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# Extending the rationals using exponentiation

The set of integers can be constructed as an equivalence relation over the natural numbers using the the binary operation of addition, and a similar process yields the rationals from integers and multiplication. I'm curious what happens if I try to continue this sequence with next logical step: extending the rationals via an equivalence relation equating pairs of exponentials of rational numbers raised to to rational powers.

I know that after suitably defining the relation to exclude indeterminate forms like 0^0, the resulting set will be isomorphic to a proper superset of the rational numbers, since it includes the square root of 2, and a proper subset of the real numbers since numbers like pi can't be represented (it occurs to me I don't actually know how to prove this, though I do know how to prove pi is irrational). So my question is just what is this set of numbers? Does it have a name? And lastly, why isn't it mentioned in less advanced mathematics classes (presumably it lacks sufficiently useful algebraic properties, maybe?)?

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I don't understand your question. You may or may not be talking about a certain subring of the algebraic numbers (en.wikipedia.org/wiki/Algebraic_number). See also en.wikipedia.org/wiki/Transcendental_number . – Qiaochu Yuan Mar 15 '13 at 5:44
Can you explain the "suitable definition"? – Alex Becker Mar 15 '13 at 5:45
@AlexBecker Sure. In constructing the integers you first form the cartesian product of the naturals, and define the equivalence relation. But to construct the rationals, an equivalence relation is defined on Z-{0} instead of Z because there isn't a way to define zero to have a multiplicative inverse. All I meant by "suitable definition" was to exclude rational numbers which can't be used as bases of logarithms (at least I think that is sufficient for what I describe). – David H Mar 15 '13 at 6:02
@DavidHammett But what equivalence relation are you using? – Alex Becker Mar 15 '13 at 6:12

You'll have to be more careful about what you mean here.

The process by which we form the integers from the natural numbers really is the same in some sense as the process by which we form the rationals from the integers. In both cases, we have a commutative monoid ($\langle\mathbb{N},0,+\rangle$ and $\langle\mathbb{Z}^*,1,\times\rangle$), i.e. an associative, commutative operation with an identity element, and we form the Grothendieck group of the monoid, which is a fancy way of saying that we formally adjoin inverses to all elements.

But exponentiation on the rationals is very different from addition and multiplication. It isn't commutative or even associative. It has a right identity, 1 ($x^1 = x$ for all $x$), but no left identity (there is not $q$ such that $q^x = x$ for all $x$). And $\mathbb{Q}$ isn't even closed under exponentiation ($2^\frac{1}{2} \notin \mathbb{Q}$).

My point is just that continuing this sequence with $\mathbb{Q}$ and exponentiation might not be as logical a next step as you think.

Edit: Here are some specific questions to consider with respect to your proposed construction.

1. In the construction of $\mathbb{Z}$ from $\mathbb{N}$ and of $\mathbb{Q}$ from $\mathbb{Z}$, we consider pairs of numbers $(a,b)$. In the first case, $(a,b)$ is supposed to represent $a-b$, and in the second case, $(a,b)$ is supposed to represent $\frac{a}{b}$. In your case, what do you intend $(a,b)$ to represent?

2. One thing that makes these constructions useful is that we can extend the algebraic operations from the original structure to the new structure. In the construction of $\mathbb{Z}$, we can let $(a,b) + (a',b') = (a + a',b+b')$, since $(a - b) + (a' - b') = (a+a') - (b+b')$. In the construction of $\mathbb{Q}$ from $\mathbb{Z}$, we can let $(a,b) \times (a',b') = (aa',bb')$, since $\frac{a}{b}\times\frac{a'}{b'} = \frac{aa'}{bb'}$. We can also extend $+$ to $\mathbb{Q}$ using a more complicated "common denominators" rule. Which algebraic operations would you like to extend to your newly constructed set, and how would you extend them?

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A cell made up of two hydrogen electrodes. The positive electrode is in contact with a solution of 10^(–4) M hydroge

### Question Asked by a Student from EXXAMM.com Team

Q 2176223176.     A cell made up of two hydrogen electrodes. The positive electrode is in contact with a solution of 10^(–4) M hydrogen ions. The emf of the cell is 0.118 V at 25^oC . The concentration of hydrogen ions at negative electrode is
A

10^(-6) M

B

10^(-4) M

C

10^(-2) M

D

None of these

#### HINT

(Provided By a Student and Checked/Corrected by EXXAMM.com Team)

#### Access free resources including

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# Is the danger associated with HC-=N solutions associated with its acidity?

There is no necessary relationship. $H C \equiv N$ is a deadly poison, but then so is basic ""^(-)C-=N.
On the other hand, most things that we eat that taste good are slightly acidic (i.e. $p H < 7$). See this old answer.


text: Dominant Diminished Scales (Diminished Blues) The Dominant Diminished Scale consists of eight notes, and therefore belongs to the category of octatonic scales. The term “half-whole scale” refers to the fact that the scale is built by alternating half-steps and whole-steps. #1 Way To Improve Your Guitar Playing. Then take that diminished arpeggio, add a half step below each note and you get the half-whole diminished scale! I don't how many scales there are, i want how many different sounding full diminished scales there are. In major scales, a diminished triad occurs only on the seventh scale degree.For instance, in the key of C, this is a B diminished triad (B, D, F). Dominant Diminished – Root On String 6 . C° or just the rather insulting shorthand, dim: Cdim. I'm still not quite following. This is different from the major modes, which have only 7 notes, again counting the root only once. If you look at the scale, every other note is a part of the chord. They don't have to stick with notes in the key. If Beethoven wants a viio7, then he can use one. The chord note spelling reflects this note is flattened twice: bb7. Half Vs Fully Diminished. The symbol for the diminished chord is a superscript circle above the letter of the chord. A more advanced diminished scale is the Fully Diminished scale (half-step/whole-step). Rank Root Scale Notes Intervals More Info; 1 : A : Half Diminished Scale : A,B,C,D,Eb,F,G : 1 2 b3 4 b5 b6 b7 Using this idea allows you to create something called a Double Diminished Chord. Enjoy our FREE full length backing tracks and improvisation video. Take note of the different 7th chords that each diminished mode builds. When heard alone, diminished chords can be perceived as eerie, goofy, or even annoying. And luckily it’s pretty straightforward, a fully diminished chord is simply a stack of minor 3rds… Because the construction of these chords is symmetrical, there are only 3 diminished options: a stack of minor 3rds from C, from C#, and from D. That’s it – the pattern just repeats itself. Learn more about how to build the fully diminished scale at: https://bit.ly/2Cxq8MS In this music video lesson, we will learn how to build the fully diminished scale. In the final two lectures, I'm going to talk about the symbols we use to describe triads and seventh chords in pop, jazz, and classical music. And we'll continue practicing spelling chords. It has a long history, and yet has managed to stay relevant even in today’s modern age of music. GuitarToneMaster is an archive of Guitar Jam Tracks and dedicated to provide guitarists with the best backing tracks. The diminished scale can be created by stacking two diminished 7th chords a half-step apart on top of one another: Notice that the scale has eight unique notes, thus the name octatonic (which literally means eight tones). You could start the diminished arpeggio from any one of those notes, so a B dim, D dim, F dim or Ab dim arpeggio. Your search for the best backing tracks ends here. The best way to approach this is probably to take a look at the scale that is the key. Diminished chords come in three varieties: diminished triads, diminished 7ths, and half-diminished chords. And so forth. For example, the seventh note in a Ddim7 chord is Cb. That doesn’t make sense to me since you can’t build an augmented chord. For example: A diminished chord built on the root of C would include Eb and Gb. The diminished seventh (or fully diminished chord) adds a minor-third above a diminished triad. The most common way you’ll use this sound in your solos is over dominant chords. Another option is the ♭13 / 13. I briefly cover those two scales in my Diminished Modes, Chords & Scale article. It is also a so-called symmetrical scale since the intervals are consistent. From there I started introducing my ears to its singular sound. Check out this essential lesson for your overall bass playing knowledge! Follow On Instagram Like On Facebook Subscribe on You Tube. Character and Mood of Diminished Chords . 3. The “half/whole diminished scale” works over a dominant seventh with a flat nine and/or sharp nine chord. These chords are used in different contexts, and for the sake of brevity we’re going to take a look at the first two and save half-diminished chords for another discussion. You may have also noticed that the Diminished Scale is really just two Diminished Chords a semitone apart, superimposed over each other. When studying diminished scales, one of the most interesting concepts that you can explore is the use of both Dim7 and dominant 7th arpeggios through this scale.. Because you can solo over Dim7 chords with the diminished scale, it is no surprise that you can use Dim7 arpeggios built from this scale. They provide a sense of musical conflict, making the resolution back to consonant chords more impactful. This means there is harmonic movement occuring around these chords. Alternate scale names: fully diminished scale, whole-half diminished scale, or WH diminished scale *Note: This scale has the same notes as the G dominant diminished scale. Typically, composers resolve the tritones by contrary motion: augmented fourths expanding outward to sixths, diminished fifths contracting inward to thirds. Some people call diminished scales “Octatonic” scales, as they have 8 notes (counting the root only once). Diminished,... scales on with our high quality, well structured, and fun Guitar Backing Tracks. A diminished chord’s character is best described as ambiguous. I prefer repeating the letters for the diminished 7th and major 7th, though some prefer the ♭5 / #5. And then you can use that scale over that backdrop. They are: C°, C#° and D°. Bb-7th: The 7th note quality of the major scale is major, and the note interval quality needed is diminished, so the 7th note scale note name - A, is adjusted 2 half-tones / semitones / 1 whole tone down to Abb. Play a G7 chord then the diminished arpeggio B-D-F-Ab and listen to how that resolves into a C7 chord. I’ll go over chord tones with you and really dig into what each scale is and how they differ. This is largely because dominant 7th chords are chords of transition. The notes in-between act as leading tones to the chord tones. 161k members in the guitarlessons community. Your search for the best backing tracks ends here. A subreddit dedicated to learning guitar. It’s the diminished chord. GuitarToneMaster is an archive of Guitar Jam Tracks and dedicated to provide guitarists with the best backing tracks. How many DIFFERENT SOUNDING full diminished scales are there. Diminished chords, also known as diminished triads or dim chords, are dissonant chords that combine a root note with two minor thirds above the root. Also: I as always love to use my delay pedal [playing: 6.31 to 6.49]. Post lessons, ask questions, and get feedback on your playing … Diminished,... scales on with our high quality, well structured, and fun Guitar Backing Tracks. Diminished Scale Arpeggios. We saw that when we raised the leading tone in minor, the seventh chord built on the seventh note of the scale changes from a dominant seventh chord to a fully diminished seventh chord. The diminished 7th chord has a 7th note that is considered a double flat 7th and fully diminished has a flat 5th and a double flat 7th, this gives the chord tension and thats the plan. Meaning the seventh note is three semitones above the flattened fifth. Enjoy our FREE full length backing tracks and improvisation video. Find guitar scales using graphic interface. It displayed countless chords alongside scales that corresponded to them and connected dominant seventh chords with the double diminished scale. Notice now that, as there are 12 notes on the tempered scale and a diminished chord corresponds to another 4 chords identical to it, we can conclude that there are only 3 different diminished chords. The diminished scale is one of the most useful and versatile tools for jazz improvisers. Diminished scales consist of alternating whole steps and half steps. F#-7th: The 7th note quality of the major scale is major, and the note interval quality needed is diminished, so the 7th note scale note name - E#, is adjusted 2 half-tones / semitones / 1 whole tone down to Eb. [MUSIC] The “whole/half diminished scale” works over a fully diminished seventh chord. #1 Way To Improve Your Guitar Playing. Both fully- and half-diminished seventh chords include a tritone between scale degrees $\hat7$ and $\hat4$ which typically resolves in the manner described in Chapter 16. Diminished scales are used in the construction of flat nine (also known as sharp nine) chords, but may also be used in conjunction with most dominant 7th chords. So using this whole half scale (diminished scale) over the dim7 works very well giving your lines the sophistication you want. This is why some people call it the “dominant diminished scale”. If you play a D♭dim7 chord in your left hand and a Cdim7 chord in your right hand, you have yourself a Double Diminished Chord. The scale that you need to use in a context not the diminished scale. Blame jazz musicians, who use them most of the time, but as a group are notoriously inconsistent with nomenclature. The dominant diminished scale can be applied to the V7 chord in a 2-5-1 chord progression, ... so make sure to take these scale shapes to all 12 keys in your practicing in order to get a full understanding of how the dominant diminished scale sits on the fretboard. Here’s an introduction to this alluring scale, along with some ways to put it into action. That will give you some ideas to make music with the diminished scale. Primarily because the diminished scale is a symetrical and synthetic construction and not really what fits any type of tonal song. The chord note spelling reflects this note is flattened twice: bb7. Most jazz standard progressions, where we find the diminished chords, are tonal. This is not to say that the diminished scale is used every day, but it provides a very specific function in music, a function that no other scale can satisfy. (Those are chords built using notes of a major or natural minor scale.) #BASSNATION in this week’s lesson I’ll be covering the difference between a half diminished and a fully diminished scale on bass. Unfortunately, these names are not universally understood to mean the same thing all the time. Therefore, the complete Ddim7 chord has the notes D – F – Ab – Cb. But “diminished”, or “fully diminished” or “diminished 7”. The half tone, whole tone formula is consistent throughout the scale, for which reason it is sometimes called the Half-tone/Whole-tone Scale. If so, there’s a chord that appears only once in the sequence. I want to focus on the diminished modes that come from the Melodic & Harmonic minor scales. The fully diminished scale is an interesting scale overall. 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# On the \$ L^ p \$-geometry of autonomous Hamiltonian diffeomorphisms of surfaces Academic Article

•
• Overview
•

### abstract

• Abstract: We prove a number of results on the interrelation between the \$ L^ p \$-metric on the group of Hamiltonian diffeomorphisms of surfaces and the subset of all autonomous Hamiltonian diffeomorphisms. More precisely, we show that there are Hamiltonian diffeomorphisms of all surfaces of genus \$ g\neq 1\$ lying arbitrarily \$ L^ p \$-far from this subset; answering a variant of a question of Polterovich for the \$ L^ p \$-metric.

• May 30, 2014


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# Tag Archives: fraction division

## Partitive fraction division

As promised, more notebook pages on fraction division. This is based on the work I did a while back on trying to write authentic partitive division problems with fractional divisors. (As I wrote that last sentence, I reminded myself what a bizarre niche market I am trying to occupy on this here blog.)

I settled on situations involving fractional values of unit rates, such as the following.

If $\frac{2}{3}$ of a lawn takes $\frac{3}{4}$ of an hour, how much can I mow in one hour?

Before we begin, remember that if the problem were about 2 lawns in 3 hours, we would easily and naturally divide by 3. Only the numbers have changed, so the mathematical structure remains the same and we need to find $\frac{2}{3} \div \frac{3}{4}$.

Click each image to see it full size. If you’re into this sort of thing.

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## Common numerator fraction division [#algorithmchat]

My future elementary teachers explore the common denominator fraction division algorithm at the end of the semester. Reading their work got me thinking about common numerator fraction division, and about what sense I could make of the symbols that result.

I tried to keep my work neat so others could follow it. If this sort of thing amuses you (as it obviously does me), then you’ll want to take a few minutes with the larger versions of these images. If it does not amuse (and I cannot begrudge anyone this), then you’ll just want to move along; there’s nothing here for you today.

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Article

Full entry | PDF   (1.9 MB)
Keywords:
measure; noncompatible observables; joint distribution; commutators; quantum logic
Summary:
This paper i a continuation of the first part under the same title. The author studies a joint distribution in $\sigma$-finite measures for noncompatible observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.
References:
[I] A. Dvurečenskij: Remark on joint distribution in quantum logics. I. Compatible observables. Apl. mat. 32, 427-435 (1987). MR 0916059
[19] T. Lutterová S. Pulmannová: An individual ergodic theorem on the Hilbert space logic. Math. Slovaca, 35, 361- 371 (1985). MR 0820633
[20] S. Pulmannová: Relative compatibility and joint distributions of observables. Found. Phys., 10, 614-653(1980). MR 0659345
[21] L. Beran: On finitely generated orthomodular lattices. Math. Nachrichten 88, 129-139 (1979). DOI 10.1002/mana.19790880111 | MR 0543398 | Zbl 0439.06005
[22] E. L. Marsden: The commutator and solvability in a generalized orthomodular lattice. Рас. J. Math., 33, 357-361 (1970). MR 0263712 | Zbl 0234.06004
[23] W. Puguntke: Finitely generated ortholattices. Colloq. Math. 33, 651-666 (1980).
[24] G. Grätzer: General Lattice Theory. Birkhauser - Verlag, Basel (1978). MR 0504338
[25] A. Dvurečenskij: On Gleason's theorem for unbounded measures. JINR, E 5-86-54, Dubna (1986).

Partner of


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The following are the formulas used for different information criteria. The first column is generally used in anything which generates a log likelihood (log L is the sample value for that), while the second can be used for models estimated by least squares. The two give identical orderings of models assuming a Normal likelihood.

k is the number of estimated parameters (or regressors) and T is the number of observations. It's also possible (and in many ways desirable) to "standardize" these by dividing the expression by T which gives the statistic a more manageable scale while not changing the relative order of models. That's what the @REGCRITS procedure does.

You're looking for the model which minimizes the chosen criterion. Note that you should always use the same sample range for all models considered. If you don't, you can be biasing your decision in favor of one model or another based mainly upon the number of data points used. Note also that you need to be careful that the values of "k" are also compatible. For instance, the IC formulas for least squares models usually don't count the variance as part of k (the variance is concentrated out in least squares), which is fine if you're comparing least squares models with each other. However, if you're comparing a least squares model with (for instance) a GARCH model, the variance needs to be included in the parameter count for the regression, since the GARCH model is explicitly including parameters to describe the variance. The RATS variable %NFREE (which is used by @REGCRITS) includes variance (or covariances for multivariate models) that are concentrated out in the estimation, so @REGCRITS will give you comparable parameter counts if you're using different general model types.

 AIC (Akaike Information Criterion) $$- 2\log L + k \times 2$$ $$T\log \,{{\hat \sigma }^2} + k \times 2$$ SBC (Schwarz Bayesian Criterion, or Bayesian Information Criterion) $$- 2\log L + k \times \log T$$ $$T\log \,{{\hat \sigma }^2} + k \times \log T$$ HQ (Hannan-Quinn) $$- 2\log L + k \times 2\log \left( {\log T} \right)$$ $$T\log \,{{\hat \sigma }^2} + k \times 2\log \left( {\log T} \right)$$ FPE (log) (Final Prediction Error) $$- 2\log L + T\log \left( {\frac{{T + k}}{{T - k}}} \right)$$ $$T\log \,{{\hat \sigma }^2} + T\log \left( {\frac{{T + k}}{{T - k}}} \right)$$


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## Journal of Symbolic Logic

### Minimal Realizability of Intuitionistic Arithmetic and Elementary Analysis

Zlatan Damnjanovic

#### Abstract

A new method of "minimal" realizability is proposed and applied to show that the definable functions of Heyting arithmetic (HA)--functions $f$ such that HA $\vdash \forall x\exists!yA(x, y)\Rightarrow$ for all $m, A(m, f(m))$ is true, where $A(x, y)$ may be an arbitrary formula of $\mathscr{L}$(HA) with only $x, y$ free--are precisely the provably recursive functions of the classical Peano arithmetic (PA), i.e., the $< \varepsilon_0$-recursive functions. It is proved that, for prenex sentences provable in HA, Skolem functions may always be chosen to be $< \varepsilon_0$-recursive. The method is extended to intuitionistic finite-type arithmetic, $HA^\omega_0$, and elementary analysis. Generalized forms of Kreisel's characterization of the provably recursive functions of PA and of the no-counterexample-interpretation for PA are consequently derived.

#### Article information

Source
J. Symbolic Logic, Volume 60, Issue 4 (1995), 1208-1241.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744873

Mathematical Reviews number (MathSciNet)
MR1367206

Zentralblatt MATH identifier
0854.03054

JSTOR


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16 July 2008

Deriving the equations in vectors

• All constant are taken in capital letters
• All variable are given in small letter

Deriving v(t) = Ui + A (t - Ti)

$\vec{a} = \frac{d \vec{v}}{dt} = \frac{d^2 \ \vec{s}}{dt^2}$

so

$\vec v (t) = \int _{T_i} ^t \vec a dt$

for zero or constant acceleration A we have

$\vec v (t) = \vec A \int _{T_i} ^t dt$
$\vec v (t) = \vec A \, [t] _{T_i} ^t$
$\vec v (t) = \vec A \, (t - {T_i}) + K$

we have one unknown K , we need to consider initial or final condition

• Lets take initially we have $\vec v (T_i) = \vec U_i$
$\vec U_i = \vec A \, ({T_i} - {T_i}) + K$

ie

$k= \vec U_i$
 $\vec v (t) = \vec U_i + \vec A \, (t - {T_i})$ eq ... (1)

If we take final condition in consideration then ie : $\vec v (T_f) = \vec V_f$

 $\vec v (t) = \vec V_f + \vec A \, ({T_f} -t )$ eq ... (2)

constant acceleration can be found using initial and final condition

 $\vec A = \frac { \vec V_f - \vec U_i }{ T_f - T_i}$ eq ... (3)

Deriving s(t) = Si + Ui(t-Ti) + (1/2)A (t - Ti)2

now we have

$\vec v (t) = \frac{d \vec{s}}{dt} = \vec U_i + \vec A \, (t - {T_i})$
$\vec{s}(t)= \int _{T_i} ^{t} \Big(\vec U_i + \vec A \, (t - {T_i}) \Big) {dt}$
$\vec{s}(t) = (\vec U_i - \vec A \, T_i ) \int _{T_i} ^{t} {dt} + \vec A \int _{T_i} ^{t} t \ {dt}$
$\vec{s}(t) = (\vec U_i - \vec A \, T_i ) \ \Big[t\Big] ^{t} _{T_i} + \vec A \Bigg[\frac { t^2} {2} \Bigg] _{T_i} ^{t} + K$

$\vec{s}(t) = (\vec U_i - \vec A \, T_i ) ( {t} - {T_i} ) + \frac {\vec A } {2 } (t^2-{T_i}^2) + K$

for eliminating K we need either final or initial condition ie $\vec{s}(T_i) = \vec S_i$

$\vec{S_i} = (\vec U_i - \vec A \, T_i ) ( {T_i} - {T_i} ) + \frac {\vec A } {2 } ({T_i}^2-{T_i}^2) + K$
$K= \vec{S_i}$
$\vec{s}(t) = \vec{S_i} + (\vec U_i - \vec A \, T_i ) ( {t} - {T_i} ) + \frac {\vec A } {2 } (t^2-{T_i}^2)$
$\vec{s}(t) = \vec{S_i} + \vec U_i ( {t} - {T_i} ) - \vec A ( {t}T_i - {T_i}^2 ) + \frac {\vec A } {2 } (t^2-{T_i}^2)$
$\vec{s}(t) = \vec{S_i} + \vec U_i ( {t} - {T_i} ) + \frac {\vec A } {2 } (t^2 - 2 {t}T_i + {T_i}^2)$
 $\vec{s}(t) = \vec{S_i} + \vec U_i ( {t} - {T_i} ) + \frac {\vec A } {2 } (t - T_i ) ^2$ eq ... (4)

If we would have taken final condition ie $\vec{s}(T_f) = \vec S_f$ then

 $\vec{s}(t) = \vec{S_f} + \vec V_f ( {T_f} - {t} ) - \frac {\vec A } {2 } (T_f - t ) ^2$ eq ... (5)

Taking t = Tf and putting eq (3) in eq (4) , we will get

$\vec{s}(T_f) = \vec{S_i} + \vec U_i ( {T_f} - {T_i} ) + \frac {\frac { \vec V_f - \vec U_i }{ T_f - T_i} } {2 } (T_f - T_i ) ^2$

Or

$\vec{S_f} = \vec{S_i} + \vec U_i ( {T_f} - {T_i} ) + \frac { { \vec V_f - \vec U_i } } {2 } (T_f - T_i )$
 $\vec{S_f} = \vec{S_i} + \frac { ( \vec V_f + \vec U_i ) (T_f - T_i ) } {2 }$ eq ... (6)

Deriving |v(t)|2 = |Ui|2 + 2 A . (s(t)-S_i)

$\vec v (t) = \vec U_i + \vec A \, (t - {T_i})$
$\vec v (t) . \vec v (t) = \big(\vec U_i + \vec A \, (t - {T_i}) \big).\big(\vec U_i + \vec A \, (t - {T_i}) \big)$
$\vec v (t) . \vec v (t) = \vec U_i . \vec U_i + \vec A . \vec A \ (t - T_i)^2 + 2 \vec A . \vec U_i (t- T_i)$
$\vec v (t) . \vec v (t) = \vec U_i . \vec U_i + \vec A . \vec A \ (t - T_i)^2 + 2 \vec A . \Bigg( \vec s(t) - \vec S_i - \frac {\vec A } {2 } (t - T_i ) ^2 \Bigg)$
$\vec v (t) . \vec v (t) = \vec U_i . \vec U_i + \vec A . \vec A \ (t - T_i)^2 + 2 \vec A . ( \vec s(t) - \vec S_i) - 2 \vec A . ( \frac {\vec A } {2 } (t - T_i ) ^2 )$

$\vec v (t) . \vec v (t) = \vec U_i . \vec U_i + 2 \vec A . ( \vec s(t) - \vec S_i)$

taking case for final velocity

 $\vec V_f . \vec V_f = \vec U_i . \vec U_i + 2 \vec A . ( \vec S_f - \vec S_i)$ eq ... (7)

or

 $\left | \vec V_f \right | ^ 2 = \left | \vec U_i \right | + 2 \vec A . ( \vec S_f - \vec S_i)$ eq ... (7)


text:
# Optimal Transport Methods Machine Learning

## Introduction

Optimal transport concepts applied to machine learning applications can also be referred to as computational Optimal Transport (OT). At its core, machine learning focuses on making comparisons between complex objects. To properly measure these similarities, a metric is needed, which is a distance function. Optimal transport is a natural tool in this case, as it is concerned with efficient distance functions (e.g. the Wasserstein metric).

In addition, optimal transport respects the underlying structure and geometry of a problem while providing a framework for comparing probability distributions. Optimal transport methods have received attention from researchers in fields as varied as economics, statistics, and quantum mechanics. The categories that applications of OT methods in a machine learning context can be divided into include learning, domain adaptation, Bayesian inference, and hypothesis testing[1].

### Learning Methods

These methods have used transport-based distances in a wide range of research contexts, ranging from language analysis to image recognition. To a certain extent, the geometry underlying optimal transport models helps to generate faster algorithms as well as establish data connections. Optimal transport models still have limitations, however, as some structure may not be captured. This structure can be described as either intrinsic or extrinsic. It is intrinsic if the distributions reflect structured objects (e.g. images with segments) and it can be characterized as extrinsic if there is additional information that creates structure (such as groupings) [2]. The following are some types of models that fall into the learning methods in the OT category.

#### Graph-based Semi-supervised Learning

Graph-based semi-supervised learning is an effective approach for classification from a large variety of domains, including image and text classification. It is possible to use graph-based algorithms with this method, and the method is often useful for unlabeled data. To imagine what data is best analyzed by this type of model, consider a graph with multiple nodes. These graph nodes are usually associated with a simple label, like a real number. Now consider more complex node labels, such as probability distributions or histograms. A concrete example of this would be a set of traffic cameras distributed across a geographical area, each of which produces a histogram or a distribution of commuter traffic over a given period of time. Many other interesting research questions exhibit a similar structure (e.g. climate data, rankings of restaurants, etc.). Optimal transport-inspired methods have used the two-Wasserstein distance between distributions to analyze models of this type [3] The work of Solomon, et. al. in this area led to additional insights by Gao, et. al, where Wassertstein propagation on graphs was used as the basis for a message-passing algorithm. This technique facilitated a generalization that could be applied to hypergraphs through Wasserstein barycenters [4].

#### Generative Adversarial Networks (GAN)

A class of frameworks for machine learning developed in 2014 by Ian Goodfellow (Goodfellow). Machine learning frameworks where two neural networks are used compete against each other in a game-theoretic sense. If a GAN is trained on a given set of training data, it can produced a new set of data that is at the surface similar to the original set of data (for example, it can create new, similar images based on an initial training set of images). These techniques have been used in semi-supervised learning. They offer increased flexibility in use of the objective function, including the use of f-divergences and Jensen-Shannon [5].

##### Wasserstein GAN (WGAN)

This is classified as a type of tool for unsupervised learning, which uses a minimization of the distance between data distribution contained in the training set and the distribution of the observed data. In certain cases this produces a more stable training process. It minimizes an approximation of the earth mover (EM) distance. The optimization problem associated with the WGAN was shown to be sound by Arjovsky. This type of GAN reduces some of the problems associated with GAN approaches, including a reduction in the need to balance the training of the generator and the discriminator [5]. The WGAN also estimates the EM distance in a continuous manner via training the discriminator [5]. The following is an algorithm implementing the WGAN approach [5].

#### Algorithm: Wasserstein Generative Adversarial Networks (WGAN)

   Require : ${\displaystyle \alpha }$ learning rate, ${\displaystyle c}$ clipping parameter, ${\displaystyle m}$ batch size, ${\displaystyle n_{\text{critic }}}$  number of iterations of critic per generator iteration.
Require : ${\displaystyle w_{0},}$ initial critic parameters, ${\displaystyle \theta _{0},}$ initial generator's parameters.
1: while ${\displaystyle \theta }$ has not converged do
2:    for ${\displaystyle t=0,\ldots ,n_{\text{critic }}}$ do
3:          Sample ${\displaystyle \left\{x^{(i)}\right\}_{i=1}^{m}\sim \mathbb {P} _{r}}$ a batch from the real data.
4:          Sample ${\displaystyle \left\{z^{(i)}\right\}_{i=1}^{m}\sim p(z)}$ a batch of prior samples.
5:          ${\displaystyle g_{w}\leftarrow \nabla _{w}\left[{\frac {1}{m}}\sum _{i=1}^{m}f_{w}\left(x^{(i)}\right)-{\frac {1}{m}}\sum _{i=1}^{m}f_{w}\left(g_{\theta }\left(z^{(i)}\right)\right)\right]}$
6:          ${\displaystyle w\leftarrow w+\alpha \cdot \operatorname {RMSProp} \left(w,g_{w}\right)}$
7:          ${\displaystyle w\leftarrow \operatorname {clip} (w,-c,c)}$
8:    end for
9:       Sample ${\displaystyle \left\{z^{(i)}\right\}_{i=1}^{m}\sim p(z)}$ a batch of prior samples.
10:      ${\displaystyle g_{\theta }\leftarrow -\nabla _{\theta }{\frac {1}{m}}\sum _{i=1}^{m}f_{w}\left(g_{\theta }\left(z^{(i)}\right)\right)}$
11:      ${\displaystyle \theta \leftarrow \theta -\alpha \cdot \operatorname {RMSProp} \left(\theta ,g_{\theta }\right)}$
12: end while

Recall that the EM distance is both continuous and differentiable, this implies that the critic should be trained to toward optimality. As the critic is trained more, the gradient (Wasserstein) becomes more reliable. The Wasserstein GAN can have problems due to the enforcement of the Lipschitz constraint the critic, which required weight clipping (reflected in algorithm) [6]

### Restricted Bolzman Machines (RBM)

Restricted Boltzman Machines (RBM) are probabilistic graphical models and can obtain hierarchical features at multiple levels. Originally created under the name "Harmonim" in 1986 by Smolensky, RBM can learn a probability distribution over a given set of inputs. These machines are able to learn complex and multi-scale distributions based on empirical data. The RBM generally has a layer of latent variables with a given probability distribution. This probability distribution is defined over a set of binary observed variables with a parameter that is to be learned.

Learning of model parameters is usually facilitated by Kullback-Lieber divergence (divergence from the training sample to the model). In research by Montavan, it is assumed that a metric can be established between observations [7]. This was used to define the Wasserstein distance between the training sample distribution that the distribution formed by the Boltzman machine.

### Entropy-regularized Wasserstein Loss

This has been used for multi-label classification. It is characterized by a relaxation of the transport problem which addresses unnormalized measure. It does this by replacing the equality constraints with soft penalties with respect to KL-divergence.[8]

In this case the goal is to learn about or extrapolate from one domain to another, often by finding domain-invariant representations [9] In cases where these sorts of models are useful the distributions are generally different but related. This is a technique that is often used to transfer information based on labelled data to unlabeled data. For example, in some cases the labels are available in the source domain, but the classification needs to be conducted on the target. If the classifier is trained on the source domain, the data will have performance deficiencies in the targeted domain [10].

By obtaining the best transportation plan connecting the probability distributions of source and target domains, estimates of learning samples are estimated. The transformation is non-linear and invertible. This allows for the use of a variety of machine learning methods that can be used on the transformed dataset. Regularized, unsupervised models have been used, as well as Joint Class Proportion and Optimal Transport (JCPOT) to address multi-source domain adaptation. [8]The following is a summary of an algorithm implemented by Redko, et al:[11]

#### Algorithm: Joint Class Proportion and Optimal (JCPOT)

   1: Parameters: ${\displaystyle \epsilon ,}$ maxIter, ${\displaystyle \forall k\left(\mathrm {C} ^{(k)}and\lambda ^{(k)}\right)}$
2: ${\displaystyle cpt\leftarrow 0}$, ${\displaystyle err\leftarrow \infty }$
3: ${\displaystyle \zeta ^{(k)}\leftarrow \exp \left(-{\frac {\mathbf {C} ^{(k)}}{\epsilon }}\right),\quad \forall k}$
4: while ${\displaystyle cpt}$ < maxIter and ${\displaystyle err}$ > threshold do
5:   ${\displaystyle \zeta ^{(k)}\leftarrow \operatorname {diag} \left({\frac {\mathbf {m} }{c(k)\mathbf {1} }}\right)\zeta ^{(k)},\quad \forall k}$
6:   ${\displaystyle \mathbf {h} ^{(cpt)}\leftarrow exp(\sum _{k=1}^{K}\lambda ^{(k)}\log(\mathbf {D} _{1}^{(k)}\zeta ^{(k)}1))}$
7:   ${\displaystyle \zeta ^{(k)}\leftarrow \zeta ^{(k)}\operatorname {diag} \left({\frac {\mathbf {D} _{2}^{(k)}\mathbf {h} }{\zeta ^{(k)}\mathbf {1} }}\right),\quad \forall k}$
8:   ${\displaystyle \operatorname {err} \leftarrow \left\|\mathbf {h} ^{(cpt)}-\mathbf {h} ^{(cpt-1)}\right\|_{2}}$
9:   ${\displaystyle \quad cpt\leftarrow cpt+1}$

This algorithm describes the optimization problem solved in Redko, et al which used operators defined such that proportions between the source and target domains are equal. A Wasserstein barycenter problem was used to achieve this [11] [12].

## Bayesian Inference

An important aspect of Bayesian inference is calculating expectations given a posterior probability function. This creates a complicated problem, mainly because of multidimensional integrals. The method usually used to solve these problems is Monte Carlo Markov Chains, which in turn introduce complications (slow convergence to empirical expectation, lack of sample independence) [1]. The need for these complicated MCMC models was bypassed in the work of El Moselhy and Marzouk [13] In a transport based method, Bayesian inference was formulated in terms of optimal transport theory, and the existence and uniqueness of a measure-preserving map was established. The map was subsequently parametrized and via the solution of an optimization problem. This approach helps bypass many of the problems associated with the MCMC models.

## Hypothesis Testing

There are many methods of hypothesis testing that connect to machine learning. This might be more colloquially described as "goodness-of-fit," or a summary of how well the actual observed data fits the data predicted by a given model. The most common tests used in this context are the chi-square, Kolmogorov-Smirnov and the Anderson-Darling tests. These tests are generally based on some distance between probability laws [14]. As such, it is not surprising that del Barrio et al described tests of goodness of fit based on the ${\displaystyle L_{2}}$-Wasserstein Distance [14]. The Wasserstein distance was also used (with an entropic smoothing) to investigate the connections between univariate (PP/QQ plots, ROC/ODC curves) and multivariate tests (energy statistics, kernel based maximum mean discrepancy) [15].

Recent research on cutting edge topics relating to optimal transport applications to machine learning can be found at MSRI Workshops.

Tutorials on machine learning methods can be found at the following: Optimal Transport for Machine Learning.

## References

1. Kolouri, Soheil, Serim Park, Matthew Thorpe, Dejan Stepcey, Gustavo Rohde. Optimal Mass Transport: Signal Processing and Machine-Learning Applications. IEEE Signal Process Mag. 2017 Jul. 34(4): 43-59.
2. Alvarez-Melis, David, Et al. Structured Optimal Transport. Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) 2018, Lanzarote, Spain. PMLR: Volume 84.
3. Solomon, J., et al. Wasserstein Propagation for Semi-Supervised Learning. Proceedings of the 31st International Conference on Machine Learning, Beijing, China, 2014. JMLR: W&CP volume 32. Copy- right 2014.
4. Gao, Tingran, et al. Wasserstein Soft Label Propagation on Hypergraphs: Algorithm and Generalization Error Bounds. Univ. of Chicago. 2019
5. Arjovsky, Martin, et al. Wasserstein GAN. arXiv:1701.07875v3 [stat.ML] 6 Dec 2017
6. Gulrajani, I., et al. Improved Training of Wasserstein GANs. arXiv:1704.00028v3 [cs.LG] 25 Dec 2017.
7. Montavon, G., Klaus-Robert Muller, Marco Cuturi. Wasserstein Training of Restricted Boltzmann Machines. 30th Conference on Neural Information Processing Systems (NIPS 2016)
8. Peyre, Gabriel, and Marco Cuturi, "Computational Optimal Transport", Github, https://optimaltransport.github.io/pdf/ComputationalOT.pdf.
9. Courty, Nicolas, et al. Optimal Transport for Domain Adaptation. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. X, No. X, Jan XX. 2018
10. Flamary, R. Domain Adaptation with Optimal Transport: From Mapping to Learning with Joint Distribution
11. Redko, I., et al. Optimal Transport for Multi-source Domain Adaptation under Target Shift. arXiv:1803.04899v1 [stat.ML] 13 Mar 2018 et al.
12. Benamou, Jean-David, Carlier, Guillaume, Cuturi, Marco, Nenna, Luca, & Peyré, Gabriel. 2015. Iterative Bregman projections for regularized Transportation problems. SIAM Journal on Scientific Computing, 37(2), A1111–A1138
13. El Moselhy, Tarek A., Youssef M Marzouk. Bayesian Inference with Optimal Maps. Journal of Computational Physics 231(23). September 2011.
14. del Barrio E, Cuesta-Albertos JA, Matrán C, et al. Tests of goodness of fit based on the l_2-Wasserstein distance. The Annals of Statistics. 1999; 27(4):1230–1239
15. Ramdas, Aaditya, Nicolás García Trillos, and Marco Cuturi. On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests. Entropy 2017, 19(2), 47


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Check your answer makes sense: If we have a negative z-score the corresponding raw score should be less than the mean, and a positive z-score must correspond to a raw score higher than the mean. How to calculate a z-score using excel. How to calculate a z-score using excel. To calculate the z-score of a specific value, x, first you must calculate the mean of the sample by using the AVERAGE formula. For example, if the range of scores in your sample begin at cell A1 and end at cell A20, the formula =AVERAGE(A1:A20) returns the average of those numbers. Next, you mush calculate the standard deviation of the sample by using the STDEV.S formula. For example, if the range of scores in your sample begin at cell A1 and end at cell A20, the formula = STDEV.S (A1:A20) returns the standard deviation of those numbers. Now to calculate the z-score type the following formula in an empty cell: = (x РІР‚вЂњ mean) / [standard deviation]. To make things easier, instead of writing the mean and SD values in the formula you could use the cell values corresponding to these values. For example, = (A12 РІР‚вЂњ B1) / [C1]. Then, to calculate the probability for a SMALLER z-score, which is the probability of observing a value less than x (the area under the curve to the LEFT of x), type the following into a blank cell: = NORMSDIST( and input the z-score you calculated). To find the probability of LARGER z-score, which is the probability of observing a value greater than x (the area under the curve to the RIGHT of x), type: =1 - NORMSDIST (and input the z-score you calculated). Algebra II : Z-scores. Study concepts, example questions & explanations for Algebra II. All Algebra II Resources. Example Questions. Example Question #1 : Z Scores. A large group of test scores is normally distributed with mean 78.2 and standard deviation 4.3. What percent of the students scored 85 or better (nearest whole percent)? If the mean of a normally distributed set of scores is and the standard deviation is , then the -score corresponding to a test score of is. From a -score table, in a normal distribution, We want the percent of students whose test score is 85 or better, so we want . This is. or about 5.7 % The correct choice is 6%. Example Question #1 : Z Scores. The salaries of employees at XYZ Corporation follow a normal distribution with mean 60,000 and standard deviation 7,500. What proportion of employees earn approximately between 69,000 and 78,000? Use the normal distribution table to calculate the probabilities. Round your answer to the nearest thousandth. Let X represent the salaries of employees at XYZ Corporation. We want to determine the probability that X is between 69,000 and 78,000: To approximate this probability, we convert 69,000 and 78,000 to standardized values (z-scores). We then want to determine the probability that z is between 1.2 and 2.4. The proportion of employees who earn between 69,000 and 78,000 is 0.107 . Example Question #1 : Graphing Data. On a statistics exam, the mean score was and there was a standard deviation of . If a student's actual score of , what is his/her z-score? The z-score is a measure of an actual score's distance from the mean in terms of the standard deviation. The formula is: Where are the mean and standard deviation, respectively. is the actual score. If we plug in the values we have from the original problem we have. which is approximately . Example Question #1 : Z Scores. A distributor manufactures a product that has an average weight of pounds. If the standard deviation is pounds, determine the z-score of a product that has a weight of pounds. The z-score can be expressed as. Therefore the z-score is: Example Question #1 : Z Scores. The mean grade on a science test was 79 and there was a standard deviation of 6. If your sister scored an 88, what is her z-score? Use the formula for z-score: Where is her test score, is the mean, and is the standard deviation. Example Question #1 : Z Scores. Your teacher tells you that the mean score for a test was a and that the standard deviation was for your class. You are given that the -score for your test was . What did you score on your test? The formula for a z-score is. where = mean and = standard deviation and =your test grade. Plugging in your z-score, mean, and standard deviation that was originally given in the question we get the following. Now to find the grade you got on the test we will solve for . Example Question #1 : Z Scores. In a normal distribution, if the mean score is 8 in a gymnastics competition and the student scores a 9.3, what is the z-score if the standard deviation is 2.5? Write the formula to find the z-score. Z-scores are defined as the number of standard deviations from the given mean. Substitute the values into the formula and solve for the z-score. Example Question #1 : Z Scores. Suppose a student scored a on a test. The mean of the tests are , and the standard deviation is . What is the student's z-score? Write the formula for z-score where is the data, is the population mean, and is the population standard deviation. Substitute the variables. Example Question #1 : Z Scores. Suppose Bob's test score is 50. Determine the z-score if the standard deviation is 3, and the mean is 75. Write the formula for z-scores. This tells how many standard deviations above the below the mean. Substitute the known values into the equation. Example Question #1 : Z Scores. Find the z-score if the result of a test score is 6, the mean is 8, the standard deviation is 2. Write the formula to determine the z-scores. Substitute all the known values into the formula to determine the z-score. Simplify this equation. All Algebra II Resources. Report an issue with this question. If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources. DMCA Complaint. Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org. Please be advised that you will be liable for damages (including costs and attorneysвЂ™ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney. Please follow these steps to file a notice: You must include the following: Send your complaint to our designated agent at: Charles Cohn Varsity Tutors LLC 101 S. Hanley Rd, Suite 300 St. Louis, MO 63105. Correct Score Prediction & Correct Score Tips For Today & Tomorrow. As an expert correct score prediction site we recommend various prediction and tips for both today and tomorrow. вљЅ Football рџЋѕ Tennis рџЋ® eSports. Correct Score for today. Brazil - Serie A Correct Score. вњ”пёЏ Expires in 14 hrs вњ”пёЏ 37 Voucher used вњ”пёЏ 22Bet - рџ’Ћ Best Bookmaker вњ”пёЏ Sports, eSports, Casino. Colombia - Primera A Correct Score. Cyprus - First Division Correct Score. вњ”пёЏ Expires in 14 hrs вњ”пёЏ 38 Voucher used вњ”пёЏ 1xBet - рџ’Ћ Best Bookmaker вњ”пёЏ Sports, eSports, Casino. Denmark - Superliga Correct Score. England - Premier League Correct Score. Greece - Greek Cup Correct Score. Italy - Serie C - Group B Correct Score. Portugal - Primeira Liga Correct Score. Romania - Liga 1 Correct Score. South Africa - Nedbank Cup Correct Score. Switzerland - Super League Correct Score. Turkey - Super Lig Correct Score. World - FIFA Club World Cup Correct Score. вњ”пёЏ Expires in 14 hrs вњ”пёЏ 38 Voucher used вњ”пёЏ 1xBet - рџ’Ћ Best Bookmaker вњ”пёЏ Sports, eSports, Casino. Correct Score Prediction & Correct Score Tips For Today & Tomorrow. Feedinco offers 100 correct score prediction for both Today and tomorrow with special focus from our Team experts for the best correct score prediction result. These can be played directly with Bet365 correct score, betway correct score and lot more of bookmakers. These exact score tips are offered for all major big leagues with four football prediction tips each sure correct score. So what is the difference between Correct Score Prediction & Correct Score Tips? Nothing, since referring to the same thing. Both suggest the same outcome with the Correct score. As the title says, Correct Score Prediction & Correct Score Tips, this website offers correctscores tips daily . correct score tips today. correct score prediction free. today correct score tips. correct score tips. correct score prediction. Feedinco is always trying to give the best tipster which in return offer free exact score tips. These prediction for correct scores today can be used with any bookmaker site, by using our betslip generation (clicking on the odds button). These correct score tips daily are as said free to use, being a sure correct score site! To navigate between the correct score of today and correct score tomorrow all you need to do is press on the top day filters. All football correct score tips and ht correct score tips are found in out tips page where one can check the statistical analysis of both teams. Feedinco is now considered to give the best correct score prediction / best site for correct score. All football correct score prediction is free to use! Feedinco is trying to be the Best correct score prediction site to offer various fixed matches as free correct score. With these fixed correct scores for today, one can win Big! Correct Score Prediction. Feedinco is now offering 100 correct score prediction free and became the correct score predictions sites which punters want to have. All the correct score tips above are all free exact score tips. All best tipster correct score is reviewed by our experts and done with mathematical analysis from previous matches and H2h analysis. We try our best to give the best correct score prediction. Correct Score. Feedinco is now one of the best betting tips sites you can find. With the daily betting tips you can create your betting tips list from the list above and create a bet tip win so that you can win Big! These correct score are offered daily so if you want correct score tomorrow all you need to do is select the preferred day!. The best football tips for today are always available for free, everyday! These online betting tips in this hot prediction site. All accurate football prediction found here are given correct score prediction also with 4 other betting tips. So to make a best football prediction site free you must be included in the top soccer prediction sites. All today match prediction and real football prediction are all given for Free. If you want the paid betting tips please visit our paid betting tips which are given daily and odds of around 2.00. The best way to understand Correct score, this research paper cover Prediction for the outcome of soccer matches. What are Correct Score Predictions? correct score predictions bets are played dry. The player indicates the exact number of goals that both teams will score in the course of ninety regular minutes. Certainly a very dangerous game in the eyes of an expert. The professional player knows correct score tips are a great asset if well placed. We basically have two different approaches. The first approach is mathematical, through the use of mathematical rules and formulas to create game systems that bring the player to the cashier. We can also call it the most nerdy approach. That point of view that only a fan tends to exploit. The second approach is a little less mathematical but certainly no less methodical. You make predictions based on your knowledge of the teams that will take the field. It is an approach that relies heavily on factor c, which is luck. In the first case, however, what the player tries to do is eliminate luck as much as possible to try to be certain on the correct score. 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Correct Score is a bet type that highlights the exact score of the outcome of a given match in soccer. Most bookmakers present it in many forms with the most being fixed to first half or full time. It usually one of he most crucial bet to predict but our Experts got you covered. Top Tips Today. Table below shows top most correct score tips and predictions for today. Time League Match Pick 02:05 Colombia 02:05 Colombia - Primera A, Apertura, Round 4 Alianza Petrolera vs Rionegro Aguilas Doradas Correct Score HT(1-0) 04:00 Costa Rica 04:00 Costa Rica - Primera Division, Clausura, Round 6 San Carlos vs CS Cartagines Correct Score HT(1-0) 13:00 Turkey 13:00 Turkey - TFF 3. Lig, Grup 1, Round 17 Bucaspor 1928 vs Diyarbekirspor Correct Score FT(2-1) 14:30 Iran 14:30 Iran - Persian Gulf Pro League, Round 14 Tractor vs Persepolis Correct Score FT(0-1) 16:00 Uruguay 16:00 Uruguay - Primera DivisiГіn, Clausura, Round 4 Club Atletico River Plate vs Danubio Correct Score HT(1-0) 18:00 Ghana 18:00 Ghana - Premier League, Round 12 Medeama vs Accra Hearts of Oak Correct Score HT(0-0) 19:15 United Arab Emirates 19:15 United Arab Emirates - UAE Pro League, Round 15 Al-Wahda FC vs Al-Sharjah Correct Score FT(1-2) 19:30 Italy 19:30 Italy - Serie C, Girone B, Round 22 Mantova vs Padova Correct Score HT(0-1) 20:00 Guatemala 20:00 Guatemala - Liga Nacional, Apertura Playoffs, Final Deportivo Guastatoya vs CSD Municipal Correct Score FT(1-1) 20:30 World 20:30 World - Club World Cup, Round 2 Al Duhail vs Al Ahly Correct Score HT(0-0) Its is usually abbreviated as "CS" and its annotation is as follows; CS HT(0-0), CS FT(2-1), CS HT(1-0), CS FT(1-1) etc. Most bookmakers have a huge number of odds placed on this bet type, hence our free tips helps your to increase your chances of winning on this bet. Tomorrow's Tips. All correct score prediction of the matches scheduled to play tomorrow. The correct score matches will be updated soon. Otherwise subscribe to our Expert tips or Premium tips. Best market for correct score tips include Egypt soccer competition, Greece Super League, Turkish Super Lig just to name a few. This is not always guaranteed due to ever changing nature of events. Use our exact score tips to boost your chances of winning. Yesterday's Tips. Outcome of correct score predictions yesterday. MathBootCamps. The z-score is a way of counting the number of standard deviations between a given data value and the mean of the data set. In this lesson, we will look at the formula for the z-score, how to calculate it, and a little more closely at this idea of counting standard deviations. Table of Contents. The formula. To calculate the z-score, you first find the distance from the mean, and then divide by the standard deviation. If we label the value $$x$$, we could write the formula as: Calculating and interpreting the z-score. LetвЂ™s look at an example to see how to use this formula. Example. The mean score on a standardized test was 508 with a standard deviation of 42. One test-takerвЂ™s score was 590. Find and interpret the z-score for this score. From the example, we have the following information: The mean is: $$\mu = 508$$ The standard deviation is: $$\sigma = 42$$ Therefore, the z-score is: When it comes to interpreting, you should note that by subtracting the mean from a data value, we will get a negative if it is smaller than the mean and a positive if it is larger. So, the sign gives the вЂњdirectionвЂќ away from the mean. By dividing this difference by the standard deviation, we are putting this distance between the mean and the data value in terms of a number of standard deviations. Therefore, we can say: вЂњThe test-score of 590 is about 1.95 standard deviations above the mean.вЂќ Interpreting the z-score: For a given data value, the z-score gives the number of standard deviations above (positive) or below (negative) the mean. More about interpretation. As you saw above, the value and the sign of the z-score gives you information about the location of the data value. Specifically: A positive z-score means the data value is larger than the mean. If a data value has a z-score of 2, that tells us that this data value is 2 standard deviations larger than the mean. A negative z-score means the data value is smaller than the mean. If a data value has a z-score of вЂ“3.1, then this data value is 3.1 standard deviations smaller than the mean. A z-score of zero means that the data value equals the mean. For example, consider a data set with a mean of 50 and a standard deviation of 2. If a data value also is 50, then the z-score is: This will be the case anytime the mean and the data value are the same. Conclusion. In your study of statistics, you will come across the z-score in a wide variety of settings. For this reason, it is important to make sure you thoroughly understand the ideas discussed here. The best way to do this is to practice the calculation so that it simply becomes second nature! This will help make future topics much easier to understand. Subscribe to our Newsletter! We are always posting new free lessons and adding more study guides, calculator guides, and problem packs. A-z correct score. A z-score describes the position of a raw score in terms of its distance from the mean, when measured in standard deviation units. The z-score is positive if the value lies above the mean, and negative if it lies below the mean. It is also known as a standard score, because it allows comparison of scores on different kinds of variables by standardizing the distribution. A standard normal distribution (SND) is a normally shaped distribution with a mean of 0 and a standard deviation (SD) of 1 (see Fig. 1). Figure 1. A standard normal distribution (SND). Why are z-scores important? Why are z-scores important? It is useful to standardized the values (raw scores) of a normal distribution by converting them into z-scores because: (a) it allows researchers to calculate the probability of a score occurring within a standard normal distribution; (b) and enables us to compare two scores that are from different samples (which may have different means and standard deviations). How do you calculate the z-score? How do you calculate the z-score? The formula for calculating a z-score is is z = (x-РћС)/РџС“, where x is the raw score, РћС is the population mean, and РџС“ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation. Figure 2. Z-score formula in a population. When the population mean and the population standard deviation are unknown, the standard score may be calculated using the sample mean (xРњвЂћ) and sample standard deviation (s) as estimates of the population values. How do you interpret a z-score? How do you interpret a z-score? The value of the z-score tells you how many standard deviations you are away from the mean. If a z-score is equal to 0, it is on the mean. A negative z-score reveals the raw score is below the mean average. For example, if a z-score is equal to -2, it is 2 standard deviations below the mean. Another way to interpret z-scores is by creating a standard normal distribution (also known as the z-score distribution or probability distribution). Fig 3 illustrates the important features of any standard normal distribution (SND). Figure 3. A standard normal distribution (SND) and normal distribution. The SND (i.e. z-distribution) is always the same shape as the raw score distribution. For example, if the distribution of raw scores if normally distributed, so is the distribution of z-scores. The mean of any SND always = 0. The standard deviation of any SND always = 1. Therefore, one standard deviation of the raw score (whatever raw value this is) converts into 1 z-score unit. Figure 4. Proportion of a standard normal distribution (SND) in percentages. How to calculate a raw score when a z-score is known. How to calculate a raw score when a z-score is known. Sometimes we know a z-score and want to find the corresponding raw score. The formula for calculating a z-score in a sample into a raw score is given below: As the formula shows, the z-score and standard deviation are multiplied together, and this figure is added to the mean. Check your answer makes sense: If we have a negative z-score the corresponding raw score should be less than the mean, and a positive z-score must correspond to a raw score higher than the mean. How to calculate a z-score using excel. How to calculate a z-score using excel. To calculate the z-score of a specific value, x, first you must calculate the mean of the sample by using the AVERAGE formula. For example, if the range of scores in your sample begin at cell A1 and end at cell A20, the formula =AVERAGE(A1:A20) returns the average of those numbers. Next, you mush calculate the standard deviation of the sample by using the STDEV.S formula. For example, if the range of scores in your sample begin at cell A1 and end at cell A20, the formula = STDEV.S (A1:A20) returns the standard deviation of those numbers. Now to calculate the z-score type the following formula in an empty cell: = (x РІР‚вЂњ mean) / [standard deviation]. To make things easier, instead of writing the mean and SD values in the formula you could use the cell values corresponding to these values. For example, = (A12 РІР‚вЂњ B1) / [C1]. Then, to calculate the probability for a SMALLER z-score, which is the probability of observing a value less than x (the area under the curve to the LEFT of x), type the following into a blank cell: = NORMSDIST( and input the z-score you calculated). To find the probability of LARGER z-score, which is the probability of observing a value greater than x (the area under the curve to the RIGHT of x), type: =1 - NORMSDIST (and input the z-score you calculated). Algebra II : Z-scores. Study concepts, example questions & explanations for Algebra II. All Algebra II Resources. Example Questions. Example Question #1 : Z Scores. A large group of test scores is normally distributed with mean 78.2 and standard deviation 4.3. What percent of the students scored 85 or better (nearest whole percent)? If the mean of a normally distributed set of scores is and the standard deviation is , then the -score corresponding to a test score of is. From a -score table, in a normal distribution, We want the percent of students whose test score is 85 or better, so we want . This is. or about 5.7 % The correct choice is 6%. Example Question #1 : Z Scores. The salaries of employees at XYZ Corporation follow a normal distribution with mean 60,000 and standard deviation 7,500. What proportion of employees earn approximately between 69,000 and 78,000? Use the normal distribution table to calculate the probabilities. Round your answer to the nearest thousandth. Let X represent the salaries of employees at XYZ Corporation. We want to determine the probability that X is between 69,000 and 78,000: To approximate this probability, we convert 69,000 and 78,000 to standardized values (z-scores). We then want to determine the probability that z is between 1.2 and 2.4. The proportion of employees who earn between 69,000 and 78,000 is 0.107 . Example Question #1 : Graphing Data. On a statistics exam, the mean score was and there was a standard deviation of . If a student's actual score of , what is his/her z-score? The z-score is a measure of an actual score's distance from the mean in terms of the standard deviation. The formula is: Where are the mean and standard deviation, respectively. is the actual score. If we plug in the values we have from the original problem we have. which is approximately . Example Question #1 : Z Scores. A distributor manufactures a product that has an average weight of pounds. If the standard deviation is pounds, determine the z-score of a product that has a weight of pounds. The z-score can be expressed as. Therefore the z-score is: Example Question #1 : Z Scores. The mean grade on a science test was 79 and there was a standard deviation of 6. If your sister scored an 88, what is her z-score? Use the formula for z-score: Where is her test score, is the mean, and is the standard deviation. Example Question #1 : Z Scores. Your teacher tells you that the mean score for a test was a and that the standard deviation was for your class. You are given that the -score for your test was . What did you score on your test? The formula for a z-score is. where = mean and = standard deviation and =your test grade. Plugging in your z-score, mean, and standard deviation that was originally given in the question we get the following. Now to find the grade you got on the test we will solve for . Example Question #1 : Z Scores. In a normal distribution, if the mean score is 8 in a gymnastics competition and the student scores a 9.3, what is the z-score if the standard deviation is 2.5? Write the formula to find the z-score. Z-scores are defined as the number of standard deviations from the given mean. Substitute the values into the formula and solve for the z-score. Example Question #1 : Z Scores. Suppose a student scored a on a test. The mean of the tests are , and the standard deviation is . What is the student's z-score? Write the formula for z-score where is the data, is the population mean, and is the population standard deviation. Substitute the variables. Example Question #1 : Z Scores. Suppose Bob's test score is 50. Determine the z-score if the standard deviation is 3, and the mean is 75. Write the formula for z-scores. This tells how many standard deviations above the below the mean. Substitute the known values into the equation. Example Question #1 : Z Scores. Find the z-score if the result of a test score is 6, the mean is 8, the standard deviation is 2. Write the formula to determine the z-scores. Substitute all the known values into the formula to determine the z-score. Simplify this equation. All Algebra II Resources. Report an issue with this question. If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources. DMCA Complaint. Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org. Please be advised that you will be liable for damages (including costs and attorneysвЂ™ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney. Please follow these steps to file a notice: You must include the following: Send your complaint to our designated agent at: Charles Cohn Varsity Tutors LLC 101 S. Hanley Rd, Suite 300 St. Louis, MO 63105. Correct Score Prediction & Correct Score Tips For Today & Tomorrow. As an expert correct score prediction site we recommend various prediction and tips for both today and tomorrow. вљЅ Football рџЋѕ Tennis рџЋ® eSports. Correct Score for today. Brazil - Serie A Correct Score. вњ”пёЏ Expires in 14 hrs вњ”пёЏ 37 Voucher used вњ”пёЏ 22Bet - рџ’Ћ Best Bookmaker вњ”пёЏ Sports, eSports, Casino. Colombia - Primera A Correct Score. Cyprus - First Division Correct Score. вњ”пёЏ Expires in 14 hrs вњ”пёЏ 38 Voucher used вњ”пёЏ 1xBet - рџ’Ћ Best Bookmaker вњ”пёЏ Sports, eSports, Casino. Denmark - Superliga Correct Score. England - Premier League Correct Score. Greece - Greek Cup Correct Score. Italy - Serie C - Group B Correct Score. Portugal - Primeira Liga Correct Score. Romania - Liga 1 Correct Score. South Africa - Nedbank Cup Correct Score. Switzerland - Super League Correct Score. Turkey - Super Lig Correct Score. World - FIFA Club World Cup Correct Score. вњ”пёЏ Expires in 14 hrs вњ”пёЏ 38 Voucher used вњ”пёЏ 1xBet - рџ’Ћ Best Bookmaker вњ”пёЏ Sports, eSports, Casino. Correct Score Prediction & Correct Score Tips For Today & Tomorrow. Feedinco offers 100 correct score prediction for both Today and tomorrow with special focus from our Team experts for the best correct score prediction result. These can be played directly with Bet365 correct score, betway correct score and lot more of bookmakers. These exact score tips are offered for all major big leagues with four football prediction tips each sure correct score. So what is the difference between Correct Score Prediction & Correct Score Tips? Nothing, since referring to the same thing. Both suggest the same outcome with the Correct score. As the title says, Correct Score Prediction & Correct Score Tips, this website offers correctscores tips daily . correct score tips today. correct score prediction free. today correct score tips. correct score tips. correct score prediction. Feedinco is always trying to give the best tipster which in return offer free exact score tips. These prediction for correct scores today can be used with any bookmaker site, by using our betslip generation (clicking on the odds button). These correct score tips daily are as said free to use, being a sure correct score site! To navigate between the correct score of today and correct score tomorrow all you need to do is press on the top day filters. All football correct score tips and ht correct score tips are found in out tips page where one can check the statistical analysis of both teams. Feedinco is now considered to give the best correct score prediction / best site for correct score. All football correct score prediction is free to use! Feedinco is trying to be the Best correct score prediction site to offer various fixed matches as free correct score. With these fixed correct scores for today, one can win Big! Correct Score Prediction. Feedinco is now offering 100 correct score prediction free and became the correct score predictions sites which punters want to have. All the correct score tips above are all free exact score tips. All best tipster correct score is reviewed by our experts and done with mathematical analysis from previous matches and H2h analysis. We try our best to give the best correct score prediction. Correct Score. Feedinco is now one of the best betting tips sites you can find. With the daily betting tips you can create your betting tips list from the list above and create a bet tip win so that you can win Big! These correct score are offered daily so if you want correct score tomorrow all you need to do is select the preferred day!. The best football tips for today are always available for free, everyday! These online betting tips in this hot prediction site. All accurate football prediction found here are given correct score prediction also with 4 other betting tips. So to make a best football prediction site free you must be included in the top soccer prediction sites. All today match prediction and real football prediction are all given for Free. If you want the paid betting tips please visit our paid betting tips which are given daily and odds of around 2.00. The best way to understand Correct score, this research paper cover Prediction for the outcome of soccer matches. What are Correct Score Predictions? correct score predictions bets are played dry. The player indicates the exact number of goals that both teams will score in the course of ninety regular minutes. Certainly a very dangerous game in the eyes of an expert. The professional player knows correct score tips are a great asset if well placed. We basically have two different approaches. The first approach is mathematical, through the use of mathematical rules and formulas to create game systems that bring the player to the cashier. We can also call it the most nerdy approach. That point of view that only a fan tends to exploit. The second approach is a little less mathematical but certainly no less methodical. You make predictions based on your knowledge of the teams that will take the field. It is an approach that relies heavily on factor c, which is luck. In the first case, however, what the player tries to do is eliminate luck as much as possible to try to be certain on the correct score. 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