Functions Definition and 1000 Threads

The trig identities for adding trig functions can be seen: But here the amplitudes are identical (i.e. A = 1). However, what do I do if I have two arbitrary, real amplitudes for each term? How would the identity change? Analysis: If the amplitudes do show up on the RHS, we would expect them...

Hi PF Given the following def f1(var1, var2): var3 = var1 + var2 return var3 def f2(var1, var2, var4) var4 = 10 var5 = f1(var1, var2)*var4 return var5 it is obvious function f2 does not explicitly need variables var1 and var2. However, it needs the result of f1, which...

Hello. I am reading about state equations from a physics textbook, Physics by Frederick J. Keller, W. Edward Gettys, Malcolm j. Skove (Volume I). I don't understand some parts but since I have the Turkish translation of the book I must translate it as good and clear as possible. "State...

Let ##\epsilon>0##. Choose ##N\in\mathbb{N}## s.t. for each integer ##n## s.t. ##n\geq N##, $$|\sup\{|f-f_n|(x):x\in D\}|<\frac{\epsilon}{3}$$ where ##D## denotes the intersection of the domains of ##f## and ##f_n##. Choose a partition ##P:=\{x_0,\ldots,x_m\}## with ##x_i<x_{i+1}## where...

Hi, I'm writting because I sort of had an idea that looks that it should work but, I did not find any paper talking about it. I was thinking about approximating something like algebraic functions. That is to say, a function of a complex variable z,(probably multivalued) that obeys something...

##f : [0,2] \to R##. ##f## is continuous and is defined as follows: $$ f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$ $$ f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$ ##V = \text{space of all such f}## What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V##...

I argue not. Let ##f:\mathbb{Q}\rightarrow\mathbb{R}## be defined s.t. ##f(r)=r^2##. Consider an increasing sequence of points, to be denoted as ##r_n##, that converges to ##\sqrt2##. It should be clear that ##\sqrt2\equiv\sup\{r_n\}_{n\in\mathbb{N}}##. Continuity defined in terms of sequences...

So I have $$f(x,y,z,t,n) = 0,g(x,y,z,t,n) = 0,h(x,y,z,t,n) = 0 $$ and i need to find the best ##[x,y,z,t]## that fit the data, where n is the variable. Now, the amount of data for each function is pretty low (2 pair for f (that is, two (n,f)), 3 pair for g and another 3 pair for h) The main...

Hello Simple question Whether the minimum of the product of two functions in one single variable, is it greater or less than the product of their minimum thanks Sarrah

R is the triangle which area is enclosed by the line x=2, y=0 and y=x. Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v## ## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as...

From what I've gather the primary benefits to worm wheels are: - their ability to provide high reduction ratios - self-locking which can be useful for hoisting and lifting applications. - Operates silently and smoothly, which reduces vibrations Feel free to add any important ones I might've...

Because it drives to contradictions. Here is a nice example from E. Rosinger Generalized solutions of nonlinear PDE. We can multiply generalized functions from ##\mathcal D'(\mathbb{R})## by functions from ##C^\infty(\mathbb{R})##. This operation is well defined. For example $$x\delta(x)=0\in...

Hello, recently I'm learning about correlation functions in the context of QFT. Correct me with I'm wrong but what i understand is that tha n-point correlation functions kinda of describe particles that are transitioning from a point in space-time to another by excitations on the field. So, what...

As is well known, almost periodic functions can be represented as a Fourier series with incommensurable (non-multiple) frequencies https://en.wikipedia.org/wiki/Almost_periodic_function. It seems to me that I came up with an integral criterion for the degree of non-periodicity. The integral of a...

*** MENTOR NOTE: This thread was moved from another forum to this forum hence no homework template. Summary:: Trying to find transfer functions to design a block diagram on simulink with a PID controller and transfer functions for a water tank system. ----EDIT--- The variables and parameters...

Humanoid Robots. Just requiring your thoughts on this. Major events,functions example Weddings, Birthday, Anniversary celebrations, Cricket, Football live match etc are captured using Video camera/s with Humans performing the function with later on the captured recorded video is edited with...

While deriving Lorentz transformation equations, my professor assumes the following: As ##\beta \rightarrow 1,## $$-c^2t^2 + x^2 = k$$ approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...

So I thought that the graph tries to tell us that the function is periodic after 2π interval. So I tried to derive its function from the graph as follows using the point slope equation form for the points (0,0) & (a,π): ##y= ({a}/{π})*x## I hope this function is alright and I just need to find...

Suppose f is a function such that f'(7) is undefined. Which of the following statements is always true? (Give evidences that supports your answer, then explain how those evidences supports your answer) a. f must be continuous at x = 7. b. f is definitely not continuous at x = 7. c. There is not...

I did only the the first three prop. And on a means we have, on pose or posons means let there be , propriétés means properties, alors meand then. I apologize i am a french native speaker and i am busy so i cannot rewrite this in entirely english

I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ... I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof...

I am not sure of the overall purpose of the concepts developed below regarding Riemann integrable functions with compact support ... nor am I sure of the details ... so I am sketching out the meaning as I understand it in 2 dimensions and depicting the relevant entities in diagrams ... I am...

There's a famous functional equation that was asked in the 2019 IMO. It looks like this: find all f: Z -> Z where f(2a)+2f(b)=f(f(a+b)). I thought of solving it using a recurrence relation where a_n=f(nx). But when I substituted values in the functional equation (after setting a and b equal...

Mathematicians will use the term "elementary functions," often in the context of integration wherein some integrals cannot be expressed in elementary functions. The elementary functions are usually listed as being arithmetic, rational, polynomial, exponential, logarithmic, trigonometric...

When I look at a range of inputs around x=c and consider the corresponding range of outputs If 0< |x-c| <δ -> |f(x)-L1|<ϵ1 and |g(x)-L2|<ϵ2 as we shrink the range of inputs the corresponding outputs f(x) and g(x) narrow on L1 and L2 respectively. |f(x)-L1||g(x)-L2|<ϵ2ϵ1 The product of the...

Can anyone out there give me a hint as to where to start with this problem? I've been looking at it for a while and can't see a way forward. What exactly is "the curvature itself" here?BTW I think the dynamic initial value equations 21.116 and 21.117 are incorrect. MTW should have inserted to...

Is it possible for Python matplotlib to plot in one graph a domain dependent function? For example, suppose there is a function where y=x from 0< x <=5, y = x sq when 5<x<=7 and y=2x+9 when x>7. Is it possible in Python to plot this with one plot on one graph? If so, how would it be done?

Suppose f1,f2... is a sequence of functions from a set X to R. This is the set T={x in X: f1(x),... has a limit in R}. I am confused about what is the meaning of the condition in the set. Is the limit a function or a number value? Why?

How can we be sure that a system on the scale of atoms can be described by a single scalar field or the wave function ##\psi##. I don't just want to do shut up and calculate, maybe using a wave function and then putting it through the time evolution of the Schrödinger equation works, but why...

I want to understand how the domain and range change upon applying transformations like (left/right shifts, up/down shifts, and vertical/horizontal stretching/compression) on functions. Let f(x)=2-x if 0 ≤x ≤2 and 0 otherwise. I want to describe the following functions 1) f(-x) 2) -f(x) 3)...

Consider the case of a real function f for which f inverse exists. 1) We we are not used to having the y-axis (vertical axis) to denote the independent variable which it does in x=f-1(y). We rotate the system through positive 90 degree and reflect about the vertical to change the sense of the...

For example: h(x)=f(x)+g(x) If f(x) and g(x) are real numbers and real numbers can be added, subtracted, multiplied and divided (except by 0). why do we insist that the x in f(x) and g(x) be {x: x∈ dom f ∩ dom g}? My thoughts: The equality of two functions requires two criteria: 1) They operate...

##f## is continuou on ##\mathbb{C}##, so for al ##\epsilon>0##, there is a ##\delta>0## such that $$|\tilde{z}-z|\leq \delta \Rightarrow |f(\tilde{z})-f(z)|\leq \epsilon$$ for all ##\tilde{z}## and ##z## in ##\mathbb{C}##. Complex conjugation is a norm preserving operation on ##\mathbb{C}##, so...

[FONT=times new roman]Problem Statement : [FONT=times new roman]Solve for ##x## : Attempt : If I take ##x=\tan\theta##, the L.H.S. reads $$\tan^{-1}\frac{1-\tan\theta}{1+\tan\theta}= \tan^{-1}\left[\tan\left(\frac{\pi}{4}-\theta \right) \right ]=\frac{\pi}{4}-\theta.$$ On going back to ##x##...

I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##. Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...

##lim_{|z|\rightarrow \infty}\frac{f}{g}=1\neq \frac{\infty}{\infty}## so ##lim_{|z|\rightarrow \infty}f\neq \infty## and ##lim_{|z|\rightarrow \infty}g\neq \infty##. Because f(z) and g(z) are bounded and entire, f(z) and g(z) are constant functions by Liouville's theorem . f(z) and g(z) are...

Hello, I wonder if it is possible to write Bloch wave functions in momentum space. To be more specific, it would calculate something like (using Sakurai's notation): $$ \phi(\vec k) = \langle \vec k | \alpha \rangle$$ Moving forward in a few steps: Expanding: $$ \phi(\vec k) = \int d^3\vec r...

Hi, a basic doubt about thermodynamic functions and state variables. Take for instance transformations I and II in the following ##(p,V)## plane. As far as I can tell, just because the transformations are drawn as continuous lines they are reversible by definition. Namely we can transform...

We show that there is a partition s.t. the upper sum and the lower sum of ##f## w.r.t. this partition converge onto one another. Let ##\epsilon>0##. Define a sequence of functions ##g_n:[a,b]\setminus(\{a_n\}_{n\in\mathbb{N}}\cup\{y_0\})## s.t. ##g_n(x)=|f(x)-f(a_n)|##. Suppose there is a...

Hi, Just curious as to whether distances 'd' , used in Knn ; K nearest neighbors, in Machine Learning, are required to be metrics in the Mathematical Sense, i.e., if they are required to satisfy, in a space A: ##d: A \times A \rightarrow \mathbb R^{+} \cup \{0\} ; d(a,a)=0 ; d(a,b)=d(b,a) ...

Consider the following theorem: Theorem: Let ##f## be a concave differentiable function and let ##g## be a concave function. Then: ##y \in argmax_{x} {f(x)+g(x)}## if and only if ##y \in argmax_{x} {f(y)+f'(y)(x-y)+g(x)}.## The intuition is that local maxima and global maxima coincide for...

If Tl;dr I am struggling in Math 171 and Physics 191 and throwing around the idea of declaring a geology major with an astronomy minor because the Physics major "juice is not worth the squeeze" at my age(29) anyone else out there who struggled with Calculus 1 when they first took it?Hello...

Let's say I want to describe a massive box in spacetime as described by the Einstein Field Equations. If one were to construct a metric in cartesian coordinates from the Minkowski metric, would it be reasonable to use a piecewise Stress-Energy Tensor to find our metric? (For example, having...

How can i prove that 6cos(x+45) cos(x-45) is equal to 3cosx?

Hi For a function f ( x , t ) = 6x + g( t ) where g( t ) is an arbitrary function of t ; then is it correct to say that f ( x , t ) is not an explicit function of t ? For the above function is it also correct that ∂f/∂t = 0 because f is not an explicit function of t ? Thanks

Hi PF I have a quote from Spanish 6th edition of "Calculus", by Robert A. Adams, and some queries. I translate it this way:"The inverse of secondary trigonometric functions can easily be calculated with the reciprocal function. For example DEFINITION 13 The inverse function of secant ##sec^{-1}...

How do I build functions by using Arithmetic Sequence, Geometric Sequence, Harmonic Sequence? Is it possible to create all the possible function by using these sequences? Thanks!

Ok in my thinking, i would say that it depends on ##x##, if ##x## belongs to the integer class, then the rational functions would be ##i ## and ##iii##...but from my reading of rational functions, i came up with this finding: I would appreciate your input on this.

(I must confess that, in spite of working through the chapter on inverse circular functions, I could barely proceed with this problem. Note what it asks to prove : ##x\sqrt{1-x^2}+\ldots## and how much is that at odds with the formula (1 above) of adding two ##sin^{-1}##'s, where you have...

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