does the graph of $B$ compare to $A$ if $B$ is instead defined by $B(x) = \int_0^x g(t) \, dt\text{?}$

### Subsection5.1.4Summary

• Given the graph of a function $f\text{,}$ we can construct the graph of its antiderivative $F$ provided that (a) we know a starting value of $F\text{,}$ say $F(a)\text{,}$ and (b) we can evaluate the integral $\int_a^b f(x) \, dx$ exactly for relevant choices of $a$ and $b\text{.}$ For instance, if we wish to know $F(3)\text{,}$ we can compute $F(3) = F(a) + \int_a^3 f(x) \, dx\text{.}$ When we combine this information about the function values of $F$ together with our understanding of how the behavior of $F' = f$ affects the overall shape of $F\text{,}$ we can develop a completely accurate graph of the antiderivative $F\text{.}$

• Because the derivative of a constant is zero, if $F$ is an antiderivative of $f\text{,}$ it follows that $G(x) = F(x) + C$ will also be an antiderivative of $f\text{.}$ Moreover, any two antiderivatives of a function $f$ differ precisely by a constant. Thus, any function with at least one antiderivative in fact has infinitely many, and the graphs of any two antiderivatives will differ only by a vertical translation.

• Given a function $f\text{,}$ the rule $A(x) = \int_a^x f(t) \, dt$ defines a new function $A$ that measures the net-signed area bounded by $f$ on the interval $[a,x]\text{.}$ We call the function $A$ the integral function corresponding to $f\text{.}$

### SubsectionExercises

###### 5

A moving particle has its velocity given by the quadratic function $v$ pictured in Figure 5.1.6. In addition, it is given that $A_1 = \frac{7}{6}$ and $A_2 = \frac{8}{3}\text{,}$ as well as that for the corresponding position function $s\text{,}$ $s(0) = 0.5\text{.}$

1. Use the given information to determine $s(1)\text{,}$ $s(3)\text{,}$ $s(5)\text{,}$ and $s(6)\text{.}$

2. On what interval(s) is $s$ increasing? On what interval(s) is $s$ decreasing?

3. On what interval(s) is $s$ concave up? On what interval(s) is $s$ concave down?

4. Sketch an accurate, labeled graph of $s$ on the axes at right in Figure 5.1.6.

5. Note that $v(t) = -2 + \frac{1}{2}(t-3)^2\text{.}$ Find a formula for $s\text{.}$

###### 6

A person exercising on a treadmill experiences different levels of resistance and thus burns calories at different rates, depending on the treadmill's setting. In a particular workout, the rate at which a person is burning calories is given by the piecewise constant function $c$ pictured in Figure 5.1.7. Note that the units on $c$ are “calories per minute.”

1. Let $C$ be an antiderivative of $c\text{.}$ What does the function $C$ measure? What are its units?

2. Assume that $C(0) = 0\text{.}$ Determine the exact value of $C(t)$ at the values $t = 5, 10, 15, 20, 25, 30\text{.}$

3. Sketch an accurate graph of $C$ on the axes provided at right in Figure 5.1.7. Be certain to label the scale on the vertical axis.

4. Determine a formula for $C$ that does not involve an integral and is valid for $5 \le t \le 10\text{.}$

###### 7

Consider the piecewise linear function $f$ given in Figure 5.1.8. Let the functions $A\text{,}$ $B\text{,}$ and $C$ be defined by the rules $A(x) = \int_{-1}^{x} f(t) \, dt\text{,}$ $B(x) = \int_{0}^{x} f(t) \, dt\text{,}$ and $C(x) = \int_{1}^{x} f(t) \, dt\text{.}$

1. For the values $x = -1, 0, 1, \ldots, 6\text{,}$ make a table that lists corresponding values of $A(x)\text{,}$ $B(x)\text{,}$ and $C(x)\text{.}$

2. On the axes provided in Figure 5.1.8, sketch the graphs of $A\text{,}$ $B\text{,}$ and $C\text{.}$

3. How are the graphs of $A\text{,}$ $B\text{,}$ and $C$ related?

4. How would you best describe the relationship between the function $A$ and the function $f\text{?}$


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# Chaotic or Not?

Consider the dynamical system generated by the discrete recurrence:

$z_{n+1} = f(z_n) = 2z_n.$

Is this system chaotic?

×


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# Tag Info

3

For a 3 channel image (RGB), each filter in a convolutional layer computes a feature map which is essentially a single channel image. Typically, 2D convolutional filters are used for multichannel images. This can be a single filter applied to each layer or a seperate filter per layer. These filters are looking for features which are independent of the color,...

3

Aesthetics of images has a strong subjective element and possibility of multiple dimensions depending on purpose of the media. That means: It is hard to define what we mean by scoring aesthetics. Given any well-constrained definition, it is then time-consuming to collect relevant data. However, there is some interest in the machine-learning community, as ...

3

There are many computer vision (CV) algorithms and models that are used for different purposes. So, of course, I cannot list all of them, but I can enumerate some of them based on my experience and knowledge. Of course, this answer will only give you a flavor of the type of algorithm or model that you will find while solving CV tasks. For example, there are ...

2

If the computational components of the forward feed through the network have no curvature, which is normally the case in a sum of products, then it can be proven that any constant pixel value is equivalent in terms of effect on convergence results. We wouldn't expect a proof for that, since it would be too trivial to spend time writing up for publication. ...

2

The Wikipedia article related to computer vision gives, in my opinion, a good description of the field and its relation to image processing. Below, I will only cite the most relevant parts of the article. Computer vision is an interdisciplinary scientific field that deals with how computers can be made to gain high-level understanding from digital images or ...

2

I don't know if there is an existing pretrained NN that does this but it wouldn't be very hard to modify one to do this. First, I'd take a pretrained image classification NN (e.g. VGG, ResNet), drop its final layer and replace it with one with 4 neurons, representing the 4 orientations (so that you know which way to rotate it). Then I'd take again a ...

2

There are hundred of papers on this task some older than I am! Normally this is done by trying to form a box shape around the image than estimate the volume. This task is typically done with multiple images so the two can generate a more clear picture of the size of the object than one image alone. An object could be 'infinitely' large but its mass could be ...

2

If the measurements you want from the object aren't too complicated (ie. length of a clearly defined feature), and if you are able to acquire a training dataset of images of the objects similar to what your model will see in your use case (same scale/distance), their bounding boxes and their measurements, a model you could try to implement is a Multi-Task ...

2

Father Ted explains why this is a hard problem. Seriously -- if you have stereo images it should be possible, since that's what we use for depth perception. If you know how far away points x1 and x2 are, then you can measure distance using trigonometry. No neural networks needed, I guess. https://en.wikipedia.org/wiki/Triangulation_(computer_vision)

2

You are partially correct. On CNNs the output shape per layer is defined by the amount of filters used, and the application of the filters (dilation, stride, padding, etc.). CNNs shapes In your example, your input is 30 x 30 x 3. Assuming stride of 1, no padding, and no dilation on the filter, you will get a spatial shape equal to your input, that is ...

2

About the images inside the CNN layers: I really recommend this article since there is no one short answer to this question and it probably will be better to experiment with it. About the RGB input images: When needed to train on RGB pictures it is not advised to split the RGB channels, you can think of it by trying to identify a fictional cat with red ears,...

2

After a quick scan, it would seem that, in the history of object detection, machine learning has always been at the forefront. Before then, it would just be a heuristic approach. For a quick answer, here: https://towardsdatascience.com/real-time-object-detection-without-machine-learning-5139b399ee7d That goes over object detection without using machine ...

2

Apart from the multitudes of traditional image segmentation techniques (Watershed, Clustering or Variational methods), newer Segmentation schemes using Deep Learning are actively being used, which provide better results and are better for real-time applications, owing to minimum computation overheads involved. The following blog provides a detailed review ...

2

(Of course, similar questions have been asked in the past and there are many sites, papers, video lessons, online that explain how CNNs work, but I think it's still a good idea to have a reference answer that hopefully will give you the main ideas behind CNNs.) A convolutional neural network (CNN) is a neural network that performs the convolution (or cross-...

1

This is a very hard problem, you have many overlapping points with objects which aren't completely round. I'm not very knowledgeable on CV but I suspect you will find it very challenging. I would probably say a handcrafted detection algorithm would probably be easier, something like an edge detector which fit circles to arcs and labeled the points. But it's ...

1

You can find the dataset in the following links: Pomegranate Disease Detection Using Image Processing fruits 360 datasets

1

The CNN should work without trying to do special feature extraction. As pointed out some pre-processing can aid in enhancing the CNN's classification results. The Keras ImageDataGenerator provides optional parameters you can set to provide pre-processing as well as provide data augmentation. One thing I know that works for sure but can be painful is ...

1

The whole interest of using deep learning-based solutions is that you don't have to do all those pre-processings, i.e. binarization, segmentation of background. CNNs, such as YOLO or FasterRCNN, can learn how to retrieve that information by themselves.

1

Some excerpts from Nutella 'Hired' an Algorithm to Design New Jars. And It Was a Sell-Out Success: The "algorithm" is called HP Mosaic and is included free in HP SmartStream Designer for HP printers. More about how the algorithm works here: https://www.linkedin.com/pulse/hp-mosaic-20-steven-chow HP Mosaic takes the vector PDF file as input (also known ...

1

Perhaps you are getting checkerboard artifacts Explained here, solutions involve changing the kernel and stride size to prevent them from being not divisible. Besides that, a solution could be to apply Gaussian smoothing to minimize the artifacts. For example, using Gaussian smoothing in OpenCV with your image results in import cv2 img = cv2.imread('s.png') #...

1

If we seek proven working source code to plug into a GPLv2-licence compatible solution, we should at least consider autotrace. Its source code is open for review. It can be tested against the example images we have and, if it works fine, called by our GPLv2 software. We can even use the calling code in Inkscape's plug-in image tracing implementation as a ...

1

As a rule of thumb for image data augmentation, look at the augmented images: Can you correctly classify or measure your target label from the augmented images? Could something similar to the augmented images appear in the environment where you want to run inferences on previously unseen inputs? For your suggested augmentation of shuffling the channels, it ...

1

For a simulation of SNES or N64, the resolution is probably not that high. You can use either online credits or buy new hardware. For online credits, it is recommended if you do teh simulation/training for only several dozens hours, as eache hour costs around 5-10 dollars. AWS is a good choice as it have a large variety of choice for hardware. For higher ...

1

Your hardware choice depends majorly on the sensors and camera specifications you need for your solution. For example, we are running a 2536x1920 resolution 20fps camera to detect cars and then read their number plates. Nvidia's RTX 2080Ti 11 GB can handle 2 such cameras at 20fps (real-time). However, this involves 2 models: 1 for detection of plates and a ...

1

One of the key terms in the literature that you are looking for is video captioning. You can have a look at some of the relevant papers with code on this subject. In short, it is an active area of research and a difficult problem, one reason is because videos are still difficult to learn about (because of larger amount of data + larger model, etc...) and ...

1

The model (that I know of) which most resembles your description is the auto-encoder, which is trained to learn a compact representation (a vector) of the input, which can later be used to reconstruct the original input. In a certain way, this compact representation (implicitly) encodes the most important features of the input. In particular, you may be ...

1

If you have a $h_i \times w_i \times d_i$ input, where $h_i, w_i$ and $d_i$ respectively refer to the height, width and depth of the input, then we usually apply $m$ $h_k \times w_k \times d_i$ kernels (or filters) to this input (with the appropriate stride and padding), where $m$ is usually a hyper-parameter. So, after the application of $m$ kernels, you ...

1

As far as I know, more than 3 channel is perfectly fine, since, 3 channels are what we use for images and that's enough since we can only see this many colors, but I don't see why more than that wouldn't work Your 2nd question is like asking whether or not you will be good at a sport... Just try it For your 3rd question, I've never seen any language AI ...

1


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Find zero-state response

Image text transcribed for accessibility: A stable LTIC system is specified by the frequency response H(omega) = 1/ 2- j omega Find the zero-state response of this system if the input is x(t) = cos(0.25 pi t). Hints: (i) Fourier transform of cos(0.25 pi t) from the Fourier transform table, (ii) convolution theorem of Fourier transform, (Hi) sifting property of the unit impulse function delta (omega).


text:
Thread: Row space of a matrix?

1. Row space of a matrix?

How do I prove:

$row(A^2)\subset row(A)$, for a matrix A

2. each row of A^2 can be shown to be linear combination of rows of A. (look at the definition of matrix multiplication). Hence row space of A^2 is a subset of row space of A.

Not sure if there is another more conceptual argument.
Thanks


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RS 422/RS 485 Limitations On Number of Multi-dropped devices

R

Richard George

Can anyone advise what, if any,is the maximum number of devices which can be multidropped using either RS422/RS485 serial interfaces employing Modbus RTU protocol?

L

Lynn August Linse

The "legal" answer is "32-unit loads", with (on average) 1 unit load per slave or 32 devices. Some vendors sell chips with 1/4 unit load, so you could have 128 devices, but I haven't seen any field devices use these yet.

However, some vendors load the RS-485/422 wires with various bias and/or surge loads, meaning you may not reach the magic 32-device number before

In general, I say you have the following variables:
1) How fast are you talking (slower is "better").
2) How long are your drops between device & the bus (shorter is "better")
3) How good of quality is your cable (more- is "better")
4) How long is your overall bus (shorter is "better")
5) How many devices are on your bus (fewer is "better")

Given these 5 variables, the more "better" you are in 3 or 4 of them, the less "better" you can be in the other 1 or 2.

For example if your baud rate is 9600, your drops are short (close to zero), your cable is average but only 300 meters long, then your risk to
assuming 32 devices on the bus will work is less.

If you do have a problem, but can drop the baud rate, that may overcome the problem. etc. Also, using isolating repeaters can segment your RS-485 and be a fall-back if your 32-devices don't work well.

Regards

Lynn A Linse, Senior Industrial Automation Apps Engineer
Lantronix, 15353 Barranca Parkway, Irvine CA 92618 (USA)
Mobile: (949)300-6337 Office (949)450-7272 Fax (949)453-7152
[email protected]

G

grantt1

Valid slave devices are in the range of 0-247 decimal. Individual slave devices are assigned addresses in the range of 1-247. Additional
information can be found at "www.modbus.org":www.modbus.org

K

kmorin

Modbus has a limit of 32 addresses per network. Its not a hardware limit just a legacy of the comm. protocol.

32 can be circumvented by subdividing into more networks. some versions support 64 -again having to do with parsing the address characters
out of the message sting composition in binary.

A

Alex Pavloff

From the Modbus Protocol Reference -- Modicon document PI-MBUS-300 Rev J:

Valid slave device addresses are in the range of 0 - 247 decimal.
The individual slave devices are assigned addresses in the range of 1 - 247.

Alex Pavloff
Software Engineer
Eason Technology

T

Tim Linnell

The general rule is 32 devices per RS485 network segment (including the master). Many driver chips now allow more, typically 64. More than this requires a segment repeater (for signal regeneration). 32 is always safe, as part of the RS485 spec: check with device vendors if you need to exceed this.

Strictly speaking, RS422 is point to point (like RS232 but using differential signalling to cancel noise) so the node count is 2. However if slave devices tristate their outputs when not transmitting (which they almost always do when using Modbus RTU), the Slave TX/Master RX channel becomes effectively an RS485 network, so the same node count as RS485 applies.

(Similarly by tristating the master TX/slave RX channel when not transmitting it is theoretically possible to create a multi-master system on RS422 using token passing or some such, though this is very infrequent in practise).

Cheers

Tim Linnell (Eurotherm)


text:
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### Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
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### Why does the Dedekind Cut work well enough to define the Reals?

I am a seventeen year old high school student and I was studying some Real Analysis on my own. In the process, I encountered the Dedekind Cut being used to construct the Reals. I just can't get the ...
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### How can an ordered pair be expressed as a set?

My book says $$(a,b)=\{\{a\},\{a,b\}\}$$ I have been staring at this for a bit and it doesn't make sense to me. I have read several others posts on this, but none made ...
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### True Definition of the Real Numbers

I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a ...
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### Building the integers from scratch (and multiplying negative numbers)

Now I understand that what I am about to ask may seem like an incredibly simple question, but I like to try and understand math (especially something as fundamental as this) at the deepest level ...
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### The real numbers and the Von Neumann Universe

So I'm going to prefix this question by saying that I probably don't have a great understanding of what I'm asking. We build the cumulative hierarchy as follows: $V_0=\emptyset$ For every $\alpha$, ...
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### How to write $\pi$ as a set in ZF?

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# How do you calculate the number of ions in a solution?

Sep 17, 2014

Find the molar concentration then look at the formula to work out the concentration of each ion.

Lets look at an example: "How many sodium ions are there in a solution of sodium chloride of concentration 58.5 g.dm^(-3) ? "

We need to convert this into $m o l . {\mathrm{dm}}^{- 3}$ . To do this we add up the ${A}_{r}$ values to get the relative formula mass. For NaCl this will be 23 +35.5 = 58.5. So 1 mole of NaCl weighs 58.5g.

So now we need to convert grams into moles by dividing mass in grams by the mass of 1 mole.

So the number of moles of NaCl = 58.5/58.5 = 1 mole

So the concentration of NaCl is $1 m o l . {\mathrm{dm}}^{- 3}$

This means that in $1 {\mathrm{dm}}^{3}$ of solution there must be 1 mole of sodium ions.

The number of particles in 1 mole is given by the Avogadro Constant which is equal to $6.02 x {10}^{23} m o {l}^{- 1}$. We usually give this the symbol L.

So the solution in question contains L sodium ions.

Check the stoichiometry of the formula. If you have $1 {\mathrm{dm}}^{3}$ of a 1 molar solution of sodium sulfate $N {a}_{2} S {O}_{4}$ then this would be 2 molar with respect to sodium ions so would contain 2L sodium ions.


text:
The CERN Document Server website will be unavailable the 21st of August at 12:00 noon CEST for a short time due to a planned intervention.

# Preprints

Nyeste elementer:
2019-08-19
04:44
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 EUDAQ2 -- A Flexible Data Acquisition Software Framework for Common Test Beams / Liu, Y. (DESY) ; Amjad, M.S. (University Coll. London) ; Baesso, P. (Bristol U.) ; Cussans, D. (Bristol U.) ; Dreyling-Eschweiler, J. (DESY) ; Ete, R. (DESY) ; Gregor, I. (DESY) ; Huth, L. (DESY) ; Irles, A. (Orsay, LAL) ; Jansen, H. (DESY) et al. The data acquisition software framework, EUDAQ, was originally developed to read out data from the EUDET-type pixel telescopes. [...] arXiv:1907.10600. - 13 p. Fulltext

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# Homework Help: Evaluating an infinite series

1. Mar 1, 2010

### Ali812

1. The problem statement, all variables and given/known data
$$\sum_{n=1}^{\infty}\frac{sinn}{2^n}$$

2. Relevant equations
Definition of a geometric series:
$$\sum_{n=0}^{\infty}x^n=\frac{1}{1-x}$$

3. The attempt at a solution
Basically I can use the geometric series idea and implement it into the denominator of the question (i.e. sub x=2 into the equation from part b and change the lower index to n=1)
$$\sum_{n=1}^{\infty}2^n=\frac{1}{1-2}$$

Taking the derivate of both sides:
$$\sum_{n=1}^{\infty}n2^{n-1}=\frac{1}{(1-2)^2}$$

Multiplying both sides by 2:
$$\sum_{n=1}^{\infty}n2^n=\frac{2}{(1-2)^2}$$

Of course I can simplify the question furthermore and get a single value as a result, my problem is I don't understand on how to incorporate the numerator (sin n) into the problem. Is my approach correct, or am I substituting the wrong x-value?

2. Mar 1, 2010

### Staff: Mentor

What exactly are you supposed to do with this series?

3. Mar 1, 2010

### Ali812

Suppose to solve for the overall sum I'm assuming, the question simply states "Find $$\sum_{n=1}^{\infty}\frac{sinn}{2^n}$$."

4. Mar 1, 2010

### Mathnerdmo

There are two problems here:

(1) The equation you gave for the geometric series only works for |x| < 1.

(2) We can't simply change the lower index to n=1.

5. Mar 1, 2010

### Ali812

Well that was the only approach I could think of, any other method to solve this question? A starting tip/hint will suffice.

6. Mar 1, 2010

### Mathnerdmo

I'm also having some difficulty with the sine in the numerator - I haven't done something like this in a while.

What class is this for, by the way? (to give me an idea of methods you have)

-----

Also, if there is a solution with geometric series, the problems I pointed out can be fixed... you just need to be careful about what you use.

7. Mar 1, 2010

### Ali812

Calculus II (First year)

8. Mar 1, 2010

### Ali812

From what I can tell, you can change the lower index to n=1, but I assume I will have to check for convergence first, if the series does converge, I can move ahead with this method, if not, I will need an alternative.

9. Mar 1, 2010

### Mathnerdmo

10. Mar 1, 2010

### Ali812

I believe we haven't learned that yet unfortunately, however I have determined that the series is absolutely convergent via the comparison test. What r value can i use so $$|r| < 1$$, involved with the numerator as well? :S

11. Mar 1, 2010

### Mathnerdmo

Hrmmm... I don't know a solution if you can't use Euler's formula. Sorry.

(doesn't mean there isn't one, but I'm not seeing it)

12. Mar 1, 2010

### Dick

I'm with Mathnerdmo. It's pretty easy if you put r=exp(i)/2 and realize your sum is related to the imaginary part of sum(r^n). So |r|<1 and you can treat it as a geometric series. Are you sure you didn't do that in Calc I and just forgot it?

13. Mar 2, 2010

### Ali812

After skimming through some info on Euler's formula, I can substitute sinn= (e^in - e^(-in)) / 2i, I'm confused as to what you are trying to state, could you perhaps elaborate? How exactly did you derive with r = exp(i)/2?

14. Mar 2, 2010

### Dick

exp(i)^n=exp(in)=cos(n)+i*sin(n). If you sum that series and take the imaginary part you get the sum of sin(n). What happens if you do the same thing with (exp(i)/2)^n?


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# How do you solve 3(p+2)-7p=18?

Feb 8, 2017

See the entire solution process below:

#### Explanation:

First, expand the terms in parenthesis on the left side of the equation:

$\left(3 \times p\right) + \left(3 \times 2\right) - 7 p = 18$

$3 p + 6 - 7 p = 18$

Next, group and combine like terms on the left side of the equation:

$3 p - 7 p + 6 = 18$

$\left(3 - 7\right) p + 6 = 18$

$- 4 p + 6 = 18$

Then, subtract $\textcolor{red}{6}$ from each side of the equation to isolate the $p$ term while keeping the equation balanced:

$- 4 p + 6 - \textcolor{red}{6} = 18 - \textcolor{red}{6}$

$- 4 p + 0 = 12$

$- 4 p = 12$

Now, divide each side of the equation by $\textcolor{red}{- 4}$ to solve for $p$ while keeping the equation balanced:

$\frac{- 4 p}{\textcolor{red}{- 4}} = \frac{12}{\textcolor{red}{- 4}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 4}}} p}{\cancel{\textcolor{red}{- 4}}} = - 3$

$p = - 3$


text:
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text:
# Number theory ultimatum by Rohith, Anirudh and Arjun!

$$a^{2} + b^{2} + c^{2} = 2(a+b+c+1)$$ and
$$ab+bc+ca = a^{2} + b^{2} + c^{2} - c$$

Let $$a, b, c$$ are positive integers which satisfy the equations above, find $$a + b + c$$.

×