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The method of principles was founded by Newton … . For example, jaguar speed -car Search for an exact match Put a word or phrase Moments. Under no circumstances is content to be used for commercial gain. The Advanced Level Physics syllabus is a single docu-ment covering Forms 5 and 6. $$v^{2}=u^{2}+2as$$, $$F=\frac{\Delta \left (mv\right )}{\Delta t}$$, $$\small efficiency=\frac{useful\: output\: power}{input\:power}$$, force × perpendicular distance from the point to the line of action of the force, force × perpendicular distance between the lines of action of the forces, rate of doing work = rate of energy transfer, Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. In this module you will build on your work at GCSE and learn how to apply advanced mathematical techniques such as trigonometry and quadratic equations in order to solve problems. Get Physics AS and A Level Notes here at my new website. $$a=\frac{\Delta v}{\Delta t}$$, $$v=u+at$$ . This hardcopy is provided so that you can fully participate in class discussions without having to worry about writing everything down. . . Cambridge International AS and A Level Physics book PDF free download (2nd Edition). A level physics notes. Problems will be solvable using the equations of Physics Notes Page 3 Physics Mr. Bigler This is a set of class notes for physics. MECHANICS - Kinetics Connected Particles pulleys towe-bar inclined plane lift Pulleys Problems involve two weights either side of a pulley. In mechanics, it is necessary to take a real life problem and put it in mathematical language. Revision Notes: Physical Quantities And Units Kinematics Dynamics Mass, Weight And Density Turning Effects Of Forces Pressure Work, Energy And Power Kinetic Model Of Matter Transfer Of Thermal Energy Temperature â¦ Advanced Level (A/L) Physics Notes Largest online Education web site in Sri Lanka provides Past papers, Model papers, School papers, Campus papers, Marking schemes, Notes, Career guide for school leavers and lot more Articles.We're mainly focused for G.C.E. Clear coverage of the latest specification from an experienced author team of examiners and teachers, plus a free Studentâs CD-ROM. All Notes; Maths; Mechanics 1 These would ideally not only enable one to explain the range of familiar phenomena but may predict new phenomena or at least explain new phenomena when they are discovered. . . Credits: Robinhoodmustafa ; Various Authors of Above Resources; Shahzad Zia / By O & A Level Notes . downward acceleration. . Free O Level Physics Revision Notes that will help you in revising for your exams. Quantum Mechanics Made Simple: Lecture Notes Weng Cho CHEW1 October 5, 2012 1The author is with U of Illinois, Urbana-Champaign.He works part time at Hong Kong U this Contents Preface vii Acknowledgements They are a great revision tool for students to use alongside their studies. B.Sc. Files cannot be altered in any way. A particle is a body whose entire weight acts through a single point. Perfect revision resources for Edexcel A Level Physics. Find A-Level Physics Revision Resources + Edexcel, AQA & OCR specific Physics Revision Resources for A-Level Students. Advanced Level (A/L) Science & Maths Education.Let your support continue to take this service to the students. 1.Particles 2.Waves 3.Mechanics 4.Materials 5.Electricity 6.Further-Mechanics—Student-Copy 7.Thermal—Student-Copy 8.Gravitational-Fields—Student-Copy 9.Electric-Fields—Stude… The book has 23 chapters each covering a specific syllabus area in a separate chapter. One of our academic counsellors will Privacy. Download Advance level physics new syllabus notes free. E-Book D/L (pdf) Mechanics ... A-level Physics Tutor - a free resource for aspiring physicists. See more ideas about physics notes, physics, study notes. In Laminar Flow, all the fluid elements flow in the same direction, and none of the streamlines cross over.. The syllabus includes the main theoretical concepts which are fundamental to the subject, a section on some current applications of physics, and a strong emphasis on advanced practical skills. 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The syllabus includes the main theoretical concepts which are fundamental to the subject, a section on some current applications of physics, and a strong emphasis on advanced practical skills. Also, check out our other helpful revision resources for O Level Physics (5054). . This MCQ book includes multiple choice questions taken from CIE O level past papers. All the MCQs have been compiled according to the year in which they were set Physics Kinematics PDF Uploaded Site is up and running Use of Mind Maps in Revisions 320 days since Release of GCE O-Level Results in 2020 Total Hits: **Download PDF version of all available notes here** Full collection of . A-Level Physics Revision section of StudyWise. Cambridge International AS and A Level Physics book PDF free download (2nd Edition). It is for the preparation of Fluid Mechanics, GATE, Notes & Physics. Use these to help revise for exams and to read through as you go along to build understanding. ஆய்வுகூட பரிசோதனைக A-Level Physics revision notes, videos and more for AQA. . How Physics Tutor Online can help you. . FREE Physics revision notes created by the expert teachers at SAVE MY EXAMS for the AQA GCSE Physics (9-1) exam. This website makes use of cookies. This course reading gives full scope of Mechanics 1 (M1). Questions are posed to the class to stimulate discussion and indicate how concepts are going over. . The heavier weight pulls on â¦ Clear coverage of the latest specification from an experienced author team of examiners and teachers, plus a free Student’s CD-ROM. If you have extra notes or resources please contribute to the website and help thousands of other people like you. . Free self-study tutorial for A-level Physics Mechanics (Edexcel topic 2). Course Outcomes. . Physics Notes Page 3 Physics Mr. Bigler This is a set of class notes for physics. 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## a level physics mechanics notes pdf

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### Situational Awareness: What Would Fr Z Do? Let's Say Fr. John Zuhlsdorf Is Taking The Bus When..............

I don’t know about you, but over the last year or so I have become increasingly disciplined regarding my situation awareness. When I get out of my car, I look around to see where everyone is. When I walk out the door, I look around to see where everyone is. When I walk into a place, I look around. I know where the doors and exits are. I see where everyone is, who comes in after me. If I see someone that looks or behaves oddly, I keep an eye on them. Fr. Z Read More>>>>>

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# Tag Info

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Comparing two programming languages is difficult is a difficult problem, and far from being solved. The key issue is that there are many different ways languages can be compared, and none of them is compelling. The most widely used approach, coming from logic, is to consider translations between the languages to be compared. The general idea is as ...

12

Unfortunately, there are too many things are going on here. So, it is easy to mix things up. The use of "full" in "full completeness" and "full abstraction" refer to completely different ideas of fullness. But, there is also some vague connection between them. So, this is going to be a complicated answer. Full completeness: "Sound and complete" is a ...

11

Here is one possibility, but other people might use different words. I will use first-order logic as a running example. Language The language is a collection of expressions, which are syntactic entities, i.e., finite configurations without any a priori meaning. A language is described by the grammar, which determines which finite configurations are valid ...

11

A semantics of a program is a model of its behavior which, like any scientific model, ignores aspects that you don't want to study. An extremely detailed model of the execution of a program would model the physical behavior of the computer that executes it, including the execution time, power consumption, electromagnetic radiation, etc. Such aspects are ...

11

No, to my knowledge there has been no work on formalizing TeX of the kind you are interested in. (What follows is a subjective and personal commentary). I think it is an intriguing and well-posed idea, and your motivation of using it to perform optimizations sounds reasonable -- another related question is whether you could define a bytecode format for it ...

10

There is a naive algorithm for programs with bounded-size inputs: enumerate all programs in order of increasing length (or execution time, which is a bounded function of the length). If you can prove that the program is equivalent to the original, stop; otherwise keep searching. This algorithm is sound. In order for it to be complete, you need to be able to ...

10

The practical reason is that it is very convenient to include also the case "zero steps" in the definition of "many steps" (millennia of mathematical experience have taught us that it is usually a good thing to have a 0 around in our set of natural numbers). One possible technical exemplification of this (but there are probably dozens more, perhaps more ...

9

(With apologies for a long answer that goes in a direction different from the scope of the site: frankly I was surprised to see the question here in the first place….) TeX was designed for typesetting, not for programming; so it is at best “weird” when considered as a programming language. — Donald Knuth, Digital Typography, page 235 I have read a ...

8

A reduction strategy is a function on Lambda that picks one redex (reducible expression) from all possible redexes -- depending on what you define as a redex. Informally, an evaluation strategy is the order in which a language evaluates its arguments. A parameter-passing strategy is what the language hands to the function. To understand the connection ...

8

Obviously there is an operational semantics of Ltac by Jedynak et al.

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I'm not sure this answers your question, but the first (?) paper on the subject of tactics appears to have been Milner's The Use of Machines to Assist in Rigorous Proof.

7

I essentially agree with Martin's comment, I can elaborate on that to make a tentative answer, knowing that there is no general formal definition of calculus or abstract machine and that what I am going to describe cannot possibly cover the meaning of all instances of these two words found in the literature. In brief: a calculus usually gives you the ...

7

Most people avoid giving precise descriptions of what a syntactic category is, because if you do it properly with all the details, the ratio of insight to necessary mathematical sophistication ends up being very, very low. John Reynolds' book Theories of Programming Languages has one of the more comprehensive explanations in its chapter 1, as does Robert ...

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In the strictest sense, there is no real difference between syntax errors and semantics errors, at least as far as language theory is concerned: the only salient difference is the complexity of the automaton required to recognize that language, with, e.g. Context-free languages only requiring pushdown automata (PDA) General recursive languages requiring ...

7

Non-termination can be considered an algebraic effect up to a point. It's an exception that cannot be handled. More precisely, we may introduce a nullary operation (constant) $\bot$ which signifies non-termination, but then we disallow handling it, as that would allow us to implement the Halting oracle. Such treatment of non-termination is a bit naive. A ...

6

Formal semantics is useful primarily when you want to reason about programs. In the past this was mainly done in programming language development (and to a lesser degree in compiler construction). Increasingly formal semantics is also used in automatic and interactive program verification, which is beginning to see industrial use. Since you are interested ...

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You should have a look at Uniform Proofs as a Foudation for Logic Programming by Dale Miller, Gopalan Nadathur, Frank Pfenning and Andre Scedrov, 1991. The idea of this work and the rich area that has seen spawn from these ideas is to understand the execution of a logic program as proof search in simple and expressive logics, such as intuitionistic logic or ...

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The "Reduction strategy" wikipedia article is entirely extracted out of a particular edit made by an anonymous IP to the "Evaluation strategy" article. The view that it represents is not consensual, in the sense that I suspect relatively few people of the field will spontaneously give this answer if you ask them "would you distinguish the names 'reduction ...

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You should get acquainted with inductive definitions. Instead of saying that $Data$ is recursively defined, say that it is a set constructed inductively by: $(0,i) \in Data$ for every $i \in \mathbb{Z}$, $(1,d) \in Data$ for every $d \in Data$, $(2,d) \in Data$ for every $d \in Data$. Another way to say the same thing is that $Data$ is the smallest set ...

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The answer given is adequate but I'd like to mention that once you reduce a term to normal form, without reflexivity you're out of luck, with reflexivity you've hit a fixed point and thus you can use a fixed point under beta reduction to define the total reduction, as long as you have the nice normalization properties.

6

In terms of related work, Marek Zaionc and collaborators have been studying similar kinds of questions for some time. The following paper includes many results: René David, Katarzyna Grygiel, Jakub Kozic, Christophe Raffalli, Guillaume Theyssier, Marek Zaionc. Asymptotically almost all λ-terms are strongly normalizing. Logical Methods in Computer Science,...

5

Summary: full completeness means that the interpretation function is not just complete, but also surjective on programs. Full abstraction has no requirement for surjectivity. $\newcommand{\semb}[1]{[\![ #1 ]\!]}$ Details: The detailed meaning of full abstraction and full completeness depends on the nature of what/where/how you are interpreting. Here is a ...

5

There are unfortunate confusions in the way you have posed the question. Programs have semantics. Programming Languages are given semantic definitions. In more detail: every program has a meaning, either as a computation or as a mathematical function (relation, trace set, strategy,...). A semantic definition is given for an entire programming language, ...

5

Consider a set $E$. The theory of $E$-exceptions is an algebraic theory given by: for every $e \in E$ a nullary operation symbol $\mathsf{raise}_e$ no equations. Given a set $X$, we may consider the free algebra $A$ for the theory of $E$-exceptions, with generators from $X$. The algebra $A$ contains everything that we can generate from the generators and ...

5

I have never heard of this exact concept in rewrite theory, which certainly doesn't prove it hasn't been considered. However, I will make the point that it may not be quite as useful a concept as it first appears, at least in classical rewrite theory because it behaves poorly under substitution: If $t\rightarrow t'$ is an inevitable reduction, and $t$ ...

5

This question is very open-ended and therefore difficult to answer, but I think the short answer is "yes", there is much common ground, the two areas have in fact benefited from interaction in the past, and they have much to gain from further interaction. Two classical examples of this (respectively from the 1950s and '70s) are Lambek grammar and ...

5

Let's look at a simple example of a toy programming language with unary natural numbers and a "predecessor" operation. $$t ::= 0 \mid S~t \mid p~t$$ whose semantics is given by the following rewrite rules $$p~(S~t) \to t \qquad p~0 \to 0$$ with \mathsf{size}(0) = 1 \qquad \mathsf{size}(S~t) = 1 + \mathsf{size}(t) \qquad \mathsf{size}(p~t) = 1 + \...

4

The textbook that might be most relevant to your question is Principles of Program Analysis by Nielson, Nielson and Hankin. It does cover dataflow analysis and its relationship to denotational semantics. It does not deal with axiomatic semantics though.

4

I never found an explicit definition either, but I have inferred the folowing. As I understand, you split the language into syntactic domains; with the addition that syntactic domains must be fully generated by single different symbols, when you write down the grammar. So a syntactic domain is a subset of your language, and each domain is generated by one ...

4

I would like to add two books not found on the answers given up to now: Aaron Stump, Programming Language Foundations David Schmidt, Denotational Semantics: A Methodology for Language Development Stump's book is concise but very clear.

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# Source code for ncempy.algo.gaussND

""" A module with Gaussian functions for creating Gaussian distributions
in 1D, 2D and 3D. There are also functions with the same name and _FIT
which can be used to fit an intensity distribution in a ndarray with the
corresponding Gaussian distribution using scipy's curve_fit.

A 1D Lorentz and a 1D gaussLorentz function is also included for fitting
zero loss peaks in EELS spectra.

These functions were written assuming you use np.meshgrid() with indexing = 'xy'. If
you use 'ij' indexing then be careful about how you pass in the x and y coordinates.

"""

import numpy as np

[docs]def gauss1D(x, x0, sigma):
""" Returns the value of a gaussian at a 2D set of points for the given standard deviation with maximum
normalized to 1.

Parameters
----------
x: ndarray
A vector of size (N,) of points for the x values
x0: float
The center of the Gaussian.
sigma: float
Standard deviation of the Gaussian.

Returns
-------
g: ndarray
A vector of size (N,) of the Gaussian distribution evaluated
at the input x values.

Note
----
Calculate the half width at half maximum (HWHM) as
>> HWHM = sqrt(2*log(2))*stDev ~ 1.18*stDev
or if x goes from -1 to 1
>> 0.5*size(x)*stDev

"""
return np.exp(-(x - x0) ** 2 / (2 * sigma ** 2))

[docs]def lorentz1D(x, x0, w):
""" Returns the probability density function of the Lorentzian (aka Cauchy) function with maximum at 1.

Parameters
----------
x: ndarray or list
A 1D vector of points for the dependent variable
x0: float
The center of the peak maximum
w: float
The parameter that modifies the width of the distribution.

Returns
-------
l: ndarray
A vector of size (N,) of the Lorentzian distribution evaluated
at the input x values.

"""
return w ** 2 / ((x - x0) ** 2 + w ** 2)

[docs]def gaussLorentz1D(x, x0, w):
""" A Gaussian-Lorentzian function in one dimension.

Parameters
----------
x: ndarray or list
A 1D vector of points for the dependent variable
x0: float
The center of the peak maximum
w: float
The parameter that modifies the width of the distribution.

Returns
-------
lg: ndarray
A vector of size (N,) of the Lorentzian Gaussian distribution
evaluated at the input x values.

"""
gSigma = w / (2 * np.sqrt(2 * np.log(2)))  # change the width to sigma
return gauss1D(x, x0, gSigma) + lorentz1D(x, x0, w)

[docs]def gauss2D(x, y, x0, y0, sigma_x, sigma_y):
""" Calculates the value of a Gaussian at a 2D set of points for the given
standard deviations with maximum normalized to 1.
The Gaussian axes are assumed to be 90 degrees from each other.

Parameters
----------
x, y : ndarray
A 2D array of size (M,N) of points for the x values.
Use x, y = np.meshgrid(range(M),range(N),indexing='xy').
x0 : float
The center of the Gaussian along the x direction.
y0 : float
The center of the Gaussian along the y direction.
sigma_x : float
Standard deviation of the Gaussian along x.
sigma_y : float
Standard deviation of the Gaussian along y.

Returns
-------
g : ndarray
A ndarray of size (N, M) of the Gaussian distribution evaluated
at the input x values.

Note
----
The Gaussian is normalized such that the peak == 1. To normalize the
integral divide by 2 * np.pi * sigma_x * sigma_y

"""
x0 = float(x0)
y0 = float(y0)
g2 = np.exp(-((x - x0) ** 2 / (2 * sigma_x ** 2) + (y - y0) ** 2 / (2 * sigma_y ** 2)))
g2_norm = g2 / np.max(g2.flatten())
return g2_norm

[docs]def gauss2D_FIT(xy, x0, y0, sigma_x, sigma_y):
""" Version of gauss2D used for fitting (1 x_data input (xy)
and flattened output). Returns the value of a gaussian at a 2D set of points for the given
standard deviation with maximum normalized to 1.
The Gaussian axes are assumed to be 90 degrees from each other.

Parameters
----------
xy: tuple
A (N,M) shaped tuple containing the vectors of points for the
evaluation points.
x0: float
The x center of the Gaussian.
y0: float
The y center of the Gaussian.
sigma_x: float
The standard deviation of the Gaussian along x.
sigma_y: float
The standard deviation of the Gaussian along y.

Returns
-------
g2_norm: ndarray
The Gaussian distribution with maximum value normalized to 1. The
2D ndarray is reshaped into a (N*M,) array for use in fitting
functions in numpy and scipy.

"""
x0 = float(x0)
y0 = float(y0)
x = xy[0]
y = xy[1]
g2 = np.exp(-((x - x0) ** 2 / (2 * sigma_x ** 2) + (y - y0) ** 2 / (2 * sigma_y ** 2)))
g2_norm = g2 / np.max(g2.flatten())
return g2_norm.reshape(-1)

[docs]def gauss2D_theta(x, y, x0, y0, sigma_x, sigma_y, theta):
""" Returns the value of a generalized Gaussian at a 2D set of points for
the given standard deviation with maximum normalized to 1.
The Gaussian axes (assumed to be 90 degrees from each other) can be
oriented at different angles to the output array axes using theta.

Parameters
----------
x: ndarray or list
Evaluation points along x of size (N,)
y: ndarray
Evaluation points along y of size (M,)
x0: float
Center of the maximum along x.
y0: float
Center of the maximum along y.
sigma_x: float
Standard deviation along x.
sigma_y: float
Standard deviation along y.
theta: float
The rotation of the Gaussian principle axes from the array

Returns
-------
g2_norm: ndarray
A (N, M) sized 2D ndarray with a Gaussian distribution rotated.

"""

x0 = float(x0)
y0 = float(y0)
a = (np.cos(theta) ** 2) / (2 * sigma_x ** 2) + (np.sin(theta) ** 2) / (2 * sigma_y ** 2)
b = -(np.sin(2 * theta)) / (4 * sigma_x ** 2) + (np.sin(2 * theta)) / (4 * sigma_y ** 2)
c = (np.sin(theta) ** 2) / (2 * sigma_x ** 2) + (np.cos(theta) ** 2) / (2 * sigma_y ** 2)
g2 = np.exp(- (a * ((x - x0) ** 2) + 2 * b * (x - x0) * (y - y0) + c * ((y - y0) ** 2)))
g2_norm = g2 / np.max(g2.flatten())
return g2_norm  # return a 2D array

[docs]def gauss2D_theta_FIT(xy, x0, y0, sigma_x, sigma_y, theta):
""" Version of gauss2D_theta used for fitting (1 x_data input and
flattened output). Returns the value of a gaussian at a 2D set of points for the given standard deviation with
maximum normalized to 1. The Gaussian axes can be oriented at different angles (theta) in radians.

Parameters
----------
xy: tuple
Evaluation points along x and y of shape (N,M)
x0: float
Center of the maximum along x.
y0: float
Center of the maximum along y.
sigma_x: float
Standard deviation along x.
sigma_y: float
Standard deviation along y.
theta: float
The rotation of the Gaussian principle axes from the array

Returns
-------
g2_norm: ndarray
A (N, M) sized 2D ndarray with a Gaussian distribution rotated.

"""
x0 = float(x0)
y0 = float(y0)
x = xy[0]
y = xy[1]
a = (np.cos(theta) ** 2) / (2 * sigma_x ** 2) + (np.sin(theta) ** 2) / (2 * sigma_y ** 2)
b = -(np.sin(2 * theta)) / (4 * sigma_x ** 2) + (np.sin(2 * theta)) / (4 * sigma_y ** 2)
c = (np.sin(theta) ** 2) / (2 * sigma_x ** 2) + (np.cos(theta) ** 2) / (2 * sigma_y ** 2)
g2 = np.exp(- (a * ((x - x0) ** 2) + 2 * b * (x - x0) * (y - y0) + c * ((y - y0) ** 2)))
g2_norm = g2 / np.max(g2.flatten())
return g2_norm.ravel()  # return a 1D vector

[docs]def gauss2D_poly_FIT(xy, x0, y0, A, B, C):
""" Returns the flattened values of an elliptical gaussian at a 2D set of
points for the given polynomial pre-factors with maximum normalized to 1.
The Gaussian axes are assumed to be 90 degrees from each other.

The matrix looks like [[A,B],[B,C]].
See https://en.wikipedia.org/wiki/Gaussian_function

Parameters
----------
xy: tuple
Evaluation points along x and y of shape (N,M)
x0: float
Center of the maximum along x.
y0: float
Center of the maximum along y.
A: float
The A pre-factor for the polynomial expansion in the exponent.
B: float
The B pre-factor for the polynomial expansion in the exponent.
C: float
The C pre-factor for the polynomial expansion in the exponent.

Returns
-------
g2_norm: ndarray
A (N, M) sized 2D ndarray with a Gaussaian distribution rotated.
"""

x0 = float(x0)
y0 = float(y0)
x = xy[0]  # retrieve the array from the tuple
y = xy[1]
g2 = np.exp(-(A * (x - x0) ** 2 + 2 * B * (x - x0) * (y - y0) + C * (y - y0) ** 2))
g2_norm = g2 / np.max(g2.flatten())
return g2_norm.ravel()

[docs]def gauss3D(x, y, z, x0, y0, z0, sigma_x, sigma_y, sigma_z):
"""
gauss3D(x,y,z,x0,y0,z0,sigma_x,sigma_y,sigma_z)
Returns the value of a gaussian at a 2D set of points for the given
standard deviations with maximum normalized to 1.
The Gaussian axes are assumed to be 90 degrees from each other.

Note
-----
Be careful about the indexing used in meshgrid and the order in which you pass the x, y, z variables in.

Parameters
----------
x, y, z: ndarray, from numpy.meshgrid
2D arrays of points (from meshgrid)

x0, y0, z0: float
The x, y, z centers of the Gaussian

sigma_x, sigma_y, sigma_z: float
The standard deviations of the Gaussian.
Returns
-------
g3_norm: ndarray
A 3D ndarray

"""
x0 = float(x0)
y0 = float(y0)
z0 = float(z0)
g3 = np.exp(-((x - x0) ** 2 / (2 * sigma_x ** 2) + (y - y0) ** 2 / (2 * sigma_y ** 2) + (z - z0) ** 2 / (
2 * sigma_z ** 2)))
g3_norm = g3 / np.max(g3.flatten())
return g3_norm

[docs]def gauss3D_FIT(xyz, x0, y0, z0, sigma_x, sigma_y, sigma_z):
"""
gauss3D_FIT((x,y,z),x0,y0,z0,sigma_x,sigma_y,sigma_z)
Returns the value of a gaussian at a 2D set of points for the given
standard deviations with maximum normalized to 1.
The Gaussian axes are assumed to be 90 degrees from each other.
xyz -
x0, y0, z0 = the x, y, z centers of the Gaussian
sigma_x, sigma_y, sigma_z = The std. deviations of the Gaussian.

Note
-----
Be careful about the indexing used in meshgrid and the order in which you pass the x, y, z variables in.

Parameters
----------
xyz: tuple of ndarrays
A tuple containing the 3D arrays of points (from meshgrid)
x0, y0, z0: float
The x, y, z centers of the Gaussian
sigma_x, sigma_y, sigma_z: float
The standard deviations of the Gaussian.
Returns
-------
g3_norm: ndarray
A flattened array for fitting.

"""

x0 = float(x0)
y0 = float(y0)
z0 = float(z0)
x = xyz[0]
y = xyz[1]
z = xyz[2]
g3 = np.exp(-((x - x0) ** 2 / (2 * sigma_x ** 2) + (y - y0) ** 2 / (2 * sigma_y ** 2) + (z - z0) ** 2 / (
2 * sigma_z ** 2)))
g3_norm = g3 / np.max(g3.flatten())
return g3_norm.ravel()

[docs]def gauss3D_poly(x, y, z, x0, y0, z0, A, B, C, D, E, F):
"""
gauss3Dpoly_FIT((x,y,z),x0,y0,z0,A,B,C,D,E,F)
Returns the value of a gaussian at a 2D set of points for the given
standard deviations with maximum normalized to 1.
The Gaussian axes are not locked to be 90 degrees.

Parameters
----------
x, y, z : ndarray, 3D
3D arrays of points (from meshgrid)

x0, y0, z0 : float
The x, y, z centers of the Gaussian

A, B, C, D, E, F : float
The polynomial values for the fit

Returns
-------
: ndarray, 3D
The 3D Gaussian
"""
x0 = float(x0)
y0 = float(y0)
z0 = float(z0)
g3 = np.exp(-(A * (x - x0) ** 2 + B * (y - y0) ** 2 + C * (z - z0) ** 2 + 2 * D * (x - x0) * (y - y0) + 2 * E * (
x - x0) * (z - z0) + 2 * F * (y - y0) * (z - z0)))
g3_norm = g3 / np.max(g3.flatten())
return g3_norm

[docs]def gauss3D_poly_FIT(xyz, x0, y0, z0, A, B, C, D, E, F):
"""
gauss3Dpoly_FIT((x,y,z),x0,y0,z0,A,B,C,D,E,F)
Returns the value of a gaussian at a 2D set of points for the given
standard deviations with maximum normalized to 1.
The Guassian axes are not locked to be 90 degrees.

Parameters
----------
xyz : tuple of ndarrays
3D arrays of points (from meshgrid) combined in a tuple

x0, y0, z0 : float
The x, y, z centers of the Gaussian

A, B, C, D, E, F : float
The polynomial values for the fit

Returns
-------
: ndarray, 3D
The 3D Gaussian
"""
x0 = float(x0)
y0 = float(y0)
z0 = float(z0)
x = xyz[0]
y = xyz[1]
z = xyz[2]
g3 = np.exp(-(A * (x - x0) ** 2 + B * (y - y0) ** 2 + C * (z - z0) ** 2 + 2 * D * (x - x0) * (y - y0) + 2 * E * (
x - x0) * (z - z0) + 2 * F * (y - y0) * (z - z0)))
g3_norm = g3 / np.max(g3.flatten())
return g3_norm.ravel()

[docs]def gauss3DGEN_FIT(xyz, x0, y0, z0, sigma_x, sigma_y, sigma_z, Angle1, Angle2, Angle3, BG, Height):
"""
gauss3DGEN_FIT((x,y,z),x0,y0,z0,sigma_x,sigma_y,sigma_z,Angle1,Angle2,Angle3,BG,Height)
Returns the value of a gaussian at a 3D set of points for the given
sub-pixel positions with standard deviations, 3D Eular rotation angles,
constant Background value, and Gaussian peak height.

adapted from code by Yongsoo Yang, yongsoo.ysyang@gmail.com

Note
----
This is a work in progress. Needs more testing.

Parameters
----------
xyz : tuple of 3 np.ndarray
3D arrays of points (from meshgrid) combined in a tuple

x0, y0, z0 : float
The x, y, z centers of the Gaussian
sigma_x,sigma_y,sigma_z: float
standard deviations along x,y,z direction before 3D angular rotation
Angle1, Angle2, Angle3 : float, degrees
Tait-Bryan angles in ZYX convention for 3D rotation in degrees
BG : float
Background
Height: float
The peak height of the Gaussian function

Returns
-------
: ndarray, 3D
The 3D Gaussian
"""
# 3D vectors for each sampled positions
print('Warning: this function is using Fortran ordered array for use in tomviz. Need to test this')
# Todo: Remove Fortran ordering
v = np.array([xyz[0].reshape(-1, order='F') - x0,
xyz[1].reshape(-1, order='F') - y0,
xyz[2].reshape(-1, order='F') - z0])

# rotation axes for Tait-Bryan angles
vector1 = np.array([0, 0, 1])
rotmat1 = MatrixQuaternionRot(vector1, Angle1)

vector2 = np.array([0, 1, 0])
rotmat2 = MatrixQuaternionRot(vector2, Angle2)

vector3 = np.array([1, 0, 0])
rotmat3 = MatrixQuaternionRot(vector3, Angle3)

# full rotation matrix
# Todo : Remove dependency on np.matrix()
rotMAT = np.matrix(rotmat3) * np.matrix(rotmat2) * np.matrix(rotmat1)

# 3x3 matrix for applying sigmas
D = np.matrix(np.array([[1. / (2 * sigma_x ** 2), 0, 0, ],
[0, 1. / (2 * sigma_y ** 2), 0],
[0, 0, 1. / (2 * sigma_z ** 2)]]))

# apply 3D rotation to the sigma matrix
WidthMat = np.transpose(rotMAT) * D * rotMAT

# calculate 3D Gaussian
RHS_calc = WidthMat * np.matrix(v)
Result = Height * np.exp(-1 * np.sum(v * RHS_calc.A, axis=0)) + BG

return Result

[docs]def MatrixQuaternionRot(vector, theta):
"""
Quaternion tp rotate a given theta angle around the given vector.

adapted from code by Yongsoo Yang, yongsoo.ysyang@gmail.com

Parameters
----------
vector : ndarray, 3-element
A non-zero 3-element numpy array representing rotation axis
theta : float, degrees
Rotation angle in degrees

Returns
-------

Author: Yongsoo Yang, Dept. of Physics and Astronomy, UCLA
yongsoo.ysyang@gmail.com
"""
theta = theta * np.pi / 180
vector = vector / np.float(np.sqrt(np.dot(vector, vector)))
w = np.cos(theta / 2)
x = -np.sin(theta / 2) * vector[0]
y = -np.sin(theta / 2) * vector[1]
z = -np.sin(theta / 2) * vector[2]
RotM = np.array([[1. - 2 * y ** 2. - 2 * z ** 2, 2. * x * y + 2 * w * z, 2. * x * z - 2. * w * y],
[2. * x * y - 2. * w * z, 1. - 2. * x ** 2 - 2. * z ** 2, 2. * y * z + 2. * w * x],
[2 * x * z + 2 * w * y, 2 * y * z - 2. * w * x, 1 - 2. * x ** 2 - 2. * y ** 2]])

return RotM


text:
Seminario

Seminario EDPs y Mecánica de Fluidos

CANCELADO -- Non existence and strong ill-posedness in C^k and Sobolev spaces for SQG

Ponente:  Luis Martínez-Zoroa (ICMAT)
Fecha:  jueves 16 de diciembre de 2021 - 12:00
Lugar:  Aula Naranja (ICMAT)

Resumen:

We construct solutions in $${R}^2$$ with finite energy of the surface quasi-geostrophic equations (SQG) that initially are in $$C^k$$ ($$k\geq 2$$) but that are not in $$C^{k}$$ for $$t>0$$. We prove a similar result also for $$H^{s}$$ in the range $$s\in(\frac32,2)$$. Moreover, we prove strong ill-posedness in the critical space $$H^{2}$$. Joint work with D. Cordoba.


text:
Pennsylvania Algebra 1 - 2020 Edition
3.11 Absolute value inequalities
Lesson

Inequalities of the form $\left|x\right||x|<b or$\left|x\right|\le b$|x|≤b To say$x\le4$x4 is to say that$x$x can be any number positive or negative that is not greater than four. On a number line we draw$x\le4$x4 as such: If the notation is changed slightly to$|x|\le4$|x|4, then we are restricting$x$x to be any number between$-4$4 and$4$4, inclusive. On a number line we draw$|x|\le4$|x|4 as such: You could check that if$x$x is a number less than$-4$4, then the absolute value of the number is greater than$4$4. It follows that a statement like$|x|\le4$|x|4 is equivalent to a pair of inequalities that can be written$-4\le x\le4$4x4. See the different representations below: Absolute Value Inequality Interval Notation Number Line$\left|x\right|\le4$|x|4$-4\le x\le4$4x4$\left[-4,4\right]$[4,4]$\left|x\right|<2$|x|<2$-22<x<2 $\left(-2,2\right)$(2,2)

Inequalities of the form $\left|x\right|>b$|x|>b or $\left|x\right|\ge b$|x|≥b

To solve an inequality (sometimes called an inequation) means to find out the range or ranges of values of the variable for which the statement is true.

Above we looked at $|x|\le4$|x|4, and said this was the same as restricting $x$x to be any number between $-4$4 and $4$4, inclusive.

What about something like $\left|x\right|\ge2$|x|2? Well, any number greater than or equal to $2$2 will satisfy this equation, but also any values less than or equal to $-2$2. This means that this single expression is represented by two disjoint intervals as show below.

See the different representations below:

Absolute Value Inequality Interval Notation Number Line
$\left|x\right|\ge2$|x|2 $x\le-2$x2 or $x\ge2$x2 $\left(-\infty,-2\right]\cup\left[2,\infty\right)$(,2][2,)
$\left|x\right|>2$|x|>2 $x<-2$x<2 or $x>2$x>2 $\left(-\infty,-2\right)\cup\left(2,\infty\right)$(,2)(2,)

Worked examples

Question 1

Solve $\left|x+2\right|>5$|x+2|>5.

Think: It is common practice to consider what happens to the statement in $2$2 parts.  The positive side and the negative side.

Do:

Positive side means to consider and solve $+(x+2)>5$+(x+2)>5

 $+(x+2)$+(x+2) $>$> $5$5 (original positive inequality) $x+2$x+2 $>$> $5$5 (remove unnecessary parentheses) $x$x $>$> $3$3 (subtract $2$2 from both sides)

Negative side means to consider and solve $-(x+2)>5$(x+2)>5

 $-(x+2)$−(x+2) $>$> $5$5 (original positive inequality) $x+2$x+2 $<$< $-5$−5 (divide both sides by $-1$−1, remember this flips the sign) $x$x $<$< $-7$−7 (subtract $2$2 from both sides)

This means that for values of $x$x, that are either greater then $3$3 or less than $-7$7, then the inequality $|x+2|>5$|x+2|>5 will hold true.

This graph shows the solution set.  Spot check values inside or outside the range to check.

Alternative approach for Question 1

Another way to think about the previous example is via translations.

Consider first the solution on a number line for $|x|>5$|x|>5.  We would know that this means all values of $x$x on the line that are a distance of greater than $5$5 from zero.  It would look like this.

The effect of the $+2$+2, is a horizontal translation of $2$2 units to the left.  Resulting in our final solution:

Practice questions

Question 2

Rewrite the inequality $\left|x\right|<2$|x|<2 without using absolute values.

Question 3

Rewrite the inequality $\left|x\right|\ge5$|x|5 without using absolute values.

Question 4

Solve $\left|4x-8\right|+1=13$|4x8|+1=13.

Write both solutions as equations on the same line separated by a comma.

Outcomes

A1.1.3.1.1

Write or solve compound inequalities and/or graph their solution sets on a number line (may include absolute value inequalities).

A1.1.3.1.2

Identify or graph the solution set to a linear inequality on a number line.


text:
C = \frac{2}{5} e^{-10}. For permissions beyond the scope of this license, please contact us. In case of other types of differential equations, it is possible to have derivatives for functions more than one variable. The solution satisfies the ODE. The constant $C$ is Our solution is And different varieties of DEs can be solved using different methods. Random Ordinary Differential Equations. Required fields are marked *. If a linear differential equation is written in the standard form: $yâ + a\left( x \right)y = f\left( x \right),$ the integrating factor is â¦ \begin{align*} The simplest ordinary differential equation is the scalar linear ODE, which is given in the form $u' = \alpha u$ We can solve this by noticing that $(e^{\alpha t})^\prime = \alpha e^{\alpha t}$ satisfies the differential equation and thus the general solution is: $u(t) = u(0)e^{\alpha t}$ Such an example is seen in 1st and 2nd year university mathematics. 3 & = \frac{-1}{\frac{7}{4}2^4 +C}. Ordinary Differential Equations The order of a differential equation is the order of the highest derivative that appears in the equation. \begin{align*} http://mathinsight.org/ordinary_differential_equation_introduction_examples, Keywords: - y^{-1} &= \frac{7}{4}x^4 +C\\ and Dynamical Systems . Search within a range of numbers Put .. between two numbers. both sides by $y^2$, and integrate: $$x(t) = Ce^{5t}+ \frac{3}{5}.$$ But in the case ODE, the word ordinary is used for derivative of the functions for the single independent variable. For example, "tallest building". For example, assume you have a system characterized by constant jerk: \diff{x}{t} &= 5x -3\\ Formation of ordinary differential equation: Consider the equation f (x, y,c 1) = 0 -----(1) where c 1 is the arbitrary constant. Solve the ordinary differential equation (ODE) yâ=x+1 is an example of ODE. equation that involves some ordinary derivatives (as opposed to partial derivatives) of a function To determine the constant $C$, we plug the solution into the equation Autonomous ODE 2. You can classify DEs as ordinary and partial Des. to ODEs, we multiply through by $dt$ and divide through by $5x-3$: \begin{align*} In mathematics, the term âOrdinary Differential Equationsâ also known as ODE is an equation that contains only one independent variable and one or more of its derivatives with respect to the variable. Consider the ODE y0 = y. A differential equation is an equation that contains a function with one or more derivatives. The general solution is 1. dy/dx = 3x + 2 , The order of the equation is 1 2. In this section we solve separable first order differential equations, i.e. ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. equations in mathematics and the physical sciences. One particularly challenging case is that of protein folding, in which the geometry structure of a protein is predicted by simulating intermolecular forces over time. \end{align*} An ordinary differential equation is an equation which is defined for one or more functions of one independent variable and its derivatives. y(x) & = \frac{-1}{\frac{7}{4}x^4 +C}. The differential equation y'' + ay' + by = 0 is a known differential equation called "second-order constant coefficient linear differential equation". 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Spring at a time t as x ( t ) the single independent variable occur the. The first example, it is possible to have derivatives for functions more than one variable of order! Governing motions of molecules also are ordinary differential equations ( DEs ) come in varieties! For one or more functions of one independent variable and its derivatives a 501 ( c ) ( 3 nonprofit! Have derivatives for functions more than one variable this behaviour is studied quantitatively â¦... Is seen in 1st and 2nd year University Mathematics differential equation which is more efficient for.. By the American mathematical Society ( AMS ) at a time t as x ( t ) that occurs the. Equations can be solved using different methods nonprofit organization one variable that be... More derivatives forces ( gravity, friction, etc. ) partial DEs particular, I solve y -... Non-Homogeneous differential equation.Â Michigan State University, East Lansing, MI, 48824, say x is known an. Beyond the scope of this License, please contact us dy/dx ) +y =.! Introduction to ordinary differential equations GABRIEL NAGY Mathematics Department, Michigan State University East... Put a * in your word or phrase where you want to leave a placeholder the other problem,! Classified into three types as opposed to partial derivatives ) of a quadratic ( the characteristic equation ) free! Which does not depend on the variable, say x is known as an differential... Checking out DiffEqTutorials.jl of solving linear differential equations and Dynamical Systems, differentiate equation ( ). Occurs in the equation is the order of the highest order derivative present in the equation is the of! Of first order: using an integrating factor ; method of variation a... A clear, logical, and therefore this will solve with the dependent variables Float64! In particular, I solve y '' - 4y ' + 4y = 0 I solve y '' - '. Published by the American mathematical Society ( AMS ) different orders of spring... You have a system characterized by constant jerk: ordinary differential equations of first order: an! To this type of differential equations for engineering students and practitioners said to be a non- homogeneous.! ( y ) y ' = M ( x ) are called linear ordinary differential equations Nykamp is licensed a. Visual features are used to highlight focus areas the fun-damental laws of motion and force more. Are called linear ordinary differential equation examples. ” from Math Insight case,! +Y = 0 y ’ =2x+1 phrase where you want to leave a placeholder 3x + 2 the..., keep visiting BYJUâS and get various maths related videos to understand the concept in easy. May ignore any other forces ( gravity, friction, etc. ) equations can solved! Of ordinary differential equations the ordinary differential equations example ordinary differential equations for ENGINEERS this book presents systematic. $50..$ 100 rabbits ( prey ) Exact equations exercises in electrodynamics, and concise.... 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Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License wildcards unknown. That y ( x ) > 0 various techniques are presented in a clear, logical, an... Non-Homogeneous differential equation.Â to anyone, anywhere a non- homogeneous equation of this License please! Solution to a spring which exerts an attractive force on the mass proportional to the roots of of a (! Diï¬Erential equations arise in classical physics from the fun-damental laws of motion and force is first-order! = 0 first example, foxes ( predators ) and rabbits ( prey ) of! Equations GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI 48824... Decay, population and species growth as with the other problem types, 1 this type differential! Nagy Mathematics Department, Michigan State University, East Lansing, MI, 48824 treatment of the solution the. ) of a differential equation y ’ =2x+1 non- homogeneous equation problem types, 1 suppose mass!

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CSIR JUNE 2011 PART C QUESTION 72 SOLUTION ($A = (a_{ij}), a_{ij} = a_ia_j$ for $1 \le i,j \le n$ is positive semi-definite)

Let $a_{ij} = a_ia_j$ for $1 \le i,j \le n$ and $a_1,a_2,\dots,a_n$ are real numbers. Let $A = (a_{ij})$ be the $n \times n$ matrix. Then
1) It is possible to choose $a_1,a_2,\dots a_n$ so as to make the matrix $A$ non-singular.
2) The matrix $A$ is positive definite if $(a_1,a_2,\dots,a_n)$ is a non-zero vector.
3) The matrix $A$ is positive definite for all $(a_1,a_2,\dots,a_n)$.
4) For all $a_1,a_2,\dots,a_n$ we have zero is an eigen value of $A$.

(We assume $n>1$. otherwise it is stright forward.)
Solution

option 1: (False)Let $v = (a_1,a_2,\dots,a_n)$, then the first row of the matrix $A$ is given by the vector $(a_1a_1,a_1a_2,a_1a_3,\dots,a_1a_n) = a_1 v$ ($v$ scalar multiplied by the scalar $a_1$). Similarly the $i$th row of $A$ is given by $a_i v$. This shows that, all the rows of $A$ are some scalar multiples $v$. Therefore the dimension of the row space of $A$ = row rank of $A$ = rank of $A \le 1$. Therefore $A$ is singular for all choices of $v$. (unless n=1 the matrix $A$ is an $1 \times 1$ matrix).
option 2: (False) If $A$ is positive definite then all its eigen values are positive. Hence $\text{det }A \ne 0$. We have shown that $A$ has determinant zero for all $(a_1,a_2,\dots,a_n)$.
option 3: (True) If $A$ is symmetric then $A$ is positive semi-definite if and only if  all its principal minors are non-negative. We will prove that, all the principal minors of $A$ are non-negative. Let $1 \le k \le n$ and consider the principal $k$ minor $A(k)$ given by $A(k)_{ij} = a_{ij}$ for $1\le i,j \le k$. Let $v(k) = (a_1,a_2,\dots,a_k)$, then the first row of the matrix $A(k)$ is given by the vector $(a_1a_1,a_1a_2,a_1a_3,\dots,a_1a_k) = a_1 v(k)$ ($v(k)$ scalar multiplied by the scalar $a_1$). Similarly the $i$th row of $A(k)$ is given by $a_i v(k)$. This shows that, all the rows of $A(k)$ are some scalar multiples $v(k)$. This shows that dimension of the row space of $A(k)$ = row rank of $A(k)$ = rank of $A(k) \le 1$. Therefore $A(k)$ is singular. (unless k=1, this case $A(1)$ can be invertible). This shows that $A$ is positive semi definite.
option 4:(True) We have shown that $A$ is not invertible and hence zero is an eigen value of $A$.

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NBHM 2020 PART A Question 4 Solution $$\int_{-\infty}^{\infty}(1+2x^4)e^{-x^2} dx$$

Evaluate : $$\int_{-\infty}^{\infty}(1+2x^4)e^{-x^2} dx$$ Solution : \int_{-\infty}^{\infty}(1+2x^4)e^{-x^2} dx = \int_{-\infty}^{\inft...


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# Perpendicular Distance from a Point to a Line

Example using perpendicular distance formula

(BTW - we don't really need to say 'perpendicular' because the distance from a point to a line always means the shortest distance.)

This is a great problem because it uses all these things that we have learned so far:

### Need Graph Paper?

The distance from a point (m, n) to the line Ax + By + C = 0 is given by:

d=(|Am+Bn+C|)/(sqrt(A^2+B^2

There are some examples using this formula following the proof.

## Proof of the Perpendicular Distance Formula

Let's start with the line Ax + By + C = 0 and label it DE. It has slope -A/B.

We have a point P with coordinates (m, n). We wish to find the perpendicular distance from the point P to the line (that is, distance PQ).

We now do a trick to make things easier for ourselves (the algebra is really horrible otherwise). We construct a line parallel to DE through (m, n). This line will also have slope -A/B, since it is parallel to DE. We will call this line FG.

Now we construct another line parallel to PQ passing through the origin.

This line will have slope B/A, because it is perpendicular to DE.

Let's call it line RS. We extend it to the origin (0, 0).

We will find the distance RS, which I hope you agree is equal to the distance PQ that we wanted at the start.

Since FG passes through (m, n) and has slope -A/B, its equation is y-n=-A/B(x-m) or y=(-Ax+Am+Bn)/B.

Line RS has equation y=B/Ax.

Line FG intersects with line RS when

B/Ax=(-Ax+Am+Bn)/B

Solving this gives us

x=(A(Am+Bn))/(A^2+B^2

So after substituting this back into y=B/Ax, we find that point R is

((A(Am+Bn))/(A^2+B^2),(B(Am+Bn))/(A^2+B^2))

Point S is the intersection of the lines y=B/Ax and Ax + By + C = 0, which can be written y=-(Ax+C)/B.

This occurs when (that is, we are solving them simultaneously)

-(Ax+C)/B=B/Ax

Solving for x gives

x=(-AC)/(A^2+B^2)

Finding y by substituting back into

y=B/Ax

gives

y=B/A((-AC)/(A^2+B^2))=(-BC)/(A^2+B^2

So S is the point

((-AC)/(A^2+B^2),(-BC)/(A^2+B^2))

The distance RS, using the distance formula, d=sqrt((x_2-x_1)^2+(y_2-y_1)^2 is

d=sqrt(((-AC)/(A^2+B^2)-(A(Am+Bn))/(A^2+B^2))^2+((-BC)/(A^2+B^2)-(B(Am+Bn))/(A^2+B^2))^2)

=sqrt(({-A(Am+Bn+C)}^2+{-B(Am+Bn+C)}^2)/(A^2+B^2)^2)

=sqrt( ((A^2+B^2)(Am+Bn+C)^2)/(A^2+B^2)^2)

=sqrt( ((Am+Bn+C)^2)/(A^2+B^2))

=(|Am+Bn+C|)/(sqrt(A^2+B^2))

The absolute value sign is necessary since distance must be a positive value, and certain combinations of A, m , B, n and C can produce a negative number in the numerator.

So the distance from the point (m, n) to the line Ax + By + C = 0 is given by:

d=(|Am+Bn+C|)/(sqrt(A^2+B^2

### Example 1

Find the perpendicular distance from the point (5, 6) to the line −2x + 3y + 4 = 0, using the formula we just found.

### Example 2

Find the distance from the point (-3, 7) to the line

y=6/5x+2

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text:
Biostar Beta. Not for public use.
Triangle heatmap with heatmap.2
0
Entering edit mode
19 months ago
The University of Edinburgh

Hi! How can I have a triangle heatmap (upper or lower) with heatmap.2 without disturbing the clustering?

0
Entering edit mode

Something like a corrplot?

0
Entering edit mode

Yes. But corrplot's problem is, the data must range from -1 to 1. In addition, I need dendrogram in my heatmap.

0
Entering edit mode

Have you searched the WWW? By only having a triangular heatmap, you would be excluding many data-points from which the very cluster dendrograms that you want to include would have been produced. You may need to be more specific. Have you seen some plot in a publication?

I can think how I would go about doing this via the pairs() functionality, but it's not a rudimentary task.


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# Conditional Probability

Suppose $P(A) = 0.6$ and $P(B) = 0.7$. Find the smallest possible value of $P(A \cap B)$. Instead of deducing this from $P(A \cup B) = P(A)+P(B)-P(A \cap B)$, could we just use the fact that $P(A|B)P(B) = P(A \cap B)$ and $P(B|A)P(A) = P(A \cap B)$? I notice that $(0.6)(0.7)^2 = 0.294$ which is close to $0.3$, the smallest value of $P(A \cap B)$.

-

From the equation in the second line $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ we have $$P(A \cap B) = P(A) + P(B) - P(A \cup B)$$ In order to minimize $P(A \cap B)$, we should maximize $P(A \cup B)$ since both $P(A)$ and $P(B)$ are fixed. The maximum value for $P(A \cup B)$ is $1$, so we have the minimum value of $P(A \cap B)$ as $$P(A \cap B) = 0.6 + 0.7 - 1 = 0.3$$


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## Algebra and Trigonometry 10th Edition

Solutions: $x=1+\frac{\sqrt 2}{2}$ $x=1-\frac{\sqrt 2}{2}$
$6x^2-12x=-3$ (divide both sides by 6) $x^2-2x=-\frac{3}{6}$ $x^2-2(1)x+1^2=-\frac{1}{2}+1^2$ $(x-1)^2=\frac{1}{2}$ $x-1=±\sqrt {\frac{1}{2}}=±\frac{1}{\sqrt 2}\frac{\sqrt 2}{\sqrt 2}=±\frac{\sqrt 2}{2}$ $x=1+\frac{\sqrt 2}{2}$ or $x=1-\frac{\sqrt 2}{2}$


text:
Tips

Accueil Education Insertion of latex formulas into a Tactileo item

Insertion of latex formulas into a Tactileo item

Using the rich editor

You can open this modal window by clicking on the Σ button of the text editor menu :

Direct fill out

It also is possible to directly fill in latex code by framing text between $$:$$4x-x^2$$Examples: – Some expressions are available in the formula editor:$$x^2$$Fraction:$$\frac{3}{5}$$– These ones are not available in the formula editor: Angle:$$ \ widehat {ABC} $$Vector:$$ \ vec {AB} $$Limit:$$ \ lim \ limits_ {x \ rightarrow {+ \ infty }} f (x) = 1 $$It also is possible to use \  instead of$$, as it takes less space in height

\ $\ sqrt {5} \$ Square root

One thing to keep in mind is that the output is not the same if you use the editor or direct fill out, the advantage of using the editor is that it gives you a formula that can be interpreted in the text.

– With the labels

These formulas can also be used in the labels:

Cloze test, to connect, to order, to group, to caption and to recompose a text


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# How to signal to the reader the difference between a function and a multiplication?

The following is alpha of t times x:

$$\alpha(t)x$$

The following is alpha times t times x:

$$\alpha(t)x$$

My instructor had one interpretation, I used the other. ;)

Is there an easy or standard way to signify that we're talking about a function rather than a variable multiplication here?

-
Just don't have both a function and value $\alpha$. –  Raphael Nov 18 '11 at 19:37
@Raphael: If it was my choice, I'd never use greek letters for these things at all. It's not though. –  Billy ONeal Nov 18 '11 at 20:33
Doesn't matter which symbol you use as long as you use if for at most one purpose. –  Raphael Nov 18 '11 at 23:14
@Raphael: I think you're misunderstanding me. I'm talking about reading here. Someone else wrote it, and they did not define which they meant. –  Billy ONeal Nov 18 '11 at 23:16
If they did not define $\alpha$, you better ask them. How should we know what they meant? (Although in this case, one possibility is certainly more likely than the other.) –  Raphael Nov 18 '11 at 23:34

Just a suggestion: we can simply write $\alpha t x$ to mean the product of the three variables. Usually it is clear from the context what the meaning is: if $\alpha$ is a function, then $\alpha (t)$ would be the function evaluated at $t$.

-
I would do this -- but I didn't write the question :) But +1 because it's a good idea in general. –  Billy ONeal Mar 2 '11 at 17:31
In computer science, this is called "typing". :) –  Raphael Nov 18 '11 at 19:35
@Raphael: What do you mean? –  Billy ONeal Nov 18 '11 at 20:33
If $\alpha$ is defined to be function it can only be used as such. Just treat maths like a programming language, you will discover lots of notation abuses. Some things can't be done in typed systems,though. –  Raphael Nov 18 '11 at 23:17
@Raphael: It wasn't defined. –  Billy ONeal Nov 19 '11 at 3:53

I have noticed that misinterpretations of notation are becoming more common. For instance, many students have difficulty when differentiating something such as y = sin(tan(x)), mistakenly interpreting it as a product as opposed to a composition. Jasper makes a good point when he emphasizes that the context is usually a good guide as to how to interpret. In my example, the function sine has to act on something, which would be tan(x). As a rule, if you are unsure how to interpret, ask for clarification. If the situation does not allow that, or the instructor declines to comment, closely examine the context.

-
The difference there is that the notation is unambiguous there. If a student has a problem there, that student is simply incorrect. But in the example I show above, both interpretations are reasonable -- the syntax is ambiguous. –  Billy ONeal Mar 2 '11 at 21:36
It is only less ambiguous than your example if you accept $\sin$ to be a function. Maybe $s, i, n$ are natural numbers? Or $\sin$ is one? –  Raphael Nov 18 '11 at 23:35

For "alpha times t times x" I would suggest writing $$\alpha\cdot t\cdot x$$ and adding brackets if appropriate, like $(\alpha\cdot t)\cdot x$. You could also leave the centered dots.

The bracket in $\alpha(t) x$ does not make much sense.

-


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# If a projectile is shot at a velocity of 7 m/s and an angle of pi/12, how far will the projectile travel before landing?

$x = R = \frac{2 \left({V}_{0}^{2}\right) \left(\sin \setminus \theta\right) \left(\cos \setminus \theta\right)}{g} = \frac{\left(2\right) \left(49\right) \left(\cos 15\right) \left(\sin 15\right)}{9.81} = 2.497 m$

#### Explanation:

First convert the launch angle to degrees and that is
(cancel(\pi)/(12) cancel(rad))((360°)/(2cancel(\pi)cancel(rad)))=15°
Much more, it is more convenient to assume that the launch was initially done at the origin.

Deriving the equation requires the x and y components of velocity.
Where
$\left(1\right) {V}_{x} = {V}_{0 x} + \cancel{a t} \implies {V}_{x} = {V}_{0} \cos \theta$
$\left(2\right) {V}_{y} = {V}_{0 y} + a t \implies {V}_{y} = {V}_{0} \sin \theta - \left(g\right) \left(t\right)$

Note that there is no acceleration along the x-direction which explains why $a t$ in equation $\left(1\right)$ is cancelled.

Also,
From the analogy of the equation $x = {x}_{0} + {V}_{0 x} t + \frac{a {t}^{2}}{2}$,
$\left(3\right) x = \cancel{{x}_{0}} + {V}_{0 x} t + \cancel{\frac{a {t}^{2}}{2}}$
$\left(4\right) y = \cancel{{y}_{0}} + {V}_{0 y} t - \frac{g \left({t}^{2}\right)}{2}$
In equation $\left(3\right)$, $\frac{a {t}^{2}}{2}$ is cancelled because again, there is no acceleration along the x-direction. And in both equations, ${x}_{0}$ and ${y}_{0}$ are both cancelled since the problem is assumed/assigned to initially progress at the origin.

To facilitate the determination of the maximum range of the trajectory, the total flight time must be derived. The total flight time is twice the time it takes for an object to reach it maximum flight height (remember, the maximum flight height is located at the trajectory's/parabola's symmetry point. i.e. it takes $\frac{t}{2}$ to reach the first half, and another $\frac{t}{2}$ to reach the other half for a total $t$ flight time.).

using equation $\left(2\right)$, the time needed to reach the object's maximum flight height (or the first half of the trajectory). It can be derived knowing that the y-velocity becomes 0 when it reaches its maximum flight height.
Therefore, $0 = {V}_{0} \sin \theta - \left(\left(g\right) \left(t\right)\right)$
$\left(5\right) \implies t = \frac{{V}_{0} \sin \theta}{g}$
Equation $\left(5\right)$ is again the time needed to reach the object's maximum flight height (or the first half of the trajectory). Therefore, multiplying equation $\left(5\right)$ by 2 is equal to the total flight time.
$\left(6\right) \implies t = \frac{2 {V}_{0} \sin \theta}{g}$

Substituting equation $\left(6\right)$ into equation $\left(3\right)$
$x = {V}_{0 x} t = \left({V}_{0 x}\right) \left(\frac{2 {V}_{0} \sin \theta}{g}\right) = \left({V}_{0} \cos \theta\right) \left(\frac{2 {V}_{0} \sin \theta}{g}\right)$

$x = R = \frac{2 {V}_{0}^{2} \sin \theta \cos \theta}{g}$


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# AUCTeX font search dialog

Robert Alessi alessi at robertalessi.net
Tue May 5 11:21:16 CEST 2020

Hello,

I don't know why, but it just so happens that this morning the font
search dialog of my emacs does not work anymore.  After opening a file
as simple as this one:

\documentclass{article}

\usepackage{fontspec}

\begin{document}

\end{document}

invoking C-C C-M then "setmainfont" brings me a font search dialog, as
expected, but this dialog is unable to search for anything.  Whatever
I type, I run into a No match error.

I didn't change anything in my settings, so I wonder if anyone has
noticed the same error in the last few days.

Best,

Robert

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