text:
# Exponential Equation 2^(x+1) + 2^(x+2) + 3(2^(x-2)) = 8(3^(x-2)) ?

## A step by step solution of the following would be greatly appreciated Thank you :D

Jun 4, 2016

$x = 5$

#### Explanation:

Consider ${2}^{x + 1} \to {2}^{x} \times 2$

Consider ${2}^{x + 2} \to {2}^{x} \times {2}^{2}$

Consider $3 \left({2}^{x - 2}\right) \to {2}^{x} \times \frac{3}{{2}^{2}}$

Consider $8 \left({3}^{x - 2}\right) \to {3}^{x} \times \frac{8}{3} ^ 2$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\text{ } {2}^{x} \left(2 + {2}^{2} + \frac{3}{2} ^ 2\right) = \frac{8}{3} ^ 2 \times {3}^{x}$

$\text{ } {2}^{x} \times \frac{27}{4} = \frac{8}{9} \times {3}^{x}$

Divide both sides by ${3}^{x}$

$\text{ } {2}^{x} / \left({3}^{x}\right) \times \frac{27}{4} = \frac{8}{9} \times {3}^{x} / \left({3}^{x}\right)$

But $\frac{{3}^{x}}{{3}^{x}} = 1$

$\text{ } {2}^{x} / \left({3}^{x}\right) \times \frac{27}{4} = \frac{8}{9}$

Multiply both sides by $\frac{4}{27}$

$\text{ } {2}^{x} / \left({3}^{x}\right) \times \frac{27}{4} \times \frac{4}{27} = \frac{8}{9} \times \frac{4}{27}$

But $\frac{27}{4} \times \frac{4}{27} = \frac{27}{27} \times \frac{4}{4} = 1$

$\text{ } {2}^{x} / \left({3}^{x}\right) = \frac{8}{9} \times \frac{4}{27}$

$\text{ } {2}^{x} / {3}^{x} = \frac{32}{243}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
From this both should be true that

$\text{ "2^x=32" and } {3}^{x} = 243$

Using logs

$\text{ } x \ln \left(2\right) = \ln \left(32\right) \implies x = \ln \frac{32}{\ln} \left(2\right)$

$\text{ } \implies x = 5$
'..........................................

" "xln(3)=ln(243)=> x=ln(243)/(ln(3)

$\text{ } \implies x = 5$

Jun 4, 2016

$x = 5$

#### Explanation:

$2 \times {2}^{x} + 4 \times {2}^{x} + \frac{3}{4} \times {2}^{x} = \left(\frac{8}{9}\right) {3}^{x}$

grouping

$\left(\frac{27}{4}\right) {2}^{x} = \left(\frac{8}{9}\right) {3}^{x}$
${\left(\frac{3}{2}\right)}^{x} = \frac{27 \times 9}{8 \times 4} = {\left(\frac{3}{2}\right)}^{5}$

$x = 5$


text:
Accelerating the pace of engineering and science

# dcgain

Low-frequency (DC) gain of LTI system

k = dcgain(sys)

## Description

k = dcgain(sys) computes the DC gain k of the LTI model sys.

### Continuous Time

The continuous-time DC gain is the transfer function value at the frequency s = 0. For state-space models with matrices (ABCD), this value is

K = D – CA–1B

### Discrete Time

The discrete-time DC gain is the transfer function value at z = 1. For state-space models with matrices (ABCD), this value is

K = D + C (I – A)–1B

## Examples

### Example 1

To compute the DC gain of the MIMO transfer function

$H\left(s\right)=\left[\begin{array}{cc}1& \frac{s-1}{{s}^{2}+s+3}\\ \frac{1}{s+1}& \frac{s+2}{s-3}\end{array}\right]$

type

```H = [1 tf([1 -1],[1 1 3]) ; tf(1,[1 1]) tf([1 2],[1 -3])];
dcgain(H)
```

to get the result:

```ans =
1.0000   -0.3333
1.0000   -0.6667
```

### Example 2

To compute the DC gain of an identified process model, type;

```load iddata1
sys = idproc('p1d');
syse = procest(z1, sys)

dcgain(syse)
```

The DC gain is stored same as syse.Kp.

format: plain


text:
### 4.3.1 Line breaking

Line breaks are normally determined automatically. They are chosen so that lines look neither cramped nor loose, and consecutive lines have similar density.

To manually force a line break at a bar line, use the \break command:

\relative c'' {
c4 c c c | \break
c4 c c c |
}

By default, a \break in the middle of a measure is ignored, and a warning is printed. To force a line break in the middle of a measure, add an invisible bar line with ‘\bar ""’:

\relative c'' {
c4 c c
\bar "" \break
c |
c4 c c c |
}

A \break occurring at a bar line is also ignored if the previous measure ends in the middle of a note, such as when a tuplet begins and ends in different measures. To allow \break commands to work in these situations, remove the Forbid_line_break_engraver from the Voice context. Note that manually forced line breaks have to be added in parallel with the music:

\new Voice \with {
\remove "Forbid_line_break_engraver"
} \relative {
<<
{ c''2. \tuplet 3/2 { c4 c c } c2. | }
{ s1 | \break s1 | }
>>
}

Similarly, line breaks are normally forbidden when beams cross bar lines. This behavior can be changed by setting \override Beam.breakable = ##t:

\relative c'' {
\override Beam.breakable = ##t
c2. c8[ c | \break
c8 c] c2. |
}

The \noBreak command forbids a line break at the bar line where it is inserted.

Within a score, automatic line breaking is prevented within music lying between \autoLineBreaksOff and \autoLineBreaksOn commands. If automatic page breaks should also be prevented, the commands \autoBreaksOff and \autoBreaksOn should be used. Manual breaks are unaffected by these commands. Note that inhibiting automatic line breaks may cause music to run over the right margin if it cannot all be contained within one line.

Automatic line breaks (but not page breaks) may be enabled at single bar lines by using \once \autoLineBreaksOn at a bar line. This identifies a permitted rather than a forced line break.

The most basic settings influencing line spacing are indent and line-width. They are set in the \layout block. They control the indentation of the first line of music, and the lengths of the lines.

If ragged-right is set to true in the \layout block, then systems end at their natural horizontal length, instead of being spread horizontally to fill the whole line. This is useful for short fragments, and for checking how tight the natural spacing is.

The option ragged-last is similar to ragged-right, but affects only the last line of the piece.

\layout {
indent = 0\mm
line-width = 150\mm
ragged-last = ##t
}

For line breaks at regular intervals use \break separated by skips and repeated with \repeat. For example, this would cause the following 28 measures (assuming 4/4 time) to be broken every 4 measures, and only there:

<<
\repeat unfold 7 {
s1 \noBreak s1 \noBreak
s1 \noBreak s1 \break
}
{ the actual music… }
>>

#### Predefined commands

\break, \noBreak, \autoBreaksOff, \autoBreaksOn, \autoLineBreaksOff, \autoLineBreaksOn.

#### Selected Snippets

Using an extra voice for breaks

Often it is easier to manage line and page-breaking information by keeping it separate from the music by introducing an extra voice containing only skips along with the \break, pageBreak and other layout information.

This pattern becomes especially helpful when overriding line-break-system-details and the other useful but long properties of NonMusicalPaperColumnGrob.

music = \relative c'' { c4 c c c }

\score {
\new Staff <<
\new Voice {
s1 * 2 \break
s1 * 3 \break
s1 * 6 \break
s1 * 5 \break
}
\new Voice {
\repeat unfold 2 { \music }
\repeat unfold 3 { \music }
\repeat unfold 6 { \music }
\repeat unfold 5 { \music }
}
>>
}

Notation Reference: \paper variables for line breaking The \layout block.

Snippets: Spacing.

Internals Reference: LineBreakEvent.

#### Known issues and warnings

Placing \autoLineBreaksOff or \autoBreaksOff before any music will cause error messages to appear. Always place these commands after some music.

Other languages: deutsch, español, français, italiano, 日本語.


text:
×
Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 1.2 - Problem 8e
Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 1.2 - Problem 8e

×

# In a certain company, every worker received a $50- ISBN: 9780073401331 38 ## Solution for problem 8E Chapter 1.2 Statistics for Engineers and Scientists | 4th Edition • Textbook Solutions • 2901 Step-by-step solutions solved by professors and subject experts • Get 24/7 help from StudySoup virtual teaching assistants Statistics for Engineers and Scientists | 4th Edition 4 5 1 344 Reviews 16 1 Problem 8E In a certain company, every worker received a$50-per-week raise. How does this affect the mean salary? The standard deviation of the salaries?

Step-by-Step Solution:
Step 1 of 3

Solution 8E

Step1 of 3:

We have company in that every worker recieved a $50 per week raise. We need to check How does this affect the mean salary? and The standard deviation of the salaries? Step2 of 3: Let be the old salary of workers. 1).Mean of the X is given by: Where, = Mean x = random variable N = sample size = 3. Now, = = = Hence, = . Therefore, the mean salary is raised by$50.

Step3 of 3:

2). Standard deviation of X is given by:

=

=

Hence, =

Therefore, standard deviation will be same.

Step 2 of 3

Step 3 of 3

## Discover and learn what students are asking

Calculus: Early Transcendental Functions : Cylindrical and Spherical Coordinates
?In Exercises 1-6, convert the point from cylindrical coordinates to rectangular coordinates. $$(2,-\pi,-4)$$

Calculus: Early Transcendental Functions : Second-Order Nonhomogeneous Linear Equations
?Verifying a Solution In Exercises 1-4,verify the solution of the differential equation. Solution D

Statistics: Informed Decisions Using Data : Testing the Significance of the Least-Squares Regression Model
?Credit Scores An economist wants to determine the relation between one’s FICO score, x, and the interest rate of a 36-month auto loan, y. The data rep

#### Related chapters

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text:
# Net (mathematics)

Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is usually any topological space. However, in the context of topology, sequences do not fully encode all information about a function between topological spaces. In particular, the following two conditions are not equivalent in general for a map ƒ between topological spaces X and Y:

1. The map ƒ is continuous
2. Given any point x in X, and any sequence in X converging to x, the composition of ƒ with this sequence converges to ƒ(x)

It is true however, that condition 1 implies condition 2, in the context of all spaces. The difficulty encountered when attempting to prove that condition 2 implies condition 1 lies in the fact that topological spaces are, in general, not first-countable. If the first-countability axiom were imposed on the topological spaces in question, the two above conditions would be equivalent. In particular, the two conditions are equivalent for metric spaces.

The purpose of the concept of a net, first introduced by E. H. Moore and H. L. Smith in 1922,[1] is to generalize the notion of a sequence so as to confirm the equivalence of the conditions (with "sequence" being replaced by "net" in condition 2). In particular, rather than being defined on a countable linearly ordered set, a net is defined on an arbitrary directed set. In particular, this allows theorems similar to that asserting the equivalence of condition 1 and condition 2, to hold in the context of topological spaces which do not necessarily have a countable or linearly ordered neighbourhood basis around a point. Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do because of the fact that collections of open sets in topological spaces are much like directed sets in behaviour. The term "net" was coined by Kelley.[2][3]

Nets are one of the many tools used in topology to generalize certain concepts that may only be general enough in the context of metric spaces. A related notion, that of the filter, was developed in 1937 by Henri Cartan.

## Definition

If X is a topological space, a net in X is a function from some directed set A to X.

If A is a directed set, we often write a net from A to X in the form (xα), which expresses the fact that the element α in A is mapped to the element xα in X.

## Examples of nets

Every non-empty totally ordered set is directed. Therefore every function on such a set is a net. In particular, the natural numbers with the usual order form such a set, and a sequence is a function on the natural numbers, so every sequence is a net.

Another important example is as follows. Given a point x in a topological space, let Nx denote the set of all neighbourhoods containing x. Then Nx is a directed set, where the direction is given by reverse inclusion, so that ST if and only if S is contained in T. For S in Nx, let xS be a point in S. Then (xS) is a net. As S increases with respect to ≥, the points xS in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are led to the idea that xS must tend towards x in some sense. We can make this limiting concept precise.

## Limits of nets

If (xα) is a net from a directed set A into X, and if Y is a subset of X, then we say that (xα) is eventually in Y (or residually in Y) if there exists an α in A so that for every β in A with β ≥ α, the point xβ lies in Y.

If (xα) is a net in the topological space X, and x is an element of X, we say that the net converges towards x or has limit x and write

lim xα = x

if and only if

for every neighborhood U of x, (xα) is eventually in U.

Intuitively, this means that the values xα come and stay as close as we want to x for large enough α.

Note that the example net given above on the neighborhood system of a point x does indeed converge to x according to this definition.

Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that (xα) is eventually in all members of the base containing this putative limit.

## Supplementary definitions

If φ is a net on X based on directed set D and A is a subset of X, then φ is frequently in (or cofinally in) A if for every α in D there exists some β ≥ α, β in D, so that φ(β) is in A.

A point x in X is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood U of x, the net is frequently in U.

A net φ on set X is called universal, or an ultranet if for every subset A of X, either φ is eventually in A or φ is eventually in X-A.

## Examples

Sequence in a topological space:

A sequence (a1, a2, ...) in a topological space V can be considered a net in V defined on N.

The net is eventually in a subset Y of V if there exists an N in N such that for every nN, the point an is in Y.

We have limxc an = L if and only if for every neighborhood Y of L, the net is eventually in Y.

The net is frequently in a subset Y of V if and only if for every N in N there exists some nN such that an is in Y, that is, if and only if infinitely many elements of the sequence are in Y. Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.

Function from a metric space to a topological space:

Consider a function from a metric space M to a topological space V, and a point c of M. We direct the set M\{c} reversely according to distance from c, that is, the relation is "has at least the same distance to c as", so that "large enough" with respect to the relation means "close enough to c". The function ƒ is a net in V defined on M\{c}.

The net ƒ is eventually in a subset Y of V if there exists an a in M\{c} such that for every x in M\{c} with d(x,c) ≤ d(a,c), the point f(x) is in Y.

We have limxc ƒ(x) = L if and only if for every neighborhood Y of L, ƒ is eventually in Y.

The net ƒ is frequently in a subset Y of V if and only if for every a in M\{c} there exists some x in M\{c} with d(x,c) ≤ d(a,c) such that f(x) is in Y.

A point y in V is a cluster point of the net ƒ if and only if for every neighborhood Y of y, the net is frequently in Y.

Function from a well-ordered set to a topological space:

Consider a well-ordered set [0, c] with limit point c, and a function ƒ from [0, c) to a topological space V. This function is a net on [0, c).

It is eventually in a subset Y of V if there exists an a in [0, c) such that for every xa, the point f(x) is in Y.

We have limxc ƒ(x) = L if and only if for every neighborhood Y of L, ƒ is eventually in Y.

The net ƒ is frequently in a subset Y of V if and only if for every a in [0, c) there exists some x in [a, c) such that f(x) is in Y.

A point y in V is a cluster point of the net ƒ if and only if for every neighborhood Y of y, the net is frequently in Y.

The first example is a special case of this with c = ω.

## Properties

Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that of limit of a sequence. The following set of theorems and lemmas help cement that similarity:

• A function ƒ:XY between topological spaces is continuous at the point x if and only if for every net (xα) with
lim xα = x
we have
lim ƒ(xα) = ƒ(x).
Note that this theorem is in general not true if we replace "net" by "sequence". We have to allow for more directed sets than just the natural numbers if X is not first-countable.
• In general, a net in a space X can have more than one limit, but if X is a Hausdorff space, the limit of a net, if it exists, is unique. Conversely, if X is not Hausdorff, then there exists a net on X with two distinct limits. Thus the uniqueness of the limit is equivalent to the Hausdorff condition on the space, and indeed this may be taken as the definition. Note that this result depends on the directedness condition; a set indexed by a general preorder or partial order may have distinct limit points even in a Hausdorff space.
• If U is a subset of X, then x is in the closure of U if and only if there exists a net (xα) with limit x and such that xα is in U for all α.
• A subset A of X is closed if and only if, whenever (xα) is a net with elements in A and limit x, then x is in A.
• The set of cluster points of a net is equal to the set of limits of its convergent subnets.
• A net has a limit if and only if all of its subnets have limits. In that case, every limit of the net is also a limit of every subnet.
• A net in the product space has a limit if and only if each projection has a limit. Symbolically, if (xα) is a net in the product X = πiXi, then it converges to x if and only if $\pi_i(x_\alpha)\to \pi_i(x)$ for each i. Armed with this observation and the above characterization of compactness in terms on nets, one can give a slick proof of Tychonoff's theorem.
• If ƒ:XY and (xα) is an ultranet on X, then (ƒ(xα)) is an ultranet on Y.

## Related ideas

In a metric space or uniform space, one can speak of Cauchy nets in much the same way as Cauchy sequences. The concept even generalises to Cauchy spaces.

The theory of filters also provides a definition of convergence in general topological spaces.

## Limit superior

Limit superior and limit inferior of a net of real numbers can be defined in a similar manner as for sequences.[4][5][6] Some authors work even with more general structures, like complete lattices.[7]

For a net $(x_\alpha)_{\alpha\in I}$ we put

$\limsup x_\alpha = \lim_{\alpha\in I} \sup_{\beta\succeq\alpha} x_\beta=\inf_{\alpha\in I} \sup_{\beta\succeq\alpha} x_\beta.$

Limit superior of a net of real numbers has many properties analogous to the case of sequences, e.g.

$\limsup (x_\alpha+y_\alpha) \le \limsup x_\alpha+\limsup y_\alpha,$

where equality holds whenever one of the nets is convergent.

## References

1. ^ Moore, E. H.; Smith, H. L. (1922). "A General Theory of Limits". American Journal of Mathematics 44 (2): 102–121. doi:10.2307/2370388. JSTOR 2370388
2. ^ (Sundström 2010, p. 16n)
3. ^ Megginson, p.143
4. ^ Aliprantis-Border, p.32
5. ^ Megginson, p.217, p.221, Exercises 2.53-2.55
6. ^ Beer, p.2
7. ^ Schechter, Sections 7.43-7.47
• Kelley, John L. (1991). General Topology. Springer. ISBN 3540901256.
• Wilard, Stephen (2004). General Topology. Dover Publications. ISBN 0486434796.
• Sundström, Manya Raman (2010). "A pedagogical history of compactness". arXiv:1006.4131v1 [math.HO].
• Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite dimensional analysis: A hitchhiker's guide (Third ed.). Berlin: Springer. pp. xxii+703 pp.. ISBN 978-3-540-32696-0, 3-540-32696-0. MR2378491.
• Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications 268. Dordrecht: Kluwer Academic Publishers Group. pp. xii+340. ISBN 0-7923-2531-1. MR1269778.
• Megginson, Robert E. (1998). An Introduction to Banach Space Theory. Graduate Texts in Mathematics. 193. New York: Springer. ISBN 0-387-98431-3.
• Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press. ISBN 126227608.

Wikimedia Foundation. 2010.

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text:
If xenon has an atomic number of 54 and a mass number of 108, how many neutrons does it have?

$108 - 54 = 54$


text:
# zbMATH — the first resource for mathematics

Cyclotomic mapping permutation polynomials over finite fields. (English) Zbl 1154.11342
Golomb, Solomon W. (ed.) et al., Sequences, subsequences, and consequences. International workshop, SSC 2007, Los Angeles, CA, USA, May 31–June 2, 2007. Revised invited papers. Berlin: Springer (ISBN 978-3-540-77403-7/pbk). Lecture Notes in Computer Science 4893, 119-128 (2007).
Summary: We explore a connection between permutation polynomials of the form $$x ^{r } f(x ^{(q - 1)/l })$$ and cyclotomic mapping permutation polynomials over finite fields. As an application, we characterize a class of permutation binomials in terms of generalized Lucas sequences.
For the entire collection see [Zbl 1130.94005].

##### MSC:
 11T06 Polynomials over finite fields 94A55 Shift register sequences and sequences over finite alphabets in information and communication theory
Full Text:


text:
Filter low quality reads in bam files
1
1
Entering edit mode
5.8 years ago
ammarsabir15 ▴ 70

I an trying to filter low quality reads in bam file using python's pysam. I have used the code from here.

I have modified this code a little and the whole code is shown below, but the code is not giving any bam file ,

import argparse,pysam,re,sys

"""
returns: True if the read is ok
"""
if any([ord(c)-33 < _BASE_QUAL_CUTOFF for c in list(read.qual)]):
return False
else:
return True

_BASE_QUAL_CUTOFF = 30

bam_in = pysam.Samfile(in_file, 'rb')
bam_out = pysam.Samfile(out_file, 'wb', template=bam_in)

out_count = 0
out_count += 1

bam_out.close()
bam_in.close()

def GetArgs():
"""
GetArgs - read the command line
returns - an input bam file name and teh output filtered bam file
"""

def ParseArgs(parser):
class Parser(argparse.ArgumentParser):
def error(self, message):
sys.stderr.write('error: %s\n' % message)
self.print_help()
sys.exit(2)

parser = Parser(description='Calculate PhiX Context Specific Error Rates.')
type=str,
required=True,
help='Input Bam file.')
type=str,
required=True,
help='Output Bam file.')
return parser.parse_args()

parser = argparse.ArgumentParser()
args = ParseArgs(parser)

return args.bam_file, args.output_file

def Main():
bam_file, output_file = GetArgs()

if __name__ == '__main__':
Main()

next-gen python pysam • 3.5k views
0
Entering edit mode

I think you need to explain in more detail what you are trying to accomplish. Are you trying to reject any read with any base that has a quality score under 30? If so, I suggest you rethink your approach, because I can't imagine a scenario where that's a good idea.

0
Entering edit mode

Thanks Bushnell, Yes I want to reject reads having quality score < 30. I want to do it using python's pysam. I you have a better approach please share it.

0
Entering edit mode

No, I don't have a better approach, because I think removing any read with any base under Q30 is a terrible idea and will lead to extreme sequence-specific coverage bias. I have written tools to remove reads with average quality (based on expected error rates from the quality scores) below a certain level, but that should be used conservatively to avoid sequence-specific bias. I cannot imagine a scenario where it would be a good idea to remove every read that has a single base below a certain quality score, so I won't suggest a way to do it unless you can explain why you want to do so. I suggest you quality-trim the reads and reject reads with length under a specified value prior to mapping. You can do so like this, with the BBMap package (assuming interleaved reads):

bbduk.sh in=reads.fq out=trimmed.fq qtrim=r trimq=12 minlen=125

You can set trimq to 30 if you want, but again, I cannot imagine a situation where that would be a good idea. For this command, minlen should be set to some number less than your read length; e.g. for 150bp reads, maybe set it to 100.

0
Entering edit mode

I am developing a variant caller for detecting heteroplasmy in ngs datasets which will take bam file as input according to workflow. As the link is not working so I have added an image , see the galaxy naive variant caller portion in that figure For calculating heteroplasmy they filtered bases having <30 score.

0
Entering edit mode

It might be a terrible idea in some cases, but when working with ancient DNA, which might have sequencing errors due to oxidative and hidrolytic damage as to contamination, you might want to trim bases below certain quality (usually <30), anyways, i just write this in order to enrich the discussion, i don't know what kind of data ammarsabir is (was) working with.

if in further data handling you have a vcf step, it might be easier to just keep all the data in the bam file, and then trim the reads you want to using vcftools, like following:

vcftools --vcf yourfile.vcf --minQ 30 --recode --recode-INFO-all --out yourtrimmedfile

there's no need to put any extention in the last file name, since the tool writes a default extention.

cheers.

1
Entering edit mode
5.8 years ago

I recommend completely ignoring the Academy of Sciences, if they suggest ignoring all reads with a single base under Q30. That's a recipe for disaster, since all platforms have sequence-related quality; so, your filtered reads would be extremely biased.

You can, if you wish, trim low-quality reads with the BBMap package like this:

bbduk.sh in=reads.fq out=trimmed.fq qtrim=r trimq=12

That will optimally trim the low-quality tails of reads, while biasing the output as little as possible. It takes no special precautions to avoid bias, but none of the operations add nearly as much bias as a simple filter for reads with a single base under Q30 would. You can, of course, set trimq to 30 and get bad data, but I don't recommend it.


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# Positive functionals on $\ell^\infty$

A continuous linear functional $\varphi: \ell^\infty \to \mathbb{R}$ is said to be positive if $$x \ge 0 \rightarrow \varphi(x) \ge 0 \quad \forall x \in \ell^\infty.$$

If $\varphi$ is in $\ell^1 \subset {(\ell^\infty)}^{*}$, then exists $(\alpha_n) \in \ell^1$ such that $\varphi(x) = \sum \alpha_n x_n$. If $(\alpha_n) \ge 0$, then $\varphi$ is positive.

If $\varphi$ is a Banach limit, then it's positive.

Are there other good/simple/notable classes of positive operators in ${(\ell^\infty)}^*$?

 As stated in the comments below, feel free to give examples and constructions based on the Axiom of Choice.

-
The dual space of $\ell^{\infty}$ consists of the (signed) finitely additive measures on $\mathbb{N}$, so you're asking about the construction of (special/notable) positive finitely-additive measures. Non-trivial examples cannot be constructed without using some choice. You may wish to look at my answer here, and also visit this blog post of Tao –  t.b. Aug 24 '11 at 11:42
Hm. I didn't tell you anything that isn't already on the wiki-page you linked to. A notable class (still a Banach limit) are so-called medial means. –  t.b. Aug 24 '11 at 11:45
I had looked for questions about "positive operator", but I forgot to look for "non-negative operators"... anyway, I'm perfectly fine with using AC, and I fact I edited my question adding the Banach limits as another class of positive operators. Thank you for the links! –  Angelo Lucia Aug 24 '11 at 11:53
Maybe this trivial observation is a more on-topic remark: Let $1$ be the constant sequence $(1,1,1,\ldots)$. Prove that a linear functional on $\ell^{\infty}$ is positive if and only if $\phi(1) = \|\phi\|$, so this gives you a means of constructing many examples if you don't care about further properties. –  t.b. Aug 24 '11 at 12:29
@Martin: I meant what I wrote: $x$ is positive if and only if $\| \|x\| \cdot 1 - x \| \leq \|x\|$. Thus if $\phi(1) = \|\phi\|$ and $x \geq 0$ we have $|\|\phi\| \cdot \|x\| - \phi(x)| = |\phi(\|x\|\cdot 1 - x)| \leq \|\phi\|\cdot \|x\|$ but this means that $\phi(x) \geq 0$. The other direction is even easier. Note also that replacing $\phi$ by $2\phi$ multiplies both sides of $\phi(1) = \|\phi\|$ by $2$. –  t.b. Aug 24 '11 at 14:36
show 5 more comments

## 2 Answers

Another interesting class of positive linear functionals on $\ell_\infty$ are $\mathcal F$-limits or limits along an ultrafilter.

Let $(x_n)$ be a real sequence and $\mathcal F$ be a filter on $\mathbb N$. A real number $L$ is $\mathcal F$-limit of this sequence if for each $\varepsilon>0$ $$\{n; |x_n-L|<\varepsilon\}\in\mathcal F.$$

It is known that if the sequence $(x_n)$ is bounded and $\mathcal F$ is an ultrafilter, then $\mathcal F$-limit exists and it is unique. See also this question for the proof of this fact and some references.

If I remember correctly, the $\mathcal F$-limits are precisely the extreme points of the set of all positive normed functionals from $\ell_\infty^*$. (By normed I mean $\lVert f \rVert =1$.) They are characterized by the property, that they are multiplicative $\varphi(x.y)=\varphi(x).\varphi(y)$. (I.e., if a linear functional $\varphi\in\ell_\infty^*$ is positive, normed and multiplicative, then there is an ultrafilter $\mathcal F$ such that $\varphi$ is $\mathcal F$-limit.)

You may notice that it is not possible to have a functional on $\ell_\infty$ which extends limits and is both shift-invariant and multiplicative.

For the sequence $x=(1,0,1,0,\ldots)$ we have $x+Sx=\overline 1$ (where $S$ denotes the shift operator). If a functional is shift-invariant than $\varphi(x)=\varphi(Sx)=\frac12$.

On the other hand, if a functional $\varphi$ is multiplicative, we get $\varphi(x.Sx)=\varphi(x).\varphi(Sx)=0$, which leads to $(1/2)^2=0$, a contradiction.

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add comment

It may be worth observing that the cone of positive functionals is weak-* closed. Combining this with the Banach-Alaoglu theorem gives another technique for producing positive functionals: any norm-bounded set of positive functionals has a weak-* limit point, which is again a positive functional.

For example, if $\phi_n$ is the evaluation functional $\phi_n(x) = x_n$, then any weak-* limit point $\phi$ of the sequence $\{\phi_n\}$ is a positive functional. These have the interesting property that $\phi(x)$ is always a limit point (in particular, a subsequential limit) of the bounded sequence $\{x_n\}$. These are rather different from Banach limits. (Consider $x = (1,0,1,0,\dots)$; a Banach limit $\psi$ must have $\psi(x) = 1/2$, but $\phi(x)$ is either 0 or 1.)

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Mathematics

# how do I solve this equation 4-2x=8 ( Algebra 1 )

$4-2x=8\ \ \ \ /-4\\\\-2x=8-4\\\\-2x=4\ \ \ \ /:(-2)\\\\x=4:(-2)\\\\x=-2$
$4-2x=8 \\ -2x=4\\ x=-2$


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# R->R function in C∞ whose set of critical points are dense

• September 29th 2010, 04:23 PM
ajskim
R->R function in C∞ whose set of critical points are dense
I need to find such a function. Any ideas about how to approach this? We have just covered Sard's theorem if that provides some sort of insight on how I should tackle this.
Thanks
• September 29th 2010, 05:14 PM
tonio
Quote:

Originally Posted by ajskim
I need to find such a function. Any ideas about how to approach this? We have just covered Sard's theorem if that provides some sort of insight on how I should tackle this.
Thanks

Sard's theorem is trivial for one variable functions, as $f(X)=\{0\}\,,\,\,X=$ the set of critical points of $f$ ,

so I don't think it'll help a lot here.

Of course, $f(x)=c=$ constant is a trivial example, but I'm not sure how to construct other examples, or even whether there exist...

Tonio
• September 29th 2010, 06:17 PM
ajskim
I was hoping to find a non-trivial example, but thank you. Yeah, I agree that Sard's probably isn't applicable here.
• September 29th 2010, 06:39 PM
ajskim
Oops! I meant critical values, not critical points
• September 29th 2010, 08:44 PM
ajskim
Solved! Sum of bump functions


text:
# statsmodels.sandbox.regression.gmm.GMMResults.cov_params¶

GMMResults.cov_params(r_matrix=None, column=None, scale=None, cov_p=None, other=None)

Returns the variance/covariance matrix.

The variance/covariance matrix can be of a linear contrast of the estimates of params or all params multiplied by scale which will usually be an estimate of sigma^2. Scale is assumed to be a scalar.

Parameters: r_matrix (array-like) – Can be 1d, or 2d. Can be used alone or with other. column (array-like, optional) – Must be used on its own. Can be 0d or 1d see below. scale (float, optional) – Can be specified or not. Default is None, which means that the scale argument is taken from the model. other (array-like, optional) – Can be used when r_matrix is specified. cov – covariance matrix of the parameter estimates or of linear combination of parameter estimates. See Notes. ndarray

Notes

(The below are assumed to be in matrix notation.)

If no argument is specified returns the covariance matrix of a model (scale)*(X.T X)^(-1)

If contrast is specified it pre and post-multiplies as follows (scale) * r_matrix (X.T X)^(-1) r_matrix.T

If contrast and other are specified returns (scale) * r_matrix (X.T X)^(-1) other.T

If column is specified returns (scale) * (X.T X)^(-1)[column,column] if column is 0d

OR

(scale) * (X.T X)^(-1)[column][:,column] if column is 1d


text:
# zbMATH — the first resource for mathematics

Cut-off for large sums of graphs. (English) Zbl 1135.05039
Summary: If $$L$$ is the combinatorial Laplacian of a graph, $$\exp(-Lt)$$ converges to a matrix with identical coefficients. The speed of convergence is measured by the maximal entropy distance. When the graph is the sum of a large number of components, a cut-off phenomenon may occur: before some instant the distance to equilibrium tends to infinity; after that instant it tends to 0. A sufficient condition for cut-off is given, and the cut-off instant is expressed as a function of the gap and eigenvectors of components. Examples include sums of cliques, stars and lines.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 60J27 Continuous-time Markov processes on discrete state spaces
##### Keywords:
Laplacian; sum of graphs; spectrum; Kullback distance; cut-off
Full Text:
##### References:
 [1] Aldous, D.; Diaconis, P., Shuffling cards and stopping times, Amer. Math. Monthly, 93, 5, 333-348, (1986) · Zbl 0603.60006 [2] Austin, D.; Gavlas, H.; Witte, D., Hamiltonian paths in Cartesian powers of directed cycles, Graphs and Combinatorics, 19, 4, 459-466, (2003) · Zbl 1032.05069 [3] Barrera, J.; Lachaud, B.; Ycart, B., Cutoff for n-tuples of exponentially converging processes, Stoch. Proc. Appl., 116, 10, 1433-1446, (2006) · Zbl 1103.60023 [4] Bellman, R., Introduction to matrix analysis, (1960), McGraw-Hill, London · Zbl 0124.01001 [5] Bezrukov, S. L.; Elsässer, R., Edge-isoperimetric problems for Cartesian powers of regular graphs, Theor. Comput. Sci., 307, 3, 473-492, (2003) · Zbl 1070.68114 [6] Chung, F.; Oden, K., Weighted graph Laplacians and isoperimetric inequalities, Pacific J. of Math., 192, 257-274, (2000) · Zbl 1009.05095 [7] Çinlar, E., Introduction to stochastic processes, (1975), Prentice Hall, New York · Zbl 0341.60019 [8] Colin de Verdière, Y., Spectres de graphes, 4, (1998), SMF · Zbl 0913.05071 [9] Colin de Verdière, Y.; Pan, Y.; Ycart, B., Singular limits of Schrödinger operators and Markov processes, J. Operator Theory, 41, 151-173, (1999) · Zbl 0990.47013 [10] Cvetković, D.; Doob, M.; Gutman, I.; Torgašev, A., Recent results in the theory of graph spectra, 36, (1988), North-Holland, Amsterdam · Zbl 0634.05054 [11] Cvetković, D.; Doob, M.; Sachs, H., Spectra of graphs - Theory and application, (1980), Academic Press, New York · Zbl 0458.05042 [12] Mohar, B., Laplace eigenvalues of graphs - a survey, Discrete Math., 109, 171-183, (1992) · Zbl 0783.05073 [13] Pollard, D., User’s guide to measure theoretic probability, (2001), Cambridge University Press · Zbl 0992.60001 [14] Saloff-Coste, L.; Kesten, H., Random walks on finite groups, Probability on discrete structures, 110, 263-346, (2004), Springer, Berlin · Zbl 1049.60006 [15] Ycart, B., Cutoff for samples of Markov chains, ESAIM Probab. Stat., 3, 89-107, (1999) · Zbl 0932.60077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
Find a polynomial with cubic values for consecutive integers.

Can we find a cubic polynomial (except the obvious ones, i.e. cubes of linear polynomials), say, $f(x)\in \Bbb{Q[x]}$ whose values are cubes for four consecutive integers? What about five consecutive integers, or more?

To find such polynomial for two consecutive integers is very obvious, i.e. $f(n)=n^3+7n$, for $n=0,1$ takes on the value $0^3,2^3$.
To find such polynomial for $3$ consecutive integers, $n(n+1)(n-1)$ also works but although we do get same value for i.e. $0^3$ when we plug in $n=-1,0,1$.
But I hope things to become more interesting from $4$ onwards. Right?

In general, given some random positive integer, say $n$ (no matter how big) can we always find a cubic polynomial whose values are cubes for $n$ consecutive integers?

It could be an open problem, very well, but if there are some examples for $n=4,5,6$ it will be interesting to know. Thanks!

On other note, my question is inspired by the following:

There is a polynomial, like, $1+24n$ which gives squares for three consecutive integers, namely $0,1,2$ (In this case these three squares ($1,25,49$) are in A.P. too, whereas it is impossible to find four squares in A.P.).
Anyways, then there is $60n^2-60n+1$ which has square values for five consecutive integers namely $n=-2,-1,0,1,2,3$

• Do you want to say $f(x)\in \mathbb{Z}[x]$? or it's as you stated "over real numbers"? – Elaqqad Jul 15 '15 at 14:09
• Well if we can find it over $\Bbb{Z}$, it will be extra cool, but I asked it over reals. – Bhaskar Vashishth Jul 15 '15 at 14:18
• If it is over $\mathbb{R}$ can you not just interpolate through $\{(n,a^3),(n+1,b^3),(n+2,c^3),(n+3),d^3\}$ ??? And continue from $x=n$ to $x=n+i-1$ for $i$ consecutive cubes? – JP McCarthy Jul 15 '15 at 14:20
• @JpMcCarthy what if the polynomial is supposed to be of degree of $3$ – Elaqqad Jul 15 '15 at 14:25
• Sorry, I m nly asking for cubics throughout the question and thus I hv edited it to make it clear. Plz see edit. Thanks – Bhaskar Vashishth Jul 15 '15 at 14:34

For $4$ consecutive integers, consider $f(x)=103\,x^3-304\,x^2+228\,x$, where $f(0)=0^3$, $f(1)=3^3$, $f(2)=4^3$, and $f(3)=9^3$. For $5$ consecutive integers, consider $f(x)=85\,x^3-84\,x$, where $f(-2)=(-8)^3$, $f(-1)=(-1)^3$, $f(0)=0^3$, $f(+1)=(+1)^3$, and $f(+2)=(+8)^3$.

Now, we assume that $k\geq 4$. Let $s_0,s_1,s_2,\ldots,s_k\in\mathbb{Z}$. There exists a cubic polynomial $f(x)\in\mathbb{Q}[x]$ such that for which $f(i)=s_i^3$ for $i=0,1,2,\ldots,k$ if and only if $$s_j^3-4\,s_{j-1}^3+6\,s_{j-2}^3-4\,s_{j-3}^3+s_{j-4}^3=0$$ for all $j=4,5,\ldots,k$. To deal with $k>5$, we have to solve cubic equations for rational numbers such as $4p^3-5=q^3$ (I believe that the only rational points on this curve are $(p,q)=(1,-1)$ and $(p,q)=(2,3)$). Personally, I don't think there is a polynomial $f(x)\in\mathbb{Q}[x]$ of degree at most $3$ other than the trivial $f(x)=a^3(x-b)^3$ and the constant polynomials such that $f$ takes perfect-cube values at more than $5$ consecutive integer points.

• What approach you took? It is really nice! – Bhaskar Vashishth Jul 15 '15 at 14:43
• I used Lagrange polynomials, and tried to adjust the values of $f$ so that the coefficients are integers. In fact, for any $s_0,s_1,s_2,s_3\in\mathbb{Z}$, there exists a cubic $f(x)\in\mathbb{Q}[x]$ such that $f(i)=s_i^3$ for $i=0,1,2,3$. If you want $f$ to have integer coefficients, you need $2\mid s_0-s_2$, $2\mid s_1-s_3$, and $3\mid s_0-s_3$. – Batominovski Jul 15 '15 at 14:47
• Yup, just recalled lagrange interpolation on wiki, this wont work for five points or more. Thanks. It works fine for four points though. :) – Bhaskar Vashishth Jul 15 '15 at 14:57

Engineers and numerical analysts know how to fit a polynomial curve passing through, e.g., the points $(1,10^3), (2, -7^3), (3, 11^3), (4,7^3)$ etc.

Given a million and one points there exists a polynomial of degree at most one million passing through them. If the given points all have integer co-ordinates, the polynomial will have rational coefficients. (Lagrange Interpolation Theorem). Can also be regarded a version of Chinese Remainder theorem.

• The OP is looking for cubics only. – JP McCarthy Jul 15 '15 at 16:12


text:
Basis in Linear Transformations

I'm pretty dim when it comes to basis and span, please take that into account :)

Help me prove this theorem:

Let $V$, $W$ be vector spaces, also $T:V \rightarrow W$ is a linear transformation.

If $\beta = \{v_1, v_2, ..., v_n\}$ is a basis for $V$ then $R(T) = span(\{T(\beta)\}) = span (\{T(v_1), T(v_2), ..., T(v_n)\})$

1 Answer

This is a simple consequence of two facts:

1) any vector in $V$ is a linear combination of the vectors $\{v_i\}$ in a basis of $V$ . this means that $\forall u \in V$ we have :$u=u_1v_1+u_2v_2+\cdots + u_nv_n$

2) $T$ is a linear transformation. This means that $\forall u \in V$ we have $T(u)=T(u_1v_1+u_2v_2+\cdots + u_nv_n)=u_1T(v_1)+u_2T(v_2)+\cdots+u_nT(v_n)$

Now, by definition, the range of $T$ is $R(T)=\{w:w\in W,w=T(u)\}$ . This means that $w\in R(T) \iff w= u_1T(v_1)+u_2T(v_2)+\cdots+u_nT(v_n)$, but this means exactly that $R(T)$ is the span of $\{v_1,v_2\cdots v_n\}$.


text:
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text:
Condensed Matter Physics, 2003, vol. 6, No. 3(35), p. 459-470, English
DOI:10.5488/CMP.6.3.459

Title: THE STRUCTURE OF CONCENTRATED Li-AMMONIA SOLUTIONS AS DERIVED FROM MD SIMULATIONS
Author(s): S.Hannongbua, T.Remsungnen (Department of Chemistry, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand), M.Kiselev (Institute of Solution Chemistry, Russian Academy of Sciences, 153045 Ivanovo, Russia), K.Heinzinger (Max-Planck-Institut f\"{u}r Chemie (Otto-Hahn-Institut), POB 3060, 55020 Mainz, Germany)

The results of Molecular Dynamics simulations of lithium-ammonia solutions over the whole concentration range from 0.5 to 19.6~MPM at 240~K are reported. The pseudopotential theory is employed at the higher concentrations and the direct contribution to the total potential which has been derived from \textit{ab initio} calculations has been supplemented by three-body terms. The resulting partial structure and radial distribution functions are compared in detail with recent X-ray and neutron diffraction studies with isotopic substitution. There is an overall good agreement between simulation and experiment. Differences are discussed. The solvation number of the lithium ion is found to be six for the dilute solution and four for the higher concentrations in analogy to the hydration of Li$^{+}$ found for various aqueous solutions.

Key words: Molecular Dynamics simulation, lithium-ammonia solutions, pseudopotential theory
PACS: 61.20.J, 61.20.Qg, 71.15.Pd

 [ps,pdf] << Contents of Vol.6 No.3(35)


text:
# Natural logarithm power notation

I am trying to understand how to use Dirichlet's test for convergence and saw an example here (example 2).

Show that $\displaystyle\sum_{i=1}^\infty \frac{2^{2n}n^2}{e^n\,n!}\frac{1}{\ln^2n}$ converges.

Could someone please explain to me what $\ln^2n$ means? Does it mean $\ln(\ln(n))$, $(\ln(n))^2$ or something else entirely?

PS: I'm a newbie in both mathematical analysis and math SE. I spent quite a bit of effort in typing this so I hope that there aren't any glaring mistakes in the formatting. Please let me know if there are any!

• I guess it's $(\ln n)^2$; it's a very ambiguous notation indeed. – egreg Apr 15 '16 at 10:10
• @egreg I'm hoping that having the context of the question will help. Thanks for helping me edit! – jessica Apr 15 '16 at 10:15
• 1. Use Stirling approximation 2. Combine $4^{n}$ and $n^{n}$ 3. Combine $n^{2}$ and $(\ln n)^2$ 4. Find some sequence which larger than that and converges – openspace Apr 15 '16 at 10:24

Without Stirling's formula: With the $n$th term being $A_n,$ show that $A_{n+1}/A_n\to 0$ as $n\to \infty,$ which is more than enough for convergence, as even the weaker result $\exists r\in (0,1)\;\exists m\;(n>m\implies$ $|A_{n+1}|\leq r|A_n|)$ guarantees absolute convergence by comparison with the geometric series $\sum_n r^n.$
1. Using Stirling : $$yoursum = \sum {\frac{4^{n}n^{2}}{n^{n}(\ln n)^{2}}}$$
2. $$\frac{4^n}{n^n} \le \frac{1}{n^2}$$ for some big $n$


text:
# Is it OK to ask a question from stackexchange again on overflow?

Recently, I read a question on difference between stackexchange and overflow and I found out that overflow is more focused on difficult questions. Some time ago, I asked on a stackexchange a question (How many pure ternary operators are there?), that is probably too difficult to solve/answer and requires a lot more energy, focus and understanding than people on stackexchange are willing to put into it.

So, learning this new information, I considered asking the same question on overflow again. Would that be Ok or would that be considered a simple duplicate?

It is true that I could try to draw more attention to it by putting a bounty on it, but that does not guarantee results and must be paid upfront, so I would rather try other ways first.

• No, I don't think so: it's not about difficulty. People ask questions equivalent to famous open problems at MSE (and the lemmings upvote those questions, slap-happily), just because they don't know they are famous open problems. Overflow is about research problems, people there can judge if a question is difficult (so I'd like to discourage your asking about "how many pure ternary operators are there", I'm not surprised you aren't willing to pay for an answer). – Professor Vector Sep 11 '17 at 8:56
• @ProfessorVector It is not famous open problem, it is designed by me and I could not solve it on my own or find any other published work on that topic. It might be open problem someone is dealing with, but definitely not famous. – TStancek Sep 11 '17 at 9:03
• @TStancek I suggest you post a question on Mathoverflow's meta asking about the appropriateness of posting the question there. Include a link to your original post, relevant references, any further details that may clarify why you think it may be appropriate there. If the response you get is positive, then go ahead with the post. Questions here disappear quickly from the front page, so not having received an answer yet is not necessarily an indication that it is of the appropriate level. – Andrés E. Caicedo Sep 11 '17 at 11:50
• @AndrésE.Caicedo Great idea, thank you. – TStancek Sep 11 '17 at 11:52
• How long ago is "some time ago"? If it's more than a week, it can't hurt to try it on MathOverflow, BUT be sure to link the question on each site to the question on the other site so everyone's in on it. – Gerry Myerson Sep 11 '17 at 13:07
• BTW I suggested this at meta.MO some time ago - but it seems that most people disagree with that idea: Thread for asking about suitability of math.SE question on MO? – Martin Sleziak Sep 11 '17 at 13:12
• I would doubt that a lack of time and energy acounts for the failure to get a response about the number of "pure" ternary operators. Perhaps if you linked to your Question, rather than merely mentioning it, we could better advise you. – hardmath Sep 11 '17 at 20:07
• – TStancek Sep 11 '17 at 20:45
• BTW if you simply write overflow, some people might think that you are asking about Stack Overflow and some people might think that you are asking about MathOverflow. (I assume it is the latter - which is why I added the (mathoverflow) tag.) – Martin Sleziak Sep 11 '17 at 22:01

I've looked at the Question at issue, and I think it not in good shape for Math.SE or MathOverflow.

It asks about "how many" functions of three variables cannot be expressed as a composition of two bivariate functions in a particularly constrained way, namely that $T(x,y,z)$ cannot be expressed as either of these forms:

$$f(x,g(y,z)) \;\text{ or }\; f(g(x,y),z)$$

So the Question is somewhat related to Hilbert's Thirteenth Problem, which also concerns the possibility of expressing a function of three variables as a composition of finitely many functions of two variables.

But in addition to restricting how the compositions are taken (preserving in particular the "order" of arguments above), the presently posed Question fails to say anything about the underlying domain, except to name it:

$$T:A\times A\times A \to A$$

There are two versions of Hilbert's Thirteenth Problem, one for (real) continuous functions and one for algebraic functions, but by asking "how many" pure ternary functions (or operators, if you prefer), the OP is probably thinking about $A$ as a finite set.

Given the fact that Edits and a posted Answer have already occurred on the original Question, my advice is to post a new Question on Math.SE with the necessary gaps filled in about the domain $A$. Such a new Question can of course link back to the old one for context.

• "Such a new Question can of course link back to the old one for context." I would say, such a new question must of course link back to the old one for context (and I would add that the old question must link forward to the new one). – Gerry Myerson Sep 11 '17 at 21:55
• @Gerry: I stand corrected ! – hardmath Sep 11 '17 at 22:05


text:
# scipy.signal.sepfir2d¶

scipy.signal.sepfir2d(input, hrow, hcol)

Convolve with a 2-D separable FIR filter.

Convolve the rank-2 input array with the separable filter defined by the rank-1 arrays hrow, and hcol. Mirror symmetric boundary conditions are assumed. This function can be used to find an image given its B-spline representation.

Parameters
inputndarray

The input signal.

hrowndarray

A row direction filter defined by the rank-1 arrays.

hcolndarray

A column direction filter defined by the rank-1 arrays.

Returns
outputndarray

The filtered signal.

Examples

Examples are given in the tutorial.

#### Previous topic

scipy.signal.correlate2d

#### Next topic

scipy.signal.choose_conv_method


text:
## Introduction

Cerium dioxide (CeO2), also known as ceria, is considered one of the most important rare earth oxides due to its multiple applications in several key areas of thin film technology1,2. In particular, nanostructured ceria thin films are very attractive due to their promising performance as catalysts for the treatment of automobile exhaust gases, where the catalytic activity is very sensitive to the exposed crystalline facets3. For instance, the (111) direction (an oxygen-rich surface) of the fluorite CeO2 structure is the most energetically stable, and, thus, the most frequently exposed facet in thin films, having an intrinsically low catalytic activity. However, the (100) direction (a cerium-rich surface) comprises a dipolar plane that is less stable, and, consequently, harder to grow in thermodynamic terms but more catalytically active than the (111) counterpart3,4. In contrast, under Pd surface modification of CeO2, the (111) facet becomes more catalytically active than the (100) facet for room-temperature CO oxidation5. Regarding prospective applications in self-cleaning windows, the (111) surface is more hydrophobic than the (100) surface, while the (110) surface is the most hydrophilic6. This theoretical study also revealed that the relatively unstable oxygen-terminated (100) surface is stabilized by full coverage of molecular adsorbed water. Therefore, the surface properties of the (100) facet are critically dependent on the deposition method6. Therefore, strict control of the CeO2 growth direction (texture) in thin films is relevant for technological applications.

Several physical and chemical processes have been employed to prepare CeO2 thin films, including flash evaporation7, electron-beam evaporation8, sputtering9, laser ablation10, spin coating11 and spray pyrolysis12,13,14,15,16. Among these techniques, spray pyrolysis is very attractive for its low cost and effectiveness in the deposition of high-quality thin oxide films17,18. This technique also shows great versatility in both small- and large-area coverage, which makes it suitable for large-scale production methods17,18. Material selection (solvents, precursors, additives, substrates) and parameter optimization (molarity of solution, distance nozzle-substrate, flow rate, and spray generation) have a great influence on the final properties of films. Thus, the precise control of these parameters, as well as knowledge of their impact on the physicochemical phenomena involved in the corresponding chemical reactions, are crucial to control the morphology and material growth. When using pneumatic-based sprays to deposit CeO2 thin films on glass substrates, low crystalline films are typically obtained, and post-thermal annealing processes are frequently required12. Additionally, there is a great challenge in the accurate control of deposition parameters to obtain a strongly textured CeO2 film, either along the (111) or (100) directions, on such an amorphous substrate12,19,20,21. On the other hand, by using ultrasonic-based sprays, a small and homogeneous droplet size is obtained compared with pneumatic-based sprays, with a potential impact on film texture control and other physical properties15,16,17. Despite the potential of the ultrasonic spray pyrolysis (USP) technique, CeO2 thin film deposition on a glass substrate has rarely been explored13.

R. Suresh et al. used the USP technique to obtain CeO2 thin films, and the effects of water, ethanol and methanol solvents and the substrate temperature on the morphology of the deposit were studied13. Fairly inhomogeneous morphology and/or poor substrate coverage were observed in all cases. It was concluded that water and ethanol favour preferential orientation along the (200) reflection ((100) direction), while methanol favours the preferential orientation along the (111) direction. Temperatures near the extremes of the range between 300 °C and 500 °C produced amorphous films, while at intermediate temperatures, the X-ray diffraction patterns showed low crystalline features. Strikingly, discussion for the physicochemical origin of this behaviour was omitted, which could eventually lead to convenient modifications of the deposition route. A deep understanding of the growth mechanism is of relevant importance for film deposition experimental design. The impact of other important parameters, such as carrying flux and precursor concentration, was not included in the study by Suresh et al.13. As ultrasonic spray pyrolysis is highly dependent on many deposition parameters, more studies are required to fill some of the gaps that remain in previous work. On the other hand, García Sánchez et al. evaluated the influence of flow rates and substrate temperature on the structural, chemical and optical properties of CeO2 thin films prepared by USP using a system with two flow rates16. They found that flow rate variations modified the substrate temperature, crystalline orientation and Ce3+/Ce4+ ratio in the material. A mix of Ce3+ and Ce4+ was reported, even in samples with thermal annealing. A photoluminescence signal in the visible region of spectra was observed, which is associated with the high concentration of Ce3+ (~ 40% in as-grown samples and 25% in treated films), probably in the grain surface.

In the present work, the influence of flow rate, solution molarity and substrate temperature on texture and morphological characteristics, as well as on optical absorption and Raman response of nanostructured CeO2 thin films deposited by USD, are evaluated. The results demonstrate that tuning the proper deposition conditions allows us to obtain highly crystalline CeO2 films with strong texture along the (100) and (111) directions, as well as randomly oriented crystallites (no texture), on an amorphous substrate such as glass. Homogeneous substrate coverage can also be attained. The obtained results are discussed based on the reaction kinetics of crystallite growth. These findings are relevant for catalytic applications.

### Results and discussion

The XRD patterns for the 0.03-HF-475, 0.03-HF-425, 0.02-HF-475, and 0.03-LF-475 samples are presented in Fig. 1. Although growth took place on an amorphous substrate, the deposited films are polycrystalline, which is signed by the multipeak patterns. In all cases, the positions of the observed peaks match well with those of the PDF 34–0394 reference, corresponding to the fluorite structure (space group Fm3m) of CeO2 with a lattice constant of 5.41 Å. However, except for the 0.03-LF-475 sample, it is noteworthy that the relative intensities of the XRD patterns fairly deviate from those of the PDF 34-0394 reference, revealing well texturized CeO2 films. Regarding the sensitivity of this technique, no secondary phase impurities were detected.

The average crystallite sizes (D) were estimated from XRD patterns for the most intense peaks using the Scherrer formula22:

$$D=\frac{0.9\lambda }{\beta cos{\theta }_{B}}$$
(1)

Here, λ is the wavelength of the incident beam, β is the intrinsic width at half maximum of the hkl line and ΘB is the Bragg diffraction angle. The preferential orientation can be evaluated by calculation of the texture coefficient (TC) using the formula22:

$$T_{{C_{{\left( {hkl} \right)}} }} = ~\frac{{{\raise0.7ex\hbox{{I\left( {hkl} \right)}} \!\mathord{\left/ {\vphantom {{I\left( {hkl} \right)} {I_{0} \left( {hkl} \right)}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{{I_{0} \left( {hkl} \right)}}}}}{{\frac{1}{N}\mathop \sum \nolimits_{N} \left[ {{\raise0.7ex\hbox{{I\left( {hkl} \right)}} \!\mathord{\left/ {\vphantom {{I\left( {hkl} \right)} {I_{0} \left( {hkl} \right)}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{{I_{0} \left( {hkl} \right)}}}} \right]}}$$
(2)

where $$I\left(hkl\right)$$ is the measured relative intensity of a plane $$\left(hkl\right)$$, $${I}_{0}\left(hkl\right)$$ is the standard intensity taken from the PDF data and N is the number of diffraction peaks. TC qualitatively evaluates the strength of the crystalline orientation along a certain (hkl) direction with respect to a randomly oriented PDF powder reference. As the TC values are closer to unity, the crystallites of the film are more randomly oriented, while larger TC values indicate stronger crystalline orientation along the corresponding (hkl) direction. The results for the crystallite size and TC along the (111), (200), (220) and (311) directions are presented in Table 1.

The obtained crystallite size along the (111) direction was observed to be the smallest in every sample, independent of the strength of the texture coefficient along this direction. For all samples, the Williamson-Hall approximation23,24 did not fit the relevant data obtained from the XRD experiment well. This fact means that crystallite size and/or lattice strain are fairly dependent of the crystalline direction in the films. This observation is supported by DFT calculations, as well as previous experimental works25,26. These works concluded that oxygen vacancies (which lead to lattice strain) become energetically favoured by Ce atom displacement along the (111) direction, resulting in a distorted Ce3+/4+-O tetrahedron. Therefore, the strain contribution to the peak broadening in the XRD pattern can significantly differ from one direction to another, being particularly strong along the (111) direction25,26. Thus, the crystallite size determination via the Scherrer formula should be read as a minimum and not necessarily as a value well adhered to the real size. When the crystallite sizes are compared with films grown on crystalline (silicon) substrates16, similar grain size values are obtained for the (111) direction, but higher values are obtained for the other directions. This could be related to the ceria tendency to grow in a columnar structure.

Table 1 shows that the 0.03-HF-475 sample is strongly texturized along the (200) direction, with a TC of 2.2. It is well known that preferential growth of thin films derived from spray-based techniques is modulated by the substrate nature, reaction atmosphere (these two factors are common for the samples in this work), deposition temperature, precursor nature and concentration and carrying fluxes. These parameters could be interrelated in some way and affect the reaction kinetics. The reaction kinetics strongly influence (via the migration energy of the deposited atoms) the chance of ad-atoms accommodating (or not) minimal energy surfaces24. According to the surface energy of the different planes of the fluorite structure of CeO2, the surface stability of the crystal follows the order of (111) > (200) > (220) > (311)24. Therefore, the fact that the 0.03-HF-475 film is strongly texturized along the (200) direction means that the reaction kinetics under these conditions are fast enough (rapid crystallization) to inhibit ad-atom accommodation in the lower energy surface (111). Taking this sample (0.03-HF-475) as a reference, if a lower substrate temperature is used during the deposition process, as is the case for the 0.03-HF-425 sample, the texture coefficients become strong along the (111) direction (Tc = 2.1) and very weak along the remaining planes (TC < 0.6). This result is indicative of a fairly slower reaction rate in these lower temperature conditions, which allows ad-atom migration to the favoured lowest surface energy facet (111) during crystal formation. In this case, the preferential growth along (111) is probably related to ordered staking of crystallites in this direction rather than large crystallite size (see Table 1). When a lower precursor concentration is used, such as that in the 0.02-HF-475 sample, the texture coefficient along the (200) direction becomes even stronger (but weaker along the remaining directions) than that of the 0.03-HF-475 counterpart (2.2 vs 3.1). This behaviour means that the reaction kinetics are even faster in this case, which can be understood since the energy available for the reaction to occur is the same for both the 0.02-HF-475 and 0.03-HF-475 samples, but the amount of organic matter to burn (giving rise to CeO2 formation) is smaller for the former. For the lower flux, such as that in the 0.03-LF-475 sample, a less texturized film is obtained, i.e., texture coefficients are close to the unit. This fact points to an intermediate kinetic reaction between 0.03-HF-475 and 0.03-HF-425. This means that lowering the flux decreases the reaction kinetics but to a lesser extent than lowering the substrate temperature (at least in the test conditions). In conclusion, in the explored parameter ranges, higher temperatures and fluxes and lower precursor concentration favour faster reaction kinetics, which trigger texture along the (200) direction, while lower temperature and flux, as well as higher precursor concentration, favour slower reaction kinetics, promoting texture along the (111) direction. This result overcomes preceding studies regarding texture control of CeO2 thin films on glass substrates by USD13, but it is consistent with ceria grown on silicon substrates16. This is an interesting result that suggests that texture is more influenced by the control of grown conditions than by the nature of the substrate.

SEM micrographs of the 0.03-HF-475, 0.03-HF-425, 0.02-HF-475, and 0.03-LF-475 samples are shown in Fig. 2. The surface morphology of the layers observed by SEM is a consequence of the degree of crystallinity and orientation (texture) resulting from the growth process. Therefore, the SEM technique allows visualization of the impact of texture on the morphological termination of the surface of the films. Correlating TCs with morphological surface characterization by SEM for films is a very useful and common characterization strategy27. Thus, the discussion delivered above for the origin of film texture can be extrapolated here to the corresponding morphology observed by SEM.

SEM micrographs of the 0.03-HF-475 sample show homogeneous coverage of the substrate (Fig. 2a). The sample has a rough surface morphology composed of irregular polygonal grains with sizes ranging from ~ 20 to 50 nm. We want to note that this film presented a (200) texture (Table 1). In contrast, the 0.03-HF-425 sample (with a strong texture along (111)) has a fairly flat surface morphology with a crack network delimiting islands of ~ 100–300 nm in size (Fig. 2b). It is known that vacancy accumulation favours film cracking while minimizing the lattice strain and energy28; thus, this morphology is in accordance with the preferred strain accumulation (probably originating from oxygen vacancy formation) along the (111) direction, as hypothesized above26. Considering the intrinsic anisotropic nature of the CeO2 structure and some related properties25, an anisotropic thermal expansion coefficient showing a larger mismatch along (111) with respect to that of the substrate cannot be discarded. There are no reports in the literature in this regard. The morphology of the 0.02-HF-475 sample (Fig. 2c) somehow resembles that of the 0.03-HF-475 sample, but in this case, the surface is more homogeneously covered, and the polygonal grains are better defined, packaged and show sharper edges. In correspondence with the analysis above, this behaviour is associated with the stronger texture along the same (200) direction. For this sample, grain sizes are also in the ~ 20–50 nm range. The morphology of the 0.03-LF-475 sample (Fig. 2d) is intermediate between that of the 0.03-HF-475 and 0.03-HF-425 samples. This result is also in accordance with the observations from the XRD analysis.

The transmittance spectra for the 0.03-HF-475, 0.03-HF-425, 0.02-HF-475, and 0.03-LF-475 samples are presented in Fig. 3a. The spectral onset (below ~ 390 nm) for all the studied samples is very similar. The spectral oscillations in the ~ 390–800 nm range obey the interference effect, as typically observed in transparent thin films. Different thicknesses lead to different oscillation amplitudes.

The corresponding Tauc plots for the data in Fig. 3a for direct and indirect band gap calculations are shown in Fig. 3b and c, respectively. All the samples have the same direct band-gap energy (EgDir) value of 3.58 eV and indirect band-gap (EgInd) of 3.23 eV. There are two gap values reported for bulk CeO2: an indirect gap in the 2.99–3.3 eV range and a direct gap at 3.57 eV29,30. Thus, the obtained Eg values are fairly coincident with those reported for bulk CeO2 and are also in accordance with the transparent appearance of the layers. It has been reported that the absorption edge of ceria films is not significantly altered up to doping levels of 15% substitution of Ce+4 by trivalent cations31. The coincidence between the obtained Eg indicates that all the samples studied in this work present similar electronic structures (at least regarding the top of the valence band and the bottom of the conduction band). This fact is the cause of the observed similar spectral onset in Fig. 3a. These Eg values define the electronic transitions from the O 2p-based top of the valence band to the partially occupied bottom of the conduction band, which is dominated by the Ce 4f orbital contribution32.

As Raman scattering spectroscopy is a powerful technique in the structural characterization of materials, it was used for further analysis. The Raman spectra for the 0.03-HF-475, 0.03-HF-425, 0.02-HF-475, and 0.03-LF-475 samples are shown in Fig. 4. The four spectra profiles are fairly coincident (within 5%) and show characteristic features corresponding to the fluorite structure (Fm3 m) of CeO2. As in XRD analysis, only CeO2 was detected in all cases.

The most intense band is located at approximately 462 cm−1 and is attributed to symmetric stretching of the Ce4+-O bonds (F2g symmetry) of the Ce-O8 octahedron5. This band is redshifted by 4 cm−1 with respect to that of the stoichiometric single crystal5. This observed redshift is ascribed to a substoichiometric composition of CeO2−δ, which can be roughly estimated as δ = 0.023 by following the methodology developed in33. By means of a semiempirical model32, the F2g band also allows an estimation of the average crystallite size, correlation length (homogeneity size), and order of magnitude of oxygen vacancy density of 30–60 nm (comparable to that obtained from XRD), 18–10 nm and 1017–1018 cm−3, respectively. The weak band at approximately 590 cm−1 corresponds to a longitudinal optical (LO) Raman-forbidden mode (F1u symmetry). This band becomes Raman-active due to oxygen vacancy defects, which leads to a lack of symmetry, and, consequently, relaxation of the selection rules34. The other weak band at approximately 1180 cm−1 is the corresponding 2LO overtone with A1g symmetry34.

The Raman spectrum of a given material is ultimately determined by its structure and symmetry and is sensitive to structural defects and crystallite size (depending on the excitation conditions and sample nature)32,33,34. Thus, the observed similarity of the four Raman spectral shapes obtained in this work points to similar bulk structure, symmetry and defect nature and densities in the studied films, at least up to the sensitivity of the excitation conditions used in this work (laser wavelength and power of 455 nm and 7 mW, respectively, for the best signal-to-noise ratio). It needs to be mentioned that to more accurately sense the presence of vacancies, crystallite frontiers, or other structural discontinuities of the lattice periodicity, it is preferable to measure the Raman active modes while electron transfer from the O 2p to Ce 4f bands takes place, i.e., in the resonance condition34. The spectra were collected with a blue laser (455 nm), and a UV laser was required (photon energy above the corresponding computed Eg). This condition allows the preferential amplification of oxygen vacancy-related modes and eventually establishes a major spectral differentiation between the studied films.

## Conclusions

The physicochemical-based selection of critical deposition parameters in the ultrasonic spray pyrolysis technique allows precise control of the morphology and texture of polycrystalline CeO2 thin films on glass substrates. These results improve those of a previous study13. Considering the following deposition parameter ranges: molarity of 0.02–0.03 M, carrier flow of 1.0–1.5 L min−1, directing flow of 1.5–2.0 L min−1 and substrate temperature of 425–475 °C, increasing the temperature and flux and decreasing the precursor concentration favours fast reaction kinetics, which promote texture along the (200) direction. On the other hand, decreasing the temperature and flux as well as increasing the precursor concentration favours slow reaction kinetics, resulting in texture along the (111) direction. This is relevant not only for being the first report of this specific concern but also for facet-dependent applications of supported CeO2. The resulting films are strongly anisotropic regarding certain structural and morphological features but fairly homogeneous in their absorption and vibrational response, given the sensitivity of the used techniques. These results are very relevant considering the increased interest in ceria for catalytic and smart window applications.

## Materials and methods

Films were grown with the experimental setup previously reported17. It is important to note the use of two different flow rates in this USP system. A carrier gas transports the mist from the nebulizer to the nozzle. The maximum flow rate allowed for this gas is related to the mist generation rate to maintain the continuity of the deposition process. However, it is very important to maintain a quasi-laminar flow, as turbulence can produce a coarsening of drops, which decreases the homogeneity of the deposition process. On the other hand, the director gas enables an increase in the kinetic energy of droplets arriving on the heated substrate but in the same way reduces the time required for droplets to approach the substrate16. Films were deposited onto glass slices, which were ultrasonically cleaned in different steps with trichloroethylene (C2HCl3), acetone (C3H6O) and isopropanol (C3H8O). The spray solution was cerium(IV) acetylacetonate [Ce(acac)4 = Ce(C5H7O2)4] from Sigma-Aldrich Chemicals dissolved in anhydrous methanol. One mL of acetic acid (CH3COOH) in 0.5 L of anhydrous methanol was added15. Table 2 summarizes the four different experimental conditions used in this work. These conditions were selected to evaluate the influence of three different growth parameters (solution molarity, flow rates and substrate temperature). The first sample (0.03-HF-475) was prepared with a precursor concentration of 0.03 M, substrate temperature of 475 °C and flow rates of 1.5 L min−1 and 2.0 L min−1 for the carrier and directing gas, respectively. For the second sample (0.02-HF-475), the molarity was varied (0.02 M) with regard to the first sample. For the third sample (0.03-LF-475), the flow rates were changed: low flow (LF) rates of 1.0 L min−1 and 1.5 L min−1 were used instead of high flow (HF) rates of 1.5 L min−1 and 2.0 L min−1 for the carrier and directing gas, respectively, in the first sample. Finally, 425 °C was used as the heating plate temperature for the four samples (0.03-HF-425), instead of the temperature of 475 °C used in the first sample. The deposition time was fixed at 60 min, and air was used as the carrier and directing gas for all thin films.

### Characterization techniques

X-ray diffraction patterns (XRD) were obtained using a D2 PHASER diffractometer (from Bruker) with a Cu source (Kα1 radiation, λ = 1.5406 Å) operating at 10 mA and 30 kV in a Bragg–Brentano configuration. The transmittance spectra for the films were collected using a Cary-5000 spectrometer from Agilent. Raman scattering spectroscopy was performed using a DXR Micro-Raman spectrometer from Thermo Fisher Scientific with an excitation laser of 455 nm at 7 mW of power by focusing the laser radiation through a 50X microscope objective lens at 0.2 mW using 5 s as the integration time and 3 scans. Films were examined with a Zeiss field emission SEM (AURIGA model) operating at 1 kV.


text:
## Quantitative analysis

I had tried different tactics and none of them seem to work. I'm very weak in Math and I'm completely confused. Please help

Given:

Company A’s demand is uniform throughout the year and totals 18,000 units per year. Ordering costs total $38 per order. The annual holding cost rate is 26% of the value of the inventory. The per-unit cost of inventory is$12.

Company B’s demand is uniform throughout the year and totals 15,000 units per year. The production setup costs total $84 per setup. The annual holding cost rate is 28% of the value of the inventory. The per-unit cost of finished product is$19. The production rate is constant and equivalent to 60,000 units per year.


text:
# How did we count this open circuit voltage?

I am supposed to solve (to find voltage and current on resistor R2) this circuit using Thévenin's theorem. How did we get the formula for open circuit voltage? And why are we treating resistances as conductances?

Moreover, I don't seem to be able to find generally applicable way to calculate Ui. I'm really new to circuits and the formulas for Ui seem to be significantly different for each different circuit.

• Why are you using "U" instead of "V" or "E" as symbols denoting voltage difference? – EM Fields Oct 24 '16 at 22:21
• @EM Fields It is written by our university lecturer, it's an actual standard in this country. – John Smith Oct 24 '16 at 22:42
• electronics.stackexchange.com/questions/99584/… – jonk Oct 24 '16 at 23:32
• "I don't seem to be able to find generally applicable way"? Really, how about Ohm's Law?! Vth = (R3)I where I = R3(U/(R3+R4)). – CroCo Oct 25 '16 at 1:48
• @EMFields In the Netherlands we learned to use "U". to indicate a potential difference. Original german "Unterschied" meaning difference. – Decapod Oct 25 '16 at 9:59

Remove $R_2$, to give an open circuit in order to determine $V_{TH}$.

$R_3$ and $R_4$ then form a voltage divider across $U$, and $U_2= V_{TH}$.

Thus, $V_{TH}=\dfrac{R_3}{R_3 +R_4}U$. I guess $U=10 \:V$.

You can change the resistances into conductances to get the formula in your question, but there's no need to do this. Some prefer to work in G, but mixing the two in one formula is really confusing. Perhaps that was the aim of the question.

I don't have the ability to comment yet, but there is some missing information here? This isn't a complete question. I think this is all correct...

Note that resistance and conductance are just reciprocals of the same quantity... $$\frac{1}{R_N} = G_N \textbf{ and } \frac{1}{G_N} = R_N$$

Anyway, let's imagine redrawing the circuit as follows just by disconnecting $R_2$, which we can do because this is essentially going to be our Thevenin terminals, which we shall open-circuit:

simulate this circuit – Schematic created using CircuitLab

We want to find the voltage across and current going through $R_2$ (in the original circuit, not the one where $R_2$ is disconnected, obviously). So, we need to find a Thevenin equivalent for everything on the right-hand side of the above figure...

So, let's start by finding a Thevenin resistance, $r$:

In order to do this, we must make $V$ a short-circuit, leaving us with:

simulate this circuit

Now (hopefully, you can find equivelant resistances for simple parallel/series networks?):

$$r_{th} = \frac{R_4 \times R_3}{R_4 + R_3} + R_1$$

We also need to find a Thevenin voltage, $V_{th}$:

simulate this circuit

Let's name the nodes as above, such that:

$$V_{th} = V_3 - V_1 = V_3 - V_2 = R_3 \times I$$

Where $I$ is the current flowing through $R_4$ and $R_3$.

Now we need an expression for $I$ (obtainable using KVL, which you're OK with, hopfeully?):

$$V - R_4 I - R_3 I = 0$$ $$V - I(R_4 + R_3) = 0$$ $$I = \frac{V}{R_3 + R_4}$$ $$\therefore V_{th} = \frac{V \times R_3}{R_4 + R_3}$$

simulate this circuit

Now we have what we want:

$$I_{th} = I_2 = \frac{V_{th}}{R_2 + r}$$ $$V_2 = R_2 \times I_{th}$$

You should check out this webpage for more detailed explanations... and do some Googling. There isn't really a general 'formula' to work out $V$... otherwise there would be no 'analysis' in circuit analysis.

• I think Vth = (R3) I where I = U/(R3 + R4). – CroCo Oct 25 '16 at 1:41
• @CroCo didn't spot that. Thanks! Updated... – smollma Oct 25 '16 at 16:16