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# Firm Historical Headquarter State from SEC 10K/Q Filings¶

## Why the need to use SEC filings?¶

In the Compustat database, a firm's headquarter state (and other identification) is in fact the current record stored in comp.company. This means once a firm relocates (or updates its incorporate state, address, etc.), all historical observations will be updated and not recording historical state information anymore.

To resolve this issue, an effective way is to use the firm's historical SEC filings. You can follow my previous post Textual Analysis on SEC filings to extract the header information, which includes a wide range of meta data. Alternatively, the University of Notre Dame's Software Repository for Accounting and Finance provides an augmented 10-X header dataset.

## Do I have to use SEC filings?¶

I'll skip the parsing procedure for now. The most important point is that using the historical SEC filings, you can ensure that you truly are using the historical headquarter state in your empirical estimation. Based on the augmented 10-X header dataset, I find that around 2-3% of Compustat firms changed their headquarter state (as indicated by their business address) each year.

Year Firms Changed State Total Firms % Firms Changed State
1995 22 4205 0.52
1996 69 7939 0.87
1997 199 8101 2.46
1998 206 8126 2.54
1999 202 8199 2.46
2000 202 8252 2.45
2001 204 7802 2.61
2002 167 7421 2.25
2003 214 6930 3.09
2004 175 6742 2.6
2005 154 6478 2.38
2006 156 6267 2.49
2007 144 6091 2.36
2008 125 5797 2.16
2009 127 5523 2.3
2010 128 5479 2.34
2011 152 5445 2.79
2012 160 5494 2.91
2013 171 5491 3.11
2014 195 5455 3.57
2015 147 5322 2.76
2016 117 5092 2.3
2017 129 4914 2.63
2018 107 4847 2.21

Moreover, 2,947 out of the 17,221 firms, or about 17% firms changed their headquarter state in the merged sample. This is by no means a small number that can be ignored. So, whenever possible, you should try to use the historical information from past SEC filings' metadata.

## How to get the actual historical firm HQ state using SEC filings?¶

### 1969 - 2003¶

I start with the firm historical HQ state provided by Bai, Fairhurst and Serfling (2020 RFS). This dataset contains the historical HQ locations from 1969 to 2003, which is based on the SEC filings post 1994 and hand-collected by the authors from the Moody’s Manuals (later Mergent Manuals) and Dun & Bradstreet’s Million Dollar Directory (later bought by Mergent).1

### 1994 - 2018¶

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 import pandas as pd filepath = "~/Downloads/LM_EDGAR_10X_Header_1994_2018.csv" if __name__ == "__main__": df = pd.read_csv( filepath, usecols=["cik", "file_date", "ba_state"], dtype={"cik": str}, parse_dates=["file_date"], ) # Some ba_stata codes are lowercase df["ba_state"] = df["ba_state"].str.upper() # Some ba_state codes are not valid US states df = df[df["ba_state"].str.isalpha() & ~pd.isnull(df["ba_state"])] df.drop_duplicates().to_stata( "~/Downloads/historical_state_1994_2018.dta", write_index=False, convert_dates={"file_date": "td"}, )

The result is a historical_state.dta Stata file like this:

### 1969 - 2018 merged¶

Finally, to merge the two datasets together, I imported them into WRDS Cloud and run the following SAS script:

• Pre-2003, use Bai, Fairhurst and Serfling (2020 RFS).
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 libname hs "~/historical_state"; /* Historical HQ state (1994 to 2018) from augmented 10-X header dataset */ proc import datafile="~/historical_state/historical_state_1994_2018.dta" out=historical_state_1994_2018 dbms=stata replace; /* Historical HQ state (1969 to 2003) from Bai, Fairhurst and Serfling (2020 RFS) */ proc import datafile="~/historical_state/hist_headquarters_Bai_et_al.dta" out=hist_headquarters_Bai_et_al dbms=stata replace; /* Build the post-1994 dataset using SEC filings */ proc sql; create table funda as select gvkey, cik, datadate, fyear from comp.funda where indfmt= 'INDL' and datafmt='STD' and popsrc='D' and consol='C' and year(datadate) between 1994 and 2018 /* "For firms that change fiscal year within a calendar year, we take the last reported date when extracting financial data. This leaves us with one set of observations for each firm (gvkey) in each year." -- Pelueger, Siriwardane and Sunderam (2020 QJE) */ group by gvkey, fyear having datadate=max(datadate); create table firm_historical_state as select a.*, b.ba_state as state_sec label="State from SEC filings" from funda as a left join historical_state as b on a.cik=b.cik and year(a.datadate)=year(b.file_date) and b.file_date<=a.datadate group by a.gvkey, a.datadate /* use the SEC filing closet to and before the Compustat datadate */ having b.file_date=max(b.file_date); create table historical_state_1994_2018 as select a.*, b.state as state_comp label="State from Compustat" from firm_historical_state as a left join comp.company as b on a.gvkey=b.gvkey order by a.gvkey, a.datadate; quit; /* Sanity check: no duplicated gvkey-fyear */ proc sort data=historical_state_1994_2018 nodupkey; by gvkey datadate; run; proc sql; create table hist_headquarters_Bai_et_al as select put(gvkeyn, z6.) as gvkey, fyear, state from hist_headquarters_Bai_et_al; quit; /* Stack together the two datasets */ data states; set hist_headquarters_Bai_et_al historical_state_1994_2018(where=(fyear>2003) keep=gvkey fyear state:); run; proc sql; create table hs.corrected_hist_state_1969_2018 as select *, coalesce(state, state_sec, state_comp) as corrected_state from states where not missing(calculated corrected_state) order by gvkey, fyear; quit; /* Sanity check: no duplicated gvkey-fyear */ proc sort data=hs.corrected_hist_state_1969_2018 nodupkey; by gvkey fyear; run;


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29.0: Prelude to Quantum Physics

Quantum mechanics is the branch of physics needed to deal with submicroscopic objects. Because these objects are smaller than we can observe directly with our senses and generally must be observed with the aid of instruments, parts of quantum mechanics seem as foreign and bizarre as parts of relativity. But, like relativity, quantum mechanics has been shown to be valid—truth is often stranger than fiction.

Certain aspects of quantum mechanics are familiar to us. We accept as fact that matter is composed of atoms, the smallest unit of an element, and that these atoms combine to form molecules, the smallest unit of a compound (Figure $$\PageIndex{2}$$). While we cannot see the individual water molecules in a stream, for example, we are aware that this is because molecules are so small and so numerous in that stream. When introducing atoms, we commonly say that electrons orbit atoms in discrete shells around a tiny nucleus, itself composed of smaller particles called protons and neutrons. We are also aware that electric charge comes in tiny units carried almost entirely by electrons and protons. As with water molecules in a stream, we do not notice individual charges in the current through a lightbulb, because the charges are so small and so numerous in the macroscopic situations we sense directly.

MAKING CONNECTIONS: REALMS OF PHYSICS

Classical physics is a good approximation of modern physics under conditions first discussed in the The Nature of Science and Physics. Quantum mechanics is valid in general, and it must be used rather than classical physics to describe small objects, such as atoms.

Atoms, molecules, and fundamental electron and proton charges are all examples of physical entities that are quantized—that is, they appear only in certain discrete values and do not have every conceivable value. Quantized is the opposite of continuous. We cannot have a fraction of an atom, or part of an electron’s charge, or 14-1/3 cents, for example. Rather, everything is built of integral multiples of these substructures. Quantum physics is the branch of physics that deals with small objects and the quantization of various entities, including energy and angular momentum. Just as with classical physics, quantum physics has several subfields, such as mechanics and the study of electromagnetic forces. The correspondence principle states that in the classical limit (large, slow-moving objects), quantum mechanics becomes the same as classical physics. In this chapter, we begin the development of quantum mechanics and its description of the strange submicroscopic world. In later chapters, we will examine many areas, such as atomic and nuclear physics, in which quantum mechanics is crucial.

Glossary

quantized
the fact that certain physical entities exist only with particular discrete values and not every conceivable value
correspondence principle
in the classical limit (large, slow-moving objects), quantum mechanics becomes the same as classical physics
quantum mechanics
the branch of physics that deals with small objects and with the quantization of various entities, especially energy

This page titled 29.0: Prelude to Quantum Physics is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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Functions Modeling Change: A Preparation for Calculus, 5th Edition

Yes, $Q(t)=2^t$ is exponential.
Using the fact that $(a^b)^n=a^{bn}$, this function becomes $$Q(t)=(8^{1/3})^t$$ Using the fact that $8^{1/3}=\sqrt[3]{8}=2$, the function becomes exponential. $$Q(t)=2^t$$


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# NAG Library Function Document

## 1Purpose

nag_zgghrd (f08wsc) reduces a pair of complex matrices $\left(A,B\right)$, where $B$ is upper triangular, to the generalized upper Hessenberg form using unitary transformations.

## 2Specification

 #include #include
 void nag_zgghrd (Nag_OrderType order, Nag_ComputeQType compq, Nag_ComputeZType compz, Integer n, Integer ilo, Integer ihi, Complex a[], Integer pda, Complex b[], Integer pdb, Complex q[], Integer pdq, Complex z[], Integer pdz, NagError *fail)

## 3Description

nag_zgghrd (f08wsc) is usually the third step in the solution of the complex generalized eigenvalue problem
 $Ax=λBx.$
The (optional) first step balances the two matrices using nag_zggbal (f08wvc). In the second step, matrix $B$ is reduced to upper triangular form using the $QR$ factorization function nag_zgeqrf (f08asc) and this unitary transformation $Q$ is applied to matrix $A$ by calling nag_zunmqr (f08auc).
nag_zgghrd (f08wsc) reduces a pair of complex matrices $\left(A,B\right)$, where $B$ is triangular, to the generalized upper Hessenberg form using unitary transformations. This two-sided transformation is of the form
 $QHAZ=H QHBZ=T$
where $H$ is an upper Hessenberg matrix, $T$ is an upper triangular matrix and $Q$ and $Z$ are unitary matrices determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices ${Q}_{1}$ and ${Z}_{1}$, so that
 $Q1AZ1H=Q1QHZ1ZH, Q1BZ1H=Q1QTZ1ZH.$

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256

## 5Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{compq}$Nag_ComputeQTypeInput
On entry: specifies the form of the computed unitary matrix $Q$.
${\mathbf{compq}}=\mathrm{Nag_NotQ}$
Do not compute $Q$.
${\mathbf{compq}}=\mathrm{Nag_InitQ}$
The unitary matrix $Q$ is returned.
${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$
q must contain a unitary matrix ${Q}_{1}$, and the product ${Q}_{1}Q$ is returned.
Constraint: ${\mathbf{compq}}=\mathrm{Nag_NotQ}$, $\mathrm{Nag_InitQ}$ or $\mathrm{Nag_UpdateSchur}$.
3:    $\mathbf{compz}$Nag_ComputeZTypeInput
On entry: specifies the form of the computed unitary matrix $Z$.
${\mathbf{compz}}=\mathrm{Nag_NotZ}$
Do not compute $Z$.
${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$
z must contain a unitary matrix ${Z}_{1}$, and the product ${Z}_{1}Z$ is returned.
${\mathbf{compz}}=\mathrm{Nag_InitZ}$
The unitary matrix $Z$ is returned.
Constraint: ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, $\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{ilo}$IntegerInput
6:    $\mathbf{ihi}$IntegerInput
On entry: ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ as determined by a previous call to nag_zggbal (f08wvc). Otherwise, they should be set to $1$ and $n$, respectively.
Constraints:
• if ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
• if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
7:    $\mathbf{a}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the matrix $A$ of the matrix pair $\left(A,B\right)$. Usually, this is the matrix $A$ returned by nag_zunmqr (f08auc).
On exit: a is overwritten by the upper Hessenberg matrix $H$.
8:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:    $\mathbf{b}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the upper triangular matrix $B$ of the matrix pair $\left(A,B\right)$. Usually, this is the matrix $B$ returned by the $QR$ factorization function nag_zgeqrf (f08asc).
On exit: b is overwritten by the upper triangular matrix $T$.
10:  $\mathbf{pdb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11:  $\mathbf{q}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array q must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$ when ${\mathbf{compq}}=\mathrm{Nag_InitQ}$ or $\mathrm{Nag_UpdateSchur}$;
• $1$ when ${\mathbf{compq}}=\mathrm{Nag_NotQ}$.
The $\left(i,j\right)$th element of the matrix $Q$ is stored in
• ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$, q must contain a unitary matrix ${Q}_{1}$.
If ${\mathbf{compq}}=\mathrm{Nag_NotQ}$, q is not referenced.
On exit: if ${\mathbf{compq}}=\mathrm{Nag_InitQ}$, q contains the unitary matrix $Q$.
Iif ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$, q is overwritten by ${Q}_{1}Q$.
12:  $\mathbf{pdq}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
• if ${\mathbf{compq}}=\mathrm{Nag_InitQ}$ or $\mathrm{Nag_UpdateSchur}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compq}}=\mathrm{Nag_NotQ}$, ${\mathbf{pdq}}\ge 1$.
13:  $\mathbf{z}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array z must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdz}}×{\mathbf{n}}\right)$ when ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$;
• $1$ when ${\mathbf{compz}}=\mathrm{Nag_NotZ}$.
The $\left(i,j\right)$th element of the matrix $Z$ is stored in
• ${\mathbf{z}}\left[\left(j-1\right)×{\mathbf{pdz}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{z}}\left[\left(i-1\right)×{\mathbf{pdz}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$, z must contain a unitary matrix ${Z}_{1}$.
If ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, z is not referenced.
On exit: if ${\mathbf{compz}}=\mathrm{Nag_InitZ}$, z contains the unitary matrix $Z$.
If ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$, z is overwritten by ${Z}_{1}Z$.
14:  $\mathbf{pdz}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
• if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, ${\mathbf{pdz}}\ge 1$.
15:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_ENUM_INT_2
On entry, ${\mathbf{compq}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{compq}}=\mathrm{Nag_InitQ}$ or $\mathrm{Nag_UpdateSchur}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{compq}}=\mathrm{Nag_NotQ}$, ${\mathbf{pdq}}\ge 1$.
On entry, ${\mathbf{compz}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{compz}}=\mathrm{Nag_UpdateZ}$ or $\mathrm{Nag_InitZ}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{compz}}=\mathrm{Nag_NotZ}$, ${\mathbf{pdz}}\ge 1$.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
On entry, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdq}}>0$.
On entry, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdz}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INT_3
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{ilo}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ihi}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ilo}}\ge 1$ and
if ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The reduction to the generalized Hessenberg form is implemented using unitary transformations which are backward stable.

## 8Parallelism and Performance

nag_zgghrd (f08wsc) is not threaded in any implementation.

This function is usually followed by nag_zhgeqz (f08xsc) which implements the $QZ$ algorithm for computing generalized eigenvalues of a reduced pair of matrices.


text:
Find Paper, Faster
Example：10.1021/acsami.1c06204 or Chem. Rev., 2007, 107, 2411-2502
Comparison of models for (∞,n)‐categories, II
Journal of Topology  (IF1.582),  Pub Date : 2020-09-07, DOI: 10.1112/topo.12167
Julia E. Bergner, Charles Rezk

In this paper we complete a chain of explicit Quillen equivalences between the model category for $Θ n + 1$‐spaces and the model category of small categories enriched in $Θ n$‐spaces. The Quillen equivalences given here connect Segal category objects in $Θ n$‐spaces, complete Segal objects in $Θ n$‐spaces, and $Θ n + 1$‐spaces.


text:
# Proving the limit laws

Addition. Let $$E\subset \mathbf{R}$$ be a subset of the real numbers, $$f, g : E \to \mathbf{R}$$ be real functions, and $$L_1, L_2\in \mathbf{R}$$ be real numbers. Now suppose $$\lim_{x\to a} f(x) = L_1$$ and $$\lim_{x\to a} g(x) = L_2$$. We would like to show that $$\lim_{x\to a} (f(x)+g(x)) = L_1+L_2$$.

Since $$\lim_{x\to a} f(x) = L_1$$ and $$\lim_{x\to a} g(x) = L_2$$, we know that $\forall \epsilon >0 \exists \delta >0 \forall x (0 < |x-a|<\delta \to |f(x)-L_1|<\epsilon)$ and $\forall \epsilon >0 \exists \delta >0 \forall x (0 < |x-a|<\delta \to |g(x)-L_2|<\epsilon)$ Now let $$\epsilon > 0$$. We need to find $$\delta > 0$$ such that $0<|x-a|<\delta \to |(f(x)+g(x))-(L_1+L_2)|<\epsilon$ But \begin{align} |f(x)+g(x)-(L_1+L_2)| &= |f(x)-L_1 + g(x)-L_2| \\ &\leq |f(x)-L_1| + |g(x)-L_2| \end{align} by the triangle inequality. So if we can show $$|f(x)-L_1| + |g(x)-L_2|<\epsilon$$, then our result follows immediately. But, $$\frac{\epsilon}{2}$$ being a positive real number, we have $$\delta_1,\delta_2 > 0$$ such that $0 < |x-a|<\delta_1 \to |f(x)-L_1|<\frac{\epsilon}{2} \\ 0 < |x-a|<\delta_2 \to |f(x)-L_2|<\frac{\epsilon}{2}$ So if we let $$\delta := \min(\delta_1,\delta_2)$$, then $$|f(x)-L_1|$$ and $$|g(x)-L_2|$$ are both less than $$\frac{\epsilon}{2}$$, so $|f(x)-L_1| + |g(x)-L_2|< \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$ which completes the proof. $$\square$$


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# Virtual particle energy-momentum

1. Mar 10, 2008

### jostpuur

A mentioning about virtual particle problem in my other thread just reminded me of some thoughts, which I now succeeded putting together.

When calculating cross sections in QFT, we encounter terms like this

$$\langle 0|a_{\textbf{k}'} a_{\textbf{p}'} a^{\dagger}_{\textbf{p}_1} a^{\dagger}_{\textbf{p}_2} a'_{\textbf{k}_1} a^{'\dagger}_{\textbf{k}_2} a_{\textbf{p}_3} a_{\textbf{p}_4} a^{\dagger}_{\textbf{k}} a^{\dagger}_{\textbf{p}}|0\rangle,$$

where we start with some excitation, operate on it with annihilation operators, create some intermediate particle, annihilate the intermediate particle, create two final particles, and then consider the inner product with some fixed outcome.

As it turns out, the calculations lead into some expressions that contain propagator of the intermediate particle, and the energy-momentum in the propagator is not on shell. This is usually interpreted as sign of the intermediate particle being off shell, but this doesn't fully make sense to me.

Firstly, the calculation is quite abstract. I don't see any obvious reason to interpret the resulting propagator as actually describing the propagation of the intermediate particle.

Secondly, isn't it impossible to create off shell particles with the usual creation operators? I mean, if you operate on vacuum with $a^{'\dagger}_{\textbf{p}}$, you get a particle whose energy momentum is $(\sqrt{|\textbf{p}|^2 + (m')^2}, \textbf{p})$, right? Here $m'$ is the constant mass, characteristic to the particular particle type.

So, what I hear everyone explaining, is this:

"Since the energy-momentum is conserved in the vertexes, the intermediate particle is clearly off shell. This, however, is not problematic, since it is a virtual particle that exists only for short period of time, and cannot be observed directly."

But why not like this:

"Since the intermediate particle is on shell, clearly the energy-momentum is not conserved in the vertexes. This, however, is not problematic, since the state, at which energy-momentum conservation is violated, exists only for a short period of time, and cannot be observed directly."?

Last edited: Mar 10, 2008
2. Mar 10, 2008

### tiny-tim

It's a trick!

The operators in $$\langle 0|a_{\textbf{k}'} a_{\textbf{p}'} a^{\dagger}_{\textbf{p}_1} a^{\dagger}_{\textbf{p}_2} a'_{\textbf{k}_1} a^{'\dagger}_{\textbf{k}_2} a_{\textbf{p}_3} a_{\textbf{p}_4} a^{\dagger}_{\textbf{k}} a^{\dagger}_{\textbf{p}}|0\rangle,$$
are all for real, not virtual, particles.

From that, we get the position Feynman diagrams, in which the integration is over $$d^3p$$.

(The momentums in the position Feynman diagrams are still all on-shell, since they still represent real operators for real particles.)

Those integrals include a step function.

Then a Fourier trick is used to convert that step function into a fourth integral, which combined with the $$d^3p$$ gives a convenient Lorentz-friendly four-variable integral (see, for example, Weinberg QFT page 276).

The four-vectors in this, and in the associated momentum Feynman diagram, have nothing to do with the real momentums in the position diagram - they only look similar. It's a good trick, isn't it?

It is convenient to call them off-shell four-vectors, and to think of them as representing the energy-momentum of "virtual particles".


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# Non-separable compact space

Off the top of my head, I can't think of a non-separable compact space.

Can you provide a good example?

• That's because any compact metric space is separable. So any example has to be non metrizable May 7, 2014 at 15:03
• May 7, 2014 at 15:07
• Thanks Léo & Giuseppe. The Alexandroff double circle is non-separable and compact. May 7, 2014 at 15:25

$\pi$-Base, an online version of the reference chart from Steen and Seebach's Counterexamples in Topology, gives the following examples of compact, non-separable spaces. You can view the search result for more details on these spaces.

Alexandroff Square

Closed Ordinal Space $[0, \Omega]$

Concentric Circles

Either-Or Topology

Lexicographic Ordering on the Unit Square

The Extended Long Line

Tychonoff Plank

Uncountable Excluded Point Topology

Uncountable Fort Space

Uncountable Modified Fort Space

• Awesome list, Austin! May 8, 2014 at 3:58
• The long line is my favourite example of this phenomenon! Dec 15, 2018 at 3:07

A not so simple, but curious example is the Alexandroff Double Circle:

Consider $C_i=\{(x,y) \in \mathbb{R}^2; x^2+y^2=i\}$ for $i=1,2$. Let $X=C_1 \cup C_2$ and $f:C_1 \to C_2$ the radial homeomorphism. Define a topology on $X$ as follows: the points in $C_2$ are all isolated; for each $x \in C_1$ and $n \in \mathbb{N}$ let $O(x,n)$ the arc in $C_1$ centered at $x$ and with length $\frac{1}{n}$. Now take $B(x,n)=O(x,n) \cup f(O(x,n)-\{x\})$ as open neighborhood. The picture below illustrate such $B(x,n)$.

This way we form a basis for a topology on $X$. $X$ provided with this topology is what we call the Alexandroff Double Circle.

Now, $X$ is not separable because all points in $C_2$ are isolated, hence no countable subset of $C_2$ is dense in $C_2$. For compactness, take an open cover $\{A_{\alpha}\}$ of $X$ which consists of basis neighborhoods. Then it is also an open cover of $C_1$. But notice that the topology of $C_1$ as a subspace of $X$ is exactly the usual one. So it is compact and we can find a finite subcollection $\{A_1,\dots,A_k\} \subset \{A_\alpha\}$ that covers $C_1$. Now, by the nature of our neighborhhods, $X-(A_1\cup \dots \cup A_k)$ is a finite set, so we take an extra $A_{\alpha}$ for each of these points, obtaining a finite subcover.

I don't know if this space is useful some some other things besides counterexamples. This and a lot of other crazy spaces can be found on "Counterexamples in topology", by L. A. Steen and J. A. Seebach.

This I think would be the simplest example. Let $(Y,\tau_d)$ be an uncountable discrete space. Consider the space $X=Y\cup \{x\}$ with topology $\tau=\tau_d \cup \{X\}$. $(X,\tau)$ is clearly compact and, since all points in $Y\subset X$ are isolated, not separable.

The space I described is $\text{T}_0$ but not $\text{T}_1$. @Léo 's answer gives an example of a compact Hausdorff space that is not separable.

A space $X$ which is product of a collection of compact Hausdorff spaces $\{C_j\}_{j \in J}$ such that the cardinality $|J|$ of the collection is very large. $X$ is compact by Tychonoff's theorem. If $X$ is separable, i.e. if there is a countable dense subset $A$, then every point of $X$ is the limit of a sequence of points in $A$. The number of such sequences is at most the cardinality of the reals, and each sequence has at most one limit because $X$ is Hausdorff, and so $X$ has cardinality at most equal to the cardinality of the reals. But you can make $X$ have larger cardinality by choosing $|J|$ sufficiently large.

Edit: As pointed out in the comment of @Henno Brendsma, I should not have used convergence of sequences since that carries an additional assumption that each point has a countable neighborhood basis. Instead one can just use convergence of nets. The cardinality of all nets in a countable subset $A$ is bounded, so the cardinality of the set of limits of convergent nets in $A$ is bounded, and every point in the closure of $A$ is the limit of a convergent net in $A$.

• It's not true that in every point of $X$ is the limit of points from a dense subset. This is true for first countable spaces, but large products of $[0,1]$, say, won't be first countable. In fact $[0,1]^I$ is separable iff $|I| \le |\mathbb{R}|$. So your idea does work, but for a different reason. May 7, 2014 at 16:13
• The cardinality of all subnets of a countable set is not bounded. You need to use filters to get the upper bound of $2^{2^{\aleph_0}}$ for the cardinality of a separable space (because a filter on $X$ is an element of $\mathcal{P}(\mathcal{P}(X))$). This is a known area of deficiency of Moore-Smith convergence. Nov 11, 2019 at 10:20

Based on this question I asked recently, the one-point compactification of a non-separable metric space.

I specifically asked about the space of cadlag real-valued functions on $$[0,1]$$ under the supremum metric, but there are lots of other non-separable metric spaces out there, and other compactifications where the same argument applies.

The metric space is an open subset of its one-point compactification, and is not separable, so the one-point compactification is also not separable.


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# Drawing Euler Diagrams with Circles: The Theory of Piercings

Stapleton, Gem, Zhang, Leishi, Howse, John, Rodgers, Peter (2011) Drawing Euler Diagrams with Circles: The Theory of Piercings. IEEE Transactions on Visualization and Computer Graphics, 17 (7). pp. 182-196. (doi:10.1109/TVCG.2010.119)

PDF
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Official URL
http://www.cs.kent.ac.uk/pubs/2011/2982

## Abstract

Euler diagrams are effective tools for visualizing set intersections. They have a large number of application areas ranging from statistical data analysis to software engineering. However, the automated generation of Euler diagrams has never been easy: given an abstract description of a required Euler diagram, it is computationally expensive to generate the diagram. Moreover, the generated diagrams represent sets by polygons, sometimes with quite irregular shapes which make the diagrams less comprehensible. In this paper we address these two issues by developing the theory of piercings, where we define single piercing curves and double piercing curves. We prove that if a diagram can be built inductively by successively adding piercing curves under certain constraints then it can be drawn with circles, which are more aesthetically pleasing than arbitrary polygons. The theory of piercings is developed at the abstract level. In addition, we present a Java implementation that, given an inductively pierced abstract description, generates an Euler diagram consisting only of circles within polynomial time.

Item Type: Article 10.1109/TVCG.2010.119 Q Science > QA Mathematics (inc Computing science) > QA 76 Software, computer programming, Faculties > Sciences > School of Computing > Computational Intelligence Group Peter Rodgers 21 Sep 2012 09:49 UTC 23 Jan 2020 04:06 UTC https://kar.kent.ac.uk/id/eprint/30750 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-4100-3596


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## Intermediate Algebra for College Students (7th Edition)

The solution is $x=3$ and $y=4$.
The given system of linear equation is $x+2y=11$ $x-y=-1$ The augmented matrix is $\left[\begin{array}{cc|c} 1& 2 &11 \\ 1&-1 &-1 \\ \end{array}\right]$ Perform $R_2\rightarrow R_2- R_1$. Subtract row one from row two as shown below. $\left[\begin{array}{cc|c} 1& 2 &11 \\ 1-1&-1-2 &-1-11 \\ \end{array}\right]$ Simplify. $\left[\begin{array}{cc|c} 1& 2 &11 \\ 0&-3 &-12 \\ \end{array}\right]$ Perform $R_2\rightarrow \frac{R_2}{-3}$. Divide row two by $-3$. $\left[\begin{array}{cc|c} 1& 2 &11 \\ \frac{0}{-3}&\frac{-3}{-3} &\frac{-12}{-3} \\ \end{array}\right]$ Simplify. $\left[\begin{array}{cc|c} 1& 2 &11 \\ 0&1 &4 \\ \end{array}\right]$ Use back substitution to solve the linear system. $x+2y=11$ $y=4$ Substitute into above equation. $x+2(4)=11$ $x+8=11$ $x=11-8$ $x=3$.


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# Talk:Closed-form expression

WikiProject Mathematics (Rated C-class, Mid-priority)
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## Fluid dynamics

In Fluid dynamics, the closure form of the governing equations is called for the boundary layer approximation or the parabolic approximation.

I'd thought that closed-form solutions were possible for quintic equations, although there is no general formula for them? Could someone more knowledgeable edit this article?

You can of course find closed form solutions to quintics such as ${\displaystyle x^{5}-1=0}$ (the five roots of one). There is even a general process to solve quintics by introducing a new radical (The Bring radical). --njh 12:12, 12 July 2006 (UTC)

Why the word "solution"?? Some cases of closed-form expressions in mathematics are in some sense "solutions", and others are not. This seems to me like one of those cases where someone picks a word like "equation" or "solution" as a sort of catch-all term to be used when they don't know the right nomenclature. This happens frequently in math, when, for example, lay persons promiscuously use the word "equation" to denote anything at all that is written in mathematical notation.

Since closed form seems to be a disambiguation page, I propose that this article be moved to closed-form expression. Michael Hardy 00:05, 26 November 2005 (UTC)

I agree with your basic point. I have a general problem with the notion of "closed form", as I think it's a bit difficult to define "closed form" in the abstract, or to differentiate "expressed in closed form" from "expressed analytically". For example, one can often solve (in)equalities by using inverse functions, but whether or not the resulting expression would be said to be "in closed form" depends on what the basic inventory of functions and expressions is. I'm not sure if e.g. Lambert's W function would be considered part of that inventory, but it's obviously very convenient for expressing solutions of certain equations in a form that can easily be evaluated by numerical software. I have the feeling that large aspects of this notion of "closed form" are very much a remnant of a past time when only a small number of expressions could be conveniently evaluated by hand or looked up in a table. --MarkSweep (call me collect) 01:27, 26 November 2005 (UTC)

I am inclined to suspect that the concept does admit some precise definition, but I am skeptical of the claims even of some fairly sophisticated mathematicians to have done that definitively. But the fact that it is not yet fully precise doesn't mean there should be no article on it. Michael Hardy 22:33, 26 November 2005 (UTC)

So, let's move it, shall we? Regarding the content, here are some external links, which may or may not shed further light on the meaning of "closed form":
Also note that these articles all talk about solutions (though I agree with you now that that's too limited). --MarkSweep (call me collect) 10:58, 29 November 2005 (UTC)
1. Are there any expressions that are not closed-form expressions? Can someone give examples?
2. Are there indeed contexts in which people call certain expressions "closed-form expressions" in which these expressions are not the solutions of equations discussed in that context? Can someone give examples?
3. The Google search term [closed-form-solution -wikipedia] gets almost twice the number of hits of [closed-form-expression -wikipedia]. In general the recommendation is to use the most common form for the article's title.
--Lambiam 14:31, 26 December 2007 (UTC)

## Opposite of closed form

Should the opposite of a closed form expression be defined on (or linked from) this page? It would seem to be a useful addition to the article. Alchemeleon 21:12, 7 June 2007 (UTC)

## The Fibonacci numbers as an example

Perhaps the Fibonacci numbers provide a good example of a closed-form solution in contrast to a definition that does not use a closed-form expression. By the way, I don't think there is a single term with the opposite meaning. —141.150.24.182 (talk) 04:36, 15 February 2010 (UTC)

## Bounded or finite?

The current lede says a closed form expression may contain a bounded number of operations. To me that is not clear, and I think it should be "finite" instead. Obviously infinite summations are not closed forms. But a summation whose number of terms varies with the argument of the expression would be bounded (for each argument), but not involve a finite number of operations because there is no bound in the expression itself. For instance I would not consider the definition ${\displaystyle \textstyle \sum _{i=1}^{n}i}$ of a triangular number to be a closed-form expression for it, while ${\displaystyle \textstyle {\frac {n^{2}+n}{2}}}$ does give a closed-form expression. By the same token there would be no closed-form expression for factorials at all, unless we explicitly place them in our repertoire of "well-known"" functions. If my interpretation is agreed upon, I think "finite" would be the correct term to use. Marc van Leeuwen (talk) 10:33, 9 March 2010 (UTC)

Re-reading the intro, I think an even more radical change is in order: "[an expression] can be expressed analytically in terms of a bounded number of certain well-known functions" makes no sense: an expression cannot be expressed, it is already expressed. It would be silly to call an expression like ${\displaystyle \textstyle \sum _{i\in \mathbb {N} }{\frac {x^{i}}{i!}}}$ a closed form just because it happens to be equivalent to (i.e., can be expressed as) a different one (guess) that is. Also an expression is always finite, although this might need stressing (in view of practices such as continued fractions and infinite summations that are written using ellipses; in fact these are improperly written expressions, corresponding in a well understood way to limit expressions over hopefully equally well understood (in spite of the ellipses) sequences). So an expression is a closed form if it only involves certain well-known operations and functions, where expressly are excluded summations (as opposed to additions which are allowed), products (again with a variable or infinite number of factors; multiplications are OK), limits, case distinctions, and maybe some more I've forgotten here. The point is one may add basic functions to the repertoire (provided this is clearly stated), but the operations excluded are always forbidden. If anybody disagrees, please explain here; otherwise I will make this change some day. Marc van Leeuwen (talk) 09:24, 16 March 2010 (UTC)

I tend to agree with your suggestion that the boundedness of a closed-form formula be required. This would exclude ${\displaystyle n!}$. However, it could also exclude ${\displaystyle k^{n}}$ as it is usually defined as ${\displaystyle \underbrace {k\times ...\times k} _{n}}$ so that its length does depend on n, just like the length of ${\displaystyle n!}$ = ${\displaystyle \underbrace {1\times ...\times n} _{n}}$. If, however, one considers products ${\displaystyle n!}$ and ${\displaystyle k^{n}}$ of de facto unbounded lengths closed forms, what would be the reason for excluding sums of unbounded lengths? For instance, in such a case, ${\displaystyle \prod _{i=0}^{n}{\frac {1}{2^{2^{i}}}}}$ will be considered a closed form so why ${\displaystyle \sum _{i=0}^{n}{\frac {1}{2^{2^{i}}}}}$ should not? After all, ${\displaystyle k\times n}$ = ${\displaystyle \underbrace {k+...+k} _{n}}$ so that products seem much less closed forms than sums are.172.88.206.28 (talk) 15:29, 22 September 2016 (UTC)

Also, whether an expression is closed-form should be effectively verifiable (as a minimum, provable/disprovable) by means of finite number of obvious steps (just like whether a sequence of formulas is a proof is supposed to be), so a phrase "... can be expressed ..." seems inappropriate as "can" may turn out true but unprovable (or false but undisprovable) in any accepted system (PA, ZFC, etc.). 172.88.206.28 (talk) 01:23, 21 September 2016 (UTC) — Preceding unsigned comment added by 172.88.206.28 (talk) 01:29, 20 September 2016 (UTC)

## Closed-form number

The first paragraph of that section needs to be clarified:

"...in increasing order of size, these are the EL numbers, Liouville numbers, and elementary numbers."
"The first, denoted L for Liouville numbers..."

what?

I might be able to work out the correct interpretation on my own, were it not for the fact that three sets are mentioned, and I see four labels in the paragraph: E, L, EL and C. —Preceding unsigned comment added by 24.28.74.115 (talk) 02:52, 16 March 2010 (UTC)

## Proposed merger from Analytical expression

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.
Clear consensus for a merger. KeithbobTalk 19:26, 27 July 2014 (UTC)

I propose that the Analytical expression article be merged into this one. The reason is that it is a very short article, a stub really though not marked as such, on basically the same or a very closely related topic, and that if there is any distinction this could be treated more clearly in a single article. I am personally neutral to the question whether the name Analytical expression or Closed-form expression should be preferred, but given the relative sizes, a merger of this article into the other seems less natural. Marc van Leeuwen (talk) 07:14, 6 May 2011 (UTC)

• Agree The article Analytical expression is short and unsourced. It is poorly written because it does not explain well what an analytical expression is. At present Analytical expression is unlikely to survive any nomination for deletion. Dolphin (t) 02:37, 24 June 2011 (UTC)
• Agree From reading the article, I couldn't understand what an Analytical expression was, until I read the article Closed-form expression. Now I finally understand what the poorly orgainzed words in "Analytical expression" are attempting to say. Wikfr (talk) 21:44, 30 October 2011 (UTC)
• Agree They basically synonyms, and I completely agree with the reasons the guys above referred. BTW, how many people agreeing does this require for an undisputed merging? JMCF125 (discussioncontribs) 14:08, 19 June 2013 (UTC)

The above discussion is closed. Please do not modify it. Subsequent comments should be made in a new section.

## Clarification needed

@CRGreathouse: Please try to be constructive here. I found the relevance of the sentence to what precedes it in the paragraph to be unclear, so I took a shot at stating what the point was. You reverted, saying the addition was invalid. Then I put in a clarification-needed tag, which you reverted, saying the point was already pretty clear and giving an informally worded version of the point in the edit box but not in the relevant place in the article. Then I tried to give a more carefully worded version of your edit-box explanation in the article, saying in my edit summary "If you don't like this version, put in something better." You responded by reverting without replacing, with the edit summary "I don't mind a clarification if it's correct, but this one is not". This is not constructive in the absence of a replacement clarification.

I'm trying to get this passage in the lede improved to a point where people who know enough math to read the article can understand the point of the sentence. If the point is not what I put in in either of my two edits, then its point is not clear to me, and if it's not clear to me then it's not clear to others as well.

Please just put in an addendum to the sentence clarifying it in a way that's satisfactory to you. Duoduoduo (talk) 20:02, 12 January 2012 (UTC)

## Unclear sentence

The first paragraph has the following:

" Unlike the broader analytic expressions, the closed-form expressions do not include infinite series or continued fractions; neither includes integrals or limits."

"Neither" refers to infinite series and continued fractions, because 'neither' means something along the lines of "not the one nor the other of two people or things". But I am guessing here that the sentence was meant to say that closed-form expressions do also not include integrals or limits? (I don't know, I'm not a mathematician). If, OTOH, it is meant to say that neither infinite series or continued fractions include integrals or limits, then I wonder why this information is given, as it feels unrelated to the definition of closed-form expressions? maye (talk) 16:49, 26 September 2013 (UTC)

## Is this grammar of the Wikipedia article correct?

Is this grammar of the Wikipedia article correct? "to numbers defined in explicitly or implicitly in terms of algebraic operations"

## Hentai

The best thing ever. — Preceding unsigned comment added by 50.109.253.250 (talk) 05:41, 4 January 2017 (UTC)

## This should at least mention analytic functions

The phrase "analytic function" is a technical term with a precise meaning. The term "analytic" is defined twice in this article; one definition is possibly consistent with that meaning, but the other is not. It is tempting to assume that an analytic expression is the value of an analytic function. The article should mention that there are such things as analytic functions, and how they relate to the expressions.

2001:388:6080:109:D578:2477:98CA:8194 (talk) 04:34, 20 March 2017 (UTC)


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# Wave propagation

Last updated

Wave propagation is any of the ways in which waves travel.

## Contents

With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves.

For electromagnetic waves, propagation may occur in a vacuum as well as in the material medium. Other wave types cannot propagate through a vacuum and need a transmission medium to exist.

## Reflection of plane waves in a half-space

The propagation and reflection of plane waves—e.g. Pressure waves (P-wave) or Shear waves (SH or SV-waves) are phenomena that were first characterized within the field of classical seismology, and are now considered fundamental concepts in modern seismic tomography. The analytical solution to this problem exists and is well known. The frequency domain solution can be obtained by first finding the Helmholtz decomposition of the displacement field, which is then substituted into the wave equation. From here, the plane wave eigenmodes can be calculated.

### SV wave propagation

The analytical solution of SV-wave in a half-space indicates that the plane SV wave reflects back to the domain as a P and SV waves, leaving out special cases. The angle of the reflected SV wave is identical to the incidence wave, while the angle of the reflected P wave is greater than the SV wave. For the same wave frequency, the SV wavelength is smaller than the P wavelength. This fact has been depicted in this animated picture. [1]

### P wave propagation

Similar to the SV wave, the P incidence, in general, reflects as the P and SV wave. There are some special cases where the regime is different.

## Wave velocity

Wave velocity is a general concept, of various kinds of wave velocities, for a wave's phase and speed concerning energy (and information) propagation. The phase velocity is given as:

${\displaystyle v_{\rm {p}}={\frac {\omega }{k}},}$

where:

The phase speed gives you the speed at which a point of constant phase of the wave will travel for a discrete frequency. The angular frequency ω cannot be chosen independently from the wavenumber k, but both are related through the dispersion relationship:

${\displaystyle \omega =\Omega (k).\,}$

In the special case Ω(k) = ck, with c a constant, the waves are called non-dispersive, since all frequencies travel at the same phase speed c. For instance electromagnetic waves in vacuum are non-dispersive. In case of other forms of the dispersion relation, we have dispersive waves. The dispersion relationship depends on the medium through which the waves propagate and on the type of waves (for instance electromagnetic, sound or water waves).

The speed at which a resultant wave packet from a narrow range of frequencies will travel is called the group velocity and is determined from the gradient of the dispersion relation:

${\displaystyle v_{\rm {g}}={\frac {\partial \omega }{\partial k}}}$

In almost all cases, a wave is mainly a movement of energy through a medium. Most often, the group velocity is the velocity at which the energy moves through this medium.

## Related Research Articles

Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as temporal frequency to emphasize the contrast to spatial frequency, and ordinary frequency to emphasize the contrast to angular frequency. Frequency is measured in hertz (Hz) which is equal to one event per second. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period, T—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.

The phase velocity of a wave is the rate at which the wave propagates in some medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance of one or more quantities, sometimes as described by a wave equation. In physical waves, at least two field quantities in the wave medium are involved. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction it is said to be a traveling wave; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

Longitudinal waves are waves in which the vibration of the medium is parallel to the direction the wave travels and displacement of the medium is in the same direction of the wave propagation. Mechanical longitudinal waves are also called compressional or compression waves, because they produce compression and rarefaction when traveling through a medium, and pressure waves, because they produce increases and decreases in pressure. A wave along the length of a stretched Slinky toy, where the distance between coils increases and decreases, is a good visualization. Real-world examples include sound waves and seismic P-waves.

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. Media having this common property may be termed dispersive media. Sometimes the term chromatic dispersion is used for specificity. Although the term is used in the field of optics to describe light and other electromagnetic waves, dispersion in the same sense can apply to any sort of wave motion such as acoustic dispersion in the case of sound and seismic waves, in gravity waves, and for telecommunication signals along transmission lines or optical fiber. Physically, dispersion translates in a loss of kinetic energy through absorption.

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in both pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the phase velocity and group velocity of waves in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency dependence of wave propagation and attenuation.

An optical medium is material through which electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it. The medium has an intrinsic impedance, given by

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

Precursors are characteristic wave patterns caused by dispersion of an impulse's frequency components as it propagates through a medium. Classically, precursors precede the main signal, although in certain situations they may also follow it. Precursor phenomena exist for all types of waves, as their appearance is only predicated on the prominence of dispersion effects in a given mode of wave propagation. This non-specificity has been confirmed by the observation of precursor patterns in different types of electromagnetic radiation as well as in fluid surface waves and seismic waves.

Seismic inversion involves the set of methods which seismologists use to infer properties through physical measurements. Surface-wave inversion is the method by which elastic properties, density, and thickness of layers in the subsurface are obtained through analysis of surface-wave dispersion. The entire inversion process requires the gathering of seismic data, the creation of dispersion curves, and finally the inference of subsurface properties.

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

Mathematical Q models provide a model of the earth's response to seismic waves. In reflection seismology, the anelastic attenuation factor, often expressed as seismic quality factor or Q, which is inversely proportional to attenuation factor, quantifies the effects of anelastic attenuation on the seismic wavelet caused by fluid movement and grain boundary friction. When a plane wave propagates through a homogeneous viscoelastic medium, the effects of amplitude attenuation and velocity dispersion may be combined conveniently into the single dimensionless parameter, Q. As a seismic wave propagates through a medium, the elastic energy associated with the wave is gradually absorbed by the medium, eventually ending up as heat energy. This is known as absorption and will eventually cause the total disappearance of the seismic wave.

In physics, a sinusoidalplane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane.

## References

1. The animations are taken from Poursartip, Babak (2015). "Topographic amplification of seismic waves". UT Austin.


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Department of Mathematics Search | Help | Login | |

Math @ Duke

 ....................... ....................... Webpage

## Publications [#303545] of John Harer

Papers Published

1. Turner, K; Mileyko, Y; Mukherjee, S; Harer, J, Fréchet Means for Distributions of Persistence diagrams (June, 2012) [1206.2790v2]
(last updated on 2017/12/14)

Abstract:
Given a distribution $\rho$ on persistence diagrams and observations $X_1,...X_n \stackrel{iid}{\sim} \rho$ we introduce an algorithm in this paper that estimates a Fr\'echet mean from the set of diagrams $X_1,...X_n$. If the underlying measure $\rho$ is a combination of Dirac masses $\rho = \frac{1}{m} \sum_{i=1}^m \delta_{Z_i}$ then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fr\'echet mean computed by the algorithm given observations drawn iid from $\rho$. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.

dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320


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# Dot product calculator program

2020-02-21 18:02

Calculating Cross Products, Dot Products, and Unit Vectors on TI89 by Aaron Quinn Cross Product 1. Turn on calculator. 2. Press Catalog 3. Scroll down to crossP 4. Enter 1st vector with this format, [x, y, z (the brackets are above the comma and divide key) 5. Insert a comma followed by the 2nd vector with the same format as theEssentially, the dot product is matrix product if we consider $$x \in \mathbbRn$$ and $$y \in \mathbbRn$$, then the dot product is defined as: \[ \langle x, y \rangle \sumi1n xi yi xt \cdot y \ Some uses of the dot product are super neat and practical: The dot product calculator and the angle. Indeed, the dot or inner product dot product calculator program

Dot Product. The dot product of two vectors u and v is formed by multiplying their components and adding. In the plane, uv u 1 v 1 u 2 v 2; in space its u 1 v 1 u 2 v 2 u 3 v 3. If you tell the TI8384 to multiply two lists, it multiplies the elements of the two lists to make a third list.

Dot product calculator calculates dot product (scalar product), magnitudes and angle between two vectors. The calculator can be used for dot product calculation of 2D or 3D vectors. Dot product calculation can be done for two different vector forms: vector components or magnitudes and angle between vectors. Jan 19, 2018 C Program to Calculate Dot Product of Two Matrices Posted on January 19, 2018 by Manish. This is a C Program to Calculate Dot Product of Two Matrices. Problem Description. The program takes two matrices and calculates the dot product. Problem Solution. 1. The program takes the size of the two matrices. dot product calculator program The Dot Product Calculator an online tool which shows Dot Product for the given input. Byju's Dot Product Calculator is a tool which makes calculations very simple and interesting. If an input is given then it can easily show the result for the given number.

Free vector scalar multiplcation calculator solve vector multiply operations stepbystep dot product calculator program The vector dot product calculator comes in handy when you are solving vector multiplication problems. Instead of calculating the scalar product by hand, you can simply input the components of two vectors into this tool and let it do the math for you. en. Follow @symbolab. Related Symbolab blog posts. Advanced Math Solutions Vector Calculator, Simple Vector Arithmetic. Vectors are used to represent anything that has a direction and magnitude, length. The most popular example of CC program to make a simple calculator; There are two vector A and B and we have to find the dot product and cross product of two vector array. Dot product is also known as scalar product and cross product also known as vector product. Khan, Salman Vector dot product and vector length , The Khan Academy, Vector Dot Product and Vector Length. Cite this content, page or calculator as: Furey, Edward Dot Product Calculator ; from Online Calculator Resource.

Rating: 4.94 / Views: 867

2020 © folklenga.cf | Sitemap


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## EE Student Information

Academic Year 20-21 FAQ and updated EE course list

# Probability Seminar presents "Optimal delocalization for generalized Wigner matrices"

Topic:
Optimal delocalization for generalized Wigner matrices
Monday, June 1, 2020 - 4:00pm
Venue:
Zoom ID 959 3090 9831 (locks at 4:10pm PST)
Speaker:
Lucas Benigni (Univ of Chicago)
Abstract / Description:

We consider eigenvector statistics of large random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, it can be shown that extremal coordinates are no larger than $C\sqrt{\log N/N}$ with high probability.

There has been an extensive amount of work on generalizing such a result, known as delocalization, to a more general entry distribution. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdös–Yau, the analysis of high moments as well as new level repulsion estimates which will be presented during the talk.

This is based on a joint work with P. Lopatto.


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Fixed-Fixed Beam with Point Load

Fixed-Fixed Beam with Point Load is a beam type that has fixed supports on both ends and transverse point load located in the middle.

Fixed-Fixed Beam with Point Load

Fixed-Fixed Beam Equations

Reactions

$latex R_A = R_B = \frac{P}{2} &s=1$

Shear Force

$latex V(x) = q (\frac{L}{2}-x) &s=1$
$latex V_A = -V_B = \frac{P}{2} &s=1$

Bending moment

$latex M(x) = \frac{P}{8} (4x – L) \quad X = (0, \frac{L}{2}) &s=1$
$latex M_{min}(ends) = – \frac{PL}{8} &s=1$
$latex M_{max}(center) = \frac{PL}{8} &s=1$

Deflection

$latex \delta(x) = \frac{Px^2}{48EI}(3L – 4x) \quad X = (0, \frac{L}{2}) &s=1$
$latex \delta_{max}(center) = \frac{PL^3}{192EI} &s=1$


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# Blind wavelet compression of the solution of a nonlinear PDE with singular forcing term within optimal order cost : stability of restricted approximation to small errors

Abstract : Level lifting of the wavelet expansion is related to an interpolation result for Sobolev $\mathrm H^s$ spaces; the nonlinear $N$-term approximation is linked with a non-linear interpolation result for Sobolev spaces $\mathrm W^{s,p}$ non-compactly included into $\mathrm L^2$. \cite{CDH} introduced the intermediate notion of restricted approximation. Based on this, we construct an optimal order resolution algorithm extending beyond the linear elliptic case of \cite{CDD98}, as we illustrate numerically. We undeline that optimal order adaptivity implies the blind compression of the unknown of the PDE. We illustrate on a univariate version of the bivariate PDE $\Delta u+\mathrm e^{cu}=0$, $c{>}0$, used to benchmark three nonadaptive multilevel methods \cite{H}.\\ %At the end, we present an efficient search of relevant coefficients of %the second derivative of the father wavelet in its dual wavelet basis; this %search enlarges the range of the Lebesgue index $p$ allowed for the %application. The adaptiveness of our algorithm is highlighted by the addition in this illustration of a singular forcing term. This term is an element of $$\mathrm H^{-1}$$ but does not belong to $$\mathrm H^{-\frac56}$$: more precisely, it is the second derivative of $t\mapsto |3t{-}1|^{2/3}$. This illustration passed the numerical implementation test ($C\varepsilon^{-0.505}$ flops). The algorithm's convergence and cost ($\varepsilon^{-\frac d{s-1}}$ where $\varepsilon$ is the final error in $\mathrm H^1$ norm) in both univariate ($d=1$) and bivariate ($d=2$) general cases is shown to have optimal order, with $$s$$ less than three.
Keywords :
Document type :
Journal articles
Domain :

https://hal.inrae.fr/hal-02677479
Contributor : Migration Prodinra Connect in order to contact the contributor
Submitted on : Sunday, May 31, 2020 - 7:26:13 PM
Last modification on : Thursday, November 19, 2020 - 4:22:03 PM

### Citation

Daniel Goujot. Blind wavelet compression of the solution of a nonlinear PDE with singular forcing term within optimal order cost : stability of restricted approximation to small errors. Journal of Scientific Computing, Springer Verlag, 2005, 29 (2), pp.257-297. ⟨10.1007/s10915-005-9003-6⟩. ⟨hal-02677479⟩

Record views


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## Invitation to Computer Science 8th Edition

First, we search for Sherman in the list. Since it's the first element of the list, it takes just one comparison to get to it. To find Jane we perform 2 comparisons. To find Ted, which is the third on the list, we need 3 comparisons. It's easy to notice that for each n-th name we need n comparisons. The average of the comparisons is the sum $1+2+3+4+5+6+7$ divided by $7 .$ We can use the formula $\frac{n *(n+1)}{2}$ for the sum. Dividing the sum by $n$ (7 in our case) gives us $\frac{n+1}{2},$ in our case $\frac{7+1}{2}=4$ We have reached the expression that we have used previously $\frac{n+1}{2}$


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# Is the inverse of the Fourier transform $L^1(\mathbb R)\to (C_0(\mathbb R),\Vert \cdot \Vert_\infty)$ bounded?

As is well-known, the Fourier-transform is an injective bounded linear operator from $$L^1(\mathbb R)$$ to $$(C_0(\mathbb R),\Vert \cdot \Vert_\infty)$$, where the latter is the space of continuous functions on $$\mathbb R$$, vanishing at infinity, equipped with the supremum norm.

My question: is the inverse operator (defined on the image) bounded?

I.e., is there a $$C>0$$ such that $$\Vert \hat f \Vert_\infty \geq C \Vert f \Vert_{L^1}\qquad \forall\; f\in L^1(\mathbb R) ?$$

Edit: Maybe I should add (in view of the first comment) that I expect the answer to be negative $$-$$ in fact, I've spent some time trying to construct simple counterexamples but without success. Therefore, a way to re-phrase this "provocative" question is:

How can we see that the answer is no? What is a nice counterexample?

2nd Edit: Perhaps it is easier to answer the analogous question for the discrete Fourier transform $$l^1(\mathbb Z)\to (C(S^1),\Vert \cdot \Vert)$$, where $$S^1$$ is the circle.

• No. It seems there should be a very simple counterexample, not sure what it is right now. If you can't find a simpler example a modification of the Rudin-Shapiro polynomials should work. Mar 8, 2020 at 11:03
• Thank you for your comment, I agree that there should be a counterexample. I edited the question accordingly.
– B K
Mar 8, 2020 at 11:11
• A simple counter-example in the discrete case is given by the Dirichlet kernel: $\|D_n\|_1 \to \infty$ but the Fourier coefficients of $D_n$ are bonded by $1$. Mar 8, 2020 at 12:30
• @ Kavi Rama Murthy: unfortunately, what you say concerns the opposite direction of my question, which would be to consider $L^1(S^1)$ and $l^\infty(\mathbb Z)$.
– B K
Mar 8, 2020 at 13:15
• ??? One can certainly regard $D_n$ as an element of $L^2(S^1)$; I don't follow your objection. In any case, I wrote out the details of using the Dirchlet kernel here in an edit below... Mar 9, 2020 at 15:11

Here we have two counterexamples. I suspect that many readers will consider the first example "simpler". I actually feel like the second is simpler, because it starts from less; I actually see every detail of why the second example works, while the first depends on mysterious previous knowledge. Anyway

## First Example

As suggested in a comment but disputed in another comment, one can use the Dirichlet kernel to give a simple counterexample.

Say $$\phi\in L^1$$, $$\delta>0$$, $$|\phi(t)|\ge \delta\quad(|t|\le\delta),$$and $$\hat \phi$$ is supported in $$(-1/2,1/2)$$. Let $$f_n(t)=\phi(t)D_n(t)=\phi(t)\sum_{j=-n}^ne^{ijt}.$$Then $$||f_n||_1\ge\delta\int_{-\delta}^\delta|D_n(t)|\,dt\to\infty$$but $$||\hat f_n||_\infty=||\hat\phi||_\infty.$$

## Second Example

Lemma. If $$f\in L^1(\Bbb R)$$ then $$\lim_{\xi\to\pm\infty}\int_{-\infty}^\infty|1-e^{i\xi t}||f(t)|\,dt=\alpha||f||_1$$, where $$\alpha=\frac1{2\pi}\int_0^{2\pi}|1-e^{it}|\,dt.$$

(WLOG $$f$$ is uniformly continuous with compact support...)

Note that $$\alpha>1$$. Fix $$\beta\in(1,\alpha)$$.

Now say $$f_0\in L^1$$ with $$||\hat f_0||_\infty=1$$ and $$\hat f_0$$ supported in $$[0,1]$$. We're going to let $$f_1(t)=(1-e^{i\xi t})f(t)$$for a suitable constant $$\xi$$. If $$\xi$$ is large enough we have $$||\hat f_1||_\infty=||f_0||_\infty=1,$$ $$||f_1||_1\ge\beta||f_0||_1.$$Repeat: $$||\hat f_n||_\infty=1$$, $$||f_n||_1\ge\beta^n||f_0||_1\to\infty$$.

• That is a nice argument. Somehow I have the feeling that one might be able to simpliy it further... I'll think about it while waiting for more answers.
– B K
Mar 8, 2020 at 14:02


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Frankfurt UAS, KMITL, VGU & SPbGUT - Combinatorics & Number Theory

#### Start

2017-03-26 21:00 AKDT

## Frankfurt UAS, KMITL, VGU & SPbGUT - Combinatorics & Number Theory

#### End

2017-03-30 10:00 AKDT
The end is near!
Contest is over.
Not yet started.
Contest is starting in -1667 days 3:17:19

85:00:00

0:00:00

# Problem SInteresting Integers

source: xkcd.com/587

Undoubtedly you know of the Fibonacci numbers. Starting with $F_1 = 1$ and $F_2 = 1$, every next number is the sum of the two previous ones. This results in the sequence $1,1,2,3,5,8,13,\ldots$.

Now let us consider more generally sequences that obey the same recursion relation

\begin{equation*} G_ i = G_{i-1} + G_{i-2} \qquad \text {for } i>2 \end{equation*}

but start with two numbers $G_1 \leq G_2$ of our own choice. We shall call these Gabonacci sequences. For example, if one uses $G_1=1$ and $G_2=3$, one gets what are known as the Lucas numbers: $1,3,4,7,11,18,29,\ldots$. These numbers are – apart from $1$ and $3$ – different from the Fibonacci numbers.

By choosing the first two numbers appropriately, you can get any number you like to appear in the Gabonacci sequence. For example, the number $n$ appears in the sequence that starts with $1$ and $n-1$, but that is a bit lame. It would be more fun to start with numbers that are as small as possible, would you not agree?

## Input

On the first line one positive number: the number of test cases, at most $100$. After that per test case:

• one line with a single integer $n$ ($2 \leq n \leq 10^9$): the number to appear in the sequence.

## Output

Per test case:

• one line with two integers $a$ and $b$ ($0 < a \leq b$), such that, for $G_1=a$ and $G_2=b$, $G_ k = n$ for some $k$. These numbers should be the smallest possible, i.e., there should be no numbers $a’$ and $b’$ with the same property, for which $b’<b$, or for which $b’=b$ and $a’<a$.

Sample Input 1 Sample Output 1
5
89
123
1000
1573655
842831057

1 1
1 3
2 10
985 1971
2 7


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LaTeX force slash fraction notation

Is it possible to force latex to use the slash notation for a fraction (i.e., separating the numerator and denominator with a slash instead of stacking them on top of one another)? Just using a slash (/) instead of \frac looks bad because the slash does not size itself properly.


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# Alternative Proofs To Euclid's Propositions

by moe darklight
Tags: alternative, euclid, proofs, propositions
 P: 411 So lately I've been trying to start practicing proving things without being given a prompt (as in actually finding something out, not answering the question: "prove so and so"), which I'd never done before. -- But then the bummer is not knowing the name of what you've just proved So after searching I found out that one is a proof of Euclid's proposition 37 from book I (Triangles which are on the same base and in the same parallels equal one another). -- Is there a website where I can find alternative proofs from the one given in the actual book? I used an algebraic proof and I want to find out who came up with it. -- I found websites with a bazillion (ok, more like 3 or 4) alternative proofs to the pythagorean theorem, but I can't find a website with proofs to the other propositions. Is there like a proofs directory somewhere out there or something?

 Related Discussions Set Theory, Logic, Probability, Statistics 3 Calculus & Beyond Homework 0 General Math 48 General Discussion 14


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# Round Up

• April 25th 2008, 03:10 PM
recca
Round Up
NOT SURE. I THINK ADAMS WAY IS NOT VALID 2ND OPION

ADAM HAS MADE UP HIS OWN METHOS OF ROUNDING. STARTING AT THE RIGHTMOST PLACE IN A DECIMAL NUMBER, HE KEEPS ROUNDING TO THE VALUE OF THE NEXT PLACE TO THE LEFT UNTIL HE REACHES THE PLACE TO WHICH THE DECIMAL NUMBER WAS TO BE ROUNDED. FOR EXAMPLE, ADAM WOULD USE THE FOLLOWING STEPS TO ROUND 11.3524 TO THE NEAREST TENTH:

11.3524-11.352-11.35-11

IS ADAMS METHOD A VALID WAY TO ROUND?

THANK YOU (Doh)
• April 25th 2008, 04:44 PM
xifentoozlerix
Quote:

Originally Posted by recca
NOT SURE. I THINK ADAMS WAY IS NOT VALID 2ND OPION

ADAM HAS MADE UP HIS OWN METHOS OF ROUNDING. STARTING AT THE RIGHTMOST PLACE IN A DECIMAL NUMBER, HE KEEPS ROUNDING TO THE VALUE OF THE NEXT PLACE TO THE LEFT UNTIL HE REACHES THE PLACE TO WHICH THE DECIMAL NUMBER WAS TO BE ROUNDED. FOR EXAMPLE, ADAM WOULD USE THE FOLLOWING STEPS TO ROUND 11.3524 TO THE NEAREST TENTH:

11.3524-11.352-11.35-11

IS ADAMS METHOD A VALID WAY TO ROUND?

THANK YOU (Doh)

Nope. It is not a good way to round. For example, look at the number $2.468$. If you round to nearest integer (ones place) , doing each digit one by one you get $2.468\implies2.47\implies2.5\implies3$, whereas if you use only the tenths digit you get $2.468\implies2$


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# Study of the maximum and minimum points of the function.

Does the function f (x) = x ^ 3 + x ^ 2 + x + 1 at x = 0 have some maximum or minimum point? In other exercises to make the 1st derivative, you can factor terms and give two values ax but in this function I could not factor. Making the first derivative is left f'(x) = 3x ^ 2 + 2x + 1, but I can not factor. I hope they help me because it is very messy.

2voto

Anon User Points 0
Bluenile, the derivative that you calculated is ok. For quadratic polynomials, there is a very direct way of finding the roots and is using the following expression:
\begin{align}&{\color{blue}a}x^2+{\color{red}b}x+{\color{green}c}=0\\&x_{1,2}=\frac{-{\color{red}b}\pm \sqrt{{\color{red}b}^2-4\cdot {\color{blue}} \cdot {\color{green}c}}}{2\cdot {\color{blue}a}}\\&in this case we have that:\\&a=3\\&b=2\\&c=1\\&\text{so the expression becomes:}\\&x_{1,2}=\frac{-{\color{red}2}\pm \sqrt{{\color{red}2}^2-4\cdot {\color{blue}3} \cdot {\color{green}1}}}{2\cdot {\color{blue}3}}=\\&\frac{-2 \pm \sqrt{-8}}{6}\end{align}As you can see from this expression became the root of a negative number which means that it does not have real roots.

Taking advantage of the use of the geogebra graph the function, and I think you will realize that you do not have ends (which if you have is a point of inflection at x=-1/3)

Salu2

2voto

Anon User Points 0
So...
\begin{align}&f(x)=x^3+x^2+x+1\\&\\&f'(x)=3x^2+2x+1\\&\\&\text{The question is if x=0 is an extreme. If we calculate f'(0) we have that is equal to 1}\\&\text{either x=0 is not an endpoint. Then you ask if the function f has extremo}\\&\\&f'(x)=3\left(x^2+\frac{2}{3}x+\frac{1}{3}\right)\\&\\&f'(x)=3\left(x^2+\frac{2}{3}x+\frac{1}{9}-\frac{1}{9}+\frac{1}{3}\right)\\&\\&f'(x)=3\left(x^2+\frac{2}{3}x+\frac{1}{9}+\frac{2}{9}\right)\\&\\&f'(x)=3\left(x^2+\frac{2}{3}x+\frac{1}{9}\right)+\frac{2}{3}\\&\\&f'(x)=3\left(x+\frac{1}{3}\right)^2+\frac{2}{3}\\&\\&f'(x)\geq \frac{2}{3} , \forall x \in\mathbb{R}\\&\\&\text{let us Remember this, if x=x_0 is an endpoint to a function f:X\subset \mathbb{R}\to \mathbb{R}, then}\\&\text{f'(x_0)=0 so by comparison, our function does not have extreme }\forall x \in \mathbb{R}, \text{is more}\\&\text{the function is always increasing}\\&\\&\end{align}

### TheXpert.space

TheXpertSpace is an online community of people who give and receive advice.
You can check other people responses or create a new question if it's not already solved


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# The Universe of Discourse

Sun, 06 Aug 2017

Yesterday I discussed an interesting failure on the part of Shazam, a phone app that can recognize music by listening to it. I said I had no idea how it worked, but I did not let that stop me from pulling the following vague speculation out of my butt:

I imagine that it does some signal processing to remove background noise, accumulates digests of short sections of the audio data, and then matches these digests against a database of similar digests, compiled in advance from a corpus of recordings.

Julia Evans provided me with the following reference: “An Industrial-Strength Audio Search Algorithm” by Avery Li-Chun Wang of Shazam Entertainment, Ltd. Unfortunately the paper has no date, but on internal evidence it seems to be from around 2002–2006.

M. Evans summarizes the algorithm as follows:

1. find the strongest frequencies in the music and times at which those frequencies happen
2. look at pairs !!(freq_1, time_1, freq_2, time_2)!! and turn those into pairs into hashes (by subtracting !!time_1!! from !!time_2!!)
3. look up those hashes in your database

She continues:

so basically Shazam will only recognize identical recordings of the same piece of music—if it's a different performance the timestamps the frequencies happen at will likely be different and so the hashes won't match

Thanks Julia!

Moving upwards from the link Julia gave me, I found a folder of papers maintained by Dan Ellis, formerly of the Columbia University Electrical Engineering department, founder of Columbia's LabROSA, the Laboratory for the Recognition and Organization of Speech and Audio, and now a Google research scientist.

In the previous article, I asked about research on machine identification of composers or musical genre. Some of M. Ellis’s LabROSA research is closely related to this. See for example:

There is a lot of interesting-looking material available there for free. Check it out.

(Is there a word for when someone gives you a URL like http://host/a/b/c/d.html and you start prying into http://host/a/b/c/ and http://host/a/b/ hoping for more goodies? If not, does anyone have a suggestion?)


text:
Preparing a .bed for Homer annotation
1
1
Entering edit mode
6.5 years ago
giroudpaul ▴ 70

Hello

I have a 5 columns bed file with Chr | Peak Start | Peak End | Unique ID | Score

I want to annotate this Bed file using HOMER.

However, and this seem logical to me, homer require a 6th column with the strand information.

I have seen some solutions on how adding a strand column with only + in it, however I am wondering if this makes sense to do this, as the annotation may not be the same depending on the strand.

So my question is, is there a way to add the correct strand information for each peaks, is there a peak calling setting for this (I used macs14, and I didn't found an option that seems to do this) ?

Thanks

Homer annotatePeaks Bed • 3.4k views
0
Entering edit mode

In BED files, a dot "." can be used successfully with many tools to represent a "NULL" value for strand (this defaults to "+" when used with some tools), Did you tried to use dot?

0
Entering edit mode
5.0 years ago
jiab ▴ 60

annotatePeaks.pl in the HOMER package seems to ignore strand information anyway.

I posted about it here: Strand information when doing peak annotation with HOMER


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# Achievements

We have spent a fruitful –and stressful- summer working on an empirical, mathematical and comprehensive approach on the engineering pillars of SynBio. We have fulfilled the Gold Medal requirements: we have documented our standards that were submitted to the Registry of Biological Parts, and also validated them experimentally; the characterization of existing parts has been enormously improved -all the ten BioBricks used in the St2OOL project are combinations of existing parts of the registry, and we believe have performed one of the most complete characterizations of existing Registry parts in terms of stability, standardization and orthogonality; and, last but not least, we have developed an original Human Practices approach dealing with Intellectual Property in Synthetic Biology in the context of Responsible Research and Innovation.

## Yes, we could!

We have performed an unprecedented research effort to determine in detail how biological parts behave under different conditions. We have used several hundreds of Petri dishes and fluorescence –among other- measurements, which have allowed us to finely characterize the behavior of ten selected SynBio devices. Our main results are the following:

1. Escherichia coli is, as expected, an extremely strong chassis for SynBio, as its is able to survive in a very wide rage of environmental stresses, of which we have tested pH, salt, Mars-like vaccum, temperature and radiation. The only factor to which E. coli proved very sensitive to was UV light. By contrast, Biobricks behavior was dependent on environmental conditions: sub-optimal temperatures and vacuum dramatically decreased Biobrick output.

2. Emergent properties. Even in 2014, many new species are discovered hidden in the jungles of our Planet. Similarly, we have been lucky to report here an unexpected emergent property in the best-known bacterium on Earth: thermal oscillations resulted in a significant increase of E. coli DH5-alfa growth . We are still finding out how. Surprisingly, fatigue of materials in biology is not always a deleterious process.

3. Standards not that standard. We were shocked to see only Biobrick 5 behave in a relatively standard way when its output was compared within six different E. coli strains. Variation in the output was the norm, not the exception in what we expected to be standard parts.

4. Orthogonality bench testing. We used a range of techniques (fluorimetry, cell citometry, protein electrophoresis and proteomics) to find out whether one of the simplest device combination possible (two different reporter proteins under the control of the same promoter and on the same plasmid) behave in a orthogonal way. The hypothesis was that, if so, the relative outputs should be approximately equal. Again, we found that engineering assumptions could not be validated: co-transformants with two Biobricks displayed an asymmetric behavior indicating a clear lack of orthogonality, and a huge cell-to-cell variation in output signal was measured .

5. What is the effect of a genetic modification in the proteome? We performed a proteomics analysis of one of our Biobricks (Bb1, GFP) and compared it with a wild-type strain and an empty plasmid-bearing one. Three replicates were analysed. The results reveal a small impact in the proteome profile of both the empty plasmid and the plasmid with Biobrick1. Only chaperones and other few proteins were affected. To the best of our knowledge, this is the first proteomic demonstration of the effect on the proteomic architecture of the introduction of a simple SynBio device.

6. Bridging gaps in IP. We wanted to contribute to the IP controversy in a new way. We propose a marriage between the concepts of Responsible Research and Innovation and the Diverse Ecology scenario, being the former a key tool to choose within the multiple ecology of IP forms. We have contributed to that goal by creating dictionaries of SynBio terms for lawyers, and of legal terms for Synthetic Biologists, and by making a politically incorrect proposal: why not using Maths to help experts on IP take their decisions?

7. Modeling and wetlab, together. We modeled the four legs of the St$^2$OOL: from the stability of Biobricks (which proved to fit the shape of our actual data) to the Human Practices equation. We studied in depth orthogonality and the behavior of plasmids co-transformed into a host cell; we have created a software to generate promoter sequences with a desired level of expression and compute the stability & standardization (St$^2$) index and proved able to both model and forecast a range of biological experimental data.

8. On weakness and strength. Our experimental results and modeling analysis confirm that engineering pillars cannot be taken for granted in SynBio. A new bioprospection point of view is thus needed. We have developed a DNA extraction robot “The Tool” to help us carry out a functional metagenomic approach directed at identifying natural DNA parts with particularly standard and stable behaviors. If engineering assumptions are not met by the Biobricks we tested, let’s isolate/build new ones with the desired properties.

## Beyond scientific achievements

We really had fun! In our “Art & More section” you’ll find Paula’s great watercolors for the wiki, the poster and the presentation; Kristie’s epigenetic music interpreted as what she calls “genomic serialism”; and videos such as the YMCA parody “The St2OOL is going to i.G.E.M.”

## And finally, the St2OOL quote

"I never use a piano stool. I always use a drum stool. Because I feel that when you're down there, you're playing in that way you're supposed to. I like to be above it." Jamie Cullum