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# Question regarding Atomic Structure (orbits)

1. May 19, 2005

### Sanosuke Sagara

I have my doubt,solution and question in the attachment that followed.Thanks for anybody that spend some time on this question.

#### Attached Files:

• ###### Doc 2.doc
File size:
31 KB
Views:
71
2. May 19, 2005

### Staff: Mentor

It's only a coincidence that you got the right number for part (a). The fact that using the same procedure for part (b) gives the wrong answer, is a good indicator that you are using the wrong procedure.

Your problem is that you are calculating $1 / \lambda$ which is not a number of revolutions. It's simply the reciprocal of the wavelength, with units of 1/meters. The answer you need doesn't have any units; it's just a "counting" number.

I suggest that you look in your textbook for equations related to the speed $v$ of the electron in an orbit with quantum number $n$, and the radius $r$ of that orbit. If you know those quantities, you can calculate the time the electron takes for one orbit (the period, $T$), and from that, the number of orbits it makes in $10^{-8}$ seconds.

3. May 19, 2005

### Dr.Brain

Ok first of all, calculate the wavelength emitted , in the same way you have calculated in your doc-attachment.then calculate the velocity of the electron in n=2 , i hope you can do that.
then v= (frequency) (wavelength)
calculate frequency with the above formula , this will be give you cycles /sec

Now you want cycles in $10^-8$ seconds , use unitary method.I am getting the correct answer.

4. May 20, 2005

### Sanosuke Sagara

Yes your both are right and I admit that I just have the 'luck' getting the first answer right and not the second.Thanks for your both detail explaination,jtbell and Dr.Brain,I really appreciate it.

format: plain


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# Calendar

DIMACS Theory of Computing Seminar

## A tale of two local models of computation

Location:  Other - CoRE 301
Date & time: Wednesday, 28 September 2016 at 11:00AM - 11:11AM

Guy Even , Tel-Aviv University: We consider two models of computation: centralized local algorithms and local distributed algorithms. Algorithms in one model are adapted to the other model to obtain improved algorithms.

Distributed vertex coloring is employed to design improved centralized local algorithms for: maximal independent set, maximal matching, and an approximation scheme for maximum (weighted) matching over bounded degree graphs. The improvement is threefold: the algorithms are deterministic, stateless, and the number of probes grows polynomially in $$\log^* n$$, where $$n$$ is the number of vertices of the input graph.

The recursive centralized local improvement technique by Nguyen and Onak is employed to obtain an improved distributed approximation scheme for maximum (weighted) matching. The improvement is twofold: we reduce the number of rounds from $$O(\log n)$$ to $$O(\log^* n)$$ and our algorithms are deterministic.

Time permitting, a centralized local algorithm for generating random preferential attachment graphs will be presented. Per query, the algorithms uses polylogarithmic random bits, time, and space.

Talk based on joint work with Reut Levy, Moti Medina, Dana Ron, and Adi Rosen.

## Special Note to All Travelers

Directions: map and driving directions. If you need information on public transportation, you may want to check the New Jersey Transit page.

Unfortunately, cancellations do occur from time to time. Feel free to call our department: 848-445-6969 before embarking on your journey. Thank you.


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# Calc question regarding the volume of my cheese

So I've just got my cheese from the cheesemonger and he's cut it along the axis $x$,$y$, and $z$ so I can do my math homework with it. The lengths of my cheese are:
$x$=6
$y$=4
$z$=2

I'm asked to find the volume using calculus, because using geometry would be cheating, evidently.

The three ways I am asked to "cut up" the cheese are perpendicular to the $y$-axis (triangles), the $x$-axis (rectangles), and $z$-axis (also rectangles).

This is easy, and it turns out the volume is 24 units cubed (all three times). E.g:

4$\int_0^6 \frac{x}{3} \mathrm{d}x = 24$

Now, I ask myself what if I want to take the other rectangle, and use that as my slice. I know that width is 4 ($y$-axis), and the length is z$\sqrt{10}$ (pythagoras with $\sqrt{z^2 + (3z)^2}$). But how do I find my limits of integration?

$4\sqrt{10}\int_a^b z \mathrm{d}z.$

I would need to know the distance from the $y$-axis to the top (in this photo) of the wedge. However, I've exhausted my math knowledge here.

I asked my calc teacher, and he told me that I find this number by "taking more calculus classes" (a good joke actually). Since I don't have time to go bug him in his office hours, can someone explain some of the different ways to find this distance?

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The volume is not $4 \sqrt{10} \int_{a}^{b} z dz$. since your integration now is not along z but along a line perpendicular to $\frac{x}{6} + \frac{z}{2} = 1$. – user17762 Feb 10 '11 at 0:45
Very interesting. I'm reading your post and gears are turning now :) . – Stephano Feb 10 '11 at 0:49
Side note: when in math will I learn how to solve this? I.e. will this be something I can do easily after I finish Calc 4? Diff eq? – Stephano Feb 10 '11 at 0:52
I dont know. I did my high-school and undergrad in a different country and I dont know how it works here. We did this in our XI which is equivalent to junior year of high school. – user17762 Feb 10 '11 at 1:04
@Stephano: the way it's done in Sivaram's answer really only involves early high-school-level geometry (albeit a creative application of it), assuming you're in the US. I guess you could also do it using vector calculus, but of course I don't know what number of calculus class that corresponds to at your school. (It does come before differential equations) – David Z Feb 10 '11 at 1:26

If we were to look at the cheese with the $Y$ axis going into the board, we will be seeing a right triangle in the $X-Z$ plane right angled at the origin.

The sides of the right triangle are given by $x=0$, $z=0$ and $\frac{x}{6} + \frac{z}{2} = 1$.

A cut parallel to the "top" side of the cheese will then be the equation of the form $\frac{x}{6} + \frac{z}{2} = c$ in the $X-Z$ plane, where $c \in (0,1)$ which is parallel to the line $\frac{x}{6} + \frac{z}{2} = 1$.

Staying in the $X-Z$ plane, the perpendicular distance, say $l$, of the line $\frac{x}{6} + \frac{z}{2} = c$ from the origin can be found as follows. We want a line passing through the origin and perpendicular to $\frac{x}{6} + \frac{z}{2} = c$. So we just need to take the negative inverse of the slope of the above line and we get $z = 3x$. The intersection of these two lines is the point where the perpendicular from the origin meets the line. The point of intersection is $(\frac{3c}{5},\frac{9c}{5})$. So $l = \sqrt{(\frac{3c}{5})^2 + (\frac{9c}{5})^2} = \frac{3c \sqrt{10}}{5}$. There are also other ways to find $l$ like equating the area of the triangle calculated by two different ways. i.e. $(6c) \times (2c) = l \times (2 \sqrt{10} c) \Rightarrow l = \frac{3c \sqrt{10}}{5}$.

Now the volume element becomes $dV = 4 \times \text{Length of the line segment} \{\frac{x}{6} + \frac{z}{2} = c\} \text{bounded by the axes} \times dl$.

Note that $4$ comes from the span along the $y$ direction which is constant for all slices.

$dV = 4 \times 2 \sqrt{10} c \times \text{ }dl$ and $dl = \frac{3 \sqrt{10}}{5} dc$.

$dV = 8 \sqrt{10} \frac{3 \sqrt{10}}{5} c dc = 48 c dc$.

Hence, the volume is given by $V = \displaystyle \int_{0}^{1} 48 c dc = 24$.

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+1 for understanding my crazy question. I'll mark as correct as soon as I understand all of this... brain... hurting... :) – Stephano Feb 10 '11 at 0:54
wow. what program can make that awesome diagram? i used photoshop as I'm clearly not a math guy. – Stephano Feb 10 '11 at 1:03


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Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English version of the page.

# gain2aperture

Convert gain to effective aperture

## Syntax

```A = gain2aperture(G,lambda) ```

## Description

`A = gain2aperture(G,lambda)` returns the effective aperture in square meters corresponding to a gain of `G` decibels for an incident electromagnetic wave with wavelength `lambda` meters. `G` can be a scalar or vector. If `G` is a vector, `A` is a vector of the same size as `G`. The elements of `A` represent the effective apertures for the corresponding elements of `G`. `lambda` must be a scalar.

## Input Arguments

 `G` Antenna gain in decibels. `G` is a scalar or a vector. If `G` is a vector, each element of `G` is the gain in decibels of a single antenna. `lambda` Wavelength of the incident electromagnetic wave. The wavelength of an electromagnetic wave is the ratio of the wave propagation speed to the frequency. For a fixed effective aperture, the antenna gain is inversely proportional to the square of the wavelength. `lambda` must be a scalar.

## Output Arguments

 `A` Antenna effective aperture in square meters. The effective aperture describes how much energy is captured from an incident electromagnetic plane wave. The argument describes the functional area of the antenna and is not equivalent to the actual physical area. For a fixed wavelength, the antenna gain is proportional to the effective aperture. `A` can be a scalar or vector. If `A` is a vector, each element of `A` is the effective aperture of the corresponding gain in `G`.

## Examples

collapse all

An antenna has a gain of 3 dB. Calculate the antenna's effective aperture when used to capture an electromagnetic wave with a wavelength of 10 cm.

`a = gain2aperture(3,0.1)`
```a = 0.0016 ```

collapse all

### Gain and Effective Aperture

The relationship between the gain, G, in decibels of an antenna and the antenna’s effective aperture is:

`${A}_{e}={10}^{G/10}\frac{{\lambda }^{2}}{4\pi }$`

where λ is the wavelength of the incident electromagnetic wave.

## References

[1] Skolnik, M. Introduction to Radar Systems, 3rd Ed. New York: McGraw-Hill, 2001.


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Neural Light Transport for Relighting and View Synthesis

Author & Institution

XIUMING ZHANG, Massachusetts Institute of Technology
SEAN FANELLO and YUN-TA TSAI, Google
TIANCHENG SUN, University of California, San Diego
TIANFAN XUE, ROHIT PANDEY, SERGIO ORTS-ESCOLANO, PHILIP DAVIDSON, CHRISTOPH RHEMANN, PAUL DEBEVEC, and JONATHAN T. BARRON, Google
RAVI RAMAMOORTHI, University of California, San Diego
WILLIAM T. FREEMAN, Massachusetts Institute of Technology & Google

Abstract

• Light transport (LT)

• The light transport (LT) of a scene describes how it appears under different lighting conditions from different viewing directions
• Complete knowledge of a scene’s LT enables the synthesis of novel views under arbitrary lighting
• In this paper

• Focus on image-based LT acquisition, primarily for human bodies within a light stage setup
• Propose a semi-parametric approach for learning a neural representation of the LT that is embedded in a texture atlas of known but possibly rough geometry
• Model all non-diffuse and global LT as residuals added to a physically-based diffuse base rendering
• Show how to fuse previously seen observations of illuminants and views to synthesize a new image of the same scene under a desired lighting condition from a chosen viewpoint
• Allows the network to learn complex material effects (such as subsurface scattering) and global illumination (such as diffuse interreflection), while guaranteeing the physical correctness of the diffuse LT (such as hard shadows)
• With this learned LT, one can relight the scene photorealistically with a directional light or an HDRI map, synthesize novel views with view-dependent effects, or do both simultaneously, all in a unified framework using a set of sparse observations

1. Introduction

• Light Transport

• Models how light interacts with objects in the scene to produce an observed image
• Inferring light transport

• Acquiring the LT of a scene from images of that scene requires untangling the myriad interconnected effects of occlusion, shading, shadowing, interreflections, scattering, etc
• Application

• Phototourism
• Telepresence
• Storytelling
• Special effects
• Generating ground truth data for machine learning task
• Many works rely on high-quality renderings of relit subjects under arbitrary lighting conditions and from multiple viewpoints
• Relighting
• View synthesis
• Re-enacting
• Alpha-matting
• Previous work has shown that it is possible to construct a light stage , plenoptic camera , or gantry that directly captures a subset of the LT function and thereby enables the image-based rendering thereof

• Widely used in film productions and within the research community
• Can only provide sparse sampling of the LT limited to the number of LEDs(~ 300 on a spherical dome) and the number of cameras (~50-100 around the subject), resulting in the inability to produce photorealistic renderings outside the supported camera/light locations
• Traditional image-based rendering approaches are usually designed for fixed viewpoints and are unable to synthesize unseen (novel) views under a desired illumination
• In this paper

• Learn to interpolate the dense LT function of a given scene from sparse multi-view, One-Light-at-A-Time (OLAT) images acquired in a light stage, through a semi-parametric technique that called Neural Light Transport (NLT)
• Many prior works have addressed similar tasks with classic works tending to rely on physics to recover analytical and interpretable models
• Recent works using neural networks to infer a more direct mapping from input images to an output image
• Rendering method

• Make simplifying assumptions when modeling geometry, BRDFs, or complex inter-object interactions
• Make the problem tractable
• Deep learning approaches
• Can tolerate geometric and reflectance imperfections
• Require many aspects of image formation (even those guaranteed by physics) be learned “from scratch,” which may necessitate a prohibitively large training set
• NLT
• Construct a classical model of the subject being imaged (a mesh and a diffuse texture atlas per Lambertian reflectance)
• Embed a neural network within the parameterization provided by that classical model
• Construct the inputs and outputs of the model in ways that leverage domain knowledge of classical graphics techniques
• Train that network to model all aspects of LT, including those not captured by a classical model
• Able to learn an accurate model of the complicated LT function for a subject from a small training dataset of sparse observations
• A key novelty: the learned model is embedded within the texture atlas space of an existing geometric model of the subject, which provides a novel framework for simultaneous relighting and view interpolation
• Express the 6D LT function at each location on the surface of the geometric model as simply the output of a deep neural network
• Works well (as neural networks are smooth and universal function approximators) and obviates the need for a complicated parameterization of spatially-varying reflectance

• Main contribution
• An end-to-end, semi-parametric method for learning to interpolate the 6D light transport function per-subject from real data using convolutional neural networks
• A unified framework for simultaneous relighting and view synthesis by embedding networks into a parameterized texture atlas and leveraging as input a set of One-Light-at-A-Time (OLAT) images
• A set of augmented texture-space inputs and a residual learning scheme on top of a physically accurate diffuse base, which together allow the network to easily learn non-diffuse, higher-order light transport effects including specular highlights, subsurface scattering, and global illumination
• Problem
• Recovering a model of light transport from a sparse set of images of some subject
• Predicting novel images of that subject from unseen views and/or under unobserved illuminations

Single observation

• The most sparse sampling is just a single image, from which one could attempt to infer a model (geometry, reflectance, and illumination) of the physical world that resulted in that image
• Via hand-crafted or learned priors
• Though practical, the quality gap between what can be accomplished by single-image techniques and what has been demonstrated by multi-image techniques is significant
• Can’t show complex light transport effects such as specular highlights or subsurface scattering
• Limited to a single task, such as relighting, and some support only a limited range of viewpoint change

Multiple views

• Multiview geometry techniques recover a textured 3D model that can be rendered using conventional graphics or photogrammetry techniques
• Have material and shading variation baked in
• Do not enable relighting
• Image-based rendering techniques such as light fields or lumigraphs
• Can be used to directly sample and render the plenoptic function
• The accuracy of these techniques is limited by the density of sampled input images
• Do not enable relighting
• Reprojection-based methods
• For unstructured inputs
• Assume the availability of a geometry proxy (so does our work), reproject nearby views to the query view, and perform image blending in that view
• Rely heavily on the quality of the geometry proxy
• A class-specific geometry prior (such as that of a human body) can be used to increase the accuracy of a geometry proxy
• Cannot synthesize pixels that are not visible in the input views
• Do not enable relighting
• Deep learning
• Been used to synthesize new images from sparse sets of input images
• Usually by training neural networks to synthesize some intermediate geometric representation that is then projected into the desired image
• Some techniques even entirely replace the rendering process with a learned “neural” renderer
• Generally do not attempt to explicitly model light transport
• Do not enable relighting
• Capable of preserving view-dependent effects for the fixed illumination condition under which the input images were acquired
• Often breaks “backwards compatibility” with existing graphics systems

Multiple illuminants

• Repeatedly imaging a subject with a fixed camera but under different illuminations and then recovering the surface normals
• Most photometric stereo solutions assume Lambertian reflectance and do not support relighting with non-diffuse light transport
• Neural networks can be applied to relight a scene captured under multiple lighting conditions from a fixed viewpoint

Multiple views and illuminant

• Utilize the symmetry of illuminations and view directions to collect sparse samples of an 8D reflectance field, and reconstruct a complete field using a low-rank assumption

• Lack an explicit geometric model

• Rendering is limited to a fixed set of viewpoints
• Supports relighting and view synthesis

• But assume pre-defined BRDFs
• Cannot synthesize more complex light transport effects present in real images
• In this paper

• Follows the convention of the nascent field of “neural rendering”, in which a separate neural network is trained for each subject to be rendered, and all images of that subject are treated as “training data.”
• These approaches have shown great promise in terms of their rendering fidelity
• But require per-subject training and are unable to generalize across subjects yet
• Paper’s approach
• Unlike prior work that focuses on a specific task
• The texture-space formulation allows for simultaneous light and view interpolation
• The model is a valuable training data generator for many works that rely on high-quality renderings of subjects under arbitrary lighting conditions and from multiple viewpoints

3. Method

• The framework is a semi-parametric model with a residual learning scheme that aims to close the gap between the diffuse rendering of the geometry proxy and the real input image

• The semi-parametric approach is used to fuse previously recorded observations to synthesize a novel, photorealistic image under any desired illumination and viewpoint
• The method relies on recent advances in computer vision that have enabled accurate 3D reconstructions of human subjects

• Such as technique that takes as input several images of a subject and produces as output a mesh of that subject and a UV texture map describing its albedo
• At first glance, this appears to address the entirety of the problem: given a textured mesh, we can perform simultaneous view synthesis and relighting by simply re-rendering that mesh from some arbitrary camera location and under some arbitrary illumination
• Simplistic model of reflectance and illumination only permits equally simplistic relighting and view synthesis
• Assuming Lambertian reflectance:
$$\tilde L_o(\pmb x,\pmb \omega_o)=\rho(\pmb x)L_i(\pmb x,\pmb \omega_i)(\pmb \omega_i\cdot \pmb n(\pmb x))$$

• $\tilde L_0(\pmb x,\pmb\omega_0)$ is the diffuse rendering of a point $\pmb x$ with a surface normal $\pmb n(\pmb x)$ and albedo $\rho(\pmb x)$, lit by a directional light $\pmb \omega_i$ with an incoming intensity $L_i(\pmb x,\pmb \omega_i)$ and view from $\pmb \omega_0$

• This reflectance model is only sufficient for describing matte surfaces and direct illumination

• More recent methods also make strong assumptions about materials by modeling reflectance with a cosine lobe model

• The shortcomings of these methods are obvious when compared to a more expressive rendering approach, such as the rendering equation, which makes far fewer simplifying assumptions:

$$L_o(\pmb x,\pmb \omega_o)=L_e(\pmb x,\pmb \omega_o)+\int_\Omega f_s(\pmb x,\pmb \omega_i,\pmb \omega_o)L_i(\pmb x,\pmb \omega_i)(\pmb \omega_i\cdot \pmb n(\pmb x))\mathrm d\pmb \omega_i$$

• Limitations in computing $\tilde L_o(\pmb x,\pmb \omega_o)$

• It assumes a single directional light instead of integrating over the hemisphere of all incident directions $\Omega$
• It approximates an object’s BRDF $f_s(\cdot)$ as a single scalar
• It ignores emitted radiance $L_e(\cdot)$ (in addition to scattering and transmittance, which this rendering equation does not model either)
• The goal of the learning-based model is to close the gap between $L_o(\pmb x,\pmb \omega_o)$ and $\tilde L_o(\pmb x,\pmb \omega_o)$, and furthermore between $L_o(\pmb x,\pmb \omega_o)$ and the observed image

• Motivation

• the geometry and texture atlas offers us a mapping from each image of a subject onto a canonical texture atlas that is shared across all views of that subject
• Approach

• Use geometry and texture atlas to map the input images of the subject from “camera space” (XY pixel coordinates) to “texture space” (UV texture atlas coordinates)
• Use a semi-parametric neural network embedded in this texture space to fuse multiple observations and synthesize an RGB texture atlas for the desired relit and/or novel-view image
• Warped back into the camera space of the desired viewpoint, thereby giving us an output rendering of the subject under the desired illumination and viewpoint

The Demo can explain everything:

4. Limitations

1. The method must be trained individually per scene, and generalizing to unseen scenes is an important future step for the field
2. The fixed 1024×1024 resolution of the texture-space model limits the model’s ability to synthesize higher-frequency contents
• Especially when the camera zooms very close to the subject, or when an image patch is allocated too few texels
• This could be solved by training on higherresolution images, but this would increase memory requirements and likely require significant engineering effort
3. Has occasional failure modes, where complex light transport effects, such as the ones on the glittery chain, are hard to synthesize, and the final renderings lack high-frequency details

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WRITTEN BY
Wenbo Chen
CG Student


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Now showing items 84-103 of 119

• #### Shear flow dynamics in the Beris-Edwards model of nematic liquid crystals

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We consider the Beris-Edwards model describing nematic liquid crystal dynamics and restrict to a shear flow and spatially homogeneous situation. We analyze the dynamics focusing on the effect of the flow. We show that in ...
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Spectral properties and the confinement phenomenon for the coupling $H + V$ are studied, where $H =-i\alpha \cdot \nabla + m\beta$ is the free Dirac operator in $\mathbb{R}^3$ and $V$ is a measure-valued potential. The ...
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(2017-12-15)
This thesis is devoted to the study of the Dirac Hamiltonian perturbed by delta-type potentials and Coulomb-type potentials. We analysed the delta-shell interaction on bounded and smooth domains and its approximation by ...
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(2017-02-02)
In this note we consider the 1-D cubic Schrödinger equation with data given as small perturbations of a Dirac-$\delta$ function and some other related equations. We first recall that although the problem for this type of ...
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(Journal of Mathematical Analysis and Applications, 2018-10)
We prove that there exist infinitely many distributional solutions with infinite kinetic energy to both the incompressible Navier-Stokes equations in $\mathbb{R}^2$ and Burgers equation in $\mathbb{R}$ with vanishing ...
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(Rendiconti Lincei - Matematica e Applicazioni, 2017-05-30)
In this note we prove an end-point regularity result on the $L^P$ integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known ...
• #### Sparse bounds for maximal rough singular integrals via the Fourier transform

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We prove a quantified sparse bound for the maximal truncations of convolution-type singular integrals with suitable Fourier decay of the kernel. Our result extends the sparse domination principle by Conde-Alonso, Culiuc, ...
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We prove sparse bounds for pseudodifferential operators associated to H\"ormander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of ...
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In this note, we show that if $T$ is a multilinear singular integral operator associated with a kernel satisfies the so-called multilinear $L^{r}$-Hörmander condition, then $T$ can be dominated by multilinear sparse operators.
• #### Spectral asymptotics for $\delta$-interactions on sharp cones

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We investigate the spectrum of three-dimensional Schr\"odinger operators with $\delta$-interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues ...
• #### Spectral asymptotics of the Dirichlet Laplacian in a conical layer

(Communications on Pure and Applied Analysis, 2015-05-01)
The spectrum of the Dirichlet Laplacian on conical layers is analysed through two aspects: the infiniteness of the discrete eigenvalues and their expansions in the small aperture limit. On the one hand, we prove that, for ...
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We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing ...
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We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary ...
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We consider an energy functional motivated by the celebrated K13 problem in the Oseen-Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an ...
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(2016-12-21)
We develop an approach to prove self-adjointness of Dirac operators with boundary or transmission conditions at a $C^2$-compact surface without boundary. To do so we are lead to study the layer potential induced by the ...
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(Communications on Pure and Applied Analysis (CPAA), 2015-07-01)
This paper focuses on surveying some recent results obtained by the author together with V. Banica on the evolution of a vortex filament with one corner according to the so-called binormal flow. The case of a regular polygon ...
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In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the ...
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text:
# Extreme Value Theorem - Find the distribution of $M_n$

This is my problem:

Consider an i.i.d sequence of random variables $$X_1, X_2,\ldots,X_n,$$ and define $$m_n = 1 - \min(X_1, \ldots, X_n)$$. Find the distribution of $$m_n$$ in terms of $$F_X$$ the CDF of X .

For context, I have solved a similar problem which is shown below:

Consider an i.i.d sequence of random variables $$X_1, X_2,\ldots, X_n,$$ and define $$M_n = \max(X_1, \ldots, X_n)$$.

Find the distribution of $$M_n$$ in terms of $$F_X$$ the CDF of X .

\begin{align} P(M_n \leq x) &= P(\max(X_1, \ldots, X_n) \leq x) \\&= P(X_1 \leq X_1,\ldots, X_n \leq x) \\&= P(X_1 \leq x)\ldots P(X_n \leq x) \\&= F_x(x) \ldots F_x(n) \\&=(F_x(x))^n \end{align}

I am just wondering if i am missing some rule between max() and min() that I could use to solve this problem? or could i use the following

$$P(\min(X,Y) \leq x) = F_x(x)+ F_y(x) - F_x(x) F_y(x)$$

, but I am not sure how to include that with the 1 - term in the probability function.

$$\{m_n\leq x\}=\{1-\min(X_1,\dots,X_n)\leq x\}=\{\min(X_1,\dots,X_n\geq1-x\}=$$$$\{X_1\geq 1-x\}\cap\dots\cap\{X_n\geq1-x\}$$
On base of independepence we find:$$P(\{X_1\geq 1-x\}\cap\dots\cap\{X_n\geq1-x\})=P(X\geq1-x)^n=(1-P(X<1-x))^n\tag1$$
Here: $$P(X<1-x)=\lim_{z\downarrow x}F_X(1-z)$$ and equals $$F_X(1-x)$$ if $$P(X=1-x)=0$$.
In that case you can write the RHS of $$(1)$$ as: $$(1-F_X(1-x))^n$$


text:
# CJA 304 Week 5

363.8k points

CJA 304 Week 5

363.8k points

Thumbnail of first page

Excerpt from file: Due Day 3 (Thursday): Submit a response to Discussion Question #1 in the Main forum and begin commenting on each others responses. Please post your response in the DQ threads. Your response to the question should be 200 to 300 words. Please post your response to the following Discussion Question

Filename: CJA 304 Week 5.zip

Filesize: 2.7M

Print Length: 4 Pages/Slides

Words: NA

Account not required

Surround your text in *italics* or **bold**, to write a math equation use, for example, $x^2+2x+1=0$ or $$\beta^2-1=0$$
Use LaTeX to type formulas and markdown to format text. See example.


text:
# Is the trace of a Lyaponov transform of a semistable matrix always nonpositive?

Let $A$ be a semistable real matrix (i.e. the real parts of all the eigenvalues of $A$ are nonnegative). Let $P$ be a positive definite matrix.

Is it always true that $\text{trace}{A^{T}P+PA} \leq 0$?

P.S. In fact, slightly more is known about $A$: that $A+A^{T}$ is negative semidefinite.

-

Here is a counterexample. Notice that $tr(A^TP)=tr(PA)$, so it suffices to calculate $tr(PA)$. Suppose $n=2$ and $A$ triangular: $$P=\begin{pmatrix} x & y \\\\ y & z \end{pmatrix},\qquad A=\begin{pmatrix} a & b \\\\ 0 & c \end{pmatrix},$$ where $x,z,xz-y^2$ positive, and $a,c\le0$. We have $tr(PA)=ax+by+cz$. Suppose that $y\ne0$. Then you may chose $b$ so that $tr(PA)>0$.


text:
# Lessons 1–5Self Assessment 1

For each statement, rate your understanding according to the continuum, and then provide evidence for your rating.

I understand and can do it accurately.I understand most of the time, but I’m still working on it.I don’t understand this yet.
I can

find the center of dilation and scale factor for two similar figures where the image is formed by a dilation of the pre-image.

Evidence Rating

I can

dilate a figure when given the center of dilation and the scale factor.

Evidence Rating

I can

define similarity in two ways: one in terms of transformations, and the other in terms of corresponding angles and sides.

Evidence Rating

I can

write and solve correct proportionality statements for segments formed on the sides of a triangle when a line is drawn parallel to the base.

Evidence Rating

## Next steps

My plan for mastering this content is…

# Lessons 6–7Self Assessment 2

For each statement, rate your understanding according to the continuum, and then provide evidence for your rating.

I understand and can do it accurately.I understand most of the time, but I’m still working on it.I don’t understand this yet.
I can

find the midpoint of a line segment on a coordinate grid.

Evidence Rating

I can

find the point that partitions a line segment in a given ratio (for example, ) on a coordinate grid.

Evidence Rating

I can

prove the Pythagorean theorem using proportionality.

Evidence Rating

I can

write and solve correct proportionality statements for segments formed when an altitude is drawn in a right triangle perpendicular to the hypotenuse.

Evidence Rating

## Next steps

My plan for mastering this content is…

# Lessons 8–11Self Assessment 3

For each statement, rate your understanding according to the continuum, and then provide evidence for your rating.

I understand and can do it accurately.I understand most of the time, but I’m still working on it.I don’t understand this yet.
I can

define the trigonometric ratios—sine, cosine and tangent—in terms of the sides of a right triangle and an acute angle.

Evidence Rating

I can

find the values of the trigonometric ratios for a right triangle given the measure of one of its acute angles and one of its sides.

Evidence Rating

I can

explain and use trigonometric relationships such as and .

Evidence Rating

I can

apply trigonometric ratios to solve practical situations, such as finding the angle of elevation given the rise and run.

Evidence Rating

## Next steps

My plan for mastering this content is…


text:
## Stream: new members

### Topic: Running the linter locally

#### Frédéric Dupuis (Jul 21 2020 at 00:15):

Is there a way to run the linter locally to make sure all the checks pass before committing?

#### Jalex Stark (Jul 21 2020 at 00:26):

#lint

#### Jalex Stark (Jul 21 2020 at 00:26):

i usually put it at the bottom of the file but maybe one can put it anywhere instead

#### Floris van Doorn (Jul 21 2020 at 05:09):

To elaborate on Jalex: #lint checks all declarations in that file (above the #lint command). There is also #lint_mathlib that checks in all files (in mathlib) that you've imported.

#### Frédéric Dupuis (Jul 21 2020 at 13:21):

Thanks!

Last updated: May 14 2021 at 05:20 UTC


text:
# Happy National Donut Day 2015

“National Donut Day 2015 will take place on Friday, June 5th. While this annual event is a great reason to enjoy this timelessly tasty treat, there’s more to the celebration than mere sweet indulgence.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


text:
Diameter of a Cylinder Calculator

Created by Mariamy Chrdileli
Reviewed by Wojciech Sas, PhD candidate
Last updated: Apr 09, 2022

Use this diameter of a cylinder calculator as an easy-to-use tool that calculates the diameter of a cylinder for you! The calculator can also come in handy if you're interested in the volume of a cylinder. If you're curious about the formula for the diameter of a cylinder and how to find the diameter of a cylinder, come along!

How do I find diameter of a cylinder? The formula for diameter of a cylinder

To find the diameter of a cylinder, you can utilize the following formula:

$d= 2×\sqrt \frac {V}{πh},$

where:

• $d$ is the diameter of a cylinder;
• $V$ is the volume of a cylinder; and
• $h$ is the height of a cylinder.

For instance, if the volume of a cylinder is 60 cm³ and the height is 8 cm, the diameter of the cylinder in centimeters would be:

$d= 2×\sqrt \frac {60}{π8} = 3.09$

Looks cumbersome? Don't worry; the diameter of a cylinder calculator will do all the computations for you!

Now that you know how to find the diameter of a cylinder, check out other cylinder related tools, similar to the diameter of a cylinder calculator, that are also convenient and easy to use:

FAQ

How do I calculate the diameter of a cylinder given the volume and height?

To calculate the diameter of a cylinder given the volume and height:

1. Multiply the height of a cylinder by pi;
2. Divide the volume of a cylinder by the number computed in Step 1;
3. Square root the number computed in Step 2;
4. Multiply your result by far by 2; and
5. Ta-da! You have calculated the diameter of a cylinder given the radius and height.

How do I find the volume of a cylinder with diameter and height?

To find the volume of a cylinder with a diameter, you need to:

1. Divide the diameter of a cylinder by two, and then square the number;
2. Take the result you computed in step 1, and multiply it by the height of a cylinder and pi; and
3. Voilà! You have calculated the volume of a cylinder.
Mariamy Chrdileli
Right / oblique full cylinder
Height (h)
in
in
Volume (V)
cu in
Diameter (d)
in
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text:
## Geometry & Topology

### Hyperbolic cusps with convex polyhedral boundary

#### Abstract

We prove that a $3$–dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthermore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp.

The proof uses the discrete total curvature functional on the space of “cusps with particles”, which are hyperbolic cone-manifolds with the singular locus a union of half-lines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced metric on the boundary and the curvatures of the singular locus.

Our main theorem is equivalent to a part of a general statement about isometric immersions of compact surfaces.

#### Article information

Source
Geom. Topol., Volume 13, Number 1 (2009), 457-492.

Dates
Revised: 7 October 2008
Accepted: 17 September 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800186

Digital Object Identifier
doi:10.2140/gt.2009.13.457

Mathematical Reviews number (MathSciNet)
MR2469522

Zentralblatt MATH identifier
1179.57026

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 53C24: Rigidity results

#### Citation

Fillastre, François; Izmestiev, Ivan. Hyperbolic cusps with convex polyhedral boundary. Geom. Topol. 13 (2009), no. 1, 457--492. doi:10.2140/gt.2009.13.457. https://projecteuclid.org/euclid.gt/1513800186

#### References

• A Alexandroff, Existence of a convex polyhedron and of a convex surface with a given metric, Rec. Math. [Mat. Sbornik] N.S. 11(53) (1942) 15–65
• A D Alexandrov, Convex polyhedra, Springer Monographs in Math., Springer, Berlin (2005) Translated from the 1950 Russian edition by N S Dairbekov, S S Kutateladze and A B Sossinsky, With comments and bibliography by V A Zalgaller and appendices by L A Shor and Yu A Volkov
• M T Anderson, Scalar curvature and the existence of geometric structures on $3$–manifolds. I, J. Reine Angew. Math. 553 (2002) 125–182
• E M Andreev, Convex polyhedra in Lobačevskiĭ spaces, Mat. Sb. $($N.S.$)$ 81 (123) (1970) 445–478
• W Blaschke, G Herglotz, Über die Verwirklichung einer geschlossenen Fläche mit vorgeschriebenem Bogenelement im Euklidischen Raum, Sitzungsber. Bayer. Akad. Wiss., Math.-Naturwiss. Abt. 2 (1937) 229–230
• A I Bobenko, I Izmestiev, Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes, Ann. Inst. Fourier $($Grenoble$)$ 58 (2008) 447–505
• F Bonsante, J-M Schlenker, AdS manifolds with particles and earthquakes on singular surfaces, To appear in Geom. Funct. Anal.
• R D Canary, D B A Epstein, P L Green, Notes on notes of Thurston [MR0903850], from: “Fundamentals of hyperbolic geometry: selected expositions”, London Math. Soc. Lecture Note Ser. 328, Cambridge Univ. Press (2006) 1–115 With a new foreword by Canary
• A-L Cauchy, Sur les polygones et polyèdres (Second Mémoire), from: “Œvres complètes, Seconde série 1” (1905) 26–38
• R Connelly, J-M Schlenker, On the infinitesimal rigidity of weakly convex polyhedra
• F Fillastre, Fuchsian polyhedra in Lorentzian space-forms
• F Fillastre, Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces, Ann. Inst. Fourier $($Grenoble$)$ 57 (2007) 163–195
• F Fillastre, Polyhedral hyperbolic metrics on surfaces, Geom. Dedicata 134 (2008) 177–196
• F Fillastre, I Izmestiev, Dual metrics of hyperbolic cusps with convex polyhedral boundary, In preparation (2008)
• C D Hodgson, Deduction of Andreev's theorem from Rivin's characterization of convex hyperbolic polyhedra, from: “Topology '90 (Columbus, OH, 1990)”, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 185–193
• C D Hodgson, I Rivin, A characterization of compact convex polyhedra in hyperbolic $3$–space, Invent. Math. 111 (1993) 77–111
• C Indermitte, T M Liebling, M Troyanov, H Clémençon, Voronoi diagrams on piecewise flat surfaces and an application to biological growth, Theoret. Comput. Sci. 263 (2001) 263–274 Combinatorics and computer science (Palaiseau, 1997)
• I Izmestiev, A variational proof of Alexandrov's convex cap theorem, To appear in Discrete Comput. Geom.
• K Krasnov, J-M Schlenker, Minimal surfaces and particles in $3$–manifolds, Geom. Dedicata 126 (2007) 187–254
• F Labourie, J-M Schlenker, Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante, Math. Ann. 316 (2000) 465–483
• F Luo, Rigidity of polyhedral surfaces
• J Milnor, Collected papers. Vol. 1. Geometry, Publish or Perish, Houston, TX (1994)
• S Moroianu, J-M Schlenker, Quasi-Fuchsian manifolds with particles
• I Rivin, On geometry of convex polyhedra in hyperbolic $3$–space, PhD thesis, Princeton University (1986)
• I Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. $(2)$ 139 (1994) 553–580
• J-M Schlenker, Hyperbolic manifolds with polyhedral boundary, Available online at http://www.picard.ups-tlse.fr/\char'176schlenker/texts/ideal.pdf
• J-M Schlenker, On weakly convex star-shaped polyhedra
• J-M Schlenker, Surfaces convexes dans des espaces lorentziens à courbure constante, Comm. Anal. Geom. 4 (1996) 285–331
• J-M Schlenker, Convex polyhedra in Lorentzian space-forms, Asian J. Math. 5 (2001) 327–363
• J-M Schlenker, Hyperbolic manifolds with convex boundary, Invent. Math. 163 (2006) 109–169
• S Sechelmann, Alexandrov polyhedron editor http://www.math.tu-berlin.de/geometrie/ps/software.shtml
• W P Thurston, Shapes of polyhedra and triangulations of the sphere, from: “The Epstein birthday schrift”, Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 511–549
• Y A Volkov, Existence of a polyhedron with a given development, PhD thesis, Leningrad State University (1955) In Russian
• Y A Volkov, Existence of convex polyhedra with prescribed development I, Vestn. Leningr. Univ. 15 (1960) 75–86


text:
Polynomials can be used to model many aspects of the physical world. Adding and subtracting polynomials can be done by finding the like terms and adding or subtracting those terms according to the symbol of operation. Like terms are the similar terms whose variables are same.

In Adding Polynomials first clear the parenthesis then combine the like terms. Finally add the combined like terms.

Below you could see examples

### Solved Examples

Question 1: Solve $(2x^{2} - 4)+ (x ^{2} +5x-5)$
Solution:
$2x^{2} -4 + x^{2} + 5x - 5$

= $2x^{2} + x^{2} + 5x - 4 - 5$

= $3x^{2} + 5x - 9$

Question 2: Solve ($2y^4 + 3y + 2y^3 + 4y^2 + 7$) + ($4y^4 + 5y^3 + 4y+ 5$)
Solution:
$2y^4 + 2y^3 + 4y^2 + 3y + 7$

$4y^4 + 5y^3 + 0y^2 + 4y + 5$

$6y^4 + 7y^3 + 4y^2 + 7y + 12$


text:
## Abstract

One of the major challenges to build a task-oriented dialogue system is that dialogue state transition frequently happens between multiple domains such as booking hotels or restaurants. Recently, the encoderdecoder model based on the end-to-end neural network has become an attractive approach to meet this challenge. However, it usually requires a sufficiently large amount of training data and it is not flexible to handle dialogue state transition. This paper addresses these problems by proposing a simple but practical framework called Multi-Domain KB-BOT (MDKB-BOT), which leverages both neural networks and rule-based strategy in natural language understanding (NLU) and dialogue management (DM). Experiments on the data set of the Chinese Human-Computer Dialogue Technology Evaluation Campaign show that MDKB-BOT achieves competitive performance on several evaluation metrics, including task completion rate and user satisfaction.

## 1. INTRODUCTION

In the past decade, dialogue systems have become an attractive topic and they can be classified into open-domain dialogue systems and task-oriented dialogue systems. One general approach to dialogue system design is to treat it as a retrieval problem by learning the relevance matching score between user queries and system responses. Inspired by the recent advances in deep learning, building an end-to-end dialogue system has been a popular approach for its flexibility and extendibility. For example, encoderdecoder models based on recurrent neural networks (RNNs) directly maximize the likelihood of the desired responses when previous dialogue history data are available. However, two major drawbacks of those systems are that multiple training corpora are required and generic responses such as “I do not know” are likely to be generated. These drawbacks limit the generalization ability, especially for a task-oriented system in which knowledge from multiple domains is needed to understand users' underlying intents.

Compared to end-to-end approach, designing a task-oriented dialogue system as modularized pipeline is feasible. And each essential component is trained individually, including 1) Natural Language Understanding (NLU), to specify task domain and user intent and extract slot-value pairs, 2) Dialogue Manager (DM), to keep tracking the dialogue state and guide users to achieve a desired goal, and 3) Natural Language Generation (NLG), to generate responses. One of the challenges for a task-oriented dialogue system is that dialogue state transition frequently happens between multiple domains. If earlier components make mistakes in slot value extraction and errors are accumulated, the entire system's functionality will be severely impaired.

To address the complex dialogue state transition problem, we adopt the architecture of modularized pipeline and propose a multi-domain KB-BOT (MDKB-BOT), which leverages both rule extraction and neural networks. We run the evaluation experiments on the data set of the Chinese Human-Computer Dialogue Technology Evaluation Campaign and experimental results show that MDKB-BOT can robustly fulfill the frequent changes of user intent among three domains (flight, train and hotel) and achieve competitive scores based on human evaluation metrics.

## 2. RELATED WORK

As mentioned before, there have been a lot of research efforts in applying deep learning to task-oriented dialogue systems. One of the most effective approaches is to build a modularized pipeline system by connecting NLU, DM and NLG together. Traditional approach to NLU is to model domain classification/intent detection as sentence classification while treating slot value pairs extraction as a sequence labeling task. A desirable NLU system should not be sensitive to intent error and slot error, especially for slot filling. For example, Xu and Sarikaya [1] applied a RNN to perform contextual domain classification and used a triangular conditional random field (CRF) based on a convolutional neural network for intent detection and slot filling. Jaech, Heck and Ostendorf [2] applied multi-task learning to achieve the goal to leverage knowledge of source domains with rich data to improve the performance on the target domain with little data. Bapna et al. [3] explored the role of context information in NLU via injecting previous dialogues into a RNN based encoder and a memory network.

On the other hand, many attempts have been made to improve the architecture of DM. Recent research indicates that reinforcement learning (DL) holds promise for planning a dialogue policy based on the current dialogue state. Williams, Asadi and Zweig [4] proposed a model called Hybrid Code Networks (HCNs), which is a mixture of supervised learning and reinforcement learning. HCNs select a dialogue action every step by optimizing the reward for completing a task with policy gradient [5]. Faced with the sparse nature of the reward signal in RL, Peng et al. [6] designed an end-to-end framework for hierarchical RL where a MANAGER is used to choose current goal (like a specific domain task) and a WORKER is used to take actions and help users finish the current subtask. Inspired by recent advances in RL, Mrkšíc et al. [7] introduced a belief tracker that can overcome the drawback of requiring a large amount of hand-crafted lexicons to capture some of the linguistic variation in users' language. Their Neural Belief Tracking (NBT) models can reason over pre-trained word embeddings of system output, user utterance and candidate pairs in databases.

As for NLG, most of the current work applied information retrieval technique to a large query-response database, or used template-based methods with a set of rules to map frames to natural language or generation models. Dušek and Jurčíček [8] encoded frames based on the syntax tree and used the seq2seq model for generation.

## 3. PROPOSED FRAMEWORK

The proposed framework is illustrated in Figure 1, which includes NLU, DM and NLG. The implementation of these components is described from Section 3.1 to Section 3.3.

Figure 1.

The overall framework of the model consists of three components: (1) NLU module, which predicts intent domain and gives slot value of user utterance, (2) DM module, which outputs the dialogue action to NLG module, and (3) NLG module, which generates the final response.

Figure 1.

The overall framework of the model consists of three components: (1) NLU module, which predicts intent domain and gives slot value of user utterance, (2) DM module, which outputs the dialogue action to NLG module, and (3) NLG module, which generates the final response.

### 3.1 Natural Language Understanding (NLU)

The main tasks for NLU involve domain classification, intent detection and slot filling as illustrated in Figure 1.

#### 3.1.1 Domain Classification

A convolutional neural network proposed by Kim [9] was adopted in domain classification. Let WRv*d be the d dimensional word embedding table, where v is the vocabulary size. Then sentence semantic representation of user query XRn*d is obtained by looking up each word in W, where n is the number of words in this query. Then 1-D convolutional layer is adopted on X to extract n-gram features. However, in a convolutional neural network (CNN) errors may happen in some cases containing several domains' descriptions. For example, “The train is cheaper, but to save time, give me airline flight schedules and flight timetables.” Thus, for our online model, some rule strategies are used to deal with this misclassification problem by constructing a keyword list from both corpora and databases, e.g., city name list.

#### 3.1.2 Slot Filling

Slot filling is treated as a name entity recognition task where the popular bigin-in-out (BIO) format is used for representing tags of each word in a query. Then Long Short-Term Memory (LSTM) scans the words and outputs the representation:

$fk=α(Wfxk+Ufhk−1+bf),$
(1)
$ik=α(Wfxk+Uihk−1+bi),$
(2)
$ok=α(Woxk+Uohk−1+bo),$
(3)
$c˜k=tanh(Wcxk+Uchk−1+bc),$
(4)
$ck=fk⊙ck−1+ik⊙c˜k,$
(5)
$hk=ok⊙tanh(ck).$
(6)

To enhance the ability to extract slot value pairs, a CRF network is connected to the output of LSTM or bidirectional LSTM (BLSTM). Then the score of a sentence X along with a path of tags Y can be calculated as the sum of the transition score A and the LSTM network score f:

$S(X,Y,θ)=∑t=1T([A]Yt−1,Yt+[fϑ]Yt,Xt),$
(7)

where θ is the trainable parameter of the LSTM network.

Figure 2 shows a bidirectional LSTM network enhanced with a CRF network on the top layer. For our online model, we apply both keyword matching and BLSTM-CRF to avoid cases like diverse or nonstandard expressions.

Figure 2.

BLSTM-CRF model for slot filling.

Figure 2.

BLSTM-CRF model for slot filling.

#### 3.1.3 Intent Detection

Based on slots extracted from BLSTM-CRF, we update the maintained dialogue state template. Then user intent is inferred by comparing the predefined dialogue template with the new state template.

### 3.2 Dialogue Management (DM)

After the NLU module, we obtain the output of NLU that includes the user intent domain and the slot value of the current turn.

In order to avoid too many unnecessary turns of dialogue on some insignificant information for many users, we divide all of the slots into two categories: required slots and extra slots. The required slots, like <departCity>, are necessary for the task, and the extra slots may make the dialogue too tedious for many users who do not care them, such as <trainValue> and <countRate>. So we only complete the required slots necessary for booking, but if a user mentions extra slots in the dialogue, we will consider it while retrieving the information from our data base.

To regulate the dialogue course when our system interacts with users, DM module is then applied to update the conversation state and the next dialogue action. We divide this module into three states. The detailed procedures are described as follows.

• Initial state At the beginning of the conversation, utterance with no explicit intention will be considered a purposeless talk. After identifying a user's intent, the system will turn into the slot filling state. Note that the system will store the slot information even before domain prediction, and the information will be distributed to the corresponding slot afterward.

• Slot filling state The main task at this state is to interactively interact with the user to obtain the required slot information for generating responses.

• Recommendation state Our bot will list the results that can adapt to users' demands by retrieval and extraction of the data base. In case of failure, we set a series of strategies for similar recommendations, such as: (1) remove the limitations of extra slots, (2) make appropriate adjustments for the departure time, (3) change the cabin or train type, and (4) increase the price range. In addition, users can change their requests and return to the slot filling state or recommendation state again.

Throughout the whole process of the dialogue, when the system finds that an intent cannot be completed, it will prompt the user in time to avoid wasting time. For example, when a user wants to book a flight to a city where no air service is available, it is unwise to continue the dialogue with the user, and the system will recommend other means of traveling. A user can also change his or her intent anytime, and our system will store the common information of slots automatically at the transformation process.

### 3.3 Natural Language Generation (NLG)

So far, we have obtained both the category of a user query's intention and the next dialogue action each turn, which guide the NLG module to generate natural language texts and replay the user's query. Given the user's slots list, we convert it into the SQL statement, then retrieve from the date base which stores information on trains, flights and accomendations, to check if there are eligible items for the user's goal, and match the appropriate template for replay.

Because of the shortage of the large-scale dialogue corpus in these domains, we generate the utterance by the template-based NLG, which means we need to capture every case of different slot states to presuppose the dialogue template. In this way, once the user dialogue actions are found in the predefined sentence templates, we will replace the slot value with user history information. One advantage is that it can ensure the controllability of the response given to users.

## 4. EXPERIMENTS

In the Chinese Human-Computer Dialogue Technology Evaluation Campaign, a task-oriented dialogue system is developed to help users book flights, trains and hotels.

### 4.1 Data Sets

Since only three databases are provided, we extend the data set of task 1 for domain classification and rule extraction. We also annotate a 300-dialogue corpus (about 1,500 sentences for training) with slot labels for evaluating LSTM-CRF and BLSTM-CRF models. Table 1 and Table 2 show the details of the data set.

Table 1.

The statistics of training corpus for domain classification.

TrainFlightHotelOthers
#sentence 533 510 512 588
TrainFlightHotelOthers
#sentence 533 510 512 588
Table 2.

The statistics of training corpus for slot filling.

#label#sentenceAvg. sentence length (word)
TrainFlightHotel
13 15 1,013 5.86
#label#sentenceAvg. sentence length (word)
TrainFlightHotel
13 15 1,013 5.86

### 4.2 Evaluation

For slot filling in NLU, the entity-level prediction F1 score of common name entity recognition is adopted. However, dialogue evaluation still remains to be a difficult task. We use the evaluation metrics of the Chinese Human-Computer Dialogue Technology Evaluation Campaign, including task completion rate, user satisfaction score, dialogue naturalness, number of turns and robustness of uncovered cases.

## 5. DISCUSSION

Table 3 shows the results of slot filling. We compare the performance of LSTM-CRF and BLSTM-CRF with unigram and unigram plus bigram separately. As illustrated, accuracy can increase by 1.7% when considering word sequence order with BLSTM. Using bigram feature can be of help for LSTM-CRF, though it is worse for BLSTM as the average length of bigram sequence is short. One possible way is to use character embedding instead of word embedding.

Table 3.

Comparison of labeling performance on NLU.

FeatureLSTM-CRFBLSTM-CRF
Unigram 85.93% 87.63%
Unigram + bigram 86.20% 87.21%
FeatureLSTM-CRFBLSTM-CRF
Unigram 85.93% 87.63%
Unigram + bigram 86.20% 87.21%

Table 4 illustrates the performance of our system according to the evaluation metrics mentioned before. Most of the metrics are annotated manually except for the average dialogue turn. our system obtained the best score on user satisfaction, dialogue naturalness and boot ability due to the reasonable dialogue intent transition template we predefined. But this leads to a decline in task performance, especially when the user intent is not identified or the important slot value is not extracted correctly.

Table 4.

Dialogue quality evaluation results between top 4 teams in the competition.

MetricShenSiKaoPuTaoWeiDuMDKB-BOTChuMenWenWen
Completion rate 31.75 19.05 19.05 11.11
#turn 64.53 72.28 78.72 71.39
User satisfaction −1 −2
Naturalness −1 −1
Boot ability
MetricShenSiKaoPuTaoWeiDuMDKB-BOTChuMenWenWen
Completion rate 31.75 19.05 19.05 11.11
#turn 64.53 72.28 78.72 71.39
User satisfaction −1 −2
Naturalness −1 −1
Boot ability

## 6. CONCLUSION

In this paper, we proposed a simple but practical framework for multi-domain task-oriented dialogue system. Our model leverages both neural network and rule-based strategy to handle the domain transition problem. It achieves competitive results on the Chinese Human-Computer Dialogue Technology Evaluation Campaign, especially for user-friendliness and utterance guidance metrics. For future work, we are going to apply end-to-end neural networks to NLG based on the information extracted and maintained in NLU and DM to improve the system performance.

## AUTHOR CONTRIBUTIONS

Y. Lao (laoyadi@bupt.edu.cn) and W. Liu (liuweijie@bupt.edu.cn) are the leaders of the MDKB-Bot system, who drew the whole framework of the system. S. Gao (gaosheng@bupt.edu.cn, corresponding author) and S. Li (lisi@bupt.edu.cn) summarized the applications and drafted the paper. All authors participated in the revision of the manuscript.

## ACKNOWLEDGEMENTS

This work was supported by Beijing Natural Science Foundation (No. 4174098), National Natural Science Foundation of China (No. 61702047) and the Fundamental Research Funds for the Central Universities (No. 2017RC02).

## REFERENCES

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P.
Xu
, &
R.
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Williams
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Hybrid code networks: Practical and efficient end-to-end dialogue control with supervised and reinforcement learning
. arXiv preprint. arXiv: 1702.03274, 2017.
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Li
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Celikyilmaz
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Composite task-completion dialogue policy learning via hierarchical deep reinforcement learning
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Sequence-to-sequence generation for spoken dialogue via deep syntax trees and strings
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This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a full description of the license, please visit https://creativecommons.org/licenses/by/4.0/legalcode.


text:
# Can measurement of the momentum of a particle can be done without observing its position?

Uncertainty principle says that one cannot measure exactly the position and momentum of a particle at same time. As per common understanding when we are measuring momentum of an object it is implicit that we aware of its position. My doubt is in Quantum world, how can we measure momentum of a particle without knowing its position. Are the two momentum and position are mutually exclusive?

• Unfortunately you are using somewhat layman's interpretation of the uncertainty principle. You can make a measurement of the position and momentum of a particle at the same time. What the uncertainty principle tells us is that If you were to run your experiment many times you would find a spread in your measurements of both position ($\Delta x$) and momentum ($\Delta p$), and that for any system you will never be able to get the product of these two below a certain amount ($\Delta x\cdot\Delta p\geq\hbar/2$) – Aaron Stevens Jun 27 at 18:19
• look atmy answer here physics.stackexchange.com/questions/479475/… where bubble chamber tracks are discussed, momentum measured by $Bqv=mv^2/r$ taking into account the ionisation loss,and the interaction point measured at the main vertex. – anna v Jun 27 at 18:45
• @AaronStevens I am not sure that your statement about the possibiiity of a measurement of the position and momentum of a particle at the same time is consistent with the request of QM that the effect of any measurement is to project the wavefunction onto the eigenstate corresponding to the measured eigenvalue. A really simultaneous measurement would imply a common eigenvecctor of two non-commuting operators. – GiorgioP Jun 28 at 3:34
• @GiorgioP No. Claiming that the results of your measurements were both eigenvectors of your initial state implies a common eigenvector of two non-commuting operators. Or if the measurements weren't simultaneous, claiming that the second measurement doesn't change the state that the system was in after the first measurement. Even though they are closely related, non-common eigenvalues and the HUP are not saying the exactly same thing. – Aaron Stevens Jun 28 at 9:58
• @AaronStevens I fully agree with such a point of view. – GiorgioP Jun 28 at 23:03

The reason this is interesting is that while the Heisenberg principle limits your precision in measuring both $$x$$ and $$p_x$$ at the same time it does not limit your precision in measuring $$y$$ and $$p_x$$ at the same time.
• Something that confuses me is the statement of "measuring $x$ and $p_x$ at the same time". Won't the HUP only manifest itself after multiple measurements of similar systems? Isn't the precision of a single measurement more about the detector? – Aaron Stevens Jun 27 at 23:38


text:
# The differential equation for the variation of amount of salt in a tank is given by:(dx/dt) + (x/20) = 10. Where x is in kg and t is in minutes. Assuming that at time zero there is no slat in the tank, find the time at which the amount of salt increases to 100 kg

This question was previously asked in
ISRO Scientist ME 2013 Paper
View all ISRO Scientist ME Papers >
1. 100 ln2
2. 50 ln2
3. 20 ln2
4. 10 ln2

Option 3 : 20 ln2
Free
ISRO Scientist ME 2020 Paper
1504
80 Questions 240 Marks 90 Mins

## Detailed Solution

Concept:

The differential equation is, (dx/dt) + (x/20) = 10.

$$\frac{{dx}}{{dt}} + \frac{x}{{20}} = 10$$

$$\frac{{dx}}{{dt}} = 10 - \frac{x}{{20}} = \frac{{200 - x}}{{20}}$$

⇒  $$\smallint \frac{{dx}}{{200 - x}} = \smallint \frac{{dt}}{{20}}$$

⇒ $$- \ln \left( {200 - x} \right) = \frac{t}{{20}} + C$$

At t = 0, x = 0 ⇒ C = -ln 200

$$- \ln \left( {200 - x} \right) = \frac{t}{{20}} - \ln \left( {200} \right)$$

⇒ $$\ln \left( {200} \right) - \ln \left( {200 - x} \right) = \frac{t}{{20}}$$

$$t = 20\ln \left( {\frac{{200}}{{200 - 100}}} \right)$$

t = 20 ln2


text:
### Abstract

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in $O (\min(n^\omega, n^2 + nh \log h + \chi^2))$ time, where $\omega<2.373$ denotes the matrix multiplication exponent and $\chi\in \Omega(n)\cap O(n^2)$ is the number of edges of the graph of oriented distances. We also provide an alternative algorithm for computing the diameter that runs in $O(n^2 \log n)$ time.

Publication
In 34th European Workshop on Computational Geometry (EuroCG 2018) and 29th International Symposium on Algorithms and Computation (ISAAC 2018)


text:
## Page Content

The Ogham alphabet was developed in Ireland during the early Christian period in Ireland and was used primarily to label gravestones (Ogham stones). The language used was a form of Irish, but one that pre-dates later Old Irish texts.

Note: This Ogham alphabet should not be confused with the Runic alphabet. Although the forms look superficially similar, the pronunciation and symbols are quite different.

### Vertical vs. Horizontal Text

On the stone inscriptions, text was written vertically bottom to top, but a manuscript tradition of horizontal Ogham from left to right has also developed. Since support for vertical text is not widespread, horizontal display is currently recommended.

## Font Recommendations

### Test Passage

If you see a question mark or a bunch of squares below, then you need to install a runic font. See the list in the following section for details.

(286) LUGUDECA
ᚂᚒᚌᚒᚇᚓᚉᚐ

## Inputting Ogham

### Inputting from Character Utilities

For short passages, it may be practical to input characters from either the Windows Character Map or the Macintosh Character Viewer/Palette. In some cases, you may need to adjust the font to a Ogham font.

### Typing Ogham Characters

Neither Microsoft or Apple provide a Ogham keyboard, but there may several freeware keyboard utilities which can be installed
Note: Not all applications have been tested.

## Web Development

This section presents information specific to the Runic script. For general information about developing non-English Web sites, see the Encoding Tutorial or the Web Layout sections.

### Unicode

Unicode (utf-8) is the required encoding to correctly display Ogham. Pages not encoded as Unicode may not display Ogham characters correctly.

### Language Codes

Language Tags allow browsers and other software to process Runic text more efficiently. For Ogham, the language is usually Primitive Irish.

• pgl (Primitive Irish)

### Unicode Entity Codes

Ogham numeric Unicode entity codes can be used for small pieces of text or when other
methods to not work.


text:
# Tuning a Telemetry Antenna with a VNA

### Help Support The Rocketry Forum:

#### OverTheTop

##### Well-Known Member
TRF Supporter
Tuning a Telemetry Antenna with a VNA

As part of my Vertical Trajectory System I am adding another telemetry downlink from the rocket, and for that I need an antenna. I want something robust and simple so I chose to just make a simple dipole that can be taped to the outside, or inside, of the rocket airframe. This was accomplished by just using a SMA pigtail and stripping the coax. This however had to be tuned to the correct frequency.

I grabbed my Nano VNA, and quickly calibrated it for a frequency range of 1-3GHz, to suit my target frequency of 1.28GHz.

Connecting the commercial antenna to it quickly confirmed the center frequency of that antenna at 1.24GHz.

Next up I grabbed my roughly-trimmed dipole and connected it, revealing a frequency of 1.1GHz in air (look for the resonance dip in the yellow trace. That is where more energy is going out into the air).

I then taped the antenna to the airframe, simulating its final mounting to some degree. The frequency was shifted to below 1GHz due to the fiberglass dielectric in proximity to the dipole.

Based on the reduction of frequency due to the fiberglass and the low frequency from the first attempt I calculated the length required and cut the dipole. The result is an antenna on the 1.28GHz I wanted.

I took the antenna and taped it to the stick of a Q-tip for now. It will be quite adequate for bench testing.

I really can’t wax lyrical enough about the VNA. Such a handy tool to have around, and the whole process of tuning the antenna took around 15 minutes. Totally amazing!

You are amazing

#### cerving

TRF Supporter
It really is worth the money. Don't hesitate to push the button. Very easy to use, but you might need to purchase some more adapters. $129USD sounds a bit expensive, unless you are talking the larger screen. Make sure you get the V2 that has native capabilities to 3GHz IIRC, rather than the earlier version that just uses harmonics to get up that high. Last edited: #### Voyager1 ##### Well-Known Member TRF Supporter Thanks for presenting this information OTT. This is really interesting stuff and close to one of my current design projects - albeit, a little inactive at present due to other commitments. I totally agree with the usefulness of the Nano VNA V2. It's great! Also, as Cris mentioned, the RF Explorer is a great little spec analyser, as we use one for confirming the frequency and power output from our member's telemetry Txs. In your case, where you form the dipole elements from the stripped coax, although the coax (looks like RG316 or similar) is, I assume 50 Ohm, the dipole feed point will be around 70+ Ohms when the dipole is straight. However, I believe that as you bend the dipole elements towards each other, the characteristic impedance reduces as the bend increases. This is useful because it might bring the feed impedance closer to the desired 50 Ohms of the coax and the VNA, but the penalty might be in a reduction in overall gain. Additionally, there might also be an increase in the SWR either side of that balance point. As for the far field radiation pattern, there is probably no significant change due to the bending of the dipole in that orientation. Curving towards or away from a ground plane might be a different case, though. It would be interesting to compare this configuration with the dipole along the main axis of the body. In this way you might be able to determine the effects of the dipole bending vs the dielectric on the characteristics. Also, what are the dipole characteristics when curved in free space vs being curved around the dielectric body cylinder. As an afterthought, would you consider using two dipoles with cross polarisation? One one each side of the body. They could be coupled through a simple 2-1 power divider/combiner. Keep up the good work. Last edited: #### OverTheTop ##### Well-Known Member TRF Supporter Good stuff Voyager. Yes the impedance does drop as a make a V with the dipole. That was likely the configuration I was going to run with in the end. What I have is good enough for bench testing for now. I seem to remember having the antenna in a V formation also means less change in signal level due to wrong polarisation matching with the receiver. Not sure the mechanism but I seem to recall reading it is so. I am considering going circular polarisation but keeping it simple initially. At the moment I have a lend of a spectrum analyser from work so I don't need to buy one just yet. There is a superfluity of them there currently so I am permitted to borrow one. #### Voyager1 ##### Well-Known Member TRF Supporter At the moment I have a lend of a spectrum analyser from work so I don't need to buy one just yet. There is a superfluity of them there currently so I am permitted to borrow one. Lucky lad! Having retired from the profession (if you ever do!) a couple of years ago, I no longer have access to the R&Ss and Agilents, etc. I just have to make the best of these cheap units, but they are very good value for money. #### OverTheTop ##### Well-Known Member TRF Supporter Work also have three Agilent VNAs, ranging from a cheap$10k FieldFox, and upwards. The NanoVNA suits my needs well, so I don't need to borrow one of those! Takes up less bench space too .

I had been looking at one of those RF Explorer units but have not bitten yet.

#### cerving

##### Owner, Eggtimer Rocketry
TRF Supporter
It really is worth the money. Don't hesitate to push the button. Very easy to use, but you might need to purchase some more adapters. $129USD sounds a bit expensive, unless you are talking the larger screen. Make sure you get the V2 that has native capabilities to 3GHz IIRC, rather than the earlier version that just uses harmonics to get up that high. I just pulled the trigger on one of the new 4" ones, got it for$90. Should be fun to play with!

TRF Supporter

#### OverTheTop

##### Well-Known Member
TRF Supporter
I paid about \$110AUD for mine IIRC. I think that was with cal kit as well.

Don't forget to buy some connector savers as shown in my original thread too.


text:
# Fraction No More!

How many integer values of $$y$$ are there such that $$\dfrac{5y^2 - 9}{2y+6}$$ is also an integer?

×


text:
# Po-Lam Yung

Title: Maximal operators associated to Hilbert transforms along variable parabolas in $\mathbb{R}^2$

Abstract: For $u \in (0,\infty)$, the Hilbert transform along the parabola $\{(t,ut^2) \colon t \in \mathbb{R}\}$ is defined by
$$H^u f(x) = \text{p.v.} \int_{\mathbb{R}} f(x_1 - t, x_2 - ut^2) \frac{dt}{t}.$$
For $U \subset (0,\infty)$ we study the maximal operator
$$f(x) \mapsto \sup_{u \in U} |H^u f(x)|.$$
Via a local smoothing estimate for certain Fourier integral operators and a Chang-Wilson-Wolff inequality for martingales, we show that for $p \in (2,\infty)$, the above maximal operator is bounded on $L^p(\mathbb{R}^2)$ if and only if $U$ can be covered by finitely many dyadic intervals. We will also briefly discuss the case when $p \in (1,2]$. This is joint work with Shaoming Guo, Joris Roos and Andreas Seeger.

Slides


text:
# Sets,Real Number System and Logic.

Sets Real number System and logic

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics.

A set can be explained in different ways:

Listing method: A = {a, b, c, .z}

Descriptive method: N = {the natural numbers from 1 to 50}

Set builder method: A – B  = A -  (A∩ B )

Venn – diagram

Universal sets

A universal set is the collection of all objects in a particular context or theory. All other sets in that framework constitute subsets of the universal set, which is denoted as letter U. The objects themselves are known as elements or members of U.

Subsets

The set made by elements of the universal sets is called subsets of the universal sets

For example

U = {1, 2, 3, 4, …………..50}

A = {even integers from 1 to 50}

B   = {odd numbers from 1 to 50}

Here, A and B are the subsets of U

Overlapping sets

Two sets are said to be overlapped if they have same element in common.

A∩ B = {6}

A and B are overlapping sets.

Disjoint sets

Two sets are said to be disjoint sets if there is no element in common.

Cardinality of the sets

The number of the elements in the given sets is known as cardinality of sets.

A = {1, 2, 5,}

B = {5, 3, 4}

AUB = {1, 2, 3, 4, 5}

n(A) = 3

n(B) = 3

n(AUB) = 5

Cardinality of the three sets

Let  A and B and C  represent three sets as a shown in the figure s

n(AUBUC) =  n(A) + n(B) + n(C) - n(A∩ B) -- n(B∩ C) - n(C∩ A) +n(A∩ B∩ C)

Operation of sets

Union of sets

The set which includes elements of A and B is called union of the sets.

A = {1, 2, 5,}

B = {5, 3, 4}

AUB = {1, 2, 3, 4, 5}

Or, AUB = {x: xϵ A or xϵ B}

n(A) = 3

n(B) = 3

n(AUB) = n(A) + n(B) - n(A∩ B)

= 3 +3 -1 = 5

Intersections of sets

If the elements of set belongs to both Sets A and B, it is called intersection of A and B.

A = {1, 2, 5}

B = {5, 3, 4}

A∩ B = {5}

n(A∩ B) = n(A) – n0 (A) = 3 -2 = 1

Or, A∩ B = {x: xϵ A and x ϵ B}

Complement of sets

The set that contains all the elements of universal sets except the given set A is called complement of the set A. It is denoted by  A̅

Difference of sets

If A and B are the two sets , the difference of the dsets is the elements of the set thst includes only in one set .

A – B  = A -  (A∩ B )

B- A    = B  -  (A∩ B )

Example

Worked out examples

Let U = { a, b, c, d, e,f,g,h}  A =  {a,b,c,d } B  = {c,d,e,f }  and C = {d,e,f,g,h }.verify the following relations

a. AUB (B ∩ C) = (AUB)∩(A∩ C)b. A∩ (BUC) = (A∩B) U (A∩ C)

c.${\rm{\: }}\overline {{\rm{AUB}}}$ = A̅ ∩ B̅

d. A – (B U C)= (A – B) ∩ (A – C).

Soln:  Here,

A U B = {a, b, c, d} U {c, d, e, f} = {a, b, c, d, e, f}

A U C = {a, b, c, d} U {d, e, f, g, h} = {a, b, c, d, e, f, g, h}

B ∩ C = {c, d, e, f} U {d, e, f, g, h} = {d, e, f}

a.

L.H.S. = A U (B ∩ C)

= {a, b, c, d, e, f} U {d, e, f} = {a, b, c, d, e, f}

R.H.S. = (A U B) ∩ (A U C)

= {a, b, c, d, e, f} ∩ {a, b, c, d, e, f, g, h} = {a, b, c, d, e, f}

Hence, A U (B ∩ C) = (A U B) ∩ (A U C)

b.

A ∩ B = {a, b, c, d} ∩ {c, d, e, f} = {c, d}

A ∩ C = {a, b, c, d} ∩ {d, e, f, g, h} = {d}

B U C = {c, d, e, f} U {d, e, f, g, h} = {c, d, e, f, g, h}

Now, LHS = A ∩ (B U C) = {a, b, c, d} ∩ {c, d, e, f, g, h} = {c, d}

RHS = (A∩B) U (A ∩ C) = {c, d} U {d} = {c, d}

Hence. A ∩ (B U C) = (A∩B) U (A ∩ C)

c.

soln:

A¯ = U – A = {a, b, c, d, e, f, g, h} – {a, b, c, d} = {e, f, g, h}

B¯ = U – B = {a, b, c, d, e, f, g, h} – {c, d, e, f} = {a, b, g, h}

Now, LHS = AUB¯¯¯¯¯¯¯¯¯¯¯AUB¯

= U – (AUB)

= {a, b, c, d, e, f, g, h} – {a, b, c, d, e, f} = {g, h}

R.H.S. = A¯∩B¯A¯∩B¯ = {e, f, g, h} ∩ {a, b, g, h} = {g, h}

Hence, AUB¯¯¯¯¯¯¯¯¯¯¯=A¯∩B¯AUB¯=A¯∩B¯.

d.

Here,

A – B = {a, b, c, d} – {c, d, e, f} = {a, b}

A – C = {a, b, c, d} – {d, e, f, g, h} = {a, b, c}

Now, LHS = A – (B U C) = {a, b, c, d} – {c, d, e, f, g, h} = {a, b}

RHS = (A – B) ∩ (A – C) = {a, b} ∩ {a, b, c} = {a, b}

Hence, A – (B U C)= (A – B) ∩ (A – C).

In a survey it was found that 8.%  people like oranges ,85% like mangoes and 75% like both But 45 people like none of them  Drawing Venn –diagram , find the number of the people who  were in the survey.

Soln:

Let O and M be the number of people who like oranges and mangoes respectively.

n (U) =  n(OUM) + $\overline {{\rm{n}}\left( {{\rm{OUM}}} \right)}$

100%   =  n (O) + n (M) - n(O∩M) + $\overline {{\rm{n}}\left( {{\rm{OUM}}} \right)}$

100% =  80% + 85% - 75% + $\overline {{\rm{n}}\left( {{\rm{OUM}}} \right)}$

$\overline {{\rm{n}}\left( {{\rm{OUM}}} \right)} {\rm{\: }}$  = 10%

According to the question,

10% of total number (say x)  = 45

Or,  x=  $\frac{{4500}}{{10}}$ = 450

Real number System

The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as  √ 2  (1.41421356…, the square root of 2 an irrational algebraic number.

Natural numbers

A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways: N = {0, 1, 2, 3,}

Integers

An integer is a whole number that can be positive, negative, or zero. Examples of integers are: -5, 1, 5, 8, 97, and 3,043. Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, .09, and 5,643.1.

Field of axiom

Closure property

If a , b ϵ R and a+b ϵ R

Commutative property

If a, b ϵ R, a + b = b + a

Associative property

a+b  c ϵ R then, a+(b+c)  = (a+b)+ c

a ϵ R  ,there is real number  0  such that a+ 0 = 0 + a =a

a  ϵ R  -a ϵ R a+ (-a) = (-a)  +a  = 0 –a is the additive inverse of a

Multiplicative Axioms

Closure property

If a , b ϵ R and a.b ϵ R

Commutative property

If a , b ϵ R and a.b  = b.a ϵ R

Associative property

If a , b ϵ R  a(bc) =  (ab)c

Multiplicative Inverse

a ϵ R  (a ≠ 0 ) a-1  ϵ R  aa-1  = a-1  a

Distributive property

If a, b, c ϵ R then, a (b+c) = ab +bc

Order axioms

Real number satisfies the order axiom.

If a > 0, b > 0 then ab> 0

Trichotomy property

a, b,  ϵ R a< b   a = b  a > b

Transitivity property

a, b, c ϵ R a >b ,  b > c ⇒   a > c

If a < b a+ c < b+ c

Multiplicative property

If a > b   $\frac{{\rm{a}}}{{\rm{c}}}{\rm{\: }} > \frac{{\rm{b}}}{{\rm{c}}}{\rm{\: \: }}$when c > 0

If a > b   $\frac{{\rm{a}}}{{\rm{c}}}{\rm{\: }} > \frac{{\rm{b}}}{{\rm{c}}}{\rm{\: \: }}$when c < 0

The axiom of the density

If a and b are the two real numbers such that a< b  , then there exists areal number c such that a <  c < b

Example

Solve the following inequalities

(a)

|x + 2| < 4.

=  – 4 < x + 2 < 4

=  – 4  – 2 < x + 2 – 2 < 4 – 2

=  – 6 < x < 2.

So, the solution is {x: –6 < x < 2} and the graph is:

(b)

|x – 1| ≤ 2.

=  – 2 ≤ x – 1 ≤ 2

=  – 2 + 1 ≤ x – 1 + 1≤2 + 1

=  – 1 ≤ x ≤ 3.

So, the solution is {x: –1 ≤ x ≤ 3} and the graph is:

(c)

|2x + 3| ≤ 1.

=  – 1 ≤ 2x + 3 ≤ 1

=  – 1 – 3 ≤ 2x +3 –3 ≤ 1 –3

=  – 4 ≤ 2x ≤  –2.

=  – 2 ≤ x ≤  – 1

So, the solution is {x: –2 ≤ x ≤ –1} and the graph is:

(d)

| x-1|>1

Here, two cases arises,

Case I:

When (x – 1) > 0 then.

= x – 1 + 1 > 1 + 1

So, x > 2, i.e. x ԑ (2, ∞)

Case II:

When (x – 1) < 0 then,

= x – 1 < – 1

= x – 1 + 1 < – 1 + 1

So, x < 0 i.e. x ԑ (–∞, 0)

Hence, the required solution of |x – 1| > 1 is {x:x< 0 or x > 2} i.e. x ԑ (–∞, 0) U (2, ∞)

The graph is:

(e)

|2x + 1| ≥ 3

Here, two cases arises

Case I:

When (2x + 1) > 0 then.

(2x + 1) ≥ 3

= 2x + 1 – 1 ≥ 3 – 1

= 2x ≥ 2

So, x ≥ 1 i.e. x ԑ [1, ∞)

Case II:

When (2x + 1) < 0 then,

– (2x + 1) > 3

= 2x + 1 ≤  – 3

= 2x + 1 – 1 ≤  – 3 – 1

= 2x ≤  – 4

So, x ≤  – 2.i.e. x ԑ (–∞,  – 2]

Thus, the required solution is {x:x ≤  – 2 or x ≥ 1}

I.e. x ԑ(–∞. – 2] U [1, ∞) and the graph is.

Logic

Logic is the process by which we arrive at the conclusion from the given statement with the valid reason.

Logic expressions are written in symbolic form.

Statement

An assertion which is expressed in words, symbols, which is either true or false but not both at same time, is known as a statement.

There are two types of statement:

Simple statement: A statement that declares only one thing

Compound statement: A statement that explains two or more than two things

Logical connectives

Compound statements are made up of simple statement by words or phrase like and or If ….then and   if and only if’’ .They are known as logical connectives or simply connectives.

Conjunctions: If p and q are two statements are joined by the conjunction to make compound statement. The symbol is denoted by   Λ  . I.e. p Λ q

Disjunction: Two simple statements combined by the word or to form a compound statement is known as disconjunction .It is denoted by V.

Negation: The statement which denies the given statement is called negation of the given statement. The symbol for the negation   is ∼. Negation of p is denoted    ∼p.

Conditional (Implication): Two simple statement combined by the word  If ……..then ‘’ to form a compound statement is known as the conditional of the given statement. The symbol used for implication is ⇒  .

Biconditional (Equivalence) :Two simple  statement combined with word   if and only if ‘ to form a compound statement , is known as biconditional .The symbol used for biconditional  is ⇔ .

Tautology: A compound statement which is always true, whatever be the truth value of the components ,is known as tautology .

Laws of logic

Let p, q and r be any three statements

a. Law of excluded middle

Only one statement is true p or ∼ p is true.

b. law of tautology

The disjunction of a statement and its negation is a tautology.

i.e  p V ∼ p

The conjunction of a statement and its negation is a contradiction

i.e  p   Λ ∼ p

d. law of involution

The negation of the negation of the statement is the given statement.

e. Law of Syllogism

If  p ⇒  q and q⇒  r then p⇒  r

( p ⇒  q)  Λ (q⇒  r) ⇒  (P ⇒  r)

f. law of contra positive

( p ⇒  q)   ≡   (∼ q ) ⇒  (∼ p )

g. Inverse Law

(∼ P)  ⇒ (∼ q)  ≡ q ⇒p

The inverse and converse of a conditional are logically equivalent .

Example

If p and q are any two statements, prove that

a. p Λ ~p ≡ c.

b. p v q ≡ p v p.

c. ~[pv(~q)] ≡~p ^ q.

d.. ~p[(~p ^ q)] ≡ p v (~q).

Solution:

(a)

We prove the given relations using truth table.

 p ~p p ^ ~p c = contradiction T T F F F F T T F F F F F F F F

So, from the above truth table, p ^ ~p ≡≡ c.

(b)

We know the given relations using truth table,

 p q pvq qvp T T F F T T T F T T T F T T T F

From the above truth table, p v q ≡ p v p.

(c)

We prove the given relations using truth table.

 p q ~p ~q pv(~q) ~[pv(~q)] ~p^q T T F F T F T F F F T T F T F T T T F T F F T F F F T F

From the above truth table, ~[pv(~q)]≡≡~p ^ q.

(d)

We prove the given relations using truth table.

 p q ~p ~q ~p ^ q ~[(~p)^q] pv(~q) T T F F T F T F F F T T F T F T F F T F T T F T T T F T

From the above truth table, ~p[(~p ^ q)] ≡≡ p v (~q).

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# Tag Info

## Hot answers tagged special-functions

7

Unfortunately, there is no single approach that will lead to robust, accurate, and high-performance implementations across the large universe of special functions. Often, two or more methods must be used for different parts of the input domain, and the necessary research and implementation work may take weeks for elementary functions and months for higher ...

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$F_{2l}(x)$ can be calculated in closed form for $\text{Im}\ x<0$ using the integral from Gradsteyn and Ryzhik $$\int_0^1u^\lambda P_{2l}(u)du=\frac{(-1)^l\Gamma\left(l-\frac{\lambda}{2}\right)\Gamma\left(\frac{1}{2}+\frac{\lambda}{2}\right)}{2\Gamma\left(-\frac{\lambda}{2}\right)\Gamma\left(l+\frac{3}{2}+\frac{\lambda}{2}\right)},\quad \text{Re}\ ... 2 Notice that: \Gamma(m) = \lim\limits_{n \to \infty} \frac{n! \; n^m}{m \times (m+1) \times \dots \times (m+n)} = \lim\limits_{n \to \infty} \frac{(m-1)! \; n! \; n^m}{(m + n)!} = (m - 1)! \times \lim\limits_{n \to \infty} \frac{n! \; n^m}{(m + n)!}. Now let's show that \lim\limits_{n \to \infty} \frac{n! \; n^m}{(m + n)!} = 1: \lim\limits_{n \to ... 2 I suspect there's a well defined special function related to this series. Is it so ? Not really. Basically, \text{Exl}(x)=-F'(1), where F(k)=\displaystyle\sum_{n\ge0}\frac{x^n}{n!^k}~.~ The only known values of F are F(0)=\dfrac1{1-x},~F(1)=e^x, and F(2)=I_0\big(2\sqrt x\big). See Bessel function for more information. Neither F, let alone ... 1 With some help from Mathematica:$$G_{2,3}^{3,1}\left(z\left| \begin{smallmatrix} 0,1 \\ 0,0,0 \\ \end{smallmatrix} \right.\right) = -2 z \, {_3F_3}\left(\begin{smallmatrix}1,1,1\\2,2,2\end{smallmatrix}\middle|\,z\right)-\frac{1}{2} \frac{\partial^2}{\partial t^2}\left.{_1F_1}\left(\begin{smallmatrix}t\\1\end{smallmatrix}\middle|\,z\right)\right|_{t=0}+ ...

1

One way to describe what makes translating sine waves special is the following: sine waves $f(t)$ have the property that the subspace of the space of functions spanned by their translates $f(t + t_0), t_0 \in \mathbb{R}$ is $2$-dimensional, spanned by $\cos t, \sin t$. What other functions have this property? (I'm not going to require periodicity yet; this ...

Only top voted, non community-wiki answers of a minimum length are eligible


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# What's the Difference Between a Vector and an Hypercomplex Number?

What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties.

Perhaps this question could be put more generally as: what's the difference between a vector space and a field?

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–  a.r. Mar 16 '11 at 6:46

In a field you can multiply any two elements, as well as add them, while in a vector space you can add vectors, but you can only multiply by scalars; you can't multiply two vectors. (Multiplication in a field has to satisfy some axioms, but I think this is the essential point for your question.)

A field containing $\mathbb R$, or more generally a system of hypercomplex numbers containing $\mathbb R$, will in particular be a vector space (since you can add, and you can multiply by elements of $\mathbb R$, i.e. by scalars), but it will have extra structure. (E.g. you can multiply quaternions, and this is extra structure which is not at all obvious if you just know about $4$-vectors.)

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They are separate concepts, it's just that hypercomplex numbers can be used to model the behavior of vectors in $\mathbb{R}^n$.

A vector is an element of a vector space. Vector spaces are (up to isomorphism) direct sums of the field $k$, in that a vector is an ordered $n$-tuple $(a_1,\ldots,a_n)$ with each $a_i \in k$ which can be multiplied by an element $b \in k$, which is called a scalar, as follows: $k*(a_1,\ldots,a_n) = (k*a_1,\ldots,k*a_n)$. In the case of real vectors, this field is $\mathbb{R}$.

Hypercomplex numbers, such as the quaternions, are different in that they are elements of certain algebras over the real numbers. Algebras are defined similarly to vector spaces, only they also allow you to multiply elements of the algebra by other elements of the algebra.

Essentially, vectors and hypercomplex numbers are different concepts and cannot be used interchangeably. However, they are similar in that often the same problem can be solved using either.

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You may mean vector space where you say vector field . –  Myself Mar 16 '11 at 10:54
I cannot believe I did that. Late night. –  Alex Becker Mar 16 '11 at 14:12

A hypercomplex number is an element of a certain kind of distributive unital algebra over $\mathbb{R}$. An algebra over $\mathbb{R}$ is, by definition, a vector space over $\mathbb{R}$ with additional structure so that you can multiply elements together. It's not very hard to turn a given finite-dimensional vector space into an algebra of hypercomplex numbers, but the question is whether you can do so in a natural, useful way.

Of course, there are vector spaces over any field at all, so it's also easy to come up with a vector space which cannot be turned into an algebra of hypercomplex numbers. For instance, a finite dimensional vector space over a finite field has only finitely many elements, so obviously cannot be turned into an algebra over $\mathbb{R}$.

The interesting thing is that there are objects which are simultaneously all of these things, and in a recursive way: for example, $\mathbb{C}$ is both a field and an algebra over $\mathbb{R}$, and a fortiori a vector space over $\mathbb{R}$. But $\mathbb{R}$ is in turn both a field and an algebra over $\mathbb{Q}$... The study of fields which are algebras over other fields is called Galois theory, which also finds applications in algebraic geometry (though probably not in the way you expect!).

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Great answer – thanks for the extra insights! –  msutherl Mar 16 '11 at 15:54


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At what point does does a departure from the normal constitute inflation (QQ plot)
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Entering edit mode
6.3 years ago

I'm trying to interpret a QQ plot I have from Matrix eQTL, to see if there is a factor I haven't accounted for. I know a departure from the x=y at low values can indicate a confounding factor, but at what point is a digression from the normal inflation, and when is it acceptable? Is there a "rule of thumb"?

Edit: uploaded the wrong QQ-plot:

qq-plot eQTL • 4.1k views
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Entering edit mode

I refer to R-fortunes:

library(fortunes)
fortune('chicken')

A sufficiently trained statistician can read the vagaries of a Q-Q plot like a shaman can read a chicken's entrails, with a similar recourse to scientific principles. Interpreting Q-Q plots is more a visceral than an intellectual exercise. The uninitiated are often mystified by the process. Experience is the key here.
-- Department of Mathematics and Statistics, Murdoch University
StatsNotes

But because I don't want my comment to be completely useless, this thread might help you, in particular the latest 3 replies. Failing that, I am inclined to agree with Jean-Karim in that visually your plot suggests non-normality and normal distribution can be tested with the mentioned tests (though I have only used the Shapiro-Wilk test).

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Entering edit mode

That is a fantastic quote.

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Entering edit mode
6.3 years ago

I don't know of any rule of thumb for deciding when a distribution is normal or not based on a QQ plot. Any visually strong departure from the diagonal line like in your plot is definitely a sign of non-normality. However, even small deviations from the line can indicate non-normality but in these cases normality can be a good approximation. If you need statistical motivation for deciding whether your data is normally distributed, there are a number of tests you could use, the most sensitive being Shapiro-Wilk and Anderson-Darling tests.