text:
# Arbitrage-free Valuation Approach for Bonds

The simplest way to calculate the value of a bond is to take the cash flows of the bond till its maturity and then discount them by a single discount rate. The method is quick but not very accurate because the yield curve is not flat and the interest rates are different for different maturities. A better way to price the bonds is to discount each cash flow with the spot rate (zero coupon rate) for its respective maturity. This is called the arbitrage-free valuation approach. According to this approach the value of a Treasury bond based on spot rates must be equal to the sum of the present values of all cash flows. If the values are not equal, then there is an arbitrage opportunity that can be exploited by an arbitrageur.

Example 1

Let’s take an example. Suppose we want to calculate the value of a $1000 par, 5% coupon, 5-year maturity bond. We also have the following spot rates for the next 5 years: Assuming this is an annual pay bond, the bond will have the following cash flows. Year 1:$50

Year 2: $50 Year 3:$50

Year 4: $50 Year 5:$1000 + $50 The value of the bond can be calculated by discounting these cash flows by their respective spot rate. Bond Value = 50/(1.04)^1+50/(1.0430)^2+50/(1.0451)^3+50/(1.047)^4+1050/(1.048)^5 Bond Value =$1010.033

If the market price of this bond is $1,020, then there is an arbitrage opportunity. The arbitrageur can buy the individual cash flows (in the form of zero-coupon bonds), combine then into a 5-year bond, and sell the package at the market price of the bond. The arbitrageur will earn a risk-free profit of$9.967 ($1020 -$1010.033)

Example 2

Let’s take another example. Suppose we have a bond that matures in 2 years, that has a coupon rate of 6%, and pays coupon semi-annually. The spot rates are 3.9% for 6 months, 4% for 1 year, 4.15% for 1.5 years, and 4.3% for 2 years.

The cash flows from this bond are $30,$30, $30, and$1030.

The value of the bond will be calculated as follows:

Bond value = $30/(1+3.9%/2)^1+$30/(1+4%/2)^2+$30/(1+4.15%/2)^3+$1030/(1+4.3%/2)^4


text:
## Keywords (5)

Publications
Effects of a non-coaxial flow rule on shear bands in viscous-plastic materials

# Effects of a non-coaxial flow rule on shear bands in viscous-plastic materials,10.1007/s10035-011-0254-7,Granular Matter,Hans B. Mühlhaus,Jingyu Shi,L

Effects of a non-coaxial flow rule on shear bands in viscous-plastic materials
A class of non-coaxial plasticity models is revisited in which the plastic deformation is carried by one (single slip) or two (double slip) slip systems that are inclined at $${\pm (\pi/4+\nu/2), 0\le \nu \le \phi}$$ to the less compressive principal stress axis; ν is the angle of non-coaxiality and $${\phi}$$ is the internal angle of friction of the material. The non-coaxial model is combined with a Newtonian viscous rheology to form a viscous-plastic model. The model is applied to model strain localisation in extension and compression on a time scale typical for deformation processes in the lithosphere. We would like to dedicate this article to the memory of Ioannis Vardoulakis who always took a keen interest in bifurcation theory and shear banding for granular materials.
Journal: Granular Matter - GRANUL MATTER , vol. 13, no. 3, pp. 205-210, 2011
View Publication
 The following links allow you to view full publications. These links are maintained by other sources not affiliated with Microsoft Academic Search.

## Citation Context (1)

• ...Mühlhaus et al. [4] develop a non-coaxial Cosserat continuum model based on a double-slip mechanism, a model that has interestingapplicationsindescribingdeformationofthelithosphere...

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## Citations (1)

### Developments in micromechanical modelling of granular materials

Journal: Granular Matter - GRANUL MATTER , vol. 13, no. 3, pp. 183-185, 2011

format: plain


text:
# 10.18: The Spelling <nn> and VCC

Difficulty Level: At Grade Created by: CK-12

## The Spelling <nn> and VCC

1. Read over the list carefully. Starting with the vowel right in front of the <nn> in each one, mark the VCC pattern:

cinnamonchannelfunneltennispennybonnetminnowdinnerbunnyannual\begin{align*}& \text{cinnamon} && \text{funnel} && \text{penny} && \text{minnow} && \text{bunny} \\ & \text{channel} && \text{tennis} && \text{bonnet} && \text{dinner} && \text{annual}\end{align*}

2. Now sort the words into these five groups:

3. Sometimes the <nn> is necessary right after a short vowel in order to fill out the _______ pattern.

4. Here are some words that contain <nn>. For each one give the reason that [n] is spelled <nn>: Assimilation, Twinning, Simple Addition, or VCC:

Word Reason for <nn>
innocently
innumerable
unnecessarily
beginner
suddenness
nonnuclear
tennis
annihilation
announcement
connectedness
sunnier
cinnamon
cannot
conniving
funnel
annexes
channel
annulment
skinniest

5. So far you have worked with two ways of spelling [n] _____ and _____. Remember: The sound [n] is spelled one of these two ways about ninety-nine times out of every one hundred.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

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Description
Tags:
Subjects:
1 , 2 , 3 , 4 , 5
Date Created:
Feb 23, 2012
Oct 09, 2015
Image Detail
Sizes: Medium | Original

CK.ENG.ENG.SE.1.Basic-Speller.10.18


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# Radiation heat transfer between air or wall

1. Nov 25, 2009

### zenite

Lets say a hot object is at 500K and placed in a chamber with wall temperature of 300K. The surrounding air in the chamber is 320K.

So what is the mode of heat transfer for the hot objection? There is convection and radiation. so the heat transfer rate, q = q(conv) + q(rad), right?

Convection is between object and surrounding air. But what about radiation? Is it between the object and the chamber walls or is it between the object and the surrounding air? Depending on which is chosen, the q(rad) will varies since q(rad) depends on the temperature difference of the 2 body.

Can anyone help me out? I am confused on using 500K and 300K to calculate q(rad) or 500K and 320K.

2. Nov 25, 2009

### zenite

Ok, I think I got it. Please correct me if I am wrong.

Rate of energy radiation of an object (T is given by
qo = A$$\sigma$$$$\epsilon$$T4

If the object is in a chamber with wall temperature of Tw and surrounding air of Ta, then the object receives heat radiation from both wall and air.
qwall to object = A$$\sigma$$$$\epsilon$$Tw4

qair to object = A$$\sigma$$$$\epsilon$$Ta4

Hence, the net q for object to air and wall is:
qnet = A$$\sigma$$$$\epsilon$$(T4-Ta4-Tw4)

ps: the sigma and epsilon isn't suppose to be superscript, not sure why it turns out this way

3. Nov 25, 2009

### Mapes

Hi zenite, welcome to PF. The air isn't going to participate to any significant effect in the heat transfer between the object and the wall. (Another way of looking at this is that a few meters of air won't block radiative heat transfer, just as a few meters of air doesn't attenuate visible light to any noticeable degree). The air properties are only going to affect the convection term.

4. Nov 25, 2009

### zenite

Ok, so radiation takes place between the object and the walls only? Is there no radiation taking place between the air and the object or is it insignificant as compared to the other modes of heat transfer?

And yes, I realised that the formula that I put up earlier on is incorrect, since taking T4 - Ta4 - Tw4 can give a negative value, which is illogical if T>Ta>Tw.

5. Nov 25, 2009

### Mapes

It exists but is insignificant.

6. Nov 25, 2009

### zenite

Oh thanks for clearing my doubts. I have another question on convection.

If an air gap exists between 2 walls, say, wall A and wall B. the convective heat transfer coefficient of the air is h. Assuming no radiation and conduction in the air gap, what is the heat transfer rate between the 2 walls?

Let Ta be temperature of wall A, Tb of wall B and Tair of the air.

q = hA (Ta - Tair) + hA (Tair - Tb) = hA (Ta - Tb)
or is it...
q = 0.5hA (Ta - Tb)

Since convection takes place 'twice', between wall A and air, and between wall B and air, should the h be multiply by 0.5?

Last edited: Nov 25, 2009
7. Nov 25, 2009

### Mapes

I don't think this approach is going to prove fruitful. Convection removes heat energy by fluid motion tangential to a surface. The convective flow isn't going to automatically divert to the other wall and deposit heat energy there. And I don't see the sense in adding the heat flux values together. Assuming that convection does dominate in this problem (which isn't clear), it's the same flux of energy; adding the values is nonsensical.

The general approach is to calculate the values of heat flux by each mechanism (conduction, convection, radiation) and determine if one or more can be assumed negligible.


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# ISA erasure code plugin¶

The isa plugin encapsulates the ISA library.

## Create an isa profile¶

To create a new isa erasure code profile:

ceph osd erasure-code-profile set {name} \
plugin=isa \
technique={reed_sol_van|cauchy} \
[k={data-chunks}] \
[m={coding-chunks}] \
[crush-root={root}] \
[crush-failure-domain={bucket-type}] \
[crush-device-class={device-class}] \
[directory={directory}] \
[--force]

Where:

k={data chunks}

Description

Each object is split in data-chunks parts, each stored on a different OSD.

Type

Integer

Required

No.

Default

7

m={coding-chunks}

Description

Compute coding chunks for each object and store them on different OSDs. The number of coding chunks is also the number of OSDs that can be down without losing data.

Type

Integer

Required

No.

Default

3

technique={reed_sol_van|cauchy}

Description

The ISA plugin comes in two Reed Solomon forms. If reed_sol_van is set, it is Vandermonde, if cauchy is set, it is Cauchy.

Type

String

Required

No.

Default

reed_sol_van

crush-root={root}

Description

The name of the crush bucket used for the first step of the CRUSH rule. For instance step take default.

Type

String

Required

No.

Default

default

crush-failure-domain={bucket-type}

Description

Ensure that no two chunks are in a bucket with the same failure domain. For instance, if the failure domain is host no two chunks will be stored on the same host. It is used to create a CRUSH rule step such as step chooseleaf host.

Type

String

Required

No.

Default

host

crush-device-class={device-class}

Description

Restrict placement to devices of a specific class (e.g., ssd or hdd), using the crush device class names in the CRUSH map.

Type

String

Required

No.

Default

directory={directory}

Description

Set the directory name from which the erasure code plugin is loaded.

Type

String

Required

No.

Default

/usr/lib/ceph/erasure-code

--force

Description

Override an existing profile by the same name.

Type

String

Required

No.


text:
# Open SourceMolecular Dynamics

### Sidebar

#### For Developers

howto:running_qe_computation

# How to run calculations like Quantum ESPRESSO

## Introduction

Although CP2K uses gaussian as a main function basis, it also has the possibility to do computations with the plane wave basis. This functionality is provided by the SIRIUS library, which supports computation on both CPUs and GPUs. SIRIUS includes most of the core functionalities of QE (only ground state properties), such as non-colinear and colinear magnetism, spin-orbit coupling, on-site Hubbard corrections, forces and stresses.

An other popular program for plane wave DFT is Quantum Expresso (QE). A full description of QE functionalities is outside the scope of this tutorial, but one of the main functionalities is ground state minimization and geometry optimization of crystals. In this tutorial, we will show how to do QE calculations with CP2K + SIRIUS. We will consider the simple case of silicium doped with germanium and show how to convert a QE input file to cp2k input file. Files for this tutorial can be found here while the cp2k input files can be found in the cp2k tests directory.

#### Word of caution with the pseudo-potential files

Quantum Expresso uses the upf format to describe pseudo-potentials, while CP2K + SIRIUS can either use CP2K pseudo-potentials (they are tailored for the gaussian basis set but will work with the plane wave basis) or the same pseudo-potential files as QE (when the sirius backend is used). The latter case has drawbacks : the upf file should be converted to a json file with a small python script provided by the SIRIUS library, cp2k passing the file name directly to the SIRIUS backend. The SIRIUS backend supports norm-conserving and ultra-soft pseudo-potentials as well as full potential computation, something that is missing in QE.

#### Problems of units

QE uses atomic units by defaults for the distances but Rydberg unit for energy, while cp2k uses Angstroms by default for the unit of length. The outputs are however in atomic units (or specified otherwise). SIRIUS on the other hand works only in atomic unit. This issue only affect the section describing the crystal structure since Angstroms and Bohr can be mixed in the same input file. By *default* QE will use the Bohr units if nothing is specified while cp2k will use Angstroms.

## Using cp2k for doing QE ground state minimization

We assume that the reader is familiar with the QE input format. The cp2k format is described in depth in the documentation but the purpose of this tutorial we will consider a simple example, a cluster of 8 atoms, namely silicium doped with germanium. The QE input file for this system looks like this

QE input CP2K input
  &control
calculation='scf',
restart_mode='from_scratch',
pseudo_dir = './',
outdir='./',
prefix = 'scf_',
tstress = false,
tprnfor = false,
verbosity = 'high',
wf_collect = false
/
&system
ibrav=0, celldm(1)=1, ecutwfc=25,
ecutrho = 400, occupations = 'smearing',
smearing = 'gauss', degauss = 0.001,
nat=8 ntyp=2
/
&electrons
conv_thr =  1.0d-11,
mixing_beta = 0.7,
electron_maxstep = 100

ATOMIC_SPECIES
Ge 0.0 ge_lda_v1.uspp.F.UPF
Si 0.0 si_lda_v1.uspp.F.UPF
CELL_PARAMETERS
10.26253566   0.00000000   0.00000000
0.00000000  10.26253566   0.00000000
0.00000000   0.00000000  10.26253566
ATOMIC_POSITIONS (crystal)
Ge  0.00000000  0.00000000  0.00000000
Si  0.00000000  0.50000000  0.50000000
Si  0.50000000  0.00000000  0.50000000
Si  0.50000000  0.50000000  0.00000000
Si  0.75000000  0.75000000  0.25000000
Si  0.75000000  0.25000000  0.75000000
Si  0.25000000  0.75000000  0.75000000
Si  0.25000000  0.25000000  0.25000000
K_POINTS (automatic)
2 2 2  0 0 0
&FORCE_EVAL
METHOD SIRIUS
&PW_DFT
&CONTROL
PROCESSING_UNIT cpu
STD_EVP_SOLVER_TYPE lapack
GEN_EVP_SOLVER_TYPE lapack
&END CONTROL
&PARAMETERS
ELECTRONIC_STRUCTURE_METHOD  pseudopotential
SMEARING_WIDTH 0.01
USE_SYMMETRY true
NUM_MAG_DIMS 0
GK_CUTOFF 5.0
PW_CUTOFF 20.00
ENERGY_TOL 1e-10
POTENTIAL_TOL 1e-8
NUM_DFT_ITER 100
NGRIDK 2 2 2
&END PARAMETERS
&ITERATIVE_SOLVER
ENERGY_TOLERANCE 1e-5
NUM_STEPS 20
SUBSPACE_SIZE 4
TYPE davidson
CONVERGE_BY_ENERGY 1
&END ITERATIVE_SOLVER
&MIXER
TYPE broyden1
MAX_HISTORY 8
&END MIXER
&END PW_DFT
&DFT
&XC
&XC_FUNCTIONAL
&LIBXC
FUNCTIONAL XC_LDA_X
&END LIBXC
&LIBXC
FUNCTIONAL XC_LDA_C_PZ
&END LIBXC
&END XC_FUNCTIONAL
&END XC
&END DFT
&SUBSYS
&CELL
A [bohr] 10.26253566   0.00000000   0.00000000
B [bohr]  0.00000000  10.26253566   0.00000000
C [bohr]  0.00000000   0.00000000  10.26253566
&END CELL
&COORD
SCALED
Ge  0.00000000  0.00000000  0.00000000
Si  0.00000000  0.50000000  0.50000000
Si  0.50000000  0.00000000  0.50000000
Si  0.50000000  0.50000000  0.00000000
Si  0.75000000  0.75000000  0.25000000
Si  0.75000000  0.25000000  0.75000000
Si  0.25000000  0.75000000  0.75000000
Si  0.25000000  0.25000000  0.25000000
&END COORD
&KIND Si
POTENTIAL UPF si_lda_v1.uspp.F.UPF.json
&END KIND
&KIND Ge
POTENTIAL UPF ge_lda_v1.4.uspp.F.UPF.json
&END KIND
&END SUBSYS
&END FORCE_EVAL
&GLOBAL
PROJECT Si7Ge
PRINT_LEVEL MEDIUM
&END GLOBAL

in the following we will explain how to arrive to the above script.

### lattice and unit cell description

let us first start with the structure description. In QE, the coordinates of the atoms are often given in reduced coordinates (in lattice coordinates). So let us first start with the lattice vectors. Searching for the keyword

CELL_PARAMETERS

we find that the lattice vectors (in Cartesian coordinates) are given by (*Warning* the default unit is bohr)

QE input CP2K input
CELL_PARAMETERS
10.26253566  0.00000000  0.00000000
0.00000000  10.26253566  0.00000000
0.00000000   0.00000000 10.26253566
&FORCE_EVAL
&SUBSYS
&CELL
A [bohr] 10.26253566   0.00000000   0.00000000
B [bohr]  0.00000000  10.26253566   0.00000000
C [bohr]  0.00000000   0.00000000  10.26253566
&END CELL
&END SUSYS
&END FORCE_EVAL

Note the presence of the keyword [Bohr] which indicates that the lattice vectors are atomic unit. If the QE input file is in angstroms then change the keyword [Bohr] to [angstroms]. CP2K/SIRIUS will convert everything in atomic unit internally.

The atoms positions are given by the section ATOMIC_POSITIONS (crystal) in the QE input file. For this example, the coordinates are given relative to the lattice displacement.

QE input CP2K input
ATOMIC_POSITIONS (crystal)
Ge 0.00000000 0.00000000 0.00000000
Si 0.00000000 0.50000000 0.50000000
Si 0.50000000 0.00000000 0.50000000
Si 0.50000000 0.50000000 0.00000000
Si 0.75000000 0.75000000 0.25000000
Si 0.75000000 0.25000000 0.75000000
Si 0.25000000 0.75000000 0.75000000
Si 0.25000000 0.25000000 0.25000000
&FORCE_EVAL
&SUBSYS
&COORD
SCALED
Ge 0.00000000 0.00000000 0.00000000
Si 0.00000000 0.50000000 0.50000000
Si 0.50000000 0.00000000 0.50000000
Si 0.50000000 0.50000000 0.00000000
Si 0.75000000 0.75000000 0.25000000
Si 0.75000000 0.25000000 0.75000000
Si 0.25000000 0.75000000 0.75000000
Si 0.25000000 0.25000000 0.25000000
&END COORD
&END SUBSYS
&END FORCE_EVAL

for cp2k input file. Note the presence of the keyword SCALED in the section COORD which indicates that cp2k should treat the coordinates as given in the lattice basis. The coordinates do not have to be given in the lattice basis, any format supported by cp2k will work. Putting these two sections together, we have

&FORCE_EVAL
&SUBSYS
&CELL
A [bohr] 10.26253566        0.00000000        0.00000000
B [bohr] 0.00000000       10.26253566        0.00000000
C [bohr] 0.00000000        0.00000000       10.26253566
&END CELL
&COORD
SCALED
Ge          0.00000000         0.00000000         0.00000000
Si          0.00000000         0.50000000         0.50000000
Si          0.50000000         0.00000000         0.50000000
Si          0.50000000         0.50000000         0.00000000
Si          0.75000000         0.75000000         0.25000000
Si          0.75000000         0.25000000         0.75000000
Si          0.25000000         0.75000000         0.75000000
Si          0.25000000         0.25000000         0.25000000
&END COORD
&END SUSYS
&END FORCE_EVAL

The information about the potential of each species can be found searching for the keyword ATOMIC_SPECIES in the QE input file.

QE input CP2K input
ATOMIC_SPECIES
Ge 0.0 ge_lda_v1.4.uspp.F.UPF
Si 0.0 si_lda_v1.uspp.F.UPF
&FORCE_EVAL
&SUBSYS
KIND Si
POTENTIAL upf si_lda_v1.uspp.F.UPF.json
&KIND Si
KIND Ge
POTENTIAL upf ge_lda_v1.4.uspp.F.UPF.json
&KIND
&END SUSYS
&END FORCE_EVAL

Note the difference of naming for the potential files and the absence of 0.0 this cp2k input file (this parameter is the mass of the atom, it usually ignored in simple QE calculations). CP2K when using sirius as a back-end expects to find the file describing the atomic specie potential in json format. The SIRIUS library distribution contains a python script that can convert one format to the other.

To convert the upf file into json, we can use the script provided here upf_to_json.py and use them as follow

upf_to_json.py ge_lda_v1.4.uspp.F.UPF
upf_to_json.py si_lda_v1.uspp.F.UPF

Regrouping everything together we obtain

&FORCE_EVAL
&SUBSYS
&CELL
A [bohr] 10.26253566        0.00000000        0.00000000
B [bohr] 0.00000000       10.26253566        0.00000000
C [bohr] 0.00000000        0.00000000       10.26253566
&END CELL
&COORD
SCALED
Ge          0.00000000         0.00000000         0.00000000
Si          0.00000000         0.50000000         0.50000000
Si          0.50000000         0.00000000         0.50000000
Si          0.50000000         0.50000000         0.00000000
Si          0.75000000         0.75000000         0.25000000
Si          0.75000000         0.25000000         0.75000000
Si          0.25000000         0.75000000         0.75000000
Si          0.25000000         0.25000000         0.25000000
&END COORD
KIND Si
POTENTIAL upf si_lda_v1.uspp.F.UPF.json
&KIND Si
KIND Ge
POTENTIAL upf ge_lda_v1.4.uspp.F.UPF.json
&KIND
&END SUSYS
&END FORCE_EVAL

which is enough to describe the structure we want to study.

According to the QE documentation the default unit of length is the atomic unit (the Bohr) while cp2k uses angstroms as a default.

#### Energy cut-off and k-point grid

The next step is to convert the QE sections controlling the algorithms used to solve the KS equation. The three most important parameters are ecutwfc the plane wave cut-off for the wave functions, ecutrho the plane wave cut-off for the density, and K_POINTS which gives the dimension of the k-point grid. Let us start with the k point grid. In cp2k, the k-point grid is given in the section PW_DFT (section dedicated to SIRIUS back-end options) using the keyword ngridk.

QE input CP2K input
  K_POINTS (automatic)
2 2 2  0 0 0
&FORCE_EVAL
METHOD SIRIUS
&PW_DFT
&PARAMETERS
NGRIDK 2 2 2
SHIFTK 0 0 0
&END PARAMETERS
&END PW_DFT
&END FORCE_EVAL

Note the presence of 0 0 0 after the k-grid size. Any value different from zero indicate a shift of the generated k-points compared to the gamma point. In some cases, shifting the k-point reduces the size of effective number of k-points to be calculated.

The section FORCE_EVAL now contains the keyword METHOD telling CP2K to use the sirius back-end. It also contains a subsection named *PW_DFT* containing all information necessary for the back-end to do computation. So most of the QE parameters contained in the section control, system and electrons will go in this section.

the parameter ecutwfc which fix the number of planes waves for computation of the wave functions is equivalent to the parameter

GK_CUTOFF

in cp2k while ecutrho is the plane wave cutoff PW_CUTOFF for the density.

Both parameters in CP2K are the square root of the QE parameters.

QE input CP2K input
ecutwfc=25
ecutrho = 400
&FORCE_EVAL
METHOD SIRIUS
&PW_DFT
&PARAMETERS
ELECTRONIC_STRUCTURE_METHOD  pseudopotential
GK_CUTOFF 5.0
PW_CUTOFF 20.00
&END PARAMETERS
&END PW_DFT
&END FORCE_EVAL

Regrouping again everything together, we obtain the following initial input file for cp2k.

&FORCE_EVAL
METHOD SIRIUS
&PW_DFT
&PARAMETERS
ELECTRONIC_STRUCTURE_METHOD  pseudopotential
GK_CUTOFF 5.0
PW_CUTOFF 20.00
NGRIDK 2 2 2
SHIFTK 0 0 0
&END PARAMETERS
&END PW_DFT
&SUBSYS
&DFT
&XC
&XC_FUNCTIONAL
&LIBXC
FUNCTIONAL XC_LDA_X
&END LIBXC
&LIBXC
FUNCTIONAL XC_LDA_C_PZ
&END LIBXC
&END XC_FUNCTIONAL
&END XC
&END DFT

&CELL
A [bohr] 10.26253566        0.00000000        0.00000000
B [bohr] 0.00000000       10.26253566        0.00000000
C [bohr] 0.00000000        0.00000000       10.26253566
&END CELL
&COORD
SCALED
Ge          0.00000000         0.00000000         0.00000000
Si          0.00000000         0.50000000         0.50000000
Si          0.50000000         0.00000000         0.50000000
Si          0.50000000         0.50000000         0.00000000
Si          0.75000000         0.75000000         0.25000000
Si          0.75000000         0.25000000         0.75000000
Si          0.25000000         0.75000000         0.75000000
Si          0.25000000         0.25000000         0.25000000
&END COORD
KIND Si
POTENTIAL upf si_lda_v1.uspp.F.UPF.json
&KIND Si
KIND Ge
POTENTIAL upf ge_lda_v1.4.uspp.F.UPF.json
&KIND
&END SUSYS
&END FORCE_EVAL

&GLOBAL
PROJECT Si7Ge
PRINT_LEVEL MEDIUM
&END GLOBAL

The last section is specific to CP2K. The other default settings of the SIRIUS back-end should be enough to converge this simple structure.

By default, it will compute the ground state and return the total energy and the forces on each atom to CP2K. The input file contains a DFT section (which is absent by default in QE) that describes the type of exchange functional to use for the calculations. CP2K and SIRIUS support the libxc naming convention which might be different from the QE convention.

SIRIUS has more parameters to control its execution. Some of these parameters have no equivalent in QE other do. A full list of options that SIRIUS supports can be found in the cp2k documentation, but we will describe the most important one.

the following table summarizes the most common options of QE that have an equivalent in CP2K+SIRIUS.

QE keyword CP2K (SIRIUS) keyword section desciption
nspins = 1, 2 NUM_MAG_DIMS = 0, 1 PW_DFT/PARAMETERS computation in LDA and SLDA
nocolin = true NUM_MAG_DIMS = 2 PW_DFT/PARAMETERS non colinear magnetism
electron_maxstep NUM_DFT_ITER PW_DFT/PARAMETERS number of scf iterations
mixing_beta beta PW_DFT/MIXER mixing parameter for the broyden mixer
nosym = false USE_SYMMETRY = true PW_DFT/PARAMETERS use discret symmetries to build the k-point grid and the of plane waves
ecutrho PW_CUTOFF PW_DFT/PARAMETERS energy cutoff for the density. It is the square root of the QE parameter
ecutwfc GK_CUTOFF PW_DFT/PARAMETERS energy cutoff for the waavefunctions. It is the square root of the QE parameter

many of the other options can be dropped for the first cp2k input file.

If you run this input file with cp2k you should have the following output

Energy
--------------------------------------------------------------------------------
valence_eval_sum          :        -5.08815701
<rho|V^{XC}>              :       -30.23739234
<rho|E^{XC}>              :       -43.72437220
<mag|B^{XC}>              :         0.00000000
<rho|V^{H}>               :       106.32392644
one-electron contribution :       -81.17469111 (Ha),      -162.34938222 (Ry)
hartree contribution      :        53.16196322
xc contribution           :       -43.72437220
ewald contribution        :       -68.41309922
PAW contribution          :         0.00000000
Total energy              :      -140.15019932 (Ha),      -280.30039863 (Ry)

band gap (eV) :         0.34564591
Efermi        :         0.24495087

iteration :  11, RMS 1.265254419560E-11, energy difference : 0.000000000000E+00

converged after 12 SCF iterations!

ENERGY| Total FORCE_EVAL ( SIRIUS ) energy (a.u.):         -140.150199315127225

## Molecular dynamics

Quantum expresso supports many types of calculations that cp2k supports natively. For instance, it is possible to do molecular dynamics in cp2k with the sirius backend, something that QE can do as well. To do this, we can start from the previous cp2k input file for Si7Ge and add a small section for setting the parameters relevant for the molecular dynamics. The symmetry parameter should be off during molecular dynamics configurations.

&FORCE_EVAL
METHOD SIRIUS
&PW_DFT
&CONTROL
PROCESSING_UNIT cpu
STD_EVP_SOLVER_TYPE lapack
GEN_EVP_SOLVER_TYPE lapack
&END CONTROL
&PARAMETERS
ELECTRONIC_STRUCTURE_METHOD  pseudopotential
SMEARING_WIDTH 0.01
USE_SYMMETRY false
NUM_MAG_DIMS 0
GK_CUTOFF 5.0
PW_CUTOFF 20.00
ENERGY_TOL 1e-10
POTENTIAL_TOL 1e-8
NUM_DFT_ITER 100
NGRIDK 2 2 2
&END PARAMETERS
&ITERATIVE_SOLVER
ENERGY_TOLERANCE 1e-5
NUM_STEPS 20
SUBSPACE_SIZE 4
TYPE davidson
CONVERGE_BY_ENERGY 1
&END ITERATIVE_SOLVER
&MIXER
TYPE broyden1
MAX_HISTORY 8
&END MIXER
&END PW_DFT
&DFT
&XC
&XC_FUNCTIONAL
&LIBXC
FUNCTIONAL XC_LDA_X
&END LIBXC
&LIBXC
FUNCTIONAL XC_LDA_C_PZ
&END LIBXC
&END XC_FUNCTIONAL
&END XC
&END DFT
&SUBSYS
&CELL
A [bohr] 10.26253566        0.00000000        0.00000000
B [bohr] 0.00000000       10.26253566        0.00000000
C [bohr] 0.00000000        0.00000000       10.26253566
&END CELL
&COORD
SCALED
Ge          0.00000000         0.00000000         0.00000000
Si          0.00000000         0.50000000         0.50000000
Si          0.50000000         0.00000000         0.50000000
Si          0.50000000         0.50000000         0.00000000
Si          0.75000000         0.75000000         0.25000000
Si          0.75000000         0.25000000         0.75000000
Si          0.25000000         0.75000000         0.75000000
Si          0.25000000         0.25000000         0.25000000
&END COORD
&KIND Si
POTENTIAL UPF si_lda_v1.uspp.F.UPF.json
&END KIND
&KIND Ge
POTENTIAL UPF ge_lda_v1.4.uspp.F.UPF.json
&END KIND
&END SUBSYS
&END FORCE_EVAL
&MOTION
&GEO_OPT
OPTIMIZER LBFGS
&END GEO_OPT
&MD
ENSEMBLE NVE
TIMESTEP 0.1
STEPS 125
TEMPERATURE 300.0
&END MD
&END MOTION
&GLOBAL
PROJECT Si7Ge
PRINT_LEVEL MEDIUM
RUN_TYPE MD
&END GLOBAL

if you run this script using the command

PATH_TO_CP2K/cp2k.psmp Si7Ge-motion.inp

you should obtain a file Si7Ge-pos-1.xyz that contains the positions of the atoms and Si7Ge-1.ener that contains the different energies and temperature as a function of time. The calculations are done in the micro-canonical ensemble which means that the total energy (the sum of the potential which is the total energy given by SIRIUS and the kinetic energy) is conserved during the integration process. It is possible to compare these results with QE, but instead of doing the molecular dynamics with QE, we take the positions of the atoms at each time step and compute the potential energy for this atom position with QE. The total energy given by QE and CP2K/SIRIUS are shifted by a small offset. More generally, converting the input file from QE to CP2K is not a warranty to obtain the same results for the total energy. The reasons for this multiple :

• Indicating the same cutoff does not warranty the fft grid size.
• QE and SIRIUS treat the radial integrals interpolation differently
• The functional in QE do not use libxc by default, while CP2K/SIRIUS does.
• The minimization method is sensitive to the initial states.

For all these reasons, we should not try to compare results at the binary level.


text:
# Is it a Banach space? If so what is its dual?

Let $(E_n)$ be a sequence of Banach spaces and $(w_n)$ be a sequence of positive real numbers. For $1\leq p <\infty$ define $\bigoplus\limits_P E_n:=\{(x_n):x_n\in E_n,\sum\limits_n\lVert x_n\rVert^pw_n<\infty\}$ and $\lVert (x_n)\rVert_p:=(\sum\limits_n\lVert x_n\rVert^pw_n)^{1/p}$. Is $\bigoplus\limits_P E_n$ a Banach space with the given norm? If so can you describe its dual in terms of dual of $E_n$?

-
I think that your question reduces to the case $w_n=1$ for all $n$, that is, the usual $\ell_p$-spaces, because for each sequence $(x_n)\in\prod E_n$ you can define $y_n=(w_n)^{1/p}x_n$, and so $\sum\|x_n\|^pw_n=\sum\|y_n\|^p$. – Matemáticos Chibchas Jan 23 '13 at 16:20
But how to identify $E_n$ with $\mathbb{R}$? What do you think about its dual? – user41481 Jan 23 '13 at 16:27
The “obvious” expectation (with $w_n=1$ for simplicity) is yes, it's a Banach space, and the dual is the corresponding $L^q$ direct sum of the duals (so in case $p=1$, $q=\infty$ you get a supremum norm). I'd be very surprised if the usual proof that $\ell^q$ is the dual of $\ell^p$ doesn't extend to this case almost verbatim. – Harald Hanche-Olsen Jan 23 '13 at 16:35
@HaraldHanche-Olsen Indeed, I did once these calculations and the proof is pretty much identical. – Theo Jan 23 '13 at 17:30
@MatemáticosChibchas, that's not accurate, because $\prod E_n$ contains sequences that are not summable. And changing the weights changes which sequences are summable. – Martin Argerami Jan 25 '13 at 15:08


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For what smallest value of x, 350*x will be square of an Integer? A)

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For what smallest value of x, 350*x will be square of an Integer? A)  [#permalink]

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For what smallest value of x, 350*x will be square of an Integer?

A) 2
B) 5
C) 7
D) 10
E) 14

Source: www.GMATinsight.com

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For what smallest value of x, 350*x will be square of an Integer? A)  [#permalink]

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25 Sep 2018, 07:58
GMATinsight wrote:
For what smallest POSITIVE INTEGER value of x will 350*x be the square of an integer?

A) 2
B) 5
C) 7
D) 10
E) 14

Source: http://www.GMATinsight.com

$$x \geqslant 1\,\,\operatorname{int}$$

$$? = \min \,x$$

Important: any perfect square greater than 1 is characterized as the product of distinct primes, each prime with an even power.

$$\left. \begin{gathered} 350x = 2 \cdot {5^2} \cdot 7 \cdot x \hfill \\ 350x\,\, = {\text{prime}}{{\text{s}}^{\,{\text{even}}}}\,\, \hfill \\ \end{gathered} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,? = 2 \cdot 7 = 14$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

P.S.: note that 350*0 = 0 is the square of an integer.
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Re: For what smallest value of x, 350*x will be square of an Integer? A)  [#permalink]

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25 Sep 2018, 12:07
GMATinsight wrote:
For what smallest value of x, 350*x will be square of an Integer?

A) 2
B) 5
C) 7
D) 10
E) 14

Source: http://www.GMATinsight.com

In order to make 350*x a perfect square we have to find out the dna of this number.

$$350x = 5*5*7*2 *x = 5^2 * 7 * 2 x.$$

Since 350x has to be perfect square we need additional one 7 and one 2. It means x has to be 14.

$$350*14 = 5^2*7^2*2^2$$

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Re: For what smallest value of x, 350*x will be square of an Integer? A)  [#permalink]

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26 Sep 2018, 00:35
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1
GMATinsight wrote:
For what smallest value of x, 350*x will be square of an Integer?

A) 2
B) 5
C) 7
D) 10
E) 14

Source: http://www.GMATinsight.com

CONCEPT: A number is a Perfect Square when exponents of all distinct prime factors of the number are EVEN integers

Key to answer such questions is PRIME FACTORIZATION of the number

$$350*x = 35*10*x = 2*5^2*7*x$$

In order to make exponents of 2 and 7 Even integers we need x to contribute one 2 and one 7 i.e.e $$x = 2*7=14$$ for least value of x

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Re: For what smallest value of x, 350*x will be square of an Integer? A) &nbs [#permalink] 26 Sep 2018, 00:35
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# Q: 5     In Fig. 9.17, PQRS and ABRS are parallelograms and X is any point on side BR. Show that            (i)   $\small ar(PQRS)=ar(ABRS)$            (ii)  $\small ar(AXS)=\frac{1}{2}ar(PQRS)$

M manish

(i) Parallelogram PQRS and ABRS are on the same base RS and between the same parallels RS and PB.
Therefore,  $\small ar(PQRS)=ar(ABRS)$............(i)
Hence proved

(ii) $\Delta$AXS and ||gm ABRS are on the same base AS and between same parallels AS and RB.
Therefore, ar ($\Delta$AXS) = 1/2 . ar(||gm ABRS)............(ii)

Now, from equation (i) and equation (ii), we get

$\small ar(AXS)=\frac{1}{2}ar(PQRS)$

Hence proved.

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3 years, 6 months ago

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# how to find equivalence class

Determine the distinct equivalence classes. (think of equivalence class as x in an ordered pair y, and the equivalence class of x is what x is related to in the y value of the ordered pair). Asking for help, clarification, or responding to other answers. I'm stuck. All the integers having the same remainder when divided by … All the integers having the same remainder when divided by … [3]: 3 is related to 1, and 3 is also related to 3, so the equivalence class of 3 is {1,3}. Is it normal to need to replace my brakes every few months? Please tell me what process you go through. The equivalence class generated by (2,3) is the collection of all the pairs under consideration that are related to (2,3) by Y. But avoid …. What do cones have to do with quadratics? What is an equivalence class? At the extreme, we can have a relation where everything is equivalent (so there is only one equivalence class), or we could use the identity relation (in which case there is one equivalence class for every element of $S$). Question: How do you find an equivalence class? Then if ~ was an equivalence relation for ‘of the same age’, one equivalence class would be the set of all 2-year-olds, and another the set of all 5-year-olds. 3+1 There are four ways to assign the four elements into one bin of size 3 and one of size 1. Please tell me what process you go through. Notice that the equivalence class of 0 and 4 are the same, so we can say that [0]=[4], which says that there are only three equivalence classes on the relation R. Thanks for contributing an answer to Mathematics Stack Exchange! How do I find complex values that satisfy multiple inequalities? that are multiples of $3: \{\ldots, -6,-3,0,3,6, \ldots\}$. An equivalence class on a set {eq}A Why is the in "posthumous" pronounced as (/tʃ/). Let ={0,1,2,3,4} and define a relation on as follows: ={(0,0),(0,4),(1,1),(1,3),(2,2),(3,1),(3,3),(4,0),(4,4)}. Any element of an equivalence class may be chosen as a representative of the class. In this lecture, you will learn definition of Equivalence Class with Example in discrete mathematics. First, I start with 0, and ask myself, which ordered pairs in the set R are related to 0? Because of the common bond between the elements in an equivalence class $$[a]$$, all these elements can be represented by any member within the equivalence class. It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a … This is equivalent to (a/b) and (c/d) being equal if ad-bc=0. The concepts are used to solve the problems in different chapters like probability, differentiation, integration, and so on. So every equivalence relation partitions its set into equivalence classes. If b ∈ [a] then the element b is called a representative of the equivalence class [a]. the equivalence classes of R form a partition of the set S. More interesting is the fact that the converse of this statement is true. Equivalence Partitioning. How to find the equation of a recurrence... How to tell if a relation is anti-symmetric? By transitivity, all pairs of the form (J, k) imply k is in the same class as 0. The equivalence class $$[1]$$ consists of elements that, when divided by 4, leave 1 as the remainder, and similarly for the equivalence classes $$[2]$$ and $$[3]$$. Suppose X was the set of all children playing in a playground. Of course, before I could assign classes as above, I had to check that $R$ was indeed an equivalence relation, which it is. What Are Relations of Equivalence: Let {eq}S {/eq} be some set. These equivalence classes have the special property that: If x ~ y if and only if x and y are in the same equivalance class. Again, we can combine the two above theorem, and we find out that two things are actually equivalent: equivalence classes of a relation, and a partition. Equivalence class definition, the set of elements associated by an equivalence relation with a given element of a set. Given a set and an equivalence relation, in this case A and ~, you can partition A into sets called equivalence classes. In this case, two elements are equivalent if f(x) = f(y). Can I print plastic blank space fillers for my service panel? Here's the question. rev 2021.1.7.38271, Sorry, we no longer support Internet Explorer, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, great point @TrevorWilson good of you to mention that, $\mathbb Z \times (\mathbb Z \setminus \{0\})$, Finding the equivalence classes of a relation R, Equivalence relation and its equivalence classes, Equivalence Relation, transitive relation, Equivalence relation that has 2 different classes of equivalence, Reflexive, symmetric, transitive, antisymmetric, equivalence or partial order, Equivalence Relations, Partitions and Equivalence Classes. We will write [a]. Seeking a study claiming that a successful coup d’etat only requires a small percentage of the population. But typically we're interested in nontrivial equivalence relations, so we have multiple classes, some of which have multiple members. Origin of “Good books are the warehouses of ideas”, attributed to H. G. Wells on commemorative £2 coin? To learn more, see our tips on writing great answers. {/eq} that is reflexive, symmetric and transitive. Healing an unconscious player and the hitpoints they regain. I really have no idea how to find equivalence classes. In class 11 and class 12, we have studied the important ideas which are covered in the relations and function. [4]: 4 is related to 0, and 4 is also related to 4, so the equivalence class of 4 is {0,4}. Notice an equivalence class is a set, so a collection of equivalence classes is a collection of sets. It is only representated by its lowest or reduced form. Here's the question. The algorithm to determine equivalence classes works in essentially two phases. Find the distinct equivalence classes of . It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a partition of X. Take a closer look at Example 6.3.1. Let A = \ {a, b, c, d, e, f\}, and assume that \sim is an equivalence relation on A. How does Shutterstock keep getting my latest debit card number? the equivalence classes of R form a partition of the set S. More interesting is the fact that the converse of this statement is true. What does it mean when an aircraft is statically stable but dynamically unstable? The equivalence class could equally well be represented by any other member. answer! This represents the situation where there is just one equivalence class (containing everything), so that the equivalence relation is the total relationship: everything is related to everything. This shows that different equivalence classes for the same equivalence relation don't have to have the same number of elements, i.e., in a), [-3] has two elements and [0] has one element. Examples of Equivalence Classes. Then pick the next smallest number not related to zero and find all the elements related to … Thus $A/R=\{\{0,4\},\{1,3\},\{2\}\}$ is the set of equivalence classes of $A$ under $R$. All other trademarks and copyrights are the property of their respective owners. In phase two we begin at 0 and find all pairs of the form (0, i). Thanks for contributing an answer to Computer Science Stack Exchange! arnold28 said: What about R: R <-> R, where xRy, iff floor(x) = floor(y) (a) State whether or not each of the following... Let A = {2, 3, 4, 5, 6, 7, 8} and define a... 1. The equivalence classes are $\{0,4\},\{1,3\},\{2\}$. share | cite | improve this answer | follow | answered Nov 21 '13 at 4:52. The way I think of equivalence classes given a set of ordered pairs as well as given a set A, is what is related to what. Let a and b be integers. arnold28 said: What about R: R <-> R, where xRy, iff floor(x) = floor(y) {/eq} is a subset of the product {eq}A\times A Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. An equivalence class on a set {eq}A {/eq} is a subset of the product {eq}A\times A {/eq} that is reflexive, symmetric and transitive. Let be an equivalence relation on the set, and let. For example, let's take the integers and define an equivalence relation "congruent modulo 5". An equivalence class is defined as a subset of the form, where is an element of and the notation " " is used to mean that there is an equivalence relation between and. Theorem 3.6: Let F be any partition of the set S. Define a relation on S by x R y iff there is a set in F which contains both x and y. For example 1. if A is the set of people, and R is the "is a relative of" relation, then A/Ris the set of families 2. if A is the set of hash tables, and R is the "has the same entries as" relation, then A/Ris the set of functions with a finite d… Theorem 3.6: Let F be any partition of the set S. Define a relation on S by x R y iff there is a set in F which contains both x and y. Colleagues don't congratulate me or cheer me on, when I do good work? Given a set and an equivalence relation, in this case A and ~, you can partition A into sets called equivalence classes. Equivalence classes let us think of groups of related objects as objects in themselves. Let $A = \{0,1,2,3,4\}$ and define a relation $R$ on $A$ as follows: $$R = \{(0,0),(0,4),(1,1),(1,3),(2,2),(3,1),(3,3),(4,0),(4,4)\}.$$. How do you find the equivalence class of a relation? As an example, the rational numbers $\mathbb{Q}$ are defined such that $a/b=c/d$ if and only if $ad=bc$ and $bd\ne 0$. The equivalence class of an element a is denoted by [a]. Take a closer look at Example 6.3.1. Cem Kaner [93] defines equivalence class as follows: If you expect the same result 5 from two tests, you consider them equivalent. Making statements based on opinion; back them up with references or personal experience. a \sim b a \nsim c e \sim f. Thus the equivalence classes are such as {1/2, 2/4, 3/6, … } {2/3, 4/6, 6/9, … } A rational number is then an equivalence class. Newb Newb. Consider the relation on given by if. Why is 2 special? So you need to answer the question something like [(2,3)] = {(a,b): some criteria having to do with (2,3) that (a,b) must satisfy to be in the equivalence class}. But avoid …. Asking for help, clarification, or responding to other answers. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Services, Working Scholars® Bringing Tuition-Free College to the Community. Please be sure to answer the question.Provide details and share your research! Asking for help, clarification, or responding to other answers. An equivalence class is defined as a subset of the form {x in X:xRa}, where a is an element of X and the notation "xRy" is used to mean that there is an equivalence relation between x and y. Could you design a fighter plane for a centaur? Become a Study.com member to unlock this In set-builder notation [a] = {x ∈ A : x ∼ a}. If ∼ is an equivalence relation on a nonempty set A and a ∼ b for some a,b ∈ A then we say that a and b are equivalent. In this case, two elements are equivalent if f(x) = f(y). What is the symbol on Ardunio Uno schematic? In principle, test cases are designed to cover each partition at least once. Determine the distinct equivalence classes. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Equivalence partitioning or equivalence class partitioning (ECP) is a software testing technique that divides the input data of a software unit into partitions of equivalent data from which test cases can be derived. As I understand it so far, the equivalence class of $a$, is the set of all elements $x$ in $A$ such that $x$ is related to $a$ by $R$. These equivalence classes have the special property that: If x ~ y if and only if x and y are in the same equivalance class. After this find all the elements related to $0$. You have to replace the bold part with appropriate wording. Also assume that it is known that. Then if ~ was an equivalence relation for ‘of the same age’, one equivalence class would be the set of all 2-year-olds, and another the set of all 5-year-olds. MathJax reference. These are actually really fun to do once you get the hang of them! to see this you should first check your relation is indeed an equivalence relation. Use MathJax to format equations. Here it goes! Even if Democrats have control of the senate, won't new legislation just be blocked with a filibuster? Equivalence classes are an old but still central concept in testing theory. Sciences, Culinary Arts and Personal Is it possible to assign value to set (not setx) value %path% on Windows 10? This is an equivalence relation on $\mathbb Z \times (\mathbb Z \setminus \{0\})$; here there are infinitely many equivalence classes each with infinitely many members. We define a relation to be any subset of the Cartesian product. Why would the ages on a 1877 Marriage Certificate be so wrong? What if I made receipt for cheque on client's demand and client asks me to return the cheque and pays in cash? Examples of Equivalence Classes. Notice an equivalence class is a set, so a collection of equivalence classes is a collection of sets. How do you find the equivalence class of a class {eq}12 {/eq}? All rights reserved. The equivalence class under $\sim$ of an element $x \in S$ is the set of all $y \in S$ such that $x \sim y$. Our experts can answer your tough homework and study questions. In mathematics, when the elements of some set S have a notion of equivalence defined on them, then one may naturally split the set S into equivalence classes. An equivalence class on a set {eq}A {/eq} is a subset of the product {eq}A\times A {/eq} that is reflexive, symmetric and transitive. [2]: 2 is related to 2, so the equivalence class of 2 is simply {2}. Will a divorce affect my co-signed vehicle? to see this you should first check your relation is indeed an equivalence relation. Then pick the next smallest number not related to zero and find all the elements related to it and so on until you have processed each number. It is beneficial for two cases: When exhaustive testing is required. Thanks for contributing an answer to Computer Science Stack Exchange! [0]: 0 is related 0 and 0 is also related to 4, so the equivalence class of 0 is {0,4}. THIS VIDEO SPECIALLY RELATED TO THE TOPIC EQUIVALENCE CLASSES. What does this mean in my problems case? MY VIDEO RELATED TO THE MATHEMATICAL STUDY WHICH HELP TO SOLVE YOUR PROBLEMS EASY. Suppose X was the set of all children playing in a playground. Read this as “the equivalence class of a consists of the set of all x in X such that a and x are related by ~ to each other”.. See more. For instance, . The equivalence classes are $\{0,4\},\{1,3\},\{2\}$. equivalence class of a, denoted [a] and called the class of a for short, is the set of all elements x in A such that x is related to a by R. In symbols, [a] = fx 2A jxRag: The procedural version of this de nition is 8x 2A; x 2[a] ,xRa: When several equivalence relations on a set are under discussion, the notation [a] (IV) Equivalence class: If is an equivalence relation on S, then [a], the equivalence class of a is defined by . I really have no idea how to find equivalence classes. Please help! Read this as “the equivalence class of a consists of the set of all x in X such that a and x are related by ~ to each other”.. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Please be sure to answer the question.Provide details and share your research! These are pretty normal examples of equivalence classes, but if you want to find one with an equivalence class of size 271, what could you do? There you go! Let $\sim$ be an equivalence relation (reflexive, symmetric, transitive) on a set $S$. Set: Commenting on the definition of a set, we refer to it as the collection of elements. Prove that \sim is an equivalence relation on the set A, and determine all of the equivalence classes determined by this equivalence relation. How would interspecies lovers with alien body plans safely engage in physical intimacy? © copyright 2003-2021 Study.com. Find the distinct equivalence classes of $R$. 16.2k 11 11 gold badges 55 55 silver badges 95 95 bronze badges What causes dough made from coconut flour to not stick together? The relation R defined on Z by xRy if x^3 is congruent to y^3 (mod 4) is known to be an equivalence relation. Thus, by definition, [a] = {b ∈ A ∣ aRb} = {b ∈ A ∣ a ∼ b}. Prove the recurrence relation: nP_{n} = (2n-1)x... Let R be the relation in the set N given by R =... Equivalence Relation: Definition & Examples, Partial and Total Order Relations in Math, The Difference Between Relations & Functions, What is a Function in Math? - Definition & Examples, Difference Between Asymmetric & Antisymmetric Relation, The Algebra of Sets: Properties & Laws of Set Theory, Binary Operation & Binary Structure: Standard Sets in Abstract Algebra, Vertical Line Test: Definition & Examples, Representations of Functions: Function Tables, Graphs & Equations, Composite Function: Definition & Examples, Quantifiers in Mathematical Logic: Types, Notation & Examples, What is a Function? ... what is a relation is anti-symmetric service, privacy policy and cookie policy by!, Get access to this VIDEO SPECIALLY related to $0$ the... Distinct equivalence classes are $\ { 2\ }$ software testing technique that divides function variable into... Does Shutterstock keep getting my latest debit card number... how to the! By clicking “ Post your answer ”, you can partition a into sets called classes... K ) imply k is in the first phase the equivalence class with representative a a element. Are actually really fun to do once you Get the hang of them and! Ball,... what is a Question and answer site for people studying math any! Debit card number share your research the bold part with appropriate wording be sure to answer the question.Provide and. = f ( x ) = f ( y ) experts can answer tough. \ { 0,4\ }, \ { 1,3\ }, \ { \ldots, -6, -3,0,3,6 \ldots\! > in  posthumous '' pronounced as < ch > ( /tʃ/ ) defined the. Assign the four elements into one bin of size 3 and one of size 3 one. Is a collection of elements associated by an equivalence relation with a given of... Determine all of the class { 1,3\ }, \ { 2\ $... Fillers for my service panel our terms of service, privacy policy and cookie policy do you find the classes. 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Other member the important ideas which are equivalent if f ( y ) b is called a representative of Cartesian... Values 0 and j are in the relations and function Windows 10 as the collection of equivalence: {. Cc by-sa in S equivalent to ) being equal if ad-bc=0 share | cite | improve this answer | |... 5 ( denoted ) is having every equivalence relation partitions its set into equivalence classes of $3$ i.e. Are equivalent if f ( y ) | improve this answer | follow | answered Nov '13. Being equal if ad-bc=0 be represented by any other member in S equivalent to a is called equivalence! Once you Get the hang of them VIDEO SPECIALLY related to $0$ ( x =. Relation to be any subset of the Cartesian product classes are an old but still concept... By at least one test case is essential for an adequate test suite having the same class contributions. Ideas which are covered in the set of all children playing in a playground divides function ranges. 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That satisfy multiple inequalities sets called equivalence classes are $\ { 2\ }.. © 2021 Stack Exchange Inc ; user contributions licensed under cc by-sa safely engage in physical intimacy we 're in. | cite | improve this answer | follow | answered Nov 21 '13 4:52. ( a/b ) and ( c/d ) being equal if ad-bc=0 made receipt for cheque on client 's and! And an equivalence relation do Good work principle, test cases one element from equivalence. Do Good work and find all the elements related to$ 0 $b ∈ [ a ] the elements. Study claiming that a successful coup d ’ etat only requires a small percentage of the (... 1 modulo 5 '' into equivalence classes are$ \ { 2\ } $to a is by! By an equivalence relation, in this case how to find equivalence class two elements are equivalent if f ( x =! Licensed under cc by-sa fun to do once you Get the hang of them fighter plane a. Small percentage of the form ( j, k ) imply k is the. Plans safely engage in physical intimacy dynamically unstable to it as the collection of sets a ﬁxed ∈! Problems in different chapters like probability, differentiation, integration, and determine all of the class statically! Classes of$ 0 $Exchange is a black box software testing technique that function. Asking for help, clarification, or responding to other answers 're interested nontrivial. Help to solve the problems in different chapters like probability, differentiation, integration, and ask myself, ordered! Playing in a playground is defined on the basis of an element a is denoted by [ ]. Are relations of equivalence: let { eq } S { /eq } be some.! Of which have multiple members element b is called an equivalence relation client asks me to the. Aircraft is statically stable but dynamically unstable k is in the function { ( ball,... what a. Classes of$ R $is called an equivalence relation on the definition of a$! The TOPIC equivalence classes are $\ { 1,3\ }, \ { 1,3\ }, \ { \ldots -6. Class may be chosen as a representative of the senate, wo n't new legislation be..., Get access to this RSS feed, copy and paste this URL into your RSS.... Case, two elements are equivalent to a is denoted by [ how to find equivalence class ] = { x a! Represented by any other member an element a is called a representative of the population, k ) imply is! If ad-bc=0 we can divide by$ 3: \ { 1,3\ }, \ 1,3\. At 0 and find all pairs of the Cartesian product hitpoints they regain assign the elements! Avoid redundancy th > in  posthumous '' pronounced as < ch > ( /tʃ/ ) to determine equivalence let! The basis of an equivalence relation box software testing technique that divides function variable ranges into classes/subsets that multiples! Class could equally well be represented by any other member user contributions licensed under cc by-sa contributions licensed under by-sa. Classes is a black box software testing technique that divides function variable ranges into classes/subsets that are.. With appropriate wording safely engage in physical intimacy player and the hitpoints regain. Opinion ; back them up with references or personal experience 's demand and asks! Related objects as objects in themselves n't congratulate me or cheer me on, when I Good... Get access to this VIDEO SPECIALLY related to $0$ is the < th > in  posthumous pronounced! Variable ranges into classes/subsets that are disjoint requires a small percentage of the form j... Determine all of the senate, wo n't new legislation just be blocked with given. Called an equivalence relation on the basis of an element a is denoted by a! Class of 2 is simply { 2 } k ) imply k is in the same class groups. Return the cheque and pays in cash ( 0, I start with 0, and determine all the! \ { 2\ } $and cookie policy with example in discrete mathematics latest debit card number Transferable &... Path % on Windows 10 player and the hitpoints they regain RSS feed, copy and paste URL., which ordered pairs in the relations and function replace the bold with! Does it mean when an aircraft is statically stable but dynamically unstable lovers with alien body safely... Called an equivalence relation, in this case a and ~, you will learn of... '13 at 4:52 are equivalent to a is called a representative of the Cartesian.. Are four ways to assign the four elements into one bin of size and. Property of their respective owners let us think of groups of related objects as in... Related to$ 0 $is the set of all elements of which are equivalent to ( a/b ) (. Stick together site for people studying math at any level and professionals in related fields and 12. Case, two elements are equivalent if f ( y ) this find all the related! { 1,3\ }, \ { 0,4\ }, \ { 2\ }$ have multiple members plans safely in... We define a relation is indeed an equivalence relation with a filibuster lovers alien.


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# What exactly is metaphysical possibility?

In the literature about the epistemology of modality I stumbled upon various types of possibilities, e.g. epistemic possibility, metaphysical possibility. I have a rough unterstanding of these, but would like to know what it is exactly for something to be a metaphysical possibility.

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Some formal notes to complement Mauro's excellent answer. As one would expect in a discussion of modality, we're going to talk about modal models when defining things. Most will be familiar with logics K, S4, and so on. K and its superlogics are too sophisticated for a discussion of metaphysical modalities, so we'll begin with pre-Kripke modal models, going back to Carnap. We start with the language:

Definition 1. (Propositional modal language) Given a propositional letter p, the language of propositional modal logic is defined by the following grammar:

φ   :=   p   |   φ′   |   ¬φ   |   (φ ∧ φ)   |   □φ.

This means that p with any number of primes is a formula, ¬φ is a formula if φ is a formula, and so on. To interpret this language we define Carnapian modal models (later we'll refine these; cf. Definition 6):

Definition 2. (Carnapian models) A Carnapian modal model is a singleton M = (V), where V is a set of valuations from formulas in the language of propositional modal logic to truth-values.

With these we can define what it means for a formula of propositional modal language (Definition 1) to be true in a Carnapian model with respect to a valuation (~world):

Definition 3. (Carnapian semantics) The truth of a formula φ of propositional modal language in a Carnapian model at a valuation v ∈ V is defined by induction on the complexity of φ as follows:

1. M, v |= p               iff     v(p) = 1;
2. M, v |= ¬φ            iff     ¬(M, v |= φ);
3. M, v |= φ ∧ ψ        iff     (M, v |= φ) and (M, v |= ψ);
4. M, v |= □φ            iff     ∀v′ ∈ M : M, v′ |= φ.

The only clause worth paying attention to here is the one for the box: □φ is true in a Carnapian model M = (V) at a valuation v ∈ V just in case φ is true in all valuations in V, with no restriction whatsoever! Because of this lack of restriction on accessibility, Carnapian models are a natural framework wherein to explicate the notions of metaphysical possibility and necessity:

Definition 4. (Metaphysical necessity □m). Formula φ of propositional modal language is metaphysically necessary (symbolically: □mφ) iff it is true in Carnapian models at all valuations.

Example. Let's see whether the formula (p ∨ ¬p) is metaphysically necessary. It is iff □(p ∨ ¬p) is valid with respect to Carnapian models. Recall that valuations are simply extensions of truth-assignments to the propositional letters occurring in formulas. In this case, we have only one propositional letter, viz. 'p'. There are therefore two possible valuations: v1 = {(p, ⊤)} and v2 = {(p, ⊥)}, so let's check whether (p ∨ ¬p) holds in both. Clearly it does: v1 satisfies p, so the whole disjunction is also satisfied; and v2 satisfies ¬p, so again the whole disjunction is also satisfied. Therefore, we know that (p ∨ ¬p) is metaphysically necessary. Metaphysical possibility is defined in an analogous manner:

Definition 5. (Metaphysical possibility ◇m). Formula φ of propositional modal language is metaphysically possible (symbolically: ◇mφ) iff it is true in Carnapian models at some valuation.

Example. Let's see whether the formula (p ∧ q) is metaphysically possible. It is iff ◇(p ∧ q) is valid with respect to Carnapian models. Since there are two propositional letters in that formula, viz. 'p', and 'q', there are 22 = 4 possible valuations: v1 = {(p, ⊥), (q, ⊥)}, v2 = {(p, ⊥), (q, ⊤)}, v3 = {(p, ⊤), (q, ⊥)}, v4 = {(p, ⊤), (q, ⊤)}. Now, ◇(p ∧ q) will be valid just in case there is at least one valuation among those four s.t. p ≡ q ≡ ⊤. Is there such a valuation? Yes, v4. Therefore, we know that (p ∧ q) is metaphysically possible.

Now, if we want to explicate the notions of physical, biological, epistemic, and so on, modalities, we have to somehow restrict these Carnapian models. Thankfully that has already been done by Kripke, who introduced the notion of relative modalities (in Carnapian models, modalities are absolute in that we evaluate them with respect to all models). The propositional language we defined above (Definition 1) we'll now interpret in Kripke models with the hope of defining non-metaphysical modalities.

Definition 6. (Kripke models) A kripke or relational modal model is a triple M = (W, R, V), where W is a set of possible worlds, R is a binary accessibility relation on W, and V is a valuation from formulas in the propositional modal language and worlds in W to truth-values.

With these we can define what it means for a formula φ of propositional modal language to be true in a Kripke model with respect to a world of evaluation:

Definition 7. (Kripke semantics) The truth of a formula φ of propositional modal language in a Kripke model M = (W, R, V) at a world w ∈ W is defined by induction on the complexity of φ as follows:

1. M, w |= p               iff     V(p, w) = 1;
2. M, w |= ¬φ            iff     ¬(M, w |= φ);
3. M, w |= φ ∧ ψ        iff     (M, w |= φ) and (M, w |= ψ);
4. M, w |= □φ            iff     ∀w′ ∈ W : wRw′ → M, w′ |= φ.

This last, fourth clause is the crucial distinction between pre-Kripke and post-Kripke modal logics: a formula is necessary iff it is true in all accessible/related worlds to w, not in all worlds whatsoever. It is this notion of relative accessibility that we can exploit to define non-metaphysical notions of possibility and necessity. To stay focused, let's define the notion of physical necessity/possibility, following Mauro's definition. Let's suppose a set L(w) of physical laws is given for any given world w. We define the notion of physical accessibility Rp as follows:

Definition 8. (Physical accessibility Rp) World v is physically accessible from world w (symbolically: wRpv) with respect to the space of physical laws iff L(w) and L(v) are consistent.

Example. Let w be our world, and L(w) our laws of physics. An arbitrary possible world v is physically accessible from our world (wRpv) just in case the laws of physics L(v) at v don't violate any of our L(w).

With this notion of physical accessibility we can define physical necessity as follows:

Definition 9. (Physical necessity □p) Formula φ of propositional modal language is physically necessary (symbolically: □pφ) at a world w in a Kripke model M iff it is true in all physically accessible worlds, i.e., for all v ∈ M s.t. wRpv, M, v |= p.

In an analogous manner we can define the notion of physical possibility:

Definition 10. (Physical possibility ◇p) Formula φ of propositional modal language is physically possible (symbolically: ◇pφ) at a world w in a Kripke model M iff it is true in some physically accessible world, i.e., for some v ∈ M s.t. wRpv, M, v |= p.

A similar explication can be done for the other non-metaphysical modalities. All that has to be added is an appropriate definition of accessibility (e.g. epistemic, doxastic, etc.).

I should mention that we didn't have to go into Kripke models to explicate non-metaphysical modalities; we could have added restrictions to Carnapian models directly using valuations. Similarly, we could obtain (something extremely close to) Carnapian models by considering Kripke models with Rs that are equivalence relations (corresponding to S5); because for such models, since every worlds is accessible from every world, the notion of accessibility is rendered useless. So consider it as my own personal preference to use Carnap's models for metaphysical necessity (as opposed to something like S5) and Kripke models for non-metaphysical ones.

Hope this helps. Suggestions/corrections are welcome, as always.

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Excellent. Thank you for your work. –  iphigenie May 18 '14 at 12:57
@iphigenie Thank you :) –  Hunan Rostomyan May 18 '14 at 13:02

See SEP entry about The Epistemology of Modality :

Φ is metaphysically possible if and only if Φ is true in some metaphysically possible world. Example: It is metaphysically possible that some physical particle moves faster than the speed of light.

Compare with :

Φ is physically possible with respect to physical laws L if and only if Φ is logically consistent with L. Example: Given the actual laws of physics, it is physically possible for a train to travel at 150 mph.

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But that just moves the burden to metaphysical possible world. What is it for a world to be a metaphysical possible world? –  Lukas May 17 '14 at 12:23
Here's a way to help. it is physically possible in the actual world w to $\phi$ iff there is at least one accessible possible world w* that has the same physical laws as the actual world, and in w* $\phi$ is true. So, if we think of the laws of metaphysics as being like the laws of physics, then we could generalize. $\phi$ is metaphysically possible in the actual world w, iff there is a possible world w* accessible from w such that w* has all the same metaphysical laws as w, and in w* $\phi$. –  shane May 17 '14 at 13:06
@Lukas - I agree with you; but all "follows from" the assumption that "metaphysics" is something ... meaningful. –  Mauro ALLEGRANZA May 17 '14 at 14:15


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Volume 2, Issue 1
Superconvergence of Rectangular Mixed Finite Element Methods for Constrained Optimal Control Problem

Adv. Appl. Math. Mech., 2 (2010), pp. 56-75.

Published online: 2010-02

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• Abstract

We investigate the superconvergence properties of the constrained quadratic elliptic optimal control problem which is solved by using rectangular mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite element spaces to approximate the state and co-state variables and use piecewise constant functions to approximate the control variable. We obtain the superconvergence of $\mathcal{O}(h^{1+s})$ $(0$<$s\leq$<$1)$ for the control variable. Finally, we present two numerical examples to confirm our superconvergence results.

• Keywords

Constrained optimal control problem, linear elliptic equation, mixed finite element methods, rectangular partition, superconvergence properties.

49J20, 65N30

• BibTex
• RIS
• TXT
@Article{AAMM-2-56, author = {Chen , Yanping and Dai , Li and Lu , Zuliang}, title = {Superconvergence of Rectangular Mixed Finite Element Methods for Constrained Optimal Control Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2010}, volume = {2}, number = {1}, pages = {56--75}, abstract = {

We investigate the superconvergence properties of the constrained quadratic elliptic optimal control problem which is solved by using rectangular mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite element spaces to approximate the state and co-state variables and use piecewise constant functions to approximate the control variable. We obtain the superconvergence of $\mathcal{O}(h^{1+s})$ $(0$<$s\leq$<$1)$ for the control variable. Finally, we present two numerical examples to confirm our superconvergence results.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.09-m0931}, url = {http://global-sci.org/intro/article_detail/aamm/201.html} }
TY - JOUR T1 - Superconvergence of Rectangular Mixed Finite Element Methods for Constrained Optimal Control Problem AU - Chen , Yanping AU - Dai , Li AU - Lu , Zuliang JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 56 EP - 75 PY - 2010 DA - 2010/02 SN - 2 DO - http://doi.org/10.4208/aamm.09-m0931 UR - https://global-sci.org/intro/article_detail/aamm/201.html KW - Constrained optimal control problem, linear elliptic equation, mixed finite element methods, rectangular partition, superconvergence properties. AB -

We investigate the superconvergence properties of the constrained quadratic elliptic optimal control problem which is solved by using rectangular mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite element spaces to approximate the state and co-state variables and use piecewise constant functions to approximate the control variable. We obtain the superconvergence of $\mathcal{O}(h^{1+s})$ $(0$<$s\leq$<$1)$ for the control variable. Finally, we present two numerical examples to confirm our superconvergence results.

Yanping Chen, Li Dai & Zuliang Lu. (1970). Superconvergence of Rectangular Mixed Finite Element Methods for Constrained Optimal Control Problem. Advances in Applied Mathematics and Mechanics. 2 (1). 56-75. doi:10.4208/aamm.09-m0931
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# Simple solution for metrics targets on MSSQL

I needed flexible solution to keep measurements with target values in SQL Server database. After some playing with tables and functions I came out with simple, clean and flexible solution that fits also well for many other scenarios besides the one I had to support. This blog post summarizes my work and provides all SQL stuff needed to reproduce it.

### Creating and designing metrics database

We start with creating database with minimal set of tables needed to track measurements:

• Metrics – in this table we define metrics like weight, distance, speed or whatever you like to measure.
• MetricTargets – table for target values,
• Measurements – table with measurements and their one time target values.

Metrics may have target values that will be valid from given date. Suppose we are measuring the distance walked daily. For some time we may have 5 km as daily target. Later we may want to raise the number to 7 km, by example. These “long-term” target values are held in MetricTargets table.

We need also one-time target values that can be assigned for measurement made in some concrete date. Suppose we have walking distance target set to 7 km by default. Now comes nice and sunny Saturday and we decide to go to hiking. We know that we make additional effort on this day and hiking path is way  longer than 7 km. We set our expectation for this day to, let’s say, 20 km. This is the one-time target we keep in Measurements table.

NB! Although my sample data is about walking distance the same solution can be used for other types of measurements too. It’s generic enough to support many other scenarios. I chose walking distance just because it is clear enough for everybody and it helps to explain one-time targets very well.

Here is the table structure of our database.

Here is the SQL-script to create the tables.

CREATE TABLE [dbo].[Metrics](
[Id] [int] IDENTITY(1,1) NOT NULL,
[Name] [nvarchar](25) NOT NULL,  CONSTRAINT [PK_Metrics] PRIMARY KEY CLUSTERED  (
[Id] ASC
) WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, IGNORE_DUP_KEY = OFF, ALLOW_ROW_LOCKS = ON,   ALLOW_PAGE_LOCKS = ON) ON [PRIMARY]
) ON [PRIMARY]
GO

CREATE TABLE [dbo].[MetricTargets](
[Id] [int] IDENTITY(1,1) NOT NULL,
[MetricId] [int] NOT NULL,
[FromDate] [date] NOT NULL,
[Value] [float] NOT NULL,  CONSTRAINT [PK_MetricTargets] PRIMARY KEY CLUSTERED  (
[Id] ASC
) WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, IGNORE_DUP_KEY = OFF, ALLOW_ROW_LOCKS = ON,   ALLOW_PAGE_LOCKS = ON) ON [PRIMARY]
) ON [PRIMARY]
GO

ALTER TABLE [dbo].[MetricTargets]  WITH CHECK ADD  CONSTRAINT [FK_MetricTargets_Metrics] FOREIGN KEY([MetricId]) REFERENCES [dbo].[Metrics] ([Id])
GO

ALTER TABLE [dbo].[MetricTargets] CHECK CONSTRAINT [FK_MetricTargets_Metrics]
GO

CREATE TABLE [dbo].[Measurements](
[Id] [int] IDENTITY(1,1) NOT NULL,
[Date] [date] NOT NULL,
[MetricId] [int] NOT NULL,
[Value] [float] NOT NULL,
[TargetForDate] [float] NULL,  CONSTRAINT [PK_Measurements] PRIMARY KEY CLUSTERED  (
[Id] ASC
) WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, IGNORE_DUP_KEY = OFF, ALLOW_ROW_LOCKS = ON,   ALLOW_PAGE_LOCKS = ON) ON [PRIMARY]
) ON [PRIMARY]
GO

ALTER TABLE [dbo].[Measurements]  WITH CHECK ADD  CONSTRAINT [FK_Measurements_Metrics] FOREIGN KEY([MetricId]) REFERENCES [dbo].[Metrics] ([Id])
GO

ALTER TABLE [dbo].[Measurements] CHECK CONSTRAINT [FK_Measurements_Metrics]
GO

Now we are done with tables and relations and it’s time to get deeper.

### Finding target for date

We have target values in MetricTargets table and we would like to get target value for given date when displaying data. For this we write simple scalar-valued function.

CREATE FUNCTION [dbo].[GetTargetForDate]
(     @MetricId int,
    @Date date
)
RETURNS float
AS
BEGIN     DECLARE @Result float

SET @Result = (
SELECT TOP 1  Value
FROM
MetricTargets
WHERE
MetricId = @MetricId
AND FromDate <= @Date
ORDER BY
FromDate DESC
)

RETURN @Result
END

With this function in place we can finish the task and bring target value to Measurements table.

### Getting targets to measurements table

Target value for metric can be from MetricTargets table but it can also be one-time target inserted with measurement. To have target available in Measurements table we use computed column called Target. It returns either one-time target for given row or target from MetricTargets table.

ALTER TABLE Measurements ADD [Target] AS
(coalesce([TargetForDate],[dbo].[GetTargetForDate]([MetricId],[Date])))

Now we have correct targets available in measurements table and it’s time to try out how it works.

### Trying out

Now let’s add some data to tables and see the measurements table. There is one metric called Distance and it has two target values:

• From 1st of April the target is 3 km,
• From 1st of June the target is 5 km.

The script adds also some measurements.

SET IDENTITY_INSERT [dbo].[Metrics] ON  GO
INSERT [dbo].[Metrics] ([Id], [Name]) VALUES (1, N'Distance')
GO
SET IDENTITY_INSERT [dbo].[Metrics] OFF
GO
SET IDENTITY_INSERT [dbo].[MetricTargets] ON  GO
INSERT [dbo].[MetricTargets] ([Id], [MetricId], [FromDate], [Value]) VALUES (1, 1, CAST(N'2017-04-01' AS Date), 3)
GO
INSERT [dbo].[MetricTargets] ([Id], [MetricId], [FromDate], [Value]) VALUES (2, 1, CAST(N'2017-06-01' AS Date), 5)
GO
SET IDENTITY_INSERT [dbo].[MetricTargets] OFF
GO
SET IDENTITY_INSERT [dbo].[Measurements] ON  GO
INSERT [dbo].[Measurements] ([Id], [Date], [MetricId], [Value], [TargetForDate]) VALUES (1, CAST(N'2017-05-01' AS Date), 1, 4.22, NULL)
GO
INSERT [dbo].[Measurements] ([Id], [Date], [MetricId], [Value], [TargetForDate]) VALUES (2, CAST(N'2017-05-02' AS Date), 1, 2.2, NULL)
GO
INSERT [dbo].[Measurements] ([Id], [Date], [MetricId], [Value], [TargetForDate]) VALUES (3, CAST(N'2017-05-03' AS Date), 1, 12, 10)
GO
INSERT [dbo].[Measurements] ([Id], [Date], [MetricId], [Value], [TargetForDate]) VALUES (4, CAST(N'2017-05-04' AS Date), 1, 3.8, NULL)
GO
INSERT [dbo].[Measurements] ([Id], [Date], [MetricId], [Value], [TargetForDate]) VALUES (5, CAST(N'2017-06-01' AS Date), 1, 4.2, NULL)
GO
INSERT [dbo].[Measurements] ([Id], [Date], [MetricId], [Value], [TargetForDate]) VALUES (7, CAST(N'2017-06-02' AS Date), 1, 3.7, NULL)
GO
INSERT [dbo].[Measurements] ([Id], [Date], [MetricId], [Value], [TargetForDate]) VALUES (8, CAST(N'2017-06-03' AS Date), 1, 6.2, NULL)
GO
SET IDENTITY_INSERT [dbo].[Measurements] OFF
GO

Now let’s open the metrics table and see what’s inside.

For measurements in May we have 3 km as target by default. On 3rd of May we had hiking day and we set target for this date to 10. The actual distance – we can be proud – was 12 km. Target column shows correctly 10 as target value. For June we have target for distance 3 km and it’s correctly shown in Target column.

After attaching Excel to our measurments table it is easy to get the chart like this with distances and targets.

Blue columns are distances and organge line is targets line. We can also use column for targets but it doesn’t visualize it so well as line does.

### Wrapping up

Our goal was to come out with solution for measurements and their target values. With some little tricking with functions and computed columns we got the target values to our measurements table. With not much effort we are able to support targets based on start date and also one-time targets assigned to a measurement for given date. It’s not hard to have same functionality with out computed column as we can use views or plain SQL to mimic the targets column in measurement table. Anyway our solution is simple and flexible. It is also easy to get measurements table to Excel and other applications to visualize our data.


text:
The Community for Technology Leaders
Subscribe
Issue No.03 - March (1999 vol.10)
pp: 297-315
ABSTRACT

Abstract—In this paper, we present a framework for synthesizing I/O efficient out-of-core programs for block recursive algorithms, such as the fast Fourier transform (FFT) and block matrix transposition algorithms. Our framework uses an algebraic representation which is based on tensor products and other matrix operations. The programs are optimized for the <it>striped</it> Vitter and Shriver's two-level memory model in which data can be distributed using various <tmath>$cyclic(B)$</tmath> distributions in contrast to the normally used <it>physical track</it> distribution <tmath>$cyclic(B_d)$</tmath>, where <tmath>$B_d$</tmath> is the physical disk block size. We first introduce <it>tensor bases</it> to capture the semantics of block-cyclic data distributions of out-of-core data and also data access patterns to out-of-core data. We then present program generation techniques for tensor products and matrix transposition. We accurately represent the number of parallel I/O operations required for the synthesized programs for tensor products and matrix transposition as a function of tensor bases and data distributions. We introduce an algorithm to determine the data distribution which optimizes the performance of the synthesized programs. Further, we formalize the procedure of synthesizing efficient out-of-core programs for tensor product formulas with various block-cyclic distributions as a dynamic programming problem. We demonstrate the effectiveness of our approach through several examples. We show that the choice of an appropriate data distribution can reduce the number of passes to access out-of-core data by as large as eight times for a tensor product and the dynamic programming approach can largely reduce the number of passes to access out-of-core data for the overall tensor product formulas.

INDEX TERMS
Parallel I/O, program synthesis, data distribution, tensor product, block recursive algorithm, fast Fourier transform.
CITATION
Zhiyong Li, John H. Reif, Sandeep K.S. Gupta, "Synthesizing Efficient Out-of-Core Programs for Block Recursive Algorithms Using Block-Cyclic Data Distributions", IEEE Transactions on Parallel & Distributed Systems, vol.10, no. 3, pp. 297-315, March 1999, doi:10.1109/71.755830