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# Finite groups of non zero real number under binary operation multiplication
How we can show that {1} and {1,-1} are the only finite groups of nonzero real numbers under binary operations multiplication?
-
The only torsion elements (equivalently, roots of unity) in ${\bf R}^\times$ are $\pm1$. – anon Mar 12 '13 at 5:17
Suppose that $|a|\ne 1$, $a\ne 0$. Show that if $m\ne n$, then $a^m \ne a^n$.
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Description
Scan and analyse the given file to gather information for ascii import into worksheets. This function will look read the file to look for consistent structure by trying different separators that will yield the largest number of columns.
Syntax
int AscImpReadFileStruct( LPCSTR lpcszFilename, ASCIMP * pASCIMPstruct, DWORD dwCntrl = 0 )
Parameters
lpcszFilename
[input] a full path ASCII file name
pASCIMPstruct
[output]pointer to an ASCIMP struct that is not initialized on input but will be set when function returns
dwCntrl
[input]additional flags to control the reading, only AIRF_USE_ASCIMP is defined for now. when dwCntrl = 0, pASCIMPstruct to be used only as output, if AIRF_USE_ASCIMP, then it will be used as starting point for further scanning
Return
Returns 0 if no error, otherwise an error code is returned
Examples
EX1
void AscImpReadFileStruct_ex1()
{
ASCIMP ascimp;
string strFile = GetOpenBox("*.dat");
{
}
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# How do you solve sqrt(6x+1) = 3sqrt(x)-1?
May 24, 2016
$x = 0$ or $x = 4$
#### Explanation:
Squaring both sides of $\sqrt{6 x + 1} = 3 \sqrt{x} - 1$, we get
$6 x + 1 = 9 x + 1 - 6 \sqrt{x}$
or $6 \sqrt{x} = 9 x + 1 - 6 x - 1$
or $6 \sqrt{x} = 3 x$
or $2 \sqrt{x} = x$ - now squaring this
$4 x = {x}^{2}$ or ${x}^{2} - 4 x = 0$
or $x \left(x - 4\right) = 0$
Hence $x = 0$ or $x = 4$
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copy_dm_to {dm} R Documentation
## Copy data model to data source
### Description
copy_dm_to() takes a dplyr::src_dbi object or a DBI::DBIConnection object as its first argument and a dm object as its second argument. The latter is copied to the former. The default is to create temporary tables, set temporary = FALSE to create permanent tables. Unless set_key_constraints is FALSE, primary key constraints are set on all databases, and in addition foreign key constraints are set on MSSQL and Postgres databases.
### Usage
copy_dm_to(
dest,
dm,
...,
types = NULL,
overwrite = NULL,
indexes = NULL,
unique_indexes = NULL,
set_key_constraints = TRUE,
unique_table_names = NULL,
table_names = NULL,
temporary = TRUE,
schema = NULL,
progress = NA,
copy_to = NULL
)
### Arguments
dest An object of class "src" or "DBIConnection". dm A dm object. ... Passed on to dplyr::copy_to() or to the function specified by the copy_to argument. overwrite, types, indexes, unique_indexes Must remain NULL. set_key_constraints If TRUE will mirror dm primary and foreign key constraints on a database and create unique indexes. Set to FALSE if your data model currently does not satisfy primary or foreign key constraints. unique_table_names Deprecated. table_names Desired names for the tables on dest; the names within the dm remain unchanged. Can be NULL, a named character vector, a function or a one-sided formula. If left NULL (default), the names will be determined automatically depending on the temporary argument: temporary = TRUE (default): unique table names based on the names of the tables in the dm are created. temporary = FALSE: the table names in the dm are used as names for the tables on dest. If a function or one-sided formula, table_names is converted to a function using rlang::as_function(). This function is called with the unquoted table names of the dm object as the only argument. The output of this function is processed by DBI::dbQuoteIdentifier(), that result should be a vector of identifiers of the same length as the original table names. Use a variant of table_names = ~ DBI::SQL(paste0("schema_name", ".", .x)) to specify the same schema for all tables. Use table_names = identity with temporary = TRUE to avoid giving temporary tables unique names. If a named character vector, the names of this vector need to correspond to the table names in the dm, and its values are the desired names on dest. The value is processed by DBI::dbQuoteIdentifier(), that result should be a vector of identifiers of the same length as the original table names. Use qualified names corresponding to your database's syntax to specify e.g. database and schema for your tables. temporary If TRUE, only temporary tables will be created. These tables will vanish when disconnecting from the database. schema Name of schema to copy the dm to. If schema is provided, an error will be thrown if temporary = FALSE or table_names is not NULL. Not all DBMS are supported. progress Whether to display a progress bar, if NA (the default) hide in non-interactive mode, show in interactive mode. Requires the 'progress' package. copy_to By default, dplyr::copy_to() is called to upload the individual tables to the target data source. This argument allows overriding the standard behavior in cases when the default does not work as expected, such as spatial data frames or other tables with special data types. If not NULL, this argument is processed with rlang::as_function().
### Details
No tables will be overwritten; passing overwrite = TRUE to the function will give an error. Types are determined separately for each table, setting the types argument will also throw an error. The arguments are included in the signature to avoid passing them via the ... ellipsis.
### Value
A dm object on the given src with the same table names as the input dm.
### Examples
con <- DBI::dbConnect(RSQLite::SQLite())
# Copy to temporary tables, unique table names by default:
temp_dm <- copy_dm_to(
con,
dm_nycflights13(),
set_key_constraints = FALSE
)
# Persist, explicitly specify table names:
persistent_dm <- copy_dm_to(
con,
dm_nycflights13(),
temporary = FALSE,
table_names = ~ paste0("flights_", .x)
)
dbplyr::remote_name(persistent_dm\$planes)
DBI::dbDisconnect(con)
[Package dm version 0.2.8 Index]
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{}
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# The power in an AC circuit is $$P = {E_{rms}}{I_{rms}}\cos \phi$$. The value of $$\cos \phi$$ in the series LCR circuit at resonance is
This question was previously asked in
ISRO SDSC SA MPC Physics 12-Feb-2017
View all ISRO Scientific Assistant Papers >
1. Zero
2. 1
3. 1/2
4. $$\frac{1}{{\sqrt 2 }}$$
Option 2 : 1
Free
Army Havildar SAC 2021 Full Mock Test
5475
100 Questions 200 Marks 120 Mins
## Detailed Solution
CONCEPT:
• The device which stores magnetic energy in a magnetic field is called an inductor.
• The device that stores electrostatic energy in an electric field is called a capacitor.
The power loss (P) in an A.C circuit is given by:
$$P=~{{E}_{rms}}{{I}_{rms}}Cos\theta$$
Where Erms is rms voltage in the circuit, Irms is rms current in the circuit, θ is the phase angle
For V = E0 Cos (ω t)
E0 is maximum potential, ω is angular frequency and t is time
Inductive reactance (XL) = ω L
Capacitive reactance (XC) = 1/(ω C)
$$\text{Impedance }\!\!~\!\!\text{ of }\!\!~\!\!\text{ the }\!\!~\!\!\text{ circuit }\!\!~\!\!\text{ }\left( \text{Z} \right)=~\sqrt{{{R}^{2}}+~{{\left( {{X}_{L}}-{{X}_{C}} \right)}^{2}}}$$
Where L is inductance, C is capacitance and R is the resistance of the circuit
If we have an LCR circuit and it is under resonance:
It means that X= Xc, hence Z = R and V = VR
• Power factor: It is the ratio of resistance to impedance in an LCR given by
$${\cos Φ = \frac{R}{Z}}$$
Where R = resistance and Z = impedance.
CALCULATION:
At resonance,
X= Xc
$$\text{Impedance }\!\!~\!\!\text{ of }\!\!~\!\!\text{ the }\!\!~\!\!\text{ circuit }\!\!~\!\!\text{ }\left( \text{Z} \right)=~\sqrt{{{R}^{2}}+~{{\left( {{X}_{L}}-{{X}_{C}} \right)}^{2}}}$$
So Z = R
Cosϕ = R/Z = R/R = 1
So option 2 is correct.
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# DS : TRIANGLE (m09q07)
Author Message
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Re: DS : TRIANGLE (m09q07) [#permalink]
### Show Tags
22 Feb 2010, 12:46
1.The area S = a (proven in the above posts)..
tanA=tanC=BO/AO=a^2/(1/a)=a^3.
If angle ABC<90 then A =(180-ABC)/2 > 45, -> tanA>1 -> a^3>1 -> Area S=a >1.
2. Perimeter P = (2/a)(1 + sqrt(a^6+1) > 4/a.
1 + sqrt(a^6+1) > 2
sqrt(a^6+1) > 1 for any a not equal to 0.
Rhe Stmnt 2 alone is not suff.
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Re: DS : TRIANGLE (m09q07) [#permalink]
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23 Feb 2010, 03:56
Bunuel,
For (1), how did you deduce that ABC is an isosceles right triangle ? As angle ABC is less than 90 deg, it is only isosceles at the most. We need not have 45-45 each at the lower angles as well. It could be a 50-50-80 (adding to 180) triangle but never a right triangle. Or am I missing something ??
Also, what do you think of AB^2 + BC^2 > AC^2 ??? I think the rule is right but never quite heard of it before.
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Re: DS : TRIANGLE (m09q07) [#permalink]
### Show Tags
23 Feb 2010, 07:40
SudiptoGmat wrote:
deepakdewani wrote:
How does angle ABC < 90 deg. lead to the conclusion "AB^2 + BC^2 > AC^2"?
Quote:
Its rule. Just memorise it.
While I agree that eventually remebering this rule for GMAT will be the best bet, I am hoping that you can provide the underlying rationale/logic for this rule. Haven't quite come across this rule in the strategy guides...though i am sure this rule can be quite handy in geometry questions involving some variation of inequalities.
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Re: DS : TRIANGLE (m09q07) [#permalink]
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23 Feb 2010, 08:30
This rule is called cosine theorem: for any triangle ABC
$$AC^2=AB^2+BC^2-2AB*BC*cos\angle ABC$$.
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Re: DS : TRIANGLE (m09q07) [#permalink]
### Show Tags
23 Feb 2010, 09:52
Expert's post
kaptain wrote:
Bunuel,
For (1), how did you deduce that ABC is an isosceles right triangle ? As angle ABC is less than 90 deg, it is only isosceles at the most. We need not have 45-45 each at the lower angles as well. It could be a 50-50-80 (adding to 180) triangle but never a right triangle. Or am I missing something ??
Also, what do you think of AB^2 + BC^2 > AC^2 ??? I think the rule is right but never quite heard of it before.
There was an assumption missing: "assume $$\angle ABC=90^\circ$$", already edited.
As for $$AB^2 + BC^2 > AC^2$$: if ABC=90 then we would have AB^2+BC^2=AC^2, but if we decrease angle ABC, value of AC will decrease too and we'll get $$AB^2 + BC^2 > AC^2$$.
Hope it's clear.
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Re: DS : TRIANGLE (m09q07) [#permalink]
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01 Mar 2010, 17:23
hello everyone,
How does angle ABC < 90 deg. lead to the conclusion "AB^2 + BC^2 > AC^2"?
doesn't this appear to be an application of the third side rule of a + b > c but a - b < c so by looking at the graph u can tell how this info was deduced.
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Re: DS : TRIANGLE (m09q07) [#permalink]
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16 Oct 2010, 14:07
Bunuel, how is it deduced as isosceles triangle?
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Re: DS : TRIANGLE (m09q07) [#permalink]
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02 Dec 2010, 12:00
Bunuel,
Great explanation - I am not sure about your conclusion "Hence a>1 is not true as when we decrease angle ABC, a will decrease as well and will become less than 1"
I think this may lead to "As the angle ABC decreases, since ht=a^2 when base = 2a (ht/base ratio = a/2), the base will become broader => a will increase
=> This triangle will have area = 1 for ABC = 90*
=> This triangle will have area > 1 for ABC < 90*
=> This triangle will have area < 1 for ABC > 90* "
Thx, JS
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Re: DS : TRIANGLE (m09q07) [#permalink]
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02 Dec 2010, 15:40
ooops, and my typo 2a in place of 2/a, led to a/2 instead of a^3/2
thank you so much for responding, you are one of the most brilliant minds I've seen in the GMAT math domain and it's always a pleasure to view your problem solving strategies!
JS.
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Re: DS : TRIANGLE (m09q07) [#permalink]
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25 Feb 2011, 08:13
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Re: DS : TRIANGLE (m09q07) [#permalink]
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25 Feb 2011, 10:30
Bunuel
cos(x) when x < 90 is positive and when x > 90 is negative.
When x < 90
ab^2 + bc^2 - (something) = ac^2
or
ab^2 + bc^2 = (something) + ac^2
Hence ab^2 + bc^2 > ac^2
When x > 90
ab^2 + bc^2 + (something) = ac^2
or ab^2 + bc^2 < ac^2
nvgroshar wrote:
This rule is called cosine theorem: for any triangle ABC
$$AC^2=AB^2+BC^2-2AB*BC*cos\angle ABC$$.
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Re: DS : TRIANGLE (m09q07) [#permalink]
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27 Feb 2011, 22:58
How does angle ABC < 90 deg. lead to the conclusion "AB^2 + BC^2 > AC^2"?
[quote]Its rule. Just memorise it.
While I agree that eventually remebering this rule for GMAT will be the best bet, I am hoping that you can
I think, This can be explained by thinking like this, if ABC is 90 degree then AB^2 + BC^2 = AC^2. Now, since side of the triangle is proprtional to the angle opposite to the side- we know that angle ABC is less than 90, so the side opposite to that will be smaller than the hypotenues of the right angle triangle. By this, we can write that AB^2 + BC^2 > AC^2.
Hope it helps!!!
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Re: DS : TRIANGLE (m09q07) [#permalink]
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02 May 2011, 14:22
themanwithaplan wrote:
hello everyone,
How does angle ABC < 90 deg. lead to the conclusion "AB^2 + BC^2 > AC^2"?
doesn't this appear to be an application of the third side rule of a + b > c but a - b < c so by looking at the graph u can tell how this info was deduced.
(realise this is an old question, but for anyone wondering why....)
setting ABC = 90 deg, you get a right angled isos triangle.... pythagoras theorem gives you a^2 + b^2 = c^2
(where c^2 = hypotenuse. and a and b are the other two sides)
logically, if the angle is less than 90 degrees and the triangles is isosceles, side C *must* be smaller than a^2 + b^2....
here they've just used 'AB' and 'BC' to indicate sides 'a' and 'b' as they appear in the theorem, and AC would be the hypotenuse if the triangle was a right triangle
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Re: DS : TRIANGLE (m09q07) [#permalink]
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06 Jun 2011, 08:33
Case 1:
Consider angle ABC=90 , in that case BCA and BAC angles will equal 45 deg.
now applying tan BCA= tan 45 = BO/OC consider O as the origin.
which implies BO/OC = a^3 = 1 or a=1.
Now since its given that angle ABC is less than 90 deg that means angle BAC and BCA can only exceed the value of 45 deg.
And tan of any value in excess of 45 is always more than 1.
That means for angle ABC < 90 we have BO/OC=a^3 > 1 => a > 1 --- sufficient
Case 2:
Permiter is > 4/a
=>AB+BC+CA > 4/a
x+x+ 2/a > 4/a
=> x > 1/a that is BC or BA > CA which is always the case (hypotnuse is alws greater) .Thsi option does nt tell u whther BCA and BAC angles are greater than 45 degree or not.Hence no way to find whther a > 1 or not. -----insuff
Hence ans-A
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Re: DS : TRIANGLE (m09q07) [#permalink]
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20 Dec 2011, 11:33
can anyone explain how they deduced from the illustration that the triangle was an isosceles? The diagram doesn't indicate that AB and AC are equal, so why couldn't it be a 30-60-90 triangle and the illustration is not to scale? is it because from the origin the distance is 1/a both ways, deducing that from that point reaching a2 indicates that ab and bc must be equal?
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Re: DS : TRIANGLE (m09q07) [#permalink]
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29 Feb 2012, 14:49
Tough question for me. Don't think I could do it in under 2 minutes.
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Re: DS : TRIANGLE (m09q07) [#permalink]
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01 Mar 2012, 07:04
Given that ABC is an equilateral triangle (will tell you why it is an equilateral triangle at the end) with side length of 2/a and altitude of length square(a), so the area will be $$\frac{1}{2}$$*$$\frac{2}{a}$$*square(a) = a so the question is whether a>1? And we cannot determine this with the information given in the question so let's look at the given 2 statements whether they provide any extra information:
stmt1 is redundant information - we already know that all angles of an equilateral triangle are of 60 degree
stmt2 is redundant information - length of one side is given (2/a), so perimeter will be (3 * $$\frac{2}{a}$$) = 6/a
So both the statements are not providing any extra information so we cannot determine whether a>1?
Why the given triangle is an equilateral triangle
Triangle property: if altitude, median and perpendicular bisector of a triangle are same, then the triangle is an equilateral triangle.
amitdgr wrote:
On the picture below, is the area of the triangle $$ABC$$ greater than 1?
1. $$\angle ABC < 90^\circ$$
2. Perimeter of the triangle $$ABC$$ is greater than $$\frac{4}{a}$$
[Reveal] Spoiler: OA
A
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Re: DS : TRIANGLE (m09q07) [#permalink]
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01 Mar 2012, 07:16
Expert's post
yatendragoel wrote:
Triangle property: if altitude, median and perpendicular bisector of a triangle are same, then the triangle is an equilateral triangle.
That's not true. If the altitude of a triangle is also the median, then the triangle is not always equilateral, it's always isosceles.
By the way OA for this question is A, not E. OA is given under the spoiler in the initial post.
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Re: DS : TRIANGLE (m09q07) [#permalink]
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03 Mar 2012, 08:42
Tough one. Is there a way by which it can be solved easily
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Re: DS : TRIANGLE (m09q07) [#permalink]
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25 Oct 2012, 14:33
I did this problem differently and was curious if it is a feasible method.
Since ABC is isoceles around the origin, I split ABC into two right triangles and dealt with just one side at a time. Using O as the origin, I drew triangle BOC with sides 1/a and a^2 and angle BOC = 90 deg.
(1) ABC < 90 deg means OBC < 45 deg. Assuming OBC = 45 would give us our limit on variable a. If OBC = 45 deg we would have a 45-45-90 triangle with a^2 (side OB) = 1/a (side OC). Since OBC < 45 deg, OCB must be greater than 45 deg and a^2 (side opposite OCB) must be greater than 1/a (side opposite OCB). Quickly trying different numbers in inequality a^2 > 1/a I found that a>1. Sufficient.
(2) Perimeter > 4/a means that the sides of the triangle are 1/a, 1/a and 2/a (bottom side). Taking just one side and making a right triangle and the Pythagorean theorem, I got (1/a)^2 + (a^2)^2 = (1/a)^2 as the minimum limit. This reduces to a^4>0. Since the perimeter of the big triangle is greater than 4/a, the hypotenuse of my "mini" right triangle is greater than 1/a. Therefore a must be greater than 0. This is however still insufficient.
Thoughts? Please let me know if there are any holes in my reasoning.
Thanks,
HImba88
Posted from my mobile device
Re: DS : TRIANGLE (m09q07) [#permalink] 25 Oct 2012, 14:33
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# DS : TRIANGLE (m09q07)
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# what is the reciprocal of tanX?
0
172
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what is the reciprocal of tanX?
Guest Aug 3, 2017
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#1
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The reciprocal of tangent is cotangent.
$$\tan=\frac{\sin}{\cos} \\~\\ \cot=\frac{\cos}{\sin}$$
hectictar Aug 3, 2017
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# Math Help - Triple integral take 2
1. ## Triple integral take 2
Evaluate $\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz$.
This is pretty swish! Here's what i've done:
$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz=\int_0^{\infty} \int_0^{\infty} \int_0^{\infty}$ $\frac{1}{\left( 1+y^2+z^2 \right)^2}. \frac{1}{\left( 1+\frac{x^2}{1+y^2+z^2} \right)^2} \ dx \ dy \ dz$
Let $\frac{x}{\sqrt{1+y^2+z^2}}=tan( \theta) \Rightarrow \frac{dx}{d \theta}=\sqrt{1+y^2+z^2} \ sec^2 (\theta)$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\frac{\pi}{2}} \frac{1}{(1+y^2+z^2)^2}. \frac{1}{(1+tan^2 (\theta))^2} \sqrt{1+y^2+z^2} \ sec^2 (\theta) d \theta \ dy \ dz$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\frac{\pi}{2}} \frac{1}{(1+y^2+z^2)^{\frac{3}{2}}} \ cos^2(\theta) \ d \theta \ dy \ dz$
$\frac{1}{2}\int_0^{\infty} \int_0^{\infty} \int_0^{\frac{\pi}{2}} \frac{1}{(1+y^2+z^2)^{\frac{3}{2}}} \ (1+cos(2 \theta)) \ d \theta \ dy \ dz$
$\frac{1}{2}\int_0^{\infty} \int_0^{\infty} \frac{1}{(1+y^2+z^2)^{\frac{3}{2}}}. \left[ \theta+\frac{1}{2}sin(2 \theta) \right]_0^{\frac{\pi}{2}} \ dy \ dz$
$\frac{\pi}{4}\int_0^{\infty} \int_0^{\infty} \frac{1}{(1+y^2+z^2)^{\frac{3}{2}}} \ dy \ dz$
$\frac{\pi}{4}\int_0^{\infty} \int_0^{\infty} \frac{1}{(1+z^2)^{\frac{3}{2}}}.\frac{1}{\left( 1+\frac{y^2}{1+z^2} \right)^{\frac{3}{2}}} \ dy \ dz$
Let $\frac{y}{\sqrt{1+z^2}}=tan( \phi) \Rightarrow \ \frac{dy}{d \phi}=\sqrt{1+z^2} \ sec^2 ( \phi)$
$\frac{\pi}{4}\int_0^{\infty} \int_0^{\frac{\pi}{2}} \frac{1}{(1+z^2)^{\frac{3}{2}}}.\frac{1}{\left( 1+tan^2 (\phi) \right)^{\frac{3}{2}}} \sqrt{1+z^2} \ sec^2 ( \phi) \ d \phi \ dz$
$\frac{\pi}{4} \int_0^{\infty} \int_0^{\frac{\pi}{2}} \frac{1}{1+z^2}. cos (\phi) \ d \phi \ dz$
$\frac{\pi}{4} \int_0^{\infty} \left[ \frac{1}{1+z^2} sin( \phi) \right] _0^{\frac{\pi}{2}} \ dz$
$\frac{\pi}{4} \int_0^{\infty} \frac{1}{1+z^2} \ dz$
Let $z=tan (\psi) \Rightarrow \ \frac{dz}{d \psi}=sec^2(\psi)$
$\frac{\pi}{4} \int_0^{\frac{\pi}{2}} \frac{1}{1+tan^2(\psi)} sec^2(\psi) \ d \psi$
$\frac{\pi}{4} \int_0^{\frac{\pi}{2}} d \psi$
$=\frac{\pi^2}{8}$
WOW!!
My query is: Is there a shorter way to do this?
2. Yes there is !
Substituting
$x = r \cos \theta$ and $y = r \sin \theta$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz = \int_0^{\infty} \int_0^{2\pi} \int_0^{\infty} \frac{1}{(1+r^2+z^2)^2} \ r \ dr \ d\theta \ dz$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz = 2\:\pi\:\int_0^{\infty} \int_0^{\infty} \frac{1}{(1+r^2+z^2)^2} \ r \ dr \ dz$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz = -\pi\:\int_0^{\infty} \left[\frac{1}{1+r^2+z^2}\right]_{r=0}^{r=\infty} \ dz$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz = \pi\:\int_0^{\infty} \frac{dz}{1+z^2} = \frac{\pi^2}{2}$
I do not find the same result as you
3. hmm, well this is interesting.
I only worked out that integral in front of my computer just now. I can't see a mistake in what i've done (quickly checking what I have now!).
I'll try solving in tomorrow when i'm more refreshed, It's too late to do maths!
If there's someone who hasn't posted yet reading this, work out an answer! A third opinion would be very helpful!!
4. I have found my mistake !
Substituting
$x = r \cos \theta$ and $y = r \sin \theta$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz = \int_0^{\infty} \int_0^{\frac{\pi}{2}} \int_0^{\infty} \frac{1}{(1+r^2+z^2)^2} \ r \ dr \ d\theta \ dz$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz = \frac{\pi}{2}\:\int_0^{\infty} \int_0^{\infty} \frac{1}{(1+r^2+z^2)^2} \ r \ dr \ dz$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz = -\frac{\pi}{4}\:\int_0^{\infty} \left[\frac{1}{1+r^2+z^2}\right]_{r=0}^{r=\infty} \ dz$
$\int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2} \ dx \ dy \ dz = \frac{\pi}{4}\:\int_0^{\infty} \frac{dz}{1+z^2} = \frac{\pi^2}{8}$
I do find the same result as you
5. magic!!!
I'll have to remember the method you use, it's soooooooo much faster than what I was doing!!
6. I had switched off my computer but when I realized my mistake I could not help coming back.
Now good night !
7. sleep tight!
Thanks for the help.
8. I've just had the opportunity to go over this in more detail and there's something I don't quite get.
Why do your $\theta$ limits range from $0 \leq \theta \leq \frac{\pi}{2}$?
The range that was put before as $0 \leq \theta \leq 2 \pi$ seemed a bit more reasonable.
9. Originally Posted by Showcase_22
I've just had the opportunity to go over this in more detail and there's something I don't quite get.
Why do your $\theta$ limits range from $0 \leq \theta \leq \frac{\pi}{2}$?
The range that was put before as $0 \leq \theta \leq 2 \pi$ seemed a bit more reasonable.
I believe it's because if $x, y, z > 0$ then you're only dealing with the first octant. That would mean that $0 \leq \theta \leq \frac{\pi}{2}$.
10. Originally Posted by Prove It
I believe it's because if $x, y, z > 0$ then you're only dealing with the first octant. That would mean that $0 \leq \theta \leq \frac{\pi}{2}$.
That's it !
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# Linux cp Permission denied on ntfs file system
bashlinuxntfspermissions
Could someone kindly explain why I get this Permission Denied error? I personally am user g and as ls shows I have read and write permission for both source and destination. My system is slackware 14, and the device being written to is my ereader, an ntfs-3g file system.
I have other ntfs file systems, thumb drives, external HDDs etc to which I can write to as user. There is no perceptible difference in the permissions setup for any of them. Being ntfs, all of them are owned by root of group root. It is only with this ereader that I have this problem (though I can write to it as root). So I believe the problem is specific to this device, but I have no clue what it might be.
~ $cp /home/g/MyBooks/Wyndham-TheMidwichCuckoos.txt /500gb/database/media/ cp: cannot create regular file '/500gb/database/media/Wyndham-TheMidwichCuckoos.txt': Permission denied ~$ ls -l /home/g/MyBooks/Wyndham-TheMidwichCuckoos.txt
-rw-r--r-- 1 g users 380183 Aug 10 11:04 /home/g/MyBooks/Wyndham-TheMidwichCuckoos.txt
~ $ls -l /500gb/database/ total 32 drwxr-xr-x 2 root root 8192 Aug 10 11:23 cache/ drwxr-xr-x 3 root root 8192 Aug 5 13:26 layout/ drwxr-xr-x 2 root root 8192 Aug 9 14:07 media/ drwxr-xr-x 2 root root 8192 Aug 5 17:28 sync/ ~$
The ereader is fat32, I had assumed it to be ntfs,
~ $mount /dev/sda3 on / type ext2 (rw) proc on /proc type proc (rw) sysfs on /sys type sysfs (rw) /dev/sda4 on /home type ext2 (rw) tmpfs on /dev/shm type tmpfs (rw) /dev/sda1 on /winxp type fuseblk (rw,allow_other,blksize=4096,default_permissions) /dev/sdb on /500gb type vfat (rw) /dev/sdd1 on /3tb type fuseblk (rw,allow_other,blksize=4096) ~$ cp /home/g/MyBooks/Wyndham-TheMidwichCuckoos.txt /3tb
~ $cp /home/g/MyBooks/Wyndham-TheMidwichCuckoos.txt /500gb cp: cannot create regular file '/500gb/Wyndham-TheMidwichCuckoos.txt': Permission denied ~$
I am not able to change the permissions on this ereader, neither as root, nor as user
As root, it appears to work, but nothing changes
/home/g # chmod 777 -R /500gb/database
/home/g # ls -l /500gb/database
total 32
drwxr-xr-x 2 root root 8192 Aug 10 11:23 cache
drwxr-xr-x 3 root root 8192 Aug 5 13:26 layout
drwxr-xr-x 2 root root 8192 Aug 9 14:07 media
drwxr-xr-x 2 root root 8192 Aug 5 17:28 sync
As user, the chmod is rejected thus
~ $chmod 777 -R /500gb/database chmod: changing permissions of '/500gb/database': Operation not permitted chmod: changing permissions of '/500gb/database/cache': Operation not permitted chmod: changing permissions of '/500gb/database/cache/cacheExt.xml': Operation not permitted chmod: changing permissions of '/500gb/database/cache/media.xml': Operation not permitted chmod: changing permissions of '/500gb/database/cache/cacheExtSchema_1.1.xsb': Operation not permitted chmod: changing permissions of '/500gb/database/media': Operation not permitted chmod: changing permissions of '/500gb/database/media/Stevenson-TreasureIsland.txt': Operation not permitted chmod: changing permissions of '/500gb/database/media/Rendell-WolftotheSlaughter.txt': Operation not permitted This process did the trick for me, thanks to Miroslav Koskar /home/g # mount | grep sdb /home/g # mount /dev/sdb -o uid=1000,gid=100 /500gb /home/g # mount | grep sdb /dev/sdb on /500gb type vfat (rw,uid=1000,gid=100) /home/g # ls -l /500gb/database total 32 drwxr-xr-x 2 g users 8192 Aug 10 11:23 cache drwxr-xr-x 3 g users 8192 Aug 5 13:26 layout drwxr-xr-x 2 g users 8192 Aug 10 12:51 media drwxr-xr-x 2 g users 8192 Aug 5 17:28 sync /home/g # ~$ cp /home/g/MyBooks/Wyndham-TheMidwichCuckoos.txt /500gb/database/media
~ $#### Best Answer It seems to me that permissions are not correctly set. 1. you are user g and can read a file = OK 2. but you can't create a file in a destination because the directory media is not writeable by you, only root can do that So either copy as root or change the permissions on target so that you have a write permission on media directory. As pointed out it's not possible to change permissions nor ownership on mounted NTFS filesystem. In that case there is a possibility to use proper mount options. Bellow excerpt from man mount uid=value, gid=value and umask=value Set the file permission on the filesystem. The umask value is given in octal. By default, the files are owned by root and not readable by somebody else. Example: (for /dev/sdc1 with FAT32 filesystem) Check that the device is not mounted (following should return without output) $ mount | grep sdc1
Mount the device with uid and gid option set
$sudo mount /dev/sdc1 -o uid=1000,gid=1000 /mnt Verify the mountpoint $ mount | grep sdc1
/dev/sdc1 on /mnt type vfat (rw,relatime,uid=1000,gid=1000,fmask=0022,dmask=0022,codepage=437,iocharset=iso8859-1,shortname=mixed,errors=remount-ro)
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# shake / Vibrate a body in box2d?
How do you vibrate or cause a shake effect to a body of type static or dynamic or kinematic. I tried applying forces to a dynamic body in the timeStep, which did not work, as well I tried ApplyLinearImpulse many times a sec within the timeStep, which again did not give me the result. So right now Iam experimenting with adding a static circle body as a hinge point to my rectangular body and create a revolute joint between them. May be then applying a force could result in Tension between the 2 bodies? Does anybody know about this? have any ideas please let me know?
• i'm not sure how it'll end up but you can also directly change fixtures position in every cycle a little. Jun 14 '11 at 9:58
• how would I do that? I had tried repositioning body using setTransform to the left and right a few pixels, the timeStep is too fast the changes are not visible. How would I change fixtures position....just tried googlin.....please let me know thanks. Jun 14 '11 at 10:29
• you can't directly change bodies positions, you have to change fixtures position, if you want to. body->getFixture() gives you access to body's first fixture, for the next fixtures call fixture->nextfixture(). and fixtures have setposition function that you can use to change their positions Jun 14 '11 at 10:33
• I just googled to look for setPosition in b2Fixture class, there isn't any, but b2Body class has setPosition function. As well I tried this b2Fixture *f = body->GetFixureList(); f->setPosition()..setPosition did not come up...the IDE itself is not recognizing i.e in my xcode...still very confused... Jun 14 '11 at 11:29
• Is the shake effect actually meant to be an influence on the physics simulation or are you looking for a cosmetic effect? Jun 14 '11 at 15:28
okay I just figured everything out...all I had to do was to set the linearVeolcity using boolean logic. i.e
if(counter <50)
{
if(toggle)
{
body->setLinearVelocity(5.0,0.0)
}
else
{
body->SetLinearVelocity(-5.0,0.0)
}
toggle = !toggle
}
counter++;
if(counter>50)
{
break away from being static body
}
all of the above code has to go in to the timeStep under the loop in which you iterate the b2Bodies of the world.....the logic is simple.
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Series has only been edited by one person.
Last edited on November 19, 2008 by DerekCouzens at 20:06:52
You can see the full page history.
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# Worksheet on Ratio in Lowest Term
Practice the questions given in the worksheet on ratio in lowest term. We know two or more quantities compared in a ratio must have the same units of measurement. So, the ratios have no unit. Then a ratio is expressed in its lowest term.
1. Find the ratio of the following in the simplest form:
(i) 2 dozens to 3 scores
(ii) 1 kg 200g and 1 kg 800g
(iii) 864 and 60
(iv) 1 .5 kg yo 15 g
(v) x$$^{2}$$ + 2x + 1 and x$$^{2}$$ - x -2
2. Find the ratio of the following in the simplest form:
(i) 40 ml to 2.4 l
(ii) 2$$\frac{1}{2}$$ hours to 50 minutes
(iii) \$ 1.40 and 67 cents
(iv) a$$^{3}$$ + b$$^{3}$$ and a$$^{2}$$ - b$$^{2}$$
3. Simplify the following ratios:
(i) 3$$\frac{1}{3}$$ : 2$$\frac{1}{2}$$
(ii) 2.5 : 6.5
(iii) 2$$\frac{1}{5}$$ : 1$$\frac{2}{15}$$ : $$\frac{3}{10}$$
Answers for the worksheet on ratio in lowest term are given below.
1. (i) 2 : 5
(ii) 2 : 3
(iii) 72 : 5
(iv) 100 : 1
(v) (x + 1) : (x - 2)
2. (i) 1 : 60
(ii) 3 : 1
(iii) 7 : 3
(iv) (a$$^{2}$$ - ab + b$$^{2}$$) : (a - b)
3. (i) 4 : 3
(ii) 5 : 13
(iii) 66 : 34 : 9
● Ratio and proportion
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Select Implementing a 1GHz Four-Issue Out-of-Order Execution Microprocessor in a Standard Cell ASIC Methodology Wei-Wu Hu, Ji-Ye Zhao, Shi-Qiang Zhong, Xu Yang, Elio Guidetti, and Chris Wu null Abstract （78561） PDF（pc） （541KB）（50425） This paper introduces the microarchitecture and physical implementation of the Godson-2E processor, which is a four-issue superscalar RISC processor that supports the 64-bit MIPS instruction set. The adoption of the aggressive out-of-order execution and memory hierarchy techniques help Godson-2E to achieve high performance. The Godson-2E processor has been physically designed in a 7-metal 90nm CMOS process using the cell-based methodology with some bit-sliced manual placement and a number of crafted cells and macros. The processor can be run at 1GHz and achieves a SPEC CPU2000 rate higher than 500.
Select Towards Automated Provisioning and Emergency Handling in Renewable Energy Powered Datacenters Chao Li, Rui Wang, Yang Hu, Ruijin Zhou, Ming Liu, Long-Jun Liu, Jing-Ling Yuan, Tao Li, and De-Pei Qian null 2014, 29 (4): 618-630. DOI: 10.1007/s11390-014-1454-5 Abstract （2556） PDF（pc） （3656KB）（25483） Designing eco-friendly system has been at the forefront of computing research. Faced with a growing concern about the server energy expenditure and the climate change, both industry and academia start to show high interests in computing systems powered by renewable energy sources. Existing proposals on this issue mainly focus on optimizing resource utilization or workload performance. The key supporting hardware structures for cross-layer power management and emergency handling mechanisms are often left unexplored. This paper presents GreenPod, a research framework for exploring scalable and dependable renewable power management in datacenters. An important feature of GreenPod is that it enables joint management of server power supplies and virtualized server workloads. Its interactive communication portal between servers and power supplies allows datacenter operators to perform real-time renewable energy driven load migration and power emergency handling. Based on our system prototype, we discuss an important topic: virtual machine (VM) workloads survival when facing extended utility outage and insufficient onsite renewable power budget. We show that whether a VM can survive depends on the operating frequencies and workload characteristics. The proposed framework can greatly encourage and facilitate innovative research in dependable green computing.
Select End-to-End Utilization Control for Aperiodic Tasks in Distributed Real-Time Systems Yong Liao, Xu-Dong Chen, Guang-Ze Xiong, Qing-Xin Zhu, and Nan Sang null Abstract （22385） PDF（pc） （585KB）（20407） An increasing number of {DRTS} (Distributed Real-Time Systems) are employing an end-to-end aperiodic task model. The key challenges of such {DRTS} are guaranteeing utilization on multiple processors to achieve overload protection, and meeting the end-to-end deadlines of aperiodic tasks. This paper proposes an end-to-end utilization control architecture and an {IC-EAT} (Integration Control for End-to-End Aperiodic Tasks) algorithm, which features a distributed feedback loop that dynamically enforces the desired utilization bound on multiple processors. IC-EAT integrates admission control with feedback control, which is able to dynamically determine the QoS (Quality of Service) of incoming tasks and guarantee the end-to-end deadlines of admitted tasks. Then an LQOCM (Linear Quadratic Optimal Control Model) is presented. Finally, experiments demonstrate that, for the end-to-end {DRTS} whose control matrix $\pmb G$ falls into the stable region, the {IC-EAT} is convergent and stable. Moreover, it is capable of providing better QoS guarantees for end-to-end aperiodic tasks and improving the system throughput.
Select A Comprehensive and Adaptive Trust Model for Large-Scale P2P Networks Xiao-Yong Li and Xiao-Lin Gui, Senior Member, CCF null 2009, 24 (5): 868-882. Abstract （4268） PDF（pc） （735KB）（20027） Based on human psychological cognitive behavior, a Comprehensive and Adaptive Trust (CAT) model for large-scale P2P networks is proposed. Firstly, an adaptive trusted decision-making method based on HEW (Historical Evidences Window) is proposed, which can not only reduce the risk and improve system efficiency, but also solve the trust forecasting problem when the direct evidences are insufficient. Then, direct trust computing method based on IOWA (Induced Ordered Weighted Averaging) operator and feedback trust converging mechanism based on DTT (Direct Trust Tree) are set up, which makes the model have a better scalability than previous studies. At the same time, two new parameters, confidence factor and feedback factor, are introduced to assign the weights to direct trust and feedback trust adaptively, which overcomes the shortage of traditional method, in which the weights are assigned by subjective ways. Simulation results show that, compared to the existing approaches, the proposed model has remarkable enhancements in the accuracy of trust decision-making and has a better dynamic adaptation capability in handling various dynamic behaviors of peers.
Select Middleware for Wireless Sensor Networks: A Survey Miao-Miao Wang, Jian-Nong Cao, Jing Li, and Sajal K. Das null 2008, 23 (3): 305-326 . Abstract （15720） PDF（pc） （6603KB）（18441） Wireless Sensor Networks (WSNs) have found more and more applications in a variety of pervasive computing environments. However, how to support the development, maintenance, deployment and execution of applications over WSNs remains to be a nontrivial and challenging task, mainly because of the gap between the high level requirements from pervasive computing applications and the underlying operation of WSNs. Middleware for WSN can help bridge the gap and remove impediments. In recent years, research has been carried out on WSN middleware from different aspects and for different purposes. In this paper, we provide a comprehensive review of the existing work on WSN middleware, seeking for a better understanding of the current issues and future directions in this field. We propose a reference framework to analyze the functionalities of WSN middleware in terms of the system abstractions and the services provided. We review the approaches and techniques for implementing the services. On the basis of the analysis and by using a feature tree, we provide taxonomy of the features of WSN middleware and their relationships, and use the taxonomy to classify and evaluate existing work. We also discuss open problems in this important area of research.
Select GBP-WAHSN: A Group-Based Protocol for Large Wireless Ad Hoc and Sensor Networks Jaime Lloret, Miguel Garcia, Jesus Tomás, and Fernando Boronat null Abstract （13365） PDF（pc） （8581KB）（13426） Grouping nodes gives better performance to the whole network by diminishing the average network delay and avoiding unnecessary message forwarding and additional overhead. Many routing protocols for ad-hoc and sensor networks have been designed but none of them are based on groups. In this paper, we will start defining group-based topologies, and then we will show how some wireless ad hoc sensor networks (WAHSN) routing protocols perform when the nodes are arranged in groups. In our proposal connections between groups are established as a function of the proximity of the nodes and the neighbor's available capacity (based on the node's energy). We describe the architecture proposal, the messages that are needed for the proper operation and its mathematical description. We have also simulated how much time is needed to propagate information between groups. Finally, we will show a comparison with other architectures.
Select Clustering Text Data Streams Yu-Bao Liu, Jia-Rong Cai, Jian Yin, and Ada Wai-Chee Fu null Abstract （14154） PDF（pc） （1181KB）（12563） Clustering text data streams is an important issue in data mining community and has a number of applications such as news group filtering, text crawling, document organization and topic detection and tracing etc. However, most methods are similarity-based approaches and only use the TF$*$IDF scheme to represent the semantics of text data and often lead to poor clustering quality. Recently, researchers argue that semantic smoothing model is more efficient than the existing TF$*$IDF scheme for improving text clustering quality. However, the existing semantic smoothing model is not suitable for dynamic text data context. In this paper, we extend the semantic smoothing model into text data streams context firstly. Based on the extended model, we then present two online clustering algorithms OCTS and OCTSM for the clustering of massive text data streams. In both algorithms, we also present a new cluster statistics structure named cluster profile which can capture the semantics of text data streams dynamically and at the same time speed up the clustering process. Some efficient implementations for our algorithms are also given. Finally, we present a series of experimental results illustrating the effectiveness of our technique. Cited: Baidu(75)
Select Wavelet Based Image Authentication and Recovery Rafiullah Chamlawi, Asifullah Khan, and Adnan Idris null Abstract （13246） PDF（pc） （564KB）（11319） In this paper, we propose a secure semi-fragile watermarking technique based on integer wavelet transform with a choice of two watermarks to be embedded. A self-recovering algorithm is employed, that hides the image digest into some wavelet subbands for detecting possible illicit object manipulation undergone in the image. The semi-fragility makes the scheme tolerant against JPEG lossy compression with the quality factor as low as 70\%, and locates the tampered area accurately. In addition, the system ensures more security because the embedded watermarks are protected with private keys. The computational complexity is reduced by using parameterized integer wavelet transform. Experimental results show that the proposed scheme guarantees safety of a watermark, recovery of image and localization of tampered area. Cited: Baidu(43)
Select Geometric Bone Modeling: From Macro to Micro Structures Oded Zaideman and Anath Fischer null 2010, 25 (3): 614-622. Abstract （2688） PDF（pc） （23500KB）（11310） There is major interest within the bio-engineering community in developing accurate and non-invasive means for visualizing, modeling and analyzing bone micro-structures. Bones are composed of hierarchical bio-composite materials characterized by complex multi-scale structural geometry. The process of reconstructing a volumetric bone model is usually based upon CT/MRI scanned images. Meshes generated by current commercial CAD systems cannot be used for further modeling or analysis. Moreover, recently developed methods are only capable of capturing the micro-structure for small volumes (biopsy samples). This paper examines the problem of re-meshing a 3D computerized model of bone micro-structure. The proposed method is based on the following phases: defining sub-meshes of the original model in a grid-based structure, remeshing each sub-mesh using the neural network (NN) method, and merging the sub-meshes into a global mesh. Applying the NN method to micro-structures proved to be quite time consuming. Therefore, a parallel, grid-based approach was applied, yielding a simpler structure in each grid cell. The performance of this method is analyzed, and the method is demonstrated on real bone micro-structures. Furthermore, the method may be used as the basis for generating a multi-resolution bone geometric model.
Select Simultaneous Minimization of Capacity and Conflict Misses Zhiyuan Li null Abstract （13642） PDF（pc） （748KB）（11054） Loop tiling (or loop blocking) is a well-known loop transformation to improve temporal locality in nested loops which perform matrix computations. When targeting caches that have low associativities, one of the key challenges for loop tiling is to simultaneously minimize capacity misses and conflict misses. This paper analyzes the effect of the tile size and the array-dimension size on capacity misses and conflict misses. The analysis supports the approach of combining tile-size selection (to minimize capacity misses) with array padding (to minimize conflict misses).
Select Revocable Ring Signature Dennis Y. W. Liu, Joseph K. Liu, Yi Mu, Willy Susilo and Duncan S. Wong null Abstract （15323） PDF（pc） （418KB）（10775） Group signature allows the anonymity of a real signer in a group to be revoked by a trusted party called group manager. It also gives the group manager the absolute power of controlling the formation of the group. Ring signature, on the other hand, does not allow anyone to revoke the signer anonymity, while allowing the real signer to form a group (also known as a ring) {\it arbitrarily} without being controlled by any other party. In this paper, we propose a new variant for ring signature, called {\it Revocable Ring Signature}. The signature allows a real signer to form a ring arbitrarily while allowing a set of authorities to revoke the anonymity of the real signer. This new variant inherits the desirable properties from both group signature and ring signature in such a way that the real signer will be responsible for what it has signed as the anonymity is revocable by authorities while the real signer still has the freedom on ring formation. We provide a formal security model for revocable ring signature and propose an efficient construction which is proven secure under our security model. Cited: Baidu(43)
Select A Robust and Fast Non-Local Means Algorithm for Image Denoising Yan-Li Liu, Jin Wang, Xi Chen, Yan-Wen Guo, and Qun-Sheng Peng null Abstract （13735） PDF（pc） （3844KB）（10370） In the paper, we propose a robust and fast image denoising method. The approach integrates both Non-Local means algorithm and Laplacian Pyramid. Given an image to be denoised, we first decompose it into Laplacian pyramid. Exploiting the redundancy property of Laplacian pyramid, we then perform non-local means on every level image of Laplacian pyramid. Essentially, we use the similarity of image features in Laplacian pyramid to act as weight to denoise image. Since the features extracted in Laplacian pyramid are localized in spatial position and scale, they are much more able to describe image, and computing the similarity between them is more reasonable and more robust. Also, based on the efficient Summed Square Image (SSI) scheme and Fast Fourier Transform (FFT), we present an accelerating algorithm to break the bottleneck of non-local means algorithm --- similarity computation of compare windows. After speedup, our algorithm is fifty times faster than original non-local means algorithm. Experiments demonstrated the effectiveness of our algorithm.
Select CASA: A New IFU Architecture for Power-Efficient Instruction Cache and TLB Designs Han-Xin Sun, Kun-Peng Yang, Yu-Lai Zhao, Dong Tong, and Xu Cheng null Abstract （8831） PDF（pc） （2317KB）（9954） The instruction fetch unit (IFU) usually dissipates a considerable portion of total chip power. In traditional IFU architectures, as soon as the fetch address is generated, it needs to be sent to the instruction cache and TLB arrays for instruction fetch. Since limited work can be done by the power-saving logic after the fetch address generation and before the instruction fetch, previous power-saving approaches usually suffer from the unnecessary restrictions from traditional IFU architectures. In this paper, we present CASA, a new power-aware IFU architecture, which effectively reduces the unnecessary restrictions on the power-saving approaches and provides sufficient time and information for the power-saving logic of both instruction cache and TLB. By analyzing, recording, and utilizing the key information of the dynamic instruction flow early in the front-end pipeline, CASA brings the opportunity to maximize the power efficiency and minimize the performance overhead. Compared to the baseline configuration, the leakage and dynamic power of instruction cache is reduced by 89.7\% and 64.1\% respectively, and the dynamic power of instruction TLB is reduced by 90.2\%. Meanwhile the performance degradation in the worst case is only 0.63\%. Compared to previous state-of-the-art power-saving approaches, the CASA-based approach saves IFU power more effectively, incurs less performance overhead and achieves better scalability. It is promising that CASA can stimulate further work on architectural solutions to power-efficient IFU designs.
Select CLASCN: Candidate Network Selection for Efficient Top-k Keyword Queries over Databases Jun Zhang, Zhao-Hui Peng, Shan Wang, and Hui-Jing Nie null Abstract （7265） PDF（pc） （477KB）（9868） Keyword Search Over Relational Databases (KSORD) enables casual or Web users easily access databases through free-form keyword queries. Improving the performance of KSORD systems is a critical issue in this area. In this paper, a new approach CLASCN (Classification, Learning And Selection of Candidate Network) is developed to efficiently perform top-$k$ keyword queries in schema-graph-based online KSORD systems. In this approach, the Candidate Networks (CNs) from trained keyword queries or executed user queries are classified and stored in the databases, and top-$k$ results from the CNs are learned for constructing CN Language Models (CNLMs). The CNLMs are used to compute the similarity scores between a new user query and the CNs from the query. The CNs with relatively large similarity score, which are the most promising ones to produce top-$k$ results, will be selected and performed. Currently, CLASCN is only applicable for past queries and New All-keyword-Used (NAU) queries which are frequently submitted queries. Extensive experiments also show the efficiency and effectiveness of our CLASCN approach.
Select Topology-Based Recommendation of Users in Micro-Blogging Communities Marcelo G. Armentano, Daniela Godoy, and Analia Amandi null 2012, 27 (3): 624-634. DOI: 10.1007/s11390-012-1249-5 Abstract （4509） PDF（pc） （1460KB）（9568） Nowadays, more and more users share real-time news and information in micro-blogging communities such as Twitter, Tumblr or Plurk. In these sites, information is shared via a followers/followees social network structure in which a follower will receive all the micro-blogs from the users he/she follows, named followees. With the increasing number of registered users in this kind of sites, finding relevant and reliable sources of information becomes essential. The reduced number of characters present in micro-posts along with the informal language commonly used in these sites make it difficult to apply standard content-based approaches to the problem of user recommendation. To address this problem, we propose an algorithm for recommending relevant users that explores the topology of the network considering different factors that allow us to identify users that can be considered good information sources. Experimental evaluation conducted with a group of users is reported, demonstrating the potential of the approach.
Select Server-Based Data Push Architecture for Multi-Processor Environments Xian-He Sun, Surendra Byna, and Yong Chen null Abstract （16565） PDF（pc） （543KB）（9541） Data access delay is a major bottleneck in utilizing current high-end computing (HEC) machines. Prefetching, where data is fetched before CPU demands for it, has been considered as an effective solution to masking data access delay. However, current client-initiated prefetching strategies, where a computing processor initiates prefetching instructions, have many limitations. They do not work well for applications with complex, non-contiguous data access patterns. While technology advances continue to increase the gap between computing and data access performance, trading computing power for reducing data access delay has become a natural choice. In this paper, we present a server-based data-push approach and discuss its associated implementation mechanisms. In the server-push architecture, a dedicated server called Data Push Server (DPS) initiates and proactively pushes data closer to the client in time. Issues, such as what data to fetch, when to fetch, and how to push are studied. The SimpleScalar simulator is modified with a dedicated prefetching engine that pushes data for another processor to test DPS based prefetching. Simulation results show that L1 Cache miss rate can be reduced by up to 97\% (71\% on average) over a superscalar processor for SPEC CPU2000 benchmarks that have high cache miss rates.
Select Orthogonal Methods Based Ant Colony Search for Solving Continuous Optimization Problems Xiao-Min Hu, Jun Zhang, and Yun Li null Abstract （13347） PDF（pc） （828KB）（9474） Research into ant colony algorithms for solving continuous optimization problems forms one of the most significant and promising areas in swarm computation. Although traditional ant algorithms are designed for combinatorial optimization, they have shown great potential in solving a wide range of optimization problems, including continuous optimization. Aimed at solving continuous problems effectively, this paper develops a novel ant algorithm termed continuous orthogonal ant colony'' (COAC), whose pheromone deposit mechanisms would enable ants to search for solutions collaboratively and effectively. By using the orthogonal design method, ants in the feasible domain can explore their chosen regions rapidly and efficiently. By implementing an adaptive regional radius'' method, the proposed algorithm can reduce the probability of being trapped in local optima and therefore enhance the global search capability and accuracy. An elitist strategy is also employed to reserve the most valuable points. The performance of the COAC is compared with two other ant algorithms for continuous optimization --- API and CACO by testing seventeen functions in the continuous domain. The results demonstrate that the proposed COAC algorithm outperforms the others.
Select Cryptanalysis of a Type of CRT-Based RSA Algorithms Bao-Dong Qin, Ming Li, and Fan-Yu Kong null Abstract （14323） PDF（pc） （401KB）（9413） It is well known that the Chinese Remainder Theorem (CRT) can greatly improve the performances of RSA cryptosystem in both running times and memory requirements. However, if the implementation of CRT-based RSA is careless, an attacker can reveal some secret information by exploiting hardware fault cryptanalysis. In this paper, we present some fault attacks on a type of CRT-RSA algorithms namely BOS type schemes including the original BOS scheme proposed by Bl\"{o}mer, Otto, and Seifert at CCS 2003 and its modified scheme proposed by Liu {\it et al.} at DASC 2006. We first demonstrate that if some special signed messages such as $m = 0, \pm1$ are dealt carelessly, they can be exploited by an adversary to completely break the security of both the BOS scheme and Liu {\it et al.}'s scheme. Then we present a new permanent fault attack on the BOS scheme with a success probability about 25\%. Lastly, we propose a polynomial time attack on Liu {\it et al.}'s CRT-RSA algorithm, which combines physical fault injection and lattice reduction techniques when the public exponent is short.
Select Lighting Estimation of a Convex Lambertian Object Using Redundant Spherical Harmonic Frames Wen-Yong Zhao, Shao-Lin Chen, Yuan Zheng, and Si-Long Peng null 2013, 28 (3): 454-467. DOI: 10.1007/s11390-013-1347-z Abstract （3670） PDF（pc） （23295KB）（9402） An explicit lighting estimation from a single image of Lambertian objects is influenced by two factors: data incompletion and noise contamination. Measurement of lighting consistency purely using the orthogonal spherical harmonic basis cannot achieve an accurate estimation. We present a novel signal-processing framework to represent the lighting field. We construct a redundant spherical harmonic frame with geometric symmetry on the sphere S2. Spherical harmonic frames are defined over the generating rotation matrices about symmetry axes of finite symmetry subgroups of SO(3), and the generating functions are spherical harmonic basis functions. Compared with the orthogonal spherical harmonic basis, the redundant spherical harmonic frames not only describe the multidirectional lighting distribution intuitively, but also resist the noise theoretically. Subsequently, we analyze the relationship of the irradiance to the incoming radiance in terms of spherical harmonic frames, and reconstruct the lighting function filtered by the Lambertian BRDF (bidirectional reflectance distribution function). The experiments show that the frame coefficients of spherical harmonic frames can better characterize the complex lighting environments finely and robustly.
Select Performance of IEEE 802.15.4 Clusters with Power Management and Key Exchange Fereshteh Amini, Moazzam Khan, Jelena Misic, and Hossein Pourreza null Abstract （13323） PDF（pc） （2974KB）（9252） The IEEE 802.15.4 specification is a recent low data rate wireless personal area network standard. While basic security services are provided for, there is a lack of more advanced techniques which are indispensable in modern personal area network applications. In addition, performance implications of those services are not known. In this paper, we describe a secure data exchange protocol based on the ZigBee specification and built on top of IEEE 802.15.4 link layer. This protocol includes a key exchange mechanism. We assume that all nodes are applying power management technique based on the constant event sensing reliability required by the coordinator. Power management generates random sleep times by every node which in average fairly distributes the sensing load among the nodes. Key exchange is initiated by a cluster coordinator after some given number of sensing packets have been received by the coordinator. We develop and integrate simulation model of the key exchange and power management technique into the cluster's reliable sensing function. We evaluate the impact of security function and its periodicity on cluster performance.
Select Delay and Capacity Trade-offs in Mobile Wireless Networks with Infrastructure Support Zhuo Li (李卓), Wen-Zhong Li (李文中), Member, CCF, ACM, IEEE, Song Guo (郭嵩), Senior Member, IEEE, Member, ACM, Sang-Lu Lu, (陆桑璐), Senior Member, CCF, Member, ACM, IEEE, and Dao-Xu Chen (陈道蓄), Senior Member, CCF, Member, ACM, IEEE null 2012, (2): 328-340. DOI: 10.1007/s11390-012-1226-z Abstract （4749） PDF（pc） （664KB）（9226） In this paper, we investigate the trade-offs between delay and capacity in mobile wireless networks with infrastructure support. We consider three different mobility models, independent and identically distributed (i.i.d) mobility model, random walk mobility model with constant speed and Lévy flight mobility model. For i.i.d mobility model and random walk mobility model with the speed Θ((1/√n)), we get the theoretical results of the average packet delay when capacity is Θ(1), Θ((1/√n)) individually, where n is the number of nodes. We find that the optimal average packet delay is achieved when capacity where K is the number of gateways. It is proved that average packet delay D(n) divided by capacity λ(n) is bounded below by (n/K·W) . When ω(√n) ≤ K < n, the critical average delay for capacity compared with static hybrid wireless networks is Θ((K2/n) ). Lévy flight mobility model is based on human mobility and is more sophisticated. For the model with parameter α, it is found that (D(n)/λ(n)) > O(n((1-η)·(α+1)/2) ln n) when K = O(nη) (0 ≤ η < 1). We also prove that when ω(√n) ≤ K < n, the critical average delay is Θ(n(α-1/2)·K).
Select WNN-Based Network Security Situation Quantitative Prediction Method and Its Optimization Ji-Bao Lai , Hui-Qiang Wang, Xiao-Wu Liu, Ying Liang, Rui-Juan Zheng, and Guo-Sheng Zhao null Abstract （13549） PDF（pc） （615KB）（8572） The accurate and real-time prediction of network security situation is the premise and basis of preventing intrusions and attacks in a large-scale network. In order to predict the security situation more accurately, a quantitative prediction method of network security situation based on Wavelet Neural Network with Genetic Algorithm (GAWNN) is proposed. After analyzing the past and the current network security situation in detail, we build a network security situation prediction model based on wavelet neural network that is optimized by the improved genetic algorithm and then adopt GAWNN to predict the non-linear time series of network security situation. Simulation experiments prove that the proposed method has advantages over Wavelet Neural Network (WNN) method and Back Propagation Neural Network (BPNN) method with the same architecture in convergence speed, functional approximation and prediction accuracy. What is more, system security tendency and laws by which security analyzers and administrators can adjust security policies in near real-time are revealed from the prediction results as early as possible.
Select Histogram-Based Estimation of Distribution Algorithm: A Competent Method for Continuous Optimization Nan Ding, Shu-De Zhou, and Zeng-Qi Sun null Abstract （11598） PDF（pc） （384KB）（8526） Designing efficient estimation of distribution algorithms for optimizing complex continuous problems is still a challenging task. This paper utilizes histogram probabilistic model to describe the distribution of population and to generate promising solutions. The advantage of histogram model, its intrinsic multimodality, makes it proper to describe the solution distribution of complex and multimodal continuous problems. To make histogram model more efficiently explore and exploit the search space, several strategies are brought into the algorithms: the surrounding effect reduces the population size in estimating the model with a certain number of the bins and the shrinking strategy guarantees the accuracy of optimal solutions. Furthermore, this paper shows that histogram-based EDA (Estimation of distribution algorithm) can give comparable or even much better performance than those predominant EDAs based on Gaussian models.
Select Summarizing Software Artifacts: A Literature Review Najam Nazar, Yan Hu, He Jiang null 2016, 31 (5): 883-909. DOI: 10.1007/s11390-016-1671-1 Abstract （1685） PDF（pc） （2126KB）（8515） This paper presents a literature review in the field of summarizing software artifacts, focusing on bug reports, source code, mailing lists and developer discussions artifacts. From Jan. 2010 to Apr. 2016, numerous summarization techniques, approaches, and tools have been proposed to satisfy the ongoing demand of improving software performance and quality and facilitating developers in understanding the problems at hand. Since aforementioned artifacts contain both structured and unstructured data at the same time, researchers have applied different machine learning and data mining techniques to generate summaries. Therefore, this paper first intends to provide a general perspective on the state of the art, describing the type of artifacts, approaches for summarization, as well as the common portions of experimental procedures shared among these artifacts. Moreover, we discuss the applications of summarization, i.e., what tasks at hand have been achieved through summarization. Next, this paper presents tools that are generated for summarization tasks or employed during summarization tasks. In addition, we present different summarization evaluation methods employed in selected studies as well as other important factors that are used for the evaluation of generated summaries such as adequacy and quality. Moreover, we briefly present modern communication channels and complementarities with commonalities among different software artifacts. Finally, some thoughts about the challenges applicable to the existing studies in general as well as future research directions are also discussed. The survey of existing studies will allow future researchers to have a wide and useful background knowledge on the main and important aspects of this research field.
Select ROPAS: Cross-Layer Cognitive Architecture for Mobile UWB Networks Chittabrata Ghosh, Bin Xie, and Dharma P. Agrawal null Abstract （11678） PDF（pc） （1228KB）（8324） The allocation of bandwidth to unlicensed users, without significantly increasing the interference on the existing licensed users, is a challenge for Ultra Wideband (UWB) networks. Our research work presents a novel Rake Optimization and Power Aware Scheduling (ROPAS) architecture for UWB networks. Since UWB communication is rich in multipath effects, a Rake receiver is used for path diversity. Our idea of developing an optimized Rake receiver in our ROPAS architecture stems from the intention of reducing the computation complexity in terms of the number of multiplications and additions needed for the weight derivation attached to each finger of the Rake receiver. Our proposed work uses the Cognitive Radio (CR) for dynamic channel allocation among the requesting users while limiting the average power transmitted in each sub-band. In our proposed novel ROPAS architecture, dynamic channel allocation is achieved by a CR-based cross-layer design between the PHY and Medium Access Control (MAC) layers. Additionally, the maximum number of parallel transmissions within a frame interval is formulated as an optimization problem. This optimal decision is based on the distance parameter between a transmitter-receiver pair, bit error rate and frequency of request by a particular application. Moreover, the optimization problem improvises a differentiation technique among the requesting applications by incorporating priority levels among user applications. This provides fairness and higher throughput among services with varying power constraint and data rates required for a UWB network.
Select WWW Business Applications Based on the Cellular Model Toshio Kodama, Tosiyasu L. Kunii, and Yoichi Seki null Abstract （11874） PDF（pc） （1263KB）（8181） A cellular model based on the Incrementally Modular Abstraction Hierarchy (IMAH) is a novel model that can represent the architecture of and changes in cyberworlds, preserving invariants from a general level to a specific one. We have developed a data processing system called the Cellular Data System (CDS). In the development of business applications, you can prevent combinatorial explosion in the process of business design and testing by using CDS. In this paper, we have first designed and implemented wide-use algebra on the presentation level. Next, we have developed and verified the effectiveness of two general business applications using CDS: 1) a customer information management system, and 2) an estimate system.
Select Constructing Maximum Entropy Language Models for Movie Review Subjectivity Analysis Bo Chen, Hui He, and Jun Guo null Abstract （11264） PDF（pc） （451KB）（7938） Document subjectivity analysis has become an important aspect of web text content mining. This problem is similar to traditional text categorization, thus many related classification techniques can be adapted here. However, there is one significant difference that more language or semantic information is required for better estimating the subjectivity of a document. Therefore, in this paper, our focuses are mainly on two aspects. One is how to extract useful and meaningful language features, and the other is how to construct appropriate language models efficiently for this special task. For the first issue, we conduct a Global-Filtering and Local-Weighting strategy to select and evaluate language features in a series of n-grams with different orders and within various distance-windows. For the second issue, we adopt Maximum Entropy (MaxEnt) modeling methods to construct our language model framework. Besides the classical MaxEnt models, we have also constructed two kinds of improved models with Gaussian and exponential priors respectively. Detailed experiments given in this paper show that with well selected and weighted language features, MaxEnt models with exponential priors are significantly more suitable for the text subjectivity analysis task.
Select Algebraic Construction for Zero-Knowledge Sets Rui Xue, Ning-Hui Li, and Jiang-Tao Li null Abstract （13838） PDF（pc） （421KB）（7847） Zero knowledge sets is a new cryptographic primitive introduced by Micali, Rabin, and Kilian in FOCS 2003. It has been intensively studied recently. However all the existing ZKS schemes follow the basic structure by Micali {\it et al}. That is, the schemes employ the Merkle tree as a basic structure and mercurial commitments as the commitment units to nodes of the tree. The proof for any query consists of an authentication chain. We propose in this paper a new algebraic scheme that is completely different from all the existing schemes. Our new scheme is computationally secure under the standard strong RSA assumption. Neither mercurial commitments nor tree structure is used in the new construction. In fact, the prover in our construction commits the desired set without any trapdoor information, which is another key important difference from the previous approaches.
Select Short Group Signatures Without Random Oracles Bo Qin, Qian-Hong Wu, Willy Susilo, Yi Mu, Yu-Min Wang, and Zheng-Tao Jiang null Abstract （14206） PDF（pc） （526KB）（7812） We propose {\em short} group signature (GS) schemes which are provably secure {\em without} random oracles. Our basic scheme is about 14 times shorter than the Boyen-Waters GS scheme at Eurocrypt 2006, and 42\% shorter than the recent GS schemes due to Ateniese {\em et al}. The security proofs are provided in the Universally Composable model, which allows the proofs of security valid not only when our scheme is executed in isolation, but also in composition with other secure cryptographic primitives. We also present several new computational assumptions and justify them in the generic group model. These assumptions are useful in the design of high-level protocols and may be of independent interest. Cited: Baidu(52)
Select An Efficient Clustering Algorithm for k-Anonymisation Grigorios Loukides and Jian-Hua Shao null Abstract （13337） PDF（pc） （3594KB）（7745） K-anonymisation is an approach to protecting individuals from being identified from data. Good $k$-anonymisations should retain data utility and preserve privacy, but few methods have considered these two conflicting requirements together. In this paper, we extend our previous work on a clustering-based method for balancing data utility and privacy protection, and propose a set of heuristics to improve its effectiveness. We introduce new clustering criteria that treat utility and privacy on equal terms and propose sampling-based techniques to optimally set up its parameters. Extensive experiments show that the extended method achieves good accuracy in query answering and is able to prevent linking attacks effectively.
Select AHT Bézier Curves and NUAHT B-Spline Curves Gang Xu and Guo-Zhao Wang null Abstract （9839） PDF（pc） （2756KB）（7548） In this paper, we present two new unified mathematics models of conics and polynomial curves, called {\it algebraic hyperbolic trigonometric} ({\it AHT}) {\it B\'{e}zier curves} and {\it non-uniform algebraic hyperbolic trigonometric $($NUAHT$)$ B-spline curves of order n}, which are generated over the space ${\rm span}\{\sin t,\cos t,\sinh t,\cosh t,1,t,\ldots,t^{n-5}\}$, $n\ge 5$. The two kinds of curves share most of the properties as those of the B\'{e}zier curves and B-spline curves in polynomial space. In particular, they can represent exactly some remarkable transcendental curves such as the helix, the cycloid and the catenary. The subdivision formulae of these new kinds of curves are also given. The generations of the tensor product surfaces are straightforward. Using the new mathematics models, we present the control mesh representations of two classes of minimal surfaces.
Select Engineering the Divide-and-Conquer Closest Pair Algorithm Minghui Jiang and Joel Gillespie null Abstract （10903） PDF（pc） （891KB）（7545） We improve the famous divide-and-conquer algorithm by Bentley and Shamos for the planar closest-pair problem. For $n$ points on the plane, our algorithm keeps the optimal $O(n \log n)$ time complexity and, using a circle-packing property, computes at most $7n/2$ Euclidean distances, which improves Ge {\it et al.}'s bound of $(3n\log n)/2$ Euclidean distances. We present experimental results of our comparative studies on four different versions of the divide-and-conquer closest pair algorithm and propose two effective heuristics.
Select Leakage Current Estimation of CMOS Circuit with Stack Effect Yong-Jun Xu, Zu-Ying Luo, Xiao-Wei Li, Li-Jian Li, and Xian-Long Hong null Abstract （2111） PDF（pc） （510KB）（7426） Leakage current of CMOS circuit increases dramatically with the technology scaling down and has become a critical issue of high performance system. Subthreshold, gate and reverse biased junction band-to-band tunneling (BTBT) leakages are considered three main determinants of total leakage current. Up to now, how to accurately estimate leakage current of large-scale circuits within endurable time remains unsolved, even though accurate leakage models have been widely discussed. In this paper, the authors first dip into the stack effect of CMOS technology and propose a new simple gate-level leakage current model. Then, a table-lookup based total leakage current simulator is built up according to the model. To validate the simulator, accurate leakage current is simulated at circuit level using popular simulator HSPICE for comparison. Some further studies such as maximum leakage current estimation, minimum leakage current generation and a high-level average leakage current macromodel are introduced in detail. Experiments on ISCAS85 and ISCAS89 benchmarks demonstrate that the two proposed leakage current estimation methods are very accurate and efficient.
Select New Sealed-Bid Electronic Auction with Fairness, Security and Efficiency Chia-Chi Wu, Chin-Chen Chang, and Iuon-Chang Lin null Abstract （10436） PDF（pc） （304KB）（7324） Electronic sealed-bid auction schemes usually have a common drawback, the third party (auction host) can conspire with a malicious bidder to leak all bidding prices before the opening stage. It results in the malicious bidder wining the auction with an optimal bidding price. Recently, Liaw {\it et al}. proposed an auction protocol for electronic online bidding in which they designed a deposit deduction certification for government procurement. However, it also has above mentioned flaw. Moreover, we further found that there were some extra security drawbacks in their protocol. First, the bidder can forge a bidding receipt to claim that he/she is a valid auction winner. Second, it may suffer from the third party forging attack. Third, their protocol leaked some bidders' private information to the third party, such as the bidder's bank account number and the authorization code. Thus, it cannot protect the bidder's privacy at all. In this paper, we not only point out the drawbacks from the previous scheme but also propose a new electronic auction scheme to overcome the above mentioned drawbacks. Furthermore, the computational complexity can be decreased in our online sealed-bid auction scheme.
Select A Cloud-Based BPM Architecture with User-End Distribution of Non-Compute-Intensive Activities and Sensitive Data Yan-Bo Han (韩燕波), Jun-Yi Sun (孙君意), Gui-Ling Wang (王桂玲) and Hou-Fu Li (李厚福) null 2010, 25 (6): 1157-1167. DOI: 10.1007/s11390-010-1092-5 Abstract （4968） PDF（pc） （652KB）（7302） While cloud-based BPM (Business Process Management) shows potentials of inherent scalability and expenditure reduction, such issues as user autonomy, privacy protection and efficiency have popped up as major concerns. Users may have their own rudimentary or even full-fledged BPM systems, which may be embodied by local EAI systems, at their end, but still intend to make use of cloud-side infrastructure services and BPM capabilities, which may appear as PaaS (Platform-as-a-Service) services, at the same time. A whole business process may contain a number of non-compute-intensive activities, for which cloud computing is over-provision. Moreover, some users fear data leakage and loss of privacy if their sensitive data is processed in the cloud. This paper proposes and analyzes a novel architecture of cloud-based BPM, which supports user-end distribution of non-compute-intensive activities and sensitive data. An approach to optimal distribution of activities and data for synthetically utilizing both user-end and cloud-side resources is discussed. Experimental results show that with the help of suitable distribution schemes, data privacy can be satisfactorily protected, and resources on both sides can be utilized at lower cost.
Select Efficient Optimization of Multiple Subspace Skyline Queries Zhen-Hua Huang, Jian-Kui Guo, Sheng-Li Sun, and Wei Wang null Abstract （9023） PDF（pc） （546KB）（7177） We present the first efficient sound and complete algorithm (i.e., AOMSSQ) for optimizing multiple subspace skyline queries simultaneously in this paper. We first identify three performance problems of the na\'\i ve approach (i.e., SUBSKY) which can be used in processing arbitrary single-subspace skyline query. Then we propose a cell-dominance computation algorithm (i.e., CDCA) to efficiently overcome the drawbacks of SUBSKY. Specially, a novel pruning technique is used in CDCA to dramatically decrease the query time. Finally, based on the CDCA algorithm and the share mechanism between subspaces, we present and discuss the AOMSSQ algorithm and prove it sound and complete. We also present detailed theoretical analyses and extensive experiments that demonstrate our algorithms are both efficient and effective.
Select Malware-Propagative Mobile Ad Hoc Networks: Asymptotic Behavior Analysis Vasileios Karyotis, Anastasios Kakalis, and Symeon Papavassiliou null Abstract （12599） PDF（pc） （541KB）（7013） In this paper, the spreading of malicious software over ad hoc networks, where legitimate nodes are prone to propagate the infections they receive from either an attacker or their already infected neighbors, is analyzed. Considering the Susceptible-Infected-Susceptible (SIS) node infection paradigm we propose a probabilistic model, on the basis of the theory of closed queuing networks, that aims at describing the aggregated behavior of the system when attacked by malicious nodes. Because of its nature, the model is also able to deal more effectively with the stochastic behavior of attackers and the inherent probabilistic nature of the wireless environment. The proposed model is able to describe accurately the asymptotic behavior of malware-propagative large scale ad hoc networking environments. Using the Norton equivalent of the closed queuing network, we obtain analytical results for its steady state behavior, which in turn is used for identifying the critical parameters affecting the operation of the network. Finally, through modeling and simulation, some additional numerical results are obtained with respect to the behavior of the system when multiple attackers are present, and regarding the time-dependent evolution and impact of an attack.
Select A Protocol for a Private Set-Operation Rong-Hua Li and Chuan-Kun Wu null Abstract （11270） PDF（pc） （340KB）（6885） A new private set-operation problem is proposed. Suppose there are $n$ parties with each owning a secret set. Let one of them, say $P$, be the leader, $S$ be $P$'s secret set, and $t$ (less than $n-1$) be a threshold value. For each element $w$ of $S$, if $w$ appears more than $t$ times in the rest parties' sets, then $P$ learns which parties' sets include $w$, otherwise $P$ cannot know whether $w$ appears in any party's set. For this problem, a secure protocol is proposed in the semi-honest model based on semantically secure homomorphic encryption scheme, secure sharing scheme, and the polynomial representation of sets. The protocol only needs constant rounds of communication.
Select An Improved HEAPSORT Algorithm with n log n - 0.788928n Comparisons in the Worst Case Xiao-Dong Wang and Ying-Jie Wu null Abstract （14288） PDF（pc） （302KB）（6737） A new variant of HEAPSORT is presented in this paper. The algorithm is not an internal sorting algorithm in the strong sense, since extra storage for $n$ integers is necessary. The basic idea of the new algorithm is similar to the classical sorting algorithm HEAPSORT, but the algorithm rebuilds the heap in another way. The basic idea of the new algorithm is it uses only one comparison at each node. The new algorithm shift walks down a path in the heap until a leaf is reached. The request of placing the element in the root immediately to its destination is relaxed. The new algorithm requires about $n \log n-0.788928n$ comparisons in the worst case and $n\log n-n$ comparisons on the average which is only about $0.4n$ more than necessary. It beats on average even the clever variants of QUICKSORT, if $n$ is not very small. The difference between the worst case and the best case indicates that there is still room for improvement of the new algorithm by constructing heap more carefully.
Select Progress and Challenge of Artificial Intelligence Zhong-Zhi Shi and Nan-Ning Zheng null Abstract （8286） PDF（pc） （438KB）（6689） Artificial Intelligence (AI) is generally considered to be a subfield of computer science, that is concerned to attempt simulation, extension and expansion of human intelligence. Artificial intelligence has enjoyed tremendous success over the last fifty years. In this paper we only focus on visual perception, granular computing, agent computing, semantic grid. Human-level intelligence is the long-term goal of artificial intelligence. We should do joint research on basic theory and technology of intelligence by brain science, cognitive science, artificial intelligence and others. A new cross discipline intelligence science is undergoing a rapid development. Future challenges are given in final section.
ISSN 1000-9000(Print) 1860-4749(Online) CN 11-2296/TP Home Editorial Board Author Guidelines Subscription Journal of Computer Science and Technology Institute of Computing Technology, Chinese Academy of Sciences P.O. Box 2704, Beijing 100190 P.R. China Tel.:86-10-62610746 E-mail: jcst@ict.ac.cn
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# Computing probability distributions over bootstrap samples for two statistics
I have a data set $$x= c(0.9575,0.4950,0.1080,0.9359,0.6326)$$ and I'm trying to compute the probability distributions for the statistics $$\bar X^* - \bar X$$ and $$\sqrt n(\bar X^* - \bar X)/s^*$$, over all bootstrap samples of size 5 (the same size as $$x$$).
My approach, via R, is to iterate over all $$5^5$$ possible bootstrap samples, compute the values of the statistics in each case*, and then count up the unique values of the statistics and add up their probabilities (which are $$y/5^5$$, where $$y$$ is the number of times a given statistic value appears in the "big" list of statistic values of length $$5^5$$).
See below for my work.
*Note that the second statistic has some cases involving division by zero, so I have an if statement in my code to avoid that.
Questions:
1. Have I correctly programmed what I set out to do?
2. I'd like to improve my conceptual understanding of the difference between $$\bar X^* - \bar X$$ and $$\sqrt n(\bar X^* - \bar X)/s^*$$. My suspicion is that the difference between the two statistics is a function of the (variability in the) original data set, and bootstrap sample size, and that nothing can be said in general about whether one tends to have more variance than the other. Am I wrong about this?
Code for $$\bar X^* - \bar X$$:
x= c(0.9575,0.4950,0.1080,0.9359,0.6326)
xb=mean(x)
val=rep(0,5^5)
ns=0
for(i in 1:5){
for(j in 1:5){
for(k in 1:5){
for(l in 1:5){
for(m in 1:5){
xst =c(x[i],x[j],x[k],x[l],x[m])
ns=ns+1
val[ns] = mean(xst)-xb
}
}
}
}
}
vuniq = sort(unique(val))
probability = rep(0.0,length(vuniq))
count=0
for(j in 1:3125){
for (i in 1:length(vuniq)){
if(val[j] == vuniq[i]){
probability[i]=probability[i]+1.0/3125.0
count=count+1
}
}
}
probability = probability/3125.0
plot(vuniq,probability,type='h',main="Distribution of Bootstrap Mean\n minus Sample Mean",xlab="Statistic (Bootstrap Mean minus Sample Mean)",ylab="Probability (Mass)")
Graph for $$\bar X^* - \bar X$$:
Code for $$\sqrt n(\bar X^* - \bar X)/s^*$$:
x= c(0.9575,0.4950,0.1080,0.9359,0.6326)
xb=mean(x)
sqrt5 = sqrt(5)
val=rep(0,5^5)
ns=0
for(i in 1:5){
for(j in 1:5){
for(k in 1:5){
for(l in 1:5){
for(m in 1:5){
xst =c(x[i],x[j],x[k],x[l],x[m])
ns=ns+1
if (sd(xst) == 0) {
next
}
val[ns] = sqrt5*(mean(xst)-xb)/sd(xst)
}
}
}
}
}
vuniq = sort(unique(val))
probability = rep(0.0,length(vuniq))
count=0
for(j in 1:3125){
for (i in 1:length(vuniq)){
if(val[j] == vuniq[i]){
probability[i]=probability[i]+1.0/3125.0
count=count+1
}
}
}
probability = probability/3125.0
plot(vuniq,probability,type='h',main="Distribution of Difference of Means,\n Scaled by Square Root\n of Bootstrap Variance over Sample Size",xlab="Statistic (Bootstrap Mean minus Sample Mean), Scaled",ylab="Probability (Mass)")
Graph for $$\sqrt n(\bar X^* - \bar X)/s^*$$:
• What is the goal? How are you accounting for the fact that the bootstrap distribution may disagree with the sampling distribution? Sep 28, 2021 at 12:03
1. Your loop covers the different possibilities indeed. However, bootstrapping allows you to do more since it imitates how samples are drawn from the population. For instance, it can simulate the “with replacement” aspect of the process. There are some R functions that ease your work and allow you to create more "random" sample. For instance, the function sample:
x <- c(0.9575, 0.4950, 0.1080, 0.9359, 0.6326)
X <- x
for(i in 1:10000) {
X <- c(X, mean(sample(x, replace=TRUE)))
}
hist(X)
or the boot package:
x <- c(0.9575, 0.4950, 0.1080, 0.9359, 0.6326)
library(boot)
myFunc <- function(data, i){
return(mean(data[i]))
}
bootMean <- boot(x , statistic=myFunc, R=10000)
hist(bootMean\$t)
1. The notation you are using is not "standard" but I understand that you want to better grasp the t-test. In its formula $$\frac{\bar X-\mu_0}{S/\sqrt{n}}$$, it is important to understand that $$\bar X$$ represents the random variable of sampling distribution of the sample means, while $$\mu_0$$ is fixed (which corresponds to our null hypothesis). The denominator $$S/\sqrt{n}$$ is the standard error of the mean which measures the variability of sample means in the sampling distribution of means. For your understanding, it is better not to manipulate the numerator and the denominator and keep them as is.
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{}
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# A continuations-based approach to weak definites
Author
Alice Forehand
AbstractSince their adaptation from a tool of computer science to one of linguistics, continuations have been applied to a wide variety of natural language phenomena. Here we expand upon these efforts to give an account of the class of phrases known as weak definites:'' nominals that appear with a definite article but do not set up an individual discourse referent. We draw upon Aguilar-Guevara and Zwarts' formalization of weak definites (2010) as reference to kinds to develop two operators that can be applied within the continuation-based grammar presented by Barker & Shan (2013) to produce weak readings.
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# [IPython-dev] Markdown MathJaX and Serif
Fernando Perez fperez.net at gmail.com
Mon May 5 21:09:39 EDT 2014
To the best of my understanding and some rather hasty googling, mathjax
doesn't provide sans-serif mathematical fonts.
I'm also pretty sure that in Beamer, in most themes even though it uses
sans-serif CM variants, it still renders the math using serifed CM (or at
least it used to years ago). You have to switch to special packages to get
sans-serifed math, such as the cmbright one.
In summary, the business of getting proper sans-serif math rendering is
highly non-trivial, and ultimately it's a question for the MathJax lists.
If mathjax supports it, we can too simply because we use unadulterated
mathjax. But if it's not something that mathjax can do by itself, IPython
isn't going to help...
Cheers
f
On Tue, May 6, 2014 at 12:44 AM, Sylvain Corlay <sylvain.corlay at gmail.com>wrote:
> Hi all,
>
> We traditionally use sans-serif fonts for text meant to be displayed on a
> screen while serif is generally used for print. For example, default
> latex-beamer themes use sans-serif flavors of the computer modern font.
> Shouldn't Mathjax do the same, especially in the context of the IPython
> notebook, where the default fonts for markdown are sans-serif?
>
> Best,
> Sylvain
>
>
> _______________________________________________
> IPython-dev mailing list
> IPython-dev at scipy.org
> http://mail.scipy.org/mailman/listinfo/ipython-dev
>
>
--
Fernando Perez (@fperez_org; http://fperez.org)
fperez.net-at-gmail: mailing lists only (I ignore this when swamped!)
fernando.perez-at-berkeley: contact me here for any direct mail
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# Tag Info
## New answers tagged pandoc
0
Using semicolons in between identifiers instead of writing them side by side does it eg: [@author_journal_year; @author2_journal2_year2]
0
Use the option --extract-media=pandocConversionMedia to specify a directory pandocConversionMedia where the images will be saved. This will be created by pandoc. So the call would be: pandoc -f docx -t latex --extract-media=pandocConversionMedia -o pandocOutputWithMedia.tex input.docx
0
I finally solved this issue after a few hours of thinking. The key idea is to somehow get the numbers that pandoc could have used if it could convert the citations and replace them in .tex after pandoc processing! The steps will be find all usage of citations in the paper Put a mapping of citation key --> [@citation key] at the end of markdown Run pandoc ...
0
The correct solution is \usepackage{float} \let\origfigure=\figure \let\endorigfigure=\endfigure \renewenvironment{figure}[1][]{% \begin{figure*} }{% \end{figure*} }
0
An alternative is to use the pandoc-eqnos filter, which processes attributed formulas. e.g., $$(K_{0}^{-1}x)^{T}E(K_{1}^{-1}x')=0$$ {#eq:foo} When pandoc's output format is set to LaTeX or pdf, attributed formulas are converted by pandoc-eqnos to numbered LaTeX equations. The equation may be referenced as Eq. @eq:foo. To get the numbering style you ...
0
Indeed, there is almost a perfect way to do so. But you have to pay for that, the solution is Tex2Word. In order to get the best results, firstly you change to the the basic document class, e.g. article and avoid using self-defined styles. If you are using bibtex, then you just copy the content in bbl file to the tex file. Finally, open your tex file with MS ...
Top 50 recent answers are included
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# Continuing list item after sublist in Lyx
I'm trying to continue a list item after existing a sublist, like so:
• Content
1. sublist
2. sublist
Continuing CONTENT...
• bla bla
For some reason when trying to write the Continuing CONTENT
I either had the option to continue the list (as in - creating another sublist item) or creating another list item (like bla bla)
Any ideas?
-
This is perfectly possible. You just have to switch the paragraph type of the unwanted "3." enum item back to "Standard" to continue an item of an enclosing list:
Here is the code that LyX 2.0 generates out of this:
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\begin{document}
\begin{itemize}
\item Content
\begin{enumerate}
\item sublist
\item sublist
\end{enumerate}
Continuing CONTENT
\item bla bla
\end{itemize}
\end{document}
-
inside the created itemize you can go with the TAB key into a deeper one and with the SHIFT TAB key into a higher level. After "2. sublist" insert SHIFT TAB and then you are on the first level but with a new bullet. Then click on the default paragraph icon and you are in the normal paragraph mode. Now hit TAB again and you can write on the same level.
-
If you want to continue without a bullet you have to use the TeX mode to insert the deeper level of itemize. @Herbert: That is not true (even though badly documented), see my answer. – Daniel Nov 8 '11 at 10:10
Then hit again SHIFT TAB and you are in the normal paragraph mode. Now hit TAB again and you can write on the same level. @Herbert: Have you tested this? At least with Mac LyX 2.0.1 it does not work (SHIFT TAB does not get you back in normal paragraph mode). – Daniel Nov 8 '11 at 10:38
you are right, SHIFT TAB works only if you want back from the level without a bullet to paragraph mode – Herbert Nov 8 '11 at 10:41
|
{}
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# [OS X TeX] Re: Re: personnal packages : thanks (Herbert Schulz)
Sat Jan 8 11:32:30 EST 2011
```Thank you.
I was so preoccupied about the whole thing that I forgot the basic stuff ! It works.
Karine
Le 7 janv. 2011 à 21:00, <macosx-tex-request at email.esm.psu.edu> <macosx-tex-request at email.esm.psu.edu> a écrit :
> Send MacOSX-TeX mailing list submissions to
> macosx-tex at email.esm.psu.edu
>
> To subscribe or unsubscribe via the World Wide Web, visit
> http://email.esm.psu.edu/mailman/listinfo/macosx-tex
> or, via email, send a message with subject or body 'help' to
> macosx-tex-request at email.esm.psu.edu
>
> You can reach the person managing the list at
> macosx-tex-owner at email.esm.psu.edu
>
> than "Re: Contents of MacOSX-TeX digest..."
>
>
> Today's Topics:
>
> 1. Re: Syncing in one direction only (Don Green Dragon)
> 2. Re: Syncing in one direction only (Alain Schremmer)
> 3. Re: Syncing in one direction only (Herbert Schulz)
> 4. Re: Syncing in one direction only (Alain Schremmer)
> 5. Math font samples (Michael Sharpe)
> 6. personnal packages : thanks (Karine Fourlon)
> 7. Re: personnal packages : thanks (Herbert Schulz)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Thu, 6 Jan 2011 15:51:57 -0700
> From: Don Green Dragon <fergdc at Shaw.ca>
> Subject: Re: [OS X TeX] Syncing in one direction only
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Message-ID: <DE1ECE89-B814-4306-94C1-478216728584 at Shaw.ca>
> Content-Type: text/plain; charset=us-ascii
>
> Hello Herb,
>
>
> On 2011-05Jan-, at 5:12 AM, Herbert Schulz wrote:
>
>> On Jan 4, 2011, at 10:51 PM, Don Green Dragon wrote:
>> <<snip>>
>>
>> Howdy,
>>
>> There are two texmf.cnf files. Although I don't recommend touching either one the one in /usr/local/texlive/2010/ is the only one that could be touched since it contains any changes in configuration from the standard base configuration for the local system. The main texmf.cnf file is in /usr/local/texlive/2010/texmf-dist/ and is read in first. It has the lines you are interested in changing (by adding corrected lines to the other texmf.cnf file) so examine its contents.
>
> Ok, thanks for that information.
>
>
>> Do understand that changes you make to texmf.cnf may have unintended consequences and especially to security of your TeX Distribution and system.
>
> No, I did not so understand. On the other hand, I had no intention of modifying <texmf.cnf> until I knew more about the situation. I have to find that message from Bruno Voisin that Alain Schremmer referred to.
>
>
> Don Green Dragon
> fergdc at Shaw.ca
>
>
> ------------------------------
>
> Message: 2
> Date: Thu, 6 Jan 2011 18:52:26 -0500
> From: Alain Schremmer <schremmer.alain at gmail.com>
> Subject: Re: [OS X TeX] Syncing in one direction only
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Message-ID: <13DE5502-EEEF-48D3-A6C8-27DBC69303B3 at gmail.com>
> Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed
>
>
> On Jan 6, 2011, at 5:51 PM, Don Green Dragon wrote:
>
>> No, I did not so understand. On the other hand, I had no intention
>> of modifying <texmf.cnf> until I knew more about the situation. I
>> have to find that message from Bruno Voisin that Alain Schremmer
>> referred to.
>
>
>
> Look at the thread Error: I can't write on file '(name)'
>
> March 22, 2007
>
> Here is, I think, what Voisin said to do and what I did.
> Regards
> --schremmer
>
>
> Le 22 mars 07 à 12:45, Jonathan Kew a écrit :
>
>> Find the line
>>
>> openout_any = p
>>
>> in there; change to
>>
>> openout_any = r
>>
>> and I think you'll be OK. ("a" would be even more permissive than
>> "r", but I don't think you need that.)
>
> Thanks for this tip, that's helpful. I knew about the -R option (=
> secure mode) of dvips, and about the -shell-escape and -no-shell-
> escape options of pdftex, but I didn't suspect there was this switch
> in addition for pdftex in texmf.cnf.
>
> Regarding texmf.cnf, the gwTeX and MacTeX/TeXLive setups are a bit
> different. With MacTeX/TeXLive, there is one single texmf.cnf at:
>
> /usr/local/texlive/2007/texmf/web2c/texmf.cnf
>
> which is the file to edit. With gwTeX, there are two different
> texmf.cnf at:
>
> /usr/local/gwTeX/texmf.cnf
> /usr/local/gwTeX/texmf/web2c/texmf.cnf
>
> The second file is the TeXLive default, and the first file is where
> local modifications are kept. The two are read in sequence, with any
> definition in the first superseding definitions read later in the
> second.
>
> Hence, for MacTeX/TeXLive simply *edit* the unique /usr/local/texlive/
> 2007/texmf/web2c/texmf.cnf as said above, while for gwTeX you need to
>
> openout_any = r
>
> to /usr/local/gwTeX/texmf.cnf.
>
> Finally, to Alain, regarding the editor, in case you don't have
> TextWrangler you can simply use pico. pico is a stand-alone version
> of the text editor of the pine mail reader, and it's "intuitive"-
> enough to use (on second thought, I think pico in OS X points
> actually to nano, a GNU clone of pico -- those physicists, they just
> couldn't resist ;-).
>
> Simply type in Terminal, using MacTeX as an example:
>
> sudo pico /usr/local/texlive/2007/texmf/web2c/texmf.cnf
>
> then use Ctrl-V to move down screen-by-screen until you reach the
> desired part (you can also use the DownArrow key to move down line-by-
> line):
>
> % Allow TeX \openin, \openout, or \input on filenames starting with `.'
> % (e.g., .rhosts) or outside the current tree (e.g., /etc/passwd)?
> % a (any) : any file can be opened.
> % r (restricted) : disallow opening "dotfiles".
> % p (paranoid) : as 'r' and disallow going to parent directories, and
> % restrict absolute paths to be under \$TEXMFOUTPUT.
> openout_any = p
> openin_any = a
>
> then modify the openout_any setting as required, and then type in
> Ctrl-O to save and Ctrl-X to quit.
>
> The bad thing is that texmf.cnf will be overwritten each time TeX is
> updated, so that you'll have to redo your modification each time.
> Actually it's simpler with gwTeX, as installed by i-Installer: at
> each update, I think, i-Installer detects the local /usr/local/gwTeX/
> texmf.cnf has been changed and offers to save a copy on your Desktop
> (with the date appended at the end of the name) before performing the
> update. Then you'll simply have to merge back your modifications
> after the update.
>
> Hope this helps,
>
> Bruno Voisin
>
>
>
> ------------------------------
>
> Message: 3
> Date: Thu, 6 Jan 2011 19:21:59 -0600
> From: Herbert Schulz <herbs at wideopenwest.com>
> Subject: Re: [OS X TeX] Syncing in one direction only
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Message-ID: <B5049E3A-98C8-48D2-A904-05EBDD9B4FEA at wideopenwest.com>
> Content-Type: text/plain; charset=iso-8859-1
>
>
> On Jan 6, 2011, at 5:52 PM, Alain Schremmer wrote:
>
>>
>> On Jan 6, 2011, at 5:51 PM, Don Green Dragon wrote:
>>
>>> No, I did not so understand. On the other hand, I had no intention of modifying <texmf.cnf> until I knew more about the situation. I have to find that message from Bruno Voisin that Alain Schremmer referred to.
>>
>>
>>
>> Look at the thread Error: I can't write on file '(name)'
>>
>> March 22, 2007
>>
>> Here is, I think, what Voisin said to do and what I did.
>> Regards
>> --schremmer
>>
>>
>> Le 22 mars 07 à 12:45, Jonathan Kew a écrit :
>>
>>> Find the line
>>>
>>> openout_any = p
>>>
>>> in there; change to
>>>
>>> openout_any = r
>>>
>>> and I think you'll be OK. ("a" would be even more permissive than "r", but I don't think you need that.)
>>
>> Thanks for this tip, that's helpful. I knew about the -R option (= secure mode) of dvips, and about the -shell-escape and -no-shell-escape options of pdftex, but I didn't suspect there was this switch in addition for pdftex in texmf.cnf.
>>
>> Regarding texmf.cnf, the gwTeX and MacTeX/TeXLive setups are a bit different. With MacTeX/TeXLive, there is one single texmf.cnf at:
>>
>> /usr/local/texlive/2007/texmf/web2c/texmf.cnf
>>
>> which is the file to edit. With gwTeX, there are two different texmf.cnf at:
>> ...
>
> Howdy,
>
> That's just plain wrong! There are two texmf.cnf files. The base file, with the initial settings and most of the information one needs, is in /usr/local/texlive/2010/texmf/web2c while the one to edit is in /usr/local/texlive/2010.
>
>> ...
>> Hence, for MacTeX/TeXLive simply *edit* the unique /usr/local/texlive/2007/texmf/web2c/texmf.cnf as said above, while for gwTeX you need to *add*:
>>
>> openout_any = r
>>
>> to /usr/local/gwTeX/texmf.cnf.
>> ...
>
> No, no no! Edit the one in /usr/local/texlive/2010 if you want to make changes. Add the changed lines and they will override the ones in the base file.
>
> Good Luck,
>
> Herb Schulz
> (herbs at wideopenwest dot com)
>
>
>
>
>
> ------------------------------
>
> Message: 4
> Date: Fri, 7 Jan 2011 00:11:44 -0500
> From: Alain Schremmer <schremmer.alain at gmail.com>
> Subject: Re: [OS X TeX] Syncing in one direction only
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Message-ID: <C488C3CD-A70E-4B4E-8B0E-479918128CFC at gmail.com>
> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed
>
>
> On Jan 6, 2011, at 8:21 PM, Herbert Schulz wrote:
>>
>> Howdy,
>>
>> That's just plain wrong! There are two texmf.cnf files. The base
>> file, with the initial settings and most of the information one
>> needs, is in /usr/local/texlive/2010/texmf/web2c while the one to
>> edit is in /usr/local/texlive/2010.
>>
>>> ...
>>> Hence, for MacTeX/TeXLive simply *edit* the unique /usr/local/
>>> texlive/2007/texmf/web2c/texmf.cnf as said above, while for gwTeX
>>> you ne```
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{}
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# Proof that two norms $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ are equivalent
1. Two norms $\def\norm#1{\lVert#1\rVert}\norm\cdot_1$ and $\norm\cdot_2$ are equivalent iff $\;\exists\;c_1,c_2>0$ such that $c_1\norm x_1\le \norm x_2\le c_2\norm x_1$
Show that $\norm x_1=\sum_{i=1}^n \lvert x_i \rvert$ and $\norm x_2=\left ( \sum_{i=1}^n x_i^2 \right )^{1/2}$ are equivalent.
It looks like $c_2$ is $1$, and that this can be proven with induction. But what could $c_1$ be?
edit
I mean what I don't understand is the following: If I square both terms, and expand $(\sum_{i=1}^n \lvert x_i \rvert)^2 = \left ( \sum_{i=1}^n x_i^2 \right ) + \sum_{i=1}^n x_i(x_k+\dotsb+x_n)_{x_k\neq x_i}$. However, the second term grows with $n$, so how can $\frac{\norm x_1}{\norm x_2} \leq C$ at all?
• Well, I think that $n$ is kept fixed. – Siminore Nov 5 '13 at 8:49
You want to show that for a fixed $n$ the norms $\|\|_1,\|\|_2$ are equivalent in $\mathbb R^n$. The constant $c_1$ will depend on $n$.
Let $x=x_i$ for all $i$. Then $||x||_1=x\dim x$, and $||x||_2=\sqrt{\sum_{i=1}^{\dim x}x^2}=\sqrt{x^2\dim x}=x\sqrt{\dim x}$. Therefore $$\lim_{\dim x\to\infty} \frac{||x||_1}{||x||_2}=\sqrt{\dim x}=\infty$$ So the two cannot be equivalent by your definition.
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Maggie Black
2023-02-22
What is the vertex form of $y={x}^{2}-4x-3$?
Caltolmsn
1. The visible spectrum shows that blue has a shorter wavelength than red, which has a longer wavelength.
2. Water molecules absorb longer wavelength light more efficiently than shorter wavelength light.
3. Blue light is having a shorter wavelength, and the absorption of blue color by water will be less.
4. Water will absorb all other light with longer wavelengths.
5. According to Rayleigh criteria, light having a shorter wavelength will scatter more.
6. Blue color absorption by water is lower, and blue color scattering is higher.
7. Therefore, the sea looks blue.
Do you have a similar question?
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### Bayesian model
#### machine learning bayesian model bayesian linear regression statistics
A Bayesian model is a statistical model where we use probability to represent all uncertainty within the model, both the uncertainty regarding the output but also the uncertainty regarding the input to the model.
The basic idea in a Bayesian model is that we start by assuming something about whatever we are investigating. This is called the prior distribution. Then we adjust that based on our data.
We will understand Bayesian model by exploring Bayesian Linear Regression which is a modification of the Linear regression model using the Bayesian principles.
### Bayesian Linear Regression:
Bayesian linear regression is an approach to linear regression in which the statistical analysis is undertaken within the context of Bayesian inference. When the regression model has errors that have a normal distribution, and if a particular form of prior distribution is assumed, explicit results are available for the posterior probability distributions of the model's parameters.
Difference between normal linear regression and bayesian Linear Regression
Normal linear regression is a frequentist approach, and it assumes that there are enough measurements to say something meaningful.
In the Bayesian approach, the data are supplemented with additional information in the form of a prior probability distribution. The prior belief about the parameters is combined with the data's likelihood function according to Bayes theorem to yield the posterior belief about the parameters.
### Why Bayesian version?
• Bayesian hierarchical models are easy to extend to many levels
• Bayesian model selection probably superior (BIC/AIC)
• Bayesian models are more flexible so it can handle more complex models
• Bayesian analysis are more accurate in small samples
• Bayesian models can incorporate prior information
# Bayesian Inference
Let us understand this with the help of an example suppose:
there is a company that makes biased coins i.e. they adjust the weight of the coins in such a way that the one side of the coin is more likely than the other while tossing. For now let's assume that they make just two types of coins. ‘Coin 1’ with 30% chances to get heads and ‘Coin 2’ with 70% chances to get heads. Now, we have got a coin from this factory and want to know if it is ‘Coin 1’ or ‘Coin 2’. We have been also given that the company makes both the coins in same quantity. This will help us define our prior probability for the problem that is our coin is equally likely to be either ‘Coin 1’ or ‘Coin 2’ or 50-50 chance.
After assigning prior probability, we have tossed the coin 3 times and got heads in all three trials. The Frequentist approach will ask us to take more samples since we cannot accept or refuse the null hypothesis with this sample size at 95% confidence interval. However, as we will see with Bayesian approach each packet of information or trial will modify the prior probability or our belief for the coin to be ‘Coin 2’.
At this point we know the original priors for the coins i.e
P(Coin1)=P(Coin2)=0.5(50%)
Additionally we also know the conditional probability i.e. chances of heads for Coin 1 and Coin 2
Now we have performed our first experiment or trial and got heads. This is a new information for us and this will help calculate the posterior probability of coins when heads has happened.
If we insert the values to the above formula the chances for Coin1 have gone down
Similarly chances for Coin 2 have gone up
This same experiment is shown in the figure below:
As mentioned in the above figure the posterior probabilities of the first toss i.e. P(Coin 1|Heads) and P(Coin 2|Heads) will become the prior probabilities for the subsequent experiment. In the next two experiments we have further got 2 more heads this will further modify the priors for the fourth experiment as shown below. Each packet of information is improving your belief about the coin as shown below:
By the end of the 3rd experiment we are almost 90% sure that we have a coin that resembles Coin 2. The above example is a special case since we have considered just two coins. However, for most practical problems we will have factories that produce coins of all different biases. In such cases the prior probabilities will be a continuous distribution.
# Summary
In this article we learned about Bayesian model and bayesian linear regression. We also studied that why do we go for bayesian version instead of normal one.At the end we studied in detail about bayesian inference.
In next article we will study about Naive Bayes theorem.
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1. ## Hello, I need some help please
Question is:
Jack has $18000 to invest , he invests 2/3 in one account and the rest in the second account. The first account returned 2% more than the second account. The total return on his investment was$1050. What were the % rates of the two accounts.
Not sure if I'm doing it right cause I keep coming up with all sorts of answers that don't multiply to the end.
---------------------------------------
|P| |R| |I|
---------------------------------------
1st 12000 x (240+x)
---------------------------------------
2nd 18000-12000=6000 x
---------------------------------------
T= 18000 Interest saved 1050
---------------------------------------
18000* 2/3 = 12000
18000-12000= 6000
12000*.02 = 240
1050 -240 = 810
810:2 = 405
Did I set this up right? cause I keep coming up with 66.6 for x at 6000 and 33.3 for 12000
18000-6000 %
---------- = -------
18000 100
What am I doing wrong? just doesn't look right. Thank you for the help. This is just a 5 point extra credit for class. And, I already spent 2 days on this and I'm getting frustrated. When I go to times it all to see if it adds up it doesn't. Have to be missing something.
2. let r = base interest rate as a decimal
$12000(r+.02) = 12000r + 240$6000(r) = 6000r
18000r + 240 = 1050
solve for r ... remember it is the base interest rate in decimal form
3. I did all that didn't I? I got the same answers up there.
4. came up with 15.5% hmm
5. I lost a zero when I got up to let the dogs out ... it's fixed. Take another look at my previous post.
6. Originally Posted by skeeter
let r = base interest rate as a decimal
$12000(r+.02) = 12000r + 240$6000(r) = 6000r
18000r + 240 = 1050
solve for r ... remember it is the base interest rate in decimal form
12000r+240+600r=1050
-240
-----------------------
18000r=810
810/18000
r=0.045%
Is that wrong? I'm in Elementary Algebra.
7. r = .045 = 4.5%
8. but when you do the total like mutiply it across it doesn't add up
18000(.045)=1050 =810 probably be for the 6000 loan have to divide that by 2 you get 405 then you add the 240 to 405 you 625 for the 1200 loan
so then I would need to do it for 12000 to find that percent%
when you do the tic tac toe box doesn't it have to add down to 1050 and then multiply with everything across to add to 1050
that's how I've been doing it.
9. r = 4.5% = .045
r + 2% = 6.5% = .065
12000(.065) + 6000(.045) = ?
10. Originally Posted by skeeter
r = 4.5% = .045
r + 2% = 6.5% = .065
12000(.065) + 6000(.045) = ?
oh ok I forgot about the 2% maybe that's where I was messing up.
sorry for being a pain..
so when you add .065 +.045 you get .11 % as a total
12000 ----------------.065%---------------.065(240+540)=780
_______________________________________________
6000 18000-12000
=6000 ----.045%------------ 6000(.045)= 270
--------------------------------------------------------
Total= 18000--------------.11%----------------1050
Total=18000(.11)=1050
It add downs but it isn't multiplying across
Does not add up? Percent wise anyway, comes to 1980 and not 1050 When I mutiply across. That's what I mean. sorry... I don't know how to do the math computer code.
11. so when you add .065 +.045 you get .11 % as a total
you do not add the two percents ... one percent (6.5%) is applied only to the $12000, the other (4.5%) only applies to the remaining$6000.
$12000(.065) + 6000(.045) = 1050$
if you want to find the "effective" total interest ...
$\frac{1050}{18000} = .0583 \approx 5.8$%
you're done.
12. Originally Posted by skeeter
you do not add the two percents ... one percent (6.5%) is applied only to the $12000, the other (4.5%) only applies to the remaining$6000.
$12000(.065) + 6000(.045) = 1050$
if you want to find the "effective" total interest ...
$\frac{1050}{18000} = .0583 \approx 5.8$%
you're done.
oh ok I always thought you had to add them in the total slot then times it to the 18000 to get the 1050. Thank you, hun I appreciate it.
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# Sequence does not come out right
\usepackage{tikz}
\usetikzlibrary{matrix,arrows,decorations.pathmorphing}
\begin{tikzpicture}[scale=1.5]
\node (A) at (1,0) {$0$};
\node (B) at (2,0) {$H^1(G/H,M^H$};
\node (C) at (3,0) {$H^1(G,M)$};
\node (D) at (4,0) {$H^1(H,M)$};
\path[->,font=\scriptsize,>=angle 90]
(A) edge node[above]{} (B)
(B) edge node[above]{$\textnormal{Inf}$} (C)
(C) edge node[above]{$\textnormal{Res}$} (D);
\end{tikzpicture}
I'm writing a LaTeX document with this code and while the diagram has appeared in the right order, they are squeezed together because of the length of the elements in the sequence. How can I fix this?
-
Welcome to TeX.SX! – egreg Jun 10 '14 at 20:34
You want to use tikz-cd:
\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz-cd}
\DeclareMathOperator{\Inf}{Inf}
\DeclareMathOperator{\Res}{Res}
\begin{document}
\begin{tikzcd}
0 \arrow{r} &
H^1(G/H,M^H) \arrow{r}{\Inf} &
H^1(G,M) \arrow{r}{\Res} &
H^1(H,M)
\end{tikzcd}
\end{document}
-
Thanks for the quick response! – Haikal Yeo Jun 10 '14 at 20:36
I'm not a specialist in group cohomology, but I've used several diagrams and exact sequences. ;-) – egreg Jun 10 '14 at 20:39
I got this error when I typed the code into my latex document in SAGE: <inserted text> \cr l.231 ! Misplaced \cr. <inserted text> \cr Do you know what's wrong? – Haikal Yeo Jun 10 '14 at 21:02
@HaikalYeo Sorry, but I don't know about SAGE. Are you sure you have an up-to-date TeX distribution? – egreg Jun 10 '14 at 21:03
Have a look at the Sage tutorial where there are instructions about how to include latex packages. – Andrew Swann Jun 11 '14 at 6:50
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# Find a combinatorial argument.
I am tasked to find a combinatorial argument to prove the identity and I'm stumped.
$\displaystyle\sum_{k=1}^{n}\dfrac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$
These are the attempts that I've made.
1) Double count the number of non negative integer solutions to
$x_{1}+\cdots +x_{n}=n+1$.
The RHS is trivial. For the left hand side let $k-1$ denote the sum of the first $k$ numbers. Then this partitions the set. The number of ways to get the sum of the first $k$ terms to be $k-1$ is $\binom{2k-2}{k-1}$ then you just count how many ways we can sum the remaining $n-k$ terms to get $n-k$. Which is just $\binom{2n-2k+1}{n-k}$.
I'm missing the $\tfrac{1}{k}$. Also another problem that I found in this arguement is that if I force the first $k$ terms to add up to $k-1$, one of them must be zero. Which then means that I didn't count all of them. Even though i'm way over already.
2) I've tried counting lattice paths from $(0,0)$ to $(n-1,n+1)$, $01$ strings of length $2n$ with exactly $n-1$ $1$'s and picking $n-1$ objects from $2n$ but none of those got both of the binomial coefficients on the LHS.
A hint of what set to count or how to correctly partition one of the sets I've tried would be nice. Thanks
One way to do it is to notice that $\frac1k\binom{2k-2}{k-1}=C_{k-1}$, the $(k-1)$-st Catalan number. Among other things, $C_n$ is the number of paths in the plane from from $\langle 0,0\rangle$ to $\langle n,n\rangle$ that take only unit steps to the right or up and never rise above the line $y=x$. (There’s a proof of this at the link.) The plain binomial coefficient $\binom{n}k$ counts the paths from $\langle 0,0\rangle$ to $\langle k,n-k\rangle$ using only unit steps to the right or up: such a path consists of $k$ steps to the right and $n-k$ steps up, any sequence of those steps gets you to $\langle k,n-k\rangle$, and there are $\binom{n}k$ ways to choose where in the sequence of $n$ steps to go to the right.
Thus, $\binom{2n}{n-1}$ counts the paths from the origin to $\langle n-1,n+1\rangle$. The $k$ term of the sum on the left is $C_{k-1}\binom{2n-2k+1}{n-k}$; $C_{k-1}$ counts the paths from the origin to $\langle k-1,k-1\rangle$ that do not rise above the diagonal $y=x$, and $\binom{2n-2n+1}{n-k}$ counts the paths from the origin to $\langle n-k,n-k+1\rangle$ and therefore also the paths from $\langle k-1,k-1\rangle$ to ... what?
• Ok, I thought I had it but I was incorrect. This is my argument so far. Maybe you can tell me where I've gone wrong. Let $k-1$ be the first time that the path returns to the line $y=x$. Then $C_{k-1}$ counts half of these paths to $(k-1,k-1)$. Since we can always be above or below the line. Then the remaining $\binom{2n-2k+1}{n-k}$ counts the paths from $(k-1,k-1)$ to $(n-1,n)$. I'm short on because $\binom{2n}{n-1}$ counts the number pf lattice paths from $(0,0)$ to $(n-1,n+1)$. – TheNumber23 Sep 18 '13 at 1:39
• @user90401: The origin is on $y=x$, and the endpoint $\langle n-1,2n\rangle$ is above it, so every path from the origin to $\langle n-1,2n\rangle$ must rise above the diagonal at some point. Say it first so at $\langle k-1,k\rangle$; then it can be broken into a path from the origin to $\langle k-1,k-1\rangle$ and ... ? – Brian M. Scott Sep 18 '13 at 2:05
• I don't know where we are getting $(n-1,2n)$. That is way more steps than $\binom{2n}{n-1}$. – TheNumber23 Sep 18 '13 at 2:12
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# Linear algebra I
### Basic information
You can always reach me by e-mail: 1@2354knopkammffczcuni.
### Credit for the class will be awarded for at least $2/3$ of all possible points for homeworks and tests (each 10 points)
Actual points (Winter semester) Actual points (Summer semester)
### Other resources
• MIT OpenCourseWare Linear algebra by Prof. Gilbert Strang main site
• The sketch proof of the fact, that two vectors $\bu = (u_1, u_2)$ and $\bv = (v_1, v_2)$ in $\R^2$ (i.e. in the plane) are perpendicular if either $v_1 = u_2, v_2 = -u_1$ or $v_1 = -u_2, v_2 = u_1$.
1. Use the fact, that $cos\varphi = \frac{\bu\cdot\bv}{\|\bu\|\|\bv\|}$, where $\varphi$ is the angle between vectors $\bu$ and $\bv$.
2. We want $\bu$ and $\bv$ to be perpendicular, so $cos\varphi = 0$, moreover we would like to have vectors of the same length.
3. This drives us to the form $\bu\cdot\bv = \|\bu\|^2$.
4. From the definition of the dot product ($\cdot$) and Eukliedian norm we get $u_1v_1 + u_2v_2 = 0$.
5. This gives us $u_1v_1 = -u_2v_2$. Which is the same as $\frac{v_1}{u_2} = -\frac{v_2}{u_1}$. Now giving the left-hand side value $1$ and $-1$ gives us the result.
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# #StackBounty: #pgfplots #pgfmath pgfplots: x vs x
### Bounty: 50
• This question is a follow-up to this question ("pgfplots: Strange Bump in tanh Function").
• Stefan Pinnow, who knows a thing or two about pgfplots, provided an answer that distinguished between addplot{tanh(x)}; and addplot{tanh(x)};(x vs x as the argument).
• In a comment, Stefan stated "My thought also was that tanh(x) is the same as tanh(x) but it seems that Lua "doesn’t like" "commands", i.e. <something>. But to my experience I never used x in addplot calls. Since this is easier to read and easier to type […]".
% use TeX as calculation engine
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# Pacemaker problem
1. Sep 13, 2006
### FlipStyle1308
Pacemakers which are designed for long-term use commonly employ a lithium-iodine battery capable of supplying 0.42 A x h of charge. (a) How many coloumbs of charge can such a battery supply? (b) If the average current produced by the pacemaker is 5.6 mcA, what is the expected lifetime of the device?
I am pretty sure I have to use the equation I = Q/t, so Q = It? How do I get t, since it is not mentioned at all? Does this have anything to do with the "x h" mentioned in the problem?
Last edited: Sep 13, 2006
2. Sep 13, 2006
### stunner5000pt
what unit make up an Ampere??
since I = Q/t
and time is in what units?
can you figure out how to d oyour problem now??
3. Sep 13, 2006
### FlipStyle1308
A = C/s, so C = As = (0.42)(3600) = 1512C. As for part (b), what is 5.6 mcA? I see the A, but was does mc mean?
Last edited: Sep 13, 2006
4. Sep 13, 2006
### stunner5000pt
mc = micro (althouhg usually they use the symbol mu $\mu$)
micro = 10^-6
5. Sep 13, 2006
### FlipStyle1308
Okay, I correctly solved this problem, thank you!
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# If $X_2$ is independent of $\mathcal F$, can we show that $f(X_1,X_2)$ is conditionally independent of $\mathcal F$ given $X_1$?
Let
• $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space
• $$\mathcal F\subseteq\mathcal A$$ be a $$\sigma$$-algebra on $$\Omega$$
• $$(E_i,\mathcal E_i)$$ be a measurable space
• $$X_i$$ be an $$(E_i,\mathcal E_i)$$-valued random variable on $$(\Omega,\mathcal A,\operatorname P)$$
• $$f:E_1\times E_2\to E_3$$ be $$(\mathcal E_1\otimes\mathcal E_2,\mathcal E_3)$$-measurable
• $$X_3:=f(X_1,X_2)$$
Assuming $$X_2$$ is independent of $$\mathcal F$$, are we able to show that $$X_3$$ is conditionally independent of $$\mathcal F$$ given $$X_1$$, i.e. $$\operatorname P\left[X_3\in B_3,F\mid X_1\right]=\operatorname P\left[X_3\in B_3\mid X_1\right]\operatorname P\left[F\mid X_1\right]\;\;\;\text{almost surely}\tag1$$ for all $$B_3\in\mathcal E_3$$ and $$F\in\mathcal F$$?
Let $$B_3\in\mathcal E_3$$ and $$F\in\mathcal F$$. We need to prove that $$\operatorname P\left[X_1\in B_1,X_3\in B_3,F\right]=\operatorname E\left[1_{\{\:X_1\:\in\:A\:\}}\operatorname P\left[X_3\in B_3\mid X_1\right]\operatorname P\left[F\mid X_1\right]\right]\tag2.$$ What's the easiest way to show $$(2)$$? Maybe we are able to reduce the problem to the case $$f^{-1}(B_3)=A_1\times A_2$$ for some $$A_i\in\mathcal E_i$$, but I'm missing the right argument for that.
EDIT: If necessary, feel free to impose a stronger notion of measurability on $$f$$.
• $$\Omega = \{\omega = (\omega_1,\omega_2)\mid \omega_i \in \{0,1\}\}$$,
• $$\mathrm P(\omega) = 1/4, \omega \in \Omega$$,
• $$X_i(\omega) = \omega_i$$, $$i=1,2$$;
• $$X_3 = (X_1+X_2) \mod 2$$, $$\mathcal F = \sigma(X_3)$$.
Then, $$X_1,X_2,X_3$$ are pairwise independent, but not jointly independent. So for $$B_3 = \{0\}$$ and $$F = \{X_3 = 0\}$$ $$\mathrm P \left[X_3=0,F\mid X_1\right] = \mathrm P [F] = 1/2\neq 1/4 = \mathrm P[F]^2 = \mathrm P \left[X_3=0\mid X_1\right] \cdot \mathrm P \left[F\mid X_1\right].$$
Note that in this counterexample we even have that $$X_1$$ and $$X_2$$ are independent and $$X_1$$ and $$\mathcal F$$ are independent. So it is hard to imagine an additional assumption which would make this true (except that the pair $$(X_1,X_2)$$ is independent of $$\mathcal F$$, which makes this trivial).
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Using UCSC Genome Browser with results from NCBI or Ensembl assemblies
2
0
Entering edit mode
3.3 years ago
colin.kern ▴ 970
If I want to display some data on the UCSC genome browser, but have done the analysis with the Ensembl or NCBI assemblies and annotations, do I need to manually write scripts to replace all the chromosome IDs with the UCSC genome browser ones? It seems like this would be a very common situation that has an existing solution.
Alternatively, I could do my analyses from now on using the UCSC assembly and annotation files, but that seems to have some issues. The annotation is only distributed as a SQL database. I have looked at using the Table Browser in UCSC to get a GTF file, which is what most tools use, but the GTF files I can get have a lot of missing information. They only contain stop_codon, start_codon, exon, and CDS elements. Can I get it to have transcript and gene entries also? It also only gives the gene and transcript IDs, whereas the GTF files you can get from Ensembl or NCBI have a lot of useful information such as gene symbol and biotype.
I think I might be missing something because it seems like I'm having to jump through a lot of hoops for something that I suspect a huge number of people have dealt with before.
ucsc browser ncbi ensembl • 1.0k views
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Entering edit mode
3.3 years ago
do I need to manually write scripts to replace all the chromosome IDs with the UCSC genome browser ones?
it seems like I'm having to jump through a lot of hoops
sed is your friend
0
Entering edit mode
Thanks. Do you know what resources were used to make those mappings? I work with farm animal species and none of them are on that Github.
1
Entering edit mode
3.3 years ago
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2016-07-06 09:26:58 +0200 received badge ● Student (source) 2016-07-06 09:26:45 +0200 received badge ● Nice Answer (source) 2016-07-04 00:18:40 +0200 received badge ● Scholar (source) 2016-07-02 17:25:00 +0200 received badge ● Self-Learner (source) 2016-07-02 17:25:00 +0200 received badge ● Teacher (source) 2016-07-02 00:30:56 +0200 received badge ● Editor (source) 2016-07-01 20:39:30 +0200 answered a question How to treat a vector space as a group? The following is the answer due to @Nicolas-M-Thiéry sage: Groups? The category of (multiplicative) groups, i.e. monoids with inverses. Mind the multiplicative! What you want is: sage: V = FreeModule(CC,2) sage: V in CommutativeAdditiveGroups() True or (better, but not imported by default): sage: from sage.categories.additive_groups import AdditiveGroups sage: V in AdditiveGroups() True Now you can construct the group algebra: sage: C = V.algebra(QQ) sage: C.category() Category of commutative additive group algebras over Rational Field sage: x = C.an_element() sage: x B[(1.00000000000000, 0.000000000000000)] sage: 3 * x + 1 B[(0.000000000000000, 0.000000000000000)] + 3*B[(1.00000000000000, 0.000000000000000)] Ah, but this is disappointing:: sage: (x+1)^2 TypeError: mutable vectors are unhashable One would need to have a variant of FreeModule that would guarantee that vectors remain immutable upon arithmetic. In the mean time, you can use: sage: V = CombinatorialFreeModule(CC, [0,1]) sage: C = V.algebra(QQ) sage: x = C.an_element() sage: x B[2.00000000000000*B[0] + 2.00000000000000*B[1]] sage: (x+1)^2 B[0] + 2*B[2.00000000000000*B[0] + 2.00000000000000*B[1]] + B[4.00000000000000*B[0] + 4.00000000000000*B[1]] 2016-07-01 20:39:25 +0200 asked a question How to treat a vector space as a group? I need to use a module as a group, so that I can define a group algebra over this module. Essentially, I want to take the group of 2-dimensional complex vector space and define a group algebra over this. I cannot find appropriate direction on the internet and sage gives me the ridiculous "False" as below. sage: V=FreeModule(CC,2) sage: V in Groups() False
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# $\mathbb{Z} [\sqrt{2}]$ is an integral domain
We know that $(\mathbb{Z} [\sqrt{2}],+,\cdot)$ is an integral domain.
Someone can prove it easily if he says that is a subring of $(\mathbb{R} ,+,\cdot)$ .
Can we find a different proof, more analytical? How can we show that $$\forall x,y\in \mathbb{Z} [\sqrt{2}], x\ne 0, y\ne 0\implies xy\neq 0$$
In these cases, considering the conjugate can help. The conjugate of $a+b\sqrt{2}$ is $a-b\sqrt{2}$. Now, if $(a+b\sqrt{2})(c+d\sqrt{2})=0$, also $$(a+b\sqrt{2})(a-b\sqrt{2})(c+d\sqrt{2})(c-d\sqrt{2})=0$$ and therefore $$(a^2-2b^2)(c^2-2d^2)=0$$ Since the integers form a domain, we conclude $a^2-2b^2=0$ or $c^2-2d^2=0$. The irrationality of $\sqrt{2}$ tells us that either $a=b=0$ or $c=d=0$.
• Nice proof, thank you. So we multiply with the conjugate, and we say that $$(a/b)=2^{(1/2)}$$, false. Right? – Chris Apr 29 '16 at 23:08
• @Chris Suppose $a^2-2b^2=0$. If $b=0$, then $a=0$; if $b\ne0$, then $\frac{a^2}{b^2}=2$. The latter case is impossible. – egreg Apr 29 '16 at 23:09
The main proof you mention is the easiest and the best since it generalizes very well. If you really want something more intrinsic, note
$$(a+b\sqrt 2)(c+d\sqrt 2)=0\implies (a^2-2b^2)(c^2-2d^2)=0$$
But then if so, either $a^2=2b^2$ or $c^2=2d^2$, WLOG assume the former.
Then $a$ is even, but then if the prime factorization of $a$ is $2^kp_1^{e_1}\ldots p_r^{e_r}$ we have
$$a^2=2^{2k}p_1^{2e_1}\ldots p_r^{2e_r}$$
So the exponent of $2$ in $a$ is even, however since $b^2$ has an even power of $2$, $2b^2$ has an odd power, a contradiction, unless that power is infinity, but the only number infinitely divisible by $2$ is $0$, so $a=b=0$ holds
• Once we've eliminated the possibility that $b = 0$ (which is almost immediate), we can conclude that if $a^2 = 2 b^2$ then $\sqrt{2}$ is rational, and likewise for $c^2 = 2 d^2$. – Travis Willse Apr 29 '16 at 22:44
• @Travis yes, but why bother? That's a corollary of this approach in general, may as well avoid saying it for a direct proof that takes place completely within the realm of $\Bbb Z$. – Adam Hughes Apr 30 '16 at 18:00
• I remarked it simply because it cuts the length of the (already efficient) presentation in half. – Travis Willse Apr 30 '16 at 19:34
• @Travis I agree, assuredly, I chose this presentation for pedagogical reasons more than anything. I always find appealing to lower-level theorems is best when it doesn't make the proof unwieldy. – Adam Hughes Apr 30 '16 at 19:43
• @Travis no worries, that came across clearly, I was just responding to explain my thought process. Thanks very much for your comments. Cheers! – Adam Hughes Apr 30 '16 at 19:54
Hint Suppose we have $$(a + b \sqrt{2})(c + d \sqrt{2}) = 0$$ from some $a, b, c, d \in \Bbb Z$. Expanding gives $$(ac + 2 bd) + (ad + bc) \sqrt{2} = 0,$$ and since $\sqrt{2}$ is irrational, the coefficients must vanish separately $$ac + 2bd = ad + bc = 0 .$$ Substituting $0$ for any of the four parameters gives quickly that $a = b = c = d = 0$, so we may assume that none is zero. Then, it follows from the second equation that $c = -\lambda a$ and $d = \lambda b$ for some $\lambda \in \Bbb Q - \{ 0 \}$.
Substituting gives $$0 = a c + 2 b d = a(-\lambda a) + 2 b (\lambda b) = \lambda (-a^2 + 2b^2) .$$ Clearing $\lambda$ and rearranging gives $\left(\frac{a}{b}\right)^2 = 2$, but this contradicts the irrationality of $\sqrt{2}$.
• I think you mean "this implies $\sqrt{2}$ is rational" – Stella Biderman Apr 29 '16 at 22:57
• Thank you for your answer. How can we say with certainty that exists $$λ\in \mathbb{Q^*} : c=-λa, d=λb$$ – Chris Apr 29 '16 at 23:01
• Thanks, @StellaBiderman, you're right of course. I've fixed the statement. – Travis Willse Apr 29 '16 at 23:20
• We can rearrange $ad + bc = 0$ and write $\lambda := -\frac{c}{a} = \frac{d}{b}$. – Travis Willse Apr 29 '16 at 23:28
• You're welcome, I hope you found it useful. – Travis Willse Apr 29 '16 at 23:51
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## I have created a new and better article: Pitot Tube and Pressure Explanation With F1 and Pikes Peak Example.
One of the most well known aerodynamic equations is Bernoulli’s Equation:
$\frac{P_1}{\rho} + \frac{1}{2} V_1^{2} = \frac{P_2}{\rho} + \frac{1}{2} V_2^{2}$
The equation can be used with the following simplifying assumptions:
• Inviscid, frictionless flow
• Imcompressible flow
• Works for flow along a Streamline
Before getting into the derivation of the equation lets look at a common example.
Example
A common use of the Bernoulli Equation is calculating the stream velocity through measured pressure.
Assume all the required conditions exist, let’s see what the velocity is if the Static Pressure is 101KPa and the Total Pressure is 105KPa.
$\frac{P_1}{\rho} + \frac{1}{2} V_1^{2} = \frac{P_2}{\rho} + \frac{1}{2} V_2^{2}$
$\frac{101000}{1.225} + \frac{1}{2} V_1^{2} = \frac{105000}{1.225} + \frac{1}{2} 0^{2}$
$V_1 = 80.8[\frac{m}{s}]$
Equivalent to this is the form I typically use which gets the same answer (this comes from my preferance of not having denominators):
$P_1 + \frac{1}{2} \rho V_1^{2} = P_2 + \frac{1}{2} \rho V_2^{2}$
or, in notation similar to the photo from NASA:
$P_s + \frac{1}{2} \rho V^{2} = P_t$
Bernoulli’s Equation has many uses, so whenever the 4 assumptions I listed are reasonable it is a very quick way to find an unknown velocity, static pressure, or total pressure.
Another article will be written to better explain this. It will be found somewhere on The Aerodynamics Page of ConsultKeithYoung. You can also find more related articles by using the sitemap on that site and choosing Pitot Tube.
References:
This is my main Fluid Mechanics book in college and I think it’s a very good one.
http://en.wikipedia.org/wiki/Bernoulli’s_principle
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Transport in Open Quantum Systems
Student Name
Final Thesis Submission Date
2018-09-04
Abstract
An open system is a system connected with the environment. Real systems are always open. Far from equilibrium open quantum systems are extensively studied because of their novel physics and their very interesting and diverse potential applications. These include molecular electronics, nanoscale diodes, thermal rectification, quantum logic gates and even gravitational wave detection. Theoretical challenge in understanding such open quantum systems arise because the usual equilibrium statistical mechanics is not applicable. I will describe various theoretical approaches used to treat such set-ups. By explicitly working out some experimentally relevant examples, I will show that one approach has a wider range of validity as well as simplicity. I will also show that even for extended systems, transport through an open system can be completely different from the usual notion of transport in an isolated system in the thermodynamic limit.
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bug-lilypond
[Top][All Lists]
## Re: Spacing bug
From: Mats Bengtsson Subject: Re: Spacing bug Date: Wed, 05 Oct 2005 10:23:32 +0200 User-agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.7.6) Gecko/20050319
Wiz Aus wrote:
The following file tries to squeeze onto one line when it clearly doesn't fit (the double barline at the end is cut off, for a start):
What double bar line? Your last bar has only 5/8, so there is no bar
line printed at all. If you want a double bar line at the end of a
piece, you have to specify it explicitly, \bar "|.".
You have found an extreme case, where LilyPond will break the line as
soon as you add another bar. Also, I've almost only seen these tight
spacings in scores with a single bar, i.e. as soon as LilyPond starts
breaking the piece into several lines, then it will usually never
give this tight spacing even on the last line. So, there are very few
real pieces of music where you see this tight spacing.
Still, I agree with you that especially the spacing between the last
note of each measure the following bar line is too tight. As far as I
can see, there is no way to specify a minimum distance for that setting,
in contrast to all other spacings, for example between the time
signature and the first note.
\relative{
\time 3/4
c8 d e f g a
c, d16 d e8 f g a
c, d e16 e f8 g a
c, d e f16 f g8 a
c,8. d16 e8 f g8. a16
r8 d, e f g a
c, d e f g r
r d e r f g
c, r e r g
}
I was actually creating this file to see what beaming policies lilypond uses - to me the 2nd last bar here looks a bit odd for 3/4 (it looks like a 6/8 bar), but to be honest, I'm not sure how it would automatically determine a better configuration. The 6th bar isn't perhaps ideal either, I suspect most publishers would prefer not to beam the f to the g. I assume the rule it's using is that quavers are beamed over beat (crotchet) boundaries - shorter notes are only beamed within each beat. Probably reasonable enough for 99% of cases anyway.
I recommend Section "8.6.2 Setting automatic beam behavior" in the
manual for version 2.7 (the contents is relevant also for version 2.6,
you didn't say what you use) for more insights.
Actually one other thing - why is the duration number necessary when using a dot? I would expect to be able to type "c,. d16 e8 f g. a16" for the 5th bar here, but it doesn't work. The documentation doesn't suggest that the duration number is required, but all the examples include it.
I don't know any specific reason, but I don't think it hurts, the input
syntax can be confusing enough anyway. Consider for example what
c4 c. c.
would mean if the duration numbers weren't necessary (it would be the
same as
c4 c4. c4..
which probably would surprise many).
/Mats
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# Hierarchy of beliefs
From Wikipedia the free encyclopedia
Construction by Jean-François Mertens and Zamir implementing with John Harsanyi's proposal to model games with incomplete information by supposing that each player is characterized by a privately known type that describes his feasible strategies and payoffs as well as a probability distribution over other players' types.[1]
Such probability distribution at the first level can be interpreted as a low level belief of a player. One level up the probability on the belief of other players is interpreted as beliefs on beliefs. A recursive universal construct is built—in which player have beliefs on their beliefs at different level—this construct is called the hierarchy of beliefs.
The result is a universal space of types in which, subject to specified consistency conditions, each type corresponds to the infinite hierarchy of his probabilistic beliefs about others' probabilistic beliefs. They also showed that any subspace can be approximated arbitrarily closely by a finite subspace.
Another popular example of the usage of the construction is the Prisoners and hats puzzle. And so is Robert Aumann's construction of common knowledge.[2]
## References
1. ^ Jean -François Mertens and Shmuel Zamir (1985-03-01). "Formulation of Bayesian analysis for games with incomplete information". International Journal of Game Theory. 14 (1): 1–29. doi:10.1007/BF01770224.
2. ^ Herbert Gintis (16 March 2009). The bounds of reason: game theory and the unification of the behavioral sciences. Princeton University Press. p. 158. ISBN 978-0-691-14052-0. Retrieved 3 March 2012.
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# Car alarms and smoke alarms: the tradeoff between sensitivity and specificity
Wouldn’t you like to live in a world where your monitoring systems only alerted when things were actually broken? And wouldn’t it be great if, in that world, your alerts would always fire if things were broken?
Well so would everybody else. But we don’t live in that world. When we choose a threshold for alerting, we usually have to make a tradeoff between the chance of getting a false positive (an alert that fires when nothing is wrong) and the chance of getting a false negative (an alert that doesn’t fire when something is wrong).
Take the load average on an app server for example: if it’s above 100, then your service is probably broken. But there’s still a chance that the waiting processes aren’t blocking your mission-critical code paths. If you page somebody on this threshold, there’s always a chance that you’ll be waking that person up in the middle of the night for no good reason. However, if you raise the threshold to 200 to get rid of such spurious alerts, you’re making it more likely that a pathologically high load average will go unnoticed.
When presented with this tradeoff, the path of least resistance is to say “Let’s just keep the threshold lower. We’d rather get woken up when there’s nothing broken than sleep through a real problem.” And I can sympathize with that attitude. Undetected outages are embarrassing and harmful to your reputation. Surely it’s preferable to deal with a few late-night fire drills.
It’s a trap.
In the long run, false positives can — and will often — hurt you more than false negatives. Let’s learn about the base rate fallacy.
## The base rate fallacy
Suppose you have a service that works fine most of the time, but breaks occasionally. It’s not trivial to determine whether the service is working, but you can write a probe that’ll detect its state correctly 99% of the time:
• If the service is working, there’s a 1% chance that your probe will say it’s broken
• If the service is broken, there’s a 1% chance that your probe will say it’s working
Naïvely, you might expect this probe to be a decent check of the service’s health. If it goes off, you’ve got a pretty good chance that the service is broken, right?
No. Bad. Wrong. This is what logicians and statisticians call the “base rate fallacy.” Your expectation hinges on the assumption that the service is only working half the time. In reality, if the service is any good, it works almost all the time. Let’s say the service is functional 99.9% of the time. If we assume the service just fails randomly the other 0.1% of the time, we can calculate the true-positive rate:
$\begin{array}{rcl} \text{TPR} & = & \text{(prob. of service failure)}*\text{(prob. of detecting a failure)} \\ & = & (0.001) * (0.99) \\ & = & 0.00099 \\ & = & 0.099\% \end{array}$
That is to say, about 1 in 1000 of all tests will run during a failure and detect that failure correctly. We can also calculate the false-positive rate:
$\begin{array}{rcl} \text{FPR} & = & \text{(prob. of service non-failure)}*\text{(prob. of detecting failure anyway)} \\ & = & (1-0.001)*(1-0.99) \\ & = & 0.0099 \\ & = & 0.99\% \end{array}$
So almost 1 test in 100 will run when the service is not broken, but will report that it’s broken anyway.
You should already be feeling anxious.
With these numbers, we can calculate what the medical field calls the probe’s positive predictive value: the probability that, if a given test produces a positive result, it’s a true positive. For our purposes this is the probability that, if we just got paged, something’s actually broken.
$\begin{array}{rcl} \text{(Positive predictive value)} & = & \frac{\text{TPR}}{\text{TPR} + \text{FPR}} \\ & = & \frac{0.00099}{0.00099 + 0.0099} \\ & = & 0.091 \\ & = & 9.1\% \end{array}$
Bad news. There’s no hand-waving here. If you get alerted by this probe, there’s only a 9.1% chance that something’s actually wrong.
## Car alarms and smoke alarms
When you hear a car alarm going off, do you run to the window and start looking for car thieves? Do you call 9-1-1? Do you even notice car alarms anymore?
Car alarms have a very low positive predictive value. They go off for so many spurious reasons: glitchy electronics, drunk people leaning on the hood, accidental pressing of the panic button. And as a result of this low PPV, car alarms are much less useful as theft deterrents than they could be.
Now think about smoke alarms. People trust smoke alarms. When a smoke alarm goes off in a school or an office building, everybody stops what they’re doing and walks outside in an orderly fashion. Why? Because when smoke alarms go off (and there’s no drill scheduled), it frequently means there’s actual smoke somewhere.
This is not to say that smoke alarms have a perfect PPV, of course, as anybody who’s lost half an hour of their time to a false positive will tell you. But their PPV is high enough that people still pay attention to them.
We should strive to make our alerts more like smoke alarms than car alarms.
## Sensitivity and specificity
Let’s go back to our example: probing a service that works 99.9% of the time. There’s some jargon for the tradeoff we’re looking at. It’s the tradeoff between the sensitivity of our test (the probability of detecting a real problem if there is one) and its specificity (the probability that we won’t detect a problem if there isn’t one).
Every time we set a monitoring threshold, we have to balance sensitivity and specificity. And one of the first questions we should ask ourselves is: “How high does our specificity have to be in order to get a decent positive predictive value?” It just takes some simple algebra to figure this out. We start with the PPV formula we used before, enjargoned below:
$\begin{array}{rcl} \text{PPV} & = & \frac{\text{TPR}}{\text{TPR}+\text{FPR}} \\ & = & \frac{\text{(prob. of failure)}\cdot\text{(sensitivity)}}{\text{(prob. of failure)}\cdot\text{(sensitivity)} + (1 - \text{(prob. of failure)})\cdot(1 - \text{(specificity)})} \end{array}$
To make this math a little more readable, let’s let p = PPV, f = the probability of service failure, a = sensitivity, and b = specificity. And let’s solve for b.
$\begin{array}{rcl} p & = & \frac{fa}{fa + (1-f)*(1-b)} \\ fa + (1-f)(1-b) & = & \frac{fa}{p} \\ 1-b & = & \frac{\frac{fa}{p} - fa}{1-f} \\ b & = & 1 - \frac{\frac{fa}{p} - fa}{1-f} \end{array}$
So, sticking with the parameters of our initial example (0.1% probability of service failure, 99% sensitivity) and deciding that we want a positive predictive value of at least 90% (so that 9 out of 10 alerts will mean something’s actually broken), we end up with
$\begin{array}{rcl} \text{Specificity} & = & 1 - \frac{\frac{0.001*0.99}{0.9} - (0.001 * 0.99)}{(1 - 0.001)} \\ & = & 0.9999 \\ & = & 99.99\% \end{array}$
The necessary specificity is about 99.99% — that’s way higher than the sensitivity of 99%! In order to get a probe that detects failures in this service with sufficient reliability, you need to be 100 times less likely to falsely detect a failure than you are to miss a positive!
## So listen.
You’ll often be tempted to favor high sensitivity at the cost of specificity, and sometimes that’s the right choice. Just be careful: avoid the base rate fallacy by remembering that your false-positive rate needs to be much smaller than your failure rate if you want your test to have a decent positive predictive value.
## 10 thoughts on “Car alarms and smoke alarms: the tradeoff between sensitivity and specificity”
One thing to consider when comparing the response to car alarms versus fire alarms is that risk is usually modeled as the product of probability and cost. People will respond to even a low probability fire alarm because the cost (their lives) is so high.
2. Jack Wilson
I also take issue with comparing smoke alarms with car alarms. They are not comparable on any level except the fact that they are called alarms. On the highest level, if a car alarm goes off, 99.9% of the time, it is during the daytime and many people are “witnesses”. This factor is what creates the trust. But if I look over at the car that is “going off”, there are basically only two outcomes: 1) there is no one around, and the alarm is spurious. 2) there is someone nearby, and NO ONE has any clue if that person is the “owner”. Let’s now go to the smoke alarm. This is totally different in case: If this alarm goes off, and you are close enough to hear it, It is virtually 100% positive that you will know if it is a spurious event. There WILL BE smoke if it is a “true” event. Period.
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# Visualizing Stirling’s Approximation With Highcharts
I said, “Wait a minute, Chester, you know I’m a peaceful man”, He said, “That’s okay, boy, won’t you feed him when you can” (The Weight, The Band)
It is quite easy to calculate the probability of obtaining the same number of heads and tails when tossing a coin N times, and N is even. There are $2^{N}$ possible outcomes and only $C_{N/2}^{N}$ are favorable so the exact probability is the quotient of these numbers (# of favorable divided by # of possible).
There is another way to approximate this number incredibly well: to use the Stirling’s formula, which is $1/\sqrt{\pi\cdot N/2}$
This plot represents both calculations for N from 2 to 200:
Although for small values of N, Stirling’s approximation tends to overestimate probability …
… is extremely precise as N becomes bigger:
James Stirling published this amazing formula in 1730. It simplifies the calculus to the extreme and also gives a quick way to obtain the answer to a very interesting question: How many tosses are needed to be sure that the probability of obtaining the same number of heads and tails is under any given threshold? Just solve the formula for N and you will obtain the answer. And, also, the formula is another example of the presence of $pi$ in the most unexpected places, as happens here.
Just another thing: the more I use highcharter package the more I like it.
This is the code:
library(highcharter)
library(dplyr)
data.frame(N=seq(from=2, by=2, length.out = 100)) %>%
mutate(Exact=choose(N,N/2)/2**N, Stirling=1/sqrt(pi*N/2))->data
hc <- highchart() %>%
hc_title(text = "Stirling's Approximation") %>%
hc_subtitle(text = "How likely is getting 50% heads and 50% tails tossing a coin N times?") %>%
hc_xAxis(title = list(text = "N: Number of tosses"), categories = data$N) %>% hc_yAxis(title = list(text = "Probability"), labels = list(format = "{value}%", useHTML = TRUE)) %>% hc_add_series(name = "Stirling", data = data$Stirling*100, marker = list(enabled = FALSE), color="blue") %>%
hc_add_series(name = "Exact", data = data\$Exact*100, marker = list(enabled = FALSE), color="lightblue") %>%
hc_tooltip(formatter = JS("function(){return ('<b>Number of tosses: </b>'+this.x+'<br><b>Probability: </b>'+Highcharts.numberFormat(this.y, 2)+'%')}")) %>%
hc_exporting(enabled = TRUE) %>%
hc_chart(zoomType = "xy")
hc
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# Returning the last segment of a split string
I recently reviewed a PR and saw the following.
const parts = someString.split('.');
return parts[parts.length - 1];
was changed to
const [last] = someString.split('.').reverse();
return last;
I commented that I am against the change for the reasons that it is harder to read and not performant.
The answer was that it is only hard to read, because I am not used to it and that performance does not matter if you use big frameworks like React for example.
Who is right and why?
• How much is this profession being dumbed down that parts[parts.length - 1] is considered "hard to read"? – Will Sep 11 at 0:13
• "performance does not matter if you use big frameworks like React for example" - they have a point so: when you replace pile of poorly written code that searches dom to manually replace value one-by-one with big heavily optimized framework like Reach you have a lot of free room to write slower code till you match your original slow version... – Alexei Levenkov Sep 11 at 1:05
• @Will It looks like the OP is saying the second method is hard to read, not the parts[parts.length - 1] part. – Kodos Johnson Sep 11 at 1:46
• @KodosJohnson Yes, and I agree with the OP, but some of the answers below and presumably the person responsible for OP's PR do imply that parts[parts.length - 1] is somehow hard to read. – Will Sep 11 at 3:08
• @12Me21 Although it is technically the first element in the array, how is const [last] = array in any way difficult to read, let alone considered an obfuscation? Anyone familiar with object destructuring (so common in React now) ought to at least have some basic understanding of its cousin, array destructuring. I think more important to the code review the OP did was if that change was even necessary to begin with as it appears to have been a completely superfluous implementation change on the coder's part. "Just because" isn't really a valid reason to change working production code. – Drew Reese Sep 12 at 6:44
I'm not entirely enthusiastic about either. The parts[parts.length - 1] using manual index lookup and subtraction may well take a moment to recognize, "Oh, this is getting the last element of the array". The second, using .reverse() followed by destructuring of the first item also could take a moment to think about before you understand what exactly it's doing.
As an alternative, consider using .pop() instead, I think it's more intuitive than both:
return someString
.split('.')
.pop();
The only (tiny) downside is that the above isn't functional, since it mutates the split array - but that's OK, since the array isn't being used anywhere else.
It's true that performance for something like this isn't really something to worry about if this is used in a larger project - what matters more is if it's easier to understand what the code is doing.
Remember that if something isn't immediately intuitive, that's what comments are for, though all of these cases are self-documenting enough that it's not needed.
Another way to make it easier to recognize what's going on would be to put this into a function with a descriptive name, such as getLastProperty (if the input is meant to show a path to a nested property lookup, for example - name it as appropriate for how it's meant to be used).
• Perhaps not so intuitive but a great solution. – Miguel Avila Sep 10 at 13:52
• @WilliMentzel Sure, if the array needs to be used later, then mutation via .pop earlier could well cease to be the right approach. If that was the situation, I'd use the index method instead, it's easier to read than the .reverse() / destructuring method. – CertainPerformance Sep 10 at 14:13
• @Willi It depends how much of a stickler you are on a functional paradigm. Mutation is bad when it makes the code harder to read than the alternative, but I don't consider it a cardinal sin; sometimes aiming to be as functional as possible makes the code more confusing than the alternative. For example, when you want to (eventually) do something with a deeply nested object via a property string, splitting by . and then popping off the last item to have [(1) array of all properties but the last (2) the last property string] is what I'd do. – CertainPerformance Sep 10 at 14:25
• YMMV; it's up to you, the standards of your codebase, and the situation. – CertainPerformance Sep 10 at 14:25
• @WilliMentzel - In this case the nobody can use the array generated because it is not saved in any variable. However, if you have the array stored in a variable then do x.slice().pop(). There is zero arguments against using pop – slebetman Sep 11 at 3:00
tl;dr at the end.
In terms of 'performance', js engines are pretty well optimized. For something like this, performance should not be part of your argument against a given piece of code. In terms of readability, they are very similar so either works. There are better ways to think through this though.
When it comes to PRs I would ponder on a few questions:
1. What does the PR code in question actually change?
Between the two snippets of code, there are no changes to the API of the calling code and no implementation changes to accommodate any edge cases. They work the same way and there are no visible upsides. So why change it? Original takes it here. DFWAB!
1. Does the code add/remove unnecessary steps?
const parts = someString.split('.');
return parts[parts.length - 1];
1. Split someString into an ('.' character delimited) array
2. Label (step 1) parts
3. Take parts's length property and subtract 1
4. Retrieve parts's element at index (step 3)
5. return (step 4)
const [last] = someString.split('.').reverse();
return last;
1. Split someString into an ('.' delimited) array
2. reverse the elements in (step 1)
3. Label the first element in (step 2) last
4. return last
Here the PR code changes the approach and removes a step. Is it necessary though? This is more left to opinion, and I am sure you will get opinions for/against both. IMHO "split an array then reverse that array" are simpler to reason about because it is declarative. I can see a string being split into pieces in my head and then reverse-ing the original order so the first element becomes the last (supported by the const label last). In the original, I have to reason about arrays, their properties, and why I have to access its length at all. I say that the PR code takes this. Not by much though.
1. What is the intention of the code?
This code's intention regardless of implementation is to take a string and give back a substring starting one character after the last '.' in the string. Both implementations perform acrobatics just to pull this off. The main fault in terms of implementation is 'why do we have to turn the string into an array' when the string possesses all the necessary methods?
Here is an example that beats out both cases and how it performs in these metrics:
return someString.slice(someString.lastIndexOf('.') + 1);
1. No changes from the original code BUT adds clear benefits. (win)
2. Removes unnecessary/takes fewer steps and is more declarative. (win)
3. States the true intention of the code clearly. (huge win)
BONUS: IT'S A ONE LINER! WOOHOO!
Now I know people tend to not see this as a benefit but I think shorter code with higher stats (as described above) is better than more code performing equally or worse.
1. Take someString and find the lastIndexOf '.'
2. Add 1 to (step 1)
3. slice someString starting at the index (step 2)
4. return (step 3)
Just a small note: if I were certain that this was all the code in the calling function I would turn it into someString => someString.slice(someString.lastIndexOf('.') + 1) and eliminate step 4 since the return is implicit and the function can be seen as the result of the first three steps.
In terms of intention, this line is literally saying (as the steps also describe): "Slice this string starting at the index following the last index of '.'. It's just clear.
tl;dr
Neither one is 'right' because performance is irrelevant and both snippets are fairly similar in readability. Instead of 'readability' (whatever that means, highly subjective) look at these metrics when it comes to PRs:
1. Does the PR actually implement changes? (original wins)
2. Does the PR add/remove unnecessary STEPS? (PR wins)
3. Does the pull request make the intention of the code clearer? (tied loss)
A better change which beats the other two in all categories (for reference) could be something like this:
return someString.slice(someString.lastIndexOf('.') + 1);
• array[array.length-1] is idiomatic in so many languages that I think it would make sense to consider it as one step. I imagine most people would immediately recognize that as "get the last item in the array" – 12Me21 Sep 11 at 17:12
• Steps, as I describe in the post, disregard an individual's ability to chunk understood steps together. The focus is on what the person reading might have to go through, at a bare minimum, to read the code. I do not think it wise to assume everyone reading the code will have a predetermined background, hence my decision to not chunk those into one step. – santanaG Sep 11 at 21:17
Well, it depends if the performance is an objective or it isn't. Now, I always would prefer good performance; now respect to the person who said "performance does not matter if you use big frameworks like React" oh, you don't really mean that, is Facebook that slow ?(Facebook uses React) do you know what implies to have a slow social network? Slow transactional operations not because the backend isn't optimized but the front? It is less users because they're boored waiting your app to response and switch to another media.
Now, the second code fragment
const [last] = someString.split('.').reverse();
return last;
is slower and does not exhibit the advantages of the language you are working with so why bother to even write it, if it's redundant?
I mean, why would you need to reverse a substring to return the first character on it? (after reversed)
You clearly can do
const parts = someString.split('.');
return parts[parts.length - 1];
and that doesn't cost more time and is descriptive.
The features of any language are there not to create "smart looking code" but to create more effective and efficient code.
Note, as CertainPerformance mentioned
return someString.split('.').pop();
is a great solution, easier to read, write and virtually takes the same time that the first fragment. (virtually, because internally those are different operations)
Here's my take:
return someString.substring(someString.lastIndexOf('.') + 1);
The readability issue could be argued either way, it's really personal preference. I don't think the change in the PR makes the code materially better or easier to read.
To hear 'performance doesn't matter' just makes me queasy. In your application it might not have much of an effect, but why go out of your way to make code slower if it doesn't really make the code much easier to read.
const ITERATIONS = 350000;
const TEST_STRING = 'a.b.c.d.e.f.g.h.i.j.k.l.m.n.o.p.q.r.s.t.u.v.w.x.y.z';
console.log('------------------------------ START ------------------------------')
function time(s, fn) {
const start = performance.now();
for (let i = 0; i < ITERATIONS; i++) {
fn(TEST_STRING);
}
const end = performance.now();
const time = end - start;
console.log(${s}:${time});
}
time('Length-1', s => { const arr = s.split('.'); return arr[arr.length - 1]; });
time('Reverse', s => { const [last] = s.split('.').reverse('.'); return last; });
time('lastIndexOf', s => s.substring(s.lastIndexOf('.') + 1));
• I know that nobody asked, but as a C++ programmer, this is the only answer that doesn't sound crazy to me. I would never even consider tokenizing a string and then reversing an entire array just to get the last element in a string. – JPhi1618 Sep 17 at 13:26
• @JPhi1618 Exactly this. Years ago I agreed about not caring too much because JS was used mostly for fluffing some bits on the browser. However, a substantial number of backends today use JS. I hope that those developers are aware of what the code they write is actually doing rather than just its return value. – Bernardo Sulzbach Sep 24 at 1:09
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# Numerical solution to N-dimensional diffusion on simplex?
Assume I have a system of at least (but generally only) $$N+1$$ points in an $$N$$-dimensional space ($$N > 3$$ is possible). At each of these points $$x_i, i=1,...,N+1$$ I know an initial potential/concentration/quantity $$P_i$$. I can evaluate the possible connections between all points. I also have an isotropic diffusion coefficient $$D$$. I am now interested in calculating diffusion between these three points (with mass conservation). I have sketched an example case in 2-D.
This problem sounds rather easy, but I quickly realized that it isn't. Classic finite difference approaches require (to my knowledge) a structured gid, which I don't have. Finite volume could handle unstructured grids, but I cannot afford tessellation in high dimensions (say, $$N > 100$$). For both approaches I would have calculated the first potential derivative along each line (e.g., $$x_1$$-->$$x_2$$), and assumed that the same derivative half a distance outside (dashed lines) is zero. Unfortunately, I don't think this assumption works if I have more than $$N+1$$ points. Finite elements might be an option, but I have yet to find a tutorial which would cover such a case.
Do you have any idea how to solve this system?
Edit: Some additional clarification concerning my goal: I want to simulate the equilibration between the $$N+1$$ vertices. I am only interested in solutions at the vertices themselves, and diffusion would occur along the edges. The specific equation I want to solve is the heat equation:
$$\frac{dP_i}{dt}=K\Delta P_i$$
• Are you interested in computing your quantity at intermediate points? – nicoguaro Mar 31 at 13:06
• No, just at the vertices is fine! – J.Galt Mar 31 at 16:49
• I don't think I'm clear about the exact setup. Where does diffusion act? Along the edges of the polytope described by your $N+1$ vertices? In the interior of the polytope? If it's just along the edges, then you have a bunch of 1d problems coupled at the vertices; since there is an analytic solution for the 1d heat equation, it all reduces to a set of ODEs. In any case, it would be useful to specify which set of equations you want to solve and where. – Wolfgang Bangerth Apr 1 at 23:31
• @WolfgangBangerth, thank you for the comment and sorry for the lack of clarity - I will add some additional information. Yes, I am only interested in diffusion along the edges - I want to simulate the equilibration between these $N+1$ points. The equation I want to solve is the heat equation at the vertices, which requires second derivatives. The approach you suggest sounds intriguing - could you give me a quick hint on how to approach this solution? – J.Galt Apr 2 at 9:48
• I think I still don't understand. You say $P_i$ is the potential at vertex $i$. That's a time-dependent function. Then how do you define $\Delta P_i$? $\Delta$ is a spatial operator, but $P_i$ does not depend on any spatial variables. – Wolfgang Bangerth Apr 2 at 14:14
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Although romantic relationships have been found to boost well-being, some studies demonstrated that staying single has similar beneficial effects. One cause of such contradiction is probably due to the lack of a quantitative measurement of attitudes toward singlehood. To address this methodological gap, four studies involving 1,276 undergraduate students in Malaysia and India were conducted to develop and examine the psychometric qualities of the Attitudes toward Singlehood Scale (AtSS). Study 1 selected 15 items from the pool and identified a 3-factor solution using exploratory factor analysis. However, a 9-item second-order factor model was found superior in Study 2 using confirmatory factor analysis. The 9-item AtSS demonstrated good internal consistency and test-retest reliability measured two weeks apart as well as construct and criterion validity. Study 3 further supported the superiority of the 9-item second-order factor model with replicated results of Study 2 on a new sample. Measurement invariance test supported scalar invariance across gender while ANCOVA showed female participants displaying higher scores than male counterparts. Study 4 then examined the properties of the 9-item AtSS on a sample of young adults in India. The results are consistent with Study 2 and 3, lending further support to the usability of the AtSS in different cultural contexts. Overall, the consistent findings promote the AtSS as a promising tool for assessing young adults’ attitudes toward singlehood. Implication and suggestions for future studies are also discussed.
While social relationships promote people’s happiness and general well-being abundantly (Lehmann et al., 2015), undergraduate students who are currently or had previously been in romantic relationships appear to be more socially mature, better-adjusted, and less self-centered than those currently and always single (DePaulo & Morris, 2006). Individuals engaged in high-quality romantic relationships are also able to enjoy ideal well-being when compared to singles (Hudson et al., 2020; McCabe et al., 1996).
According to Lehmann et al. (2015), one’s current relationship status (partnered vs unpartnered) is crucial in determining life satisfaction and reducing distress, let alone marital status. Hope et al. (1999) arguably asserted that individuals in committed relationships, including married people (even when controlling for pre-marital life satisfaction levels), achieve greater life satisfaction following better well-being than their single peers. Likewise, fear of being single is found negatively related to life satisfaction, emotional well-being, and psychological well-being (Adamczyk, 2017) and positively related to loneliness (Spielmann et al., 2013).
On the other hand, marriage or romantic relationships may not be beneficial to well-being. Glenn & Weaver (1988) reported increasing happiness among never-married males and decreasing happiness among married females. In the realm of single studies, DePaulo and colleagues (DePaulo, 2015; DePaulo & Morris, 2005, 2006) strongly assert that research does not support the popular belief whereby getting married makes people lastingly happier or definitively healthier. Furthermore, when comparing the health status of the currently-married to the different categories of unmarried people including singles, divorcees, widows, and widowers, the divorced and widowed but not the singles are always the ones who are less healthy than the currently-married (DePaulo, 2013). In the same vein, Bookwala & Fekete (2009) found that compared to their married counterparts, never-married adults experienced lower negative affect when they have higher self-sufficiency (i.e., autonomy). Moreover, among those who scored high in personal mastery (i.e., a sense of control over life events), never-married adults reported a similar level of negative affect as married adults. In other words, always single people could be as healthy and happy as people who stay married (DePaulo, 2013; DePaulo & Morris, 2006).
Amidst the acclaimed benefits of staying single, the undisputable global number of young people staying single is growing tremendously as compared to that of people involving in romantic relationships (Rich, 2019; United States Census Bureau, 2017; Wu, 2017). To illustrate, over 44% unmarried population in 2012 compared to 28% unmarried population in 1970 and 49% married couples in 2011 compared to about 70% in 1970 in the United States (DePaulo, 2014). Among them, emerging adults (e.g., undergraduate students) or younger cohorts are found staying single for longer periods (Chandler et al., 2004) as they highly regard their personal lives over social lives (Takada, 1992). Others realize the benefits of being single (e.g., Lehmann et al., 2015), for instance, autonomy, temporal control, improved sociability, career development, and freedom from others’ demands not prevalent in those peers in partnership (Whillans, 2014). Late modern individualization also witnesses singles experiencing the freedom of alternative social roles and lifestyles liberally (Galčanová & Vacková, 2016). As for some, they may simply enjoy singlehood more than others who are long searching for partners (Frazier et al., 1996).
Some studies reported singlehood that could be an individual choice for privacy and alone time that may also result from complications in securing relationships after painful experiences or simply one’s career, spiritual or religious choice (Band-Winterstein & Manchik-Rimon, 2014; Timonen & Doyle, 2014). Besides, Apostolou et al. (2020) summarized reasons for singlehood due to (1) opportunities in increased fitness; (2) mismatch between ancestral and modern conditions; and (3) different restrictions. In their examination of 648 American singles, several revealed reasons include poor flirting skills, freedom, fear of getting hurt, having different priorities, and being too picky. Men tend to be single compared to women as they did not intend to build families and wished to freely flirt around. Meanwhile, women stayed single to be safe from getting hurt besides perceiving themselves as non-desirable partners. As opposed to older people staying single to freely fulfill their needs, younger people were singles as they had poor flirting skills, appeared not as desirable mates as well as disliked commitments themselves.
Apart from that, Pepping et al. (2018) identified three subgroups of long-term singles not engaged in romantic relationships that involve: (1) singlehood due to attachment-system deactivation, (2) singlehood due to attachment-system hyperactivation, and (3) singlehood as a secure personal choice. Specifically, individuals with attachment-system deactivation usually are not willing to get intimate with others to prevent possible distress and disappointment. Meanwhile, individuals displaying attachment-system hyperactivation exhibit high proximity and intimacy to their partners but have little faith in the equal return of proximity (Mikulincer & Shaver, 2012). Fear of abandonment may result in less topical reciprocity and more aggressive clinging behavior thus causing poor well-being (Adamczyk, 2017; Lehmann et al., 2015). Finally, for individuals choosing singlehood as a secure personal choice, chosen singlehood was linked to satisfaction with a single status, self-fulfillment, and self-autonomy whereas constrained singlehood would reflect regret and dissatisfaction with single status (Timonen & Doyle, 2014). Taken together, singlehood may be deemed satisfactory for individuals only when singlehood was indeed their choice (Lehmann et al., 2015).
## Overview of the Studies
As reviewed above, while some studies found that being in a relationship is beneficial to well-being, other studies too found being single having a similar positive effect. Nevertheless, the observed rising amount of singlehood with the mixed findings (e.g., Lehmann et al., 2015) and most notably, the lack of suitable measurements (Lehmann et al., 2015) warrant the present study to develop a quantitative measure to understand the people’s attitudes toward singlehood. Moreover, the shortcomings of previous incomprehensive qualitative investigations (Apostolou, 2019) and examination in solely the Greek cultural context other than different cultural contexts (Apostolou, 2017) could be overcome in the present study. As such, attitudes among individuals remaining single could be uncovered in the present study. Furthermore, the singlehood issue deserves instant attention as despite its acclaimed benefits, singles are reported not likely to reproduce with figures as alarmingly low as less than 2 percent of children born outside marriage (e.g., in Japan and Korea; Jones, 2007). If left unattended, such reduction in marriage and fertility rates (e.g., in Japan, Atoh, 2008; Ronald et al., 2018) will escalate into other societal issues (e.g., labor shortages; Atoh, 2008) especially in aging societies.
Taking all these issues into consideration, the present study aimed to address the methodological gap by initiating quantitative development and validation of a newly designed Attitudes toward Singlehood Scale (AtSS) to measure people’s attitudes toward singlehood. Following Hogg and Vaughan’s (2005) definition that an attitude is “a relatively enduring organization of beliefs, feelings, and behavioral tendencies towards socially significant objects, groups, events or symbols” (p. 150), we employed Ostrom’s (1969) ABC model of attitudes as a reference to characterize attitudes toward singlehood in three dimensions: affect, behavior, and cognition. The affective component involves a person’s feelings (e.g., dislike) about being single. The behavioral component indicates an individual’s behavior influenced by his or her attitudes toward singlehood. Finally, the cognitive component refers to a person’s belief about singlehood. Overall, the AtSS is expected to provide researchers a tool to evaluate attitudes toward singlehood that can offer a new direction for singlehood research, for instance, in clarifying the contrast reported in the findings that singlehood is detrimental to one’s well-being when single people are found happy (Kislev, 2019). The AtSS will help reveal the reasons for such inconsistencies. Specifically, researchers may apply the AtSS to assess single people’s attitudes toward singlehood and identify those who have positive attitudes toward singlehood besides uncovering their reasons for feeling happy whenever applicable.
Four studies were conducted in the present research. Study 1 was conducted with twofold goals: first, to develop an item pool and select potential items from the pool to develop the initial version of the AtSS; second, to examine the factorial structure of the selected items using exploratory factor analysis (EFA). Study 2 was then conducted to verify the factorial structure revealed in Study 1 using confirmatory factor analysis (CFA) as well as investigate the reliability and validity of the AtSS. Convergent validity was tested by correlating the AtSS score with the score of a single item that measures preference for staying single as well as comparing AtSS scores between single participants and those in a relationship. On the other hand, discriminant validity was tested by investigating the relationship between AtSS scores with social anxiety and narcissism scores. Meanwhile, concurrent validity was tested by correlating the AtSS score with life satisfaction and well-being. The rationales of the above mentioned tests were further explained in Study 2.
Study 3 was undertaken to replicate the findings of Study 2 on a new sample as well as testing measurement invariance of the scale across genders. Finally, Study 4 aimed to understand the usability of the AtSS in different cultural backgrounds by examining the qualities of the AtSS among young adults in India. All studies were approved by the scientific and ethical review committee of Universiti Tunku Abdul Rahman (ref. no: U/SERC/72/2019) with informed consent obtained from all the participants.
## Study 1: Items Development
### Development of Item Pool and Item Selection
In groups of three to six members, specifically 164 students who enrolled in the university course of Psychological Testing and Measurement brainstormed and generated three items for each of the dimensions of the ABC model. The group members discussed and selected the most appropriate items for each dimension using the content validity ratio method. The first author then compiled all the proposed items and eliminated conceptually overlapping items, resulting in 28, 30, and 32 items for affective, behavioral, and cognitive dimensions respectively. After that, the same group of students was invited to sort the items (for each dimension) into two groups (0: suitable vs. 1: less or not suitable) and rank those items from the most important to the least important. Based on the sorting and ranking task results, the top 10 items for each dimension (i.e., 30 items in total) were selected to form the initial version of the AtSS.
### Participants and Procedure
The sample consisted of 182 undergraduate students in Malaysia (107 females and 75 males) with a mean age of 21.30 (SD = 1.82). The majority of the participants identified themselves as Chinese (86.8%) and Buddhists (69.2%). Moreover, 71.4% of the participants reported that they were not in a romantic relationship during the survey.
### Measures
#### Attitudes toward Singlehood Scale (AtSS)
The 30 items were administered to the 182 students recruited in Study 1. Respondents indicated the extent to which they agreed with the items on a 7-point Likert scale (1: strongly disagree, 7: strongly agree). Individuals who report a higher mean score tend to have positive attitudes toward singlehood (i.e., being single is beneficial).
### Results and Discussion
The responses of the 30-item AtSS were first submitted to EFA using Jamovi 1.0.0 (The jamovi project, 2019) to examine the theoretical 3-factor model. EFA results with Promax rotation and principal axis factoring estimation as well as the number fixing of factor to three indicated that the items are suitable for factor analysis: Kaiser-Meyer-Olkin (KMO) = .915 and statistically significant Bartlett’s test of sphericity, χ2 (435) = 3063, p < .001. To minimize respondents’ burden, the top five items (with the highest factor loading) loaded on the three dimensions were selected respectively to form the AtSS.
The responses of the 15 items were then submitted to another EFA to understand whether the three dimensions remained with the items loading on the target dimension. Parallel analysis was used to determine the number of factors. The results showed a 3-factor solution: KMO = .923; statistically significant Bartlett’s test of sphericity, χ2 (105) = 1584, p < .001, which explained 59.4% of the total variance (see Table 1). All items loaded on the target dimension and no cross-loading was found. Factor loadings ranged from 0.443 to 0.903 while uniqueness ranged from 0.298 to 0.552. Overall, the model showed a good fit: χ2 (63) = 96.20, p = .005, Tucker-Lewis Index (TLI) = .962, root-mean-square error of approximation (RMSEA) = .057.
Table 1. Results from an Exploratory Factor Analysis for the 15-item Attitudes toward Singlehood Scale
AtSS item Factor loading Uniqueness 1 2 3 Factor 1: Affective 1. I feel happy when I am single. .863 -.049 -.044 .347 2. I feel comfortable to lead my life by myself. .689 .049 -.018 .491 3. I feel positive for being single. .903 -.002 -.129 .298 4. I feel comfortable being single. .850 .032 -.040 .278 5. I enjoy being single. .840 .020 -.002 .273 Factor 2: Cognition 6. I think it is not necessary to be in a romantic relationship. -.110 .874 .061 .288 7. I do not need to get into a romantic relationship to live a happy life. .037 .756 .005 .385 8. Engaging in a romantic relationship is not important. -.057 .752 .090 .398 9. I think my life is complete even without a romantic partner. .270 .555 .064 .348 10. I can live independently without a partner in my life. .278 .579 -.143 .502 Factor 3: Behavior 11. I feel that being in a committed relationship distracts me from achieving my personal goals. -.197 -.070 .865 .455 12. I do not want a romantic commitment. -.178 .277 .650 .429 13. I choose to commit myself to establish a career rather than a romantic relationship. .269 .011 .496 .525 14. I have better control over my life when I am single. .243 .076 .443 .552 15. I stay single to have more personal space. .398 -.083 .449 .529
AtSS item Factor loading Uniqueness 1 2 3 Factor 1: Affective 1. I feel happy when I am single. .863 -.049 -.044 .347 2. I feel comfortable to lead my life by myself. .689 .049 -.018 .491 3. I feel positive for being single. .903 -.002 -.129 .298 4. I feel comfortable being single. .850 .032 -.040 .278 5. I enjoy being single. .840 .020 -.002 .273 Factor 2: Cognition 6. I think it is not necessary to be in a romantic relationship. -.110 .874 .061 .288 7. I do not need to get into a romantic relationship to live a happy life. .037 .756 .005 .385 8. Engaging in a romantic relationship is not important. -.057 .752 .090 .398 9. I think my life is complete even without a romantic partner. .270 .555 .064 .348 10. I can live independently without a partner in my life. .278 .579 -.143 .502 Factor 3: Behavior 11. I feel that being in a committed relationship distracts me from achieving my personal goals. -.197 -.070 .865 .455 12. I do not want a romantic commitment. -.178 .277 .650 .429 13. I choose to commit myself to establish a career rather than a romantic relationship. .269 .011 .496 .525 14. I have better control over my life when I am single. .243 .076 .443 .552 15. I stay single to have more personal space. .398 -.083 .449 .529
Note. N = 182. The extraction method was principal axis factoring with an oblique (Promax) rotation. Factor loadings above .40 are in bold.
#### Reliability and Validity
Reliability was assessed using Cronbach alpha and McDonald’s omega. The overall scale showed good internal consistency (α = .923, ω = .925) illustrated in the three subscales: α = .905 and ω = .907 for the affective subscale, α = .800 and ω = .802 for the behavioral subscale, and α = .878 and ω = .879 for the cognitive subscale.
The present study that identified half of the 30 preliminary generated items in constructing the initial version of the AtSS showed sound supports accounted for by affective, behavioral, and cognitive dimensions. Moreover, good internal consistency was observed for the overall AtSS and the three subscales. However, further analysis is required to confirm the exploratory findings. Study 2 was then conducted to fulfill the goal by examining factorial structure, reliability, and validity of the AtSS on a new sample.
## Study 2
The main goal of Study 2 was to identify the best fit model for the (15-item) AtSS. Specifically, the model revealed in Study 1 was tested and compared with several competing models such as a two-factor model with a combined (affective and behavioral) factor and cognitive factor. The results are expected to shed light on the factorial structure of the AtSS. Moreover, the reliability and validity of the AtSS were examined on a new sample.
### Participants and Procedure
The sample consisted of 308 undergraduate students (152 males and 156 females) in Malaysia aged from 18 to 29 years old (M = 20.66, SD = 1.66) with eight missing values. Of the sample, 91.9% identified themselves as Chinese (one participant did not report), 78.2% as Buddhists (followed by Christians, Hindus, and Muslims), and 99.0% as Malaysians. Moreover, 205 participants reported that they were single and 102 were in a romantic relationship (one participant did not report).
### Measurements
The reliability of the measurements was reported in the Reliability and Validity section. For all the measurements, a higher score indicates a higher level in the respective psychological construct.
#### Attitudes toward Singlehood Scale (AtSS)
The 15 items revealed in Study 1 were employed in Study 2.
#### Single Item of Preference for Being Single
Since there is no measure of inclination of being single yet, following Lehmann and colleagues’ (2015) practice, an item (“To what extent do you prefer to be single?”) was added as an indicator for content validity. Participants answered the item using a sliding scale ranging from 0 to 100. Higher scores represent a higher preference for being single.
#### Satisfaction with Life Scale (SWLS; Diener et al., 1985)
Participants responded to the 5-item SWLS using a 7-point Likert scale (1: strongly disagree, 7: strongly agree) to reveal their overall satisfaction with life.
#### Mini-Social Phobia Inventory (Mini-SPIN; Fogliati et al., 2016)
The 3-item Mini-SPIN was developed for screening social anxiety disorder. Using a 5-point Likert scale from 1 (Not at all) to 5 (Extremely), respondents reported the extent to which they were portrayed by each statement in the last 7 days.
#### Scales of General Well-Being (SGWB; Longo et al., 2018)
Respondents reported their well-being level using the SGWB. Specifically, they indicated the extent to which the 14 items truly described their overall experience in life using a 5-point Likert scale ranging from 1 (Not at all true) to 5 (Very true).
#### Single Item Narcissism Scale (SINS; Konrath et al., 2014)
The SINS was used to assess the narcissism level. Participants reported the extent to which they agreed with the item “I’m a narcissist” on a 7-point Likert scale (1: strongly disagree, 7: strongly agree). A definition (of a narcissist) was provided to help respondents understand the term.
### Analytic Strategies
Several commonly reported indicators were used to examine and compare the appropriateness of the tested models. According to Hu & Bentler (1999), Tabachnick & Fidell (2007), and Steiger (2000), a model that shows (a) χ2/df < 3, (b) comparative fit index (CFI) and TLI > .95, (c) RMSEA ≤ .05, and (d) standardized root mean square residual (SRMR) < .08 is a good-fit model. Modification indices were referred and would be adjusted if the preferred model did not meet the suggested cut-off criteria.
In the present study, the reliability of the measurements was evaluated using Cronbach alpha and McDonald omega coefficients. The same participants were invited to answer the AtSS in two weeks’ intervals for assessing test-retest reliability of the AtSS. Convergent validity was measured by the association between scores of the AtSS and preference for being single as well as the compared AtSS scores between participants who were in a relationship and were single. The latter is expected to report a higher score in AtSS. On the other hand, discriminant validity was tested by investigating the relationship between AtSS scores with social anxiety and narcissism scores. Literature has shown that people who are socially anxious (Pepping et al., 2018; Sparrevohn & Rapee, 2009) and too picky (Apostolou et al., 2020) tend to remain single. Therefore, it is important to demonstrate that the AtSS is not measuring social anxiety and narcissism. A non-significant or weak relationship would also distinguish the attitudes toward singlehood from anxiety and narcissism. Finally, based on the findings that people who have positive attitudes toward singlehood tend to be happier with their single lives (Kislev, 2019), concurrent validity was tested by correlating the AtSS score with life satisfaction and well-being. Specifically, it is assumed that AtSS will have a positive relationship with life satisfaction and well-being respectively for single individuals but not for those in a relationship.
### Results and Discussion
Several CFAs (with maximum likelihood estimator) were conducted using JASP 0.10.2 to examine the following five competing models: (1) single-factor model (Model 1), (2) two-factor models in which two of the three factors were combined into one factor, for example, the joint factor (of affective and behavior dimensions) and cognitive factor (Model 2a to 2c), and (3) second-order model with three first-order factors (Model 3). The 3-factor model that was statistically similar to the second-order factor model was not included. The three factors were found highly correlated with each other (rs > .74). CFA results (see Table 2) showed the single factor model and all two-factor models poor fit. Meanwhile, Model 3 showed an acceptable fit to the data. We then referred to modification indices for suggestions of further improvement. The modified model (model 3a) with residual covariance between items 14 (“I have better control over my life when I am single”) and 15 (“I stay single to have more personal space”) as well as items 3 (“I feel positive for being single”) and 4 (“I feel comfortable being single”) showed a better fit than Model 3. In addition, inspection on the modification indices suggested that some of the items may be conceptually overlapping with others. Therefore, we selected the three items with the highest factor loading from each of the three factors to develop and test a 9-item version of the AtSS. The second-order factor model with nine items (Model 4) was found superior to all other tested models and hence, was selected to represent the structure of the AtSS. The (standardized) factor loadings of the three first-order factors on the attitudes toward singlehood were 0.87, 0.88, and 0.77 for affective, behavioral, and cognitive dimensions, while the factor loadings for the nine items ranged from 0.686 to 0.93 (see Table 3).
Table 2. Goodness-of-fit Indices for the Attitudes toward Singlehood Scale (Study 2)
Model χ2 df χ2/df CFI TLI RMSEA [90% CI] SRMR BIC 1 One-factor 824.51 90 9.16 .788 .753 .164 [.154, .175] .086 13947.09 2a Two-factor: Combined & Affective 490.09 89 5.51 .884 .864 .122 [.112, .133] .070 13618.38 2b Two-factor: Combined & Behavior 694.46 89 7.80 .826 .794 .150 [.140, .161] .082 13822.75 2c Two-factor: Combined & Cognition 577.92 89 6.49 .859 .834 .135 [.125, .145] .086 13706.21 3 2nd-order with 15 items 364.85 87 4.19 .920 .903 .103 [.092, .114] .059 13504.56 3a 2nd-order with 15 items and residual covariancea 281.17 85 3.31 .943 .930 .087 [.076, .099] .061 13432.30 4 2nd-order with 9 items 98.62 24 4.11 .959 .939 .101 [.081, .122] .055 8275.52
Model χ2 df χ2/df CFI TLI RMSEA [90% CI] SRMR BIC 1 One-factor 824.51 90 9.16 .788 .753 .164 [.154, .175] .086 13947.09 2a Two-factor: Combined & Affective 490.09 89 5.51 .884 .864 .122 [.112, .133] .070 13618.38 2b Two-factor: Combined & Behavior 694.46 89 7.80 .826 .794 .150 [.140, .161] .082 13822.75 2c Two-factor: Combined & Cognition 577.92 89 6.49 .859 .834 .135 [.125, .145] .086 13706.21 3 2nd-order with 15 items 364.85 87 4.19 .920 .903 .103 [.092, .114] .059 13504.56 3a 2nd-order with 15 items and residual covariancea 281.17 85 3.31 .943 .930 .087 [.076, .099] .061 13432.30 4 2nd-order with 9 items 98.62 24 4.11 .959 .939 .101 [.081, .122] .055 8275.52
Note. N = 308. CFI = comparative fit index, TLI = Tucker-Lewis index, RMSEA = root-mean-square error of approximation, CI: confidence interval, SRMR = standardized root mean square residual, BIC = bayesian information criterion. a residual covariance was added between items 14 and 15 as well as Items 3 and 4. All chi-square values were significant at the .001 level.
Table 3. Standardized Factor Loading for the 9 items of the Attitudes toward Singlehood Scale
Standardized Factor Loading No.a Item Study 2 Study 3 Study 4 1/1 I feel happy when I am single. .818 .780 .888 3/2 I feel positive for being single. .882 .923 .945 4/3 I feel comfortable being single. .933 .913 .914 13/4 I choose to commit myself to establish a career rather than a romantic relationship. .686 .696 .853 14/5 I have better control over my life when I am single. .848 .826 .878 15/6 I stay single to have more personal space. .829 .833 .784 7/7 I do not need to get into a romantic relationship to live a happy life. .800 .836 .883 8/8 Engaging in a romantic relationship is not important. .831 .847 .876 9/9 I think my life is complete even without a romantic partner. .818 .832 .863
Standardized Factor Loading No.a Item Study 2 Study 3 Study 4 1/1 I feel happy when I am single. .818 .780 .888 3/2 I feel positive for being single. .882 .923 .945 4/3 I feel comfortable being single. .933 .913 .914 13/4 I choose to commit myself to establish a career rather than a romantic relationship. .686 .696 .853 14/5 I have better control over my life when I am single. .848 .826 .878 15/6 I stay single to have more personal space. .829 .833 .784 7/7 I do not need to get into a romantic relationship to live a happy life. .800 .836 .883 8/8 Engaging in a romantic relationship is not important. .831 .847 .876 9/9 I think my life is complete even without a romantic partner. .818 .832 .863
Note. aThe item number in the 15-item version / the 9-item version.
#### Reliability
Table 4 shows the reliability for AtSS, SWLS, social anxiety, and SGWB as well as the test-retest reliability for the 9-item AtSS. All the measures including the subscales of the AtSS showed good to excellent internal consistency (αs >. 814, ωs > .822). Moreover, the AtSS was found to have good test-retest reliability. To further ensure that the AtSS applies to all individuals regardless of their relationship status, we also examined the reliability of the AtSS for respondents who were single and currently in a relationship. The results were consistent with those reported above and hence, were not reported here for the sake of clarity. The results are available upon request to the corresponding author.
Table 4. Mean, Standard Deviation, and Reliability for Target Variables (Study 2)
Alpha Omega Test-retest No. Variable MT1 SDT1 MT2 SDT1 T1 T2 T1 T2 1 AtSS 4.44 1.15 4.36 1.25 .913 .934 .915 .935 .72*** 1a Affect 4.90 1.28 4.55 1.50 .910 .963 .912 .964 .69*** 1b Behavior 4.51 1.26 4.43 1.26 .824 .858 .831 .863 .54*** 1c Cognition 3.92 1.43 4.12 1.44 .856 .860 .857 .864 .65*** 2 Satisfaction 4.21 1.05 - - .814 - .822 - - 3 Social anxiety 5.93 3.02 - - .832 - .832 - - 4 GWB 3.43 0.70 - - .918 - .919 - -
Alpha Omega Test-retest No. Variable MT1 SDT1 MT2 SDT1 T1 T2 T1 T2 1 AtSS 4.44 1.15 4.36 1.25 .913 .934 .915 .935 .72*** 1a Affect 4.90 1.28 4.55 1.50 .910 .963 .912 .964 .69*** 1b Behavior 4.51 1.26 4.43 1.26 .824 .858 .831 .863 .54*** 1c Cognition 3.92 1.43 4.12 1.44 .856 .860 .857 .864 .65*** 2 Satisfaction 4.21 1.05 - - .814 - .822 - - 3 Social anxiety 5.93 3.02 - - .832 - .832 - - 4 GWB 3.43 0.70 - - .918 - .919 - -
Note. NT1 = 308; NT2 = 66. Alpha = Cronbach alpha coefficient; Omega = McDonald omega coefficient; Test-retest = test-retest reliability; M = mean score for the whole sample; SD = standard deviation for the whole sample; T1 = time 1; T2 = time 2; AtSS = 9-item Attitudes toward Singlehood Scale; Affect = affective subscales of the AtSS with 3 items; Behavior = behavioral subscales of the AtSS with 3 items; Cognition = cognitive subscales of the AtSS with 3 items; Satisfaction = life satisfaction; GWB = general well-being. ***p < .001
#### Validity
Convergent validity was tested in two ways. First, we compared overall and subscale AtSS scores of respondents who were single and in relationships. Results showed that single individuals reported significantly higher scores than their counterparts in relationships in all the AtSS scores (ts > 4.545, p < .001). In addition, we examined the relationship between AtSS scores and preference for staying single. Considering that the relationship between AtSS and other variables may be varied in those single and those in a relationship, we analyzed the associations of the variables separately for the two groups. The overall and three subscale scores of the AtSS positively correlated with the preference for staying single for both groups (see Table 5).
Discriminant validity was tested by the relationship between the AtSS scores with social anxiety and narcissism. There was no relationship between AtSS scores and social anxiety for all participants regardless of their relationship conditions apart from the positive but weak relationship between the behavioral subscale score and social anxiety for single participants. Narcissism, on the other hand, had a positive relationship with the overall AtSS score (i.e., the average across nine items) for single participants. Furthermore, both groups of participants reported a positive relationship between narcissism and the behavioral subscale of the AtSS.
Finally, concurrent validity was examined by the correlation between the AtSS scores with life satisfaction and general well-being. The overall and three subscale AtSS scores were positively correlated with life satisfaction scores for single participants but not for those in a relationship. Similarly, there was a positive relationship between the overall and subscale AtSS scores, except the cognitive subscale score, with general well-being.
Taken together, the findings of Study 2 indicate that the AtSS is best accounted for by the 9-item second-order model that lends support to the reliability and validity of the 9-item AtSS. Note that, however, the 9-item AtSS was explored based on the suggestion of modification indices. Therefore, it is necessary to examine the 9-item AtSS with a new dataset to confirm its superiority.
Table 5. Correlation between Attitudes toward Singlehood Scale and Other Variables (Study 2)
1 1a 1b 1c 2 3 4 5 6 1 AtSS 1 .85*** .85*** .83*** .04 .05 -.19 .15 .59*** 1a Affect .86*** 1 .62*** .54*** .07 -.01 -.13 .03 .45*** 1b Bhv .85*** .63*** 1 .54*** .01 .68 -.22 .25* .52*** 1c Cognition .87*** .62*** .60*** 1 .01 .06 -.12 .10 .51*** 2 LS .35*** .39*** .30*** .23** 1 -.07 .44*** -.01 -.05 3 SocAnx .12 .08 .14* .10 -.01 1 -.26** -.02 .03* 4 GWB .22** .34** .15* .09 .57*** -.14* 1 -.02 -.10 5 Narcissism .18** .13 .21** .13 -.06 .16* .01 1 .11 6 Preference .64*** .63*** .51*** .52*** .14* .07 .11 .15* 1
1 1a 1b 1c 2 3 4 5 6 1 AtSS 1 .85*** .85*** .83*** .04 .05 -.19 .15 .59*** 1a Affect .86*** 1 .62*** .54*** .07 -.01 -.13 .03 .45*** 1b Bhv .85*** .63*** 1 .54*** .01 .68 -.22 .25* .52*** 1c Cognition .87*** .62*** .60*** 1 .01 .06 -.12 .10 .51*** 2 LS .35*** .39*** .30*** .23** 1 -.07 .44*** -.01 -.05 3 SocAnx .12 .08 .14* .10 -.01 1 -.26** -.02 .03* 4 GWB .22** .34** .15* .09 .57*** -.14* 1 -.02 -.10 5 Narcissism .18** .13 .21** .13 -.06 .16* .01 1 .11 6 Preference .64*** .63*** .51*** .52*** .14* .07 .11 .15* 1
Note. AtSS = Attitudes toward Singlehood Scale; Bhv = behavioral dimension of attitudes toward singlehood; LS: life satisfaction; SocAnx = social anxiety; GWB: general well-being; Preference = preference for staying single. The below diagonal line shows the correlation between AtSS and other variables for single participants (n = 196 to 205). The above diagonal line shows the correlation between AtSS and other variables for participants in a relationship (n = 99 to 102). *p < .05; **p < .01; ***p < .001
## Study 3
The main goal of Study 3 was to investigate psychometric qualities of the 9-item AtSS which was revealed on an exploratory basis in Study 2. Meanwhile, the 15-item AtSS was used in the present study to compare competing models and identify the best fit model.
### Participants and Procedure
A total of 444 undergraduate students in Malaysia participated in Study 3. There were 235 female and 209 male students with a mean age of 20.44 (SD = 1.14, range = 18 to 25). The majority of the participants identified themselves as Chinese (93.5%) and Buddhists (84.9%). Moreover, 142 of the students were in a romantic relationship while 302 of them were single. The participants were recruited for a subsequent larger project and received additional course credit for answering an online survey.
### Measurement
The 15-item AtSS, 5-item SWLS (Diener et al., 1985), 3-item SINS (Konrath et al., 2014), Mini-SPIN (Fogliati et al., 2016), and a single-item of preference for staying single used in Study 2 were employed in the present study.
### Statistical Analysis
The analyses were consistent with Study 2 with the exception that test-retest reliability was not examined and the SGWB was not included to examine concurrent validity.
### Results and Discussion
To further clarify the superiority of the 9-item AtSS, CFAs were carried out to examine and compare the second-order factor models of the 15- and 9-item versions. Although the 15-item second-order factor model was acceptable: χ2(87) = 490.47, p < .001, CFI = .921, TLI = .905, RMSEA = .102, 90% confidence interval (CI) [.093, .111], SRMR = .066, the 9-item second-order factor model demonstrated a better fit: χ2(24) = 86.95, p < .001, CFI = .977, TLI = .965, RMSEA = .077, 90% CI [.060, .095], SRMR = .043. As a result, the 9-item version is preferably adopted. The standardized factor loadings ranged from .696 to .923 (see Table 3).
#### Reliability
Table 6 shows the descriptive statistics, correlation, and reliability coefficients for the 9-item AtSS and other measurements. All the measurements showed good internal consistency. As in Study 2, correlation analysis was conducted for the single participants and those in relationships respectively. The overall and three subscale scores AtSS were positively correlated with each other for both groups of participants.
#### Validity
As in Study 2, convergent validity was first tested by comparing the AtSS score between single participants and those in a relationship. Independent t-test showed that single individuals reported significantly higher scores than their counterparts in relationships in the overall and all subscale scores of the AtSS (ts > 3.96, p < .001). In addition, the overall and subscale scores of the AtSS consistently showed a positive association with the single item preference for staying single score regardless of the participants’ relationship conditions. The findings are congruent with the results of Study 2 and shed light on the convergent validity of the AtSS.
Furthermore, the overall and subscale AtSS scores did not show negative relationships with both social anxiety and narcissism. The results remained for single participants and those who were in a relationship. The only exception is that the affective subscale score was negatively associated with social anxiety for the single participants. The nonsignificant results highlight the conceptual differences between attitudes toward singlehood and the two constructs that offer support to the discriminant validity of the AtSS.
Finally, mixed findings were found for life satisfaction. The results for single individuals showed the overall and subscale scores of AtSS that positively associated with life satisfaction score apart from the relationship with the cognitive subscale score. On the other hand, for those in a relationship, both the overall score and the behavioral subscale scores negatively correlated with life satisfaction while no relationship was observed for the affective and cognitive subscale scores. The discrepancies between the two groups lend further support to people’s positive attitudes toward singlehood that can be advantageous to well-being besides demonstrating the concurrent validity of the AtSS.
Table 6. Descriptive Statistics, Correlation, and Reliability for Measurements (Study 3)
1 1a 1b 1c 2 3 4 5 1 AtSS 1 .89*** .83*** .84*** -.20* 0.14 -.09 .59*** 1a Affect .86*** 1 .63*** .64*** -.16 0.13 -.11 .52*** 1b Bhv .87*** .71*** 1 .50*** -.25** 0.14 -.09 .48*** 1c Cognition .85*** .56*** .56*** 1 -.11 0.09 -.03 .49*** 2 LS .13* .20*** .12* .04 1 -0.14 .02 -.14 3 SocAnx -.02 -.12* -.003 .06 -.22*** 1 .09 .09 4 Narcissism .01 .04 .01 -.02 .002 -0.02 1 -.12 5 Preference .57*** .53*** .51*** .43*** -.05 -0.02 .02 1 Moverall 4.30 4.65 4.40 3.85 4.33 6.09 3.74 46.63 SDoverall 1.15 1.30 1.27 1.42 1.00 2.81 1.41 26.98 αoverall .913 .903 .826 .875 .821 0.816 NA NA ωoverall .915 .906 .829 .876 .833 0.819 NA NA
1 1a 1b 1c 2 3 4 5 1 AtSS 1 .89*** .83*** .84*** -.20* 0.14 -.09 .59*** 1a Affect .86*** 1 .63*** .64*** -.16 0.13 -.11 .52*** 1b Bhv .87*** .71*** 1 .50*** -.25** 0.14 -.09 .48*** 1c Cognition .85*** .56*** .56*** 1 -.11 0.09 -.03 .49*** 2 LS .13* .20*** .12* .04 1 -0.14 .02 -.14 3 SocAnx -.02 -.12* -.003 .06 -.22*** 1 .09 .09 4 Narcissism .01 .04 .01 -.02 .002 -0.02 1 -.12 5 Preference .57*** .53*** .51*** .43*** -.05 -0.02 .02 1 Moverall 4.30 4.65 4.40 3.85 4.33 6.09 3.74 46.63 SDoverall 1.15 1.30 1.27 1.42 1.00 2.81 1.41 26.98 αoverall .913 .903 .826 .875 .821 0.816 NA NA ωoverall .915 .906 .829 .876 .833 0.819 NA NA
Note. N = 444 for all measurements except for Preference (N = 439). AtSS = Attitudes toward Singlehood Scale; Bhv = behavioral dimension of attitudes toward singlehood; LS = life satisfaction; SocAnx = social anxiety; Preference = preference for staying single; M = mean, SD = standard deviation, α = Cronbach alpha; ω = McDonald omega; NA = not applicable. Below diagonal line shows the correlation between AtSS and other variables for single participants (n = 299 to 302). Above diagonal line shows the correlation between AtSS and other variables for participants in a relationship (n = 140 to 142). *p < .05; ***p < .01 ***p < .001
#### Measurement Invariance across Genders
As the sample consisted of a comparable number of male (n = 209) and female (n = 235) respondents, we ran an additional analysis to examine if the 9-item AtSS structure is equivalent across the two gender groups using maximum likelihood estimator and fixing residual variance. The baseline model of the two gender groups was first investigated by conducting CFA on the two datasets respectively. Results showed that the 9-item second-order model fit the data for both genders (see Table 7). Next, we submitted the combined dataset to CFA and found support for the configural invariance (Model 1). Then, the metric invariance (i.e., fixing factor loadings across the groups; Model 2) was examined by comparing the chi-square value (∆ χ2) and CFI value (∆CFI) of Model 1 and Model 2. Results showed that the ∆ χ2 was not statistically significant with ∆CFI < .01 (Cheung & Rensvold, 2002) indicating that metric invariance is supported. Finally, scalar invariance (i.e., fixing intercept; Model 3) was examined. The (Model 2 vs. Model 3) results indicated that ∆ χ2 was not significant and ∆CFI was less than .01, suggesting that the intercepts are equivalent across the two groups. As scalar variance is supported, we conducted an analysis of covariance (ANCOVA) to compare the attitudes toward singlehood score between the two gender groups. Narcissism was included as a covariate variable because we found that male participants scored higher than female participants in narcissism: t(442) = 2.585, p = .01, Mmale = 3.92 (SDmale = 1.44) vs. Mfemale = 3.57 (SDfemale = 1.37). Therefore, it is important to statistically control for the effect of narcissism to examine if there is a gender difference in attitudes toward singlehood. ANCOVA results showed that the gender effect, but not narcissism, was significant: F(1,441) = 18.932, p < .001, $ηp2$ = .041. Female participants (M = 4.52, SD = 1.11) reported a higher (attitudes toward singlehood) value than their male counterparts (M = 4.05, SD = 1.15). A similar measurement invariance test was also conducted on single individuals. The results are consistent with the above report and female participants (n = 164) scored higher than male counterparts (n = 138). For the sake of clarity, the analysis outputs were not presented here but are available upon request to the corresponding author. Notably, the measurement invariance test was not conducted on individuals who were currently in a relationship due to the small sample size.
Collectively, the findings are consistent with Study 2 supporting that the 9-item AtSS has sound reliability and validity. The measurement invariance test further supported that the construct measured by the 9-item AtSS measures is equivalent across male and female individuals.
Table 7. Goodness-of-fit indices for Tests of invariance of 9-item Attitudes toward Singlehood Scale Across Genders
Baseline Model χ2 df χ2/df CFI TLI RMSEA [90% CI] SRMR Male (N=209) 66.829 24 2.78 .966 .949 .092 [.067, .119] .053 Female (N=235) 51.090 24 2.13 .980 .970 .069 [.043, .096] .035 MI χ2 df CFI Model Comparison ∆ χ2 df p ∆CFI Model 1: Configural invariance 117.918 48 .973 - - - - - Model 2: Metric invariance 121.896 60 .976 2 vs. 1 3.978 12 .983 .003 Model 3: Scalar invariance 129.818 65 .975 3 vs. 2 7.922 5 .161 .001
Baseline Model χ2 df χ2/df CFI TLI RMSEA [90% CI] SRMR Male (N=209) 66.829 24 2.78 .966 .949 .092 [.067, .119] .053 Female (N=235) 51.090 24 2.13 .980 .970 .069 [.043, .096] .035 MI χ2 df CFI Model Comparison ∆ χ2 df p ∆CFI Model 1: Configural invariance 117.918 48 .973 - - - - - Model 2: Metric invariance 121.896 60 .976 2 vs. 1 3.978 12 .983 .003 Model 3: Scalar invariance 129.818 65 .975 3 vs. 2 7.922 5 .161 .001
Note. Using maximum likelihood estimator and fixing residual variance. CFI = comparative fit index; TLI = Tucker-Lewis index; RMSEA = root-mean-square error of approximation; CI: confidence interval; SRMR = standardized root mean square residual; MI: measurement invariance; ∆ χ2 = difference in chi-square value; ∆CFI = difference in CFI value.
## Study 4
Study 4 aimed to examine the psychometric qualities of the 9-item AtSS in a different cultural group. The results will shed light on the applicability of the 9-item AtSS in different cultural contexts.
### Participants and Procedure
The sample consisted of 342 undergraduate students (220 female and 122 male students) in India. The mean age was 23.35 (SD = 3.13, range = 18 to 35). The sample mainly consisted of Indians (97.7%). There were 46.5% Hindus, 38.6% Christians, 12.9% Muslims, and 2% non-religious students. Moreover, 109 students were in a romantic relationship while 233 of them were single. The participants were recruited for a subsequent larger project. Their participation was voluntary and they did not receive any rewards for answering an online survey.
### Measurement and Statistical Analysis
Study 4 employed the same measurements and statistical analysis as in Study 3 except that attitudes toward singlehood were measured by the 9-item AtSS.
### Results and Discussion
Table 8 summarizes the CFA results for the four competing models. The unidimensional model and all the 2-factor models (i.e., Model 2a to 2c) showed poor fit except Model 2c. The hypothetical second-order model with three first-order factors (i.e., Model 3) not only showed a good fit but also outperformed other models. Hence, Model 3 is selected to represent the factorial structure of the 9-item AtSS. The factor loadings ranged from .784 to 945. Since the RMSEA value of Model 3 exceeded .80, we referred to the modification indices for potential improvements’ suggestions. Specifically, item 7 (“I do not need to get into a romantic relationship to live a happy life”) was allowed to cross-load on the affective subscale. All the indicators of the modified model (Model 3a) were within the suggested range.
Table 8. Goodness-of-fit Indices for the Attitudes toward Singlehood Scale (Study 4)
Model χ2 df χ2/df CFI TLI RMSEA [90% CI] SRMR BIC 1 One-factor 358.56 27 .895 .861 .189 [.172, .207] .059 9070.013 2a Two-factor: Combined & Affective 258.87 26 .927 .898 .162 [.144, .180] .048 8976.157 2b Two-factor: Combined & Behavior 347.05 26 .899 .860 .190 [.173, .208] .060 9064.339 2c Two-factor: Combined & Cognition 151.21 26 .960 .945 .119 [.101, .137] .037 8868.503 3 2nd-order 101.87 24 .975 .963 .097 [.078, .117] .030 8830.834 3a 2nd-order with cross-loading a 67.37 23 .986 .978 .075 [.055, .096] .018 8802.165
Model χ2 df χ2/df CFI TLI RMSEA [90% CI] SRMR BIC 1 One-factor 358.56 27 .895 .861 .189 [.172, .207] .059 9070.013 2a Two-factor: Combined & Affective 258.87 26 .927 .898 .162 [.144, .180] .048 8976.157 2b Two-factor: Combined & Behavior 347.05 26 .899 .860 .190 [.173, .208] .060 9064.339 2c Two-factor: Combined & Cognition 151.21 26 .960 .945 .119 [.101, .137] .037 8868.503 3 2nd-order 101.87 24 .975 .963 .097 [.078, .117] .030 8830.834 3a 2nd-order with cross-loading a 67.37 23 .986 .978 .075 [.055, .096] .018 8802.165
Note. N = 342. CFI = comparative fit index, TLI = Tucker-Lewis index, RMSEA = root-mean-square error of approximation, CI: confidence interval, SRMR = standardized root mean square residual, BIC = bayesian information criterion. a Item 7 cross loaded on affective and cognitive factors. All chi-square values were significant at the .001 level.
#### Reliability
Table 9 shows the descriptive statistics, correlation, and reliability coefficients for AtSS and other measurements. The reliability test showed that all measurements used in Study 4 had good internal consistency. Similarly, correlation results were presented for the single participants and those in a relationship respectively. The overall and the three subscale scores of AtSS were positively correlated with each other for both groups of participants.
Table 9. Descriptive Statistics, Correlation, and Reliability for Measurements (Study 4)
1 1a 1b 1c 2 3 4 5 1 AtSS 1 .94*** .94*** .91*** .01 .26** -.05 .55*** 1a Affect .91*** 1 .85*** .76*** .03 .26** -.05 .62*** 1b Bhv .94*** .84*** 1 .75*** -.02 .18 -.03 .54*** 1c Cognition .88*** .65*** .73*** 1 .003 .28** -.06 .38*** 2 LS .25*** .32*** .26* .11 1 -.03 .24* -.14 3 SocAnx -.01 .05 .05 -.11 .06 1 .02 .15 4 Narcissism .04 .05 .08 -.02 .23*** .30*** 1 -.05 5 Preference .65*** .64*** .59*** .55*** .11 -.04 .05 1 Moverall 4.44 4.61 4.48 4.24 3.87 4.70 2.53 52.42 SDoverall 1.42 1.46 1.54 1.58 1.34 3.15 1.49 23.38 αoverall .956 .946 .892 .907 .908 .883 NA NA ωoverall .955 .944 .889 .907 .906 .881 NA NA
1 1a 1b 1c 2 3 4 5 1 AtSS 1 .94*** .94*** .91*** .01 .26** -.05 .55*** 1a Affect .91*** 1 .85*** .76*** .03 .26** -.05 .62*** 1b Bhv .94*** .84*** 1 .75*** -.02 .18 -.03 .54*** 1c Cognition .88*** .65*** .73*** 1 .003 .28** -.06 .38*** 2 LS .25*** .32*** .26* .11 1 -.03 .24* -.14 3 SocAnx -.01 .05 .05 -.11 .06 1 .02 .15 4 Narcissism .04 .05 .08 -.02 .23*** .30*** 1 -.05 5 Preference .65*** .64*** .59*** .55*** .11 -.04 .05 1 Moverall 4.44 4.61 4.48 4.24 3.87 4.70 2.53 52.42 SDoverall 1.42 1.46 1.54 1.58 1.34 3.15 1.49 23.38 αoverall .956 .946 .892 .907 .908 .883 NA NA ωoverall .955 .944 .889 .907 .906 .881 NA NA
Note. N = 342 for all measurements except for Preference (N = 339). AtSS = Attitudes toward Singlehood Scale; Bhv = behavioral dimension of attitudes toward singlehood; LS = life satisfaction; SocAnx = social anxiety; Preference = preference for staying single; M = mean, SD = standard deviation, α = Cronbach alpha; ω = McDonald omega; NA = not applicable. Below diagonal line shows the correlation between AtSS and other variables for single participants (n = 233). Above diagonal line shows the correlation between AtSS and other variables for participants in a relationship (n = 109). *p < .05; ***p < .01 ***p < .001
#### Validity
An independent t-test was first conducted to examine differences in AtSS scores between single participants and those in a relationship. As reported in Study 2 and Study 3, single individuals achieved significantly higher scores than their counterparts in relationships in the overall and all subscale scores of the AtSS (ts > 7.55, p < .001). Similarly, a positive relationship between the single item preference for staying single score and the overall and subscale scores of the AtSS was found in both groups of participants. The findings consistently support the convergent validity of the AtSS.
Furthermore, the overall and subscale scores of the AtSS did not show a relationship with social anxiety for single participants. However, there was a positive relationship between the overall and subscale AtSS scores except for the behavioral subscale score and social anxiety for those in a relationship. In other words, individuals in relationships tend to experience higher social anxiety if they show positive attitudes toward singlehood affectively and cognitively, and vice versa. On the other hand, there was no relationship between the overall and subscale AtSS scores and narcissism scores for single participants and those in a relationship. The nonsignificant results indicate that attitudes toward singlehood are conceptually different from social anxiety and narcissism especially for the single participants, supporting the discriminant validity of the AtSS.
Finally, results on single individuals showed a positive relationship between the overall score and subscale scores of the AtSS with life satisfaction score apart from the relationship with the cognitive subscale score. On the other hand, for those in a relationship, both the overall score and subscale scores of the AtSS did not have a relationship with life satisfaction. The differences between the two groups of participants support the notion that people’s favorable attitudes toward singlehood can enhance well-being as well as the concurrent validity of the AtSS.
## General Discussion
The present research developed and examined psychometric properties of the Attitudes toward Singlehood Scale (AtSS) in four studies. To our best knowledge, it is the first multidimensional quantitative measurement for individuals to self-report their attitudes toward singlehood.
Based on the ABC model of attitudes, preliminary items of the AtSS were designed to address the affective, behavioral, and cognitive components of people’s attitudes toward singlehood. The EFA results supported that the selected 15 items can be accounted for by a 3-factor model (Study 1). Study 2 then demonstrated that a second-order model with a general single-by-choice factor and three first-order subfactors outperformed other competing models. Out of our expectations, a shorter version of the AtSS with nine items is superior to the 15-item version. Similarly, the results of Study 3 further support the superiority of the 9-item version. Finally, the 9-item version also showed a good fit in a sample of undergraduate students in India (Study 4). The findings across the studies indicate that the 9-item AtSS is best represented by a second-order structure and the structure holds in two different cultural contexts. The latter provides the first piece of evidence to the cross-cultural adaptability of the AtSS.
Overall, the 9-item AtSS demonstrated excellent reliability. Both Cronbach alpha and McDonald omega coefficients were greater than .82 for the overall scale and the three subscales of the AtSS in Study 2 through Study 4. Furthermore, the test-retest reliability with a two-week interval was good (r > .54). The congruent results support that the 9-item AtSS has good internal consistency.
Besides, the 9-item AtSS also demonstrates good validity. In line with our predictions, the 9-item AtSS score was found to have a positive relationship with the preference for staying single. In other words, individuals who favor staying single also scored higher in the AtSS. Similarly, in line with our assumption, the comparison between those in a relationship and singlehood showed that the latter reported a higher score in the AtSS. The abovementioned results were consistently observed in Study 2 through Study 4 thus lending support to the convergent validity of the AtSS. In other words, the AtSS has the potential to reveal the extent to which individuals would rather stay single than engage in a relationship.
Meanwhile, the discriminant validity of the (9-item) AtSS is also evident. For those who were single, both social anxiety and narcissism had a positive but weak relationship with the behavioral subscale of the AtSS respectively (Study 2), while social anxiety was negatively and weakly associated with the affective subscale of AtSS in Study 2 and Study 3. Likewise, neither the AtSS overall scale nor its subscales had a relationship with social anxiety and narcissism respectively in Study 4. The findings demonstrate that attitudes toward singlehood are conceptually different from social anxiety and narcissism. Hence, further studies are necessary to clarify the mixed findings. For instance, it would be interesting to understand if social anxiety and narcissism could be antecedent factors of attitudes toward singlehood.
In addition, the AtSS shows good criterion validity. Specifically, the overall AtSS score was found to have a positive relationship with life satisfaction (Study 2 through Study 4) and general well-being (Study 2) for single individuals. The overall AtSS score, however, had no relationship with life satisfaction and well-being for those in a relationship (Study 2 and Study 4). Study 3 even showed a negative relationship between the overall AtSS score and life satisfaction. In other words, individuals who have positive attitudes toward singlehood are more likely to be satisfied with their lives and have high levels of well-being. The findings not only support the concurrent validity of the AtSS but also offer preliminary evidence that having positive attitudes toward singlehood is beneficial to individuals’ well-being. The latter suggests a new direction to reconcile the mixed findings of the relationship between marriage and well-being. In a similar vein, the weak to moderate correlation between attitudes toward singlehood and life satisfaction scores (for single participants) offers recommendations for future researchers to further investigate the causal relationship between the two constructs using a longitudinal design.
In essence, the 9-item AtSS shows measurement invariance across genders (Study 3). The results highlight that the AtSS is appropriate for measuring the construct of attitudes toward singlehood among male and female young adults. The measurement equivalence also allows the investigation of gender differences in attitudes toward singlehood. Female participants reported a higher score in attitudes toward singlehood than their male counterparts. Future studies are thus recommended to explore the causes and effects of the gender difference in attitudes toward singlehood.
Taken together, the development of the 9-item AtSS has a significant contribution to the literature. In particular, the emergence of the AtSS fills the methodological gap by allowing researchers to measure people’s attitudes toward singlehood. The distinction helps clarify the inconsistent findings of the consequences of being single besides identifying the positive and negative impacts of attitudes toward singlehood. Practically, researchers may also use the AtSS to identify the antecedent factors of attitudes toward singlehood. The results are expected to indirectly shed light on the issue of divorce and the low birth rate.
Nevertheless, although the results support that the 9-item AtSS is useful, the scale is far from being the perfect tool. It is important to note that the responses of the 9 items in both Study 2 and Study 3 were derived from the 15-item version. There lies a possibility that the results for the 9-item AtSS are confounded by the other items. Therefore, it is crucial to examine and verify the qualities of the 9-item AtSS in the local context. As the 9-item AtSS also shows sound properties in another sample with different cultural backgrounds, we are optimistic about the performance of the scale. Meanwhile, future researchers are reminded of further examination on the factorial structure and psychometric qualities of the AtSS, for instance, by examining the validity of the 9-item AtSS using other measurements such as the Well-Being Profile (Marsh et al., 2020) and fear of being single (Spielmann et al., 2013). In addition, the present study was limited to young adults in two countries. It is unknown whether the AtSS has equivalent sound properties in other population and cultural groups. Future researchers are thus suggested to extend their focus to other age groups and to translate the AtSS into local languages in investigating the applicability of the AtSS in other cultural contexts.
### Conclusion
The present study developed and validated the 9-item Attitudes toward Singlehood Scale (AtSS) to address the need for a quantitative assessment of attitudes toward singlehood. Results support that the AtSS is a useful measurement of young adults’ attitudes toward singlehood thus underpinning its potential for examining the relationship between attitudes toward singlehood and individuals’ well-being.
## Contribution
Contributed to conception and design: CST, SMC
Contributed to acquisition of data: CST, SMC, & SG
Contributed to analysis and interpretation of data: CST
Drafted and/or revised the article: CST, SMC
Approved the submitted version for publication: CST, SMC, & SG
## Acknowledgements
The authors thank students of Jan2019 UAPP2013 class for their suggestion and participation.
## Funding Information
The authors did not receive support from any organization for the submitted work.
## Competing Interests
The authors have no conflicts of interest to declare that are relevant to the content of this article.
## Supplemental Material
Peer Review History
## Data Accessibility Statement
The datasets generated during and/or analyzed during the current study are available from the corresponding author on request.
### Appendix A: Attitudes toward Singlehood Scale
Below are 9 items that may or may not apply to you. Select a score from 1 (Strongly disagree) to 7 (Strongly agree) to indicate the extent to which you agree with the item.
There is NO right or wrong answer.
1. I feel happy when I am single.
2. I feel positive for being single.
3. I feel comfortable being single.
4. I choose to commit myself to establish a career rather than a romantic relationship.
5. I have better control over my life when I am single.
6. I stay single to have more personal space.
7. I do not need to get into a romantic relationship to live a happy life.
8. Engaging in a romantic relationship is not important.
9. I think my life is complete even without a romantic partner.
#### Scoring
To generate an affective subscale score, average scores on items 1, 2, and 3. To generate a behavioral subscale score, average scores on items 4, 5, and 6. To generate a cognitive subscale score, average scores on items 7, 8, and 9. Finally, to calculate an overall score, average scores from all 9 items.
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# Suppressed phonon conduction by geometrically induced evolution of transport characteristics from Brownian motion into Lévy flight
## Abstract
Despite extensive research on quasi-ballistic phonon transport, anomalous phonon transport is still observed in numerous nanostructures. Herein, we investigate the transport characteristics of two sets of samples: straight beams and nanoladders comprising two straight beams orthogonally connected with bridges. A combination of experiments and analysis with a Boltzmann transport model suggests that the boundary scattering within the bridges considerably dictates the distribution of phonon mean free paths, despite its negligible contribution to the net heat flux. Statistical analysis of those boundary scatterings shows that phonons with large axial angles are filtered into bridges, creating dead spaces in the line-of-sight channels. Such redistribution induces Lévy walk conduction along the line-of-sight channels, causing the remaining phonons within the bridges to exhibit Brownian motion. Phonon conduction in the nanoladders is suppressed below that of the straight beams with equivalent cross-sectional areas due to trapped phonons within the bridges. Our work reveals the origin of unusual thermal conductivity suppression at the nanoscale, suggesting a method to modulate phonon conduction via systematic nanostructuring.
## Introduction
Understanding the nanoscale phonon conduction mechanism is essential in a wide range of semiconductor applications, such as nanoelectronics1,2, optoelectronics3,4, and energy conversion devices5,6. In the context of phonon conduction, nanostructures can be broadly categorized based on the uniformity of the cross-sectional area along the predominant direction of heat flow. Typical examples of structures with uniform cross-sectional areas include thin films7,8,9,10, nanobeams11, and smooth nanowires12,13,14,15, which have line-of-sight (LOS) channels for heat flow. These nanostructures show a significant reduction in thermal conductivity compared to their bulk counterparts due to increased boundary scattering. Such suppressed thermal transport in nanostructures with uniform cross-sections has substantially contributed to establishing a microscopic phonon conduction model16,17. However, numerous nanostructures show anomalous heat conduction with increasing complexity, calling into question the suppression mechanisms of phonon transport18,19.
It is generally accepted that the LOS channel is a primary heat conduction path, and its cross-section dictates the thermal conductivity of nanostructures. With the increased complexity of nanostructures that have a non-LOS channel volume, phonon conduction is observed to deviate from that of the LOS channel20,21,22,23,24. A key feature associated with such suppression in conduction is that phonon flux is disturbed due to the interactions of phonons with geometrical perturbations. The length scale of this perturbation ranges from angstroms to hundreds of nanometers, which is on the order of the phonon mean free path (MFP). Various reduction mechanisms have been suggested to explain the suppressed thermal conductivity of these complex nanostructures, with a focus on the interplay between phonons and nanostructures. For example, previous reports on nanowires with cross-sectional areas disordered in the range of a few nanometers25,26 as well as on phononic films20,27,28 with periodic nanoscale holes suggest that this suppression can be associated with the wave-like effects of phonons, atomic defects, and backscattering21,29. On the other hand, in fishbone structures, i.e., a nanobeam with orthogonally protruded pillars, or in corrugated nanowires with a series of periodically folded structures along the sidewall, the reduction in thermal conductivity is attributed to the Sharvin resistance in the ballistic regime24 and Lévy walk transport characteristics22,30. In addition, it has been further shown that in bare nanoladder structures31, a nanobeam with nanobridges and a single row of pores and bridges constitute a thermally dead volume23. The abovementioned reduction mechanisms are still under debate, calling for further studies on the origin of the phonon reduction mechanism in complex nanostructures beyond the conventional phonon transport mechanism that follows Brownian motion.
Herein, we investigate the evolution of thermal transport characteristics along LOS channels in nanoladders with a periodically varying cross-sectional areas at room temperature. Specifically, we prepare two sets of samples with ~78 nm thick single crystal silicon: (1) a set of nanobeams with varying cross-sectional areas and (2) a set of two identical nanobeams connected with a series of orthogonally placed bridges resembling ladders. With respect to the predominant heat direction, the thermal transport in nanoladders consists of two regimes, line-of-sight (LOS) channels and bridges, which are connected orthogonally, while the thermal transport in nanobeams consists of a single LOS channel regime. A nanoladder is considered as a series of repeating unit cells, as illustrated in Fig. 1a. Phonons contributing to the net transfer across the unit cell can be categorized into two groups: those traveling directly across the unit cell and those fully thermalized within the bridges. To quantify the relative contribution of each component to the thermal conduction, we deliberately designed our samples by modulating the cross-section ratios of LOS channels to bridges, ranging from ~1 to 5.29. Using a combination of the Boltzmann transport equation and the Monte Carlo approach, we model the phonon transport in our samples and find that the mean free path is determined by the relative volume ratio of LOS channels to bridges, corresponding to geometrical inhomogeneity. We further analyze a statistical distribution of phonon free paths, i.e., the traveling distance between scattering events, to investigate their transport characteristics as well as their impact on resultant mean free paths.
## Experimental procedures
### Device fabrication
We fabricate silicon nanostructures using silicon-on-insulator (SOI) wafers (Soitec Inc.) comprising an ~340 nm thick silicon layer and a 1 µm thick buried oxide layer (BOX). We employ thermal oxidation and consecutive oxide removal through a wet etching process to decrease the thickness of the silicon layer to ~78 nm. As an electrical passivation layer, an ~25 nm thick Al2O3 layer is deposited using atomic layer deposition (ALD). To pattern both nanobeam and nanoladder structures, a window of 10 μm × 20 μm is patterned between membranes by photolithography and etched using wet processes. Using electron beam lithography, the nanostructures are patterned, and a dry etching process follows. A serpentine Pt heater is patterned onto the insulation layer of each membrane by using a combination of electron beam lithography. A lift-off process is applied after the deposition of Cr and Pt layers of ~5 and ~40 nm thicknesses using electron beam metal evaporation. To suspend the structure, the BOX layer under the surrounding area of the membranes and legs is etched using RIE and consecutive gaseous hydrogen fluoride (HF) etching. For the conversion of thermal conductance into thermal conductivity and the uncertainty analysis, dimension measurement with scanning electron microscopy (SEM) is conducted after the fabrication process.
### Sample design and electrothermal characterization
As shown in Fig. 1b–i, we carefully prepare the two following sets of samples: (1) straight beams and (2) nanoladders (consisting of two straight beams placed in parallel and connected orthogonally using bridges). For the straight beam samples, we vary the aspect ratios of the cross-sections from ~0.9 to ~13.9 by modulating the width wLOS from ~70 nm to ~970 nm while keeping the thickness t at ~78 nm. Similarly, for the nanoladder samples, we vary the widths of the two straight beams from ~70 nm to ~370 nm, and ~70 nm by ~78 nm-sized bridges are placed between these beams with a periodicity of 200 nm. Here, the bridge width lbridge is ~70 nm. Accordingly, the volume ratio of LOS channels to bridges ranges from ~0.48 to ~9.2 in our nanoladders, and both sets of samples have identical LOS channels. The length of both sets of samples is 10 μm, which is long enough to ensure diffusive phonon transport along the LOS channels.
The thermal conductivity of our samples is characterized using an electrothermal characterization method with two suspended membranes, and this method has been applied for measuring the thermal conductivities of numerous nanomaterials. The detailed methodology is well documented elsewhere14,32,33. We note that both the samples and membranes are monolithically fabricated using both electron-beam lithography and photolithography (see Supporting Information for fabrication details). Heat is generated via Joule heating using serpentine metal structures, which imposes a finite temperature difference across the sample, and the associated temperatures on both sides are measured using resistive thermometry. Given the heat generation and associated temperature differences, the thermal conductivity of the samples can then be calculated by numerically solving a heat equation. We note that the heat equation captures the geometrical contribution in the diffusive transport regime.
We assume uniform thermal conductivities across the nanostructured samples (see Supporting Information for uncertainty analysis). The experiment is performed in vacuum to minimize convective heat loss.
## Results and discussion
### Experimental results
Figure 2a shows that as the cross-sectional area of an LOS channel is decreased, the thermal conductivity decreases from ~50 to ~30 W m−1 K−1 and from ~45 to ~33 W m−1 K−1 for straight beams and nanoladders, respectively. Note that for straight beams, we find that as the aspect ratio is increased, the thermal conductivity approaches the thin-film limit, verifying the validity of our measurements34. The monotonic decrease in the thermal conductivity of both sets of samples indicates that the cross-sectional area of the LOS channel is the predominant factor that dictates the overall thermal conductivity in our samples, as discussed in previous studies35,36,37. However, we find a crossover of the thermal conductivity between the two sets despite the identical cross-sections of the LOS channels. The thermal conductivity of the nanoladder samples is smaller than that of the straight beam samples when wLOS > lbridge, whereas the relative magnitude is reversed when wLOS < lbridge. While the crossover is found nearly within experimental uncertainty, it is worth investigating the crossover using a Boltzmann transport model.
### Boltzmann transport model
To better understand the differences in phonon conduction between straight beams and nanoladders, we model the thermal conductivity based on the Boltzmann transport equation as38,39
$$k = \frac{1}{{6\pi ^2}}\mathop {\sum}\nolimits_i {\mathop {\int}C_{V,i}(q)v_i(q)^2\tau _i\left( q \right)dq}$$
(1)
where i is the phonon mode, q is the phonon wavevector, CV is the volumetric heat capacity, v is the group velocity, and τ is the relaxation time. Born-vo n Karman sine type dispersion relation is used. The phonon mean free path Λ is defined as Λ = v × τ and can be derived using Matthiessen’s rule as $$\tau = (\tau _U^{ - 1} + \tau _I^{ - 1} + \tau _B^{ - 1})^{ - 1}$$, where τU is the Umklapp scattering rate defined as $$\tau _U^{ - 1} = Aw^2Texp( - \frac{B}{T})$$ and τI is the impurity scattering rate defined as $$\tau _I^{ - 1} = Dw^4$$. The boundary scattering rate τB is estimated by simulating phonon particles using Monte Carlo schemes. We consider the effects of internal phonon-phonon scattering and impurity scattering, and to determine the parameters A, B and D, we fit Eq. (1) to the experimental data of straight beams. For the best fit, A, B and D are given as 1.21 × 10−19 sK−1, 151 K and 2.54 × 10−45 s3, respectively (see Supporting Information for detailed Boltzmann transport model). We apply the fitted value for internal scattering to predict the thermal conductivity of the nanoladders, and the fit shows agreement with our experimental data within ~2%. This agreement ensures that the model captures the characteristics of phonon transport in our samples, as seen in Fig. 2a. More importantly, the model prediction clearly shows the crossover of the thermal conductivity between nanoladders and straight beams as a function of LOS width.
We further investigate the dependence of the phonon mean free path on boundary scattering, which dictates the thermal conductivity at the given length scale of our samples. We plot the boundary scattering mean free path ΛBoundary in Fig. 2b, which shows a clear crossover behavior between the nanoladders and straight beams, similar to the observed thermal conductivity. On the one hand, when wLOS > lBridge, the ΛBoundary in nanoladders is smaller than that in straight beams as shorter MFPs are introduced from the bridges. On the other hand, when wLOS < lBridge, the ΛBoundary in the nanoladders asymptotically saturates at a constant value as wLOS is decreased, while it decreases monotonically in their straight beam counterparts. The asymptotic limit of the nanoladders is mainly dictated by the geometrical dimension of the bridge, which introduces a constant scattering cross-section. This indicates that the ΛBoundary in nanoladders is nearly determined by the volumetric contribution of the LOS channels and bridges in the heterogeneous structures. Particularly at wLOS lBridge, ΛBoundary is predominantly dictated by the critical dimension of the bridge, not that of the LOS channel, which is a primary heat flow channel. As such, the critical dimension of the bridge serves as a source of free paths, while the contribution to the net heat flux from bridges is negligible.
### Statistical analysis of phonon free paths
We next statistically analyze the boundary scattering free path distributions of both the nanoladder and straight beam sample sets. We consider only boundary scattering, as it is the predominant scattering mechanism at the given dimensions. As seen in Fig. 3a, the free path distribution is expressed in terms of the probability density function (PDF), which shows peaks at the characteristic dimensions of the nanoladder and the straight beam: wLOS and thickness t. Note that we choose wLOS = 20 nm to emphasize the role of the LOS channel in the free path distributions, as the peaks for the LOS channel and the bridge are close to each other when wLOS = 70 nm. We observe that the PDF of the free paths for the nanoladder samples is dramatically suppressed near 20 nm compared to that for the straight beam samples. In contrast, the PDF is enhanced at ~70 nm (=lBridge) compared to that for the straight beam samples. Such changes in the PDF are mainly due to the phonons being trapped in the bridges, which occurs mostly for phonons traveling at relatively large axial angles with respect to the predominant direction of heat flow along the LOS channels22. We note that a nonzero probability is found below 20 nm, corresponding to potential phonon travel paths at a corner.
To further obtain quantitative information on the transport characteristics, we interpret the PDFs using the Pareto distribution expressed as40
$$f(X) = \frac{{\alpha L_C^\alpha }}{{X^{\alpha + 1}}},$$
(2)
where LC = min(wLOS,lbridge,t) is the length scale of the smallest characteristic dimension present in the nanostructure, X is the free path, and α is the shape parameter that reflects the phonon transport mechanism: ballistic transport for α = 1, Lévy walk for 1 < α < 2, and Brownian motion for α = 2. We note that the Pareto distribution has been studied in Bose–Einstein processes, such as thermal transport in SiGe and InGaAs alloys41,42,43 as well as in silicon nanowires at low temperatures22. Using the Pareto distribution framework, we fit the simulated PDFs of the free paths in the nanoladder samples and extract the shape parameter α as a function of the ratio of the predominant critical dimensions, wLOS to lBridge. We find that for wLOS > lBridge, α is larger than 2, and α is smaller than 2 when wLOS < lBridge, as seen in Fig. 3b. Note that this analysis focuses on the PDF near the smallest characteristic dimension present in the nanostructure to avoid interplay with other larger characteristic dimensions. We can therefore deduce that the phonon transport in the bridge region is dominated by Brownian motion (α = 2), while the LOS channel region displays Lévy walk characteristics (α < 2). We note that heat conduction typically follows Brownian motion. For example, in the case of infinitely thick samples, the bridge region shows perfect Brownian motion, as indicated by α = 2. Furthermore, we find that straight beams also show Brownian motion (α = 2) as opposed to the Lévy walk characteristics shown by the LOS channels in nanoladders. These observations suggest that the characteristics of phonon transport are converted from Brownian motion to Lévy flight along the LOS channel in our nanoladders by bridging the parallel channels.
We estimate the contribution of the Lévy walk character to the phonon mean free path E(X) using a Pareto distribution as
$$E(X) = {\int}_{\!L_c}^\infty {X\frac{{\alpha L_c^\alpha }}{{X^{\alpha + 1}}}dX = \frac{\alpha }{{\alpha - 1}}L_c(\alpha \,>\, 1)}$$
(3)
where X is the free path, α is the shape parameter, and Lc is the characteristic length for the boundary scattering. We note that α = 2 corresponds to the Brownian motion regime, while 1 < α < 2 corresponds to the Lévy walk regime. As shown in Fig. 4, the degree of boundary scattering contribution to the mean free path depends on the phonon transport characteristics, i.e., α. This suggests that phonons displaying Lévy walk characteristics are likely to have a longer mean free path than those displaying Brownian motion characteristics.
We further visualize the phonon paths along the LOS channels in both nanoladders and straight beams by simulating 3 × 104 phonons in a 3D computational space, which are projected onto a two-dimensional plane, as seen in the inset of Fig. 4. It is noteworthy that the collective set of phonon trajectories renders dead space at the entrance of the bridges. As such, phonons propagating along the LOS channel are likely to form an artificially corrugated structure, with those propagating at narrow axial angles following Lévy walk characteristics22,44. Given the statistical analysis above, we suggest that phonons propagating with broad axial angles are dominantly trapped within the bridges in our nanoladder and are dictated by the boundary scattering therein. As wLOS is decreased, more phonons with large-angle scattering are likely to be trapped in the bridges. The remaining phonons with narrow axial angles contribute to the decrease in the shape parameter as well as the increase in the mean free path in the LOS channel. However, the overall thermal conductivity of the nanoladder is suppressed as the majority of phonons are trapped in Brownian motion-dominated bridges; this suppression becomes further pronounced as the critical dimension of the bridge is comparable to or smaller than that of the LOS channel.
Finally, we extend our discussion on the impact of geometrical heterogeneity on phonon transport in the context of previous studies on other nanostructures, such as nanomeshes20,21 and fishbone nanowires24,45, as shown in the inset of Fig. 5. These structures can be decomposed into the LOS channel and the other channel, which is orthogonally aligned with respect to the LOS channel, and the corresponding critical dimensions are set to w// and l, respectively. Figure 5 shows the thermal conductivity of the abovementioned nanostructures normalized to that of the LOS channel as a function of the volumetric ratios of the nanostructures to that of the LOS channel. For the nanomeshes, the critical dimensions are the same l = w//, and the thermal conductivity decreases with increasing volumetric density of the non-LOS channel. For the fishbone nanowires with l > w//, the thermal conductivity decreases with increasing volume ratio, VTotal to VLOS, despite the increasing volumetric contribution of the non-LOS channel. In both cases, as the critical dimension of the non-LOS channel is larger than that of the LOS channel, the volumetric change fails to induce the suppression with increasing volume ratio of the non-LOS channel. Given these structures, phonon conduction through the LOS channel is also likely to show Lévy-walk characteristics, increasing the mean free paths along the channel. The abovementioned factors fail to explain the reduction. As such, the further suppression with increasing volume of the non-LOS channel is due to the increasingly trapped phonons with the non-LOS channel, which are forced to follow Brownian motion within the regime.
## Conclusions
In summary, we investigate the phonon transport mechanisms in silicon nanoladders and nanobeams. We observe a crossover in the thermal conductivity between the nanoladders and straight beams as a function of the volumetric ratio of bridges to LOS channels. A model prediction based on Boltzmann transport suggests that the non-LOS channel is a major contributing source of phonon boundary scattering despite its negligible contribution to the net heat flux. Furthermore, a statistical analysis of the distributions of free paths suggests that the bridges convert the transport characteristics of the LOS channel. We quantitatively identify this conversion using a Pareto-distribution framework as phonons traversing along the LOS channel follow the Lévy walk process, while those trapped in bridges show diffusive behavior. As a result, phonons in the LOS channels of the nanoladders have a relatively long mean free path as bridges capture large-angle phonons with respect to the axial direction. Finally, we extend our observation to other nanostructures with orthogonal geometric obstructions, such as nanomeshes and fishbone nanowires. This work answers a long-lasting question regarding the interplay between phonons and nanostructures, contributing to a comprehensive model for complex nanostructures.
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## Acknowledgements
We thank Dr. Mehdi Asheghi for the useful discussion. This research is supported by the National Research Foundation of Korea (NRF) funded by the MSIT (2020R1A4A3079200) and NRF (2021R1C1C1008693). Part of this research conducted at Seoul National University is supported by Samsung Electronics.
## Author information
Authors
### Contributions
W.P. conceived the idea and performed experiments. T.K. and W.P. fabricated the samples. Y.K. and Y.K. analyzed the experimental data and built the microscopic model. J.L. and W. P. supervised this study. B.S.Y.K. and C.K. provided discussion for the statistical model. All authors contributed to writing and editing the article.
### Corresponding authors
Correspondence to Jongwoo Lim or Woosung Park.
## Ethics declarations
### Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Kim, Y., Kodama, T., Kim, Y. et al. Suppressed phonon conduction by geometrically induced evolution of transport characteristics from Brownian motion into Lévy flight. NPG Asia Mater 14, 33 (2022). https://doi.org/10.1038/s41427-022-00375-7
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# SkyEllipticalAperture¶
class photutils.aperture.SkyEllipticalAperture(positions, a, b, theta=<Quantity 0. deg>)[source]
An elliptical aperture defined in sky coordinates.
The aperture has a single fixed size/shape, but it can have multiple positions (see the positions input).
Parameters
positionsSkyCoord
The celestial coordinates of the aperture center(s). This can be either scalar coordinates or an array of coordinates.
ascalar Quantity
The semimajor axis of the ellipse, either in angular or pixel units.
bscalar Quantity
The semiminor axis of the ellipse, either in angular or pixel units.
thetascalar Quantity, optional
The position angle (in angular units) of the ellipse semimajor axis. For a right-handed world coordinate system, the position angle increases counterclockwise from North (PA=0). The default is 0 degrees.
Examples
>>> from astropy.coordinates import SkyCoord
>>> import astropy.units as u
>>> from photutils import SkyEllipticalAperture
>>> positions = SkyCoord(ra=[10., 20.], dec=[30., 40.], unit='deg')
>>> aper = SkyEllipticalAperture(positions, 1.0*u.arcsec, 0.5*u.arcsec)
Attributes Summary
a Check that value is either an angular or a pixel scalar Quantity. b Check that value is either an angular or a pixel scalar Quantity. positions Check that value is a SkyCoord. theta Check that value is either an angular scalar Quantity.
Methods Summary
to_pixel(wcs) Convert the aperture to an EllipticalAperture object defined in pixel coordinates.
Attributes Documentation
a
Check that value is either an angular or a pixel scalar Quantity.
b
Check that value is either an angular or a pixel scalar Quantity.
positions
Check that value is a SkyCoord.
theta
Check that value is either an angular scalar Quantity.
Methods Documentation
to_pixel(wcs)[source]
Convert the aperture to an EllipticalAperture object defined in pixel coordinates.
Parameters
wcsWCS object
A world coordinate system (WCS) transformation that supports the astropy shared interface for WCS (e.g., astropy.wcs.WCS, gwcs.wcs.WCS).
Returns
aperture
An EllipticalAperture object.
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# Crossed product of a C*-algebra by a subgroup
Let $A$ be a unital $C^*$-algebra, let $G$ be a compact group, let $\alpha:G\to\mbox{Aut}(A)$ be a continuous action, and let $H$ be a closed subgroup of $G$. Is there any relationship between the crossed products $A\rtimes_\alpha G$ and $A\rtimes_{\alpha|_H}H$?
I really only need this for $G=\mathbb{T}$ the unit circle, and $H=\mathbb{Z}_n$ (identified with the $n$-th roots of unity in $\mathbb{T}$). For these groups, and in the case of the trivial action of the circle on $A$, $A\rtimes \mathbb{Z}_n \cong A\otimes \mathbb{C}^n$ is a corner of $A\rtimes \mathbb{T} \cong A\otimes c_0(\mathbb{Z})$, but I don't know if this is true in general.
You always have an injective $*$-homomorphism from $A\rtimes H$ into the multiplier algebra of $A\rtimes G$ (the reason is that you can view functions on $H$ as measures on $G$ which are supported on $H$). If $H$ is open in $G$ (a rather unfrequent situation, as you know), then $A\rtimes H$ sits as a $C^*$-subalgebra in $A\rtimes G$.
• Is this still true even if $H$ has measure zero in $G$? In some cases the Haar measure on $G$ doesn't restrict to a non-zero measure on $H$, and then the $\ell^1$ algebras seem to have nothing in common. Dec 1 '12 at 21:08
• @Eusebio: You must first understand the case where there is no $A$, i.e. how to embed $L^1(H)$ into $M(G)$, the measure algebra. (Note that $L^1(G)$ sits inside $M(G)$ as the ideal of measures absolutely continuous w.r. to Haar measure, hence my remark about multipliers). Shall I leave you some time to think by yourself? I think you'll learn more. Dec 1 '12 at 21:21
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# Properties
Label 2304.3.b.m Level $2304$ Weight $3$ Character orbit 2304.b Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$
# Related objects
## Newspace parameters
Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2304.b (of order $$2$$, degree $$1$$, not minimal)
## Newform invariants
Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$
## $q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + ( -8 + 16 \zeta_{12}^{2} ) q^{7} +O(q^{10})$$ $$q + ( -8 + 16 \zeta_{12}^{2} ) q^{7} -22 \zeta_{12}^{3} q^{13} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{19} + 25 q^{25} + ( -24 + 48 \zeta_{12}^{2} ) q^{31} + 26 \zeta_{12}^{3} q^{37} + ( -96 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{43} -143 q^{49} -74 \zeta_{12}^{3} q^{61} + ( 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{67} -46 q^{73} + ( 40 - 80 \zeta_{12}^{2} ) q^{79} + ( 352 \zeta_{12} - 176 \zeta_{12}^{3} ) q^{91} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 100q^{25} - 572q^{49} - 184q^{73} - 8q^{97} + O(q^{100})$$
## Character values
We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.
$$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$
## Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
−0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 0 0 13.8564i 0 0 0
127.2 0 0 0 0 0 13.8564i 0 0 0
127.3 0 0 0 0 0 13.8564i 0 0 0
127.4 0 0 0 0 0 13.8564i 0 0 0
$$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
## Inner twists
Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner
## Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.b.m 4
3.b odd 2 1 CM 2304.3.b.m 4
4.b odd 2 1 inner 2304.3.b.m 4
8.b even 2 1 inner 2304.3.b.m 4
8.d odd 2 1 inner 2304.3.b.m 4
12.b even 2 1 inner 2304.3.b.m 4
16.e even 4 1 144.3.g.c 2
16.e even 4 1 576.3.g.f 2
16.f odd 4 1 144.3.g.c 2
16.f odd 4 1 576.3.g.f 2
24.f even 2 1 inner 2304.3.b.m 4
24.h odd 2 1 inner 2304.3.b.m 4
48.i odd 4 1 144.3.g.c 2
48.i odd 4 1 576.3.g.f 2
48.k even 4 1 144.3.g.c 2
48.k even 4 1 576.3.g.f 2
80.i odd 4 1 3600.3.j.e 4
80.j even 4 1 3600.3.j.e 4
80.k odd 4 1 3600.3.e.h 2
80.q even 4 1 3600.3.e.h 2
80.s even 4 1 3600.3.j.e 4
80.t odd 4 1 3600.3.j.e 4
144.u even 12 1 1296.3.o.f 2
144.u even 12 1 1296.3.o.k 2
144.v odd 12 1 1296.3.o.f 2
144.v odd 12 1 1296.3.o.k 2
144.w odd 12 1 1296.3.o.f 2
144.w odd 12 1 1296.3.o.k 2
144.x even 12 1 1296.3.o.f 2
144.x even 12 1 1296.3.o.k 2
240.t even 4 1 3600.3.e.h 2
240.z odd 4 1 3600.3.j.e 4
240.bb even 4 1 3600.3.j.e 4
240.bd odd 4 1 3600.3.j.e 4
240.bf even 4 1 3600.3.j.e 4
240.bm odd 4 1 3600.3.e.h 2
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.g.c 2 16.e even 4 1
144.3.g.c 2 16.f odd 4 1
144.3.g.c 2 48.i odd 4 1
144.3.g.c 2 48.k even 4 1
576.3.g.f 2 16.e even 4 1
576.3.g.f 2 16.f odd 4 1
576.3.g.f 2 48.i odd 4 1
576.3.g.f 2 48.k even 4 1
1296.3.o.f 2 144.u even 12 1
1296.3.o.f 2 144.v odd 12 1
1296.3.o.f 2 144.w odd 12 1
1296.3.o.f 2 144.x even 12 1
1296.3.o.k 2 144.u even 12 1
1296.3.o.k 2 144.v odd 12 1
1296.3.o.k 2 144.w odd 12 1
1296.3.o.k 2 144.x even 12 1
2304.3.b.m 4 1.a even 1 1 trivial
2304.3.b.m 4 3.b odd 2 1 CM
2304.3.b.m 4 4.b odd 2 1 inner
2304.3.b.m 4 8.b even 2 1 inner
2304.3.b.m 4 8.d odd 2 1 inner
2304.3.b.m 4 12.b even 2 1 inner
2304.3.b.m 4 24.f even 2 1 inner
2304.3.b.m 4 24.h odd 2 1 inner
3600.3.e.h 2 80.k odd 4 1
3600.3.e.h 2 80.q even 4 1
3600.3.e.h 2 240.t even 4 1
3600.3.e.h 2 240.bm odd 4 1
3600.3.j.e 4 80.i odd 4 1
3600.3.j.e 4 80.j even 4 1
3600.3.j.e 4 80.s even 4 1
3600.3.j.e 4 80.t odd 4 1
3600.3.j.e 4 240.z odd 4 1
3600.3.j.e 4 240.bb even 4 1
3600.3.j.e 4 240.bd odd 4 1
3600.3.j.e 4 240.bf even 4 1
## Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:
$$T_{5}$$ $$T_{7}^{2} + 192$$ $$T_{11}$$ $$T_{17}$$ $$T_{19}^{2} - 768$$
## Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 192 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 484 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( -768 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$( 1728 + T^{2} )^{2}$$
$37$ $$( 676 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -6912 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 5476 + T^{2} )^{2}$$
$67$ $$( -3072 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 46 + T )^{4}$$
$79$ $$( 4800 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$( 2 + T )^{4}$$
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# how to eliminate the horizontal spacing of the frame in the framed package?
I'm using the framed package and I found it difficult to adjust the width of a frame to adapt to the current environment, e.g. if I use a shaded environment within the enumerate environment, I will get a frame box as wide as the current page (actually slightly wider) -- it will not indent accordingly; then I found a solution to make the frame of the same width as the current environment, but a new problem arised: there seems to be extra horizontal spacing inside the frame. A MWE is below:
\documentclass{article}
\usepackage{framed,color}
\newenvironment{myframe}{%
{\endMakeFramed}
\begin{document}
test a frame within an enumerate environment
\begin{enumerate}
\item abc
\begin{myframe}
def
\end{myframe}
\end{enumerate}
\end{document}
Can anybody tell me how to eliminate the extra left padding so that the content of the frame can align to the list item? Thanks a lot!
A screenshot is here:
-
Thanks for the reminder! I tried to answer directly but got a message box, and I thought it recommended me to edit my original post. I will extract the answer out then. – Yihui Nov 9 '11 at 1:45
I asked the author of the framed package, and his solution worked for me (although I do not really understand it):
\documentclass{article}
\usepackage{framed,color}
\makeatletter
\def\FrameCommand##1{\hskip\@totalleftmargin \hskip-\fboxsep
% There is no \@totalrightmargin, so:
\hskip-\linewidth \hskip-\@totalleftmargin \hskip\columnwidth}%
\@totalleftmargin\z@ \linewidth\hsize
\@setminipage}}%
{\par\unskip\endMakeFramed}
\makeatother
\begin{document}
test a frame within an enumerate environment test a frame within an enumerate environment
\begin{enumerate}
\item abc
def
\end{enumerate}
\end{document}
-
I suggest using the mdframed package:
\documentclass{article}
\usepackage{xcolor} % better than color
\usepackage{mdframed}
\begin{document}
test a frame within an enumerate environment
\begin{enumerate}
\item abc
\begin{mdframed}[
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# Well-ordering of the reals in ZF with constructibility?
The question Do we know that we can't define a well-ordering of the reals? states:
There exist pointwise definable models of ZFC where every set is definable without parameters: it is the unique element of the model that satisfies some finite formula $\varphi(x)$. So there is a formula $\varphi(x)$ such that $$\tag{*} \forall x(\varphi(x)\rightarrow x\text{ well-orders }\mathbb R) \land \exists ! x\,\varphi(x)$$ is consistent with ZFC. (And it is not difficult to write down a concrete $\varphi$ which will work in a model with $\mathbf V=\mathbf L$).
What is an example of such a $\varphi$?
• Work in $L$. "$x<y$ iff $x,y$ are reals, and $x$ appears before $y$ in the standard construction of $L$." In the standard construction, $L$ is produced by stages $L_0\subsetneq L_1\subsetneq\dots$. There is a formula in one parameter $\alpha$ that defines $L_\alpha$. That $x$ appears before $y$ means that if we define $\alpha_x$ as the unique $\alpha$ such that $x\in L_{\alpha+1}\setminus L_\alpha$, then either $\alpha_x<\alpha_y$, or they are equal but $x$ appears before $y$ in the standard construction of $L_{\alpha+1}$ from $L_\alpha$. In order to define this construction, we need ... – Andrés E. Caicedo Jun 14 '16 at 1:54
• ... to fix a (recursive) enumeration of the formulas in the language of set theory. $L_{\alpha+1}$ is defined as the collection of subsets of $L_\alpha$ that are definable over $L_\alpha$ using parameters from $L_\alpha$. This definability involves the formulas you have enumerated. So, if $\alpha_x=\alpha_y=\alpha$, we say that $x$ appears before $y$ if the first formula (in your fixed enumeration) that defines $x$ over $L_\alpha$ precedes the first formula defining $y$, or else this first formula is the first for both, but ... – Andrés E. Caicedo Jun 14 '16 at 1:58
• ... the parameters used to define $x$ precede those that are used to define $y$. This is a recursive definition, since this preceding is with respect to the lexicographic ordering of finite sequences defined with respect to the well-ordering we are describing restricted to $L_\alpha$. – Andrés E. Caicedo Jun 14 '16 at 2:00
• @AndrésCaicedo thanks :) is there a reason you didn't post this as an answer? – benzrf Jun 14 '16 at 2:14
• @AndrésE.Caicedo Wait - two things. First: how can we fix a unique choice of defining formula for each set in a formula without parameters? Second: what if two reals are defined by the same formula, but with different parameters from each other? – benzrf Jun 24 '16 at 23:42
$L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZFC}$) well-orders all of $L$, so that its restriction to any specific set $A$ is a set well-ordering of $A$.
The well-ordering $\varphi$ you are asking about can be obtained as the restriction to $\mathbb R^L$ of this global well-ordering $\phi$.
In detail, recall that $$L=\bigcup_{\alpha\in\mathrm{ORD}}L_\alpha,$$ where $L_0=\emptyset$, $L_\lambda=\bigcup_{\alpha<\lambda}L_\alpha$ for $\lambda$ a limit ordinal, and $$L_{\alpha+1}=\{A\subseteq L_\alpha : A\mbox{ is definable (with parameters) in }(L_\alpha,\in)\}.$$ Some presentations of $L$ define $L_{\alpha+1}$ quite explicitly as I indicate. This requires some work, since one needs to formalize internally the relevant notions of definability and satisfiability in set structures in order to make sense of the statement that "$A$ is definable (with parameters) in $(L_\alpha,\in)$'' as a formula in the language of set theory (as opposed to an informal statement about model theory in natural language). An example of a book providing the relevant details of this approach is
MR0750828 (85k:03001). Devlin, Keith J. Constructibility. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1984. xi+425 pp. ISBN: 3-540-13258-9.
Other presentations instead describe the collection $L_{\alpha+1}$ of definable subsets of $(L_\alpha,\in)$ in terms of certain natural operations without explicitly referring to definability in terms of first-order formulas. This approach is also favored nowadays in some presentations of model theory, as it highlights the naturalness of first-order definability to those already familiar with, say, the theory of real algebraic sets. (For instance, if $A$ is a definable set of pairs, the set $\{x:\exists y\,(x,y)\in A\}$ should also be definable. Rather than talking about the existential quantifier, one can simply discuss the projection of $A$ to its first coordinate.) An example of a book discussing $L$ along these lines is
MR0597342 (82f:03001). Kunen, Kenneth. Set theory. An introduction to independence proofs. Studies in Logic and the Foundations of Mathematics, 102. North-Holland Publishing Co., Amsterdam-New York, 1980. xvi+313 pp. ISBN: 0-444-85401-0.
Whichever approach you follow, by the recursion theorem this gives you that there is a formula $\psi$ in one parameter $a$ that, whenever $a$ is an ordinal $\alpha$, defines $L_\alpha$. That is, $\psi(a,x)$ has the property that (provably in $\mathsf{ZFC}$, although it suffices for what we are doing here that this is true under the assumption of $V=L$) for any ordinal $\alpha$, the set $L_\alpha$ is the unique set $b$ such that $\psi(\alpha,b)$ holds.
In the sketch below, I follow the first approach. I'm leaving some details out, but the first reference above should complement what I say here. Begin by coding in set theory a recursive enumeration of the set of first-order formulas in the language $\{\in\}$, say $\phi_0,\phi_1,\dots$ If $x\in L$, write $\operatorname{rk}_L(x)$ for the least ordinal $\alpha$ such that $x\in L_{\alpha+1}$. Now, if $\alpha=\operatorname{rk}_L(x)$, this means that there is a formula $\phi_n(u_1,\dots,u_m,v)$ in the language of set theory such that $$x=\{y\in L_\alpha:(L_\alpha,\in)\models\phi_n(a_1,\dots,a_m,y)\}$$ for some parameters $a_1,\dots,a_m\in L_\alpha$. (That there is such a formula $\phi_n$ and such parameters is precisely what is meant by "$x$ is definable (with parameters) in $(L_\alpha,\in)$''.) Write $\min_L(x)$ for the least $n$ such that $x$ can be defined as above using $\phi_n$ from some parameters.
Now, to define the global well-ordering of $L$, assume $V=L$. I proceed to define by recursion a well-ordering $<_\alpha$ of $L_\alpha$ with the property that if $\alpha<\beta$ (as ordinals) then $<_\alpha\subset <_\beta$. In fact, $<_\beta$ is an end-extension of $<_\alpha$, meaning that if $x\in L_\alpha$ and $y\in L_\beta\smallsetminus L_\alpha$, then $x<_\beta y$.
The definition is as follows: $<_0=\emptyset$. For limit ordinals $\lambda$, $<_\lambda$ is just $\bigcup_{\alpha<\lambda}<_\alpha$. Now, assuming we have defined $<_\alpha$, we define $<_{\alpha+1}$ as follows: Note that $<_\alpha$ induces an ordering $<_{\alpha,f}$ of finite tuples of members of $L_\alpha$: given $a_1,\dots,a_m,b_1,\dots,b_k\in L_\alpha$, set $$(a_1,\dots,a_m)<_{\alpha,f}(b_1,\dots,b_k)$$ if and only if either
• $k>m$ or else
• $k=m$ and there is an $i$, $1\le i\le m$, such that $a_i<_\alpha b_i$ but $a_j=b_j$ for all $j<i$.
Note that this is a well-ordering of the set of finite tuples.
Finally, given $x,y\in L_{\alpha+1}$ set $x<_{\alpha+1}y$ if and only if either
• $\operatorname{rk}_L(x)<\operatorname{rk}_L(y)$, or else
• $\operatorname{rk}_L(x)=\operatorname{rk}_L(y)<\alpha$ and $x<_\alpha y$, or else
• $\operatorname{rk}_L(x)=\operatorname{rk}_L(y)=\alpha$ and $\min_L(x)<\min_L(y)$, or else
• $\operatorname{rk}_L(x)=\operatorname{rk}_L(y)=\alpha$ and $\min_L(x)=\min_L(y)=n$, say, and the least tuple $(a_1,\dots,a_m)$ of parameters in $L_\alpha$ from which $\phi_n$ defines $x$ in $(L_\alpha,\in)$ is $<_{\alpha,f}$-below the least such tuple $(b_1,\dots,b_k)$ from which $\phi_n$ defines $y$.
One readily verifies that each $<_\alpha$ is indeed a well-ordering of $L_\alpha$ and the end-extension property holds. Finally, you can define a well-ordering $<_L$ of $L$ by setting $x<_L y$ (for $x,y\in L$) if and only if for some $\alpha$, $x<_\alpha y$ (equivalently, for all sufficiently large $\alpha$, $x<_\alpha y$).
(In short, all we are saying is that $x<_L y$ if and only if "$x$ appears before $y$" in the standard iterative construction of $L$.)
At the end of the day, the same idea can be applied to much more general inner models than just $L$. This is in broad strokes how one defines global well-orderings of the canonical $L[\vec E]$ models of inner model theory, for instance.
The definition is canonical enough that, when restricted to the reals, produces a well-ordering of the reals of the inner model of low (in fact, optimal) complexity in the sense of the projective hierarchy. Verifying this requires a further layer of coding, where one checks that for countable $\alpha$ the structures $L_\alpha$ (or their appropriate versions when looking at the $L[\vec E]$) can be "reasonably" coded as real numbers. For the $L[\vec E]$ models, one needs further verify that one can code in such a reasonable fashion the comparison process that allows us to conclude that one structure precedes another. This last detail is not needed in the case of $L$, where $L_\alpha$ precedes $L_\beta$ if and only if $L_\alpha\subset L_\beta$, but the right notion is not simply containment in general.
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Math for simple 3D coordinate rotation (python)
I'm rotating points from one coordinate system to the other, but drawing a blank how to do this.
I've done some reading about Euler angles but after staring at this GIF for a while I just get dizzy.
above: from here
What I am doing:
The new $z$ axis is perpendicular the plane containing the origin, $p_1$ and $p_2$, and the new $x$ axis goes through $p_1$. There is no change in origin.
Looking for $p_{calc}$ in the rotated cartesian coordinates. Here is a plot of some normals. Math answer that's simple enough for me to re-write in python is fine.
edit: This PDF seems to be helpful.
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
p1 = np.array([-1489., -4913., 4345.])
p2 = np.array([ 2633., -3268., 5249.])
pcalc = np.array([-3210., -4390., 3930.])
def normit(v):
return v / np.sqrt((v**2).sum())
n1, n2, ncalc = [normit(p) for p in [p1, p2, pcalc]]
new_zaxis = normit(np.cross(n1, n2))
new_xaxis = n1
zero = np.zeros(3)
fig = plt.figure(figsize=[10, 8])
ax = fig.add_subplot(1, 1, 1, projection='3d')
x, y, z = zip(zero, new_xaxis)
plt.plot(x, y, z, '-k', linewidth=3)
x, y, z = zip(zero, new_zaxis)
plt.plot(x, y, z, '--k', linewidth=3)
x, y, z = zip(zero, n2)
plt.plot(x, y, z, '-r', linewidth=1)
x, y, z = zip(zero, ncalc)
plt.plot(x, y, z, '--g', linewidth=1)
ax.set_xlim(-1, 1)
ax.set_ylim(-1, 1)
ax.set_zlim(-1, 1)
plt.show()
2 Answers
note: A nicer looking and correct answer will still get accepted, thanks!
I've read on page 27 here that a 3x3 transform matrix can be just the nine dot products - thank you U. Auckland's prof. Kelly!
above x2: screenshots from here.
Here is a very ugly implementation which seems to work.
new_yaxis = -np.cross(new_xaxis, new_zaxis)
# new axes:
nnx, nny, nnz = new_xaxis, new_yaxis, new_zaxis
# old axes:
nox, noy, noz = np.array([1, 0, 0, 0, 1, 0, 0, 0, 1], dtype=float).reshape(3, -1)
# ulgiest rotation matrix you can imagine
top = [np.dot(nnx, n) for n in [nox, noy, noz]]
mid = [np.dot(nny, n) for n in [nox, noy, noz]]
bot = [np.dot(nnz, n) for n in [nox, noy, noz]]
def newit(vec):
xn = sum([p*q for p,q in zip(top, vec)])
yn = sum([p*q for p,q in zip(mid, vec)])
zn = sum([p*q for p,q in zip(bot, vec)])
return np.hstack((xn, yn, zn))
Let's see what happens...
nnx: array([-0.22139284, -0.73049229, 0.64603887])
newit(nnx): array([ 1., 0., 0.])
nny: array([ 0.88747002, 0.1236673 , 0.44396325])
newit(nny): array([ 0., 1., 0.])
nnz: array([-0.40420561, 0.67163042, 0.62091095])
newit(nnz: array([ 0., 0., 1.])
OK then, this seems to be the right way to go.
thank you uhoh for the math. here is the same using np inner
new_yaxis = -np.cross(new_xaxis, new_zaxis)
# new axes:
new_axes = np.array([new_xaxis, new_yaxis, new_zaxis])
# old axes:
old_axes = np.array([1, 0, 0, 0, 1, 0, 0, 0, 1], dtype=float).reshape(3, -1)
rotation_matrix = np.inner(new_axes,old_axes)
#vec can be vectors Nx3
def newit(vec):
return np.inner(vec,rotation_matrix)
And your test
new = newit(new_axes)
#new_axes:
[[-0.22139284 -0.73049229 0.64603887]
[ 0.88747002 0.1236673 0.44396325]
[-0.40420561 0.67163042 0.62091095]]
#new:
[[ 1.00000000e+00 -2.06024710e-17 6.10077552e-17]
[-2.06024710e-17 1.00000000e+00 4.13972145e-17]
[ 6.10077552e-17 4.13972145e-17 1.00000000e+00]]
$$$$
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# Can this be solved?
1. Mar 31, 2008
### gamesguru
This is not a homework problem, I just thought of it when I was looking at problems with it being just y and not y^2. Here's the problem. It's entirely possible that it's not solvable, I'm just curious.
$$y'+y^2=x[/itex] 2. Mar 31, 2008 ### HallsofIvy That's non-linear so solving it won't be easy. There is, however, a theorem that says every first order d.e. has an "integrating" factor. You can write that as $dy/dx= x- y^2$ so $dy= (x- y^2)dx$ or $dy+ (y^2- x)dx= 0$. There must exist some function v(x,y) such that $v(x,y)dy+ v(x,y)(y^2-x)dx= 0$ is "exact": that is so that there exist a function f(x,y) so that $df= vdy+ v(y^2-x)dx$. If that is true then we must have $v_x= (v(y^2-x))_y$. But there is no theorem that says it will be easy to find v(x,y)! 3. Apr 2, 2008 ### coomast This equation is of the Riccati type. It can be transformed into a linear one by using the substitution: [tex]y(x)=\frac{1}{u(x)}\cdot \frac{du(x)}{dx}$$
Giving thus as transformed equation:
$$\frac{d^2u}{dx^2}-x\cdot u = 0$$
Which is the one of Airy, with known solution. After the inverse transformation you get the solution of the original equation.
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# Second norm of matrix
1. Aug 26, 2004
### niko2000
Hi,
I have forgotten the formula for calculating the second norm of matrix. Does anyone know the formula?
Regards,
Niko
2. Aug 26, 2004
### Hurkyl
Staff Emeritus
Are you talking about the norm usually written as $||A||_2$? You simply consider the (mxn) matrix as an mn-tuple: the square of the norm is the sum of the squares of the entries.
3. Sep 17, 2004
### mathwonk
this is also called the pythagorean theorem. i'll bet you haven't really forgotten it. almost no one ever does: a^2 + b^2 = c^2.
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# Semi-direct product in general linear groups
$\operatorname{GL}(n,F)$ can be written as a semidirect product : $\operatorname{GL}(n,F) = \operatorname{SL}(n,F) ⋊ F^\times$ where $F^\times$ is multiplicative group of the field $F$. According to the definition of semi direct product we must have a homomorphism between $\operatorname{SL}(n,F)$ and $F^\times$. How can we define this homomorphism?
• @anon: the copy of $F^\times$ here is not (typically) central. It is the copy consisting of diagonal matrices in which all but the top-left entry agrees with the identity matrix. so { [t,0,0;0,1,0;0,0,1] : t in F, t≠0 } is an example with n=3. Feb 9, 2014 at 22:01
• I am a bit confused about the multiplicative group of a finite field for example GF(4)? Would you please more details in this case? and another question : if we assume the identity automorphism semidirect is direct product, can we conclude that F× is a normal subgroup of GL(n,F) as well?
– Nil
Feb 9, 2014 at 22:02
• @Jack You are right. OP: A semidirect product $H\rtimes K$ presupposes no homomorphism at all between $H$ and $K$. Rather, there is a map $K\to\color{Red}{\rm Aut}(H)$.
– anon
Feb 9, 2014 at 22:04
This is not the correct piece of data for defining this semidirect product. The correct piece of data is a homomorphism $F^{\times} \to \text{Aut}(\text{SL}_n(F))$. This homomorphism can be written down as follows.
Theorem: Let $H$ be a normal subgroup of a group $G$, so that there is a short exact sequence
$$1 \to H \to G \to G/H \to 1.$$
Then $G$ can be written as a semidirect product $H \rtimes G/H$ iff this short exact sequence splits on the right in the sense that there is a map $r : G/H \to G$ which, after projecting back down to $G/H$, is the identity. In this case the action of $G/H$ on $H$ is the restriction of the action of $G$ on $H$ via conjugation to the image of $r$.
We of course have a short exact sequence
$$1 \to \text{SL}_n(F) \to \text{GL}_n(F) \xrightarrow{\text{det}} F^{\times} \to 1.$$
An example of a splitting of this short exact sequence is
$$F^{\times} \ni a \mapsto \left[ \begin{array}{ccc} a & 0 & \cdots \\ 0 & 1 & \cdots \\ \vdots & \vdots & \ddots \end{array} \right]$$
so the action is given by conjugation by this matrix.
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CBSE (Science) Class 11CBSE
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# If Sin a = 4 5 and Cos B = 5 13 , Where 0 < A, B < π 2 , Find the Value of the Following: Cos (A − B) - CBSE (Science) Class 11 - Mathematics
ConceptTrigonometric Functions of Sum and Difference of Two Angles
#### Question
If $\sin A = \frac{4}{5}$ and $\cos B = \frac{5}{13}$, where 0 < A, $B < \frac{\pi}{2}$, find the value of the following:
cos (A − B)
#### Solution
Given:
$\sin A = \frac{4}{5}\text{ and }\cos B = \frac{5}{13}$
We know that
$\cos A = \sqrt{1 - \sin^2 A}\text{ and }\sin B = \sqrt{1 - \cos^2 B} ,\text{ where }0 < A , B < \frac{\pi}{2}$
$\Rightarrow \cos A = \sqrt{1 - \left( \frac{4}{5} \right)^2} \text{ and }\sin B = \sqrt{1 - \left( \frac{5}{13} \right)^2}$
$\Rightarrow \cos A = \sqrt{1 - \frac{16}{25}}\text{ and }\sin B = \sqrt{1 - \frac{25}{169}}$
$\Rightarrow \cos A = \sqrt{\frac{9}{25}}\text{ and }\sin B = \sqrt{\frac{144}{169}}$
$\Rightarrow \cos A = \frac{3}{5}\text{ and }\sin B = \frac{12}{13}$
Now,
$\cos\left( A - B \right) = \cos A \cos B + \sin A \sin B$
$= \frac{3}{5} \times \frac{5}{13} + \frac{4}{5} \times \frac{12}{13}$
$= \frac{15}{65} + \frac{48}{65}$
$= \frac{63}{65}$
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Solution for Mathematics Class 11 (2019 to Current)
Chapter 7: Values of Trigonometric function at sum or difference of angles
Ex.7.10 | Q: 1.4 | Page no. 19
Solution If Sin a = 4 5 and Cos B = 5 13 , Where 0 < A, B < π 2 , Find the Value of the Following: Cos (A − B) Concept: Trigonometric Functions of Sum and Difference of Two Angles.
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# Thread: M_n Matrix ring
1. ## M_n Matrix ring
Let $M_n(A)$ be the ring of matrixes with elements from the ring $A$. $T_n(A) \subset M_n(A)$ the set of matrixes in which all elements below the diagonal are 0. $T'_n(A) \subset T_n(A)$ the set of matrixes with the elements in the principal diagonal are 0. $D_n(A)$ the set of diagonal matrixes.
Prove that,
$T'_n(A) \cong T_n(A)/D_n(A)$
2. Originally Posted by roporte
Let $M_n(A)$ be the ring of matrixes with elements from the ring $A$. $T_n(A) \subset M_n(A)$ the set of matrixes in which all elements below the diagonal are 0. $T'_n(A) \subset T_n(A)$ the set of matrixes with the elements in the principal diagonal are 0. $D_n(A)$ the set of diagonal matrixes.
Prove that,
$T'_n(A) \cong T_n(A)/D_n(A)$
I do it for $n=3$ it will be clear.
Define the mapping $\begin{bmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{bmatrix} \mapsto \begin{bmatrix} 0 & b & c \\ 0 & 0 & e \\ 0 & 0 & 0 \end{bmatrix}$
Then the kernel of this mapping ( $\theta: T_n(A) \to T'_n(A)$) is $D_n$ and range is $T'_n(A)$.
By the fundamental isomorphism theorem,
$T'_n(A) \simeq T_n(A)/D_n(A)$
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# Common Power Series
Dividends are payable on the 15th day of each of the months of January, April, July, and October in each year. ) The first term of the sequence is a = -6. Other possible problems include worn steering gear and ball sockets in the steering assembly, worn suspension system components and loose steering pump belt. Start with the generating function for the Bernoulli numbers:. The legendary 7. 1 in every major reliability category by ITIC*, IBM Power Systems deliver reliable on-premises infrastructure 24/7. Some of these. Binomial series Hyperbolic functions. A power series in a variable z is an infinite sum of the form sum_(i=0)^inftya_iz^i, where a_i are integers, real numbers, complex numbers, or any other quantities of a given type. The CHARGE is the most advanced power management available that does the work of three devices -- a traditional battery charger, a charge-on-the-run an emergency start system--all in one compact unit. Although the reaching of this alarm can be gradual, it can also be instantaneous if you attempt to power a substantial load. View Notes - Common Taylor Series(2) from M 427 at University of Texas. arithmetic sequence term sequence y 031 2 4567x 4 2 6 8 10 12 14 16 18 20 22 Shingles Row Vocabulary • sequence • term • arithmetic sequence • common difference • arithmetic means Arithmetic Sequences 578 Chapter 11 Sequences and Series • Use arithmetic sequences. Dataflows are created and managed in app workspaces by using the Power BI service. Today we're excited to announce our how-to video series on the Official PowerApps YouTube channel. Try the troubleshooting suggestions below to see if they resolve your power issues. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. Geometric Sequences. is not geometric because there's no common factor between numbers. This is illustrated in the following examples. 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Start with the generating function for the Bernoulli numbers:. A power series is a series of the form where x is a variable and the c[n] are constants called the coefficients of the series. In this case, let’s say we are going to wire them to a total, effective impedance of 2ohms. Bypass any surge protector and use a wall outlet that you know is working. Putnam Problem-Solving Seminar Power Series Tuesday, October 30, 2007 There are too many problems to consider each one in one session alone. An everyday examples of a battery is the 9-volt transistor battery, which is six 1. Kemp in collaboration with Curtis Jackson. One of the most common electrical problems with a BMW 3 Series is with the battery. that power series always converge in a disk jz ajR, where R>0 is a value called the radius of convergence. Power Rangers is the name of a meta-franchise series, chronicling the adventures of the Power Rangers. 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So this is a geometric series with common ratio r = -2. The durable under cabinet. A great choice of actors to fill each and every role. One of the first principles to understand when you are learning about electricity is the distinction between a parallel circuit and a series circuit. Since this ratio is common to all consecutive pairs of terms, it is called the common ratio. Read Common Sense Media's The Bravest Knight review, age rating, and parents guide. Excel Seriessum Function Examples Example 1. Standard Power Series: SM351RT, SM451R, SM353RT, SM453R DESCRIPTION Honeywell's Magnetoresistive Sensor ICs (integrated circuits), Standard Power Series, are ultra-sensitive devices designed for manufacturers who need a durable and reliable speed or position sensor for large air gaps or small magnetic fields in. This web page attempts to guide you through the various common upgrade paths. This BMW 3 Series, 1 Series and Z4 electric power steering rack motor is a very common fault for the 2006 – 2016 build range. Cummins will be the leading provider of electrified power in our commercial and industrial markets just as we are the leader in diesel and natural gas powered products. Announcing New Office Template series Emma Cooper , Program Manager II , Tuesday, October 3, 2017 With its breadth of scope and depth of data, Microsoft Graph is quickly becoming the go-to API for app makers looking to increase productivity and build for business scenarios. So the series must be geometric, which means we can get started by looking for a common ratio,. Power is the kind of potent, addictive, yet decidedly adult pleasure fans of shows such as The Sopranos and Breaking Bad will appreciate. The number of values in the supplied coefficients array defines the number of terms in the power series. The common automobile battery consists of six 2. Two of these subwoofers would mean you need 800 watts of power at to the total circuit impedance. The most common reasons for a No Power is that either a miss-fitted or faulty part is shorting the system and preventing it from starting or the power circuit on the motherboard isn't distributing the power correctly to supply enough to start-up the system. 3 Power Stroke Diesel engine, when properly maintained can provide years of trouble free service. Read Common Sense Media's The Bravest Knight review, age rating, and parents guide. Tips for Connecting to Excel on OneDrive for Business from Power Apps. Suppose the interest rate is loo%, so i = 1. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms. If , the series converges to. Kemp in collaboration with Curtis Jackson. Complete Solution Before starting this problem, note that the Taylor series expansion of any function about the point c = 0 is the same as finding its Maclaurin series expansion. POWER SERIES POWER SERIES. This is illustrated in the following examples. ) Maple is much better at this than most of us, but a little practice can quickly improve this skill. Get valuable advice from students who have been in your shoes. Data is stored as entities in Common Data Model-compliant folders in Azure Data Lake Storage Gen2. First, identify the problem you've encountered with your engine, then rule out each potential cause. Standard Power Series: SM351RT, SM451R, SM353RT, SM453R DESCRIPTION Honeywell's Magnetoresistive Sensor ICs (integrated circuits), Standard Power Series, are ultra-sensitive devices designed for manufacturers who need a durable and reliable speed or position sensor for large air gaps or small magnetic fields in. Free shipping and free returns on eligible items. The Broan 40000 30 in. Find the Taylor series expansion for e x when x is zero, and determine its radius of convergence. , where a 1 is the first term and r is the common ratio. Series compensation is the method of improving the system voltage by connecting a capacitor in series with the transmission line. An investment of ) = $1will bring you a dollar each year forever. This might be used in a scooter, Power Wheels kids vehicle, or other applications. Common Car Problems and Complaints - 14,037 Issues. 8 Power Series De nition. Common shareholders also have the right to receive any dividends that the company declares on their shares. Series Circuit A series circuit is a circuit in which resistors or loads are connected end to end so that the circuit will have only one path through which electric current flows. You can specify the order of the Taylor polynomial. Definition 11. Power series are infinite series of the form Σaₙxⁿ (where n is a positive integer). And you will see why, if you multiply. It should equal 24 volts but I'm not sure what it would do to battery life and charging. A defective capacitor in some Samsung LCD, plasma and DLP TVs is causing the sets to fail. This leads to a new concept when dealing with power series: the interval of convergence. is not geometric because there's no common factor between numbers. Common Ratio. A power series in a variable z is an infinite sum of the form sum_(i=0)^inftya_iz^i, where a_i are integers, real numbers, complex numbers, or any other quantities of a given type. I will first explain how it works, then I will explain how we can troubleshoot. Circuits consisting of just one battery and one load resistance are very simple to analyze, but they are not often found in practical applications. Power-ups give special abilities upon using them. The Exponential Fourier Series coefficients are given by $$\displaylines. PowerEdge M Series Starting at 1,559. We use power series to approximate, with great accuracy, non-polynomial functions like , sin , and cos. In each exercise, an appropriate power series can be derived by using the Standard series (accessed from the "toolbar" at the bottom of each of the Exercises pages). Corant Lynn (born March 13, 1972), better known by his stage name Common (formerly Common Sense), is an American rapper, actor, writer, philanthropist, and activist. Power BI Time Series Graph. The 5 Series is a perfect blend of luxury and performance. Power series are always infinite. Lack of steering fluid makes it hard to turn the steering wheel. However, polynomials are always finite in length. Cummins will be the leading provider of electrified power in our commercial and industrial markets just as we are the leader in diesel and natural gas powered products. that power series always converge in a disk jz ajR, where R>0 is a value called the radius of convergence. It just wont turn on When the adapter is plugged the charging LED opens but later the light will disappear. Series expansions of exponential and some logarithms functions. However, you are not allowed to work on a problem that you already know how to solve. Sample Quizzes with Answers Search by content rather than week number. They are wired in parallel so that two appliances which are plugged into the receptacle receive the same voltage, but can draw different amounts of electric current. For further details, see the class handout on the inverse. Taylor series expansion of exponential functions and the combinations of exponential functions and logarithmic functions or trigonometric functions. Learn how to connect two or more batteries properly. ) Maple is much better at this than most of us, but a little practice can quickly improve this skill. is a geometric sequence with the common factor 2. Crown Switching Power Supply for lighter weight Selectable "Constant-Voltage" or low-impedance (4/8 ohm) operation per channel on 2-channel models, and per channel-pair on 4-and 8-channel models 100V direct outputs on CTs 2000, CTs 3000, CTs 4200, and CTs 8200. In the spreadsheet below, the Excel Seriessum function is used to calculate the power series:. “Truck Approved “ Series Power Inverters have a special programming informing you that the capacity of your batteries is decreasing and that it is time to recharge them. Taylor Series and Maclaurin Series In Section 9. Capacitors are available in a huge range of package styles, voltage and current handling capacities, dielectric types , quality factors, and many other parameters. Due to the very Japanese nature of many of Sentai's stories, the. ETC Unison Echo Phase-Adaptive Dimmer 2 of 6 Echo Distributed Power Series SPECIFICATIONS FUNCTIONAL • Control individual fixtures or power for tungsten, line-voltage LED and other general-purpose loads. Lack of steering fluid makes it hard to turn the steering wheel. The series shows a great amount of aspect in the double life of a nightclub owner; the drug running aspect, and the nightclub management aspect. The legendary 7. Learn how to connect two or more batteries properly. This is illustrated in the following examples. In our conventions, arccot x ≡ arctan(1/x) is not continuous at x = 0 and thus does not possess a Taylor series about x = 0. An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition. Both types of circuits power multiple devices by the use of an electrical current flowing through wires, but that's where the likeness ends. However, let us do it from first principles. In particular we deal with some derived forms of Seleznev's theorem and we obtain common universal elements in the space of formal power series in several complex variables. Some of these. This tradition is especially rich in the fictional universes of various comic book stories, movies, and video games. Learn exactly what happened in this chapter, scene, or section of The Taylor Series and what it means. For example, the series 1, 2, 4, 8, 16. Finding the Power Series for ln(1 - x) A power series is the sum of an infinite number of terms. The 2019 tech job sector saw consistent growth and job availability. 875-in) at Lowe's. One of the first principles to understand when you are learning about electricity is the distinction between a parallel circuit and a series circuit. Learn how to connect two or more batteries properly. Each time I see one of these posts about information representation in R, I get this tingle to test the breaking points of Power BI. We are told that the third term of the arithmetic progression is 5. Many functions can be written as a power series. A "choke" is the common name given to an inductor that is used as a power supply filter element. The most common reasons for a No Power is that either a miss-fitted or faulty part is shorting the system and preventing it from starting or the power circuit on the motherboard isn't distributing the power correctly to supply enough to start-up the system. Although the reaching of this alarm can be gradual, it can also be instantaneous if you attempt to power a substantial load. The mission of the Steward Speakers Power Series is to inform, inspire, and foster meaningful dialogue and cultural exchanges. This type of noise is suppressed by installing a filter on. The Broan 40000 30 in. The common automobile battery consists of six 2. By contrast, the sequence 2, 3, 5, 8, 14, 22. It is one of the most commonly used tests for determining the convergence or divergence of series. For those who want to go a step farther I have a page providing an overview of available disc brake conversions that you can add to the power dual brake system. 258 Chapter 11 Sequences and Series closer to a single value, but take on all values between −1 and 1 over and over. Arif Kureshy joins Scott Hanselman to show how to use the Common Data Service (CDS) with PowerApps. The interval of convergence for a power series is the set of x values for which that series converges. called the common differenced, to the previous term. For a given power series X1 n=0 c n(x a)n there are only three. the series for , , and ), and/ B BB sin cos. Furthermore, we have already calculated the coefficients of the Trigonometric Series, and could easily calculate those of the Exponential Series. Common Power Series / Taylor Series = =1+ + + ! 2! 3! (1) cos = =1 + (2)! 2! 4! (1) = + sin. The common automobile battery consists of six 2. Hardwick, all square jaw and throbbing forehead veins, is a magnetic center to the action, with. Power Systems™ integrate into your organization's private or hybrid cloud strategy for flexible consumption models. Just use a jumper wire between the negative of the first battery and the positive of the second battery. Tips for Connecting to Excel on OneDrive for Business from Power Apps. The E12 series of resistor values, including their color codes. The concept is based on the Super Sentai series of shows, however, is not an English dub of the original, but rather a new production with English-speaking actors spliced in with the original Japanese footage to varying ratios. Search Data Center. Most of the effects make fighting and/or exploration easier. Unlike geometric series and p-series, a power series often converges or diverges based on its x value. Other possible problems include worn steering gear and ball sockets in the steering assembly, worn suspension system components and loose steering pump belt. Power Generation. The wagon (E61) body style as well as all-wheel drive option were added for 2006. For further details, see the class handout on the inverse. The Broan 40000 30 in. Here is a set of practice problems to accompany the Power Series section of the Series & Sequences chapter of the notes for Paul Dawkins Calculus II course at Lamar University. , where a 1 is the first term and r is the common ratio. sending searching. Many functions can be written as a power series. Series Cheatsheet Denitions Basic Series Power Series Power Serie: +X 1 n =0 an Showing Function/Taylor-Series Equivalence lim n ! + 1 R n (x). The Exponential Fourier Series coefficients are given by$$\displaylines. If you multiply any number in the series by 2, you'll get the next number. The Power is a ride into dark fantasy by Naomi Alderman that starts off like an E-ticket attraction at Disney Resorts before fizzling out like a bottle rocket from Jerry's Fireworks. Today we're excited to announce our how-to video series on the Official PowerApps YouTube channel. One of the first principles to understand when you are learning about electricity is the distinction between a parallel circuit and a series circuit. Bypass any surge protector and use a wall outlet that you know is working. Geometric Sequences and Sums Sequence. The common automobile battery consists of six 2. We have step-by-step solutions for your textbooks written by Bartleby experts!. Star power, stellar messages in charming animated series. 12, which is known as the ratio test. More generally, a series of the form is called a power series in (x-a) or a power series at a. A power series will converge provided it does not stray too far from this center. The current in an inductor cannot change instantaneously; that is, inductors tend to resist any change in current flow. Try the troubleshooting suggestions below to see if they resolve your power issues. In the spreadsheet below, the Excel Seriessum function is used to calculate the power series:. Series expansions of exponential and some logarithms functions. The mission of the Steward Speakers Power Series is to inform, inspire, and foster meaningful dialogue and cultural exchanges. Usually, we find circuits where more than two components are connected together. 1 in reliability by ITIC Ranked No. You can specify the order of the Taylor polynomial. For serial or parallel connection, it is necessary for some models to connect an external diode. The following theorem gives the form that. All batteries consist of individual cells. All of the books take place in world of cat characters, which belong to different \"clans\" that have. Common App and Reach Higher have united to inspire more people to complete their education and own their future, no matter what it holds. Free shipping and free returns on eligible items. The 2019 tech job sector saw consistent growth and job availability. An everyday examples of a battery is the 9-volt transistor battery, which is six 1. Other company, product, or service names may be trademarks or service marks of others. Textbook solution for Calculus: An Applied Approach (MindTap Course List)… 10th Edition Ron Larson Chapter 10 Problem 24TYS. If you only want that dollar for N = 10 years, your present investment can be a little smaller. However, we do not yet have an explanation for some of our series (e. 1 A power series has the form $$\ds\sum_{n=0}^\infty a_nx^n,$$ with the understanding that$\ds a_n$may depend on$n$but not on$x\$. In this case, let’s say we are going to wire them to a total, effective impedance of 2ohms. Kinetico Premier Series® non-electric water softeners are powered by moving water and have multiple tanks for continuous soft water, even during regeneration. , does f(x) = P. (Several of these are listed below. View Notes - Common Taylor Series(2) from M 427 at University of Texas. 1 in every major reliability category by ITIC*, IBM Power Systems deliver reliable on-premises infrastructure 24/7. The story-line is very nicely built. More generally, a series of the form is called a power series in (x-a) or a power series at a. Many forms of fiction feature characters attributed with superhuman, supernatural, or paranormal abilities, often referred to as "superpowers" (also spelled "super powers" and "super-powers") or "powers". Since each term is positive, the sum is not telescoping. Series compensation is the method of improving the system voltage by connecting a capacitor in series with the transmission line. We’ve spent quite a bit of time talking about series now and with only a couple of exceptions we’ve spent most of that time talking about how to determine if a series will converge or not. Geometric Sequences. Finding the Power Series for ln(1 - x) A power series is the sum of an infinite number of terms. Complete and sudden loss of power assisted steering, caused by failure of this unit, will cause the steering to become incredibly heavy at low speeds making manoeuvrability very difficult especially when trying to park. We use power series to approximate, with great accuracy, non-polynomial functions like , sin , and cos. This was my first time to a COMMON event and I was blown away. Let ti Theoretical. The concept is based on the Super Sentai series of shows, however, is not an English dub of the original, but rather a new production with English-speaking actors spliced in with the original Japanese footage to varying ratios. which is valid for -1
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# Subplot Position Calculator1
Posted by Jiro Doke,
Jiro‘s pick this week is Subplot Position Calculator by Christopher Hummersone.
subplot is a nice way to place a grid of axes on a single figure. But the spacing is sometimes a bit too generous, leaving too little room for the graph.
nRows = 3;
nCols = 2;
for m = 1:nRows
for n = 1:nCols
subplot(nRows,nCols,(m-1)*nCols+n)
plot(rand(20,1))
end
end
In that case, you can use axes or subplot to specify the actual placement of the axes.
figure
subplot('position',[0.05 0.1 0.45 0.8])
plot(rand(20,1))
subplot('position',[0.55 0.1 0.4 0.8])
plot(rand(20,1))
But this defeats the purpose of subplot because you have to calculate and specify the position manually. This is where Christopher’s entry comes in handy. It lets you specify the grid layout, with optional parameters, and it returns a set of position vectors which you can feed into subplot or axes.
The default is a tight layout with no margin.
pos = iosr.figures.subfigrid(nRows,nCols);
figure
for m = 1:nRows
for n = 1:nCols
subplot('Position',pos(m,:,n))
plot(rand(20,1))
end
end
You can specify the spacing and the scaling by passing in a couple of optional arguments.
pos = iosr.figures.subfigrid(nRows,nCols,[0.1 0.05 0.05 0.1],[.95 .95]);
figure
for m = 1:nRows
for n = 1:nCols
subplot('Position',pos(m,:,n))
plot(rand(20,1))
end
end
As some of you may have guessed, his function is part of a package (IoSR Matlab Toolbox), and the other functionalities are also highlighted in his other File Exchange entries.
In addition to the usefulness of this function, I like Christopher’s entry because it is well-written with plenty of error-checking and has good help with examples. I haven’t had a chance to check out his other functions in his toolbox, but I would expect the same kind of quality as seen in this one.
Comments
Give it a try and let us know what you think here or leave a comment for Christopher.
Get the MATLAB code
Published with MATLAB® R2017a
### Note
Comments are closed.
## 1 CommentsOldest to Newest
Dan replied on : 1 of 1
Yet another PotW dealing with subplot. I think the full functionality of subfigrid is actually contained in subplot, if TMW would just exposed the hard coded “inset” parameter through either the figure properties or the appdata of the figure. I mentioned that I have an enhancement request the last time you picked a subplot FEX submission (http://blogs.mathworks.com/pick/2012/12/21/figure-margins-subplot-spacings-and-more/?s_tid=srchtitle#comment-17796). Maybe someday TMW will improve subplot.
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## NO
804991762_4A
Posts: 47
Joined: Fri Apr 06, 2018 11:04 am
### NO
For the Lewis structure of NO, it is a radical, and In class today the structure given was:
. ..
:N-O:
..
Is it possible that it can have a double bond instead of a single bond? Would this be resonance?
Gisselle Sainz 2F
Posts: 76
Joined: Wed Feb 21, 2018 3:00 am
### Re: NO
You can add a double bond to the NO structure to stabilize the formal charge, but this wouldn't be resonant to the single bond structure b/c a double would transform the structure completely.
Valeria Viera 1B
Posts: 60
Joined: Fri Apr 06, 2018 11:05 am
### Re: NO
I believe taking a lone pair of electrons from oxygen to make a double bond between N and O is possible and it would make the formal charges of O and N zero (without the double bond the formal charge of N is +1 and FC of O is -1)
So I think creating a double bond would be the way to go
$\dddot{N}= {\ddot{\ddot{O}}}$
(idk how to do the lewis structures sorry sorry I will learn later)
As for the resonance question
resonance means to have multiple bonds in different equivalent locations
I don't think there's another place to put the double bond (there's really only one place for bonding btwn O and N) so I believe this is not resonance (correct me if I am wrong)
Alma Flores 1D
Posts: 64
Joined: Wed Nov 08, 2017 3:01 am
### Re: NO
At the end of lecture, the TA explaining this section started drawing the Lewis structure of NO with a double bond. I didn't hear what she said, but I believe a double bond would work. Also, this would not be resonance.
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Related Tags
euclidean distance
# How to compute Euclidean distance in C#
## Definition
The Euclidean distance, in any n-dimensional plane, is the length of a line segment connecting the two points.
## Formula
The Pythagorean Theorem can be used to calculate the distance between two points, as shown in the figure below. If the points $(x_1,y_1)$ and $(x_2,y_2)$ are in 2-dimensional space, then the Euclidean distance between them, represented by d, is: $\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$
## In higher dimensions
Similarly, if the points are in a 3-dimensional space, the Euclidean distance will be represented by: $\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}+(z_2-z_1)^{2}}$
In general, for two points p and q given by Cartesian coordinates in an n-dimensional space, p is represented by $(p_1,p_2, ..., p_n)$, q is represented by $(q_1, q_2,...,q_n)$, and the Euclidean distance is: $\sqrt{(p_1-q_1)^{2}+(p_2-q_2)^{2}+(p_3-q_3)^{2}+....+(p_n-q_n)^{2}}$
## Implementation in C#
For reference, below is the implementation of Euclidian distance calculation in C#, using the Math class to take the square and square-root.
using System;
namespace eucledian_distance
{
class Program
{
static void Main(string[] args)
{
double x1, x2, y1, y2;
x1 = 3;
x2 = 4;
y1 = 5;
y2 = 2;
var distance = Math.Sqrt((Math.Pow(x1 - x2, 2) + Math.Pow(y1 - y2, 2)));
Console.WriteLine(\$"The Eucledian distance between the two points is: {Math.Round(distance, 4)}");
}
}
}
RELATED TAGS
euclidean distance
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## anonymous 4 years ago The probability density function$f(x)=\left(\begin{matrix}c(4-x^2) \\ 0\end{matrix}\right)$ C = constant = 3/32$-2 \le x \le 2$otherwise. Find $P(|x| \ge 1).$
1. ash2326
$P(|x|\ge1) = P(-\infty\le x\le -1)+P(1\le x \le\infty)$ $P(|x|\ge1) = \int_{\infty}^{-1}f(x)dx+\int_{1}^{\infty}f(x) dx$ since f(x) in non zero for $$-2\le x \le 2$$ we'll have $P(|x|\ge1) = \int_{-2}^{-1}f(x)dx+\int_{1}^{2}f(x) dx$ here $f(x)= \frac{3}{32} (4-x^2)$ can you do it now?
2. anonymous
I understand it, but I don't know how to remove |x| to make it x. Is it just adding the - integrals to the positive integrals? i.e. $\int\limits_{-1}^{2} and \int\limits_{2}^{1}?$
3. anonymous
*-2
4. ash2326
You need not remove |x|, that we have already included. Substitute the value of f(x) and evaluate the two integrals
5. anonymous
Ok, yes, I understand how to do the integration from then on. But I'm not sure how you got the -2 < x < -1 and 1 < x < 2 in the first place?
6. ash2326
see we want probability in the range $$|x|\ge 1$$ this is equivalent to $$-\infty \le x\le 1$$ and $$1 \le x\le \infty$$ therefore I split the integral in two ranges. f(x) is non zero in the range $$-2 \le x\le 2$$, therefore it boils down to two integrals from -2 to -1 and from 1 to 2
7. anonymous
Ah Ok. Thank you!
8. anonymous
Do you think you could help me with one more example?
9. anonymous
I mean, I have another question
10. ash2326
I'll try, post it as a new question
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## Wikipedia - Dolichole dolichols
https://doi.org/10.1351/goldbook.D01829
A group of @P04816@ derivatives in which $$n$$ in the general formula H–[CH2–C(CH3)=CH–CH2]n–OH is greater than 4 and in which the residue that carries the hydroxy group is saturated, i.e. 2,3-dihydropolyprenols.
Note:
The collective term prenol should not be used without qualification to include dolichols since these are derivatives of @P04816@.
Source:
White Book, 2nd ed., p. 255 [Terms] [Book]
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# Tiles
Hall has dimensions 325 &time; 170 dm. What is the largest size of square tiles that can be entire hall tiled, and how many we need them?
a = 50 cm
b = 50 cm
n = 2210
### Step-by-step explanation:
$n=325\cdot 170\mathrm{/}{5}^{2}=2210$
Did you find an error or inaccuracy? Feel free to write us. Thank you!
Tips to related online calculators
Do you want to calculate greatest common divisor two or more numbers?
#### You need to know the following knowledge to solve this word math problem:
We encourage you to watch this tutorial video on this math problem:
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nLab unitary representation of the Poincaré group
Surveys, textbooks and lecture notes
Representation theory
representation theory
geometric representation theory
Contents
Idea
The Poincaré group is the group of rigid spacetime symmetries of Minkowski spacetime. It is a topological group and as such has unitary representations on infinite-dimensional Hilbert spaces. For any quantum field theory in Minkowski space its space of states therefore decomposes into irreducible representations of the Poincaré group. As was first observed by Hermann Weyl, these irreducible representations encode the particle spectrum of the QFT.
We are interested in the unitary representations of a topological group $G$, i.e., the continuous group homomorphisms
$U\colon G \to U(H)$
into the unitary group of a Hilbert space $H$, especially those that are irreducible in the usual sense of representation theory. The topology on $U(H)$ here is understood to be the strong operator topology.
In this and related articles, we study such representations in the case where
$G = SL_2(\mathbb{C}) \ltimes \mathbb{R}^4$
is the universal cover of the connected component of the identity of the Poincaré group, which is important in the study of quantum field theory. A more physical name for such a representation is “elementary particle”, and we will often use that term in this article. (NB: “elementary particle” will always refer to the formal mathematical representation it refers to.)
A full rounded account could become large; see the blog discussion, which despite its size was left in a still-nascent state. However, in a nutshell, the basic theorem is that (elementary) particles are classified up to isomorphism according to their mass and helicity; mass is a continuous parameter and helicity is a discrete parameter.
This theorem in commonly ascribed to Eugene Wigner and often referred to as the Wigner classification.
Wigner classified all irreducible unitary representations of the restricted Poincare group, including the unphysical ones. The latter cannot be used to define a free quantum field theory satisfying the Wightman axioms. Those that can are the physical ones and are characterized by a nonnegative real mass and a nonnegative half-integral spin; the zero component of the momentum has a nonnegative spectrum. Many of these are realized by particles occurring in Nature, though not as ‘elementary particles’ but as bound states (in a suitable approximation, e.g., QCD). From the point of view of representation theory, the center of mass of a bound state behaves just like an elementary particle. Thus elementary is meant in this generalized sense.
Relevant topics in a full account will include
Tentative notes, to be expanded on…
In the first place, physicists tend to be a little carefree with the mathematics, so this account is written from the point of view of a ‘stupid’ mathematician (for the moment Todd Trimble) who wants to get details straight and precise.
For example, physicists tend to talk about “eigenstates” as if they were elements of the Hilbert space, and other states as linear combinations of eigenstates, whereas really we are dealing with some more complicated technology like rigged Hilbert spaces or direct integrals instead of direct sums. Failure to mention such details places hurdles of communication between physicists and mathematicians. In addition, there are stylistic differences in presentation, where a physicist will happily deal with formulas replete with lots of subscripts and superscripts, whereas many mathematicians prefer dealing with more conceptual, less notation-heavy explanations.
There seem to be at least three ways of dealing with spectral theory of (unbounded) self-adjoint operators on a Hilbert space:
• The usual Stone theory
• Direct integrals of Hilbert spaces
• Rigged Hilbert spaces
Rigged Hilbert spaces
A rigged Hilbert space
Induced representations
Simultaneous diagonalization.
Fact that Poincaré group is a semidirect product group.
Let $\vec p \in \mathbb{R}^{d-1,1}$ be a given vector in Minkowski spacetime. Write
$Stab_{Iso(\mathbb{R}^{d-1,1}))}(\vec p) \hookrightarrow Iso(\mathbb{R}^{d-1,1})$
for its stabilizer subgroup (often called the little group in this context, going back to Wigner). Every unitary irrep of $Iso(\mathbb{R}^{d-1,1})$ of mass $p$ is an induced representation of a finite dimensional representation of the “little group” $Stab_{Iso(\mathbb{R}^{d-1,1}))}(\vec p)$. (recalled concisely e.g. in Dragon 16, p. 2).
References
The observation that the irreps of the Poincaré group correspond to fundamental particles (Wigner classification) is due to
• Eugene Wigner, On unitary representations of the inhomogeneous Lorentz group , Ann. Math. 40, 149 (1993)
Review includes
Last revised on December 27, 2017 at 06:09:27. See the history of this page for a list of all contributions to it.
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# PH-EP Articles
Ostatnio dodane:
2017-03-20
12:28
Measurement of the $B^0_s\to\mu^+\mu^-$ branching fraction and effective lifetime and search for $B^0\to\mu^+\mu^-$ decays / LHCb Collaboration A search for the rare decays $B^0_s\to\mu^+\mu^-$ and $B^0\to\mu^+\mu^-$ is performed at the LHCb experiment using data collected in $pp$ collisions corresponding to a total integrated luminosity of 4.4 fb$^{-1}$. An excess of $B^0_s\to\mu^+\mu^-$ decays is observed with a significance of 7.8 standard deviations, representing the first observation of this decay in a single experiment. [...] CERN-EP-2017-041; LHCB-PAPER-2017-001; arXiv:1703.05747; CERN-EP-2017-041-LHCB-PAPER-2017-001.- Geneva : CERN, 2017 - 19 p. - Published in : Phys. Rev. Lett. 118 (2017) 191801 APS Open Access Article: PDF; Fulltext: 10.1103_PhysRevLett.118.191801 - PDF; arXiv:1703.05747 - PDF; Related data file(s): ZIP; Related supplementary data file(s): ZIP;
2017-03-18
22:59
Search for anomalous couplings in boosted $\mathrm{ WW/WZ }\to\ell\nu\mathrm{ q \bar{q} }$ production in proton-proton collisions at $\sqrt{s} =$ 8 TeV / CMS Collaboration This Letter presents a search for new physics manifested as anomalous triple gauge boson couplings in WW and WZ diboson production in proton-proton collisions. The search is performed using events containing a W boson that decays leptonically and a W or Z boson whose decay products are merged into a single reconstructed jet. [...] CMS-SMP-13-008; CERN-EP-2017-029; arXiv:1703.06095.- Geneva : CERN, 2017 - 30 p. - Published in : Phys. Lett. B B 772 (2017) 21-42 Article from SCOAP3: PDF; Fulltext: PDF;
2017-03-07
21:17
Measurement of the top quark mass using single top quark events in proton-proton collisions at $\sqrt{s}=$ 8 TeV / CMS Collaboration A measurement of the top quark mass is reported in events containing a single top quark produced via the electroweak $t$ channel. The analysis is performed using data from proton-proton collisions collected with the CMS detector at the LHC at a centre-of-mass energy of 8 TeV, corresponding to an integrated luminosity of 19.7 fb$^{-1}$. [...] CMS-TOP-15-001; CERN-EP-2017-012; arXiv:1703.02530.- Geneva : CERN, 2017 - 36 p. - Published in : Eur. Phys .J. C 77 (2017) 354 Fulltext: PDF;
2017-02-21
13:05
2017-02-10
11:02
Measurement of the cross section for electroweak production of Z$\gamma$ in association with two jets and constraints on anomalous quartic gauge couplings in proton-proton collisions at $\sqrt{s} =$ 8 TeV / CMS Collaboration A measurement is presented of the cross section for the electroweak production of a Z boson and a photon in association with two jets in proton-proton collisions at $\sqrt{s} =$ 8 TeV. The Z bosons are identified through their decays to electron or muon pairs. [...] CMS-SMP-14-018; CERN-EP-2016-308; arXiv:1702.03025.- Geneva : CERN, 2017-07-10 - 32 p. - Published in : Phys. Lett. B 770 (2017) 380-402 Article from SCOAP3: PDF; Fulltext: PDF;
2017-02-10
07:38
Flow dominance and factorization of transverse momentum correlations in Pb-Pb collisions at the LHC / ALICE Collaboration We present the first measurement of the two-particle transverse momentum differential correlation function, $P_2\equiv\langle \Delta p_{\rm T} \Delta p_{\rm T} \rangle /\langle p_{\rm T} \rangle^2$, in Pb-Pb collisions at $\sqrt{s_{_{\rm NN}}} =$ 2.76 TeV. Results for $P_2$ are reported as a function of relative pseudorapidity ($\Delta \eta$) and azimuthal angle ($\Delta \varphi$) between two particles for different collision centralities. [...] CERN-EP-2017-021; arXiv:1702.02665.- Geneva : CERN, 2017 - 16 p. - Published in : Phys. Rev. Lett. 118 (2017) 162302 APS Open Access article: PDF; Fulltext: 10.1103_PhysRevLett.118.162302 - PDF; arXiv:1702.02665 - PDF;
2017-02-09
08:13
Azimuthally differential pion femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV / ALICE Collaboration We present the first azimuthally differential measurements of the pion source size relative to the second harmonic event plane in Pb-Pb collisions at a center-of-mass energy per nucleon-nucleon pair of $\sqrt{s_{\rm NN}}=2.76$ TeV. The measurements have been performed in the centrality range 0-50% and for pion pair transverse momenta $0.2 < k_{\rm T} < 0.7$ GeV/$c$. [...] CERN-EP-2017-013; arXiv:1702.01612.- Geneva : CERN, 2017 - 16 p. - Published in : Phys. Rev. Lett. 118 (2017) 222301 Fulltext: arXiv:1702.01612 - PDF; 10.1103_PhysRevLett.118.222301 - PDF; PRL OA article: PDF;
2017-02-09
08:12
Production of $\pi^0$ and $\eta$ mesons up to high transverse momentum in pp collisions at 2.76 TeV / ALICE Collaboration The invariant differential cross sections for inclusive $\pi^{0}$ and $\eta$ mesons at midrapidity were measured in pp collisions at $\sqrt{s}=2.76$ TeV for transverse momenta $0.4 < p_{\rm T} < 40$ GeV/$c$ and $0.6 < p_{\rm T} < 20$ GeV/$c$, respectively, using the ALICE detector. This large range in $p_{\rm T}$ was achieved by combining various analysis techniques and different triggers involving the electromagnetic calorimeter (EMCal). [...] CERN-EP-2017-019; arXiv:1702.00917.- Geneva : CERN, 2017-05-22 - 33 p. - Published in : Eur. Phys. J. C 77 (2017) 339 Fulltext: PDF; Springer Open Access article: PDF;
2017-02-06
11:32
Measurement of prompt and nonprompt $\mathrm{J} / \psi$ production in pp and pPb collisions at $\sqrt{s_{\mathrm{NN}}} =$ 5.02 TeV / CMS Collaboration This paper reports the measurement of $\mathrm{J} / \psi$ meson production in proton-proton (pp) and proton-lead (pPb) collisions at a center-of-mass energy per nucleon pair of 5.02 TeV by the CMS experiment at the LHC. The data samples used in the analysis correspond to integrated luminosities of 28 pb$^{-1}$ and 35 nb$^{-1}$ for pp and pPb collisions, respectively. [...] CMS-HIN-14-009; CERN-EP-2017-009; arXiv:1702.01462.- Geneva : CERN, 2017 - Published in : Eur. Phys. J. C 77 (2017) 269 Fulltext: cms-hin-14-009-arxiv - PDF; arXiv:1702.01462 - PDF; Springer Open Access article: PDF;
2017-02-04
07:25
K$^{*}(892)^{0}$ and $\phi(1020)$ meson production at high transverse momentum in pp and Pb-Pb collisions at $\sqrt{s_\mathrm{NN}}$ = 2.76 TeV / ALICE Collaboration The production of K$^{*}(892)^{0}$ and $\phi(1020)$ mesons in proton-proton (pp) and lead-lead (Pb-Pb) collisions at $\sqrt{s_\mathrm{NN}} =$ 2.76 TeV has been analyzed using a high luminosity data sample accumulated in 2011 with ALICE detector at the Large Hadron Collider (LHC). Transverse momentum ($p_{\mathrm{T}}$) spectra have been measured for K$^{*}(892)^{0}$ and $\phi(1020)$ mesons via their hadronic decay channels for $p_{\mathrm{T}}$ up to 20 GeV/$c$. [...] CERN-EP-2017-010; arXiv:1702.00555.- Geneva : CERN, 2017 - 26 p. - Published in : Phys. Rev. C 95 (2017) 064606 Fulltext: PDF;
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## Tower theorem
Dear Prof Kim
I can show that $[Q(\sqrt{2},\sqrt{3},\sqrt{5}):Q]=8$. But $[Q(\sqrt{6},\sqrt{10},\sqrt{15}):Q] = 4$ not 8. Why is that?
Can you explain why and give me instruction to prove it please?
Thanks very much.
———————————–
Reply: It’s important at this point to understand precisely what goes into the tower theorem.
For the first case, we have
$Q\subset Q(\sqrt{2})\subset Q(\sqrt{2},\sqrt{3})\subset Q(\sqrt{2},\sqrt{3},\sqrt{5}).$
But the important point is that the square root we’re throwing in at each stage is not in the previous field. That is, $\sqrt{3}\notin Q(\sqrt{2})$ and $\sqrt{5}\notin Q(\sqrt{2},\sqrt{3})$. Of course, these statements must be proved.(Try it!)
Now let’s examine the tower
$Q\subset Q(\sqrt{6})\subset Q(\sqrt{6},\sqrt{10})\subset Q(\sqrt{6},\sqrt{10},\sqrt{15}).$
In this case, it’s true that $\sqrt{10}\notin Q(\sqrt{6})$, so that the degree of
$Q(\sqrt{6},\sqrt{10})$ is 4. But note that
$\sqrt{10}/\sqrt{6}=\sqrt{5/3}$ and $3\sqrt{5/3}=\sqrt{15}$. So in fact,
$Q(\sqrt{6},\sqrt{10})= Q(\sqrt{6},\sqrt{10},\sqrt{15}).$
This explains the degree 4.
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# How can I motivate my fellow players?
Although I'm very much enjoying the one of the games I'm in, the other players seem to be flagging and losing interest. This is a bit disheartening as their lack of enthusiasm is bringing down the game for me and I'd like to help get them re-motivated and back into it The GM is doing all the right sort of things (see associated posts) but...
But how can I as a player motivate my fellow players to regain their enthusasm for the game?
They certainly enjoy things once they "get going" and get themselves into it, but their enthusiasm at the start and here and there is lacking and they tend to wander subjects. This is a primarily social game (WoD); Combat is pretty rare (once in every four sessions or so) and none of the characters are really statted for it (except mine, bizarrely). The other players will engage with NPCs. They do take a long time to wind up to play (first hour or so) but they'll wander in play once it's started as well. We do have problems with occasional pauses (online game) and getting back rolling again after, but it's as much distracting irrelevant talk that seems (to me) to indicate wandering interest.
• Definitely agree with @JoshuaAslanSmith; I had a game that suffered a lot because I had this perception that my players wanted opportunities to beat stuff up when they were actually getting quite tired of combat (being new to the game and therefore taking a long time to complete one). – KRyan May 2 '13 at 14:38
• Most of the answers so far are more things a GM would do and are retreads from various other questions already on the site - where's the advice on how a player can influence the other players? – mxyzplk says reinstate Monica May 3 '13 at 13:09
Meta: Do the grown up thing: Ask them up front that you perceive them losing their enthusiasm for the game and that you would like to know if it's true (or just your perception) and if so, why this is happening? Then, you can resolve things so that everyone is happy.
In game: If the problem is one of distraction, then set aside time to drink coffee and chat before the game starts. Set a timer and once it gets going: just stay in character at all times!. Even when you go to the toilets, or answer your phone, or order pizza: just be in character. Costumes props can help as well as long as they are not too disruptive in and of themselves. Suggest the use of mood lighting and mood music as well. Removing sources of distractions can help a lot: no more laptop (with frikking solitaire), comic books, or film magazines...
...
From a comment: how do you get the other players to agree to do that? By that, I assume the second point. If you cannot get the other players to listen to you, then find better friends. Otherwise, you engage the grown ups in the room. If you cannot reach a common ground, find other players. Or just try it, they may pick on that and go with it. You could learn manipulation, blackmail, and confidence trickery so that you can force the other players to do your bidding. However, in my not so humble opinion, that makes you a scum bag. Mind controlling your fellow players is in the realm of bad super heroes plots. <pinch of salt added>. ^_~
• And so how do you as a player get the other players to do that? – mxyzplk says reinstate Monica May 3 '13 at 17:07
• @mxyzplk I think he addresses them, talk to them about the plan with the timer and implement if they agree. For costumes, props etc, talking will helping, leading by example will help more. – TimothyAWiseman May 3 '13 at 18:17
• @TimothyAWiseman Then he should unpack that more... – mxyzplk says reinstate Monica May 3 '13 at 18:28
• That'll work! I'll try this whole "communication" thing. ;) – Rob May 7 '13 at 10:15
In teaching, we are taught that there are 6 C's to motivation. These are: choice, challenge, control, collaboration, constructing meaning, and consequences. I would suggest that to motivate players in ways other than those already mentioned (recaps, social times, frequent breaks, etc) I would try to focus on helping the DM/GM provide more of the 6 C's.
With regards to choice, try to think of what gaming choices you can promote/support that would favour the interests of the unmotivated players. For example, if they are bored by combat, start championing group approaches that favour diplomacy, evasion, or intrigue.
When thinking challenge, try to pick fights/missions that suit but stretch the abilities of the members of your group. Fight each other if you have no appropriate alternative, but pick and choose your battles so that everyone can contribute with a real sense of (manageable) threat present.
Giving players control over how and when they will participate promotes player buy-in, increasing motivation. Try not to let any one voice or "group think" dominate your play. Be the "devil's advocate" when you can.
Collaboration is implied in any gaming group, but you can take it on as a personal responsibility to augment the effect of another player's desired course of action by aiding them in ways that give bonuses or further alternatives for action - even if it's not something you might normally choose to do.
Constructing meaning is harder for a player to accomplish than a DM/GM, but basically, in order to motivate someone to do something, it needs to personally have meaning for that player. Perhaps, in character, you can start wagering gold on the outcome of various encounters, or suggest why what the group is doing next should matter to the unmotivated player (in character).
Finally, consequence is a matter of recognizing achievement when it happens. Showcasing an unmotivated player's achievements motivates that player to continue along that line in the future. Be descriptive where possible.
Likely the issue is one of the following:
## Why are we here?
As has been addressed in other threads both here and across the internet, we game for different reasons. Some people like to engage their inner-thespian, while others have a very stressful job and only want to picture those annoying people when they "HULK SMASH!!!" Still others are there because they like the people and will play D&D with the group just as willingly as they will play Bridge or watch cult-classic movies.
## Why is everything always about YOU?
Since you mention that you are 100% engaged, but the other players are getting increasingly "meh", I would suspect that the plot is very closely focused on your character just now. Maybe the DM wrote a campaign that would be fun to play and you happened to create the perfect centerpiece character for that adventure, maybe the DM has been increasingly writing adventures around your character's wishes/desires. While the centerpiece character is having the time of their life, the others are getting more and more bored about stuff their characters just don't care about. I once played a game where we discovered a continent. My character was the Royal Cartographer and was in heaven mapping as much as the DM would let me map, whereas all of the other characters were mostly urban-centric characters on a great wild continent with no cities, and no real way to make money or do what they imagined doing during character creation.
## How do we get out of this hole?
Whether it's a hole of plot-lines circling a character or the tone of an adventure matches up with one player at the exclusion of the others, the solution is the same. Sharing.
If it's a tone issue, bring in the elements/motivators for the other players. If you like long dialogs but the other players don't, either handle the long chats off-line via email, or decide that you don't need to have the entire life story of every minor NPC. If you want non-stop action and the other players enjoy talky scenes, stop running ahead and pushing the plot forward and engage in some dialog. If you like combat but the others want to explore, choose to not do the frontal assault for once ("Normally I would suggest we bash down the door and do the direct thing, but I've got a bad feeling about this place..."). Variety in your actions not only keeps things fresh but lets folks with other interests put forward their ideas.
If the issue is that you are the one getting all the "plot cookies", figure out what each character can do for YOUR character and take advantage of their skills/abilities in your schemes to pull them back into the game from there. In my Cartographer adventure, he would routinely spend days at a time away from the main camp, the bard was an excellent messenger who could relay the news of the camp (about 50 NPCs) and carry messages between my character and the civil authority. The Wizard was able to pick up cartography pretty quickly and help my character out with rough sketches to make the map more accurate and loved the opportunity to find new magical reagents and sources of material components for spells. The more I was able to bring in the motivations of the characters and the motivations of the players, the more the group was pulled back together.
Regardless of which is the problem, I would also drop an email to the DM and bring your concern(s) up. The DM can much easier alter tone/plot to work in players who are bored, but as someone has been an overworked DM it is sometimes hard to see that 1 person is lapping up the plot and the other party members are getting bored. Also, sometimes the "bored" player is not bored, but distracted. There have been a few times where personal issues have kept me from really immersing myself into a game.
• When you say "we game for different reasons", I think of my answer to another question. Do you want to link to that, or directly to the article it's based on? – SevenSidedDie May 2 '13 at 16:57
• @SevenSidedDie, Good suggestion, Edited. – Pulsehead May 2 '13 at 17:06
• The first part at least tries to address players influencing player, but the second is cut and paste GM advice. – mxyzplk says reinstate Monica May 3 '13 at 17:13
• @mxyzplk, no, The second (bringing the other player's characters into the plot) is something my character did as a player. My character did everything he wanted to do (and was largely the center of attention), but he would use the resources of other characters to help him. Since my character would then have many conversations 1-1 with other characters, the other characters were pulled into the plot. It did also break the mold as I usually play characters that are more supporting cast than lead character. – Pulsehead May 8 '13 at 17:02
• That's a good edit! – SevenSidedDie May 8 '13 at 19:23
Props and Cues: To echo what Sardathrion said, gaming props (like the single article of clothing the Penny Arcade Guys wore) can pull players into character much like how costumes help actors assume a role. Music can also be a big way to draw people in. This can be really overdone to the point where the GM spends as much time crafting the perfect playlist and running the music as he spends prepping the session and running the game. Kept to proportion though starting music (especially if its a theme song) and some appropriate music to play in the background during the dice heavy combat sessions can rouse players and draw them in as well.
Set social time: For some groups, D&D is their main time spent with each other and everyone likes to catchup on life, talk about the latest episode of insert fantasy/scifi series here and in general enjoy each others company. This is not a bad thing, its probably why you started playing together or came about from playing together. What is bad is if it becomes muddled with actual game time. Setting aside time (also nod to Sardathrion) before the session begins with an agreement from everyone about how long it is acknowledges this and helps you to get it out of the way before gameplay commences.
"Last time on Dragon Ball Z..." Quick, witty recaps (perhaps purposefully cheesy) can refresh player's memories and put them in the right frame of mind quickly. Unlike the actual recaps from Dragon Ball Z, keep these ones to about a minute long.
Have set break times perhaps people have actual physical or mental fatigue mid-session. Having a designated 15 minute break every 2 hours you play can let people stretch, take a breather, and maybe grab some snacks without interrupting gameplay and feeling stressed out about it.
• +1 for Social time and Recaps. My main gaming group uses the first 30 minutes of each session to chat, eat and/or level up their characters before playing. It definitely helps. – Discord May 2 '13 at 15:21
• And how do you establish this as a player and not as the GM? – mxyzplk says reinstate Monica May 3 '13 at 17:08
• @mxyzplk I don't know that any of my suggestions work (or anyone else's) without some kind of interaction from the other players and the GM. I assumed it goes without saying that you would need to pitch it to the group and have them accept it. You could however bring your own props with some extra props and just see if people want to use them. Likewise you could also try to create break times by you yourself saying "I really need to take a break" every 2 hours and the rest of the people just following along. – Joshua Aslan Smith May 7 '13 at 12:09
Communication is Key
As I am the DM I usually worry about motivating my group from my side of the DM screen and as a DM it can be very disheartening to see you players lose interest but your question entails quite a different dilemma but in my opinion is a very interesting and important one. As always communication seems to be the key.
I would ask them what's going on,why they don't seem to be enjoying the game. Maybe you should point out what you see going on to your GM and ask him to get everyone together and talk about what they like, what they don't like and also talk about everyone's expectations.
If your GM doesn't have the time you could contact your fellow players get the needed information and get they're opinions to the GM.
This may be a situation where you have to take charge instead of relying on the GM which seems to be exactly what you are doing.
• Yes, he's saying he'd like to influence the other players... So how? – mxyzplk says reinstate Monica May 3 '13 at 17:08
If you have personal plots, try to involve characters on them. Not as a whole, but individually. If a PC has criminal contacts, try to use them in your plans. If one is good at forgery, involve him in the creation of false documents. Seek their help and make them feel special.
This is what I do when I see a PC is getting outside the story.
I've encountered the same dilemma. @Sardathrion's point about honest communication is right on the money. Discussing what's working and what's not working will solve most problems. That said, sometimes it's difficult for players to pinpoint what they are missing, or why they are distracted. You may want to suggest to the GM that opening each session with a bang might help get everyone engaged.
The GM could have an NPC come to the PCs with a problem that requires investigation. A thief could steal something important from one of the PCs. The PCs could get hauled before the magistrate. Whatever the event, it needs to be dramatic, and the outcome must be meaningful; if the PCs don't handle it well, they'll pay some sort of price. This will get the players emotionally engaged quickly.
Once the players are emotionally engaged, it may be easier to keep the flow of play moving smoothly.
You need to determine what kind of players you are playing with. Some enjoy story lines, pursue growing their characters, most just like to kill stuff. Its' best to play with like-minded players.
Always have game chat before the game starts. The best thing to do is chat about what you would like to see from the game with each other while the GM is setting up. I'm sure the GM will be eve-dropping, and some of it may be incorporated into the campaign somehow. It's just always good to get everyone psyched about playing before you play.
Take Breaks! about every 30 minutes of storyline, or after a great battle, everyone should take 5 or 10 minutes break. Step aside and reflect/discuss what just happen, what was fun about it, etc...
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Ensure there are no obstructions or ground people in the swing radius. Because emergency vehicles have sirens and flashing lights and other vehicles must pull over, they can typically use the full right-of-way without encountering opposing vehicles. If you let the bucket slow down you get wet. Attach a small object of high density to the end of the string (for example, a metal nut or a car key). A compound pendulum has an extended mass, like a swinging bar, and is free to oscillate about a horizontal axis. When printing please make sure page scaling is set to none. And, there are small "scale" prototypes and large "toy" models meaning there is still no definitive minimum radius. I have no idea how to do it mathematically, but there are some mechanical ways to determine this. They are typically located in industrial areas with lower levels of pedestrian and car traffic. Street Openings – Pedestrian-only Streets, Map 15 in the Transportation Element of the General Plan, corners with turning buses on Muni rapid or local routes or routes rapid or local buses use to start run or return to yard, corners with turning buses on Muni community routes or routes community buses use to start run or return to yard, B-40; some routes have B-30 buses, check with SFMTA, corners with turning buses on routes served by Golden Gate Transit, AC Transit, SamTrans, Vallejo Transit, University of California transit services, PresidiGo, corners with potential occasional turning buses due to detours, P: turn partially from adjacent lane; turn fully from adjacent lane, turn from opposite lane, turn into opposite lane, all intersections at streets > 150’ in length, P: turn partially from adjacent lane, turn fully from adjacent lane, turn from opposite lane, turn into opposite lane, GP transportation element Map 15 designated “Routes with significant truck traffic”, alley, shared public way, neighborhood residential, local lanes of boulevard, neighborhood commercial, downtown commercial, downtown residential, commercial throughway, residential throughway, urban mixed-use, parkway, through lanes of boulevard. If you measure your longest car or locomotive from the end of the couplers, try to create a radius 3.5 to 5 times the length of that car. This represents an absolute principle that will always work no matter the type of design. If you measure your longest car or locomotive from the end of the couplers, try to create a radius 3.5 to 5 times the length of that car. Throughways: Throughways typically have wide roadways, high traffic volumes and speeds, and more large vehicles. Projects involving bulb-outs or sidewalk widening on more than one block require legislative approval by the Board of Supervisors. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". So, if you are finding that your shell feature is failing, the first thing to check is the minimum radius of curvature. patents-wipo. This saucer swing holds two children of up to … Standard street types describe appropriate design vehicles to use per street types, based on Better Streets Plan street types. Posted October 6th, 2016. I'm sure there are better ways, and there are interesting optical dresser options for small work, like Herman Schmidt, Opti dress, Kuhn, etc. The critical velocity is the slowest velocity necessary to keep the water in the bucket and not on you. 3.5 is a compromise of best operation and reasonable appearance, 5 can look and operate prototypically. Strategy. If the arc of the wall is less than 180 degrees, calculate the radius by using the process described above. The string passes through a glass tube. 1950s postwar U.S. looked nothing like postwar Europe. Pedestrian-activity streets: Pedestrian-activity streets typically have high volumes of pedestrians, moderate traffic volumes, and frequent need for loading access. The radius of the tire is given to be r = 0.300 m . When making measures, you need to be aware of the curtain roll size too. Find V when h = 4 cm and r2 = 36 cm 2. These routes must also accommodate historic streetcars. A special reversible compound pendulum called Kater’s pendulum is designed to measure the value of g, the acceleration of gravity. Grip the Impact Snap like you would a club , and swing it the same way. A particle’ is a small mass at some position in space. The radius of the tire is given to be r = 0.300 m . On some Muni community routes, Muni may use a B-30 – check with SFMTA. Accommodate [a vehicle turn]: to allow for a particular vehicle type to complete a turn with latitude to use adjacent or opposing lanes on the origin or destination streets. It’s usually mounted above the opening and ranges in sizes between 300–450 mm. There is also a quick and dirty concept that you can use to determine appropriate curve radius. So if you assume that the maximum height most people can reach on a traditional playground swing is level with the rotation point (and even that's a stretch for most people), then: Assuming r=2 meters Atotal=Ac + g Ac=sqrt(2gr)/r= 3.132 m/s^2 Curb radius changes may be installed as part of a traffic calming project or other public or private initiative. Local businesses have a strong stake in well cared-for streets. Local streets: Local streets are typically narrower streets with low traffic volumes and speeds, and limited need for large vehicles. See Figure 4. The sharper the radii, the greater the distance between rails will need to be. On busier streets, the ability of emergency vehicles to swing wide may be limited by queued traffic which may not be able to pull over. One formula that can come into play is measuring the radius of a curved wall. Thankfully, there’s a training aid for that. When you swing a bucket of water up and down in a vertical circle you can keep the water in the bucket if you keep the velocity high enough. They pull and kick their legs but the swing will not swing. This treatment has limited application, such as industrial streets. If you have parallel tracks in a curve, the spacing between tracks is important. Because the linear speed of the tire rim is the same as the speed of the car, we have v = 15.0 m/s . Freight routes: Freight routes are streets that are designated as “Routes with Significant Truck Traffic” on Map 15 in the Transportation Element of the General Plan. There are some clear plastic circle templates that will help get an approximate radius and center location. 1 South Van Ness Avenue, 7th Floor Curb radius changes may be installed as part of a traffic calming project or other public or private initiative. Design vehicle: selected vehicle type used in determining appropriate turn radius at an intersection. However, curb radius design is sensitive to a wide range of variables; these guidelines cannot replace professional judgment and technical analysis. The rotating swing rides have seats attached below a metal structure. A chaining pin is used first to mark point C. If the radius fits when the radius distance is swung from this point, a hub with a nail may be set to mark the point. There may be smaller, more compact roller curtain sizes, but they usually don’t go below 205 mm. If the piece of metal is small enough to attempt to balance on a finger. At intersections where buses make designated turns, streets should be designed for a B-40 bus. Where large vehicles are occasional users of a street, there are low traffic volumes, or other characteristics such as high pedestrian volumes necessitate taking greater measures for pedestrian safety and comfort, designers may consider “accommodating” these vehicles. ... Machine swing radius Swing radius … Restricted access: Where there is a desire to keep curb radii small, restrictions on large vehicles making the turn may be considered. The volume V of a cylinder varies jointly with the height h and the radius squared r 2, and V = 150.72 cm 3 when h = 3 cm and r2 = 16 cm 2. Attach a small object of high density to the end of the string (for example, a metal nut or a car key). Conversely the shorter the pendulum the faster the swing rate. Methods: We identified various stages of radius and ulna epiphysis maturity, which were graded as R1-R11 for the radius and U1-U9 for the ulna. Apparently easy to set up and use, but there are regimes where the form is not quite true due to the way they swing. For nuclear sizes, a different way of measuring size becomes convenient. Try to estimate with a meter stick the diameter of the circle it is swinging around. there’s a big force acting on them. With some exceptions, fronting property owners are responsible for the on-going maintenance and upkeep of sidewalk paving as well as all sidewalk elements directly fronting their property, such as trees, landscaping, and streetscape furnishings. There is a large “gap” in physical sizes between the typical atomic dimension of about $10^{-10}$ meter and the nuclear dimensions $10^{-15}$ meter, $10^{-5}$ times smaller. Compound radii effectively shorten crossing distances and make pedestrians visible while accommodating larger vehicles to turn; because they allow more sweeping turns, they do not slow turning vehicles. There is also a quick and dirty concept that you can use to determine appropriate curve radius. A series of strict measures to fight the coronavirus outbreak has been introduced by the government. Or, to request to be added to our weekly email blast, provide your information below. Effective radius: The radius available for the design vehicle to make the vehicle turn, accounting for the presence of parking, bike lanes, medians, or other features. The ride is powered by a motor that makes the swing ride spin around the center axis of the ride. The sharper the radii, the greater the distance between rails will need to be. Emergency vehicles: All streets greater than 150’ in length should accommodate emergency vehicle (WB-40) turns within the full right-of-way of the intersection. Specifically, forces are defined through Newton’s laws of motion 0. Curb radius: the actual radius proscribed by the curb line at an intersection. For a more detailed description of maintenance responsibilities, see Maintenance, A guide to making street improvements in San Francisco. Arc… Livable Streets Division The driver was allowed to stay in the secured area because he was a friend and knew the operator. On streets where other transit providers are present, the curb radius should be designed for their transit vehicles as well. Ensure there is room for the swing radius and bucket operation. Swing Radius found in: OSHA DANGER Keep Clear Swing Radius Cranes Bilingual Sign ODB-4035, OSHA CAUTION Keep Clear Of Swing Radius Sign OCE-28311,.. US-made signs and labels Starting at an angle of less than 10º, allow the pendulum to swing and measure the pendulum’s period for 10 oscillations using a stopwatch. If the arc of the wall is 180 degrees or greater, measure the distance across the wall at it's widest point. A centripetal force (from Latin centrum, "center" and petere, "to seek") is a force that makes a body follow a curved path.Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Starting at an angle of less than 10º, allow the pendulum to swing and measure the pendulum’s period for 10 oscillations using a stopwatch. A centripetal force (from Latin centrum, "center" and petere, "to seek") is a force that makes a body follow a curved path.Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. At-grade paving treatments: To accommodate occasional trucks in very low traffic areas, consider a corner design in which the area between the large and the small curb returns is at street level, and is textured to discourage high-speed turns but allow low-speed use by larger vehicles. 4) Measure the radius distance from point B perpendicularly (Figure 7-2) and mark this as point C, which is the radius point. Outcome measures: Radius and ulna plain radiographs, and various anthropomorphic parameters were assessed. For curb radius changes as part of a traffic calming project, see Traffic Calming Overview Official sidewalk curb lines are established by Board of Supervisors Ordinance #1061, “Regulating the Width of Sidewalks.” Bulb-outs or sidewalk widening on one block or less can be administratively approved by DPW, with input from other agenci… Swing Radius For Fall Protection. If you have a 10mm radius and you offset this 9mm the newly created internal radius will be 1mm: Therefore, if you try to create a shell thickness (or offset) of 11mm it will fail. A swing is suspended by nylon ropes and connected to a limb that is 30 feet in the air. Each project should consider the particular characteristics of the site, and adjust the design as necessary. They may have significant pedestrian volumes and/or concerns about pedestrian safety or wide crossing distances. Painted median: Where there is sufficient lane width on the destination street, a painted median can enable a large vehicle to complete a turn without turning into opposing traffic. Side Hinged Door. Although he later devised more satisfactory clocks (though not like the ones we know), Galileo’s first experiments on motion were done by using his pulse to count off equal intervals of time. Small curb radii are more pedestrian friendly because they decrease crossing distances and slow vehicles at turns. That means that if … The dimensions measure 5.5 feet by 5.9 feet by 6 feet for a compact swing set that fits in small yards. Each seat is suspended by metal chains. Calculate the angular velocity of a 0.300 m radius car tire when the car travels at 15.0 m/s (about 54 km/h). When rigging for fall protection (or fall restraint when working on roofs), keep the tie-back in a straight line (or within 15 degrees) from the anchor point, which must be capable of supporting a load of at least 5,000 pounds. ... Machine swing radius Swing radius … Determining a design vehicle should consider and balance the needs of the various users of a street, from pedestrians and bicyclists to emergency vehicles and large trucks, considering the volume and frequency of these various users. Cut a piece of a string or dental floss so that it is about 1 m long. 1. School of Golf: Establish your swing radius May 21, 2015 School of Golf's Martin Hall gets his tape measure out to help viewers find the radius in their golf swing. Parallel Tracks . Smaller turning radii increase pedestrian safety by shortening crossing distances, increasing pedestrian visibility, and decreasing vehicle turning speed. They function as the central public space of San Francisco neighborhoods. On trolley bus routes, overhead wire locations determine the turning envelope for the bus. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". It should measure out to exactly 1″. Parallel Tracks . Larger WB-60 trucks may also be present on City streets, particularly on designated state highways and in industrial areas. The swing radius is the entire circle from a given point that parts of the equipment may move within. Calculate g. How accurate is this measurement? This should be considered in light of the overall street network. If they don’t move (or move at constant velocity), then there is no force. Industrial streets: Similar to freight routes, industrial streets are used for loading, shipping and deliveries. Before increasing curb radius dimensions to accommodate necessary design vehicles, consider the alternative measures described here: Compound radius: A compound radius changes the curb radius over the length of the turn, such that it has a smaller radius at the crosswalks, and a larger radius in the center where vehicles are turning. * Accommodations include: turning partially or entirely from adjacent lanes, turning from opposing lanes, or turning into opposing lanes. We can’t see a force; we can only deduce its existence by observing its effect. To verify you have printed the tool to the correct scale measure the 1″ Scale Line at the bottom of the page. The effective turning radius, not the curb return radius, should always be used to determine the ability of vehicles to negotiate a turn. There is no solution because the original equation is undefined at x = 5. Try to measure the length 1 of the string that is actually swinging rotating, by measuring the length of the string that is hanging and subtract this length from the total length L to get l. 1 = 3. The only force keeping the object on its circular track is the force of gravity, which means that at the apex, the speed of the object has to be such that the centripetal force equals the object’s weight to keep it going in a circle whose radius is the same as the radius of the loop. Because the linear speed of the tire rim is the same as the speed of the car, we have v = 15.0 m/s . People who use them swear by J & S Fluid-motion style. Freight routes should be designed for WB-50 trucks. Here is an example: All unnecessary personnel shall be prohibited from the work area. Transit routes: Transit routes include transit service routes as well as routes transit vehicles use to start their run and return to the yard. Keep workers outside swing areas by marking them with rope, tape, or other barriers. The only force keeping the object on its circular track is the force of gravity, which means that at the apex, the speed of the object has to be such that the centripetal force equals the object’s weight to keep it going in a circle whose radius is the same as the radius of the loop. If the radius does not fit, point C may be adjusted by For curb radius changes as part of a traffic calming project, see Traffic Calming Overview, Official sidewalk curb lines are established by Board of Supervisors Ordinance #1061, “Regulating the Width of Sidewalks.”. Download the tool by clicking here. The longer the pendulum, whether it is a string, metal rod or wire, the slower the pendulum swings. On other corners along Muni routes, where buses may have to make occasional detours, turns should accommodate a Muni vehicle using the entire roadway, similar to an emergency vehicle. There are only two forces in the vertical direction, the downward gravitational force with a magnitude of M 2 g (where g = 9.8 N/kg) and the upward tension (T). If the project will only change the radius of the curb (without bulb-out or extension), it does not require legislative action. For example, on designated transit or freight routes with frequent large turning vehicles, streets should be “designed for” these vehicles. Advance stop lines: Advance stop lines on the destination street can increase the space available for large vehicles to make a turn by enabling them to swing into opposing lanes on the destination street while opposing traffic is stopped. You can do this through the ‘check’ tool: General conditions apply to all streets where conditions are present as described below. The problem is that the kids cannot build any momentum in order to swing. There’s arguably no more important move in the swing than the release. Private development is a significant contributor of street improvements. That means that if … On the industrial side of fabrication, the mechanical press brake was becoming the standard. The machines performed bottoming; that is, the punch “bottomed” to the bottom of the V die, forcing the sheet metal against the die angle. ANSWER: 30 ft is a very long pendulum. 3.5 is a compromise of best operation and reasonable appearance, 5 can look and operate prototypically. Generally, the City is responsible to maintain roadway paving and other features in the roadway, such as medians. If you have parallel tracks in a curve, the spacing between tracks is important. You can subscribe to The Informer by adding the RSS feed to your feed reader. An opening portion of a door is enlarged, so that the protrusion of the door beyond a radius of swing can be avoided during opening of the door and an operator cab can … These may need to be accommodated in certain instances, though they are not practical in most of San Francisco. The way to measure a distance was well known long before Galileo, but there were no accurate ways of measuring time, particularly short times. There is also a practical section where the student will operate the machine to confirm understanding of key controls and functions. Basically, this is a mass on a string attached to a rubber stopper. No curb radius should be constructed that forces the bus to deviate more than nine feet on center from the middle of the overhead wires. It is simple to measure a radius by using our radius measurement tool. Effective radius: Where a curbside parking and/or bicycle lane is present, the effective radius of the turn is increased. San Francisco, CA 94103 However, no barricades were in place to stop workers from coming within the crane's swing radius. These guidelines provide a general overview of the bulb-out design process. Generally, when we think of swing radius we picture the superstructure of a crane but the movement of any boom, bucket, or cab is also included. Curb radii should be designed to maximize pedestrian space and shorten pedestrian crossing distance to the greatest extent feasible; the smallest possible curb radius should be used while allowing vehicle movements as described below. livablestreets@sfmta.com. There is a piece of tape to help you keep the length of the string outside the glass tube constant. See Figure 4. Your maximum g load at the bottom of the swing is dependent on velocity and radius of the swing. As a new superpower, the U.S. had a massive manufacturing base, built up to feed the war machine, and it had the people needed to do it. Calculate the angular velocity of a 0.300 m radius car tire when the car travels at 15.0 m/s (about 54 km/h). For curb radius changes initiated through a traffic calming project, see Traffic Calming Overview: Maintenance. One of the most deadly work practices is being within the swing radius of heavy equipment. Your comment will be posted after it is approved. The shape and dimensions of curb radii vary based on street type and transportation context. How To Measure Roller Shutter Doors. Is there a way to fix this swing? There is also a practical section where the student will operate the machine to confirm understanding of key controls and functions. There are various other kinds of pendulums. Compound radii may be considered where there are high pedestrian volumes, or a desire to make pedestrians visible, but a need for frequent large turning vehicles such as right-turning buses. Cut a piece of a string or dental floss so that it is about 1 m long. On Muni LRV routes, the curb radius should be constructed such that no part of the sidewalk is closer than two feet from the dynamic envelope of a turning LRV. Divide that distance by 2 to calculate the radius. Print the tool. 311 is San Francisco's 24x7 Customer Service Center, San Francisco Municipal Transportation Agency To determine whether a particular intersection is used by transit vehicles to start their run or return to the yard, check with SFMTA. 75% of construction’s caught-in or struck by injuries involve heavy equipment. The shape of a corner curb radius (the radius defined by two sidewalks on perpendicular streets that come together at a corner) has a significant effect on the overall operation and safety of an intersection. The swing rate, or frequency, of the pendulum is determined by its length. Strategy. Consideration should also be given to private transit operators in areas where large tourist buses and vans are likely to conduct business on a regular, ongoing basis. Side Hinged Door: Benefits The way you use this toy is to spin the rubber stopper around in a circle. Before increasing curb radius dimensions to accommodate necessary design vehicles, consider the alternative measures described here: Compound radius: A compound radius changes the curb radius over the length of the turn, such that it has a smaller radius at the crosswalks, and a larger radius in the center where vehicles are turning. Bulb-outs or sidewalk widening on one block or less can be administratively approved by DPW, with input from other agencies. Design for [a vehicle turn]: to allow for a particular vehicle type to complete a turn fully within its designated travel lane or lanes. And, there are small "scale" prototypes and large "toy" models meaning there is still no definitive minimum radius. 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Proscribed by the curb radius changes may be considered of pedestrians, moderate traffic volumes and speeds and., a guide to making street improvements weekly email blast, provide your information.! Be posted after it is swinging around tracks in a curve, the curb without! Swing rate, or turning into opposing lanes bucket slow down you get wet How... A force ; we can ’ t move ( or move at constant velocity ), it does not legislative... Coronavirus outbreak has been introduced by the government attached to a rubber stopper around a. Swing is suspended by nylon ropes and connected to a wide range of variables ; these can... That the kids can not build any momentum in order to swing function as the public... Swing than the release is approved and large ` toy '' models there! Bus routes, Muni may use a B-30 – check with SFMTA but usually... Yard, check with SFMTA distinguish between “ designing for ” these vehicles remains constant all the way you this... 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Roadways, high traffic volumes and speeds, and various anthropomorphic parameters were assessed ( definitions! Mechanical press brake was becoming the standard than 180 degrees, calculate the angular velocity of a 0.300 radius! Measure Roller Shutter Doors Shutter Doors public space of San Francisco radius proscribed by the curb ( bulb-out... It the same way the Board of Supervisors r = 0.300 m matter type... Of San Francisco, provide your information below called the effective cross section measures radius. Rate, or frequency, of the tire is given to be aware of the circle it is about m. The minimum radius to your feed reader of a string or dental floss so that it is a mass a... As medians tire is given to be accommodated in certain instances, though they are typically narrower streets low! Distance across the wall is less than 180 degrees, calculate the radius of.... Because he was a friend and knew the operator wire locations determine the turning envelope the. The industrial side of fabrication, the spacing between tracks is important radius remains constant the. Radius by using the process described above, and frequent need for loading access force acting on them be... To measure the apparent area, $\sigma$, called the effective radius of the turn is increased,... And in industrial areas … Outcome measures: radius and center location, barricades... Considered in light of the site, and frequent need for large vehicles ( see definitions above ) most!, whether it is about 1 m long the coronavirus outbreak has been introduced by government. Move in the swing than the release swing it the same as the speed the! And dirty concept that you can use to determine appropriate curve radius don... Moderate traffic volumes and speeds, and is free to oscillate about a horizontal axis streets typically wide. And technical analysis km/h ) be considered in light of the turn is increased to be aware of site. Plastic circle templates that will always work no matter the type of.... To our weekly email blast, provide your information below need for large vehicles ( see definitions above ) scale! Point that parts of the turn is increased the longer the pendulum the faster the swing will not..
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# Why does the proof of undecidability of $A_{TM}$ require the universal TM to take input $\langle M,\langle M\rangle\rangle$?
I've read a proof explaining why $A_{\mathrm{TM}}$ is undecidable, and I don't seem to understand why we need to give the opposite of $H$ function $D$ itself as input.
Here's the copy-paste of that proof (link: https://courses.cs.washington.edu/courses/cse322/04au/Lect10.pdf):
$A_{\mathrm{TM}} = \{\langle M,w\rangle\mid M\text{ is a TM that accepts }w\}$ is undecidable! Proof (by contradiction):
1. Assume $A_{\mathrm{TM}}$ is decidable there’s a decider $H$, $L(H)$ = $A_{\mathrm{TM}}$.
2. $H$ on $\langle M,w\rangle = \mathrm{ACC}$ if $M$ accepts $w$ $\mathrm{REJ}$ if $M$ rejects $w$ (halts in $q_{\mathrm{REJ}}$ or loops on $w$).
3. Construct new TM $D$: On input $\langle M\rangle$: Simulate $H$ on $\langle M,\langle M\rangle\rangle$ (here, $w = \langle M\rangle$). If $H$ accepts, then reject input $\langle M\rangle$; If $H$ rejects, then accept input $\langle M\rangle$.
4. What happens when $D$ gets $\langle D\rangle$ as input? $D$ rejects $\langle D\rangle$ if $H$ accepts $\langle D,\langle D\rangle\rangle$ if $D$ accepts $\langle D\rangle$; $D$ accepts $\langle D\rangle$ if $H$ rejects $\langle D,\langle D\rangle\rangle$ if $D$ rejects $\langle D\rangle$. Either way: Contradiction! $D$ cannot exist, so $H$ cannot exist. Therefore, $A_{\mathrm{TM}}$ is not a decidable language.
So, I don't understand why we need to give $H$ the input of $\langle D,\langle D\rangle\rangle$, why not give it the input of $\langle D,w\rangle$? And give $D$ the same input, isn't that the same thing? Or what's the logic behind the second $\langle D\rangle$? For some reason it's very confusing for me. Thanks.
The problem is that your proposed $D$ either isn't a TM or doesn't lead to a contradiction. Suppose we did as you suggested, defining a machine $D$ that required as input a pair $(\langle\,M\,\rangle, w)$ where $\langle\,M\,\rangle$ was the description of a TM $M$ and $w$ was an arbitrary word. Then, using $H$ as a subroutine we'd have $$D(\langle\,M\,\rangle,w)=\begin{cases} \text{ACC} & \text{if M doesn't accept w}\\ \text{REJ} & \text{if M accepts w} \end{cases}$$ Now what happens if you give $D$ the input pair $(\langle\,D\,\rangle, w)$? We'd have the behavior $$D(\langle\,D\,\rangle,w)=\begin{cases} \text{ACC} & \text{if D doesn't accept w}\\ \text{REJ} & \text{if D accepts w} \end{cases}$$ and here's where the problem crops up. The conditions "$D$ doesn't accept $w$" and "$D$ accepts $w$" make no sense, since $D$ was defined on inputs that were required to be pairs: a TM description and a word. It might happen that $w$ in the condition was interpretable as a (TM, word) pair, but the result would have nothing to do with the original input $(\langle\,D\,\rangle,w)$. In any case, we wouldn't get to the contradiction we needed, which is why the original construction needed to define $D$ with a single argument, $\langle\,M\,\rangle$.
Why not give it the input of $\langle D,w\rangle$?
What's $w$? The machine $D$ is given some input, which it interprets as the coding of a Turing machine $M$. It's not given any $w$: only $\langle M\rangle$.
Imagine instead that you're given a function $f$ which is defined as $f(x)=x^x$. I ask you to calculate $f(10)$ and your question is a bit like asking, "Why does it have to be $10^{10}$? Why can't it be $10^y$?"
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Thread: Iverse variation of a function WP
1. Iverse variation of a function WP
The weight of a body in space varies inversely as the square of its distance from the center of the earth. If something weighs 5 lbs on the surface of the earth, how much does it weigh 1000 miles from the surface of the earth? The radius of the earth is 4,000 miles.
Would it be W= k/d^2 ? Could someone work this out for me.
Thanks
2. Originally Posted by na300zx
The weight of a body in space varies inversely as the square of its distance from the center of the earth. If something weighs 5 lbs on the surface of the earth, how much does it weigh 1000 miles from the surface of the earth? The radius of the earth is 4,000 miles.
Would it be W= k/d^2 ? Could someone work this out for me.
Thanks
$5 = \frac{k}{4000^2}$
solve for $k$, then determine $W$ when $d = 5000$
3. Thanks for your reply. So the correct equation would be k= 80,000,000 W=80,000,000/5000^2 which equals 3.2 lbs?
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# First Order Logic Consequence Relation
I have been reading my text book for a bit now and am still puzzled on how to show something like in the following example problems:
For arbitrary formulas $\phi$, $\psi$, and $\chi$ show that:
1. $(\phi \lor \psi) \models \chi$ iff $\phi \models \chi$ and $\psi \models \chi$
2. $\emptyset \models (\phi \to \psi)$ iff $\phi \models \psi$
Thank you for helping me understand in advance.
The notion $\models$ is saying that every model which satisfies the formula on the left side satisfies the formulas on the right side.
1. Assume that $\phi\lor\psi\models \chi$. If $M\models \phi$, then $M\models \phi\vee \psi$ and thus $\phi\lor\psi\models \chi$ implies that $M\models \chi$, thus any arbitrary model which satisfies $\phi$ also satisfy $\chi$ i.e. $\phi\models \chi$. If we now assume $N\models\psi$ then we may in the same way show $N\models \chi$ and thus $\psi\models \chi$.
For the second direction assume that $\phi \models \chi$ and $\psi\models \chi$. If $M\models \phi\vee\psi$, then by the definition of $\vee$, we know that $M\models \phi$ or $M\models \psi$. If $M\models \phi$ then $\phi \models \chi$ implies that $M\models \chi$ on the other hand if $M\models\psi$ then $\psi \models \chi$ implies that $M\models \chi$. Thus we conclude that $M\models \chi$ and thus $\phi\lor\psi\models \chi$ hold.
1. Assume $\emptyset\models (\phi\rightarrow \psi)$. Thus each structure satisfies $\phi\rightarrow\psi$. Assume that $M\models \phi$. Since each structure satisfy $\phi\rightarrow\psi$ we know that $M\models \phi\rightarrow\psi$. Thus we may especially draw the conclusion that $M\models \psi$, hence $phi\models \psi$.
For the second direction assume $\phi\models \psi$ and let $M$ be any structure. If $M\not\models \phi$ then $M\models \phi\rightarrow \psi$ since the first part of the implication is false. On the other hand if $M\models \phi$ then $\phi\models \psi$ implies that $M\models \psi$, thus $M\models \phi\rightarrow \psi$ hold, as both parts of the implication are true. Hence any model satisfies $(\phi\rightarrow \psi)$, which make us able to conclude that $\emptyset\models (\phi\rightarrow \psi)$.
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# Schreiber stable infinity-stacks -- homological algebra
previous: principal infinity-bundles
home: sheaves and stacks
• We had found a concrete cocycle description of hom-sets in the homotopy category of infinity-stacks and had announced that these hom-sets describe very general notions of cohomology.
• As a first application of this, we looked at the special case of Cech cohomology. This provides a concrete component formula for cocycles in nonabelian cohomology.
• More simplifications occur when we restrict attention to special coefficient objects, i.e. to special simplicial sheaves. Namely it turns out that whenever the infinity-groupoids that we deal with have a “maximally nice” abelian group structure on them – in which case they are called stable infinity-groupoids, great simplifications kick in and we obtain strong algebraic models for computing with these beasts.
• At the heart of this convenient reformulation of stably abelian $\infty$-groupoids is the Dold-Kan correspondence. This says that and how precisely certain particularly nice stable $\infty$-groupoids are modeled by the methods of homological algebra.
• Indeed, conversely, it is helpful to understand the origin and purpose of homological algebra as such as being the study of stably abelian $\infty$-groupoids. This clarifies many structures and constructions in this old subject.
• Indeed, the very notion of chain complex is seen, by the classical Dold-Kan correspondence, to be precisely equivalent to nothing but a more efficient encoding of the structure given by a Kan complex with strict abelian group structure.
• Therefore we now
• In the next installment we will then use these tools to describe the stably abelian part of general cohomology theory: abelian sheaf cohomology.
# The Dold-Kan correspondence
Last revised on September 9, 2009 at 01:00:41. See the history of this page for a list of all contributions to it.
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math handbook calculator - Fractional Calculus Computer Algebra System software
+ + + =
# Java Math
## Math class in Java
Math class
### The operators in Java
Java operators +, -, *, /, is as same as JavaScript
• Summary of Operators in Java
#### Math Object Properties (Math.Constants)
Constants in Java Math class should used as Math.constant
Property Description
Math.E Returns Euler's number (approx. 2.718)
Math.PI Returns PI (approx. 3.14)
#### Math Object Methods (Math.Functions)
Function in Java Math class should used as Math.function
Method Description
abs(x) Returns the absolute value of x
acos(x) Returns the arccosine of x, in radians
acosh(x) Returns the hyperbolic arccosine of x
asin(x) Returns the arcsine of x, in radians
asinh(x) Returns the hyperbolic arcsine of x
atan(x) Returns the arctangent of x as a numeric value between -PI/2 and PI/2 radians
atan2(y, x) Returns the arctangent of the quotient of its arguments
atanh(x) Returns the hyperbolic arctangent of x
cbrt(x) Returns the cubic root of x
ceil(x) Returns x, rounded upwards to the nearest integer
cos(x) Returns the cosine of x (x is in radians)
cosh(x) Returns the hyperbolic cosine of x
exp(x) Returns the value of Ex
Math.expm1(x) Returns the value of Ex minus 1
floor(x) Returns x, rounded downwards to the nearest integer
log(x) Returns the natural logarithmof x
log10(x) Returns the base-10 logarithm of x
Math.log1p(x) Returns the natural logarithm of 1 + x
max(x, y, z, ..., n) Returns the number with the highest value
min(x, y, z, ..., n) Returns the number with the lowest value
pow(x, y) Returns the value of x to the power of y
random() Returns a random number between 0 and 1
round(x) Rounds x to the nearest integer
sign(x) Returns the sign of a number (checks whether it is positive, negative or zero)
sin(x) Returns the sine of x (x is in radians)
sinh(x) Returns the hyperbolic sine of x
sqrt(x) Returns the square root of x
tan(x) Returns the tangent of an angle
tanh(x) Returns the hyperbolic tangent of a number
trunc(x) Returns the integer part of a number (x)
## Math class in mathHand
### API
• Java API
• Internal Function 内部函数
### operator in mathHand
• = is equation sign
• ! is factorial, e.g. 2! is factorial(2)
• assignment operator is :=
• Power operators are ^ and **
• operators +, -, *, /, ^, ** also are complex operators.
### Math Constants
Property Description
e Returns Euler's number (approx. 2.718)
pi Returns PI (approx. 3.14)
degree Returns PI/180 (approx. 0.017453292519943295)
eulerGamma Returns euler Gamma (approx. 0.5772156649015329)
## Real math
Java Math class is real math. Math in mathHand included Java Math class, and complex math as following. More is in help.
## Complex math
complex math also work in real domain.
Complex numbers are represented by a dictionary of real and imaginary attributes:
complex( x, y ) — returns a complex number as { re: x, im: y }
complex( x ) — returns a complex number as { re: x, im: 0 }
#### Real and imaginary part
The separate attributes can be accessed through their respective names,
or with convenience functions:
re( x ) — real part of a real or complex number
real( x ) — real part of a real or complex number
im( x ) — imaginary part of a real or complex number
imag( x ) — imaginary part of a real or complex number
• complex operators in mathHand are +, -, *, /, ^, **
### example
e.g. 1+2i is complex(1,2)
more is in example
## calculator
math calculator calculate and plot functions.
## Coding
• Front-end webpage is coding with HTML + JavaScript.
• Back-end server is coded with Java.
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# Transversals of Longest Paths
Let (G) be the minimum cardinality of a set of vertices that intersects all longest paths in a graph G. Let ω(G) be the size of a maximum clique in G, and (G) be the treewidth of G. We prove that (G) ≤{1,ω(G)-2} when G is a connected chordal graph; that (G) =1 when G is a connected bipartite permutation graph or a connected full substar graph; and that (G) ≤(G) for any connected graph G.
## Authors
• 1 publication
• 9 publications
• 2 publications
• 7 publications
• 16 publications
12/27/2019
### Transversals of Longest Cycles in Partial k-Trees and Chordal Graphs
Let lct(G) be the minimum cardinality of a set of vertices that intersec...
06/08/2021
### On the Average (Edge-)Connectivity of Minimally k-(Edge-)Connected Graphs
Let G be a graph of order n and let u,v be vertices of G. Let κ_G(u,v) d...
01/15/2020
### Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach
It can be shown that each permutation group G S_n can be embedded, in a...
11/19/2017
### Approximating geodesics via random points
Given a `cost' functional F on paths γ in a domain D⊂R^d, in the form F(...
08/10/2021
### Tangled Paths: A Random Graph Model from Mallows Permutations
We introduce the random graph 𝒫(n,q) which results from taking the union...
08/17/2021
### A heuristic for listing almost-clique minimal separators of a graph
Bodlaender and Koster (Discrete Mathematics 2006) introduced the notion ...
03/31/2019
### Pebble Exchange Group of Graphs
A graph puzzle Puz(G) of a graph G is defined as follows. A configurati...
##### This week in AI
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## I Introduction
It is a well-known fact that, in a connected graph, any two longest paths have a common vertex. In 1966, Gallai raised the following question: Does every connected graph contain a vertex that belongs to all of its longest paths? The answer to Gallai’s question is already known to be negative. Figure 1 shows the smallest known negative example, on 12 vertices, which was independently found by Walther and Voss [19] and Zamfirescu [20]. However, when we restrict ourselves to some specific classes of graphs, the answer to Gallai’s question turns out to be positive. For example, it is well known that any set of subtrees of a tree satisfies the Helly property. If we consider the set of subtrees consisting of the longest paths of the tree, since they are pairwise intersecting, we conclude that there is a vertex that belongs to all of them.
There are other graph classes which are known to have a positive answer to Gallai’s question. Klavžar and Petkovšek [16] proved that this is the case for split graphs, cacti, and graphs whose blocks are Hamilton-connected, almost Hamilton-connected or cycles. Balister et al. [2] and Joos [15] proved the same for the class of circular arc graphs. De Rezende et al. [7] proved that the answer to Gallai’s question is positive for 2-trees and Chen et al. [6] extended this result for series-parallel graphs, also known as partial 2-trees. Chen [5] proved the same for graphs with matching number smaller than three, while Cerioli and Lima [4, 17] proved it for -sparse graphs, -free graphs, graphs that are the join of two other graphs and starlike graphs, a superclass of split graphs. Finally, Jobson et al. [14] proved it for dually chordal graphs and Golan and Shan [11] for -free graphs.
A more general approach to Gallai’s question is to ask for the size of the smallest transversal of longest paths of a graph, that is, the smallest set of vertices that intersects every longest path. Given a graph , we denote the size of such a set by . In this direction, Rautenbach and Sereni [18] proved that for every connected graph on vertices, that for every connected planar graph on vertices, and that for every connected graph of treewidth at most .
In this work, we provide exact results and upper bounds on the value of when belongs to some specific classes of graphs. More specifically, we prove that:
• for every connected chordal graph , where is the size of a maximum clique of .
• for every connected bipartite permutation graph .
• for every connected graph of treewidth at most .
• for every connected full substar graph .
This paper is organized as follows. In the next section, we state the definitions and basic results that are going to be used throughout the text. In Sections III, IV, V, and VI, we consider, respectively, the class of chordal graphs, the class bipartite permutation graphs, the class of graphs of treewidth at most and the class of full substar graphs. Finally, in Section VII, we state the open problems to be considered in future work.
## Ii Definitions and notation
All graphs considered are simple. Let be a vertex in a graph , we denote by the set of neighbors of in , and by the cardinality of . If the context is clear, we write simply and respectively. Let be a path in a graph . We denote by the length of , that is, the number of edges in . Given a path such that the only vertex it shares with is an extreme of both of them, we denote by the concatenation of and . For a vertex in , let and be the paths such that with . We refer to these two paths as the -tails of . Given a path that contains vertices and , we denote by the -tail of that does not contain and by the -tail of that does not contain . Also, if the context is clear, we denote by the subpath of that has and as its extremes. Thus .
Let be a set of vertices of . Let be a path in that does not contain all vertices of and contains a vertex not in . We say that fences if all the vertices of are in a single component of , otherwise we say that crosses . Given a path that crosses and has both extremes not in , we say that is extreme-separated by when the extremes of are in different components of , and that is extreme-joined by if its extremes are in the same component of .
For an integer , we say that -touches if intersects at exactly vertices. A path is an -corner path if 1-touches . Let be an -corner path. If is fenced by , we say that is an -corner-fenced path. If crosses , we say that is an -corner-crossing path. If two paths and touch at the same set of vertices, we say they are -equivalent, otherwise they are -nonequivalent.
If is fenced by , we denote by the set of vertices of the component of where lies. For a set of vertices not contained in , we denote by the set of vertices of the components of where lies. Two fenced paths and are -component-disjoint if . If is clear from the context, we just say they are component-disjoint.
From now on, we use for the length of a longest path in . Also, remember that is the size of a maximum clique of .
A graph is called a minor of the graph if can be formed from by deleting edges and vertices and by contracting edges.
A tree decomposition [8, p. 337] of a graph is a pair , conformed by a tree and a collection of bags , that satisfies the following three conditions:
1. for every , there exists a bag such that
2. if a vertex is in two different bags , then is also in any bag such that is on the (unique) path from to in .
The width of is the number
max{|Vt|−1:t∈V(T)},
and the treewidth of is the minimum width of any tree decomposition of .
A graph is called chordal if every induced cycle has length three. Next we present some basic properties on tree decompositions for general and chordal graphs. We fix a graph and a tree decomposition of . Proposition 1 is due to Bodlaender [3]. Gross [12] presented a proof for it and refers to tree decompositions such as in Proposition 1 as full tree decompositions. The tree decomposition mentioned in Proposition 2 is also called clique tree and it was introduced by Gavril [10]. Proposition 3 is a direct consequence of Corollary 12.3.12 of the book of Diestel [8].
###### Proposition 1.
If is the treewidth of a graph , then has a tree decomposition of width such that for every , and for every .
###### Proposition 2.
Every chordal graph has a tree decomposition such that the bags of are the maximal cliques of .
###### Proposition 3.
For every chordal graph , .
Given two different nodes , of , we denote by the component of where lies. We say that such component is a branch of at and that the components of are the branches of at [13]. Similarly, for a vertex , we denote by the branch of at such that . We also say that is in . Moreover, we can extend the notation and say that, if is a path fenced by for some , then , where is a vertex of . We also say that is in . Next we show some basic propositions of branches. Propositions 4 to 6 are used to justify that the previous two definitions are coherent. The first two of them appear in the work of Heinz [13].
###### Proposition 4.
Let be a node of and be a vertex of such that . Let and be nodes of . If , then and are in the same branch of at .
###### Proposition 5.
Let and be two vertices of , and let be a node of . If , , and and are not separated by in , then .
###### Proposition 6.
Let be a node of and be a path fenced by . For every two vertices and in , .
###### Proof.
By definition of fenced paths, and lie in the same component of , so they are not separated by in and we can apply Proposition 5. ∎
###### Proposition 7.
If is a path fenced by for some , then there exists a neighbor of in such that .
###### Proof.
Let be a vertex of (that exists by the definition of fenced). As , there exists a bag that contains . Let be the neighbor of in such that is in the (unique) path from to in . Then . ∎
Proposition 8 appears in the book of Diestel [8] as Lemma 12.3.1. Proposition 9 is a corollary of Proposition 8.
###### Proposition 8.
Let . Let and be the components of , with and . Then separates from in .
###### Proposition 9.
Let . Let and be vertices of with . If and , then and are not adjacent.
###### Proof.
Observe that is in a bag of because is in and . As is in a bag of , by Proposition 8, separates from . Hence, and are not adjacent. ∎
###### Proposition 10.
Let . Let be a path fenced by that 1-touches such that , where . Then .
###### Proof.
Suppose by contradiction that there exists a vertex . As is fenced by , there exists a vertex . Moreover, . Let be the subpath of with and as extremes. Since 1-touches , the subpath also 1-touches . This implies that is internally disjoint from . As , , the subpath is disjoint from . But then we contradict Proposition 8, which says that separates , which is in a bag of , from , which is in a bag of . ∎
###### Proposition 11.
Let and . Let be a path fenced by that 1-touches such that . Let be a path fenced by that 1-touches at a vertex in . Then .
###### Proof.
Suppose by contradiction that . As , we must have that . By Proposition 10, , a contradiction. ∎
## Iii Chordal graphs
We start by proving a lemma that is valid for every graph.
###### Lemma 12.
Let be a graph with a clique . Let be the set of all longest paths in that cross , 2-touch , and are extreme-joined by . There are at most two -nonequivalent paths in .
###### Proof.
Suppose by contradiction that there are (at least) three -nonequivalent longest paths , and in . Say , and , where , and are pairwise distinct but not necessarily pairwise disjoint. We may assume that either or . If is component-disjoint from (and from ), and is component-disjoint from (and from ), then and are paths whose lengths sum more than , a contradiction, as at least one of them would have length greater than . So,
CompK(~P)=CompK(Qc) \ or \ CompK(~Q)=CompK(Pa), (1)
Applying the same reasoning to paths and , and to paths and , we conclude that
CompK(~P)=CompK(Re) \ or \ CompK(~R)=CompK(Pa), (2)
and that
CompK(~Q)=CompK(Re) \ or \ CompK(~R)=CompK(Qc). (3)
Also, as , , and cross , from (1), (2), and (3), we have that
(4)
Without loss of generality, we may assume that . (Otherwise, interchange with , and with .) See Figure 2(a). Now, if , then, by (4), , and thus, by (3), . But then , and we contradict (4). Hence, , and, by (2), . Similarly, one can deduce that . Thus, by (3), , and, again, we can deduce that . As , , and are extreme-joined, we conclude that
CompK(Pa)=CompK(Pb)≠CompK(~Q), (5)
CompK(Qc)=CompK(Qd)≠CompK(~R), (6)
CompK(Re)=CompK(Rf)≠CompK(~P). (7)
See Figure 2(b).
Hence, by (5), (6), and (7), we have three paths, , , and , whose lengths sum more than , which leads to a contradiction. ∎
The previous lemma examines how longest paths that are extreme-joined by a clique behave. The following lemma examines the case in which the longest paths are extreme-separated. Observe that, in both cases, we are only considering longest paths that cross the clique and touch it at most twice.
###### Lemma 13.
Let be a graph with a clique and let be the set of all longest paths in that are extreme-separated and touch at most twice. Every two elements of have a common vertex of .
###### Proof.
Let and be two arbitrary paths in . Suppose by contradiction that . As is extreme-separated by , path crosses and therefore either 1-touches or 2-touches . To address these two possibilities at once, let and be such that touches at and , with if 2-touches . Also, if , then let and be different -tails of and let be the path consisting of only the vertex . Let and , and possibly , , and , be defined similarly for .
As both and are extreme-separated by , the tail is component-disjoint from at least one in . Analogously, is component-disjoint from at least one in , is component-disjoint from at least one in and is component-disjoint from at least one in . We may assume without loss of generality that and are component-disjoint and that and are component-disjoint. (Otherwise interchange and .) Observe also that is component-disjoint from at least one in . Without loss of generality, assume that is component-disjoint from . (Otherwise interchange and , and and simultaneously.)
Note that is component-disjoint from at least one in . First suppose that is component-disjoint from . (See a representation of the interactions between the parts of and in Figure 3(a).) Then, one of the paths or is longer than , a contradiction. Now suppose that is not component-disjoint from , that is, , and thus is component-disjoint from . If and are component-disjoint (see Figure 3(b)), then one of or is longer than , a contradiction. If and are not component-disjoint, that is, , then, as and and are component-disjoint, we have that is component-disjoint from (see Figure 3(c)). Thus, one of the paths or is longer than , a contradiction. ∎
The following lemma synthesizes the two previous lemmas. It says that, for every clique, when the transversal is not in it, we would have a longest path that is fenced by the clique. Observe that the lemma is valid only for chordal graphs. Remember that is the size of a maximum clique in . A -clique is a subset of vertices in that are pairwise adjacent.
###### Lemma 14.
Let be a connected chordal graph with a -clique . One of the following is true:
• .
• There exists a longest path that does not touch .
• There exists a vertex of such that there is a longest path that is fenced by and 1-touches at . Moreover, no longest path that 1-touches at crosses .
• There exists an edge of such that there is a longest path that is fenced by and 2-touches at the ends of . Moreover, no longest path that 2-touches at the ends of crosses .
###### Proof.
We will prove that the negation of , , and implies . So, suppose that no clique of size is a longest path transversal in , that every longest path touches at least once, and that, if a vertex is such that some longest path -touches at , then there exists a longest path that -touches at and crosses . If , then either or holds. So we may assume that . If , then by Proposition 3, and holds by Chen et al. [6]. We conclude that and .
Case 1. There is a longest path that 1-touches .
If then, as and do not hold, for every vertex in , there exists a longest path that 1-touches at that vertex. Also, as is false, we may assume that each such path crosses , a contradiction to Lemma 13, because . So . As does not hold, and we are assuming that there is a longest path that 1-touches , there exists a longest path that 1-touches at a vertex and crosses . As does not apply, for every -clique in containing , there exists a longest path that does not contain any vertex in that clique. If any of these longest paths 1-touches at a vertex , then, as does not hold, there is a longest path that crosses at , contradicting Lemma 13. Hence, for every edge in not incident to , there exists a longest path that 2-touches at the ends of that edge. Again, by Lemma 13, as crosses at , none of these paths is extreme-separated by . As , there are at least three such paths. By Lemma 12, one of these edges, call it , is such that no longest path crosses and 2-touches at the ends of . Moreover, we know that there is a longest path that 2-touches at the ends of and, by the previous discussion, that path is fenced by . So holds.
Case 2. Every longest path touches at least twice.
If , then any subset of vertices of of size is a longest path transversal, and would hold. Thus we may also assume that . As , for every edge of , there exists a longest path that 2-touches at the ends of that edge. As , there are at least six nonequivalent longest paths that 2-touch . Suppose by contradiction that does not hold. Hence, we may assume that these six paths cross . By Lemma 12, four of these longest paths are extreme-separated by . As at least two of the corresponding four edges of are disjoint, by Lemma 13, we have a contradiction. ∎
We can finally prove our main result.
###### Theorem 15.
For every connected chordal graph
###### Proof.
Suppose by contradiction that . Then, for every clique in , there exists a longest path fenced by as in , or of Lemma 14. We create a directed graph , that admits antiparalell arcs, as follows. Let be a tree decomposition of . The nodes of are exactly the nodes of . Let be a node in and let be a longest path in fenced by that satisfies one of the conditions , , or of Lemma 14. By Proposition 7, there exists a neighbor of in such that . Hence is an arc in . Thus every node of is the tail of at least one arc in .
Let be the last arc of a maximal directed path in . As is a tree, is also an arc in , which implies that there exist two longest paths and in such that and , where is fenced by and is fenced by , and both satisfy one of the conditions , , or of Lemma 14.
From now on, we assume that is a tree decomposition of as in Proposition 2. Note that the bags containing vertices of are only in , and the bags containing vertices of are only in . As and are disjoint, . Let be a vertex such that
. Suppose for a moment that
contains and let be a neighbor of in . By Proposition 9, vertex cannot be in , so . This implies that is an edge in and, as is a clique, contains all vertices of , contradicting the fact that is fenced. So does not contain vertices in . By a similar reasoning, does not contain vertices in . Thus . As is connected,
P∩Q = P∩Q∩Vt∩Vt′ ≠ ∅. (8)
This implies that and , therefore none of and satisfies condition of Lemma 14.
Suppose for a moment that . Then, as , there exists a longest path that does not contain any vertex of . As is connected, intersects . As does not contain vertices in and does not contain vertices in , we have that . As the bags containing vertices of are only in , has a vertex in a bag of . A similar reasoning, with instead of , shows that also has a vertex in a bag of . This is a contradiction to Proposition 8, as contains no vertex in . Hence . Moreover, as both and are maximal (and different), we conclude that and .
Remember that none of and satisfies condition of Lemma 14. So touches at least once and touches at least once. First suppose that 1-touches at a vertex . That is, satisfies condition of Lemma 14. By (8),
∅ ≠ P∩Q = P∩Vt∩Q∩Vt′ = {v}∩Vt′∩Q = {v}∩Q.
So . That is, and only intersect each other at , which implies that divides both longest paths in half. Let and be the two -tails of , and let and be the two -tails of . Let and . As 1-touches , we may assume without loss of generality that . Suppose that also 1-touches . Then, we may assume without loss of generality that . But then is a longest path that 1-touches at and crosses . As exists, condition of Lemma 14 is not satisfied, a contradiction. Now suppose that 2-touches at . Note that or . If then is a longest path that 1-touches at and crosses , again a contradiction. Hence, . But then is a longest path that 2-touches and crosses . As exists, condition of Lemma 14 is not satisfied, again a contradiction. Therefore touches at least twice.
By a similar reasoning, we may conclude that touches at least twice. So both and must satisfy condition of Lemma 14. Suppose that 2-touches at the ends of edge . First suppose that also 2-touches at the same vertices. Then, , , and . If then is a longest path that 2-touches and crosses . As exists, condition of Lemma 14 is not satisfied, a contradiction. Hence, and . Then is a longest path that 2-touches and crosses , again a contradiction. Hence, we may assume that 2-touches at the ends of an edge with . Then and are paths, yielding the final contradiction.
The previous theorem implies the following results.
###### Corollary 16.
If is a tree or a 2-tree, then
###### Corollary 17.
If is a 3-tree or a connected chordal planar graph, then .
## Iv Bipartite permutation graphs
Let and be two parallel lines in the plane. Consider two sets and of segments that joins a point in with a point in , such that the extremes of every two elements in are pairwise disjoint. Moreover, every two elements in do not intersect each other and every two elements in do not intersect each other.
Let be the function that maps the extreme in of a segment to the other extreme. That is, if is the extreme in of a segment in , then the other extreme is ; and if is the extreme in of a segment in , then the other extreme is . Consider an associated bipartite graph where if and only if the segments and intersect each other. We call the tuple ( a line representation of and a graph is called a bipartite permutation graph if it has a line representation. (See Figure 4.)
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leedcode4
# 114. Flatten Binary Tree to Linked List
## Description
Given a binary tree, flatten it to a linked list in-place.
For example, given the following tree:
1
/
2 5
/ \
3 4 6
The flattened tree should look like:
1
2
3
4
5
6
## Hints
If you notice carefully in the flattened tree, each node’s right child points to the next node of a pre-order traversal.
# 4. Median of Two Sorted Arrays
There are two sorted arrays nums1 and nums2 of size m and n respectively.
Find the median of the two sorted arrays. The overall run time complexity should be$O(log (m+n))$.
You may assume nums1 and nums2 cannot be both empty.
## Method
cut A and B into two parts:
left_part right_part
A[0], A[1], …, A[i-1] A[i], A[i+1], …, A[m-1]
B[0], B[1], …, B[j-1] B[j], B[j+1], …, B[n-1]
there is two conditions we should make sure:
• len(left_part) == len(right_part)
• max(left_part) <= min(right_part)
so, for the aboving condition, the equivalent should be：
$i + j = m - i + n - j \quad and \quad m \leq n\\ B[j -1] \leq A[i] \quad and \quad A[i - 1] \leq B[j]$
the condition $m \leq n$ means $j \geq 0$
then the median of two arrays is:
the mission can be solved by the following steps:
• find min(len(array1), len(array2)) and set it to m
• searching i in [0, m] to find the ‘i’ that:
• B[j - 1] <= A[i] and A[i -1] <= B[j] where j = (m + n + 1)/2 - i
if the time complexity is $O(log (min(m+n)))$,we should use binary search to find the ‘i’
## Pseudo-code
when it comes to eadge values i=0,i=m,j=0,j=n, where** A[i-1],B[j-1],A[i],B[j]** may not exist.
Author: NYY
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## Statistical Science
### R. A. Fisher in the 21st century (Invited paper presented at the 1996 R. A. Fisher Lecture)
#### Abstract
Fisher is the single most important figure in 20th century statistics. This talk examines his influence on modern statistical thinking, trying to predict how Fisherian we can expect the 21st century to be. Fisher's philosophy is characterized as a series of shrewd compromises between the Bayesian and frequentist viewpoints, augmented by some unique characteristics that are particularly useful in applied problems. Several current research topics are examined with an eye toward Fisherian influence, or the lack of it, and what this portends for future statistical developments. Based on the 1996 Fisher lecture, the article closely follows the text of that talk.
#### Article information
Source
Statist. Sci., Volume 13, Number 2 (1998), 95-122.
Dates
First available in Project Euclid: 9 August 2002
https://projecteuclid.org/euclid.ss/1028905930
Digital Object Identifier
doi:10.1214/ss/1028905930
Mathematical Reviews number (MathSciNet)
MR1647499
Zentralblatt MATH identifier
1074.01536
#### Citation
Efron, Bradley. R. A. Fisher in the 21st century (Invited paper presented at the 1996 R. A. Fisher Lecture). Statist. Sci. 13 (1998), no. 2, 95--122. doi:10.1214/ss/1028905930. https://projecteuclid.org/euclid.ss/1028905930
#### References
• SAVAGE, L. J. H. 1976. On rereading R. A. Fisher with discus. Z sion. Ann. Statist. 4 441 500. Savage say s that Fisher's work greatly influenced his seminal book on subjective Bayesianism. Fisher's great ideas are examined lovingly. here, but not uncritically.Z.
• YATES, F. and MATHER K. 1971. Ronald Ay lmer Fisher. In Z. Collected Papers of R. A. Fisher K. Mather, ed. 1 23 52. Z Univ. Adelaide Press. Reprinted from a 1963 Roy al Statistical Society memoir. Gives a nontechnical assessment of. Fisher's ideas, personality and attitudes toward science. Z. Z
• BOX, J. F. 1978. The Life of a Scientist. Wiley, New York. This is both a personal and an intellectual biography by Fisher's daughter, a scientist in her own right and also an historian of science, containing some unforgettable vignettes of precocious mathematical genius mixed with a difficulty in ordinary human interaction. The sparrow quote in Section 4 is. put in context on page 130.
• FISHER, R. A. 1925. Theory of statistical estimation. Proc. Z Cambridge Philos. Soc. 22 200 225. Reprinted in the Mather collection, and also in the 1950 Wiley Fisher collection Contributions to Mathematical Statistics. This is my choice for the most important single paper in statistical theory. A competitor might be Fisher's 1922 Philosophical Society paper, but as Fisher himself points out in the Wiley collection, the 1925 paper is more compact and businesslike. than was possible in 1922, and more sophisticated as well. Z. EFRON B. 1995. The statistical century. Roy al Statistical SociZ. Z ety News 22 5 1 2. This is mostly about the postwar boom in statistical methodology and uses a different statistical. triangle than Figure 8.
• FISHER, R. A. 1934. Two new properties of mathematical likeliZ hood. Proc. Roy. Soc. Ser. A 144 285 307. Concerns two situations when fully efficient estimation is possible in finite samples: one-parameter exponential families, where the MLE is a sufficient statistic, and location scale families, where there are exhaustive ancillary statistics. Reprinted in. the Mather and the Wiley collections. Section 4 Z.
• EFRON, B. 1978. Controversies in the foundations of statistics. Z Amer. Math. Monthly 85 231 246. The Bay es Frequentist Fisherian argument in terms of what kinds of averages should the statistician take. Includes Fisher's famous circle. example of ancillarity. Z.
• EFRON, B. 1982. Maximum likelihood and decision theory. Ann. Z Statist. 10 240 356. Examines five questions concerning maximum likelihood estimation: What kind of theory is it? How is it used in practice? How does it look from a frequentistic decision-theory point of view? What are its principal virtues and defects? What improvements have been sug. gested by decision theory? Z.
• CIFARELLI, D. and REGAZZINI, E. 1996. De Finetti's contribution to probability and statistics. Statist. Sci. 11 253 282. The second half of the quote in my Section 4, their Section 3.2.2, goes on to criticize the Ney man Pearson school. De Finetti is less kind to Fisher in the discussion following Savage's Z. 1976 article.
• DICICCIO, T. and EFRON, B. 1996. Bootstrap confidence interZ. Z vals with discussion. Statist. Sci. 11 189 228. Presents and discusses the cd4 data of Figure 2. The bootstrap confi-. dence limits in Table 1 were obtained by the BC method. a
• REID, N. 1995. The roles of conditioning in inference. Statist. Sci. 10 138 157. This is a survey of the p formula, what I called the magic formula following Ghosh's terminology, and many other topics in conditional inference; see also the Z. discussion following the companion article on pages 173 199, in particular McCullagh's commentary. Gives an extensive bibliography.Z.
• EFRON, B. and HINKLEY, D. 1978. Assessing the accuracy of the maximum likelihood estimator: observed versus expected Z Fisher information. Biometrika 65 457 487. Concerns ancillarity, approximate ancillarity and the assessment of ac. curacy for a MLE.
• EFRON, B. 1993. Bay es and likelihood calculations from conZ fidence intervals. Biometrika 80 3 26. Shows how approximate confidence intervals can be used to get good approximate confidence densities, even in complicated prob. lems with a great many nuisance parameters.
• EFRON, B. and GONG, G. 1983. A leisurely look at the bootstrap, the jackknife, and cross-validation. Amer. Statist. 37 36 48. ZThe chronic hepatitis example is discussed in Section 10 of. this bootstrap jackknife survey article. Z. O'HAGAN, A. 1995. Fractional Bay es factors for model compariZ. son with discussion. J. Roy. Statist. Soc. Ser. B 57 99 138. ZThis paper and the ensuing discussion, occasionally rather heated, give a nice sense of Bayesian model selection in the. Jeffrey s tradition. Z.
• KASS, R. and RAFTERY, A. 1995. Bay es factors. J. Amer. Statist. Z Soc. 90 773 795. This review of Bayesian model selection features five specific applications and an enormous bibliog. raphy.
• EFRON, B. 1996. Empirical Bay es methods for combining likelihoods. J. Amer. Statist. Assoc. 91 538 565.
• KASS, R. and WASSERMAN, L. 1996. The selection of prior distributions by formal rules. J. Amer. Statist. Assoc. 91 Z 1343 1370. Begins Subjectivism has become the dominant philosophical tradition for Bayesian inference. Yet in practice, most Bayesian analyses are performed with so-called. noninformative priors.... ''
• LINDLEY, D. V. 1974. The future of statistics a Bayesian 21st century. In Proceedings of the Conference on Directions for Mathematical Statistics. Univ. College, London. Z.
• SMITH, A. 1995. A conversation with Dennis Lindley. Statist. Z Sci. 10 305 319. This is a nice view of Bayesians and. Bayesianism. The 2020 prediction is attributed to de Finetti.
• , A so that density and likelihood have the form Z. Z. f, A and L ;, A ; the amplified formulas would then appear as
• , for example. The confidence density approach gives sensible results in such situations. My 1993 Biometrika paper argues, a la Kass, that this is a way of using frequentist methods to aid Bayesian calculations. Z. 4 Professor Hinkley notes the continued vitality of Tukey-sty le data analysis. In its purest form this line of work is statistics without probability theory Zsee, e.g., Mosteller and Tukey's 1977 book Data. Analy sis and Regression'' and as such I could not place it any where in the statistical triangle of Section 11. This is my picture's fault of course, not Tukey's. Problem-driven areas like neural networks often begin with a healthy burst of pure data analysis before settling down to an accommodation with statistical theory. Z. 5 I am grateful to Professor Fraser for presenting a more intelligible version of the magic formula. This was the spot in the talk where avoiding technicalities'' almost avoided coherency. Trick'' is a positive word in my vocabulary, reflecting a Caltech education, and I only wish I could think of some more Fisher-level tricks. Fisherian statistics was
• BENNETT, J. H. 1972. Collected Papers of R. A. Fisher. Univ. Adelaide Press. Z.
• BENNETT, J. H., ed. 1990. Statistical Inference and Analy sis. Selected Correspondence of R. A. Fisher. Oxford Univ. Press. Z.
• COX, D. R. 1982. A remark on randomization in clinical trials. Z. Utilitas Math. 21A 245 252. Birthday volume for F. Yates. Z.
• DEMPSTER, A. P. 1971. Model searching and estimation in the logic of inference. In Foundations of Statistical Inference Z. V. P. Godambe and D. A. Sprott, eds. 56 78. Holt, Rinehart and Winston, Toronto. Z.
• EFRON, B. 1971. Forcing a sequential experiment to be balanced. Biometrika 58 403 417. Z. Z
• EFRON, B. 1987. Better bootstrap confidence intervals with. discussion. J. Amer. Statist. Assoc. 82 171 200. Z.
• FISHER, R. A. 1928. Correlation coefficients in meteorology. Nature 121 712. Z.
• FISHER, R. A. 1929. Statistics and biological research. Nature 124 266 267. Z.
• FISHER, R. A. 1935. Design of Experiments. Oliver and Boy d, Edinburgh.Z.
• FISHER, R. A. 1956. Statistical Methods and Scientific InferZ ence. Oliver and Boy d, Edinburgh. Slightly revised versions. appeared in 1958 and 1960. Z.
• FISHER, R. A. 1958. The nature of probability. Centennial Review 2 261 274. Z.
• GELMAN, A., MENG, X.-L. and STERN, H. 1996. Posterior predicZ. tive assessments of model fitness with discussion. Statist. Sinica 6 773 807. Z.
• HALD, A. 1981. T. N. Thiele's contributions to statistics. Internat. Statist. Rev. 49 1 20. Z.
• LANE, D. A. 1980. Fisher, Jeffrey s and the nature of probability. R. A. Fisher: An Appreciation Lecture Notes in Statist. 1. Springer, New York. Z.
• MAHALANOBIS, P. C. 1938. Professor Ronald Ay lmer Fisher. Sankhy a 4 265 272. Z.
• NEy MAN, J. 1977. Frequentist probability and frequentist statistics. Sy nthese 36 97 131. Z.
• RIPLEY, B. D. 1996. Pattern Recognition and Neural Networks. Cambridge Univ. Press. Z.
• SAVAGE, L. J. 1976. On rereading R. A. Fisher. Ann. Statist. 4 441 483. Z.
• YATES, F. 1970. Experimental Design. Selected Papers of Frank Yates, C.B.E., F.R.S. Griffin, London.
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# How is the redshift of starlight precisely measured?
When galaxies move away from us (caused by the expanding spacetime) their light seems to show a redshift. But what is really needed for this?
What time lap is for example needed to see a difference in frequency, and how does this precisely work?
Is this also possible for gravitational waves, or are amplitudes the only solution to derive the redshift from?
• Can you do some research on what "redshift" is en.wikipedia.org/wiki/Redshift There is no "time lap" involved in measuring a redshift, and the way your question is worded makes me wonder if you have done this kind of research. Oct 28 '17 at 20:54
Redshift is measured by comparing the wavelengths of a redshifted pattern of absorption or emission lines with the wavelengths they would have in an object at rest. All lines are shifted by the same factor of $1+z$, where $z$ is the redshift.
• @marjinn The redshift (a cosmological redshift is not a "speed") is found from $(1+z) = 500/122$. Distance has to be measured in some other way to derive the Hubble parameter. If you think you already know the Hubble parameter then a redshift can be used to estimate a distance. Gravitational waves do not come from galaxies, they come from merging binary systems. If you can identify which galaxy the merger takes place in, then you can use the redshift of that galaxy and the distance to the gravitational wave source to independently derive the Hubble parameter. Oct 29 '17 at 9:31
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+0
# math
+21
895
9
how do you get volume
Guest Jan 27, 2012
#3
+92781
+17
That is assuming you are talking about the volume of a rectangular prism!
Melody Apr 30, 2014
#1
+14
WidthxLengthxHeight
Guest Aug 23, 2012
#2
+104
+14
Length*Width*Height
BluePhoenix912 Apr 30, 2014
#3
+92781
+17
That is assuming you are talking about the volume of a rectangular prism!
Melody Apr 30, 2014
#4
+8258
+8
If you are finding the volume of a cube, the answer is 33. If you are finding the volume of a rectangular prism, it is length times width times height.
DragonSlayer554 Jul 17, 2014
#5
+121
+8
LxWxH
e.g. $${\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{2}} = {\mathtt{40}}$$
L W H
Hope i Helped
robtre Oct 27, 2014
#6
+5
Guest Mar 10, 2015
#7
+31
0
You do: lengthXwidthXheight!
FUNKYGAL Apr 14, 2015
#8
0
With too much volume you go deaf haha🔊🆗
Guest Apr 24, 2015
#9
+8
0
$${\mathtt{LengthWidthHeight}} = ABC$$
JimBob May 15, 2015
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Today we're going to talk about Copying, Renaming, Creating Text Files, and displaying what is inside files when using DOS. These tutorials assume that a general knowledge of DOS is already known. The first tutorial and second tutorial are excellent resources when learning DOS.
### Diskcopy
Diskcopy is a command that allows us to copy the contents from one floppy disk to another. This can be useful to create backups of the contents on a floppy disk. This can be useful in easily backing up content on a floppy that contains old finical records and other important documents from the golden days of 3.5" floppy disks.
C:\> diskcopy a:
This would make a copy of the disk that is currently in the a: (default floppy) drive. You should not use diskcopy for anything other than floppy drives, however.
### Rename or Ren
Rename or Ren allows us to rename files and folders. Let us start by navigating to our directory that we want to test our files in. The default directory for vista and XP will suffice for this example.
Vista/7 C:\Users\Dennis>mkdir test
XP C:\Documents and Settings\Dennis>mkdir test
The mkdir command creates a new directory in the current directory that we're in so no need to put the entire path to the new directory.
The syntax for the ren/rename is a little strange and nothing like any command previously mention before in this series. Be sure the path points to the parent directory (one level above the test directory), otherwise an error will occur. The syntax is the following:
ren [drive]
path]
!current name] [new name]
Example: ren C:\users\dennis\test test2
To verify the name change actually occurred, use the dir command to view the contents of the directory.
### Edit
The edit command is a very basic text editor in DOS. This allows us to type basic .txt files without word wrap, without spell checker, without any configuration whatsoever.
There are two flavors of the edit command. There is the default blue background\white text option and then we can add a switch that makes the window and text the same as the default DOS window (black background and a light-gray text).
In order to activate the edit window, all we need to do is simply type "edit".
C:\users\dennis\test> edit
Type a few lines of text and then press file and save as.
Once the file is saved (be sure to include the file extension ".txt"), double check to make sure the file was successfully created by using the dir command.
### Type
The type command allows us to view any files created using the edit command regardless of file extension. Files encoded with Microsoft Office such as .doc or .docx cannot be viewed using the type command. The syntax of the type command is as follows:
[Drive:]!path]\type [filename.file extension] C:\users\dennis\test>type dosstuff.txt
Notice that we are able to view any file extension as long as it was encoded using the edit command in DOS. If we were to view a file encoded in Microsoft Office 2007 it would look like the following:
Notice the arbitrary characters that occur when we try to view documents encoded using MS Word. This is because the encoding format is something that DOS cannot read.
Be Sociable, Share!
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# How do you find the domain of y =sqrt(x + 4)?
Since the domain talks about real numbers only, we need to find when the √ ≥ 0.
0 ≤ x+ 4
-4 ≤ x
Therefore , the domain is {x| x ≥ -4, x in RR}.
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# Verlet list
This method may easily be applied to Monte Carlo simulations. For short-range interactions, a cut-off radius is typically used, beyond which particle interactions are considered "close enough" to zero to be safely ignored. For each particle, a Verlet list is constructed that lists all other particles within the potential cut-off distance, plus some extra distance so that the list may be used for several consecutive Monte Carlo "sweeps" before being updated. If we wish to use the same Verlet list n times before updating, then the cut-off distance for inclusion in the Verlet list should be ${\displaystyle R_{c}+2nd}$, where ${\displaystyle R_{c}}$ is the cut-off distance of the potential, and ${\displaystyle d}$ is the maximum Monte Carlo step of a single particle. Thus, we will spend of order ${\displaystyle N^{2}}$ time to compute the Verlet lists (${\displaystyle N}$ is the total number of particles), but are rewarded with ${\displaystyle n}$ Monte Carlo "sweeps" of order ${\displaystyle Nn^{2}}$ (instead of ${\displaystyle NN}$). Optimizing our choice of ${\displaystyle n}$, it can be shown that the ${\displaystyle O(N^{2})}$ problem of Monte Carlo sweeps has been converted to an ${\displaystyle O(N^{5/3})}$ problem by using Verlet lists.
Using cell lists to identify the nearest neighbors in ${\displaystyle O(N)}$ further reduces the computational cost.
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## Editorial for Добри числа
Remember to use this editorial only when stuck, and not to copy-paste code from it. Please be respectful to the problem author and editorialist.
Submitting an official solution before solving the problem yourself is a bannable offence.
by: donchominkov
static void solveGoodNumbers() {
Scanner in = new Scanner(System.in);
int n = in.nextInt();
int m = in.nextInt();
// Holds the counter of good numbers
// each new good number will increase this counter
int counter = 0;
// Iterate over all the possible good numbers
for (int number = n; number < m + 1; number++) {
// initially, we presume the number is good
boolean isGoodNumber = true;
// We copy the number, so we can split
// its digits
int x = number;
// when x becomes 0, there are no digits left
// This is result of integer division
// i.e. 1 / 0 = 0
// Example with 123
// Get digit 3, the number becomes 12
// Get digit 2, the number becomes 1
// Get digit 1, the number becomes 0
// The loop stops
while (isGoodNumber && x > 0) {
// extract the last digit
int digit = x % 10;
// If the digit is not 0
// and the digit does not divide the number
// the number is not good, so we mark isGoodNumber as false
if (digit != 0 && number % digit != 0) {
isGoodNumber = false;
}
x /= 10;
}
// If isGoodNumber has remained true,
// the number is good
if (isGoodNumber) {
++counter;
}
}
System.out.println(counter);
}
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# QualByDepth
Variant call confidence normalized by depth of sample reads supporting a variant
## Overview
This annotation puts the variant confidence QUAL score into perspective by normalizing for the amount of coverage available. Because each read contributes a little to the QUAL score, variants in regions with deep coverage can have artificially inflated QUAL scores, giving the impression that the call is supported by more evidence than it really is. To compensate for this, we normalize the variant confidence by depth, which gives us a more objective picture of how well supported the call is.
### Statistical notes
The QD is the QUAL score normalized by allele depth (AD) for a variant. For a single sample, the HaplotypeCaller calculates the QD by taking QUAL/AD. For multiple samples, HaplotypeCaller and GenotypeGVCFs calculate the QD by taking QUAL/AD of samples with a non hom-ref genotype call. The reason we leave out the samples with a hom-ref call is to not penalize the QUAL for the other samples with the variant call.
#### Here is a single-sample example:
2 37629 . C G 1063.77 . AC=2;AF=1.00;AN=2;DP=31;FS=0.000;MLEAC=2;MLEAF=1.00;MQ=58.50;QD=34.32;SOR=2.376 GT:AD:DP:GQ:PL:QSS 1/1:0,31:31:93:1092,93,0:0,960
QUAL/AD = 1063.77/31 = 34.32 = QD
#### Here is a multi-sample example:
10 8046 . C T 4107.13 . AC=1;AF=0.167;AN=6;BaseQRankSum=-3.717;DP=1063;FS=1.616;MLEAC=1;MLEAF=0.167;QD=11.54
GT:AD:DP:GQ:PL:QSS 0/0:369,4:373:99:0,1007,12207:10548,98 0/0:331,1:332:99:0,967,11125:9576,27 0/1:192,164:356:99:4138,0,5291:5501,4505
QUAL/AD = 4107.13/356 = 11.54 = QD
### Caveat
This annotation can only be calculated for sites for which at least one sample was genotyped as carrying a variant allele.
### Related annotations
• AS_QualByDepth outputs an allele-specific version of this annotation.
• Coverage gives the filtered depth of coverage for each sample and the unfiltered depth across all samples.
• DepthPerAlleleBySample calculates depth of coverage for each allele per sample (AD).
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# Water pressure and density relationship
### How does pressure affect density of fluid?
The pressure exerted by a static fluid depends only upon the depth of the fluid, the density of the fluid, and the acceleration of gravity. The pressure in a static fluid Discussion. Fluid column height in the relationship inches water= cm water. Density is related to volume by the fact that density is defined as mass divided by volume: $\displaystyle \rho ={\frac {m}{V}}$ So a relationship between. as the depth increases, wouldn't the density of the liquid increase because of the This table gives the compressibility of some liquids, including water. However when applying a physics relation it is important to understand.
The force down is going to be equal to the mass of the liquid times gravity. What is that mass of the liquid? Well, now I'll introduce you to a concept called density, and I think you understand what density is-- it's how much there is of something in a given amount of volume, or how much mass per volume. That's the definition of density.
## What is pressure?
The letter people use for density is rho-- let me do that in a different color down here. The units are kilograms per meter cubed-- that is density. I think you might have an intuition that if I have a cubic meter of lead-- lead is more dense than marshmallows.
Because of that, if I have a cubic meter of lead, it will have a lot more mass, and in a gravitational field, weigh a lot more than a cubic meter of marshmallows. Of course, there's always that trick people say, what weighs more-- a pound of feathers, or a pound of lead?
### How Are Density, Mass & Volume Related? | Sciencing
Those, obviously, weigh the same-- the key is the volume. A cubic meter of lead is going to weigh a lot more than a cubic meter of feathers. Making sure that we now know what the density is, let's go back to what we were doing before.
We said that the downward force is equal to the mass of the liquid times the gravitational force, and so what is the mass of the liquid? We could use this formula right here-- density is equal to mass times volume, so we could also say that mass is equal to density times volume.
I just multiply both sides of this equation times volume. In this situation, force down is equal to-- let's substitute this with this. The mass of the liquid is equal to the density of the liquid times the volume of the liquid-- I could get rid of these l's-- times gravity. What's the volume of the liquid?
The volume of the liquid is going to be the cross-sectional area of the cylinder times the height. So let's call this cross-sectional area A.
Pressure and Density - A level Physics
A for area-- that's the area of the cylinder or the foil that's floating within the water. We could write down that the downward force is equal to the density of the fluid-- I'll stop writing the l or f, or whatever I was doing there-- times the volume of the liquid.
## Water's Unexpected Properties
The volume of the liquid is just the height times the area of the liquid. So that is just times the height times the area and then times gravity.
We've now figured out if we knew the density, this height, the cross-sectional area, and the gravitational constant, we would know the force coming down. That's kind of vaguely interesting, but let's try to figure out what the pressure is, because that's what started this whole discussion. What is the pressure when you go to deep parts of the ocean? This is the force-- what is the pressure on this foil that I have floating?
• What does pressure mean?
• Pressure-density relationship
• Units of Pressure
It's the force divided by the area of pressure on this foil. So I would take the force and divide it by the area, which is the same thing as A, so let's do that.
Let's divide both sides of this equation by area, so the pressure coming down-- so that's P sub d. The downward pressure at that point is going to be equal to-- keep in mind, that's going to be the same thing as the upward pressure, because the upward force is the same. The area of whether you're going upwards or downwards is going to be the same thing.
The downward pressure is going to be equal to the downward force divided by area, which is going to be equal to this expression divided by area. Essentially, we can just get rid of the area here, so it equals PhAg divided by A-- we get rid of the A's in both situations-- so the downward pressure is equal to the density of the fluid, times the depth of the fluid, or the height of the fluid above it, times the gravitational constant Phg.
As I said, the downward pressure is equal to the upward pressure-- how do we know that? Because we knew that the upward force is the same as the downward force. Scales are often employed to determine the mass of a substance since weight is a function of mass and gravity.
Since gravity is very nearly the same over the surface of the Earth, weight becomes a good indicator of mass.
Increasing and decreasing the amount of material measured increases and decreases the mass of the substance. Volume Volume describes how much space a substance occupies and is given in liters SI or gallons English.
The volume of a substance is determined by how much material is present and how closely the particles of the material are packed together. As a result, temperature and pressure can greatly affect the volume of a substance, especially gases.
As with mass, increasing and decreasing the amount of material also increases and decreases the volume of the substance. For example, 10 grams of freshwater has a volume of 10 milliliters.
Unlike mass and volume, increasing the amount of material measured does not increase or decrease density. This makes density a useful property in identifying many substances.
However, since volume deviates with changes in temperature and pressure, density can also change with temperature and pressure. Specific Gravity One derivative measurement of density is specific gravity. Specific gravity compares the density of a substance with the density of a reference material. In the case of gases, the reference material is standard dry air, or air without water.
In the case of liquids and solids, the reference material is fresh water. Specific gravity is calculated by dividing the density of a substance by the reference substance's density. For example, gold has a density of
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# Multiple browser instances through different OpenVPN servers
browserPROXYvpn
How can I config multiple instances of browser to connect through different VPN servers?
I'm working in a Linux server and I need that every browser instance uses different VPN servers.
Edited
The goal is navigate simultaneously the same web page with 5 different Selenium instances, when every instance should have IP from different countries.
So, the solution that I thought was using different proxies when every proxy use a VPN.
Does that make sense?
I'm very newbie to these topics, so if the SSH is a good solution, I would like receive more information about how to get a right connection with the SSH solution. Please help me to understand what does it mean to use every param on the connection and how to config a proxy for that.
• # Per browser window proxy
It is unlikely that there is a solution for this on a per tab basis. However, you could use the profile feature (or incognito) to allow multiple instances of the browser. Each browser window then can manage its own extensions, thus, can manage its own proxy settings. Here is how I did it in Chrome.
• In the upper right corner of the browser window, click the button for the current user. It may show that person's name, email, or an icon shaped like a person.
• Click Switch Person.
• At the bottom of the window, click Add Person. Create a new account.
• Install Proxy SwitchOmega. There are many proxy management extensions in the chrome store, this is the one I tried and it worked for me. Configure it for this window. The settings will not be shared to other personas.
# Per tab proxie (but not really)
I don't think I have a possible workaround without using separate browsing instances. You mentioned that you are doing this to one page. In that case, you could do this by using port redirection on your local host.
## Set two SSH Port Redirects
For example, two set up two SSH port redirects through two different servers, you could use the following commands.
ssh -L 8888:example.com:80 user@1.2.3.4
ssh -L 9999:example.com:80 user@4.5.6.7
## Each browser instance hits a different port.
Have one browser point to 127.0.0.1:8888 and another to 127.0.0.1:9999
## Problems
• This isn't a VPN. You can't guarantee all of the browser traffic will go through the SSH connection. For example, any resource fetching images or scripts on external domains will travel through your host connection. Only resources for the targeted 127.0.0.1:8888 or 127.0.0.1:9999 will be routed through the tunnel. If the webpage has absolute urls, you will have to manually edit the links to reflect the port.
• The server may be picky about the Host: header. In this case, you could use a header editing extension to set the header statically.
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# Law Of Sines And Cosines Pdf
Supply answer and units. 7 Law of Sines and Law of Cosines 509 Using the Law of Sines (SSA Case) Solve the triangle. In mathematics, the sine is a trigonometric function of an angle. Add a comment. Application:. On this page you can read or download law of cosines and sines all things algebra gina wilson in PDF format. In Section 2, we list some basic trigonometric identities and in Section 3 we prove a lemma which is used in Section4. This calculator uses the Law of Sines: $~~ \frac{\sin\alpha}{a} = \frac{\cos\beta}{b} = \frac{cos\gamma}{c}~~$ and the Law of Cosines: $~~ c^2 = a^2 + b^2 - 2ab \cos\gamma ~~$ to solve oblique triangle i. Unit 8 Test Right Triangles And Trigonometry Answer Key. The Law of Cosines does not use ratios, as the Law of Sines does. converting g hz to g 2 hz mechanical acoustics vibration. c 2 = a 2 + b 2 − 2ab cos C. I show examples of triangles that can be solved using the Sine Law or Cosine Law, and how to tell which Law to use. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). From the definitions of sine and cosine, D. Law of Sines and Cosines Worksheet with Key (pdf). 1 Use Law of Cosines to find a. Solve mathematical problems using the Law of Sines. 4014257: sin 0. the Laws of Sines and Cosines so that we can study non-right triangles. The law of cosines. so some distance as i know, those 2 regulations have no particular exceptions. The angle between the coastline and the line between the. ASA Triangle: In ∆ , ∠ 85°, ∠ 25° and 7. Law of Sines Substitute. Refer to the Homework Help note for Lesson 13. The law of sines and the law of cosines, including Heron's formula Polar coordinates, including converting coordinates and equations Graphing polar curves, including circles, roses, cardioids, limacons, and lemniscates. Chapter 8 39 Glencoe Geometry NAME DATE PERIOD HOMEWORK SCORE The Law of Sines and Law of Cosines Find x. Juan and Romella are standing at the seashore 10 miles apart. Law of Sines Problem: A helicopter is hovering between two helicopter pads. -1-Solve each triangle. 1 Analysis of Circuits (2017-10213) Phasors: 10 – 3 / 11 A useful way to think of a cosine wave is as the. Determine whether the Law of Cosines or the Law of Sines is the best choice. 64279 = opposite side ÷ hypotenuse. pdf from MATH 200 at Cambridge College. Jackie skates for 5 meters and Peter skates for 7 meters. Here are the requirements: •! You need to include TWO story problems, one that involves the Law of Sines and the other that involves the Law of. It's important to abide by the law. Both can see the same ship in the water. 5--Law of Sines and Cosines + Review. Another plane leaves 20 minutes later and travels 22° West of North at the rate of 585 mi/h. Sine and cosine rule. 1) 7 yd 2) 33. 15 Solve a triangle M. What do they have in common?. Ô æ Ü á º L Õ æ Ü á » L Ö æ Ü á ¼ Law of Cosines – The law of cosines is stated below. belcher math 4. 93, and c = 18, use the Law of Sines to solve the triangle (if possible) for the value of b. The values in that term are side-by-side so they are multiplied together! When using Law of Cosines, be. Law of Cosines. add example. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). It is important to identify which tool is appropriate. a2 = b2 + c2 — 2bc cos A. Complete all the problems. Write down known. 2 = 62 + 92_2(6)(9)cos93° x1 ~ 36 + 81-108(-0. Assuming that a, b and c are the 3 sides of the triangle opposite to the angles A, B and C as shown in the figure below, the law of sines states that: For the calculation of the three sides (a, b and c) these formulas are applicable: a = b*sin(A)/sin(B) a = c*sin(A)/sin(C) b = a*sin(B)/sin(A) b = c*sin(B)/sin(C) c = a*sin(C)/sin(A) c = b*sin(C)/sin(B). Law of Sines & Cosines Maze! Directions: Determine whether the Law of Sines or Law of Cosines can be applied, then find each missing side or angle. a 13, b 18, c 19 10. \displaystyle \sin {\theta}=\frac {y} { {r}} sinθ = ry. The law of cosines. Algebra and Trigonometry provides a comprehensive and multi-layered exploration of algebraic principles. 1 Oblique Tri angles 11. Law of Sines and Law of Cosines - Big Ideas Learning. Round your answers to the nearest tenth. To derive the law of sines, let us take the area of a triangle whose sides are a, b, c and the angles opposite to the respective sides are A, B, and C. In general, the side […]. AC c is side opposite to C i. The Law of Sines and the Law of Cosines 1. But in that case, the cosine is negative. If a, b, and c are the lengths of the sides and C is the angle opposite side c, then c2 = a2 + b2 − 2ab cos. If more than one solution exists, find both solutions. If you want to read Also, sine and cosine functions are fundamental for describing periodic phenomena - thanks to them, we can describe oscillatory movements (as simple. Law of Sines and Cosines Review Name_____ ID: 1 Date_____ Period____ ©i G2r0L1J8b iKEuFtoaP lSXorfRtJwyadrneW ALULyCs. Law of Sines The Conceptualizer!. sin(A) sin(B) Prove: sin(A) = and sin(B) b sin(A) = h and asln(B) bsin(A = asin B sin(A) sin(B) sin(A) sln(B) *Note that we can prove Proofs Law of Cosines Given: AACB with altitude h drawn. Step #1: Start by graphing the parent function y = sin O if there is no period change (b). Notice this arrangement is SAS. FBI Project. Application problems in the last lesson show the utility of each of these formulas. Law of sines in vector - formula. The angle between the. Most of the problems will give key insights into new ideas and so you are encouraged to do as many as possible by yourself before going for help. A parallelogram has sides of 55 cm and 71 cm. Law of Sines and Law of Cosines Mini-Project In order to practice and refine your skill at using the Law of Sines and the Law of Cosines, you will be creating and working through two story problems. 2 – The Cosine Law for Oblique Triangles (Concept #22/23) Use the Sine Law to write the relationship of the three pairs of sides and opposite angles for each triangle, then solve for the unknown values. Determining the measures of all the sides and angles of a triangle is referred to as solving the triangle. Hedrick's Class 1630 views. zip: 3k: 04-05-10: Laws of Trigonometry 83+ v1. Sine (sin) - The sine of an angle is equal to the. com for an interactive tool to investigate this exploration. 1) 7 yd 2) 33. Download. a 17, b 22, m B 49 12. tan 117 Use the Law of Sines to find each measure. AB Let s look at its proof. Download PDF for free. Find the length of each diagonal to the nearest tenth if the largest angle measures 106q. A = 35; B = 15; c = 5. The Law of Cosines Use the given measurements to solve each triangle. In the triangle BCD, from the definition of sine:or. 84, c = 10 1) Two sides and an angle are given. 8 sin 42o sin 107o Algebra 2 Section 13. Grieser Page 5 Example 6: Given a triangle with m L a 2 2= b 2+ c - 2bc cos A. 93, and c = 18, use the Law. Mathematics Learner's Module,Department of. A guy-wire is to be attached to the top of the tower and to the ground, 165 m downhill from the base of the tower. sin(α + β) = sin α cos β + cos α sin βsin(α − β) = sin α cos β. 50 feet by 3. This preview shows page 1 - 16 out of 16 pages. Law of Cosines -. Law of Sines and Cosines Word Problems. The law of sines and the law of cosines, including Heron's formula Polar coordinates, including converting coordinates and equations Graphing polar curves, including circles, roses, cardioids, limacons, and lemniscates. 2 Apply the laws of sines and cosines to solving problems. Write "none" for transformations that do not exist. Thus, we need to examine the possibility of no solution, one or two solutions. The Law of Cosines does not use ratios, as the Law of Sines does. side gou are solving for is across from the angle gou are using the Cosine of. Part of Trigonometry For Dummies Cheat Sheet. We have the following triangle: We know two angles and one side of this triangle. This calculators will solve three types of 'work' word problems. In Section 8, we used trigonometric functions to solve right triangles. If you want to read Also, sine and cosine functions are fundamental for describing periodic phenomena - thanks to them, we can describe oscillatory movements (as simple. Answer Explanation: To answer this question correctly, you need to both understand trigonometry and radians. Algebra and Trigonometry guides and supports students. Round to the Or, if student uses the Law of Sines, check the work. c 2 = a 2 + b 2 − 2ab cos(C). You can use the Law of Cosines discussed in the last section to solve general triangles, but only under certain conditions. An efficient law-of-cosine-based search for vector quantization. While most of trigonometry is based on the relationships of right triangles, the law of sines can apply to any triangle, whether or not it has a right angle. Some of the worksheets for this concept are Law of cosines work, Law of cosines work, Find each measurement round your answers to the, Law of sines and cosines work answers, Find each measurement round your answers to the, Extra practice, Law of sines and cosines review work date, 25 the law of cosines and its. Area of a triangle. In symbols, Case 2: SSA or The Ambiguous Case. 2682 - 1 - Page 1 Name: _ Weekly work Law of Sines and Cosines 1) The sides of a triangle are 5, 10, and 14. Solving Triangles - using Law of Sine and Law of Cosine. 1) Find mB 14 in 39 in C B A 115° 2) Find mA 13 yd 9 yd A C B 75° 3) Find mB 13 in B29 in C A 115° 4) Find AC C18 A B 32° 30° 5) Find AB 17 B C A 31° 27° 6) Find BC 20 CB A 28° 15°. This handout is a teaching resource. Solve for side a and side c. mLB 1010 L 77' 6. Round to the nearest tenth. You can use the Law of Cosines discussed in the last section to solve general triangles, but only under certain conditions. Law of sines: the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. For any , let the lengths of the sides opposite angles A, B, and C be a, b, and c respectively. Law of Sines Law of Cosines Example In ABC, BC = a = 16, AC = b = 10, and m = 22°. Law of Sines. Law of Cosines - How Does it Work? It is easy to show how this law works. 7 Cosine Law with examples May 11:2D/3D word problems May 14: Review for unit 5 Test. Law of Sines and Cosines Word Problems. A communications tower is located at the top of a steep hill, as shown. State whether the Law of Sines or Law of Cosines is the best choice to solve for x for. Juan and Romella are standing at the seashore 10 miles apart. The proof involves using right triangle trigonometry. 1 - Law of Cosines and Sines Author: James Bonnar. The Law of Sines and Law of Cosines can be used to find angle and side measures in any angle. Law of Sines and Cosines Practice Answers. 97MB LAW OF COSINE WORD PROBLEMS WITH ANSWERS As Pdf, WORD. Here's how the former could be translated into plain English [Euclid, p. Find side a. So the law of cosines is now used with the angle 360◦ − 2α instead of 2α. In mathematics, the sine is a trigonometric function of an angle. Input a combination of three sides or angles, and Law of Sines and Cosines solves for the rest. These laws are used when you don't have a right triangle — they work in any triangle. Complete cell phone triangulation and sketch diagrams using the Law of Cosines and Law of Sines. Practice deciding which formula (law of sines/cosines) to use to solve problems. Search Search. Choose your answers to the questions and click 'Next' to see the next set of questions. cos ∠ A C B \displaystyle AB^2=AC^2+BC^2-2. Example 1: Find the length of b. Law Of Cosines And Sines Worksheet Worksheets For School - Roostanama Law Of Sines Worksheets Photos - Roostanama Law Of Sines Cosines Worksheet Free Worksheets Library | Download L W Of S Es Cos Es W Ksheet Free W Ksheets Libr Ry Downlo D. ) C A B 5 4 C 4 73° 5 12 5 62° 123° 28° B 20° 5. 2c 2 2= a + b - 2ab cos C. A post is supported by two wires (one on each side going in opposite directions) creating an angle of 80° between the wires. Each of the six trig functions is equal to its co-function evaluated at the complementary angle. Problem 1 gives students the opportunity to review the Law of Sines and Cosine. The formulas also give the tangent of a difference formula, for tan (alpha − beta). (Draw the diagram – it will make the problem clearer. Law of Sines and Cosines Word Problems. Application:. These laws are used when you don’t have a right triangle — they work in any triangle. Determine whether the Law of Cosines or the Law of Sines is the best choice. 3: Law of Sines and Cosines] | 3 EXPERIENCE COLLEGE BEFORE COLLEGE 4. 8000 5000 + 6042. The cosine laws are given by: 1) a 2 = b 2 + c 2 - 2 b c cos (A) which gives A = arccos [ (b 2 + c 2 - a 2) / 2 b c ] 2) b 2 = a 2 + c 2 - 2 a c cos (B) which gives B = arccos [ (a 2 + c 2 - b 2) / 2 a c ] 3) c 2 = b 2 + a 2 - 2 b a cos (C) which gives C = arccos [ (b 2 + a 2 - c 2) / 2 a b ]. Law of Sines Law of Cosines Example In ABC, BC = a = 16, AC = b = 10, and m = 22°. 93, and c = 18, use the Law of Sines to solve the triangle (if possible) for the value of b. Step 3: Repeat step 1, but don’t assume that is a right angle. View Law_of_Sines_Cosines_Test_. Write "none" for transformations that do not exist. pdf from MATH 200 at Cambridge College. It should be noted, that by using the formulas given above and corresponding. The angle between the. Note: The Law of Sines is used with triangles that aren't right triangles. Refer to the Homework Help note for Lesson 13. 2 Law of Sines 11. Law of Sines and Cosine. Cell Phone Triangulation. I show examples of triangles that can be solved using the Sine Law or Cosine Law, and how to tell which Law to use. 052) 2 117 + 5. Using the law of sines makes it possible to find unknown angles and sides of a triangle given enough information. Complete the steps to prove the Law of Sines. View Alg2_M10_10. pdf: File Size: 169 kb: File Type: pdf. Product to Sum Ident. Find each triangles missing sides and/or angles. For part (a) the formula gives us the equation and solution shown below. Example 1: Solve the triangle in Figure 2 given θ = 32°, χ = 77°, and d = 12. File Type PDF Law Of Cosine Word Problems With Solutions. In Problem 2, students prove the Law of Sine. Relations between cosine, sine and exponential functions. The Sine Rule, also called the law of sines, is a rule of trigonometry that relates the sides of a triangle and its angle measurements. c o s ∠ A C B. The scripts calculate the k-coverage probability (based on SINR* values) in a single-tier cellular network using a method based on a homogeneous Poisson process model. 8 sin 65o sin 42o B. Label the sides and angles given. Download and install Law of Sines and Cosines Mod (paid) 1. Use the Law of Sines to solve, if possible, the triangle or triangles in the ambiguous case. 1) Find BC 8 BA C 61° 30° 7 2) Find mA 2528 C BA 62°52°. Use the Law of sine and the Law of cosine to find the missing sides and. 02, Use trigonometric and inverse trigonometric functions to model and solve problems. The coastline is a straight line between them. Law Of Sines And Cosines Worksheet With Answers Doc. In this case the tool is useful when you know two sides and their included angle. Remember, the law of cosines is all about included angle (or knowing 3 sides and wanting to find an angle). Tangent, we haven't learned the details of tangent yet but we're going to learn later that tangent is sine over cosine. Check out two popular trigonometric laws: law of sines and law of cosines calculators, helping to solve any kind of a triangle. AB 1310 10. It should be noted, that by using the formulas given above and corresponding. State whether the Law of Sines or Law of Cosines is the best choice to solve for x for. Math 1149: The Laws of Sines and Cosines. Law of Sines and Cosines Word Problems. This unit develops the three main formulas, the area, the Law of Sines, and the Law of Cosines, in relationship to classic right triangle trigonometry. Round to the nearest hundredth. Explore professional development books with Scribd. If a, b and c represent the sides lengths opposite LA. In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. Problem 1 gives students the opportunity to review the Law of Sines and Cosine. Law of Cosines. The Law of Cosines (also called the Cosine Rule ) says The Law of Cosines is useful for finding: the third side of a triangle when we know two sides and the angle between them (like the example above). This is ASA problem which uses the law of sines: The trick to this type is that you have to find the third angle, C: We know that the sum of the triangles. 5 Trig Identities 2018 May 8: Quiz and Practice Identities May 9: 5. Model Problems In the following example you will find the length of a side of a triangle using Law of Cosines. [4] X Research source. 3) Find BC. Law of Cosines -. Law of Cosines Sines Law of. In Problem 2, students prove the Law of Sine. The law of sines is one of two trigonometric equations which is used to find lengths and angles in scalene triangles. Law of Sines and Cosines Word Problems. Displaying top 8 worksheets found for - Law Of Cosines And Sines. c = √a2 + b2 − 2ab cosC. A B a c b C a, b, c, A, B, C, 430 Chapter 6 Additional. 8) Tell what additional information, if any, is needed to find BC using the Law of Sines. c 2 = a 2 + b 2 − 2ab cos(C). -1-Solve each triangle. 5 Solve for the angle Pete needs to navigate to the island. Each section offers a diagram or an example of the formulas. The Law of Sines is also known as the sine rule, sine law, or sine formula. • Use Heron’s formula to find the area of a triangle. For any , let the lengths of the sides opposite angles A, B, and C be a, b, and c respectively. Cofunction Ident. Problem 1 gives students the opportunity to review the Law of Sines and Cosine. From that, you can use the Law of Cosines to find the third side. The proof involves using right triangle trigonometry. Identify whether you would use the Law of Sines or Law of Cosines as the first step when solving the given triangle. 37 980 23 280 18 mLC = 12 150, 17 35 530 6) mLC= 11 10) 24 980 5) mZB = 1270 650, 29 32 1300, 17 1130 14 mZB — 10 13 20. Law of Sines and Cosines Hint Cards: Law of Sines Hint 1A: Find sin(A) and sin(B). Cell Phone Triangulation. 052) 2 117 + 5. Another Expression ab a b c C ca c a b B bc b c a A 2, cos 2, cos 2 cos + −2 = + − = + − = We can find angles from three. And Law of Cosine Functions, Great for use on tests (b/c they also display the formulas) 3/11/03. 73: Law of Cosines 5: Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines Answer Section 1 ANS: 1 13 2=15 +142 −2(15)(14)cosC 169 =421−420cosC −252 =−420cosC 252 420 =cosC 53 ≈C REF: 061110a2 2 ANS: 1 REF: 010233siii 3 ANS: 41. Law Of Sines And Cosines - Displaying top 8 worksheets found for this concept. 9° Use a calculator. θ r x y (x, y) (0, 0) x- axis. A post is supported by two wires (one on each side going in opposite directions) creating an angle of 80o between the wires. Homophone: lore (in non-rhotic accents with the horse-hoarse merger). cos θ = x r. Module 10 Law of Sines & Cosines Assignment You have been learning about the Law of Sines and the Law of Cosines. the, Find each measurement round your answers to the, Solving oblique triangles the law of Law Of Cosine Word Problems Worksheets - Kiddy Math Law of Sines and Cosines Word Problems. a 2 = b 2 + c 2 − 2 b c ⋅ c o s ( ∠ a) a 2 = 20 2 + 13 2 − 2 ⋅ 20 ⋅ 13 ⋅ c o s ( 66) Problem 3. 9° Use a calculator. ) The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the middle-valued angle is opposite the intermediate side. 3) Find BC. Given a triangle ABC, in which $$AB=5$$ and $$\angle ACB=30^{\circ}$$, determine the length of the circumradius (radius of the circumscribed circle). Law Of Sines And Cosine - Displaying top 8 worksheets found for this concept. Practice Solutions. 5² C A B Answer: 13. Label the sides and angles given. Thus, we need to examine the possibility of no solution, one or two solutions. Download Sine Law And Cosine Law Worksheet pdf. Solutions are on the back. A a b q 130 , 50, 30 2. 1) 70° 10 40° A) B = 70°, a = 10, c = 6. Then using the law of cosines with the third angle gives the magnitude of the resultant. It could be either two angles and one side known as AAS or ASA, or it could be two sides The Law of Cosines comes in handy when you need to find the third side of a triangle. Round your answers to the nearest tenth. Use Law of Cosines when given: Two sides and the included angle. From Wikipedia, the free encyclopedia. Author: TiaCoble Created Date:. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). 1) 7 yd 2) 33. 9 42° x 88° 20 13. Complete the steps to prove the Law of Sines. Example 1: Find the length of a. Step 1: Draw a diagonal, dividing quadrilateral into 2 triangles. How far apart are the skaters? 2. Note: The Law of Sines is used with triangles that aren't right triangles. + - Continue ESC. Then solve each triangle. H M GA]lnlU YrFiagNhWt[sp IrheCsFevrOvLeXdq. View Weekly_work_2_law_of_sines_and_cosines. Law of Sines and Cosines Why do I need them? To solve non‐right triangles. each person should work out every problem on. 1 Use Law of Cosines to find a. For part (a) the formula gives us the equation and solution shown below. You could not only going taking into consideration book accrual Just invest tiny mature to entre this on-line proclamation law of sines and cosines kuta answers as with ease as evaluation them wherever you. Is there a problem?. Use law of sines/cosines to solve for required values. Law of Sines/Law of Cosines worksheet Set up and label a diagram. Law of tangents. Trig Ident. ASA Triangle: In ∆ , ∠ 85°, ∠ 25° and 7. Trig Laws Math. ) Law of Sines: Law of Cosines: c 2 = a 2 + b 2 ‐ 2ab cos C b 2 = a 2 + c 2 ‐ 2ac cos B a 2 = b 2 + c 2 ‐ 2bc cos A. That is, given some information about the triangle we can find more. Round angle measures to the nearest degree and side measure to the nearest tenth. Use the Law of Sines to solve, if possible, the triangle or triangles in the ambiguous case. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). The formula you’ll get is the spherical law of cosines,. A guy-wire is to be attached to the top of the tower and to the ground, 165 m downhill from the base of the tower. A new model of stegasvm-shifted lsb in discrete cosine transform domain on image steganography approach. 2 - Class Practice Law of Sines and Law of Cosines (1). This gives a value for the angle θ as the difference in wind direction and bearing. Law of Sines Problem: A helicopter is hovering between two helicopter pads. We have two laws that tell us what to do in a triangle with no right angles. pdf: File Size: 292 kb: File Type: pdf Modern usage is to consider only the 3 preferred trigonometric functions (sine, cosine and tangent) whereas. book of trigonometry (note there are several inexpensive problem books available for trigonometry to help supplement the text of this book if you find the problems lacking in number). Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a ratio of. In symbols, Case 2: SSA or The Ambiguous Case. Solutions are on the back. 93, and c = 18, use the Law. Use the Law of Cosines to Solve Problems You can use the Law of Cosines to solve some problems involving triangles. A communications tower is located at the top of a steep hill, as shown. Use the Law of Sines to find the measure of the angle that is opposite of the shorter of the. In order to find out what the equivalent to $\cos({3π}/{10})$ is, you need to change ${3π}/{10}$ into the form ${π}/{2}-x$. sin A a sin B b sin C c. The Trigonometry of Triangles Definitions and formulas for basic trigonometry, sine, cosine, tangent, cosecant, secant, cotangent, the law of sines and the law of cosines Just scroll down or click on what you want and I'll scroll down for you!. A guy-wire is to be attached to the top of the tower and to the ground, 165 m downhill from the base of the tower. 8 sin 42o sin 107o Algebra 2 Section 13. Depending on which side one chooses to be the base, the area can. The formulas also give the tangent of a difference formula, for tan (alpha − beta). It can be used to derive the third side given two sides and the included angle. How far apart are the skaters? 2. geometry_-_030414_-_8. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. com Section 9. Law of Cosines, you Qet 92. Calculator that shows work to solve oblique triangle using sine and cosine law. Remember, the law of cosines is all about included angle (or knowing 3 sides and wanting to find an angle). side gou are solving for is across from the angle gou are using the Cosine of. Worksheet on the Law of Sines and the Law of Cosines. Law of Cosines Worksheet Free pdf with answer key visual aides from Law Of Sines And Cosines Worksheet, source:mathwarehouse. Another plane leaves 20 minutes later and travels 22° West of North at the rate of 585 mi/h. Find side a. A L x£ääÂÇ Step 1 Find the length of the third side. Oblique Triangles Law of Sines, Cosines, Area Study Guide Name_____ MULTIPLE CHOICE Solve the triangle. The proof involves using right triangle trigonometry. Plug the sine of the angle into the formula. 7 Use the law or sines or cosines to solve the following problems. So Area(UABC ) = sin γ 2 1 ⋅a⋅b⋅. 4-3 Trigonometric Functions on the Unit Circle. A post is supported by two wires (one on each side going in opposite directions) creating an angle of 80o between the wires. 84 B) B = 65°, a = 10, c = 6. Where currently learning Geometry. 1 - Law of Cosines and Sines Author: James Bonnar. 4º or 115º20’ REF: 060140siii. It takes three coats of paint to make a barn quilt. Law of Sines and Area of Triangle Using Trig. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). The Law of Cosines Date_____ Period____ Find each measurement indicated. Find the perimeter of the triangle. Finan 24 The Law of Sines One important use of trigonometry is to solve problems that can be mod-eled by a triangle. To find the remaining angles, it is easiest to now use the Law of Sines. 7 Law of Sines and Law of Cosines 509 Using the Law of Sines (SSA Case) Solve the triangle. a2 = b2 + c2 — 2bc cos A. Math · High school geometry · Non-right triangles & trigonometry (Advanced) · Law of sines Solve triangles using the law of sines CCSS. These two important laws. But cos(2α) = cos(360 − 2α), so the proof is still valid. Each law has its own particular information that is necessary in order to utilize the formula. Sketch the triangle Find the measure of angle B. Grieser Page 5 Example 6: Given a triangle with m L a 2 2= b 2+ c - 2bc cos A. Law of cosines - Wikipedia Sine Law and Cosine Law Find each measurement indicated. This concept teaches students to find missing sides and angles in non-right triangles using the Laws of Sines and Cosines. 1 Use Law of Cosines to find a. Find each measurement indicated. 11] If know SAS or SSS of a triangle, then we use the Law of. Write down known. Input a combination of three sides or angles, and Law of Sines and Cosines solves for the rest. In a right triangle, the trigonometric functions are: sine θ = opposite / hypothenuse cosine θ = adjacent / hypothenuse tangent θ = opposite / adjacent 10. 6 Sine Law May 10: 5. McDougal Littell Geometry Chapter 9: Right Triangles and Trigonometry Chapter Exam Instructions. Find the training resources you need for all your activities. Law of Cosines Substitute. Type 2 worksheets feature exercises in the word format. Forensic Practice. A guy-wire is to be attached to the top of the tower and to the ground, 165 m downhill from the base of the tower. Solve each triangle. a 2 = − 2bc cos b 2 = − 2ac cos. GCSE is the qualification taken by 15 and 16 year olds to mark their graduation from the Key Stage 4 phase of secondary education in England, Northern Ireland and Wales. Use the Law of Cosines to solve a triangle if you know the length of two sides and the measure of the angle between them or the length of all the sides. LAW OF SINES and LAW OF COSINES PRACTICE OBJECTIVE: To build an understanding of the Law of Sines and the Law of Cosines for Algebra 2 Honors, Pre-Calculus, Trigonometry, and College Algebra students by providing concentrated practice. When I began to look for activities or examples that use the law of cosines in this way, it was frequently not in everyday life. en With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. Read PDF Law Of Sines And Cosines Kuta Answers Law of Sines and Law of Cosines Law of Sines: or Law of Cosines: Law of Cosines is the best choice if: Case1: The length of all three sides of a triangle are know and you are trying to find an angle: Case 2: Two sides and an enclosed angle are know and you are trying to find the side opposite the. Use of the law of sines with wind velocity and airspeed gives the angle of offset for the aircraft, β. Play this game to review Trigonometry. Press this key. This is why we present the book compilations in this website. The Law of Sines states that. The law of cosines can be used in conjunction with the law of sines to find all angles and sides of any triangle. Wsidoersk(bb oanodkcH) aonldt the included Right Triangles and Trigonometry: Law of Sines and Law of. 1) 2) LEVEL: PROFICIENT Directions: Solve the following. Known are wind velocity , airspeed, and the desired bearing angle. No; there are an infinite number of possibilities for the side lengths. (Remember that the Law of Cosines is used to solve triangles given other configurations of known sides and angles). 2 Special Triangles used May 3: 5. 4º or 115º20’ REF: 060140siii. Law of Sines / Cosines II. 3 Developing CAST rule May 4: 5. Round angle measures to the nearest degree and side lengths to the nearest tenth. Log InorSign Up. The Pythagorean formula for sines and cosines. Geometry unit 8 right triangles and trigonometry. Write "none" for transformations that do not exist. Law of Cosines Worksheets Answer to the nearest tenth. who's law is it, anyway?. 1-7-19 Practice. The Sine Rule states that the sides of a triangle are proportional to the sines of the opposite angles. AB Let s look at its proof. Law of Sines. Then use the above formula to get the value of sin 0. Пример предложения с "law of sines", памяти переводов. Problem 1 gives students the opportunity to review the Law of Sines and Cosine. 2) Find AB. 1 - Law of Cosines and Sines Author: James Bonnar. 4-7 The Law of Sines and the Law of Cosines PC. geometry_-_030414_-_8. A communications tower is located at the top of a steep hill, as shown. 1) 28 yd 17. Quite the same Wikipedia. The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines. 02, Use trigonometric and inverse trigonometric functions to model and solve problems. Determine the measurement indicated. From this, choose to use the Law of Sines or Law of Cosines. Ws 7 Law Of Cosines 10 5. By the Law of Cosines we have c 2= p a + b2 2abcosC = p 388 2+ 212 2(388)(212)cos42:4 ˇ416:8ft: Applications of the Law of Cosines and Law of Sines The Law of Cosines can be used to derive a formula for nding the area of a triangle given two sides and the included angle. pdf - 0 downloads. 2b 2 = 2a + c - 2ac cos B. Law of Sines and Cosines Word Problems - onlinemath4all 12 Law of Sine and Cosine Word Problems - Free download as PDF File (. Original filename: lawcos. Derivatives of the Sine, Cosine and Tangent Functions. Name_____ C B A 5 70° 1. For an arbitrary triangle ΔABC the law of sines states: a / sinA = b / sinB = c / sinC = 2 ∙ R. This calculators will solve three types of 'work' word problems. ART Local artists are painting barn quilts at a Community Center. The Law of Sines is also known as the sine rule, sine law, or sine formula. 4 x ° 19 24 22. Law of Sines Calculator. 2 – The Cosine Law for Oblique Triangles (Concept #22/23) Use the Sine Law to write the relationship of the three pairs of sides and opposite angles for each triangle, then solve for the unknown values. Represent the frequency with the variable b. Use your solutions to navigate through the maze. Law of Sines Handout: This practice sheet includes the law of sines formula, steps for solving oblique triangles, and 2 practice problems with solutions. 18 Law of Cosines Y. Law of Sines and Cosine. Identities expressing trig functions in terms of their complements. It's important to abide by the law. 4* and 87 Careful!! works. Then using the law of cosines with the third angle gives the magnitude of the resultant. Applying the law of sines: For any angle θ of a triangle, 0 < sin θ ≤ 1. Round to the. Given C = 130°, a = 9. AB Let s look at its proof. In the triangle ADB, applying the Pythagorean Theorem. Part of Trigonometry For Dummies Cheat Sheet. Round answer(s) to two decimal places. 3: Law of Sines and Cosines] | 3 EXPERIENCE COLLEGE BEFORE COLLEGE 4. Define the Law of Cosines. Determine whether the Law of Cosines or the Law of Sines is the best choice. Law of Sines and Area of Triangle Using Trig. pdf: File Size: 169 kb: File Type: pdf. 2c 2 2= a + b - 2ab cos C. If 0 < sin B < 1, then either one or two triangles satisfy the given conditions. [ l QAmlUlW qr_iFgvhMtss` irmelsle\rlvCeSds. Example 1: Find the length of b. The law of cosines is the relationship of a triangle’s sides, angles, and cosine. 6 Using the Law of Sines in an Application (ASA) Two stations are on an east-west line 110 miles apart. This lab requires you to: • Use the Law of Sines to solve oblique triangles. Round decimal answers to the nearest tenth. Trigonometry Law of Sines and Cosines Review Worksheet Name_____ Date_____ Period____ ©p H2[0i1P6_ eKnu_tsaw uSlotfKtIwya\rJec HLoLXCJ. Law of Sines and Cosines. A triangle has side lengths of 3, 8, and 9. A guy-wire is to be attached to the top of the tower and to the ground, 165 m downhill from the base of the tower. Rather, it uses a foimula somewhat similar to the Pythagorean Theorem. Two angles are needed to use the law of Sines. 2² 79° 41° 5 C B A 10 7 3. Law of Sines and Cosines Worksheet (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle). The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). pdf), Text File (. Algebra and Trigonometry guides and supports students. SSS is when we know the lengths of the three sides a, b, and c. Find the perimeter of the triangle. Find each measurement indicated. For triangle ABC Sine Law is Here A, B, C are vertices of ABC a is side opposite to A i. 2² 79° 41° 5 C B A 10 7 3. Download Sine Law And Cosine Law Worksheet pdf. Find the area of an oblique triangle using the sine function. Law of Cosines Notes Just like the Law of Sines, the Law of Cosines can be used to find side lengths and angle measurements for any triangle. Vocabulary: Law of Sines, Law of Cosines. Keyword-suggest-tool. Law of Sines and Cosines Worksheet (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle) Law of Sines; Ambiguous Case of the Law of Sines Trigonometry Worksheets (pdf) with answer keys. But in that case, the cosine is negative. 7 Law of Sines and Law of Cosines 509 Using the Law of Sines (SSA Case) Solve the triangle. Video and other resources. Supply answer and units. The Law of Cosines (also called the Cosine Rule ) says The Law of Cosines is useful for finding: the third side of a triangle when we know two sides and the angle between them (like the example above). Consider a triangle ABC inscribed in a circle with center O and radius r. It works on any triangle, not just right triangles. pdf Title: 000. Tangent of an acute ∠ increases as measure of the ∠ increases. Dodaj komentarz. 4 Law of Sines and osines Oblique Triangle A triangle that is not a right triangle, either acute or obtuse. Round to the nearest tenth 650 12 25 1080 20 17 340 13 23 520 30 z I qq_ 6S I 1 cos Z ŒII — COS 1000 108. Solve for MN. Let B stands for the angle at B. First, let's take the triangle from above and drop a vertical line to the side marked c. pdf: Download File. This technique is also known as triangulation. Law of Slnes Given: AACB with altitude h drawn. 2 Drawing an altitude to prove the Law of Sines Study Tip The Law of Sines can be expressed with the sines in. Law of Sines/Cosines Word Problems. Given: qA 40, qB 70 cm, cm. 7 Law of Sines and Law of Cosines 509 Using the Law of Sines (SSA Case) Solve the triangle. Download. To flip the card, click on the 'Flip' button. 50 feet by 3. In this case, you already know two sides and one angle. If the sides of the triangle are A, B and C and the angles opposite to those sides are a, band c, respectively, then the law of sines states that:. ) C A B 5 4 C 4 73° 5 12 5 62° 123° 28° B 20° 5. Press this key. ) sin θ = sin (180°– θ) (Supplementary angles have the same sine value. Law of Sines and Cosines Level 3 Ambiguous Case. Unit 4 – Applying the Laws of Sines and Cosines Page 1 Objective In this lesson, you will understand and apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles. Cosine Law Problems The Law of Cosines (also called the Cosine Rule) says:. net Sine, Cosine and Tangent. Law of Cosines PDF (Free Printable) which includes the formula, detailed steps to solve oblique triangles, and 2 practice problems. With the law of cosine, you can use the Pythagorean theorem to calculate triangle sides and angles. The law of sines and the law of cosines, including Heron's formula Polar coordinates, including converting coordinates and equations Graphing polar curves, including circles, roses, cardioids, limacons, and lemniscates. Law of Sines Law of Cosines Law of Tangents Mollweid's Formula. It can be used to derive the third side given two sides and the included angle. ) Given: To Find: 1. Law Of Sines Cosines Applications Key ixl kentucky high school math standards. Find third angle A + B + C = 180º 1. Reading Strategy. Here we are going to see some example problems based on law of sines and cosines. Model Problems In the following example you will find the length of a side of a triangle using Law of Sines. Get OrganizedTell which law you would use to solve each given triangle and then draw an example. 7 Law of Sines and Law of Cosines 509 Using the Law of Sines (SSA Case) Solve the triangle. Law of Sines Calculator. Area of a triangle. 6 LAw of Sines and Cosines. For an angle in standard position, we define the trigonometric ratios in terms of x, y and r: sin θ = y r. When to Consider the Ambiguous Case a b A a b A Acute triangle Obtuse triangle C S A T all cosine sine. (If sin θ = 1, then θ = 90° and the triangle is a right triangle. 21 Law of Sines, Law of Cosines Notes Mrs. We get cosh(a)cosh(c) = cosh2(h)cosh(b 1)cosh(b 2). -1-Decide if Law of Sines or Law of Cosines, why? Write the formula. 8 sin 42o sin 107o Algebra 2 Section 13. Then the following equations are true. The law of cosines can be used in conjunction with the law of sines to find all angles and sides of any triangle. BIf sin B = 1, then one triangle satisfies the given conditions and = 90°. c2 = a2 + b2 – 2 a b Cos C. Using notation as in Fig. Applications of Sine/Cosine Laws J. Find the angle measurement for the angle across from the side with length of 8. Topics: Definitions of sine, cosine, tangent Definitions of arcsin, arccos, arctan Law of Sines Law of Cosines Trigonometric Identities Pythagorean Identity (sin2ϴ + cos2ϴ =1) tan(ϴ) = sin(ϴ)/cos(ϴ) Complementary Angle Identity Area with Trigonometry Trigonometry Word Problems. Write "none" for transformations that do not exist. Area of a triangle. The cosine formula applies to all triangles, which includes right triangles. Wsidoersk(bb oanodkcH) aonldt the included Right Triangles and Trigonometry: Law of Sines and Law of. Add a comment. The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines. To solve for all the parts of a triangle, you need to know at least three parts, at least one of which must be a side. The Law of Sines Students will utilize the Law of Sines to find the missing sides and angles of acute b. Find the inverse. Complete all the problems. θ r x y (x, y) (0, 0) x- axis. The Law of Cosines Date_____ Period____ Find each measurement indicated. Law of Sines and Cosines Law of Sines Apply the Law of Sines to solve real-world problems. Graphing Trigonometric Functions (Sine, Cosine, Tangent) with Translations Flip BookThis flip book reviews graphing the sine, cosine, and tangent functions. The Law of Cosines The Law of Cosines c a b ab C b c a ca B a b c bc A 2 cos 2 cos 2 cos 2 2 2 = + − = + − = + − If we know two sides and the included angle, we can find the side which is opposite to this angle. Math 1149: The Laws of Sines and Cosines. 21 Law of Sines, Law of Cosines Notes Mrs. Quizlet flashcards, activities and games help you improve your grades. Introduction. The Law of Cosines. The Law of Sines - MATH law of sines and cosines word problems Problem 1 : A farmer wants to purchase a triangular shaped land with sides 120 feet and Sines Practice Geometry. Show all use of formulas and show what you are typing into your calculator. The law of tangents, although not as commonly known as the law of sines or the law of cosines, is equivalent to the law of sines, and can be used in any case where two sides and the included angle. This PDF 1. 25 feet by 4. Right Triangle. mLB 1010 L 77' 6. Round your answers to the nearest tenth. Step 1: Draw a diagonal, dividing quadrilateral into 2 triangles. Problem 1 gives students the opportunity to review the Law of Sines and Cosine. Determine whether the Law of Cosines or the Law of Sines is the best choice. Law Of Sines And Cosines - Displaying top 8 worksheets found for this concept. A guy-wire is to be attached to the top of the tower and to the ground, 165 m downhill from the base of the tower. Law of Cosines - How Does it Work? It is easy to show how this law works. Here we are going to see some example problems based on law of sines and cosines. PDF | The role of Mathematics in the understanding of the foundations and structure of Science, technological advancement, economic In this study, the researcher improvised a low cost apparatus which is the sine/cosine apparatus as an instructional material to be used on the lesson of the Laws. You can use the law of sines to find both missing side lengths.
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# Rower center of mass
Here’s the result of some other home experiment. In the previous post, I wrote about the angular velocity of the scull, and in an earlier post I discussed the velocity of the rower’s center of mass with respect to the boat, and how this velocity affects boat speed. Only in emergency situations does the handle leave the rower’s hand, so in all practical situations there should be a relation between handle speed and the speed of the rower’s center of mass.
In the picture, I show a measurement on my own body. For various positions in the stroke, I measure the distance of the hands, feet, shoulder, and slide position from the hands position at the catch. All values are in centimeters. On an internet discussion forum for crime writers (!), I found some weight percentages for the weight of different body parts: Head 9%, legs, 33%, arms 11%, trunk 46%. I used these weight percentages to calculate the center of mass (marked CM in the picture).
The center of mass velocity is roughly half the handle velocity. However, at the start of the stroke, both velocities are almost equal. I approximate the relationship between handle and center of mass position by, using relative coordinates, $x_{\rm handle}=1$ corresponding to the finish position:
$x_{\rm CM} = x_{\rm CM, 0} + x_{\rm handle} - \frac{x_{\rm handle}^2}{2 }$
This approximation is a little too fast at the start of the stroke and a little too slow at the end of the stroke. I expect this to be of minor influence. On the other hand, Kleshnev is discussing these differences here, and indeed I try to row without the “shoulder swing” myself, so perhaps I should improve the model at this point.
The position of the handle is given by
$x_{\rm handle} = L_{\rm in} (\sin\phi_0) - \sin\phi)$
and, thus:
$v_{\rm handle} = \dot{\phi} L_{\rm in} \cos\phi$
The crew velocity is given by:
$v_c = v_{\rm handle} \left( 1 - x_{\rm handle}\right)$
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# 10th PHYSICS (ENGLISH MEDIUM NEW)
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Welcome to your 10th PHYSICS (ENGLISH MEDIUM NEW)
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மாவட்டம்
மின்னஞ்சல்
1. The value of variation of accelaration due to gravity (g) is ______ at the centre of the earth.
2. The principle involved in the working of a jet plane is:
3. _____ of a body is defined as the quantity of matter contained in the object.
4. When a bus starts suddenly the passengers in the standing position are pushed backwards, this action is due to:
5. When a body at rest breaks into two pieces of equal masses, then the parts will move:
6. An athlete runs a long path before taking a long jump to increase:
7. The weight of a person is 50 kg. The weight of that person on the surface
8. Which is incorrect statement about the action and reaction referred to Newton’s third law of motion?
9. The unit of gravitational constant is:
10. In vacuum, all freely failing objects have the same:
11. When an object is thrown up, the force of gravity:
12. The refractive index of four substances A, B, C and D are 1.31,1.43,1.33, 2.4 respectively. The speed of light is maximum in:
13. Where should an object be placed so that a real and inverted image of the same size is obtained by a convex lens ______.
14. A convex lens forms a real, diminished point sized image at focus. Then the position of the object is at:
15. The eye defect ‘presbyopia’ can be corrected by:
16. Which of the following lens would you prefer to use while reading small letters found in a dictionary?
17. If VB, VG, VR be the velocity of blue, green and red light respectively in a glass prism, then which of the following statement gives the correct relation?
18. Assertion: If the refractive index of the medium is high (denser medium) the velocity of the light in that medium will be small Reason: Refractive index of the medium is inversely proportional to the velocity of the light.
19. The scattering of sun light by the atoms or molecules of the gases in the Earth’s atmosphere is known as:
20. In an inelastic scattering the energy of the incident beam of light is ……….. that of scattering beam.
21. The value of universal gas constant:
22. Assertion: There is no effects on other end when one end of the rod is only heated. Reason: Heat always flows from a region of lower temperature to higher temperature of the rod.
23. Find the final temperature of a copper rod whose area of cross section changes from 10 m² to 11 m² due to heating. The copper rod is initially kept at 90 K. (Coefficient of superficial expansion is 0.0021 /K).
24. The value of 27° C in the kelvin scale:
25. Kelvin scale has zero reading at temperature _____.
26. Linear expansion is related to _____.
28. If a temperature of 327°C is equivalent to ………. in kelvin scale.
29. A piece of wire of resistance 10 ohm is drawn out so that its length is increased to three times its original length. Calculate the new resistance.
30. The work done in moving a charge of 2 C across two points in a circuit is 2 J. What is the potential difference between the points?
31. The amount of work done to move a unit charge from one point to the other is:
32. The potential difference required to pass a current 0.2 A in a wire of resistance 20 ohm is:
33. The resistivity of a material is 4 × 10-8 Ωm and its conductivity ______.
34. The value of one horse power is:
35. What is the amount of current, when 20 C of charges flows in 4 s through a conductor? [l = $$\frac{q}{v}$$]
36. The heat produced in an electric heater of resistance 2 Ω is connected to an electric source, when a current of 6 A flows for 5 minutes _____.
37. The value of resistivity of nichrome is:
38. if a conductor has a length of 1 m and area of 1 m² then its resistivity is equal to its:
39. LED TV screen was developed by James P. Mitchell in _____.
40. Heat developed across a conductor H =
41. Assertion: Sound waves cannot be propagated through vacuum but light can be transmitted. Reason: Sound waves cannot be polarised but light wave can be polarised.
42. Velocity of sound at a temperature T is given by:
43. Which of the following property of sound waves is used in ultrasonography?
44. Velocity of sound ……… as the density of the solid increases.
45. The direction of compression is reversed during:
46. In which of the following, no change in mass number of the daughter nuclei takes place _____. (i) α decay (ii) β decay (iii) γ decay (iv) neutron decay
47. Gaming radiations are dangerous because of _____.
48. Which of the following statements is/are correct? (i) a particles are photons. (ii) Penetrating power of γ radiation is very low. (iii) Ionization power is maximum for a rays. (iv) Penetrating power of γ radiation is very high.
49. Which of the following is/are correct? (i) Chain reaction takes place in a nuclear reactor and an atomic bomb. (ii) The chain reaction in a nuclear reactor is controlled (iii) The chain reaction in a nuclear reactor is not controlled (iv) No chain reaction takes place in an atom bomb
50. The elements that undergo spontaneous radioactivity:
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This paper implements all the well-known advances in deep learning segmentation methods and make it work together on the CamVid and Gatech datasets.
The proposed network implements the following things:
• Dense blocks
• U-Net architecture
## Dense blocks
The dense block provides a low increase of parameters while keeping a good feature extraction.
## U-Net
The U-Net is well-known to have a good performance on various image segmentation datasets.
## Architecture
The network architecture is a straight U-Net with more convolution stages in each step.
In the following table, $$m$$ represents the number of feature maps output, DB means dense block, TD means transition down and TU means transition up.
### Layers
Here is the internal description of all the blocks defined above.
## Results
Overall the model is as accurate than the other state-of-the-art models while having much fewer parameters (100x less for some of them).
### Issues
Even if the model has fewer parameters than the others models, you should keep in mind that it still has far more layers than a regular U-Net so the memory footprint is huge when you want to train it.
If you run it on theano you should use some THEANO_FLAGS like optimizer_including=fusion to decrease the memory footprint and use a smaller batch_size.
#### Implementations
This original Theano/Lasagne implementations of the paper can be found here FC-DenseNet.
This is my personal Keras/notebook contribution implementation.
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for Journals by Title or ISSN for Articles by Keywords help
Publisher: Cambridge University Press (Total: 365 journals)
Compositio Mathematica [SJR: 2.965] [H-I: 37] [1 followers] Follow Subscription journal ISSN (Print) 0010-437X - ISSN (Online) 1570-5846 Published by Cambridge University Press [365 journals]
• Motivic and real étale stable homotopy theory
• Authors: Tom Bachmann
Pages: 883 - 917
Abstract: Let $S$ be a Noetherian scheme of finite dimension and denote by $\unicode[STIX]{x1D70C}\in [\unicode[STIX]{x1D7D9},\mathbb{G}_{m}]_{\mathbf{SH}(S)}$ the (additive inverse of the) morphism corresponding to $-1\in {\mathcal{O}}^{\times }(S)$ . Here $\mathbf{SH}(S)$ denotes the motivic stable homotopy category. We show that the category obtained by inverting $\unicode[STIX]{x1D70C}$ in $\mathbf{SH}(S)$ is canonically equivalent to the (simplicial) local stable homotopy category of the site $S_{\text{r}\acute{\text{e}}\text{t}}$ , by which we mean the small real étale site of $S$ , comprised of étale schemes over $S$ with the real étale topology. One immediate application is that $\mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]$ is equivalent to the classical stable homotopy category. In particular ...
PubDate: 2018-05-01T00:00:00.000Z
DOI: 10.1112/S0010437X17007710
Issue No: Vol. 154, No. 5 (2018)
• ++++++++++++++++++ +++++++++++++++++++++ +++++++++++++++++++++ ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$(1,1)$ ++++++++++++ ++++++++ ++++ ++++++++++++++++++ ++++++++++++++++L-space+knots&rft.title=Compositio+Mathematica&rft.issn=0010-437X&rft.date=2018&rft.volume=154&rft.spage=918&rft.epage=933&rft.aulast=Greene&rft.aufirst=Joshua&rft.au=Joshua+Evan+Greene&rft.au=Sam+Lewallen,+Faramarz+Vafaee&rft_id=info:doi/10.1112/S0010437X17007989">$(1,1)$ L-space knots
• Authors: Joshua Evan Greene; Sam Lewallen, Faramarz Vafaee
Pages: 918 - 933
Abstract: We characterize the $(1,1)$ knots in the $3$ -sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, $1$ -bridge braids in these manifolds admit non-trivial L-space surgeries. We also recover a characterization of the Berge manifold among $1$ -bridge braid exteriors.
PubDate: 2018-05-01T00:00:00.000Z
DOI: 10.1112/S0010437X17007989
Issue No: Vol. 154, No. 5 (2018)
• Abelian varieties isogenous to a power of an elliptic curve
• Authors: Bruce W. Jordan; Allan G. Keeton, Bjorn Poonen, Eric M. Rains, Nicholas Shepherd-Barron, John T. Tate
Pages: 934 - 959
Abstract: Let $E$ be an elliptic curve over a field $k$ . Let $R:=\operatorname{End}E$ . There is a functor $\mathscr{H}\!\mathit{om}_{R}(-,E)$ from the category of finitely presented torsion-free left $R$ -modules to the category of abelian varieties isogenous to a power of $E$ , and a functor $\operatorname{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories. We also prove a partial generalization in which $E$ is replaced by a suitable higher-dimensional abelian variety over $\mathbb{F}_{p}$ .
PubDate: 2018-05-01T00:00:00.000Z
DOI: 10.1112/S0010437X17007990
Issue No: Vol. 154, No. 5 (2018)
• Pair correlation for quadratic polynomials mod 1
• Authors: Jens Marklof; Nadav Yesha
Pages: 960 - 983
Abstract: It is an open question whether the fractional parts of non-linear polynomials at integers have the same fine-scale statistics as a Poisson point process. Most results towards an affirmative answer have so far been restricted to almost sure convergence in the space of polynomials of a given degree. We will here provide explicit Diophantine conditions on the coefficients of polynomials of degree two, under which the convergence of an averaged pair correlation density can be established. The limit is consistent with the Poisson distribution. Since quadratic polynomials at integers represent the energy levels of a class of integrable quantum systems, our findings provide further evidence for the Berry–Tabor conjecture in the theory of quantum chaos.
PubDate: 2018-05-01T00:00:00.000Z
DOI: 10.1112/S0010437X17008028
Issue No: Vol. 154, No. 5 (2018)
• The Hodge diamond of O’Grady’s six-dimensional example
• Authors: Giovanni Mongardi; Antonio Rapagnetta, Giulia Saccà
Pages: 984 - 1013
Abstract: We realize O’Grady’s six-dimensional example of an irreducible holomorphic symplectic (IHS) manifold as a quotient of an IHS manifold of K3 $^{[3]}$ type by a birational involution, thereby computing its Hodge numbers.
PubDate: 2018-05-01T00:00:00.000Z
DOI: 10.1112/S0010437X1700803X
Issue No: Vol. 154, No. 5 (2018)
• A mass transference principle for systems of linear forms and its
applications
• Authors: Demi Allen; Victor Beresnevich
Pages: 1014 - 1047
Abstract: In this paper we establish a general form of the mass transference principle for systems of linear forms conjectured in 2009. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine–Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine–Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira.
PubDate: 2018-05-01T00:00:00.000Z
DOI: 10.1112/S0010437X18007121
Issue No: Vol. 154, No. 5 (2018)
• Sums of three squares and Noether–Lefschetz loci
• Authors: Olivier Benoist
Pages: 1048 - 1065
Abstract: We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in $\mathbb{P}^{3}$ whose function field has level $2$ is dense in the set of those that have no real points.
PubDate: 2018-05-01T00:00:00.000Z
DOI: 10.1112/S0010437X18007017
Issue No: Vol. 154, No. 5 (2018)
• Diffeomorphism groups of tame Cantor sets and Thompson-like groups
• Authors: Louis Funar; Yurii Neretin
Pages: 1066 - 1110
Abstract: The group of ${\mathcal{C}}^{1}$ -diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$ -smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.
PubDate: 2018-05-01T00:00:00.000Z
DOI: 10.1112/S0010437X18007066
Issue No: Vol. 154, No. 5 (2018)
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## Wednesday, November 15, 2017
### Influence of DNA Lesions on Polymerase-Mediated DNA Replication at Single-Molecule Resolution
Hailey L. Gahlon, Louis J. Romano, and David Rueda
Faithful replication of DNA is a critical aspect in maintaining genome integrity. DNA polymerases are responsible for replicating DNA, and high-fidelity polymerases do this rapidly and at low error rates. Upon exposure to exogenous or endogenous substances, DNA can become damaged and this can alter the speed and fidelity of a DNA polymerase. In this instance, DNA polymerases are confronted with an obstacle that can result in genomic instability during replication, for example, by nucleotide misinsertion or replication fork collapse. It is important to know how DNA polymerases respond to damaged DNA substrates to understand the mechanism of mutagenesis and chemical carcinogenesis. Single-molecule techniques have helped to improve our current understanding of DNA polymerase-mediated DNA replication, as they enable the dissection of mechanistic details that can otherwise be lost in ensemble-averaged experiments. These techniques have also been used to gain a deeper understanding of how single DNA polymerases behave at the site of the damage in a DNA substrate. In this review, we evaluate single-molecule studies that have examined the interaction between DNA polymerases and damaged sites on a DNA template.
DOI
### A review on optical actuators for microfluidic systems
Tie Yang, Yue Chen and Paolo Minzioni
During the last few decades microfluidic systems have become more and more popular and their relevance in different fields is continually growing. In fact, the use of microchannels allows a significant reduction of the required sample-volume and opens the way to a completely new set of possible investigations, including the study of the properties of cells, the development of new cells' separation techniques and the analysis of single-cell proteins. One of the main differences between microscopic and macroscopic systems is obviously dictated by the need for suitable actuation mechanisms, which should allow precise control of microscopic fluid volumes and of micro-samples inside the fluid. Even if both syringe-pump and pneumatic-pump technologies significantly evolved and they currently enable sub-μL samples control, completely new approaches were recently developed for the manipulation of samples inside the microchannel. This review is dedicated to describing different kinds of optical actuators that can be applied in microfluidic systems for sample manipulation as well as for pumping. The basic principles underlying the optical actuation mechanisms will be described first, and then several experimental demonstrations will be reviewed and compared.
DOI
### Stochastic Optical Trapping and Manipulation of Micro Object with Neural-Network Adaptation
Xiang Li ; Chien Chern Cheah
Optical tweezers are capable of manipulating micro/nano objects without any physical contact, and therefore widely used in biomedical engineering and biological science. While much progress has been achieved in automated optical manipulation of micro objects, Brownian motion is commonly ignored in the stability analysis in order to simplify the control problem. However, random Brownian perturbations exist in micromanipulation problem and therefore may result in failure of optical trapping due to the escape of micro object from the trap. In addition, it is usually assumed in the development of controller that the model of trapping stiffness is known, but the model is difficult to obtain because of its spatially varying feature around the centre of laser beam and variations with laser power and dimensions of objects. In this paper, a neural-network control method is proposed for optical trapping and manipulation of micro object, in the presence of stochastic perturbations and unknown trapping stiffness. The unknown trapping stiffness and dynamic parameters of micro objects, which vary with different laser power settings and sizes of the objects, are approximated by using adaptive neural networks. The stability analysis is carried out from stochastic perspectives, by considering the effect of Brownian motion in the dynamic model. Both experimental results and simulation results are presented.
DOI
### Radiation forces of beams generated by Gaussian mirror resonator on a Rayleigh dielectric sphere
Bin Tang, Kai Chen, Lirong Bian, Xin Zhou, Li Huang & Yi Jin
Optical trapping and manipulating of micron-sized particles have attracted enormous interests due to the potential applications in biotechnology and nanoscience. In this work, we investigate numerically and theoretically the radiation forces acting on a Rayleigh dielectric particle produced by beams generated by Gaussian mirror resonator (GMR) in the Rayleigh scattering regime. The results show that the focused beams generated by GMR can be used to trap and manipulate the particles with both high and low index of refractive near the focus point. The influences of optical parameters of the beams generated by GMR on the radiation forces are analyzed in detail. Furthermore, the conditions for trapping stability are also discussed in this paper.
DOI
### Ferdinando Borghese (26 May 1940–19 January 2017)
M.A. Iatì, R. Saija, O.M. Maragò, P. Denti
Here we summarize the life and scientific legacy of Ferdinando Borghese (1940–2017). He has been a pioneer in the theory and modeling of light scattering by nonspherical particles and clusters in the framework of the transition matrix approach. His work has found applications in many research fields ranging from interstellar dust to aerosol science, plasmonics, and optical trapping.
DOI
### Dimerization regulates both deaminase-dependent and deaminase-independent HIV-1 restriction by APOBEC3G
Michael Morse, Ran Huo, Yuqing Feng, Ioulia Rouzina, Linda Chelico & Mark C. Williams
APOBEC3G (A3G) is a human enzyme that inhibits human immunodeficiency virus type 1 (HIV-1) infectivity, in the absence of the viral infectivity factor Vif, through deoxycytidine deamination and a deamination-independent mechanism. A3G converts from a fast to a slow binding state through oligomerization, which suggests that large A3G oligomers could block HIV-1 reverse transcriptase-mediated DNA synthesis, thereby inhibiting HIV-1 replication. However, it is unclear how the small number of A3G molecules found in the virus could form large oligomers. Here we measure the single-stranded DNA binding and oligomerization kinetics of wild-type and oligomerization-deficient A3G, and find that A3G first transiently binds DNA as a monomer. Subsequently, A3G forms N-terminal domain-mediated dimers, whose dissociation from DNA is reduced and their deaminase activity inhibited. Overall, our results suggest that the A3G molecules packaged in the virion first deaminate viral DNA as monomers before dimerizing to form multiple enzymatically deficient roadblocks that may inhibit reverse transcription.
## Wednesday, November 8, 2017
### Emulsified and Liquid–Liquid Phase-Separated States of α-Pinene Secondary Organic Aerosol Determined Using Aerosol Optical Tweezers
Kyle Gorkowski, Neil M. Donahue, and Ryan C. Sullivan
We demonstrate the first capture and analysis of secondary organic aerosol (SOA) on a droplet suspended in an aerosol optical tweezers (AOT). We examine three initial chemical systems of aqueous NaCl, aqueous glycerol, and squalane at ∼75% relative humidity. For each system we added α-pinene SOA—generated directly in the AOT chamber—to the trapped droplet. The resulting morphology was always observed to be a core of the original droplet phase surrounded by a shell of the added SOA. We also observed a stable emulsion of SOA particles when added to an aqueous NaCl core phase, in addition to the shell of SOA. The persistence of the emulsified SOA particles suspended in the aqueous core suggests that this metastable state may persist for a significant fraction of the aerosol lifecycle for mixed SOA/aqueous particle systems. We conclude that the α-pinene SOA shell creates no major diffusion limitations for water, glycerol, and squalane core phases under humid conditions. These experimental results support the current prompt-partitioning framework used to describe organic aerosol in most atmospheric chemical transport models and highlight the prominence of core–shell morphologies for SOA on a range of core chemical phases.
DOI
### Elliptical orbits of microspheres in an evanescent field
Lulu Liu, Simon Kheifets, Vincent Ginis, Andrea Di Donato, and Federico Capasso
We examine the motion of periodically driven and optically tweezed microspheres in fluid and find a rich variety of dynamic regimes. We demonstrate, in experiment and in theory, that mean particle motion in 2D is rarely parallel to the direction of the applied force and can even exhibit elliptical orbits with nonzero orbital angular momentum. The behavior is unique in that it depends neither on the nature of the microparticles nor that of the excitation; rather, angular momentum is introduced by the particle’s interaction with the anisotropic fluid and optical trap environment. Overall, we find this motion to be highly tunable and predictable.
DOI
### Mechanically switching single-molecule fluorescence of GFP by unfolding and refolding
Green fluorescent protein (GFP) variants are widely used as genetically encoded fluorescent fusion tags, and there is an increasing interest in engineering their structure to develop in vivo optical sensors, such as for optogenetics and force transduction. Ensemble experiments have shown that the fluorescence of GFP is quenched upon denaturation. Here we study the dependence of fluorescence on protein structure by driving single molecules of GFP into different conformational states with optical tweezers and simultaneously probing the chromophore with fluorescence. Our results show that fluorescence is lost during the earliest events in unfolding, 3.5 ms before secondary structure is disrupted. No fluorescence is observed from the unfolding intermediates or the ensemble of compact and extended states populated during refolding. We further demonstrate that GFP can be mechanically switched between emissive and dark states. These data definitively establish that complete structural integrity is necessary to observe single-molecule fluorescence of GFP.
DOI
### Optical tweezing and binding at high irradiation powers on black-Si
Tatsuya Shoji, Ayaka Mototsuji, Armandas Balčytis, Denver Linklater, Saulius Juodkazis & Yasuyuki Tsuboi
Nowadays, optical tweezers have undergone explosive developments in accordance with a great progress of lasers. In the last decade, a breakthrough brought optical tweezers into the nano-world, overcoming the diffraction limit. This is called plasmonic optical tweezers (POT). POT are powerful tools used to manipulate nanomaterials. However, POT has several practical issues that need to be overcome. First, it is rather difficult to fabricate plasmonic nanogap structures regularly and rapidly at low cost. Second, in many cases, POT suffers from thermal effects (Marangoni convection and thermophoresis). Here, we propose an alternative approach using a nano-structured material that can enhance the optical force and be applied to optical tweezers. This material is metal-free black silicon (MFBS), the plasma etched nano-textured Si. We demonstrate that MFBS-based optical tweezers can efficiently manipulate small particles by trapping and binding. The advantages of MFBS-based optical tweezers are: (1) simple fabrication with high uniformity over wafer-sized areas, (2) free from thermal effects detrimental for trapping, (3) switchable trapping between one and two - dimensions, (4) tight trapping because of no detrimental thermal forces. This is the NON-PLASMONIC optical tweezers.
DOI
### Improved generation of periodic optical trap arrays using noniterative algorithm
Anita Dalal; Aniket Chowdhury; Raktim Dasgupta; Shovan Kumar Majumder
In a holographic optical tweezers setup, although the use of noniterative algorithms can result in the fast generation of multiple traps array, the performance of these algorithms is often inferior compared to iterative types of algorithms. Particularly in the case of symmetric trap arrays, the performance of noniterative algorithms is very poor. Suitability of the use of a noniterative superposition algorithm for generating symmetric trap arrays has been investigated after introducing small position disorders for the individual traps. It could be seen that the introduction of small disorders in the positions of the individual traps can significantly improve the quality of the generated trap array pattern over the case when an ideal symmetric pattern is targeted.
DOI
## Tuesday, November 7, 2017
### Manufacturing with light - micro-assembly of opto-electronic microstructures
Shuailong Zhang, Yongpeng Liu, Yang Qian, Weizhen Li, Joan Juvert, Pengfei Tian, Jean-Claude Navarro, Alasdair W Clark, Erdan Gu, Martin D. Dawson, Jonathan M. Cooper, and Steven L. Neale
Optical micromanipulation allows the movement and patterning of discrete micro-particles within a liquid environment. However, for manufacturing applications it is desirable to remove the liquid, leaving the patterned particles in place. In this work, we have demonstrated the use of optoelectronic tweezers (OET) to manipulate and accurately assemble Sn62Pb36Ag2 solder microspheres into tailored patterns. A technique based on freeze-drying technology was then developed that allows the assembled patterns to be well preserved and fixed in place after the liquid medium in the OET device is removed. After removing the liquid from the OET device and subsequently heating the assembled pattern and melting the solder microspheres, electrical connections between the microspheres were formed, creating a permanent conductive bridge between two isolated metal electrodes. Although this method is demonstrated with 40 µm diameter solder beads arranged with OET, it could be applied to a great range of discrete components from nanowires to optoelectronic devices, thus overcoming one of the basic hurdles in using optical micromanipulation techniques in a manufacturing micro-assembly setting.
DOI
### KiloHertz Bandwidth, Dual-Stage Haptic Device Lets You Touch Brownian Motion
Tianming Lu; Cécile Pacoret; David Hériban; Abdenbi Mohand-Ousaid; Stéphane Régnier; Vincent Hayward
This paper describes a haptic interface that has a uniform response over the entire human tactile frequency range. Structural mechanics makes it very difficult to implement articulated mechanical systems that can transmit high frequency signals. Here, we separated the frequency range into two frequency bands. The lower band is within the first structural mode of the corresponding haptic device while the higher one can be transmitted accurately by a fast actuator operating from conservation of momentum, that is, without reaction forces to the ground. To couple the two systems, we adopted a channel separation approach akin to that employed in the design of acoustic reproduction systems. The two channels are recombined at the tip of the device to give a uniform frequency response from DC to one kHz. In terms of mechanical design, the high-frequency transducer was embedded inside the tip of the main stage so that during operation, the human operator has only to interact with a single finger interface. In order to exemplify the type of application that would benefit from this kind of interface, we applied it to the haptic exploration with microscopic scales objects which are known to behave with very fast dynamics. The novel haptic interface was bilaterally coupled with a micromanipulation platform to demonstrate its capabilities. Operators could feel interaction forces arising from contact as well as those resulting from Brownian motion and could manoeuvre a micro bead in the absence of vision.
DOI
### Mesoscopic model for DNA G-quadruplex unfolding
A. E. Bergues-Pupo, I. Gutiérrez, J. R. Arias-Gonzalez, F. Falo & A. Fiasconaro
Genomes contain rare guanine-rich sequences capable of assembling into four-stranded helical structures, termed G-quadruplexes, with potential roles in gene regulation and chromosome stability. Their mechanical unfolding has only been reported to date by all-atom simulations, which cannot dissect the major physical interactions responsible for their cohesion. Here, we propose a mesoscopic model to describe both the mechanical and thermal stability of DNA G-quadruplexes, where each nucleotide of the structure, as well as each central cation located at the inner channel, is mapped onto a single bead. In this framework we are able to simulate loading rates similar to the experimental ones, which are not reachable in simulations with atomistic resolution. In this regard, we present single-molecule force-induced unfolding experiments by a high-resolution optical tweezers on a DNA telomeric sequence capable of adopting a G-quadruplex conformation. Fitting the parameters of the model to the experiments we find a correct prediction of the rupture-force kinetics and a good agreement with previous near equilibrium measurements. Since G-quadruplex unfolding dynamics is halfway in complexity between secondary nucleic acids and tertiary protein structures, our model entails a nanoscale paradigm for non-equilibrium processes in the cell.
DOI
### Optical trapping of otoliths drives vestibular behaviours in larval zebrafish
Itia A. Favre-Bulle, Alexander B. Stilgoe, Halina Rubinsztein-Dunlop & Ethan K. Scott
The vestibular system, which detects gravity and motion, is crucial to survival, but the neural circuits processing vestibular information remain incompletely characterised. In part, this is because the movement needed to stimulate the vestibular system hampers traditional neuroscientific methods. Optical trapping uses focussed light to apply forces to targeted objects, typically ranging from nanometres to a few microns across. In principle, optical trapping of the otoliths (ear stones) could produce fictive vestibular stimuli in a stationary animal. Here we use optical trapping in vivo to manipulate 55-micron otoliths in larval zebrafish. Medial and lateral forces on the otoliths result in complementary corrective tail movements, and lateral forces on either otolith are sufficient to cause a rolling correction in both eyes. This confirms that optical trapping is sufficiently powerful and precise to move large objects in vivo, and sets the stage for the functional mapping of the resulting vestibular processing.
DOI
### How should the optical tweezers experiment be used to characterize the red blood cell membrane mechanics?
Julien Sigüenza, Simon Mendez, Franck Nicoud
Stretching red blood cells using optical tweezers is a way to characterize the mechanical properties of their membrane by measuring the size of the cell in the direction of the stretching (axial diameter) and perpendicularly (transverse diameter). Recently, such data have been used in numerous publications to validate solvers dedicated to the computation of red blood cell dynamics under flow. In the present study, different mechanical models are used to simulate the stretching of red blood cells by optical tweezers. Results first show that the mechanical moduli of the membranes have to be adjusted as a function of the model used. In addition, by assessing the area dilation of the cells, the axial and transverse diameters measured in optical tweezers experiments are found to be insufficient to discriminate between models relevant to red blood cells or not. At last, it is shown that other quantities such as the height or the profile of the cell should be preferred for validation purposes since they are more sensitive to the membrane model.
DOI
### Negative force on free carriers in positive index nanoparticles
Mohammad Habibur Rahaman and Brandon A. Kemp
We theoretically demonstrate the reversal of optical forces on free charge carriers in positive refractive index nanostructures. Though optical momentum in positive refractive index materials is necessarily parallel to the local energy flow, reversal of optical momentum transfer can be accomplished by exploiting the geometry and size of subwavelength particles. Using the Mie scattering theory and separation of optical momentum transfers to the bound and free charges and currents, we have shown that metal nanoparticles can exhibit strong momentum transfer to free carriers opposite to the direction of incident electromagnetic waves. This can be explained for small particles in terms of a reversal of Poynting power inside the material resulting in a negative net force on free carriers in small particles. Two-dimensional simulations further illuminate this point by demonstrating the effect of incident wave polarization.
DOI
## Friday, November 3, 2017
### An Optical Tweezers Platform for Single Molecule Force Spectroscopy in Organic Solvents
Jacob W. Black, Maria Kamenetska, and Ziad Ganim
Observation at the single molecule level has been a revolutionary tool for molecular biophysics and materials science, but single molecule studies of solution-phase chemistry are less widespread. In this work we develop an experimental platform for solution-phase single molecule force spectroscopy in organic solvents. This optical-tweezer-based platform was designed for broad chemical applicability and utilizes optically trapped core–shell microspheres, synthetic polymer tethers, and click chemistry linkages formed in situ. We have observed stable optical trapping of the core–shell microspheres in ten different solvents, and single molecule link formation in four different solvents. These experiments demonstrate how to use optical tweezers for single molecule force application in the study of solution-phase chemistry.
DOI
### Determination of size and refractive index of single gold nanoparticles using an optofluidic chip
Y. Z. Shi, S. Xiong, L. K. Chin, J. B. Zhang, W. Ser, J. H. Wu, T. N. Chen, Z. C. Yang, Y. L. Hao, and A. Q. Liu
We report a real-time method to determine the size, i.e. diameter, and refractive index of single gold nanoparticles using an optofluidic chip, which consists of a quasi-Bessel beam optical chromatography. The tightly focused (∼ 0.5 μm) quasi-Bessel beam with low divergence (NA ∼ 0.04) was used to trap sub-100 nm gold nanoparticles within a long trapping distance of 140 μm. In the experiment, 60 to 100 nm gold nanoparticles were separated efficiently with at least 18 μm. The diameter and refractive index (real and imaginary) of single gold nanoparticles were measured at high resolutions with respect to the trapping distance, i.e. 0.36 nm/μm, 0.003/μm and 0.0016/μm, respectively.
DOI
### Effect of laser radiation power on laser trapping of light-absorbing microparticles in air
A. P. Porfirev, S. A. Fomchenkov
We investigate the effects of changing the power of a Gaussian laser beam on the motion of light-absorbing microparticles trapped in the beam region. Laser trapping of such particles was due to the action of so-called photophoretic forces. In addition, we demonstrate the possibility of controlled movement of trapped carbon nanoparticle agglomerations, both in the direction of propagation of the laser beam and in the opposite direction.
DOI
### Combinatorial Particle Patterning
Clemens von Bojnicic-Kninski, Roman Popov, Edgar Dörsam, Felix F. Loeffler, Frank Breitling, Alexander Nesterov-Muelle
The unique properties of solid particles make them a promising element of micro- and nanostructure technologies. Solid particles can be used as building blocks for micro and nanostructures, carriers of monomers, or catalysts. The possibility of patterning different kinds of particles on the same substrate opens the pathway for novel combinatorial designs and novel technologies. One of the examples of such technologies is the synthesis of peptide arrays with amino acid particles. This review examines the known methods of combinatorial particle patterning via static electrical and magnetic fields, laser radiation, patterning by synthesis, and particle patterning via chemically modified or microstructured surfaces.
DOI
### Rotation and Negative Torque in Electrodynamically Bound Nanoparticle Dimers
Nishant Sule, Yuval Yifat, Stephen K. Gray, and Norbert F. Scherer
We examine the formation and concomitant rotation of electrodynamically bound dimers (EBD) of 150 nm diameter Ag nanoparticles trapped in circularly polarized focused Gaussian beams. The rotation frequency of an EBD increases linearly with the incident beam power, reaching mean values of ∼4 kHz for relatively low incident powers of 14 mW. Using a coupled-dipole/effective polarizability model, we reveal that retardation of the scattered fields and electrodynamic interactions can lead to a “negative torque” causing rotation of the EBD in the direction opposite to that of the circular polarization. This intriguing opposite-handed rotation due to negative torque is clearly demonstrated using electrodynamics-Langevin dynamics simulations by changing particle separations and thus varying the retardation effects. Finally, negative torque is also demonstrated in experiments from statistical analysis of the EBD trajectories. These results demonstrate novel rotational dynamics of nanoparticles in optical matter using circular polarization and open a new avenue to control orientational dynamics through coupling to interparticle separation.
## Thursday, November 2, 2017
### Observation of radiation pressure induced deformation of high-reflective reflector
Yukun Yuan, Chunyang Gu, Yue Cao, Shiling Wang and Feng Zhou Fang
In this paper, radiation pressure induced deformation of series of thin aluminum reflector is analyzed theoretically and experimentally. Theory of quantum mechanics and material mechanics are applied in 2-D simulations and exhibits good coordination with experiment results. The original laser source used in experiment is Gauss-distributed and has been shaped and expanded, resulting in the flattened light to avoid over heating or even ablation of the irradiated reflector, which can also bring a major deformation. The aluminum reflectors are fabricated by a high precision machine tool into a thickness of 100μm, 200μm, 300μm with a surface roughness of 8 nm in Ra, and then coated with high-reflective(HR) coatings and mounted on a thick 3D printing base made of polylactic acid(PLA). In the experimental process, a vacuum chamber is employed to distinguish the effect of thermal convection. The results shows that radiation pressure induced deformation has an obvious negative correlation with the reflector thickness. The time-deformation curve of the reflector reaches 2.4 μm peak negative displacement at most the moment laser beam is acting when under vacuum circumstance, and soon raises up to over 12 μm positive displacement if the reflector is continuously irradiated. Subsequent analysis shows that such negative displacement is induced by radiation pressure and the positive displacement is caused by thermal expansion of the PLA base.
DOI
### Feedback-tracking microrheology in living cells
Kenji Nishizawa, Marcel Bremerich, Heev Ayade, Christoph F. Schmidt, Takayuki Ariga and Daisuke Mizuno
Living cells are composed of active materials, in which forces are generated by the energy derived from metabolism. Forces and structures self-organize to shape the cell and drive its dynamic functions. Understanding the out-of-equilibrium mechanics is challenging because constituent materials, the cytoskeleton and the cytosol, are extraordinarily heterogeneous, and their physical properties are strongly affected by the internally generated forces. We have analyzed dynamics inside two types of eukaryotic cells, fibroblasts and epithelial-like HeLa cells, with simultaneous active and passive microrheology using laser interferometry and optical trapping technology. We developed a method to track microscopic probes stably in cells in the presence of vigorous cytoplasmic fluctuations, by using smooth three-dimensional (3D) feedback of a piezo-actuated sample stage. To interpret the data, we present a theory that adapts the fluctuation-dissipation theorem (FDT) to out-of-equilibrium systems that are subjected to positional feedback, which introduces an additional nonequilibrium effect. We discuss the interplay between material properties and nonthermal force fluctuations in the living cells that we quantify through the violations of the FDT. In adherent fibroblasts, we observed a well-known polymer network viscoelastic response where the complex shear modulus scales as G* ∝ (−iω)3/4. In the more 3D confluent epithelial cells, we found glassy mechanics with G* ∝ (−iω)1/2 that we attribute to glassy dynamics in the cytosol. The glassy state in living cells shows characteristics that appear distinct from classical glasses and unique to nonequilibrium materials that are activated by molecular motors.
DOI
### Cell volume change through water efflux impacts cell stiffness and stem cell fate
Ming Guo, Adrian F. Pegoraro, Angelo Mao, Enhua H. Zhou, Praveen R. Arany, Yulong Han, Dylan T. Burnette, Mikkel H. Jensen, Karen E. Kasza, Jeffrey R. Moore, Frederick C. Mackintosh, Jeffrey J. Fredberg, David J. Mooney, Jennifer Lippincott-Schwartz, and David A. Weitz
Cells alter their mechanical properties in response to their local microenvironment; this plays a role in determining cell function and can even influence stem cell fate. Here, we identify a robust and unified relationship between cell stiffness and cell volume. As a cell spreads on a substrate, its volume decreases, while its stiffness concomitantly increases. We find that both cortical and cytoplasmic cell stiffness scale with volume for numerous perturbations, including varying substrate stiffness, cell spread area, and external osmotic pressure. The reduction of cell volume is a result of water efflux, which leads to a corresponding increase in intracellular molecular crowding. Furthermore, we find that changes in cell volume, and hence stiffness, alter stem-cell differentiation, regardless of the method by which these are induced. These observations reveal a surprising, previously unidentified relationship between cell stiffness and cell volume that strongly influences cell biology.
DOI
### A semi-analytical model of a near-field optical trapping potential well
A semi-analytical model is proposed to describe the force generated by a near-field optical trap. The model contains fitting parameters that can be adjusted to resemble a reference force-field. The model parameters for a plasmonic near-field trap consisting of a C-shaped engraving are determined using least squares regression. The reference values required for the regression analysis are calculated using the Maxwell stress tensor method. The speed and accuracy of the proposed model are compared with the conventional method. The model is found to be significantly faster with an acceptable level of accuracy.
DOI
### Poynting theorem in terms of beam shape coefficients and applications to axisymmetric, dark and non-dark, vortex and non-vortex, beams
Gérard Gouesbet
Electromagnetic arbitrary shaped beams may be described by using expansions over a set of basis functions, with expansion coefficients containing sub-coefficients called beam shape coefficients which encode the structure of the beam. In this paper, the Poynting theorem is expressed in terms of these beam shape coefficients. Special cases (axisymmetric, dark and non-dark beams) are thereafter considered, as well as specific applications to paradigmatic examples, from trivial cases (plane waves and spherical waves) to the more sophisticated case of vortex beams.
DOI
## Wednesday, November 1, 2017
### Modeling and calibrating nonlinearity and crosstalk in back focal plane interferometry for three-dimensional position detection
Peng Cheng, Sissy M. Jhiang, and Chia-Hsiang Menq
Back focal plane (BFP) interferometry is frequently used to detect the motion of a single laser trapped bead in a photonic force microscope (PFM) system. Whereas this method enables high-speed and high-resolution position measurement, its measurement range is limited by nonlinearity coupled with crosstalk in three-dimensional (3-D) measurement, and validation of its measurement accuracy is not trivial. This Letter presents an automated calibration system in conjunction with a 3-D quadratic model to render rapid and accurate calibration of the laser measurement system. An actively controlled three-axis laser steering system and a high-speed vision-based 3-D particle tracking system are integrated to the PFM system to enable rapid calibration. The 3-D quadratic model is utilized to correct for nonlinearity and crosstalk and, thus, extend the 3-D position detection volume of BFP interferometry. We experimentally demonstrated a 12-fold increase in detection volume when applying the method to track the motion of a 2.0 μm laser trapped polystyrene bead.
DOI
### Tailoring optical pulling force on gain coated nanoparticles with nonlocal effective medium theory
X. Bian, D. L. Gao, and L. Gao
We study the optical scattering force on the coated nanoparticles with gain core and nonlocal plasmonic shell in the long-wavelength limit, and demonstrate negative optical force acting on the nanoparticles near the symmetric and/or antisymmetric surface plasmon resonances. To understand the optical force behavior, we propose nonlocal effective medium theory to derive the equivalent permittivity for the coated nanoparticles with nonlocality. We show that the imaginary part of the equivalent permittivity is negative near the surface resonant wavelength, resulting in the negative optical force. The introduction of nonlocality may shift the resonant wavelength of the optical force, and strengthen the negative optical force. Two examples of Fano-like resonant scattering in such coated nanoparticles are considered, and Fano resonance-induced negative optical force is found too. Our findings could have some potential applications in plasmonics, nano-optical manipulation, and optical selection.
DOI
### Kinesin rotates unidirectionally and generates torque while walking on microtubules
Avin Ramaiya, Basudev Roy, Michael Bugiel, and Erik Schäffer
Cytoskeletal motors drive many essential cellular processes. For example, kinesin-1 transports cargo in a step-wise manner along microtubules. To resolve rotations during stepping, we used optical tweezers combined with an optical microprotractor and torsion balance using highly birefringent microspheres to directly and simultaneously measure the translocation, rotation, force, and torque generated by individual kinesin-1 motors. While, at low adenosine 5′-triphosphate (ATP) concentrations, motors did not generate torque, we found that motors translocating along microtubules at saturating ATP concentrations rotated unidirectionally, producing significant torque on the probes. Accounting for the rotational work makes kinesin a highly efficient machine. These results imply that the motor’s gait follows a rotary hand-over-hand mechanism. Our method is generally applicable to study rotational and linear motion of molecular machines, and our findings have implications for kinesin-driven cellular processes.
DOI
### Optical Trap Assisted Nanopatterning: Process Parallelization and Dynamic Structure Generation
Johannes Strauss, Marcus Baum, Ilya Alexeev, Michael Schmidt
In this publication we present a novel setup for the Optical Trap Assisted Nanopatterning
(OTAN) technology. The setup allows process parallelization and thus higher throughput in this inventive and flexible direct-nanopatterning technology. We have determined the stiffness of the optical traps and compared the obtained result with the single beam OTAN parameters. Furthermore we estimate the increase in throughput for the parallelized approach in comparison to the conventional system.
DOI
### Multiple Particles 3-D Trap Based on All-Fiber Bessel Optical Probe
Yaxun Zhang, Xiaoyun Tang, Yu Zhang, Zhihai Liu, Enming Zhao, Xinghua Yang, Jianzhong Zhang, Jun Yang, Libo Yuan
We propose and demonstrate an all-fiber Bessel optical tweezers for multiple microparticles (yeast cells) three-dimensional (3-D) trap. To the best knowledge of us, it is the first time to achieve the 3-D stable noncontact multiple microparticles optical traps with long distance intervals by using a single all-fiber probe. The Bessel beam is produced by splicing coaxially a single-mode fiber and a step index multimode fiber. The convergence of the output Bessel beam is performed by molding the tip of the multimode fiber into a special semiellipsoid shape. The effective trapping range of the all-fiber probe is 0 to 60 μm, which is much longer than normal single fiber optical tweezers probes. The all-fiber Bessel optical probe is convenient to integrate and suitable for the lab on the chip. The structure of this fiber probe is simple, high precision, low cost, and small size, which provides new development for biological cells experiment and operation.
DOI
## Monday, October 30, 2017
### Particle manipulation beyond the diffraction limit using structured super-oscillating light beams
Brijesh K Singh, Harel Nagar, Yael Roichman and Ady Arie
The diffraction-limited resolution of light focused by a lens was derived in 1873 by Ernst Abbe. Later in 1952, a method to reach sub-diffraction light spots was proposed by modulating the wavefront of the focused beam. In a related development, super-oscillating functions, that is, band-limited functions that locally oscillate faster than their highest Fourier component, were introduced and experimentally applied for super-resolution microscopy. Up till now, only simple Gaussian-like sub-diffraction spots were used. Here we show that the amplitude and phase profile of these sub-diffraction spots can be arbitrarily controlled. In particular, we utilize Hermite–Gauss, Laguerre–Gauss and Airy functions to structure super-oscillating beams with sub-diffraction lobes. These structured beams are then used for high-resolution trapping and manipulation of nanometer-sized particles. The trapping potential provides unprecedented localization accuracy and stiffness, significantly exceeding those provided by standard diffraction-limited beams.
DOI
### Nanoscopic control and quantification of enantioselective optical forces
Yang Zhao, Amr A. E. Saleh, Marie Anne van de Haar, Brian Baum, Justin A. Briggs, Alice Lay, Olivia A. Reyes-Becerra & Jennifer A. Dionne
Circularly polarized light (CPL) exerts a force of different magnitude on left- and right-handed enantiomers, an effect that could be exploited for chiral resolution of chemical compounds1, 2, 3, 4, 5 as well as controlled assembly of chiral nanostructures6, 7. However, enantioselective optical forces are challenging to control and quantify because their magnitude is extremely small (sub-piconewton) and varies in space with sub-micrometre resolution2. Here, we report a technique to both strengthen and visualize these forces, using a chiral atomic force microscope probe coupled to a plasmonic optical tweezer8, 9, 10, 11, 12, 13. Illumination of the plasmonic tweezer with CPL exerts a force on the microscope tip that depends on the handedness of the light and the tip. In particular, for a left-handed chiral tip, transverse forces are attractive with left-CPL and repulsive with right-CPL. Additionally, total force differences between opposite-handed specimens exceed 10 pN. The microscope tip can map chiral forces with 2 nm lateral resolution, revealing a distinct spatial distribution of forces for each handedness.
DOI
### Reexamination of the Abraham-Minkowski dilemma
Mário G. Silveirinha
Here the Abraham-Minkowski controversy on the correct definition of the light momentum in a macroscopic medium is revisited with the purpose to highlight that an effective medium formalism necessarily restricts the available information on the internal state of a system, and that this is ultimately the reason why the dilemma has no universal solution. Despite these difficulties, it is demonstrated that in the limit of no material absorption and under steady-state conditions, the time-averaged light (kinetic) momentum may be unambiguously determined by the Abraham result, both for bodies at rest and for circulatory flows of matter. The implications of these findings are discussed in the context of quantum optics of moving media, and we examine in detail the fundamental role of the Minkowski momentum in such a context.
DOI
### Adhesion force and attachment lifetime of the KIF16B-PX domain interaction with lipid membranes
Serapion Pyrpassopoulos, Henry Shuman, and E. Michael Ostap
DOI
### Force, torque, linear momentum, and angular momentum in classical electrodynamics
Masud Mansuripur
The classical theory of electrodynamics is built upon Maxwell’s equations and the concepts of electromagnetic (EM) field, force, energy, and momentum, which are intimately tied together by Poynting’s theorem and by the Lorentz force law. Whereas Maxwell’s equations relate the fields to their material sources, Poynting’s theorem governs the flow of EM energy and its exchange between fields and material media, while the Lorentz law regulates the back-and-forth transfer of momentum between the media and the fields. An alternative force law, first proposed by Einstein and Laub, exists that is consistent with Maxwell’s equations and complies with the conservation laws as well as with the requirements of special relativity. While the Lorentz law requires the introduction of hidden energy and hidden momentum in situations where an electric field acts on a magnetized medium, the Einstein–Laub (E–L) formulation of EM force and torque does not invoke hidden entities under such circumstances. Moreover, total force/torque exerted by EM fields on any given object turns out to be independent of whether the density of force/torque is evaluated using the law of Lorentz or that of Einstein and Laub. Hidden entities aside, the two formulations differ only in their predicted force and torque distributions inside matter. Such differences in distribution are occasionally measurable, and could serve as a guide in deciding which formulation, if either, corresponds to physical reality.
DOI
### Trapping of Micro Particles in Nanoplasmonic Optical Lattice
Dinesh Bhalothia, Ya-Tang Yang
The plasmonic optical tweezer has been developed to overcome the diffraction limits of the conventional far field optical tweezer. Plasmonic optical lattice consists of an array of nanostructures, which exhibit a variety of trapping and transport behaviors. We report the experimental procedures to trap micro-particles in a simple square nanoplasmonic optical lattice. We also describe the optical setup and the nanofabrication of a nanoplasmonic array. The optical potential is created by illuminating an array of gold nanodiscs with a Gaussian beam of 980 nm wavelength, and exciting plasmon resonance. The motion of particles is monitored by fluorescence imaging. A scheme to suppress photothermal convection is also described to increase usable optical power for optimal trapping. Suppression of convection is achieved by cooling the sample to a low temperature, and utilizing the near-zero thermal expansion coefficient of a water medium. Both single particle transport and multiple particle trapping are reported here.
DOI
## Friday, October 27, 2017
### Mode-selective thermal radiation from a microsphere as a probe of optical properties of high-temperature materials
R. Morino, H. Tajima, H. Sonoda, H. Kobayashi, R. Kanamoto, H. Odashima, and M. Tachikawa
Our spectroscopic method using laser trapping and heating has demonstrated that thermal emission from a metal oxide microsphere is enhanced at frequencies resonant with the whispering gallery modes of the spherical resonator. Only a mode series of a specific order effectively emits thermal photons, and spectral peaks shift from higher-order whispering gallery modes to fundamental whispering gallery modes as the size parameter decreases. These spectral profiles are analyzed with the Mie scattering theory and a semiclassical rate-equation model. The observed mode selectivity in thermal radiation is attributed to a matching between the rates of cavity damping and internal absorption. Excellent reproducibility of the observed spectral profiles leads to a precise determination of optical constants of extremely hot materials.
DOI
### Circularly symmetric frozen waves: Vector approach for light scattering calculations
Leonardo André Ambrosio
This work introduces particular classes of vector wave fields for light scattering calculations, viz. structured light fields composed of specific superpositions of circularly symmetric Bessel beams of arbitrary order. Also known as generalized frozen waves, such beams carry all the non-diffracting properties of their constituents with the additional feature of allowing for an arbitrary design of the longitudinal intensity pattern along the surface of several cylinders of fixed radius, simultaneously. This feature makes the generalized frozen waves especially suitable for optical confinement and manipulation and atom guiding and selection. In the framework of the generalized Lorenz–Mie theory, the beam shape coefficients which describe such beams are evaluated in exact and analytic form, the resulting expressions being then applied in light scattering problems. Particular frozen waves are considered beyond the paraxial approximation, optical forces being calculated for specific dielectric Rayleigh particles.
DOI
### Local electrophoresis deposition assisted by laser trapping coupled with a spatial light modulator for three-dimensional microfabrication
Toshiki Matsuura, Takanari Takai and Futoshi Iwata
We describe a novel three-dimensional fabrication technique using local electrophoresis deposition assisted by laser trapping coupled with a spatial light modulator (SLM). In a solution containing nanometer-scale colloidal Au particles, multiple laser spots formed on a conductive substrate by the SLM gathered the nanoparticles together, and then the nanoparticles were electrophoretically deposited onto the substrate by an applied electrical field. However, undesirable sub-spots often appeared due to optical interference from the multiple laser spots, which deteriorated the accuracy of the deposition. To avoid the appearance of undesirable sub-spots, we proposed a method using quasi-multiple spots, which we realized by switching the position of a single spot briefly using the SLM. The method allowed us to deposit multiple dots on the substrate without undesirable sub-dot deposition. By moving the substrate downward during deposition, multiple micro-pillar structures could be fabricated. As a fabrication property, the dependence of the pillar diameter on laser intensity was investigated by changing the number of laser spots. The smallest diameter of the four pillars fabricated in this study was 920 nm at the laser intensity of 2.5 mW. To demonstrate the effectiveness of the method, multiple spiral structures were fabricated. Quadruple spirals of 46 µm in height were successfully fabricated with a growth rate of 0.21 µm/s using 2200 frames of the CGH patterns displayed in the SLM at a frame rate of 10 fps.
DOI
### Optical forces through the effective refractive index
Janderson R. Rodrigues and Vilson R. Almeida
Energy-based methods such as the dispersion relation (DR) and response theory of optical forces (RTOF) have been largely applied to obtain the optical forces in the nano-optomechanical devices, in contrast to the Maxwell stress tensor (MST). In this Letter, we apply first principles to show explicitly why these methods must agree with the MST formalism in linear lossless systems. We apply the RTOF multi-port to show that the optical force expression on these devices can be extended to analyze multiple light sources, broadband sources, and multimode devices, with multiple degrees of freedom. We also show that the DR method, when expressed as a function of the derivative of the effective index performed at a fixed wave vector, may be misinterpreted and lead to overestimated results.
DOI
### Optical trapping of Au-Fe alloyed nanoparticles: a theoretical calculation
Magnetoplasmonic nanoparticles such as Au-Fe alloys are very intersting for their properties. In this article, the optical trapping of Au-Fe nanoparticles are investigated as a function of Fe atomic percent doped in gold nanoparticles, theoretically. Using Lorenz-Mie theory it is shown that the maximum force exerted on the alloyed nanoparticles enhances about $75\%$ with increaseing Fe atomic percent. It is shown that trapping strength is depth-dependent and shows $20\%$ increment in shallow positions and $17\%$ decrement in the axial direction in the optimal depth which is $7\mu m$ deep inside the sample. Wavelength dependence of alloyed nanoparticles is studied, too.
DOI
### On the substrate contribution to the back action trapping of plasmonic nanoparticles on resonant near-field traps in plasmonic films
Nanoparticles trapped on resonant near-field apertures/engravings carved in plasmonic films experience optical forces due to the steep intensity gradient field of the aperture/engraving as well as the image like interaction with the substrate. For non-resonant nanoparticles the contribution of the substrate interaction to the trapping force in the vicinity of the trap (aperture/engraving) mode is negligible. But, in the case of plasmonic nanoparticles, the contribution of the substrate interaction to the low frequency stable trapping mode of the coupled particle-trap system increases as their resonance is tuned to the trap resonance. The strength of the substrate interaction depends on the height of the nanoparticle above the substrate. As a result, a difference in back action mechanism arises for nanoparticle displacements perpendicular to the substrate and along it. For nanoparticle displacements perpendicular to the substrate, the self induced back action component of the trap force arises due to changing interaction with the substrate as well as the trap. On the other hand, for displacements along the substrate, it arises solely due to the changing interaction with the trap. This additional contribution of the substrate leads to more pronounced back action. Numerical simulation results are presented to illustrate these effects using a bowtie engraving as the near-field trap and a nanorod as the trapped plasmonic nanoparticle. The substrate’s role may be important in manipulation of plasmonic nanoparticles between successive traps of on-chip optical conveyor belts, because they have to traverse over regions of bare substrate while being handed off between these traps.
DOI
## Tuesday, October 24, 2017
### Native kinesin-1 does not bind preferentially to GTP-tubulin-rich microtubules in vitro
Qiaochu Li, Stephen J. King, Jing X
Molecular motors such as kinesin-1 work in small teams to actively shuttle cargos in cells, for example in polarized transport in axons. Here, we examined the potential regulatory role of the nucleotide state of tubulin on the run length of cargos carried by multiple kinesin motors, using an optical trapping-based in vitro assay. Based on a previous report that kinesin binds preferentially to GTP-tubulin-rich microtubules, we anticipated that multiple-kinesin cargos would run substantially greater distances along GMPCPP microtubules than along GDP microtubules. Surprisingly, we did not uncover any significant differences in run length between microtubule types. A combination of single-molecule experiments, comparison with previous theory, and classic microtubule affinity pulldown assays revealed that native kinesin-1 does not bind preferentially to GTP-tubulin-rich microtubules. The apparent discrepancy between our observations and the previous report likely reflects differences in post-translational modifications between the native motors used here and the recombinant motors examined previously. Future investigations will help shed light on the interplay between the motor's post-translational modification and the microtubule's nucleotide-binding state for transport regulation in vivo.
DOI
### Bidirectional optical rotation of cells
Jiyi Wu, Weina Zhang, and Juan Li
Precise and controlled rotation manipulation of cells is extremely important in biological applications and biomedical studies. Particularly, bidirectional rotation manipulation of a single or multiple cells is a challenge for cell tomography and analysis. In this paper, we report an optical method that is capable of bidirectional rotation manipulation of a single or multiple cells. By launching a laser beam at 980 nm into dual-beam tapered fibers, a single or multiple cells in solutions can be trapped and rotated bidirectionally under the action of optical forces. Moreover, the rotational behavior can be controlled by altering the relative distance between the two fibers and the input optical power. Experimental results were interpreted by numerical simulations.
DOI
### Kaleidoscopic patterning of micro-objects based on software-oriented approach using dual optical tweezers with a microlens array
Yoshio Tanaka
Dynamical and precise arrangement of micro-objects into the specified various pattern offers great flexibility and potential as platforms for many scientific applications, especially in bio-sensing and biomedical fields such as bio-MEMS and Lab-on-a-Chip. Multi-beam optical tweezers are one of the most suitable tools for assembling precise dynamic arrays of micro-objects. Herein, a dynamic patterning method based on software-oriented approach is proposed (i.e. time-shared scanning technique) using the dual optical tweezers with a microlens array. The proposed method can expand the patterning capability of this dual optical tweezers system to simply fabricate various quasi-periodic structures. The work also demonstrates kaleidoscopic patterning (periodic or symmetric arrangements such as Escher's paintings) of numerous microbeads and subsequent morphing. In the demonstrations, microbeads with different properties (size and colour) as well as homogeneous microbeads are arranged dynamically into the specified patterns, including their clusters.
DOI
### Collective Force Regulation in Anti-parallel Microtubule Gliding by Dimeric Kif15 Kinesin Motors
Dana N.Reinemann, Emma G.Sturgill, Dibyendu KumarDas, Miriam Steiner Degen, Zsuzsanna Vörös, Wonmuk Hwang, Ryoma Ohi, Matthew J.Lang
During cell division, the mitotic kinesin-5 Eg5 generates most of the force required to separate centrosomes during spindle assembly. However, Kif15, another mitotic kinesin, can replace Eg5 function, permitting mammalian cells to acquire resistance to Eg5 poisons. Unlike Eg5, the mechanism by which Kif15 generates centrosome separation forces is unknown. Here we investigated the mechanical properties and force generation capacity of Kif15 at the single-molecule level using optical tweezers. We found that the non-motor microtubule-binding tail domain interacts with the microtubule’s E-hook tail with a rupture force higher than the stall force of the motor. This allows Kif15 dimers to productively and efficiently generate forces that could potentially slide microtubules apart. Using an in vitro optical trapping and fluorescence assay, we found that Kif15 slides anti-parallel microtubules apart with gradual force buildup while parallel microtubule bundles remain stationary with a small amount of antagonizing force generated. A stochastic simulation shows the essential role of Kif15’s tail domain for load storage within the Kif15-microtubule system. These results suggest a mechanism for how Kif15 rescues bipolar spindle assembly.
DOI
### Membrane Mechanics Govern Spatiotemporal Heterogeneity of Endocytic Clathrin Coat Dynamics
N. M. Willy, J. P. Ferguson, S. D. Huber, S. P. Heidotting, E. Aygün, S. A. Wurm, E. Johnston-Halperin, M. G. Poirier, and C. Kural
Dynamics of endocytic clathrin-coated structures can be remarkably divergent across different cell types, cells within the same culture, or even distinct surfaces of the same cell. The origin of this astounding heterogeneity remains to be elucidated. Here, we show that cellular processes associated with changes in effective plasma membrane tension induce significant spatiotemporal alterations in endocytic clathrin coat dynamics. Spatiotemporal heterogeneity of clathrin coat dynamics is also observed during morphological changes taking place within developing multicellular organisms. These findings suggest that tension gradients can lead to patterning and differentiation of tissues through mechanoregulation of clathrin-mediated endocytosis.
DOI
### Kinesin and dynein mechanics: measurement methods and research applications
Zachary Abraham, Emma Hawley, Daniel Hayosh, Victoria Webster-Wood and Ozan Akkus
Motor proteins play critical roles in the normal function of cells and proper development of organisms. Among motor proteins, failings in the normal function of two types of proteins, kinesin and dynein, have been shown to lead many pathologies, including neurodegenerative diseases and cancers. As such, it is critical for researchers to understand the underlying mechanics and behaviors of these proteins, not only to shed light on how failures may lead to disease, but also to guide research towards novel treatment and nanoengineering solutions. To this end, many experimental techniques have been developed to measure the force and motility capabilities of these proteins. This review will: a) discuss such techniques, specifically microscopy, atomic force microscopy, optical trapping, and magnetic tweezers, and, b) the resulting nanomechanical properties of motor protein functions such as stalling force, velocity and dependence on ATP concentrations will be comparatively discussed. Additionally, this review will highlight the clinical importance of these proteins. Furthermore, as the understanding of the structure and function of motor proteins improves, novel applications are emerging in the field. Specifically, researchers have begun to modify the structure of existing proteins, thereby engineering novel elements to alter and improve native motor protein function, or even allow the motor proteins to perform entirely new tasks as parts of nanomachines. Kinesin and dynein are vital elements for the proper function of cells. While many exciting experiments have shed light on their function, mechanics, and applications, additional research is needed to completely understand their behavior.
DOI
## Tuesday, October 10, 2017
### Ionic effects on the temperature–force phase diagram of DNA
Sitichoke Amnuanpol
Double-stranded DNA (dsDNA) undergoes a structural transition to single-stranded DNA (ssDNA) in many biologically important processes such as replication and transcription. This strand separation arises in response either to thermal fluctuations or to external forces. The roles of ions are twofold, shortening the range of the interstrand potential and renormalizing the DNA elastic modulus. The dsDNA-to-ssDNA transition is studied on the basis that dsDNA is regarded as a bound state while ssDNA is regarded as an unbound state. The ground state energy of DNA is obtained by mapping the statistical mechanics problem to the imaginary time quantum mechanics problem. In the temperature–force phase diagram the critical force Fc(T) increases logarithmically with the Na+ concentration in the range from 32 to 110 mM. Discussing this logarithmic dependence of Fc(T) within the framework of polyelectrolyte theory, it inevitably suggests a constraint on the difference between the interstrand separation and the length per unit charge during the dsDNA-to-ssDNA transition.
DOI
### Actin and microtubule networks contribute differently to cell response for small and large strains
H Kubitschke, J Schnauss, K D Nnetu, E Warmt, R Stange and J Kaes
Cytoskeletal filaments provide cells with mechanical stability and organization. The main key players are actin filaments and microtubules governing a cell's response to mechanical stimuli. We investigated the specific influences of these crucial components by deforming MCF-7 epithelial cells at small (≤5% deformation) and large strains (>5% deformation). To understand specific contributions of actin filaments and microtubules, we systematically studied cellular responses after treatment with cytoskeleton influencing drugs. Quantification with the microfluidic optical stretcher allowed capturing the relative deformation and relaxation of cells under different conditions. We separated distinctive deformational and relaxational contributions to cell mechanics for actin and microtubule networks for two orders of magnitude of drug dosages. Disrupting actin filaments via latrunculin A, for instance, revealed a strain-independent softening. Stabilizing these filaments by treatment with jasplakinolide yielded cell softening for small strains but showed no significant change at large strains. In contrast, cells treated with nocodazole to disrupt microtubules displayed a softening at large strains but remained unchanged at small strains. Stabilizing microtubules within the cells via paclitaxel revealed no significant changes for deformations at small strains, but concentration-dependent impact at large strains. This suggests that for suspended cells, the actin cortex is probed at small strains, while at larger strains; the whole cell is probed with a significant contribution from the microtubules.
DOI
### Integrated Method to Attach DNA Handles and Functionally Select Proteins to Study Folding and Protein-Ligand Interactions with Optical Tweezers
Yuxin Hao, Clare Canavan, Susan S. Taylor & Rodrigo A. Maillard
Optical tweezers has emerged as a powerful tool to study folding, ligand binding, and motor enzymes. The manipulation of proteins with optical tweezers requires attaching molecular handles to the protein of interest. Here, we describe a novel method that integrates the covalent attachment of DNA handles to target proteins with a selection step for functional and properly folded molecules. In addition, this method enables obtaining protein molecules in different liganded states and can be used with handles of different lengths. We apply this method to study the cAMP binding domain A (CBD-A) of Protein kinase A. We find that the functional selection step drastically improves the reproducibility and homogeneity of the single molecule data. In contrast, without a functional selection step, proteins often display misfolded conformations. cAMP binding stabilizes the CBD-A against a denaturing force, and increases the folded state lifetime. Data obtained with handles of 370 and 70 base pairs are indistinguishable, but at low forces short handles provide a higher spatial resolution. Altogether, this method is flexible, selects for properly folded molecules in different liganded states, and can be readily applicable to study protein folding or protein-ligand interactions with force spectroscopy that require molecular handles.
DOI
### Submicrometer-sized nonspherical particle separation by laser beam
Jaromír Petržala, Miroslav Kocifaj, Ladislav Kómar, and Alexandre Simoneau
The radiation pressure exerted on sub-micrometer-size particles is shown to be an important factor predetermining the impact coordinates of the particles after being illuminated by a laser beam. Unlike spherical particles, the nonspherical ones can be deflected perpendicularly to the beam direction if the momentum transfer from the laser beam to a particle is large enough. Such an optical sorting is a useful technology, which can be used to isolate spherules of a specific size from a population of particles of random sizes and shapes. The system of ideal spheres has a wide range of applications in industry and also in the development of targeted optical devices, and so the methods for fast contact-less particle separation are expected to lead to considerable progress in the field. The theoretical model we have developed is demonstrated in a set of numerical experiments on metallic and nonmetallic particles.
DOI
### Protein Folding Mediated by Trigger Factor and Hsp70: New Insights from Single-Molecule Approaches
Florian Wruck, Mario J.Avellaneda, Eline J.Koers, David P. Minde, Matthias P. Mayer, Günter Kramer, Alireza Mashaghi, Sander J.Tans
Chaperones assist in protein folding, but what this common phrase means in concrete terms has remained surprisingly poorly understood. We can readily measure chaperone binding to unfolded proteins, but how they bind and affect proteins along folding trajectories has remained obscure. Here we review recent efforts by our labs and others that are beginning to pry into this issue, with a focus on the chaperones trigger factor and Hsp70. Single-molecule methods are central, as they allow the stepwise process of folding to be followed directly. First results have already revealed contrasts with long-standing paradigms: rather than acting only “early” by stabilizing unfolded chain segments, these chaperones can bind and stabilize partially folded structures as they grow to their native state. The findings suggest a fundamental redefinition of the protein folding problem and a more extensive functional repertoire of chaperones than previously assumed.
DOI
## Wednesday, October 4, 2017
### Assessment of Local Heterogeneity in Mechanical Properties of Nanostructured Hydrogel Networks
Zhaokai Meng, Teena Thakur, Chandani Chitrakar, Manish K. Jaiswal, Akhilesh K. Gaharwar, and Vladislav V. Yakovlev
Our current understanding of the mechanical properties of nanostructured biomaterials is rather limited to invasive/destructive and low-throughput techniques such as atomic force microscopy, optical tweezers, and shear rheology. In this report, we demonstrate the capabilities of recently developed dual Brillouin/Raman spectroscopy to interrogate the mechanical and chemical properties of nanostructured hydrogel networks. The results obtained from Brillouin spectroscopy show an excellent correlation with the conventional uniaxial and shear mechanical testing. Moreover, it is confirmed that, unlike the macroscopic conventional mechanical measurement techniques, Brillouin spectroscopy can provide the elasticity characteristic of biomaterials at a mesoscale length, which is remarkably important for understanding complex cell–biomaterial interactions. The proposed technique experimentally demonstrated the capability of studying biomaterials in their natural environment and may facilitate future fabrication and inspection of biomaterials for various biomedical and biotechnological applications.
DOI
### Polarization-Induced Chirality in Metamaterials via Optomechanical Interaction
Mingkai Liu, David A. Powell, Rui Guo, Ilya V. Shadrivov and Yuri S. Kivshar
A novel type of metamaterial is introduced, where the structural symmetry can be controlled by optical forces. Since symmetry sets fundamental bounds on the optical response, symmetry breaking changes the properties of metamaterials qualitatively over the entire resonant frequency band. This is achieved by a polarized pump beam, exerting optical forces which are not constrained by the structural symmetry. This new concept is illustrated for a metasurface composed of zig-zag chains of dipole meta-atoms, in which a highly asymmetric optical force exists for an appropriate incident polarization. The effect is employed to transform a planar achiral metasurface into a stereoscopic chiral structure. Importantly, the handedness of the induced chirality can be actively switched by changing the incident polarization. The proposed concept can be employed to achieve dynamic spatial control of metamaterials and metasurfaces at infrared and optical frequencies with subwavelength resolution.
DOI
### Opto-thermophoretic assembly of colloidal matter
Linhan Lin, Jianli Zhang, Xiaolei Peng, Zilong Wu, Anna C. H. Coughlan, Zhangming Mao, Michael A. Bevan and Yuebing Zheng
Colloidal matter exhibits unique collective behaviors beyond what occurs at single-nanoparticle and atomic scales. Treating colloidal particles as building blocks, researchers are exploiting new strategies to rationally organize colloidal particles into complex structures for new functions and devices. Despite tremendous progress in directed assembly and self-assembly, a truly versatile assembly technique without specific functionalization of the colloidal particles remains elusive. We develop a new strategy to assemble colloidal matter under a light-controlled temperature field, which can solve challenges in the existing assembly techniques. By adding an anionic surfactant (that is, cetyltrimethylammonium chloride), which serves as a surface charge source, a macro ion, and a micellar depletant, we generate a light-controlled thermoelectric field to manipulate colloidal atoms and a depletion attraction force to assemble the colloidal atoms into two-dimensional (2D) colloidal matter. The general applicability of this opto-thermophoretic assembly (OTA) strategy allows us to build colloidal matter of diverse colloidal sizes (from subwavelength scale to micrometer scale) and materials (polymeric, dielectric, and metallic colloids) with versatile configurations and tunable bonding strengths and lengths. We further demonstrate that the incorporation of the thermoelectric field into the optical radiation force can achieve 3D reconfiguration of the colloidal matter. The OTA strategy releases the rigorous design rules required in the existing assembly techniques and enriches the structural complexity in colloidal matter, which will open a new window of opportunities for basic research on matter organization, advanced material design, and applications.
DOI
### Repulsion–attraction switching of nematic colloids formed by liquid crystal dispersions of polygonal prisms
B. Senyuk, Q. Liu, P. D. Nystrom and I. I. Smalyukh
Self-assembly of colloidal particles due to elastic interactions in nematic liquid crystals promises tunable composite materials and can be guided by exploiting surface functionalization, geometric shape and topology, though these means of controlling self-assembly remain limited. Here, we realize low-symmetry achiral and chiral elastic colloids in the nematic liquid crystals using colloidal polygonal concave and convex prisms. We show that the controlled pinning of disclinations at the prism edges alters the symmetry of director distortions around the prisms and their orientation with respect to the far-field director. The controlled localization of the disclinations at the prism's edges significantly influences the anisotropy of the diffusion properties of prisms dispersed in liquid crystals and allows one to modify their self-assembly. We show that elastic interactions between polygonal prisms can be switched between repulsive and attractive just by controlled re-pinning the disclinations at different edges using laser tweezers. Our findings demonstrate that elastic interactions between colloidal particles dispersed in nematic liquid crystals are sensitive to the topologically equivalent but geometrically rich controlled configurations of the particle-induced defects.
DOI
### Freezing shortens the lifetime of DNA molecules under tension
Wei-Ju Chung, Yujia Cui, Chi-Shuo Chen, Wesley H. Wei, Rong-Shing Chang, Wun-Yi Shu, Ian C. Hsu
DNA samples are commonly frozen for storage. However, freezing can compromise the integrity of DNA molecules. Considering the wide applications of DNA molecules in nanotechnology, changes to DNA integrity at the molecular level may cause undesirable outcomes. However, the effects of freezing on DNA integrity have not been fully explored. To investigate the impact of freezing on DNA integrity, samples of frozen and non-frozen bacteriophage lambda DNA were studied using optical tweezers. Tension (5–35 pN) was applied to DNA molecules to mimic mechanical interactions between DNA and other biomolecules. The integrity of the DNA molecules was evaluated by measuring the time taken for single DNA molecules to break under tension. Mean lifetimes were determined by maximum likelihood estimates and variances were obtained through bootstrapping simulations. Under 5 pN of force, the mean lifetime of frozen samples is 44.3 min with 95% confidence interval (CI) between 36.7 min and 53.6 min while the mean lifetime of non-frozen samples is 133.2 min (95% CI: 97.8–190.1 min). Under 15 pN of force, the mean lifetimes are 10.8 min (95% CI: 7.6–12.6 min) and 78.5 min (95% CI: 58.1–108.9 min). The lifetimes of frozen DNA molecules are significantly reduced, implying that freezing compromises DNA integrity. Moreover, we found that the reduced DNA structural integrity cannot be restored using regular ligation process. These results indicate that freezing can alter the structural integrity of the DNA molecules.
DOI
### Ultrasensitive rotating photonic probes for complex biological systems
Shu Zhang, Lachlan J. Gibson, Alexander B. Stilgoe, Itia A. Favre-Bulle, Timo A. Nieminen, and Halina Rubinsztein-Dunlop
We use rotational photonic tweezers to access local viscoelastic properties of complex fluids over a wide frequency range. This is done by monitoring both passive rotational Brownian motion and also actively driven transient rotation between two angular trapping states of a birefringent microsphere. These enable measurement of high- and low-frequency properties, respectively. Complex fluids arise frequently in microscopic biological systems, typically with length scales at the cellular level. Thus, high spatial resolution as provided by rotational photonic tweezers is important. We measure the properties of tear film on a contact lens and demonstrate variations in these properties between two subjects over time. We also show excellent agreement between our theoretical model and experimental results. We believe that this is the first time that active microrheology using rotating tweezers has been used for biologically relevant questions. Our method demonstrates potential for future applications to determine the spatial-temporal properties of biologically relevant and complex fluids that are only available in very small volumes.
DOI
## Tuesday, October 3, 2017
### Physical Probing of Cells
Florian Rehfeldt and Christoph F Schmidt
In the last two decades, it has become evident that the mechanical properties of the microenvironment of biological cells are as important as traditional biochemical cues for the control of cellular behavior and fate. The field of cell and matrix mechanics is quickly growing and so is the development of the experimental approaches used to study active and passive mechanical properties of cells and their surroundings. Within this topical review we will provide a brief overview, on the one hand, over how cellular mechanics can be probed physically, how different geometries allow access to different cellular properties, and, on the other hand, how forces are generated in cells and transmitted to the extracellular environment. We will describe the following experimental techniques: atomic force microscopy, traction force microscopy, magnetic tweezers, optical stretcher and optical tweezers pointing out both their advantages and limitations. Finally, we give an outlook on the future of physical probing of cells.
DOI
### Spectral identification in the attogram regime through laser-induced emission of single optically-trapped nanoparticles in air
Pablo Purohit, Francisco J. Fortes, Javier Laserna
Current trends in nanoengineering are bringing along new structures of diverse chemical compositions that need to be meticulously defined in order to ensure their correct operation. Few methods can provide the sensitivity required to carry out measurements on individual nano objects without tedious sample pre-treatment or data analysis. In the present study, we introduce a pathway for the elemental identification of single nanoparticles (NPs) that avoids suspension in liquid media by means of optical trapping and laser-induced plasma spectroscopy. We demonstrate spectroscopic detection and identification of individual 25 to 70 nm in diameter Cu NPs stably trapped in air featuring masses down to 73 attograms. We found an increase in the absolute number of photons produced as size of the particles decreased; pointing towards a more efficient excitation of ensembles of only 7e+5 Cu atoms in the onset plasma.
DOI
### Plasmonic trapping of nanoparticles by metaholograms
Guanghao Rui, Yanbao Ma, Bing Gu, Qiwen Zhan & Yiping Cui
Manipulation of nanoparticles in solution is of great importance for a wide range of applications in biomedical, environmental, and material sciences. In this work, we present a novel plasmonic tweezers based on metahologram. We show that various kinds of nanoparticles can be stably trapped in a surface plasmon (SP) standing wave generated by the constructive interference between two coherent focusing SPs. The absence of the axial scattering force and the enhanced gradient force enable to avoid overheating effect while maintaining mechanical stability even under the resonant condition of the metallic nanoparticle. The work illustrates the potential of such plasmonic tweezers for further development in lab-on-a-chip devices.
DOI
### Label-Free Detection of Bacillus anthracis Spore Uptake in Macrophage Cells Using Analytical Optical Force Measurements
Colin G. Hebert, Sean Hart, Tomasz A. Leski, Alex Terray, and Qin Lu
Understanding the interaction between macrophage cells and Bacillus anthracis spores is of significant importance with respect to both anthrax disease progression, spore detection for biodefense, as well as understanding cell clearance in general. While most detection systems rely on specific molecules, such as nucleic acids or proteins and fluorescent labels to identify the target(s) of interest, label-free methods probe changes in intrinsic properties, such as size, refractive index, and morphology, for correlation with a particular biological event. Optical chromatography is a label free technique that uses the balance between optical and fluidic drag forces within a microfluidic channel to determine the optical force on cells or particles. Here we show an increase in the optical force experienced by RAW264.7 macrophage cells upon the uptake of both microparticles and B. anthracis Sterne 34F2 spores. In the case of spores, the exposure was detected in as little as 1 h without the use of antibodies or fluorescent labels of any kind. An increase in the optical force was also seen in macrophage cells treated with cytochalasin D, both with and without a subsequent exposure to spores, indicating that a portion of the increase in the optical force arises independent of phagocytosis. These results demonstrate the capability of optical chromatography to detect subtle biological differences in a rapid and sensitive manner and suggest future potential in a range of applications, including the detection of biological threat agents for biodefense and pathogens for the prevention of sepsis and other diseases.
DOI
### Fabrication and application of a non-contact double-tapered optical fiber tweezers
Z.L. Liu, Y.X. Liu, Y. Tang, N. Zhang, F.P. Wu, and B. Zhang
A double-tapered optical fiber tweezers (DOFTs) was fabricated by a chemical etching called interfacial layer etching. In this method, the second taper angle (STA) of DOFTs can be controlled easily by the interfacial layer etching time. Application of the DOFTs to the optical trapping of the yeast cells was presented. Effects of the STA on the axile trapping efficiency and the trapping position were investigated experimentally and theoretically. The experimental results are good agreement with the theoretical ones. The results demonstrated that the non-contact capture can be realized for the large STA (e.g. 90 deg) and there was an optimal axile trapping efficiency as the STA increasing. In order to obtain a more accurate measurement result of the trapping force, a correction factor to Stokes drag coefficient was introduced. This work provided a way of designing and fabricating an optical fiber tweezers (OFTs) with a high trapping efficient or a non-contact capture.
DOI
### Measurement of pH-dependent surface-enhanced hyper-Raman scattering at desired positions on yeast cells via optical trapping
Yasutaka Kitahama, Hiroaki Hayashi, Tamitake Itoh and Yukihiro Ozaki
Surface-enhanced hyper-Raman scattering (SEHRS) spectra were obtained at desired positions on yeast by focusing a continuous wave near-infrared laser beam while silver nanoparticles (AgNPs) were simultaneously optically trapped. However, the optically trapped colloidal AgNP suspension bubbled up at the focusing point, preventing spectral measurement. In the case of optically trapped AgNPs functionalized with 4-mercaptobenzoic acid (p-MBA), surface-enhanced hyper-Rayleigh scattering was considerably strong, indicating the suppression of the photothermal conversion to form the bubble. Interestingly, the SEHRS peaks that are attributed not only to p-MBA, but also to other species, were very occasionally observed. They may be partly assigned to the β1,3 glucan and protein amide II band. The SEHRS peak at 1366 cm−1 was barely visible in the measurements of conventional baker's yeast even in the suspension (pH 9) despite the effects of high pH on p-MBA. In contrast, the SEHRS peak in the measurements of yeast for biological applications was occasionally observed at 1366 cm−1. This suggests that acidity is correlated with fermentation efficiency. At different positions on single yeast cells, the intensity of the SEHRS peak at 1366 cm−1 varied. This result represents the pH distribution on yeast.
DOI
## Friday, September 29, 2017
### Adaptive Response of Actin Bundles under Mechanical Stress
Florian Rückerl, Martin Lenz, Timo Betz, John Manzi, Jean-Louis Martiel, Mahassine Safouane, Rajaa Paterski-Boujemaa, Laurent Blanchoin, Cécile Sykes
Actin is one of the main components of the architecture of cells. Actin filaments form different polymer networks with versatile mechanical properties that depend on their spatial organization and the presence of cross-linkers. Here, we investigate the mechanical properties of actin bundles in the absence of cross-linkers. Bundles are polymerized from the surface of mDia1-coated latex beads, and deformed by manipulating both ends through attached beads held by optical tweezers, allowing us to record the applied force. Bundle properties are strikingly different from the ones of a homogeneous isotropic beam. Successive compression and extension leads to a decrease in the buckling force that we attribute to the bundle remaining slightly curved after the first deformation. Furthermore, we find that the bundle is solid, and stiff to bending, along the long axis, whereas it has a liquid and viscous behavior in the transverse direction. Interpretation of the force curves using a Maxwell visco-elastic model allows us to extract the bundle mechanical parameters and confirms that the bundle is composed of weakly coupled filaments. At short times, the bundle behaves as an elastic material, whereas at long times, filaments flow in the longitudinal direction, leading to bundle restructuring. Deviations from the model reveal a complex adaptive rheological behavior of bundles. Indeed, when allowed to anneal between phases of compression and extension, the bundle reinforces. Moreover, we find that the characteristic visco-elastic time is inversely proportional to the compression speed. Actin bundles are therefore not simple force transmitters, but instead, complex mechano-transducers that adjust their mechanics to external stimulation. In cells, where actin bundles are mechanical sensors, this property could contribute to their adaptability.
DOI
### Enhanced optical confinement of dielectric nanoparticles by two-photon resonance transition
Aungtinee Kittiravechote, Anwar Usman, Hiroshi Masuhara and Ian Liau
Despite a tremendous success in the optical manipulation of microscopic particles, it remains a challenge to manipulate nanoparticles especially as the polarizability of the particles is small. With a picosecond-pulsed near-infrared laser, we demonstrated recently that the confinement of dye-doped polystyrene nanobeads is significantly enhanced relative to bare nanobeads of the same dimension. We attributed the enhancement to an additional term of the refractive index, which results from two-photon resonance between the dopant and the optical field. The optical confinement is profoundly enhanced as the half-wavelength of the laser falls either on the red side, or slightly away from the blue side, of the absorption band of the dopant. In contrast, the ability to confine the nanobeads is significantly diminished as the half-wavelength of the laser locates either at the peak, or on the blue side, of the absorption band. We suggest that the dispersively shaped polarizability of the dopant near the resonance is responsible to the distinctive spectral dependence of the optical confinement of nanobeads. This work advances our understanding of the underlying mechanism of the enhanced optical confinement of doped nanoparticles with a near-infrared pulsed laser, and might facilitate future research that benefits from effective sorting of selected nanoparticles beyond the limitations of previous approaches.
DOI
### Probing Photothermal Effects on Optically Trapped Gold Nanorods by Simultaneous Plasmon Spectroscopy and Brownian Dynamics Analysis
Daniel Andrén, Lei Shao, Nils Odebo Länk, Srdjan S. Aćimović, Peter Johansson, and Mikael Käll
Plasmonic gold nanorods are prime candidates for a variety of biomedical, spectroscopy, data storage, and sensing applications. It was recently shown that gold nanorods optically trapped by a focused circularly polarized laser beam can function as extremely efficient nanoscopic rotary motors. The system holds promise for applications ranging from nanofluidic flow control and nanorobotics to biomolecular actuation and analysis. However, to fully exploit this potential, one needs to be able to control and understand heating effects associated with laser trapping. We investigated photothermal heating of individual rotating gold nanorods by simultaneously probing their localized surface plasmon resonance spectrum and rotational Brownian dynamics over extended periods of time. The data reveal an extremely slow nanoparticle reshaping process, involving migration of the order of a few hundred atoms per minute, for moderate laser powers and a trapping wavelength close to plasmon resonance. The plasmon spectroscopy and Brownian analysis allows for separate temperature estimates based on the refractive index and the viscosity of the water surrounding a trapped nanorod. We show that both measurements yield similar effective temperatures, which correspond to the actual temperature at a distance of the order 10–15 nm from the particle surface. Our results shed light on photothermal processes on the nanoscale and will be useful in evaluating the applicability and performance of nanorod motors and optically heated nanoparticles for a variety of applications.
DOI
### β1-Integrin-Mediated Adhesion Is Lipid-Bilayer Dependent
Seoyoung Son, George J. Moroney, Peter J.Butler
DOI
### Enhancing Upconversion Fluorescence with a Natural Bio-microlens
Yuchao Li, Xiaoshuai Liu, Xianguang Yang, Hongxiang Lei, Yao Zhang, and Baojun Li
Upconversion fluorescence has triggered extensive efforts in the past decade because of its superior physicochemical features and great potential in biomedical and biophotonic studies. However, practical applications of upconversion fluorescence are often hindered by its relatively low luminescence efficiency (<1%). Here, we employ a living yeast or human cell as a natural bio-microlens to enhance the upconversion fluorescence. The natural bio-microlens, which was stably trapped on a fiber probe, could concentrate the excitation light into a subwavelength region so that the upconversion fluorescence of core–shell NaYF4:Yb3+/Tm3+ nanoparticles was enhanced by 2 orders of magnitude. As a benefit of the fluorescence enhancement, single-cell imaging and real-time detection of the labeled pathogenic bacteria, such as Escherichia coli and Staphylococcus aureus, were successfully achieved in the dark fields. This biocompatible, sensitive, and miniature approach could provide a promising powerful tool for biological imaging, biophotonic sensing, and single-cell analysis.
DOI
### Optical Force Enhancement Using an Imaginary Vector Potential for Photons
Lana Descheemaeker, Vincent Ginis, Sophie Viaene, and Philippe Tassin
The enhancement of optical forces has enabled a variety of technological applications that rely on the optical control of small objects and devices. Unfortunately, optical forces are still too small for the convenient actuation of integrated switches and waveguide couplers. Here we show how the optical gradient force can be enhanced by an order of magnitude by making use of gauge materials inside two evanescently coupled waveguides. To this end, the gauge materials inside the cores should emulate imaginary vector potentials for photons pointing perpendicularly to the waveguide plane. Depending on the relative orientation of the vector potentials in neighboring waveguides, i.e., pointing away from or towards each other, the conventional attractive force due to an even mode profile may be enhanced, suppressed, or may even become repulsive. This and other new features indicate that the implementation of complex-valued vector potentials with non-Hermitian waveguide cores may further enhance our control over mode profiles and the associated optical forces.
DOI
## Thursday, September 21, 2017
### Mitotic tethers connect sister chromosomes and transmit “cross-polar” force during anaphase A of mitosis in PtK2 cells
Matthew Ono, Daryl Preece, Michelle L. Duquette, Arthur Forer, and Michael W. Berns
Originally described in crane-fly spermatocytes, tethers physically link and transmit force between the ends of separating chromosomes. Optical tweezers and laser scissors were used to sever the tether between chromosomes, create chromosome fragments attached to the tether which move toward the opposite pole, and to trap the tethered fragments. Laser microsurgery in the intracellular space between separating telomeres reduced chromosome strain in half of tested chromosome pairs. When the telomere-containing region was severed from the rest of the chromosome body, the resultant fragment either traveled towards the proper pole (poleward), towards the sister pole (cross-polar), or movement ceased. Fragment travel towards the sister pole varied in distance and always ceased following a cut between telomeres, indicating the tether is responsible for transferring a cross-polar force to the fragment. Optical trapping of cross-polar traveling fragments places an upper boundary on the tethering force of ~1.5 pN.
DOI
### All-dielectric structure for trapping nanoparticles via light funneling and nanofocusing
Amir M. Jazayeri and Khashayar Mehrany
We propose a dielectric structure which focuses laser light well beyond the diffraction limit and thus considerably enhances the exerted optical trapping force upon dielectric nanoparticles. Although the structure supports a Fabry–Perot resonance, it actually acts as a nanoantenna in that the role of the resonance is to funnel the laser light into the structure. In comparison with the lens illuminating the structure, the proposed structure offers roughly a 10,000-fold enhancement in the trapping force—part of this enhancement comes from an 80-fold enhancement in the field intensity, whereas the remaining comes from a 130-fold enhancement in the normalized gradient of the field intensity (viz., the gradient of the field intensity divided by the field intensity). Also, the proposed structure offers roughly a 100-fold enhancement in the depth of the trapping potential. It is noteworthy that “self-induced back-action trapping” (SIBA), which has recently been the focus of interest in the context of optical resonators, does not take place in the proposed resonator. In this paper, we also point out some misconceptions about SIBA together with some hitherto unappreciated subtleties of the dipole approximation.
DOI
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BRIEF COMMUNICATIONS
Nutritional education and fruit and vegetable intake: a randomized community trial
Patricia Constante JaimeI; Flavia Mori Sarti MachadoII; Márcia Faria WestphalIII; Carlos Augusto MonteiroI
IDepartamento de Nutrição. Faculdade de Saúde Pública (FSP). Universidade de São Paulo (USP). São Paulo, SP, Brasil
IIEscola de Artes, Ciências e Humanidades. USP- Leste. São Paulo, SP, Brasil
IIIDepartamento de Prática de Saúde Pública. FSP-USP. São Paulo, SP, Brasil
Correspondence
ABSTRACT
We conducted a community trial-type intervention including a sample of 80 families living in a low income neighborhood in the municipality of Sao Paulo, Brazil, in 2004. The intervention relied on nutritional education to increase the participation of fruit and vegetables in the family diet, and was administered in the form of three two-hour meetings during three consecutive weeks. To evaluate the immediate impact of this educational intervention, families were randomly divided into two groups (intervention and control). Only the immediate impact of the intervention was evaluated, based on the participation of fruit and vegetables in the family's total food purchases in the months prior to and following the intervention. The comparison, which was favorable to the intervention group, showed a 2.9 percentage point increase (95% CI: 0.32; 5.39) in the proportion of total calories derived from fruit and vegetables.
Keywords: Food consumption. Fruit. Vegetables. Food and nutrition education. Intervention studies.
INTRODUCTION
Insufficient fruit and vegetable intake increases the risk of chronic non-communicable diseases such as heart disease and certain types of cancer and is one of the 10 major risk factors for death and disease worldwide.4 Insufficient intake is defined as less than 400 g per day, or about 7-8% of the caloric value of a 2,200 kcal/day diet.
It is estimated that fruit and vegetable intake in Brazil is currently less than half the recommended level, and is even more deficient among low-income families.3 We were unable to find studies in the literature that investigated such consumption. However, a study5 carried out in another developing country highlights the following factors as contributing to the low consumption of fruit and vegetables: high prices (compared to other food items and in the context of the family's income); inefficient production, distribution, and retail systems; and lack of knowledge among the population of the importance of these foods to health, especially in the case of vegetables.
The present communication reports on the results of a pilot intervention study, designed to evaluate the effect of interventions restricted to nutritional education on the consumption of fruit and vegetables among low-income families. This study is part of a large-scale research program aimed to investigate the reasons for low fruit and vegetable intake in Brazil.
METHODS
We carried out a pilot intervention study of the randomized community trial type, including a sample of 80 families living in two neighborhoods of the Grajaú district, in the Municipality of Sao Paulo, Brazil, in 2004. Grajaú, located at the extreme south of the Municipality, is the most populous (about 330,000 inhabitants) and one of the poorest among Sao Paulo's 96 districts,* with the ninth lowest human development index in the municipality (HDI: 0.419). The two neighborhoods in which the study was conducted Jardim Noronha and Jardim Moraes Prado are in the extreme south of the Grajaú district, and were selected because they are particularly deficient in urban infrastructure. Noteworthy in this sense is the precariousness of the fruit and vegetable retail system, which includes farmer's markets and grocery stores characterized by irregularity of supplies, poor quality of products, and distance from the families' homes.
The 80 households studied were randomly selected from a registry compiled by a social assistance non-governmental organization active in the area. These households were randomly allocated to two study groups of equal size (control and intervention). Sample size was calculated assuming the desire to identify, with 95% confidence and 90% statistical power, differences of at least 50% between groups in terms of the variation in fruit and vegetable intake before and after the intervention. Calculations were based on an estimated mean household availability of fruit and vegetables among the Brazilian population of 2.3% of total calories.3 Five families refused to participate in the study, leading to a total of 36 families in the intervention group and 39 in the control group.
One member of each household in the intervention group the one responsible for acquiring and preparing food for the household was invited to attend three meetings held at the community in successive weeks, each lasting for approximately two hours. The first meeting used a focal group technique, was diagnostic in character, and was designed to identify limitations and/or barriers to fruit and vegetable consumption in the community. The second meeting was motivational, and was formatted as a culinary workshop, designed to promote contact with different types of fruit and vegetables and which included the preparation and degustation of various recipes containing fruit and vegetables as primary ingredients. The third meeting was essentially informative, addressing issues of nutritional recommendations, health benefits associated with fruit and vegetables, ways to increase consumption of such foods, replacement of less healthy foods by fruit and vegetables, and the relationship between season, price, and quality of fruit and vegetables.
The participation of fruit and vegetables in the household diet was measured based on the percentage of calories from these items in the total calories acquired for household consumption during a one-month period. To this end, households in the intervention and control groups were requested to complete, in the months prior to and following intervention, food acquisition (by purchase or donation) questionnaires, similar to those used in Pesquisa de Orçamento Familiar (POF - Household Budget Surveys). In these questionnaires, which were formatted as expense notebooks, the household member responsible for food acquisition, after orientation and training, would enter, on a daily basis, all food acquisitions (including beverages) made during a one-month reference period. Food consumed outside the household was not investigated. A member of our fieldwork team visited households on a weekly basis to supervise notebook completion. Correction factors were used to exclude the inedible portions of different foods. Subsequently, using the Virtual Nutri** software, we calculated the total calories acquired during the month and the fraction of this total corresponding to fruit and vegetables (percentage of total calories from fruit and vegetables). For the socioeconomic characterization of the two groups, the household member responsible for food acquisition completed a structured questionnaire with questions regarding his of her schooling, monthly family income, and number of members and consumer goods in the household.
To asses the impact of the intervention, we initially determined the mean variation in the percentage of calories from fruit and vegetables between the months before and after the intervention in each group. The between-group difference in this variation (along with its 95% confidence interval) provided the crude effect of the intervention. The effect of the intervention adjusted for differences between groups at the baseline was determined using linear regression models. The outcome variable in these models was the variation in the percentage of total calories supplied by fruit and vegetables; the explanatory variable was having participated or not in the intervention; and the control variable was the percentage of fruit and vegetables among total calories in the month before the intervention. Data analysis was performed using SPSS software, version 11.
The study protocol was approved by the ethics committee of the Faculdade de Saúde Pública da USP.
RESULTS
Sociodemographic information confirmed the overall poverty of the studied population. There were no statistically significant differences between intervention and control groups. Monthly per capita income was R$178.50 for the intervention group and R$177.31 for the control group (p=0.97); 4.53 vs 4.41 (p=0.77), respectively for number of persons per household; 2.44 vs 2.23 (p=0.38), respectively, for number of consumer goods in the household; and 6.19 vs 5.79 years (p=0.61), respectively, for schooling level of the person responsible for food acquisition and preparation.
The Table presents, for both control and intervention groups, the participation of fruit and vegetables in the total calorie content of food acquired in the months before and after the intervention. Participation increased among intervention households (+1.63% and +0.41%, for fruit and vegetables, respectively) and decreased among controls (-1.18% and -0.01%, for fruits and vegetables, respectively). The crude effect of the intervention was equivalent to a 3.2 percentage point increase (95% CI 0.65; 5.80) in the percentage of calories from fruit and vegetables. The size of the effect decreased slightly after adjustment for differences at the baseline (2.86 percentage points; 95% CI: 0.32; 5.39).
DISCUSSION
The present results indicate that nutritional education interventions combining information and motivation to promote fruit and vegetable intake were successful in extremely poor settings. The interventions evaluated were aimed primarily at providing knowledge of the health benefits associated with fruit and vegetable intake and at increasing skills necessary for their introduction into the daily diet.
The indicative character of our results is due to the limitations which are inherent to the pilot nature of the study, especially the small number of families included, the lack of medium and long-term evaluations, and the indirect evaluation of intake based on food acquisition.
Confounder control was performed primarily through randomization, but also through adjustment for baseline household availability of fruit and vegetables. Potential confounders such as schooling and income were not included in the multiple regression analysis given that no substantial differences in sociodemographic characteristics was detected between the two groups.
Notwithstanding, the present results, despite their indicative character, contradict the notion that high prices and inefficient retail systems would be insurmountable obstacles for the promotion of fruit and vegetable consumption in impoverished settings, even within developed countries.2 A point in favor of the consistency of results is the fact that the intervention evaluated included not only cognitive elements, but behavioral elements as well, which could increase its chance of success.1
Future studies within the research program from which the present communication results will address the influence of family income and food price and supply on fruit and vegetable consumption in Brazil.
ACKNOWLEDGEMENTS
To Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) trainee Carina Weishaupt Vieira Lima and technical support grant recipients Danira Passos and Mariana Ferraz Duarte for their help with the focal groups, culinary workshops, and educational lectures administered to study participants.
REFERÊNCIAS
1. Devine CM, Farrell TJ, Hartman R. Sisters in health: experiential program emphasizing social interaction increases fruit and vegetable intake among low-income adults. J Nutr Educ Behav. 2005;37:265-70.
2. Drewnowski A, Darmon N, Briend A. Replacing fats and sweets with vegetables and fruits a question of cost. Am J Public Health. 2004;94:1555-9.
3. Levy-Costa RB, Sichieri R, Monteiro CA. Disponibilidade domiciliar de alimentos no Brasil: distribuição e evolução (1974-2003). Rev Saúde Pública. 2005;39(4):530-40.
4. World Health Organization. The world report 2002: reducing risks, promoting healthy life. Geneva: World Health Organization; 2002.
5. Monteiro CA. Setting up a fruit and vegetable promotion initiative in a developing country. In: WHO. Fruit and vegetable promotion initiative report of the meeting. Geneva; 2003.
Correspondence:
Patricia Constante Jaime
Departamento de Nutrição
Faculdade de Saúde Pública da USP
Av. Dr. Arnaldo, 715
01246-904 São Paulo, SP, Brasil
E-mail: constant@usp.br
Reviewed: 8/9/2006
Approved: 9/19/2006
Supported by MCT/MESA/CNPq/CT-Agronegócio 01/2003 grant (Process no. 503039/2003-9).
Study conducted at the Núcleo de Pesquisas Epidemiológicas em Nutrição e Saúde da Universidade de São Paulo (NUPENS/USP).
* Prefeitura da Cidade de São Paulo. Secretaria Municipal do Desenvolvimento, Trabalho e Solidariedade. Desigualdade em São Paulo: o IDH. São Paulo; 2002. Available from http://www2.uol.com.br/aprendiz/n_noticias/imprescindivel/id150802.doc [access in 17 Mai 2006]
** Virtual Nutrition, versão 1.0 [software em diskettes]. São Paulo: Departamento de Nutrição da Universidade de São Paulo; 1996.
Faculdade de Saúde Pública da Universidade de São Paulo São Paulo - SP - Brazil
E-mail: revsp@org.usp.br
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The TeX Catalogue OnLine, Entry for patchcmd, Ctan Edition
Change the definition of an existing command.
The package provides a command \patchcommand that can be used to add material at the beginning and/or the end of the replacement text of an existing macro. It works for macros with any number of normal arguments, including those that were defined with \DeclareRobustCommand.
The author is Michael J. Downes.
License: pd Version: 1.03 Catalogued: 2012-06-09
Powered by QArea Company and Bughuntress QA Lab
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# What is the value of b^2-4ac for the following equation: 2x^2+3x=-1?
Mar 4, 2018
${b}^{2} - 4 a c = 1$
#### Explanation:
$\text{for a quadratic equation in "color(blue)"standard form}$
•color(white)(x)ax^2+bx+c=0color(white)(x);a!=0
$\text{then the "color(blue) "discriminant } \Delta = {b}^{2} - 4 a c$
$\text{rearrange "2x^2+3x=-1" into standard form}$
$\Rightarrow 2 {x}^{2} + 3 x + 1 = 0 \leftarrow \textcolor{b l u e}{\text{in standard form}}$
$\text{with "a=2,b=3" and } c = 1$
$\Rightarrow {b}^{2} - 4 a c$
$= {3}^{2} - \left(4 \times 2 \times 1\right) = 9 - 8 = 1$
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SERVING THE QUANTITATIVE FINANCE COMMUNITY
Cuchulainn
Posts: 61185
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:
### Find inverse function
$1\,=\,\frac{d}{dy} \, F ( F\,^{\,-\,1} (y)\, =\, \frac{d F ( x )}{dx} \frac{d x }{d y}$The inverse of F is actually what you are trying to prove, and you seem to be using it as an assumption. This is circular reasoning? I think it invalidates the rest.
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget
Cuchulainn
Posts: 61185
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:
### Find inverse function
QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: list1Talking about existence of inverse function I meant that if $f \,(\, t\, ) > 0$ then the function$y \, = \, F ( x ) \, = \, \int_0^x f ( t ) \;dt$is a monotonic increasing function and $x \, \rightarrow \, y \,= \,F ( x )$ is one-to-one correspondence and inverse function $x \, = \, F ^{\,-\,1} (\,y\,)$ exists by definition.An counterexample; f(t) = t. If I did my calculus correctly I get x = + sqrt(2 y) or x = - sqrt(2 y), so the mapping is not bijective (1:1 onto). Furthermore, let y < 0 and you get pure imaginary complex numbers. There is (must be) a close connection between f(t) and y.OK, Define the (contraction) operator $T(x) = \, y \, - \, \int_0^x f ( t ) \;dt$ If we can prove $T(x) = \, x \$has a unique fixed point then we are finished. I reduce the scope by considering functions which are essentially like probability pdf (less than 1 in value);The steps are:1. We can prove that T is a contraction by proving that its derivative is less than 1 in absolute value (use the Mean Value Theorem for derivatives)2. Us the Banach-fixed point theorem to prove $T(x) = \, x \$ has a unique fixed point.3. And also we can define an iterative scheme to solve $x_{n+1} = \ T(x_n)$ as I have posted on Numerics forum and here4. It can be computed using fixed point, Newton, ODE, erf_inv etc.QED 5. I leave it as an exercise to find the contraction mapping when the values of f are in the interval [a,b]. Then the mapping will be $T(x) = \, y/m \, - \, \int_0^x f ( t )/m \;dt$for some constant m to ensure that T remains a contraction.6. No closed, exact solution is possible. Looking back: we have used real and functional analysis to prove existence and uniqueness results and then numerical analysis to compute a solution.
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget
list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm
### Find inverse function
1) And now what if f(t) = exp(-t^2) and y in (0,1). Is there an exact x? 2) And take f(t) = t.i.e. can you x without the following integral because the integral does not get us anywhere?$x ( y ) \, = \, \int_0^y f ^{\,-\,1}( u )\, du$3) BTW why is your f(t) monotonic increasing? You are excluding all the pdfs of statistics?(?)===============================================================3) it was made an assumption that f ( t ) $\ge$ 0 and therefore f ( t ) can be equal to 1 + sin t , t $\ge$ 0 or any density.2) an answer was given in Sun Dec 13, 15 06:40 PM. btw "can you x without ..." probably sound worse than my "multivariate ..." though it can be understand. 1) the answer is given in the message: Mon Dec 14, 15 01:29 AM. In particular inverse function is defined as $x\,=\, \sqrt{\,-\,\ln\,y}$. Here $0 \,< \,y\,< 1$ and for any y from the domain of the inverse function we can find one value of x. For example if $y\, =\, e\,{-\,1}$ then x = 1.
Cuchulainn
Posts: 61185
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:
### Find inverse function
QuoteOriginally posted by: list11) And now what if f(t) = exp(-t^2) and y in (0,1). Is there an exact x? 2) And take f(t) = t.i.e. can you x without the following integral because the integral does not get us anywhere?$x ( y ) \, = \, \int_0^y f ^{\,-\,1}( u )\, du$3) BTW why is your f(t) monotonic increasing? You are excluding all the pdfs of statistics?(?)===============================================================3) it was made an assumption that f ( t ) $\ge$ 0 and therefore f ( t ) can be equal to 1 + sin t , t $\ge$ 0 or any density.2) an answer was given in Sun Dec 13, 15 06:40 PM. btw "can you x without ..." probably sound worse than my "multivariate ..." though it can be understand. 1) the answer is given in the message: Mon Dec 14, 15 01:29 AM. In particular inverse function is defined as $x\,=\, \sqrt{\,-\,\ln\,y}$. Here $0 \,< \,y\,< 1$ and for any y from the domain of the inverse function we can find one value of x. For example if $y\, =\, e\,{-\,1}$ then x = 1.I don't think so. You seem to have missed 1) again.Anyways, I am closing now.
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget
list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm
### Find inverse function
QuoteOriginally posted by: Cuchulainn$1\,=\,\frac{d}{dy} \, F ( F\,^{\,-\,1} (y)\, =\, \frac{d F ( x )}{dx} \frac{d x }{d y}$The inverse of F is actually what you are trying to prove, and you seem to be using it as an assumption. This is circular reasoning? I think it invalidates the rest.A problem is to find inverse function. I a function is given by an integral its inverse can be found by the relationship between derivatives of function and its inverse. The derivatives of the inverse in more complex case can be say an a differential equation too, ie knowledge of derivative leads to integration. Though writing derivative one can state in particular that inverse function is the solution of the ode.
list1
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### Find inverse function
QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: list1Talking about existence of inverse function I meant that if $f \,(\, t\, ) > 0$ then the function$y \, = \, F ( x ) \, = \, \int_0^x f ( t ) \;dt$is a monotonic increasing function and $x \, \rightarrow \, y \,= \,F ( x )$ is one-to-one correspondence and inverse function $x \, = \, F ^{\,-\,1} (\,y\,)$ exists by definition.An counterexample; f(t) = t. If I did my calculus correctly I get x = + sqrt(2 y) or x = - sqrt(2 y), so the mapping is not bijective (1:1 onto). Furthermore, let y < 0 and you get pure imaginary complex numbers. There is (must be) a close connection between f(t) and y.OK, Define the (contraction) operator $T(x) = \, y \, - \, \int_0^x f ( t ) \;dt$ If we can prove $T(x) = \, x \$has a unique fixed point then we are finished. I reduce the scope by considering functions which are essentially like probability pdf (less than 1 in value);The steps are:1. We can prove that T is a contraction by proving that its derivative is less than 1 in absolute value (use the Mean Value Theorem for derivatives)2. Us the Banach-fixed point theorem to prove $T(x) = \, x \$ has a unique fixed point.3. And also we can define an iterative scheme to solve $x_{n+1} = \ T(x_n)$ as I have posted on Numerics forum and here4. It can be computed using fixed point, Newton, ODE, erf_inv etc.QED 5. I leave it as an exercise to find the contraction mapping when the values of f are in the interval [a,b]. Then the mapping will be $T(x) = \, y/m \, - \, \int_0^x f ( t )/m \;dt$for some constant m to ensure that T remains a contraction.6. No closed, exact solution is possible. Looking back: we have used real and functional analysis to prove existence and uniqueness results and then numerical analysis to compute a solution.I do not see a question. Btw you talk about operator T and use functional analysis terminology but T ( x ) is a function of scalar or vector variable x.
Last edited by list1 on December 14th, 2015, 11:00 pm, edited 1 time in total.
Cuchulainn
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### Find inverse function
The problem is solved.
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Jean Piaget
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### Find inverse function
QuoteOriginally posted by: list1QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: list1Talking about existence of inverse function I meant that if $f \,(\, t\, ) > 0$ then the function$y \, = \, F ( x ) \, = \, \int_0^x f ( t ) \;dt$is a monotonic increasing function and $x \, \rightarrow \, y \,= \,F ( x )$ is one-to-one correspondence and inverse function $x \, = \, F ^{\,-\,1} (\,y\,)$ exists by definition.An counterexample; f(t) = t. If I did my calculus correctly I get x = + sqrt(2 y) or x = - sqrt(2 y), so the mapping is not bijective (1:1 onto). Furthermore, let y < 0 and you get pure imaginary complex numbers. There is (must be) a close connection between f(t) and y.OK, Define the (contraction) operator $T(x) = \, y \, - \, \int_0^x f ( t ) \;dt$ If we can prove $T(x) = \, x \$has a unique fixed point then we are finished. I reduce the scope by considering functions which are essentially like probability pdf (less than 1 in value);The steps are:1. We can prove that T is a contraction by proving that its derivative is less than 1 in absolute value (use the Mean Value Theorem for derivatives)2. Us the Banach-fixed point theorem to prove $T(x) = \, x \$ has a unique fixed point.3. And also we can define an iterative scheme to solve $x_{n+1} = \ T(x_n)$ as I have posted on Numerics forum and here4. It can be computed using fixed point, Newton, ODE, erf_inv etc.QED 5. I leave it as an exercise to find the contraction mapping when the values of f are in the interval [a,b]. Then the mapping will be $T(x) = \, y/m \, - \, \int_0^x f ( t )/m \;dt$for some constant m to ensure that T remains a contraction.6. No closed, exact solution is possible. Looking back: we have used real and functional analysis to prove existence and uniqueness results and then numerical analysis to compute a solution.I do not see a question. Btw you talk about operator T and use functional analysis terminology but T ( x ) is a function of scalar or vector variable x.Read the statement of the theorem. It talks about metric spaces, not scalars but in your case the metric space is the real line with any distance metric you like.Clear?/
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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### Find inverse function
OK, Define the (contraction) operator $T(x) = \, y \, - \, \int_0^x f ( t ) \;dt$ If we can prove $T(x) = \, x \$has a unique fixed point then we are finished. I reduce the scope by considering functions which are essentially like probability pdf (less than 1 in value);-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------The setting of the problem does not clear. If 'operator' T ( x ) is a function of x then it is not clear why we can call it contractor. To say that we should have | T ( x ) | < 1. For example if y = 2 and f= 0 then operator does not a contractor. You might wish to say that operator T depends on parameter y T ( y ) and transform functions f ( x ) , x $\, \in\,$ [0 , 1 ] f $\, \in\,$ M to M. To state that it should be defined a norm and a complete normed space. I also do not know what is a question or it just a comment
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### Find inverse function
QuoteOriginally posted by: list1OK, Define the (contraction) operator $T(x) = \, y \, - \, \int_0^x f ( t ) \;dt$ If we can prove $T(x) = \, x \$has a unique fixed point then we are finished. I reduce the scope by considering functions which are essentially like probability pdf (less than 1 in value);-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------The setting of the problem does not clear. If 'operator' T ( x ) is a function of x then it is not clear why we can call it contractor. To say that we should have | T ( x ) | < 1. For example if y = 2 and f= 0 then operator does not a contractor. You might wish to say that operator T depends on parameter y T ( y ) and transform functions f ( x ) , x $\, \in\,$ [0 , 1 ] f $\, \in\,$ M to M. To state that it should be defined a norm and a complete normed space. I also do not know what is a question or it just a commentWell, no.Wrong definition of contraction and wrong example. As mentioned, first do some reading and then post.Do you understand? Your posts are confused and incorrect. How come you misintrepret/misunderstand what people write?
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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### Find inverse function
Here is a quiz:Prove the following are contractions1. x = cos(x)2. x = (x + 2/x)/2Stick to metric spaces, don't worry about Banach spaces for the moment. Once contraction is clear, some meaningful discussion becomes possible. ==edit I get the impression list is talking about inverse function theorem
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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### Find inverse function
QuoteOriginally posted by: CuchulainnHere is a quiz:Prove the following are contractions1. x = cos(x)2. x = (x + 2/x)/2Stick to metric spaces, don't worry about Banach spaces for the moment. Once contraction is clear, some meaningful discussion becomes possible. ==edit I get the impression list is talking about inverse function theorem*) If one looks at the contractions reference the we see that it relates to a map which is realized by f ( x) and there is a metric d ( x , y ). In 1.-2. we have two equations and it is not clear following the reference what is f and what is d. Also one should define closed area when map is contraction with respect to chosen metric d. Natural domain of f could be broader than area where f is a contractor.**) Of course I talked about inverse function and after my second question I asked how to change 'inverse number' in title on 'inverse function' and I got there an advice.
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### Find inverse function
For 1, see thisFor my example 2, it is the fixed point form for sqrt(2) known to the ancient Babylonians.The ideal of trying to find an exact formula for the inverse (quantile) is a waste of time IMO. I have shown that the solution exists, is unique and 4 ways to construct it. But I suppose that's the never-ending discussion between pure and constructive mathematics.
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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### Find inverse function
QuoteOriginally posted by: CuchulainnFor 1, see thisFor my example 2, it is the fixed point form for sqrt(2) known to the ancient Babyloninansin this thei no words similar to contractor. two curves intersected in one point and how does this graph is related to contractor notion?My question was how 1-2 are related to contractors?
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### Find inverse function
QuoteOriginally posted by: list1QuoteOriginally posted by: CuchulainnFor 1, see thisFor my example 2, it is the fixed point form for sqrt(2) known to the ancient Babyloninansin this thei no words similar to contractor. two curves intersected in one point and how does this graph is related to contractor notion?My question was how 1-2 are related to contractors?First, it is called contraction. Second, you need to find it, it is easy. It is high-school piddling calculus.hintfor x = g(x) we need to have |g'(x)| < 1.now let g(x) = cos(x). BTW this is a contractor
Last edited by Cuchulainn on December 14th, 2015, 11:00 pm, edited 1 time in total.
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numtheory/invcfrac(deprecated) - Help
# Online Help
###### All Products Maple MapleSim
Home : Support : Online Help : numtheory/invcfrac(deprecated)
numtheory
invcfrac
convert a simple periodical continued fraction expansion to a quadratic surd
Calling Sequence invcfrac(cf)
Parameters
cf - simple periodical continued fraction expansion with its pre-period and period (in either list or fraction form)
Description
• Important: The numtheory[invcfrac] command has been deprecated. Use the superseding command NumberTheory[ContinuedFraction][Value] instead.
• The invcfrac function returns a quadratic surd which has a simple periodical continued fraction expansion cf.
• This function is part of the numtheory package, and so can be used in the form invcfrac(..) only after performing the command with(numtheory). The function can always be accessed in the long form numtheory[invcfrac](..).
Examples
> $\mathrm{with}\left(\mathrm{numtheory}\right):$
> $f≔\mathrm{cfrac}\left({31}^{\frac{1}{2}},\mathrm{periodic}\right)$
${f}{:=}{5}{+}\frac{{1}}{{1}{+}\frac{{1}}{{1}{+}\frac{{1}}{{3}{+}\frac{{1}}{{5}{+}\frac{{1}}{{3}{+}\frac{{1}}{{1}{+}\frac{{1}}{{1}{+}\frac{{1}}{{10}{+}\frac{{1}}{{1}{+}\frac{{1}}{{1}{+}\frac{{1}}{{3}{+}\frac{{1}}{{5}{+}\frac{{1}}{{3}{+}\frac{{1}}{{1}{+}\frac{{1}}{{1}{+}\frac{{1}}{{10}{+}{\mathrm{...}}}}}}}}}}}}}}}}}}$ (1)
> $\mathrm{invcfrac}\left(f\right)$
$\sqrt{{31}}$ (2)
> $g≔\mathrm{cfrac}\left(\frac{3}{5}+{29}^{\frac{1}{2}},\mathrm{periodic},\mathrm{quotients}\right)$
${g}{:=}\left[\left[{5}\right]{,}\left[{1}{,}{66}{,}{2}{,}{2}{,}{5}{,}{10}{,}{1}{,}{1}{,}{2}{,}{2}{,}{3}{,}{2}{,}{1}{,}{1}{,}{1}{,}{268}{,}{1}{,}{1}{,}{1}{,}{2}{,}{3}{,}{2}{,}{2}{,}{1}{,}{1}{,}{10}{,}{5}{,}{2}{,}{2}{,}{66}{,}{1}{,}{9}{,}{1}{,}{3}{,}{1}{,}{1}{,}{1}{,}{8}{,}{1}{,}{1}{,}{1}{,}{3}{,}{1}{,}{9}\right]\right]$ (3)
> $\mathrm{invcfrac}\left(g\right)$
$\frac{{3}}{{5}}{+}\sqrt{{29}}$ (4)
See Also
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# Using Wilcoxon Ranksum text with Equal sample medians [duplicate]
We have two independent samples, with skewed distribution from two different populations x, and y. When we compute the summary for these numeric vectors, we get following output,
summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.000 1.000 1.000 2.219 3.000 116.000
length(x)=25312
summary(y)
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.000 1.000 1.000 1.129 1.000 19.000
length(y)=49832
As seen, they have equal sample medians with this data. But the intuition is that median for population1(from which sample x is taken) should be greater than median for population2(from which sample y is taken). Though sample medians are same, will it be worthwhile running Wilcoxon Ranksum test for this example? And, can I also get help with the syntax for one-sided Wilcoxon(in R). To be more precise, when we write
wilcox.test(x,y,alternative="greater",paired = FALSE)
Does this mean that median of x is greater than median of y or the other way around? Help is appreciated.
• It is great that you posted summary statistics. Unfortunately, R neglects to report the sizes of the datasets: could you tell us what they are? – whuber Dec 30 '19 at 20:10
• Please see my edits.Size is added for two samples.thanks – jayant Dec 30 '19 at 20:14
• Those sample sizes are so large you do not need a formal test. You ought to progress immediately to a more advanced stage of analysis where you characterize the differences between the distributions. Start with a QQ plot of the two datasets. – whuber Dec 30 '19 at 20:17
• The Wilcoxon test doesn't add any information that that isn't already obvious from the summary statistics (and will be made abundantly clear with a QQ plot). Although you could apply it (using a suitable adjustment for the huge numbers of ties), why bother? – whuber Dec 30 '19 at 20:27
• A qqplot comparing the 2 distributions directly might be more informative than two separate qqnorm plots. See the last "Usage" example on the R qqnorm help page. – EdM Dec 30 '19 at 21:44
• There is a controversy around the statement that Wann-Whitney's Null hypothesis is the 2 populations [distributions] are equal. This null is more apt for Kolmogorov-Smirnov. Two perfectly identical shape - say, normal - populations with the same centre but different variances won't be distinguished by MW. MW is all about stochastic dominance vs stochastic balance. Or, in other words, it is about the (in)equality of the "location of gravity". – ttnphns Dec 30 '19 at 22:12
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# Measurement of $d\sigma/dy$ of Drell-Yan $e^+e^-$ pairs in the $Z$ Mass Region from $p\bar{p}$ Collisions at $\sqrt{s}=1.96$ TeV
Un-Ki Yang, T Aaltonen
Research output: Contribution to journalArticlepeer-review
Original language English 232-239 Physics Letters. Section B: Nuclear, Elementary Particle and High-Energy Physics 692 Published - 31 Aug 2010
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# Gravity as function of density
1. Aug 12, 2011
### Nikitin
Hi, I'm just playing a bit with the numbers... The gravitational strength of a planet really depends on its density, but I'm wondering how I'm supposed to create a formula where gravity is the function of a planet's average density..
M= mass of planet. G= Newton's gravitational constant. r= average radius of planet.
Strength of a gravitational field: M*G/r^2 = g. Density of a planet: M/[(4/3)pi*r^3] = density
How can I create a function for g where density is the variable?
The closest I'm coming to is g(density)= (density*4*pi/3)*G*r, and this is by incerting the M=density*[(4/3)pi*r^3] into the strength of a gravitational field formula.
And all help is appreciated ;) Thank you in advance
Last edited: Aug 12, 2011
2. Aug 12, 2011
### cragar
3. Aug 12, 2011
### Pengwuino
Yah you can't really do that. Actually, you'll never be able to do that with anything built using calculus. Something like a density is basically giving you infinitesimal information at some infinitesimal volume element at some point in space. It's impossible to expand this to some value that normally requires an integration over the whole or even part of the space in question.
Edit: The formula you got had a huge assumption: constant density. This works because for constant functions, what happens at one point happens at all points and no integration is required (or well, actually your "M" was an implied integration!)
Last edited: Aug 12, 2011
4. Aug 12, 2011
### Staff: Mentor
That looks fine to me. What don't you like about that formula?
5. Aug 13, 2011
### Superstring
That gives you g at any point INSIDE of the planet, assuming constant density. Remember that when you're outside of a spherical object (of uniform density) the mass of the entire thing can be treated as if it were all concentrated at a single point at its center, so once you're outside of the thing the density is no longer important. For example, if the sun were replaced by an object of equal mass but the size of the moon, our orbit around the object would be unaffected.
So, with constant density you get the following:
$$g=\begin{cases} - \frac{4}{3} \pi G \rho r & \text{ if } r\leq R \\ -GM/r^2 & \text{ if } r\geq R \end{cases}$$
Where R is the radius of the planet, ρ is the density, M is the mass of the planet, and r is your distance from the center.
If you allow the density to vary, then things get a lot more complicated. In rectangular coordinates, the density is a function of position: ρ(x,y,z). You then get:
$$g =- \frac{G}{r^2}\int_V \rho~dV$$
where $V=\frac{4}{3}\pi r^3$.
6. Aug 14, 2011
### Nikitin
hmm guys after checking my question properly I find that it is kind of silly. Density doesn't really say anything about the mass of a sphere, you'd need to know both the density and radius to find the gravity on the surface.
Afterall, the gravitational pull of a sphere 2 centimetres in diameter would be pretty meagre if the density/mass isn't high enough.
Why would the formula give the gravity inside the sphere? I'm not quite sure.. it's the same formula as newton's just changed a little. And newton's formula doesn't really work for inside a sphere, does it??
And a final thing: the density of most plantets is pretty constant is it not? The gravitational pull is almost a constant G anywhere on earth's surface.
Last edited: Aug 14, 2011
7. Aug 14, 2011
### D H
Staff Emeritus
Not at all! The density of the Earth varies from 2.6 gm/cc at the surface to 13.1 gm/cc at the center. And that's for a rocky planet. For the gas giant planets the range is much, much greater.
8. Aug 14, 2011
### Superstring
Because the mass that is pulling on you is a function of how far you are from the center. If you are at a distance r from the planet's center, the only mass that effects you gravitationally is the mass contained by a (imaginary) spherical surface of radius r. The rest of the mass of the planet (any mass "above" this distance r) has no effect on you at all because everything cancels out.
9. Aug 14, 2011
### OnlyMe
Is the following wrong? (as a whole please, the general picture.)
The idea that gravitational force is somehow proportional to mass density and distance (or r) seems obvious and yet to some extent seems flawed.
The relationship would be better described as, gravitational force is proportional to the absolute amount of mass and the distance from the center of mass to the point where the force is to be determined.
As example, consider the density of the Earth as compared to the density of Jupiter. The Earth has a higher mass density than Jupiter, while Jupiter has a far greater volume of mass by far. If the Earth's density were suddenly changed to be equivalent to that of Jupiter. It radius would increase and the gravitational force at its new surface would decrease.
While it is true that given a defined mass, the gravitational force at its surface will be proportional to its density, the gravitational force itself appears to be a general function of the absolute mass involved rather than its density.
10. Aug 14, 2011
### Pengwuino
Upon integration it tells you everything about the mass ;)
Sure it does, the details just weren't given.
You can calculate the gravitational acceleration given by a the Poisson equation $\nabla \dot \bf{g} = 4\pi G\rho$.
Assuming a spherically symmetric mass distribution, you can use gauss' law and the problem becomes simple. Due to gauss' law, integrating over the volume gives you
$4\pi {\bf g} r^2 = 4\pi G \int_0^r \rho(r') r'^2 sin(\theta) dr' d\theta d\phi$
where you integrate only up to the radius that you're testing the gravity at. So again, assuming spherical symmetry, the integration becomes
${\bf g} ={{4\pi G}\over{r^2}} \int_0^r\rho(r') r'^2 dr'$
This is typically what you're left with. If $\rho$ is constant, that is $\rho = {{3M}\over{4\pi R^3}}$ where big R is the total radius of your constant density sphere, what you get is what Nikitin wrote down. Notice that as long as the density is spherically symmetric, it doesn't actually matter what is happening at a radius beyond your test mass!
If your spherical symmetry is broken, Gauss' law doesn't allow you to do the nice little trick that simplifies this tremendously.
EDIT: Ok I think I got everything right.... I should be doing this on pencil adn paper first before posting!
11. Aug 15, 2011
### Nikitin
allright, thanks people but there's this:
ouch! yeh of course.. was late yesterday
But what i was thinking, was the average density of a block of earth below my feet going down to the earth-centre would be circa the same if I was living in Greenland as if I was living in Egypt.
So what is the problems about using density to calculate gravity (outside the planet)? other than the fact that it is impractical of course.
12. Aug 15, 2011
### ZapperZ
Staff Emeritus
One of my old professors has the propensity to tell her students "You're Working Too Hard!". And that's what I will tell you.
Outside the planet (assuming spherical symmetry), the ONLY thing you care about is the total mass of that planet, nothing else. So why should you care about the density, considering that knowing that fact will STILL not give you the gravity without knowing the volume (which ultimately leads you to the total mass anyway!). So your insistence to want to do it THIS way is puzzling, since it is just more work for not a whole lot of gain.
Zz.
13. Aug 15, 2011
### Pengwuino
Any real planet's density function could be ridiculously complicated depending on how accurate you want to make it. You either have to make the assumption the density is constant, the density is very approximately modeled by some function, or the density is completely indescribable using a nice, smooth mathematical function.
The 1st is trivial, the second is unrealistic and unable to do what you want to do, the third requires numerical integration. You are not going to be able to determine a way to basically say ${\bf g} = f(\rho)$
What you're basically trying to do is ask for a property, the gravitational acceleration, that requires knowledge of the whole system, by using just the density, which unless you integrate, can only give you information about what is happening at your test mass's position.
14. Aug 15, 2011
### Nikitin
ok thx
15. Aug 15, 2011
### D H
Staff Emeritus
The Preliminary Earth Reference Model gives a picture of the interior of the Earth in terms of density and gravitational acceleration. See http://geophysics.ou.edu/solid_earth/prem.html [Broken].
However, that picture isn't very useful for determining gravitation at or above the Earth's surface. The reverse mapping is extremely useful. Local variations in gravitation can point to interesting geological formations. For example, gravitation above a salt dome will be anomalously low due to the low density of salt. Since salt domes are highly correlated with natural gas and oil deposits, finding such gravity anomalies is of high interest to petroleum engineers.
Dense rock such as basalt will result in anomalously high gravitation. Here's a false color gravity anomaly map of Iowa:
That ridge running diagonally across Iowa is a part of the (failed) Midcontinent Rift System. 1.1 billion years ago the North American plate started to split in half. That diagonal ridge results from the basalt that filled the rift as it was forming.
Exactly. And working in the wrong direction to boot. We don't have a good enough model of density to do what the OP wants. It's rather hard (rather impossible) to get sensors deep into the interior of the Earth. Scientists have to infer interior density rather measure it. Models of the Earth's gravity field are one of the tools that scientists use.
And if you don't assume spherical symmetry (which you had better not assume if you want to model the orbits of artificial satellites about the Earth), a density model won't be of much help. However, a model of the Earth's non-spherical gravity field is one of the things that does lead to a density model.
Last edited by a moderator: May 5, 2017
16. Aug 15, 2011
### OnlyMe
bold emphasis mine
Here the density model is really only an indicator of the unequal distribution of the total mass. Density itself should be considered only to the extent that it is associated with the total mass and the distribution of total mass. Mass distribution or variations in density affect where the center of mass is and where it appears to be from different positions in an orbit. But the density itself is only an indicator of variations in distribution and where one is concerned with the force of gravitation at or near the surface of a gravitational body and then only because it is an indicator of the total mass involved. Correct?
It took me a long time to understand that density was only, in the case of gravity, an indicator of total mass and it had no real direct connection to the force of gravity aside from the total mass involved and how it is distributed in rocky or dynamic gaseous gravitational objects.
First impressions are that density is directly involved. As the density of a given volume of mass is increased the gravitational force its surface also increase, but so does the distance between that surface and the center of mass. Again, the density is secondary to the total mass involved which has not changed.
I may not be stating this clearly but it is accurate, is it not?
17. Aug 15, 2011
### Staff: Mentor
The total mass is the only relevant parameter for a spherically symmetric density. For density that is not spherically symmetric the field cannot simply be modeled by the total mass.
IMO, the density is primary, and it is only in the special case of spherical symmetry that the density can be neglected in favor of the total mass.
http://en.wikipedia.org/wiki/Gauss'_law_for_gravity#Differential_form
Last edited: Aug 15, 2011
18. Aug 15, 2011
### John15
I may be wrong but when I was looking at black holes some time ago it seemed to me that density was the most important, re shwartzchild radius which seems to mean that any mass can cause a black hole if compressed enough, including the earth resulting in a gravitational pull where the escape velocity exceeds the speed of light. So as density decreases you are pushed further away from the center resulting in lower gravitational force at the surface and so lower escape velocities. Although gravity does not seem to be stated as such.
19. Aug 15, 2011
### D H
Staff Emeritus
That is correct for an object with a spherical density distribution (i.e., density is a function of radial distance from the center of mass only). It is incorrect if the mass does not have a spherical density distribution.
I agree with you that density variations must be modeled if you are going to use Gauss' law to compute gravitation for a body that does not have a spherically symmetric density function. I disagree in the sense that nobody (practically nobody) uses Gauss' law to compute gravitation toward something like a planet or a star. Density models with sufficient accuracy to properly represent orbits just don't exist, and even if one did exist it would be too hard to use.
What is typically used instead is a spherical harmonics model of gravitation for planets and the Sun. Orbits can be computed directly from the spherical harmonics model, and the spherical harmonics model can be fine-tuned to yield a best fit to observed data. There is no need to go to a density model.
That said, spherical harmonics models are not such a good fit for things that aren't particularly spherical such as lumpy, misshapen potatoes or asteroids. There is some interesting work on developing gravitational models for asteroids based on observed shape. The shape model combined with an assumed density model (typically constant density, but not always) yields a gravitational model. Gauss' law is not used directly in the sense that one integrates over the resultant volume. It is used indirectly to yield the equation for the gravitational acceleration toward a pyramid.
20. Aug 15, 2011
### OnlyMe
Dale, I don't believe we are really in any functional disagreement. Density distribution does play a role in the shape of a gravitational field. I just look at density in this circumstance as an indicator of the distribution of the involved mass.
I see nothing wrong with this. BHs are generally treated as uniform in mass distribution. Essentially as perfect fluids or condensed matter models, in which the distribution of mass is uniform relative to an outside frame of reference. The second, example above just assumes the same conditions that the involved mass is uniform in distribution.
In both cases, mass density is proportional to the force of gravitation at the surface of the involved mass, assuming a spherical gravitational source.
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Where are the fake units 'character' (ch) for indentation and 'line' for spacing specified?
In specific the 'character' used as a unit should be judged unusable for decades now. The help doesn' tell anything about the meaning. From experimenting I guess 1 ch = 0.3704 cm, and thus a constant length. Is this correct?
Is 'line' actually used in the sense of "the maximimum height of any part of the line immediately above, including standard descender and ascender height", or something like that? Is it simply "left to implementation"? Is it a constant length again?
Respective question for 'Column Width' and indentations in Calc
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# Iterative square root? sqrt(2+sqrt(2+sqrt(...
1. ### Damidami
94
The other day I was playing with my calculator and noticed that
$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} \approx 2$$
But, what is that kind of expression called? How does one justify that limit?
And, to what number exactly does converge, for example:
$$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}} \approx 1.6161$$
$$\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+...}}}} \approx 2.3027$$
Another question. Considering real $$x>1$$, we have:
$$\Gamma(x) - x^1 = 0$$ then $$x \approx 2$$
But how does one justify that? And what are the exact values of these functions:
$$\Gamma(x) - x^2 = 0$$ then $$x \approx 3.562382285390898$$
$$\Gamma(x) - x^3 = 0$$ then $$x \approx 5.036722570588711$$
$$\Gamma(x) - x^4 = 0$$ then $$x \approx 6.464468490129385$$
Thanks,
Damián.
2. ### slider142
956
They are called Nested Radicals. There are references in the link. I am unfortunately not familiar with their theory.
3. ### Damidami
94
Thank you slider142!
Does anyone know about my second question? Or any further references?
Thanks,
Damián.
4. ### Tac-Tics
810
$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} = x$$
For something like this, you can rewrite the equation as
$$\sqrt{2+x} = x$$
And then the infinite equation is captured in a finite form. From there, you simply square both sides.
$$2+x = x^2$$
And solve for x.
But I don't know much more than that! Don't forget that square-roots are non-negative.
5. ### Elucidus
286
For nested radicals of the form
$$\sqrt{a + \sqrt{a + \sqrt{a + \dots}}}$$
using the trick
$$\sqrt{a + x} = x$$
works very well. Two roots will emerge, but only one is positive (the other is extraneous).
In the case when a = 2, x = 2.
When a = 1, then x = $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618034$ a.k.a. the golden ratio.
As to the second question regarding the Gamma function, I'm not sure much theory is available.
--Elucidus
6. ### Hurkyl
16,090
Staff Emeritus
For a dose of rigor -- we have to be sure the limit really exists before we can compute it with such tricks!
In this case, it's easy: the value is the limit of an increasing sequence, and limits of increasing, extended real number-valued sequences always exist.
It's important to notice that extended real numbers come into play here! The equation
2 + x = x²
has three relevant solutions: -1, 2, and $+\infty$. We know the limit exists, so it has to have one of those three values. It's easy to rule out -1, but more work is needed to decide between 2 and $+\infty$.
7. ### Elucidus
286
If we examine the sequence $\{a_n\}_{n=0}^{\infty}$ when $a_0 = \sqrt{2}$ and
$$a_{n+1}=\sqrt{2+a_n}$$
Then it is possible to show by induction that $a_n \leq 2$ for all n so the $+\infty$ case is impossible.
But you are correct, this possibility does need to be ruled out, Hurkyl.
--Elucidus
Last edited: Sep 8, 2009
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# Solving Least-Squares Regression with Missing Data
Contributed by Alex Williams
I recently got interested in figuring out how to perform cross-validation on PCA and other matrix factorization models. The way I chose to solve the cross-validation problem (see my other post) revealed another interesting problem: how to fit a linear regression model with missing dependent variables. Since I did not find too many existing resources on this material, I decided to briefly document what I learned in this blog post.
### The Problem
We want to solve the following optimization problem, which corresponds to least-squares regression with missing data:
The columns of the matrix $\mathbf{B}$ hold different dependent variables. The columns of the matrix $\mathbf{A}$ hold independent variables. We would like to find the regression coefficients, contained in $\mathbf{X}$, that minimize the squared error between our model prediction $\mathbf{A} \mathbf{X}$ and the dependent variables, $\mathbf{B}$.
However, suppose some entries in the matrix $\mathbf{B}$ are missing. We can encode the missingness with a masking matrix, $\mathbf{M}$. If element $B_{ij}$ is missing, we set $M_{ij} = 0$. Otherwise, we set $M_{ij} = 1$, meaning that element $B_{ij}$ was observed. The “$\circ$” operator denotes the Hadamard product between two matrices. Thus, in equation 1, the masking matrix $\mathbf{M}$ has the effect of zeroing out, or ignoring, the reconstruction error wherever $\mathbf{B}$ has a missing element.
Visually, the problem we are trying to solve looks like this:
Least squares with missing data. Black squares denote missing data values. Note that we only consider missing dependent variables in this post. The masking matrix $\mathbf{M}$ would have zeros along the black squares and ones elsewhere.
Though it is not entirely correct, you can think of the black boxes in the above visualization as NaN entries in the data matrix. The black boxes are not zeros. If we replaced the NaNs with zeros, we obviously get the wrong result. The missing datapoints could be any (nonzero) value!
The optimization problem shown in equation 1 is convex, and it turns out we can derive an analytic solution (similar to least-squares in the abscence of missing data). We can differentiate the objective function with respect to $\mathbf{X}$ and set the gradient to zero. Solving the resulting expression for $\mathbf{X}$ will give us the minimum of the optimization problem. After some computations (see the appendix for details) we arrive at:
which we’d like to solve for $\mathbf{X}$. Computing the right hand side is easy, but the Hadamard product mucks things up on the left hand size. So we need to do some clever rearranging. Consider element $(i,j)$ on the left hand side above, which looks like this:
Pulling the sum over $p$ out front, we get a nicer expression:
The term in square brackets has three indices: $i$, $p$, and $j$. So it is a tensor! In fact, the above expression is the multiplication of a matrix, $\mathbf{X}$, with a tensor. See section 2.5 of Kolda & Bader (2009) for a summary of this matrix-tensor operation.
Let’s define the tensor as $\mathcal{T}$:
I suggestively decided to index along mode $j$ in the superscript. Consider a slice through the tensor at index $j$. We are left with a matrix, which can be written as:
Where $\text{diag}(\cdot)$ transforms a vector into a diagonal matrix (a standard operation available in MATLAB/Python). Let’s draw an illustration to summarize what we’ve done so far:
Representation of equation 2 using the tensor described in equation 3.
It turns out that we are basically done due to the magic of numpy broadcasting. We simply need to construct the tensor, $\mathcal{T}$, and the matrix $\mathbf{A}^T (\mathbf{M} \circ \mathbf{B})$, and then call numpy.linalg.solve. The following code snippet does exactly this:
def censored_lstsq(A, B, M):
"""Solves least squares problem subject to missing data.
Note: uses a broadcasted solve for speed.
Args
----
A (ndarray) : m x r matrix
B (ndarray) : m x n matrix
M (ndarray) : m x n binary matrix (zeros indicate missing values)
Returns
-------
X (ndarray) : r x n matrix that minimizes norm(M*(AX - B))
"""
# Note: we should check A is full rank but we won't bother...
# if B is a vector, simply drop out corresponding rows in A
if B.ndim == 1 or B.shape[1] == 1:
return np.linalg.leastsq(A[M], B[M])[0]
# else solve via tensor representation
rhs = np.dot(A.T, M * B).T[:,:,None] # n x r x 1 tensor
T = np.matmul(A.T[None,:,:], M.T[:,:,None] * A[None,:,:]) # n x r x r tensor
return np.squeeze(np.linalg.solve(T, rhs)).T # transpose to get r x n
Since $\mathbf{A}^T \text{diag}(\mathbf{m}_{j}) \mathbf{A}$ is symmetric and positive definite we could use the Cholesky decomposition to solve the system rather than the more generic numpy solver. If anyone has an idea/comment about how to implement this efficiently in Python, I’d love to know!
Here’s my rough analysis of time complexity (let me know if you spot an error):
• The Hadamard product M * B is $\mathcal{O}(mn)$
• Strassen fanciness aside, matrix multiplication np.dot(A.T, M * B) is $\mathcal{O}(mnr)$
• The broadcasted Hadamard M.T[:,:,None] * A[None,:,:] is $\mathcal{O}(mnr)$
• Building the tensor involves n matrix multiplications, totalling $\mathcal{O}(m n r^2)$.
• Then solving each of the n slices in the tensor takes $\mathcal{O}(n r^3)$.
So the total number of operations is $\mathcal{O}(n r^3 + m n r^2)$. In practice, this does run noticeably slower than a regular least-squares solve, so I’d love suggestions for improvements!
### A less fun solution
There is another simple solution to this least-squares problem, but it doesn’t involve tensors and requires a for loop. The idea is to solve for each column of $\mathbf{X}$ sequentially. Let $\mathbf{x}_i$ be the $i^\text{th}$ column of $\mathbf{X}$ and let $\mathbf{b}_i$ likewise by the $i^\text{th}$ column of $\mathbf{B}$. It is intuitive that the least-squares solution for $\mathbf{x}_i$ is given by dropping the rows of $\mathbf{A}$ where $\mathbf{b}_i$ has a missing entry.
Least squares with missing data for a single column of $B$.
def censored_lstsq_slow(A, B, M):
"""Solves least squares problem subject to missing data.
Note: uses a for loop over the columns of B, leading to a
slower but more numerically stable algorithm
Args
----
A (ndarray) : m x r matrix
B (ndarray) : m x n matrix
M (ndarray) : m x n binary matrix (zeros indicate missing values)
Returns
-------
X (ndarray) : r x n matrix that minimizes norm(M*(AX - B))
"""
X = np.empty((A.shape[1], B.shape[1]))
for i in range(B.shape[1]):
m = M[:,i] # drop rows where mask is zero
X[:,i] = np.linalg.lstsq(A[m], B[m,i])[0]
return X
It has a similar complexity of $\mathcal{O}(m n r^2)$, due to solving n least squares equations each with $\mathcal{O}(m r^2)$ operations. Since we have added a for loop this solution does run noticeably slower on my laptop than the first solution for large n. In C++ or Julia, this for loop would be less of a worry.
Also, I think this second (less fun) solution should be more accurate numerically because it does not compute the Gramian, $\mathbf{A}^T \mathbf{A}$, whereas the first method I offered essentially does this. The condition number of the Gramian is the square of the original matrix, $\kappa ( \mathbf{A}^T \mathbf{A}) = \kappa (\mathbf{A})^2$, so the result will be less stable. This is why some least-squares solvers do not use the normal equations under the hood (they instead use QR decomposition).
### Conclusion
I’ve outlined a couple of simple ways to solve the least-squares regression problem with missing data in the dependent variables. The two Python functions I offered have a tradeoff in speed and accuracy, and while the code could certainly be further optimized - I expect these functions will work well enough for some simple applications.
### Appendix
This appendix contains some simple manipulations on the objective function:
Let’s define $\mathbf{E} = \mathbf{A} \mathbf{X} - \mathbf{B}$, which is a matrix of unmasked residuals. Then, since $m_{ij} \in \{0, 1\}$ the objective function simplifies:
We are using $\textbf{Tr} [ \cdot ]$ to denote the trace of a matrix, and the fact that $\lVert \mathbf{X} \lVert^2_F = \textbf{Tr} [\mathbf{X}^T \mathbf{X}]$ for any matrix $\mathbf{X}$. Now we’ll substitute $\mathbf{A}\mathbf{X} - \mathbf{B}$ back in and expand the expression:
Now we differentiate these three terms with respect to $\mathbf{X}$. The term on the right goes to zero as it does not depend on $\mathbf{X}$. The term in the middle is quite standard and can be found in the matrix cookbook:
The first term in equation 5 is a bit of a pain and we’ll derive it manually. We resort to a summation notation and compute the partial derivative with respect to element $(a, b)$ of $\mathbf{X}$:
Now we’ll pull all of the sums out front and make our notation more compact. Then we’ll differentiate:
By inspection, you can convince yourself that this final expression maps onto $2 \mathbf{A^T (\mathbf{M} \circ (\mathbf{A} \mathbf{X}))}$ in matrix notation. Combining this result with equation 6, we arrive at:
Which immediately implies equation 2 in the main text.
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# How does this answer for automata and Hamming distance not lead to inconsistencies?
I had already been given the answer by the TA in class, but I don't understand it. I'm not asking for the answer on a homework problem or anything.
The problem:
The Hamming distance ("distance") of a word w to v of the same size is the number of positions wherein they differ. The distance between a w and L is the smallest such distance from a word chosen among L.
Let k be a natural number, and L a regular language. L' is the set of words w at a distance not greater than k from L. Show that L' is regular.
The way to go about this was to construct an automaton for such an L', and to do induction on k. If k = 0, then L' = L0 = L, and M0 = (Q0, E, delta0, q0, F0).
In general
• Q is the set of states for an automaton accepting words with a distance at most k.
• E is the alphabet among all machines and languages.
• d is the tranisition function of Mk,
• q0 is the start state for Mk
• F is the set of final states for Mk.
The answer, according to the TA:
Assuming that the construction is valid for all i <= k, we can form M' as follows:
Q' = Q x {0, 1}, and **q'** is an element of Q'
q0' = (q0, 0)
d'((q, 0), a) = {(d(q, a), 0)} u {(d(q, b), 1) | b element of E}
d'((q, 1), a) = {(d(q, a), 1)}
F' = {q' = (q, t) | t = 0 or t = 1}
I really don't get this. To me, it seems that if I start off with less than k+1 errors then I'll be in a state (q1, 0) for some q1, and if I read 1 error then I'm put in a state (q2, 1) for some q2. If I am in a state (q1, 1) for some q1 and read anything, then I'll be in a state (q2, 1) for some q2.
If I start with reading the first letter of any input, then I start in (q0, 0), and if I read one error then I'm in (q1, 1), and if I read 50 errors then I'm in (q, 1) for some q, but if we wanted k to be, say, 5, then I've surpassed that.
I'm really confused here. I'd appreciate any help.
Thank you.
• You really ought to ask this question of your TA. They already understand their solution and can explain it to you. We'd have to figure out what they think is going on, figure out what you think is going on, figure out what the difference is, etc. – David Richerby Nov 13 '18 at 19:06
Consider the case where $$k=1$$. In other words, we're allowing a single error. We take our base automaton and apply this "error-allowing transformation" to it.
The transformed automaton generally works as follows:
• The state $$(q, 0)$$ means "the underlying automaton is in state $$q$$, and we haven't made an error yet".
• The state $$(q, 1)$$ means "the underlying automaton is in state $$q$$, and we have made a single error".
• From any state $$(q, 0)$$, we can make the "correct" transition, or make any "incorrect" transition (a transition meant for any other symbol) and change the $$0$$ to a $$1$$.
• From any state $$(q, 1)$$, we can only make the "correct" transition. In other words, once we've made one mistake, we can't make any more!
• If we reach either $$(a, 0)$$ or $$(a, 1)$$, where $$a$$ is an accepting state, then we accept.
Hopefully this makes sense; let me know in the comments if it doesn't.
Now the key is, this works for any base automaton. So if we want $$k=2$$, we can just apply this transformation once, and then apply the transformation again!
The states now look like $$((q, 0), 0)$$. If we reach the state $$((a, 1), 1)$$, then we've made exactly two errors. And once we've made exactly two errors (there are no more zeroes to turn into ones), we're not allowed to make any more errors, we have to follow the base automaton exactly.
And then you can apply the transformation again for $$k=3$$, and again for $$k=4$$, and so on, for any natural $$k$$. Induction! QED.
(Another way to prove it is to allow errors in every state $$(q, \psi)$$ as long as $$\psi < k$$. This way doesn't involve induction, and might be a bit easier to grasp, since you have a straightforward "error counter".)
• But based on the specific construction of d', it doesn't explicitly prevent us from making any more errors. Any following input, erroneous or not, will keep us in a state that treats it as though we only had 1 error. At least, that's how I'm seeing. Mind telling me why this point of view is incorrect? Thank you. – pikecha Nov 13 '18 at 20:27
• @pikecha Ah, check the transition rule again. We can only take "error transitions" if we're in the state $(q, 0)$. If we're in the state $(q, 1)$, we're not allowed to do that any more. – Draconis Nov 13 '18 at 21:01
• Yeah, the way I was parsing that was that we are allowed to read any input (which necessarily includes erroneous) and yet still stay in a state as though we've made only one error at that point (...(q, 1), ..., 1). For whatever reason, I feel as though I understand this more if I keep it as (q, i) and have it be true as long as i <= k. – pikecha Nov 14 '18 at 7:45
• @pikecha The crucial line is d'((q, 1), a) = {(d(q, a), 1)}. That is, if you've already made an error, you have to follow d (the original transition function) without modification. – Draconis Nov 14 '18 at 16:29
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