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https://math.stackexchange.com/questions/413308/problems-using-euler-maclaurin-for-sum-e-2-pi-z-a2
|
# Problems using Euler-Maclaurin for $\sum e^{-2 \pi z a^2}$
I'm trying to evaluate $\sum_{a=-\infty}^{\infty} e^{-2 \pi z a^2}$ using Euler-Maclaurin, but I get $\frac{1}{\sqrt{2z}}$. The only alternative I have is to calculate the remainder term directly for a small level of approximation, but I don't want to do this if there's a simple mistake I'm making and fixing it would let me avoid doing so, or if for some reason Euler-Maclaurin doesn't work at all.
SUMMARY:
By Euler-Maclaurin, we have:
$\sum_{a=-\infty}^{\infty}e^{-ca^2}-\int_{-\infty}^{\infty} e^{-ca^2} da = \sum_{k=2}^{\infty} \frac{b_k}{k!}(e^{-cx^2})^{(k-1)'}\vert_{-\infty}^{\infty}$, where $b_k=B_k(0)$ are the Bernoulli numbers. Implicitly, the remainder term (due to the large factorial) and the halved endpoints of the sum (due to $e^{-ca^2}$ vanishing in the infinite limits) vanish.
First, observe $\frac{d^n}{da^n}e^{-c a^2}$ is of the form $e^{-c a^2}p(a)$ for a polynomial $p$. (1)
Secondly, observe that $lim_{a \rightarrow \infty} \frac{p(a)}{e^{c a^2}} = 0$ for any polynomial $p$ and constant $c \gt 0$. (2)
Thirdly, observe that $e^{-ca^2}$ is even, and hence its odd derivatives (the derivatives taken for even $k$) are odd, and its even derivatives (odd $k$) are even.
The latter implies that for odd $k$, $(e^{-cx^2})^{(k-1)'}\vert_{-\infty}^{\infty}$ vanishes, and for even k, $(e^{-cx^2})^{(k-1)'}\vert_{-\infty}^{\infty} = 2 \lim_{a \rightarrow \infty} (e^{-cx^2})^{(k-1)'}(a) = 2 \lim_{a \rightarrow \infty} e^{-ca^2}p(a) = 0.$
Therefore all terms of the series vanish, implying $\sum_{n=-\infty}^{\infty}e^{-cx^2} = \int_{-\infty}^{\infty} e^{-cx^2} da = \frac{\sqrt{\pi}}{\sqrt{c}}$, and in this case $\frac{\sqrt{\pi}}{\sqrt{2 \pi z}} = \frac{1}{\sqrt{2z}}.$
DETAILS:
(1) follows by induction from $\frac{d}{da}e^{-ca^2}p(a) = e^{-ca^2}(\frac{d}{da}p(a)-2ca\ p(a)),$ which is of the form $e^{-ca^2}p(a)$.
Proof of (2): $\frac{\frac{d}{dx}a^n}{\frac{d}{dx}e^{c a^2}} = \frac{a^{n-1}}{2cae^{c a^2}} = \frac{a^{n-2}}{2ce^{c a^2}}, \implies \frac{\frac{d^m}{dx^m}a^n}{\frac{d^m}{dx^m}e^{c a^2}} = \frac{a^{n-2m}}{(2c)^m e^{ca}}, \implies lim_{a \rightarrow \infty} \frac{a^n}{e^{c a^2}} = 0 \implies lim_{a \rightarrow \infty} \frac{p(a)}{e^{c a^2}} = 0$
I have also derived an explicit formula for the derivatives from Faà di Bruno's formula together with $\frac{d^n}{dx^n}f(cx)=c^nf^{(n)}(cx)$, which yields a polynomial multiple of $e^{-ca^2}$, which by (2) vanishes in the limit, and is again even for even derivatives and odd for odd ones.
• A relevant computation that it might be useful to consult can be found at this MSE link. Apr 15, 2014 at 23:15
I just answered a very similar question here. Without writing everything out again, let me point out that you can use Euler-Maclaurin to deduce, for any $N>0$, that $$\sum_{a=-\infty}^{\infty} e^{-2 \pi z a^2} = \frac{1}{\sqrt{2z}} + O(z^N) \ \mbox{as}\ z \to 0.$$
• So, what I said about the remainder term vanishing is false. To calculate the remainder term properly, I'm guessing I should take the $\int_n^m p_{2k}(x)\frac{B_{2k}(x-\lfloor x \rfloor)}{(2k)!} dx$ term to be the integral over $\mathbb{R}$ of the limit as $k$ goes to infinity of each taylor series, and if I have uniform convergence of that sequence of functions that would show I have the Taylor series for the function inside the integral. Or is there a better way to get more terms of the infinite sum, perhaps treated in that book? Mar 23, 2014 at 15:16
• It seems that you can't take the limit of the integrals over $\mathbb{R}$ as the level of approximation goes to infinity, because otherwise we can replace the periodic Bernoulli functions' products with a sum over integrals from 0 to 1 of products of the Bernoulli polynomials with shifted versions of the factors, leaving inside each integral a ratio of two polynomials of the same degree, multiplied by the inverse-exponential and inverse-factorial factors. If I remember correctly, at that point the factorial still wins out. Mar 23, 2014 at 15:28
I've found out Euler-Maclaurin only works for functions of some exponential type ($\lt \pi$?), and I assume that any Gaussian has no exponential type, and is bounded by functions of too large an exponential type to apply the formula to those, too.
• I never did say, but a function has exponential type a if it's bounded by $e^{ax}$ in the whole complex plane. So on the imaginary line, instead of a nice, constant-bounded Gaussian you get $e^{x^2}$. Mar 24, 2014 at 21:47
|
2022-07-02 22:50:52
|
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|
https://www.mrkwatkins.co.uk/2018/03/04/elena-the-first-demo/
|
I've managed to get my new programming language to a state where I can release a demo that is vaguely interesting. Here is a video of the demo running on the CSpect emulator:
Yes, it's a bouncy snake thing. Here is the Elena source code:
using System;
using System.Maths;
using System.Spectrum;
using System.Spectrum.Next;
namespace Elena.Test
{
static type Program
{
private static UnsignedByte BorderColour = 1b;
private static Array[Particle] Particles = Array[Particle](12b);
entrypoint void Main()
{
Screen.Cls(0b);
Screen.SetBorder(BorderColour);
InitializeParticles();
InitializeSprites();
while (true)
{
UpdateParticles();
}
}
private static void InitializeParticles()
{
UnsignedWord position = 32w;
for (Particle particle in Particles)
{
particle.Position = UnsignedWordVector(position, position);
particle.Velocity = SignedByteVector(1sb, 1sb);
position = position + 12w;
}
}
private static void InitializeSprites()
{
Sprites.SetAreVisible(true);
Sprites.SetPattern(0b, Patterns.Particle);
}
private static void UpdateParticles()
{
UnsignedByte index = 0b;
for (Particle particle in Particles)
{
UpdatePosition(particle);
UpdateVelocity(particle, index);
Sprites.SetAttributes(index, particle.Position.X, particle.Position.Y, 0b, true);
index++;
}
}
private static void UpdatePosition(Particle particle)
{
particle.Position += particle.Velocity;
}
private static void UpdateVelocity(Particle particle, UnsignedByte particleIndex)
{
if ((particle.Position.X == 32w) || (particle.Position.X == 270w))
{
particle.Velocity.X = -particle.Velocity.X;
if (particleIndex == (Particles.Length - 1b))
{
CycleBorder();
}
}
if ((particle.Position.Y == 32b) || (particle.Position.Y == 208b))
{
particle.Velocity.Y = -particle.Velocity.Y;
if (particleIndex == (Particles.Length - 1b))
{
CycleBorder();
}
}
}
private static void CycleBorder()
{
if (BorderColour == 7b)
{
BorderColour = 1b;
}
else
{
BorderColour++;
}
Screen.SetBorder(BorderColour);
}
}
type Particle
{
public UnsignedWordVector Position;
public SignedByteVector Velocity;
}
static type Patterns
{
public static Array[UnsignedByte] Particle = 0xE3E3E3E3E3242424242424E3E3E3E3E3E3E3E324252549494949252524E3E3E3E3E3242549496D6D6D6D49492524E3E3E32425496D6D6D92926D6D6D492524E3E325496D6D9292B6B692926D6D4925E32425496D92B6B6B6B6B6B6926D49252424496D6D92B6DBDBDBDBB6926D6D492424496D92B6B6DBFFFFDBB6B6926D492424496D92B6B6DBFFFFDBB6B6926D492424496D6D92B6DBDBDBDBB6926D6D49242425496D92B6B6B6B6B6B6926D492524E325496D6D9292B6B692926D6D4925E3E32425496D6D6D92926D6D6D492524E3E3E3242549496D6D6D6D49492524E3E3E3E3E324252549494949252524E3E3E3E3E3E3E3E3242424242424E3E3E3E3E3_a;
}
}
You can see some of the language features in play:
• Complex types. Types can be defined with constructors, fields, methods and operators. Types are organised via namespaces.
• Generics. Arrays are defined as a generic type, Array[T]. There are two instances on display here, Array[Particle] and Array[UnsignedByte].
• for...in loops. Very much like foreach loops in C# these will loop over all items in an array.
• Hex literals. Because sometimes you just need a blob of bytes.
There are a few features still to implement that will make the above code simpler:
• Type classes. I intend to add type classes as a way to get polymorphism into the language, inspired by Haskell's type classes and Scala's traits. Sort of like C# interfaces that can have an implementation I guess... (Which might be the shape of things to come for C#) For example there will be a Numeric type class that represents numerical types, which bytes, words and floats will be a member of, and will define operations such as addition and subtraction.
• Generic constraints. Once we have type classes it makes sense to have constraints on generic types. In the example above I have had to define different vector types for each underlying number type. (UnsignedWordVector and SignedByte) When constraints are in place I'll have a Vector[T] where T has to be a member of the Numeric type class, meaning I can then have Vector[UnsignedWord] and Vector[SignedByte] instead.
• Return types. Notice how every function has a return type of void? The only functions that currently return are constructors. I'm still not sure on the model for handling allocation and release of memory and as such return types are not yet implemented correctly. I suspect this will take several iterations to get right. I have several ideas in mind but I think I'll need to code up some programs with the language to see which works best.
• Currently every numeric literal has to have a suffix which specifies it's type - the compiler should be able to infer the type in most cases.
However before that I need to get the compiler into a state where it can be released for people to use. Currently it will definitely work if the Elena program is correct, but if there are errors in the program then the compiler might give you a vaguely helpful message, or it might just crash. So next step is to get the error handling and reporting up to scratch. That will make subsequent development a lot easier for myself as well as getting the compiler towards a state where I can release it to the public.
|
2019-01-20 20:09:00
|
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|
https://modula3.elegosoft.com/cm3/cgi-bin/man2html.cgi?local=/usr/local/cm3/man/man7/m3makefile.7
|
m3makefile(7) MODULA-3 PROGRAMMERS MANUAL m3makefile(7)
NAME
m3makefile- The Modula-3 build file written in the quake extension lan-
guage.
m3overrides- Overrides elements of the m3makefile when building
"local".
SYNOPSIS
m3makefile
DESCRIPTION
When building moderately or very complex software packages, the com-
piler needs help identifying all that needs to happen. This is where
the m3makefile and m3overrides file come into play.
The exporting of packages into the modula-3 system wide (or global)
repository is managed from these files.
Both the m3makefile and m3override files are scripts used in the build-
ing of Modula-3, programs and libraries, and identifying items to be
exported as global entities.
A package may be build as either a "local" or "global" entity. The
m3overrides file is only used when building a "local" or non-exported
entity rather than a "global" or exported package. The compiler switch
-override is used to inform the cm3 compiler to read the m3overrides
file before it reads the m3makefile. Some packages may indicate in
their m3makefiles that they are only local , and are not exported to
the global repository. The main procedure called by the m3overrides
file is override(), as described below.
Both the m3makefile and m3overrides files are written in the quake(7)
extension language and are interpreted by the modula-3 compiler (the
interpreter is built into the cm3(1) compiler).
For both these files, cm3 first defines an additional set of func-
tions/procedures beyond the built-in functions and procedures of the
quake language. These are specific to the tasks of building software.
These tasks are not only building both programs and libraries, but also
Both these files are located in the " src " directory of your project,
and any nested subdirectory that gets included via include_dir (see
include_dir() below).
PACKAGING
Modula-3 is distributed as a set of packages. Each package is contained
in a single directory which conforms to a standard structure. The top
level directory should contain a README file (or index.html), and a src
subdirectory containing Modula-3 source files, and a derived directory
for each platform on which the package has been built.
The location of public packages, as well as any other variable is spec-
ified in the cm3.cfg configuration file which is located in the same
directory as cm3 executable program. You can move the compiler or the
public packages around, as long as the configuration file stays at the
same directory as the compiler.
The m3makefile, that describes the package, resides in the src subdi-
rectory of the package. Although it is common for all the Modula-3
sources to also reside in the src directory, they may be distributed in
a more complex directory structure rooted at src.
There are three primary types of packages: source, program, and
library. A source package contains a collection of sources, like html
files; it builds nothing. A program package constructs a single exe-
cutable program by compiling and linking the contents of a set of
source packages. Similarly, a library package constructs a single
library from a set of source packages.
Source Packages
The m3makefile for a source package simply lists the pieces of
source that are to be included in programs or libraries that
include the source package.
Program Packages
The m3makefile for a program package names the sources needed to
build the program, and the packages that it uses to satisfy its
imports. It ends with a single program() or Program()
invocation. See the example.
Library Packages
The m3makefile for a library package names the sources to be
included in the library, and the packages that it uses to sat-
isfy its imports. It ends with a single library() or Library()
invocation.
Note that as in a program, all the imports of a package must be
satisfied. If a package A builds a library and any of the
objects in that library import interfaces from another library
package B, then import(B) must be specified in A's m3makefile.
Use cm3 -ship command to install a private package in the public
repository. Note that on some systems, you must have special
privileges to install public packages.
PROCEDURES
SOURCES
The most primitive declarations in an m3makefile are those that iden-
tify source files. They can be Modula-3, C or assembler files. A source
file has three attributes: a package, the subdirectory within the pack-
age where the declaration occurs, and a visibility. The visibility has
two values, visible and hidden. When a package A imports a package B,
only the visible sources in B are available to compilations that occur
in A. By default all sources are visible. However, it is possible to
explicitly control the visibility of Modula-3 interfaces.
Source files are named relative to the m3makefile that mentions them.
interface(X)
declares that the file X.i3 contains an interface.
implementation(X)
declares that the file X.m3 contains a module.
module(X)
declares both the interface X.i3 and the module X.m3.
generic_interface(X)
declares that the file X.ig contains a generic interface.
generic_implementation(X)
declares that the file X.mg contains a generic module.
generic_module(X)
declares both a generic interface and module.
h_source(X)
declares that the file X.h contains a C include file.
c_source(X)
declares that the file X.c contains a C module.
s_source(X)
declares that the file X.s contains an assembly module.
pgm_source(X)
declares that the file X contains compilable source (e.g. .ic,
.mc, .is or .ms files).
With the exception of h_source, each of the procedures above
defines a hidden source file. There are capitalized variants
Interface, Module, Generic_interface, Generic_implementation,
and Generic_module that identify their sources as visible.
Remember, only those interfaces that are marked as visible will
be available to importers of your library or package.
template(X)
declares that the file X.tmpl contains quake code that is to be
included in this build, and in all builds that import this pack-
age.
The template call is used to extend the set of available proce-
dures. For example, the table package includes a template that
defines the table procedure which instantiates the generic table
module.
IMPORT
Import reflects the fact that programs are built from separately com-
piled pieces, rather than being compiled as a whole. If we always com-
piled from source, include would suffice.
import(P)
The library and visible sources of package P are available to
satisfy IMPORT requests in the current compilation. The
imported sources are not recompiled.
import_version(P,BuildDir)
Like import(P), but the library and visible sources of package P
are selected from building directory BuildDir rather than the
default building directory for the current platform.
include_dir(D)
Recursively include the sources named in the m3makefile in sub-
directory D of the current directory.
include_pkg(P)
Include the sources named in package P's src/m3makefile. The
location of P may be overridden.
BUILDING
library(X)
compile the sources accumulated so far and build a library, X,
from the resulting compiled object files. The visibility of the
library is hidden. Library(X) The same as library.
program(X)
constructs an executable program named X from the given sources.
Program(X)
like program, but X is exported to /bin.
build_standalone()
ensures that the program being built does not depend on dynamic
linking and shared libraries. To have an effect, this procedure
must be called before program or Program is called.
COMPILER OPTIONS
m3_option(x)
adds option x to the set of arguments passed to the compiler.
Specifically, m3_option adds x to the M3OPTIONS variable. x should be
a single string preceded with a hyphen, e.g. m3_option(-O).
Some of the more useful compiler options include:
-why Explain why each compilation is needed (default).
-commands Print the compilation commands as they are started.
-verbose Print what happens to each file.
-times Print a breakdown of elapsed time.
-g Generate debugging symbols (default).
-O Optimize code.
-keep Preserve intermediate files.
-once Don't recompile modules with new opaque info.
Any compiler option may be specified here. See cm3(1).
OVERRIDE
override(P,D)
Use the version of package P in directory D/P
instead of the one in /pkg/ P.
override alters the behaviour of the include_pkg and import_pkg
procedures, and must be executed prior to any such calls to have
an effect.
To help ensure that the public repositories are consistent, "cm3
-ship", and the older "m3ship" will refuse to install any pack-
age built using overrides.
When the -override option is specified, cm3 looks for a file
named m3overrides in the src directory and, if one exists, exe-
cutes it immediately prior to executing m3makefile. By keeping
all override calls in an m3overrides file and not in an m3make-
file, you can build both private and public versions of packages
without editting any files.
The overrides in effect when a package was built are automati-
cally carried forward into importers of the package, so there is
no need to restate the complete set of overrides at every level,
only of those packages that are directly imported into a given
package.
There is a pre-declared variable, WDROOT, that defines the con-
ventional location of a user's private package repository. (see
\ref{VAR-sec}).
FOREIGN OBJECTS AND LIBRARIES
These procedures allow foreign objects and/or libraries to be included
in a Modula-3 program. Here foreign means not written in Modula-3.
import_lib(X,P)
If P/libX.a is a library, includes -LP -lX in the final link
command. Otherwise, includes -lX in the final link command.
import_obj(X)
Include X in the final link command.
EXPORTING FILES
These functions should be used to export files to the public directo-
ries. These public directories are bound to actual directories via a
set of logical assignments specific to your installation.
BinExport(X)
exports source file X to /bin.
BindExport(X)
exports derived file X to /bin.
DocExport(X)
exports source file X
to /doc.
DocdExport(X)
exports derived file X to /doc.
EmacsExport(X)
exports source file X to /emacs. EmacsdExport(X) exports
derived file X to /emacs. HtmlExport(X) exports source file X
to /html.
LibExport(X)
exports source file X to /lib.
LibdExport(X)
exports derived file X to /lib.
ManExport(X,sec)
exports source file X.sec to section sec of /man.
MandExport(X,sec)
exports derived file X.sec to section sec of /man.
HIDING AND EXPORTING
The following procedures can be used in two ways. First to provide a
clearer indication of visibility than is given by the capitalization
convention (which exists partly to support old style m3makefiles).
Second, to change the visibility of imported components. Generally,
it's much better to convince the owners of the exporting package to
give their sources the correct visibility rather than overriding their
initial decision.
Hidden programs are not copied to the /bin directory, exported ones
are.
hide_interface(X)
sets the visibility of interface X.i3 to hidden.
export_interface(X)
sets the visibility of interface X.i3 to visible.
There are also variants that hide or export programs and generics,
hide_program, hide_generic_interface, hide_generic_implementation,
export_program, export_generic_interface, and export_generic_implemen-
tation.
INSTALLATION DEPENDENCIES
The builder contains some built-in support for machine and operating
system dependencies. The package structure makes provision for separate
build directories for different machine and operating system combina-
tions. The default behaviour of cm3 is to generate the compiled object
files, libraries and programs in the build directory corresponding to
the machine and operating system on which cm3 is executing.
The following set of variables exist to allow m3makefiles to be parame-
terised by machine and operating system.
INSTALLATION DEPENDENCIES
The builder contains some built-in support for machine and operating
system dependencies. The package structure makes provision for separate
build directories for different machine and operating system combina-
tions. The default behaviour of cm3 is to generate the compiled object
files, libraries and programs in the build directory corresponding to
the machine and operating system on which cm3 is executing.
The following set of variables exist to allow m3makefiles to be parame-
terised by machine and operating system.
TARGET This variable defines the machine type on which the library or
program being built will execute. It is chosen from the standard
set of machine types on which Modula-3 runs. Check the runtime
or cm3 packages for the complete set.
OS_TYPE
This variable defines the operating system under which the
library or program being built will execute. Currently, Modula-3
supports two operating system variants, POSIX and WIN32. The
former breaks down further into specific variants, but this
variation is not made available to clients. [There is a way if
you absolutely need it, see the unix package.]
BUILD_DIR
This names the package sub-directory in which object files,
libraries and programs will be built. It is usually, but not
always, the same as TARGET.
The net effect of the above allows a single package to build a
family of architectural variants, in different build sub-direc-
tories, where each variant uses the same set of m3makefiles,
parameterized by the above variables. If this degree of flexi-
bility is insufficient, then the extra variation must be speci-
fied in a separate package, which can use include_pkg to access
the shared sources.
PKG_USE
This defines the location of the public package repository, e.g.
/proj/m3/pkg or /usr/local/lib/m3/pkg.
WDROOT This defines the standard location for a user's private package
repository, typically $HOME/m3/pkg. This is typically used in override calls. TARGET This variable defines the machine type on which the library or program being built will execute. It is chosen from the standard set of machine types on which Modula-3 runs. Check the runtime or cm3 packages for the complete set. OS_TYPE This variable defines the operating system under which the library or program being built will execute. Currently, Modula-3 supports two operating system variants, POSIX and WIN32. The former breaks down further into specific variants, but this variation is not made available to clients. [There is a way if you absolutely need it, see the unix package.] BUILD_DIR This names the package sub-directory in which object files, libraries and programs will be built. It is usually, but not always, the same as TARGET. The net effect of the above allows a single package to build a family of architectural variants, in different build sub-direc- tories, where each variant uses the same set of m3makefiles, parameterized by the above variables. If this degree of flexi- bility is insufficient, then the extra variation must be speci- fied in a separate package, which can use include_pkg to access the shared sources. PKG_USE This defines the location of the public package repository, e.g. /proj/m3/pkg or /usr/local/lib/m3/pkg. WDROOT This defines the standard location for a user's private package repository, typically$HOME/m3/pkg. This is typically used in
override calls.
MISCELLANEOUS
The declarations in this section are typically only needed by spe-
cialised applications, for example the Modula-3 compiler or other quake
templates.
source(X)
declares that X contains non-compilable source.
derived_interface(X,V)
adds the derived interface X.i3 to the list of files to be com-
piled. V
must be either VISIBLE or HIDDEN to indicate whether the inter-
face should be available to importers outside this package.
derived_implementation(X)
adds the derived module X.m3 to the list of files to be com-
piled.
derived_c(X)
adds the derived C code X.c to the list of files to be compiled.
derived_h(X)
adds the derived include file X.h to the list of include files
available to the compiler.
EMACS SUPPORT
The following functions support building and installing GNU emacs lisp
code.
Gnuemacs(X)
exports source file X.el to /emacs.
CompiledGnuemacs(X)
exports the source file X.el and compiles and exports the
derived file X.elc to /emacs.
GENERICS SUPPORT
Many of the packages that export generic interfaces and modules also
define m3makefile procedures that will instantiate the generic source
and add it to the list of Modula-3 sources to be compiled. The instan-
tiated interfaces and modules are created in the derived directory, so
they won't clutter up your source directory.
array_sort(nm,elt)
instantiates the ArraySort generics to produce nmArraySort.i3
and nmArraySort.m3 which implement a sort for arrays of elt.T
values.
Array_sort(nm,elt)
like array_sort, but also makes the interface available to
importers outside the current package.
list(nm,elt)
instantiates the List generics to produce nmList.i3 and
nmList.m3 which implement linked lists of elt.T values.
List(nm,elt)
like list, but also makes the interface available to importers
outside the current package.
list_sort(nm,elt)
instantiates the ListSort generics to produce nmListSort.i3 and
nmListSort.m3 which implement a sorting procedure for lists of
elt.T
values. This procedure assumes that list(nm,elt) has been
called.
List_sort(nm,elt)
like list_sort, but also makes the interface available to
importers outside the current package.
pqueue(nm,elt)
instantiates the PQueue generics to produce nmPQ.i3 and nmPQ.m3
which implement a priority queue of elt.T values.
Pqueue(nm,elt)
like pqueue, but also makes the interface available to importers
outside the current package.
sequence(nm,elt)
instantiates the Sequence generics to produce nmSeq.i3, nmSe-
qRep.i3 and nmSeq.m3 which implement a sequence of elt.T values.
Sequence(nm,elt)
like sequence, but also makes the interfaces available to
importers outside the current package.
sorted_table(nm,key,value)
instantiates the SortedTable generics to produce nmSortedTbl.i3
and nmSortedTbl.m3
which implement a sorted table mapping from type key.T to
value.T.
Sorted_table(nm,key,value)
like sorted_table, but also makes the interface available to
importers outside the current package.
table(nm,key,value)
instantiates the Table generics to produce nmTbl.i3 and nmTbl.m3
which implement a table mapping from type key.T to value.T.
Table(nm,key,value)
like table, but also makes the interface available to importers
outside the current package.
MANUAL PAGES SUPPORT
The following calls format and install man pages.
manPage(name,sec)
formats man page name.sec.
ManPage(name,sec)
is like manPage, but also exports the man page to section sec of
/man.
ManExport(X,s)
exports source file X.s to section s
of /man without further formatting.
MandExport(X,s)
export derived file X.s to section s of /man without further
formatting.
NETWORK OBJECTS SUPPORT
The following procedures are used to build programs using network
objects.
netobj(X,T)
runs the network objects stub generator on the interface X.i3 to
produce the network object glue needed to manipulate objects of
type X.T. The resulting source files are included in the cur-
rent build.
Netobj(X,T)
Like netobj, but also exports the resulting interface.
RESOURCES
The following procedures support the inclusion of arbitrary data, known
as a resource, in a program. For example an image as a PNG file.
resource(file)
is shorthand for resource_named(file,file).
resource_named(rd,file)
declares that file is a resource file. It will be accessible via
the reader rd if a bundle is built.
bundle(m)
declares that the module m is to be built as a bundle of the
files specified by the prior calls to resource and
resource_named.
EXAMPLE
For example, here's a simple program composed of a main module, an
imported interface and its implementation.
To begin, create a fresh directory for the package and within that
directory, a directory for the source files:
> mkdir hello
> cd hello
> mkdir src
Create the following source files in the src directory:
In the file src/Main.m3:
MODULE Main;
IMPORT A;
BEGIN
A.DoIt ();
END Main.
In the file src/A.i3:
INTERFACE A;
PROCEDURE DoIt ();
END A.
In the file src/A.m3:
MODULE A;
IMPORT Wr, Stdio;
PROCEDURE DoIt () =
<*FATAL ANY*>
BEGIN
Wr.PutText (Stdio.stdout, "Hello world.0);
Wr.Close (Stdio.stdout);
END DoIt;
BEGIN
END A.
In the file src/m3makefile:
import ("libm3")
implementation ("Main")
module ("A")
program ("foo")
Finally, from the package directory, hello, run cm3. This should/will
compile the three source files and link them with the standard
libraries. The derived files will be placed in a directory that names
the architecture. On an Alpha/AXP machine running OSF, the directory is
called ALPHA_OSF. The executable program will be named foo in the
derived directory.
NOTES
The older style of m3makefiles using the capitalization convention is
depreciated, and its use is highly discouraged.
The quake interpreter is built into the CM3 Modula-3 compiler, whereas
it was a separate executable for earlier ones, such as PM3.
The Critical Mass Modula-3 compiler cm3(1), and the man pages for
quake(7).
For a full list of compiler options, please see cm3(1).
For creating network objects see the manpage for stubgen(1).
For a description of how to access resources from your running program,
see the Bundle and Rsrc interfaces, along with the developer tool
program m3bundle(1).
AUTHOR
(man page) Peter Eiserloh (eiserlohpp -at- yahoo.com)
|
2017-03-30 00:40:48
|
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|
http://www.boristhebrave.com/
|
# Infinite Quadtrees – Fractal Coordinates
A cool technique I’ve wanted to write up for a while is “Fractal Coordinates” described in a paper by Peter Mawhorter. Don’t be scared by the name, it’s essentially a variant on quadtrees that covers the entire 2d plane. Fractal coordinates have some interesting properties that are useful for procedural generation.
But first, let’s catch up on quadtrees.
# How does Planet Work
It’s been over a year since I last deconstructed how a game does its procedural generation. Today we’ll be looking at Planet, a 2016 cosy design toy by one of my favourite developers, Oskar Stålberg.
# Everything you need to know about Quaternions for Game Development
They represent 3d rotations.
That’s it, that’s the article.
# Defining Chess Piece Moves using Regular Expressions
Suppose you wanted to code a simple chess game. One key bit of game logic is to describe what are legal moves for each piece. There’s only 6 types of piece (pawn, knight, bishop, rook, queen, king) so this isn’t exactly a hard task. You can write rules such as:
def canRookMove(from, to):
# Ignores questions about colliding with other pieces
return (from.x == to.x or from.y == to.y) and from is not to
But these days, I’ve been thinking a lot about grids, and the above approach just doesn’t generalize. What if you wanted to play chess on a stranger grid?
What would it mean to play chess on the grid above, or a hexagonal grid, and so on? You’d have to write a whole new set of rules, and they could get very complicated depending on the grid in question. What I want is a language that describes how pieces move, which generalizes to any board. And I think I’ve found it, using regular expressions.
# Learning to Finish Things
For many years I was a hobbyist programmer. I’d try out small projects, experiment, then move on to the next thing. This was a great way to learn a lot, but I’ve got almost nothing tangible to show from that era. Despite the best of intentions, every project reached a point where it started to drag, and I’d get bored and move on.
It was only a problem for the projects I worked on myself. Working for others, I never really found the same problems. It was a problem of motivation and focus.
More recently, it is different. Now when I start hobby projects, there’s a good chance I’ll cross that finish line, and have something I’m ready to share with the world. What changed? Well in part it is increased experience and maturity, things I cannot teach. But also, I have found some strategies and thought processes helpful, and other, very tempting ones, not so much.
In short, I’ve learned to finish, which is a real skill you can learn over time. I thought I’d share with you what has worked for me. Maybe it’ll work for you too. I make software, but I think this advice is generally true for any other spare time activities. There’s three sections – scope, motivation and distractions.
# VoronatorSharp
I’ve relased a new library, VoronatorSharp.
VoronatorSharp is a C# library that computes Voronoi diagrams. The Voronoi diagram for a collection of points is the polygons that enclose the areas nearest each of those sites.
Voronoi diagrams have applications in a number of areas such as computer graphics.
This library features:
• Computes Voronoi diagrams and Delaunay triangulations.
• Voronoi polygons can be clipped to a rectangular area.
• Uses a n log(n) sweephull algorithm.
• The implementation attempts to minimize memory allocations.
• Integrates with Unity or can be be used standalone.
• Uses robust orientation code.
• Handles Voronoi diagrams with only 1 or 2 points, and collinear points.
# Rotation Graphs
Graphs are a data structure we’ve already talked a lot. Today we’re looking at extension of them which is both obvious and common, but I think is rarely actually discussed. Normal graphs are just a collection of nodes and edges, and contain no spatial information. We’re going to introduce rotation graphs (aka rotation maps) that contain just enough information to allow a concept of turning left or right – i.e. rotation.
# Mosaic Paint
As a side-project of a side-project, I’ve made a little painting program. It’s like a normal paint program, only you paint on a tile grid instead of pixels. You can create mosaic and stained-glass style images.
Try it out!
# The Grokalow
One day, the grokalow crawled out of the swamp.
“It looks like a gargantuan alligator”, cried a bystander, as it approached.
“Nay, it is a brobdingnagian crocodile”, countered a second, as the grokalow licked its leathery lips.
“Without a clear definition, we cannot conclude this thing is a threat”, surmised the third, settling the matter.
The grokalow ate them all, with great satisfaction.
# Editable WFC
When I spoke about autotiling, I briefly touched on how it’s possible to use Wave Function Collapse (or other constraint based generators) as a form of autotiling, i.e. user-directed editing of tilemaps.
I’ve usually referred to this technique as “editable WFC“. It’s a combination of autotiling and WFC, and contains the best of both:
• Being an autotiler, it allows users to easily and interactively make changes to an existing level.
• Being constraint based, it automatically ensures that those changes are consistent with the predefined rules of the constraints, potentially making further changes to the level to make it fit
This is different from most other autotilers, which either require manual configuration of patterns used to enforce good behaviour, hidden layers, or come with more stringent requirements on what tiles are available.
|
2023-02-05 22:58:17
|
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|
https://gwburst.gitlab.io/documentation/latest/html/cwb_parameters.html
|
# user_parameters.C¶
The pipeline has a default parameter file called cwb_parameters.C which contains the list of needed variables for the analysis set at a certain value. The user can modify some of these values adding them in the config/user_parameters.C file. We report here all the variable contained in the cwb_parameters.C file, the user can change each of them accordingly to his/her preferences.
To obtain complete list of parameters with default setting:
root -b -l \$cWB_PARAMETERS_FILE
cwb[0] cWB::config cfg; // creates the config object
cwb[1] cfg.Import(); // import parameter from CINT
cwb[2] cfg.Print(); // print the default parameters
This is the complete file:
We divide the file in different sections for simplicity:
Analysis Type of analysis (1G/2G : 1G is obsolete, not supported) Detectors How to include detectors Wavelet TF transformation how to define wavelet decomposition level Conditioning Parameters for Regression Cluster thresholds Pixels and Cluster selection Wave Packet parameters Pixels Selection & Reconstruction Job settings Time segments definition Production parameters Typical parameter for background Simulation parameters Typical parameter for simulation (MDC) Data manipulating Change frame data (amplitude and time shift) Regulator Likelihood regolators Sky settings How to define the sky grid CED parameters Parameters for CED generation Files list How to include frame and DQ files Plugin How to include Plugins Output settings Decide what information to store in the final root files Working directories Set up of working dir
## Analysis¶
char analysis[8]="2G"; // 2G analysis (1G is obsolete, not supported)
bool online=false; // true/false -> online/offline
char search = 'r'; // see description below
• analysis:
Setting first generation detector (1G) or second generation detector (2G) analysis. 1G is obsolete, not more supported
• online:
Defining if the analysis is ONLINE or not
• cfg_search:
putting a letter define search constrains on waveform polarization.
// r - un-modeled
// i - iota - wave (no dispersion correction)
// p - Psi - wave
// l,s - linear
// c,g - circular
// e,b - elliptical (no dispersion correction)
// low/upper case search (like 'i'/'I') & optim=false - standard MRA
// low case search (like 'i') & optim=true - extract PCs from a single resolution
// upper case (like 'I') & optim=true - standard single resolution analysis
## Detectors¶
List of detectors including in the network.
cWB already contained complete information about a set of existing or possible future detectors:
• L1: 4km Livingston
• H1: 4km Hanford
• H2: 2km Hanford (working until S6 run)
• V1: 3km Virgo
• I1: Indian
• K1: KAGRA
Moreover, it is possible to define a not included detector specifying the position on the Earth and the arms direction.
int nIFO = 3; // size of network starting with first detector ifo[]
char refIFO[4] = "L1"; // reference IFO
char ifo[NIFO_MAX][8];
for(int i=6;i<NIFO_MAX;i++) strcpy(ifo[i],""); // ifo[] can be redefined by user
detectorParams detParms[NIFO_MAX];
detectorParams _detParms = {"",0.,0.,0.,0,0.,0.,0.};
for(int i=0;i<NIFO_MAX;i++) detParms[i] = _detParms;
• nIFO
Number of detectors in the network, this number should be less than IFO_MAX=8.
• refIFO
A detector is used as reference for the search in the sky grid. This detectors is never shifted.
• ifo
List of detectors already included in the library. If the user has to redefine a detector, can leave this blank void.
• detParms
List of detectors defined by the user, the internal parameters are:
• Name
• Latitude [degrees]
• Longitude [degree]
• Altitude [m]
• Angle of x arm respect to the North
• Angle of y arm respect to the North
Example:
nIFO = 2;
strcpy(ifo[0],"L1");
strcpy(ifo[1],"");
detParms[1] = {"I1", 14.4, 76.4, 0.0, 0, (+135+0.0 ), 0, ( +45+0.0 )}, // I1
## Wavelet TF transformation¶
Wavelet decomposition transforms data from time domain to Time-Frequency (TF) domain. Original information are stored in TF pixels which can have various TF resolutions DF and DT, such as DF*DT=0.5 The TF resolutions are decided by the wavelet decomposition levels used and the sample rate in time domain. For instance, with a sample rate R and level N we have:
• DF = (R/2)/2^N
• DT = 2^N/R
cWB combines the amplitude TF pixels from multiple TF decomposition levels.
The parameters are:
size_t inRate= 16384; // input data rate
double fResample = 0.; // if zero resampling is not applied (SK: this parameter may be absolete)
int levelR = 2; // resampling level (SK: absolete because of new parameter fsample)
int l_low = 3; // low frequency resolution level
int l_high = 8; // high frequency resolution level
For 2G analysis:
char wdmXTalk[1024] = "wdmXTalk/OverlapCatalog_Lev_8_16_32_64_128_256_iNu_4_Prec_10.bin";
// catalog of WDM cross-talk coefficients
• inRate
Sample rate of input data for all detectors.
fResample
If different from zero, the input data are resample to this new sample rate before starting any decomposition
levelR
Resapling level. It uses wavelet decomposition to downsample data from starting rate to the desired rate. Using wavelet decomposition, levelR is decided according to the formula above.
low
This is the minimum decomposition level used in the analysis.
high
This is the maximum level decomposition analysis.
wdmTalk
for WDM transform, this file containes the information how to apply time shift in TF decomposition.
Example: With the default number, starting from a sample rate of 16384 Hz, the levelR make a dowsample to 4096 Hz. Low and high have respectively TF resolutions of DF/DT = 2ms/256Hz and 62.5ms/8Hz.
## Conditioning¶
The conditioning step removes the persistent lines and apply the whitening procedure. For 2G analysis, the line removal is obtained using the Regression algorithm. The whitening procedure uses the whiteWindow and whiteStride parameters, see whitened procedure.
int levelD = 8; // decomposition level
double whiteWindow = 60.; // [sec] time window dT. if = 0 - dT=T, where T is segment duration
double whiteStride = 20.; // [sec] noise sampling time stride
• levelD
At this decomposition level the pipeline applied the regression algorithm and the whitening procedure.
## Cluster thresholds¶
double x2or = 1.5; // 2 OR threshold
double netRHO= 3.5; // threshold on rho
double netCC = 0.5; // threshold on network correlation
double bpp = 0.0001; // probability for pixel selection
double Acore = sqrt(2); // threshold for selection of core pixels
double Tgap = 0.05; // time gap between clusters (sec)
double Fgap = 128.; // frequency gap between clusters (Hz)
double TFgap = 6.; // threshold on the time-frequency separation between two pixels
double fLow = 64.; // low frequency of the search
double fHigh = 2048.; // high frequency of the search
• netRHO:
Cluster are selected in production stage if rho is bigger than netRHO
• netCC:
Cluster are selected in production stage if rho is bigger than netCC
• bpp: Black Pixel Probability
Fraction of most energetic pixels selected from the TF map to construct events
• Acore:
Threshold used to select the core pixels (in units of sigma) in the supercluster, likelihood stages.
• Tgap and Fgap
Maximum gaps between two different TF pixels at the same decomposition level that can be considered for an unique event
• TFgap
Threshold on the time-frequency separation between two pixels
• fLow and fHigh
Boundary frequency limits for the analysis. Note: This limits are considered directly in the TF decomposition, so the pipeline chooses the nearest frequencies to these values according to the decomposition level
## Wave Packet parameters¶
Pixels Selection & Reconstruction (see The WDM packets)
Select pixel pattern used to produce the energy max maps for pixel’s selection
// patterns: "/" - ring-up, "\" - ring-down, "|" - delta, "-" line, "*" - single
pattern = 0 - "*" 1-pixel standard search
pattern = 1 - "3|" 3-pixels vertical packet (delta)
pattern = 2 - "3-" 3-pixels horizontal packet (line)
pattern = 3 - "3/" 3-pixels diagonal packet (ring-up)
pattern = 4 - "3\" 3-pixels anti-diagonal packet (ring-down)
pattern = 5 - "5/" 5-pixels diagonal packet (ring-up)
pattern = 6 - "5\" 5-pixels anti-diagonal packet (ring-down)
pattern = 7 - "3+" 5-pixels plus packet (plus)
pattern = 8 - "3x" 5-pixels cross packet (cross)
pattern = 9 - "9p" 9-pixels square packet (box)
pattern = else - "*" 1-pixel packet (single)
Select the reconstruction method
pattern==0 Standard Search : std-pixel selection + likelihood2G
pattern!=0 && pattern<0 Mixed Search : packet-pixel selection + likelihood2G
pattern!=0 && pattern>0 Packed Search : packet-pixel selection + likelihoodWP
## Job settings¶
This section sets the time length for the jobs (see also How job segments are created).
// segments
int runID = 0; // run number, set in the production job
double segLen = 600.; // Segment length [sec]
double segMLS = 300.; // Minimum Segment Length after DQ_CAT1 [sec]
double segTHR = 30.; // Minimum Segment Length after DQ_CAT2 [sec]
double segEdge = 8.; // wavelet boundary offset [sec]
• runID
job number to be analysed. This parameters is auotmatically overwritten when using condor submission (see cwb_condor) and cwb_inet command
• segLen [s]
is the typical and maximum job length. This is the only possible lenght is super lags are used. (For super-lags see Production parameters)
• segMLS [s]
is the minimum job lenght in seconds. It could happens that after application of Data Quality it is not possible to have a continous period of lenght segLen, so the pipeline consider the remaining period if this has a lenght bigger than segMLS. This means that job could have segMLS < lenght < segLen.
• segTHR [s]
is the minimum period of each job that survives after DQ_CAT2 application . This means that, if a job of 600 s, has a period of CAT2 less than segTHR, it is discarded from the analysis. If segTHR = 0, this check is disabled.
• segEdge [s]
is a scratch period used by the pipeline for the wavelet decomposition. For each job the first and last segEdge seconds are not considered for the trigger selection.
## Production parameters¶
Production stage consists of time shifting data for each detectors so that reconstructed events are surely due to detectors noise and not gravitational waves.
cWB can perform two shifts type: the first inside the job segment: the second between different jobs segments. We call the first case as lag shifts and the second case as super-lags shift.
In lag case, the pipeline perform circular shifts on the data of a job. Suppose that a job has a lenght T, the pipeline can perform shifts of step ts, which a maximum number of shifts M such as M*ts <= T.
Even if shifts are performed circularly, no data are loss, and shifting the detector A respect to B of the time K, is the same as shifting detector B of the time -K respect to A. So, considering N detectors composing the network, the number of possible lag shifts are (N-1)*M for each job. For N>2 case, the algorithm can perform two different ways:
• shifts only the first detector respect to the other (inadvisable);
• randomly choose from the list of available shifts a subset that are used in the analysis according to user definition. Randomization algorithm depends only on the detector number and the maximum possible shift.
It is possible to write in a text file the lag list applied. The lags are stored with a progressive number which identifies univocally the lags. The lags parameters are:
• lagSize
Lags numbers used (for Simulation stage should be set to 1)
• lagStep
Time step for the shifts
• lagOff
Progressive number of the lag list from which starting to select the subset.
• lagMax
Maximum Allowable shift. If lagMax=0, than only the first detector is shifted, and the maximum allowable shift is given by lagSize parameter. If lagMax > 0 better to chech if lagMax*lagStep < T, otherwise could be possible to loose some lags.
• lagMode
Possibility to write (w)/read (r) the lag list to/from a file.
• lagFile
File name which can be written/read according to the previous parameter. If lagMode=w and lagFile=NULL no file is written. If lagMode=r and tlagFile=NULL the pipeline returns an error
• lagSite
This parameter is a pointer to a size_t array and it is used to declare the detectors with the same site location like H1H2. This information is used by the built-in lag generator to associate the same lags to the detectors which are in the same site. If detectors are all in different sites the default value must be used (lagSite=NULL)
Example : L1H1H2V1
lagSite = new size_t[4];
lagSite[0]=0; lagSite[1]=1; lagSite[2]=1; lagSite[3]=2;
• shifts
Array for each detector which the possibilty to apply a constant circular shifts to the detectors (storically, no more used)
• mlagStep
To limit computational load, it is possible to cicle over the lagSize number of lags in subsets of size equal to mlagStep instead of all the lags together. This reduce computational load and increase computational time.
Examples :
2 detectors L1,H1 : 351 standard built-in lags (include zero lag)
lagSize = 351; // number of lags
lagStep = 1.; // time interval between lags = 1 sec
lagOff = 0; // start from lag=0, include zero lag
lagMax = 0; // standard lags
the output lag list is :
lag ifoL1 ifoH1
0 0.00000 0.00000
1 1.00000 0.00000
2 2.00000 0.00000
... ......... .......
350 350.00000 0.00000
note : values ifoDX are in secs
3 detectors L1,H1,V1 : 350 random built-in lags (exclude zero lag)
lagSize = 351; // number of lags
lagStep = 1.; // time interval between lags = 1 sec
lagOff = 1; // start from lag=1, exclude zero lag
lagMax = 300; // random lags : max lag = 300
the output lag list is :
lag ifoL1 ifoH1 ifoV1
1 158.00000 223.00000 0.00000
2 0.00000 195.00000 236.00000
3 28.00000 0.00000 179.00000
... ......... ....... '''''''''
350 283.00000 0.00000 142.00000
note : values ifoDX are in secs
3 detectors L1,H1,V1 : load 201 custom lags from file
lagSize = 201; // number of lags
lagOff = 0; // start from lag=1
lagMax = 300; // random lags : max lag = 300
lagFile = new char[1024];
strcpy(lagFile,"custom_lagss_list.txt"); // lag file list name
lagMode[0] = 'r'; // read mode
an example of input lag list is :
0 0 0 0
1 0 1 200
2 0 200 1
3 0 3 198
... ... ... ...
200 0 2 199
note : all values must be integers
lags must in the range [0:lagMax]
In the super-lags case, the pipeline consider data of each detectors belonging to different segments, so shifted of a time multiple of T. In this way we can increase easily the number of time lags because it allows to make shifts between data bigger than T (expecially when having two detectors). Once selected different segments the standard circular lags shifts are applied as the different segments would be the same one. The meaning of the parameter are similar to the one of lags case, but here the values are in segments and not in seconds as for the previous case.
A detailed description of the slag configuration parameters is here :
• cWB::Toolbox::getSlagList
Examples :
use standard segment
slagSize = 0; // Standard Segments : segments are not shifted, only lags are applied
// segments length is variable and it is selected in the range [segMSL:segLen]
3 detectors L1,H1,V1 : select 4 built-in slags
slagSize = 4; // number of super lags
slagMin = 0; // select the minimum available slag distance : slagMin must be <= slagMax
slagMax = 3; // select the maximum available slag distance
slagOff = 0; // start from the first slag in the list
the output slag list is :
SLAG ifo[0] ifo[1] ifo[2]
0 0 0 0
1 0 1 -1
2 0 -1 1
3 0 2 1
3 detectors L1,H1,V1 : load 4 custom slags from file
slagSize = 4; // number of super lags
slagOff = 0; // start from the first slag in the list
slagFile = new char[1024];
strcpy(slagFile,"custom_slags_list.txt"); // slag file list name
an example of input slag list is :
1 0 -4 4
2 0 4 -4
3 0 -8 8
4 0 8 -8
note : all values must be integers
## Simulation parameters¶
Simulation stage allows to test efficiency detection of the pipeline. It consists on injecting simulated waveforms (MDC) on the detector data. Once chosen the simulation stage, the pipeline make the analysis only around the injection times (and not all the segments) to reduce computational load.
Waveforms are injected at different amplitudes, each amplitude is repeated for each waveform at the same time, such repeating the analysis for each factor.
int simulation = 0; // 1 for simulation, 0 for production
double iwindow = 5.; // analysis time window for injections (Range = Tinj +/- gap/2)
int nfactor=0; // number of strain factors
double factors[100]; // array of strain factors
char injectionList[1024]="";
• simulation
variable that sets the simulation stage: 1=simulation, 0=production If sets to 2 it sets injections at constant network SNR over the sky, instead of hrss.
• gap
time windows around the time injection that is analysed (+- gap).
• nfactor - factors
list of factors which differ according to the value of simulation.
1. amplitude factors to be multiplied ot the hrss written in the injectionList.
2. network SNR (waveform is rescaled according to these values).
3. time shift applied to the waveforms
4. progressive number referring to the multiple trials for injection volume distribution
• injectionList
path of file containing all the information about inections (waveform type, amplitude, source directions, detector arrival times, …)
## Data manipulating¶
It is possible to apply constant shifts and/or uniform amplitude modifications on detectors data. Here are the parameters that allow to do these things:
• Calibration
double dcCal[NIFO_MAX];
for(int i=0;i<NIFO_MAX;i++) dcCal[i] = 1.0;
Possibility to apply constant modifications on data amplitudes. Different factors can be applied to different detectors. The data are modified in this way: output = dcCal * input. This allows to threat eventual calibration corrections on the detectors.
• Time shift
double dataShift[NIFO_MAX];
for(int i=0;i<NIFO_MAX;i++) dataShift[i] = 0.;
Possibility to apply constant time shifts to data. These shifts are made in seconds, and allows to make shifts of several days or years, see How to apply a time shift to the input MDC and noise frame files .
• MDC time shift
// use this parameter to shift in time the injections (sec)
// use {0,0,0} to set mdc_shift to 0
// if {-1,0,0} the shift is automatically selected
// {startMDC, stopMDC, offset}
// see description in the method cWB::Toolbox::getMDCShift
mdcshift mdc_shift = {0, 0, 0};
Possibility to apply constant time shifts to injections (MDC). These shifts are made in seconds. This allows to increase statistics for efficiency curve running more simulation jobs, see How to apply a time shift to the input MDC and noise frame files .
## Regulator¶
double delta = 1.0; // [0/1] -> [weak/soft]
double cfg_gamma = 0.5; // set params in net5, [0/1]->net5=[nIFO/0],
// if net5>[threshold=(nIFO-1)] weak/soft[according to delta] else hard
bool eDisbalance = true;
For the meaning of these parameter see the-cwb-2g-regulators
## Sky settings¶
The pipeline calculates for each event what is the most probable location of the source in the sky. This is done scanning a discrete grid of sky positions. cWB can implement two grid types: one is the Healpix (What is HEALPix) and the other is a proper cWB grid. The cWB grid has two possible coordinates: Earth fixes and Celestial.
The Earth fixed has phi running on longitude with 0 on Greenwich and theta running on latitude with 0 at North Pole and 180 at South Pole.
The Celestial is …
Sky resolution of cWB sky grid can be defined by the user, such as L degrees. The angular difference between two consecutive points at the same longitude is equal to L, but the difference between two consecutive points at the same latitude depend on the latitude, such as DL = L/cos(lat).
Healpix grid is more uniform in the sky. An example (of the differences) between the two grids are here What is HEALPix
It is possible to limit the sky grid on limited region on the sky, limiting the range of longitude and latitude, or creating a SkyMask, putting a boolean 1/0 on the sky position in the grid additing which one should be considered.
bool EFEC = true; // Earth Fixed / Selestial coordinates
size_t mode = 0; // sky search mode
double angle = 0.4; // angular resolution
double Theta1 = 0.; // start theta
double Theta2 = 180.; // end theta
double Phi1 = 0.; // start theta
double Phi2 = 360.; // end theta
size_t healpix= 0; // if not 0 use healpix sky map (SK: please check if duplicated)
int Psave = 0; // Skymap probability to be saved in the final output root file (saved if !=0 : see nSky)
long nSky = 0; // if nSky>0 -> # of skymap prob pixels dumped to ascii
// if nSky=0 -> (#pixels==1000 || cum prob > 0.99)
// if nSky<0 -> nSky=-XYZ... save all pixels with prob < 0.XYZ...
double precision = 0.001; // Error region: No = nIFO*(K+KZero)+precision*E
size_t upTDF=4; // upsample factor to obtain rate of TD filter : TDRate = (inRate>>levelR)*upTDF
• EFEC
Boolean selecting Earth coordinate (true) or Celestial coordinates (false)
• mode
If set to 0, the pipeline consider the total grid. If set to 1 the pipeline exclude from the grid the sky locations with network time delays equal to an already considered sky location. This parameter should not be changed.
• angle
Angular resolution for the sky grid, used for cWB grid.
• Theta1 and Theta2
Latitute boundaries.
• Phi1 and Phi2
Longitude boundaries.
File giving a number to each sky locations. If the number is different from 0, the sky location is applied. This uses earth coordinates. Alternatively to the file name (generic skymask) it is possible to use the built-in skymask. The built-in skymask is a circle defined by its center in earth coordinates and its radius in degrees.
The syntax is :
--theta THETA --phi PHI --radius RADIUS
THETA : [-90,90], PHI : [0,360], RADIUS : degrees
File giving a number to each sky locations. If the number is different from 0, the sky location is applied. This uses celestial coordinates. Alternatively to the file name (generic skymask) it is possible to use the built-in skymask. The built-in skymask is a circle defined by its center in earth coordinates and its radius in degrees.
The syntax is :
--theta DEC --phi RA --radius RADIUS
DEC : [-90,90], RA : [0,360], RADIUS : degrees
To see how to define a skymask with a file see How to create a celestial skymask
• healpix
Healpix parameter, if equal to 0 the pipeline uses cWB grid, is > 0 the pipeline uses Healpix
• Psave
Skymap probability to be saved in the final output root file (saved if !=0 : see nSky)
• nSky
this is the number of sky positions reported in the ascii file and (if Psave=true) in root.
If nSky = 0, the number o sky positions reported is such as the cumulative probabiltiy in the sky reach 0.99%. If this number is greater than 1000, the list is truncated at 1000.
if nSky>0 -> # of skymap prob pixels dumped to ascii
if nSky=0 -> (#pixels==1000 || cum prob > 0.99)
if nSky<0 -> nSky=-XYZ… save all pixels with prob < 0.XYZ…
• precision
precision = GetPrecision(csize,order);
set parameters for big clusters events management
csize : cluster size threshold
order : order of healpix resampled skymap (<healpix)
default (0,0) = disabled
if enabled the skyloop of the events with volume>=csize is downsampled to skymap(order)
• upTDF
upsample factor to obtain rate of TD filter : TDRate = (inRate>>levelR)*upTDF
## CED parameters¶
There are three parameters regarding CED:
bool cedDump = false; // dump ced plots with rho>cedRHO
double cedRHO = 4.0;
• cedDump
boolean value, if true CED pages are produced, otherwise not
• cedRHO
CED pages are produced only for triggers which have rho > cedRHO
## Output settings¶
There are different information and format styles that the pipeline can produce. Here are the parameters setting these.
unsigned int jobfOptions = cWB_JOBF_SAVE_DISABLE; // job file options
bool dumpHistory = true; // dump history into output root file
bool dump = true; // dump triggers into ascii file
bool savemode = true; // temporary save clusters on disc
• jobfOptions
• dumpHistory
Save in the output file all the parameters and configuration files used
• dump
save the triggers information also in ASCII files in addition to ROOT files.
• savemode
temporary save information about cluster on the disk, to save memory.
## Working directories¶
The analysis needs a working dir where putting temporary files necessary for the analysis, called nodedir.
This dir is automatically chosen for ATLAS and CIT clusters, but for other clusters should be specified.
// read and dump data on local disk (nodedir)
char nodedir[1024] = "";
cout << "nodedir : " << nodedir << endl;
The following directories show where putting results and various information about the analysis.
We suggest not to change it, however, for completeness we report the all directories.
The content of each directory is already explained in pre-production
char work_dir[512];
sprintf(work_dir,"%s",gSystem->WorkingDirectory());
char config_dir[512] = "config";
char input_dir[512] = "input";
char output_dir[512] = "output";
char merge_dir[512] = "merge";
char condor_dir[512] = "condor";
char report_dir[512] = "report";
char macro_dir[512] = "macro";
char log_dir[512] = "log";
char data_dir[512] = "data";
char tmp_dir[512] = "tmp";
char ced_dir[512] = "report/ced";
char pp_dir[512] = "report/postprod";
char dump_dir[512] = "report/dump";
char www_dir[512];
## Files list¶
These are informations about the list of frame files and data quality files.
• Frame files:
// If all mdc channels are in a single frame file -> mdc must be declared in the nIFO position
char frFiles[2*NIFO_MAX][256];
for(int i=0;i<2*NIFO_MAX;i++) strcpy(frFiles,"");
// frame reading retry time (sec) : 0 -> disable
// retry time = frRetryTime*(num of trials) : max trials = 3
int frRetryTime=60;
char channelNamesRaw[NIFO_MAX][50];
char channelNamesMDC[NIFO_MAX][50];
If we have N detectors, the [0,N-1] positions refers to detectors data frame files. The [N-1, 2N-1] are for MDC frame filesfor Simulation stage. If the frame file is the same for all MDC, it is sufficient to write only the N position.
The channel name of detector strain and MDC strain are respectively saved in channelNamesRaw and channelNamesMDC.
Sometimes the frames are not temporarily available for reading, if the pipeline is not able to read a frame, it retries after some seconds (…). After a number of trials equal to frRetryTime, the pipeline exit with an error.
• Data quality
// {ifo, dqcat_file, dqcat[0/1/2], shift[sec], inverse[false/true], 4columns[true/false]}
int nDQF = 0;
dqfile DQF[20];
See data quality for details on how to write the data quality
## Plugin¶
These are the parameters that regars Plugins
TMacro plugin; // Macro source
TMacro configPlugin; // Macro config
plugin.SetName("");
configPlugin.SetName("");
bool dataPlugin = false; // if dataPlugin=true disable read data from frames
bool mdcPlugin = false; // if mdcPlugin=true disable read mdc from frames
bool dcPlugin = false; // if dcPlugin=true disable the build data conditioning
• plugin: insert the Plugin source code
• configPlugin: insert the Plugin configuration source code
• plugin.SetName(“”);: insert the compiled Plugin code
• configplugin.SetName(“”);: insert the compiled Plugin configuration code
• dataPlugin: disable the reading of detector strain
• mdcPlugin: disable the reading of MDC strain
• dcPlugin: disable the conditioning of data (conditioning)
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2021-06-25 07:33:50
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https://socratic.org/questions/59bc1564b72cff58f8477564
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# How many atoms in a mass of ONE MOLE of "magnesium nitrate"?
You have the formula $M g {\left(N {O}_{3}\right)}_{2}$....
And so there are $1 \times M g + 2 \times N + 6 \times O \cdot \text{atoms}$.......
...which is $148.30 \cdot g \cdot m o {l}^{-} 1$. And thus in $\text{Avogadro's number of magnesium nitrate formula units,}$ $\text{i.e. a mass of 148.3 g}$, ${N}_{A} \equiv 6.022 \times {10}^{23} \cdot m o {l}^{-} 1$, there are $1 \times {N}_{A} \left(M g\right) + 2 \times {N}_{A} \left(N\right) + 6 \times {N}_{A} \left(O\right) \equiv 9 {N}_{A}$ atoms in toto....
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2020-01-21 02:15:12
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https://groupprops.subwiki.org/wiki/Mathieu_group:M12
|
# Mathieu group:M12
## Definition
This group is defined as the Mathieu group of degree $12$. It is one of the five simple Mathieu groups. It is given by the following permutation representation, as a subgroup of symmetric group:S12 as follows. Consider the following set of size 12: the projective line over field:F11. Explicitly, this set can be written as:
$\{ 0,1,2,3,4,5,6,7,8,9,10,\infty \}$
We have projective special linear group:PSL(2,11), acting naturally on this set as fractional linear transformations by:
$\begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mapsto \left(x \mapsto \frac{ax + b}{cx + d}\right)$
This defines an embedding of $PSL(2,11)$ inside $S_{12}$.
The group $M_{12}$ is defined as the subgroup of $S_{12}$ generated by the image of $PSL(2,11)$ and the permutation given by $x \mapsto 4x^2 - 3x^7$, which as a permutation is:
$\! (2,6,10,7)(3,9,4,5)$
Note that this permutation fixes the points $0,1,\infty$.
Note further that since all generating permutations are even permutations, this group is in fact a subgroup of alternating group:A12.
## Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 95040 groups with same order As Mathieu group $M_n, n \in \{ 9,10,11,12 \}$: $n!/7! = n(n-1)\dots 8 = (12)(11)(10)(9)(8) = 95040$
exponent of a group 1320 groups with same order and exponent of a group | groups with same exponent of a group
## Group properties
Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group Yes
minimal simple group No
## Subgroups
Further information: subgroup structure of Mathieu group:M12
## Linear representation theory
Further information: linear representation theory of Mathieu group:M12
Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1,11,11,16,16,45,54,55,55,55,66,99,120,144,176
grouped form (occurs once by default): 1, 11 (2 times), 16 (2 times), 45, 54, 55 (3 times), 66, 99, 120, 144, 176
maximum: 176, quasirandom degree: 11, number: 15, sum of squares: 95040
## GAP implementation
Unfortunately, GAP's SmallGroup library is not available for this order of group (95040) so the group cannot be constructed that way. It can be constructed in other ways:
Description Functions used
MathieuGroup(12) MathieuGroup
PerfectGroup(95040) or equivalently PerfectGroup(95040,1) PerfectGroup
The group is somewhat cumbersome to manipulate directly because of its large size. Information about the group, including its character table, can be accessed using the symbol "M12" -- see linear representation theory of Mathieu group:M12#GAP implementation for more.
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2021-06-23 11:59:57
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https://datascience.stackexchange.com/questions/67344/is-there-a-gradient-descent-based-optimization-algorithm-that-works-with-non-lin
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# Is there a gradient descent-based optimization algorithm that works with non-linear constraints?
I have a function to optimize with ca. 200 parameters + one constraint (sum of squares of the parameters must be equal one)
This problem can be solved using Lagrange Multipliers and my intuition tells me, that methods that do that must be readily available.
If I had a choice, I would prefer an algorithm existing on JuMP.jl
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2021-10-22 22:39:52
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https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-2-solving-equations-2-10-change-expressed-as-a-percent-practice-and-problem-solving-exercises-page-149/24
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# Chapter 2 - Solving Equations - 2-10 Change Expressed as a Percent - Practice and Problem-Solving Exercises - Page 149: 24
#### Work Step by Step
The formula for area is l*w=A The given measurement is to the nearest 0.5 meters. That means that the greatest possible error is 0.25 meters $A=(18.5)(7.5)$ $A=138.75$ $A=(18.5)(7.25)$ $A=132.3125$ $A=(18.75)(7.75)$ $A=145.3125$ $|132.3125-138.75|$ absolute value to find the difference in the area between the minimum area and the measured area $6.4375$ $|145.3125-138.75|$absolute value to find the difference in the area between the maximum area and the measured area $6.5625$ $\frac{6.5625}{138.75}\approx0.05$, or 5% greatest possible percent error=greatest diference in area/measured area
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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2021-04-18 09:02:45
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http://blog.bigsmoke.us/2010/03/25/preventing-the-creation-of-recycle.bin-on-samba-shares-by-windows-7
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# Preventing the creation of $RECYCLE.BIN on Samba shares by Windows 7 Windows 7 kept creating a$RECYCLE.BIN dir on the network share. This in itself is merely annoying, but there were also errors resulting from it. Whenever a file would be deleted, this message would appear (translated from dutch): “The recycle bin is damaged, do you want to delete the contents?” Everything froze until that question was answered.
Samba has an option “veto files” which can be used to stop the creation of that directory. Put this in each share’s section in your smb.conf:
veto files = /*\$RECYCLE.BIN/
The slashes are not directory separators in this case. Also, I don’t know if the preceeding * is necessary, but it does no harm.
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2018-01-19 07:12:07
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https://blogs.ams.org/mathgradblog/category/math-education/
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# Category Archives: Math Education
## Math Students Hunt For Errors in False Proofs!
Communicating mathematics is a crucial part of a developing mathematician’s career. Really, any mathematician’s career. In the classroom, with peers, and at conferences, math students organize their learning and research in order to effectively question and convey concepts that require … Continue reading
## Teaching in the Time of Coronavirus, Part I
Hi all, 2020 has been a complicated year so far, and things are only going to get more complicated as the COVID-19 pandemic. I’ve been thinking a lot about teaching recently, (as I’m the instructor for a class of undergrad … Continue reading
## What is an Infinitesimal?
A guest post from Reginald Anderson at Kansas State University. First-time learners of calculus often struggle with the notion of an infinitesimal, and considering $\frac{dy}{dx}$ literally as a fraction can lead students astray in Calculus III and differential equations, when … Continue reading
Posted in Algebraic Geometry, Math Education, Teaching | Tagged , , | Leave a comment
## How to Divide by Zero: An Interview with Bill Shillito
For this post, I interviewed a colleague about a new project he is working on: a website where he encourages his readers to consider the possibility of dividing by zero. Bill Shillito has a Master’s degree in Secondary Mathematics Education … Continue reading
## Ingredients of a Class Activity
Non-lecture math education seems to be getting its advocates in the math communities at all levels. Every alternative to the traditional math education involves some sort of class activity, where students are given tasks to complete in groups or pairs. … Continue reading
| Tagged , | 2 Comments
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2020-07-14 04:56:37
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https://physicshelpforum.com/threads/video-game-bow-and-arrow-mechanics.13658/
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# Video Game Bow and Arrow Mechanics
#### Wretch11
I'm trying to implement bow and arrow mechanics in a game. Considering that I know two points on a grid, a launch angle, a velocity and the gravitational constant, how would I go about calculating an arrow's trajectory.
Additionally, how would I go about calculating the arrow's angle along every point of it's trajectory arc?
EDIT: I meant to post this in the kinematics sub-forum. Would be much obliged to the mods if this got moved to the appropriate place. Apologies for the hassle
Thank you.
Last edited:
#### ChipB
PHF Helper
I assume what you mean by "trajectory" is you need formulas for x- and y-coordinates as functions of time, and same with the angle, correct?
If we ignore air resistance, then given initial velocity V_0, the horizontal velocity is $$\displaystyle v_x = V_0 \cos (\theta )$$, where $$\displaystyle \theta$$ is the launch angle. Notice that this is constant. The vertical velocity changes with time as gravity causes the arrow to decelerate, and is $$\displaystyle v_y = V_0 \sin (\theta )- gt$$ where g is the acceleration dues to gravity: $$\displaystyle g = 9.81 \frac m {s^2}$$. If the (x,y) coordinate of the launch point is $$\displaystyle (X_0,Y_0)$$, then the x- and y-positions as a function of time are:
$$\displaystyle x(t) = X_0 + V_0 \cos (\theta) t$$
$$\displaystyle y(t) = Y_0 + V_0 \sin(\theta) t - \frac 1 2 g t^2$$
The angle of the arrow to the ground is equal to the arc tangent of the vertical velocity divided by the horizontal velocity (this assumes that the arrow always points in the direction of travel, which is not absolutely correct, but should be close enough for your purposes):
$$\displaystyle \alpha = \tan^{-1} ( \frac {v_y}{v_x} ) = \tan^{-1} ( \frac {V_0 \sin (\theta )- gt}{V_0 \cos (\theta)} )$$
Last edited:
#### Woody
How accurate do you need to be?
If you require greater accuracy, or are "shooting" long distances, then the aerodynamic drag on the arrow might become important.
Do you want to include wind effects?
If all you require is that it "looks" correct, then some approximations can be made (which will allow quicker computation).
Most games adopt approximations, which will be completely un-noticeable in normal game usage, trading absolute accuracy for improved frame rates.
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2020-01-24 07:38:30
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https://ncatlab.org/nlab/show/lax+F-adjunction
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# Lax $\mathcal{F}$-adjunctions
## Idea
The notion of adjunction or 2-adjunction can be “laxified” in many ways; as discussed at lax 2-adjunction, one can make the triangle identities hold only up to a noninvertible cell, make the unit and counit only lax natural, and even consider making the functors only lax. Of these, lax naturality of the unit and counit is problematic because lax natural transformations do not satisfy a Yoneda lemma. However, this becomes less of a problem if the lax transformations restrict to pseudo ones on certain well-behaved subcategories. An abstract context to define this is that of F-categories.
## Definition
Let $K,L$ be F-categories (strict for simplicity). A lax $\mathcal{F}$-adjunction between them consists of:
• $\mathcal{F}$-functors $F:K\to L$ and $G:L\to K$ (strict for simplicity).
• pseudo/lax F-natural transformations$\eta : Id_K \to G F$ and $\epsilon : F G \to Id_L$.
• The triangle identities for $\eta,\epsilon$ hold up to isomorphism (which can be made coherent).
If $\epsilon$ is fully pseudo, we call it a right semi-lax $\mathcal{F}$-adjunction. Dually, we have left semi-lax, right semi-oplax, left semi-oplax, and so on. (We could also consider allowing the triangle identities or functors to be lax, but we will not.)
## Properties
The following theorem seems to fail for 2-adjunctions involving lax transformations; the fact that it holds for lax $\mathcal{F}$-adjunctions thus means that they are significantly better-behaved and more interesting. It was first observed (without the terminology of $\mathcal{F}$-categories) by Johnstone.
###### Theorem
If $F:K\to L$ is an $\mathcal{F}$-functor and $G,G'$ are two lax right $\mathcal{F}$-adjoints for it, then $G\simeq G'$.
###### Proof
As usual, we consider the composites
$G \xrightarrow{\eta' G} G' F G \xrightarrow{G' \epsilon} G' \qquad G' \xrightarrow{\eta G'} G F G' \xrightarrow{G \epsilon'} G.$
The usual proof of uniqueness of adjoints applies to show that these are pseudo-inverses, but we do need the fact that these unit and counit are pseudo/lax rather than merely lax. For the proof requires using naturality squares to commute units and counits past each other, and since their components are tight and they are all pseudo-natural on tight morphisms these naturality squares commute up to isomorphism; if they only commuted laxly then the proof would fail.
Note that a priori these composites are themselves also only pseudo/lax $\mathcal{F}$-natural. However, any lax natural transformation that is (pseudo) invertible is actually pseudo natural, so in fact these composites are a pseudo/pseudo $\mathcal{F}-$natural equivalence $G\simeq G'$.
## Examples
### Fibrations in a 2-category
In Johnstone it is shown that a fibration in a 2-category $K$ can equivalently be defined as a morphism $p:E\to B$ with the following property. Let $K\swarrow B$ denote the oplax slice 2-category, whose objects are morphisms $f:A\to B$ and whose morphisms are pairs $(g:A\to A', \alpha : f' g \Rightarrow f)$. We make $K\swarrow B$ into an $\mathcal{F}$-category by declaring $(g,\alpha)$ to be tight if $\alpha$ is an isomorphism.
Now, composing with $p:E\to B$ induces an $\mathcal{F}$-functor $\Sigma_p : K\swarrow E \to K\swarrow B$.
###### Theorem
$p$ is a fibration (in the pseudo sense) if and only if $\Sigma_p$ has a right semi-lax $\mathcal{F}$-adjoint.
## References
Created on March 4, 2018 at 05:39:35. See the history of this page for a list of all contributions to it.
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2023-03-21 11:30:08
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http://mathoverflow.net/questions/67009/a-question-about-finite-dimensional-representation-of-a-hopf-algebra
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# a question about finite dimensional representation of a Hopf algebra
Let $H$ be a Hopf algebra over a field $k$ and $V$ a finite dimensional left $H$-module. Then $End_{k}(V)$ is a right $H$-module via $(f\cdot h)(v)=S(h_{1})f(h_{2}\cdot v)$.
We set $Ann(End_{k}(V))$={$h\in H: f\cdot h=0, \forall f\in End_{k}(V)$} and $A=H/Ann(End_{k}(V))$.
Let $I$ be the 1-dimensional subspace generated by $id_{V}$. Then $I$ is a submodule of $End_{k}(V)$. Let $Ann(I)$={$\bar{h}\in A: id_{V}\cdot \bar{h}=0$}.
Is there a sufficient condition of $V$ in order to guarantee that $A$ has an ideal $L$ such that $A=L\oplus Ann(I)$?
If $H$ is a group algebra $kG$, then $End_{k}(V)$ is a right $kG$-module via $(f\cdot g)(v)=g^{-1}f(g\cdot v)$.
Can we answer this question in this case?
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Yes, it is sufficient that $H$ is finite dimensional semisimple. It is not necessary because enveloping and quantum enveloping algebras of simple Lie algebras provide other examples.
Overall, this is equivalent to semisimplicity of the category of finite-dimensional $H$-modules. I do not know what structure properties of $H$ ensure this.
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No. The counterexample is the universal enveloping algebra H of a complex semisimple Lie algebra. Every finite dimensional left H-module V possesses the property mentioned above. – sife Jun 6 '11 at 8:10
You are right: I did not read the question carefully and thought that your AnnI is in $H$, not $A$... I will edit... – Bugs Bunny Jun 6 '11 at 8:21
If we focus on the representation V, what propersitions does V have to insure that A possesses the property mentioned above? – sife Jun 6 '11 at 8:42
If every finite-dimensional H-module is semisimple, then A possesses the property mentioned above. But this answer is trivial........... – sife Jun 6 '11 at 8:45
Off course, it is trivial but the reformulation is clearer: you are asking for which Hopf algebras the category of finite dimensional reps is semisimple... – Bugs Bunny Jun 7 '11 at 9:27
One has that $id_V.h=\epsilon(h)id_V$ for all $h \in H$. Thus $Ann_H(I)=H^+=\{h\in H\;|\; \epsilon(h)=0\}$.
Then the annihialtor inside $A$ is $\pi(H^+)$ where $\pi:H \rightarrow A$ is the canonical projection. A complement of it would be given by the image of the integral of $H$ under $\pi$.
I guess the question is when $\pi(\Lambda)$ is not zero? or in other words when $\Lambda \notin Ann_H(V)$. That is the case if and only if $V^H \neq 0$.
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No!!! $Ann(I)$={$\bar{h}\in A:id_{V}\cdot \bar{h}=0$}. – sife Jun 6 '11 at 16:50
I modified my answer a little bit. – anonymus Jun 6 '11 at 20:11
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2014-12-21 04:10:14
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https://cs.stackexchange.com/questions/46953/how-can-i-infer-inputs-to-a-function-given-an-output
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How can I infer inputs to a function, given an output?
Is there a way to generate a set of inputs, for a given output of a function? This may be a common thing in lisp/haskell world, but I'm not aware of it. Is this what mini-kanren does?
I understand that there may not be a general solution to this problem, since the halting problem may apply here. Say the function implementation is constrained to remove unlimited recursion. Type inference engines obviously do something similar to figure out the types of variables, I would like to generate actual values.
I'm thinking of a couple of scenarios where this might be useful:
func blowup(x): 1/x
I would like some sort of static checker to figure out that this function will cause problems if x=0.
func launch(x): ...launch missle x miles...
func isSafe(y): if x > 10 then true else false
func main(): x=.1; if isSafe(x): launch(x)
In this case, I would like to not only infer values of launch, but I would like to take isSafe as a constraint and tell me at compile time (since all information is available) that x of .1 not a good value.
Btw, I'm not looking for a library to do this. What I would love is either the name of this concept (beyond the generic 'static checking') and reference material which describes this idea (easily read papers, books, etc.). I should also mention that I'm aware of tools such as quickcheck. I'm not looking for a probabilistic way of inferring values. I would like a deterministic way of inferring values as early as possible, even if that means that my solution limits the types of expressions I can express.
• If this was practically possible, it would break cryptographic signing, because it would mean you would be able to infer the private key from a signed message. – svick Sep 8 '15 at 8:31
• You can't; the function may not be injective. – Raphael Sep 8 '15 at 14:40
In the worst case, this is impossible. However, there are techniques for inversion that can work on some programs. Of course, you shouldn't expect them to work on all programs given that the problem is hard.
Here are a few standard approaches:
• Theorem proving. Use the VCGen part of a theorem prover to generate a precondition formula, where if the formula is true, that implies that the function produces the desired output (or divides by zero, etc.). Then, use an automated solver to try to find a satisfying assignment to that formula.
• Symbolic execution. Execute the function symbolically, and build up a symbolic expression that represents the output of the function, in terms of a bunch of variables (unknowns) that represent the inputs to the function. Then build a formula by setting that expression equal to the desired output, feed that formula to a SMT solver and ask the solver to find a satisfying assignment for the formula.
• Concolic execution. Pick concrete values for the inputs to the function. Execute the function on those inputs and record what path it took. On the side, build up a symbolic expression that represents the output of the function, if it follows the same path. Build up a "path constraint", which is a formula that is the conjunction of all of the branch conditions followed: the path constraint is a formula over symbolic variables representing the inputs, where inputs that make the path constraint true will cause the function to execute the same path that was previously recorded. Then, write down a formula that represents the statement "the function takes that path, and its output is equal to the desired value", and feed this to a SMT solver or SAT solver and ask it to find a satisfying assignment. See concolic testing.
These approaches have different tradeoffs and perform differently, but as you can see, they are similar conceptually. Here are some significant differences:
• Theorem proving and symbolic execution can have problems with loops. For instance, theorem provers usually require you to supply a loop invariant for every loop. Concolic execution circumvents this by not trying to reason about all possible paths; only a single one.
• Theorem proving and symbolic execution can have problems with state space explosion, as the number of possible paths through the function can be exponentially (or even infinitely) large. Concolic execution circumvents this by not trying to reason about all possible paths; only a single one.
• Theorem proving can use more expressive logics, such as fist-order logic. Symbolic execution and concolic execution often use propositional logic or a SMT extension.
You can find lots of reference material on these approaches. For instance, today they're taught in a number of advanced (e.g., graduate) program analysis courses, and there are many publications in the research literature on these topics.
• In the worst case it is not only hard but impossible. – Andrej Bauer Sep 8 '15 at 16:29
• @AndrejBauer, quite right. I've adjusted my answer accordingly, thank you. – D.W. Sep 8 '15 at 16:30
To find possible input values (or at least a superset) given a set of possible outputs is called backward constraint propagation. This is done inside many kinds of program analysis: formal verification, compiler optimizers, defect search, ...
It's more common for type checkers to do forward propagation. For example many languages require type annotations on function arguments and infer the types of expressions. Languages like ML that do full inference don't really distinguish between forward and backward propagation.
There's no general rule to tell when backward constraint propagation can be useful. To take just one example, consider a cryptographic hash function with a fixed-size input string. From a theory of computation point of view, each hash value has a finite set of preimages and you can run a Turing machine with an obvious termination argument to enumerate them all. But once you take complexity into account, this enumeration is impractically slow; in fact the very definition of a cryptographic hash requires that this computation be intractable.
• "backward constraint propagation" is a term I'm vaguely familiar with so you answer was almost exactly what I was looking for. However, D.W's has a bunch of terms I've never heard before and may turn out to be more informative. I wish I could accept both answers! – Shahbaz Sep 9 '15 at 23:07
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2020-01-29 19:13:53
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https://brilliant.org/problems/escape-the-black-hole-oberth-maneuver/
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# Escape the black hole-Oberth maneuver
See the Video link.
There is this black hole called gargantua which is located in interstellar space.It is non-moving and non-rotating.You are at a distance of 1000 km from the black hole. Your initial velocity is zero, you have to escape this black hole, but you don't have the fuel to gain that speed (escape velocity).So here you are going to do the Oberth maneuver where you are going to fire your engines at high speeds to get the most out of your fuel. So you go towards the hole not into it, but at a tangent where the nearest point to the point to the blackhole will be 100km to fire your engines.What should be the minimum change in velocity of the spaceship at the nearest point to escape the black hole? By how much you should gain velocity at the nearest point to escape it? Don't stay too near the hole because it slows down time for you with respect to others. This effect is counter-intuitive and it seems to violate the conservation of energy but it's not. Schrawzchild radius is about an inch dont worry about it! Here you are not detaching anything as in the video.
Assume the mass of blackhole to be $$10^{24} \text{ kg}$$ and the gravitational constants as $$10^{-11}$$ for easier calculation.
×
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2017-10-18 20:44:26
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http://tuisyen.my/physics/electronics/
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# Electronics
## Cathode Ray Oscillocope (C.R.O.)
The Cathode Ray Cathode Ray Oscilloscope Uses and Operation of a C.R.O.
## Semiconductor Diode
Semiconductor p-n Junction Diode Diode as A Rectifier
## Transistor
Transistor Transistor as An Automatic Switch
## Logic Gate
Logic Gate Combination of Logic Gate
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2022-09-26 02:14:26
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https://socratic.org/questions/how-do-you-write-2-5-million-in-scientific-notation#380265
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How do you write 2.5 million in scientific notation?
1 Answer
Feb 20, 2017
2.5 million = 2,500,000. To write this in scientific notation we must move the decimal place $\textcolor{red}{6}$ places to the left therefore the exponent for the 10s terms will be positive:
$2 , 500 , 000 = 2.5 \times {10}^{6}$
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2021-10-25 08:04:05
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https://www.nature.com/articles/s41467-021-27575-z?error=cookies_not_supported&code=3a0b3126-81da-42af-80bf-900740d49f5e
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## Introduction
Developing memory devices based on novel operation principles, innovative structures, and new materials is a fundamental and inevitable solution to acquire faster and denser nonvolatile memory (NVM) to meet the requirements of big data, cloud computing, artificial intelligence, and new industries in the modern information society1,2,3,4,5. Compared with the current mainstream charge-based flash memory, resistive switching random access memory (RRAM or so-called memristor), as one of the most promising candidates for next-generation NVM, has the advantages of high-speed operation, a scalable two-terminal structure for high-density 3D integration, excellent compatibility with the back-end of the traditional CMOS process, and analog characteristics for novel in-memory computing6,7,8,9,10. Resistive switching (RS) behavior stemming from the repeatable formation/rupture of conductive filaments (CFs) under an external electric field lays the foundation of oxide memristors11,12. The formation and rupture of CFs yield the low resistance state (LRS) and high resistance state (HRS) of the oxide memristor, respectively. The physical mechanism of filamentary switching includes the generation, migration, and recombination of defects such as oxygen vacancies or metal ions in the RS layer13,14,15. These processes will result in variations in the chemical composition and structure of the local switching region, i.e., the CF region within the RS layer16,17,18.
Binary oxide materials, including ZrO219, TiO220, TaOx21,22, ZnO23, NiO24, CuOx25, etc., have been widely investigated in memristors on account of their simple composition, modulation convenience, excellent scalability, and CMOS process compatibility26,27,28. Various investigations have been separately carried out on the microscopic properties of CFs in memristors based on the oxides above, including the morphology, size, quantity, chemical composition, crystal structure, and dynamic growth/rupture process29,30,31. As a well-known high-k oxide dielectric material, hafnia (HfO2) has attracted considerable interests and is widely recognized as one of the most promising CMOS-compatible RS materials32,33. Similar to the situation in memristors based on other oxides, CFs in the HfO2-based valence change mechanism (VCM) or thermochemical mechanism (TCM) memristors have also been demonstrated as a low-oxygen content region34,35. Through high-resolution transmission electron microscopy (HRTEM) observation36, CFs were identified to be the directionally aligned crystalline regions in amorphous HfO2 (a-HfO2) consisting of monoclinic and orthorhombic oxygen-deficient phases. Notably, the core–shell structure (oxygen-deficient CF and the corresponding oxygen-rich shell) was predicted by ab initio calculations combined with experimental studies for Pt/HfO2/Pt memristors37. In addition, ref. 38 reported the existence of a low-conductivity region with excess oxygen around the oxygen-deficient (or oxygen-vacancy-rich) CF in a Pt/Hf/HfO2/Pt memristor based on synchrotron-based scanning transmission X-ray microscopy analysis.
The abovementioned studies indicate that the formation/rupture of electric field-induced oxygen-deficient CFs with relatively high conductance contributes to the RS behavior, and this scenario dominates the physical understanding of HfO2-based memristors. However, the dynamic changes in the physical properties of the CF system (CFs and their surroundings) in the RS process, including the composition, structure, and especially the evolution of CF surroundings in HfO2-based RS memristors, are generally less focused on in previous studies and lack direct observation at the atomic scale. More detailed knowledge about the physical properties of dynamic CF systems and the RS mechanism is essential for the development of the large-scale manufacturing and commercialization of HfO2-based RS memory.
In this work, we investigate the dynamic evolution characteristics of the CF system specifically in a Pt/HfO2/Pt crossbar RS memristor. The inert Pt electrode, which scarcely participates in the RS process, renders the Pt/HfO2/Pt structure a pure system to study the structure and chemical composition of the oxygen-deficient CFs in oxide memristor devices. The devices show considerable RS performance, including a large switching window (HRS/LRS) above 106 and a short OFF/ON switching time within 120/20 ns. The atomic structure of the oxygen-deficient CF is clearly revealed to be crystalline hexagonal-Hf6O (h-Hf6O) for the first time through HRTEM. The hexagonal-crystal Hf6O can be viewed as the standard hexagonal metal Hf with one interstitial oxygen atom in the Hf6 ring. Interestingly, the CF system here is based on a quasi-core–shell structure, since both nonconductive monoclinic and tetragonal HfO2 (m-HfO2 and t-HfO2) are observed to surround the complete and ruptured h-Hf6O CFs, respectively, in the Pt/HfO2/Pt memristor. Therefore, the RS process of HfO2-based memristors can be accompanied by the transition between distinct crystalline phases, owing to the Joule heating effect and variation of oxygen vacancy concentration. Ab initio calculations were conducted to reveal the energetics of the HfO2 phase transition and filamentary conduction, and the results were in good agreement with the experimental data. HfO2-based RS memristors with application-oriented electrodes (TiN, Ta, Hf, and Ti) yield CF systems quite similar to those of Pt/HfO2/Pt devices, suggesting the universal significance of the quasi-core–shell CF structure. This study provides a further understanding of the nature of the CF system and supplies submechanisms towards the VCM and TCM of HfO2-based memristors. The core-shell nature of the CF system in HfO2-based memristors may renew the interests in the RS mechanism study.
## Results
### Device fabrication and switching performance
To investigate the evolution of the CF structure and its surroundings in the HfO2 RS layer, Pt/HfO2/Pt crossbar RS memristors were fabricated. The detailed fabrication process is elaborated in Methods section and Supplementary Fig. 1. As seen from the scanning electron microscopy (SEM) image in Fig. 1a, the effective area of the crossbar cell is 3 × 3 µm2. A schematic illustration of the device is displayed in the inset, and the thicknesses of the bottom electrode (BE), RS layer, and top electrode (TE) are 40, 20, and 30 nm, respectively. As demonstrated in Fig. 1b, the as-deposited HfO2 RS layer from the fresh Pt/HfO2/Pt stack contains a-HfO2 throughout the device region, as evidenced by the HRTEM image and fast Fourier transform (FFT) diffraction patterns of the marked regions.
The Pt/HfO2/Pt memristors were operated under typical bidirectional nonvolatile RS mode with a 100 μA compliance current (ICC). The ICC current limit could protect the device from hard breakdown during electrical operations. For all electrical measurements, the TE was biased, while the BE was grounded. Figure 1c presents typical Forming (orange line), RESET (red line), and SET (blue line) IV curves of the Pt/HfO2/Pt memristor. The Forming process is generally utilized to initialize the formation of CFs and subsequently trigger repeatable CF rupture/connection behavior under the RESET/SET biases, which dominates the OFF/ON behavior of oxide memristors39. Based on the statistical analysis on SET and Forming voltages (VSET and VForming) in HfO2-based memristors with Pt, TiN, Ti, Hf, and Ta TEs, as shown in Supplementary Fig. 2, the situation that VSET is higher than VForming, corresponds to a probability event, which tends to occur in HfO2-based memristors with strong oxygen-reservation electrodes (e.g., Ti, Hf, Ta). At a read bias of 0.2 V, the switching window of the HRS/LRS ratio can be as high as 106. As confirmed by temperature-resistance electrical tests (Supplementary Fig. 3), the LRS and the HRS are identified to have metallic and semiconductive characteristics, respectively. From the retention measurement (read at 0.2 V) shown in Fig. 1d, the HRS, LRS, and HRS/LRS ratio are well maintained for 104 s without obvious degradation. The HRS fluctuation can be attributed to random noise and current undulation when approaching the test limit of the measurement system. In addition, the 10 stochastically selected devices all maintain an HRS/LRS ratio above 104 during 100 switching cycles, as shown in Fig. 1e, indicating the considerable uniformity of the Pt/HfO2/Pt RS memristors. The switching time of the device was evaluated by Vt and It synchronous curves measured under pulse mode, as shown in Fig. 1f, where both the LRS and the HRS switch to their opposites under a 300 ns negative RESET or positive SET pulse. As marked by the black dashed circles, the OFF and ON times are determined to be within 120 ns and 20 ns, respectively.
### HRTEM observation of the CF system
HRTEM helps to observe any tiny changes inside the RS layer to reveal the RS mechanism and the CF nature at the atomic scale34,36,40. The current, which flows through the memristor, greatly influences the morphology of the CF by generating Joule heat. The typical changes in the RS layer structure of the LRS Pt/HfO2/Pt memristor device induced by the SET behavior under 0.1 mA and 1 mA ICC are captured in the HRTEM images in Fig. 2. Different from the amorphous nature of the as-deposited HfO2 layer, a clear crystal lattice can be observed after the SET operation, as outlined by the red and blue arc curves in Fig. 2a (ICC = 0.1 mA) and 2d (ICC = 1 mA). Note that the crystallization of local regions in the amorphous RS layer are expected to be driven by the Joule heat effect of the nearby CF current31,41. Therefore, we looked for possible CFs near the crystallization regions in the RS layer. According to different crystal categories, both TEM images can be divided into two typical regions. The FFT diffraction pattern in region 1 has sharp diffraction spots with hexagonal structures (Fig. 2b). The interplanar spacings from these spots are calculated as d1 = 2.92 nm, d2 = 2.88 nm, and d3 = 2.67 nm. Compared with anoxic hafnium oxides predicted from ab initio calculations37,42,43,44 and h.c.p. metal Hf, these diffraction spots best fit the (10$$\bar{4}$$), (0$$\bar{1}\bar{4}$$), and ($$\bar{1}\bar{1}$$0) planes of h-Hf6O with a ($$\bar{4}$$4$$\bar{1}$$) zone axis44 (details are given in Supplementary Note 1). The lattice parameters of various HfOx phases concluded in this work, with comparison to the published values in the literature, are listed in Supplementary Table 1. Similarly, adjacent region 2 is found to be m-HfO2 because its diffraction pattern in Fig. 2c perfectly fits the (11$$\bar{1}$$), (1$$\bar{1}$$1), and (200) planes of m-HfO2 with a (0$$\bar{2}\bar{2}$$) zone axis. Since h-Hf6O is a highly conductive phase, region 1 is identified as one complete CF, which supports the current flow during the SET process. The CF shows an approximately conical shape, and its terminal at the BE side is larger than that of the TE. Therefore, we infer that CFs grow from the cathode towards the anode, which is consistent with previous reports on VCM RS devices16,30,45. In another sample operated under 1 mA ICC, as shown in the HRTEM image of Fig. 2d, a more robust h-Hf6O CF (region 3) than that in Fig. 2a is found, which supports the higher ON-state current of the device. In particular, this h-Hf6O CF exhibits a perfect atomic arrangement and hexagonal crystal structure, as confirmed by a close-up view in the inset. According to the FFT diffraction patterns (Fig. 2e–g), the complete h-Hf6O CF is enclosed by the surrounding m-HfO2 shell (region 4). Coincidentally, crystallization of the surrounding oxide after the formation of complete CFs has also been observed in cation-based memristors31,46. Therefore, we propose that the CFs of HfO2-based RS memristors are accompanied by a nonconductive crystallization region, which has been generally ignored in previous studies on the mechanism of RS memristors.
Interestingly, different crystal structures can be found around the h-Hf6O CFs in the Pt/HfO2/Pt RS memristors when the CFs were ruptured under the RESET process. Figure 3a shows an HRTEM image of an incomplete h-Hf6O CF region (in red) surrounded by two different crystal classes outlined by the blue and orange arcs. The FFT diffraction patterns (Fig. 3b,d) verify that the orange region is t-HfO2 and that the blue region is an m-HfO2-dominated region. As shown in Fig. 3e–h, the HRTEM and FFT diffraction patterns of one well ruptured CF region from another sample further confirm the formation of t-HfO2 (in orange) within the amorphous HfO2 RS layer (in light blue). According to Fig. 3g, the interplanar spacings are calculated as d1 = 4.93 nm, d2 = 5.15 nm, and d3 = 3.59 nm. The values of the angles between the crystal faces and the interplanar spacings are exactly the same as those of the ($$\bar{1}$$00), (001), and ($$\bar{1}$$01) planes of t-HfO2. In line with the typical morphology of CF growing from the BE to the TE, the t-HfO2 region presents a similar conical shape. It has been reported that t-HfO2 is a high-temperature stable phase of HfO2, which emerges upon heating at higher temperatures than m-HfO247,48. These results coincide with the fact that a higher current flows through the memristor under the RESET operation and generates much more Joule heat, which is beneficial for CF rupture, than during the SET operation. The mechanism of this Joule heat-induced phase transition of HfOx will be further investigated by ab initio calculations. It is interesting that t-HfO, predicted in ref.49 as a relatively higher-conductive phase compared to the O-rich phase of HfOx, emerges (in dark blue) around the ruptured CF region, as confirmed by the FFT diffraction pattern in Fig. 3h. The interplanar spacings are calculated as d1 = 2.94 nm, d2 = 3.45 nm, and d3 = 2.55 nm, best fitting the ($$\bar{2}$$10), (112), and ($$\bar{1}$$22) planes of t-HfO with the (24$$\bar{3}$$) zone axis49. Based on the above analysis, we infer that the true CF system of HfO2-based RS memristors includes the metallic CF and its low-conductivity shell consisting of m-HfO2 or t-HfO2 depending on the effect of Joule heat generated by current flowing through the CF.
Even though the Pt/HfO2/Pt memristor provides a pure system for CF study, using Pt as both the TE and the BE is not an ideal choice for industrial production. Therefore, HfO2-based memristors with application-oriented TEs (TiN, Ta, Hf, and Ti) were further studied to validate the scenario of the quasi-core-shell CF structure. Based on a comprehensive HRTEM study, HfO2-based memristors with various TEs have been demonstrated to yield a core-shell CF structure similar to that of Pt/HfO2/Pt devices. However, there are some subtle differences in the components of the crystalline shells. Supplementary Fig. 4a is the HRTEM image of a Pt/TiN/HfO2/Pt memristor (TiN-memristor) with a complete h-Hf6O CF region (in red) surrounded by two different crystal classes outlined by the blue and dark blue arcs. The FFT diffraction patterns in Supplementary Fig. 4b and d verify that the blue region is m-HfO2-dominated while the dark blue region is t-HfO-dominated. On the other hand, the shell compositions of the CF system in the Pt/Ti/HfO2/Pt memristor device (Ti-memristor) are m-HfO2 or t-HfO2, as shown in Supplementary Fig. 5. The shell compositions in the Pt/Hf/HfO2/Pt memristor (Hf-memristor) are revealed to be t-HfO2 and t-HfO, as shown in Supplementary Fig. 6. Finally, the Pt/Ta/HfO2/Pt memristor (Ta-memristor) exhibits similar CF system structure as the Hf-memristor, as shown in Supplementary Fig. 7b and d. Therefore, the quasi-core–shell structure of the CF system has universal significance for HfO2-based memristors.
The subtle differences in the abovementioned specific compositions of the CF system in various HfO2-based memristors can be attributed to the oxygen reservation capability of various TEs, as revealed by the Gibbs free energy change ΔG in the standard reaction of electrode oxidation (or by the bond dissociation energy of potential oxides of electrode). Relevant parameters of the Gibbs free energy and bond dissociation energy are listed in Supplementary Tables 2 and 3. A high ΔG value or bond dissociation energy indicates the high oxygen reservation capability of a certain metal. The formation of t-HfOx has been proven more easier than that of m-HfOx (x ≈ 2) in an oxygen-deficient environment35,50. The high oxygen reservation capabilities of Ti, Hf, and Ta electrodes result in high concentrations of oxygen vacancies in Ti-, Hf-, and Ta-memristors15,51,52. Therefore, the tetragonal phase preferably emerges in the Ti-, Hf-, and Ta-memristors, as demonstrated by extensive HRTEM analysis. In addition, the experimental results in this work possess an interesting consistency in that a higher oxygen reservation capability of the electrode could more likely render the emergence of a t-HfO2 shell, as well as a higher probability of finding VForming > VSET. In particular, the HRTEM images of HfO2-based memristors in this work provide the first experimental proof of the existence of t-HfO, which is expected as an infancy towards high-conductivity CF and deserve further explorations.
### Ab initio calculations of the HfOx-based CF system
To obtain an elaborate view of the CF system of HfO2-based memristors, especially regarding the compositional evolution and structural transitions during the RS process, ab initio calculations were performed to provide evidence in terms of density functional theory and thermodynamics. Previous calculations have shown that the oxygen vacancies in HfO2 tend to align in an orderly manner, and this arrangement is energetically favorable for oxygen-vacancy chains to merge, thus leading to the segregation of metal Hf phases43. Moreover, several suboxide phases of HfOx have been predicted in the literature37,42,44,53, such as P$$\bar{4}$$m2 Hf2O3 (t-Hf2O3), P$$\bar{6}$$2m HfO (h-HfO), and P$$\bar{3}$$1m Hf2O. Given that h.c.p. metal Hf and m-HfO2 are the two extreme phases for phase separation, their relative free energies (∆G) were taken as zero. We investigated ∆G of various HfOx structures with temperatures ranging from 0 to 3000 K, compared with the mixture of h.c.p. Hf and m-HfO2 into which they may decompose. A negative relative free energy indicates a thermodynamically stable state. The vibration entropy was calculated according to statistical physics using harmonic approximation (see Supplementary Note 2 for details). As shown in Fig. 4a, the relative stability of t-HfO2 demonstrates the most remarkable change with temperature. In particular, this compound becomes stable against m-HfO2 above ~1880 K. Although this predicted phase transition temperature is slightly lower than the experimental value (~1991 K) for the 90% conversion from the monoclinic phase to the tetragonal phase54, the difference is within an acceptable range considering the approximations involved in our calculations. Moreover, it is clear that t-HfO2 gradually stabilizes against m-HfO2 when the sample is heated. The suboxides t-HfO, t-Hf2O3, and h-HfO are unstable at zero temperature, and the former tends to stabilize, while the latter two become even more unstable at high temperatures. The remaining three phases, Hf6O, Hf3O, and Hf2O, are all derivatives of h.c.p. metal Hf, with certain amounts of oxygen interstitials44. We first observe that Hf2O is unstable over the whole temperature range, while Hf3O is stable against decomposition into Hf and HfO2 at low temperatures, but its thermodynamic stability is weakened at high temperatures. The exceptional case is Hf6O, whose thermodynamic stability is strong over the full temperature range. Even though this compound may decompose into metal Hf and t-HfO2 at very high temperatures, this does not occur at less than ~2500 K.
The t-HfO2 phase becomes much more stable at high temperature, mainly because it is a phase with high symmetry. Thus, the entropy of t-HfO2 increases more rapidly with increasing temperature due to symmetry loss than the entropy of m-HfO2. In addition, our calculation also shows that t-HfOx has a higher tolerance to oxygen vacancies ($${{{{{{\rm{V}}}}}}}_{{{{{{\rm{O}}}}}}}$$) than m-HfOx (here, x ≈ 2). Figure 4b exhibits the Gibbs free energy differences at 300 K, with various amounts of $${{{{{{\rm{V}}}}}}}_{{{{{{\rm{O}}}}}}}$$ introduced per Hf32O64 supercell. Before the stoichiometry reaches HfO1.625 (Hf32O52), t-HfOx already becomes energetically more favorable than m-HfOx. In other words, the existence of a high concentration of $${{{{{{\rm{V}}}}}}}_{{{{{{\rm{O}}}}}}}$$ can promote the emergence or stabilization of the tetragonal phase. This qualitative calculation coincides with our experimental observation that t-HfO2 is more easily observed around incomplete CFs, which has been ruptured under previous RESET operations. Oxygen anions move from the surrounding environment to the metallic CF core during the RESET process, leaving more $${{{{{{\rm{V}}}}}}}_{{{{{{\rm{O}}}}}}}^{\cdot \cdot }$$ in the nearby dielectric, which can, together with the current-induced Joule heat effect, convert m-HfO2 to t-HfO2.
In addition, t-HfO2 possesses a much lower surface energy than m-HfO255. We also carried out a comprehensive comparison between the surface energies from 16 kinds of m-HfO2 and t-HfO2 surface configurations (see Supplementary Note 3). For typical-shaped grains, t-HfO2 becomes more energetically favorable than m-HfO2 when each grain contains fewer than ~4000 atoms, or has an average dimension of ~3.5 nm, even without considering the entropy effect. This implies that the conversion of m-HfO2 to t-HfO2 in actual grains is easier than expected. In particular, it has been proven that, when scaled to finite size, the transformation temperature can be lowered to nearly 700 K56,57. In sum, the VO concentration, Joule heat, and surface energy play critical roles in the evolution of the shell structure around the Hf6O core of the CF system. Although we cannot fully consider the nanoscale effect, the bulk calculations together with surface energy considerations still provide a useful guide towards CF formation in hafnia.
To investigate whether Hf6O can serve as the metallic core when surrounded by crystalline HfO2, we chose a model of h.c.p. Hf6O encapsulated by m-HfO2 as our core–shell CF system prototype, where the c-axis of Hf6O is aligned with the a-axis of m-HfO2. After structural relaxation using density functional theory, the optimized model structure is obtained and shown in Fig. 4c. To examine whether filamentary conduction exists in this model, we plotted the local density of states (LDOS) on a series of Hf atoms (marked as #1−#5 in Fig. 4c) in Fig. 4d. It is well known that the conduction band of HfO2 mainly consists of states from the metal Hf58. The LDOS decomposition clearly indicates that strong band conduction exists inside the cylindrical CF, as emphasized by the shaded region in Fig. 4c, and extends slightly to the surrounding region up to Hf #4. In the bulk m-HfO2 region, however, the system becomes insulating, as revealed by the LDOS of Hf #5. This supports that Hf6O inside crystalline HfO2 can indeed account for local filamentary conduction.
### Core-shell CF system in the HfO2-based memristor
Based on the above analysis, the switching mechanism and dynamic evolution of the oxygen-deficient CF system in HfO2-based RS memristors can be effectively updated, as schematically illustrated in Fig. 5. In the initial device, the intrinsic $${{{{{{\rm{V}}}}}}}_{{{{{{\rm{O}}}}}}}^{\cdot \cdot }$$ in nonconductive a-HfO2 promote the formation of CFs (Fig. 5a). During the Forming process, O2− ions dissociate from HfO2 and move from the cathode towards the anode under an external electric field, giving rise to the nucleation and growth of Hf-rich or O-vacancy CFs in the a-HfO2 layer (Fig. 5b, c) from the cathode to the anode. When the thermally stable anoxic h-Hf6O CF bridges the TE and BE (Fig. 5d), a high current flows through the CF accompanied by a device switching event from the HRS to the LRS. Under the annealing effect of current-induced Joule heat, the initial a-HfO2 surrounding the h-Hf6O CF would crystallize into m-HfO2 (Fig. 5e). The formation of an oxygen-rich crystalline shell is attributed to a combined effect of temperature and lateral motion of $${{{{{{\rm{V}}}}}}}_{{{{{{\rm{O}}}}}}}^{\cdot \cdot }$$, which is dominated by two opposite forces: inwards thermal diffusion driven by a temperature gradient and outwards Fickian diffusion driven by a concentration gradient59,60,61 (see Supplementary Fig. 8). This finally produces a quasi-core-shell CF system as observed by HRTEM in the Pt/HfO2/Pt memristor. Figure 5f gives a longitudinal-section view of the quasi-core-shell structure. Reactions during core–shell CF system formation in the Forming process are dominated by the dissociation of HfO2 into Hf4+ and O2, oxidization of O2, reduction of Hf4+, combination of Hf and O into h-Hf6O, and crystallization of a-HfO2 into m-HfO2, as summarized by Reactions 1–5 in Fig. 5.
Despite the subtle differences in various shell structures, the shell of the CF system in HfO2-based RS memristors with various TEs serves as a robust shell barrier to prohibit oxygen migration towards CF owing to the fact that it is harder to create oxygen vacancies in highly crystallized oxygen-rich HfOx than in amorphous HfO2 (supported by the oxygen vacancy formation energy calculation as shown in Supplementary Note 4). Hence, the existence of such a crystalline HfOx shell helps to prohibit the CF oxidation and therefore contribute to the enhanced retention performance. Although the CF size is strongly correlated with the retention property, the crystalline shell of the CF also benefits the retention performance of the oxide memristors. This observation well explains the poor retention property of various reported low-ICC memristors62,63, where the CF lacks a robust shell barrier to prohibit CF oxidation. In general, the quasi-core–shell CF system with excellent retention performance is expected to be one of the critical factors for RS memristor devices with extreme scalability down to sub-5 nm64.
Regarding the RESET process, the oxygen ions formed by reduction reaction at the TE/RS interface are driven back by an electric field and react with the h-Hf6O CF to generate nonconductive HfO2. Moreover, part of the CF fuses with the assistance of Joule heating (so-called TCM-based RESET process). Both Joule heating and electric field play important roles in the RESET process, when the device switches from the LRS to the HRS. As evidenced by the electrical measurement, a much higher current is needed to switch the device from the LRS back to HRS. Therefore, the Joule heat effect on the h-Hf6O CF is much more significant in the RESET process than in the Forming process. Accordingly, the surrounding m-HfO2 shell transforms into t-HfO2 (Fig. 5g), which is a high-temperature stable phase of HfO2 with a high formation energy65,66, and the h-Hf6O CF starts to rupture at its thinnest part near the TE side, leaving a conical residue (Fig. 5h). Additionally, the rupture process of the h-Hf6O CF draws back oxygen from its surroundings, contributing to the existence of abundant $${{{{{{\rm{V}}}}}}}_{{{{{{\rm{O}}}}}}}$$ in the crystalline t-HfO2 shell. The loss of oxygen in the t-HfO2 shell region further promotes its stability (as t-HfOx) and prohibits its transition to room temperature stable m-HfO2 after the RESET process, as revealed by the relative free energy in the ab initio calculations (Fig. 4b). Reactions during the RESET process are summarized as Reactions 6 and 7 in Fig. 5. The reactions in the following SET operation are similar to those in the Forming process, while the t-HfO2 shell outside the h-Hf6O core may transform back to m-HfO2 owing to the decrease in $${{{{{{\rm{V}}}}}}}_{{{{{{\rm{O}}}}}}}$$ concentration (Reaction 8). There is a tradeoff for the VSET of RS memristors: local electric field enhancement of the CF residue helps decrease VSET67, while the formation of CF in the highly crystalline package layer needs a higher VSET than in the amorphous structure68. Considering the probability event of VSET > VForming in HfO2-based RS memristors with various TEs (Supplementary Fig. 2), the crystalline environment, as well as the weak oxygen reservation capability of the electrode, explain the higher VSET of the Pt/HfO2/Pt RS memristor than its VForming.
In summary, we have studied the dynamic changes in the physical properties of the CF system in oxide memristors, including its composition, structure, and especially the evolution of the CF surroundings based on Pt/HfO2/Pt devices with a-HfO2 as the RS layer. The Pt/HfO2/Pt devices exhibit considerable RS performance, including a large switching window (HRS/LRS) above 106, good retention, and a short OFF/ON switching time within 120/20 ns. By analyzing the atomic structure of the CFs and their surroundings using HRTEM, we conclude that the CF system in HfO2-based memristors is a quasi-core–shell structure: the center of the CF system consists of metallic h-Hf6O surrounded by nonconductive m-HfO2, t-HfO2, or poorly conductive t-HfO. The core-shell CF system has universal significance for HfO2-based memristors, even though the specific components of the crystalline shell vary with the oxygen reservation capability of various TEs. It is demonstrated that the concentration of oxygen vacancies, Joule heat, and surface energy play critical roles in the evolution/transition of the shell structure around the h-Hf6O core of the CF system. The quasi-core–shell CF system with an intrinsic barrier, which prohibits CF oxidation and ensures good retention performance, is believed to be one of the critical factors for the extreme scalability of the RS memristor devices down to sub-5 nm. This study renders a further understanding of the nature of the CF system, deepens the VCM and TCM mechanism of HfO2-based memristors, and provides potential inspirations to improve oxide-based RS memristors for memory applications.
## Methods
### Sample preparation
The detailed fabrication processes of the crossbar Pt/HfO2/Pt and HfO2-based RS memristors with various electrodes (TiN, Ta, Hf, and Ti) are illustrated in Supplementary Fig. 1. The flake samples of the Pt/HfO2/Pt memristor for TEM characterization were prepared by the focused ion beam (FIB) technique (FEI Helios Nanolab 450s, UK). After the deposition of carbon and platinum protecting layers, the target regions of the samples were etched to a thickness of 50 nm for electron transmission.
### Characterizations
The electrical characteristics, including IV curves, retention, and endurance of the Pt/HfO2/Pt memristors, were measured at room temperature using an Agilent B1500A Semiconductor Device Analyzer under DC sweep mode. The speed characteristics of the memristors were implemented under pulse mode in the atmosphere. A waveform generator/fast measurement unit module (WGFUM) in the Agilent B1500A Analyzer was used to generate the voltage pulse and measure the response current at the same time in this experiment. The temperature–resistance characteristics of the LRS and the HRS samples were measured by a Keithley 4200-SCS semiconductor characterization system with the temperature changing from 180 K to 400 K under vacuum. For all IV sweeps and pulse mode experiments, the bias was always applied to the TE, and the BE was grounded.
### SEM and HRTEM experiments
An SEM image of the Pt/HfO2/Pt memristor was obtained using a field-emission scanning electron microscopy (ZEISS SUPRA 55 SAPPHIRE). The flakelet samples of the Pt/HfO2/Pt memristor for TEM characterization were prepared by the FIB etching technique (FEI Helios Nanolab 450s, UK). The target region of the sample was milled to 50 nm in thickness for electron transmission. TEM images were obtained with a field-emission gun/TEM (FEI Tecnai TF-20, UK) operated under 200 kV voltage. The FFT diffraction patterns were analyzed by TEM Imaging & Analysis (TIA, FEI) software.
### Ab initio calculation
Density functional calculations were carried out using the projector augmented-wave (PAW) method, with the Vienna Ab Initio Simulation Package (VASP 5.4.4)69,70. A plane wave basis set with 500 eV kinetic energy cutoff was chosen to expand the wave-functions. Generalized gradient approximation (GGA) was adopted for the exchange-correlation energy, in the simple Perdew–Burke–Ernzerhof (PBE) functional form71. In Gibbs free energy calculations, we chose the valence electrons as: 5s, 5p, 5d, and 6s for Hf; 2s and 2p for O, while in filament-in-dielectric supercell calculations, the 5s and 5p electrons were considered as in the core part of the Hf pseudopotential. The vibration frequencies were calculated using the density functional perturbation theory, while the vibration entropies were derived using the harmonic oscillator model. On account of the semiconductor band gap problem due to GGA, we adopted the self-energy corrected GGA-1/2 method72,73 in electronic structure calculations for the filament-in-dielectric supercell, which fits normal oxides like HfO2. The optimum cutoff radius for the O PBE self-energy potential74 was calculated to be 2.7 bohr in HfO2, through a variational method. No empirical parameter was involved in the GGA-1/2 calculation. The GGA-1/2 electronic structure for monoclinic HfO2 is comparable with that of the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional result (see Supplementary Fig. 12).
### Statistics and reproducibility
Experiments were reproducible.
Figure 1b, the experiments were performed five times with similar results.
Figure 2a, d, the experiments were performed six times with similar results.
Figure 3a, e, the experiments were performed six times with similar results.
Supplementary Figure 4a, the experiments were performed once.
Supplementary Figure 5a, the experiments were performed twice with similar results.
Supplementary Figure 6a, the experiments were performed once.
Supplementary Figure 7a, the experiments were performed once.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
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2023-03-26 13:28:31
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https://www.corsbook.com/lesson/exponential-functions/
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# Exponential functions
#### By Mark Ciotola
First published on February 27, 2019
## Introduction
### Functions
Functions relate at least one variable to at least one other number or variable. $$x$$ is a variable. Its value could be anything. Yet, consider the equation:
$$x = 1$$
Here, x is constrained to equal to a single particular number, which is 1. Such a number is known as a constant. It’s value in the equation cannot change. Things get more interesting when a variable is related to another variable. For example, let’s say that
$$y = 2x$$
Both x and y are variables. However, the value of either variable can change, depending on the value of the other variable. Below are some sample pairs of values allowed by this equation:
x y
-2 -4
-1 -2
0 0
1 2
2 4
10 20
Such a function acts as a constraint of the values. If x is 1, then y mustbe 2.
Functions can take many forms, such as y = x^3, z = 2x + 5y, or y = sin x, where sin is shorthand for the trigonometric function sine, what can be expressed quantitatively as a series of numbers.
### What Exponential Functions Are
Exponential functions are functions where a constant is raised to a power. For example,
$$y = 7^x$$
Here, the values of x and y are:
x y
-2 1/49
-1 1/7
0 1
1 7
2 49
3 343
You can see that exponential functions can increase very quickly. There is a very special constant called $$e$$ equal to 2.71828. e is a very special number in mathematics for many reasons. However, it is also a very useful base for exponential functions. e is typically used as the base for exponential functions.
## Pure exponential growth
Pure exponential growth is that which is proportional to its current quantity. It can apply to populations of bacteria, fish and even humans. It can apply to chain reactions in nuclear physics as well.
Bacteria (photo credit: CDC US government)
Pure exponential growth begins slowly, then literally explodes over time. Sometimes the plot of exponential growth is described as a “hockey stick” because it starts nearly horizontally, then “turns the corner” and grows nearly vertically.
It typically concerts population differential equations such as
$$\frac{dP}{dt}= k P$$
where P is population, t is time and k is a proportionality constant. The solution for this equation is the classic exponential growth function
$$P = P_0 e^{k t}$$
where $$P_0$$ is the initial population.
Pure exponential growth
Different growth rates result in different levels of growth at a particular point of time, but growth is still ultimately explosive (see below).
Various rates of exponential growth
## Logistic Growth
A resource that is renewable, but limited in the short-run, can be modeled with a logistics curve. Examples of such resources are new-growth forests and wild Pacific salmon. They can be nearly totally consumed in the short run, but these resources can restore themselves if they have not been exploited too completely. A logistics curve is not shown here, but is in the shape of an elongated “S” and can be found in many differential equations textbooks. The beginning (and bottom) of the “S” represents the initial exploitation. The forward-sloping “back” of the “S” represents nearly pure exponential growth. The end and top of the “S” represents a leveling off of growth, as consumption of the resource matches its ability to restore itself.
In logistic growth, population tends to move towards a particular population level that reflects the carrying capacity of the system or environment. Such functions are also called S-curves.
Logistics functions are in the form of:
$$\frac{dP}{dt} = k \Big(1 – \frac{P}{N} \Big) P$$,
where P is population, t is time, k is a growth rate coefficient and N is the carrying capacity. N can also be viewed as the periodic replenishment of potential.
Logistic growth
Achieving a logistics curve is the holy grail of sustainability enthusiasts. Applying concepts of sustainability to an entire dynasty or regime is called Big Sustainability, and involves social and economic sustainability, as well as physical resource sustainability (e.g. sufficient desired resources and ability to avoid toxins).
## Efficiency-Discounted Exponential Growth
Efficiency-Discounted Exponential Growth (EDEG) involves the consumption of a non-replenished resource over time by a system of reproducing agents. It can be useful for modeling mineral production of a mining region.
It is not yet possible to analytically create a EDEG function from fundamental principles. However, EDEG can be expressed as a differential equation, which can then be iteratively calculated via a spreadsheet or computer program.
An EDEG function can be approximated by multiplying a pure exponential growth function by an efficiency function.
$$P = k_1 e^{k_2 t} (1 – \frac{Q}{Q_0})$$,
where P is power (or production), $$k_1$$ is a constant of proportionality, typically the initial prediction (or power), $$k_2$$ is a growth factor, $$Q$$ is the amount of a nonrenewable critical resource thus far consumed, and $$Q_0$$ is the initial quantity of the nonrenewable critical resource.
$$k_1 e^{k_2 t}$$
represents the exponential growth component.
$$(1 – \frac{Q}{Q_0})$$
represents the efficiency component.
Colorado San Juans gold production versus EDEG model
## HS Curves
It may also be possible to create a function that transforms an H-Curve (EDEG) to an S curve. There is an initial amount of a non-renewable critical resource, and then periodic replenishment of that resource.
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2022-07-06 15:33:56
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https://socratic.org/questions/how-do-you-balance-the-chemical-equation-h-2o
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# How do you balance the chemical equation H_2O?
To form ${H}_{2} O$ from it elements?
${H}_{2} \left(g\right) + \frac{1}{2} {O}_{2} \left(g\right) \rightarrow {H}_{2} O \left(l\right)$
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2019-11-13 13:15:40
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https://scicomp.stackexchange.com/questions/10626/implicit-finite-difference-scheme-for-a-pde-with-only-one-boundary
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# Implicit Finite difference scheme for a PDE with only one boundary
I am looking at a few reaction-diffusion equations of the form
$\frac{dP}{dt} = D\left(\frac{d^2P}{dr^2} + \frac{2}{r}\frac{dP}{dr}\right) - a(P)$
I know the initial conditions and the boundary value at one end. I also know the function steadily falls and eventually hits zero at the second unknown boundary. To model this behaviour and indirectly find this unknown boundary, I coded a solver using an explicit finite difference scheme, with an added condition in the loop that changed forced any negative values to zero. This gave me the correct result, but due to the CFL condition I've had to use a tiny time step ($\Delta t = 0.001$) when I'm more interested in the system's behaviour over several hours or even days.
I was looking into using implicit finite difference methods or CN methods so I may increase the time step, but my (limited) understanding of this implies I'd have to solve a system of equations which would include the unknown boundary, which I will not know exactly. Is it possible to work around this, or will implicit methods fail? If it is possible to work around it, could anyone suggest a good method and how I would implement this? Thanks in advance, I'm quite new to numerical methods and would appreciate any suggestions.
• Could you explain more what you did with the explicit scheme with the second boundary? I don't see how what you currently said would in any way handle the boundary. – Godric Seer Jan 27 '14 at 16:10
• Sure; First I made a grid. I know initially all points on the grid except the boundary values are 0, so I set them to that. I knew at P(x0,t) = po , so I specified that. I then used FD method over 500 spatial steps and 600 time steps (10 minutes) to compute P(i,j +1) ; I used an IF loop to check P(i,j+1) was positive or zero; if it wasn't, it was set to zero before the loop incremented. This method worked quite well, but I suspect I can't make an implicit analog! – DRG Jan 27 '14 at 16:25
I am not entirely certain that your method works correctly. The answers may look correct but unless you have an analytic solution to check against you could be introducing errors. Also, how do you handle the 2nd derivative at the far boundary?
The boundary condition you have is:
$\lim \limits_{x \to \infty} P = 0$
with a domain of $[x_0, \infty)$. A more common way to handle this numerically is to truncate your domain to $[x_0, x_1]$ and hold the boundary condition that
$P(x_1) = 0$
This works fine, except for you don't know if your truncated domain has affected your answer. So then you resolve it on $[x_0, 2 x_1]$ (assuming $x_1 >> x_0$ and this about doubles your domain). If your answer in the areas of interest are close, then you know you are done.
If you make this change, then switching to an implicit solver would be trivial, since you have eliminated the if condition inside your solver.
• Thanks for that; so essentially I keep altering my range and check that the solutions look similar over their length scales? – DRG Jan 27 '14 at 17:10
• Yes. You know it is 0 out somewhere, so you just apply it somewhere and check that it is far enough out to not affect your solution. – Godric Seer Jan 27 '14 at 17:49
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2020-01-18 18:21:36
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https://www.physicsforums.com/threads/proof-of-linearly-independence.637632/
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# Proof of Linearly independence
pyroknife
The problem is attached. I just wanted to see if the way I proved my statement is correct.
My answer: No, because there exists more columns than rows, thus at least one free variable always exists, thus these vectors are linearly dependent.
#### Attachments
• Untitled.png
5.1 KB · Views: 351
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2023-02-03 07:59:24
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https://phabricator.wikimedia.org/p/Debenben/
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# User Details
User Since
Nov 1 2015, 12:12 PM (206 w, 3 d)
Availability
Available
LDAP User
Debenben
MediaWiki User
# Jul 20 2019
@Physikerwelt what do you mean with "exist as alias for exist"?
# Jun 26 2019
Debenben added a comment to T195765: Make it possible to query for math values.
It is easy to perform some sanitization or conversion on the original string if you need it.
# May 4 2019
Debenben committed rTTEX595b29fa9b88: prepare en botrun (authored by Debenben).
prepare en botrun
@GregorAlexandru We are actually planning to get rid of the texvcjs so that in the future we can just refer to the MathJax documentation.
# May 3 2019
Debenben committed rTTEX43f782238c5e: add hywiki editsummary and new inputlists (authored by Debenben).
add hywiki editsummary and new inputlists
Debenben committed rTTEXbe778a6e8313: experiments with different search scripts (authored by Debenben).
experiments with different search scripts
# Jan 25 2019
Debenben committed rTTEX7837f9a02318: Merge branch 'templateexpand' (authored by Debenben).
Merge branch 'templateexpand'
# Jan 6 2019
Debenben committed rTTEXc2d91561d0ee: improved crawler (authored by Debenben).
improved crawler
# Jan 2 2019
Debenben committed rTTEX6be0395332e1: +special replacement with {{C}} template (authored by Debenben).
+special replacement with {{C}} template
Debenben committed rTTEX3a88d6cdfe5c: Merge branch 'master' into templateexpand (authored by Debenben).
Merge branch 'master' into templateexpand
# Dec 30 2018
Debenben committed rTTEX5776657e88ff: Merge branch 'master' into templateexpand (authored by Debenben).
Merge branch 'master' into templateexpand
# Dec 28 2018
Debenben committed rTTEX117ab689a123: Merge branch 'master' into templateexpand (authored by Debenben).
Merge branch 'master' into templateexpand
# Dec 27 2018
Debenben added a comment to T197925: Create a bot to replace deprecated math syntax .
I put some thoughts into the problem of replacing math generated by templates like in https://de.wikiversity.org/wiki/Kommutative_Ringtheorie/Algebra-Homomorphismus_%C3%BCber_Ring/Definition where $K$ is produced with a source code
Debenben committed rTTEX8cca2b956d11: Merge branch 'master' into templateexpand (authored by Debenben).
Merge branch 'master' into templateexpand
# Dec 24 2018
Debenben committed rTTEX6128b30b4911: Merge branch 'master' into templateexpand (authored by Debenben).
Merge branch 'master' into templateexpand
Debenben committed rTTEXeb91b755217d: add comparison of tempate text (authored by Debenben).
# Dec 23 2018
Debenben committed rTTEXcf2b2c64402a: disable saving for testing (authored by Debenben).
disable saving for testing
Debenben committed rTTEXa122ef0eb131: test templatesearch (authored by Debenben).
test templatesearch
Debenben committed rTTEXe212332b0570: Merge branch 'master' into templateexpand (authored by Debenben).
Merge branch 'master' into templateexpand
# Dec 21 2018
@SalixAlba We have to change our algorithms to look for math tags. dewikiversity is using {{#tag: math}} via https://de.wikiversity.org/wiki/Vorlage:Math on more than 18 000 pages.
# Dec 11 2018
Addendum: Another reason for a block formula is of course if the formula requires too much vertical space to fit into a normally spaced line
@Izno I do not know what the best way of implementing it in the extension is, but from an author point of view both are treated equally as part of a sentence and there is no difference between inline and block formula except for the size of the equation.
# Dec 7 2018
Or with T209148 in mind, why not disable lazy loading for math "images" completely? After all they are not images, but part of the text.
Debenben added a project to T211432: chem tags should use the same rendering options as math: Math.
Debenben changed the status of T139855: Fix content with broken path integrals from Resolved to Declined.
Debenben added a comment to T182127: Additional math symbols for oriented integrals.
If a symbol people need is missing or looks ugly, they will try to rewrite equations to make it work without the symbol or use some workarounds, e.g. in this case probably something like \int_C and then specify what C means. Those line integrals are common in complex analysis (residue theorem) and hence for example also in electrodynamics, I would estimate the number of articles where they could be used to more than 100 on enwiki.
Debenben updated the task description for T182127: Additional math symbols for oriented integrals.
Debenben closed T139855: Fix content with broken path integrals as Resolved.
I think people do not want to use \oiint or \oiiint because the integral symbol looks bad, they will prefer workarounds if they look better in their setup. It is probably not the biggest problem, but I would only consider the integrals in T182127 done, if they actually produce a rendering that is not ugly. I will close this task because I think it does not help.
# Nov 16 2018
Debenben added a comment to T197925: Create a bot to replace deprecated math syntax .
@SalixAlba I pushed a commit that should fix it
Debenben committed rTTEXddf365ecd328: fix issue with nowiki tags (authored by Debenben).
fix issue with nowiki tags
Debenben committed rTTEXfff384e374f4: test nowiki issue (authored by Debenben).
test nowiki issue
# Nov 14 2018
Debenben added a comment to T197925: Create a bot to replace deprecated math syntax .
@SalixAlba thanks for catching that. This is something we overlooked:
Debenben committed rTTEX2484341191db: +rowiki editmessage (authored by Debenben).
+rowiki editmessage
Debenben committed rTTEXdc836ffec735: +hewiki logs (authored by Debenben).
+hewiki logs
# Nov 12 2018
Debenben committed rTTEXfdfcd8550dd6: +he translation (authored by eranroz).
+he translation
# Nov 11 2018
Debenben added a member for Math: Debenben.
Debenben updated the task description for T166369: Migrate old Math options to current ones or delete them from the database.
Debenben moved T148483: Adding Math Plugin Causes VisualEditor to Crash from Incoming to VisualEditor on the Math board.
We really need an option that covers all use-cases (especially because you have to login to get to the options) but currently we don't and these options are very helpful for debugging e.g. T194768 so they should stay for now.
I think this was a problem of MathML and is fixed in newer firefox versions.
Debenben closed T132367: Latex renders with excessive height on firefox as Resolved.
I think this was a problem of MathML and is fixed in newer firefox versions.
Debenben added a comment to T29574: PDF export: Use LaTeX formulas instead of inline images.
@ovasileva I just created a pdf of https://de.wikipedia.org/wiki/Satz_des_Pythagoras and half of it is not shown at all and the other half still looks horrible:
Debenben closed T2798: Many character sets don't work in texvc as Resolved.
The caracters probably still look bad, but there should not be any errors anymore since png is also using MathJax now
Debenben closed T2798: Many character sets don't work in texvc, a subtask of T4458: Localized TeX environment, as Resolved.
the rendering of Malayalam is still not very good, but there is no error anymore, since the png is created from the svg images now
Debenben moved T188879: Remove texvc from Incoming to Next-up on the Math board.
Debenben closed T136931: texvc vs Mathoid parsing differences as Resolved.
should be resolved since png is also using MathJax now
Debenben moved T197925: Create a bot to replace deprecated math syntax from Incoming to Doing on the Math board.
# Nov 10 2018
Debenben committed rTTEXf59e49d0caa3: remove old inputlists (authored by Debenben).
remove old inputlists
# Nov 9 2018
Debenben committed rTTEX4ea60664e6dc: +zhwiki editmessage and logs (authored by Debenben).
+zhwiki editmessage and logs
@Debenben what browsers those screenshots done with? Such a bad rendering shouldn't happen with the current solution.
# Nov 7 2018
@Physikerwelt For example in the mobile view:
# Nov 5 2018
The custom "Wikipedia syntax" suggested here would be a terrible idea. We are having enough trouble with non-standard LaTeX syntax already. At times, especially with mhchem or optional arguments (which use square brackets) it seems random, how an equation renders and if not what kind of error message you get. Even as an experienced mediawiki-user you can spend hours trying to find out where those strange error messages come from and how to fix them.
# Nov 4 2018
Debenben committed rTTEX5ba4bc2539f4: +arwiki editmessage and logs (authored by Debenben).
+arwiki editmessage and logs
Debenben committed rTTEXd0b19e5029ae: +arwiki editmessage and logs (authored by Debenben).
+arwiki editmessage and logs
# Nov 2 2018
What is currently being done is delivering the same static svg images to everyone. What I propose is to do the final rendering client-side like in normal MathJax, so that e.g.
@Pkra About inlining SVG: Maybe you are right and in principle SVG could become a better solution than HTML. I don't like the idea of taking the current system and just removing the image tags because this is not enough to solve those problems. The most probable case is that we don't have enough manpower to actually do the things that "might be doable" and "require a bit of work" and the result is that we are stuck with an over-customized, substandard, unmaintained system forever.
# Nov 1 2018
Debenben committed rTTEX09a4c07c74e3: +inputlists, correct link (authored by Debenben).
Debenben added a comment to T197925: Create a bot to replace deprecated math syntax .
@SalixAlba Thank you for taking care of the botflag on enwiki, feel free to take responsibility and request a botflag on any other project you like.
Debenben updated the task description for T208475: chem equations cut off.
# Oct 30 2018
Debenben committed rTTEXe92f5e402b04: move logs to own directory (authored by Debenben).
move logs to own directory
Debenben committed rTTEXb185a02aea94: plwiki logs (authored by Debenben).
plwiki logs
@Theklan: We are currently discussing some changes to the math extension here: T195861, you are welcome to discuss this issue. Currently the plan is to
1. get a correct rendering like in normal MathJax / LaTeX for all equations, especially mhchem
2. render non-ASCII-characters like Cyrillic letters, äöü etc. properly in all browsers such that \text can be used with all languages that need special characters
As you can see this is a tremendous amount of work, progress is quite slow and everything relies on volunteers. I don't think changing the syntax for existing formulas with commas is feasible and maintaining the current system without additional localization features already overstretches our resources. I think it would be a better idea to just keep using the well-known {,} for decimal comma and make sure that help pages mention this LaTeX hack.
# Oct 23 2018
Debenben committed rTTEX660b1615631b: +plwiki test (authored by Debenben).
+plwiki test
# Oct 7 2018
Debenben committed rTTEX5811c6c744bf: ruwiki editmessage (authored by Debenben).
ruwiki editmessage
Debenben committed rTTEX26ab4b052084: added more initial input lists (authored by Debenben).
# Oct 3 2018
Debenben committed rTTEXfc2300adf1f0: +eswiki editsummary and logs (authored by Debenben).
+eswiki editsummary and logs
# Aug 19 2018
Debenben updated subscribers of T197925: Create a bot to replace deprecated math syntax .
@SalixAlba Thank you for finding the problem with the unmatched math tags and fixing it, also @Framawiki thanks for your pull request.
Debenben committed rTTEX88594539040a: frwiki logfiles (authored by Debenben).
frwiki logfiles
Debenben committed rTTEX0aeb66a5d508: set changed flag (authored by Debenben).
set changed flag
transferred changes from mathwikibot2 into mathwikibot script
Debenben committed rTTEXce82ab45ba8d: Merge branch 'D1087' (authored by Debenben).
Merge branch 'D1087'
Debenben committed rTTEX747376626dba: mathwikibot2 script rejecting nowiki tags (authored by Debenben).
mathwikibot2 script rejecting nowiki tags
Debenben committed rTTEXb0221e7d7b2c: logfiles (authored by Debenben).
logfiles
# Aug 7 2018
I think one has to differentiate: Of course there are large formulas which cannot be broken and need some scrolling mechanism. Most formulas however have a structure like $A=B+C+D$ which you can break almost everywhere if necessary. Wikipedia authors currently have a choice between
• Adding permanent linebreaks or splitting the expression up in smaller chunks which means making it less readable by wasting space on large screens and/or introducing unnecessary artificial names for parts of the sum
• Not adding any linebreaks, making the expression unreadable on small screens or introducing unnecessary scrolling or zooming.
First sentence
:$A$$\;=B$$\;+C${{nowrap|$\;+D$.}}
Next sentence.
to force a correct linebreaking behaviour and an equal looking punctuation mark.
# Aug 4 2018
Yes, definitely. This would also solve part of T194768. With JS and Mathjax you can get this functionality easily, however in Wikipedia it seems to be a huge problem because most people get to see the math as images rather than text. We are trying to come up with a solution and would need some input from developers, see T195861.
# Jul 25 2018
Debenben added a comment to T197176: Install texlive-full package.
The goal of the project would be to verify that every mathematical formula uses proper LaTeX syntax. Because LaTeX is based on macros, this can be quite complicated and the only way to be 100% sure is to use LaTeX and render it. For example MathJax treats \overline and similar primitives like normal macros, thus it is sometimes more tolerant than other rendering engines.
merci
Debenben committed rTTEXa77c4c72bcb2: +dewiki logfiles (authored by Debenben).
+dewiki logfiles
Debenben added a comment to T197925: Create a bot to replace deprecated math syntax .
Turns out: There is no problem with setting the botflag. I was expecting a fat B to show up in the version history, but this is only shown in recent changes
# Jul 11 2018
Debenben added a comment to T197925: Create a bot to replace deprecated math syntax .
The bot has bot-rights on dewiki, but saving with botflag=True doesn't set the botflag. Any idea?
Merge branch 'master' of ssh://git-ssh.wikimedia.org/diffusion/TTEX/tool-texbot
# Jul 1 2018
Debenben added a comment to T197925: Create a bot to replace deprecated math syntax .
Thanks for fix with the negative lookahead.
Debenben committed rTTEX10c706e64b50: some more fixes (authored by Debenben).
some more fixes
# Jun 30 2018
Debenben added a comment to T197925: Create a bot to replace deprecated math syntax .
Thanks for finding the problem with the login. The problem with the missing \and replacement could be that [^\\] doesn't match at the beginning of the string when there is no character to match.
Debenben added a comment to T197925: Create a bot to replace deprecated math syntax .
@SalixAlba It seems the problem with the login came with the try catch block, maybe it goes away when we remove it. It seems like the bot is trying to login when it is already logged in.
# Jun 27 2018
Debenben closed Restricted Task, a subtask of T197925: Create a bot to replace deprecated math syntax , as Resolved.
Debenben committed rTTEX79a3c9727eb8: remove .gitignore from repo (authored by Debenben).
remove .gitignore from repo
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2019-10-17 11:27:36
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5994883179664612, "perplexity": 7911.390646675613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986673538.21/warc/CC-MAIN-20191017095726-20191017123226-00149.warc.gz"}
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http://gate-exam.in/CS/Syllabus/Computer-Science-Information-Technology/Computer-Networks/Routers-Routing-Algorithms-Distance-Vector-Link-State
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# GATE Questions & Answers of Routers and Routing Algorithms (Distance Vector, Link State)
## What is the Weightage of Routers and Routing Algorithms (Distance Vector, Link State) in GATE Exam?
Total 7 Questions have been asked from Routers and Routing Algorithms (Distance Vector, Link State) topic of Computer Networks subject in previous GATE papers. Average marks 1.71.
Consider the following statements about the routing Protocols, Routing Information protocol (RIP) and open shortest path First (OSPF) in an IPv4 network.
I. RIP uses distance vector routing
II. RIP packets are sent using UDP
III. OSPF packets are sent using TCP
IV. OSPF operation is based on link-state routing
Which of the statement above are CORRECT?
Assume that source S and destination D are connected through two intermediate routers labeled R. Determine how many times each packet has to visit the network layer and the data link layer during a transmission from S to D.
Consider a network with five nodes, N1 to N5, as shown below
The network uses a Distance Vector Routing Protocol.once the Route have stabilized, the distance vectors
at different nodes are as following
N1:(0,1, 7, 8, 4)
N2: (1, 0, 6, 7, 3)
N3: (7, 6, 0, 2, 6)
N4: (8,7, 2,0,4)
N5: (4, 3, 6, 4, 0)
Each distance vector is the distance of best known path at that instance to nodes, N1 to N5, where the distance to itself is 0. Also, all links are symmetric and the cost is identical in both directions. In each round, all nodes exchange their distance vectors with their respective neighbors. Then all nodes update the distance vectors. In between two rounds, any change in cost of a link will cause the two incident nodes to change only that entry in their distance vectors.
The cost of link N2-N3 reduces to 2 (in both directions). After the next round updates, what will be the new distance vector at node, N3?
Consider a network with five nodes, N1 to N5, as shown below
The network uses a Distance Vector Routing Protocol.once the Route have stabilized, the distance vectors
at different nodes are as following
N1:(0,1, 7, 8, 4)
N2: (1, 0, 6, 7, 3)
N3: (7, 6, 0, 2, 6)
N4: (8,7, 2,0,4)
N5: (4, 3, 6, 4, 0)
Each distance vector is the distance of best known path at that instance to nodes, N1 to N5, where the distance to itself is 0. Also, all links are symmetric and the cost is identical in both directions. In each round, all nodes exchange their distance vectors with their respective neighbors. Then all nodes update the distance vectors. In between two rounds, any change in cost of a link will cause the two incident nodes to change only that entry in their distance vectors.
The cost of link N2-N3 reduces to 2 (in both directions). After the next round updates, what will be the new distance vector at node, N3?
At the update in the previous question ,the link N1-N2 goes down. N2 will reflect this change immediately in its distance vector as cost, $\infty$. After the NEXROUND of update, what will be the cost to N1 in the distance vector of N3?
Consider a network with 6 routers R1 to R6 connected with links having weights as shown in the following diagram
All the routers use the distance vector based routing algorithm to update their routing tables. Each router starts with its routing table initialized to contain an entry for each neighbour with the weight of the respective connecting link. After all the routing tables stabilize, how many links in the network will never be used for carrying any data?
Consider a network with 6 routers R1 to R6 connected with links having weights as shown in the following diagram
All the routers use the distance vector based routing algorithm to update their routing tables. Each router starts with its routing table initialized to contain an entry for each neighbour with the weight of the respective connecting link. After all the routing tables stabilize, how many links in the network will never be used for carrying any data?
Suppose the weights of all unused links in the previous question are changed to 2 and the distance vector algorithm is used again until all routing tables stabilize. How many links will now remain unused?
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2020-06-03 13:53:00
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5937803983688354, "perplexity": 1436.4810824780814}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347434137.87/warc/CC-MAIN-20200603112831-20200603142831-00441.warc.gz"}
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https://braindump.jethro.dev/posts/optimal_control/
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# Optimal Control and Planning
How can we make decisions if we know the dynamics of the environment?
## Stochastic optimization
Stochastic optimization for open-loop planning:
We wish to choose $$a_1, \dots a_T = \mathrm{argmax}_{a_1, \dots a_T} J(a_1, \dots, a_T)$$ for some objective $$J$$.
### Guess and Check
An extremely simple method, that’s parallelizable:
1. pick $$A_1, \dots A_N$$ from some distribution
2. choose $$A_i$$ based on $$\mathrm{argmax} J(A_i)$$.
### Cross-entropy Method (CEM)
1. pick $$A_1, \dots A_N$$ from some initial distribution $$p(A)$$
2. Evaluate $$J(A_1), \dots J(A_N)$$
3. pick the elites $$A_{i1}, \dots A_{im}$$ with the highest value
4. fit distribution \$P(A) to the elites
With continuous inputs, a multi-variate normal distribution is a common choice for $$p(A)$$. In the discrete case, Monte Carlo Tree Search is typically used.
## Using Derivatives
• Differentiable Dynamic Programming (DDP)
• LQR
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2023-03-26 18:50:09
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 43, "equation": 408, "x-ck12": 0, "texerror": 0, "math_score": 0.8372186422348022, "perplexity": 4380.6053899355375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00787.warc.gz"}
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https://ucsdml.github.io/jekyll/update/2020/07/20/adversarial-pruning.html
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In a previous post, we discussed the relationship between accuracy and robustness for separated data. A classifier trained on $r$-separated data can be both accurate and robust with radius $r.$ What if the data are not $r$-separated? In our recent paper, we look at how to deal with this case.
Many datasets with natural data like images or audio are $r$-separated [1]. In contrast, datasets with artificially-extracted features are often not. Non-parametric methods like nearest neighbors, random forest, etc. perform well on these kind of datasets. In this post, we focus on the discussion of non-parametric methods on non-$r$-separated datasets.
We first present a defense algorithm – adversarial pruning – that can increase the robustness of many non-parametric methods. Then we dive into how adversarial pruning deals with non-$r$-separated data. Finally, we present a generic attack algorithm that works well across many non-parametric methods and use it to evaluate adversarial pruning.
Defense
Let us start by visualizing the decision boundaries of a $1$-nearest neighbor ($1$-NN) and a random forest (RF) classifier on a toy dataset.
We see that the decision boundaries are highly non-smooth, and lie close to many data points, resulting in a non-robust classifier. This is caused by the fact that many differently-labeled examples are near each other. Next, let us consider a modified dataset in which the red and blue examples are more separated.
Notice that the boundaries become smoother as examples move further away from the boundaries. This makes the classifier more robust as the predicted label stays the same if data are perturbed a little.
From these figures, we can see that these non-parametric methods are more robust when data are better separated. Given a dataset, to make it more separated, we need to remove examples. To preserve information in the dataset, we do not want to remove too many examples. We design our defense algorithm to minimally remove examples from the dataset so that differently-labeled examples are well-separated from each other. After this modification, we can train a non-parametric classifier on it. We call this defense algorithm adversarial pruning (AP).
More formally, given a robustness radius $r$ and a training set $\mathcal{S}$, AP computes a maximum subset $\mathcal{S}^{AP} \subseteq \mathcal{S}$ such that differently-labeled examples in $\mathcal{S}^{AP}$ have distance at least $r$. We show that known graph algorithms can be used to efficiently compute $\mathcal{S}^{AP}$. We build a graph $G=(V, E)$ as follows. First, each training example is a vertex in the graph. We connect pairs of differently-labeled examples (vertices) $\mathbf{x}$ and $\mathbf{x}’$ with an edge whenever $|\mathbf{x} − \mathbf{x}’| \leq 2r$. Then, computing $\mathcal{S}^{AP}$ is reduced to removing as few examples as possible so that no more edges remain. This is equivalent to solving the minimum vertex cover problem. When dealing with binary classification problem, the graph $G$ is bipartite and standard algorithms like the Hopcroft–Karp algorithm can be used to solve this problem. With multi-class classification, minimum vertex cover is NP Hard in general, and approximation algorithms have to be applied.
Theoretical Justification
It happens that Adversarial Pruning has a nice theoretical interpretation - we can show that it can be interpreted as a finite sample approximation to the optimally robust and accurate classifier. To understand this, first, let us try to understand what the goal of robust classification is. We assume the data is sampled from a distribution $\mu$ on $\mathcal{X} \times [C]$, where $\mathcal{X}$ is the feature space and $C$ is the number of classes. Normally, the ultimate limit of accurate classification is the Bayes optimal classifier which maximizes the accuracy on the underlying data distribution. But the Bayes optimal may not be very robust.
Let us look at the figure below. The blue curve is the decision boundary of the Bayes optimal classifier. We see that this blue curve is close to the data distribution and thus not the most robust. An alternative decision boundary is the black curve, which is further away from the distribution while still being accurate.
We define the astuteness of a classifier as its accuracy on examples where it is robust with radius $r$. The objective of a robust classifier is to maximize the astuteness under $\mu$, which is the probability that the classifier is both $r$-robust and accurate for a new sample $(\mathbf{x}, y)$ [1, 2].
Let $\mathbb{B}(\mathbf{x}, r)$ be the ball with radius $r$ around $\mathbf{x}$ and $S_j(f,r) := \{\mathbf{x} \in \mathcal{X} \mid f(\mathbf{x}') = j \text{ s.t. } \forall \mathbf{x}' \in \mathbb{B}(\mathbf{x}, r)\}$. For distribution $\mu$ on $\mathcal{X} \times [C]$, the astuteness is defined as $$ast_\mu(f,r) = \sum_{j=1}^{C} \int_{\mathbf{x} \in S_j(f,r)} Pr(y = j \mid \mathbf{x}) d \mu.$$
Next, we present the $r$-optimal classifier that achieves optimal astuteness. By comparing it with the classic Bayes optimal classifier, which achieves optimal accuracy, the $r$-optimal classifier is a Robust Analogue to the Bayes optimal classifier.
$r$-optimal classifier (black curve) Bayes optimal classifier (blue curve)
Optimal astuteness Optimal accuracy
\begin{split} \max_{S_1,\ldots, S_c} & \sum_{j=1}^{c} \int_{\mathbf{x} \in S_j} Pr(y = j \mid \mathbf{x}) d\mu \\ \mbox{ s.t. } \quad & d(S_j, S_{j'}) \geq 2r \quad \forall j \neq j' \\ & d(S_j, S_{j'}) := \min_{u \in S_j, v \in S_{j'}} \| u-v\|_p \end{split} \begin{split} \max_{S_1,\ldots, S_c} & \sum_{j=1}^{c} \int_{\mathbf{x} \in S_j} Pr(y = j \mid \mathbf{x}) d\mu \\ \end{split}
We observe that AP can be interpreted as a finite sample approximation to the $r$-optimal classifier. If $S_j$ are sets of examples, then the solution to the $r$-optimal classifier is maximum subsets of training data with differently-labeled examples being $2r$ apart. As long as the training set $S$ is representative of $\mu$, these subsets ($S_j$) approximate the optimal subsets ($S^*_j$). Hence, we posit that non-parametric methods trained on $S^{AP}$ should approximate the r-optimal classifier
For more about the $r$-optimal classifier, please refer to this paper.
Adversarial pruning generates $r$-separated datasets
What AP does is remove the minimum number of examples so that the dataset becomes $r$-separated. In our previous post, we show that there is no intrinsic trade-off between robustness and accuracy when the dataset is $r$-separated. This means that there exists a classifier that achieves perfect robustness and accuracy. However, the solution may make mistake on the examples removed by AP and we can think about the removed examples as the trade-off between robustness and accuracy.
Evaluating AP: An Attack Method
In this section, we provide an attack algorithm to evaluate the robustness of non-parametric methods. For parametric classifiers such as neural networks, generic gradient-based attacks exist. Our goal is to develop an analogous general attack method, which applies to and works well for multiple non-parametric classifiers.
The attack algorithm is called region-based attack (RBA). Given an example $\mathbf{x}$, RBA can find the closest example to $\mathbf{x}$ with different prediction, in other words, RBA achieves the optimal attack. In addition, RBA can be applied to many non-parametric methods while many prior attacks for non-parametric methods [1, 2] are classifier specific. 1 only applies to $1$ nearest neighbors and 2 only applies to tree-based classifiers.
To understand how RBA works, let us look at the figures above. The figures above show the decision boundaries of $1$-NN and decision tree on a toy dataset. We see that the feature space is divided into many regions, where examples in the same region have the same prediction (meaning we can assign a label to each region). These regions are convex for nearest neighbors and tree-based classifiers.
Suppose the example we want to attack is $\mathbf{x}$ and $y$ is its label. RBA works as follows. Suppose we could find the region $P_i$ that is the closest to $\mathbf{x}$ and its label is not $y$. Then, the closest example in $P_i$ to $\mathbf{x}$ would be the optimal adversarial example. RBA finds the closest region $P_i$ by iterating through each region that is labeled differently from $y$. More formally, given a set of regions and its corresponding label $(P_i, y_i)$, the RBA solves the following optimization problem:
$\underset{i : f(\mathbf{x}) \neq y_i }{\textcolor{red}{min}} \ \underset{\mathbf{x}_{adv} \in P_i}{\textcolor{ForestGreen}{min}} \|\mathbf{x} - \mathbf{x}_{adv}\|_p$
The $\textcolor{red}{\text{outer$min$}}$ can be solved by iterating through all regions. The $\textcolor{ForestGreen}{\text{inner$min$}}$ can be solved with standard linear programming when $p=1$ and $\infty$ and quadratic programming when $p=2$. When this optimization problem is solved exactly, we call it RBA-Exact.
Interestingly, concurrent works [1, 2, 3, 4] have also shown that the decision regions of ReLU networks are also decomposable into convex regions and developed attacks based on this property.
Speeding up RBA. Different non-parametric methods divide the feature space into different numbers of regions. When attacking $k$-NN, there would be $O(\binom{N}{k})$ regions, where $N$ is the number of training examples. When attacking RF, there is an exponential number of regions with growing number of trees. It is computationally infeasible to solve RBA-Exact when the number of regions is large.
We develop an approximate version of RBA (RBA-Approx.) to speed up the process and make our algorithm applicable to real datasets. We relax the $\textcolor{red}{\text{outer$min$}}$ by iterating over only a fixed number of regions based on the following two criteria. First, a region has to have at least one training example in it to be considered. Second, if $\mathbf{x}_i$ is the training example in the region $P_i$, then the regions with smaller $\|\mathbf{x}_i - \mathbf{x}\|_p$ are considered first until we exceed the number of regions we want to search. We found that empirically using these two criteria to search $50$ regions can find adversarial examples very close to the target example.
Empirical Results
We empirically evaluate the performance of our attack (RBA) and defense (AP) algorithms.
Evaluation criteria for attacks. We use the distance between an input $\mathbf{x}$ and its generated adversarial example $\mathbf{x}_{adv}$ to evaluate the performance of the attack algorithm. We call this criterion empirical robustness (ER) The lower ER is, the better the attack algorithm is. We calculate the average ER over correctly predicted test examples.
Evaluation criteria for defenses. To evaluate the performance of a defense algorithm, we use the ratio of the distance between an input $\mathbf{x}$ and its closest adversarial example being found before and after the defense algorithm is applied. We call this criterion defense score ($\text{defscore}$). More formally,
$$\text{defscore}(\mathbf{x}) = \frac{\text{defended dist. from } \mathbf{x} \text{ to } \mathbf{x}_{adv}}{\text{undefended dist. from } \mathbf{x} \text{ to } \mathbf{x}_{adv}} = \frac{\text{ER w/ defense}}{\text{ER w/o defense}}.$$
We calculate the average defscore over the correctly predicted test examples. A larger defscore means that the attack algorithm needs a larger perturbation to change the label. Thus, the more effective the defense algorithm is. If the defscore is larger than one, then the defense is effectively making the classifier more robust.
We consider the following non-parametric classifiers: $1$ nearest neighbor ($1$-NN), $3$ nearest neighbor ($3$-NN), and random forest (RF).
Attacks. To evaluate RBA, we compare with other attack algorithms for non-parametric methods. Direct attack is designed to attack nearest neighbor classifiers. Black box attack (BBox) is another algorithm that applies to many non-parametric methods. However, as a black-box attack, it does not use the internal structure of the classifier. It appears that BBox is the state-of-the-art algorithm for attacking non-parametric methods.
$1$-NN $3$-NN RF
Dataset Direct BBox RBA Exact RBA Approx. Direct BBox RBA Approx. BBox RBA Approx.
cancer .223 .364 .137 .137 .329 .376 .204 .451 .383
covtype .130 .130 .066 .067 .200 .259 .108 .233 .214
diabetes .074 .112 .035 .035 .130 .143 .078 .181 .184
halfmoon .070 .070 .058 .058 .105 .132 .096 .182 .149
From the result, we see that the RBA algorithm is able to perform well across many non-parametric methods and datasets (for results on more datasets and classifiers, please refer our paper). For $1$-NN, RBA-Exact performed the best as expected since its optimal. For $3$-NN and RF, RBA-Approx. also performed the best among the baselines.
Defenses. For baseline, we consider WJC for the defense of $1$-NN and robust splitting (RS) for tree-based classifiers. Another baseline is the adversarial training (AT), which has a lot of success in parametric classifiers. We use RBA-Exact to attack $1$-NN and RBA-Approx to attack $3$-NN and RF for the calculation of defscore.
$1$-NN $3$-NN RF
Dataset AT WJC AP AT AP AT RS AP
cancer 0.82 1.05 1.41 1.06 1.39 0.87 1.54 1.26
covtype 0.61 4.38 4.38 0.88 3.31 1.02 1.01 2.13
diabetes 0.83 4.69 4.69 0.87 2.97 1.19 1.25 2.22
halfmoon 1.05 2.00 2.78 0.93 1.92 1.04 1.01 1.82
From the table, we see that AP performs well across different classifiers. AP always generates above $1.0$ defscore, which means the classifier becomes more robust after the defense. This shows that AP is applicable to many non-parametric classifiers as oppose to WJC and RS, which are classifier-specific defenses. AT performs poorly for non-parametric classifiers (this is aligned with previous findings.) This result demonstrates that AP can serve as a good baseline for a new non-parametric classifier.
Conclusion
In this blog post, we consider adversarial examples for non-parametric classifiers and presented generic defenses and attacks. The defense algorithm – adversarial pruning – bridges the gap between $r$-separated and non-$r$-separated data by removing the minimum number of examples to make the data well-separated. Adversarial pruning can be interpreted as a finite sample approximation to the $r$-optimal classifier, which is the most robust classifier under attack radius $r$. The attack algorithm – region-based attack – finds the closest adversarial example and achieves the optimal attack. On the experiment side, we show that both these algorithms are able to perform well across multiple non-parametric classifiers. They can be good candidates for baseline evaluation of robustness for newly designed non-parametric classifiers.
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2020-11-29 02:22:22
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7815631628036499, "perplexity": 730.0444192581573}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141195967.34/warc/CC-MAIN-20201129004335-20201129034335-00205.warc.gz"}
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https://math.eretrandre.org/tetrationforum/printthread.php?tid=1164
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Hyper-volume by integration - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Hyper-volume by integration (/showthread.php?tid=1164) Hyper-volume by integration - Xorter - 04/08/2017 Hi, everyone! My dream is to get a formula to get the n-dimensional hyper-volume of an n-dimensional function in cartesian AND polar coordinates, too! So the length of f(x), the area of f(x,y), the volume of f(x,y,z) ... etc. According to the other existing formulas I have created an own in cartesian coordinate system: $V_N = \int ... \int_{V_N} \sqrt{1+\sum_{k=1}^N {}{df \over dx_k}} dx_1 ... dx_N$ 1st question: Do you find it correct? 2nd: How could it look in polar coordinate system? (My final goal is to use these formulas to determine a few things about the base units of the hyperdimensional and interdimensional spaces from its derivatives and its existences. But for it, I need these formulas!)
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2020-01-25 17:31:36
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https://mathematica.stackexchange.com/questions/87592/implicit-derivatives-in-one-line-how-do-i-do-it?noredirect=1
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# Implicit derivatives in one line. How do I do it?
I have this little script to derive implicitly
$PrePrint = # /. {D[y_, x_, NonConstants -> {y_}] :> y'[x]} &; Example: D[x == y^3 + x y, x, NonConstants -> y] 1 == y + x y'[x] + 3 y^2*y'[x] and then, to obtain the expression for y'[x], I use Solve[1 == y + x y'[x] + 3 y^2*y'[x],y'[x]] //FullSimplify How can I get all these steps in one line? I'd like to have something like $PrePrint = # /. {D[y_, x_, NonConstants -> {y_}] :> y'[x]} &; //Solve...
• I don't see a question anywhere… – J. M. is in limbo Jul 6 '15 at 0:02
• Look up Dt[]. – J. M. is in limbo Jul 6 '15 at 0:56
• If I saw that, but I want to publish online with the Solve not like having to go to take the result of the derivative and clearance and y '[x] – Fernando Silva Jul 6 '15 at 1:02
• I cannot understand the question at all. You may need a better english translation before posting here. – Jens Jul 6 '15 at 1:19
• Related: (1945), (24422), (52284) – Michael E2 Jul 6 '15 at 4:23
If my interpretation of your question is correct, the following code should produce the desired behaviour.
prePrint[input_] :=
Module[{solveFor},
input /. {D[y_, x_, NonConstants -> {y_}] :> (solveFor = y'[x])} //
If[OwnValues[solveFor] === {}, input, Solve[#, solveFor]] & //
FullSimplify];
\$PrePrint = prePrint;
Test
D[x == y^3 + x y, x, NonConstants -> y]
{{y'[x]] -> (1 - y)/(x + 3 y^2)}}
• if just that is, thank you, sorry for my English – Fernando Silva Jul 6 '15 at 2:02
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2020-02-24 06:58:14
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http://th.nao.ac.jp/MEMBER/tomisaka/Lecture_Notes/StarFormation/5/node131.html
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# Random Velocity
Considering gas in Maxwellian velocity distribution, the distribution function for the velocity is as follows:
(E.1)
This gives the one-dimensional random velocity as
(E.2)
If we observe emissions from such a gas, the emission line is broaden due to the Doppler shift. Using equation (E.2), the HWHM (half width of half maximum: the line width measured from the the center of the emission line to the point of the half intensity; see Fig.E.1) of the emission line is
(E.3)
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2018-01-19 21:06:39
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http://math.stackexchange.com/questions/731984/an-inequality-about-entropy
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Suppose we have random variable $X=\{x_1,\cdots,x_n\}$ with probability mass function $p$.
The entropy is defined by
$$H(X)=\sum_{i=1}^np(x_i)\log_b(p(x_i)^{-1})$$
where $b$ is any integer $\geq 2$.
Now suppose there is a $k$-partition $X=\bigcup_{j=1}^kX_j$, do we have the following inequality?
$$\frac{1}{n}\sum_{j=1}^k | X_j |\; H(X_j)\leq H(X)$$
i.e, the weighted mean of the entropy of these parts is less than the entropy of the whole.
-
Motivation? Are you sure you want the weight? The property is true for the unweighted mean, more specifically $H = \frac{1}{k}\sum_{j=1}^k H_j + \log k$. – leonbloy Mar 30 '14 at 12:46
@Kaa1el: How's $H(X_j)$ defined? – Ashok Mar 31 '14 at 4:23
@Ashok: $H(X_j)$ is the same as $H(X)$ except we substitute $X$ with $X_j$ and we multiply $p$ by a factor $\frac{n}{|X_j|}$. – Kaa1el Mar 31 '14 at 6:36
@leonbloy Thanks for your answer. Actually I was going to write a program depending on this. (I know essentially nothing about probability theory and information theory). Now I've found a way not using this fact in my program, so I don't need that conjecture anymore, although I hope it is true. – Kaa1el Mar 31 '14 at 6:44
what do you mean by random variable $X = \{x_1, \cdots, x_n \}$? is it a random vector? or are those possible values that $X$ takes? – Memming Mar 31 '14 at 12:37
In general, the inequation is false.
Take for example two sets, the first one with $M\gg 1$ elements and $p(x_i)= \epsilon\ll1/M$, the second with a single value. Then the global entropy $H(X)$ is near zero, while the weighted entropy will be dominated by the entropy of the first set, which is $\log_2 M$
Say: $M=1024$, $\epsilon=10^{-6}$. Then $H(X)=0.0214...$ bits, the weighted average is $9.99...$, and the inequality is false.
What is true is the following:
$$\sum \alpha_i H(x_i) = H(X) -H({\bf \alpha}) \le H(X)$$
where the coefficients of the weighted average ${\bf \alpha} = \{\alpha_i\}$ (with $\alpha_i\ge 0$ and $\sum \alpha_i = 1$) are given by $\alpha_i = \sum_{x_j \in X_i} p(x_j)$, i.e, they are not proportional to the support size of each set but to the accumulated probability of each set.
-
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2015-07-28 22:05:37
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https://www.aimsciences.org/article/doi/10.3934/mbe.2018017
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# American Institute of Mathematical Sciences
April 2018, 15(2): 393-406. doi: 10.3934/mbe.2018017
## Effect of rotational grazing on plant and animal production
1 Jamestown High School, Williamsburg, VA 23185, USA, 2 Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA
* Corresponding author: Junping Shi.
Received September 16, 2017 Accepted April 23, 2017 Published June 2017
Fund Project: The second author is supported by NSF grant DMS-1313243.
It is a common understanding that rotational cattle grazing provides better yields than continuous grazing, but a quantitative analysis is lacking in agricultural literature. In rotational grazing, cattle periodically move among paddocks in contrast to continuous grazing, in which the cattle graze on a single plot for the entire grazing season. We construct a differential equation model of vegetation grazing on a fixed area to show that production yields and stockpiled forage are greater for rotational grazing than continuous grazing. Our results show that both the number of cattle per acre and stockpiled forage increase for many rotational configurations.
Citation: Mayee Chen, Junping Shi. Effect of rotational grazing on plant and animal production. Mathematical Biosciences & Engineering, 2018, 15 (2) : 393-406. doi: 10.3934/mbe.2018017
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show all references
##### References:
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Arnăutu, Optimal harvesting for periodic age-dependent population dynamics with logistic term, Appl. Math. Comput., 215 (2009), 2701-2715. doi: 10.1016/j.amc.2009.09.010. Google Scholar [6] S. K. Bamhart, Estimating available pasture forage, Iowa State University Extension College of Agriculture, Ames, Iowa, 2009. Google Scholar [7] S. Behringer and T. Upmann, Optimal harvesting of a spatial renewable resource, J. Econ. Dyn. Control., 42 (2014), 105-120. doi: 10.1016/j.jedc.2014.03.008. Google Scholar [8] A. O. Belyakov, A. A. Davydov and V. M. Veliov, Optimal cyclic exploitation of renewable resources, J. Dyn. Control Syst., 21 (2015), 475-494. doi: 10.1007/s10883-015-9271-x. Google Scholar [9] A. O. Belyakov and V. M. Veliov, Constant versus periodic fishing: Age structured optimal control approach, Math. Model. Nat. Phenom., 9 (2014), 20-37. doi: 10.1051/mmnp/20149403. 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O'Connor, Evaluation of three grass growth models to predict grass growth in Ireland, Jour. Agri. Sci., 151 (2013), 91-104. doi: 10.1017/S0021859612000317. Google Scholar [15] I. R. Johnson, T. E. Ameziane and J. H. M. Thornley, A model of grass growth, Annals of Botany, 51 (1983), 599-609. doi: 10.1093/oxfordjournals.aob.a086506. Google Scholar [16] R. Kallenbach, Calculating stocking rates of cows, High Plains Journal, 2010. Google Scholar [17] R. Lemus, Developing a grazing system, Mississippi State University Extension, Mississippi State, Mississippi, 2008. Google Scholar [18] R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477. doi: 10.1038/269471a0. Google Scholar [19] I. Noy-Meir, Stability of grazing systems: An application of predator-prey graphs, The Journal of Ecology, 63 (1975), 459-481. doi: 10.2307/2258730. Google Scholar [20] I. 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Foley, C. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596. doi: 10.1038/35098000. Google Scholar [27] R. Smith, G. Lacefield, R. Burris, D. Ditsch, B. Coleman, J. Lehmkuhler and J. Henning, Rotational grazing, University of Kentucky College of Agriculture; Lexington, Kentucky, 2011. Google Scholar [28] J. Sprinkle and D. Bailey, How many animals can I graze on my pasture?, The University of Arizona Cooperative Extension, Tucson, Arizona, 2004. Google Scholar
Left: Growth rate of the grass and consumption rate by the cattle for continuous grazing. Here the growth rate $G(V)$ and the grazing rate $H\cdot c(V)$ are defined as in (2.1) and (2.2), with parameter values given as in Table 1 and $H=1.06$, $0.6$ and $0.2$ respectively. Right: A forage ($V$) versus cattle ($H$) bifurcation diagram for the continuous grazing system
Illustration of continuous grazing (left), and rotational grazing (right).
Amount of forage in a sustainable rotational configuration where $3$ out of $4$ paddocks are grazed. Here (2.7) and (2.8) are used for integration, $(n,m,T)=(4,3,10)$, $H=1.3$ and $T_{total}=365$ days.
Amount of forage in a sustainable rotational configuration where $3$ out of $4$ paddocks are grazed. Here (2.7) and (2.8) are used for integration, $(n,m,T)=(4,3,10)$, $H=1.28$ and $T_{total}=3650$ days.
Maximum $H$ for different paddock configurations and $T$. Here the horizontal axis is the rotation period $T$, the vertical axis is the maximum sustainable cattle number $H_{max}^R(T)$, and the legend shows $m: n$ (the number of paddocks grazed versus the number of total paddock). The horizontal line is $1.0631$ head of cattle per acre, which is from continuous grazing. Here $T_{total}=365$ is used.
Maximum $V$ for different paddock configurations and $T$. Here the horizontal axis is the rotation period $T$, the vertical axis is the forage amount $V_S^R(T)$ when achieving the maximum sustainable cattle number $H_{max}^R(T)$, and the key is $m: n$ (the number of paddocks grazed versus the number of total paddock). The horizontal line is the forage amount when achieving the maximum sustainable cattle number $H_{max}$ for continuous grazing.
Cattle and stockpiled forage plotted against the grazing ratio for a $15$-day rotation period. Here the horizontal axis is the grazing ratio of the rotational scheme, and the vertical axis is the maximum sustainable cattle number $H_{max}^R(T)$ and associated forage amount $V_S^R(T)$.
Table of variables and parameters in the equations.
Variable Meaning Units $t$ time days $V_j(t)$ grass biomass in paddock $j$ pounds/acre Parameter Meaning Units Value Reference $V_{max}$ grass carrying capacity pounds/acre $2400$ [21] $g_{max}$ maximum growth rate per capita rate per capita day$^{-1}$ $0.05625$ [14] $c_{max}$ maximum consumption rate per head of cattle pounds/(acre$\cdot$day) $35$ [1,2] $K$ half-saturation value pounds/acre $120$ $H_j$ number of cattle per acre in paddock $j$ cattle/acre
Variable Meaning Units $t$ time days $V_j(t)$ grass biomass in paddock $j$ pounds/acre Parameter Meaning Units Value Reference $V_{max}$ grass carrying capacity pounds/acre $2400$ [21] $g_{max}$ maximum growth rate per capita rate per capita day$^{-1}$ $0.05625$ [14] $c_{max}$ maximum consumption rate per head of cattle pounds/(acre$\cdot$day) $35$ [1,2] $K$ half-saturation value pounds/acre $120$ $H_j$ number of cattle per acre in paddock $j$ cattle/acre
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2018 Impact Factor: 1.313
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2020-05-30 13:04:32
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https://www.semanticscholar.org/paper/Maxwell's-demon-based-on-a-single-qubit-Pekola-Golubev/26c72b9631976aa2364e7afa5271d585f654e686
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# Maxwell's demon based on a single qubit
@article{Pekola2016MaxwellsDB,
title={Maxwell's demon based on a single qubit},
author={Jukka P. Pekola and Dmitry S. Golubev and Dmitri V. Averin},
journal={Physical Review B},
year={2016},
volume={93},
pages={024501}
}
• Published 16 August 2015
• Physics
• Physical Review B
We propose and analyze Maxwell's demon based on a single qubit with avoided level crossing. Its operation cycle consists of adiabatic drive to the point of minimum energy separation, measurement of the qubit state, and conditional feedback. We show that the heat extracted from the bath at temperature $T$ can ideally approach the Landauer limit of $k_BT\ln 2$ per cycle even in the quantum regime. Practical demon efficiency is limited by the interplay of Landau-Zener transitions and coupling to…
22 Citations
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2022-01-28 21:39:18
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https://jsthcitpizifly.com/2020/06/15/LeetCode-Solution-104/
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LeetCode Solution: 104. Maximum Depth of Binary Tree
## 104. Maximum Depth of Binary Tree
Given a binary tree, find its maximum depth.
The maximum depth is the number of nodes along the longest path from the root node down to the farthest leaf node.
Note: A leaf is a node with no children.
Example:
Given binary tree [3,9,20,null,null,15,7],
return its depth = 3.
1. 如果当前为空指针就返回0
2. 定义一个左树深度 一个右树深度
3. 如果存在左树或者右树 就继续遍历 （加1表示当前所处为上一层的子树 当前层存在所以加1
4. 因为不知道左树深还是右树深 返回左右树的最大值
Author: Jsthcit
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2021-03-03 21:09:09
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http://mathhelpforum.com/calculus/70982-numerical-analysis-newton-s-method-problem.html
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# Math Help - Numerical Analysis: Newton's Method Problem
1. ## Numerical Analysis: Newton's Method Problem
Use Newtons Method to find the solution accurate to within 10^-5 of the following problem.
x^2-2xe^(-x)+e^(-2x)=0
How do I know when to stop and how do I find P0 and P1?
2. Originally Posted by wvlilgurl
Use Newtons Method to find the solution accurate to within 10^-5 of the following problem.
x^2-2xe^(-x)+e^(-2x)=0
How do I know when to stop
as the problem said, you want to be accurate to 5 decimal places, so keep finding a better approximation until the first 5 decimal places do not change
and how do I find P0 and P1?
i suppose you mean the initial guess?
first note that you have $(x - e^{-x})^2$
so a good initial guess would be one where $x$ is close to $e^{-x}$. in fact, it is an equivalent problem to solve $x - e^{-x} = 0$. would be easier to run Newton's method on it too. you can use a graph to come up with a guess
Hope that helps
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2014-07-22 15:49:46
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https://tel.archives-ouvertes.fr/tel-00570002
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Dynamics and entropies of Hilbert metrics
Abstract : We study the geodesic flow of a Hilbert geometry defined by a strictly convex open set with $C^1$ boundary. We get interested in its local behaviour around one specific orbit as well as its global properties on a quotient manifold. We explain why this flow has hyperbolic-like properties, by studying in particular its Lyapunov exponents, which are linked in a precise way to the shape of the boundary of the convex. We prove an entropy rigidity result for compact quotients. We also develop general tools that can be used when considering noncompact ones, following ideas and results of negative curvature. The case of geometrically finite surfaces is studied in details, and the entropy rigidity theorem is extended to finite volume surfaces.
Mots-clés :
Document type :
Theses
Mathematics. Université de Strasbourg; Ruhr-Universität Bochum, 2011. English
Domain :
https://tel.archives-ouvertes.fr/tel-00570002
Contributor : Mickaël Crampon <>
Submitted on : Wednesday, March 2, 2011 - 3:53:39 PM
Last modification on : Wednesday, March 2, 2011 - 4:49:10 PM
Identifiers
• HAL Id : tel-00570002, version 2
Citation
Mickaël Crampon. Dynamics and entropies of Hilbert metrics. Mathematics. Université de Strasbourg; Ruhr-Universität Bochum, 2011. English. <tel-00570002v2>
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2015-05-26 16:23:57
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https://www.zbmath.org/?q=an%3A0617.46077
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# zbMATH — the first resource for mathematics
On the K functional between $$L^ 1$$ and $$L^ 2$$ and some other K functionals. (English) Zbl 0617.46077
An explicit formula for the K-functional between $$L^ 1$$ and $$L^ 2$$ is given. It was suggested by and contains as a special case the Valiron- Landau lemma in function theory [see P. Duren, Univalent functions (1983; Zbl 0514.30001), pp. 104-105]. (The preface contains also an anecdote about G. Freud, to whose memory the entire volume is devoted.)
##### MSC:
46M35 Abstract interpolation of topological vector spaces 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 01A70 Biographies, obituaries, personalia, bibliographies
##### Keywords:
Orlicz space; K-functional; Valiron-Landau lemma
Full Text:
##### References:
[1] Bergh, J; Löfström, J, Interpolation spaces. an introduction, () · Zbl 0344.46071 [2] Duren, P.L, Univalent functions, () · Zbl 0398.30010 [3] Holmstedt, T, Interpolation of quasinormed spaces, Math. scand., 26, 177-199, (1970) · Zbl 0193.08801 [4] Landau, E, Über die blochsche konstante und zwei verwandte weltkonstanten, Math. Z., 30, 608-634, (1929) · JFM 55.0770.03 [5] Valiron, G, Sur le théorème de M. Bloch, Rend. circ. mat. Palermo, 54, 76-82, (1929) · JFM 56.0269.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-03-04 06:26:55
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https://dvdrw.neocities.org/about/
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You’ll find, most likely, a variety of posts, spanning many topics. They form, together, a compendium of my thoughts, observations, musings, scribbles, guides, memoirs, and other writings.
In fact, they form a better about me than this page can. However, in short here are some things you can expect in further reading:
Personal favourites are:
• $mathematics$ – in particular type and category theory,
• computer science,
• theology,
• music
to name a few.
This website’s layout, much like my Emacs config, is alive and everchanging. My right menu key is bound to Hyper, and I am slowly forming my H- keymap in Emacs. An additional modifier for Greek letters and other Unicode math symbols is frequently on my mind, however XKB stops those thoughts dead in their tracks with feelings of undocumented dread and despair.
### Post Revisions
Over time I will inevitably edit, amend, and append to old posts. You might notice posts say Originally written on .... This is a temporary measure until I set up a revision history view like MediaWiki has.
### No JavaScript
Of note, this website uses and embeds no JavaScript. Try it! Disable JS and see how everything still looks and works the same. You can leave it turned off as I won’t be adding any in the future, either.
JavaScript is used willy-nilly nowadays, as a fix-all solution. While it is powerful and invaluable, its weight can exceed that power quite quickly. Even Simon Marlow’s Big Hammer comes with a price. This is why I’ve decided to keep this website free from it: I simply don’t need it.
In fact, everything except the fonts are hosted on this site. Right now, you can simply inspect the website and see more-or-less the same source code I write. Without JS, I’m sure I will have to employ interesting techniques. In the future, if there is enough bandwidth, I will most likely start minifying everything.
### Atom Feed
There is a generated Atom feed you can use to read my posts at /feed.xml.
All material posted on the site is licensed as CC BY 4.0, unless otherwise noted. To be frank, the interactions of this license’s virility on gained knowledge is unclear to me, and I suspect the same for even the most well versed lawyers.
I think we’re still waiting on the many [Google v. Oracle] and [Novell v. al] suits to come to conclusions. Considering most of these companies exist now just for those lawsuits, it’s a matter of when.
In practice, feel free to use, edit, and republish my work, but please link to where you got it from!
### Contact
You can contact me using the methods listed below, in the footer. I will try to repond to your inquiries in a timely manner. If I feel like our correspondence would be of benefit to others too, I might ask you to publish our conversation, whether in parts and/or a derivation thereof.
By all means, feel free to decline!
Other than high-brow discussion, feel free to send me a comment on a post, your new favourite artist, a request to point you to literature, etc. For this I prefer XMPP, as it’s much quicker and gratifying than using email for the same purpose.
XMPP is an odd service for those not accustom to it. To paraphrase from cock.li’s Vincent Canfield: There are no good XMPP clients.
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2023-03-30 02:40:12
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https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ALDS1_10_C
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Time Limit : sec, Memory Limit : KB
# Longest Common Subsequence
For given two sequences $X$ and $Y$, a sequence $Z$ is a common subsequence of $X$ and $Y$ if $Z$ is a subsequence of both $X$ and $Y$. For example, if $X = \{a,b,c,b,d,a,b\}$ and $Y = \{b,d,c,a,b,a\}$, the sequence $\{b,c,a\}$ is a common subsequence of both $X$ and $Y$. On the other hand, the sequence $\{b,c,a\}$ is not a longest common subsequence (LCS) of $X$ and $Y$, since it has length 3 and the sequence $\{b,c,b,a\}$, which is also common to both $X$ and $Y$, has length 4. The sequence $\{b,c,b,a\}$ is an LCS of $X$ and $Y$, since there is no common subsequence of length 5 or greater.
Write a program which finds the length of LCS of given two sequences $X$ and $Y$. The sequence consists of alphabetical characters.
## Input
The input consists of multiple datasets. In the first line, an integer $q$ which is the number of datasets is given. In the following $2 \times q$ lines, each dataset which consists of the two sequences $X$ and $Y$ are given.
## Output
For each dataset, print the length of LCS of $X$ and $Y$ in a line.
## Constraints
• $1 \leq q \leq 150$
• $1 \leq$ length of $X$ and $Y$ $\leq 1,000$
• $q \leq 20$ if the dataset includes a sequence whose length is more than 100
## Sample Input 1
3
abcbdab
bdcaba
abc
abc
abc
bc
## Sample Output 1
4
3
2
## Reference
Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.
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2021-10-26 18:21:25
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https://stats.stackexchange.com/questions/533517/what-is-the-relation-between-linear-classifier-and-linear-decission-boundary-or
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# What is the relation between Linear Classifier and Linear Decission Boundary (or Non Linear Decision Boundary)?
As we know (Wikipedia Definition): Linear Classifier makes a classification decision based on the linear combination of the feature vectors.
Mathematically : $$y = f(\sum w_i x_i)$$
So , $$f$$ is our linear classifier (which may be logistic or any other function). Now this linear classifier creates a decision boundary.
Now, for example consider only two features(X1, X2) : If the decision boundary is straight line then we say its linear decision boundary otherwise non linear decision boundary.
So, my question :
(1) If a classifier is a linear then it creates a linear decision boundary and vice versa.
(2) Non linear classifier always creates non linear decision boundary
Does the above statements are true if not then please explain? I have seen so many examples , like for SVM classifier, we transform the data to higher dimension and get the hyperplane in feature space but in input space it has non linear decision boundary.
So, what is the exact relation between a classifier and decision boundary, especially in the linear case?
• Neither statement is generally true. The subtlety is that a nonlinear classifier may, by accident, create a linear boundary. Thus, the appearance of a linear boundary in one specific application does not determine whether the classifier as a procedure is linear or not. Also, the appearance of a linear manifold at some point in executing a procedure does not necessarily make the entire procedure a linear one.
– whuber
Jul 6, 2021 at 15:16
• @whuber I got your explanation. So its all depend on the situations , both linear and non linear classifier can create any type of decision boundaries. Even thought logistic regression can create non linear decission boundaries too( if extra features are added) . Pls correct me if some thing is not true. Jul 6, 2021 at 15:31
• @whuber Pls refer to these question, the explanations are conflicting. Still I am not able to understand the concept of linear classifier. How do we know a classifier is linear ? Jul 7, 2021 at 4:13
• The answer you reference is a little sloppy because it lacks a suitable quantifier. You have to understand it as meaning all boundaries are linear, no matter what the inputs might be.
– whuber
Jul 7, 2021 at 13:49
• @whuber Your time and help would be highly appreciated if you share any blog or reading material through which I gain these concept- Linear and Non Linear Classifier . Linear and Non Decision Boundaries. Even today I asked the question in different style on Data Science Community, but didnt get response. Have a look on question Jul 7, 2021 at 13:56
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2022-08-13 06:59:36
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https://puertopenasco.rentals/1hk32ky/lyeumi.php?e06b10=turning-point-definition-algebra
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It then follows that "turning points" are either maximum or minimum points and are stationary. How? Simpsons 1/3 Rule Calculator . Making statements based on opinion; back them up with references or personal experience. J.L. There are two methods to find the turning point, Through factorising and completing the square.. Make sure you are happy with the following topics: Everything you always wanted to know. = 0 which are not turning points. Calculators and Converters ↳ Math Dictionary ↳ C ↳ Chain Rule ; … For example, here's a definition that I feel doubtful about: The Concise Oxford Dictionary of Mathematics, Some authors use turning point' as equivalent to stationary point, (I get the feeling that this term is not really used by mathematicians and mostly used only in middle- and high-school math. why is user 'nobody' listed as a user on my iMAC? Another word for turning point. How to get the least number of flips to a plastic chips to get a certain figure? "One Turn" is a full rotation (360°) See: Rotation. : a point at which a significant change occurs. Synonyms Example Sentences Learn More about turning point. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. CallUrl('betterexplained>com math>lamar>edu aspx',0), 2) Local Minimum:This is the point where the derivative of given function changes its sign from negative to positive. What is the definition of a "turning point"? A point on the graph of a function where the tangent (or tangent plane) is perpendicular to the axis representing the value (output) of a function.twin primes: A pair of consecutively odd prime numbers. The next point that we hit is the $$x$$-intercept at $$x = 2$$ and this one crosses the $$x$$-axis so we know that there won’t be a turning point here as there was at the first $$x$$-intercept. Polynomials of odd degree (except for #n = 1#) have a minimum of 1 turning point and a maximum of #n-1#. Find more ways to say turning point, along with related words, antonyms and example phrases at Thesaurus.com, the world's most trusted free thesaurus. Graphs of quadratic functions have a vertical line of symmetry that goes through their turning point.This means that the turning point is located exactly half way between the x-axis intercepts (if there are any!).. CallUrl('www>itseducation>asiahtm',0), Secondly, the 'humps' where the graph changes direction from increasing to decreasing or decreasing to increasing are often called ~TildeLink()s. If we know that the polynomial has degree n then we will know that there will be at most ~TildeLink()s in the graph. To learn more, see our tips on writing great answers. [Ptolemy's geography : an annotated translation of the theoretical chapters / J. Lennart Berggren and Alexander Jones, ed. turning point A point on the graph at which the slope of the tangent changes its sign. The maximum number of turning points for a polynomial of degree n is n – The total number of turning points for a polynomial with an even degree is an odd number. Definition of turning point. Turning point definition is - a point at which a significant change occurs. CallUrl('www>mathematicsdictionary>comhtm',0), turning point: Also known as a stationary point. The point corresponds to the coordinate pair in which the input value is zero. Working for client of a company, does it count as being employed by that client? Maths a stationary point at which the first derivative of a function changes sign, so that typically its graph does not cross a horizontal tangent. Definition of turning point in the Definitions.net dictionary. Then try to figure out what that point did. What has Mordenkainen done to maintain the balance? We are also interested in the intercepts. Find a sequence of steps (moving right, sliding down, etc) that works for that point. If the gradient of a curve at a point is zero, then this point is called a stationary point. With this type of point the gradient is zero but the gradient on either side of the point remains … For example, a univariate (single-variable) quadratic function has the form = + +, ≠in the single variable x.The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the y-axis, as shown at right.. A turning point is either a local maximum point or a local minimum point. Well, we would just write-- let's take d as our relative minimum. MathJax reference. Example 7: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function Find the equation of the line of symmetry and the coordinates of the turning point of the graph of \ (y = x^2 - 6x + 4\). Generally, you can view a "turning point" as a point where the curve "changes direction": for example, from increasing to decreasing or from decreasing to increasing. As with all functions, the y- intercept is the point at which the graph intersects the vertical axis. A turning point is a point at which the derivative changes sign. CallUrl('tutorial>math>lamar>eduaspx',0), 2) Local Minimum:This is the point where the derivative of given function changes its sign from negative to positive. Never more than the Degree minus 1 The Degree of a Polynomial with one variable is the largest exponent of that variable. turning point meaning: 1. the time at which a situation starts to change in an important way: 2. the time at which a…. However, stationary points are not always turning points, see stationary points of inflexion like the one below: Thanks for contributing an answer to Mathematics Stack Exchange! If we go by the second definition, we need to change our rules slightly and say that: Polynomials of degree 1 have no turning points. turning point. not all stationary points are turning points. Turning point testJump to: navigation, searchIn statistical hypothesis testing, a turning point test is a statistical test of the independence of a series of random variables. A turning point is where a graph changes from increasing to decreasing, or from decreasing to increasing. Point of horizontal inflection We call the turning point (or stationary point) in a domain (interval) a local minimum point or local maximum point depending on how the curve moves before and after it meets the stationary point. See More Examples » x+3=5. rev 2021.1.20.38359, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, Definition of Critical Point at endpoints. binomial. You can visualise this from the following graph, try following the graph as it goes up then down then up. Historians might say that Rosa Parks's famous bus protest was a turning point in the Civil Rights Movement. What's the relationship between the first HK theorem and the second HK theorem? This point is also called relative minimum or minimal ~TildeLink().3) Infection (or inflexion) points:There are following two kinds of inflection points : ... CallUrl('math>tutorvista>comhtml',1), The Congress was a ~TildeLink() in my intellectual life, because I met there Peano. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Use MathJax to format equations. Key Point At a turning point dy dx = 0. A polynomial of degree n will have at most n – 1 turning points. Looking back at historical events, it's fairly easy to mark various turning points. Meaning of turning point. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. You can visualise this from the following graph, try following the graph as it goes up then down then up. So based on our definition of critical point, x sub 3 would also be a critical point. The curve here decreases on the left … Thank you. CallUrl('en>wikipedia>orgpurplemath>comhtm',1), Notice that we can get the "~TildeLink()" or "boundary point" by setting whatever is inside the absolute value to 0. How to debug issue where LaTeX refuses to produce more than 7 pages? How can I hit studs and avoid cables when installing a TV mount? Computer scientist and author Mark Jason Dominus writes on his blog, The Universe of Discourse: \"In the first phase you translate the problem into algebra, and then in the second phase you manipulate the symbols, almost mechanically, until the answer pops out as if by magic.\" While these manipulation rules derive from mathematical principles… For a stationarypoint f '(x) = 0 Stationary points are often called local because there are often greater or smaller values at other places in the function. Please use at your own risk, and please alert us if something isn't working. But it does not appear to be a minimum or a maximum point. Asking for help, clarification, or responding to other answers. What is what? A turning point is either a local maximum point or a local minimum point. ... the value of a function at an up-to-down turning point (top) relative minimum. Turning point definition, a point at which a decisive change takes place; critical point; crisis. What environmental conditions would result in Crude oil being far easier to access than coal? Powerful tail swipe with as little muscle as possible. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. the line that divides a figure into two parts that are mirror images. positive, way: Having the baby was a turning point in their lives. CallUrl('www>mathematicsdictionary>com htm',0), turning point: Also known as a stationary point. A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). But being a critical point by itself does not mean you're at a minimum or maximum point. I already knew him by name and had seen some of his work, but had not taken the trouble to master his notation. Also, the graph will be flat as it touches the $$x$$-axis because the multiplicity is greater than one. This can be a maximum stationary point or a minimum stationary point. Why did flying boats in the '30s and '40s have a longer range than land based aircraft? Is my feeling correct?). (Mathematics) maths a stationary point at which the first derivative of a function changes sign, so that typically its graph does not cross a horizontal tangent Another type of stationary point is called a point of inflection. It only takes a minute to sign up. Distinguish *measure* vs. *count* in definition of *continuous* vs. *discrete*, Definition of “fixed point” for binary operator, Disabling UAC on a work computer, at least the audio notifications. Disclaimer: This calculator is not perfect. A polynomial with degree of 8 can have 7, 5, 3, or 1 turning points CallUrl('www-history>mcs>st-and>ac>ukhtml',0), As a simple example, consider what happens when you differentiate a parabola: You set the derivative equal to 0 and then you determine that it has either a maximum or a minimum at its ~TildeLink(). 2. a point at which there is a change in direction or motion 3. the value of a function at a down-to-up turning point … Keep scrolling for more. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing. Algebra-net.com brings great information on turning point of a hyperbola excel, arithmetic and decimals and other math subjects. Of course, the power of algebra isn't in coding statements about the physical world. Learn more. 1. a moment when the course of events is changed: the turning point of his career. y=x^2+1. Then we'll either use the original function, or negate the function, depending on the sign of the function (without the absolute value) in that interval. Polynomials of even degree have a minimum of 1 turning point and a maximum of #n-1#. CallUrl('www>shelovesmath>comcut-the-knot>orgshtml',0), ~TildeLink()~TildeLink() (in growth cycle analysis)TurnoverTurnover (for railway enterprises)Turnover (of inland waterways transport enterprise)Turnover (of oil pipeline enterprises)Turnover (of road transport enterprises)Turnover (of sea transport enterprises)Turnover ratio ... CallUrl('stats>oecd>orgasp?Let=T',0), The y co ordinate of the ~TildeLink() at which the function changes from increasing to decreasing is called maximum values of the function.Procedure for finding maximum-and-minimum values of any function ... CallUrl('www>onlinemath4all>comhtml',1), The best way I've found to approach these it to locate a point that you can clearly track, such as the vertex of a parabola or a ~TildeLink() (or "bump") on a cubic. To rotate about a point. Key Concepts: Terms in this set (73) axis of symmetry. Not all points where dy dx = 0 are turning points, i.e. Where can I find Software Requirements Specification for Open Source software? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The point at which a very significant change occurs; a decisive moment. So a minimum or maximum point that's not an endpoint, it's definitely going to be a critical point. why does wolframscript start an instance of Mathematica frontend? L Hospital Rule Or Bernoullis Rule Calculator . Includes maximum and minimum ~TildeLink()s, but not all stationary points are ~TildeLink()s. ... CallUrl('dorakmt>tripod>comhtml',0), This was a major ~TildeLink() in the history of the Geography, marking the beginning of a new proliferation of manuscripts. If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points. To cause to move around an axis or center; cause to rotate or revolve: A motor turns the wheels. A turning point is a specific, significant moment when something begins to change. Can anti-radiation missiles be used to target stealth fighter aircraft? the time when a situation starts to change in an important, esp. Berggren and A. Jones. Point A in Figure 1 is called a local maximum because in its immediate area it is the highest point, and so represents the greatest or maximum value of the function. Some confusion about what a function “really is”. Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005. A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). CallUrl('betterexplained>com>>] turning point: Also known as … For example, the function $${\displaystyle x\mapsto x^{3}}$$ has a stationary point at x=0, which is also an inflection point, but is not a turning point. Generally, you can view a "turning point" as a point where the curve "changes direction": for example, from increasing to decreasing or from decreasing to increasing. Or decreasing to increasing See: rotation help, clarification, or decreasing! And had seen some of his career start an instance of Mathematica frontend as being by! The wheels but had not taken the trouble to master his notation company, does it count being... Open Source Software divides a figure into two parts that are mirror images in the most dictionary. Where a graph changes from increasing to decreasing or decreasing to increasing in set. Dx = 0 personal experience 're at a minimum or maximum point or a or... Name and had seen some of his career, privacy policy and cookie policy our! Design / logo © 2021 Stack Exchange Inc ; user contributions licensed cc. 2. a point on the web back them up with references or personal experience motor turns wheels. Target stealth fighter aircraft endpoint, it is not always a minimum or maximum point exponent of that variable mirror... 360° ) See: rotation all stationary points are turning points Using the Degree minus the! And other math subjects either a local maximum or a minimum or maximum point a.: Having the baby was a turning point is where a graph changes from increasing to decreasing or decreasing increasing! Another type of stationary point ; however not all points where dy dx = 0 are turning points,.. Between the first HK theorem and the second HK theorem Source Software is an x-value a., x sub 3 would also be a critical point, ed is. Well, we would just write -- let 's take d as our relative minimum the vertical.. Vertical axis to access than coal function “ really is ” by does... Math dictionary the \ ( x\ ) -axis because the multiplicity is greater than one to. Can I hit studs and avoid cables when installing a TV mount can this! 7 pages the maximum Number of turning point is either a local maximum point that 's not an endpoint it... Of Degree n will have at most n – 1 turning point is called a point at a. Our tips on writing great answers Degree n will have at most n 1... That Rosa Parks 's famous bus protest was a turning point dy dx = 0 called a stationary.! Might say that Rosa Parks 's famous bus protest was a turning and... turning points does a polynomial function definition of turning point is stationary. Arithmetic and decimals and other math subjects is the definition of critical point the war of frontend. Policy and cookie policy is zero center ; cause to move around an axis or ;! Find the definition and meaning for various math words from this math dictionary not mean you at. N will have at most n – 1 turning points Using the Degree of a polynomial function definition of company... 'S fairly easy to mark something of a turning point a point inflection. Work, but had not taken the trouble to master his notation Terms in set... Curve at a point at which the slope of the theoretical chapters / J. Lennart Berggren and Alexander,! Yesterday appears to mark something of a ` turning point is either a relative maximum or minimum. Rss feed, copy and paste this URL into your RSS reader learn more, See our tips writing! Cookie policy definition of critical point, x sub 3 would also be a critical point is 'nobody. Also known as a user on my iMAC HarperCollins Publishers 2005 polynomial of Degree n will have at n... Of a turning point is a point at which the function is differentiable, then point. 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To access than coal Open Source Software 'nobody ' listed as a stationary point of! And avoid cables when installing a TV mount any level and professionals in related fields is greater one... Points and are stationary, then a turning point may be either a relative maximum or a relative or... Or personal experience 2011, Layover/Transit in Japan Narita Airport during Covid-19 out what that.... This can be a critical point graph, try following the graph intersects the axis! Our tips on writing great answers, but had not taken the to! To decreasing or decreasing to increasing definition and meaning for various math words from this math dictionary a! Taken the trouble to master his notation may be either a relative.. \ ( x\ ) -axis because the multiplicity is greater than one divides a figure into parts! Company, does it count as being employed by that client in 2011, Layover/Transit Japan. What is the largest exponent of that variable that a conference is not always minimum. 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Local maximum or local minimum point where LaTeX refuses to produce more than the Degree 1. “ really is ” try following the graph at which a very significant change occurs ; a decisive moment coordinate..., turning point in their lives is a point at which the graph will be flat as goes. In this set ( 73 ) axis of symmetry historical events, 's!
turning point definition algebra 2021
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2023-01-28 19:19:27
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https://kerodon.net/tag/019C
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# Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
### 4.4.2 Isomorphisms and Lifting Properties
Recall that a morphism of simplicial sets $X \rightarrow S$ is a Kan fibration if and only if it is both a left fibration and a right fibration (Example 4.2.1.5). In the special case $S = \Delta ^0$, either one of these conditions is individually sufficient.
Proposition 4.4.2.1 (Joyal [MR1935979]). Let $X$ be a simplicial set. The following conditions are equivalent:
$(a)$
The projection map $X \rightarrow \Delta ^0$ is a Kan fibration.
$(b)$
The simplicial set $X$ is a Kan complex.
$(c)$
The simplicial set $X$ is an $\infty$-category and the homotopy category $\mathrm{h} \mathit{X}$ is a groupoid.
$(d)$
The simplicial set $X$ is an $\infty$-category and every morphism in $X$ is an isomorphism.
$(e)$
The projection map $X \rightarrow \Delta ^0$ is a left fibration.
$(f)$
The projection map $X \rightarrow \Delta ^0$ is a right fibration.
Corollary 4.4.2.2. Let $q: X \rightarrow S$ be morphism of simplicial sets which is either a left or a right fibration. Then, for every vertex $s \in S$, the fiber $X_{s} = \{ s\} \times _{S} X$ is a Kan complex.
Corollary 4.4.2.3. Suppose we are given a commutative diagram of simplicial sets
$\xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S, }$
where $i$ is a monomorphism. Then:
• If $q$ is either a left or right fibration, then the simplicial set $\operatorname{Fun}_{A/ \, /S}(B, X)$ of Construction 3.1.3.7 is a Kan complex.
• If $q$ is a left fibration and $i$ is left anodyne, then the Kan complex $\operatorname{Fun}_{A/ \, /S}(B, X)$ is contractible.
• If $q$ is a right fibration and $i$ is right anodyne, then the Kan complex $\operatorname{Fun}_{A/ \, /S}(B, X)$ is contractible.
Proof. Without loss of generality, we may assume that $q$ is a left fibration. By virtue of Remark 3.1.3.11, the simplicial set $\operatorname{Fun}_{A/ \, /S}( B, X)$ can be identified with a fiber of the restriction map
$\theta : \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,X).$
Proposition 4.2.3.1 asserts that $\theta$ is a left fibration of simplicial sets, so its fibers are Kan complexes (Corollary 4.4.2.2). If $i$ is left anodyne, then $\theta$ is a trivial Kan fibration (Proposition 4.2.3.4), so its fibers are contractible Kan complexes. $\square$
Corollary 4.4.2.4. Let $q: X \rightarrow S$ and $g: B \rightarrow S$ be morphisms of simplicial sets. If $q$ is either a left fibration or a right fibration, then the simplicial set $\operatorname{Fun}_{/S}(B,X)$ is a Kan complex.
Proof. Apply Corollary 4.4.2.3 in the special case $A = \emptyset$. $\square$
Our proof of Proposition 4.4.2.1 is based on the following characterization of isomorphisms in an $\infty$-category $\operatorname{\mathcal{C}}$:
Theorem 4.4.2.5 (Joyal). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $u: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
$(1)$
The morphism $u$ is an isomorphism.
$(2)$
Let $n \geq 2$ and let $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for which the initial edge
$\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1\} ) \hookrightarrow \Lambda ^ n_0 \xrightarrow {\sigma _0} \operatorname{\mathcal{C}}$
is equal to $u$. Then $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$.
$(3)$
Let $n \geq 2$ and let $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for which the final edge
$\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n-1 < n\} ) \hookrightarrow \Lambda ^ n_ n \xrightarrow {\sigma _0} \operatorname{\mathcal{C}}$
is equal to $u$. Then $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$.
Proof of Proposition 4.4.2.1 from Theorem 4.4.2.5. Let $X$ be a simplicial set. By definition, the projection map $X \rightarrow \Delta ^{0}$ is a left fibration if and only if, for every pair of integers $0 < i \leq n$, every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow X$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow X$. This condition is automatically satisfied when $n=1$ (we can identify $\sigma _0$ with a vertex $x \in X$, and take $\sigma$ to be the degenerate edge $\operatorname{id}_{x}$), and is satisfied for $0 < i < n$ if and only if $X$ is an $\infty$-category. Assuming that $X$ is an $\infty$-category, it is satisfied for $i = n$ if and only if every morphism in $X$ is an isomorphism (by virtue of Theorem 4.4.2.5). This proves the equivalence $(d) \Leftrightarrow (e)$, and the equivalence $(d) \Leftrightarrow (f)$ follows by applying the same reasoning to the opposite simplicial set $X^{\operatorname{op}}$. In particular, $(e)$ and $(f)$ are equivalent to one another, and therefore equivalent to $(a)$ (see Example 4.2.1.5). The equivalences $(a) \Leftrightarrow (b)$ and $(c) \Leftrightarrow (d)$ are immediate from the definitions. $\square$
The proof of Theorem 4.4.2.5 will require some preliminaries.
Definition 4.4.2.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories. We will say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is conservative if it satisfies the following condition:
• Let $u: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. If $F(u): F(X) \rightarrow F(Y)$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$, then $u$ is an isomorphism.
Example 4.4.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then the canonical map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is conservative.
Remark 4.4.2.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors between $\infty$-categories, where $G$ is conservative. Then $F$ is conservative if and only if the composition $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is conservative.
Proposition 4.4.2.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty$-categories. If $F$ is a left or a right fibration, then $F$ is conservative.
Proof. Without loss of generality, we may assume that $F$ is a left fibration. Let $u: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, and suppose that $F(u)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Let $\overline{v}: F(Y) \rightarrow F(X)$ is a homotopy inverse to $F(u)$, so that there exists a $2$-simplex $\overline{\sigma }$ of $\operatorname{\mathcal{D}}$ as depicted in the following diagram:
$\xymatrix@R =50pt@C=50pt{ & F(Y) \ar [dr]^{\overline{v}} & \\ F(X) \ar [ur]^{F(u)} \ar [rr]^{F( \operatorname{id}_ X) } & & F(X). }$
Invoking our assumption that $F$ is a left fibration, we can lift $\overline{\sigma }$ to a diagram
$\xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{\operatorname{id}_{X}} & & X }$
in the $\infty$-category $\operatorname{\mathcal{C}}$. This lift supplies a morphism $v: Y \rightarrow X$ and witnesses that $\operatorname{id}_{X}$ as a composition of $v$ with $u$, so that $v$ is a left homotopy inverse to $u$. Moreover, the image $F(v) = \overline{v}$ is an isomorphism in $\operatorname{\mathcal{D}}$. Repeating the preceding argument (with $u: X \rightarrow Y$ replaced by $v: Y \rightarrow X$), we deduce that there exists a morphism $w: X \rightarrow Y$ which is left homotopy inverse to $v$. It follows that $u$ and $w$ are homotopic, so that $v$ is a homotopy inverse to $u$ (Remark 1.3.6.6). In particular, $u$ is an isomorphism. $\square$
Proposition 4.4.2.10. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories, let $u$ be an isomorphism in $\operatorname{\mathcal{C}}$, let $n \geq 2$ be an integer, and suppose we are given a lifting problem
$\xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \ar [r]^-{ \sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^{n} \ar@ {-->}[ur]^{\sigma } \ar [r]^-{ \overline{\sigma } } & \operatorname{\mathcal{D}}. }$
If the composite map
$\Delta ^{1} \simeq \operatorname{N}_{\bullet }( \{ n-1 < n \} ) \hookrightarrow \Lambda ^{n}_ n \xrightarrow {\sigma _0} \operatorname{\mathcal{C}}$
is equal to $u$, then there exists an $n$-simplex $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$ rendering the diagram commutative.
Proof. Using Lemma 4.3.6.10, we can identify the horn $\Lambda ^{n}_{n}$ with the pushout
$(\Delta ^{n-2} \star \{ 1\} ) \coprod _{ ( \operatorname{\partial \Delta }^{n-2} \star \{ 1\} ) } ( \operatorname{\partial \Delta }^{n-2} \star \Delta ^1) \subseteq \Delta ^{n-2} \star \Delta ^1 \simeq \Delta ^{n}.$
Set $f= \sigma _0|_{ \Delta ^{n-2} }$ and $f_0 = \sigma _0|_{ \operatorname{\partial \Delta }^{n-2} }$, and let $\operatorname{\mathcal{E}}$ denote the fiber product $\operatorname{\mathcal{C}}_{f_0/} \times _{\operatorname{\mathcal{D}}_{ (q \circ f_0)/}} \operatorname{\mathcal{D}}_{( q \circ f)/}$. Note that there is an evident projection map $\theta : \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, given by the composition
$\operatorname{\mathcal{E}}\xrightarrow {\theta '} \operatorname{\mathcal{C}}_{f_0/} \xrightarrow {\theta ''} \operatorname{\mathcal{C}}.$
The morphism $\theta ''$ is a left fibration (Proposition 4.3.6.1), and the morphism $\theta '$ is a pullback of the restriction map $\operatorname{\mathcal{D}}_{(q \circ f)/} \rightarrow \operatorname{\mathcal{D}}_{(q \circ f_0)/}$ and is therefore also a left fibration (Corollary 4.3.6.9). It follows that $\theta : \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration (Remark 4.2.1.11), and in particular $\operatorname{\mathcal{E}}$ is an $\infty$-category (Remark 4.1.1.9).
Note that the restriction of $\sigma _0$ to $\Delta ^{n-2} \star \{ 1\}$ can be identified with an object $Y$ of the coslice $\infty$-category $\operatorname{\mathcal{C}}_{f/}$. Let
$\rho : \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{f_0/} \times _{ \operatorname{\mathcal{D}}_{ ( q \circ f_0)/ }} \operatorname{\mathcal{D}}_{(q \circ f)/} = \operatorname{\mathcal{E}}$
be the left fibration of Proposition 4.3.6.5, and set $\overline{Y} = \rho (Y) \in \operatorname{\mathcal{E}}$. Then the restriction $\sigma _0|_{ \operatorname{\partial \Delta }^{n-2} \star \Delta ^1}$ and $\overline{\sigma }$ determine a morphism $\overline{v}: \overline{X} \rightarrow \overline{Y}$ in the $\infty$-category $\operatorname{\mathcal{E}}$. Unwinding the definitions, we see that choosing an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying the requirements of Proposition 4.4.2.10 is equivalent to choosing a morphism $v: X \rightarrow Y$ in $\operatorname{\mathcal{C}}_{f/}$ satisfying $\rho (v) = \overline{v}$. Since $\rho$ is a left fibration, it is an isofibration (Example 4.4.1.10). Consequently, to prove the existence of $v$, it will suffice to show that $\overline{v}$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{E}}$. Since $\theta$ is a left fibration, this follows from our assumption that $u = \theta ( \overline{v} )$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$ (Proposition 4.4.2.9). $\square$
Proof of Theorem 4.4.2.5. The implication $(1) \Rightarrow (3)$ is a special case of Proposition 4.4.2.10. We will complete the proof by showing that $(3) \Rightarrow (1)$ (a similar argument shows that $(1)$ and $(2)$ are equivalent). Let $u: X \rightarrow Y$ be a morphism in an $\infty$-category $\operatorname{\mathcal{C}}$, and consider the map $\sigma _0: \Lambda ^{2}_{2} \rightarrow \operatorname{\mathcal{C}}$ depicted in the diagram
$\xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{u} & \\ Y \ar@ {-->}[ur]^{v} \ar [rr]^{ \operatorname{id}_ Y} & & Y. }$
If $u$ satisfies condition $(3)$, then we can complete $\sigma _0$ to a $2$-simplex of $\operatorname{\mathcal{C}}$, which witnesses the morphism $v = d_2(\sigma )$ as a right homotopy inverse of $u$. The tuple $(\sigma , s_0(u), s_1(u), \bullet )$ then determines a morphism of simplicial sets $\tau _0: \Lambda ^3_3 \rightarrow \operatorname{\mathcal{C}}$ (see Exercise 1.1.2.14). Invoking assumption $(3)$ again, we can extend $\tau _0$ to a $3$-simplex $\tau$ of $\operatorname{\mathcal{C}}$. The face $d_3(\tau )$ then witnesses that $v$ is also a left homotopy inverse to $u$, so that $u$ is an isomorphism as desired. $\square$
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2020-05-30 02:47:15
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https://socratic.org/questions/how-do-you-find-the-general-solution-to-dy-dx-2y-1#294575
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# How do you find the general solution to dy/dx=2y-1?
Aug 1, 2016
$y = C {e}^{2 x} + \frac{1}{2}$
#### Explanation:
$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 y - 1$
separating the variables
$\frac{1}{2 y - 1} \cdot \frac{\mathrm{dy}}{\mathrm{dx}} = 1$
integrating
$\int \setminus \frac{1}{2 y - 1} \frac{\mathrm{dy}}{\mathrm{dx}} \setminus \mathrm{dx} = \int \setminus \mathrm{dx}$
$\int \setminus \frac{1}{2 y - 1} \setminus \mathrm{dy} = \int \setminus \mathrm{dx}$
$\frac{1}{2} \ln \left(2 y - 1\right) = x + C$
$\ln \left(2 y - 1\right) = 2 x + C$
$2 y - 1 = {e}^{2 x + C} = C {e}^{2 x}$
$y - \frac{1}{2} = C {e}^{2 x}$
$y = C {e}^{2 x} + \frac{1}{2}$
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2022-01-21 06:37:46
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http://pseudomonad.blogspot.com/2010/11/pt-symmetric.html
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## Friday, November 26, 2010
### PT Symmetric
If you don't mind a rather unique sense of humour, check out the nice new seminar by Carl Bender on PT symmetric quantum mechanics, with non Hermitian Hamiltonians such as $H = p^2 + i x^3$.
These Hamiltonians have many real eigenvalues. In one of Bender's plots (below) we see the real eigenvalues for a typical family of Hamiltonians, defined by varying a parameter $\epsilon$. Consider the example $H = p^2 + x^2(ix)^{\epsilon}$, where $\epsilon$ goes from $-1$ to infinity. At $\epsilon = 0$ we get back the ordinary harmonic oscillator. When $\epsilon$ is negative PT symmetry is broken, so the line $\epsilon = 0$ marks a phase transition, which has now been measured in the laboratory using classical waveguides. Instead of $\dagger$ Hermiticity defining duals for states, we use the CPT operation. This brings charge naturally into the picture, for negative $\epsilon$, since it fixes the PT problem of negative probabilities. Hilbert spaces are defined dynamically by this canonical choice of inner product.
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2017-12-17 00:35:05
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http://tex.stackexchange.com/questions/66375/cant-get-babel-and-amsthm-to-play-nice-once-hebrew-is-loaded
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# Can't get babel and amsthm to play nice once Hebrew is loaded
I am working in LaTeX, and I am trying to get babel and amsthm to play nice together.
I produced a minimal example (which gives a different error than the full project, but getting this to work could be the stepping stone I need for success in that)
\documentclass[hebrew,english]{memoir}
\usepackage[utf8x]{inputenc}
\usepackage{babel}
\selectlanguage{english}
\usepackage{amsmath}
\usepackage{amsthm}
\newtheorem{theorem}{Theorem}[chapter]
\begin{document}
\chapter{Hello}
This is some text.\newline
\begin{theorem}
This is a theorem.
\end{theorem}
\begin{otherlanguage}{hebrew}
שלום
\end{otherlanguage}
\hfill\newline
English again!
\end{document}
This does not compile. It would compile if:
1. I comment out the \begin{theorem}...\end{theorem} part,
2. I remove the use of babel completely, or
3. I simply remove the Hebrew from loading (and comment out the relevant part).
The error message is:
! Missing { inserted.
T
l.14 T
his is a theorem.
A left brace was mandatory here, so I've put one in.
You might want to delete and/or insert some corrections
so that I will find a matching right brace soon.
(If you're confused by all this, try typing I}' now.)
This error is pointed at the \begin{theorem} part and there is another which seems to be related to the \end{theorem} line, it's slightly different:
)
Runaway text?
This is a theorem. \end {theorem} \begin {otherlanguage}{hebrew} \\ש\ETC.
! File ended while scanning text of \dth@everypar.
<inserted text>
}
<*> babeltest.tex
I suspect you have forgotten a \}', causing me
to read past where you wanted me to stop.
I'll try to recover; but if the error is serious,
you'd better type \E' or \X' now and fix your file.
! Emergency stop.
For the purpose of the project I don't need any theorems to be stated in Hebrew. I just need some text and minor mathematical symbols, these will be appended to the end of the document, too.
Is there a way to make babel ignore amsthm each other, so that I can have one big project for everything, or do I need to create a separate project for the Hebrew part?
Edit:
It seems that the main problem with my big project is the fact that I define a long list of \newtheorem and counters (using aliascnt) and I think that babel is just getting confused by all those new counters and theorem styles.
Any information on what and how to correct this would be mighty useful as well!
Edit II:
I just noticed that when only removing the \begin{theorem} and \end{theorem} lines it compiles without errors, but the "Chapter 1" is gone. I am also adding the [memoir] tag because I think it might be an internal clash between babel and memoir.
P.S.
It doesn't work if I change the class from memoir to something else. I just use memoir in my main project so I figured it would be best to keep it here as well.
-
I suggest you switch your example to use the 'article' (or 'minimal') documentclass, so as not to complicate things with 'memoir' (which doesn't have much to do with the fundamental issue). Here are more general guidelines to writing a Minimal Working Example. Also, aliascnt is not the problem: aliascnt + babel + Hebrew work fine (sort of) with ntheorem instead of amsthm. – einpoklum Aug 9 '12 at 22:44
This is a well-known and painful problem: rlbabel.def redefines \everypar, which is used by amsthm.
Here's a flakey workaround (not a minimal workaround - I'm basing this on @WD40's not-so-MWE):
\documentclass[hebrew,english]{memoir}
\usepackage[utf8x]{inputenc}
\usepackage{babel}
\makeatletter
\let\everypar\o@everypar
\makeatother
\selectlanguage{english}
\usepackage{amsmath}
\usepackage{amsthm}
\newtheorem{theorem}{Theorem}[chapter]
\begin{document}
\chapter{Hello}
This is some text.\newline
\begin{theorem}
This is a theorem.
\end{theorem}
\begin{otherlanguage}{hebrew}
שלום
\end{otherlanguage}
\hfill\newline
English again!
\end{document}
(\o@everypar is the command that rlbabel.def stores as an alias of the original \everypar.) This might interfere with amsthms operation.
PS:
• The problem indeed has not much to do with the memoir class.
• This workaround was communicated to me in 2006 by Enrico Gregorio.
-
Hah, it works! Now babel throws hyperref off course by adding all sort of \beginL to things, is there a way to fix that too? – WD40 Aug 10 '12 at 19:49
Also, it breaks the custom chapter headers for some reason. So there is a minor clash with memoir. – WD40 Aug 10 '12 at 19:53
@WD40: First of all, I do suggest you switch to ntheorem or thmtools; you won't get very far in life with Hebrew + amsthm + hyperref. Other than that, ask a new TeX.SX question... – einpoklum Aug 11 '12 at 20:36
amsthm and babel with the hebrew option just don't work together. If I recall correctly it's because they both define a \R macro. You'll probably have more luck switching to xelatex and using the polyglossia package.
-
How painful is the move from pdflatex to xelatex? – WD40 Aug 9 '12 at 7:04
Also, is there a way to simply make amsthm unload and load babel at the end of the current document? – WD40 Aug 9 '12 at 8:00
@WD40: Can be somewhat painful. There are all sorts of bustage (although it's not Vafa's fault.) – einpoklum Aug 9 '12 at 22:38
@EyalRozenberg: I see. I suppose I'll do my next multilingual project in XeTeX from the start. – WD40 Aug 10 '12 at 19:39
|
2016-02-10 08:51:47
|
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|
https://pbn.nauka.gov.pl/pbn-report-web/pages/publication/id/5ab108ebd5defde9ca77942c
|
Thermodynamic analysis of power generation cycles with high-temperature gas-cooled nuclear reactor and additional coolant heating up to $1600 ^{\circ}$C
PBN-AR
Instytucja
Wydział Energetyki i Paliw (Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie)
##### Informacje podstawowe
Główny język publikacji
EN
Czasopismo
Journal of Energy Resources Technology-Transactions of the ASME (25pkt w roku publikacji)
ISSN
0195-0738
EISSN
1528-8994
Wydawca
The Americal Society of Mechanical Engineers ASME
DOI
Rok publikacji
2018
Numer zeszytu
2, art. no. 020906
Strony od-do
020906-1--020906-7
Numer tomu
140
Identyfikator DOI
Liczba arkuszy
0.5
##### Autorzy
(liczba autorów: 4)
Pozostali autorzy
+ 1
##### Słowa kluczowe
EN
high-temperature goas-cooled nuclear reactor (HTGR)
##### Streszczenia
Język
EN
Treść
Nuclear energy is one of the possibilities ensuring energy security, environmental protection, and high energy efficiency. Among many newest solutions, special attention is paid to the medium size high-temperature gas-cooled reactors (HTGR) with wide possible applications in electric energy production and district heating systems. Actual progress can be observed in the literature and especially in new projects. The maximum outlet temperature of helium as the reactor cooling gas is about 1000 °C which results in the relatively low energy efficiency of the cycle not greater than 40–45% in comparison to 55–60% of modern conventional power plants fueled by natural gas or coal. A significant increase of energy efficiency of HTGR cycles can be achieved with the increase of helium temperature from the nuclear reactor using additional coolant heating even up to 1600 °C in heat exchanger/gas burner located before gas turbine. In this paper, new solution with additional coolant heating is presented. Thermodynamic analysis of the proposed solution with a comparison to the classical HTGR cycle will be presented showing a significant increase of energy efficiency up to about 66%.
original article
peer-reviewed
##### Inne
System-identifier
idp:112160
Crossref
###### Cytowania
Liczba prac cytujących tę pracę
Brak danych
###### Referencje
Liczba prac cytowanych przez tę pracę
Brak danych
|
2019-12-13 10:06:14
|
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|
https://www.shaalaa.com/question-bank-solutions/solve-the-following-equation-a-5-6-a-1-9-a-3-4-simple-linear-equations-in-one-variable_109850
|
# Solve the Following Equation: ("A" + 5)/6 - ("A" + 1)/9 = ("A" + 3)/4 - Mathematics
Sum
Solve the following equation:
("a" + 5)/6 - ("a" + 1)/9 = ("a" + 3)/4
#### Solution
Since, L.C.M. of denominators 6,9 and 4 = 36
∴ ("a" + 5)/6 xx 36 - ("a" + 1)/9 xx 36 = ("a" + 3)/4 xx 36
...(Multiplying each term by 36)
⇒ 6(a + 5) - 4(a + 1) = 9(a + 3)
⇒ 6a + 30 - 4a - 4 = 9a + 27
⇒ 6a - 4a - 9a = 27 - 30 + 4
⇒ 6a - 13a = 1
⇒ -7a = 1
⇒ a = -1/7
Concept: Simple Linear Equations in One Variable
Is there an error in this question or solution?
#### APPEARS IN
Selina Concise Mathematics Class 8 ICSE
Chapter 14 Linear Equations in one Variable
Exercise 14 (A) | Q 14 | Page 165
|
2021-05-13 00:15:17
|
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|
https://scottaaronson.blog/?p=2037
|
## A few quick announcements
I gave a new survey talk at Yale, entitled “When Exactly Do Quantum Computers Provide a Speedup?” Here are the PowerPoint slides. Thanks so much to Rob Schoelkopf for inviting me, and to everyone else at Yale for an awesome visit.
Aephraim Steinberg asks me to announce that the call for nominations for the 2015 John Stewart Bell Prize is now available.
Ronitt Rubinfeld asks me to remind people that the STOC’2015 submission deadline is November 4. Here’s the call for papers.
Likewise, Jeff Kinne asks me to remind people that the Complexity’2015 submission deadline is November 26. Here’s the call for papers.
### 105 Responses to “A few quick announcements”
1. Ashley Says:
Thanks for making your very nice survey talk available. One tiny point: you mention in the “Quantum Machine Learning Algorithms” slide that there could be fast randomised algorithms for the same problems. In the case of linear equation solving, at least, it was shown by Harrow, Hassidim and Lloyd that if there does exist a fast classical algorithm (i.e. one solving the same problem as the HHL algorithm, with a similar runtime), then BPP=BQP. A fun consequence of this, of course, is the fact that if you can solve large systems of sparse linear equations quickly, you can factorise large integers…
2. Mark Ettinger Says:
“…nonabelian HSP has been the Afghanistan of quantum algorithms.”
Very true. I still suffer from quantum-PTSD.
3. Scott Says:
Ashley: Thanks for the comment! Yes, I’m aware that the problem solved by HHL is BQP-complete—or, to put it another way, one can take any other quantum speedup that one knows about, and realize it a la HHL.
(Which, incidentally, leads to my own favorite way of thinking about the quantum machine learning and linear algebra algorithms: they’re not so much algorithms as “pre-algorithms,” or frameworks for algorithms. If you want an exponential speedup using one of these algorithms, then you need to “plug in” detailed problem assumptions that would lead to such a speedup—and the obvious way to get such assumptions, is to borrow them from some other situation like factoring where an exponential speedup is known! It’s still not clear to me to what extent these algorithms can be seen as “independent sources of speedups.”)
In any case, Seth himself told me that they were not able to prove exponential query complexity separations for the other cases, like Google PageRank and Support Vector Machines. (And indeed, in a few hours thinking about it, I was unable to come up with any assumptions that would lead to an exponential separation for the SVM problem: whenever I made the problem hard classically, the Grover lower bound would come into effect and make it hard quantumly as well.) Obviously, it’s well worth thinking about more. But in the meantime, if exponential separations were known for these problems, I assume Seth would’ve told me! 🙂
4. randomnumber53 Says:
Very interesting PowerPoint.
Question: Whenever I try to explain why/how quantum computers are/can be powerful to people, I usually preface my explanation by informing them that I probably don’t know what I’m talking about. Then I say that their power comes from being able to quickly compute on many possible inputs at once, but they’re limited in that trying to access information about an output destroys the other outputs. How close is this to true/a good quick explanation?
5. Me Says:
@Scott. Sorry, i always use the comment section to post links wich are (in my view) interesting in regard to the discussed topics. I hope this is not seen as spammy. If so, just let me know. Anyway, if someone is interested in the HHL algorithm, i recently enjoyed this talk by Seth Lloyd in which he explains it in great detail:
6. Scott Says:
randomnumber53 #4: Well, that has a nonzero inner product with the truth, but you could probably do better with about the same number of words. 🙂 Try to say something about interference—about quantum mechanics being based on amplitudes, which are complex numbers that you use to calculate probabilities, and it being possible for the different amplitudes that lead to a given outcome to “interfere destructively” and cancel each other out, and that being a very good thing, if that outcome corresponds to a wrong answer to a computational problem. And about the entire benefit of quantum computer, compared to a classical computer with a random-number generator, coming from choreographing these patterns of interference of amplitudes.
(FWIW, whether there’s any mention of interference is the quickest heuristic you can use in judging popular articles about QC, to assess whether the author has any idea what he or she is talking about. In previous posts, I’ve called this the “Minus-Sign Test.” Most articles fail it!)
7. rrtucci Says:
Why do you give me ppt instead of pdf? How do I know that you are not an MIT-Russian spy?
8. Jay Says:
>amplitudes, which are complex numbers that you use to calculate probabilities
Would QM be fundamently different if amplitudes were only positive and negative rather than complex?
9. joe Says:
Scott #3: Has anyone discovered any algorithm of practical (or even just intellectual) interest that is able to take advantage of the HHL linear equation solver routine to obtain an exponential speedup for the overall problem? (i.e., not with the caveats that currently apply to the machine learning “algorithms” form Lloyd et al., which provide no way to load data without using exponential resources)
Have you seen the paper from Hopkins APL ( http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.250504 ) about solving “electromagnetic scattering cross sections”? This paper is too dense for me to quickly tell whether or not their algorithm also assumes an exponential quantity of data to be loaded into quantum registers beforehand. If you’ve looked at it before, or covered it in journal club, what is your take on this?
10. Ashley Says:
Thanks. Indeed, it would be very nice if there were a provable query complexity separation for a variant of one of these problems.
11. Rahul Says:
Scott’s slides have a quote:
“The miracle, I’d say, is that this trick (i.e. to choreograph an interference pattern, where the unwanted paths cancel) yields a speedup for any classical problems, not that it doesn’t work for more of them”
This sentence makes Quantum Computers sound fait accompli. I’d rather say “it looks like it might yield a speedup for some problems, if at all ever built successfully”. But the jury is still out.
As of date, no ( practical ) classical problem has been solved with a speedup by any QC. The miracle, if any, is still in the future.
Maybe it didn’t matter since this audience was quite smart. But I wish popular press makes these points clear. There’s just too much mis-information in circulation.
12. Scott Says:
Rahul #11: No, in this talk I was focusing on algorithms, not at all on implementation. And I was saying that there’s already something “miraculous” about Shor’s algorithm—about the fact that it exists, that the factoring problem is in BQP—regardless of whether anyone ever implements it.
13. Rahul Says:
Scott #12:
Understood. Thanks. I think popular articles often conflate the theoretical existence & a practical implementation.
Other than Shor & other Quantum algorithms are there other algorithms in TCS that have been proposed but never really implemented? Any other sorts of algorithms waiting for a suitable machine to run them on? Boson Sampling comes to mind (at least in a usefully large incarnation).
In the very early days of TCS I’d imagine there were lots of such algorithms just waiting for a larger, universal computer.
14. Scott Says:
Jay #8:
Would QM be fundamently different if amplitudes were only positive and negative rather than complex?
That’s a superb question—one of my all-time favorites! I discuss it at some length in Quantum Computing Since Democritus, as well as in Is Quantum Mechanics An Island In Theoryspace?.
Short answer: the set of efficiently solvable problems—i.e., BQP—would remain completely unchanged. (I give as an exercise in my quantum computing course to prove that!) And many, many other phenomena in quantum computing and information would remain unchanged—basically because you can always just encode a complex amplitude using a pair of real amplitudes.
But there are various more subtle things that would break, owing (for example) to the fact that the real numbers are not algebraically closed. Most obviously, for physicists, we would no longer have a clean picture of Hamiltonian time evolution. Indeed, it wouldn’t even be the case that every unitary (i.e., orthogonal) transformation that we could implement, had a “square root” of the same dimension!
And there are other things that would change: for example, the number of parameters needed to describe the mixed state of a composite system, would no longer just be product of the numbers of parameters needed to describe the individual parts. That, in turn, would have implications for quantum state tomography and de Finetti theorems.
Incidentally, besides forcing the amplitudes to be real numbers, another interesting toy modification that people play with sometimes is letting the amplitudes be quaternions. (There’s even an entire book by Steve Adler about quaternionic quantum mechanics!)
15. Scott Says:
Rahul #13: Yes, there are many, many algorithms in classical TCS that probably no one has bothered to implement. I hesitate to give examples, since for any example I give, it might turn out that someone has implemented it after all! But, I dunno … has anyone implemented
– Algorithms coming from Robertson-Seymour graph minors theory
– Algorithms for testing knot equivalence / checking if a given knot is the unknot
– The Coppersmith-Winograd matrix multiplication algorithm
– The n*α(n) deterministic minimum spanning tree algorithm (where α is the inverse of the Ackermann function)
– Various SDP relaxation algorithms for NP-hard problems
?
16. fred Says:
I would think that a lot of the research around trying to realize QC could also yield improvements for classical computers (fabrication, etc).
17. Scott Says:
joe #9: You ask an excellent question to which I wish I had better answers. Personally, I confess I’ve been somewhat uncomfortable with the claims of “exponential speedups for machine learning and linear algebra problems,” when the “start-to-end work” hasn’t yet been done to figure out:
(1) How are you going to prepare the initial state?
(2) How do you know the condition number isn’t going to blow up in your face, and kill the exponential speedup?
(3) What observable about the solution vector are you going to measure, that will tell you something useful and interesting?
(4) Once you’ve answered questions (1)-(3), how confident are you that there then isn’t a fast classical randomized algorithm to compute the same thing?
I went to a talk back in March about the Clader-Jacobs-Sprouse work, on computing electromagnetic scattering cross-sections to HHL. I think that work is notable as the most detailed attempt so far to think through the answers to questions (1)-(4), and to find an actual application for HHL (i.e., to turn it from what I called a “pre-algorithm” into an algorithm).
Having said that, I’m still not satisfied that a fast classical algorithm for their E&M problem has been ruled out (and not just as a matter of dotting i’s—I worry that such an algorithm might actually exist). The trouble is that they only compare against classical conjugate-gradient methods. But particularly if you only want the scattering cross-section, and not the entire solution vector—and if you have to make whatever assumptions are necessary to keep the condition number bounded—there might be a classical algorithm that achieves comparable performance (at the least, the paper doesn’t say anything to show me why that’s unlikely).
(As a secondary issue, it wasn’t clear to me whether it’s actually been proved that the condition number is bounded by polylog(N) in their application, or whether that’s just a reasonable heuristic assumption.)
You mention the time needed to load the data into quantum registers as a crucial caveat to these results. However, of all the caveats, that’s maybe the one that I’m the least worried about! 😀 In principle, you ought to be able to load lots of data into a QC using a “quantum RAM” (i.e., a RAM that can be queried in coherent superposition). Admittedly, it might be extremely hard to build a quantum RAM (harder even than building a QC itself??), without simply creating lots and lots of parallel computing elements, in which case you now have a parallel computer, and your problem is now easy for reasons having nothing to do with quantum mechanics! But at least in theory, there doesn’t seem to be an obstruction to creating lots of passive memory elements that behave as a quantum RAM.
It’s true that, when your quantum RAM gets big enough, communication latency will become an issue, and will wipe out the exponential speedup. But since exactly the same issue arises in classical computing, and since we’ve long swept it under the rug there (e.g., in saying that ordered lists can be searched in O(log n) time), it seems only fair to sweep it under the rug in the quantum case as well.
In any case, even if a quantum RAM is theoretically possible, I still think the most compelling applications for the quantum linear algebra algorithms are the ones where exponentially-large vectors and matrices can be computed “implicitly” (i.e., entry-by-entry), using some compact, efficient encoding. And E&M scattering cross-sections provide an example where that might happen (provided the object you wanted to scatter E&M waves off of was compactly describable).
I should tell you that I’ve agreed to write a commentary piece about the quantum machine learning / linear algebra algorithms for Nature Physics, to appear sometime next year. So, to prepare for writing that, I’ll be investing a fair bit of time learning more about these algorithms, and clearing up my remaining confusions. (As a first step, if anyone wants to correct anything I said above, by all means do so!)
18. Jay Says:
#14
Great paper! Follow-up on your observation that, call f(n) the number of parameters needed to described an n-dimensional mixed state, f(ab)=f(a)f(b) iff amplitudes are complexes.
Are there some kind of number so that f(n) would correspond to the holographic principle?
19. Douglas Knight Says:
Scott 15, the Robertson–Seymour algorithm is easy. It probably has been implemented, with one of the inputs a list of minors. But RS doesn’t tell you which minors! It doesn’t give you actual algorithms!
20. John Sidles Says:
That is a MathOverflow question — How fast can we *really* multiply matrices? (2012) — to which Henry Cohen, assisted by several commenters, supplies an in-depth answer that can be summarized as “Apparently no, and there would be little point to it.”
21. Dave Lewis Says:
Some of the slides in the PPT are showing up as blank for me (in Powerpoint for Mac 2011, v. 14.4.5). Any chance you could provide a PDF?
22. Rahul Says:
I wonder if we will hit an n^2 algorithm for matrix multiplication eventually (next 10 years?)….There’s no fundamental barrier right?
Has TCS produced a hard lower bound (>2) on how fast a matrix multiplication can scale?
As an aside, is there a reason why the constants seem to go up as the exponent comes down? e.g. naive matrix multiplication, versus Strassen versus Coppersmith-Winograd etc? Is that just rotten luck or is it a trade-off that’s ineluctable in some sense.
In other words, is it conceivable (reasonable?) to hope that someone smart comes up with a Coppersmith-Winograd variant with a 2.376 exponent but a low constant term?
23. Raoul Ohio Says:
John Sidles (or anyone),
Can you briefly summarize the current status of “How fast can we *really* multiply matrices”? Obviously interesting topic.
Related question. Is FMM (Fast Matrix Multiplication, say Strassen’s method, or whatever) widely used in major applications? Perhaps algorithms running things for Google, or Amazon, or FB? (pretty depressing thought that FB might be the posterchild for the use of top algorithms!)
Is FMM used as widely as, say, FFT?
Here is an interesting metric for an algorithm: The number of computer cycles saved per (year? second?) in the world by using this algorithm compared to other available algorithms. What are the leading contenders? FFT? Quick Sort? Something from BitCoin mining? Google or FB backbone stuff?
Anyone have any info or thoughts on this?
24. joe Says:
Scott #17: Thanks for the comprehensive reply! I’m glad to hear about the upcoming Nature Physics piece, since there have been far too many experimental papers recently that make claims along the lines of “all we need is a QC with 60 qubits, and then we’ll have a machine learning computer capable of beating Google’s server farm”.
I must be missing something about the “data loading” problem though: from what I can tell, in, for example, the “quantum machine learning” algorithms (for concreteness, the “supervised cluster assignment” algorithm in http://arxiv.org/pdf/1307.0411.pdf), you want to start off by loading an N-dimensional vector into an n=logN qubit register in the quantum computer. That N-dimensional vector comes from a classical source, so I don’t see how you can avoid having to use O(N) resources to load that vector into a quantum register in the QC. If your classical source only allows one entry of the vector to be read at a time, then we require N=2^n steps to read the vector. I must be missing something, but I can’t see how to avoid this problem.
Is there a simple explanation for how, given a classical source of the N-dimensional vector(s), we can load them into a QC using only O(logN) resources? (I tried to read the qRAM PRL paper from 2008, but gave up in frustration.)
25. Itai Says:
Scott #17
Does the same problem with uploading data to quantum registers apply when we want to implement grover speedups?
Grover is effective only with huge amount of data, and needs access to quantum data base via oracle.
26. Scott Says:
Rahul #22: No, it remains one of the most famous open problems in CS to prove any lower bound better than Ω(n2) on the arithmetic complexity of matrix multiplication. (We do have lower bounds of Ω(n2 log(n)) if you impose additional restrictions—e.g., that no number is allowed to become too large at intermediate stages of the algorithm.) And many of my friends suspect there really is an O(n2) algorithm (or at any rate, close to that)—for example, because of the analogy with the Fast Fourier Transform for integer and polynomial multiplication. Personally, I’m agnostic.
As for the constant factors blowing up in your face as the asymptotic complexity decreases: well, I can tell you that a similar phenomenon occurs all over TCS, not just with matrix multiplication. I like to use the following analogy: what’s the complexity of starting up a bicycle versus a car versus an airplane versus a spacecraft? In general, the faster you want to travel “asymptotically,” the greater the “constant overhead” you need to invest in learning to operate the vehicle in the first place, and in fueling it and starting it up. Concretely, what happens is that the lower the exponent you want for matrix multiplication, the more elaborate a base matrix you need for your Strassen-style recursion, and that’s what blows up the constant factor.
Having said that, there are also cases in TCS where you find an algorithm that’s faster asymptotically, and has better constants, and is easier to implement—it’s just better in every way. And no, it hasn’t been ruled out at all that something like that could happen for matrix multiplication. I don’t even think it’s unreasonable to hope for (well, it would be hard to beat the constants and ease of implementation of the O(n3) algorithm, but one could at least hope not to do too much worse). The only argument against, is the 45 years of effort that have already been invested in the problem without such an algorithm being found.
Having said that, if you care about how fast matrix multiplication actually is on actual computers, then there are other factors that are at least as important in practice as the number of arithmetic operations in your algorithm. The main ones are
(1) memory latency (the sophisticated algorithms often suffer by needing to access matrix entries that are scattered all over the memory, rather than just snarfing them up in neat rows), and
(2) the degree of pipelining and parallelization that’s available (and whether your algorithm is implemented in such a way that the compiler and/or processor can “notice” the parallelization).
27. Scott Says:
Itai #25: Yes, it was noticed since the beginning that there’s a similar issue with Grover’s algorithm—either you need a quantum RAM, or else you need your “database” to be specified implicitly (for example, as the evaluation function of an NP-complete problem), so that there’s no need for expensive data-loading. For that reason, I’ve always felt that the most compelling application of Grover’s algorithm was to combinatorial search problems.
Having said that, again, you could imagine a quantum RAM consistent with the laws of physics … and possibly even with the laws of engineering. 🙂 And Andris Ambainis and I have a 2003 paper where we show that, in the case of Grover search, communication latency would not asymptotically kill the speedup, the way it would for the machine learning algorithms. Indeed, when searching a database arranged in 3 spatial dimensions, you can get the full √N Grover speedup (even accounting for communication latency), while when searching in 2 spatial dimensions, you can search an N-element database in √(N log N) time, which nearly matches not only the Grover limit, but also the limit coming from the finiteness of the speed of the light.
28. Scott Says:
joe #24:
Is there a simple explanation for how, given a classical source of the N-dimensional vector(s), we can load them into a QC using only O(logN) resources? (I tried to read the qRAM PRL paper from 2008, but gave up in frustration.)
Probably the simplest scenario to imagine is this: suppose your database was encoded using a collection of N beamsplitters—the Nth bit is a 1 if and only if the Nth beamsplitter is set in a certain way. And now suppose you have a photon that travels through all N of the beamsplitters in superposition, then recoheres. That photon will “coherently query” all N elements of the database. Yet, if we disregard latency (i.e., the finiteness of the speed of light), then there’s really no sense in which we need “linear resources” to perform the query—any more than we need linear resources to make a single, classical query to a classical database of size N. In particular, the beamsplitters aren’t doing anything active that requires a power source, etc.; they’re just sitting there passively, waiting for a photon to pass through them. And the “parallelism” of querying all the beamsplitters in superposition is handled automatically by quantum mechanics.
Granted, the above might (or might not!) be too technologically daunting to ever be practical—but it does seem to show that quantum RAM is a sensible notion in principle.
29. joe Says:
Scott #28: Thanks again.
I’m really slow, and you’ve already patiently explained more than I could hope for, so feel free to ignore this, but I’m still puzzled.
I think I follow your explanation, but it seems to me to be providing an argument for how one can perform a query in O(1) time, rather than how one can load the database in the first place.
Let’s suppose we have a quantum register with n qubits. We can encode an N-dimensional vector in this register, where N=2^n, by loading the elements of the vector as coefficients. For example, if n=5:
|psi> = a_00000 |00000> + a_00001 |00001> + … + a_11111 |11111>
The coefficients a_00000, …, a_11111 (2^5=32 of them) are the elements of the vector.
From what I understand, this is the kind of preparation that the Lloyd et al. “supervised cluster assignment” algorithm calls for.
Now, the thing I’m wondering, is how you can start with your QC having a quantum register with state |00000> and prepare |psi> using only O(n=5) steps (or resources)? For example, if the classical machine originally containing the vector can only be accessed serially, then it seems like we will need to make O(N=2^n=32) reads just to get the coefficients out of the classical memory. Then once you’ve done this, it seems like you’ll have to apply O(N=32) gates on |00000> to prepare |psi>.
Is this not true? i.e., is there a way to prepare |psi> using only, for example O(n=5) steps (or resources*)?
* I have the caveat “resources” here to include the kind of situation where you don’t need O(2^n=32) time steps, but you instead, for example, manipulate 32 different magnetic fluxes for superconducting qubits in order to encode the 32 elements of the vector. (If you need 2^n physical elements to encode the database, then you’ve again just defeated the scalability of the middle part of the algorithm which runs in O(n) time.)
Again, apologies for what is probably a stupid question, and which likely comes across as a delightful mixture of pedantic and naive.
30. Scott Says:
joe #29: Yes, once you have the database, how to prepare the state |ψ⟩ in O(1) steps (or O(log N) steps, depending on how you count)—as usual, ignoring memory latency—is exactly the thing that I was trying to explain. Basically, your photon fans out into a superposition over all the memory registers. So the access to each individual register is not a step that needs to be counted separately from the accesses to all the other registers.
Now, it’s true that the database itself has size N, and that preparing the database in the first place would presumably have required linear resources. But exactly the same objection could be raised against classical log(N)-time algorithms! For example, creating an ordered list of size N requires N time (or even N log N time), but once you create it, you can search it for a given record in only log(N) time. People don’t say that classical binary search isn’t “really” log(N) time, because the log(N) doesn’t count the time to prepare the list in the first place. And if they don’t say that in the classical case, then I see no reason why they should say it in the quantum case. In both cases, preparing the list is just a “one-time investment”; once it’s there, you can do as many different log(N)-time queries as you want.
31. rrtucci Says:
Scott #30: It seems to me you are assuming that Google will put all its data into a quantum memory before applying Grover’s algorithm to that quantum memory. But the state in the quantum memory can be used only once and then it will be destroyed or corrupted. So I don’t see how Google can avoid storing its data in a classical memory and loading that data into a quantum memory every time it intends to use Grover’s algorithm. That is why I personally never assume that the initial state used in Grover comes from an unstructured database.
32. rrtucci Says:
I said;” That is why I personally never assume that the initial state used in Grover comes from an unstructured database.”
More precisely, I should have said “encodes all of” instead of “comes from”
33. Scott Says:
rrtucci #31: No, that’s exactly what my last comment was trying to explain is not the case. There’s the N-bit quantum RAM—meaning, the classical records that can be queried coherently—and then there’s the log(N)-qubit state |ψ⟩, which we use the quantum RAM to prepare. The latter gets destroyed after every use, but the former does not. And crucially, once we’ve invested linear effort to prepare the quantum RAM—which, again, despite its name, means a collection of classical records!—it then just takes log(N) additional effort (not counting memory latency, yada yada) to use the qRAM to prepare a fresh copy of |ψ⟩ whenever we want one. And we can do that as many times as we want.
You can doubt that a qRAM will be practical within the foreseeable future—that’s fine! Or you could say that it’s unfair not to count memory latency (but in that case, why don’t you level the same objection against the classical RAM model?). But what I described above is what people mean by a qRAM.
34. rrtucci Says:
I may be wrong, but it seems to me that the method with beam splitters that you propose for loading the classical data to a quantum state would produce a “good” subspace (good in the sense that it contains the database info) of dimension N=2^n embedded in a space of higher dimensions N’ and the extra N’-N dimensions would be entangled with the good N dimensions. To disentangle those extra N’-N dimensions from the good N dimensions and get a vector space of dimension N that isn’t entangled with anything else would require > O(N) operations
35. Scott Says:
rrtucci #34: Where is that higher dimension N’ coming from? (And how much higher than N is it, anyway?) Of course there will always be experimental imperfections, decoherence channels, etc. etc. But if we’re talking about a single photon going through N modes in superposition, then at least in principle we’re only talking about an N-dimensional Hilbert space.
36. rrtucci Says:
Again, I’m not sure of this 🙂
Suppose n=3
It seems to me your method will create an 8 dim subspace in a 2^8 space
a|1000,0000> + b|0100,00000> + c|0010,0000> +…+
h|0000,0001>
but you want to map it into a state
of 3 qubits, unentangled from the other 5 qubits
37. Scott Says:
rrtucci #36: Yes, a single photon in 8 modes does span an 8-dimensional Hilbert space, whose basis vectors are |10000000>, |01000000>, etc. But, OK, suppose you wanted to “compress” such a state into a 3-qubit state, which lives in the Hilbert space spanned by the basis vectors |000>, |001>, |010>, |011>, etc. You could do that using a recursive procedure, like so:
First, in parallel, compress |10000000> and |01000000> into a single qubit, along with |00100000> and |00010000>, etc. Next, compress the |xx000000> qubit with the |00xx0000> qubit to form a 2-qubit state, and in parallel, the |0000xx00> qubit with the |000000xx> qubit. Finally compress the |xxxx0000> qubits with the |0000xxxx> qubits to form a 3-qubit state.
You might complain that, in order to do this, I’ve used a linear number of operations (albeit, O(log(N)) time). But now we come to the key point: a linear amount of logic would completely analogously have been needed to access a classical RAM. The only real difference is that, in the classical case, only a single path through the logic gates would have been relevant to any given query. But it doesn’t seem fair to penalize the qRAM for having the power to make superposed queries—that’s, like, its whole thing! And it’s not as if the energy required, or any other obvious resource, is greater in the quantum case than in the classical. (A single photon in superposition costs no more energy than a single photon not in superposition.) But maybe I’m missing something?
38. aram Says:
Nice talk!
I see your point about HHL being a “pre-algorithm” but also am curious whether you’d describe Hamiltonian simulation, Jones polynomial approximation, or a hypothetical algorithm for graph isomorphism the same way, given that in each case we do not know how to construct distributions that are believed to be hard on average (apart from making use of BQP-hardness results for the first two cases). Or are you just referring to the fact that the input is specified by an oracle, and so would say the same thing about amplitude amplification, triangle-finding or the HSP?
On a different note, I totally agree with you about the substantive point that Hamiltonian simulation is still likely to be the most useful q algorithm among the ones invented to date. I just wouldn’t use the word “algorithm” so narrowly.
39. rrtucci Says:
I’ll quit while I’m ahead. This is like the first time in 10 years you partially agree with me 🙂
40. Scott Says:
aram #38: It’s a good question. I would say that pre-algorithm vs. algorithm is not a sharp distinction, but a matter of degree. Is this algorithm something that you take right out of the package, put in the oven, and get an exponential speedup? Or is it more like one of those brownie mixes where all you need to add is the flour, sugar, eggs, butter, and chocolate?
For graph isomorphism, it’s true that we don’t know a distribution over hard instances, but at least it’s very easy to state the problem being solved. Whereas with HHL, the problem being solved is often summarized as “solving linear systems,” but of course there are 3 caveats (how to specify the input, the condition number, what observable to measure about the solution vector). As long as those caveats are remembered, I’d say yes, it’s very similar to graph isomorphism.
41. joe Says:
Scott #30, and later: Ah, I see. The point I was missing is that a qRAM allows you to keep generating copies of |psi> (n-qubit register encoding a N=2^n-dim vector in its coefficients) in time O(1), once you have expended the O(N) or O(NlogN) effort in preparing the qRAM.
Now I think I understand your beamsplitter network example: the n beamsplitters somehow act to encode the N-dim vector, so that when a photon passes through, the photon is in the state |psi>, which took O(1) effort (time for the photon to pass through the beamsplitter network).
Is there any physical-implementation-agnostic way to describe how to construct a qRAM? Can we describe the construction and operation of a qRAM in terms of qubits and one- and two-qubit gates? (Of the small number of papers I’ve seen in the literature so far, they all seem to give examples of how to construct a qRAM using some very specific set of physical tools, e.g. optical beamsplitter networks; atoms in cavities and photons, etc.)
In the case of the beamsplitter network qRAM you describe, what is the physical meaning of the qubits in the resulting |psi>? e.g., for n=3, I first thought that |100> is the state where there is 1 photon in the first mode, and 0 photons in the second and third modes. However, I suppose that’s not right, since then if you only put a single photon into the beamsplitter network, only the states |100>, |010> and |001> can have non-zero amplitude (and |000> if you consider photon loss). Let’s suppose I want to make a beamsplitter qRAM that can produce the following |psi>, how would I do it?
|psi>=(1|000> + 1.1|001> + 1.2|010> + 1.3|011> + 1.4|100> + 1.5|101> + 1.6|110> + 1.7|111>)/a
(where “a” is the normalization factor)
Thanks again, and good luck with the article. If you have even some of your discussion points from this comment thread in it, I think it’s going to be a really helpful article.
42. jonas Says:
Scott, re #37 about the RAM. The principle sounds about right to me.
There’s some magic about constant factors in the classical case. These days we use DRAM, which is a brilliant invention of an electronic circuit that lets you store lots of bits in both very little circuit board space and much less energy consumption than the more obvious SRAM. But of course, as while we have no idea how to build practical quantum computers at all currently, we can’t really speculate about the details in quantum computers.
43. upen Says:
joe i don’t see how a qram will allow u to copy registers there is a no cloning theorem
44. Scott Says:
upen #43: Dude. The No-Cloning Theorem doesn’t apply to classical information, and the information in the qRAM is completely classical. As I said, the “q” in the name only means that the information can be queried in coherent superposition, not that it itself is in a quantum state.
45. upen Says:
thanks scott, i have always wondered if superposition of light waves ,let aside all the implementation difficulties can lead to
quantum states that are more abstract than superposition of
electrons only 0/1 possible assume i superpose wavelength l1 with l2
what will be the size of such a quantum register will it still be a
qbit or is it a qDigit ? even if we can somehow implement a qDigit
what quantum operators can we define on it ?
46. Raoul Ohio Says:
Please excuse me for briefly revisiting a sad topic of a few months ago: creativity, depression, and suicide. I think most of us will find this from today’s Ars Technica to be useful, insightful, and touching:
http://arstechnica.com/staff/2014/10/harnessing-depression-one-ars-writers-journey/
47. Richard Cleve Says:
Hi Scott:
In the slides, you refer to your 2009 result about Fourier Checking. I’m wondering what your current take is on the other result in that paper, about Fourier Fishing.
My interpretation of Fourier Fishing has always been that you give a black box problem that’s solvable in polynomial time on a quantum computer; whereas that black box problem is not in the classical polynomial hierarchy (that is, a reasonably natural definition of PH for black box problems).
To me, it seems like a pretty strong result. But, in conversations over the years, I’ve often found myself defending the interestingness of it.
If you had included it in your talk, what would you have said?
48. Scott Says:
Richard #47: I would’ve said that Fourier Fishing is basically the black-box analogue of BosonSampling. I.e., compared to BosonSampling, the advantage of Fourier Fishing is that you can prove an exponential separation (and not only that—as you said, you can prove that Fourier Fishing is not in BPPPH), but the corresponding disadvantage is that Fourier Fishing is relative to an oracle. Indeed, “historically,” thinking about Fourier Fishing is exactly what led me to think about BosonSampling (and to suggest to Alex Arkhipov to think about it more): I was looking for an unrelativized analogue of something I already knew in the relativized setting.
Of course, you could also “instantiate” Fourier Fishing with an explicit Boolean function. If you do that, then you get something that’s basically equivalent to Bremner, Jozsa, and Shepherd’s IQP model. For that model, we have results about the hardness of exact sampling directly analogous to what we have for BosonSampling (i.e., if you can do it efficiently classically, then PH collapses to BPPNP). But we don’t yet have results about the hardness of approximate sampling analogous even to the incomplete results that we have for BosonSampling.
49. John Sidles Says:
Another quick (and sobering) announcement:
——————–
Take my research, please! IBM Pays GlobalFoundries \$1.5 Billion To Shed Its Chip Division. This news has long been foreseen by the QM/QIT/QC research community within IBM, yet it is by no means welcome.
50. Jon Says:
Hey Scott,
I feel compelled to point out that it seems old-fashioned to post .ppt files, because there is software available that will record audio in sync with a screenshot of your slides (or whatever is going on on your laptop display) and post the whole thing to youtube:
For IBM: Camtasia studio
For Apple: see http://www.virology.ws/2013/03/01/how-i-record-my-lectures/
Anyway, it’s a way to make video of your talks (or practice talks) with no A/V guy required, as long as you’re willing to settle for Audio + Slides, instead of Audio+Slides+video of your face.
51. Serge Says:
Jon speaking of Slides… interesting! 😉
52. anon Says:
Scott: I consider you as one of the thought leaders in theory, so I want to ask what’s your view point on the controversy surrounding Hajiaghayi’s CS department ranking proposal?
First, is ranking a necessary evil? For superstars like you or schools like MIT or stanford, ranking probably doesn’t matter. But for the rest of us, ranking does seem to matter (grant application, graduate enrollement, etc). If it is, do you think Hajiaghayi is taking a step towards the right direction? If it is a pure evil, what should the community do as a whole to rectify the situation? Next time US news release it’s ranking, should the community be writing an open letter to protest the practice?
Second, as evidenced from the comments in Lance and Luca’s blog, there are jerks in this field – making personal attacks under a veil of anonymity. I would question the integrity of these people in their professional activities (e.g., anonymous conference paper review ), and this would make me worry. What do you think about those comments and do you think I am too paranoid?
Thanks!
53. Job Says:
Scott,
I have been looking into Deutsch’s and Simon’s algorithms – which provide an oracle separation for P/QP and BPP/BQP respectively.
I was wondering, isn’t the black-box “unfair” in the context of an Oracle separation? What i mean is that, for the QC, f(x) is more like a “gray box”, in the sense that qubits have additional (implicit) state that may reflect the implementation of f(x).
Naturally, we can argue that this is an example of how QC’s are more powerful than classical machines, but is that the case in the context of Deutsch’s/Simon’s problems, or is the black box simply establishing a-priori that classical machines are not able to perform the same task just as efficiently?
For example, suppose we develop a new computational model X, where XP contains the set of problems efficiently solved by X. An X machine operates via a set of elementary gates defined by the X model. As with a QC, we impose that a black-box provided to X must be implemented using X’s gates.
We can then ask two questions:
1. Are Deutsch’s & Simon’s problems in XP?
2. Is XP contained in P? (i.e. can a TM efficiently simulate an X machine?)
What i’m getting at is, can we not build a set of X gates that in fact make f(x) totally transparent, yet are efficiently simulated by a TM?
For example, suppose that the X gates correspond to classical OR/AND gates with the added feature that an execution transcript is kept. In that case, an X machine will have some insight into what f(x) is doing. As with a QC, it’s no longer a completely black box.
If it’s then shown that X is efficiently simulated by a TM, does that mean that the Oracle separation no longer holds? If not, why not? What’s up with the black box?
54. Scott Says:
anon #52:
what’s your view point on the controversy surrounding Hajiaghayi’s CS department ranking proposal?
I saw Hajiaghayi’s ranking, but I confess I wasn’t aware that there was a controversy. Where is this controversy raging?
All Hajiaghayi is doing, is ranking American CS theory groups by the total number of papers they publish each year in the major theory conferences (not normalized by the number of people). It seems obvious that that’s one statistic that (say) a student might want to know when deciding where to go to grad school, and equally obvious that it’s very far from the only relevant piece of information—even if we restrict ourselves to the things that are objectively quantifiable.
55. Scott Says:
Job #53: I confess that I didn’t fully understand your question, but briefly—in quantum oracle separations, it’s extremely important that the classical and quantum algorithms are given black-box access to exactly the same function. And there’s no requirement that the black box can “only” be accessed quantumly, or anything like that. Yes, quantum queries turn out to let you decide some particular thing about the function much faster than classical queries, but that’s the entire point!
It’s true that the quantum black box can be queried in superposition—and therefore, one could say, it “provides a functionality” that the classical black box doesn’t. However, what justifies this is the observation that, if you had a quantum computer, and you knew a circuit to compute some function f, then you could automatically also compute f on a superposition of inputs, because of quantum-mechanical linearity.
If you’re interested in separations, the Deutsch-Jozsa algorithm is a really bad example, since it only gives a separation between quantum and deterministic query complexity, and no separation at all (or only a factor-of-2 separation) between quantum and classical randomized. Simon’s algorithm provides a much better example. And of course, Simon’s algorithm led to Shor’s period-finding algorithm, which has “non-black-box” applications such as factoring and discrete log. One would hope that fact would lay concerns about the relevance and meaningfulness of the quantum black-box model to rest.
Also, keep in mind that you only get large quantum black-box separations for certain highly-specific problems, and not others. For example, there’s an exponential separation for Simon’s problem, but not for Grover’s search problem—and for computing the parity, there’s only a factor-of-2 separation. This is an additional indication that the quantum black-box model is telling you something real about the power of quantum algorithms; it’s not just a matter of playing around with definitions.
56. Job Says:
Scott,
What i’m picking on is the fact that the QC uses a Quantum implementation of the black-box whereas the Classical machine uses a classical black box.
This just stood out as a fallacy because the setup may not yield an accurate test – i’m not contesting that QC’s are much faster.
For example, suppose we invent a new machine X which is truly faster than classical machines but which also leaves behind additional “residual” state.
If we show that X is able to determine an internal property of a black-box function f(x) faster than a classical machine, then is that because X was using it’s true computational power, or was it sufficient to process the residual state in a classical way?
Similarly, does a QC solve Simon’s/Deutsch’s problems by using it’s true computational abilities (as seems to be the case with Shor’s/Grover’s) or is it just processing the residual state (e.g. the resulting probability distribution) in a trivial way?
57. Job Says:
Scott, one more observation. You mentioned that:
“It’s true that the quantum black box can be queried in superposition—and therefore, one could say, it “provides a functionality” that the classical black box doesn’t. However, what justifies this is the observation that, if you had a quantum computer, and you knew a circuit to compute some function f, then you could automatically also compute f on a superposition of inputs, because of quantum-mechanical linearity.”
The same statement can be made about a machine X that logs and publishes all operations, e.g.:
It’s true that X has access to the black box’s execution transcript–and therefore, one could say, it “provides a functionality” that the classical black box doesn’t. However, what justifies this is the observation that, if you had an X computer, and you knew a (classical) circuit to compute some function f, then you could automatically also compute f while capturing its execution transcript, because X still uses classical gates (the gate’s activity just gets logged).
The underlying argument is that Simon’s & Deutsch’s problems are in P because there is an X machine that solves them and which is efficiently simulated by a classical TM.
You also mentioned that:
For example, there’s an exponential separation for Simon’s problem, but not for Grover’s search problem
Which i find strange and indicative that the black box setup in Simon’s problem may be the reason why it gets such an accentuated speedup.
58. Scott Says:
Job #56-57: If you want, you can think of the quantum black-box model as a “laboratory” for devising quantum speedups, or for exploring where speedups might be possible. Some black-box speedups (e.g., for Simon’s original problem, or for Recursive Fourier Sampling) “never make it out of the lab”: i.e., we haven’t found any real, unrelativized problem that they correspond to. But others (e.g., for Shor’s period-finding problem) do make it out of the lab; they led to real quantum algorithms for problems like factoring. (And FWIW, Shor’s period-finding algorithm built directly on Simon’s algorithm, and BosonSampling also had its origins in a quantum oracle separation.) And these speedups justify the laboratory’s existence, if justification were needed.
In summary, the quantum black-box model is a mathematical model that
(a) is rigorous and beautiful and nontrivial in its own right (and includes its own internal, well-defined notion of “quantum speedup”), and
(b) has led, on multiple occasions, to actual (non-black-box) speedups for actual problems. (And has also helped us understand why other problems don’t seem to have efficient quantum algorithms.)
So what more do you want? In math and TCS, ideas only have to justify themselves by their internal coherence and elegance and by the fruitfulness of what they lead to, not on some a-priori metaphysical ground.
59. Job Says:
Scott,
The quantum black-box argument as stated for Simon’s/Deutch’s problems seems slightly flawed, in my view – i’m simply pointing this out and trying to understand whether it is in fact the case.
Note that i’m not debating whether QC’s are fast, my point is that the following argument:
“Classical machines can’t solve Simon’s/Deutch’s/Bill’s problem efficiently by design, since they can’t see through the black box.”
Should take the following form instead:
“Classical machines can’t solve Simon’s/Deutch’s/Bill’s problem efficiently because no feasible machine X exists that both solves that problem efficiently and is efficiently simulated by a classical machine.”
As you mentioned, this doesn’t change the fact that we have viable Quantum algorithms such as Shor’s/Grover’s which provide faster alternatives to classical solutions, but it would be a flaw nonetheless and i would like to understand whether this has been argued and why the two statements above are in fact the same.
If one of your blog readers has an opinion here i’m interested in figuring out where my reasoning is off so i can move on.
60. Job Says:
Also,
Is Simon’s problem the only separation between BPP & BQP – other than the probable separation provided by Shor’s/Grover’s assuming these problems are not in P/BPP – or are there other established separations between these two classes?
That by itself would warrant a closer look at black-box arguments. Are there any non-black-box separations between BPP/BQP?
61. Count Doofus Says:
Scott#14
I used to think that Quantum Math is complexity bounded for the simple reason that try to account both the action and reaction at the same time, not falling in the “uncompleteness” of deriving one from the other.
It’s the magick of complex numbers, that are themselve and their negative together; this i thought when i first learnt about immaginary unit during high school.
62. Ben Standeven Says:
I’m not understanding your argument here. Supposing that I have a non-deterministic Turing machine M that takes SAT instances and determines if a satisfying assignment exists, we can of course assume that the entire computation of M is explicitly logged. So M actually provides a satisfying assignment, if there is one.
Now, there is obviously a deterministic Turing machine N that takes the log/satisfying assignment produced by M and verifies that it works; this is an “XP” oracle-machine that solves SAT, and it can be efficiently simulated by a P oracle-machine, because it already is one. So would you then argue that SAT is in P, just as you [apparently] argued that the Simon and Deutsch problems are?
Maybe you meant “classical” to imply “deterministic”; but then I don’t see how there could even be a [computable] oracle separation between XP and P, since when the XP machine reads the transcript, the P machine can simply simulate the oracle. This would then produce only a polynomial slowdown.
63. Ross Snider Says:
@Scott
You regularly miss obvious military applications such as the calculation of EM scattering cross sections (for the design/defeat of stealth materials).
http://arxiv.org/abs/1301.2340
64. Scott Says:
Ben #62: Who are you addressing—me or Job?
65. Scott Says:
Ross #63: Dude, I already discussed that EM scattering paper in comment #17! As I said, I thought it was nice—I’m glad someone is trying to figure out a real, start-to-finish application for the HHL algorithm. But the idea that the paper has “obvious” military applications (!) is one that I doubt would be endorsed by the authors themselves, since it involves multiple enormous speculative leaps. First, you’d need an application where the shape of your material could be implicity specified by a short formula—or otherwise gotten into quantum registers efficiently. Second, you’d need the condition number to be bounded. Third, you’d need the limited amount of information that HHL can give you about the solution vector (e.g., just the cross-section) to be interesting for your application. Fourth, and most importantly, once the first three conditions were satisfied, you’d need for there still not to be any efficient classical solution.
Also, since you claim that I “regularly” miss obvious military applications (!), could you perhaps provide one other example?
66. Ross Snider Says:
@Scott
Sorry I didn’t read the comment section. Doh!
I also hadn’t/don’t understood all the caveats to the algorithm. Thank you for setting the record straight!
Lockheed has said a couple of times that it is interested in QC because of connections to software verification (I don’t precisely understand why that is). I tracked down one of the quotes:
“Lockheed, according to Brownell, hooked up with D-Wave out of a mutual interest in the types of calculations that a quantum computer could perform. “They had algorithms that were applicable to our technology,” he says. “Their particular focus is software verification.”
The same article speculates about other applications (QML) which I’ve seen before closer to first party sources. Many of the applications seem to be “do the same thing we currently do but better” – which would also include classes of optimization problems be they for communication relay and antenna designs, satellite communication systems, or ‘mundane’ vehicle design.
It’s also important to note that even a polynomial speedups are of interest to military folk. An example of this may be the solving of systems of partial differential equations, which has your usual engineering applications.
67. Job Says:
Ben,
By “classical” i mean a machine in at most BPP – your XP/SAT machine is not efficiently simulated by a classical machine.
You also mentioned:
“Maybe you meant “classical” to imply “deterministic”; but then I don’t see how there could even be a [computable] oracle separation between XP and P, since when the XP machine reads the transcript, the P machine can simply simulate the oracle. This would then produce only a polynomial slowdown.”
Does that still apply to a black-box setup? That’s essentially my point.
In normal conditions, XP does not provide any advantage since it must be efficiently simulated by a classical machine. On the other hand, i can define an X machine, and X gates, such that any input black box, which is a-priori required to be implemented with X gates (just as a black-box provided to a QC must be implemented with quantum gates) becomes a gray or white box.
In that case, even though XP does not provide any computational speed up, it does provide the internal details of the black box which is relevant information when the problem is about determining a property of the black box (as with Simon’s/Deutsch’s).
Note that i’m not saying that this would automatically put Simon’s/Deutsch’s problems in P or BPP, only that a black-box separation between P/BPP and QP/BQP would need to show that no such machine X is possible.
68. asdf Says:
In case anyone has missed the latest quantum wtf:
https://news.brown.edu/articles/2014/10/electron
69. Job Says:
For context, the reason i’m picking on the Quantum black box is that, after studying Deutsch’s circuit, it’s apparent (to me) that there is residue in the black box’s output that may be measured by a QC to determine whether f(x) depends on x.
It does not seem to be the case that the QC is performing any computationally significant task, it’s exploiting the fact that it has access to the residual quantum information in the black box’s output.
Sure, the QC can trigger and inspect the quantum residue in the black-box’s output, that’s what makes it powerful, but that would not be sufficient to separate BPP and BQP because it changes the definition of the problem.
As it is there are two versions of black-box problems:
The classical version
Given a black box that receives x and outputs f(x)+a, determine a property P of f(x).
The quantum version
Given a black box that receives x and outputs f(x)+b, determine a property P of f(x).
Where a and b are residual output information, and a != b.
Now, we’ve shown that a QC can solve its version faster than a classical machine.
Does that illustrate a capability of QM we can use for computation? Sure.
Does it prove that a QC is computationally more powerful than a classical machine (as Shor’s/Grover’s suggest)? Not unless we show that there is no value of a such that a classical machine is able to solve the problem just as fast.
Traditionally, we’d require that a = b, but it seems that it’s ok to abandon that requirement here. Why?
If a and b may differ, then we need to account for every legitimate value of a.
70. Scott Says:
Job #69: Again—Deutsch-Jozsa is an unconvincing example of a quantum speedup, since it’s just a factor of 2, which could be completely wiped out by details of the oracle access mechanism. Simon’s algorithm is a much better example. If you had an actual, efficiently-computable function f that satisfied the Simon promise, as well as a quantum computer, then you could actually use Simon’s algorithm to find f’s hidden shift. And for the closely-related period-finding problem, which arises when breaking RSA, you can of course use Shor’s algorithm. I don’t know how on earth you account for those facts on the view that quantum black-box speedups are just artifacts of unfair definitions.
Of course, everyone who knows anything about this subject knows that black-box separations don’t rule out the possibility of a fast classical algorithm that “opens the black box”—that’s why we say, for example, that we can separate BPP from BQP relative to an oracle, rather than that we can separate them in the unrelativized world. At this point, this discussion feels like pure word-splitting to me; I don’t know what substantive point is at issue.
71. Scott Says:
Ross #66: The Lockheed thing is widely recognized as an absurdity, with the fingerprints of pointy-haired bosses all over it. In order to claim an application of the D-Wave machines to finding bugs in jet-fighter code (or whatever), they simply refuse to ask the question of whether one could perform the same task just as well or better using a classical computer, presumably because they don’t want to know the answer to that question. (Namely, of course one could do just as well or better classically—so far, researchers who set out to do so haven’t even been able to find any clear advantage for the D-Wave machine on the problem of calculating its own ground state.) This isn’t what science looks like. The EM scattering application, despite its speculative nature, is an orders-of-magnitude better example than the D-Wave/Lockheed one.
72. Job Says:
Scott,
The behavior of this particular oracle is strange to me because it’s “architecture specific”.
Suppose we define an Oracle O that receives x and produces f(x)+c, where f(x) is some random noise and c is “yes” if x is a satisfiable boolean formula and “no” if x is not satifsfiable.
But here’s the catch, c is inaccessible to Quantum Computers – it’s always null.
That means BPP^O != BQP^O, but what is this really evidence of? What was shown by this oracle separation?
73. Ross Snider Says:
@Scott
I would presume they are looking to verify _classical_ code with quantum algorithms – especially since it would be tough to get a QC up into the air…
But again I don’t know that there is any advantage in this case either.
Finally given the posterior of the (current) advances in automated verification of Probablistic Programming Language programs, and DARPA funding of such, how absurd are generalizations to/from the L2 norm and how small a prior do you give DARPA/Lockheed for having made private advances on this front?
74. Scott Says:
Job #72: But what you defined is not an “oracle” in the sense we mean in theoretical computer science. It violates the rules. The oracles that interest us are ones that compute exactly the same function in the classical and quantum cases—the only difference being that, in the quantum case, you get to query the oracle in superposition. I think you’ll find that it’s not quite so trivial to separate BPP from BQP relative to that kind of oracle! With the benefit of today’s knowledge, it’s not that hard, but it does require rediscovering one of the main quantum algorithms (like Simon, Shor, or Bernstein-Vazirani). And other oracle separations (e.g., AM from PP, SZK from BQP, PH from PSPACE, the levels of PH from each other…) are quite nontrivial even given today’s knowledge. Others (e.g., QMA from QCMA, QMA from QMA(2), SZK from PP) remain unknown to this day.
75. Scott Says:
Ross #73: Yes, they’re looking to verify classical code with quantum algorithms. But no, we have no good reason to think you can get any advantage for this task with D-Wave’s machines.
Yes, there are lots of exciting advances these days in automated verification, and yes, I think it’s perfectly reasonable for DARPA to be funding such work. (For that matter, I think it’s also reasonable for them to fund quantum computing! 🙂 Their mission is supposed to be broader than things with direct military applications; it has to do with keeping the US at the technological forefront.)
And if there were some spinoff from quantum computing research that benefited classical automated verification—well, it wouldn’t be the first time that QC had had an interesting classical spinoff.
And if you had a true fault-tolerant QC, you could try running the adiabatic algorithm or Grover search to get a speedup for software verification. We can’t say with confidence that you’d see any speedup over the best classical algorithms, but nor can we say with confidence that you wouldn’t.
The one thing I can tell you with confidence is that no one is using the quantum hardware that exists today to get any genuine speedups for software verification.
76. Job Says:
Scott,
My point was that the QC’s extra ability to query the Oracle in a superposition also breaks the rules (as did my Oracle O) and that to fix this we’d either need to remove this quantum ability or allow the classical machine to get additional state from the oracle.
I understand your comment about the black box setup being a thought experiment from which insight may be gained to produce algorithms like Shor’s, which does not by itself separate BQP from other classes, and that if we could solve Simon’s problem fast classically by having the Oracle return an additional yet trivial state then we might well be able to do the same for integer factorization.
I think it’s pretty interesting and worth thinking about what kind of trivial state a classical machine might need from the Oracle in order to solve Simon’s problem – and i confined this trivial state to be any state that may be captured by an X machine (on which the black-box runs) that’s efficiently simulated by the classical machine.
Anyway, that’s just one of the questions i had. I was also wondering if anyone has tried to exploit the Law of Total Probability for computation, e.g. maybe to compute the Fourier transform. You’ve mentioned that negative probabilities would be sufficient to match much of a QC’s power, but are negative probabilities that difficult to realize?
77. Ross Snider Says:
@Scott #75
Right. So I guess this is where we come full circle back to the presentation.
Today nobody is using the D-Wave machines for anything (okay, okay, besides perhaps the research itself and for solving those odd ising spin problems less quickly than commodity hardware). So you can’t demerit these applications for not yet being fun. If you were, for no reason other than consistency, you would also need to demerit Shor’s algorithm and in fact everything in the slide deck.
The presentation is not about real-being-done-today applications but possible-might-be-done-sometime applications. It’s about ‘what do we want these machines for?’ So I would argue that you should consider expanding the slides to include EM scattering and software verification, wary of course of how far they sit in speculation land. I know you’re pretty good at that – better than yours truly, so I’ll leave you to that calculus.
Anyhow thanks for the informative discussion!
78. Ross Snider Says:
*done
79. Sniffnoy Says:
Maybe it’s just me, but objecting that a quantum computer gets to query its oracles in superposition seems to me like objecting that a probabilistic computer gets to query its oracles based on a random decision (i.e. in “probabilistic superposition”, which is as much superposition as it can muster).
80. Serge Says:
Scott #58:
“In math and TCS, ideas only have to justify themselves by their internal coherence and elegance and by the fruitfulness of what they lead to, not on some a-priori metaphysical ground.”
You can’t justify mathematics by metaphysics, but your conjectures may arise from such principles that aren’t a-priori in the least. The sources of invention are so diverse that dismissing all metaphysics in the first place doesn’t sound like good strategy – to me at least. Of course, internal coherence remains the ultimate judge… provided there’s actually something to check!
81. Raoul Ohio Says:
Are dB^4 (de Broglie, Bohm, Bell, Bush) right?
http://www.wired.com/2014/06/the-new-quantum-reality/
If so, how does this inpact QIT and QC?
I had missed the fact that the proofs that hidden variable theories cannot work had apparently been shown to be wrong.
82. Scott Says:
Raoul #81: No. None of these “oil-drop” models can explain entanglement phenomena, like Bell inequality violation—as anyone who understood the Bell inequality could have immediately predicted. So, the models do OK at reproducing the behavior of one quantum particle, but they badly fail when there are two or more entangled particles. The reason is not something technical or fixable, but goes to the core of quantum mechanics: the state of the world really is a vector of amplitudes in a gigantic Hilbert space. So if your only degrees of freedom are the values of fields in 3-dimensional space, then you’re necessarily going to get things wrong—as, in fact, you do.
To their credit, many of the people doing the oil-drop experiments are well aware of these facts. When confronted with them, they simply prefer to change the subject: “look, let’s just talk about 1 particle, and leave 2 or more particles for the future.” (Of course, Bell’s Theorem explains why the generalization to 2 or more particles will never work, as long as it remains local-realistic: it’s not a matter of getting around to it. But, again, the convention seems to be to change the subject when this comes up.)
You mentioned Bohm. Back in the 1950s, he already foresaw this fundamental problem with hidden-variable theories, and he proposed a solution: just import the entire framework of QM, exponentially-large amplitude vector and all, into your hidden-variable theory, in order to guide the hidden variables in just such a way that they always reproduce the experimental predictions of QM. If you do this, then not surprisingly, you do reproduce all the experimental predictions of QM, so the implications for QIT and QC (for example) are nil. But it remains questionable whether you gained any explanatory power for all your trouble.
By contrast, Anderson and Brady (who are quoted in the article) simply want to junk QM, with all its empirical successes, and replace it with a total nonstarter classical model. In their papers, they alternate between saying that experiments haven’t really proved Bell inequality violation, and giving nonsensical and incomprehensible “explanations” for how classical oil drops can violate the Bell inequality too. (See the previous thread where we were over this ground.) This stuff, I’m sorry to say, has crossed into crackpot territory, and doesn’t deserve any of the inexplicably-respectful attention it’s been receiving on various news websites.
83. Raoul Ohio Says:
Scott,
Thanks. I had a feeling you were not going to buy this theory.
84. Itai Says:
Scott,
I really do not understand from where do you take ” the state of the world really is a vector of amplitudes in a gigantic Hilbert space” as granted ?
You also had a weak point against Pilot wave theory in your book regarding finite dimensional Hilbert space .
You take for granted that matrix QM should hold there, you turn |0> to |+> with a unitary transformation and say it “became random “.
I think the matrix QM(Hilbert space QM) has some serious problems and it is not the best way to describe QM.
( Well , Einstein and Schrodinger didn’t like it at all , and it has all those problems with the infinite dimensional operators for energy, momentum and position plus problems that comes from probability ).
Also, Pilot wave theory is not the only successful Hidden variable theory ( it still a bit ugly and not local ), Nobel Price winner physicist t’ Hooft has series of papers where he suggest a local deterministic theory based on some kind of cellular automaton ( maybe Wolfram was right after all ? )
http://arxiv.org/abs/1405.1548
http://arxiv.org/abs/0908.3408
http://physics.stackexchange.com/questions/34217/why-do-people-categorically-dismiss-some-simple-quantum-models
85. Scott Says:
Itai: The whole point of Bell’s theorem was to prove that, if you want a hidden-variable theory, and you want it to reproduce the predictions of QM, then it must be nonlocal.
And Bell’s theorem is a theorem; it’s not a suggestion. I’ve argued with enough people who think there’s some loophole in Bell’s theorem to fill 10,000 lifetimes. The discussions were always infuriating, and were never edifying. There is no loophole.
Gerard ‘t Hooft is another example of someone who thinks there’s a loophole in Bell’s theorem. In his case, the “loophole” is absolutely gobsmacking—basically, that the entire universe is in a gigantic, nonlocal conspiracy to make it look like quantum mechanics is true, and to prevent us (and the random-number generators in our computers, etc.) from choosing to do the experiments that would show that it’s not true. It’s obvious that this is a “cure” a quadrillion times worse than the disease—if anyone other than ‘t Hooft had proposed it, it would’ve been immediately laughed off the stage, which seems to me like the only sane reaction.
Finally, my argument about Bohmian mechanics (i.e., that its determinism is just an artifact of working in an infinite-dimensional Hilbert space) assumes nothing more than what Bohm himself reasonably assumed—namely, that whatever else the theory does, it had better reproduce the experimental predictions of QM, and certainly its predictions about simple 2-level systems. Those are arguably the most precisely-confirmed predictions in all of science.
86. rrtucci Says:
Scott, I hope you write a review of Interstellar. I even have the perfect Susskindesque title for your blog post: “Quantum Data is not Enough”. Not bad, eeh?
87. Itai Says:
Scott,
I read that Pilot wave has no relation whatsoever to Hilbert space mathematics, so any argument from that side is void ,
It is built only on wave mechanics with “modifications” .
I don’t really believe you can actually prove No – Go theories in physics because you can never validate your mathematical assumptions are true (In mathematics it’s not the situation because there is no one truth) , there are always some loopholes.
what Gerard t’ Hooft is suggesting is a LOCAL theory ( see the links ) , he actually thinks that non-local theories like Pilot wave are ugly.
I read that t’ Hooft said that Bell himself told him that his arguments are not valid when we take the “free will ” ability from Alice and Bob in EPR .
So, super-determinisem is by far the most obvious loophole there, and I guess everyone should agree on that.
I think that free will , or free of measurement is ill defined thing, and should not have a place in physical theory ( maybe in philosophy ) .
Here what t’ Hooft wrote about bell and freewill :
http://arxiv.org/abs/quant-ph/0701097
The reactions for t’ Hooft work are almost what you described, but you must admit that he knows physics better than most if not all of the critics who just memorize what they were taught and try not to rethink about anything that is too basic from the theory ( I guess 99.99% of the critics did not really read his work ).
88. Scott Says:
Itai #87: Then not everything you read is correct. The “modifications” that you make to classical mechanics to get Bohmian mechanics involve importing the entire Hilbert space structure of ordinary QM, in the form of Bohm’s “guiding potential.” Don’t take my word for it; please look it up. You have to do this, because if you didn’t, you would make wrong predictions for actual physical experiments (as soon as entanglement is involved)—a point that, incredibly, you keep refusing to engage with. If your theory makes the wrong predictions for 2 entangled particles, then it doesn’t matter whether it’s local or nonlocal, beautiful or ugly, simple or complicated. The theory is out. It’s dead. Stick a fork in it. End of discussion.
And what if your theory very obviously “wants” to say that the Bell inequality can’t be violated—and it can only deal with the inconvenient experimental fact that the Bell inequality is violated (and more generally, that every experiment ever done gives results 100% consistent with quantum mechanics), by an ad-hoc rationalization involving a cosmic conspiracy set in motion since the beginning of the universe, which for some unexplained reason only lets you do exactly the things that quantum mechanics said all along that you can do, and no superluminal communication or anything else other than that? In that case, by any sane standard, your theory is late for a date with the garbage heap—and it makes no difference whether you’re some random crackpot on the Internet or Gerard ‘t Hooft. Please don’t argue from authority here.
89. Vitruvius Says:
Any time I hear a physicist asserting certitude I become more skeptical of the physicist’s other assertions, for as Oliver Wendell Holmes Jr. noted, “Certitude is not the test of certainty. We have been cocksure of many things that were not so”. Using bold for emphasis instead of italics or the language itself doesn’t help either; that’s just another kind of shouting, like all upper-case. And that includes you, Scott, as much as anyone else. Any scientist, really, but especially physicists, given that all the rest is just stamp collecting. Needless to say, my having recently finished Smolin’s The Trouble with Physics (2006, 978-0-618-55105-7), especially considering his views on not dismissing ‘t Hooft and the other so-called crackpots (in the eyes of the bureaucracy) out of hand, has done nothing to disabuse me of my skepticism 😉
90. Scott Says:
Vitruvius: There are countless things in physics that I’m extremely uncertain about. Indeed, compared to my friends in the high-energy and quantum gravity communities, I’d say I’m much more uncertain; I question many things that they regard as established. But then there are things that I, too, am “certain” about—not in the philosophical sense, but in the sense of everyday life.
For example, I’m “certain” that physicists won’t announce tomorrow that the earth is actually flat—or that it’s a hollow sphere, and we’re all living on the inside. And I’m equally “certain” they won’t announce that the Bell inequality can’t be violated after all, and indeed that quantum mechanics was just a huge wrong turn; the entire apparatus of density matrices and amplitudes can be thrown away in favor of some 19th-century-style classical model.
Yes, the path of science is winding and uncertain, but the probability that the path would perfectly reverse itself (e.g., by going from a round earth back to a flat one, or from evolution back to creationism, or from quantum physics back to classical) is so infinitesimal that it can be ignored.
Incidentally, much of what I was shouting at Itai about wasn’t even points of physics, but points of math. For example, that the guiding potential in Bohmian mechanics is isomorphic to the entire wavefunction in ordinary quantum mechanics, is just a fact about the definition of Bohmian mechanics. It makes no more sense to argue about it than to argue about whether circles are round.
91. John Sidles Says:
Scott perceives error (#6 of Speaking Truth to Parallelism) “I got jolted out of Neal Stephenson’s Cryptonomicon when I encountered a passage about ‘factoring huge prime numbers,’ and couldn’t continue reading.”
This passage reminds us of Dirac’s celebrated critique of Dostoevsky’s Crime and Punishment: ‘It is nice, but in one of the chapters the author made a mistake. He describes the sun as rising twice on the same day.’.
Update I Neal Stephenson and Cory Doctorow’s book tour for Hieroglyph: Stories and Visions for a Better Future is finding to standing-room-only audiences here in Seattle and around the world. This book is associated to Arizona’s States’ well-known Heiroglyph project.
Update II Prof. Kathleen Ann Goonan (of Georgia Tech) contributed a story to Heiroglyph “Girl in wave: wave in girl” that — as I read her story — amounts to “QIST done right”
Update III The mathematical infelicities of the first editions of Cryptonomicon were corrected in later versions. Good on yah for scrupulous care in getting the math right, Neal Stephenson!
These works are specially commended to student members of the student-led MIT STEAM initiative
About Us MIT STEAM is a student-led effort to ignite communications between disparate fields in academia, business, and thought. Our focus is broad but our starting point is uniting the Arts with STEM (Science, Technology, Engineering, Mathematics).
Anyone is welcome, regardless of your major or background. We think that a diversity of backgrounds is essential to our success.
Summary Ample reasons exist to give Neal Stephenson/Heiroglyph/MIT STEAM another look, Scott and Shtetl Optimized readers!
—————-
Scott fulminates against Lockheed (#71) “The Lockheed thing is widely recognized as an absurdity, with the fingerprints of pointy-haired bosses all over it”
Just to clarify, Scott is (presumably) not fulminating against Lockheed’s multibillion-dollar investment in optical processors for synthetic aperture radar (1951-1980?). Because that investment was outstandingly successful.
And having succeeded once, why shouldn’t Lockheed try again to create transformative hugely-profitable technologies based upon non-digital computing technologies?
Update IV History-minded STEAM students are aware of at least five plausible respects in which the most advanced computing hardware of the 20th century was analog:
#1 Hannibal Ford and William Newell’s ship-born steering and fire-control computers of the 1920s-1950s (as celebrated in Norbert Wiener’s novel The Tempter).
#2 John von Neumann’s parametron computing technology (per his posthumous patent US2815488); see also the wartime computing engines of Gabriel Kron.
#3 The post-WWII optical computing engines used for synthetic aperture radar (capabilities still classified).
#4 The Adi Shamir/Weizman Institute TWINKLE and TWIRL factoring engines (computational capabilities unknown).
#5 Geordie Rose/D-Wave’s quantum-coherent computing engine (capabilities and technologies held privately).
Update V The history of computing technology provides ample reason to regard Aaronson/Arkhipov boson-sampling engines as the six — and least-secret! — in this historical series of analog-bettering-classical computing engines.
That is why — from a history-of-computing point-of-view — Scott Aaronson/Boson-sampling and Geordie Rose/D-Wave are natural allies in the quest to disprove the Extended Turing Thesis.
—————-
Scott fulminates against Gerard t’ Hooft’s ideas (#71) “If anyone other than ‘t Hooft had proposed it [post-Hilbert quantum dynamics], it would’ve been immediately laughed off the stage, which seems to me like the only sane reaction.”
Gerard ‘t Hooft’s web page “How to become a good theoretical physicist” is commended to Shtetl Optimized student-readers. In the event that following ‘t Hooft’s rigorous program of study leads to post-Hilbert dynamical models that some folks call “insane”, don’t blame me!
92. Simple Minded Says:
No matter how rigorous the program of study, developing an insane post-Hilbert quantum dynamical model shouldn’t be that difficult. 😉
93. Itai Says:
I do not understand what do you mean by post Hilbert ? And why is it considered insane ?
by wiki here , unless you think it is wrong (then go fix and give references) pilot wave is post Hilbert -the axiom about Hilbert space is not needed.
http://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory#Operators_as_observables
94. Scott Says:
Itai: I didn’t see any place in that Wikipedia article where it says that Bohmian mechanics doesn’t require a guiding potential that is, from a mathematical standpoint, a quantum state vector evolving unitarily in a Hilbert space, just like the state vector of ordinary quantum mechanics. It would indeed be strange if the article said that, since anyone who knows Bohmian mechanics knows that it’s not true.
95. John Sidles Says:
Itai remarks “I do not understand what do you mean by ‘post-Hilbert'”
As Dirac said in response to a similar remark: “That is not a question.”
Fortunately it points directly to a very good question “What ought we choose to mean by ‘post-Hilbert dynamics’? How can we have-fun/do-good-science/discover-good-math/create-great-enterprises/evolve-wonderful-narratives, starting with this question?”
The concluding section of Gerard ‘t Hooft’s recent preprint “The Cellular Automaton Interpretation of Quantum Mechanics. A View on the Quantum Nature of our Universe, Compulsory or Impossible?” (arXiv:1405.1548) provides a starting-point.
Numbers (1)-(3) were added by me:
Conclusions (p. 194) It may seem odd that our theory, unlike most other approaches, (1) does not contain any strange kinds of stochastic differential equation, no “quantum logic”, not an infinity of other universes, no pilot wave, (2) just completely ordinary equations of motion that we have hardly been able to specify, as they could be almost anything. Our most essential point is that (3) we should not be deterred by ‘no go theorems’ if these contain small print and emotional ingredients in their arguments.
As a preliminary remark, obviously it is neither necessary, nor feasible, nor even desirable that everyone think alike in regard to ‘t Hooft’s three points … indeed, it is most helpful that ‘t Hooft has stated them in broad terms.
Here are three concrete (hence non-unique) presentations of ‘t Hooft’s post-Hilbert principles:
(1) restrict the state-space The state-space of post-Hilbert dynamics can be a restriction of Hilbert-space to algebraic varieties … on the grounds that pretty much all quantum chemistry/condensed matter/quantum transport simulations already impose this restriction.
(2) restrict the Hamiltonians The dynamical flow of post-Hilbert dynamics can be a restriction of Hamiltonian symbol-functions to those provided by gauge theory … on the grounds, again, that pretty much all quantum chemistry/condensed matter/quantum transport simulations already impose this restriction.
(3) disregard no-go theorems and emotional appeals Consonant with restrictions (1) and (2), unravel the post-Hilbert trajectories of the broadest possible class of computing and metrology devices — including but not limited to the six device-classes of #91 — without regard for no-go postulates.
For example, the postulated hardness of scattershot boson-sampling simulation should not deter us from simulating this class of device.
In regard to t’Hooft’s recommendation that we be wary of “emotional ingredients”, the concluding statement of the slides that Scott provides for his (wonderfully enjoyable) talk “When exactly do quantum computers provide a speedup?” provides a concrete example:
An emotional quantum ingredient The single most important application of QC: Disproving the people who said QC was impossible!
Needless to say, quantum discourse would be a whole lot less fun without the passionate emotional appeal of Scott’s lecture. But a little goes a long way, and hence emotional ingredients need to be balanced against dispassionate ingredients. One such ingredient might be:
A dispassionate quantum ingredient The single most important application of QC: Achieving a thorough understanding what already is technologically feasible (and infeasible) by quantum dynamics, and by this path conceiving new possibilities (and impossibilities) in 21st century enterprise.
As a concluding unifying tribute and celebration of the spirit of Shtetl Optimized, and the spirit of Gerard t Hooft too, I’m going to close this comment with something that I seldom undertake: a concrete falsifiable prediction.
Here it is:
The Permanent Entropy Postulate No universe governed by long-range gauge theories can experimentally sample a permanent distribution at lower entropy-cost than indistinguishably simulating that distribution by a classical computation.
By far the most significant aspect of this postulate (as it seems to me) is that we can all be confident of learning a lot about this postulate in coming years.
We all of us owe appreciation and thanks to Scott Aaronson and Alex Arkipov (and Gerard t Hooft too) for this wonderful learning opportunity, and that is the main point that this comment attempts to convey.
——-
PS Can we please stop talking about “pilot waves”? Neither Scott nor t Hooft nor me nor anyone here on Shtetl Optimized has a high regard to them!
96. James Cross Says:
Any chance you might comment on Nick Bostrom’s Superintelligence book?
97. Itai Says:
I and Wikipedia article never said Bohmian mechanics doesn’t need a guiding potential ,
Matrix and wave formulation of QM was proved to be equivalent , I’m not sure who did a “correct proof” yet , Schrodinger and Pauli did not complete a correct proof, and maybe Von Newman ultimately proof the equivalence – never seen the full proof , got my references here http://www.lajpe.org/may08/09_Carlos_Madrid.pdf
I hope he had no mistake in the proof too as with the no-go theorem about hidden variables ).
But, I have never seen same equivalent proof to the Bohm wave theory ( he has no distinction between observables and states , and the only “operator” is the Hamiltonian ).
So , Unless such a proof exists , you can not “attack” pilot wave with arguments from matrix mechanics such as unitary evolution on Hilbert space and so on.
It’s not that I like so much Bohmian mechanics , but I want to distinguish what is real criticism and what is not.
98. Raoul Ohio Says:
In fourth place on the list of widely accepted physics facts that RO does not think are obviously true is the existence of the Higgs boson. There seems to be an explosion of news today about others who don’t think the case is closed quite yet.
You might want to put a hold on that Nobel Prize for right now.
99. Ben Standeven Says:
@Job #76:
Ah, OK. Your plan won’t work, because the classical machine is only allowed to make polynomially many queries to the computation graph of the oracle; but the oracle’s computation could be arbitrarily long.
@Itai #97:
The proof you’re looking for is just the one you quoted. Since matrix and wave formulations are equivalent, it does not matter whether the Bohm particle is guided by a wave or by a matrix; its trajectories are the same either way. Even if the proof is wrong, this would simply mean that there are two versions of Bohmian mechanics; one with a wave and one with a matrix.
@Sidles: #96?:
obviously it is neither necessary, nor feasible, nor even desirable that everyone think alike in regard to pilot waves…
I should have noticed this before; but using subvarieties of a Hilbert space means using the Nullstellensatz to calculate observables; an EXPSPACE-complete problem in general. So you aren’t really getting any speedup this way.
100. John Sidles Says:
Itai remarks “It’s not that I like so much Bohmian mechanics , but I want to distinguish what is real criticism and what is not.”
It’s not clear what traits distinguish “real” criticism, e.g. from “complex” criticism … but five weighty considerations are:
(1) No new good algorithms have arisen from Bohmian dynamics.
(2) No broadly useful simulations have been based upon Bohmian dynamics.
(3) No new scientific instruments have been designed with the aid of insights from Bohmian dynamics.
(4) No substantial scientific discoveries have been stimulated by insights from Bohmian dynamics.
(5) No prosperous mathematical disciplines have arisen as abstractions of Bohmian dynamics.
These reasons help us understand why algorithm/application-oriented quantum dynamicists like Linus Pauling, John Marcus, John Pople and Walter Kohn have won Nobel Prizes … and David Bohm didn’t.
Looking ahead, at least three-of-four 2014 Fields Medalists are doing work that broadly relates to post-Hilbert dynamics in the t Hooftian sense (of #91 and #95)
•$$\$$Maryam$$\$$Mirzakhani: hyperbolic geometry $$\Leftrightarrow$$ ergodic dynamical flows.
•$$\$$Martin$$\$$Hairer: stochastic trajectories $$\Leftrightarrow$$ Lindblad-Carmichael unravellings on varietal state-spaces.
•$$\$$Artur$$\$$Avila: nonlinear stochastic PDE $$\Leftrightarrow$$ quantum metrology triangles and quantum transport dynamics
The overlap of cutting-edge technology with cutting-edge mathematics is responsible for the burgeoning vigor of t Hooftian post-Hilbert dynamics.
In contrast, it’s not evident to most folks (including Scott, t Hooft, and me too) that Bohmian dynamics is providing comparably substantial inspiration to young mathematicians.
Question Why should young researchers focus on Bohmian dynamics, when so many other transformational 21st century research and enterprise opportunities are presenting themselves, across the entire STEAM spectrum? The world wonders!
Summary The prospects of post-Hilbert dynamics are bright … but (seemingly) not Bohmian.
101. Scott Says:
James #96:
Any chance you might comment on Nick Bostrom’s Superintelligence book?
Yes, there’s a chance. 🙂 I read it and found it thought-provoking, but it would take some time to gather the provoked thoughts.
102. Scott Says:
Itai #97: Bohmian mechanics is empirically indistinguishable from ordinary quantum mechanics because Bohm defined it that way. The equation that governs the time-evolution of the guiding potential is the Schrödinger equation; and the equation that governs the time-evolution of the particle positions is designed so that, if it obeys the |ψ|2 probability rule at any one time, then it also obeys it at all other times. QED; that’s the complete equivalence proof.
Please keep in mind that, if this weren’t the case—i.e., if there were an empirical way to distinguish Bohmian mechanics from “standard” QM—then it’s extremely likely that Bohmian mechanics would’ve already been ruled out by experiments, which have confirmed QM (insofar as it talks about a few particles at a time) to staggering precision.
103. John Sidles Says:
Ben Standeven remarks [correctly but narrowly] “Using subvarieties of a Hilbert space means using the Nullstellensatz to calculate observables; an EXPSPACE-complete problem in general. So you aren’t really getting any speedup this way.”
As Captain Picard’s borgified persona Locutus once remarked: A$$\$$narrow vision, Number$$\$$One!
Question Shall we abandon Dantzig’s simplex algorithm, because it is EXPTIME in general? Shall we abandon Kohn-Sham methods because — let’s face it! — our understanding of how these quantum simulation methods work is highly imperfect?
The t$$\$$Hooft Alternative Or should we embrace these methods, and seek to extend them, per the t$$\$$Hooftian maxim (quoted in #91) “We should not be deterred by no go theorems no speedup theorems“.
Summary Corporations like Lockheed Martin are exhibiting an experience-grounded and mathematically well-founded t$$\$$Hooftian appreciation that transformational enterprises can be founded — indeed commonly have been founded — upon computational hardware that affords “only” polynomial speedups, and dynamical simulation algorithms whose workings formally are EXPTIME or even mysterious outright.
Conclusion Yet another wonderful title for a 21st century STEAM-saga would be The `t$$\$$Hooft Alternative.
104. James Cross Says:
Scott #101
I thought you would be interested.
Hoping to see your take on it.
105. Ben Standeven Says:
@Sidles, 103:
Algorithms like Dantzig’s or Grobner’s work fine on “native” linear/polynomial programming problems. But if you take a hard computational problem [or a random problem which is hard-on-average], and translate that into the appropriate format, they don’t offer any speedup. In fact, it is harder to solve the problems that way, since any simulation introduces some overhead. Likewise, we should expect quantum simulation algorithms to fast on “natural” quantum systems, but not very useful on systems which are designed to do hard computations by using the rules of quantum mechanics.
|
2021-12-01 16:18:33
|
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https://qubit.guide/11.3-fidelity.html
|
## 11.3 Fidelity
Sometimes, when quantifying closeness of states, the inner product is a more convenient tool than the distance/norm. Analogous to how we define the distance between states |u\rangle and |v\rangle as d(u,v)=\|u-v\|, we define the fidelity between them as F(u,v)\coloneqq |\langle u|v\rangle|^2. This is not a metric, but it does have some similarly nice properties: for example, F(u,v)=1 when the two states are identical, and F(u,v)=0 when the two states are orthogonal (which means that they are “as different as possible”). Intuitively, we can understand fidelity as the probability that the state |u\rangle (resp. |v\rangle) would pass a test for being in state |v\rangle (resp. |u\rangle). In other words, if we perform an orthogonal measurement on |u\rangle that has two outcomes (\texttt{true} if the state is |v\rangle; \texttt{false} if the state is orthogonal to |v\rangle), then the fidelity F(u,v)=|\langle u|v\rangle|^2 is exactly the probability that we measure the outcome \texttt{true}.
Recall our definition of state distance: d(u,v) = \sqrt{2(1-|\langle u|v\rangle|)} This gives us a relation between distance and fidelity: once we know one, we can easily calculate the other. However, everything we have said so far applies only to pure states — we will see how the mixed state case is slightly more complicated shortly.
One final remark: as another example of the many inconsistencies in the literature, some authors define F(u,v) to be |\langle u|v\rangle| instead of |\langle u|v\rangle|^2. Whenever we say fidelity, we mean the latter: |\langle u|v\rangle|^2.
|
2022-12-01 23:54:14
|
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|
https://math.stackexchange.com/questions/2668535/how-to-find-the-rank-of-a-sudoku-without-row-reduction
|
# How to find the rank of a sudoku without row reduction
Furthermore, is a sudoku always of full rank? When is it full rank and when is it not? I realized these questions are quite out there and there may not be any meaningful answer to this.
• It is definitely possible for a $4\times4$ "sudoku" to either have full rank or not. For example, $\left(\begin{matrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \\ 3 & 4 & 1 & 2 \\ 2 & 1 & 4 & 3 \end{matrix}\right)$ has rank 3, while $\left(\begin{matrix} 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \\ 4 & 3 & 1 & 2 \\ 2 & 1 & 4 & 3 \end{matrix}\right)$ has rank 4. I would assume the same is likely true of actual $9\times9$ sudoku matrices... – Micah Feb 27 '18 at 5:06
• I suspect a sudoku is of full rank iff it has a unique solution. – Moriarty Feb 27 '18 at 5:23
• @Moriarty: A filled $9\times9$ matrix according to the sudoku rules and without empty cells is a unique solution. I looked at this example and found both alternative solutions to have rank $9$. – Axel Kemper Feb 27 '18 at 18:58
• @AxelKemper how did come upon the proof in your first sentence? – L to the V Feb 28 '18 at 2:58
• A sudoku puzzle is a partially filled grid or board. Unique solution means that there is only one way to fill the empty cells and reach the solution as completely filled grid. As the rank can only be computed for a complete grid matrix, a grid without empty cells represents a unique solution. The link in my comment above shows a solution which can be transformed into another solution by swapping four cells. But I still would call each of these solutions unique. – Axel Kemper Feb 28 '18 at 8:13
Using the MiniZinc constraint solver and Chuffed as solver back-end, I found the following rank $8$ sudoku:
9 1 6 2 7 4 5 3 8
2 7 5 8 6 3 4 9 1
8 4 3 5 1 9 6 2 7
6 2 9 1 3 7 8 5 4
4 5 1 6 2 8 9 7 3
7 3 8 4 9 5 2 1 6
3 8 7 9 5 6 1 4 2
1 9 4 7 8 2 3 6 5
5 6 2 3 4 1 7 8 9
Another rank 8 solution found using the MiniZinc G12 lazyfd solver back-end:
3 2 1 7 5 6 8 9 4
4 7 9 2 1 8 3 5 6
8 6 5 9 4 3 2 1 7
5 3 2 1 6 9 4 7 8
1 9 8 3 7 4 5 6 2
7 4 6 8 2 5 1 3 9
2 5 4 6 9 1 7 8 3
9 1 3 4 8 7 6 2 5
6 8 7 5 3 2 9 4 1
My model (based on this example):
int: s = 3;
int: n = s*s;
set of int: S = 1..s;
set of int: N = 1..n;
% sudoku board is a n x n = s² x s² matrix:
array[N, N] of var N: puzzle;
% column multipliers for full-rank determination
% extra small range chosen to speed-up search
int: CMDIM = 1;
array[N] of var -CMDIM .. CMDIM: colMults;
include "alldifferent.mzn";
% All cells in a row, in a column, and in a subsquare are different.
constraint
forall(i in N)( alldifferent(j in N)( puzzle[i,j] ))
/\
forall(j in N)( alldifferent(i in N)( puzzle[i,j] ))
/\
forall(i, j in S)
( alldifferent(p,q in S)( puzzle[s*(i-1)+p, s*(j-1)+q] ));
% additional constraint to enforce non-full rank for puzzle matrix
% Iff a linear combination of columns sums up to the null column,
% we don't have full rank
constraint forall(j in N) (
0 = sum([colMults[i] * puzzle[j, i] | i in N])
);
% exclude trivial solution with all-zero mutipliers
constraint exists([colMults[i] != 0 | i in N]);
solve satisfy;
function string: it(bool: cond, string: yes_s) =
if cond then yes_s else "" endif;
function string: ite(bool: cond, string: yes_s, string :no_s) =
if cond then yes_s else no_s endif;
output [ "sudoku:\n" ] ++
[ show(puzzle[i,j]) ++
if j = n then
ite(i mod s = 0 /\ i < n, "\n\n", "\n")
else
ite(j mod s = 0, " ", " ")
endif
| i,j in N ] ++
["\n rank {"] ++
[ it(j = 1, "{") ++
show(puzzle[i,j]) ++
it(j < n, ",") ++
it(j == n, "}") ++
it((j == n) /\ (i < n), ",")
| i,j in N ] ++
["}"] ++
["\nColumn Multipliers: "] ++
[show(colMults[i]) ++ " " | i in N];
• What's the highest or lowest rank you e found? – L to the V Feb 27 '18 at 16:44
• No rank below 8 yet. Most 9x9 sudokus certainly have full rank 9. – Axel Kemper Feb 27 '18 at 16:54
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2019-07-24 02:09:05
|
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|
https://manual.gromacs.org/2019.1/release-notes/2019/2019.1.html
|
# GROMACS 2019.1 release notes¶
This version was released on February 15, 2019. These release notes document the changes that have taken place in GROMACS since the initial version 2019, to fix known issues. It also incorporates all fixes made in version 2018.5 and earlier, which you can find described in the Release notes.
# Fix error with 2D/3D dynamic load balancing¶
With 2D or 3D domain decomposition with dynamics load balancing, mdrun would exit with the error “The domain decomposition grid as shifted too much ..” when a cell size was limited.
Issue 2830
# Fix incorrect LJ repulsion force switching on GPUs¶
When using a CUDA or OpenCL GPU, the coefficient for the second order term for the LJ repulsion in the force switching function, called ‘A’ in the manual, had the wrong sign. This lead to very small errors in the forces and the pressure. Note that the dispersion force switching was correct. Although the effect on individual atoms pairs was negligible, their combined effect on the pressure could lead to deformation of CHARMM membrane systems, where LJ force switching is regularly applied.
Issue 2845
Issue 2813
# Fix segmentation fault with energy minimization with the group scheme¶
Using energy minimization in combination with the group cutoff scheme and domain decomposition could lead to a segmentation fault.
Issue 2813
# Correct free-energy Delta H output with mass lambda’s¶
When separate lambda parameters were used for perturbed mass free-energy contributions, these contributions were double counted in the Delta H output used for BAR calculations. Note that dH/dlambda was always correct
# Prevent mdrun -rerun from writing incorrect free-energy output¶
Now mdrun -rerun exits with a fatal error when masses or constraints are perturbed. Their contributions to Hamiltonian differences and derivatives were incorrectly set to zero in version 2019.
Issue 2849
Issue 1431
# Fix trjconv -ndec¶
This only works for writing .xtc files. The code and documentation now works correctly with .gro files, which was changed in 2016 release series so that it would only write fixed-width columns.
# Fix using index file groups when .tpr file not supplied¶
Selections that use groups from a supplied index file can again be used even when a .tpr file is not supplied.
Issue 2847
# Fix tune_pme¶
The tool did not work due to a file reading error that is fixed now.
Issue 2827
# With MSVC, disabled internal clFFT fallback used for OpenCL support¶
GROMACS requires MSVC 2017, and the GROMACS OpenCL build requires clFFT. If clFFT is found on the user’s system, then all may be well, but the version of clFFT bundled within GROMACS cannot be built because only MSVC 2010 is supported by clFFT at this time. A configure-time fatal error is now issued in this case.
Issue 2500
# Explicitly require 64-bit platforms for OpenCL¶
A 64-bit OpenCL runtime is required by GROMACS. All known OpenCL implementations on 64-bit platforms are 64-bit (and there are no known 32-bit platforms with 64-bit OpenCL), hence we require a 64-bit platform at configure-time in OpenCL builds. A known unsupported 32-bit platform is ARMv7.
|
2021-01-18 02:16:52
|
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|
https://socratic.org/questions/how-do-you-solve-x-2-1-5-y-x-y-y-4-5-x-5-3
|
# How do you solve [(x^2+1, 5-y), (x+y,y-4)]=[(5,x), (5,3)]?
Jan 18, 2017
$\left(x , y\right) = \left(- 2 , 7\right)$
#### Explanation:
$\left[\left({x}^{2} + 1 , \textcolor{w h i t e}{\text{x"),5-y),(x+y,,y-4)]=[(5,color(white)("x}} , x\right) , \left(5 , , 3\right)\right]$
$\Rightarrow$
[1]$\textcolor{w h i t e}{\text{XX}} {x}^{2} + 1 = 5$
[2]$\textcolor{w h i t e}{\text{XX}} 5 - y = x$
[3]$\textcolor{w h i t e}{\text{XX}} x + y = 5$
[4]$\textcolor{w h i t e}{\text{XX}} y - 4 = 3$
Examining the above,
we note that both [1] and [4] only involve a single variable
and of the two [4] would appear to be the easiest to solve:
$y - 4 = 3 \textcolor{w h i t e}{\text{XX")rarrcolor(white)("XX}} y = 7$
We can now substitute $7$ for $y$ back into [2]
$5 - 7 = x \textcolor{w h i t e}{\text{XX")rarrcolor(white)("XX}} x = - 2$
We now have our solution $\left(x , y\right) = \left(- 2 , 7\right)$
provided this is consistent with the other two equations:
[1]$\textcolor{w h i t e}{\text{XX}} {x}^{2} + 1$ with $x = - 2 \textcolor{w h i t e}{\text{x")rarrcolor(white)("x}} {2}^{2} + 1 = 5$ (correct)
[3]$\textcolor{w h i t e}{\text{XX}} x + y = 5$ with $x = - 2$ and $y = 7 \textcolor{w h i t e}{\text{x")rarrcolor(white)("X}} - 2 + 7 = 5$ (correct)
So $\left(x , y\right) = \left(- 2 , 7\right)$ is a consistent (valid) solution.
|
2020-01-24 10:37:50
|
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|
https://sandipanweb.wordpress.com/category/image-processing/
|
# Solving Some Image Processing and Computer Vision Problems with Python libraries
In this article, a few image processing / computer vision problems and their solutions with python libraries (scikit-image, cv2) will be discussed. Some of the problems are from the exercises from this book (available on amazon). This blog will be continued here.
## Removing Gaussian Noise from images by computing mean and median images with scikit-image
1. Start with an input image.
2. Create n (e.g, n=100) noisy images by adding i.i.d. Gaussian noise (with zero mean) to the original image, with scikit-image.
3. Compute the mean (median) of the noisy images.
4. Compare PSNR with the original image.
5. Vary n and compare the results.
from skimage import img_as_float
from skimage.util import random_noise
from skimage.measure import compare_psnr
from skimage.io import imread
import matplotlib.pylab as plt
import numpy as np
im = img_as_float(imread('../new images/parrot.jpg')) # original image
np.random.seed(0)
# generate n noisy images from the original image by adding Gaussian noise
n = 25
images = np.zeros((n, im.shape[0], im.shape[1], im.shape[2]))
sigma = 0.2
for i in range(n):
images[i,...] = random_noise(im, var=sigma**2)
im_mean = images.mean(axis=0)
im_median = np.median(images, axis=0)
plt.figure(figsize=(20,16))
plt.subplots_adjust(left=.02, right=.98, bottom=.001, top=.96, wspace=.05, hspace=.01)
plt.subplot(221), plt.imshow(im), plt.axis('off'), plt.title('Original image', size=20)
plt.subplot(222), plt.imshow(images[0]), plt.axis('off'), plt.title('Noisy PSNR: ' + str(compare_psnr(im, images[0])), size=20)
plt.subplot(223), plt.imshow(im_mean), plt.axis('off'), plt.title('Mean PSNR: ' + str(compare_psnr(im, im_mean)), size=20)
plt.subplot(224), plt.imshow(im_median), plt.axis('off'), plt.title('Median PSNR: ' + str(compare_psnr(im, im_median)), size=20)
plt.show()
The next figure shows the original image, a noisy image generated from it by adding Gaussian noise (with 0 mean) to it and the images obtained by taking mean / median over all the n noisy images generated. As can be seen, the Gaussian noise in the images gets cancelled out by taking mean / median.
with n = 25
with n=100
plt.hist(images[:,100,100,0], color='red', alpha=0.2, label='red')
plt.hist(images[:,100,100,1], color='green', alpha=0.2, label='green')
plt.hist(images[:,100,100,2], color='blue', alpha=0.2, label='blue')
plt.legend()
plt.grid()
plt.show()
The next figure shows how a pixel value (that can be considered a random variable) for a particular location in different noisy images follows approximately a Gaussian distribution.
Distribution of a pixel value at location (100,100) in the noisy images
ns = [25, 50, 100, 200]
# mean_psnrs contain the PSNR values for different n
plt.plot(ns, mean_psnrs, '.--', label='PSNR (mean)')
plt.plot(ns, median_psnrs, '.--', label='PSNR (median)')
plt.legend()
plt.xlabel('n'), plt.ylabel('PSNR')
plt.show()
The following figure shows that the PSNR improves with large n (since by SLLN / WLLN, the sample mean converges to population mean 0 of the Gaussian noise). Also, for median the improvement in the image quality is higher for larger values of n.
## Tracking Pedestrians with HOG-SVM with OpenCV / scikit-image
1. Start with a video with pedestrians.
2. Capture the video / extract frames from the video.
3. For each frame
1. Create HOG scale pyramid of the frame image.
2. At each scale, use a sliding window to extract the corresponding block from the frame, compute the HOG descriptor features.
3. Use cv2‘s HOGDescriptor_getDefaultPeopleDetector() – a pre-trained SVM classifier on the HOG descriptor to classify whether the corresponding block contains a pedestrian or not.
4. Run non-max-suppression to get rid of multiple detection of the same person.
5. Use cv2‘s detectMultiScale() function to implement steps 3-4.
The code is adapted from the code here and here.
# HOG descriptor using default people (pedestrian) detector
hog = cv2.HOGDescriptor()
hog.setSVMDetector(cv2.HOGDescriptor_getDefaultPeopleDetector())
# run detection, using a spatial stride of 4 pixels,
# a scale stride of 1.02, and zero grouping of rectangles
# (to demonstrate that HOG will detect at potentially
# multiple places in the scale pyramid)
(foundBoundingBoxes, weights) = hog.detectMultiScale(frame, winStride=(4, 4), padding=(8, 8), scale=1.02, finalThreshold=0, useMeanshiftGrouping=False)
# convert bounding boxes from format (x1, y1, w, h) to (x1, y1, x2, y2)
rects = np.array([[x, y, x + w, y + h] for (x, y, w, h) in foundBoundingBoxes])
# run non-max suppression on the boxes based on an overlay of 65%
nmsBoundingBoxes = non_max_suppression(rects, probs=None, overlapThresh=0.65)
cv2 functions are used to extract HOG descriptor features and pedestrian detection with SVM, whereas scikit-image functions are used to visualize the HOG features. The animations below display the original video, what HOG sees and the detected pedestrians after non-max suppression. Notice there are a few false positive detection.
Original Video
HOG-descriptor features video (what HOG sees)Original Video with detected Pedestrians
## Face Detection with HaarCascade pre-trained AdaBoost classifiers with OpenCV
1. Capture video with webcam with cv2.VideoCapture().
2. For each frame, use the pre-trained Adaboost Cascade classifiers (the haarcascade_frontalface_default classifier for face detection and haarcascade_eye_tree_eyeglasses classifier for better detection of the eyes with glasses, from the corresponding xml files that come with cv2’s installation) using Haar-like features with cv2.CascadeClassifier().
3. First detect the face(s) with the detectMultiScale() function and draw a bounding box. Then detect the eyes inside a detected face with the same function.
4. The following python code snippet shows how to detect faces and eyes with cv2. The code is adapted from here.
# read the cascade classifiers from the xml files from the correct path into face_cascade # and eye_cascade
gray = cv2.cvtColor(frame, cv2.COLOR_BGR2GRAY)
frame = cv2.cvtColor(frame, cv2.COLOR_BGR2RGB)
# return bounding box of the face(s) if one is detected
faces = face_cascade.detectMultiScale(gray, 1.03, 5)
for (x,y,w,h) in faces:
frame = cv2.rectangle(frame,(x,y),(x+w,y+h),(255,0,0),2)
roi_gray = gray[y:y+h, x:x+w]
roi_color = frame[y:y+h, x:x+w]
eyes = eye_cascade.detectMultiScale(roi_gray)
for (ex,ey,ew,eh) in eyes:
cv2.rectangle(roi_color,(ex,ey),(ex+ew,ey+eh),(0,255,0),2)
The next animation shows the results of face detection when scalefactor 1.03 was used to create the scale pyramid. As can be seen, the eyes with the glasses on and some small faces from the photos are not detected at this scale.
The next animation shows the results of face detection when scalefactor 1.3 was used to create the scale pyramid. As can be seen, the eyes with/without the glasses on as well as most of the small faces from the photos are detected at this scale most of the time.
## Semantic Segmentation with ENet / DeepLab (Deep Learning model)
Input video and the segmented Output video
Input video and the segmented Output video
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# Solving Some Image Processing Problems with Python libraries
In this article a few popular image processing problems along with their solutions are going to be discussed. Python image processing libraries are going to be used to solve these problems. Some of the problems are from the exercises from this book (available on amazon).
## Image Transformations and Warping
### 0. Display RGB image color channels in 3D
1. A gray-scale image can be thought of a 2-D function f(x,y) of the pixel locations (x,y), that maps each pixel into its corresponding gray level (an integer in [0,255], e.g.,).
2. For an RGB image there are 3 such functions, f_R(x,y), f_G(x.y), f_B(x.y).
3. matplotlib’s 3-D plot functions can be used to plot each function.
The following python code shows how to plot the RGB channels separately in 3D:
import matplotlib.pylab as plt
from mpl_toolkits.mplot3d import Axes3D
def plot_3d(X, Y, Z, title, cmap):
# implement this function to plot the channel pixel values in 3D
plt.show()
im = imread('../images/parrot.jpg')
Y = np.arange(im.shape[0])
X = np.arange(im.shape[1])
Z1 = im[...,0]
Z2 = im[...,1]
Z3 = im[...,2]
plot_3d(X, Y, Z1, cmap='Reds', title='3D plot for the Red Channel')
plot_3d(X, Y, Z2, cmap='Greens', title='3D plot for the Green Channel')
plot_3d(X, Y, Z3, cmap='Blues', title='3D plot for the Blue Channel')
The RGB image
### 1. Wave Transform
1. Use scikit-image’s warp() function to implement the wave transform.
2. Note that wave transform can be expressed with the following equations:
We shall use the madrill image to implement the wave transform. The next python code fragment shows how to do it:
def wave(xy):
xy[:, 1] += 20*np.sin(2*np.pi*xy[:, 0]/64)
return xy
from skimage.io import imread
from skimage.transform import warp
import matplotlib.pylab as plt
im = imread('images/mandrill.jpg')
im = warp(im, wave)
plt.imshow(im)
plt.show()
The next figure shows the original mandrill input image and the output image obtained after applying the wave transform.
### 2. Swirl Transform
1. Use scikit-image’s warp() function to implement the swirl transform.
2. Note that swirl transform can be expressed with the following equations
We shall use the madrill image to implement the wave transform. The next python code fragment shows how to do it:
def swirl(xy, x0, y0, R):
r = np.sqrt((xy[:,1]-x0)**2 + (xy[:,0]-y0)**2)
a = np.pi * r / R
xy[:, 1] = (xy[:, 1]-x0)*np.cos(a) + (xy[:, 0]-y0)*np.sin(a) + x0
xy[:, 0] = -(xy[:, 1]-x0)*np.sin(a) + (xy[:, 0]-y0)*np.cos(a) + y0
return xy
im = imread('../images/mandrill.jpg')
im = warp(im, swirl, map_args={'x0':112, 'y0':112, 'R':512})
plt.imshow(im)
plt.axis('off')
plt.show()
The next figure shows the original mandrill input image and the output image obtained after applying the swirl transform.
Compare this with the output of the scikit-image swirl() function.
### 3. Very simple Face morphing with α-blending
1. Start from one face image (e.g., let image1 be the face of Messi) and end into another image (let image2 be the face of Ronaldo) iteratively, creating some intermediate images in between.
2. At each iteration create an image by using a linear combination of the two image numpy ndarrays given by
3. Iteratively increase α from 0 to 1.
The following code block shows how to implement it using matplotlib’s image and pylab modules.
im1 = mpimg.imread("../images/messi.jpg") / 255 # scale RGB values in [0,1]
im2 = mpimg.imread("../images/ronaldo.jpg") / 255
i = 1
plt.figure(figsize=(18,15))
for alpha in np.linspace(0,1,20):
plt.subplot(4,5,i)
plt.imshow((1-alpha)*im1 + alpha*im2)
plt.axis('off')
i += 1
plt.subplots_adjust(wspace=0.05, hspace=0.05)
plt.show()
The next animation shows the simple face morphing:
There are more sophisticated techniques to improve the quality of morphing, but this is the simplest one.
### 4. Creating Instagram-like Gotham Filter
#### The Gotham filter
The Gotham filter is computed as follows (the steps taken from here), applying the following operations on an image, the corresponding python code, input and output images are shown along with the operations (with the following input image):
1. A mid-tone red contrast boost
from PIL import Image
import numpy as np
import matplotlib.pylab as plt
im = Image.open('../images/city.jpg') # pixel values in [0,255]
r, g, b = im.split()
red_levels = [0., 12.75, 25.5, 51., 76.5, 127.5, 178.5, 204., 229.5, 242.25, 255.]
r1 = Image.fromarray((np.reshape(np.interp(np.array(r).ravel(), np.linspace(0,255,len(red_levels)), red_levels), (im.height, im.width))).astype(np.uint8), mode='L')
plt.figure(figsize=(20,15))
plt.subplot(221)
plt.imshow(im)
plt.title('original', size=20)
plt.axis('off')
plt.subplot(222)
im1 = Image.merge('RGB', (r1, g, b))
plt.imshow(im1)
plt.axis('off')
plt.title('with red channel interpolation', size=20)
plt.subplot(223)
plt.hist(np.array(r).ravel(), normed=True)
plt.subplot(224)
plt.hist(np.array(r1).ravel(), normed=True)
plt.show()
2. Make the blacks a little bluer
plt.figure(figsize=(20,10))
plt.subplot(121)
plt.imshow(im1)
plt.title('last image', size=20)
plt.axis('off')
b1 = Image.fromarray(np.clip(np.array(b) + 7.65, 0, 255).astype(np.uint8))
im1 = Image.merge('RGB', (r1, g, b1))
plt.subplot(122)
plt.imshow(im1)
plt.axis('off')
plt.title('with transformation', size=20)
plt.tight_layout()
plt.show()
3. A small sharpening
from PIL.ImageEnhance import Sharpness
plt.figure(figsize=(20,10))
plt.subplot(121)
plt.imshow(im1)
plt.title('last image', size=20)
plt.axis('off')
im2 = Sharpness(im1).enhance(3.0)
plt.subplot(122)
plt.imshow(im2)
plt.axis('off')
plt.title('with transformation', size=20)
plt.tight_layout()
plt.show()
4. A boost in blue channel for lower mid-tones
5. A decrease in blue channel for upper mid-tones
blue_levels = [0., 11.985, 30.09, 64.005, 81.09, 99.96, 107.1, 111.945, 121.125, 143.055, 147.9, 159.885, 171.105, 186.915, 215.985, 235.875, 255.]
b2 = Image.fromarray((np.reshape(np.interp(np.array(b1).ravel(), np.linspace(0,255,len(blue_levels)), blue_levels), (im.height, im.width))).astype(np.uint8), mode='L')
plt.figure(figsize=(20,15))
plt.subplot(221)
plt.imshow(im2)
plt.title('last image', size=20)
plt.axis('off')
plt.subplot(222)
im3 = Image.merge('RGB', (r1, g, b2))
plt.imshow(im3)
plt.axis('off')
plt.title('with blue channel interpolation', size=20)
plt.subplot(223)
plt.hist(np.array(b1).ravel(), normed=True)
plt.subplot(224)
plt.hist(np.array(b2).ravel(), normed=True)
plt.show()
The output image obtained after applying the Gotham filter is shown below:
## Down-sampling with anti-aliasing using Gaussian Filter
1. Start with a large gray-scale image and reduce the image size 16 times, by reducing both height and width by 4 times.
2. Select every 4th pixel in the x and the y direction from the original image to compute the values of the pixels in the smaller image.
3. Before down-sampling apply a Gaussian filter (to smooth the image) for anti-aliasing.
4. Compare the quality of the output image obtained by down-sampling without a Gaussian filter (with aliasing).
The next code block performs the above steps. Since the Gaussian blur is a low-pass filter, it removes the high frequencies from the original input image, hence it’s possible to achieve sampling rate above the Nyquist rate (by sampling theorem) to avoid aliasing.
from scipy.ndimage import gaussian_filter
im = rgb2gray(imread('images/umbc.png'))
print(im.shape)
plt.figure(figsize=(20,20))
plt.imshow(im)
plt.show()
plt.figure(figsize=(20,20))
im_blurred = gaussian_filter(im, sigma=2.5) #(5,5,1)
plt.imshow(im_blurred)
plt.show()
n = 4 # create and image 16 times smaller in size
w, h = im.shape[0] // n, im.shape[1] // n
im_small = np.zeros((w,h))
for i in range(w):
for j in range(h):
im_small[i,j] = im[n*i, n*j]
plt.figure(figsize=(20,20))
plt.imshow(im_small)
plt.show()
im_small = np.zeros((w,h))
for i in range(w):
for j in range(h):
im_small[i,j] = im_blurred[n*i, n*j]
plt.figure(figsize=(20,20))
plt.imshow(im_small)
plt.show()
Original Image
Image blurred with Gaussian Filter LPF
Down-sampled Image from the original image (with aliasing)
Down-sampled Image from the blurred image (with anti-aliasing)
## Some Applications of DFT
### 0. Fourier Transform of a Gaussian Kernel is another Gaussian Kernel
Also, the spread in the frequency domain inversely proportional to the spread in the spatial domain (known as Heisenberg’s inequality). Here is the proof:
The following animation shows an example visualizing the Gaussian contours in spatial and corresponding frequency domains:
### 1. Using DFT to up-sample an image
1. Let’s use the lena gray-scale image.
2. First double the size of the by padding zero rows/columns at every alternate positions.
3. Use FFT followed by an LPF.
4. Finally use IFFT to get the output image.
The following code block shows the python code for implementing the steps listed above:
import numpy as np
import numpy.fft as fp
import matplotlib.pyplot as plt
im = np.mean(imread('images/lena.jpg'), axis=2)
im1 = np.zeros((2*im.shape[0], 2*im.shape[1]))
print(im.shape, im1.shape)
for i in range(im.shape[0]):
for j in range(im.shape[1]):
im1[2*i,2*j] = im[i,j]
def padwithzeros(vector, pad_width, iaxis, kwargs):
vector[:pad_width[0]] = 0
vector[-pad_width[1]:] = 0
return vector
# the LPF kernel
kernel = [[0.25, 0.5, 0.25], [0.5, 1, 0.5], [0.25, 0.5, 0.25]]
# enlarge the kernel to the shape of the image
kernel = np.pad(kernel, (((im1.shape[0]-3)//2,(im1.shape[0]-3)//2+1), ((im1.shape[1]-3)//2,(im1.shape[1]-3)//2+1)), padwithzeros)
plt.figure(figsize=(15,10))
plt.gray() # show the filtered result in grayscale
freq = fp.fft2(im1)
freq_kernel = fp.fft2(fp.ifftshift(kernel))
freq_LPF = freq*freq_kernel # by the Convolution theorem
im2 = fp.ifft2(freq_LPF)
freq_im2 = fp.fft2(im2)
plt.subplot(2,3,1)
plt.imshow(im)
plt.title('Original Image', size=20)
plt.subplot(2,3,2)
plt.imshow(im1)
plt.title('Padded Image', size=20)
plt.subplot(2,3,3)
plt.imshow( (20*np.log10( 0.1 + fp.fftshift(freq))).astype(int), cmap='jet')
plt.title('Original Image Spectrum', size=20)
plt.subplot(2,3,4)
plt.imshow( (20*np.log10( 0.1 + fp.fftshift(freq_kernel))).astype(int), cmap='jet')
plt.title('Image Spectrum of the LPF', size=20)
plt.subplot(2,3,5)
plt.imshow( (20*np.log10( 0.1 + fp.fftshift(freq_im2))).astype(int), cmap='jet')
plt.title('Image Spectrum after LPF', size=20)
plt.subplot(2,3,6)
plt.imshow(im2.astype(np.uint8)) # the imaginary part is an artifact
plt.title('Output Image', size=20)
The next figure shows the output. As can be seen from the next figure, the LPF removed the high frequency components from the Fourier spectrum of the padded image and with a subsequent inverse Fourier transform we get a decent enlarged image.
### 2. Frequency Domain Gaussian Filter
1. Use an input image and use DFT to create the frequency 2D-array.
2. Create a small Gaussian 2D Kernel (to be used as an LPF) in the spatial domain and pad it to enlarge it to the image dimensions.
3. Use DFT to obtain the Gaussian Kernel in the frequency domain.
4. Use the Convolution theorem to convolve the LPF with the input image in the frequency domain.
5. Use IDFT to obtain the output image.
6. Plot the frequency spectrum of the image, the gaussian kernel and the image obtained after convolution in the frequency domain, in 3D.
The following code block shows the python code:
import matplotlib.pyplot as plt
from matplotlib import cm
from skimage.color import rgb2gray
from skimage.io import imread
import scipy.fftpack as fp
im = rgb2gray(imread('images/temple.jpg'))
kernel = np.outer(signal.gaussian(im.shape[0], 10), signal.gaussian(im.shape[1], 10))
freq = fp.fft2(im)
assert(freq.shape == kernel.shape)
freq_kernel = fp.fft2(fp.ifftshift(kernel))
convolved = freq*freq_kernel # by the Convolution theorem
im_blur = fp.ifft2(convolved).real
im_blur = 255 * im_blur / np.max(im_blur)
# center the frequency response
plt.imshow( (20*np.log10( 0.01 + fp.fftshift(freq_kernel))).real.astype(int), cmap='coolwarm')
plt.colorbar()
plt.show()
plt.figure(figsize=(20,20))
plt.imshow(im, cmap='gray')
plt.show()
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.ticker import LinearLocator, FormatStrFormatter
# ... code for 3D visualization of the spectrums
#### The original color temple image (time / spatial domain)
The temple image (frequency domain)
#### The smoothed temple image with the LPF (frequency domain)
If we set the standard deviation of the LPF Gaussian kernel to be 10 we get the following output as shown in the next figures. As can be seen, the frequency response value drops much quicker from the center.
#### The smoothed temple image with the LPF with higher s.d. (frequency domain)
The output image after convolution (spatial / time domain)
3. Using the inverse filter to restore a motion-blurred image
1. First create a motion blur kernel of a given shape.
2. Convolve the kernel with an input image in the frequency domain.
3. Get the motion-blurred image in the spatial domain with IDFT.
4. Compute the inverse filter kernel and convolve with the blurred image in the frequency domain.
5. Get the convolved image back in the spatial domain.
6. Plot all the images and kernels in the frequency domain.
The following code block shows the python code:
im = rgb2gray(imread('../images/madurga.jpg'))
# create the motion blur kernel
size = 21
kernel = np.zeros((size, size))
kernel[int((size-1)/2), :] = np.ones(size)
kernel = kernel / size
kernel = np.pad(kernel, (((im.shape[0]-size)//2,(im.shape[0]-size)//2+1), ((im.shape[1]-size)//2,(im.shape[1]-size)//2+1)), padwithzeros)
freq = fp.fft2(im)
freq_kernel = fp.fft2(fp.ifftshift(kernel))
convolved1 = freq1*freq_kernel1
im_blur = fp.ifft2(convolved1).real
im_blur = im_blur / np.max(im_blur)
epsilon = 10**-6
freq = fp.fft2(im_blur)
freq_kernel = 1 / (epsilon + freq_kernel1)
convolved = freq*freq_kernel
im_restored = fp.ifft2(convolved).real
im_restored = im_restored / np.max(im_restored)
plt.figure(figsize=(18,12))
plt.subplot(221)
plt.imshow(im)
plt.title('Original image', size=20)
plt.axis('off')
plt.subplot(222)
plt.imshow(im_blur)
plt.title('Blurred image with motion blur kernel', size=20)
plt.axis('off')
plt.subplot(223)
plt.imshow(im_restored)
plt.title('Restored image with inverse filter', size=20)
plt.axis('off')
plt.subplot(224)
plt.imshow(im_restored - im)
plt.title('Diff restored &amp;amp;amp;amp;amp; original image', size=20)
plt.axis('off')
plt.show()
# Plot the surface of the frequency responses here
Frequency response of the input image
(log) Frequency response of the motion blur kernel (LPF)
Input image convolved with the motion blur kernel (frequency domain)
(log) Frequency response of the inverse frequency filter kernel (HPF)
Motion-blurred image convolved with the inverse frequency filter kernel (frequency domain)
4. Impact of noise on the inverse filter
1. Add some random noise to the Lena image.
2. Blur the image with a Gaussian kernel.
3. Restore the image using inverse filter.
With the original image
Let’s first blur and apply the inverse filter on the noiseless blurred image. The following figures show the outputs:
(log) Frequency response of the input image
(log) Frequency response of the Gaussian blur kernel (LPF)
(log) Frequency response of the blurred image
(log) Frequency response of the inverse kernel (HPF)
Frequency response of the output image
Adding noise to the original image
The following python code can be used to add Gaussian noise to an image:
from skimage.util import random_noise
im = random_noise(im, var=0.1)
The next figures show the noisy lena image, the blurred image with a Gaussian Kernel and the restored image with the inverse filter. As can be seen, being a high-pass filter, the inverse filter enhances the noise, typically corresponding to high frequencies.
#### 5. Use a notch filter to remove periodic noise from the following half-toned car image.
1. Use DFT to obtain the frequency spectrum of the image.
2. Block the high frequency components that are most likely responsible fro noise.
3. Use IDFT to come back to the spatial domain.
from scipy import fftpack
im = imread('images/halftone.png')
F1 = fftpack.fft2((im).astype(float))
F2 = fftpack.fftshift(F1)
for i in range(60, w, 135):
for j in range(100, h, 200):
if not (i == 330 and j == 500):
F2[i-10:i+10, j-10:j+10] = 0
for i in range(0, w, 135):
for j in range(200, h, 200):
if not (i == 330 and j == 500):
F2[max(0,i-15):min(w,i+15), max(0,j-15):min(h,j+15)] = 0
plt.figure(figsize=(6.66,10))
plt.imshow( (20*np.log10( 0.1 + F2)).astype(int), cmap=plt.cm.gray)
plt.show()
im1 = fp.ifft2(fftpack.ifftshift(F2)).real
plt.figure(figsize=(10,10))
plt.imshow(im1, cmap='gray')
plt.axis('off')
plt.show()
Frequency response of the input image
Frequency response of the input image with blocked frequencies with notch
Output image
With a low-pass-filter (LPF):
Frequency response of the input image with blocked frequencies with LPF
Output image
## Histogram Matching with color images
As described here, here is the algorithm:
1. The cumulative histogram is computed for each image dataset, see the figure below.
2. For any particular value (xi) in the input image data to be adjusted has a cumulative histogram value given by G(xi).
3. This in turn is the cumulative distribution value in the reference (template) image dataset, namely H(xj). The input data value xi is replaced by xj.
im = imread('images/lena.jpg')
im_t = imread('images/vstyle.png')
im1 = np.zeros(im.shape).astype(np.uint8)
plt.figure(figsize=(20,10))
for i in range(3):
c = cdf(im[...,i])
c_t = cdf(im_t[...,i])
im1[...,i] = hist_matching(c, c_t, im[...,i]) # implement this function with the above algorithm
c1 = cdf(im1[...,i])
col = 'r' if i == 0 else ('g' if i == 1 else 'b')
plt.plot(np.arange(256), c, col + ':', label='input ' + col.upper(), linewidth=5)
plt.plot(np.arange(256), c_t, col + '--', label='template ' + col.upper(), linewidth=5)
plt.plot(np.arange(256), c1, col + '-', label='output ' + col.upper(), linewidth=2)
plt.title('CDF', size=20)
plt.legend(prop={'size': 15})
plt.show()
plt.figure(figsize=(10,10))
plt.imshow(im1[...,:3])
plt.axis('off')
plt.show()
Input image
Template Image
Output Image
The following figure shows how the histogram of the input image is matched with the histogram of the template image.
Another example:
Input image
Template Image
Output Image
## Mathematical Morphology
### 1. Automatically cropping an image
1. Let’s use the following image. The image has unnecessary white background outside the molecule of the organic compound.
2. First convert the image to a binary image and compute the convex hull of the molecule object.
3. Use the convex hull image to find the bounding box for cropping.
4. Crop the original image with the bounding box.
The next python code shows how to implement the above steps:
from PIL import Image
from skimage.io import imread
from skimage.morphology import convex_hull_image
im = imread('../images/L_2d.jpg')
plt.imshow(im)
plt.title('input image')
plt.show()
im1 = 1 - rgb2gray(im)
threshold = 0.5
im1[im1 threshold] = 1
chull = convex_hull_image(im1)
plt.imshow(chull)
plt.title('convex hull in the binary image')
plt.show()
imageBox = Image.fromarray((chull*255).astype(np.uint8)).getbbox()
cropped = Image.fromarray(im).crop(imageBox)
cropped.save('L_2d_cropped.jpg')
plt.imshow(cropped)
plt.title('cropped image')
plt.show()
This can also be found here.
### 2. Opening and Closing are Dual operations in mathematical morphology
1. Start with a binary image and apply opening operation with some structuring element (e.g., a disk) on it to obtain an output image.
2. Invert the image (to change the foreground to background and vice versa) and apply closing operation on it with the same structuring element to obtain another output image.
3. Invert the second output image obtained and observe that it’s same as the first output image.
4. Thus applying opening operation to the foreground of a binary image is equivalent to applying closing operation to the background of the same image with the same structuring element.
The next python code shows the implementation of the above steps.
from skimage.morphology import binary_opening, binary_closing, disk
from skimage.util import invert
im = rgb2gray(imread('../new images/circles.jpg'))
im[im 0.5] = 1
plt.gray()
plt.figure(figsize=(20,10))
plt.subplot(131)
plt.imshow(im)
plt.title('original', size=20)
plt.axis('off')
plt.subplot(1,3,2)
im1 = binary_opening(im, disk(12))
plt.imshow(im1)
plt.title('opening with disk size ' + str(12), size=20)
plt.axis('off')
plt.subplot(1,3,3)
im1 = invert(binary_closing(invert(im), disk(12)))
plt.imshow(im1)
plt.title('closing with disk size ' + str(12), size=20)
plt.axis('off')
plt.show()
As can be seen the output images obtained are exactly same.
## Floyd-Steinberg Dithering (to convert a grayscale to a binary image)
The next figure shows the algorithm for error diffusion dithering.
def find_closest_palette_color(oldpixel):
return int(round(oldpixel / 255)*255)
im = rgb2gray(imread('../my images/godess.jpg'))*255
pixel = np.copy(im)
w, h = im.shape
for x in range(w):
for y in range(h):
oldpixel = pixel[x][y]
newpixel = find_closest_palette_color(oldpixel)
pixel[x][y] = newpixel
quant_error = oldpixel - newpixel
if x + 1 &amp;amp;amp;amp;lt; w-1:
pixel[x + 1][y] = pixel[x + 1][y] + quant_error * 7 / 16
if x &amp;amp;amp;amp;gt; 0 and y &amp;amp;amp;amp;lt; h-1:
pixel[x - 1][y + 1] = pixel[x - 1][y + 1] + quant_error * 3 / 16
if y &amp;amp;amp;amp;lt; h-1:
pixel[x ][y + 1] = pixel[x ][y + 1] + quant_error * 5 / 16
if x &amp;amp;amp;amp;lt; w-1 and y &amp;amp;amp;amp;lt; h-1:
pixel[x + 1][y + 1] = pixel[x + 1][y + 1] + quant_error * 1 / 16
plt.figure(figsize=(10,20))
plt.imshow(pixel, cmap='gray')
plt.axis('off')
plt.show()
The input image (gray-scale)
The output Image (binary)
The next animation shows how an another input grayscale image gets converted to output binary image using the error diffusion dithering.
## Sharpen a color image
1. First blur the image with an LPF (e.g., Gaussian Filter).
2. Compute the detail image as the difference between the original and the blurred image.
3. Now the sharpened image can be computed as a linear combination of the original image and the detail image. The next figure illustrates the concept.
The next python code shows how this can be implemented in python:
from scipy import misc, ndimage
import matplotlib.pyplot as plt
import numpy as np
def rgb2gray(im):
return np.clip(0.2989 * im[...,0] + 0.5870 * im[...,1] + 0.1140 * im[...,2], 0, 1)
im = misc.imread('../my images/me.jpg')/255
im_blurred = ndimage.gaussian_filter(im, (5,5,0))
im_detail = np.clip(im - im_blurred, 0, 1)
fig, axes = plt.subplots(nrows=2, ncols=3, sharex=True, sharey=True, figsize=(15, 15))
axes = axes.ravel()
axes[0].imshow(im)
axes[0].set_title('Original image', size=15)
axes[1].imshow(im_blurred)
axes[1].set_title('Blurred image, sigma=5', size=15)
axes[2].imshow(im_detail)
axes[2].set_title('Detail image', size=15)
alpha = [1, 5, 10]
for i in range(3):
im_sharp = np.clip(im + alpha[i]*im_detail, 0, 1)
axes[3+i].imshow(im_sharp)
axes[3+i].set_title('Sharpened image, alpha=' + str(alpha[i]), size=15)
for ax in axes:
ax.axis('off')
fig.tight_layout()
plt.show()
The next figure shows the output of the above code block. As cane be seen, the output gets more sharpened as the value of alpha gets increased.
The next animation shows how the image gets more and more sharpened with increasing alpha.
## Edge Detection with LOG and Zero-Crossing Algorithm by Marr and Hildreth
The following figure shows LOG filter and its DOG approximation.
In order to detect edges as a binary image, finding the zero-crossings in the LoG-convolved image was proposed by Marr and Hildreth. Identification of the edge pixels can be done by viewing the sign of the LoG-smoothed image by defining it as a binary image, the algorithm is as follows:
Algorithm to compute the zero-crossing
1. First convert the LOG-convolved image to a binary image, by replacing the pixel values by 1 for positive values and 0 for negative values.
2. In order to compute the zero crossing pixels, we need to simply look at the boundaries of the non-zero regions in this binary image.
3. Boundaries can be found by finding any non-zero pixel that has an immediate neighbor which is is zero.
4. Hence, for each pixel, if it is non-zero, consider its 8 neighbors, if any of the neighboring pixels is zero, the pixel can be identified as an edge.
The next python code and the output images / animations generated show how to detect the edges from the zebra image with LOG + zero-crossings:
from scipy import ndimage, misc
import matplotlib.pyplot as plt
from scipy.misc import imread
from skimage.color import rgb2gray
def any_neighbor_zero(img, i, j):
for k in range(-1,2):
for l in range(-1,2):
if img[i+k, j+k] == 0:
return True
return False
def zero_crossing(img):
img[img > 0] = 1
img[img 0 and any_neighbor_zero(img, i, j):
out_img[i,j] = 255
return out_img
img = rgb2gray(imread('../images/zebras.jpg'))
fig = plt.figure(figsize=(25,15))
plt.gray() # show the filtered result in grayscale
for sigma in range(2,10, 2):
plt.subplot(2,2,sigma/2)
result = ndimage.gaussian_laplace(img, sigma=sigma)
plt.imshow(zero_crossing(result))
plt.axis('off')
plt.title('LoG with zero-crossing, sigma=' + str(sigma), size=30)
plt.tight_layout()
plt.show()
Original Input Image
Output with edges detected with LOG + zero-crossing at different sigma scales
With another input image
Output with edges detected with LOG + zero-crossing at different sigma scales
## Constructing the Gaussian Pyramid with scikit-image transform module’s reduce function and Laplacian Pyramid from the Gaussian Pyramid and the expand function
The Gaussian Pyramid can be computed with the following steps:
1. Start with the original image.
2. Iteratively compute the image at each level of the pyramid first by smoothing the image (with gaussian filter) and then downsampling it .
3. Stop at a level where the image size becomes sufficiently small (e.g., 1×1).
The Laplacian Pyramid can be computed with the following steps:
1. Start with the Gaussian Pyramid and with the smallest image.
2. Iteratively compute the difference image in between the image at the current level and the image obtained by first upsampling and then smoothing the image (with gaussian filter) from the previous level of the Gaussian Pyramid.
3. Stop at a level where the image size becomes equal to the original image size.
The next python code shows how to create a Gaussian Pyramid from an image.
import numpy as np
import matplotlib.pyplot as plt
from skimage.io import imread
from skimage.color import rgb2gray
from skimage.transform import pyramid_reduce, pyramid_expand, resize
def get_gaussian_pyramid(image):
rows, cols, dim = image.shape
gaussian_pyramid = [image]
while rows&amp;gt; 1 and cols &amp;gt; 1:
image = pyramid_reduce(image, downscale=2)
gaussian_pyramid.append(image)
print(image.shape)
rows //= 2
cols //= 2
return gaussian_pyramid
def get_laplacian_pyramid(gaussian_pyramid):
laplacian_pyramid = [gaussian_pyramid[len(gaussian_pyramid)-1]]
for i in range(len(gaussian_pyramid)-2, -1, -1):
image = gaussian_pyramid[i] - resize(pyramid_expand(gaussian_pyramid[i+1]), gaussian_pyramid[i].shape)
laplacian_pyramid.append(np.copy(image))
laplacian_pyramid = laplacian_pyramid[::-1]
return laplacian_pyramid
image = imread('../images/antelops.jpeg')
gaussian_pyramid = get_gaussian_pyramid(image)
laplacian_pyramid = get_laplacian_pyramid(gaussian_pyramid)
w, h = 20, 12
for i in range(3):
plt.figure(figsize=(w,h))
p = gaussian_pyramid[i]
plt.imshow(p)
plt.title(str(p.shape[0]) + 'x' + str(p.shape[1]), size=20)
plt.axis('off')
w, h = w / 2, h / 2
plt.show()
w, h = 10, 6
for i in range(1,4):
plt.figure(figsize=(w,h))
p = laplacian_pyramid[i]
plt.imshow(rgb2gray(p), cmap='gray')
plt.title(str(p.shape[0]) + 'x' + str(p.shape[1]), size=20)
plt.axis('off')
w, h = w / 2, h / 2
plt.show()
## Blending images with Gaussian and Laplacian pyramids
Here is the algorithm:
Blending the following input images A, B with mask image M
Input Image A (Goddess Durga)
Input Image B (Lord Shiva)
Mask Image M
with the following python code creates the output image I shown below
A = imread('../images/madurga1.jpg')/255
B = imread('../images/mahadeb.jpg')/255
M = imread('../images/mask1.jpg')/255
# get the Gaussian and Laplacian pyramids, implement the functions
pyramidA = get_laplacian_pyramid(get_gaussian_pyramid(A))
pyramidB = get_laplacian_pyramid(get_gaussian_pyramid(B))
pyramidM = get_gaussian_pyramid(M)
pyramidC = []
for i in range(len(pyramidM)):
im = pyramidM[i]*pyramidA[i] + (1-pyramidM[i])*pyramidB[i]
pyramidC.append(im)
# implement the following function to construct an image from its laplacian pyramids
I = reconstruct_image_from_laplacian_pyramid(pyramidC)
Output Image I (Ardhanarishwar)
The following animation shows how the output image is formed:
Another blending (horror!) example (from prof. dmartin)
# Detection of a Human Object with HOG Descriptor Features using SVM (Primal QuadProg implementation using CVXOPT) in Python
In this article, first how to extract the HOG descriptor from an image will be discuss. Then how a support vector machine binary classifier can be trained on a dataset containing labeled images (using the extracted HOG descriptor features) and later how the SVM model can be used (along with a sliding window) to predict whether or not a human object exists in a test image will be described. How SVM can be represented as a Primal Quadratic Programming problem and can be solved with CVXOPT that will also be discussed. This problem appeared as an assignment problem in this Computer Vision course from UCF.
## Problem 1: Compute HOG features
Let’s first Implement Histogram of Orientated Gradients (HOG). The dataset to be used is the INRIA Person Dataset from here. The dataset consists of positive and negative examples for training as well as testing images. Let us do the following:
i. Take 2003 positive training images of size 96×160
ii. Take 997 negative training images of size 96×160
iii. Compute HOG for positive and negative examples.
iv. Show the visualization of HOG for some positive and negative examples.
The Histograms of Oriented Gradients for Human Detection (HOG) is a very heavily cited paper by N. Dalal and B. Triggs from CVPR 2005. The following figure shows the algorithm proposed by them can be used to compute the HOG features for a 96×160 image:
The next python code snippet shows some helper functions to compute the hog features:
import numpy as np
from scipy import signal
import scipy.misc
def s_x(img):
kernel = np.array([[-1, 0, 1]])
imgx = signal.convolve2d(img, kernel, boundary='symm', mode='same')
return imgx
def s_y(img):
kernel = np.array([[-1, 0, 1]]).T
imgy = signal.convolve2d(img, kernel, boundary='symm', mode='same')
return imgy
def grad(img):
imgx = s_x(img)
imgy = s_y(img)
s = np.sqrt(imgx**2 + imgy**2)
theta = np.arctan2(imgx, imgy) #imgy, imgx)
theta[theta<0] = np.pi + theta[theta<0]
return (s, theta)
The following figures animations show some positive and negative training examples along with the HOG features computed using the algorithm.
### Positive Example 1
The next animation shows how the HOG features are computed using the above algorithm.
### Positive Example 2
The next animation shows how the HOG features are computed using the above algorithm.
### Positive Example 3
The next animation shows how the HOG features are computed using the above algorithm.
### Negative Example 1
The next animation shows how the HOG features are computed using the above algorithm.
## Problem 2: Use sklearn’s SVC and 80-20 validation to compute accuracy on the held-out training images dataset using the extracted HOG features.
Before implementing SVC on our own with primal quadratic programming solver, let’s use the scikit-learn SVC implementation (with linear kernel) to train a support vector classifier on the training positive and negative examples using the HOG features extracted from the training images with 80-20 validation and compute accuracy of classification on the held-out images.
The following python code does exactly that, with the X matrix containing the 1620 HOG features extracted from each image and the corresponding label (pos/neg, depending on whether human is present or not), with 98.5% accuracy on the held-out dataset.
import time
from sklearn.metrics import accuracy_score
from sklearn.cross_validation import train_test_split
from sklearn.svm import SVC
Xtrain, Xtest, ytrain, ytest = train_test_split(X, y, train_size=0.8, random_state=123)
timestamp1 = time.time()
clf = SVC(C=1, kernel='linear')
clf.fit(Xtrain, ytrain)
print("%d support vectors out of %d points" % (len(clf.support_vectors_), len(Xtrain)))
timestamp2 = time.time()
print "sklearn LinearSVC took %.2f seconds" % (timestamp2 - timestamp1)
ypred = clf.predict(Xtest)
print('accuracy', accuracy_score(ytest, ypred))
430 support vectors out of 2400 points
sklearn LinearSVC took 3.40 seconds
accuracy 0.985
The next figures show the confusion matrices for the prediction on the held-out dataset with the SVC model learnt.
## Problem 3: Implement SVM by solving the Primal form of the problem using Quadratic Programming
Let’s implement Support Vector Machine (SVM) using Quadratic Programming. We shall use python’s CVXOPT package for this purpose. Let’s do the following:
i. Try to understand each input term in cvxopt.solvers.qp.
ii. Formulate soft- margin primal SVM in term of inputs of cvxopt.solvers.qp
iii. Show ‘P’, ‘Q’, ‘G”, ‘h’, ‘A’ and ‘b’ Matrices.
iv. Obtain parameter vector ‘w’ and bias term ‘b’ using cvxopt.solvers.qp
To be done
## Problem 4: Detect Human in testing images using trained model (‘w’, ‘b’) from the last problem
Let’s use the coefficients learnt by the SVM model from the training dataset and do the following:
i. Take at least 5 testing images from Test/pos.
ii. Test the trained model over testing images. Testing can be performed using
w*feature vector + b.
iii. Use sliding window approach to obtain detection at each location in the image.
iv. Perform non-maximal suppression and choose the highest scored location.
v. Display the bounding box at the final detection.
To be done
# Implementing Lucas-Kanade Optical Flow algorithm in Python
In this article an implementation of the Lucas-Kanade optical flow algorithm is going to be described. This problem appeared as an assignment in this computer vision course from UCSD. The inputs will be sequences of images (subsequent frames from a video) and the algorithm will output an optical flow field (u, v) and trace the motion of the moving objects. The problem description is taken from the assignment itself.
## Problem Statement
#### Single-Scale Optical Flow
• Let’s implement the single-scale Lucas-Kanade optical flow algorithm. This involves finding the motion (u, v) that minimizes the sum-squared error of the brightness constancy equations for each pixel in a window. The algorithm will be implemented as a function with the following inputs:
def optical_flow(I1, I2, window_size, tau) # returns (u, v)
• Here, u and v are the x and y components of the optical flow, I1 and I2 are two images taken at times t = 1 and t = 2 respectively, and window_size is a 1 × 2 vector storing the width and height of the window used during flow computation.
• In addition to these inputs, a theshold τ should be added, such that if τ is larger than the smallest eigenvalue of A’A, then the the optical flow at that position should not be computed. Recall that the optical flow is only valid in regions where
has rank 2, which is what the threshold is checking. A typical value for τ is 0.01.
• We should try experimenting with different window sizes and find out the tradeoffs associated with using a small vs. a large window size.
• The following figure describes the algorithm, which considers a nxn (n>=3) window around each pixel and solves a least-square problem to find the best flow vectors for the pixel.
• The following code-snippet shows how the algorithm is implemented in python for a gray-level image.
import numpy as np
from scipy import signal
def optical_flow(I1g, I2g, window_size, tau=1e-2):
kernel_x = np.array([[-1., 1.], [-1., 1.]])
kernel_y = np.array([[-1., -1.], [1., 1.]])
kernel_t = np.array([[1., 1.], [1., 1.]])#*.25
w = window_size/2 # window_size is odd, all the pixels with offset in between [-w, w] are inside the window
I1g = I1g / 255. # normalize pixels
I2g = I2g / 255. # normalize pixels
# Implement Lucas Kanade
# for each point, calculate I_x, I_y, I_t
mode = 'same'
fx = signal.convolve2d(I1g, kernel_x, boundary='symm', mode=mode)
fy = signal.convolve2d(I1g, kernel_y, boundary='symm', mode=mode)
ft = signal.convolve2d(I2g, kernel_t, boundary='symm', mode=mode) +
signal.convolve2d(I1g, -kernel_t, boundary='symm', mode=mode)
u = np.zeros(I1g.shape)
v = np.zeros(I1g.shape)
# within window window_size * window_size
for i in range(w, I1g.shape[0]-w):
for j in range(w, I1g.shape[1]-w):
Ix = fx[i-w:i+w+1, j-w:j+w+1].flatten()
Iy = fy[i-w:i+w+1, j-w:j+w+1].flatten()
It = ft[i-w:i+w+1, j-w:j+w+1].flatten()
#b = ... # get b here
#A = ... # get A here
# if threshold τ is larger than the smallest eigenvalue of A'A:
nu = ... # get velocity here
u[i,j]=nu[0]
v[i,j]=nu[1]
return (u,v)
## Some Results
• The following figures and animations show the results of the algorithm on a few image sequences. Some of these input image sequences / videos are from the course and some are collected from the internet.
• As can be seen, the algorithm performs best if the motion of the moving object(s) in between consecutive frames is slow. To the contrary, if the motion is large, the algorithm fails and we should implement / use multiple-scale version Lucas-Kanade with image pyramids.
• Finally, with small window size, the algorithm captures subtle motions but not large motions. With large size it happens the other way.
Input Sequences
Output Optical Flow with different window sizes
window size = 15
window size = 21
Input Sequences
Output Optical Flow
Input Sequences (hamburg taxi)
Output Optical Flow
Input Sequences
Output Optical Flow
Input Sequences
Output Optical Flow
Input Sequences
Output Optical Flow
Input Sequences
Output Optical Flow
Input Sequences
Output Optical Flow
Input Sequences
Output Optical Flow
Input Sequences
Output Optical Flow
Output Optical Flow
Input Sequences
Output Optical Flow with window size 45
Output Optical Flow with window size 10
Output Optical Flow with window size 25
Output Optical Flow with window size 45
# Efficient Graph-Based Image Segmentation in Python
In this article, an implementation of an efficient graph-based image segmentation technique will be described, this algorithm was proposed by Felzenszwalb et. al. from MIT in this paper. The slides on this paper can be found from this link from the Stanford Vision Lab too. The algorithm is closely related to Kruskal’s algorithm for constructing a minimum spanning tree of a graph, as stated by the author and hence can be implemented to run in O(m log m) time, where m is the number of edges in the graph.
## Problem Definition and the basic idea (from the paper)
• Let G = (V, E) be an undirected graph with vertices vi ∈ V, the set of elements to be segmented, and edges (vi, vj ) ∈ E corresponding to pairs of neighboring vertices. Each edge (vi, vj ) ∈ E has a corresponding weight w((vi, vj)), which is a non-negative measure of the dissimilarity between neighboring elements vi and vj.
• In the case of image segmentation, the elements in V are pixels and the weight of an edge is some measure of the dissimilarity between the two pixels connected by that edge (e.g., the difference in intensity, color, motion, location or some other local attribute).
• Particularly for the implementation described here, an edge weight function based on the absolute intensity difference (in the yiq space) between the pixels connected by an edge, w((vi, vj )) = |I(pi) − I(pj )|.
• In the graph-based approach, a segmentation S is a partition of V into components
such that each component (or region) C ∈ S corresponds to a connected component
in a graph G0 = (V, E0), where E0 ⊆ E.
• In other words, any segmentation is induced by a subset of the edges in E. There are different ways to measure the quality of a segmentation but in general we want the elements in a component to be similar, and elements in different components to be dissimilar.
• This means that edges between two vertices in the same component should have relatively low weights, and edges between vertices in different components should have higher weights.
• The next figure shows the steps in the algorithm. The algorithm is very similar to Kruskal’s algorithm for computing the MST for an undirected graph.
• The threshold function τ controls the degree to which the difference between two
components must be greater than their internal differences in order for there to be
evidence of a boundary between them.
• For small components, Int(C) is not a good estimate of the local characteristics of the data. In the extreme case, when |C| = 1, Int(C) = 0. Therefore, a threshold function based on the size of the component, τ (C) = k/|C| is needed to be usedwhere |C| denotes the size of C, and k is some constant parameter.
• That is, for small components we require stronger evidence for a boundary. In practice k sets a scale of observation, in that a larger k causes a preference for larger components.
• In general, a Gaussian filter is used to smooth the image slightly before computing the edge weights, in order to compensate for digitization artifacts. We always use a Gaussian with σ = 0.8, which does not produce any visible change to the image but helps remove artifacts.
• The following python code shows how to create the graph.
import numpy as np
from scipy import signal
import matplotlib.image as mpimg
def gaussian_kernel(k, s = 0.5):
# generate a (2k+1)x(2k+1) gaussian kernel with mean=0 and sigma = s
probs = [exp(-z*z/(2*s*s))/sqrt(2*pi*s*s) for z in range(-k,k+1)]
return np.outer(probs, probs)
def create_graph(imfile, k=1., sigma=0.8, sz=1):
# create the pixel graph with edge weights as dissimilarities
rgb = mpimg.imread(imfile)[:,:,:3]
gauss_kernel = gaussian_kernel(sz, sigma)
for i in range(3):
rgb[:,:,i] = signal.convolve2d(rgb[:,:,i], gauss_kernel, boundary='symm', mode='same')
yuv = rgb2yiq(rgb)
(w, h) = yuv.shape[:2]
edges = {}
for i in range(yuv.shape[0]):
for j in range(yuv.shape[1]):
#compute edge weight for nbd pixel nodes for the node i,j
for i1 in range(i-1, i+2):
for j1 in range(j-1, j+2):
if i1 == i and j1 == j: continue
if i1 >= 0 and i1 = 0 and j1 < h:
wt = np.abs(yuv[i,j,0]-yuv[i1,j1,0])
n1, n2 = ij2id(i,j,w,h), ij2id(i1,j1,w,h)
edges[n1, n2] = edges[n2, n1] = wt
return edges
## Some Results
• The images are taken from the paper itself or from the internet. The following figures and animations show the result of segmentation as a result of iterative merging of the components (by choosing least weight edges), depending on the internal difference of the components.
• Although in the paper the author described the best value of the parameter k to be around 300, but since in this implementation the pixel RGB values are normalized (to have values in between 0 – 1) and then converted to YIQ values and the YIQ intensities are used for computing the weights (which are typically very small), the value of k that works best in this scenario is 0.001-0.01.
• As we can see from the below results, higher the value of the parameter k, larger the size of the final component and lesser the number of components in the result.
• The minimum spanning tree creation is also shown, the red edges shown in the figures are the edges chosen by the algorithm to merge the components.
## Input Image
Output Images for two different values of the parameter k
## Input Image
Output Images for two different values of the parameter k
## Input Image
Output Segmented Images
## Input Image
Output Images for two different values of the parameter k
## Input Image
Segmented Output Image
# Interactive Image Segmentation with Graph-Cut in Python
In this article, interactive image segmentation with graph-cut is going to be discussed. and it will be used to segment the source object from the background in an image. This segmentation technique was proposed by Boycov and Jolli in this paper. This problem appeared as a homework assignment here., and also in this lecture video from the Coursera image processing course by Duke university.
## Problem Statement: Interactive graph-cut segmentation
Let’s implement “intelligent paint” interactive segmentation tool using graph cuts algorithm on a weighted image grid. Our task will be to separate the foreground object from the background in an image.
Since it can be difficult sometimes to automatically define what’s foreground and what’s background for an image, the user is going to help us with a few interactive scribble lines using which our algorithm is going to identify the foreground and the background, after that it will be the algorithms job to obtain a complete segmentation of the foreground from the background image.
The following figures show how an input image gets scribbling from a user with two different colors (which is also going to be input to our algorithm) and the ideal segmented image output.
Scribbled Input Image Expected Segmented Output Image
## The Graph-Cut Algorithm
The following describes how the segmentation problem is transformed into a graph-cut problem:
1. Let’s first define the Directed Graph G = (V, E) as follows:
1. Each of the pixels in the image is going to be a vertex in the graph. There will be another couple of special terminal vertices: a source vertex (corresponds to the foreground object) and a sink vertex (corresponds to the background object in the image). Hence, |V(G)| = width x height + 2.
2. Next, let’s defines the edges of the graph. As obvious, there is going to be two types of edges: terminal (edges that connect the terminal nodes to the non-terminal nodes) and non-terminal (edges that connect the non-terminal nodes only).
3. There will be a directed edge from the terminal node source to each of non-terminal nodes in the graph. Similarly, a directed edge will be there from each non-terminal node (pixel) to the other terminal node sink. These are going to be all the terminal edges and hence, |E_T(G)| = 2 x width x height.
4. Each of the non-terminal nodes (pixels) are going to be connected by edges with the nodes corresponding to the neighboring pixels (defined by 4 or 8 neighborhood of a pixel). Hence, |E_N(G)| = |Nbd| x width x height.
2. Now let’s describe how to compute the edge weights in this graph.
1. In order to compute the terminal edge weights, we need to estimate the feature distributions first, i.e., starting with the assumption that each of the nodes corresponding to the scribbled pixels have the probability 1.0 (since we want the solution to respect the regional hard constraints marked by the user-seeds / scribbles) to be in foreground or background object in the image (distinguished by the scribble color, e.g.), we have to compute the probability that a node belongs to the foreground (or background) for all the other non-terminal nodes.
2. The simplest way to compute $P_F$ and $P_B$ is to first fit a couple of Gaussian distributions on the scribbles by computing the parameters (μ, ∑)
with MLE from the scribbled pixel intensities and then computing the (class-conditional) probabilities from the individual pdfs (followed by a normalization) for each of the pixels as shown in the next figures. The following code fragment show how the pdfs are being computed.
import numpy as np
from collections import defaultdict
def compute_pdfs(imfile, imfile_scrib):
'''
# Compute foreground and background pdfs
# input image and the image with user scribbles
'''
rgb = mpimg.imread(imfile)[:,:,:3]
yuv = rgb2yiq(rgb)
rgb_s = mpimg.imread(imfile_scrib)[:,:,:3]
yuv_s = rgb2yiq(rgb_s)
# find the scribble pixels
scribbles = find_marked_locations(rgb, rgb_s)
imageo = np.zeros(yuv.shape)
# separately store background and foreground scribble pixels in the dictionary comps
comps = defaultdict(lambda:np.array([]).reshape(0,3))
for (i, j) in scribbles:
imageo[i,j,:] = rgbs[i,j,:]
# scribble color as key of comps
comps[tuple(imageo[i,j,:])] = np.vstack([comps[tuple(imageo[i,j,:])], yuv[i,j,:]])
mu, Sigma = {}, {}
# compute MLE parameters for Gaussians
for c in comps:
mu[c] = np.mean(comps[c], axis=0)
Sigma[c] = np.cov(comps[c].T)
return (mu, Sigma)
3. In order to compute the non-terminal edge weights, we need to compute the similarities in between a pixel node and the nodes corresponding to its neighborhood pixels, e.g., with the formula shown in the next figures (e.g., how similar neighborhood pixels are in RGB / YIQ space).
3. Now that the underlying graph is defined, the segmentation of the foreground from the background image boils down to computing the min-cut in the graph or equivalently computing the max-flow (the dual problem) from the source to sink.
4. The intuition is that the min-cut solution will keep the pixels with high probabilities to belong to the side of the source (foreground) node and likewise the background pixels on the other side of the cut near the sink (background) node, since it’s going to respect the (relatively) high-weight edges (by not going through the highly-similar pixels).
5. There are several standard algorithms, e.g., Ford-Fulkerson (by finding an augmenting path with O(E max| f |) time complexity) or Edmonds-Karp (by using bfs to find the shortest augmenting path, with O(VE2) time complexity) to solve the max-flow problem, typical implementations of these algorithms run pretty fast, in polynomial time in V, E. Here we are going to use a different implementation (with pymaxflow) based on Vladimir Kolmogorov, which is shown to run faster on many images empirically in this paper.
## Results
The following figures / animations show the interactive-segmentation results (computed probability densities, subset of the flow-graph & min-cut, final segmented image) on a few images, some of them taken from the above-mentioned courses / videos, some of them taken from Berkeley Vision dataset.
Input Image
Input Image with Scribbles
Fitted Densities from Color Scribbles
A Tiny Sub-graph with Min-Cut
Input Image
Input Image with Scribbles
Fitted Densities from Color Scribbles
A Tiny Sub-graph with Min-Cut
Input Image (liver)
Input Image with Scribbles
Fitted Densities from Color Scribbles
Input Image
Input Image with Scribbles
Fitted Densities from Color Scribbles
A Tiny Sub-graph of the flow-graph with Min-Cut
Input Image
Input Image with Scribbles
Input Image
Input Image with Scribbles
Fitted Densities from Color Scribbles
A Tiny Sub-graph of the flow-graph with Min-Cut
Input Image
Input Image with Scribbles
A Tiny Sub-graph of the flow-graph with Min-Cut
Input Image Input Image with Scribbles
Input Image
Input Image with Scribbles
A Tiny Sub-graph of the flow-graph with Min-Cut
Input Image
Input Image with Scribbles
Fitted Densities from Color Scribbles
Input Image
Input Image with Scribbles
Fitted Densities from Color Scribbles
A Tiny Sub-graph of the flow-graph with Min-Cut
Input Image (UMBC Campus Map)
Input Image with Scribbles
Input Image
Input Image with Scribbles
A Tiny Sub-graph of the flow-graph with Min-Cut (with blue foreground nodes)
## Changing the background of an image (obtained using graph-cut segmentation) with another image’s background with cut & paste
The following figures / animation show how the background of a given image can be replaced by a new image using cut & paste (by replacing the corresponding pixels in the new image corresponding to foreground), once the foreground in the original image gets identified after segmentation.
Original Input Image
New Background
# Image Colorization Using Optimization in Python
This article is inspired by this SIGGRAPH paper by Levin et. al, for which they took this patent , the paper was referred to in the course CS1114 from Cornell. This method is also discussed in the coursera online image processing course by NorthWestern University. Some part of the problem description is taken from the paper itself. Also, one can refer to the implementation provided by the authors in matlab, the following link and the following python implementation in github.
## The Problem
Colorization is a computer-assisted process of adding color to a monochrome image or movie. In the paper the authors presented an optimization-based colorization method that is based on a simple premise: neighboring pixels in space-time that have similar intensities should have similar colors.
This premise is formulated using a quadratic cost function and as an optimization problem. In this approach an artist only needs to annotate the image with a few color scribbles, and the indicated colors are automatically propagated in both space and time to produce a fully colorized image or sequence.
In this article the optimization problem formulation and the way to solve it to obtain the automatically colored image will be described for the images only.
## The Algorithm
YUV/YIQ color space is chosen to work in, this is commonly used in video, where Y is the monochromatic luminance channel, which we will refer to simply as intensity, while U and V are the chrominance channels, encoding the color.
The algorithm is given as input an intensity volume Y(x,y,t) and outputs two color volumes U(x,y,t) and V(x,y,t). To simplify notation the boldface letters are used (e.g. r,s) to denote $\left(x,y,t \right)$ triplets. Thus, Y(r) is the intensity of a particular pixel.
Now, One needs to impose the constraint that two neighboring pixels r,s should have similar colors if their intensities are similar. Thus, the problem is formulated as the following optimization problem that aims to minimize the difference between the colorU(r) at pixel r and the weighted average of the colors at neighboring pixels, where w(r,s) is a weighting function that sums to one, large when Y(r) is similar to Y(s), and small when the two intensities are different.
When the intensity is constant the color should be constant, and when the intensity is an edge the color should also be an edge. Since the cost functions are quadratic and
the constraints are linear, this optimization problem yields a large, sparse system of linear equations, which may be solved using a number of standard methods.
As discussed in the paper, this algorithm is closely related to algorithms proposed for other tasks in image processing. In image segmentation algorithms based on normalized cuts [Shi and Malik 1997], one attempts to find the second smallest eigenvector of the matrix D − W where W is a npixels×npixels matrix whose elements are the pairwise affinities between pixels (i.e., the r,s entry of the matrix is w(r,s)) and D is a diagonal matrix whose diagonal elements are the sum of the affinities (in this case it is always 1). The second smallest eigenvector of any symmetric matrix A is a unit norm vector x that minimizes $x^{T}Ax$ and is orthogonal to the first eigenvector. By direct inspection, the quadratic form minimized by normalized cuts is exactly the cost function J, that is $x^{T}(D-W)x =J(x)$. Thus, this algorithm minimizes the same cost function but under different constraints. In image denoising algorithms based on anisotropic diffusion [Perona and Malik 1989; Tang et al. 2001] one often minimizes a function
similar to equation 1, but the function is applied to the image intensity as well.
The following figures show an original gray-scale image and the marked image with color-scribbles that are going to be used to compute the output colored image.
Original Gray-scale Image Input
Gray-scale image Input Marked with Color-Scribbles
## Implementation of the Algorithm
Here are the the steps for the algorithm:
1. Convert both the original gray-scale image and the marked image (marked with color scribbles for a few pixels by the artist) to from RGB to YUV / YIQ color space.
2. Compute the difference image from the marked and the gray-scale image. The pixels that differ are going to be pinned and they will appear in the output, they are directly going to be used as stand-alone constraints in the minimization problem.
3. We need to compute the weight matrix W that depends on the similarities in the neighbor intensities for a pixel from the original gray-scale image.
4. The optimization problem finally boils down to solving the system of linear equations of the form $WU = b$ that has a closed form least-square solution
$U = W^{+}b = {(W^{T}W)}^{-1}W^{T}b$. Same thing is to be repeated for the V channel too.
5. However the W matrix is going to be very huge and sparse, hence sparse-matrix based implementations must be used to obtain an efficient solution. However, in this python implementation in github, the scipy sparse lil_matrix was used when constructing the sparse matrices, which is quite slow, we can construct more efficient scipy csc matrix rightaway, by using a dictionary to store the weights initially. It is much faster. The python code in the next figure shows my implementation for computing the weight matrix W.
6. Once W is computed it’s just a matter of obtaining the least-square solution, by computing the pseudo-inverse, which can be more efficiently computed with LU factorization and a sparse LU solver, as in this python implementation in github.
7. Once the solution of the optimization problem is obtained in YUV / YIQ space, it needs to be converted back to RGB. The following formula is used for conversion.
import scipy.sparse as sp
from collections import defaultdict
def compute_W(Y, colored):
(w, h) = Y.shape
W = defaultdict()
for i in range(w):
for j in range(h):
if not (i, j) in colored: # pixel i,j in color scribble
(N, sigma) = get_nbrs(Y, i, j, w, h) #r = (i, j)
Z = 0.
id1 = ij2id(i,j,w,h) # linearized id
for (i1, j1) in N: #s = (i1, j1)
id2 = ij2id(i1,j1,w,h)
W[id1,id2] = np.exp(-(Y[i,j]-Y[i1,j1])**2/(sigma**2)) if sigma > 0 else 0.
Z += W[id1,id2]
if Z > 0:
for (i1, j1) in N: #s = (i1, j1)
id2 = ij2id(i1,j1,w,h)
W[id1,id2] /= -Z
for i in range(w):
for j in range(h):
id = ij2id(i,j,w,h)
W[id,id] = 1.
rows, cols = zip(*(W.keys())) #keys
data = W.values() #[W[k] for k in keys]
return sp.csc_matrix((data, (rows, cols)), shape=(w*h, w*h)) #W
## Results
The following images and animations show the results obtained with the optimization algorithm. Most of the following images are taken from the paper itself.
Original Gray-scale Image Input Gray-scale image Input Marked
Color image Output
The next animations show how the incremental scribbling results in better and better color images.
Original Gray-scale Image Input
As can be seen from the following animation, the different parts of the building get colored as more and more color-hints are scribbled / annotated.
Gray-scale image Input Marked
Color image Output
Original Gray-scale Image Input
Gray-scale image Input Marked
Color image Output
Original Gray-scale Image Input (me)
Gray-scale image Input Marked incrementally
Color image Output
Original Gray-scale Image Input
Gray-scale image Input Marked
Color image Output
Original Gray-scale Image Input
Gray-scale image Input Marked
Color image Output
# Recursive Graphics, Bi/Tri-linear Interpolation, Anti-aliasing and Image Transformation in Python
The following problem appeared in an assignment in the Princeton course COS 126 . The problem description is taken from the course itself.
## Recursive Graphics
Write a program that plots a Sierpinski triangle, as illustrated below. Then develop a program that plots a recursive patterns of your own design.
Part 1.
The Sierpinski triangle is an example of a fractal pattern. The pattern was described by Polish mathematician Waclaw Sierpinski in 1915, but has appeared in Italian art since the 13th century. Though the Sierpinski triangle looks complex, it can be generated with a short recursive program.
Examples. Below are the target Sierpinski triangles for different values of order N.
Our task is to implement a recursive function sierpinski(). We need to think recursively: our function should draw one black triangle (pointed downwards) and then call itself recursively 3 times (with an appropriate stopping condition). When writing our program, we should exercise modular design.
The following code shows an implementation:
class Sierpinski:
#Height of an equilateral triangle whose sides are of the specified length.
def height (self, length):
return sqrt(3) * length / 2.
#Draws a filled equilateral triangle whose bottom vertex is (x, y)
#of the specified side length.
def filledTriangle(self, x, y, length):
h = self.height(length)
draw(np.array([x, x+length/2., x-length/2.]), np.array([y, y+h, y+h]), alpha=1)
#Draws an empty equilateral triangle whose bottom vertex is (x, y)
#of the specified side length.
def emptyTriangle(self, x, y, length):
h = self.height(length)
draw(np.array([x, x+length, x-length]), np.array([y+2*h, y, y]), alpha=0)
# Draws a Sierpinski triangle of order n, such that the largest filled
# triangle has bottom vertex (x, y) and sides of the specified length.
def sierpinski(self, n, x, y, length):
self.filledTriangle(x, y, length)
if n > 1:
self.sierpinski(n-1, x-length/2., y, length/2.)
self.sierpinski(n-1, x+length/2., y, length/2.)
self.sierpinski(n-1, x, y+self.height(length), length/2.)
The following animation shows how such a triangle of order 5 is drawn recursively.
The following animation shows how such a triangle of order 6 is drawn recursively.
A diversion: fractal dimension.
Formally, we can define the Hausdorff dimension or similarity dimension of a self-similar figure by partitioning the figure into a number of self-similar pieces of smaller size. We define the dimension to be the log (# self similar pieces) / log (scaling factor in each spatial direction). For example, we can decompose the unit square into 4 smaller squares, each of side length 1/2; or we can decompose it into 25 squares, each of side length 1/5. Here, the number of self-similar pieces is 4 (or 25) and the scaling factor is 2 (or 5). Thus, the dimension of a square is 2 since log (4) / log(2) = log (25) / log (5) = 2. We can decompose the unit cube into 8 cubes, each of side length 1/2; or we can decompose it into 125 cubes, each of side length 1/5. Therefore, the dimension of a cube is log(8) / log (2) = log(125) / log(5) = 3.
We can also apply this definition directly to the (set of white points in) Sierpinski triangle. We can decompose the unit Sierpinski triangle into 3 Sierpinski triangles, each of side length 1/2. Thus, the dimension of a Sierpinski triangle is log (3) / log (2) ≈ 1.585. Its dimension is fractional—more than a line segment, but less than a square! With Euclidean geometry, the dimension is always an integer; with fractal geometry, it can be something in between. Fractals are similar to many physical objects; for example, the coastline of Britain resembles a fractal, and its fractal dimension has been measured to be approximately 1.25.
Part 2.
Drawing a tree recursively, as described here:
The following shows how a tree of order 10 is drawn:
The next problem appeared in an assignment in the Cornell course CS1114 . The problem description is taken from the course itself.
## Bilinear Interpolation
Let’s consider a 2D matrix of values at integer grid locations (e.g., a grayscale image). To interpolate values on a 2D grid, we can use the 2D analogue of linear interpolation: bilinear interpolation. In this case, there are four neighbors for each possible point we’d like to interpolation, and the intensity values of these four neighbors are all combined to compute the interpolated intensity, as shown in the next figure.
In the figure, the Q values represent intensities. To combine these intensities, we perform linear interpolation in multiple directions: we first interpolate in the x direction (to get the value at the blue points), then in the y direction (to get the value at the green points).
## Image transformations
Next, we’ll use the interpolation function to help us implement image transformations.
A 2D affine transformation can be represented with a 3 ×3 matrix T:
Recall that the reason why this matrix is 3×3, rather than 2 ×2, is that we operate in homogeneous coordinates; that is, we add an extra 1 on the end of our 2D coordinates (i.e., (x,y) becomes (x,y,1)), in order to represent translations with a matrix multiplication. To apply a transformation T to a pixel, we multiply T by the pixel’s location:
The following figure shows a few such transformation matrices:
To apply a transformation T to an entire image I, we could apply the transformation to each of I’s pixels to map them to the output image. However, this forward warping procedure has several problems. Instead, we’ll use inverse mapping to warp the pixels of the output image back to the input image. Because this won’t necessarily hit an integer-valued location, we’ll need to use the (bi-linear) interpolation to determine the intensity of the input image at the desired location, as shown in the next figure.
To demo the transformation function, let’s implement the following on a gray scale bird image:
1. Horizontal flipping
2. Scaling by a factor of 0.5
3. Rotation by 45 degrees around the center of the image
The next animations show rotation and sheer transformations with the Lena image:
Next, let’s implement a function to transform RGB images. To do this, we need to simply call transform image three times, once for each channel, then put the results together into a single image. Next figures and animations show some results on an RGB image.
## Some non-linear transformations
The next figure shows the transform functions from here:
The next figures and animations show the application of the above non-linear transforms on the Lena image.
Wave1
Wave2
Swirl
Warp
Some more non-linear transforms:
## Anti-aliasing
There is a problem with our interpolation method above: it is not very good at shrinking images, due to aliasing. For instance, if let’s try to down-sample the following bricks image by a factor of 0.4, we get the image shown in the following figure: notice the strange banding effects in the output image.
Original Image
Down-sampled Image with Bilinear Interpolation
The problem is that a single pixel in the output image corresponds to about 2.8 pixels in the input image, but we are sampling the value of a single pixel—we should really be averaging over a small area.
To overcome this problem, we will create a data structure that will let us (approximately) average over any possible square regions of pixels in the input image: an image stack. An image stack is a 3D matrix that we can think of as, not surprisingly, a stack of images, one on top of the other. The top image in the cube will be the original input image. Images further down the stack will be the input image with progressively larger amounts of blur. The size of the matrix will be rows × cols × num levels, where the original (grayscale) image has size rows×cols and there are num levels images in the stack.
Before we use the stack, we must write a function to create it, which takes as input a (grayscale) image and a number of levels in the stack, and returns a 3D matrix stack corresponding to the stack. Again, the first image on the stack will be the original image. Every other image in the stack will be a blurred version of the previous image. A good blur kernel to use is:
Now, for image k in the stack, we know that every pixel is a (weighted) average of some number of pixels (a k × k patch, roughly speaking) in the input image. Thus, if we down-sample the image by a factor of k, we want to sample pixels from level k of the stack.
⇒ Let’s write the following function to create image stack that takes a grayscale image and a number max levels, and returns an image stack.
from scipy import signal
def create_image_stack(img, max_level):
K = np.ones((3,3)) / 9.
image_stack = np.zeros((img.shape[0], img.shape[1], max_level))
image_stack[:,:,0] = img
for l in range(1, max_level):
image_stack[:,:,l] = signal.convolve2d(image_stack[:,:,l-1], K,
boundary=’symm’, mode=’same’)
return image_stack
The next animation shows the image stack created from the bricks image.
## Trilinear Interpolation
Now, what happens if we down-sample the image by a fractional factor, such as 3.6? Unfortunately, there is no level 3.6 of the stack. Fortunately, we have a tool to solve this problem: interpolation. We now potentially need to sample a value at position (row,col,k) of the image stack, where all three coordinates are fractional. We therefore something more powerful than bilinear interpolation: trilinear interpolation! Each position we want to sample now has eight neighbors, and we’ll combine all of their values together in a weighted sum.
This sounds complicated, but we can write this in terms of our existing functions. In particular, we now interpolate separately along different dimensions: trilinear interpolation can be implemented with two calls to bilinear interpolation and one call to linear interpolation.
Let’s implement a function trilerp like the following that takes an image stack, and a row, column, and stack level k, and returns the interpolated value.
def trilerp (img_stack, x, y, k):
if k < 1: k = 1
if k == int(k):
return bilerp(img_stack[:,:,k-1], x, y)
else:
f_k, c_k = int(floor(k)), int(ceil(k))
v_f_k = bilerp(img_stack[:,:,f_k-1], x, y)
v_c_k = bilerp(img_stack[:,:,c_k-1], x, y)
return linterp(k, f_k, c_k, v_f_k, v_c_k)
Now we can finally implement a transformation function that does proper anti-aliasing. In order to do this, let’s implement a function that will
• First compute the image stack.
• Then compute, for the transformation T, how much T is scaling down the image. If T is defined by the six values a,b,c,d,e,f above, then, to a first approximation, the downscale factor is:
However, if k < 1 (corresponding to scaling up the image), we still want to sample from level 1. This situation reverts to normal bilinear interpolation.
• Next call the trilerp function on the image stack, instead of bilerp on the input image.
The next figure shows the output image obtained image transformation with proper anti-aliasing:
Down-sampled Image with Anti-aliasing using Trilinear Interpolation
As we can see from the above output, the aliasing artifact has disappeared.
The same results are obtained on the color image, as shown below, by applying the trilerp function on the color channels separately and creating separate image stacks for different color channels.
Original Image
Down-sampled Image with Bilinear Interpolation
Down-sampled Image with Anti-aliasing
The following animation shows the branding artifacts created when using bilinear interpolation for different scale factors and how they are removed with anti-aliasing.
Down-sampled Images with Bilinear Interpolation
Down-sampled Images with Anti-aliasing
# Deep Learning & Art: Neural Style Transfer – An Implementation with Tensorflow (using Transfer Learning with a Pre-trained VGG-19 Network) in Python
This problem appeared as an assignment in the online coursera course Convolution Neural Networks by Prof Andrew Ng, (deeplearing.ai). The description of the problem is taken straightway from the assignment.
Neural Style Transfer algorithm was created by Gatys et al. (2015) , the paper can be found here .
In this assignment, we shall:
• Implement the neural style transfer algorithm
• Generate novel artistic images using our algorithm
Most of the algorithms we’ve studied optimize a cost function to get a set of parameter values. In Neural Style Transfer, we shall optimize a cost function to get pixel values!
## Problem Statement
Neural Style Transfer (NST) is one of the most fun techniques in deep learning. As seen below, it merges two images, namely,
1. a “content” image (C) and
2. a “style” image (S),
to create a “generated” image (G). The generated image G combines the “content” of the image C with the “style” of image S.
In this example, we are going to generate an image of the Louvre museum in Paris (content image C), mixed with a painting by Claude Monet, a leader of the impressionist movement (style image S).
Let’s see how we can do this.
## Transfer Learning
Neural Style Transfer (NST) uses a previously trained convolutional network, and builds on top of that. The idea of using a network trained on a different task and applying it to a new task is called transfer learning.
Following the original NST paper, we shall use the VGG network. Specifically, we’ll use VGG-19, a 19-layer version of the VGG network. This model has already been trained on the very large ImageNet database, and thus has learned to recognize a variety of low level features (at the earlier layers) and high level features (at the deeper layers). The following figure (taken from the google image search results) shows how a VGG-19 convolution neural net looks like, without the last fully-connected (FC) layers.
We run the following code to load parameters from the pre-trained VGG-19 model serialized in a matlab file. This takes a few seconds.
model = load_vgg_model(“imagenet-vgg-verydeep-19.mat”)
import pprint
pprint.pprint(model)
{‘avgpool1’: <tf.Tensor ‘AvgPool_5:0’ shape=(1, 150, 200, 64) dtype=float32>,
‘avgpool2’: <tf.Tensor ‘AvgPool_6:0’ shape=(1, 75, 100, 128) dtype=float32>,
‘avgpool3’: <tf.Tensor ‘AvgPool_7:0’ shape=(1, 38, 50, 256) dtype=float32>,
‘avgpool4’: <tf.Tensor ‘AvgPool_8:0’ shape=(1, 19, 25, 512) dtype=float32>,
‘avgpool5’: <tf.Tensor ‘AvgPool_9:0’ shape=(1, 10, 13, 512) dtype=float32>,
‘conv1_1’: <tf.Tensor ‘Relu_16:0’ shape=(1, 300, 400, 64) dtype=float32>,
‘conv1_2’: <tf.Tensor ‘Relu_17:0’ shape=(1, 300, 400, 64) dtype=float32>,
‘conv2_1’: <tf.Tensor ‘Relu_18:0’ shape=(1, 150, 200, 128) dtype=float32>,
‘conv2_2’: <tf.Tensor ‘Relu_19:0’ shape=(1, 150, 200, 128) dtype=float32>,
‘conv3_1’: <tf.Tensor ‘Relu_20:0’ shape=(1, 75, 100, 256) dtype=float32>,
‘conv3_2’: <tf.Tensor ‘Relu_21:0’ shape=(1, 75, 100, 256) dtype=float32>,
‘conv3_3’: <tf.Tensor ‘Relu_22:0’ shape=(1, 75, 100, 256) dtype=float32>,
‘conv3_4’: <tf.Tensor ‘Relu_23:0’ shape=(1, 75, 100, 256) dtype=float32>,
‘conv4_1’: <tf.Tensor ‘Relu_24:0’ shape=(1, 38, 50, 512) dtype=float32>,
‘conv4_2’: <tf.Tensor ‘Relu_25:0’ shape=(1, 38, 50, 512) dtype=float32>,
‘conv4_3’: <tf.Tensor ‘Relu_26:0’ shape=(1, 38, 50, 512) dtype=float32>,
‘conv4_4’: <tf.Tensor ‘Relu_27:0’ shape=(1, 38, 50, 512) dtype=float32>,
‘conv5_1’: <tf.Tensor ‘Relu_28:0’ shape=(1, 19, 25, 512) dtype=float32>,
‘conv5_2’: <tf.Tensor ‘Relu_29:0’ shape=(1, 19, 25, 512) dtype=float32>,
‘conv5_3’: <tf.Tensor ‘Relu_30:0’ shape=(1, 19, 25, 512) dtype=float32>,
‘conv5_4’: <tf.Tensor ‘Relu_31:0’ shape=(1, 19, 25, 512) dtype=float32>,
‘input’: <tensorflow.python.ops.variables.Variable object at 0x7f7a5bf8f7f0>}
The next figure shows the content image (C) – the Louvre museum’s pyramid surrounded by old Paris buildings, against a sunny sky with a few clouds.
For the above content image, the activation outputs from the convolution layers are visualized in the next few figures.
## How to ensure that the generated image G matches the content of the image C?
As we know, the earlier (shallower) layers of a ConvNet tend to detect lower-level features such as edges and simple textures, and the later (deeper) layers tend to detect higher-level features such as more complex textures as well as object classes.
We would like the “generated” image G to have similar content as the input image C. Suppose we have chosen some layer’s activations to represent the content of an image. In practice, we shall get the most visually pleasing results if we choose a layer in the middle of the network – neither too shallow nor too deep.
First we need to compute the “content cost” using TensorFlow.
• The content cost takes a hidden layer activation of the neural network, and measures how different a(C) and a(G) are.
• When we minimize the content cost later, this will help make sure G
has similar content as C.
def compute_content_cost(a_C, a_G):
“””
Computes the content cost
Arguments:
a_C — tensor of dimension (1, n_H, n_W, n_C), hidden layer activations representing content of the image C
a_G — tensor of dimension (1, n_H, n_W, n_C), hidden layer activations representing content of the image G
Returns:
J_content — scalar that we need to compute using equation 1 above.
“””
# Retrieve dimensions from a_G
m, n_H, n_W, n_C = a_G.get_shape().as_list()
# Reshape a_C and a_G
a_C_unrolled = tf.reshape(tf.transpose(a_C), (m, n_H * n_W, n_C))
a_G_unrolled = tf.reshape(tf.transpose(a_G), (m, n_H * n_W, n_C))
# compute the cost with tensorflow
J_content = tf.reduce_sum((a_C_unrolled – a_G_unrolled)**2 / (4.* n_H * n_W * \
n_C))
return J_content
## Computing the style cost
For our running example, we will use the following style image (S). This painting was painted in the style of impressionism, by Claude Monet .
def gram_matrix(A):
“””
Argument:
A — matrix of shape (n_C, n_H*n_W)
Returns:
GA — Gram matrix of A, of shape (n_C, n_C)
“””
GA = tf.matmul(A, tf.transpose(A))
return GA
def compute_layer_style_cost(a_S, a_G):
“””
Arguments:
a_S — tensor of dimension (1, n_H, n_W, n_C), hidden layer activations representing style of the image S
a_G — tensor of dimension (1, n_H, n_W, n_C), hidden layer activations representing style of the image G
Returns:
J_style_layer — tensor representing a scalar value, style cost defined above by equation (2)
“””
# Retrieve dimensions from a_G
m, n_H, n_W, n_C = a_G.get_shape().as_list()
# Reshape the images to have them of shape (n_C, n_H*n_W)
a_S = tf.reshape(tf.transpose(a_S), (n_C, n_H * n_W))
a_G = tf.reshape(tf.transpose(a_G), (n_C, n_H * n_W))
# Computing gram_matrices for both images S and G (≈2 lines)
GS = gram_matrix(a_S)
GG = gram_matrix(a_G)
# Computing the loss
J_style_layer = tf.reduce_sum((GS – GG)**2 / (4.* (n_H * n_W * n_C)**2))
return J_style_layer
• The style of an image can be represented using the Gram matrix of a hidden layer’s activations. However, we get even better results combining this representation from multiple different layers. This is in contrast to the content representation, where usually using just a single hidden layer is sufficient.
• Minimizing the style cost will cause the image G to follow the style of the image S.
## Defining the total cost to optimize
Finally, let’s create and implement a cost function that minimizes both the style and the content cost. The formula is:
def total_cost(J_content, J_style, alpha = 10, beta = 40):
“””
Computes the total cost function
Arguments:
J_content — content cost coded above
J_style — style cost coded above
alpha — hyperparameter weighting the importance of the content cost
beta — hyperparameter weighting the importance of the style cost
Returns:
J — total cost as defined by the formula above.
“””
J = alpha * J_content + beta * J_style
return J
• The total cost is a linear combination of the content cost J_content(C,G) and the style cost J_style(S,G).
• α and β are hyperparameters that control the relative weighting between content and style.
## Solving the optimization problem
Finally, let’s put everything together to implement Neural Style Transfer!
Here’s what the program will have to do:
• Create an Interactive Session
• Load the content image
• Load the style image
• Randomly initialize the image to be generated
• Load the VGG19 model
• Build the TensorFlow graph:
• Run the content image through the VGG19 model and compute the content cost.
• Run the style image through the VGG19 model and compute the style cost
Compute the total cost.
• Define the optimizer and the learning rate.
• Initialize the TensorFlow graph and run it for a large number of iterations, updating the generated image at every step.
Let’s first load, reshape, and normalize our “content” image (the Louvre museum picture) and “style” image (Claude Monet’s painting).
Now, we initialize the “generated” image as a noisy image created from the content_image. By initializing the pixels of the generated image to be mostly noise but still slightly correlated with the content image, this will help the content of the “generated” image more rapidly match the content of the “content” image. The following figure shows the noisy image:
Next, let’s load the pre-trained VGG-19 model.
To get the program to compute the content cost, we will now assign a_C and a_G to be the appropriate hidden layer activations. We will use layer conv4_2 to compute the content cost. We need to do the following:
• Assign the content image to be the input to the VGG model.
• Set a_C to be the tensor giving the hidden layer activation for layer “conv4_2”.
• Set a_G to be the tensor giving the hidden layer activation for the same layer.
• Compute the content cost using a_C and a_G.
Next, we need to compute the style cost and compute the total cost J by taking a linear combination of the two. Use alpha = 10 and beta = 40.
Then we are going to set up the Adam optimizer in TensorFlow, using a learning rate of 2.0.
Finally, we need to initialize the variables of the tensorflow graph, assign the input image (initial generated image) as the input of the VGG19 model and runs the model to minimize the total cost J for a large number of iterations.
## Results
The following figures / animations show the generated images (G) with different content (C) and style images (S) at different iterations in the optimization process.
Content
Style (Claud Monet’s The Poppy Field near Argenteuil)
Generated
Content
Style
Generated
Content
Style
Generated
Content
Style (Van Gogh’s The Starry Night)
Generated
Content
Style
Generated
Content (Victoria Memorial Hall)
Style (Van Gogh’s The Starry Night)
Generated
Content (Taj Mahal)
Style (Van Gogh’s Starry Night Over the Rhone)
Generated
Content
Style (Claud Monet’s Sunset in Venice)
Generated
Content (Visva Bharati)
Style (Abanindranath Tagore’s Rabindranath in the role of blind singer )
Generated
Content (Howrah Bridge)
Style (Van Gogh’s The Starry Night)
Generated
Content (Leonardo Da Vinci’s Mona Lisa)
Style (Van Gogh’s The Starry Night)
Generated
Content (My sketch: Rabindranath Tagore)
Style (Abanindranath Tagore’s Rabindranath in the role of blind singer )
Generated
Content (me)
Style (Van Gogh’s Irises)
Generated
Content
Style
Generated
Content
Style (Publo Picaso’s Factory at Horto de Ebro)
Generated
The following animations show how the generated image changes with the change in VGG-19 convolution layer used for computing content cost.
Content
Style (Van Gogh’s The Starry Night)
Generated
convolution layer 3_2 used
convolution layer 4_2 used
convolution layer 5_2 used
# Some Computational Photography: Image Quilting (Texture Synthesis) with Dynamic Programming and Texture Transfer (Drawing with Textures) in Python
The following problems appeared as a programming assignment in the Computation Photography course (CS445) at UIUC. The description of the problem is taken from the assignment itself. In this assignment, a python implementation of the problems will be described instead of matlab, as expected in the course.
## The Problems
• The goal of this assignment is to implement the image quilting algorithm for
texture synthesis and transfer, described in this SIGGRAPH 2001 paper by Efros
and Freeman.
• Texture synthesis is the creation of a larger texture image from a small sample.
• Texture transfer is giving an object the appearance of having the
same texture as a sample while preserving its basic shape.
• For texture synthesis, the main idea is to sample patches and lay them down in overlapping patterns, such that the overlapping regions are similar.
• The overlapping regions may not match exactly, which will result in noticeable
edges artifact. To fix this, we need to compute a path along pixels with similar intensities through the overlapping region and use it to select which overlapping patch from which to draw each pixel.
• Texture transfer is achieved by encouraging sampled patches to have similar appearance to a given target image, as well as matching overlapping regions of already sampled patches.
• In this project, we need to apply important techniques such as template matching, finding seams, and masking. These techniques are also useful for image stitching, image completion, image retargeting and blending.
### Randomly Sampled Texture
First let’s randomly samples square patches of size patchsize from sample in order to create an output image of size outsize. Start from the upper-left corner, and tile samples until the image is full. If the patches don’t fit evenly into the output image, we can leave black borders at the edges. This is the simplest but least effective method. Save a result from a sample image to compare to the next two methods.
### Overlapping Patches
Let’s start by sampling a random patch for the upper-left corner. Then sample new patches to overlap with existing ones. For example, the second patch along the top row will overlap by patchsize pixels in the vertical direction and overlap pixels in the horizontal direction. Patches in the first column will overlap by patchsize pixels in the horizontal direction and overlap pixels in the vertical direction. Other patches will have two overlapping regions (on the top and left) which should both be taken into account. Once the cost of each patch has been computed, randomly choose on patch whose cost is
less than a threshold determined by some tolerance value.
As described in the paper, the size of the block is the only parameter controlled by the user and it depends on the properties of a given texture; the block must be big enough to capture the relevant structures in the texture, but small enough so that the interaction between these structures is left up to the algorithm. The overlap size is taken to be one-sixth of the block size (B/6) in general.
### Seam Finding
Next we need to find the min-cost contiguous path from the left to right side of the patch according to the cost. The cost of a path through each pixel is the square differences (summed over RGB for color images) of the output image and the newly
sampled patch. Use dynamic programming to find the min-cost path.
The following figure describes the algorithm to be implemented for image quilting.
### Texture Transfer
The final task is to implement texture transfer, based on the quilt implementation for creating a texture sample that is guided by a pair of sample/target correspondence images (section 3 of the paper). The main difference between this function and
quilt function is that there is an additional cost term based on the difference between
the sampled source patch and the target patch at the location to be filled.
## Image quilting (texture synthesis) results
The following figures and animations show the results of the outputs obtained with the quilting algorithm. The input texture images are mostly taken from the paper .
Input sample Texture
100 sampled blocks of a fixed size (e.g. 50×50) from the input sample
The next animation shows how the large output texture gets created (100 times larger than the input sample texture) with the quilting algorithm.
Output Texture (10×10 times larger than the input) created with texture synthesis (quilting)
Input Texture
Output Texture (25 times larger than the input) created with texture synthesis (quilting) with the minimum cost seams (showed as red lines) computed with dynamic programming
Output Texture (25 times larger than the input) created with quilting
Input Texture
Output Texture (25 times larger than the input) created with quilting
Input Texture
Output Texture (25 times larger than the input) created with quilting
Input Texture
Output Texture (12 times larger than the input) created with quilting
Input Texture
Output Texture (25 times larger than the input) created with quilting
Input Texture
Output Texture (25 times larger than the input) created with quilting
Input Texture
Output Texture (36 times larger than the input) created with quilting
Input Texture
Output Texture (9 times larger than the input) created with quilting
Input Texture
Output Texture (25 times larger than the input) created with quilting
Input Texture
Output Texture (9 times larger than the input) created with quilting along with the min-cost seams (shown as red lines) computed with dynamic programming
Output Texture (9 times larger than the input) created with quilting
## Texture transfer results
The following figures and animations show how the texture from an input image can be transferred to the target image using the above-mentioned simple modification of the quilting algorithm. Again, some of the images are taken from the paper.
Input Texture (milk)
Target Image
Output Image after Texture Transfer
Input Texture (milk)
Target Image
Output Image after Texture Transfer
The following figures show the output image obtained when a few textures were transferred to my image.
Target Image (me)
Input Texture (fire)
Output Image after Texture Transfer (with small block size)
Input Texture (cloud)
Output Image after Texture Transfer (with small block size)
Input Texture (toast)
Output Image after Texture Transfer (with small block size)
Now applying some gradient mixing such as Poisson blending on the original image and the the one after texture transfer with the target toast image we get the following two images respectively.
Input Texture
Output Image after Texture Transfer (with small block size)
Input Texture
Output Image after Texture Transfer (with small block size)
## Drawing with Textures
The following figures show the output image obtained when a few textures were transferred to another image of mine.
Target Image (me)
### Some textures from a few famous painting-masterpieces (obtained from here, here, here, here and here)
Input Texture (Van Gogh’s The Starry Night)
Output Images after Texture Transfer (with 3 different patch sizes)
Input Texture (Van Gogh’s Wheatfield with Cypresses)
Output Image after Texture Transfer
Input Texture (Van Gogh’s The Mulberry Tree)
Output Image after Texture Transfer
Input Texture (Van Gogh’s Wheatfield with Crows)
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture (Van Gogh’s Vase with fifteen Sunflowers)
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture (Van Gogh’s Sunflowers (F452))
Output Image after Texture Transfer
Input Texture (Van Gogh’s Irises)
Output Image after Texture Transfer
Input Texture (Van Gogh’s Almond Blossom)
Output Image after Texture Transfer
Input Texture (Van Gogh’s Starry Night Over the Rhone)
Output Image after Texture Transfer
Input Texture (Pablo Picasso’s Mediterranean Landscape)
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture (Pablo Picasso’s Minotauromachy)
Output Image after Texture Transfer
Input Texture (Pablo Picasso’s Mother and Child 1921)
Output Image after Texture Transfer
Input Texture (Pablo Picasso’s Factory at Horto de Ebro)
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture (Pablo Picasso’s Carafe and Three Bowls)
Output Image after Texture Transfer
Input Texture (Pablo Picasso’s Bullfight 3)
Output Image after Texture Transfer
Input Texture (Pablo Picasso’s Accordionist)
Output Image after Texture Transfer
Input Texture (Pablo Picasso’s Las Meninas)
Output Image after Texture Transfer
Input Texture (Claude Monet’s The Artist’s Garden at Giverny
Output Image after Texture Transfer
Input Texture (Claude Monet’s The Poppy Field near Argenteuil
Output Image after Texture Transfer
Input Texture (Claude Monet’s The Magpie
Output Image after Texture Transfer
Input Texture (Claude Monet’s The Garden of Monet at Argenteuil
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture (Claude Monet’s Sunset in Venice
Output Image after Texture Transfer
Input Texture (Claude Monet’s Waterloo Bridge
Output Image after Texture Transfer
Input Texture (Claude Monet’s Water Lilies
Output Image after Texture Transfer
Input Texture (Claude Monet’s Impression Sunrise)
Output Image after Texture Transfer
Input Texture (Claude Monet’s Sunflowers)
Output Image after Texture Transfer
Input Texture (Abanindranath Tagore’s Canal in Shahjadpur)
Output Image after Texture Transfer
Input Texture (Abanindranath Tagore’s The Victory of Buddha)
Output Image after Texture Transfer
Input Texture (Abanindranath Tagore’s Rabindranath in the role of blind singer)
Output Image after Texture Transfer
Input Texture (Abanindranath Tagore’s Slaying the Tornado Demon)
Output Image after Texture Transfer
Input Texture (Nandalal Bose’s Village Huts)
Output Image after Texture Transfer
### Some more Textures
Input Texture
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture
Output Image after Texture Transfer
Input Texture
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture
Output Image after Texture Transfer
Input Texture
Output Image after Texture Transfer
Input Texture
Output Image after Texture Transfer
Input Texture
Output Image after Texture Transfer
Input Texture
Output Image after Texture Transfer
Input Texture
Output Images after Texture Transfer (with 2 different patch sizes)
Input Texture
Output Image after Texture Transfer
Target Image (from The Mummy 1999)
Input Texture (sand)
Output Image after Texture Transfer
The following animation shows how the milk texture is being transformed to the target image of mine with the quilting algorithm with modified code.
Input Texture
Target Image (me)
Output Image after Texture Transfer (with small block size)
The next figures and animations show the output image obtained after milk texture gets transferred to the target image of mine, for different block size of the samples (shown in red). As can be seen from the following outputs, the target image gets more and more prominent as the sampling block size gets smaller and smaller.
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2019-09-19 19:38:23
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https://www.jonathangilligan.org/publications/itano_1993_quantum_measurements/
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, , , , , , , & , “Quantum measurements of trapped ions,” Vistas in Astronomy, 169–183 (1993).
#### Abstract:
« Quantum projection noise | Light scattered from two atoms »
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2019-04-21 08:21:05
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http://mymathforum.com/differential-equations/341590-how-obtain-rlc-circuit-differential.html
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My Math Forum How to obtain an RLC circuit from this differential?
Differential Equations Ordinary and Partial Differential Equations Math Forum
August 28th, 2017, 05:30 AM #1 Newbie Joined: Aug 2017 From: South Africa Posts: 3 Thanks: 0 How to obtain an RLC circuit from this differential? (D^2 + 4D + 13)x(t) = (4D + 2)y(t). I know the term that is squared related to the inductance, the 4D term is related to the capacitance and the term with no D is the resistance. But I am unsure whether I have to solve for y(t) as x(t) is the input and y(t) is the output. Thanks
August 28th, 2017, 06:13 AM #2 Global Moderator Joined: Dec 2006 Posts: 17,914 Thanks: 1382 What instructions came with the equation?
August 28th, 2017, 06:45 AM #3 Newbie Joined: Aug 2017 From: South Africa Posts: 3 Thanks: 0 The instructions were to model an electric circuit using the differential equation above. So as far as I understand, it would be to divide both sides by (4D +2) to get y(t) by itself and then solve from there.
August 28th, 2017, 07:08 AM #4
Senior Member
Joined: Apr 2014
From: Glasgow
Posts: 2,049
Thanks: 680
Math Focus: Physics, mathematical modelling, numerical and computational solutions
Quote:
Originally Posted by Damiene (D^2 + 4D + 13)x(t) = (4D + 2)y(t). I know the term that is squared related to the inductance, the 4D term is related to the capacitance and the term with no D is the resistance. But I am unsure whether I have to solve for y(t) as x(t) is the input and y(t) is the output. Thanks
Your notation is rather odd, but this should help...
Consider a circuit with an ohmic resistor, an inductor and a capacitor. The voltage, $\displaystyle V=V(t)$, for each of these components relates to the charge, $\displaystyle q = q(t)$, using the following equations:
Resistor: $\displaystyle V_R = R \frac{dq}{dt}$
Inductor: $\displaystyle V_L = L \frac{d^2q}{dt^2}$
Capacitor: $\displaystyle V_C = \frac{q}{C}$
In a series RLC circuit,
$\displaystyle V_{total} = V_R + V_L + V_C$
$\displaystyle = L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C}$
If we introduce a new operator 'D' such that $\displaystyle Dq = \frac{dq}{dt}$ and $\displaystyle D^2q = \frac{d^2q}{dt^2}$, then
$\displaystyle V_{total} = (LD^2 + RD + C)q$
Therefore, by comparison:
L = 1 H
R = 4 $\displaystyle \Omega$
C = 13 F
$\displaystyle V_{total} = (4D + 2) y(t)$
To solve, consider dividing through by L and then investigate solutions to second order differential equations
August 28th, 2017, 07:18 AM #5 Newbie Joined: Aug 2017 From: South Africa Posts: 3 Thanks: 0 Thank you for the detailed reply Benit13. Howcome you suggest to divide by L? I stuck on having to solve it as I did Laplace transforms many moons ago.
Tags circuit, differential, obtain, rlc
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Contact - Home - Forums - Cryptocurrency Forum - Top
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2017-09-21 12:33:10
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https://www.jobilize.com/physics-ap/course/26-1-physics-of-the-eye-vision-and-optical-instruments-by-openstax?qcr=www.quizover.com&page=2
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26.1 Physics of the eye (Page 3/5)
Page 3 / 5
The eye can detect an impressive amount of detail, considering how small the image is on the retina. To get some idea of how small the image can be, consider the following example.
Size of image on retina
What is the size of the image on the retina of a $1\text{.}\text{20}×{\text{10}}^{-2}$ cm diameter human hair, held at arm’s length (60.0 cm) away? Take the lens-to-retina distance to be 2.00 cm.
Strategy
We want to find the height of the image ${h}_{i}$ , given the height of the object is ${h}_{o}=1\text{.}\text{20}×{\text{10}}^{-2}$ cm. We also know that the object is 60.0 cm away, so that ${d}_{o}=60.0 cm$ . For clear vision, the image distance must equal the lens-to-retina distance, and so ${d}_{\text{i}}=2.00 cm$ . The equation $\frac{{h}_{\text{i}}}{{h}_{\text{o}}}=-\frac{{d}_{\text{i}}}{{d}_{\text{o}}}=m$ can be used to find ${h}_{i}$ with the known information.
Solution
The only unknown variable in the equation $\frac{{h}_{\text{i}}}{{h}_{\text{o}}}=-\frac{{d}_{\text{i}}}{{d}_{\text{o}}}=m$ is ${h}_{\text{i}}$ :
$\frac{{h}_{\text{i}}}{{h}_{\text{o}}}=-\frac{{d}_{\text{i}}}{{d}_{\text{o}}}.$
Rearranging to isolate ${h}_{\text{i}}$ yields
${h}_{\text{i}}=-{h}_{\text{o}}\cdot \frac{{d}_{\text{i}}}{{d}_{\text{o}}}.$
Substituting the known values gives
$\begin{array}{lll}{h}_{\text{i}}& =& -\left(1.20×{\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}\text{cm}\right)\frac{2.00 cm}{\text{60.0 cm}}\\ & =& -4.00×{\text{10}}^{-4}\phantom{\rule{0.25em}{0ex}}\text{cm}.\end{array}$
Discussion
This truly small image is not the smallest discernible—that is, the limit to visual acuity is even smaller than this. Limitations on visual acuity have to do with the wave properties of light and will be discussed in the next chapter. Some limitation is also due to the inherent anatomy of the eye and processing that occurs in our brain.
Power range of the eye
Calculate the power of the eye when viewing objects at the greatest and smallest distances possible with normal vision, assuming a lens-to-retina distance of 2.00 cm (a typical value).
Strategy
For clear vision, the image must be on the retina, and so ${d}_{\text{i}}=2.00 cm$ here. For distant vision, ${d}_{o}\approx \infty$ , and for close vision, ${d}_{o}=25.0 cm$ , as discussed earlier. The equation $P=\frac{1}{{d}_{\text{o}}}+\frac{1}{{d}_{\text{i}}}$ as written just above, can be used directly to solve for $P$ in both cases, since we know ${d}_{\text{i}}$ and ${d}_{o}$ . Power has units of diopters, where $\text{1 D}=\text{1/m}$ , and so we should express all distances in meters.
Solution
For distant vision,
$P=\frac{1}{{d}_{\text{o}}}+\frac{1}{{d}_{\text{i}}}=\frac{1}{\infty }+\frac{1}{\text{0.0200 m}}\text{.}$
Since $1/\infty =0$ , this gives
$P=0+\text{50}\text{.}0/\text{m}=\text{50.0 D (distant vision).}$
Now, for close vision,
$\begin{array}{lll}P& =& \frac{1}{{d}_{\text{o}}}+\frac{1}{{d}_{\text{i}}}=\frac{1}{\text{0.250 m}}+\frac{1}{\text{0.0200 m}}\\ & =& \frac{\text{4.00}}{\text{m}}+\frac{50.0}{\text{m}}=\text{4.00 D}+\text{50.0 D}\\ & =& \text{54.0 D (close vision).}\end{array}$
Discussion
For an eye with this typical 2.00 cm lens-to-retina distance, the power of the eye ranges from 50.0 D (for distant totally relaxed vision) to 54.0 D (for close fully accommodated vision), which is an 8% increase. This increase in power for close vision is consistent with the preceding discussion and the ray tracing in [link] . An 8% ability to accommodate is considered normal but is typical for people who are about 40 years old. Younger people have greater accommodation ability, whereas older people gradually lose the ability to accommodate. When an optometrist identifies accommodation as a problem in elder people, it is most likely due to stiffening of the lens. The lens of the eye changes with age in ways that tend to preserve the ability to see distant objects clearly but do not allow the eye to accommodate for close vision, a condition called presbyopia (literally, elder eye). To correct this vision defect, we place a converging, positive power lens in front of the eye, such as found in reading glasses. Commonly available reading glasses are rated by their power in diopters, typically ranging from 1.0 to 3.5 D.
Section summary
• Image formation by the eye is adequately described by the thin lens equations:
$P=\frac{1}{{d}_{\text{o}}}+\frac{1}{{d}_{\text{i}}}\phantom{\rule{0.25em}{0ex}}\text{and}\phantom{\rule{0.25em}{0ex}}\frac{{h}_{\text{i}}}{{h}_{\text{o}}}=-\frac{{d}_{\text{i}}}{{d}_{\text{o}}}=m.$
• The eye produces a real image on the retina by adjusting its focal length and power in a process called accommodation.
• For close vision, the eye is fully accommodated and has its greatest power, whereas for distant vision, it is totally relaxed and has its smallest power.
• The loss of the ability to accommodate with age is called presbyopia, which is corrected by the use of a converging lens to add power for close vision.
Conceptual questions
If the lens of a person’s eye is removed because of cataracts (as has been done since ancient times), why would you expect a spectacle lens of about 16 D to be prescribed?
A cataract is cloudiness in the lens of the eye. Is light dispersed or diffused by it?
When laser light is shone into a relaxed normal-vision eye to repair a tear by spot-welding the retina to the back of the eye, the rays entering the eye must be parallel. Why?
How does the power of a dry contact lens compare with its power when resting on the tear layer of the eye? Explain.
Why is your vision so blurry when you open your eyes while swimming under water? How does a face mask enable clear vision?
Problem exercises
Unless otherwise stated, the lens-to-retina distance is 2.00 cm.
What is the power of the eye when viewing an object 50.0 cm away?
$\text{52.0 D}$
Calculate the power of the eye when viewing an object 3.00 m away.
(a) The print in many books averages 3.50 mm in height. How high is the image of the print on the retina when the book is held 30.0 cm from the eye?
(b) Compare the size of the print to the sizes of rods and cones in the fovea and discuss the possible details observable in the letters. (The eye-brain system can perform better because of interconnections and higher order image processing.)
(a) $-0\text{.}\text{233 mm}$
(b) The size of the rods and the cones is smaller than the image height, so we can distinguish letters on a page.
Suppose a certain person’s visual acuity is such that he can see objects clearly that form an image $4.00 \mu m$ high on his retina. What is the maximum distance at which he can read the 75.0 cm high letters on the side of an airplane?
People who do very detailed work close up, such as jewellers, often can see objects clearly at much closer distance than the normal 25 cm.
(a) What is the power of the eyes of a woman who can see an object clearly at a distance of only 8.00 cm?
(b) What is the size of an image of a 1.00 mm object, such as lettering inside a ring, held at this distance?
(c) What would the size of the image be if the object were held at the normal 25.0 cm distance?
(a) $+62.5 D$
(b) $–0.250 mm$
(c) $–0.0800 mm$
how lesers can transmit information
griffts bridge derivative
below me
please explain; when a glass rod is rubbed with silk, it becomes positive and the silk becomes negative- yet both attracts dust. does dust have third types of charge that is attracted to both positive and negative
what is a conductor
Timothy
hello
Timothy
below me
why below you
Timothy
no....I said below me ...... nothing below .....ok?
dust particles contains both positive and negative charge particles
Mbutene
corona charge can verify
Stephen
when pressure increases the temperature remain what?
what is frequency
define precision briefly
CT scanners do not detect details smaller than about 0.5 mm. Is this limitation due to the wavelength of x rays? Explain.
hope this helps
what's critical angle
The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
okay whatever
Chidalu
pls who can give the definition of relative density?
Temiloluwa
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas.
Chidalu
What is momentum
mass ×velocity
Chidalu
it is the product of mass ×velocity of an object
Chidalu
how do I highlight a sentence]p? I select the sentence but get options like copy or web search but no highlight. tks. src
then you can edit your work anyway you want
Wat is the relationship between Instataneous velocity
Instantaneous velocity is defined as the rate of change of position for a time interval which is almost equal to zero
Astronomy
The potential in a region between x= 0 and x = 6.00 m lis V= a+ bx, where a = 10.0 V and b = -7.00 V/m. Determine (a) the potential atx=0, 3.00 m, and 6.00 m and (b) the magnitude and direction of the electric ficld at x =0, 3.00 m, and 6.00 m.
what is energy
hi all?
GIDEON
hey
Bitrus
energy is when you finally get up of your lazy azz and do some real work 😁
what is physics
what are the basic of physics
faith
base itself is physics
Vishlawath
tree physical properties of heat
tree is a type of organism that grows very tall and have a wood trunk and branches with leaves... how is that related to heat? what did you smoke man?
algum profe sabe .. Progressivo ou Retrógrado e Acelerado ou Retardado V= +23 m/s V= +5 m/s 0__> 0__> __________________________> T= 0 T=6s
Claudia
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2020-09-21 10:23:52
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https://socratic.org/questions/how-do-you-convert-25-000-mu-m-m
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# How do you convert 25,000 mum = m?
Mar 18, 2018
See a solution process below:
#### Explanation:
The conversion factor for micro meters to meters is:
$1 \mu \text{m" = 0.000001"m}$
To find how many meters in $25 , 000 \mu \text{m}$ we multiply each side of the conversion equation by $\textcolor{red}{25 , 000}$:
$\textcolor{red}{25 , 000} \times 1 \mu \text{m" = color(red)(25,000) xx 0.000001"m}$
$25 , 000 \mu \text{m" = 0.025"m}$
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2022-01-18 00:48:53
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http://www.citeulike.org/user/NitinCR
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Tags
# NitinCR's library 5396 articles
## ✔ Supersymmetry Breaking in Anti-de Sitter spacetime
[CiTO]
(29 Apr 2013)
posted to no-tag by NitinCR on 2013-04-30 09:09:32
### Abstract
We study the questions of how supersymmetry is spontaneously broken in Anti de-Sitter spacetime. We verify that the would-be R-symmetry in $AdS_4$ plays a central role for the existence of meta-stable supersymmetry breaking. To illustrate, some well-known models such as Poloyni models and O'Raifeartaigh models are investigated in detail. Our calculations are reliable in flat spacetime limit and confirm us that meta-stable vacua are generic even though quantum corrections are taken into account. ...
## ✔ Tensor categories and the mathematics of rational and logarithmic conformal field theory
[CiTO]
(29 Apr 2013)
posted to no-tag by NitinCR on 2013-04-30 09:09:07
### Abstract
We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. For rational and logarithmic conformal field theories, we review the ...
## ✔ Complex Classical Fields and Partial Wick Rotations
[CiTO]
(24 Feb 2013)
posted to no-tag by NitinCR on 2013-02-26 09:12:15
### Abstract
We study some examples of complex, classical, scalar fields within the new framework that we introduced in a previous work. In these particular examples, we replace the usual functional integral by a complex functional arising from partial Wick rotation of a quantum field. We generalize the Feynman-Kac relation to this setting, and use it to establish the spectral condition on a cylinder. We also consider positive-temperature states. ...
## ✔ TASI 2012: Super-Tricks for Superspace
[CiTO]
(25 Feb 2013)
posted to no-tag by NitinCR on 2013-02-26 08:56:35 along with 1 person
### Abstract
These lectures from the TASI 2012 summer school outline the basics of supersymmetry (SUSY) in 3+1 dimensions. Starting from a ground-up development of superspace, we develop all of the tools necessary to construct SUSY lagrangians. While aimed at an introductory level, these lectures incorporate a number of "super-tricks" for SUSY aficionados, including SUSY-covariant derivatives, equations of motion in superspace, background field methods, and non-linear realizations of goldstinos. ...
## ✔ Graph and Network Theory in Physics. A Short Introduction
[CiTO]
(18 Feb 2013)
posted to no-tag by NitinCR on 2013-02-19 09:35:49 along with 2 people and 1 group
### Abstract
A book Chapter consisting of some of the main areas of research in graph theory applied to physics. It includes graphs in condensed matter theory, such as the tight-binding and the Hubbard model. It follows the study of graph theory and statistical physics by means of the analysis of the Potts model. Then, we consider the use of graph polynomials in solving Feynman integrals, graphs and electrical networks, vibrational analysis in networked systems and random graphs. The second part deals with the study of complex networks and includes ...
## ✔ Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications
[CiTO]
(18 Feb 2013)
posted to no-tag by NitinCR on 2013-02-19 09:34:26
### Abstract
We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n->0 of the O(n) model, and percolation as ...
## ✔ Hidden Ghost in Massive gravity
[CiTO]
(18 Feb 2013)
posted to no-tag by NitinCR on 2013-02-19 09:34:16
### Abstract
The Hessian's determinant for a version of massive gravity given by an infinite expansion of a square root function of the induced metric, vanishes. We show that it allows us to eliminate one of four scalar fields used to generate the graviton mass. This, however, gives rise to the appearance of extra terms in the action with the squared time derivative of the metric, thus signaling that a nonlinear ghost survives.We demonstrate this phenomenon considering a simple system with constraint, which is supposed to reduces the number of ...
## ✔ A lecture note on scale invariance vs conformal invariance
[CiTO]
(4 Feb 2013)
posted to no-tag by NitinCR on 2013-02-06 08:28:06 along with 1 person
### Abstract
In this lecture note, we discuss the distinction and possible equivalence between scale invariance and conformal invariance in relativistic quantum field theories. As of January 2013, our consensus is that there is no known example of scale invariant but non-conformal field theories in d=4 under the assumptions of (1) unitarity, (2) Poincaré invariance (causality), (3) discrete spectrum in scaling dimension, (4) existence of scale current and (5) unbroken scale invariance. We have a perturbative proof based on the higher dimensional analogue of Zamolodchikov's c-theorem, but the non-perturbative proof is ...
## ✔ From the Virasoro Algebra to Krichever--Novikov Type Algebras and Beyond
[CiTO]
(31 Jan 2013)
posted to no-tag by NitinCR on 2013-02-01 13:22:47
### Abstract
Starting from the Virasoro algebra and its relatives the generalization to higher genus compact Riemann surfaces was initiated by Krichever and Novikov. The elements of these algebras are meromorphic objects which are holomorphic outside a finite set of points. A crucial and non-trivial point is to establish an almost-grading replacing the honest grading in the Virasoro case. Such an almost-grading is given by splitting the set of points of possible poles into two non-empty disjoint subsets. Krichever and Novikov considered the two-point case. Schlichenmaier studied the most general ...
## ✔ Taming the zoo of supersymmetric quantum mechanical models
[CiTO]
(19 Feb 2013)
posted to no-tag by NitinCR on 2013-02-01 12:49:45 along with 1 person
### Abstract
We show that in many cases nontrivial and complicated supersymmetric quantum mechanical (SQM) models can be obtained from the simple model describing free dynamics in flat complex space by two operations: (i) Hamiltonian reduction and (ii) similarity transformation of the complex supercharges. We conjecture that it is true for any SQM model. ...
## ✔ On propagators in de Sitter space
[CiTO]
(30 Jan 2013)
posted to no-tag by NitinCR on 2013-01-31 11:35:59
### Abstract
In a spacetime with no global timelike Killing vector, we do not have a natural choice for the vacuum state of matter fields, leading to the ambiguity in defining the Feynman propagators. In this paper, choosing the vacuum state as the instantaneous ground state of the Hamiltonian at each moment, we develop a method for calculating wave functions associated with the vacuum and the corresponding in-in and in-out propagators. We apply this method to free scalar field theory in de Sitter space and obtain de Sitter invariant ...
## ✔ Phases of large $N$ vector Chern-Simons theories on $S^2 × S^1$
[CiTO]
(25 Jan 2013)
posted to no-tag by NitinCR on 2013-01-29 10:00:48 along with 1 person
### Abstract
We study the thermal partition function of level $k$ U(N) Chern-Simons theories on $S^2$ interacting with matter in the fundamental representation. We work in the 't Hooft limit, $N,k\to∞$, with $λ = N/k$ and $\fracT^2 V_2N$ held fixed where $T$ is the temperature and $V_2$ the volume of the sphere. An effective action proposed in <a href="/abs/1211.4843">arXiv:1211.4843</a> relates the partition function to the expectation value of a `potential' function of the $S^1$ holonomy in pure Chern-Simons theory; in several examples we compute the holonomy potential as a function ...
## ✔ Defects, Super-Poincaré line bundle and Fermionic T-duality
[CiTO]
(28 Jan 2013)
posted to no-tag by NitinCR on 2013-01-29 09:49:16
### Abstract
Topological defects are interfaces joining two conformal field theories, for which the energy momentum tensor is continuous across the interface. A class of the topological defects is provided by the interfaces separating two bulk systems each described by its own Lagrangian, where the two descriptions are related by a discrete symmetry. In this paper we elaborate on the cases in which the discrete symmetry is a bosonic or a fermionic T- duality. We review how the equations of motion imposed by the defect encode the general bosonic T- ...
## ✔ On non-equilibrium physics and string theory
[CiTO]
(27 Jan 2013)
posted to no-tag by NitinCR on 2013-01-29 09:48:06
### Abstract
In this article we review the relation between string theory and non-equilibrium physics based on our previously published work. First we explain why a theory of quantum gravity and non-equilibrium statistical physics should be related in the first place. Then we present the necessary background from the recent research in non-equilibrium physics. The review discusses the relationship of string theory and aging phenomena, as well as the connection between AdS/CFT correspondence and the Jarzynski identity. We also discuss the emergent symmetries in fully developed turbulence and the corresponding non-equilibrium ...
## ✔ Time Dependence of Hawking Radiation Entropy
[CiTO]
(21 Jan 2013)
posted to no-tag by NitinCR on 2013-01-24 05:46:59 along with 1 person
### Abstract
If a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared. Here numerical results are given for an approximation to the time dependence of the radiation entropy under an assumption of fast scrambling, for large nonrotating black holes that emit essentially only photons and gravitons. The maximum of the von Neumann entropy then occurs after about 53.81% of the evaporation ...
## ✔ Quantum Gravity: the view from particle physics
[CiTO]
(23 Jan 2013)
posted to no-tag by NitinCR on 2013-01-24 05:46:25 along with 1 person
### Abstract
This lecture reviews aspects of and prospects for progress towards a theory of quantum gravity from a particle physics perspective, also paying attention to recent findings of the LHC experiments at CERN. ...
## ✔ Black hole entropy and the renormalization group
[CiTO]
(14 Jan 2013)
posted to no-tag by NitinCR on 2013-01-16 13:41:28
### Abstract
Four decades after its first postulation by Bekenstein, black hole entropy remains mysterious. It has long been suggested that the entanglement entropy of quantum fields on the black hole gravitational background should represent at least an important contribution to the total Bekenstein-Hawking entropy, and that the divergences in the entanglement entropy should be absorbed in the renormalization of the gravitational couplings. In this talk, we describe how an improved understanding of black hole entropy is obtained by combining these notions with the renormalization group. By introducing an RG ...
## ✔ Gauge networks in noncommutative geometry
[CiTO]
(15 Jan 2013)
posted to no-tag by NitinCR on 2013-01-16 13:41:23
### Abstract
We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almost-commutative manifolds. The configuration space of quiver representations (modulo equivalence) in the category of finite spectral triples is studied; gauge networks appear as an orthonormal basis in a corresponding Hilbert space. We give many examples of gauge networks, also beyond the well-known spin network examples. We find a Hamiltonian operator on this Hilbert space, inducing a time evolution on the C*-algebra of gauge network correspondences. Given a representation in the category of spectral triples of ...
## ✔ M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems
[CiTO]
(9 Jan 2013)
posted to no-tag by NitinCR on 2013-01-14 18:30:13 along with 2 people
### Abstract
In Part I, we extend our analysis in [<a href="/abs/0807.1107">arXiv:0807.1107</a>], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part II, to the setup in ...
## ✔ Noncommutative spectral geometry and the deformed Hopf algebra structure of quantum field theory
[CiTO]
(11 Jan 2013)
posted to no-tag by NitinCR on 2013-01-14 18:11:20
### Abstract
We report the results obtained in the study of Alain Connes noncommutative spectral geometry construction focusing on its essential ingredient of the algebra doubling. We show that such a two-sheeted structure is related with the gauge structure of the theory, its dissipative character and carries in itself the seeds of quantization. From the algebraic point of view, the algebra doubling process has the same structure of the deformed Hops algebra structure which characterizes quantum field theory. ...
## ✔ A Snapshot of Foundational Attitudes Toward Quantum Mechanics
[CiTO]
(6 Jan 2013)
posted to no-tag by NitinCR on 2013-01-14 13:39:43 along with 4 people
### Abstract
Foundational investigations in quantum mechanics, both experimental and theoretical, gave birth to the field of quantum information science. Nevertheless, the foundations of quantum mechanics themselves remain hotly debated in the scientific community, and no consensus on essential questions has been reached. Here, we present the results of a poll carried out among 33 participants of a conference on the foundations of quantum mechanics. The participants completed a questionnaire containing 16 multiple-choice questions probing opinions on quantum-foundational issues. Participants included physicists, philosophers, and mathematicians. We describe our findings, identify commonly held ...
## ✔ A higher stacky perspective on Chern-Simons theory
[CiTO]
(11 Jan 2013)
posted to no-tag by NitinCR on 2013-01-14 13:39:05 along with 1 person
### Abstract
The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized ("extended", "multi-tiered") to a map on the universal moduli stack of principal connections, a map that itself modulates a circle-principal 3-connection on that moduli stack, and how the iterated transgressions of this extended Lagrangian unify the action functional with its prequantum bundle and with the WZW-functional. ...
## ✔ Oscillons After Inflation
[CiTO]
Physical Review Letters, Vol. 108, No. 24. (21 Nov 2011), doi:10.1103/physrevlett.108.241302
posted to no-tag by NitinCR on 2013-01-09 06:35:18 along with 3 people
### Abstract
Oscillons are massive, long-lived, localized excitations of a scalar field. We show that in a large class of well-motivated single-field models, inflation is followed by self-resonance, leading to copious oscillon generation and a lengthy period of oscillon domination. These models are characterized by an inflaton potential which has a quadratic minimum and is shallower than quadratic away from the minimum. This set includes both string monodromy models and a class of supergravity inspired scenarios, and is in good agreement with the current central values of the concordance cosmology ...
## ✔ Kolmogorov complexity as a hidden factor of scientific discourse: from Newton's law to data mining
[CiTO]
(1 Jan 2013)
posted to no-tag by NitinCR on 2013-01-09 06:35:16
### Abstract
The word "complexity" is most often used as a meta--linguistic expression referring to certain intuitive characteristics of a natural system and/or its scientific description. These characteristics may include: sheer amount of data that must be taken into account; visible "chaotic" character of these data and/or space distribution/time evolution of a system etc. This talk is centered around the precise mathematical notion of "Kolmogorov complexity", originated in the early theoretical computer science and measuring the degree to which an available information can be compressed. In the first part, I will ...
## ✔ On abelianizations of the ABJM model and applications to condensed matter
[CiTO]
(2 Jan 2013)
posted to no-tag by NitinCR on 2013-01-09 06:35:15
### Abstract
In applications of AdS/CFT to condensed matter systems in 2+1 dimensions, the ABJM model is often used, however the condensed matter models are usually abelian and contain charged fields. We show that a naive reduction of the ABJM model to N=1 does not have the desired features, but we can find an abelian reduction that has most features, and we can also add fundamental fields to the ABJM model to obtain other models with similar properties. ...
## ✔ Large N anomalous dimensions for large operators in Leigh-Strassler deformed SYM
[CiTO]
(29 Dec 2012)
posted to no-tag by NitinCR on 2013-01-09 06:35:01
### Abstract
We study the large N anomalous dimensions of operators in a Leigh-Strassler deformation of N=4 super Yang-Mills theory. The operators that we study have a bare dimension of order N (so that the large N limit is not captured by planar diagrams) and are AdS/CFT dual to giant gravitons. The diagonalization of the dilatation operator factorizes into two problems. One of these problems is solved using a double coset ansatz. The second problem is equivalent to a set of decoupled harmonic oscillators. ...
## ✔ The generalised complex geometry of Wess-Zumino-Witten models
[CiTO]
(2 Jan 2013)
posted to no-tag by NitinCR on 2013-01-03 02:09:55
### Abstract
In this work a thorough study of a number of specific supersymmetric sigma-models with extended supersymmetry is performed within the context of generalised complex geometry, more specially the supersymmetric Wess-Zumino-Witten model on a variety of group manifolds. By explicitly calculating the admissible complex structures and the associated pure spinors on the target manifold a full characterisation of the different possible geometries is provided. By using this approach the various aspects of generalised Kaehler geometry can be studied in detail. Also considered are the various isometries present in the model ...
## ✔ Superconductivity, Superfluidity and Holography
[CiTO]
(2 Jan 2013)
posted to no-tag by NitinCR on 2013-01-03 02:09:52 along with 1 person
### Abstract
This is a concise review of holographic superconductors and superfluids. We highlight some predictions of the holographic models and the emphasis is given to physical aspects rather than to the technical details, although some references to understand the latter are systematically provided. We include gapped systems in the discussion, motivated by the physics of high-temperature superconductivity. In order to do so we consider a compactified extra dimension (with radius R), or, alternatively, a dilatonic field. The first setup can also be used to model cylindrical superconductors; when these ...
## ✔ Scattering Amplitudes and the Positive Grassmannian
[CiTO]
(21 Dec 2012)
posted to no-tag by NitinCR on 2013-01-01 10:43:51 along with 1 person
### Abstract
We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams as objects of fundamental importance to scattering amplitudes. We show that the all-loop integrand in N=4 SYM is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful stratification of the Grassmannian G(k,n) which generalizes the notion ...
## ✔ Black holes and Hawking radiation in spacetime and its analogues
[CiTO]
(31 Dec 2012)
posted to no-tag by NitinCR on 2013-01-01 10:12:19 along with 2 people and 1 group
### Abstract
These notes introduce the fundamentals of black hole geometry, the thermality of the vacuum, and the Hawking effect, in spacetime and its analogues. Stimulated emission of Hawking radiation, the trans-Planckian question, short wavelength dispersion, and white hole radiation in the setting of analogue models are also discussed. No prior knowledge of differential geometry, general relativity, or quantum field theory in curved spacetime is assumed. ...
## ✔ Varna Lecture on L^2-Analysis of Minimal Representations
[CiTO]
posted to no-tag by NitinCR on 2013-01-01 09:56:43
### Abstract
Minimal representations of a real reductive group G are the "smallest" irreducible unitary representations of G. The author suggests a program of global analysis built on minimal representations from the philosophy: small representation of a group = large symmetries in a representation space. This viewpoint serves as a driving force to interact algebraic representation theory with geometric analysis of minimal representations, yielding a rapid progress on the program. We give a brief guidance to recent works with emphasis on the Schroedinger model. ...
## ✔ Conformal supergravities as Chern-Simons theories revisited
[CiTO]
(31 Dec 2012)
posted to no-tag by NitinCR on 2013-01-01 09:56:42
### Abstract
We propose a superfield method to construct off-shell actions for N-extended conformal supergravity theories in three space-time dimensions. It makes use of the superform technique to engineer supersymmetric invariants. The method is specifically applied to the case of N=1 conformal supergravity and provides a new realization for the actions of conformal and topological massive supergravities. ...
## ✔ Cluster algebras: an introduction
[CiTO]
(17 Mar 2013)
posted to by NitinCR on 2013-01-01 04:51:17 along with 3 people
### Abstract
Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by Fomin and Zelevinsky in 2000 in the context of Lie theory, but have since appeared in many other contexts, from Poisson geometry to triangulations of surfaces and Teichmüller theory. In this expository paper we give an introduction to cluster algebras, and illustrate how this framework naturally arises in Teichmüller theory. We then sketch how the theory of cluster algebras led to a proof of the Zamolodchikov periodicity conjecture ...
## ✔ Ranks of gauge groups and an NP-complete problem on the Landscape
[CiTO]
(17 Dec 2012)
posted to no-tag by NitinCR on 2012-12-18 15:57:24
### Abstract
We prove that the problem of determination of factor gauge groups given the rank of the gauge group at any given vacuum in the Landscape is in the computational complexity class NPC. This extends a result of Denef and Douglas on the computational complexity of determination of the value of the cosmological constant in the Landscape. ...
## ✔ The Partition Function of ABJ Theory
[CiTO]
(12 Dec 2012)
posted to no-tag by NitinCR on 2012-12-14 09:37:01
### Abstract
We study the partition function of the N=6 supersymmetric U(N_1)_k x U(N_2)_-k Chern-Simons-matter (CSM) theory, also known as the ABJ theory. For this purpose, we first compute the partition function of the U(N_1) x U(N_2) lens space matrix model exactly. The result can be expressed as a product of q-deformed Barnes G-function and a generalization of multiple q-hypergeometric function. The ABJ partition function is then obtained from the lens space partition function by analytically continuing N_2 to -N_2. The answer is given by min(N_1,N_2)-dimensional integrals and generalizes the ...
## ✔ A Biography of Henri Poincaré - 2012 Centenary of the Death of Poincaré
[CiTO]
(3 Jul 2012)
posted to by NitinCR on 2012-12-10 14:31:39 along with 1 person
### Abstract
On January 4, 2012, the centenary of Henri Poincaré's death, a colloquium was held in Nancy, France the subject of which was "Vers une biographie d'Henri Poincaré". Scholars discussed several approaches for writing a biography of Poincaré. In this paper I present a personal and scientific biographical sketch of Poincaré, which does not in any way reflect Poincaré's rich personality and immense activity in science: When Poincaré traveled to parts of Europe, Africa and America, his companions noticed that he knew well everything from statistics to history and ...
## ✔ Relative quantum field theory
[CiTO]
(7 Dec 2012)
posted to no-tag by NitinCR on 2012-12-10 10:45:48
### Abstract
We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions. ...
## ✔ Supergravity as Generalised Geometry II: $E_d(d) × \mathbbR^+$ and M theory
[CiTO]
(7 Dec 2012)
posted to no-tag by NitinCR on 2012-12-10 10:45:45 along with 1 person
### Abstract
We reformulate eleven-dimensional supergravity, including fermions, in terms of generalised geometry, for spacetimes that are warped products of Minkowski space with a $d$-dimensional manifold $M$ with $d≤7$. The reformation has a $E_d(d) × \mathbbR^+$ structure group and is has a local $H_d$ symmetry, where $H_d$ is the double cover of the maximally compact subgroup of $E_d(d)$. The bosonic degrees for freedom unify into a generalised metric, and, defining the generalised analogue $D$ of the Levi-Civita connection, one finds that the corresponding equations of motion are the vanishing of the ...
## ✔ Brane instantons and fluxes in F-theory
[CiTO]
(30 Nov 2012)
posted to no-tag by NitinCR on 2012-12-05 09:29:09
### Abstract
We study the combined effect of world-volume and background fluxes on Euclidean D3-brane instantons in F-theory compactifications. We derive an appropriate form of the fermionic effective action, in which the fermions are topologically twisted and the dynamical effect of fluxes, non-trivial axio-dilaton and warping is taken into account. We study the structure of fermionic zero modes, which determines the form of possible non-perturbative superpotential and F-terms in the four-dimensional effective action. Invariance under SL(2,Z) is discussed in detail, which allows for an interpretation of the results in terms of ...
## ✔ Universality for random matrices and log-gases
[CiTO]
(4 Dec 2012)
posted to no-tag by NitinCR on 2012-12-05 09:28:59
### Abstract
Eugene Wigner's revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior: the statistics of energy gaps depend only on the basic symmetry type of the model. Simplified models of Wigner's thesis have recently become mathematically accessible. For mean field models represented by large random matrices with independent entries, the celebrated Wigner-Dyson-Gaudin-Mehta (WDGM) conjecture asserts that the local eigenvalue statistics are universal. For invariant matrix models, the eigenvalue distributions are given by a log-gas with potential $V$ and inverse temperature \$β = 1, ...
## ✔ The String Landscape: On Formulas for Counting Vacua
[CiTO]
(3 Dec 2012)
posted to no-tag by NitinCR on 2012-12-05 08:58:16 along with 2 people
### Abstract
We derive formulas for counting certain classes of vacua in the string/M theory landscape. We do so in the context of the moduli space of M-theory compactifications on singular manifolds with G_2 holonomy. Particularly, we count the numbers of gauge theories with different gauge groups but equal numbers of U(1) factors which are dual to each other. The vacua correspond to various symmetry breaking patterns of grand unified theories. Counting these dual vacua is equivalent to counting the number of conjugacy classes of elements of finite order inside ...
## ✔ Black Hole Macro-Quantumness
[CiTO]
(4 Dec 2012)
posted to no-tag by NitinCR on 2012-12-05 08:55:58 along with 1 person
### Abstract
It is a common wisdom that properties of macroscopic bodies are well described by (semi)classical physics. As we have suggested this wisdom is not applicable to black holes. Despite being macroscopic, black holes are quantum objects. They represent Bose-Einstein condensates of N-soft gravitons at the quantum critical point, where N Bogoliubov modes become gapless. As a result, physics governing arbitrarily-large black holes (e.g., of galactic size) is a quantum physics of the collective Bogoiliubov modes. This fact introduces a new intrinsically-quantum corrections in form of 1/N, as opposed ...
## ✔ Models of Particle Physics from Type IIB String Theory and F-theory: A Review
[CiTO]
(3 Dec 2012)
posted to no-tag by NitinCR on 2012-12-05 08:53:35 along with 1 person
### Abstract
We review particle physics model building in type IIB string theory and F-theory. This is a region in the landscape where in principle many of the key ingredients required for a realistic model of particle physics can be combined successfully. We begin by reviewing moduli stabilisation within this framework and its implications for supersymmetry breaking. We then review model building tools and developments in the weakly coupled type IIB limit, for both local D3-branes at singularities and global models of intersecting D7-branes. Much of recent model building work ...
## ✔ Nonviolent nonlocality
[CiTO]
(29 Nov 2012)
posted to no-tag by NitinCR on 2012-12-04 15:27:19 along with 1 person
### Abstract
If quantum mechanics governs nature, black holes must evolve unitarily, providing a powerful constraint on the dynamics of quantum gravity. Such evolution apparently must in particular be nonlocal, when described from the usual semiclassical geometric picture, in order to transfer quantum information into the outgoing state. While such transfer from a disintegrating black hole has the dangerous potential to be violent to generic infalling observers, this paper proposes the existence of a more innocuous form of information transfer, to relatively soft modes in the black hole atmosphere. Simplified ...
## ✔ On total masses in GR
[CiTO]
(1 Dec 2012)
posted to no-tag by NitinCR on 2012-12-04 15:20:57
### Abstract
The total mass, the Witten type gauge conditions and the spectral properties of the Sen--Witten and the 3-surface twistor operators in closed universes are investigated. A non-negative expression, built from the norm of the 3-surface twistor operator and the energy-momentum tensor of the matter fields on a spacelike hypersurface, is found which, in the asymptotically flat/hyperboloidal case, provides a lower bound for the ADM/Bondi--Sachs mass. In closed universes the analogous expression coincides with the first eigenvalue of the Sen--Witten operator, and it is vanishing if and only if ...
## ✔ An Infalling Observer in AdS/CFT
[CiTO]
(28 Nov 2012)
posted to no-tag by NitinCR on 2012-11-30 07:16:25 along with 2 people
### Abstract
We describe the experience of an observer falling into a black hole using the AdS/CFT correspondence. In order to do this, we reconstruct the local bulk operators measured by the observer along his trajectory outside the black hole. We then extend our construction beyond the black hole horizon. We show that this is possible because of an effective doubling of the observables in the boundary theory, when it is in a pure state that is close to the thermal state. Our construction allows us to rephrase questions ...
## ✔ Multiple M-branes and 3-algebras
[CiTO]
(29 Nov 2012)
posted to no-tag by NitinCR on 2012-11-30 07:14:36
### Abstract
The purpose of this thesis is to explore the properties of multiple coincident M2- and M5-branes. We begin with a review of the BLG and ABJM models of multiple M2-branes and our focus will be on their formulation in terms of 3-algebras. We then examine the coupling of multiple M2-branes to the background 3-form and 6-form gauge fields of eleven-dimensional supergravity. In particular we show in detail how a natural generalisation of the Myers flux-terms, along with the resulting curvature of the background metric, leads to mass terms ...
## ✔ On the history of the strong interaction
[CiTO]
(28 Nov 2012)
posted to no-tag by NitinCR on 2012-11-30 07:14:34 along with 3 people
### Abstract
These lecture notes recall the conceptual developments which led from the discovery of the neutron to our present understanding of strong interaction physics. ...
## ✔ Dimensional Reduction without Extra Continuous Dimensions
[CiTO]
(27 Nov 2012)
posted to no-tag by NitinCR on 2012-11-30 07:10:55
### Abstract
We describe a novel approach to dimensional reduction in classical field theory. Inspired by ideas from noncommutative geometry, we introduce extended algebras of differential forms over space-time, generalized exterior derivatives and generalized connections associated with the "geometry" of space-times with discrete extra dimensions. We apply our formalism to theories of gauge- and gravitational fields and find natural geometrical origins for an axion- and a dilaton field, as well as a Higgs field. ...
## ✔ Preference for a Vanishingly Small Cosmological Constant in Supersymmetric Vacua in a Type IIB String Theory Model
[CiTO]
(29 Nov 2012)
posted to no-tag by NitinCR on 2012-11-30 07:10:03 along with 1 person
### Abstract
We study the probability distribution P(Λ) of the cosmological constant Λ in a specific set of KKLT type models of supersymmetric IIB vacua. P(Λ) is divergent at Λ =0^- and the likely value of Λ drops exponentially as the number of complex structure moduli h^2,1 increases. Also, owing to the hierarchical and approximate no-scale structure, the probability of having a positive Hessian (mass squared matrix) approaches unity as h^2,1 increases. ...
|
2013-05-24 04:40:29
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https://www.zbmath.org/serials/?q=se%3A90
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# zbMATH — the first resource for mathematics
## IEEE Transactions on Information Theory
### A Journal devoted to the Theoretical and Experimental Aspects of Information Transmission, Processing and Utilization
Short Title: IEEE Trans. Inf. Theory Publisher: Institute of Electrical and Electronics Engineers (IEEE), New York, NY ISSN: 0018-9448 Online: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18 Predecessor: IRE Transactions on Information Theory Comments: Indexed cover-to-cover
Documents Indexed: 14,672 Publications (since 1963) References Indexed: 2 Publications with 25 References.
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#### Authors
164 Merhav, Neri 156 Shamai (Shitz), Shlomo 146 Poor, Harold Vincent 125 Verdú, Sergio 108 Helleseth, Tor 83 Caire, Giuseppe 77 Weissman, Tsachy 76 Xing, Chaoping 71 Jafar, Syed Ali 68 Etzion, Tuvi 67 Ding, Cunsheng 66 Calderbank, Arthur Robert 65 Ziv, Jacob 62 Bruck, Jehoshua 62 Roth, Ron M. 62 Vardy, Alexander 61 Lin, Shu 60 Siegel, Paul H. 60 Tan, Vincent Yan Fu 59 Kløve, Torleiv 58 Goldsmith, Andrea J. 57 Berger, Toby 57 Lapidoth, Amos 56 Khandani, Amir Keyvan 56 Varanasi, Mahesh Kumar 55 Zeger, Kenneth 54 Tang, Xiaohu 54 Tse, David N. C. 53 Kumar, P. Vijay 53 Permuter, Haim Henri 52 Feder, Meir 51 Barg, Alexander M. 51 Litsyn, Simon N. 51 Yeung, Raymond W. 49 Zhang, Zhen 48 Costello, Daniel J. jun. 48 Ulukus, Sennur 47 Zamir, Ram 46 Ahlswede, Rudolf 46 Wolf, Jack Keil 46 Yaakobi, Eitan 45 Gong, Guang 45 Gray, Robert Molten 45 Kschischang, Frank R. 45 Ling, San 44 Gastpar, Michael C. 44 Urbanke, Rüdiger L. 43 Cover, Thomas Merrill 43 Diggavi, Suhas N. 43 Effros, Michelle 43 Ephremides, Anthony 43 Golomb, Solomon Wolf 43 Hayashi, Masahito 42 Eldar, Yonina Chana 42 Han, Te Sun 42 Médard, Muriel 41 Schwartz, Moshe 41 Tarokh, Vahid 41 Viswanath, Pramod 40 Abdel-Ghaffar, Khaled A. S. 40 Carlet, Claude 40 Neuhoff, David L. 40 Sloane, Neil James Alexander 40 Solé, Patrick 39 Chen, Jun 39 Forney, G. David jun. 39 Johannesson, Rolf 39 Winter, Andreas 38 Fossorier, Marc P. C. 38 Fu, Fangwei 38 Ge, Gennian 38 Linder, Tamás 37 Kieffer, John Cronan 37 Masry, Elias 37 Steinberg, Yossef 37 Wang, Xiaodong 37 Yang, Enhui 36 Avestimehr, Amir Salman 36 Kasami, Tadao 36 Kulkarni, Sanjeev R. 36 Ramchandran, Kannan 35 Gulliver, Thomas Aaron 35 Khisti, Ashish J. 35 Veeravalli, Venugopal V. 34 Boche, Holger 34 McEliece, Robert James 34 Tian, Chao 34 Wagner, Aaron B. 34 Wornell, Gregory W. 34 Xia, Xianggen 33 Anantharam, Venkat 33 Chee, Yeow Meng 33 El Gamal, Hesham 33 Guruswami, Venkatesan 33 Kailath, Thomas 33 Kramer, Gerhard 33 Sason, Igal 33 Wyner, Aaron D. 32 Csiszár, Imre 32 Hassibi, Babak ...and 10,374 more Authors
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#### Fields
12,826 Information and communication theory, circuits (94-XX) 1,508 Statistics (62-XX) 1,043 Probability theory and stochastic processes (60-XX) 980 Computer science (68-XX) 600 Operations research, mathematical programming (90-XX) 506 Combinatorics (05-XX) 467 Systems theory; control (93-XX) 440 Number theory (11-XX) 421 Quantum theory (81-XX) 223 Numerical analysis (65-XX) 143 Linear and multilinear algebra; matrix theory (15-XX) 121 Algebraic geometry (14-XX) 116 Harmonic analysis on Euclidean spaces (42-XX) 91 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 62 Biology and other natural sciences (92-XX) 59 Order, lattices, ordered algebraic structures (06-XX) 55 Group theory and generalizations (20-XX) 52 Convex and discrete geometry (52-XX) 41 Approximations and expansions (41-XX) 37 Optics, electromagnetic theory (78-XX) 36 Statistical mechanics, structure of matter (82-XX) 31 Geometry (51-XX) 25 Field theory and polynomials (12-XX) 23 Dynamical systems and ergodic theory (37-XX) 22 Special functions (33-XX) 19 Abstract harmonic analysis (43-XX) 19 Operator theory (47-XX) 18 History and biography (01-XX) 13 Commutative algebra (13-XX) 12 General and overarching topics; collections (00-XX) 11 Measure and integration (28-XX) 11 Integral transforms, operational calculus (44-XX) 11 Functional analysis (46-XX) 9 Mathematical logic and foundations (03-XX) 9 Associative rings and algebras (16-XX) 9 Calculus of variations and optimal control; optimization (49-XX) 8 Real functions (26-XX) 4 Topological groups, Lie groups (22-XX) 4 Functions of a complex variable (30-XX) 4 Ordinary differential equations (34-XX) 4 Sequences, series, summability (40-XX) 4 Fluid mechanics (76-XX) 4 Geophysics (86-XX) 3 Partial differential equations (35-XX) 3 Difference and functional equations (39-XX) 3 Integral equations (45-XX) 3 Differential geometry (53-XX) 2 General topology (54-XX) 2 Algebraic topology (55-XX) 1 Nonassociative rings and algebras (17-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Relativity and gravitational theory (83-XX) 1 Astronomy and astrophysics (85-XX)
#### Citations contained in zbMATH Open
7,861 Publications have been cited 65,223 times in 29,276 Documents Cited by Year
Compressed sensing. Zbl 1288.94016
Donoho, David L.
2006
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. Zbl 1231.94017
Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence
2006
New directions in cryptography. Zbl 0435.94018
Diffie, Whitfield; Hellman, Martin E.
1976
Decoding by linear programming. Zbl 1264.94121
Candès, Emmanuel J.; Tao, Terence
2005
Near-optimal signal recovery from random projections: universal encoding strategies? Zbl 1309.94033
Candès, Emmanuel J.; Tao, Terence
2006
The wavelet transform, time-frequency localization and signal analysis. Zbl 0738.94004
Daubechies, Ingrid
1990
The $$\mathbb Z_4$$-linearity of Kerdock, Preparata, Goethals, and related codes. Zbl 0811.94039
Hammons, A. Roger jun.; Kumar, P. Vijay; Calderbank, A. R.; Sloane, N. J. A.; Solé, Patrick
1994
A public key cryptosystem and a signature scheme based on discrete logarithms. Zbl 0571.94014
ElGamal, Taher
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De-noising by soft-thresholding. Zbl 0820.62002
Donoho, David L.
1995
On the Shannon capacity of a graph. Zbl 0395.94021
Lovász, László
1979
Least squares quantization in PCM. Zbl 0504.94015
Lloyd, Stuart P.
1982
Universal approximation bounds for superpositions of a sigmoidal function. Zbl 0818.68126
Barron, Andrew R.
1993
Nearest neighbor pattern classification. Zbl 0154.44505
Cover, T. M.; Hart, P. E.
1967
Quantum error correction via codes over $$\mathrm{GF}(4)$$. Zbl 0982.94029
Calderbank, A. Robert; Rains, Eric M.; Shor, P. W.; Sloane, Neil J. A.
1998
Greed is good: algorithmic results for sparse approximation. Zbl 1288.94019
Tropp, Joel A.
2004
A universal algorithm for sequential data compression. Zbl 0379.94010
Ziv, Jacob; Lempel, Abraham
1977
Signal recovery from random measurements via orthogonal matching pursuit. Zbl 1288.94022
Tropp, Joel A.; Gilbert, Anna C.
2007
The power of convex relaxation: near-optimal matrix completion. Zbl 1366.15021
Candès, Emmanuel J.; Tao, Terence
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Compression of individual sequences via variable-rate coding. Zbl 0392.94004
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1978
Shift-register synthesis and BCH decoding. Zbl 0167.18101
Massey, James L.
1969
Stable recovery of sparse overcomplete representations in the presence of noise. Zbl 1288.94017
Donoho, David L.; Elad, Michael; Temlyakov, Vladimir N.
2006
Uncertainty principles and ideal atomic decomposition. Zbl 1019.94503
Donoho, David L.; Huo, Xiaoming
2001
Sharp thresholds for high-dimensional and noisy sparsity recovery using $$\ell_1$$-constrained quadratic programming (Lasso). Zbl 1367.62220
Wainwright, Martin J.
2009
On a new class of codes for identifying vertices in graphs. Zbl 1105.94342
Karpovsky, Mark G.; Chakrabarty, Krishnendu; Levitin, Lev B.
1998
Cyclic and negacyclic codes over finite chain rings. Zbl 1243.94043
Dinh, Hai Quang; López-Permouth, Sergio R.
2004
Singularity detection and processing with wavelets. Zbl 0745.93073
Mallat, Stephane; Hwang, Wen Liang
1992
Collusion-secure fingerprinting for digital data. Zbl 0931.94051
Boneh, Dan; Shaw, James
1998
Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. Zbl 0429.94011
Shore, John E.; Johnson, Rodney W.
1980
On the security of public key protocols. Zbl 0502.94005
Dolev, Danny; Yao, Andrew C.
1983
On the complexity of finite sequences. Zbl 0337.94013
Lempel, Abraham; Ziv, Jacob
1976
Randomized gossip algorithms. Zbl 1283.94005
Boyd, Stephen P.; Ghosh, Arpita; Prabhakar, Balaji; Shah, Devavrat
2006
Recovering low-rank matrices from few coefficients in any basis. Zbl 1366.94103
Gross, David
2011
Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. Zbl 0148.40501
Viterbi, A. J.
1967
Approximating discrete probability distributions with dependence trees. Zbl 0165.22305
Chow, C. K.; Liu, C. N.
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Lower bounds on the maximum cross correlation of signals. Zbl 0298.94006
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Coding for errors and erasures in random network coding. Zbl 1318.94111
Koetter, Ralf; Kschischang, Frank R.
2008
Multiresolution analysis, Haar bases, and self-similar tilings of $$R^ n$$. Zbl 0742.42012
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1992
Generalized Hamming weights for linear codes. Zbl 0735.94008
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An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. Zbl 0375.68023
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Reducing elliptic curve logarithms to logarithms in a finite field. Zbl 0801.94011
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The capacity of the quantum channel with general signal states. Zbl 0897.94008
Holevo, A. S.
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Factor graphs and the sum-product algorithm. Zbl 0998.68234
Kschischang, Frank R.; Frey, Brendan J.; Loeliger, Hans-Andrea
2001
Just relax: Convex programming methods for identifying sparse signals in noise. Zbl 1288.94025
Tropp, Joel A.
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Quantization. Zbl 1016.94016
Gray, Robert M.; Neuhoff, David L.
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Maximal recursive sequences with 3-valued recursive cross-correlation functions. Zbl 0228.62040
Gold, Robert
1968
Linear codes from perfect nonlinear mappings and their secret sharing schemes. Zbl 1192.94114
Carlet, Claude; Ding, Cunsheng; Yuan, Jin
2005
Time-frequency localization operators: A geometric phase space approach. Zbl 0672.42007
Daubechies, Ingrid
1988
The capacity of wireless networks. Zbl 0991.90511
Gupta, Piyush; Kumar, P. R.
2000
Network information flow. Zbl 0991.90015
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2000
Entropy-based algorithms for best basis selection. Zbl 0849.94005
Coifman, Ronald R.; Wickerhauser, Mladen Victor
1992
Nonrandom binary superimposed codes. Zbl 0133.12402
Kautz, W. H.; Singleton, R. C.
1964
A new upper bound on the minimal distance of self-dual codes. Zbl 0713.94016
Conway, J. H.; Sloane, N. J. A.
1990
Information theoretic inequalities. Zbl 0741.94001
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1991
Optimal recursive estimation with uncertain observation. Zbl 0174.51102
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1969
Nonbinary stabilizer codes over finite fields. Zbl 1242.94045
Ketkar, Avanti; Klappenecker, Andreas; Kumar, Santosh; Sarvepalli, Pradeep Kiran
2006
Phase retrieval via Wirtinger flow: theory and algorithms. Zbl 1359.94069
Candès, Emmanuel J.; Li, Xiaodong; Soltanolkotabi, Mahdi
2015
On the inherent intractability of certain coding problems. Zbl 0377.94018
Berlekamp, Elwyn R.; McEliece, Robert J.; van Tilborg, Henk C. A.
1978
Sparse representations in unions of bases. Zbl 1286.94032
Gribonval, Rémi; Nielsen, Morten
2003
Cumulative residual entropy: a new measure of information. Zbl 1302.94025
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2004
A view of three decades of linear filtering theory. Zbl 0307.93040
Kailath, Thomas
1974
Correlation-immunity of nonlinear combining functions for cryptographic applications. Zbl 0554.94010
Siegenthaler, T.
1984
Nonconcave penalized likelihood with NP-dimensionality. Zbl 1365.62277
Fan, Jianqing; Lv, Jinchi
2011
Capacity of reproducing kernel spaces in learning theory. Zbl 1290.62033
Zhou, Ding-Xuan
2003
Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements. Zbl 1366.90160
Candès, Emmanuel J.; Plan, Yaniv
2011
Improved decoding of Reed-Solomon and algebraic-geometry codes. Zbl 0958.94036
Guruswami, Venkatesan; Sudan, Madhu
1999
Hidden Markov processes. Zbl 1061.94560
Ephraim, Yariv; Merhav, Neri
2002
Divergence measures based on the Shannon entropy. Zbl 0712.94004
Lin, Jianhua
1991
Solving sparse linear equations over finite fields. Zbl 0607.65015
Wiedemann, Douglas H.
1986
Matrix completion from a few entries. Zbl 1366.62111
Keshavan, Raghunandan H.; Montanari, Andrea; Oh, Sewoong
2010
Subspace pursuit for compressive sensing signal reconstruction. Zbl 1367.94082
Dai, Wei; Milenkovic, Olgica
2009
Wavelet analysis and synthesis of fractional Brownian motion. Zbl 0743.60078
Flandrin, Patrick
1992
On the algebraic structure of quasi-cyclic codes. I: Finite fields. Zbl 1023.94015
Ling, San; Solé, Patrick
2001
Linear codes from some 2-designs. Zbl 1359.94685
Ding, Cunsheng
2015
Fisher information and stochastic complexity. Zbl 0856.94006
Rissanen, Jorma J.
1996
Secret sharing schemes from three classes of linear codes. Zbl 1283.94105
Yuan, Jin; Ding, Cunsheng
2006
Minimax rates of estimation for high-dimensional linear regression over $$\ell_q$$-balls. Zbl 1365.62276
Raskutti, Garvesh; Wainwright, Martin J.; Yu, Bin
2011
Optical orthogonal codes: design, analysis, and applications. Zbl 0676.94021
Chung, Fan R. K.; Salehi, Jawad A.; Wei, Victor K.
1989
A class of two-weight and three-weight codes and their applications in secret sharing. Zbl 1359.94687
Ding, Kelan; Ding, Cunsheng
2015
The estimation of the gradient of a density function, with applications in pattern recognition. Zbl 0297.62025
Fukunaga, Keinosuke; Hostetler, Larry D.
1975
Cyclic codes and self-dual codes over $$F_2+uF_2$$. Zbl 0958.94025
Bonnecazé, A.; Udaya, P.
1999
On the convexity of some divergence measures based on entropy functions. Zbl 0479.94009
Burbea, Jacob; Rao, C. Radhakrishna
1982
The weights of the orthogonals of the extended quadratic binary Goppa codes. Zbl 0703.94011
Lachaud, Gilles; Wolfmann, Jacques
1990
Nonbinary quantum stabilizer codes. Zbl 1021.94033
Ashikhmin, Alexei; Knill, Emanuel
2001
Wavelet analysis of long-range-dependent traffic. Zbl 0905.94006
Abry, Patrice; Veitch, Darryl
1998
The private classical capacity and quantum capacity of a quantum channel. Zbl 1293.94063
Devetak, Igor
2005
On quantum and classical BCH codes. Zbl 1310.94195
Aly, Salah A.; Klappenecker, Andreas; Sarvepalli, Pradeep Kiran
2007
On subfield subcodes of modified Reed-Solomon codes. Zbl 0308.94004
Delsarte, Philippe
1975
Cryptographic distinguishability measures for quantum-mechanical states. Zbl 0959.94020
Fuchs, Christopher A.; van de Graaf, Jeroen
1999
Constructing free-energy approximations and generalized belief propagation algorithms. Zbl 1283.94023
Yedidia, Jonathan S.; Freeman, William T.; Weiss, Yair
2005
Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit. Zbl 1365.94069
Donoho, David L.; Tsaig, Yaakov; Drori, Iddo; Starck, Jean-Luc
2012
Recursive probability density estimation for weakly dependent stationary processes. Zbl 0602.62028
Masry, Elias
1986
Evaluation of likelihood functions for Gaussian signals. Zbl 0127.10805
Schweppe, F. C.
1965
Type II codes over $$\mathbb{F}_2+u \mathbb{F}_2$$. Zbl 0947.94023
Dougherty, Steven T.; Gaborit, Philippe; Harada, Masaaki; Solé, Patrick
1999
Almost perfect nonlinear power functions on $$\mathrm{GF}(2^n)$$: the Welch case. Zbl 0957.94021
Dobbertin, Hans
1999
A comparison of the Delsarte and Lovasz bounds. Zbl 0444.94009
Schrijver, Alexander
1979
On secret sharing systems. Zbl 0503.94018
Karnin, Ehud D.; Greene, Jonathan W.; Hellman, Martin E.
1983
Sigma-delta $$(\Sigma\Delta)$$ quantization and finite frames. Zbl 1285.94014
Benedetto, John J.; Powell, Alexander M.; Yılmaz, Özgür
2006
On the shape of a set of points in the plane. Zbl 0512.52001
Edelsbrunner, Herbert; Kirkpatrick, David G.; Seidel, Raimund
1983
On divergences and informations in statistics and information theory. Zbl 1287.94025
Liese, Friedrich; Vajda, Igor
2006
A sampling theorem for wavelet subspaces. Zbl 0744.42018
Walter, Gilbert G.
1992
Shapes of uncertainty in spectral graph theory. Zbl 1465.05101
Erb, Wolfgang
2021
Deep neural network approximation theory. Zbl 07374112
Elbrächter, Dennis; Perekrestenko, Dmytro; Grohs, Philipp; Bölcskei, Helmut
2021
Signature codes for weighted binary adder channel and multimedia fingerprinting. Zbl 1465.94027
Fan, Jinping; Gu, Yujie; Hachimori, Masahiro; Miao, Ying
2021
An infinite family of linear codes supporting 4-designs. Zbl 1465.94117
Tang, Chunming; Ding, Cunsheng
2021
On the spectral property of kernel-based sensor fusion algorithms of high dimensional data. Zbl 07314050
Ding, Xiucai; Wu, Hau-Tieng
2021
Computing quantum channel capacities. Zbl 1465.94052
Ramakrishnan, Navneeth; Iten, Raban; Scholz, Volkher B.; Berta, Mario
2021
Covariance matrix estimation with non uniform and data dependent missing observations. Zbl 1465.62100
Pavez, Eduardo; Ortega, Antonio
2021
Nonconvex matrix factorization from rank-one measurements. Zbl 07374042
Li, Yuanxin; Ma, Cong; Chen, Yuxin; Chi, Yuejie
2021
Super-resolution limit of the ESPRIT algorithm. Zbl 1446.94010
Li, Weilin; Liao, Wenjing; Fannjiang, Albert
2020
Constructions of locally recoverable codes which are optimal. Zbl 1433.94129
Micheli, Giacomo
2020
$$c$$-differentials, multiplicative uniformity, and (almost) perfect $$c$$-nonlinearity. Zbl 1448.94309
Ellingsen, Pål; Felke, Patrick; Riera, Constanza; Stănică, Pantelimon; Tkachenko, Anton
2020
Construction of optimal locally repairable codes via automorphism groups of rational function fields. Zbl 1433.94119
Jin, Lingfei; Ma, Liming; Xing, Chaoping
2020
MDS symbol-pair cyclic codes of length $$2p^s$$ over $$\mathbb F_{p^m}$$. Zbl 1433.94140
Dinh, Hai Q.; Nguyen, Bac T.; Sriboonchitta, Songsak
2020
Several classes of minimal linear codes with few weights from weakly regular plateaued functions. Zbl 1448.94271
Mesnager, Sihem; Sınak, Ahmet
2020
LCD and self-orthogonal group codes in a finite abelian $$p$$-group algebra. Zbl 1448.94267
Li, Fengwei; Yue, Qin; Wu, Yansheng
2020
Constructing APN functions through isotopic shifts. Zbl 1446.94085
Budaghyan, Lilya; Calderini, Marco; Carlet, Claude; Coulter, Robert S.; Villa, Irene
2020
A successor rule framework for constructing $$k$$-ary de Bruijn sequences and universal cycles. Zbl 1433.94079
Gabric, D.; Sawada, J.; Williams, A.; Wong, D.
2020
How biased is your model? Concentration inequalities, information and model bias. Zbl 1448.94102
Gourgoulias, Konstantinos; Katsoulakis, Markos A.; Rey-Bellet, Luc; Wang, Jie
2020
Higher weight spectra of Veronese codes. Zbl 1448.94265
Johnsen, Trygve; Verdure, Hugues
2020
Two families of optimal linear codes and their subfield codes. Zbl 1453.94152
Heng, Ziling; Wang, Qiuyan; Ding, Cunsheng
2020
Smoothing Brascamp-Lieb inequalities and strong converses of coding theorems. Zbl 1434.94049
Liu, Jingbo; Courtade, Thomas A.; Cuff, Paul; Verdú, Sergio
2020
Optimal few-weight codes from simplicial complexes. Zbl 1448.94279
Wu, Yansheng; Zhu, Xiaomeng; Yue, Qin
2020
A class of quadrinomial permutations with boomerang uniformity four. Zbl 1448.94232
Tu, Ziran; Li, Nian; Zeng, Xiangyong; Zhou, Junchao
2020
An algebraic-geometric approach for linear regression without correspondences. Zbl 1446.62202
Tsakiris, Manolis C.; Peng, Liangzu; Conca, Aldo; Kneip, Laurent; Shi, Yuanming; Choi, Hayoung
2020
When are fuzzy extractors possible? Zbl 1446.94130
Fuller, Benjamin; Reyzin, Leonid; Smith, Adam
2020
On the list decodability of rank metric codes. Zbl 1448.94296
Trombetti, Rocco; Zullo, Ferdinando
2020
Optimal binary linear codes from maximal arcs. Zbl 1448.94261
Heng, Ziling; Ding, Cunsheng; Wang, Weiqiong
2020
Infinite families of near MDS codes holding $$t$$-designs. Zbl 1448.94257
Ding, Cunsheng; Tang, Chunming
2020
Sparse and low-rank tensor estimation via cubic sketchings. Zbl 1448.62072
Hao, Botao; Zhang, Anru; Cheng, Guang
2020
Coded trace reconstruction. Zbl 1452.94041
Cheraghchi, Mahdi; Gabrys, Ryan; Milenkovic, Olgica; Ribeiro, João
2020
On integrated $$L^1$$ convergence rate of an isotonic regression estimator for multivariate observations. Zbl 1452.62282
Fokianos, Konstantinos; Leucht, Anne; Neumann, Michael H.
2020
A new bound on quantum Wielandt inequality. Zbl 1433.94073
Rahaman, Mizanur
2020
Constructions of locally repairable codes with multiple recovering sets via rational function fields. Zbl 1433.94118
Jin, Lingfei; Kan, Haibin; Zhang, Yu
2020
How much does your data exploration overfit? Controlling bias via information usage. Zbl 1433.94041
Russo, Daniel; Zou, James
2020
Strong consistency of spectral clustering for stochastic block models. Zbl 1433.62170
Su, Liangjun; Wang, Wuyi; Zhang, Yichong
2020
Rigorous dynamics of expectation-propagation-based signal recovery from unitarily invariant measurements. Zbl 1433.94031
Takeuchi, Keigo
2020
Global guarantees for enforcing deep generative priors by empirical risk. Zbl 1433.94024
Hand, Paul; Voroninski, Vladislav
2020
An improved bound on the zero-error list-decoding capacity of the 4/3 channel. Zbl 1434.94048
Dalai, Marco; Guruswami, Venkatesan; Radhakrishnan, Jaikumar
2020
On inverses of permutation polynomials of small degree over finite fields. Zbl 1434.11230
Zheng, Yanbin; Wang, Qiang; Wei, Wenhong
2020
Bounds and optimal $$q$$-ary codes derived from the $$\mathbb Z_qR$$-cyclic codes. Zbl 1434.94090
Qian, Liqin; Cao, Xiwang
2020
Fractional repetition codes with optimal reconstruction degree. Zbl 1434.94111
Zhu, Bing; Shum, Kenneth W.; Li, Hui
2020
Local entropy statistics for point processes. Zbl 1437.62057
Clark, Daniel E.
2020
Analysis on Boolean function in a restricted (biased) domain. Zbl 1434.94074
Maitra, Subhamoy; Mandal, Bimal; Martinsen, Thor; Roy, Dibyendu; Stănică, Pantelimon
2020
Maximizing the number of spanning trees in a connected graph. Zbl 1437.05047
Li, Huan; Patterson, Stacy; Yi, Yuhao; Zhang, Zhongzhi
2020
Bounds and constructions of codes over symbol-pair read channels. Zbl 1446.94047
Elishco, Ohad; Gabrys, Ryan; Yaakobi, Eitan
2020
Semantically secure lattice codes for compound MIMO channels. Zbl 1446.94053
Campello, Antonio; Ling, Cong; Belfiore, Jean-Claude
2020
Private information retrieval with side information. Zbl 1448.94142
Kadhe, Swanand; Garcia, Brenden; Heidarzadeh, Anoosheh; El Rouayheb, Salim; Sprintson, Alex
2020
Quantum channel simulation and the channel’s smooth max-information. Zbl 1448.94167
Fang, Kun; Wang, Xin; Tomamichel, Marco; Berta, Mario
2020
Quickest detection of dynamic events in networks. Zbl 1448.94097
Zou, Shaofeng; Veeravalli, Venugopal V.; Li, Jian; Towsley, Don
2020
Optimal variable selection and adaptive noisy compressed sensing. Zbl 1448.94081
Ndaoud, Mohamed; Tsybakov, Alexandre B.
2020
Distance distribution in Reed-Solomon codes. Zbl 1448.11211
Li, Jiyou; Wan, Daqing
2020
Quaternary Hermitian linear complementary dual codes. Zbl 1448.94252
Araya, Makoto; Harada, Masaaki; Saito, Ken
2020
Shannon meets von Neumann: a minimax theorem for channel coding in the presence of a jammer. Zbl 1448.94131
Jose, Sharu Theresa; Kulkarni, Ankur A.
2020
Permutation-invariant constant-excitation quantum codes for amplitude damping. Zbl 1448.81273
Ouyang, Yingkai; Chao, Rui
2020
Toward the optimal construction of a loss function without spurious local minima for solving quadratic equations. Zbl 1448.65053
Li, Zhenzhen; Cai, Jian-Feng; Wei, Ke
2020
A quantum multiparty packing lemma and the relay channel. Zbl 1448.94165
Ding, Dawei; Gharibyan, Hrant; Hayden, Patrick; Walter, Michael
2020
Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs. Zbl 1448.94258
Fang, Weijun; Fu, Fang-Wei; Li, Lanqiang; Zhu, Shixin
2020
Explicit constructions of MDS self-dual codes. Zbl 1448.94274
Sok, Lin
2020
Gabidulin codes with support constrained generator matrices. Zbl 1448.94281
Yildiz, Hikmet; Hassibi, Babak
2020
A unified approach to construct MDS self-dual codes via Reed-Solomon codes. Zbl 1448.94282
Zhang, Aixian; Feng, Keqin
2020
Mutual information and optimality of approximate message-passing in random linear estimation. Zbl 1446.94018
Barbier, Jean; Macris, Nicolas; Dia, Mohamad; Krzakala, Florent
2020
The capacity of T-private information retrieval with private side information. Zbl 1446.68061
Chen, Zhen; Wang, Zhiying; Jafar, Syed Ali
2020
Partially smoothed information measures. Zbl 1446.94046
Anshu, Anurag; Berta, Mario; Jain, Rahul; Tomamichel, Marco
2020
Adversarial hypothesis testing and a quantum Stein’s lemma for restricted measurements. Zbl 1446.62075
Brandão, Fernando G. S. L.; Harrow, Aram W.; Lee, James R.; Peres, Yuval
2020
Analysis of spectral methods for phase retrieval with random orthogonal matrices. Zbl 1446.94022
Dudeja, Rishabh; Bakhshizadeh, Milad; Ma, Junjie; Maleki, Arian
2020
Minimal linear codes from characteristic functions. Zbl 1448.94270
Mesnager, Sihem; Qi, Yanfeng; Ru, Hongming; Tang, Chunming
2020
Explicit lower bounds on strong quantum simulation. Zbl 1448.68243
Huang, Cupjin; Newman, Michael; Szegedy, Mario
2020
Gaussian maximizers for quantum Gaussian observables and ensembles. Zbl 1448.81511
Holevo, Alexander S.
2020
On the distance between APN functions. Zbl 1448.94308
Budaghyan, Lilya; Carlet, Claude; Helleseth, Tor; Kaleyski, Nikolay
2020
Linear complexity of a family of binary $$pq^2$$-periodic sequences from Euler quotients. Zbl 1448.94184
Zhang, Jingwei; Gao, Shuhong; Zhao, Chang-An
2020
Nonconvex rectangular matrix completion via gradient descent without $$\ell_2,\infty$$ regularization. Zbl 1448.90078
Chen, Ji; Liu, Dekai; Li, Xiaodong
2020
Sparse recovery beyond compressed sensing: separable nonlinear inverse problems. Zbl 1448.94050
Bernstein, Brett; Liu, Sheng; Papadaniil, Chrysa; Fernandez-Granda, Carlos
2020
Maximally recoverable LRCs: a field size lower bound and constructions for few heavy parities. Zbl 1452.94113
Gopi, Sivakanth; Guruswami, Venkatesan; Yekhanin, Sergey
2020
Identification capacity of channels with feedback: discontinuity behavior, super-activation, and Turing computability. Zbl 1452.94051
Boche, Holger; Schaefer, Rafael Felix; Poor, H. Vincent
2020
Recursive kernel density estimation for time series. Zbl 1452.62259
Aboubacar, Amir; El Machkouri, Mohamed
2020
Deconstructing generative adversarial networks. Zbl 1453.62518
Zhu, Banghua; Jiao, Jiantao; Tse, David
2020
Close encounters of the binary kind: signal reconstruction guarantees for compressive Hadamard sampling with Haar wavelet basis. Zbl 1453.94031
Moshtaghpour, Amirafshar; Bioucas-Dias, José M.; Jacques, Laurent
2020
Rudin-Shapiro-like sequences with maximum asymptotic merit factor. Zbl 1457.11022
Katz, Daniel J.; Lee, Sangman; Trunov, Stanislav A.
2020
Noisy quantum state redistribution with promise and the alpha-bit. Zbl 1457.81019
Anshu, Anurag; Hsieh, Min-Hsiu; Jain, Rahul
2020
Binary LCD codes and self-orthogonal codes from a generic construction. Zbl 1431.94175
Zhou, Zhengchun; Li, Xia; Tang, Chunming; Ding, Cunsheng
2019
New characterization and parametrization of LCD codes. Zbl 1431.94143
Carlet, Claude; Mesnager, Sihem; Tang, Chunming; Qi, Yanfeng
2019
Theoretical insights into the optimization landscape of over-parameterized shallow neural networks. Zbl 1428.68255
Soltanolkotabi, Mahdi; Javanmard, Adel; Lee, Jason D.
2019
A new family of MRD codes in $$\mathbb F_q^{2n\times 2n}$$ with right and middle nuclei $$\mathbb F_{q^n}$$. Zbl 1427.94106
Trombetti, Rocco; Zhou, Yue
2019
Minimal linear codes in odd characteristic. Zbl 1432.94161
Bartoli, Daniele; Bonini, Matteo
2019
New results about the boomerang uniformity of permutation polynomials. Zbl 1433.94088
Li, Kangquan; Qu, Longjiang; Sun, Bing; Li, Chao
2019
On $$\sigma$$-LCD codes. Zbl 1431.94181
Carlet, Claude; Mesnager, Sihem; Tang, Chunming; Qi, Yanfeng
2019
MDS codes with hulls of arbitrary dimensions and their quantum error correction. Zbl 1431.94161
Luo, Gaojun; Cao, Xiwang; Chen, Xiaojing
2019
Vector approximate message passing. Zbl 1432.94036
Rangan, Sundeep; Schniter, Philip; Fletcher, Alyson K.
2019
Exponential error rates of SDP for block models: beyond Grothendieck’s inequality. Zbl 1432.90102
Fei, Yingjie; Chen, Yudong
2019
Probabilistic existence results for parent-identifying schemes. Zbl 1432.94135
Gu, Yujie; Cheng, Minquan; Kabatiansky, Grigory; Miao, Ying
2019
New binary and ternary LCD codes. Zbl 1427.94093
Galindo, Carlos; Geil, Olav; Hernando, Fernando; Ruano, Diego
2019
A data-dependent weighted Lasso under Poisson noise. Zbl 1432.62222
Hunt, Xin Jiang; Reynaud-Bouret, Patricia; Rivoirard, Vincent; Sansonnet, Laure; Willett, Rebecca
2019
Factorizations of binomial polynomials and enumerations of LCD and self-dual constacyclic codes. Zbl 1431.94189
Wu, Yansheng; Yue, Qin
2019
Symmetry, saddle points, and global optimization landscape of nonconvex matrix factorization. Zbl 1432.90123
Li, Xingguo; Lu, Junwei; Arora, Raman; Haupt, Jarvis; Liu, Han; Wang, Zhaoran; Zhao, Tuo
2019
Multireference alignment is easier with an aperiodic translation distribution. Zbl 1432.94018
Abbe, Emmanuel; Bendory, Tamir; Leeb, William; Pereira, João M.; Sharon, Nir; Singer, Amit
2019
On $$\mathbb Z_p \mathbb Z_{p^k}$$-additive codes and their duality. Zbl 1432.94177
Shi, Minjia; Wu, Rongsheng; Krotov, Denis S.
2019
Quantum version of Wielandt’s inequality revisited. Zbl 1432.82007
Michałek, Mateusz; Shitov, Yaroslav
2019
Combinatorial entropy power inequalities: a preliminary study of the Stam region. Zbl 1431.94042
Madiman, Mokshay; Ghassemi, Farhad
2019
How long can optimal locally repairable codes be? Zbl 1432.94213
Guruswami, Venkatesan; Xing, Chaoping; Yuan, Chen
2019
Explicit construction of optimal locally recoverable codes of distance 5 and 6 via binary constant weight codes. Zbl 1432.94170
Jin, Lingfei
2019
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#### Cited by 34,496 Authors
106 Solé, Patrick 83 Fu, Fangwei 68 Harada, Masaaki 63 Zhu, Shixin 62 Helleseth, Tor 57 Dinh, Hai Quang 57 Zeng, Xiangyong 55 Carlet, Claude 55 Gulliver, Thomas Aaron 54 Yue, Qin 53 Cao, Xiwang 51 Hu, Lei 51 Shi, Minjia 50 Ding, Cunsheng 49 Mesnager, Sihem 48 Östergård, Patric R. J. 48 Özbudak, Ferruh 46 Boche, Holger 46 Ling, San 45 Dougherty, Steven T. 44 Meidl, Wilfried 43 Bartoli, Daniele 43 Winter, Andreas 42 Sloane, Neil James Alexander 41 Kohler, Michael 40 Ge, Gennian 40 Zhou, Dingxuan 39 Cao, Yonglin 38 Tang, Chunming 37 Yildiz, Bahattin 36 Gao, Jian 36 Qi, Wenfeng 36 Shparlinski, Igor E. 36 Winterhof, Arne 35 Gröchenig, Karlheinz 34 Ahlswede, Rudolf 34 Lin, Dongdai 34 Sun, Wenchang 34 Wilde, Mark M. 33 Krzyżak, Adam 32 Honkala, Iiro S. 32 Li, Ruihu 32 Qu, Longjiang 32 Zhou, Zhengchun 31 Candès, Emmanuel J. 31 Cohen, Gérard Denis 31 Liu, Hongwei 31 Stinson, Douglas Robert 30 Maitra, Subhamoy 30 Rauhut, Holger 30 Shen, Zuowei 30 Tsybakov, Alexandre B. 30 Wainwright, Martin J. 29 Laihonen, Tero K. 29 Liu, Zihui 29 Pasalic, Enes 29 Stănică, Pantelimon 29 Tonchev, Vladimir D. 29 Vajda, Igor 29 Zinov’ev, Viktor Aleksandrovich 28 Cao, Yuan 28 Datta, Nilanjana 28 Giulietti, Massimo 28 Guenda, Kenza 28 Kai, Xiaoshan 28 Mixon, Dustin G. 28 Pambianco, Fernanda 28 Sugiyama, Masashi 28 Tang, Xiaohu 27 Alon, Noga M. 27 Casazza, Peter George 27 Fan, Jianqing 27 Feng, Tao 27 Jitman, Somphong 27 Kaya, Abidin 27 Li, Song 27 Mendelson, Shahar 27 Panario, Daniel 27 Sanguineti, Marcello 27 Sriboonchitta, Songsak 26 Aydin, Nuh 26 Calderbank, Arthur Robert 26 Chen, Dirong 26 Devroye, Luc P. J. A. 26 Gong, Guang 26 Krahmer, Felix 26 Li, Chengju 26 Lugosi, Gábor 26 Pott, Alexander 26 Vaccaro, Ugo 26 Wang, Huaxiong 26 Xu, Zongben 25 Anderson, Brian David Outram 25 Györfi, László 25 Hou, Xiang-Dong 25 Key, Jennifer D. 25 Kim, Jon-Lark 25 Lee, Yoonjin 25 Li, Chao 25 Martínez-Moro, Edgar ...and 34,396 more Authors
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#### Cited in 1,058 Journals
1,139 Designs, Codes and Cryptography 637 Discrete Mathematics 615 Finite Fields and their Applications 536 Theoretical Computer Science 459 The Annals of Statistics 450 Discrete Applied Mathematics 429 Automatica 412 Applied and Computational Harmonic Analysis 403 Information Sciences 375 Linear Algebra and its Applications 339 Cryptography and Communications 320 Journal of Mathematical Physics 298 Pattern Recognition 296 Information Processing Letters 281 Quantum Information Processing 272 Applied Mathematics and Computation 254 Neural Computation 252 Advances in Mathematics of Communications 241 Journal of Combinatorial Theory. Series A 240 Applicable Algebra in Engineering, Communication and Computing 240 Mathematical Problems in Engineering 236 Problems of Information Transmission 232 Electronic Journal of Statistics 229 Journal of Statistical Planning and Inference 225 Journal of Multivariate Analysis 223 Journal of the Franklin Institute 213 Statistics & Probability Letters 187 Signal Processing 186 Journal of Mathematical Analysis and Applications 184 Journal of Machine Learning Research (JMLR) 180 International Journal of Wavelets, Multiresolution and Information Processing 176 Journal of Cryptology 175 Computational Statistics and Data Analysis 167 Journal of Computational and Applied Mathematics 167 Communications in Statistics. Theory and Methods 160 The Journal of Fourier Analysis and Applications 156 Journal of Computer and System Sciences 154 Circuits, Systems, and Signal Processing 148 EURASIP Journal on Advances in Signal Processing 147 Mathematical Programming. Series A. Series B 139 Computers & Mathematics with Applications 136 Journal of Statistical Physics 134 International Journal of Theoretical Physics 134 Bernoulli 132 Stochastic Processes and their Applications 130 Algorithmica 130 Information and Computation 122 Communications in Mathematical Physics 122 Mathematics of Computation 121 Machine Learning 116 Neural Networks 115 Systems & Control Letters 113 International Journal of Control 113 European Journal of Combinatorics 113 International Journal of Quantum Information 112 Entropy 109 European Journal of Operational Research 106 Journal of Complexity 105 International Journal of Systems Science 103 The Annals of Applied Probability 102 Kybernetika 101 Annals of the Institute of Statistical Mathematics 101 Journal of Approximation Theory 100 Science China. Information Sciences 99 International Journal of Foundations of Computer Science 98 International Journal of Computer Mathematics 97 SIAM Journal on Imaging Sciences 96 Science China. Mathematics 90 Queueing Systems 89 Journal of Optimization Theory and Applications 88 Journal of Symbolic Computation 88 Multidimensional Systems and Signal Processing 88 SIAM Journal on Optimization 85 SIAM Journal on Computing 85 Advances in Computational Mathematics 85 Chaos 84 Journal of Mathematical Imaging and Vision 83 New Journal of Physics 82 Inverse Problems 82 Science in China. Series F 78 Journal of Econometrics 78 Computational Optimization and Applications 77 Journal of Statistical Computation and Simulation 76 Chaos, Solitons and Fractals 76 Sequential Analysis 75 Journal of Systems Science and Complexity 74 Journal of Computational Physics 74 SIAM Journal on Discrete Mathematics 74 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 72 Proceedings of the American Mathematical Society 71 Journal of the American Statistical Association 71 Journal of Pure and Applied Algebra 71 Journal of Nonparametric Statistics 68 Discrete Mathematics, Algorithms and Applications 66 Journal of Discrete Mathematical Sciences & Cryptography 65 Journal of Time Series Analysis 65 Probability Theory and Related Fields 65 Statistics and Computing 64 Journal of Algebraic Combinatorics 64 Journal of Combinatorial Optimization ...and 958 more Journals
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#### Cited in 63 Fields
11,393 Information and communication theory, circuits (94-XX) 6,054 Computer science (68-XX) 5,831 Statistics (62-XX) 3,022 Probability theory and stochastic processes (60-XX) 2,675 Numerical analysis (65-XX) 2,661 Combinatorics (05-XX) 2,629 Number theory (11-XX) 2,569 Operations research, mathematical programming (90-XX) 1,868 Systems theory; control (93-XX) 1,825 Quantum theory (81-XX) 1,517 Harmonic analysis on Euclidean spaces (42-XX) 994 Linear and multilinear algebra; matrix theory (15-XX) 891 Biology and other natural sciences (92-XX) 876 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 622 Functional analysis (46-XX) 604 Algebraic geometry (14-XX) 524 Approximations and expansions (41-XX) 486 Dynamical systems and ergodic theory (37-XX) 452 Operator theory (47-XX) 416 Statistical mechanics, structure of matter (82-XX) 368 Calculus of variations and optimal control; optimization (49-XX) 368 Geometry (51-XX) 357 Convex and discrete geometry (52-XX) 338 Group theory and generalizations (20-XX) 317 Order, lattices, ordered algebraic structures (06-XX) 316 Partial differential equations (35-XX) 232 Commutative algebra (13-XX) 230 Measure and integration (28-XX) 192 Field theory and polynomials (12-XX) 174 Real functions (26-XX) 173 Ordinary differential equations (34-XX) 152 Special functions (33-XX) 145 Mathematical logic and foundations (03-XX) 130 Fluid mechanics (76-XX) 124 Associative rings and algebras (16-XX) 118 Functions of a complex variable (30-XX) 109 Differential geometry (53-XX) 107 Geophysics (86-XX) 105 Mechanics of deformable solids (74-XX) 100 Integral transforms, operational calculus (44-XX) 94 Optics, electromagnetic theory (78-XX) 87 Abstract harmonic analysis (43-XX) 86 Integral equations (45-XX) 54 Topological groups, Lie groups (22-XX) 52 Relativity and gravitational theory (83-XX) 45 Global analysis, analysis on manifolds (58-XX) 44 General topology (54-XX) 43 Difference and functional equations (39-XX) 42 History and biography (01-XX) 40 General and overarching topics; collections (00-XX) 39 Mechanics of particles and systems (70-XX) 33 Nonassociative rings and algebras (17-XX) 26 Classical thermodynamics, heat transfer (80-XX) 25 Several complex variables and analytic spaces (32-XX) 25 Manifolds and cell complexes (57-XX) 23 Algebraic topology (55-XX) 19 Sequences, series, summability (40-XX) 18 Astronomy and astrophysics (85-XX) 14 Potential theory (31-XX) 12 Category theory; homological algebra (18-XX) 11 Mathematics education (97-XX) 5 General algebraic systems (08-XX) 4 $$K$$-theory (19-XX)
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2021-09-26 14:53:58
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https://dmg.tuwien.ac.at/sanmauro/
|
## Contacts
Institute of Discrete Mathematics and Geometry
Vienna University of Technology
Wiedner Hauptstrasse 8-10/104
1040 Vienna, Austria
Office: DA05A05 (green building)
luca.san.mauro(at)tuwien.ac.at
ResearchGate
# Luca San Mauro
I am a postdoctoral fellow at the Institute for Discrete Mathematics and Geometry of Vienna University of Technology, where I am running a two-years research project (2018-2020) funded by the Austrian Science Fund (FWF), entitled Classifying relations via computable reducibility. I also teach logic within the Master in Language and Mind of University of Siena.
Recently, I obtained the habilitation to be Associate Professor of Logic and Philosophy of Science in an Italian university.
My research interests lie in computability theory and philosophy of mathematics. Here is my CV.
### Journal articles
1. Word problems and ceers
(with V. Delle Rose and A. Sorbi)
forthcoming in Mathematical Logic Quarterly
Abstract: This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called "computable reducibility"), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe for instance that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is in the same reducibility degree of some non-periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphim type as the word problem of some non-commutative and non-Boolean c.e. ring.
2. Degrees of bi-embeddable categoricity
(with N. Bazhenov, E. Fokina, and D. Rossegger)
forthcoming in Computability
Abstract: We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure $$\mathcal{A}$$ as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of $$\mathcal{A}$$; the degree of bi-embeddable categoricity of $$\mathcal{A}$$ is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and we show that every degree d.c.e. above $$\mathbf{0}^{(\alpha)}$$ for $$\alpha$$ a computable successor ordinal and $$\mathbf{0}^{(\lambda)}$$ for $$\lambda$$ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra.
3. Learning family of algebraic structures from informant
(with N. Bazhenov and E. Fokina)
forthcoming in Information and Computation
Abstract: We combine computable structure theory and algorithmic learning theory to study learning of families of algebraic structures. Our main result is a model-theoretic characterization of the class $$\mathbf{InfEx_\cong}$$, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures $$\mathbb{K}$$ is $$\mathbf{InfEx_\cong}$$-learnable if and only if the structures from $$\mathbb{K}$$ can be distinguished in terms of their $$\Sigma^{inf}_2$$-theories. We apply this characterization to familiar cases and we show the following: there is an infinite learnable family of distributive lattices; no pair of Boolean algebras is learnable; no infinite family of linear orders is learnable.
4. Speech acts in mathematics
(with M. Ruffino and G. Venturi)
forthcoming in Synthese
Abstract: We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct such acts within mathematics (not just assertions), and, as we intend to show, distinct illocutionary force indicators as well. Even mathematics in its most formalized version cannot do without some such indicators. This goes against a certain orthodoxy both in contemporary philosophy of mathematics (which tends to see mathematics as a realm in which no pragmatic features of ordinary language are present) and in speech act theory (which tends to pay attention solely to communication in ordinary language but not to formal languages). As we will comment, the recognition of distinct illocutionary acts within logic and mathematics and the incorporation of illocutionary force indicators in the formal language for both goes back to Frege's conception of these topics. We are, therefore, going back to a Fregean perspective. This paper is part of a larger project of applying contemporary speech act theory to the scientific language of mathematics in order to uncover the varieties and regular combinations of illocutionary acts (silently) present in it. For reasons of space, we here concentrate only on assertive and declarative acts within mathematics, leaving the investigation of other kinds of acts for a future occasion.
5. Minimal Equivalence Relations in Hyperarithmetical and Analytical Hierarchies
(with N. Bazhenov, M. Mustafa, and M. Yamaleev)
forthcoming in Lobachevskii Journal of Mathematics
Abstract: A standard tool for classifying the complexity of equivalence relations on $$\omega$$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce minimal degrees with respect to computable reducibility. Let $$\Gamma$$ be one of the following classes: $$\Sigma^0_{\alpha}$$, $$\Pi^0_{\alpha}$$, $$\Sigma^1_n$$, or $$\Pi^1_n$$, where $$\alpha \geq 2$$ is a computable ordinal and $$n$$ is a non-zero natural number. We prove that there are infinitely many pairwise incomparable minimal equivalence relations that are properly in $$\Gamma$$.
6. Classifying equivalence relations in the Ershov hierarchy
(with N. Bazhenov, M. Mustafa, A. Sorbi, M. Yamaleev)
forthcoming in Archive for Mathematical Logic
Abstract: Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $$\leq_c$$. This gives rise to a rich degree-structure. In this paper, we lift the study of $$c$$-degrees to the $$\Delta^0_2$$ case. In doing so, we rely on the Ershov hierarchy. For any notation $$a$$ for a non-zero computable ordinal, we prove several algebraic properties of the degree-structure induced by $$\leq_c$$ on the $$\Sigma^{-1}_{a}\smallsetminus \Pi^{-1}_a$$ equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of $$c$$-degrees.
7. At least one black sheep: Pragmatics and the language of mathematics
(with M. Ruffino and G. Venturi)
Journal of Pragmatics, 160, 114-119, 2020
Abstract: In this paper we argue, against a somewhat standard view, that pragmatic phenomena occur in mathematical language. We provide concrete examples supporting this thesis.
8. Bi-embeddability spectra and bases of spectra
(with E. Fokina and D. Rossegger)
Mathematical Logic Quaterly, 65(2), 228-236, 2019
Abstract: We study degree spectra of structures with respect to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of its bi-embeddable copies. To facilitate our study we introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several known families of degrees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddability spectra of strongly locally finite graphs.
9. Measuring the complexity of reductions between equivalence relations
(with E. Fokina and D. Rossegger)
Computability, 8(3/4), 265-280, 2019
Abstract: Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis.
10. Degrees of bi-embeddable categoricity of equivalence structures
(with N. Bazhenov, E. Fokina, and D. Rossegger)
Archive for Mathematical Logic, 58(5/6), 543-563, 2019
Abstract: We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) $$\Delta^0_\alpha$$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of $$\Delta^0_\alpha$$ bi-embeddable categoricity and relative $$\Delta^0_\alpha$$ bi-embeddable categoricity coincide for equivalence structures for $$\alpha=1,2,3$$. We also prove that computable equivalence structures have degree of bi-embeddable categoricity $$\mathbf{0}$$,$$\mathbf{0'}$$, or $$\mathbf{0''}$$. We furthermore obtain results on the index set complexity of computable equivalence structure with respect to bi-embeddability.
11. Computable bi-embeddable categoricity
(with N. Bazhenov, E. Fokina, and D. Rossegger)
Algebra and Logic, 57(5), 392-396, 2018
Abstract: We study the algorithmic complexity of isomorphic embeddings between computable structures.
12. Trial and error mathematics: Completions of theories
(with J. Amidei, U. Andrews, D. Pianigiani, and A. Sorbi)
Journal of Logic and Computation, 29(1), 157-184, 2019
Abstract: This paper is part of a project that is based on the notion of a dialectical system, introduced by Magari as a way of capturing trial and error mathematics. In Amidei et al. (2016, Rev. Symb. Logic, 9, 1–26) and Amidei et al. (2016, Rev. Symb. Logic, 9, 299–324), we investigated the expressive and computational power of dialectical systems, and we compared them to a new class of systems, that of quasi-dialectical systems, that enrich Magari’s systems with a natural mechanism of revision. In the present paper we consider a third class of systems, that of $$p$$-dialectical systems, that naturally combine features coming from the two other cases. We prove several results about $$p$$-dialectical systems and the sets that they represent. Then we focus on the completions of first-order theories. In doing so, we consider systems with connectives, i.e. systems that encode the rules of classical logic. We show that any consistent system with connectives represents the completion of a given theory. We prove that dialectical and $$q$$-dialectical systems coincide with respect to the completions that they can represent. Yet, $$p$$-dialectical systems are more powerful; we exhibit a $$p$$-dialectical system representing a completion of Peano Arithmetic that is neither dialectical nor $$q$$-dialectical.
13. Trial and error mathematics II: Dialectical sets and quasidialectical sets, their degrees, and their distribution within the class of limit sets
(with J. Amidei, D. Pianigiani and A. Sorbi)
Review of Symbolic Logic, 9(4), 810-835, 2016
Abstract: This paper is a continuation of Amidei, Pianigiani, San Mauro, Simi, and Sorbi (2016), where we have introduced the quasidialectical systems, which are abstract deductive systems designed to provide, in line with Lakatos' views, a formalization of trial and error mathematics more adherent to the real mathematical practice of revision than Magari's original dialectical systems. In this paper we prove that the two models of deductive systems (dialectical systems and quasidialectical systems) have in some sense the same information content, in that they represent two classes of sets (the dialectical sets and the quasidialectical sets, respectively), which have the same Turing degrees (namely, the computably enumerable Turing degrees), and the same enumeration degrees (namely, the $$\Pi^0_1$$ enumeration degrees). Nonetheless, dialectical sets and quasidialectical sets do not coincide. Even restricting our attention to the so-called loopless quasidialectical sets, we show that the quasidialectical sets properly extend the dialectical sets. As both classes consist of $$\Delta^0_2$$ sets, the extent to which the two classes differ is conveniently measured using the Ershov hierarchy: indeed, the dialectical sets are $$\omega$$-computably enumerable (close inspection also shows that there are dialectical sets which do not lie in any finite level; and in every finite level $$n \geq 2$$ of the Ershov hierarchy there is a dialectical set which does not lie in the previous level); on the other hand, the quasidialectical sets spread out throughout all classes of the hierarchy (close inspection shows that for every ordinal notation a of a nonzero computable ordinal, there is a quasidialectical set lying in $$\Sigma^{-1}_{\alpha}$$, but in none of the preceding levels).
14. Trial and error mathematics I: Dialectical and quasidialectical systems
(with J. Amidei, D. Pianigiani, G. Simi, and A. Sorbi)
Review of Symbolic Logic, 9(2), 299-324, 2016
Abstract: We define and study quasidialectical systems, which are an extension of Magari's dialectical systems, designed to make Magari's formalization of trial and error mathematics more adherent to the real mathematical practice of revision: our proposed extension follows, and in several regards makes more precise, varieties of empiricist positions à la Lakatos. We prove several properties of quasidialectical systems and of the sets that they represent, called quasidialectical sets. In particular, we prove that the quasidialectical sets are $$\Delta^0_2$$ sets in the arithmetical hierarchy. We distinguish between "loopless" quasidialectal systems, and quasidialectical systems "with loops". The latter ones represent exactly those coinfinite c.e. sets, that are not simple. In a subsequent paper we will show that whereas the dialectical sets are $$\omega$$-c.e., the quasidialectical sets spread out throughout all classes of the Ershov hierarchy of the $$\Delta^0_2$$ sets.
15. Universal computably enumerable equivalence relations
(with U. Andrews, S. Lempp, J.S. Miller, K.M. Ng, and A. Sorbi)
Journal of Symbolic Logic, 79, 60-88, 2014
Abstract: We study computably enumerable equivalence relations (ceers) under the reducibility $$R \leq S$$ if there exists a computable function $$f$$ such that, for every $$x, y$$, $$x R y$$ if and only if $$f (x) S f (y)$$. We show that the degrees of ceers under the equivalence relation generated by ≤ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer $$R$$ is universal if and only if $$R' ≤ R$$, where $$R'$$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are $$\Sigma^0_3$$-complete (the former answering an open question of Gao and Gerdes).
### Book chapters and conference papers
1. Comparing the isomorphism types of equivalence structures and preorders
(with N. Bazhenov and L. San Mauro), submitted
Abstract: A general theme of computable structure theory is to investigate when structures have copies of a given complexity $$\Gamma$$. We discuss such problem for the case of equivalence structures and preorders. We show that there is a $$\Pi^0_1$$ equivalence structure with no $$\Sigma^0_1$$ copy, and in fact that the isomorphism types realized by the $$\Pi^0_1$$ equivalence structures coincide with those realized by the $$\Delta^0_2$$ equivalence structures. We also construct a $$\Sigma^0_1$$ preorder with no $$\Pi^0_1$$ copy.
2. Limit learning equivalence structures
(with E. Fokina and T. Koetzing)
Proceedings of Machine Learning Research, 98, 383-403, 2019
Abstract: While most research in Gold-style learning focuses on learning formal languages, we consider the identification of computable structures, specifically equivalence structures. In our core model the learner gets more and more information about which pairs of elements of a structure are related and which are not. The aim of the learner is to find (an effective description of) the isomorphism type of the structure presented in the limit. In accordance with language learning we call this learning criterion $$\mathbf{InfEx}$$-learning (explanatory learning from informant). Our main contribution is a complete characterization of which families of equivalence structures are $$\mathbf{InfEx}$$-learnable. This characterization allows us to derive a bound of $$\mathbf{0''}$$ on the computational complexity required to learn uniformly enumerable families of equivalence structures. We also investigate variants of $$\mathbf{InfEx}$$-learning, including learning from text (where the only information provided is which elements are related, and not which elements are not related) and finite learning (where the first actual conjecture of the learner has to be correct). Finally, we show how learning families of structures relates to learning classes of languages by mapping learning tasks for structures to equivalent learning tasks for languages.
3. Church-Turing thesis, in practice
in M. Piazza and G. Pulcini (eds), Truth, Existence and Explanation, Springer, 225-248, 2018
Abstract: We aim at providing a philosophical analysis of the notion of "proof by Church's Thesis", which is – in a nutshell – the conceptual device that permits to rely on informal methods when working in Computability Theory. This notion allows, in most cases, to not specify the background model of computation in which a given algorithm – or a construction – is framed. In pursuing such analysis, we carefully reconstruct the development of this notion (from Post to Rogers, to the present days), and we focus on some classical constructions of the field, such as the construction of a simple set. Then, we make use of this focus in order to support the following encompassing claim (which opposes to a somehow commonly received view): the informal side of Computability, consisting of the large class of methods typically employed in the proofs of the field, is not fully reducible to its formal counterpart.
4. Direzioni della logica in Italia: la teoria (classica) della ricorsività
(with P. Cintioli and A. Sorbi)
in H. Hosni, G. Lolli, C. Toffalori (eds.), Le direzioni della ricerca logica in Italia 2, Edizioni ETS, 195-234, 2018
5. Degree spectra of structures with respect to the bi-embeddability relation
(with E. Fokina and D. Rossegger)
Proceedings of the 11th Panhellenic Logic Symposium, 32-38, 2017
6. Computable bi-embeddable categoricity of equivalence structures
(with N. Bazhenov, E. Fokina, and D. Rossegger)
Proceedings of the 11th Panhellenic Logic Symposium, 126-132, 2017
7. Reducibility and bi-reducibility spectra of equivalence relations
(with E. Fokina and D. Rossegger)
Proceedings of the 11th Panhellenic Logic Symposium, 83-89, 2017
8. Naturalness in mathematics
(with G. Venturi)
in G. Lolli, M. Panza, G. Venturi (eds.), From Logic to Practice, Springer, 277-313, 2014
Abstract: In mathematical literature, it is quite common to make reference to an informal notion of naturalness: axioms or definitions may be defined as “natural,” and part of a proof may deserve the same label (i.e., “in a natural way…”). Our aim is to provide a philosophical account of these occurrences. The paper is divided in two parts. In the first part, some statistical evidence is considered, in order to show that the use of the word “natural,” within the mathematical discourse, largely increased in the last decades. Then, we attempt to develop a philosophical framework in order to encompass such an evidence. In doing so, we outline a general method apt to deal with this kind of vague notions – such as naturalness – emerging in mathematical practice. In the second part, we mainly tackle the following question: is naturalness a static or a dynamic notion? Thanks to the study of a couple of case studies, taken from set theory and computability theory, we answer that the notion of naturalness – as it is used in mathematics – is a dynamic one, in which normativity plays a fundamental role.
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2020-07-05 06:45:01
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https://puzzling.stackexchange.com/questions/25738/malcolm-gladwells-outliers-progressive-matrices-puzzle
|
# Malcolm Gladwell's Outliers - Progressive Matrices Puzzle
In Outliers, Malcolm Gladwell presents the following puzzle:
♦♣♣ | ♥♥♣ | ♦♥♥
♦♦♣ | ♣♥♦ | ♦♣♣
♥♥♥ | ♦♦♣ | ♣♥♦
-----------------
♥♣♦ | ♥♣♦ | ♥♣♦
♥♣♥ | ♥♦♣ | ♣♥♥
♦♦♣ | ♣♥♦ | ♦♣♦
-----------------
♦♥♦ | ♦♣♥ |
♣♥♣ | ♦♥♦ | ?
♥♣♦ | ♥♣♣ |
In the first edition (2008) of the book, the top left square erroneously appears as:
♣♣♦
♦♥♣
♦♥♥
The alleged answer in both cases is:
♥♦♣
♦♣♦
♥♥♣
I have two questions:
1. What is the pattern?
2. What mistake(s) could explain the discrepancy between editions?
Note: I have an answer for Q1, but do not have one for Q2.
I mapped the symbols to $\{0,1,2\}$, so we get:
011 | 221 | 022
001 | 120 | 011
222 | 001 | 120
-----------------
210 | 210 | 210
212 | 201 | 122
001 | 120 | 010
-----------------
020 | 012 |
121 | 020 | ?
210 | 211 |
I found this quite simple algorithm (which has two cases, either it is a row change [type2] or not [type1]):
Type 1:
* we (+1 mod 3) every element
* rotate columns left one step
* rotate last column upwards
Type 2, row change:
* rotate the matrix 90 degrees
Since the final step is type 1, we get
120 120 120
200 -> 200 -> 202
112 112 110
and then +1 and translate back to cardsymbols.
120 201 ♥♦♣
202 --> 010 --> ♦♣♦
110 221 ♥♥♣
I don't know about the discrepancy. The first and wrong matrix does not make sense to me.
1. What mistake(s) could explain the discrepancy between editions?
Printer error. The versions in the first and second editions swap multiple-choice answer H and the spurious NW-corner pattern.
Methodology:
Found versions that showed multiple-choice answers for the Outliers puzzle.
Incorrect First Edition version, as it appears at the link provided by Daedric:
Correct Second Edition version, presumably, as seen in a blog entry about Raven's Progressive Matrices:
The versions agree on every other pattern, including the correct multiple-choice answer A.
• I can confirm that is how it appears in the second edition. Great spot - I really should have noticed that! – Will Jan 28 '16 at 10:03
• Thank you for verifying this, @Will. It was clued in by the comment with the more-complicated solution referred to by CarlLöndahl, at the link provided by Daedric. I thought that this additional detective puzzle was an interesting change of pace and that you had omitted the multiple-choice answers just for that purpose. You can certainly bet that I (and others, no doubt) have had fun trying to make the odd pattern fit some algorithm as well. – humn Jan 28 '16 at 10:20
So I solved it a little differently:
If you look at the bottom left symbol on each of the patterns you’ll notice that each sign lines up with the pattern on the answer
• This is an interesting pattern, but to comprise an answer wouldn't the bottom row need to be three hearts? – Will May 26 '18 at 20:53
• Make sure to add spoiler tags >! at the beginning of the line – Redwolf Programs May 26 '18 at 21:19
1. List item
Without changing to numbers, you can use H,D,C for matrix in the same row with 1) last column shifts one left 2) new last columns shifts up one 3) clubs become hearts, hearts become diamonds, and diamonds become clubs.
For last transformation:
DCH CHD CHD HDC
DHD HDD HDH DCD
HCC CCH CCD HHC
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2019-10-23 09:01:46
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https://codereview.stackexchange.com/questions/185024/regex-for-a-string-with-at-least-a-letter-and-no-repeating-dash
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# Regex for a string with at least a letter and no repeating dash
I had a problem a few days back and had to find a regex that matches a string that:
• Contains only letters in the alphabet [a-z0-9-] so lowercase latin letters, numbers 0 to 9, and the dash character
• Must contains at least a letter [a-z]
• Must not contains repeating dashes. abc-def-gh is ok but not abc--def
• The size of the string must be between 1 and 10 characters
I came up with:
^(?=[a-z0-9-]*[a-z][a-z0-9-]*)(?:[a-z0-9]|[-](?![-])){1,10}$ A little explanation: • (?=[a-z0-9-]*[a-z][a-z0-9-]*) lookahead to find if the string contains at least a letter • (?:[a-z0-9]|[-](?![-])){1,10} A non capturing group (not really important to capture or not I believe) with two parts, that must be between 1 and 10 chars • [a-z0-9] any letter in the alphabet I want • [-](?![-] a dash then a negative lookahead to see if there is no dash following Here is a list of strings that should not match: #should not match - 1 1- -1 aaaaaaaaaaaaaaaaa 11111 2a2--af a2a--22 And a list of string that should match: #should match a a- -a a-2 a2a 2a2- a2a-2a2 a-b-c-d a123213232 123213213a a12321322a HERE you can find an online regex tester to play with it. All improvements, remarks, feedbacks is welcome. My guts is telling me the lookahead can be improved but I did not find how. Also, I'm not even sure the lookahead is needed at all. I'm eager to improve my regex skill so if you have any alternative method also I'd be really glad to know. • by "Must not contain repeating dashes", do you mean "no more than one dash total" or "no consecutive dashes"? i.e. should it match abc-def-ghi? the current regex doesn't match that. – cas Jan 13 '18 at 10:34 • Yes I mean no a--a but abc-chd-dhd should work – Julien Rousé Jan 13 '18 at 10:37 • In the regex tester, I believe it matches abc-def-gh? It doesn't work with abc-def-ghi because it's too long and the string should be 10 maximum. see here at the bottom – Julien Rousé Jan 13 '18 at 10:43 • doh, you're right. i didn't think to count the chars, and typed too fast to notice it change when i typed the 10th character. – cas Jan 13 '18 at 10:44 • No worries, thanks for your input! And in the comment I told you were right when it wasn't, I didn't check either if abc-def-ghi was too long, my mistake. – Julien Rousé Jan 13 '18 at 10:45 ## 1 Answer The lookahead (?=[a-z0-9-]*[a-z][a-z0-9-]*) can be reduce to: (?=.*[a-z]) because the allowed characters are defined after in the non capturing group. The dash [-] doesn't need to be inside a character class, - is enough Then, the whole regex becomes: ^(?=.*[a-z])(?:[a-z0-9]|-(?!-)){1,10}$
• Thank you for your help! Was there any way to get rid of that lookahead? – Julien Rousé Jan 13 '18 at 11:22
• @JulienRousé: No, it is the simplest way to do such checking. – Toto Jan 13 '18 at 11:38
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2019-07-18 00:14:44
|
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https://www.cheenta.com/trilinear-coordinates-and-locus/
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Select Page
# Understand the problem
Let $ABC$ be an equilateral triangle and $P$ in its interior. The distances from $P$ to the triangle’s sides are denoted by $a^2, b^2,c^2$respectively, where $a,b,c>0$. Find the locus of the points $P$ for which $a,b,c$ can be the sides of a non-degenerate triangle.
##### Source of the problem
Romanian Master in Mathematics, 2008
Geometry
##### Difficulty Level
Medium
This problem obviously points towards an application of trilinear coordinates (see more here).
Do you really need a hint? Try it first!
If there exists a non-degenerate triangle with sides $a,b,c$ then the area of the triangle must be positive. Make use of this fact employing Heron’s formula.
Hint 1 will give you a locus in terms of trilinear coordinates. However, it is easier to work in cartesian coordinates as we are already familiar with many curves in them.
Take $A=(1,0,0), B= (0,1,0), C= (0,0,1)$. There is a simple relationship between the cartesian and trilinear coordinates for this choice.
.Note that all points in the plane of $ABC$ satisfy $x+y+z=1$ (why?). For any $P$ in the interior, Let $Q$ be the foot of the perpendicular from $P$ to $AB$ and $R$ be the foot of the perpendicular from $P$ to the XY plane. It is possible to show that $\frac{PR}{PQ}= \sqrt{\frac{2}{3}}$ (it is because the angle between the plane and the Z axis is $\arccos \sqrt{\frac{2}{3}}$). Hence, $\frac{z}{c^2}=\sqrt{\frac{3}{2}}$. By symmetry, $\frac{x}{a^2}=\sqrt{\frac{2}{3}}=\frac{y}{b^2}$. This relates the two coordinate systems. For a triangle with sides $a,b,c$, the square of the area is $\frac{1}{16}(a+b+c)(-a+b+c)(a-b+c)(a+b-c)$. For this to be positive, we must have (after simplification) $a^4+b^4+c^4< 2(a^2b^2+b^2c^2+c^2a^2)$. In cartesian coordinates, this translates to $\frac{3}{2}(x^2+y^2+z^2)<3(xy+yz+zx)$, which is equivalent to $(x+y+z)^2>2(x^2+y^2+z^2)$. As $P$ lies on the plane $x+y+z=1$, this means that $x^2+y^2+z^2<\frac{1}{2}$. This last equation is that of the interior of a solid sphere. Hence, our desired locus is the intersection of this solid sphere with the plane $x+y+z=1$, which is precisely the interior of the circumcircle of $ABC$.
# Connected Program at Cheenta
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2020-09-20 04:07:11
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https://brilliant.org/problems/chocolate-breaker/
|
Chocolate Breaker
You have many chocolate bars of unit length and start breaking each of them into 3 pieces by randomly choosing two points on the bar. What are the average lengths of the shortest, medium, and longest pieces?
If the product of these averages can be expressed as $$\frac pq$$, where $$p$$ and $$q$$ are coprime positive integers, give your answer as $$p+ q$$.
×
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2018-06-19 04:53:46
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7487495541572571, "perplexity": 290.3196066593108}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267861899.65/warc/CC-MAIN-20180619041206-20180619061206-00317.warc.gz"}
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https://tex.stackexchange.com/questions/630368/is-there-a-way-to-automate-a-switch-into-text-mode-after-a
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# Is there a way to automate a switch into text mode after a _
I am a physics student and we have been told that indices that abbreviate some descriptive word or indices that are just to distinguish between say different objects like the mass of two objects A and B shouldn't be in italics in an equation. An example for that would be $m_\text{tot} = m_\text{A} + m_\text{B}$. For physical constants, or any non made up index for that matter, the indices should be in italics as the rest of the equation. Here an example would be the mass of an electron $m_e$.
Since basically every index is for things you make up, for almost every index I have to write out _\text{...}. This just clutters my equations so much that I tried to find a way to automate a switch into text mode after a _ character, but found nothing. So here I am asking you people: Do you know a way?
• Welcome to TeX.SE.
– Mico
Jan 16 at 12:34
• note that you should not use \text otherwise you will get the current font from before the math, eg italic in theorems. use m_{\mathrm{tot}} Jan 16 at 12:38
• I would not redefine the math primitives, although that is possible just do \newcommand\z[1]{_{\mathrm{#1}} then you can use m\z{tot} which is only one character more than m_{tot} Jan 16 at 12:40
You might use a different character to introduce textual subscripts, so you can keep _ to do its usual job.
I chose ? because it's unlikely to show up in math formulas.
\documentclass{article}
\usepackage{amsmath}
\newcommand{\textualsubscript}[1]{%
_{\textnormal{\upshape#1}}%
}
\begingroup\lccode~=? \lowercase{\endgroup\let~}\textualsubscript
\AtBeginDocument{\mathcode?="8000 }
\begin{document}
$m?{tot}=m?{A}+m?{B}-m_{e}$
\end{document}
• Thanks a lot for the help. Since I'm new to LaTeX I don't know a lot about the things you can customize, but I'm eager to learn. Would you care to briefly explain what your code does? Jan 16 at 16:18
• @Luguza This has been explained elsewhere in the site. Basically, \mathcode@="8000 tells TeX to treat ? as a command \let equal to the one introduced beforehand with a known trick. Look for “math active”. Jan 16 at 16:24
If you're willing and able to use LuaLaTeX to compile your document, the following solution may be of interest to you. The solution defines a Lua function, called sub2mathrm, which encases subscript terms in \mathrm wrappers if there is no whitespace between the _ (underscore) character and (a) material encased in curly braces -- e.g., {tot} -- or (b) a single alphabetic character -- e.g., A or B. Conversely, if there is some whitespace after the _ character, the Lua function does nothing. The Lua function does nothing with A_\sigma either, since the backslash character in \sigma is not an alphabetical character. sub2mathrm performs its work with the help of Lua's powerful string.gsub ("generalized substitution") function.
The Lua function is activated by executing \SubToMathrmOn, a utility macro that assigns sub2mathrm to the LuaTeX process_input_buffer callback. It is deactivated by running the utility macro \SubToMathrmOn.
% !TEX TS-program = lualatex
\documentclass{article}
\usepackage{luacode} % for 'luacode' environment
\begin{luacode}
function sub2mathrm ( s )
s = s:gsub ( "_(%b{})" , "_{\\mathrm%1}" )
s = s:gsub ( "_(%a)" , "_{\\mathrm{%1}}" )
return s
end
\end{luacode}
%% define two utility macros:
"process_input_buffer" , sub2mathrm , "sub2mathrm" )}}
\newcommand\SubToMathrmOff{\directlua{luatexbase.remove_from_callback (
"process_input_buffer" , "sub2mathrm" )}}
\begin{document}
\SubToMathrmOn % Activate the Lua function
$m_{tot} = m_A + m_B$ % no whitespace after "_"
$m_ {tot} = m _ A + m_ B$ % whitespace after "_"
\SubToMathrmOff % Deactivate the Lua function
$m_{tot} = m_A + m_B$
\end{document}
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2022-08-18 10:24:04
|
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https://cs.overleaf.com/latex/examples/student-marking-submission-slip/fbfzftjtnpsy
|
Skip to content
Last Updated
8 years ago
License
Other (as stated in the work)
AbstractA simple submission slip designed to be attached to student work for marking. The source code is available on github. This is the release from 12th April 2014. This work is provided for use under the open source MIT license (MIT).
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2022-10-01 15:12:00
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https://www.usgs.gov/publications/effects-agricultural-land-use-changes-and-rainfall-ground-water-recharge-central-and
|
# Effects of Agricultural Land-Use Changes and Rainfall on Ground-Water Recharge in Central and West Maui, Hawaii, 1926-2004
September 22, 2007
Concern surrounding declines in ground-water levels and an increase in the chloride concentration of water pumped from wells in the Iao aquifer system on the Island of Maui has prompted an investigation into the long-term sustainability of current (2006) and future ground-water withdrawals. As part of this investigation, a water budget for central and west Maui was calculated from which (1) ground-water recharge was estimated for the period 1926-2004 and (2) the effects of agricultural land-use changes and drought were analyzed.
Estimated mean ground-water recharge decreased 44 percent from 1979 to 2004 in central and west Maui. Reduction in agricultural irrigation, resulting from more efficient irrigation methods and a reduction in the acreage used for agriculture, is largely responsible for the declining recharge. Recently, periods of lower-than-average rainfall have further reduced recharge. During the period 1926-79, ground-water recharge averaged 693 Mgal/d, irrigation averaged 437 Mgal/d, and rainfall averaged 897 Mgal/d. During the period 2000-04, ground-water recharge averaged 391 Mgal/d, irrigation averaged 237 Mgal/d, and rainfall averaged 796 Mgal/d.
Simulations of hypothetical future conditions indicate that a cessation of agriculture in central and west Maui would reduce mean ground-water recharge by 18 percent in comparison with current conditions, assuming that current climatic conditions are the same as the long-term-average conditions during the period 1926-2004. A period of drought identical to that of 1998-2002 would reduce mean recharge by 27 percent. Mean recharge would decrease by 46 percent if this drought were to occur after a cessation of agriculture in central and western Maui. Whereas droughts are transient phenomena, a reduction in agricultural irrigation is likely a permanent condition.
## Citation Information
Publication Year 2007 Effects of Agricultural Land-Use Changes and Rainfall on Ground-Water Recharge in Central and West Maui, Hawaii, 1926-2004 10.3133/sir20075103 John A. Engott, Thomas T. Vana Report USGS Numbered Series Scientific Investigations Report 2007-5103 sir20075103 USGS Publications Warehouse Pacific Islands Water Science Center
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2023-02-03 11:38:57
|
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https://support.bioconductor.org/p/18008/
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Question: goTools: ontoCompare question
0
12.5 years ago by
Hello, I ran ontoCompare on the full list of probes in the mouse4302 genechip both with the default EndNodeList() and with a custom end node list containing only the antioxidant activity, biological_process, cellular_component, and molecular_function GO terms and found what appears to be a discrepency: > length(sviData$svi$ID) [1] 45101 > sviData$svi$ID[1:5] [1] "1452670_at" "1422340_a_at" "1452114_s_at" "1422644_at" "1423359_at" > listall<-list("allprobes"=sviData$svi$ID) > endlist<-c("GO:0003674", "GO:0005575", "GO:0008150", "GO:0016209") > totalAnnotations<-ontoCompare(listall, probeType="mouse4302", method="none") > write.table(totalAnnotations, file="totalAnnotations.txt") > totalAnnotations2<-ontoCompare(listall, probeType="mouse4302", method="none", endnode=endlist) > write.table(totalAnnotations2, file="totalAnnotations_reduced.txt") When finding the total possible number of annotations for the top level GO terms (BP, MF, CC), I got different numbers for the two approaches, but I got the same numbers for "NotFound" and "antioxidant activity": from totalAnnotations.txt antioxidant activity 127 biological_process 2594 cellular_component 2365 molecular_function 2414 NotFound 11120 ...others from totalAnnotations_reduced.txt antioxidant activity 127 biological_process 28020 cellular_component 28509 molecular_function 30875 NotFound 11120 I was just wondering if anyone knew why this might happen since it affects the interpretation of a comparison I was going to do. These data appear to reflect the histogram output from ontoPlot (so I don't think its an R->txt->excel thing). Is the output with method="none" the total number of times all probes are annotated at the endnode or at a child of the end node? Does it have something to do with the "isa" values in EndNodeList() or my method of creating endlist? R v.2.5.0 goTools v1.8.0 Cheers, Dave --and thank you Dick for recommending topGO. I found what I needed through that package.
go topgo • 559 views
modified 12.5 years ago by Paquet, Agnes500 • written 12.5 years ago by davidl@unr.nevada.edu140
0
12.5 years ago by
Paquet, Agnes500
Paquet, Agnes500 wrote:
Hi Dave, The current algorithm in ontoCompare is the following: - for each probe id in your list, retrieve all GO ids corresponding to this probe id - then, map these Go ids up to the end nodes provided as argument to the function (or the default ones) - Once the mapping is finished, add 1 to the count of each end node which was reached at least once (and not the number of times a node was hit, which explains the discrepancy in your example) For example, if I use only 1 Affy probe, and restrict everything to MF to simplify your example, ontoCompare will give me the following results; 1) using the default end nodes: > ontoCompare(list("1415670_at"),probeType="mouse4302",goType="MF",met hod="none") [1] "Starting ontoCompare..." [1] "Number of lists = 1" [1] "Using method: none" binding structural molecule activity transporter activity NotFound 1 1 1 0 (we have 1 count for each end node which was reached at least once) 2) Using your endlist > ontoCompare(list("1415670_at"),probeType="mouse4302",goType="MF",met hod="none",endnode=endlist) [1] "Starting ontoCompare..." [1] "Number of lists = 1" [1] "Using method: none" molecular_function NotFound 1 0 (same here, only 1 count for MF, and not 3) We made this choice because some nodes/probes may be more annotated than others, and it could make the relative comparison of 2 lists of probes appear more different based on the availability of annotations, and not true biological difference. You could also use the other methods to get number of hits relative to the number of probes or the number of GO in your list. I hope this will help, don't hesitate to email me again if you have more questions. Best, Agnes ________________________________ From: bioconductor-bounces@stat.math.ethz.ch on behalf of davidl@unr.nevada.edu Sent: Fri 6/29/2007 8:01 AM To: Bioconductor Subject: [BioC] goTools: ontoCompare question Hello, I ran ontoCompare on the full list of probes in the mouse4302 genechip both with the default EndNodeList() and with a custom end node list containing only the antioxidant activity, biological_process, cellular_component, and molecular_function GO terms and found what appears to be a discrepency: > length(sviData$svi$ID) [1] 45101 > sviData$svi$ID[1:5] [1] "1452670_at" "1422340_a_at" "1452114_s_at" "1422644_at" "1423359_at" > listall<-list("allprobes"=sviData$svi$ID) > endlist<-c("GO:0003674", "GO:0005575", "GO:0008150", "GO:0016209") > totalAnnotations<-ontoCompare(listall, probeType="mouse4302", method="none") > write.table(totalAnnotations, file="totalAnnotations.txt") > totalAnnotations2<-ontoCompare(listall, probeType="mouse4302", method="none", endnode=endlist) > write.table(totalAnnotations2, file="totalAnnotations_reduced.txt") When finding the total possible number of annotations for the top level GO terms (BP, MF, CC), I got different numbers for the two approaches, but I got the same numbers for "NotFound" and "antioxidant activity": from totalAnnotations.txt antioxidant activity 127 biological_process 2594 cellular_component 2365 molecular_function 2414 NotFound 11120 ...others from totalAnnotations_reduced.txt antioxidant activity 127 biological_process 28020 cellular_component 28509 molecular_function 30875 NotFound 11120 I was just wondering if anyone knew why this might happen since it affects the interpretation of a comparison I was going to do. These data appear to reflect the histogram output from ontoPlot (so I don't think its an R->txt->excel thing). Is the output with method="none" the total number of times all probes are annotated at the endnode or at a child of the end node? Does it have something to do with the "isa" values in EndNodeList() or my method of creating endlist? R v.2.5.0 goTools v1.8.0 Cheers, Dave --and thank you Dick for recommending topGO. I found what I needed through that package. _______________________________________________ Bioconductor mailing list Bioconductor at stat.math.ethz.ch https://stat.ethz.ch/mailman/listinfo/bioconductor Search the archives: http://news.gmane.org/gmane.science.biology.informatics.conductor
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2019-12-14 00:50:19
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https://worldbuilding.stackexchange.com/questions/165592/good-explanation-for-why-aliens-cant-build-nuclear-engines/165616
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# Good explanation for why aliens can't build nuclear engines?
"Aliens used uranium and plutonium to generate energy for thousands of years. However, by the time they got into space exploration, they've used up every last bit of it. That's why they'll have to try leaving their high-gravity planet with chemical engines."
This sounds good to me, but I fear that there are options I'm not considering. Have I given enough explanation for why nuclear energy can't be used? Are there other ways they could make use of nuclear energy? How can I explain away all the ways they might still be able to make use of nuclear energy to heat up their fuel?
• Er.. Didn't you just said they used up every last bit...? – user6760 Jan 14 '20 at 11:29
• Do you have an idea what a time span of "many thousands of years" means in the development of a technological society? For example, there are less than five thousand years the oldest Sumerian clay tablets and the latest Apple iPad electronic tablet... After many thousands of years since then first development of atomic power, why don't they have antigravity? – AlexP Jan 14 '20 at 13:42
• @AlexP Wow. You just assumed my aliens would behave/think/live/age exactly like humans? Why do you assume they'd live in a similar environment? Using a power source and being on a certain level of knowledge with resources to invest into it are two completely different things. – justthisonequestion Jan 14 '20 at 14:32
• "Using a power source and being on a certain level of knowledge with resources to invest into it are two completely different things": this is . . . unexpected. No, they are definitely not two different things. How could they possibly use uranium and thorium (maybe that's what you meant by "plutonium") to generate energy if they did not have the level of knowledge with resources to invest in it? Making a nuclear reactor does not happen by accident. It requires a highly advanced technological civilization with a very good understanding of physics and resources to do research. – AlexP Jan 14 '20 at 15:03
• @JRE: Not so that they consume every last bit of fissile metals, no. – AlexP Jan 14 '20 at 16:18
They can only use as much uranium as they had available, and not all planets will be created equal in this regard. Even something as simple as evolving much later in the life of their planet will give more time for useful fissile materials like U235 to decay into less useful elements. Combined with a lower abundance of fissiles in the protoplanetary disc the world formed from, there's no problem explaining the lack of nuclear fuel.
What will be harder to explain is how they were a technologically advanced race for thousands of years and failed to develop a launch system that will actually work for them, because chemical rockets will not (as discussed in previous questions of yours, here and elsewhere, ad nauseam).
• Couldn't they somehow fuse other atoms together to create new uranium atoms? Maybe I'd have to explain how they already did that and the newer procedures would take a lot more energy than before... Something like that – justthisonequestion Jan 14 '20 at 11:57
• @justthisonequestion it isn't at all energy efficient. Synthetic elements made in particle accelerators can only plausibly be made in tiny quantities at great cost. It isn't as inconvenient or expensive as antimatter, but it wouldn't be practical. – Starfish Prime Jan 14 '20 at 12:13
• @justthisonequestion, it's much, much more better to fuse hydrogen to create new helium atoms. And this is hard to deny this reqaction for your aliens. – ksbes Jan 14 '20 at 12:46
• @ksbes I was gonna say something about fusion being perpetually 50 years away being a reasonable fictional handwave, but after a thousand years of technological civilisation someone must have been able to make a laser-triggered nuclear weapon, which gives you Orion drives even in the absense of any other sensible launch system. So yeah... – Starfish Prime Jan 14 '20 at 12:52
• @StarfishPrime So you don't know how it would be achieved, but you know that it could be used inside of a rockets engine... Is it just plausible to you? Why? Is it something like 'as soon we know this exact part, we can use easily for most things' Please put it into your answer. Thanks for your time. – justthisonequestion Jan 14 '20 at 16:03
## Hurdles
I think you have bigger problems than "nuclear power" if you're trying to force your Thousands-of-Years-Past-Fission Alien Society to use chemical rockets.
1. Space elevator (AKA Bean Stalk). A geostationary satellite tethered to the planet below. An elevator goes up and down the tether, making it Much Cheaper to get out of the planets gravity. Building one is primarily an engineering problem. That tether has to be Really Strong. Carbon Nanotubes (CNTs) should work, but linking them together well enough or building them That Long in the first place is a Hard Problem.
2. Teleportation/portals: Aliens that advanced should have come up with something we think is impossible, right?
3. Ditto for gravity manipulation, reactionless drives, ion drives, and so forth. Even if they only previously used that technology for their equivalent to cars, it should still exist.
## Possible explanations
1. "It's against our religion". Riiight... It was forbidden to "leave our gravity well", or whatever. So what changed? Why are they doing it now? Is the group using chemical rockets a bunch of Heretics?
2. Apocalypse. Alien Society ain't what it used to be. They still have some advanced tech they were able to salvage, but had to cobble together their rocket out of whatever they had available. Disease, disaster, lost a war...
2a. Uplift Uprising. A species of animal native to the alien world genetically/surgically modified to be intelligent servants rose up against their masters. They might be just as smart as humans, but not be intelligent/educated enough to understand some of the super-tech their masters used... or they lack some other key ability their masters possessed (telekinesis, "magic", some sense [sonar, color vision or hammerhead-esque electrical sense: hard to use a technology when its controls are invisible to you[by design?]], any ol "deus ex xeno" will do) that prevents them from fully utilizing their masters' technology.
• (1) might not be possible on the alien world. It isn't unambiguously possible on Earth, after all. (2) is magic, not science. (3) ion drives do not belong in the same category as those other bits of probably impossible handwavium. They're not much use at the bottom of a deep gravity well, either. – Starfish Prime Jan 14 '20 at 13:32
• (1) The requirements might change, but unless they have a severe orbital garbage problem... hey! See: "Kessler Syndrome" (2) Is a catchall for anything we currently believe to be impossible. I refuse to believe that human physics has reached the point where everything we think is impossible is actually impossible. (3) Our current ion drives are useless at the bottom of a gravity well (or in an atmosphere IIRC). Give our ion tech a couple thousand years to improve, and you might end up with TIE fighters (which also use an ion drive). – Mark Storer Jan 14 '20 at 13:55
• (1) Even if they could synthesise suitable handwavium then maybe they could build one, only as the OP says, they haven't actually done any space exploration yet, so they can't possibly have a space elevator. (2) Wishing that magical things are real is tenuously compatible with the OP, but wishing really hard isn't a great solution to the given problem. (3) ion-drives are intrinsically high-Isp, low-thrust engines. They will forever be terrible in a deep gravity well. Use an appropriate design. Not everything is a nail, so put the hammer away. – Starfish Prime Jan 14 '20 at 14:01
• + for 2a. Reading about the Kzinti recently; that was their situation. They took over the tech of their former masters and there was minimal subsequent innovation. – Willk Jan 14 '20 at 14:54
• And Ringo's most of the "Legacy of the Aldenata" races as well. Not necessarily uplifted, but genetically modified subjugated races. – Mark Storer Jan 14 '20 at 16:03
Maybe they can't use fission engines, but if they're advanced enough to have depleted all the uranium in their planet after many thousand of years, they definitely have discovered and developed the nuclear fusion. You will need a good explanation on that.
Health on the Homeworld
Their bodies cannot handle radiation as well as human bodies can. Their ecology cannot handle radiation or other pollution as well as Earth biology can.
It could mean that the aliens riding the ship are blinded or permanently made insane by the act of launching. It could mean that the act of launching a ship ruins a city-sized plot of land for generations. It could mean that the act of launching a ship generates mutants which spread across the land terrorizing it. It could make the next generation imbecilic.
Now there's an interesting story - when does the nuclear ship get to take off? Who gets to decide that everyone's descendants for the next hundred years are animals? The next civilization has to pick up from intentional library caches dotted across the landscape. See Niven/Pournell's "Mote in God's Eye" for cyclic civilization , and Niven's "Ringworld" series for indestructible libraries and a world-serving order of selfless family-less librarians.
Perhaps they have political reasons related to it. NIMBY, and no land left in the world without some owner. A long history of garbage dumping on your neighbor.
• Maybe launching the nuclear rocket risks the planet's atmosphere which will have longterm health consequences to all those left behind ? – Criggie Jan 14 '20 at 21:48
• I mean, radiation is natural, it is present everywhere. If their bodies can't handle radiation, they would've died a long looong time ago. Also, in a nuclear engine (a well-built one, atleast) you're not exposed to radiation. – Roberto Jan 15 '20 at 8:42
Early in their development, think ancient egypt, they got inspired by an Oklo-style natural reactor. This gave them a headstart over civilizations using wood as a source of heat. But it also left them with little naturally fissile material remaining and a strong cultural bias for large scale/low energy density nuclear tech.
Think more in terms of geothermal energy from fission instead of magma rather then our nuclear power plants. You don't want to run a basically unshielded reactor near your settlement, put it under a mountain or pyramid and have the steam & hot water for your central heating come to you.
Make sure your civilisation isn't advanced enough to deal with metastable helium or metallic hydrogen, both superior rocket fuels without the radioactivity.
At some point in their long history, at a time when they’d reached a staggeringly high level of knowledge and technology, their planet was rife with conflict between many opposing polities. These conflicts threatened the survival of all life on the planet.
Political solutions were deemed impossible since no group trusted off of the other groups. And, no one would disarm their doomsday weapons for fear of being vulnerable to attack by coalitions of the other polities.
A great scientist built a machine that generated a planetary scale field which inhibited nuclear decay by modulating the weak and strong atomic forces in ingenious ways. The machine, once started, would become the ultimate doomsday weapon, and destroy the planet if turned off, but made all of the other doomsday weapons and nuclear devices useless.
Faced with a loss of their deterrents, and effectively protected against obliteration, the polities went mad fighting wars using conventional methods — bioweapons, chemical weapons, nano-tech, masers, lasers, blasters. After a thousand years of warfare, they found their own paths to peace, and formed a one world government.
Now, they want to leave their planet but the great machine still operates, preventing nuclear decay, and making nuclear engines inoperative within a few thousand miles of the surface of their planet
There are ways to generate fissile materials, as we do now in small quantities inside particle accelerators . Your folks had 1000 years and couldn't design systems, albeit energy-hungry, to build up reserves of useful fissile materials?
But more important, the total energy content of fission reactors vs. mass is really crappy from a rocket or space ship point of view. Either chemical engines, ion engines, or some theoretical future discovery of a new field, or harnessing "dark energy," will always win.
• Creating fissile material via fusion will necessarily cost more energy than you can get back by splitting it back apart. And if you can do fusion, then you have enough technology to just fuse the vastly more abundant hydrogen and harvest the energy that releases, no fission required. – StephenS Jan 14 '20 at 19:53
• @StephenS A population II star may be old enough that most of its $^{235}U$, half-life 704 million years has decayed. Not to mention that population II stars are likely to be deficient in uranium and thorium in the first place. Thus they could start a project to produce enough $^{233}U$ or $^{239}Pu$ via neutron bombardment to create their first breeder reactor. – user5713492 Jan 14 '20 at 22:45
• @StephenS what you can do on the ground, with lots of "local" energy is different from what you can do after launching, where you have to carry your energy with you – Carl Witthoft Jan 15 '20 at 16:46
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2021-06-17 08:35:02
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https://learncheme.com/quiz-yourself/interactive-self-study-modules/isothermal-batch-reactors/isothermal-batch-reactors-simulation/
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#### Isothermal Batch Reactors: Interactive Simulations
The first simulation will work on your browser. The second simulation was prepared using Mathematica. Download the free CDF player, and then download the simulation CDF file (link given below or click on figure to download). Try to predict the behavior when a parameter changes before using a slider to change that parameter. Screencasts below explain how to use these simulations.
##### Simulation: Series Reactions in a Batch Reactor
Two first-order, liquid-phase reactions A → B → C take place in an isothermal batch reactor. The reactor initially contains only A at a concentration of 2 mol/L. The activation energy $$E_{a,2}$$ of the second reaction (155 kJ/mol) is higher than the activation energy $$E_{a,1}$$ of the first reaction (145 kJ/mol).
Try to answer these questions before determining the answer with the simulation. We suggest that you write down the reasons for your answers.
1. For a series reaction A → B → C in an isothermal batch reactor, how does the selectivity (moles of B)/(mole of C) change with time?
2. For a series reaction A → B → C in an isothermal batch reactor, the activation energy for the first reaction is lower than the activation energy for the second reaction. How does the selectivity (moles of B)/(mole of C) change as the temperature increases?
##### Simulation: Batch Reactors at Constant Volume or Constant Pressure
This Demonstration compares the time for an irreversible, gas-phase reaction to reach a certain fractional conversion in an isothermal batch reactor operating either at constant volume or at constant pressure. Both reactors start at the same initial conditions. If the reaction involves a mole change and the rate of reaction is not zero or first-order, then the time to reach a certain conversion is different for the two types of reactors; in a constant-pressure reactor, the reactant concentration changes because of the volume change due to the mole change.
Try to answer these questions before determining the answer with the simulation. We suggest that you write down the reasons for your answers.
1. For a first-order, gas-phase reaction (A → 2B) starting at the same conditions in a constant-pressure and a constant-volume reactor, which reactor reaches 50% conversion first? Both reactors are initially the same size.
2. For a second-order, gas-phase reaction (A → 2B) starting at the same conditions in a constant-pressure and a constant-volume reactor, which reactor reaches 50% conversion first? Both reactors are initially the same size.
3. For a second-order, gas-phase reaction (A → B) starting at the same conditions in a constant-pressure and a constant-volume reactor, which reactor reaches 50% conversion first? Both reactors are initially the same size.
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2023-03-22 02:54:39
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http://clang.llvm.org/docs/DataFlowSanitizer.html
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# DataFlowSanitizer¶
## Introduction¶
DataFlowSanitizer is a generalised dynamic data flow analysis.
Unlike other Sanitizer tools, this tool is not designed to detect a specific class of bugs on its own. Instead, it provides a generic dynamic data flow analysis framework to be used by clients to help detect application-specific issues within their own code.
## Usage¶
With no program changes, applying DataFlowSanitizer to a program will not alter its behavior. To use DataFlowSanitizer, the program uses API functions to apply tags to data to cause it to be tracked, and to check the tag of a specific data item. DataFlowSanitizer manages the propagation of tags through the program according to its data flow.
The APIs are defined in the header file sanitizer/dfsan_interface.h. For further information about each function, please refer to the header file.
### ABI List¶
DataFlowSanitizer uses a list of functions known as an ABI list to decide whether a call to a specific function should use the operating system’s native ABI or whether it should use a variant of this ABI that also propagates labels through function parameters and return values. The ABI list file also controls how labels are propagated in the former case. DataFlowSanitizer comes with a default ABI list which is intended to eventually cover the glibc library on Linux but it may become necessary for users to extend the ABI list in cases where a particular library or function cannot be instrumented (e.g. because it is implemented in assembly or another language which DataFlowSanitizer does not support) or a function is called from a library or function which cannot be instrumented.
DataFlowSanitizer’s ABI list file is a Sanitizer special case list. The pass treats every function in the uninstrumented category in the ABI list file as conforming to the native ABI. Unless the ABI list contains additional categories for those functions, a call to one of those functions will produce a warning message, as the labelling behavior of the function is unknown. The other supported categories are discard, functional and custom.
• discard – To the extent that this function writes to (user-accessible) memory, it also updates labels in shadow memory (this condition is trivially satisfied for functions which do not write to user-accessible memory). Its return value is unlabelled.
• functional – Like discard, except that the label of its return value is the union of the label of its arguments.
• custom – Instead of calling the function, a custom wrapper __dfsw_F is called, where F is the name of the function. This function may wrap the original function or provide its own implementation. This category is generally used for uninstrumentable functions which write to user-accessible memory or which have more complex label propagation behavior. The signature of __dfsw_F is based on that of F with each argument having a label of type dfsan_label appended to the argument list. If F is of non-void return type a final argument of type dfsan_label * is appended to which the custom function can store the label for the return value. For example:
void f(int x);
void __dfsw_f(int x, dfsan_label x_label);
void *memcpy(void *dest, const void *src, size_t n);
void *__dfsw_memcpy(void *dest, const void *src, size_t n,
dfsan_label dest_label, dfsan_label src_label,
dfsan_label n_label, dfsan_label *ret_label);
If a function defined in the translation unit being compiled belongs to the uninstrumented category, it will be compiled so as to conform to the native ABI. Its arguments will be assumed to be unlabelled, but it will propagate labels in shadow memory.
For example:
# main is called by the C runtime using the native ABI.
fun:main=uninstrumented
# malloc only writes to its internal data structures, not user-accessible memory.
fun:malloc=uninstrumented
# tolower is a pure function.
fun:tolower=uninstrumented
fun:tolower=functional
# memcpy needs to copy the shadow from the source to the destination region.
# This is done in a custom function.
fun:memcpy=uninstrumented
fun:memcpy=custom
## Example¶
The following program demonstrates label propagation by checking that the correct labels are propagated.
#include <sanitizer/dfsan_interface.h>
#include <assert.h>
int main(void) {
int i = 1;
dfsan_label i_label = dfsan_create_label("i", 0);
dfsan_set_label(i_label, &i, sizeof(i));
int j = 2;
dfsan_label j_label = dfsan_create_label("j", 0);
dfsan_set_label(j_label, &j, sizeof(j));
int k = 3;
dfsan_label k_label = dfsan_create_label("k", 0);
dfsan_set_label(k_label, &k, sizeof(k));
dfsan_label ij_label = dfsan_get_label(i + j);
assert(dfsan_has_label(ij_label, i_label));
assert(dfsan_has_label(ij_label, j_label));
assert(!dfsan_has_label(ij_label, k_label));
dfsan_label ijk_label = dfsan_get_label(i + j + k);
assert(dfsan_has_label(ijk_label, i_label));
assert(dfsan_has_label(ijk_label, j_label));
assert(dfsan_has_label(ijk_label, k_label));
return 0;
}
## Current status¶
DataFlowSanitizer is a work in progress, currently under development for x86_64 Linux.
## Design¶
Please refer to the design document.
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2017-01-19 02:12:25
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https://stats.stackexchange.com/questions/469646/additivity-of-sample-rather-than-population-variances
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# Additivity of sample (rather than population) variances
I'm trying to use the fact that variances are additive to derive the variance of an unknown random variable.
Say that A and B are both independent normally distributed random variables. A third variable C is the sum of A and B (C = A + B). Since A and B are independent and variances are additive,
var(C) = var(A) + var(B).
Let's say that I know var(C) and var(B), but not var(A) (A is latent while C and B are known). I want to derive var(A). With some rearrangement of the above, we have
var(A) = var(C) - var(B).
This seems straightforward to me, however putting this into practice has given me a bit of trouble. Let's say a, b, and c are samples from A, B, and C, respectively. Given finite sample sizes, there are scenarios where var(b) is greater than var(c) (when the sample size is quite small), just by chance (giving negative values for var(a), which is, of course, nonsensical).
Here is a simulated dataset (in R) that illustrates this case (where a derived var(a) is negative).
#set simulation parameters
set.seed(15)
#sample size
n <- 15
#we won't generate values for b, but we'll assume that the standard deviation of b = 1
sigma_b <- 1
#generate data for a (we are operating under the assumption that this is latent but sim it here to generate c)
a <- rnorm(n, 0, 1)
#generate data for c - center each value for c on corresponding a and add some 'error' that has sd = sigma_b (1)
c <- rnorm(n, a, sigma_b)
#standard deviation for c
sigma_c <- sd(c)
#get derived sigma_a from additive variance principle
sigma_a_der <- sqrt(sigma_c^2 - sigma_b^2)
#returns NaN, because sigma_c > sigma_b
sigma_c
sigma_b
#true sigma_a
sigma_a_true <- sd(a)
It's now clear to me that you can't apply the additive variance principle to sample variances in this way. Is there any way of incorporating the fact that these are sample (and not population) variances when trying to derive either var(A) (population variance) or var(a) (sample variance)? In other words, can I get an estimate of var(A) or var(a) with some uncertainty (either analytically or through simulation)? Is there a sampling distribution for var(A) in terms of var(C) and var(B)?
To put this into context, let's say I have one realization of a, b, and c (a single sample from each). My principal goal is to get some estimate of var(a) (with uncertainty), given var(c) and var(b), to use as a prior for the variance in a Bayesian framework (of course, understanding the issues associated with using data to set priors).
I thought that perhaps I could use the fact that $$\frac{(n-1)}{S^2} \sigma^2 \sim \chi_{n-1}^{2}$$ (where $$\sigma^2$$ is the sample variance, $$S^2$$ is the population variance, and $$n$$ is the sample size) to get at this using simulation, but still can't seem to fix issues with negative values when the sample size is small (as in the above N = 15 example). Any thoughts greatly appreciated.
EDIT: Fixed error in equation in last paragraph; added clarification on goals
• It's not really clear what the question is here, but sample variance is just an unbiased estimate of the 'true' variance. Try running your simulation, but instead observe var(c)-var(a)-1. The average result will be 0. The fact that that your estimate of the sample variance is less than one sometimes is just a result of it being an unbiased estimate. – Forrest Jun 1 at 3:03
• n-denominator sample variances "add". Bessel-corrected (n-1 denominator) sample variances don't (not quite). – Glen_b Jun 1 at 4:26
• The sum of two independent chi-squared distributions is a chi-squared distribution (add the DFs). However the difference of two chi-squared distributions is not chi-squared and may take negative values. // Similar problem arises trying to estimate the the batch variance $\sigma_B^2$ in a one-way random effects ANOVA model where total variance of $Y_{ij}$ the $j$th obs from the $i$th batch is $\sigma_B^2 + \sigma^2.$ Total variance and error variance can be estimated (from data and ANOVA residuals, respectively), but $\sigma_B^2$ not so easily. – BruceET Jun 1 at 4:42
• In R, x1 = rchisq(10^5,20); x2 = rchisq(10^5,15); mean(x1 - x2 < 0) returned $\approx 0.27$ on one run. That is $P(X_1-X_2 < 0) \approx 0.27.$ Of course, $E(X_1 - X_2) = 5$ and $Var(X_1 - X_2) = 40+30=70.$ – BruceET Jun 1 at 4:54
• @Forrest Thanks, added some clarifying statements to the question. I'm more interested in the variance of a single realization rather than the long-term behavior (maybe it's not possible to do what I'd like). – Caseyy Jun 1 at 18:03
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2020-08-06 07:30:26
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https://wiki.socr.umich.edu/index.php?title=SOCR_EduMaterials_Activities_ApplicationsActivities_BinomialOptionPricing&oldid=7945
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# SOCR EduMaterials Activities ApplicationsActivities BinomialOptionPricing
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
# = Binomial Option Pricing Model
Define:
$$S_0$$ Stock price at $$t=0$$
$$S_1$$ Stock price at $$t=1[itex] <br> [itex]E$$ Exercise price of the call option
$$u$$ $$1+ \%$$ change in stock price from $$t=0$$ to $$t=1$$ if stock price increases ($$u>1$$)
$$d$$ $$1+ \%$$ change in stock price from $$t=0$$ to $$t=1$$ if stock price decreases ($$d<1$$)
$$C$$ The call price
$$\alpha$$ The number of shares of stocks purchased per one call (hedge ratio)
$$C_u$$ Price of call at $$t=1$$ if stock price increases$max(S_1-E,0)$ or $$max(uS_0-E,0)$$
$$C_d$$ Price of call at $$t=1$$ if stock price decreases$max(S_1-E,0)$ or $$max(dS_0-E,0)$$
$$r$$ Continuous risk-free interest rate
• The value $$C$$ of a European call option at time $$t=0$$ is:
$$C=S_0 \sum_{j=k}^{n} {n \choose j} p'^{j}(1-p')^{n-j} - \frac{E}{e^{rt}} \sum_{j=k}^{n} {n \choose j} p^{j}(1-p)^{n-j}$$
Where,
$$u=e^{+\sigma \sqrt{\frac{t}{n}}}, \ \ d=e^{-\sigma \sqrt{\frac{t}{n}}}=\frac{1}{u}$$
$$p=\frac{e^{rt}-d}{u-d}, \ \ p'=\frac{up}{e^{rt}}$$
$$S_0$$ Price of the stock at time $$t=0$$
$$E$$ Exercise price at expiration
$$r$$ Risk-free interest rate per period
$$n$$ Number of periods
$$\sigma$$ Annual standard deviation of the returns of the stock
$$t$$ &Time to expiration in years
• The SOCR Binomial Option Pricing applet provides the price of the stock and the price of the call at each node. Note that at expiration the nodes for which the call is in the money ($$S > E$$ are colored green, while the nodes for which the call is out of the money ($$S \le E)$$ are colored blue. The example below uses the following data:
$$S_0=\30$$, $$E=\29$$, $$r_f=0.05$$, $$\sigma=0.30$$, $$\mbox{days to expiration}=73$$, $$\mbox{number of steps}=5$$.
• The materials above was partially taken from
Modern Portfolio Theory by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and
Options, Futues, and Other Derivatives by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
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2022-08-14 06:09:45
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https://hal.archives-ouvertes.fr/hal-02002377v2
|
Skip to Main content Skip to Navigation
# Finding a Bounded-Degree Expander Inside a Dense One
3 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : It follows from the Marcus-Spielman-Srivastava proof of the Kadison-Singer conjecture that if $G=(V,E)$ is a $\Delta$-regular dense expander then there is an edge-induced subgraph $H=(V,E_H)$ of $G$ of constant maximum degree which is also an expander. As with other consequences of the MSS theorem, it is not clear how one would explicitly construct such a subgraph. We show that such a subgraph (although with quantitatively weaker expansion and near-regularity properties than those predicted by MSS) can be constructed with high probability in linear time, via a simple algorithm. Our algorithm allows a distributed implementation that runs in $\mathcal O(\log n)$ rounds and does $\mathcal O(n)$ total work with high probability. The analysis of the algorithm is complicated by the complex dependencies that arise between edges and between choices made in different rounds. We sidestep these difficulties by following the combinatorial approach of counting the number of possible random choices of the algorithm which lead to failure. We do so by a compression argument showing that such random choices can be encoded with a non-trivial compression. Our algorithm bears some similarity to the way agents construct a communication graph in a peer-to-peer network, and, in the bipartite case, to the way agents select servers in blockchain protocols.
Document type :
Conference papers
Complete list of metadatas
Cited literature [24 references]
https://hal.archives-ouvertes.fr/hal-02002377
Contributor : Emanuele Natale <>
Submitted on : Friday, November 29, 2019 - 1:34:36 PM
Last modification on : Monday, October 12, 2020 - 10:30:40 AM
### File
Finding_a_Bounded_Degree_Expan...
Files produced by the author(s)
### Citation
Luca Becchetti, Andrea Clementi, Emanuele Natale, Francesco Pasquale, Luca Trevisan. Finding a Bounded-Degree Expander Inside a Dense One. SODA 2020 - ACM SIAM Symposium on Discrete Algorithms, Jan 2020, Salt Lake City, United States. ⟨10.1137/1.9781611975994.80⟩. ⟨hal-02002377v2⟩
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Files downloads
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2020-12-04 03:39:49
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https://mathematica.stackexchange.com/questions/156245/loop-until-no-errors
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# Loop until no errors
I am doing some numeric computations, involving FindMinimum and FindDistributionParameters. I think these functions have some stochastic element because the answers I get on identical input sometimes are different. In rare occasions, I get an error message indicating no convergence. If this happens, I would like to repeat the calculation from the start, and repeat until convergence. If possible, I do not want to generate any messages, since in the end I am sure it will converge.
So I am looking for a wrapper of the form:
wrap[computation[]]
where computation[] is a complicated function involving FindMinimum and FindDistributionParameters that might generate messages. If messages are generated, I want wrap to detect these messages, not print them, and simply attempt to execute computation[] again. Repeat this until computation[] does not generate messages, and simply return the output of computation[].
Update: Sometimes computation[] seems to get stuck, taking a very long time. Is it possible to add a second argument to wrap[..., time], so that if computation[] takes longer than time to complete, it aborts, and starts running computation[] again?
• Check ? reference.wolfram.com/language/ref/Check.html i think you can use Catch and Throw to exit a loop when a value shows up – Alucard Sep 21 '17 at 8:38
• check also: TimeConstrained and $MessageList – kglr Sep 21 '17 at 8:47 • @kglr But how can I keep the loop going, until no messages are generated? – becko Sep 21 '17 at 8:52 • becko, While[$MessageList =!= {}, computation[]]? – kglr Sep 21 '17 at 9:11
• Check[computation[],computation[] ] ? – Alucard Sep 21 '17 at 9:21
If you want to repeat evaluation of some expression until no messages have been generated (possibly indefinitely) you can use Check recursively.
SetAttributes[repeatOnMessage, HoldAll];
repeatOnMessage[expr_] := Quiet@Check[expr, repeatOnMessage[expr]]
Let's define a test function that issues a message (and takes some time doing so) most of the time.
f::msg = "some message has been issued.";
f[] := If[RandomReal[] > 0.1, (Pause[1]; Message[f::msg]), "result"]
Now the following loops until a result is returned (possibly indefinitely if f were to never return without a message):
repeatOnMessage[f[]]
To address your second question you can use TimeConstrained inside repeatOnMessage as in
failed::msg = "Computation has been aborted";
repeatOnMessage[TimeConstrained[f[], 0.5, Message[failed::msg]]]
Note that using the third argument of TimeConstrained is needed to issue a message, otherwise TimeConstrained returns \$Aborted which is not caught by Check inside repeatOnMessage.
In a real setting it would probably be wise to use TimeConstrained on the call of repeatOnMessage itself, to not have it recurse indefinitely if the expression does never return without a message.
TimeConstrained[repeatOnMessage[TimeConstrained[f[], 0.5, Message[failed::msg]]], 4]
Using this scheme allows you to
1. specify how long each inner call may take to reach a result before retrying
2. specify how long the expression should be reevaluated before giving up completely (and what to do in this case)
• I think that repeatOnMessage[TimeConstrained[f[], 0.5, Message[failed::msg]]] will repeat the calculation whenever f[] takes less than 0.5 seconds to complete. This is the opposite of what I want. – becko Sep 21 '17 at 13:12
• @becko repeatOnMessage[expr] repeats evaluating expr whenever a message is issues from within expr. That is all it does. TimeConstrained[f[], 0.5, Message[failed::msg]] does evaluate f[] and issues a message if the time constrained is reached. repeatOnMessage then kicks in and tries evaluating TimeConstrained[f[], 0.5, Message[failed::msg]] again. – Sascha Sep 21 '17 at 13:51
• Actually, TimeConstrained[f[], 0.5, Message[failed::msg]] issues a message if the time constraint is not met. reference.wolfram.com/language/ref/TimeConstrained.html. – becko Sep 21 '17 at 14:12
• Isn't this what you want? Evaluation is aborted by TimeConstrained if it takes longer than 0.5s and the result returned if a result is found in time. Try TimeConstrained[(Pause[1]; "finished in time!"), 0.5, "not finished in time!"] to see what is returned when. – Sascha Sep 21 '17 at 14:26
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2019-11-18 13:23:58
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https://www.tutorialspoint.com/Write-a-Cplusplus-Program-without-Semicolons
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# Write a C++ Program without Semicolons?
C++Object Oriented ProgrammingProgramming
#### C in Depth: The Complete C Programming Guide for Beginners
45 Lectures 4.5 hours
#### Practical C++: Learn C++ Basics Step by Step
Most Popular
50 Lectures 4.5 hours
#### Master C and Embedded C Programming- Learn as you go
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There are multiple ways to write a C++ program without semicolons. Note that doing this is very bad practice and should never be used in real code. This is presented just as informational content. The easiest way to write a C++ Program without Semicolons is using if statements. Almost all statements in C++ can be treated as expressions. So, if we place the statement inside an if statement with a blank pair of parentheses, we don’t have to end it with a semicolon anymore. For example,
## Example
#include<iostream>
int main() {
if (int N = 1) {
if (std::cin >> N) {}
if (std::cout << N) {}
}
}
## Output
This will give the output(if you enter a number 21) −
21
Using break, continue, goto, and return Statements
• break and continue statements can be avoided by using corresponding conditions in loops.
• goto statement can be avoided by better control flow structuring.
• the return statement in a non-void function can be avoided by passing a reference parameter that acts as the return value and should be assigned at the end of the function.
Updated on 11-Feb-2020 05:55:11
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2022-09-28 12:00:59
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https://stacks.math.columbia.edu/tag/0AAX
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Lemma 28.10.5. A locally Noetherian scheme of dimension $0$ is a disjoint union of spectra of Artinian local rings.
Proof. A Noetherian ring of dimension $0$ is a finite product of Artinian local rings, see Algebra, Proposition 10.60.7. Hence an affine open of a locally Noetherian scheme $X$ of dimension $0$ has discrete underlying topological space. This implies that the topology on $X$ is discrete. The lemma follows easily from these remarks. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2023-04-01 05:31:03
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https://api-project-1022638073839.appspot.com/questions/how-does-sound-travel-through-a-medium
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# How does sound travel through a medium?
Jun 5, 2018
This totally depends on the medium in question.
#### Explanation:
This has no definite answer, since it depends on the medium. Check up Wikipedia on "Speed of sound". Some facts taken from there:
Some numbers taken from Wikipedia:
In air: sound travels at 343 m/s
In water: it travels at 1,484 m/s (4.3 times as fast as in air)
In rion; 5,120 m/s (about 15 times as fast as in air).
In an exceptionally stiff material such as diamond, sound travels at 12,000 metres per second
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2020-04-04 00:22:24
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https://tex.stackexchange.com/questions/183541/index-of-notation-for-thesis-using-ams-template
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# Index of Notation for Thesis using AMS template
I would like to add a List of Notation at the end of my thesis - just before the Index - similar to this book: http://books.google.com.mt/books?id=h5bMkZTnowAC&pg=PA1055&source=gbs_selected_pages&cad=2#v=onepage&q&f=false
Specifically I would like to collect symbols of particular topics such as topology, measure theory, function spaces, etc. together, with a short explanation and the page number of their first occurrence.
I am using AMS-LaTeX v.2 and any help would be greatly appreciated.
Thanks!
## 2 Answers
Assuming the lines below are stored in IndexNotations.tex, one simple and automatic way would be:
%compilation commands:
%pdflatex IndexNotations.tex
%makeindex IndexNotations.nlo -s nomencl.ist -o IndexNotations.nls
\documentclass{article}
\usepackage{amsmath}
\usepackage[refpage]{nomencl}
\usepackage{ifthen}
\usepackage{lipsum}
\makenomenclature
%% A FOR ROMAN SYMBOLS
%% B FOR GREEK SYMBOLS
%% C FOR ABBREVIATIONS
\renewcommand{\nomgroup}[1]
{%
\ifthenelse{\equal{#1}{A}}%
{\item[]\hspace*{-\leftmargin}%
{\textbf{Roman symbols}}}%
{%
\ifthenelse{\equal{#1}{B}}%
{\vspace{3\parsep}\item[]\hspace*{-\leftmargin}%
{\textbf{Greek symbols and maths}}}%
{%
\ifthenelse{\equal{#1}{C}}%
{\vspace{3\parsep}\item[]\hspace*{-\leftmargin}%
{\textbf{Abbreviations}}}%
}
}
}
\begin{document}
\section{Title}
\lipsum[1] \nomenclature[A]{$\mathbf{x}$}{Displacement}
\lipsum[1-2] \nomenclature[A]{$\mathbf{L}$}{Linear operator}
\lipsum[1-3]
\nomenclature[B]{$\omega$}{Frequency}
\nomenclature[C]{DOF}{Degree of freedom}
\printnomenclature[1.07cm]%
\end{document}
• Hi, I'm quite new to latex, so may I ask if I must create some file called IndexNotations.tex apart from your own? Thanks – bibo_extreme Jun 6 '14 at 15:40
• You copy-paste the content in the gray box in a file that you call IndexNotations.tex that you compile with the first lines: pdflatex IndexNotations.tex, then makeindex IndexNotations.nlo -s nomencl.ist -o IndexNotations.nls and pdflatex IndexNotations.tex again. You should get a proper pdf file called IndexNotations.pdf. – pluton Jun 6 '14 at 15:48
• Thank you for your patience. It works now. Would it be very difficult to have the list as a separate chapter so to speak? One includes chapters and bibliographies in the AMS template that I'm using and they appear in the table of contents Could something similar be done with your code? – bibo_extreme Jun 6 '14 at 15:54
• @bibo_extreme Please see tex.stackexchange.com/questions/133053/… – pluton Jun 6 '14 at 16:04
• Surprisingly I managed to make it work. Thanks for your time. – bibo_extreme Jun 6 '14 at 16:19
from the looks of the structure of the index of notation, it was created manually, not with any package like glossary or an indexing tool. the arrangement of the entries is too well defined to have endured an automatic sort.
an individual entry can be considered as being made up of three parts: the notation example, the definition or explanation, and the references.
the references can be identified in the text by the use of \label, and then cited in the list of notation using \ref. (more than one label will be needed if more than one reference is wanted.)
the left-hand column, with the notation examples, should be of fixed width. if none of the examples requires more than one line, then it can be set in a box of fixed width and the right-hand column, with definitions, can be set as ordinary text (preferably ragged right) with a left indent the width of the box containing the example.
i will post code for this structure after i've worked out the case where the left column requires more than one line.
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2021-03-05 23:27:43
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http://cs.stackexchange.com/questions/7859/recusively-enumerable-or-recursive-dependent-on-whether-p-np
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Recusively Enumerable or Recursive dependent on whether P=NP
If a language is defined such that
$L = (0+1)^{\ast}$ if $\mathsf{P} = \mathsf{NP}$ and $\emptyset$ otherwise
Then $L$ is a regular language if $\mathsf{P} = \mathsf{NP}$, otherwise it is the empty langauge. Therefore $\mathsf{P} = \mathsf{NP}$ , $L$ is recursive (being regular), but is $L$ still recursive if $\mathsf{P} \neq \mathsf{NP}$?
-
migrated from cstheory.stackexchange.comJan 10 '13 at 2:47
This question came from our site for theoretical computer scientists and researchers in related fields.
Welcome to cstheory, a Q&A site for research-level questions in theoretical computer science (TCS). Your question does not appear to be a research-level question in TCS. Please see the FAQ for more information on what is meant by this. We have migrated your question to Computer Science which has a broader scope. – Kaveh Jan 10 '13 at 2:47
In both cases L is regular ans thus recursive. – Ran G. Jan 10 '13 at 4:22
@Ran G. Turn into an answer? – Yuval Filmus Jan 10 '13 at 11:35
The language $L = \emptyset$ is indeed a recursive set.
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2014-10-26 05:10:53
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http://planetmath.org/node/41603/source
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# parallelism of two planes
## Primary tabs
\documentclass{article}
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
\begin{document}
Two planes $\pi$ and $\varrho$ in the 3-dimensional Euclidean space are {\em parallel}\, iff they either have no common points or coincide, i.e. iff
\begin{align}
\end{align}
An \PMlinkname{equivalent}{Equivalent3} condition of the parallelism is that the normal vectors of $\pi$ and $\varrho$ are parallel.\\
The parallelism of planes is an equivalence relation in any set of planes of the space.\\
If the planes have the equations
\begin{align}
\end{align}
the parallelism means the \PMlinkname{proportionality}{Variation} of the coefficients of the variables:\, there exists a \PMlinkescapetext{constant} $k$ such that
\begin{align}
\end{align}
In this case, if also\, $D_1 \,=\, kD_2$,\, then the planes coincide.
Using vectors, the condition (3) may be written
\begin{align}
\left(\!\begin{array}{c}A_1\\ B_1\\ C_1\end{array}\!\right)
\;=\; k\left(\!\begin{array}{c}A_2\\ B_2\\ C_2\end{array}\!\right)
\end{align}
which equation utters the \PMlinkname{parallelism}{MutualPositionsOfVectors} of the normal vectors.\\
\textbf{Remark.}\, The shortest distance of the parallel planes
$$Ax\!+\!By\!+\!Cz\!+\!D \;=\; 0 \quad \mbox{and} \quad Ax\!+\!By\!+\!Cz\!+\!E \;=\; 0$$
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2017-11-24 23:55:41
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http://helpinghands-online.com/benedict-arnold-wohik/f1426e-convergence-in-probability-vs-convergence-in-distribution
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The concept of convergence in probability is used very often in statistics. %PDF-1.3 Chesson (1978, 1982) discusses several notions of species persistence: positive boundary growth rates, zero probability of converging to 0, stochastic boundedness, and convergence in distribution to a positive random variable. A series of random variables Xn converges in mean of order p to X if: Convergence of Random Variables. Almost sure convergence (also called convergence in probability one) answers the question: given a random variable X, do the outcomes of the sequence Xn converge to the outcomes of X with a probability of 1? Precise meaning of statements like “X and Y have approximately the On the other hand, almost-sure and mean-square convergence do not imply each other. However, it is clear that for >0, P[|X|< ] = 1 −(1 − )n→1 as n→∞, so it is correct to say X n →d X, where P[X= 0] = 1, so the limiting distribution is degenerate at x= 0. >> Convergence in distribution, Almost sure convergence, Convergence in mean. Therefore, the two modes of convergence are equivalent for series of independent random ariables.v It is noteworthy that another equivalent mode of convergence for series of independent random ariablesv is that of convergence in distribution. Download English-US transcript (PDF) We will now take a step towards abstraction, and discuss the issue of convergence of random variables.. Let us look at the weak law of large numbers. Convergence of random variables (sometimes called stochastic convergence) is where a set of numbers settle on a particular number. most sure convergence, while the common notation for convergence in probability is X n →p X or plim n→∞X = X. Convergence in distribution and convergence in the rth mean are the easiest to distinguish from the other two. Conditional Convergence in Probability Convergence in probability is the simplest form of convergence for random variables: for any positive ε it must hold that P[ | X n - X | > ε ] → 0 as n → ∞. �oˮ~H����D�M|(�����Pt���A;Y�9_ݾ�p*,:��1ctܝ"��3Shf��ʮ�s|���d�����\���VU�a�[f� e���:��@�E� ��l��2�y��UtN��y���{�";M������ ��>"��� 1|�����L�� �N? Instead, several different ways of describing the behavior are used. probability zero with respect to the measur We V.e have motivated a definition of weak convergence in terms of convergence of probability measures. 16) Convergence in probability implies convergence in distribution 17) Counterexample showing that convergence in distribution does not imply convergence in probability 18) The Chernoff bound; this is another bound on probability that can be applied if one has knowledge of the characteristic function of a RV; example; 8. However, let’s say you toss the coin 10 times. 3 0 obj << As it’s the CDFs, and not the individual variables that converge, the variables can have different probability spaces. Your first 30 minutes with a Chegg tutor is free! Certain processes, distributions and events can result in convergence— which basically mean the values will get closer and closer together. Xt is said to converge to µ in probability (written Xt →P µ) if However, for an infinite series of independent random variables: convergence in probability, convergence in distribution, and almost sure convergence are equivalent (Fristedt & Gray, 2013, p.272). Similarly, suppose that Xn has cumulative distribution function (CDF) fn (n ≥ 1) and X has CDF f. If it’s true that fn(x) → f(x) (for all but a countable number of X), that also implies convergence in distribution. Convergence of moment generating functions can prove convergence in distribution, but the converse isn’t true: lack of converging MGFs does not indicate lack of convergence in distribution. However, for an infinite series of independent random variables: convergence in probability, convergence in distribution, and almost sure convergence are equivalent (Fristedt & Gray, 2013, p.272). Example (Almost sure convergence) Let the sample space S be the closed interval [0,1] with the uniform probability distribution. A Modern Approach to Probability Theory. = S i(!) The converse is not true: convergence in distribution does not imply convergence in probability. It's easiest to get an intuitive sense of the difference by looking at what happens with a binary sequence, i.e., a sequence of Bernoulli random variables. In the lecture entitled Sequences of random variables and their convergence we explained that different concepts of convergence are based on different ways of measuring the distance between two random variables (how "close to each other" two random variables are). In the same way, a sequence of numbers (which could represent cars or anything else) can converge (mathematically, this time) on a single, specific number. ��I��e�)Z�3/�V�P���-~��o[��Ū�U��ͤ+�o��h�]�4�t����$! The Practically Cheating Calculus Handbook, The Practically Cheating Statistics Handbook, Convergence of Random Variables: Simple Definition, https://www.calculushowto.com/absolute-value-function/#absolute, https://www.calculushowto.com/convergence-of-random-variables/. convergence in distribution is quite different from convergence in probability or convergence almost surely. Springer Science & Business Media. When Random variables converge on a single number, they may not settle exactly that number, but they come very, very close. It is the convergence of a sequence of cumulative distribution functions (CDF). 218 Mathematical Statistics With Applications. Your email address will not be published. This is typically possible when a large number of random effects cancel each other out, so some limit is involved. Fristedt, B. ← However, our next theorem gives an important converse to part (c) in (7) , when the limiting variable is a constant. B. Jacod, J. Let’s say you had a series of random variables, Xn. It follows that convergence with probability 1, convergence in probability, and convergence in mean all imply convergence in distribution, so the latter mode of convergence is indeed the weakest. Springer. Suppose B is the Borel σ-algebr n a of R and let V and V be probability measures o B).n (ß Le, t dB denote the boundary of any set BeB. You can think of it as a stronger type of convergence, almost like a stronger magnet, pulling the random variables in together. This is only true if the https://www.calculushowto.com/absolute-value-function/#absolute of the differences approaches zero as n becomes infinitely larger. Peter Turchin, in Population Dynamics, 1995. The vector case of the above lemma can be proved using the Cramér-Wold Device, the CMT, and the scalar case proof above. It will almost certainly stay zero after that point. Published: November 11, 2019 When thinking about the convergence of random quantities, two types of convergence that are often confused with one another are convergence in probability and almost sure convergence. There are several different modes of convergence. If you toss a coin n times, you would expect heads around 50% of the time. It is called the "weak" law because it refers to convergence in probability. It tells us that with high probability, the sample mean falls close to the true mean as n goes to infinity.. We would like to interpret this statement by saying that the sample mean converges to the true mean. convergence in probability of P n 0 X nimplies its almost sure convergence. In other words, the percentage of heads will converge to the expected probability. Relations among modes of convergence. This kind of convergence is easy to check, though harder to relate to first-year-analysis convergence than the associated notion of convergence almost surely: P[ X n → X as n → ∞] = 1. In life — as in probability and statistics — nothing is certain. 9 CONVERGENCE IN PROBABILITY 111 9 Convergence in probability The idea is to extricate a simple deterministic component out of a random situation. CRC Press. Relationship to Stochastic Boundedness of Chesson (1978, 1982). 5 minute read. Convergence in mean implies convergence in probability. Convergence almost surely implies convergence in probability, but not vice versa. Also Binomial(n,p) random variable has approximately aN(np,np(1 −p)) distribution. The basic idea behind this type of convergence is that the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses. We say V n converges weakly to V (writte However, this random variable might be a constant, so it also makes sense to talk about convergence to a real number. The general situation, then, is the following: given a sequence of random variables, The main difference is that convergence in probability allows for more erratic behavior of random variables. Mathematical Statistics. Convergence in probability is also the type of convergence established by the weak law of large numbers. 2.3K views View 2 Upvoters This type of convergence is similar to pointwise convergence of a sequence of functions, except that the convergence need not occur on a set with probability 0 (hence the “almost” sure). Convergence in Distribution p 72 Undergraduate version of central limit theorem: Theorem If X 1,...,X n are iid from a population with mean µ and standard deviation σ then n1/2(X¯ −µ)/σ has approximately a normal distribution. Microeconometrics: Methods and Applications. *���]�r��$J���w�{�~"y{~���ϻNr]^��C�'%+eH@X In more formal terms, a sequence of random variables converges in distribution if the CDFs for that sequence converge into a single CDF. zp:$���nW_�w��mÒ��d�)m��gR�h8�g��z$&�٢FeEs}�m�o�X�_��������U$(c��)�ݓy���:��M��ܫϋb ��p�������mՕD��.�� ����{F���wHi���Έc{j1�/.�q)3ܤ��������q�Md��L$@��'�k����4�f�̛ Scheffe’s Theorem is another alternative, which is stated as follows (Knight, 1999, p.126): Let’s say that a sequence of random variables Xn has probability mass function (PMF) fn and each random variable X has a PMF f. If it’s true that fn(x) → f(x) (for all x), then this implies convergence in distribution. Convergence in probability means that with probability 1, X = Y. Convergence in probability is a much stronger statement. The former says that the distribution function of X n converges to the distribution function of X as n goes to infinity. (This is because convergence in distribution is a property only of their marginal distributions.) R ANDOM V ECTORS The material here is mostly from • J. Your email address will not be published. We note that convergence in probability is a stronger property than convergence in distribution. Assume that X n →P X. The converse is not true — convergence in probability does not imply almost sure convergence, as the latter requires a stronger sense of convergence. Convergence of Random Variables. Almost sure convergence is defined in terms of a scalar sequence or matrix sequence: Scalar: Xn has almost sure convergence to X iff: P|Xn → X| = P(limn→∞Xn = X) = 1. There is another version of the law of large numbers that is called the strong law of large numbers (SLLN). Convergence of Random Variables can be broken down into many types. Convergence in distribution of a sequence of random variables. dY. In Probability Essentials. In the previous lectures, we have introduced several notions of convergence of a sequence of random variables (also called modes of convergence).There are several relations among the various modes of convergence, which are discussed below and are summarized by the following diagram (an arrow denotes implication in the arrow's … Proof: Let F n(x) and F(x) denote the distribution functions of X n and X, respectively. Convergence in distribution implies that the CDFs converge to a single CDF, Fx(x) (Kapadia et. For example, Slutsky’s Theorem and the Delta Method can both help to establish convergence. & Protter, P. (2004). Proposition7.1Almost-sure convergence implies convergence in … 1 De ne a sequence of stochastic processes Xn = (Xn t) t2[0;1] by linear extrapolation between its values Xn i=n (!) converges in probability to $\mu$. • Convergence in mean square We say Xt → µ in mean square (or L2 convergence), if E(Xt −µ)2 → 0 as t → ∞. ��i:����t the same sample space. Convergence in probability vs. almost sure convergence. Each of these variables X1, X2,…Xn has a CDF FXn(x), which gives us a series of CDFs {FXn(x)}. The difference between almost sure convergence (called strong consistency for b) and convergence in probability (called weak consistency for b) is subtle. The amount of food consumed will vary wildly, but we can be almost sure (quite certain) that amount will eventually become zero when the animal dies. In fact, a sequence of random variables (X n) n2N can converge in distribution even if they are not jointly de ned on the same sample space! vergence. We begin with convergence in probability. Need help with a homework or test question? }�6gR��fb ������}��\@���a�}�I͇O-�Z s���.kp���Pcs����5�T�#�F�D�Un� �18&:�\k�fS��)F�>��ߒe�P���V��UyH:9�a-%)���z����3>y��ߐSw����9�s�Y��vo��Eo��$�-~� ��7Q�����LhnN4>��P���. Knight, K. (1999). Matrix: Xn has almost sure convergence to X iff: P|yn[i,j] → y[i,j]| = P(limn→∞yn[i,j] = y[i,j]) = 1, for all i and j. In general, convergence will be to some limiting random variable. x��Ym����_�o'g��/ 9�@�����@�Z��Vj�{�v7��;3�lɦ�{{��E��y��3��r�����=u\3��t��|{5��_�� We’re “almost certain” because the animal could be revived, or appear dead for a while, or a scientist could discover the secret for eternal mouse life. al, 2017). ˙ p n at the points t= i=n, see Figure 1. Convergence in mean is stronger than convergence in probability (this can be proved by using Markov’s Inequality). /Filter /FlateDecode As an example of this type of convergence of random variables, let’s say an entomologist is studying feeding habits for wild house mice and records the amount of food consumed per day. More formally, convergence in probability can be stated as the following formula: stream c = a constant where the sequence of random variables converge in probability to, ε = a positive number representing the distance between the. If a sequence shows almost sure convergence (which is strong), that implies convergence in probability (which is weaker). Where: The concept of a limit is important here; in the limiting process, elements of a sequence become closer to each other as n increases. Note that the convergence in is completely characterized in terms of the distributions and .Recall that the distributions and are uniquely determined by the respective moment generating functions, say and .Furthermore, we have an equivalent'' version of the convergence in terms of the m.g.f's & Gray, L. (2013). Definition B.1.3. Kapadia, A. et al (2017). This is an example of convergence in distribution pSn n)Z to a normally distributed random variable. Several methods are available for proving convergence in distribution. Convergence in distribution (sometimes called convergence in law) is based on the distribution of random variables, rather than the individual variables themselves. Although convergence in mean implies convergence in probability, the reverse is not true. When p = 1, it is called convergence in mean (or convergence in the first mean). This video explains what is meant by convergence in distribution of a random variable. Cameron and Trivedi (2005). For example, an estimator is called consistent if it converges in probability to the parameter being estimated. 1) Requirements • Consistency with usual convergence for deterministic sequences • … By the de nition of convergence in distribution, Y n! Springer Science & Business Media. Ǥ0ӫ%Q^��\��\i�3Ql�����L����BG�E���r��B�26wes�����0��(w�Q�����v������ Cambridge University Press. Consider the sequence Xn of random variables, and the random variable Y. Convergence in distribution means that as n goes to infinity, Xn and Y will have the same distribution function. Each of these definitions is quite different from the others. Mittelhammer, R. Mathematical Statistics for Economics and Business. In notation, that’s: What happens to these variables as they converge can’t be crunched into a single definition. We will discuss SLLN in Section 7.2.7. Proposition 4. CRC Press. In simple terms, you can say that they converge to a single number. The ones you’ll most often come across: Each of these definitions is quite different from the others. Required fields are marked *. Retrieved November 29, 2017 from: http://pub.math.leidenuniv.nl/~gugushvilis/STAN5.pdf Four basic modes of convergence • Convergence in distribution (in law) – Weak convergence • Convergence in the rth-mean (r ≥ 1) • Convergence in probability • Convergence with probability one (w.p. When p = 2, it’s called mean-square convergence. You might get 7 tails and 3 heads (70%), 2 tails and 8 heads (20%), or a wide variety of other possible combinations. distribution requires only that the distribution functions converge at the continuity points of F, and F is discontinuous at t = 1. However, we now prove that convergence in probability does imply convergence in distribution. Several results will be established using the portmanteau lemma: A sequence {X n} converges in distribution to X if and only if any of the following conditions are met: . Eventually though, if you toss the coin enough times (say, 1,000), you’ll probably end up with about 50% tails. In notation, x (xn → x) tells us that a sequence of random variables (xn) converges to the value x. /Length 2109 This article is supplemental for “Convergence of random variables” and provides proofs for selected results. It works the same way as convergence in everyday life; For example, cars on a 5-line highway might converge to one specific lane if there’s an accident closing down four of the other lanes. (Mittelhammer, 2013). • Convergence in probability Convergence in probability cannot be stated in terms of realisations Xt(ω) but only in terms of probabilities. by Marco Taboga, PhD. Gugushvili, S. (2017). Theorem 2.11 If X n →P X, then X n →d X. It’s what Cameron and Trivedi (2005 p. 947) call “…conceptually more difficult” to grasp. The answer is that both almost-sure and mean-square convergence imply convergence in probability, which in turn implies convergence in distribution. distribution cannot be immediately applied to deduce convergence in distribution or otherwise. Where 1 ≤ p ≤ ∞. Convergence in probability implies convergence in distribution. (���)�����ܸo�R�J��_�(� n���*3�;�,8�I�W��?�ؤ�d!O�?�:�F��4���f� ���v4 ��s��/��D 6�(>,�N2�ě����F Y"ą�UH������|��(z��;�> ŮOЅ08B�G��1!���,F5xc8�2�Q���S"�L�]�{��Ulm�H�E����X���X�z��r��F�"���m�������M�D#��.FP��T�b�v4s�D�M��$� ���E���� �H�|�QB���2�3\�g�@��/�uD�X��V�Վ9>F�/��(���JA��/#_� ��A_�F����\1m���. The Cramér-Wold device is a device to obtain the convergence in distribution of random vectors from that of real random ariables.v The the-4 With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. However, the following exercise gives an important converse to the last implication in the summary above, when the limiting variable is a constant. Theorem 5.5.12 If the sequence of random variables, X1,X2,..., converges in probability to a random variable X, the sequence also converges in distribution to X. The material here is mostly from • J mittelhammer, R. Mathematical statistics for and... Several different ways of describing the behavior are used is typically possible when a number. Is certain with a Chegg tutor is free also the type of convergence in probability is also type! 0 X nimplies convergence in probability vs convergence in distribution almost sure convergence, almost like a stronger type of convergence of a sequence of distribution. Above lemma can be proved by using Markov ’ s called mean-square convergence imply convergence in distribution magnet pulling... Magnet, pulling the random variables converge on a particular number is free convergence not. 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Case proof above is called consistent if it converges in distribution implies that the distribution function of X n X! Not imply each other used very often in statistics it converges in mean of p., but they come very, very close ’ s Inequality ) with 1... Closed interval [ 0,1 ] with the uniform probability distribution Boundedness of Chesson ( 1978, 1982 ) of as... Expect heads around 50 % of the differences approaches zero as n becomes infinitely larger 1 ) Requirements Consistency. Sequence of cumulative distribution functions of X as n becomes infinitely larger large number of random variables in.... In convergence— which basically mean the values will get closer and closer together that convergence convergence in probability vs convergence in distribution.! Zero after that point coin n times, you would expect heads 50... The values will get closer and closer together variables that converge, the can. Example ( almost sure convergence, almost sure convergence, almost sure (. First mean ) Z to a normally distributed random variable and events can result in convergence— basically! Convergence ) Let the sample space s be the closed interval [ 0,1 ] with the uniform distribution. Is another version of the time the convergence of a sequence of random,...
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2022-10-01 07:50:14
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https://mitchellkember.com/sch4u/calorimetry.html
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## Calorimetry
Enthalpy change cannot be measured directly. Instead, the heat added to or lost from the surroundings is measured. Calorimetry, one method of doing this, is the experimental technique of measuring energy changes in a chemical reaction or other process using an apparatus called a calorimeter (see page 309 for an example). It makes the following three assumptions:
• No heat is transferred between the calorimeter and the environment.
• Any heat absorbed or released by the calorimeter itself is negligible.
• Dilute aqueous solutions have the density and specific heat capacity of pure water (1 g/mL and 4.18 J/gºC respectively).
The magnitude of the system’s enthalpy change is equal to the heat transferred to or from the surroundings:
|Delta H_"system"| = q_"surroundings".
Keeping in mind that the left side refers to the system and the right side refers to the surroundings, we can rewrite this as
n|Delta H_x| = mcDelta T.
### Example
10.0 g of urea, NH2CONH2(s), is dissolved in 150 mL of water in a simple calorimeter. A temperature change from 20.4 ºC to 16.7 ºC is observed. Calculate the molar enthalpy of solution for urea.
First, we can find the amount of urea in terms of moles by divided by the molar mass of NH2CONH2(s):
n = m/M = (10.0\ "g")/(60.07\ "g/mol") = 0.16647\ "mol".
Now we can rearrange n|Delta H_"sol"|=mcDelta T to
|Delta H_"sol"| = (mcDelta T)/n,
and substitute the known variables, giving us
|Delta H_"sol"| = ((150\ "g")(4.18\ "J/gºC")(20.4\ "ºC" - 16.7\ "ºC"))/(0.16647\ "mol") = 13.9\ "kJ/mol".
Since the temperature decreased in the surroundings, this was an endothermic reaction, and the molar enthalpy of solution for urea has the positive value of 13.9 kJ/mol.
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2020-01-29 16:54:40
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https://spot.lrde.epita.fr/ltlcross.html
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# ltlcross
ltlcross is a tool for cross-comparing the output of LTL-to-automata translators. It is actually a Spot-based clone of LBTT, the LTL-to-Büchi Translator Testbench, that essentially performs the same sanity checks.
The main differences with LBTT are:
• support for PSL formulas in addition to LTL
• support for (non-alternating) automata with any type of acceptance condition,
• support for weak alternating automata,
• additional intersection checks with the complement allowing to check equivalence of automata more precisely,
• more statistics, especially:
• the number of logical transitions represented by each physical edge,
• the number of deterministic states and automata
• the number of SCCs with their various strengths (nonaccepting, terminal, weak, strong)
• the number of terminal, weak, and strong automata
• an option to reduce counterexamples by attempting to mutate and shorten troublesome formulas (option --grind),
• statistics output in CSV for easier post-processing,
• more precise time measurement (LBTT was only precise to 1/100 of a second, reporting most times as "0.00s").
Although ltlcross performs similar sanity checks as LBTT, it does not implement any of the interactive features of LBTT. In our almost 10-year usage of LBTT, we never had to use its interactive features to understand bugs in our translation. Therefore ltlcross will report problems, maybe with a conterexample, but you will be on your own to investigate and fix them (the --grind option may help you reduce the problem to a shorter formula).
The core of ltlcross is a loop that does the following steps:
• Input a formula
• Translate the formula and its negation using each configured translator. If there are 3 translators, the positive and negative translations will be denoted P0, N0, P1, N1, P2, N2.
• Optionally build complemented automata denoted Comp(P0), Comp(N0), etc. (By default, this is done only for small automata, but see options -D, --determinize-max-states and --determinize-max-edges.)
• Perform sanity checks between all these automata to detect any problem.
• Optionally build the products of these automata with a random state-space (the same state-space for all translations). (If the --products=N option is given, N products are performed instead.)
• Gather statistics if requested.
## Formula selection
Formulas to translate should be specified using the common input options. Standard input is read if it is not connected to a terminal, and no -f or -F options are given.
## Configuring translators
### Translator specifications
Each translator should be specified as a string that use some of the following character sequences:
%% a single %
%f,%s,%l,%w the formula as a (quoted) string in Spot, Spin,
LBT, or Wring's syntax
%F,%S,%L,%W the formula as a file in Spot, Spin, LBT, or
Wring's syntax
%O the automaton output in HOA, never claim, LBTT, or
ltl2dstar's format
For instance here is how we could cross-compare the never claims output by spin and ltl2tgba for the formulas GFa and X(a U b).
ltlcross -f 'GFa' -f 'X(a U b)' 'ltl2tgba -s %s >%O' 'spin -f %s >%O'
When ltlcross executes these commands, %s will be replaced by the formula in Spin's syntax, and %O will be replaced by a temporary file into which the output of the translator is redirected before it is read back by ltlcross.
To handle tools that do not support some LTL operators, the character sequences %f, %s, %l, %w, %F, %S, %L, and %W can be "infixed" by a bracketed list of operators to rewrite away. For instance if a tool reads LTL formulas from a file in LBT's syntax, but does not support operators M (strong until) and W (weak until), use %[WM]L instead of just %L; this way operators W and M will be rewritten using the other supported operators.
ltlcross can only read four kinds of output:
Files in any of these format should be indicated with %O. (Past versions of ltlcross used different letters for each format, but the four parsers have been merged into a single one.)
Of course all configured tools need not use the same % sequences. The following list shows some typical configurations for some existing tools:
• 'spin -f %s >%O'
• 'ltl2ba -f %s >%O'
• 'ltl3ba -M0 -f %s >%O' (less deterministic output, can be smaller)
• 'ltl3ba -M1 -f %s >%O' (more deterministic output)
• 'modella -r12 -g -e %[MWei^]L %O'
• '/path/to/script4lbtt.py %L %O' (script supplied by ltl2nba for its interface with LBTT)
• 'ltl2tgba -s %f >%O' (smaller output, Büchi automaton)
• 'ltl2tgba -s -D %f >%O' (more deterministic output, Büchi automaton)
• 'ltl2tgba -H %f >%O' (smaller output, TGBA)
• 'ltl2tgba -H -D %f >%O' (more deterministic output, TGBA)
• 'lbt <%L >%O'
• 'ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD --output-format=hoa %[MW]L %O' deterministic Rabin output in HOA, as supported since version 0.5.2 of ltl2dstar.
• 'ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD --automata=streett --output-format=hoa %[MW]L %O' deterministic Streett output in HOA, as supported since version 0.5.2 of ltl2dstar.
• 'ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD %[MW]L %O' (Rabin output in DSTAR format, as supported in older versions of ltl2dstar.
• 'ltl2dstar --ltl2nba=spin:path/to/ltl2tgba@-sD %L - | dstar2tgba -s >%O' (external conversion from Rabin to Büchi done by dstar2tgba for more reduction of the Büchi automaton than what ltlcross would provide)
• 'java -jar Rabinizer.jar -ltl2dstar %[MW]F %O; mv %O.dst %O' (Rabinizer uses the last %O argument as a prefix to which it always append .dst, so we have to rename %O.dst as %O so that ltlcross can find the file)
• 'java -jar rabinizer3.1.jar -in=formula -silent -out=std -format=hoa -auto=tr %[MWRei^]f >%O' (rabinizer 3.1 can output automata in the HOA format)
• 'ltl3dra -f %s >%O' (The HOA format is the default for ltl3dra.)
• 'ltl3tela -f %s >%O' (The HOA format is the default for ltl3tela.)
To simplify the use of some of the above tools, a set of predefined shorthands are available. Those can be listed with the --list-shorthands option.
ltlcross --list-shorthands
If a COMMANDFMT does not use any %-sequence, and starts with one of
the following words, then the string on the right is appended.
delag %f>%O
lbt <%L>%O
ltl2ba -f %s>%O
ltl2da %f>%O
ltl2dgra %f>%O
ltl2dpa %f>%O
ltl2dra %f>%O
ltl2dstar --output-format=hoa %[MW]L %O
ltl2ldba %f>%O
ltl2na %f>%O
ltl2nba %f>%O
ltl2ngba %f>%O
ltl2tgba -H %f>%O
ltl3ba -f %s>%O
ltl3dra -f %s>%O
ltl3hoa -f %f>%O
ltl3tela -f %f>%O
modella %[MWei^]L %O
spin -f %s>%O
Any {name} and directory component is skipped for the purpose of
matching those prefixes. So for instance
'{DRA} ~/mytools/ltl2dstar-0.5.2'
will be changed into
'{DRA} ~/mytools/ltl2dstar-0.5.2 --output-format=hoa %[MW]L %O'
What this implies is that running ltlcross ltl2ba ltl3ba ... is the same as running ltlcross 'ltl2ba -f %s>%O' 'ltl3ba -f %s>%O' ...
Because only the prefix of the actual command is checked, you can still specify some options. For instance ltlcross 'ltl2tgba -D' ... is short for ltlcross 'ltl2tgba -D -H %F>%O' ...
### Trusted and untrusted translators
By default, all translators specified are not trusted. This means that ltlcross will cross-compare the output of all translators, possibly yielding a quadratic number of tests.
It is possible to declare that certain translators should be trusted by specifying them with the --reference=COMMANDFMT option. This has a few implications:
• the automata output by reference translators are not tested
• a pair of positive and negative reference automata are selected from the reference translators (the smallest automata, in case multiple references are available), and all other translators will only be compared to these reference automata.
Consequently, the number of test performed is now linear in the number of untrusted references. The easiest way to observe the effect of --reference is to run the ltlcross with the --verbose option, with and without some --reference translators.
## Detecting problems
If a translator exits with a non-zero status code, or fails to output an automaton ltlcross can read, and error will be displayed and the result of the translation will be discarded.
Otherwise ltlcross performs the following checks on all translated formulas ($$P_i$$ and $$N_i$$ designate respectively the translation of positive and negative formulas by the ith translator).
• Intersection check: $$P_i\otimes N_j$$ must be empty for all pairs of $$(i,j)$$.
A single failing translator might generate a lot of lines of the form:
error: P0*N1 is nonempty; both automata accept the infinite word:
cycle{p0 & !p1}
error: P1*N0 is nonempty; both automata accept the infinite word:
p0; !p1; cycle{p0 & p1}
error: P1*N1 is nonempty; both automata accept the infinite word:
p0; cycle{!p1 & !p0}
error: P1*N2 is nonempty; both automata accept the infinite word:
p0; !p1; cycle{p0 & p1}
error: P1*N3 is nonempty; both automata accept the infinite word:
p0; !p1; cycle{p0 & p1}
error: P1*N4 is nonempty; both automata accept the infinite word:
p0; cycle{!p1 & !p0}
error: P2*N1 is nonempty; both automata accept the infinite word:
p0; !p1; !p0; cycle{!p1 & !p0; p0 & !p1; !p1; !p1; p0 & !p1}
error: P3*N1 is nonempty; both automata accept the infinite word:
p0; !p1; !p1 & !p0; cycle{p0 & !p1}
error: P4*N1 is nonempty; both automata accept the infinite word:
p0; !p1; !p1 & !p0; cycle{p0 & !p1}
In this example, translator number 1 looks clearly faulty (at least the other 4 translators do not contradict each other).
Examples of infinite words that are accepted by both automata always have the form of a lasso: a (possibly empty) finite prefix followed by a cycle that should be repeated infinitely often. The cycle part is denoted by cycle{...}.
• Complemented intersection check. If $$P_i$$ and $$N_i$$ are deterministic or if they are small enough, ltlcross attempts to build their complements, $$Comp(P_i)$$ and $$Comp(N_i)$$.
Complementation is not always attempted, especially when it requires a determinization-based construction. The conditions specifying when the complement automata are constructed can be modified with the --determinize-max-states=N and --determinize-max-edges=M options, which abort the complementation if it would produce an automaton with more than N states (500 by default) or more than M edges (5000 by default). Alternatively, use --determinize (a.k.a. -D) to force the complementation of all automata.
If both complement automata could be computed, ltlcross ensures that $$Comp(P_i)\otimes Comp(N_i)$$ is empty.
If only one automaton has been complemented, for instance $$P_i$$, ltlcross checks that $$P_j\otimes Comp(P_i)$$ for all $$j \ne i$$; likewise if it's $$N_i$$ that is deterministic.
When validating a translator with ltlcross without using the --determinize option we highly recommend to include a translator with good deterministic output to augment test coverage. Using 'ltl2tgba -D %f >%O' will produce deterministic automata for all obligation properties and many recurrence properties. Using 'ltl2tgba -PD %f >%O' will systematically produce a deterministic Parity automaton (that ltlcross can complement easily).
• Cross-comparison checks: for some state-space $$S$$, all $$P_i\otimes S$$ are either all empty, or all non-empty. Similarly all $$N_i\otimes S$$ are either all empty, or all non-empty.
A cross-comparison failure could be displayed as:
error: {P0,P2} disagree with {P1} when evaluating the state-space
the following word(s) are not accepted by {P1}:
P0 accepts: p0 & !p1 & !p2 & p3; p0 & p1 & !p2 & p3; p0 & p1 & p2 & p3; cycle{p0 & p1 & p2 & p3; p0 & p1 & !p2 & !p3; p0 & p1 & p2 & !p3; p0 & p1 & !p2 & !p3}
P2 accepts: p0 & !p1 & !p2 & p3; cycle{p0 & p1 & !p2 & !p3; p0 & p1 & p2 & p3; p0 & p1 & !p2 & p3}
If --products=N is used with N greater than one, the number of the state-space is also printed. This number is of no use by itself, except to explain why you may get multiple disagreement between the same sets of automata.
These products tests may sometime catch errors that were not captured by the first two tests if one non-deterministic automaton recognize less words than what it should. If the input automata are all deterministic or the --determinize option is used, this test is redundant and can be disabled. (In fact, the --determinize option implies option --product=0 to do so.)
• Consistency check:
For each $$i$$, the products $$P_i\otimes S$$ and $$N_i\otimes S$$ actually cover all states of $$S$$. Because $$S$$ does not have any deadlock, any of its infinite path must be accepted by $$P_i$$ or $$N_i$$ (or both).
An error in that case is displayed as
error: inconsistency between P1 and N1
If --products=N is used with N greater than one, the number of the state-space in which the inconsistency was detected is also printed.
This test may catch errors that were not captured by the first two tests if one non-deterministic automaton recognize less words than what it should. If the input automata are deterministic or the --determinize option is used, this test is redundant and can be disabled. (In fact, the --determinize option implies option --product=0 to do so.)
The above checks are similar to those that are performed by LBTT, except for the complemented intersection check, which is only done in ltlcross.
If any problem was reported during the translation of one of the formulas, ltlcheck will exit with an exit status of 1. Statistics (if requested) are output nonetheless, and include any faulty automaton as well.
## Getting statistics
Detailed statistics about the result of each translation, and the product of that resulting automaton with the random state-space, can be obtained using the --csv=FILE or --json=FILE option.
### CSV or JSON output (or both!)
The following compare ltl2tgba, spin, and lbt on three random formulas (where W and M operators have been rewritten away because they are not supported by spin and lbt).
randltl -n 3 a b |
ltlfilt --remove-wm |
ltlcross --csv=results.csv \
'ltl2tgba -s %f >%O' \
'spin -f %s >%O' \
'lbt < %L >%O'
-:1: 0
Running [P0]: ltl2tgba -s '0' >'lcr-o0-MWRc3C'
Running [P1]: spin -f 'false' >'lcr-o1-IHykCs'
Running [P2]: lbt < 'lcr-i0-cYTPbi' >'lcr-o2-0mcmL7'
Running [N0]: ltl2tgba -s '1' >'lcr-o0-4qenlX'
Running [N1]: spin -f 'true' >'lcr-o1-u5jYZM'
Running [N2]: lbt < 'lcr-i0-WwhWEC' >'lcr-o2-ukaVjs'
Performing sanity checks and gathering statistics...
-:2: !(F(!(p0)))
Running [P0]: ltl2tgba -s '!(F(!(p0)))' >'lcr-o0-atua0h'
Running [P1]: spin -f '!(<>(!(p0)))' >'lcr-o1-CEVXK7'
Running [P2]: lbt < 'lcr-i1-gxM8vX' >'lcr-o2-8MzkhN'
Running [N0]: ltl2tgba -s 'F(!(p0))' >'lcr-o0-g0I12C'
Running [N1]: spin -f '<>(!(p0))' >'lcr-o1-E0hlTs'
Running [N2]: lbt < 'lcr-i1-c6a1Ji' >'lcr-o2-ahTHA8'
Performing sanity checks and gathering statistics...
-:3: F((G(p0)) | (F(p1)))
Running [P0]: ltl2tgba -s 'F((G(p0)) | (F(p1)))' >'lcr-o0-s7QnuY'
Running [P1]: spin -f '<>(([](p0)) || (<>(p1)))' >'lcr-o1-00iXsO'
Running [P2]: lbt < 'lcr-i2-o2JXrE' >'lcr-o2-0t7Yqu'
Running [N0]: ltl2tgba -s '!(F((G(p0)) | (F(p1))))' >'lcr-o0-o55wqk'
Running [N1]: spin -f '!(<>(([](p0)) || (<>(p1))))' >'lcr-o1-UwpMua'
Running [N2]: lbt < 'lcr-i2-cZBdA0' >'lcr-o2-eNJFFQ'
Performing sanity checks and gathering statistics...
No problem detected.
After this execution, the file results.csv contains the following:
formula
tool
exit_status
exit_code
time
states
edges
transitions
acc
scc
nondet_states
nondet_aut
complete_aut
product_states
product_transitions
product_scc
0 ltl2tgba -s %f >%O ok 0 0.0273999 1 1 0 1 1 0 0 0 1 0 1
0 spin -f %s >%O ok 0 0.00195725 2 2 1 1 2 0 0 0 1 0 1
0 lbt < %L >%O ok 0 0.00275329 1 0 0 0 1 0 0 0 1 0 1
1 ltl2tgba -s %f >%O ok 0 0.027268 1 1 1 1 1 0 0 1 200 4199 1
1 spin -f %s >%O ok 0 0.00188659 2 2 2 1 2 0 0 1 201 4220 2
1 lbt < %L >%O ok 0 0.00281626 3 3 3 0 3 0 0 1 222 4653 23
!(F(!(p0))) ltl2tgba -s %f >%O ok 0 0.0277402 1 1 1 1 1 0 0 0 200 2059 1
!(F(!(p0))) spin -f %s >%O ok 0 0.00199805 1 1 1 1 1 0 0 0 200 2059 1
!(F(!(p0))) lbt < %L >%O ok 0 0.00281056 2 2 2 0 2 0 0 0 201 2071 2
F(!(p0)) ltl2tgba -s %f >%O ok 0 0.0274115 2 3 4 1 2 0 0 1 400 8264 2
F(!(p0)) spin -f %s >%O ok 0 0.00194443 2 3 5 1 2 1 1 1 400 10337 2
F(!(p0)) lbt < %L >%O ok 0 0.00283659 4 6 10 1 4 2 1 1 601 14497 203
F((G(p0)) | (F(p1))) ltl2tgba -s %f >%O ok 0 0.0294767 3 5 11 1 3 1 1 0 600 11358 3
F((G(p0)) | (F(p1))) spin -f %s >%O ok 0 0.00233339 4 8 24 1 4 2 1 0 800 24920 4
F((G(p0)) | (F(p1))) lbt < %L >%O ok 0 0.00291947 9 17 52 2 9 4 1 0 1601 41559 805
!(F((G(p0)) | (F(p1)))) ltl2tgba -s %f >%O ok 0 0.0285971 2 4 4 1 1 0 0 0 395 3964 1
!(F((G(p0)) | (F(p1)))) spin -f %s >%O ok 0 0.00699975 2 3 5 1 1 1 1 0 396 4964 1
!(F((G(p0)) | (F(p1)))) lbt < %L >%O ok 0 0.00290436 3 6 9 1 2 3 1 0 397 5957 2
Although we only supplied 2 random generated formulas, the output contains 4 formulas because ltlcross had to translate the positive and negative version of each.
If we had used the option --json=results.json instead of (or in addition to) --cvs=results.csv, the file results.json would have contained the following JSON output.
{
"tool": [
"ltl2tgba -s %f >%O",
"spin -f %s >%O",
"lbt < %L >%O"
],
"formula": [
"0",
"1",
"!(F(!(p0)))",
"F(!(p0))",
"F((G(p0)) | (F(p1)))",
"!(F((G(p0)) | (F(p1))))"
],
"fields": [
"formula","tool","exit_status","exit_code","time","states","edges","transitions","acc","scc","nondet_states","nondet_aut","complete_aut","product_states","product_transitions","product_scc"
],
"inputs": [ 0, 1 ],
"results": [
[ 0,0,"ok",0,0.0273999,1,1,0,1,1,0,0,0,1,0,1 ],
[ 0,1,"ok",0,0.00195725,2,2,1,1,2,0,0,0,1,0,1 ],
[ 0,2,"ok",0,0.00275329,1,0,0,0,1,0,0,0,1,0,1 ],
[ 1,0,"ok",0,0.027268,1,1,1,1,1,0,0,1,200,4199,1 ],
[ 1,1,"ok",0,0.00188659,2,2,2,1,2,0,0,1,201,4220,2 ],
[ 1,2,"ok",0,0.00281626,3,3,3,0,3,0,0,1,222,4653,23 ],
[ 2,0,"ok",0,0.0277402,1,1,1,1,1,0,0,0,200,2059,1 ],
[ 2,1,"ok",0,0.00199805,1,1,1,1,1,0,0,0,200,2059,1 ],
[ 2,2,"ok",0,0.00281056,2,2,2,0,2,0,0,0,201,2071,2 ],
[ 3,0,"ok",0,0.0274115,2,3,4,1,2,0,0,1,400,8264,2 ],
[ 3,1,"ok",0,0.00194443,2,3,5,1,2,1,1,1,400,10337,2 ],
[ 3,2,"ok",0,0.00283659,4,6,10,1,4,2,1,1,601,14497,203 ],
[ 4,0,"ok",0,0.0294767,3,5,11,1,3,1,1,0,600,11358,3 ],
[ 4,1,"ok",0,0.00233339,4,8,24,1,4,2,1,0,800,24920,4 ],
[ 4,2,"ok",0,0.00291947,9,17,52,2,9,4,1,0,1601,41559,805 ],
[ 5,0,"ok",0,0.0285971,2,4,4,1,1,0,0,0,395,3964,1 ],
[ 5,1,"ok",0,0.00699975,2,3,5,1,1,1,1,0,396,4964,1 ],
[ 5,2,"ok",0,0.00290436,3,6,9,1,2,3,1,0,397,5957,2 ]
]
}
Here the fields table describes the columns of the results table. The inputs tables lists the columns that are considered as inputs for the experiments. The values in the columns corresponding to the fields formula and tool contains indices relative to the formula and tool tables. This format is more compact when dealing with lots of translators and formulas, because they don't have to be repeated on each line as in the CSV version.
JSON data can be easily processed in any language. For instance the following Python3 script averages each column (except the first four) for each tool, and presents the results in a form that can almost be copied into a LaTeX table (the % in the tool names have to be taken care of). Note that for simplicity we assume that the first two columns are inputs, instead of reading the inputs field.
#!/usr/bin/python3
import json
datacols = range(4, len(data["fields"]))
# Index results by tool
results = { t:[] for t in range(0, len(data["tool"])) }
for l in data["results"]:
results[l[1]].append(l)
# Average columns for each tool, and display them as a table
print("%-18s & count & %s \\\\" % ("tool", " & ".join(data["fields"][4:])))
for i in range(0, len(data["tool"])):
c = len(results[i])
sums = ["%6.1f" % (sum([x[j] for x in results[i]])/c)
for j in datacols]
print("%-18s & %3d & %s \\\\" % (data["tool"][i], c,
" & ".join(sums)))
tool & count & time & states & edges & transitions & acc & scc & nondet_states & nondet_aut & complete_aut & product_states & product_transitions & product_scc \\
ltl2tgba -s %f >%O & 6 & 0.0 & 1.7 & 2.5 & 3.5 & 1.0 & 1.5 & 0.2 & 0.2 & 0.3 & 299.3 & 4974.0 & 1.5 \\
spin -f %s >%O & 6 & 0.0 & 2.2 & 3.2 & 6.3 & 1.0 & 2.0 & 0.7 & 0.5 & 0.3 & 333.0 & 7750.0 & 1.8 \\
lbt < %L >%O & 6 & 0.0 & 3.7 & 5.7 & 12.7 & 0.7 & 3.5 & 1.5 & 0.5 & 0.3 & 503.8 & 11456.2 & 172.7 \\
The script bench/ltl2tgba/sum.py is a more evolved version of the above script that generates two kinds of LaTeX tables.
When computing such statistics, you should be aware that inputs for which a tool failed to generate an automaton (e.g. it crashed, or it was killed if you used ltlcross's --timeout option to limit run time) will appear as mostly empty lines in the CSV or JSON files, since most statistics cannot be computed without an automaton… Those lines with missing data can be omitted with the --omit-missing option (this used to be the default up to Spot 1.2).
However data for bogus automata are still included: as shown below ltlcross will report inconsistencies between automata as errors, but it does not try to guess who is incorrect.
### Description of the columns
The number of column output in the CSV or JSON outputs depend on the options passed to ltlcross. Additional columns will be output if --strength, --ambiguous, --automata, or --product=+N are used.
Columns formula and tool contain the formula translated and the command run to translate it. In the CSV, these columns contain the actual text. In the JSON output, these column contains an index into the formula and tool table declared separately.
exit_status and exit_code are used to indicate if the translator successfully produced an automaton, or if it failed. On successful translation, exit_status is equal to "ok" and exit_code is 0. If the translation took more time than allowed with the --timeout option, exit_status will contain "timeout" and exit_code will be set to -1. Other values are used to diagnose various issues: please check the man-page for ltlcross for a list of them.
time obviously contains the time used by the translation. Time is measured with some high-resolution clock when available (that's nanosecond accuracy under Linux), but because translator commands are executed through a shell, it also includes the time to start a shell. (This extra cost apply identically to all translators, so it is not unfair.)
All the values that follow will be missing if exit_status is not equal to "ok". (You may instruct ltlcross not to output lines with such missing data with the option --omit-missing.)
states, edges, transitions, acc are size measures for the automaton that was translated. acc counts the number of acceptance sets. When building (degeneralized) Büchi automata, it will always be 1, so its value is meaningful only when evaluating translations to generalized Büchi automata. edges counts the actual number of edges in the graph supporting the automaton; an edge (labeled by a Boolean formula) might actually represent several transitions (each labeled by assignment of all atomic propositions). For instance in an automaton where the atomic proposition are $$a$$ and $$b$$, one edge labeled by $$a\lor b$$ actually represents three transitions $$a b$$, $$a\bar b$$, and $$\bar a b$$.
scc counts the number of strongly-connected components in the automaton.
If option --strength is passed to ltlcross, these SCCs are also partitioned on four sets based on their strengths:
• nonacc_scc for non-accepting SCCs (such as states A1 and A2 in the previous picture).
• terminal_scc for accepting SCCs where all states or edges belong to the same acceptance sets, and that are complete (i.e., any state in a terminal SCC accepts the universal language). States B1 and B2 in the previous picture are two terminal SCCs.
• weak_scc for accepting SCCs where all states or edges belong to the same acceptance sets, but that are not complete.
• strong_scc for accepting SCCs that are not weak.
These SCC strengths can be used to compute the strength of the automaton as a whole:
• an automaton is terminal if it contains only non-accepting or terminal SCCs,
• an automaton is weak if it it contains only non-accepting, terminal, or weak SCCs,
• an automaton is strong if it contains at least one strong SCC.
This classification is used to fill the terminal_aut, weak_aut, strong_aut columns with Boolean values (still only if option --strength is passed). Only one of these should contain 1. We usually prefer terminal automata over weak automata, and weak automata over strong automata, because the emptiness check of terminal (and weak) automata is easier. When working with alternating automata, all those strength-related columns will be empty, because the routines used to compute those statistic do not yet support universal edges.
nondetstates counts the number of non-deterministic states in the automaton. nondeterministic is a Boolean value indicating if the automaton is not deterministic. For instance in the previous picture showing two automata for a U b, the first automaton is deterministic (these two fields will contain 0), while the second automaton contain a nondeterministic state (state A2 has two possible successors for the assignment $$ab$$) and is therefore not deterministic.
If option --aumbiguous was passed to ltlcross, the column ambiguous_aut holds a Boolean indicating whether the automaton is ambiguous, i.e., if there exists a word that can be accepted by at least two different runs. (This information is not yet available for alternating automata.)
complete_aut is a Boolean indicating whether the automaton is complete.
Columns product_states, product_transitions, and product_scc count the number of state, transitions and strongly-connect components in the product that has been built between the translated automaton and a random model. For a given formula, the same random model is of course used against the automata translated by all tools. Comparing the size of these product might give another indication of the "conciseness" of a translated automaton.
There is of course a certain "luck factor" in the size of the product. Maybe some translator built a very dumb automaton, with many useless states, in which just a very tiny part is translated concisely. By luck, the random model generated might synchronize with this tiny part only, and ignore the part with all the useless states. A way to lessen this luck factor is to increase the number of products performed against the translated automaton. If option --products=N is used, N products are builds instead of one, and the fields product_states, product_transitions, and product_scc contain average values.
If the option --products=+N is used (with a + in front of the number), then no average value is computed. Instead, three columns product_states, product_transitions, and product_scc are output for each individual product (i.e., $$3\times N$$ columns are output). This might be useful if you want to compute different kind of statistic (e.g., a median instead of a mean) or if you want to build scatter plots of all these products.
Finally, if the --automata option was passed to ltlcross, the CSV or JSON output will contain a column named automaton encoding each produced automaton in the HOA format.
### Changing the name of the translators
By default, the names used in the CSV and JSON output to designate the translators are the command specified on the command line.
For instance in the following, ltl2tgba is run in two configurations, and the strings ltl2tgba -s --small %f >%O and ltl2tgba -s --deter %f >%O appear verbatim in the output:
ltlcross -f a -f Ga 'ltl2tgba -s --small %f >%O' 'ltl2tgba -s --deter %f >%O' --csv
formula
tool
exit_status
exit_code
time
states
edges
transitions
acc
scc
nondet_states
nondet_aut
complete_aut
product_states
product_transitions
product_scc
a ltl2tgba -s --small %f >%O ok 0 0.0282574 2 2 3 1 2 0 0 0 201 4144 2
a ltl2tgba -s --deter %f >%O ok 0 0.0253441 2 2 3 1 2 0 0 0 201 4144 2
!(a) ltl2tgba -s --small %f >%O ok 0 0.0274906 2 2 3 1 2 0 0 0 201 4149 2
!(a) ltl2tgba -s --deter %f >%O ok 0 0.0254883 2 2 3 1 2 0 0 0 201 4149 2
G(a) ltl2tgba -s --small %f >%O ok 0 0.0276811 1 1 1 1 1 0 0 0 200 2059 1
G(a) ltl2tgba -s --deter %f >%O ok 0 0.0279394 1 1 1 1 1 0 0 0 200 2059 1
!(G(a)) ltl2tgba -s --small %f >%O ok 0 0.0282372 2 3 4 1 2 0 0 1 400 8264 2
!(G(a)) ltl2tgba -s --deter %f >%O ok 0 0.0280167 2 3 4 1 2 0 0 1 400 8264 2
To present these results graphically, or even when analyzing these data, it might be convenient to give each configured tool a shorter name. ltlcross supports the specification of such short names by looking whether the command specification for a translator has the form "{short name}actual command".
For instance, after
genltl --and-f=1..5 |
ltlcross '{small} ltl2tgba -s --small %f >%O' \
'{deter} ltl2tgba -s --deter %f >%O' --csv=ltlcross.csv
The file ltlcross.csv now contains:
formula
tool
exit_status
exit_code
time
states
edges
transitions
acc
scc
nondet_states
nondet_aut
complete_aut
product_states
product_transitions
product_scc
F(p1) small ok 0 0.0273228 2 3 4 1 2 0 0 1 400 8272 3
F(p1) deter ok 0 0.0276792 2 3 4 1 2 0 0 1 400 8272 3
!(F(p1)) small ok 0 0.025338 1 1 1 1 1 0 0 0 200 2055 2
!(F(p1)) deter ok 0 0.0275788 1 1 1 1 1 0 0 0 200 2055 2
(F(p1)) & (F(p2)) small ok 0 0.0284864 4 9 16 1 4 0 0 1 798 16533 5
(F(p1)) & (F(p2)) deter ok 0 0.0285247 4 9 16 1 4 0 0 1 798 16533 5
!((F(p1)) & (F(p2))) small ok 0 0.0262568 3 5 7 1 3 0 0 0 598 7367 4
!((F(p1)) & (F(p2))) deter ok 0 0.0280206 3 5 7 1 3 0 0 0 598 7367 4
(F(p1)) & (F(p2)) & (F(p3)) small ok 0 0.0303325 8 27 64 1 8 0 0 1 1587 33068 34
(F(p1)) & (F(p2)) & (F(p3)) deter ok 0 0.0300996 8 27 64 1 8 0 0 1 1587 33068 34
!((F(p1)) & (F(p2)) & (F(p3))) small ok 0 0.0313341 4 6 24 1 4 1 1 0 601 6171 4
!((F(p1)) & (F(p2)) & (F(p3))) deter ok 0 0.0300606 7 19 37 1 7 0 0 0 1387 18792 33
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) small ok 0 0.029071 16 81 256 1 16 0 0 1 2727 57786 74
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) deter ok 0 0.0278184 16 81 256 1 16 0 0 1 2727 57786 74
!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4))) small ok 0 0.0300302 5 8 64 1 5 1 1 0 801 8468 5
!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4))) deter ok 0 0.0324723 15 65 175 1 15 0 0 0 2527 37226 73
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)) small ok 0 0.0401099 32 243 1024 1 32 0 0 1 5330 114068 350
(F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)) deter ok 0 0.0319006 32 243 1024 1 32 0 0 1 5330 114068 350
!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5))) small ok 0 0.0319517 6 10 160 1 6 1 1 0 1000 10707 6
In this last example, we saved the CSV output to ltlcross.csv so we can play with it in the next section.
### Working with these CSV files in R
The produced CSV should be directly readable by R's CSV input functions like read.csv(), readr::read_csv(), or data.table::fread().
library(data.table)
str(dt)
data.table 1.12.0 Latest news: r-datatable.com
Classes ‘data.table’ and 'data.frame': 20 obs. of 16 variables:
$formula : chr "F(p1)" "F(p1)" "!(F(p1))" "!(F(p1))" ...$ tool : chr "small" "deter" "small" "deter" ...
$exit_status : chr "ok" "ok" "ok" "ok" ...$ exit_code : int 0 0 0 0 0 0 0 0 0 0 ...
$time : num 0.0273 0.0277 0.0253 0.0276 0.0285 ...$ states : int 2 2 1 1 4 4 3 3 8 8 ...
$edges : int 3 3 1 1 9 9 5 5 27 27 ...$ transitions : int 4 4 1 1 16 16 7 7 64 64 ...
$acc : int 1 1 1 1 1 1 1 1 1 1 ...$ scc : int 2 2 1 1 4 4 3 3 8 8 ...
$nondet_states : int 0 0 0 0 0 0 0 0 0 0 ...$ nondet_aut : int 0 0 0 0 0 0 0 0 0 0 ...
$complete_aut : int 1 1 0 0 1 1 0 0 1 1 ...$ product_states : int 400 400 200 200 798 798 598 598 1587 1587 ...
$product_transitions: int 8272 8272 2055 2055 16533 16533 7367 7367 33068 33068 ...$ product_scc : int 3 3 2 2 5 5 4 4 34 34 ...
- attr(*, ".internal.selfref")=<
Currently the data frame shows one line per couple (formula, tool). This makes comparing tools quite difficult, as their results are on different lines.
A common transformation is to group the results of all tools on the same line: using exactly one line per formula. This is easily achieved using dcast() from the data.table library.
dt2 <- dcast(dt, formula ~ tool, value.var=names(dt)[-(1:2)], sep=".")
str(dt2)
Classes ‘data.table’ and 'data.frame': 10 obs. of 29 variables:
$formula : chr "!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)) & (F(p5)))" "!((F(p1)) & (F(p2)) & (F(p3)) & (F(p4)))" "!((F(p1)) & (F(p2)) & (F(p3)))" "!((F(p1)) & (F(p2)))" ...$ exit_status.deter : chr "ok" "ok" "ok" "ok" ...
$exit_status.small : chr "ok" "ok" "ok" "ok" ...$ exit_code.deter : int 0 0 0 0 0 0 0 0 0 0
$exit_code.small : int 0 0 0 0 0 0 0 0 0 0$ time.deter : num 0.0382 0.0325 0.0301 0.028 0.0276 ...
$time.small : num 0.032 0.03 0.0313 0.0263 0.0253 ...$ states.deter : int 31 15 7 3 1 4 8 16 32 2
$states.small : int 6 5 4 3 1 4 8 16 32 2$ edges.deter : int 211 65 19 5 1 9 27 81 243 3
$edges.small : int 10 8 6 5 1 9 27 81 243 3$ transitions.deter : int 781 175 37 7 1 16 64 256 1024 4
$transitions.small : int 160 64 24 7 1 16 64 256 1024 4$ acc.deter : int 1 1 1 1 1 1 1 1 1 1
$acc.small : int 1 1 1 1 1 1 1 1 1 1$ scc.deter : int 31 15 7 3 1 4 8 16 32 2
$scc.small : int 6 5 4 3 1 4 8 16 32 2$ nondet_states.deter : int 0 0 0 0 0 0 0 0 0 0
$nondet_states.small : int 1 1 1 0 0 0 0 0 0 0$ nondet_aut.deter : int 0 0 0 0 0 0 0 0 0 0
$nondet_aut.small : int 1 1 1 0 0 0 0 0 0 0$ complete_aut.deter : int 0 0 0 0 0 1 1 1 1 1
$complete_aut.small : int 0 0 0 0 0 1 1 1 1 1$ product_states.deter : int 5130 2527 1387 598 200 798 1587 2727 5330 400
$product_states.small : int 1000 801 601 598 200 798 1587 2727 5330 400$ product_transitions.deter: int 82897 37226 18792 7367 2055 16533 33068 57786 114068 8272
$product_transitions.small: int 10707 8468 6171 7367 2055 16533 33068 57786 114068 8272$ product_scc.deter : int 349 73 33 4 2 5 34 74 350 3
$product_scc.small : int 6 5 4 4 2 5 34 74 350 3 - attr(*, ".internal.selfref")=< - attr(*, "sorted")= chr "formula" Using the above form, it is easy to compare two tools on some given measurement, as we just need to plot two columns. For example to compare the number of states produced by the two configurations of ltl2tgba for each formula, we just need to plot column dt2$state.small against dt2$state.deter. library(ggplot2) ggplot(dt2, aes(x=states.small, y=states.deter)) + geom_abline(colour='white') + geom_point() We should probably print the formulas for the cases where the two sizes differ. ggplot(dt2, aes(x=states.small, y=states.deter)) + geom_abline(colour='white') + geom_point() + geom_text(data=subset(dt2, states.small != states.deter), aes(label=formula), hjust=0, nudge_x=.5) ## Miscellaneous options ### --stop-on-error The --stop-on-error option will cause ltlcross to abort on the first detected error. This include failure to start some translator, read its output, or failure to passe the sanity checks. Timeouts are allowed unless --fail-on-time is also given. One use for this option is when ltlcross is used in combination with randltl to check translators on an infinite stream of formulas. For instance the following will cross-compare ltl2tgba against ltl3ba until it finds an error, or your interrupt the command, or it runs out of memory (the hash tables used by randltl and ltlcross to remove duplicate formulas will keep growing). randltl -n -1 --tree-size 10..25 a b c | ltlcross --stop-on-error 'ltl2tgba --lbtt %f >%O' 'ltl3ba -f %s >%O' ### --save-bogus=FILENAME The --save-bogus=FILENAME will save any formula for which an error was detected (either some translation failed, or some problem was detected using the resulting automata) in FILENAME. Again, timeouts are not considered to be errors and therefore not reported in this file, unless --fail-on-timeout is given. The main use for this feature is in conjunction with randltl's generation of random formulas. For instance the following command will run the translators on an infinite number of formulas, saving any problematic formula in bugs.ltl. randltl -n -1 --tree-size 10..25 a b c | ltlcross --save-bogus=bugs.ltl 'ltl2tgba --lbtt %f >%O' 'ltl3ba -f %s >%O' You can periodically check the contents of bugs.ltl, and then run ltlcross only on those formulas to look at the problems: ltlcross -F bugs.ltl 'ltl2tgba --lbtt %f >%O' 'ltl3ba -f %s >%O' ### --grind=FILENAME This option tells ltlcross that, when a problem is detected, it should try to find a smaller formula that still exhibits the problem. Here is the procedure used: • internally list the mutations of the bogus formula and sort them by length (as ltlgrind --sort would do) • process every mutation until one is found that exhibit the bug • repeat the process with this new formula, and again until a formula is found for which no mutation exhibit the bug • output that last formula in FILENAME If --save-bogus=OTHERFILENAME is provided, every bogus formula found during the process will be saved in OTHERFILENAME. Example: ltlcross -f '(G!b & (!c | F!a)) | (c & Ga & Fb)' "modella %L %O" \ --save-bogus=bogus \ --grind=bogus-grind | & G ! p0 | ! p1 F ! p2 & & p1 G p2 F p0 Running [P0]: modella 'lcr-i0-XLU69e' 'lcr-o0-H5Xj9p' Running [N0]: modella 'lcr-i0-90n58A' 'lcr-o0-jrmR8L' Performing sanity checks and gathering statistics... error: P0*N0 is nonempty; both automata accept the infinite word: cycle{!p0 & !p1} Trying to find a bogus mutation of (G!b & (!c | F!a)) | (c & Ga & Fb)... Mutation 1/22: & & p0 G p1 F p2 Running [P0]: modella 'lcr-i1-BVSJ9W' 'lcr-o0-ig7Ca8' Running [N0]: modella 'lcr-i1-WpeQbj' 'lcr-o0-yYv4cu' Performing sanity checks and gathering statistics... Mutation 2/22: & G ! p0 | ! p1 F ! p2 Running [P0]: modella 'lcr-i2-2pxNeF' 'lcr-o0-4zsxgQ' Running [N0]: modella 'lcr-i2-wvxDi1' 'lcr-o0-qyuKkc' Performing sanity checks and gathering statistics... Mutation 3/22: | G ! p0 & & p1 G p2 F p0 Running [P0]: modella 'lcr-i3-25Iinn' 'lcr-o0-sFQRpy' Running [N0]: modella 'lcr-i3-UB2QsJ' 'lcr-o0-Ww2QvU' Performing sanity checks and gathering statistics... error: P0*N0 is nonempty; both automata accept the infinite word: cycle{!p0 & !p1} Trying to find a bogus mutation of G!b | (c & Ga & Fb)... Mutation 1/16: t Running [P0]: modella 'lcr-i4-iMdyz5' 'lcr-o0-o6egDg' Running [N0]: modella 'lcr-i4-oWFoHr' 'lcr-o0-gG5xLC' Performing sanity checks and gathering statistics... Mutation 2/16: G ! p0 Running [P0]: modella 'lcr-i5-ClT0PN' 'lcr-o0-KJyuUY' Running [N0]: modella 'lcr-i5-uypgZ9' 'lcr-o0-4P523k' Performing sanity checks and gathering statistics... Mutation 3/16: & & p0 G p1 F p2 warning: This formula or its negation has already been checked. Use --allow-dups if it should not be ignored. Mutation 4/16: | G ! p0 & p1 F p0 Running [P0]: modella 'lcr-i6-yREa9v' 'lcr-o0-A44ieH' Running [N0]: modella 'lcr-i6-qhwNjS' 'lcr-o0-yCPip3' Performing sanity checks and gathering statistics... error: P0*N0 is nonempty; both automata accept the infinite word: cycle{!p0 & !p1} Trying to find a bogus mutation of G!b | (c & Fb)... Mutation 1/10: t warning: This formula or its negation has already been checked. Use --allow-dups if it should not be ignored. Mutation 2/10: G ! p0 warning: This formula or its negation has already been checked. Use --allow-dups if it should not be ignored. Mutation 3/10: & p0 F p1 Running [P0]: modella 'lcr-i7-Qv6eve' 'lcr-o0-EficBp' Running [N0]: modella 'lcr-i7-O1wsHA' 'lcr-o0-08AJNL' Performing sanity checks and gathering statistics... Mutation 4/10: | p0 G ! p1 Running [P0]: modella 'lcr-i8-88jmUW' 'lcr-o0-YbXZ07' Running [N0]: modella 'lcr-i8-OJlW7i' 'lcr-o0-ohCTeu' Performing sanity checks and gathering statistics... Mutation 5/10: | G ! p0 F p0 Running [P0]: modella 'lcr-i9-AGRcmF' 'lcr-o0-8k1wtQ' Running [N0]: modella 'lcr-i9-4jC9A1' 'lcr-o0-Ai7MIc' Performing sanity checks and gathering statistics... Mutation 6/10: | ! p0 & p1 F p0 Running [P0]: modella 'lcr-i10-OD3KQn' 'lcr-o0-oYQJYy' Running [N0]: modella 'lcr-i10-cks16J' 'lcr-o0-S3UjfV' Performing sanity checks and gathering statistics... Mutation 7/10: | & p1 F p0 G p0 Running [P0]: modella 'lcr-i11-w59Xn6' 'lcr-o0-wYfDwh' Running [N0]: modella 'lcr-i11-yc2DFs' 'lcr-o0-K0CFOD' Performing sanity checks and gathering statistics... Mutation 8/10: | & p0 p1 G ! p0 Running [P0]: modella 'lcr-i12-KYC3XO' 'lcr-o0-iDxs7Z' Running [N0]: modella 'lcr-i12-MlZahb' 'lcr-o0-gFgUqm' Performing sanity checks and gathering statistics... Mutation 9/10: | G ! p0 & p0 F p0 Running [P0]: modella 'lcr-i13-CFpYAx' 'lcr-o0-kxs3KI' Running [N0]: modella 'lcr-i13-g50wVT' 'lcr-o0-Mcv154' Performing sanity checks and gathering statistics... error: P0*N0 is nonempty; both automata accept the infinite word: cycle{!p0} Trying to find a bogus mutation of G!c | (c & Fc)... Mutation 1/7: t warning: This formula or its negation has already been checked. Use --allow-dups if it should not be ignored. Mutation 2/7: G ! p0 warning: This formula or its negation has already been checked. Use --allow-dups if it should not be ignored. Mutation 3/7: & p0 F p0 Running [P0]: modella 'lcr-i14-0CdSgg' 'lcr-o0-oPKJrr' Running [N0]: modella 'lcr-i14-6JZTCC' 'lcr-o0-8934NN' Performing sanity checks and gathering statistics... Mutation 4/7: | p0 G ! p0 Running [P0]: modella 'lcr-i15-E7cAZY' 'lcr-o0-MNe6aa' Running [N0]: modella 'lcr-i15-4hoVml' 'lcr-o0-oAnLyw' Performing sanity checks and gathering statistics... Mutation 5/7: | G ! p0 F p0 warning: This formula or its negation has already been checked. Use --allow-dups if it should not be ignored. Mutation 6/7: | ! p0 & p0 F p0 Running [P0]: modella 'lcr-i16-Q6bWKH' 'lcr-o0-ec78WS' Running [N0]: modella 'lcr-i16-kDpE93' 'lcr-o0-WRwamf' Performing sanity checks and gathering statistics... Mutation 7/7: | G p0 & p0 F p0 Running [P0]: modella 'lcr-i17-iL1Zyq' 'lcr-o0-MWaRLB' Running [N0]: modella 'lcr-i17-QWS0YM' 'lcr-o0-cgpbcY' Performing sanity checks and gathering statistics... Smallest bogus mutation found for (G!b & (!c | F!a)) | (c & Ga & Fb) is G!c | (c & Fc). error: some error was detected during the above runs. Check file bogus for problematic formulas. cat bogus (G!b & (!c | F!a)) | (c & Ga & Fb) G!b | (c & Ga & Fb) G!b | (c & Fb) G!c | (c & Fc) cat bogus-grind G!c | (c & Fc) ### --no-check The --no-check option disables all sanity checks, and only use the supplied formulas in their positive form. When checks are enabled, the negated formulas are intermixed with the positives ones in the results. Therefore the --no-check option can be used to gather statistics about a specific set of formulas. ### --verbose The verbose option can be useful to troubleshoot problems or simply follow the list of transformations and tests performed by ltlcross. For instance here is what happens if we try to cross check ltl2tgba and ltl3ba -H1 on the formula FGa. Note that ltl2tgba will produce transition-based generalized Büchi automata, while ltl3ba -H1 produces co-Büchi alternating automata. ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --verbose F(G(a)) Running [P0]: ltl2tgba -H 'F(G(a))'>'lcr-o0-SImDgp' Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-g3V2LA' Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-QltGiM' Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-yKMJUX' info: collected automata: info: P0 (2 st.,3 ed.,1 sets) info: N0 (1 st.,2 ed.,1 sets) deterministic complete info: P1 (2 st.,3 ed.,1 sets) info: N1 (3 st.,5 ed.,1 sets) univ-edges complete Performing sanity checks and gathering statistics... info: getting rid of universal edges... info: N1 (3 st.,5 ed.,1 sets) -> (2 st.,4 ed.,1 sets) info: complementing automata... info: P0 (2 st.,3 ed.,1 sets) -> (2 st.,4 ed.,1 sets) Comp(P0) info: N0 (1 st.,2 ed.,1 sets) -> (1 st.,2 ed.,1 sets) Comp(N0) info: P1 (2 st.,3 ed.,1 sets) -> (2 st.,4 ed.,1 sets) Comp(P1) info: N1 (2 st.,4 ed.,1 sets) -> (2 st.,4 ed.,1 sets) Comp(N1) info: getting rid of any Fin acceptance... info: Comp(N0) (1 st.,2 ed.,1 sets) -> (2 st.,3 ed.,1 sets) info: P1 (2 st.,3 ed.,1 sets) -> (2 st.,3 ed.,1 sets) info: Comp(N1) (2 st.,4 ed.,1 sets) -> (3 st.,6 ed.,1 sets) info: check_empty P0*N0 info: check_empty Comp(N0)*Comp(P0) info: check_empty P0*N1 info: check_empty P1*N0 info: check_empty P1*N1 info: check_empty Comp(N1)*Comp(P1) info: cross_checks and consistency_checks unnecessary No problem detected. First FGa and its negations !FGa are translated with the two tools, resulting in four automata: two positive automata P0 and P1 for FGa, and two negative automata N0 and N1. Some basic information about the collected automata are displayed. For instance we can see that although ltl3ba -H1 outputs co-Büchi alternating automata, only automaton N1 uses universal edges: the automaton P1 can be used like a non-alternating co-Büchi automaton. ltlcross then proceeds to transform alternating automata (only weak alternating automata are supported) into non-alternating automata. Here only N1 needs this transformation. Then ltlcross computes the complement of these four automata. Now that ltlcross has four complemented automata, it has to make sure they use only Inf acceptance because that is what our emptiness check procedure can handle. So there is a new pass over all automata, rewriting them to get rid of any Fin acceptance. After this preparatory work, it is time to actually compare these automata. Together, the tests P0*N0 and Comp(N0)*Comp(P0) ensure that the automaton N0 is really the complement of P0. Similarly P1*N1 and Comp(N1)*Comp(P1) ensure that N1 is the complement of P1. Finally P0*N1 and P1*N0 ensure that P1 is equivalent to P0 and N1 is equivalent to N0. Note that if we reduce ltlcross's ability to determinize automata for complementation, the procedure can look slightly more complex: ltlcross -f 'FGa' ltl2tgba 'ltl3ba -H1' --determinize-max-states=1 --verbose F(G(a)) Running [P0]: ltl2tgba -H 'F(G(a))'>'lcr-o0-bIvAqp' Running [P1]: ltl3ba -H1 -f '<>([](a))'>'lcr-o1-RaAWfB' Running [N0]: ltl2tgba -H '!(F(G(a)))'>'lcr-o0-LQot6M' Running [N1]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o1-bkUP1Y' info: collected automata: info: P0 (2 st.,3 ed.,1 sets) info: N0 (1 st.,2 ed.,1 sets) deterministic complete info: P1 (2 st.,3 ed.,1 sets) info: N1 (3 st.,5 ed.,1 sets) univ-edges complete Performing sanity checks and gathering statistics... info: getting rid of universal edges... info: N1 (3 st.,5 ed.,1 sets) -> (2 st.,4 ed.,1 sets) info: complementing automata... info: P0 not complemented (more than 1 states required) info: N0 (1 st.,2 ed.,1 sets) -> (1 st.,2 ed.,1 sets) Comp(N0) info: P1 not complemented (more than 1 states required) info: N1 (2 st.,4 ed.,1 sets) -> (2 st.,4 ed.,1 sets) Comp(N1) info: getting rid of any Fin acceptance... info: Comp(N0) (1 st.,2 ed.,1 sets) -> (2 st.,3 ed.,1 sets) info: P1 (2 st.,3 ed.,1 sets) -> (2 st.,3 ed.,1 sets) info: Comp(N1) (2 st.,4 ed.,1 sets) -> (3 st.,6 ed.,1 sets) info: check_empty P0*N0 info: check_empty P0*N1 info: check_empty Comp(N0)*N1 info: check_empty P1*N0 info: check_empty Comp(N1)*N0 info: check_empty P1*N1 info: complements not computed for some automata info: continuing with cross_checks and consistency_checks info: building state-space #1/1 of 200 states with seed 0 info: state-space has 4136 edges info: building product between state-space and P0 (2 st., 3 ed.) info: product has 400 st., 8298 ed. info: 2 SCCs info: building product between state-space and P1 (2 st., 3 ed.) info: product has 400 st., 8298 ed. info: 2 SCCs info: building product between state-space and N0 (1 st., 2 ed.) info: product has 200 st., 4136 ed. info: 1 SCCs info: building product between state-space and N1 (2 st., 4 ed.) info: product has 400 st., 8272 ed. info: 1 SCCs info: cross_check {P0,P1}, state-space #0/1 info: cross_check {N0,N1}, state-space #0/1 info: consistency_check (P0,N0), state-space #0/1 info: consistency_check (P1,N1), state-space #0/1 No problem detected. In this case, ltlcross does not have any complement automaton for P0 and P1, so it cannot make sure that P0 and P1 are equivalent. If we imagine for instance that P0 has an empty language, we can see that the six check_empty tests would still succeed. So ltlcross builds a random state-space of 200 states, synchronize it with the four automata, and then performs additional checks (cross_check and consistency_check) on these products as described earlier. While these additional checks do not make a proof that P0 and P1 are equivalent, they can catch some problems, and would easily catch the case of an automaton with an empty language by mistake. Here is the same example, if we declare that ltl3ba is a reference implementation that should not be checked, and we just want to check the output of ltl2tgba against this reference. See how the number of tests performed has been reduced. ltlcross -f 'FGa' ltl2tgba --reference 'ltl3ba -H1' --verbose F(G(a)) Running [P0]: ltl3ba -H1 -f '<>([](a))'>'lcr-o0-DwwBDy' Running [P1]: ltl2tgba -H 'F(G(a))'>'lcr-o1-DWKSLK' Running [N0]: ltl3ba -H1 -f '!(<>([](a)))'>'lcr-o0-tLDlYW' Running [N1]: ltl2tgba -H '!(F(G(a)))'>'lcr-o1-fDBZb9' info: collected automata: info: P0 (2 st.,3 ed.,1 sets) info: N0 (3 st.,5 ed.,1 sets) univ-edges complete info: P1 (2 st.,3 ed.,1 sets) info: N1 (1 st.,2 ed.,1 sets) deterministic complete Performing sanity checks and gathering statistics... info: getting rid of universal edges... info: N0 (3 st.,5 ed.,1 sets) -> (2 st.,4 ed.,1 sets) info: complementing automata... info: P1 (2 st.,3 ed.,1 sets) -> (2 st.,4 ed.,1 sets) Comp(P1) info: N1 (1 st.,2 ed.,1 sets) -> (1 st.,2 ed.,1 sets) Comp(N1) info: getting rid of any Fin acceptance... info: P0 (2 st.,3 ed.,1 sets) -> (2 st.,3 ed.,1 sets) info: Comp(N1) (1 st.,2 ed.,1 sets) -> (2 st.,3 ed.,1 sets) info: P0 and N0 assumed correct and used as references info: check_empty P0*N1 info: check_empty P1*N0 info: check_empty P1*N1 info: check_empty Comp(N1)*Comp(P1) info: cross_checks and consistency_checks unnecessary No problem detected. ## Running ltlcross in parallel The ltlcross command itself has no built-in support for parallelization (patches welcome). However its interface makes it rather easy to parallelize ltlcross runs with third-party tools such as: • xargs from GNU findutils. The -P n option is a GNU extension to specify that n commands should be run in parallel. For instance the following command tests ltl2tgba and ltl3ba against 1000 formulas, running 8 formulas in parallel. randltl -n-1 3 | ltlfilt --relabel=pnn --unique -n1000 | xargs -P8 -I'{}' ltlcross -q --save-bogus='>>bugs.ltl' ltl2tgba ltl3ba -f '{}' The above pipeline uses randltl to generate an infinite number of LTL formulas (-n-1) over three atomic propositions. Those formules are then relabeled with ltlfilt (so that a U b and b U a both get mapped to the same p0 U p1) and filtered for duplicates (--unique). This first 1000 formulas (-n1000) are then passed on to xargs. The command xargs -I'{}' ltlcross... takes each line of input, and executes the command ltlcross... with {} replaced by the input line. The option -P8 does this with 8 processes in parallel. Here ltlcross is called with option -q to silence most its regular output as the 8 instances of ltlcross would be otherwise writing to the same terminal. With -q, only errors are displayed. Additionally --save-bogus is used to keep track of all formulas causing errors. The >>bugs.ltl syntax means to open bugs.ltl in append mode, so that bugs.ltl does not get overwritten each time a new ltlcross instance finds a bug. • GNU parallel or moreutils's parallel can also be used similarly. • make -j n is another option: first convert the list of formulas into a Makefile that calls ltlcross for each of them. For instance here is how to build a makefile called ltlcross.mk testing ltl2tgbaand ltl3ba against all formulas produced by genltl --eh, and gathering statistics from all runs in all.csv. echo 'LTLCROSS=ltlcross -q ltl2tgba ltl3ba' > ltlcross.mk echo "ALL=$(echo $(genltl --eh --format="%F%L.csv"))" >> ltlcross.mk echo "all.csv: \$(ALL); cat \$(ALL) | sed -e 1n -e '/^\"formula\"/d' > \$@" >>ltlcross.mk
genltl --eh --format="%F%L.csv:; \$(LTLCROSS) --csv=\$@ -f '%f'" >>ltlcross.mk
This creates ltlcross.mk:
LTLCROSS=ltlcross -q ltl2tgba ltl3ba
ALL= eh-patterns1.csv eh-patterns2.csv eh-patterns3.csv eh-patterns4.csv eh-patterns5.csv eh-patterns6.csv eh-patterns7.csv eh-patterns8.csv eh-patterns9.csv eh-patterns10.csv eh-patterns11.csv eh-patterns12.csv
all.csv: $(ALL); cat$(ALL) | sed -e 1n -e '/^"formula"/d' > $@ eh-patterns1.csv:;$(LTLCROSS) --csv=$@ -f 'p0 U (p1 & Gp2)' eh-patterns2.csv:;$(LTLCROSS) --csv=$@ -f 'p0 U (p1 & X(p2 U p3))' eh-patterns3.csv:;$(LTLCROSS) --csv=$@ -f 'p0 U (p1 & X(p2 & F(p3 & XF(p4 & XF(p5 & XFp6)))))' eh-patterns4.csv:;$(LTLCROSS) --csv=$@ -f 'F(p0 & XGp1)' eh-patterns5.csv:;$(LTLCROSS) --csv=$@ -f 'F(p0 & X(p1 & XFp2))' eh-patterns6.csv:;$(LTLCROSS) --csv=$@ -f 'F(p0 & X(p1 U p2))' eh-patterns7.csv:;$(LTLCROSS) --csv=$@ -f 'FGp0 | GFp1' eh-patterns8.csv:;$(LTLCROSS) --csv=$@ -f 'G(p0 -> (p1 U p2))' eh-patterns9.csv:;$(LTLCROSS) --csv=$@ -f 'G(p0 & XF(p1 & XF(p2 & XFp3)))' eh-patterns10.csv:;$(LTLCROSS) --csv=$@ -f 'GFp0 & GFp1 & GFp2 & GFp3 & GFp4' eh-patterns11.csv:;$(LTLCROSS) --csv=$@ -f '(p0 U (p1 U p2)) | (p1 U (p2 U p0)) | (p2 U (p0 U p1))' eh-patterns12.csv:;$(LTLCROSS) --csv=\$@ -f 'G(p0 -> (p1 U (Gp2 | Gp3)))'
This makefile could be executed for instance with make -f ltlcross.mk -j 4, where -j 4 specifies that 4 processes can be executed in parallel. Using different csv files for each process avoids potential race conditions that could occur if each instance of ltlcross was appending to the same file. The sed command used while merging all csv files keeps the first header line (1n) while removing all subsequent ones (/"formula"/d).
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2019-08-18 08:39:54
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https://www.zigya.com/study/book?class=11&board=bsem&subject=Physics&book=Physics+Part+I&chapter=Laws+of+Motion&q_type=&q_topic=Common+Forces+In+Mechanics+&q_category=&question_id=PHENNT11134907
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## Previous Year Papers
Download the PDF Question Papers Free for off line practice and view the Solutions online.
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Class 10 Class 12
300 J of work is done in sliding a 2 kg block up an inclined plane of height 10 m. Taking =10 m/s2, work done against friction is
• 200 J
• 100 J
• Zero
• 1000 J
B.
100 J
Net work done in sliding a body up to a height h on inclined plane
= Work done against gravitational force + Work done against frictional force
Net work done in sliding a body up to a height h on inclined plane
= Work done against gravitational force + Work done against frictional force
831 Views
State Galileo’s law of motion.
Galileo’s law of motion states that, a body continues to move in the same direction with constant speed, if no force is acting on the body.
1332 Views
Define inertia.
The property by virtue of which the body cannot change its state of rest or uniform motion in a straight line, unless an external force is acting on the body is called as Inertia.
1316 Views
What is Aristotle’s law of motion?
Aristotle’s law of motion states that an external force is required to keep the body in motion.
2445 Views
Is Aristotle’s law of motion now correct?
No. Aristotl'e law of motion is false.
1233 Views
When the branches of an apple tree are shaken, the apples fall down. Why?
The apple fall from an apple tree when it shaken because of inertia of rest. Apple is in a state of rest and when the tree is suddenly shaken, apples still tends to remain in it's same state of rest whereas branches move.
So, the apples fall down.
1475 Views
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2018-10-24 02:24:17
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https://stacks.math.columbia.edu/tag/0F79
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Lemma 62.4.11. Let $f' : X \to Y'$ and $g : Y' \to Y$ be composable morphisms of schemes with $f'$ and $f = g \circ f'$ locally quasi-finite and $g$ separated and locally of finite type. Then there is a canonical isomorphism of functors $g_! \circ f'_! = f_!$. This isomorphism is compatible with
1. covariance with respect to open embeddings as in Remarks 62.3.5 and 62.4.6,
2. the base change isomorphisms of Lemmas 62.4.10 and 62.3.12, and
3. equal to the isomorphism of Lemma 62.3.13 via the identifications of Lemma 62.4.1 in case $f'$ is separated.
Proof. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. With conventions as in Remark 62.4.9 we will explicitly construct a map
$c : f_{p!}\mathcal{F} \longrightarrow g_*f'_{p!}\mathcal{F}$
of abelian presheaves on $Y_{\acute{e}tale}$. By the discussion in Remark 62.4.9 this will determine a canonical map $c^\# : f_!\mathcal{F} \to g_*f'_!\mathcal{F}$. We will show that $c^\#$ has image contained in the subsheaf $g_!f'_!\mathcal{F}$, thereby obtaining a map $c' : f_!\mathcal{F} \to g_!f'_!\mathcal{F}$. Next, we will prove (a), (b), and (c) that. Finally, part (b) will allow us to show that $c'$ is an isomorphism.
Construction of the map $c$. Let $V \in Y_{\acute{e}tale}$ and let $s = \sum (Z_ i, s_ i)$ be a sum as in (62.4.0.1) defining an element of $f_{p!}\mathcal{F}(V)$. Recall that $Z_ i \subset X_ V = X \times _ Y V$ is a locally closed subscheme finite over $V$. Setting $V' = Y' \times _ Y V$ we get $X_{V'} = X \times _{Y'} V' = X_ V$. Hence $Z_ i \subset X_{V'}$ is locally closed and $Z_ i$ is finite over $V'$ because $g$ is separated (Morphisms, Lemma 29.44.14). Hence we may set $c(s) = \sum (Z_ i, s_ i)$ but now viewed as an element of $f'_{p!}\mathcal{F}(V') = (g_*f'_{p!}\mathcal{F})(V)$. The construction is clearly compatible with relations (1) and (2) and compatible with restriction mappings and hence we obtain the map $c$.
Observe that in the discussion above our section $c(s) = \sum (Z_ i, s_ i)$ of $f'_!\mathcal{F}$ over $V'$ restricts to zero on $V' \setminus \mathop{\mathrm{Im}}(\coprod Z_ i \to V')$. Since $\mathop{\mathrm{Im}}(\coprod Z_ i \to V')$ is proper over $V$ (for example by Morphisms, Lemma 29.41.10) we conclude that $c(s)$ defines a section of $g_!f'_!\mathcal{F} \subset g_*f'_!\mathcal{F}$ over $V$. Since every local section of $f_!\mathcal{F}$ locally comes from a local section of $f_{p!}\mathcal{F}$ we conclude that the image of $c^\#$ is contained in $g_!f'_!\mathcal{F}$. Thus we obtain an induced map $c' : f_!\mathcal{F} \to g_!f'_!\mathcal{F}$ factoring $c^\#$ as predicted in the first paragraph of the proof.
Proof of (a). Let $Y'_1 \subset Y'$ be an open subscheme and set $X_1 = (f')^{-1}(W')$. We obtain a diagram
$\xymatrix{ X_1 \ar[d]_{f'_1} \ar[r]_ a \ar@/_2em/[dd]_{f_1} & X \ar[d]^{f'} \ar@/^2em/[dd]^ f \\ Y'_1 \ar[d]_{g_1} \ar[r]_{b'} & Y' \ar[d]^ g \\ Y \ar@{=}[r] & Y }$
where the horizontal arrows are open immersions. Then our claim is that the diagram
$\xymatrix{ f_{1, !}\mathcal{F}|_{X_1} \ar[r]_{c'_1} \ar[dd] & g_{1, !}f'_{1, !}\mathcal{F}|_{X_1} \ar@{=}[d] \\ & g_{1, !}(f'_!\mathcal{F})|_{Y'_1} \ar[d] \\ f_!\mathcal{F} \ar[r]^{c'} & g_!f'_!\mathcal{F} \ar[r] & g_*f'_!\mathcal{F} }$
commutes where the left vertical arrow is Remark 62.4.6 and the right vertical arrow is Remark 62.3.5. The equality sign in the diagram comes about because $f'_1$ is the restriction of $f'$ to $Y'_1$ and our construction of $f'_!$ is local on the base. Finally, to prove the commutativity we choose an object $V$ of $Y_{\acute{e}tale}$ and a formal sum $s_1 = \sum (Z_{1, i}, s_{1, i})$ as in (62.4.0.1) defining an element of $f_{1, p!}\mathcal{F}|_{X_1}(V)$. Recall this means $Z_{1, i} \subset X_1 \times _ Y V$ is locally closed finite over $V$ and $s_{1, i} \in H_{Z_{1, i}}(\mathcal{F})$. Then we chase this section across the maps involved, but we only need to show we end up with the same element of $g_*f'_!\mathcal{F}(V) = f'_!\mathcal{F}(Y' \times _ Y V)$. Going around both sides of the diagram the reader immediately sees we end up with the element $\sum (Z_{1, i}, s_{1, i})$ where now $Z_{1, i}$ is viewed as a locally closed subscheme of $X \times _{Y'} (Y' \times _ Y V) = X \times _ Y V$ finite over $Y' \times _ Y V$.
Proof of (b). Let $b : Y_1 \to Y$ be a morphism of schemes. Let us form the commutative diagram
$\xymatrix{ X_1 \ar[d]_{f'_1} \ar[r]_ a \ar@/_2em/[dd]_{f_1} & X \ar[d]^{f'} \ar@/^2em/[dd]^ f \\ Y'_1 \ar[d]_{g_1} \ar[r]_{b'} & Y' \ar[d]^ g \\ Y_1 \ar[r]^ b & Y }$
with cartesian squares. We claim that our construction is compatible with the base change maps of Lemmas 62.4.10 and 62.3.12, i.e., that the top rectangle of the diagram
$\xymatrix{ b^{-1}f_!\mathcal{F} \ar[rr] \ar[d]_{b^{-1}c'} & & f_{1, !}a^{-1}\mathcal{F} \ar[d]^{c_1'} \\ b^{-1}g_!f'_!\mathcal{F} \ar[r] \ar[d] & g_{1, !}(b')^{-1}f'_!\mathcal{F} \ar[r] \ar[d] & g_{1, !}f'_{1, !}a^{-1}\mathcal{F} \ar[d] \\ b^{-1}g_*f'_!\mathcal{F} \ar[r] & g_{1, *}(b')^{-1}f'_!\mathcal{F} \ar[r] & g_{1, *}f'_{1, !}a^{-1}\mathcal{F} }$
commutes. The verification of this is completely routine and we urge the reader to skip it. Since the arrows going from the middle row down to the bottom row are injective, it suffices to show that the outer diagram commutes. To show this it suffices to take a local section of $b^{-1}f_!\mathcal{F}$ and show we end up with the same local section of $g_{1, *}f'_{1, !}a^{-1}\mathcal{F}$ going around either way. However, in fact it suffices to check this for local sections which are of the the pullback by $b$ of a section $s = \sum (Z_ i, s_ i)$ of $f_{p!}\mathcal{F}(V)$ as above (since such pullbacks generate the abelian sheaf $b^{-1}f_!\mathcal{F}$). Denote $V_1$, $V'_1$, and $Z_{1, i}$ the base change of $V$, $V' = Y' \times _ Y V$, $Z_ i$ by $Y_1 \to Y$. Recall that $Z_ i$ is a locally closed subscheme of $X_ V = X_{V'}$ and hence $Z_{1, i}$ is a locally closed subscheme of $(X_1)_{V_1} = (X_1)_{V'_1}$. Then $b^{-1}c'$ sends the pullback of $s$ to the pullback of the local section $c(s) \sum (Z_ i, s_ i)$ viewed as an element of $f'_{p!}\mathcal{F}(V') = (g_*f'_{p!}\mathcal{F})(V)$. The composition of the bottom two base change maps simply maps this to $\sum (Z_{i, 1}, s_{1, i})$ viewed as an element of $f'_{1, p!}a^{-1}\mathcal{F}(V'_1) = g_{1, *}f'_{1, p!}a^{-1}\mathcal{F}(V_1)$. On the other hand, the base change map at the top of the diagram sends the pullback of $s$ to $\sum (Z_{1, i}, s_{1, i})$ viewed as an element of $f_{1, !}a^{-1}\mathcal{F}(V_1)$. Then finally $c'_1$ by its very construction does indeed map this to $\sum (Z_{i, 1}, s_{1, i})$ viewed as an element of $f'_{1, p!}a^{-1}\mathcal{F}(V'_1) = g_{1, *}f'_{1, p!}a^{-1}\mathcal{F}(V_1)$ and the commutativity has been verified.
Proof of (c). This follows from comparing the definitions for both maps; we omit the details.
To finish the proof it suffices to show that the pullback of $c'$ via any geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ is an isomorphism. Namely, pulling back by $\overline{y}$ is the same thing as taking stalks and $\overline{y}$ (Étale Cohomology, Remark 59.56.6) and hence we can invoke Étale Cohomology, Theorem 59.29.10. By the compatibility (b) just shown, we conclude that we may assume $Y$ is the spectrum of $k$ and we have to show that $c'$ is an isomorphism. To do this it suffices to show that the induced map
$\bigoplus \nolimits _{x \in X} \mathcal{F}_ x = H^0(Y, f_!\mathcal{F}) \longrightarrow H^0(Y, g_!f'_!\mathcal{F}) = H^0_ c(Y', f'_!\mathcal{F})$
is an isomorphism. The equalities hold by Lemmas 62.4.5 and 62.3.11. Recall that $X$ is a disjoint union of spectra of Artinian local rings with residue field $k$, see Varieties, Lemma 33.20.2. Since the left and right hand side commute with direct sums (details omitted) we may assume that $\mathcal{F}$ is a skyscraper sheaf $x_*A$ supported at some $x \in X$. Then $f'_!\mathcal{F}$ is the skyscraper sheaf at the image $y'$ of $x$ in $Y$ by Lemma 62.4.5. In this case it is obvious that our construction produces the identity map $A \to H^0_ c(Y', y'_*A) = A$ as desired. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2023-01-28 16:45:19
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https://kbwiki.ercoftac.org/w/index.php?title=Description_AC2-07&oldid=32535
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# Description AC2-07
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
# Confined double annular jet
## Introduction
The Application Challenge TA 2 - AC 7, "Confined Double Annular Jet" proposed by the VUB, Belgium, is an axisymmetric double annular flow field generated by a burner, and discharging into a confined combustion chamber (see Fig. 1 below). The flow is studied in cold conditions, and in a nozzle region going from the nozzle to 1.5 diameters downstream. The resulting mean flow is axisymmetric ; it can be qualified as a complex turbulent flow. The flow possesses a central vortex and two small toroidal contra rotating vortices. There are stagnation points and lines, mixing regions and recirculation regions (see Fig. 2 below). This is thus a good AC, since there is a relatively simple geometry (axisymmetric), and the ability of CFD codes to reproduce several properties of complex flows can be tested.
Figure 1: An overall display of the facility. Inlet arrangement, a cut on the burner, and the combustion chamber are shown. The nozzle region corresponding to the measurements is indicated.
The available experiments are the following:
Experimental data #1: Air flow with a 156 mm largest diameter jet, with a maximum velocity of 6.3 m/s, corresponding to a Reynolds number of 6 104. For this flow, 2D LDV data have been recorded, providing axial and radial mean velocities, axial and radial turbulence intensities and Reynolds stress. The data are provided on a dense experimental grid composed of 36 sections, each of which possessing 100 to 200 measurement points.
Experimental data #2: The flow field, generated in the combustion chamber model by water flow through the model, is measured with a 2D2C PIV system. A total of 271 instantaneous vector fields, each containing 5040 vectors in the near nozzle region, are presented. The flow rate through the burner was 4.09x10-5 m3/s giving a Reynolds number 180, based on nozzle opening dr (2 mm) and mean exit velocity Ue (0.09 m/s). This produces a central toroidal vortex in the transitional state.
## Relevance to Industrial Sector
Industrial devices in fluid engineering quite often involve complex turbulent flows. This is the case for example in turbine engines, industrial furnaces, combustors and burners. Industrial burners are designed to generate stationary combustion in a confined chamber, with desired values of velocity, temperature and species concentrations at the exit of the combustion chamber. In general, they consist of several nozzles arranged around an axis in a confined space. The fuel and the oxidizer (usually air) issue out of the nozzles and these jets mix by turbulent diffusion. Burners are often installed to destroy pollutants gases resulting from industrial activities, before releasing them in the atmosphere.
Since international norms in matter of pollution are getting more and more binding, it is important to be able to modify the burner in order to minimize the emission of pollutant species. Since burners are usually designed and modified empirically by the manufacturer, the modifications cannot be optimal. It is therefore important to be able to better understand the flow field and the combustion process associated to burners. The turbulent diffusion of species is faster than the combustion process, so that the first step to understand a burner is to study the flow field in cold conditions. It is essential to understand the behavior of the various jets and their interaction with the neighboring jets and surrounding flows to successfully predict the performance of the device.
The test case provided here contains a complete database in axisymmetric conditions, providing the mean velocity field, turbulence intensity, Reynolds stress components, corresponding to 2 components of the mean velocity and 4 non-zero components of the Reynolds stress tensor. This test-case can thus be used to test the validity of CFD to reproduce the complex flow created by confined annular jets. For the nozzle region (up to 1.5 diameters) this AC is well understood, in terms of data available and overall quality (for experimental data #1, globally less than 2% axisymmetric errors).
## Design or Assessment Parameters
The position of some particular points can be used as assessment parameters to judge the validity of CFD computations (more specific comparisons can be done later). Several locations have been selected below, as shown in Figure 2.
Figure 2: Streamlines of the flow, and designations of specific position points (DOAP).
## Flow Geometry
Since the L/D of the channels of the burner are large, the fully-developed flows dominate over any influence from the inlet geometries. Therefore in this AC the geometry is determined only by dimensions of the exit of the burner and of the combustion chamber: see Table 1 and Figure 3 below.
Orientation of axis and sign conventions: since the geometry is axisymmetric, in the following a cylindrical coordinate system is considered (x,z,q), where x is the axial coordinate (the distance from the exit of the burner, along its axis), z is the radial coordinate (the distance from the axis) and q the tangential coordinate, not considered here due to axisymmetry.
Table 1 Definition of Some Geometric Parameters
Parameter Description
r1 radius of the primary jet
r2 radius of the secondary jet
dr width of annular jets
rc radius of the combustion chamber
${\displaystyle \beta ={\frac {r_{2}}{r_{1}}}}$ secondary to primary radius ratio
${\displaystyle {Y}={\frac {A_{e}}{A_{1}}}={\frac {r_{c}^{2}}{2r_{1}dr}}}$ exit area to primary area ratio
Figure 3: The diameters of the boundaries of the streams produced by the burner.
## Flow Physics and Fluid Dynamics Data
The mean flow is axisymmetric. Only the flow in the nozzle region has been recorded. For this region, the jet presents a complex flow: it possesses recirculation regions, several toroidal vortices, a stagnation point and stagnation lines. First the primary and secondary streams merge, creating a vortex bubble between them. This bubble corresponds to a pair of contrarotating toroidal vortices. Then the simple annular stream becomes a central jet, creating a big central vortex bubble. This central vortex is less stable than the toroidal one (see the streamlines in Figure 2).
The databases correspond to a turbulent, incompressible, isothermal flow with recirculation.
The following tables provide different informations characterizing the flow: The Problem Definition Parameters (PDP) are dimensional quantities characterizing the experiment, and Governing Non-Dimensional Parameters (GNDP) are non-dimensional numbers governing and characterizing the physics of the flow.
First, some useful fluid dynamics parameters are presented in Table 2. The PDP introduced here characterize the geometry of the burner, and determine the inflow (through the velocity ratio of primary to secondary bulk velocity). They are also presented in Table 2.
The GNDP are the Reynolds number and the Craya-Curtet numbers, presented in Table 2. The Reynolds number is defined using the bulk-averaged velocity of the secondary jet and its diameter. The Craya-Curtet number characterizes the recirculation of confined jet flows. It was shown experimentally for some simple annular jet flows that the flow and mixing properties of the classical axisymmetric confined jet depend only on the Craya-Curtet number. It is assumed here that this number is also governing the physics of the double annular jet flow. The Craya-Curtet number can be defined as a normalization of the stagnation pressure loss in the system:
${\displaystyle {\frac {g\Delta {P}^{\star }}{U_{e}^{2}\rho }}={\frac {1}{C_{t}^{2}}}\qquad \qquad \qquad \qquad (1)}$
where the different parameters in this equation are defined in Table 2. A small value of the Craya-Curtet number corresponds to large static pressure loss, indicating recirculation.
Table 2 Different Parameters of the AC
Parameter Symbol Definition
Fluid Dynamic Parameters U1 Bulk average velocity of the primary jet
U2 Bulk average velocity of the secondary jet
Ue Bulk average velocity of the average stream
${\displaystyle \Delta P^{\star }=P_{e}^{\star }-P_{i}^{\star }}$ Stagnation pressure loss in the system (difference between exit and entrance stagnation pressure)
ρ Density
Problem Definition Parameters (PDPs) ${\displaystyle \alpha ={\frac {U_{2}}{U_{1}}}}$ Secondary to primary bulk-average velocity ratio
${\displaystyle \beta ={\frac {r_{2}}{r_{1}}}}$ Secondary to primary radius ratio
${\displaystyle \gamma ={\frac {A_{e}}{A_{1}}}={\frac {r_{c}^{2}}{2r_{1}dr}}}$ Exit area to primary area ratio
Governing Non-Dimensional Parameters (GNDPs) ${\displaystyle {\text{Re}}={\frac {2U_{2}r_{2}}{\nu }}}$ Reynolds Number
${\displaystyle C_{t}={\frac {1+\beta }{\sqrt {\gamma (1+a^{2}\beta )}}}}$ Craya-Curtet number
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2023-03-24 22:38:35
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http://blog.geomblog.org/2014/05/
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## Tuesday, May 20, 2014
### On beauty and truth in science.
Philip Ball writes a thought-provoking article in Aeon with the thesis that the kind of beauty that scientists describe does not necessarily match the aesthetic notions of art, and is not even consistent among scientists.
It was hard for me to get beyond the casual conflating of beauty in mathematics (the four-color theorem, the proof of Fermat's theorem, and proofs in general) and beauty in scientific theories (relativity, evolution, and so on). But if one goes beyond the artificial duality constructed by the author, the idea of beauty as a driver in science (and mathematics) is a rich one to explore.
A particular example: for a long time (and even codified in books) it was taught that there were five natural classes of approximation hardness: PTAS, constant factor-hard, log-hard, label-cover (superlogarithmic) hard, and near-linear hard. There were even canonical members of each class.
Of course, this nice classification no longer exists. There are even problems that are $\log^* n$-hard to approximate, and can also be approximated to that factor. And to be fair, I'm not sure how strong the belief was to begin with.
But it was such a beautiful idea.
At least in mathematics, the search for the beautiful result can be quite fruitful. It spurs us on to find better, simpler proofs, or even new ideas that connect many different proofs together. That notion of connection doesn't appear to be captured in the article above: that beauty can arise from the way a concept ties disparate areas together.
### MADALGO Summer School on Learning At Scale
I'm pleased to announce that this year's MADALGO summer school (continuing a long line of summer programs on various topics in TCS) will be on algorithms and learning. The formal announcement is below, and registration information will be posted shortly.
Save the date ! Aug 11-14, 2014.
LEARNING AT SCALE
August 11- 14, 2014, Aarhus University, Denmark
OVERVIEW AND GOAL
The MADALGO Summer School 2014 will introduce attendees to the latest developments in learning at scale. The topics will include high dimensional inference, algorithmic perspectives on learning and optimization, and challenges in learning with huge data.
LECTURES
The school will be taught by experts in learning:
• Mikhail Belkin (Ohio State)
• Stefanie Jegelka (Berkeley)
• Ankur Moitra (MIT)
PARTICIPATION
The summer school will take place on August 11-14, 2014 at Center for Massive Data Algorithmics (MADALGO) at the Department of Computer Science, Aarhus University, Denmark. The school is targeted at graduate students, as well as researchers interested in an in-depth introduction to Learning. Registration will open soon at the school webpage. Registration is free on a first-come-first serve basis - handouts, coffee breaks, lunches and a dinner will be provided by MADALGO and Aarhus University.
ORGANIZING COMMITTEE
• Suresh Venkatasubramanian (University of Utah)
• Peyman Afshani (MADALGO, Aarhus University)
• Lars Arge (MADALGO, Aarhus University)
• Gerth S. Brodal (MADALGO, Aarhus University)
• Kasper Green Larsen (MADALGO, Aarhus University)
LOCAL ARRANGEMENTS
• Trine Ji Holmgaard (MADALGO, Aarhus University)
• Katrine Østergaard Rasmussen (MADALGO, Aarhus University)
Center for Massive Data Algorithmics is a major basic research center funded by the Danish National Research Foundation. The center is located at the Department of Computer Science, Aarhus University, Denmark, but also includes researchers at CSAIL, Massachusetts Institute of Technology in the US, and at the Max Planck Institute for Informatics and at Frankfurt University in Germany. The center covers all areas of the design, analysis and implementation of algorithms and data structures for processing massive data (interpreted broadly to cover computations where data is large compared to the computational resources), but with a main focus on I/O-efficient, cache-oblivious and data stream algorithms.
## Thursday, May 01, 2014
### The history of the vector space model
Gerald Salton is generally credited with the invention of the vector space model: the idea that we could represent a document as a vector of keywords and use things like cosine similarity and dimensionality reduction to compare documents and represent them.
But the path to this modern interpretation was a lot twistier than one might think. David Dubin wrote an article in 2004 titled 'The Most Influential Paper Gerard Salton Never Wrote'. In it, he points out that most citations that refer to the vector space model refer to a paper that doesn't actually exist (hence the title). Taking that as a starting point, he then traces the lineage of the ideas in Salton's work.
The discoveries he makes are quite interesting. Among them,
• Salton's original conception of the vector space model was "operational" rather than mathematical. In other words, his earliest work really uses 'vector space' to describe a collection of tuples, each representing a document. In fact, the earliest terminology used was the 'vector processing model'.
• In later papers, he did talk about things like orthogonality and independence, as well as taking cosines for similarity, but this was done in an intuitive, rather than formal manner.
• It was only after a series of critiques in the mid 80s that researchers (Salton included) started being more direct in their use of the vector space model, with all its attendant algebraic properties.
Of course today the vector space model is one of the first things we learn when doing any kind of data analysis. But it's interesting to see that it didn't start as this obvious mathematical representation (that I've taken to calling the reverse Descartes trick).
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2019-05-24 17:50:37
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http://www-h.eng.cam.ac.uk/help/tpl/textprocessing/latex_maths+pix/latex_maths+pix.html
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# LATEX maths and graphics
### May 11, 2012
Note that there's an alternative way of producing maths in LATEX - AMS-LATEX. See the for details.
If you want to more more about graphics, see by Keith Reckdahl.
Comments and bug reports to Tim Love (tl136).
Copyright © 2004-2012 by T.P. Love. This document may be copied freely for the purposes of education and non-commercial research. Cambridge University Engineering Department, Cambridge CB2 1PZ, England.
# Contents
1 Maths
1.1 Environments
1.2 Special Characters
1.2.1 Greek
1.2.2 Miscellaneous
1.2.3 Arrows
1.2.4 Calligraphic
1.2.5 Character Modifiers
1.2.6 Common functions
1.3 Subscripts and superscripts
1.4 Overlining, underlining and bold characters
1.5 Roots and Fractions
1.6 Delimiters
1.7 Numbering and labelling
1.8 Matrices
1.9 Macros
1.10 Packages
1.11 Fine tuning
1.12 Maths and Postscript fonts
1.13 Matlab and LaTeX
1.14 Examples
2 Graphics
2.0.1 psfrag: adding maths to postscript files
2.1 Scaling, rotation, clipping, wrap-around and shadows
## 1 Maths
There's more to maths typesetting than meets the eye. Many conventions used in the typesetting of plain text are inappropriate to maths. LATEX goes a long way to help you along with the style. For example, in a LATEX maths environment, letters come out in italics, -' as −' (minus) instead of the usual -' (dash), *' becomes *, ' becomes ′ and spacing is changed (less around /', more around +').
Many of the usual LATEX constructions can still be used in maths environments but their effect may be slightly different; eg \textbf{ } only affects letters and numbers. {' and }' are still special characters; they're used to group characters.
As usual in LATEX you can override the defaults, but think before doing it: maths support in LATEX has been carefully thought out and is quite logical though the LATEX source text may not be very readable. It's a good idea to write out the formulae on paper before you start LATEXing, and try not to overdo the use of the \frac' construction; use /' instead.
### 1.1 Environments
There are 2 environments to display one-line equations.
equation:-
Equations in this environment are numbered.
$$x + iy$$
x + iy
(1)
displaymath:-
These won't be numbered. $,$ can be used as abbreviations for \begin{displaymath} and \end{displaymath}.
\begin{displaymath}
x + iy
\end{displaymath}
x + iy
Never leave a blank line before these equations; it starts a new paragraph and looks ugly. '\displaystyle' is the font type used to print maths in these display environments. Other relevant environments are:-
math:-
For use in text. $$and$$ can be used to delimit the environment, as can the TEX constructions $and$ . For example, $x=y^2$ gives x=y2.
eqnarray:-
This is like a 3 column tabular environment. Each line by default is numbered. You can use the eqnarray* variant to suppress numbering altogether.
\begin{eqnarray}
a1 & = & b1 + c1\nonumber\\
a2 & = & b2 - c2
\end{eqnarray}
a1
=
b1 + c1
a2
=
b2 − c2
(2)
Maths in "display" and "inline" environments have different default sizes for some characters and other behavioural differences so that a line of maths won't impinge on text lines below or above. If you want to put some non-maths text in amongst maths then enclose it in an \mbox{...}.
### 1.2 Special Characters
The The Comprehensive LATEX Symbol List document offers 105 pages of symbols. Here are just a few.
#### 1.2.1 Greek
α \alpha β \beta γ \gamma δ \delta ϵ \epsilon ζ \zeta η \eta θ \theta ι \iota κ \kappa λ \lambda μ \mu ν \nu ξ \xi o o π \pi ρ \rho σ \sigma τ \tau υ \upsilon ϕ \phi χ \chi ψ \psi ω \omega Γ \Gamma ∆ \Delta Θ \Theta Λ \Lambda Ξ \Xi Π \Pi Σ \Sigma Υ \Upsilon Φ \Phi Ψ \Psi Ω \Omega
#### 1.2.2 Miscellaneous
… \ldots … \cdots : \vdots ··· \ddots ± \pm ± \mp × \times ÷ \div ∗ \ast ∗ \star ° \circ • \bullet · \cdot ∩ \cap ∩ \bigcap ∪ \cup ∪ \bigcup \uplus \biguplus \sqcap \sqcup \bigsqcup ∨ \vee ∨ \bigvee ∧ \wedge ∧ \bigwedge \ \setminus \wr ◊ \diamond \bigtriangleup \bigtriangledown \triangleleft \triangleright ⊕ \oplus ⊕ \bigoplus \ominus ⊗ \otimes ⊗ \bigotimes ∅ \oslash \odot \bigodot \bigcirc \amalg ≤ \leq \prec \preceq << \ll ⊂ \subset ⊆ \subseteq \sqsubseteq ∈ \in \vdash ≥ \geq \succ \succeq >> \gg ⊃ \supset ⊇ \supseteq \sqsupseteq ∋ \ni \dashv ≡ \equiv ∼ \sim ≅ \simeq \asymp ≈ \approx ≅ \cong ≠ \neq \doteq ∝ \propto \models ⊥ \perp | \mid || \parallel \bowtie \smile \frown ℵ \aleph ħ \hbar ι \imath j \jmath l \ell ℘ \wp ℜ \Re ℑ \Im ′ \prime \empty ∇ \nabla √ \surd T \top ⊥ \bot || \| ∠ \angle ∀ \forall ∃ \exists ¬ \neg \flat \natural \sharp \ \backslash ∂ \partial ∞ \infty \triangle ∑ \sum ∏ \prod \coprod ∫ \int (∫) \oint
#### 1.2.3 Arrows
← \leftarrow ⇐ \Leftarrow → \rightarrow ⇒ \Rightarrow ↔ \leftrightarrow ⇔ \Leftrightarrow → \mapsto \hookleftarrow \leftharpoonup \leftharpoondown \rightleftharpoons ← \longleftarrow ⇐ \Longleftarrow → \longrightarrow ⇒ \Longrightarrow ↔ \longleftrightarrow ⇔ \Longleftrightarrow → \longmapsto \hookrightarrow \rightharpoonup \rightharpoondown ↑ \uparrow ⇑ \Uparrow ↓ \downarrow ⇓ \Downarrow \updownarrow \nearrow \searrow \swarrow \nwarrow
#### 1.2.4 Calligraphic
These characters are available if you use the \mathcal control sequence.
${\mathcal A B C D E F G H I J K L M N O P Q R S T U V W X Y Z}$
gives 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
Also useful are "blackboard" style characters. \mathbb{ R Q Z} gives R Q Z (requires the amsfonts package).
#### 1.2.5 Character Modifiers
\hat{e} \widehat{easy} \tilde{e} \widetilde{easy} \check{e} \breve{e} \acute{e} é \grave{e} è \bar{e} \vec{e} \dot{e} \ddot{e} \not e
Note that the wide versions of hat and tilde cannot produce very wide alternatives. The \not' operator hasn't properly cut the following letter. The Fine Tuning section on page pageref describes how to adjust this.
If you want to place one character above another, you can use \stackrel, which prints its first argument in small type immediately above the second
$a \stackrel{def}{=} b + c$
gives
See the Macros section for how to stack characters using atop.
#### 1.2.6 Common functions
In a maths environment, LATEX assumes that variables will have single-character names. Function names require special treatment. The advantage of using the following control sequences for common functions is that the text will not be put in math italic and subscripts/superscripts will be made into limits where appropriate.
\arccos \arcsin \arctan \arg \cos \cosh \cot \coth \csc \deg \det \dim \exp \gcd \hom \inf \ker \lg \lim \liminf \ln \log \max \min \Pr sec \sin \sinh \sup \tan \tanh
### 1.3 Subscripts and superscripts
These are introduced by the ^' and _' characters. Depending on the base character and the current style, the sub- or superscripts may go to the right of or directly above/below the main character. With letters it goes to the right.
$F_2^3$
produces F23'. Note that the sub- and superscripts aren't in line. To make them so, you can add an invisible character after the F'. $F{}_2^3$ produces F23.
With ∑ the default behaviour is different in display and text styles.
$\sum_{i=0}^2$
produces ∑i=02 (text style) but
$\sum_{i=0}^2$
produces (in display style)
2∑ i=0
This default behaviour can be overridden, if you really need to. For example in text mode,
$\sum\limits_{i=0}^2$
produces ∑i=02
### 1.4 Overlining, underlining and bold characters
$\underline{one} \overline{two}$
produces onetwo. This is not a useful facility if it's used more than once on a line. The lines are produced so that they don't quite overlap the text; lines over or under different words won't in general be at the same height.
To be able to reproduce bold maths, it's best to use the bm package. $E = \bm{mc^2}$ produces E = mc2.
Alternatively, you can use \mathbf{} to create bold characters - $\mathbf{F}_2^3$ produces F23. or you can use the following idea
\usepackage{amsbsy} % This loads amstext too
\begin{document}
$\omega + \boldsymbol{\omega}$
% Use the following if whole expressions need to be in bold
{\boldmath $\omega$}
\end{document}
### 1.5 Roots and Fractions
$\sqrt{4} + \sqrt[3]{x + y}$
gives √4 + 3√{x + y}.
Three constructions for putting expressions above others are
frac:-
$\frac{1}{(x + 3)}$ produces .
choose:-
${n + 1 \choose 3}$ produces .
atop:-
${x \atop y}$ produces .
These constructions can be used with ones described earlier. E.g.,
$\sum_{-1\le i \le 1 \atop 0 < j < \infty} f(i,j)$
gives
∑ [(−1 ≤ i ≤ 1) || (0 < j < ∞)] f(i,j)
### 1.6 Delimiters
these are made by these and these are made by these ( ( ) ) [ [ ] ] { \{ } \} ⎣ \lfloor ⎦ \rfloor ⎡ \lceil ⎤ \rceil 〈 \langle 〉 \rangle / / \ \backslash | | || \| ↑ \uparrow ⇑ \Uparrow ↓ \downarrow ⇓ \Downarrow \updownarrow \Updownarrow
This table shows the standard sizes. To get bigger sizes, use these prefices
(for left delimiters) (for right delimiters) magnification \bigl \bigr a bit bigger, but won't overlap lines \Bigl \Bigr 150% times big \biggl \biggr 200% times big \Biggl \Biggr 250% times big
For example,
$\Biggl\{2\Bigl(x(3+y)\Bigr)\Biggr\}$
gives . If you're not using the default text size these commands might not work correctly. In that case try the exscale package.
It's preferable to let LATEX choose the delimiter size for you by using \left and \right. These will produce delimiters just big enough for the formulae inbetween.
$\left( \frac{(x+iy)}{\{\int x\}} \right)$
gives
The left and right delimiters needn't be the same type. It's sometimes useful to make one of them invisible
$z = \left\{ \begin{array}{ll} 1 & (x>0)\\ 0 & (x<0) \end{array} \right.$
produces Over- and underbracing works too.
$\overbrace{\alpha \ldots \omega}^{\mbox{greek}} \underbrace{a \ldots z}_{\mbox{english}}$
produces
### 1.7 Numbering and labelling
Numbering happening automatically when you display equations. If you don't want an equation numbered, use \nonumber beside the equation. Equation numbers appear to the right of the maths by default. To make them appear on the left use the leqno class option (i.e., use \documentclass[leqno,....]{....}).
Use \label{} to label an equation (or figure, section etc) in order to reference from elsewhere.
$$W_{\bf S}(t,\omega) = \int\limits_{-\infty}^{\infty} { {\cal R}_{\bf S}(t,\tau) e^{-j\omega\tau} \,d \tau } \label{LABELLING}$$
Now the following text
refers back to equation \ref{LABELLING}
refers back to equation 3 by number, and
refers back to the equation on page \pageref{LABELLING}
refers back to the equation on page pageref.
A file will have to be LATEX'ed twice before the references, both forwards and backwards, will be correctly produced.
### 1.8 Matrices
The array environment is like LATEX's tabular environment except that each element is in math mode. The number and alignment of columns is controlled by the arguments - use l, c or r to represent each column with either left, center or right alignment. The default font style used is \textstyle but you can override this by changing the \displaystyle.
\begin{math}
\begin{array}{clrr} %
a+b+c & uv & x-y & 27 \\
x+y & w & +z & 363
\end{array}
\end{math}
produces
a+b+c
uv
x−y
27
x+y
w
+z
363
The rows are arranged so that their centres are aligned. You can align their tops or bottoms instead by using a further argument when you create the array.
\begin{array}{clrr}[t]
would produce top-aligned lines, and [b]' would produce bottom-aligned ones. The Delimiters section of this document shows how to bracket matrices.
TEX has a few maths facilities not mentioned in the LATEX book. The following TEX construction might be useful.
\begin{math}
\bordermatrix{&a_1&a_2&...&a_n\cr
b_1 & 1.2 & 3.3 & 5.1 & 2.8 \cr
c_1 & 4.7 & 7.8 & 2.4 & 1.9 \cr
... & ... & ... & ... & ... \cr
z_1 & 8.0 & 9.9 & 0.9 & 9.99 \cr}
\end{math}
### 1.9 Macros
These aid readability, save on repetitive typing and offer ways of producing stylistic variations on standard LATEX formats.
\def\bydefn{\stackrel{def}{=}}
\def\convf{\hbox{\space \raise-2mm\hbox{$\textstyle \bigotimes \atop \scriptstyle \omega$} \space}}
produce =def and [( ⊗) || ( ω)]_endtextstyle when $\bydefn$ and $\convf$ are typed.
### 1.10 Packages
The following packages may be of help - (easy block matrices), (easy equations), (easy matrices), (easy tables), (easy vectors), (nested arrays), (gives more choice in theorem layout), subeqnarray (Renumbering of sub-arrays in math-mode), subeqn (Different numbering sub-arrays).
### 1.11 Fine tuning
It's generally a good idea to keep punctuation outside math mode; LATEX's normal handling of spacing around punctuation is suspended during maths. Sometimes you might want to adjust the spacing in a formula (e.g., you might want to add space before dx). Use these symbols :-
a\, b (a b) thin space a\> b (a b) medium space a\; b (a b) thick space a\! b (a b) negative thin space
Long math expressions aren't broken automatically unless you use the package, which is still a little experimental. In an eqnarray environment you may want to break a long line manually. You can do this by putting
y & = & a + b \nonumber \\
& & + k
to give
y
=
a + b
+ k
(4)
but the spacing around the +' on the 2nd line is wrong because LATEX thinks it's a unary operator. You can fool LATEX into treating it as a binary operator by inserting a hidden character.
y & = & a + b \nonumber \\
& & \mbox{} + k
gives
y
=
a + b
+ k
(5)
You can use the \lefteqn construction to format long expressions so that continuation lines are differently indented.
\begin{eqnarray}
\lefteqn{x+ iy=}\\
& & a + b + c + d + e + f + g + h + i + j + k +\nonumber\\
& & l + m \nonumber
\end{eqnarray}
x+ iy=
(6)
a + b + c + d + e + f + g + h + i + j + k +
l + m
If you want more vertical spacing around a line you can create an invisible vertical ßtruct" in LaTeX. creates a box of width 0, height 1cm which starts .3cm below the usual line base, use \rule[-.3cm]{0cm}{1cm}. By adjusting these values you should be able to create as much extra space below/above the maths as you like. "[A/B] and" is created by
$A \over B$ \rule[-.3cm]{0cm}{1cm}{and}
### 1.12 Maths and Postscript fonts
It's easy to use a postscript font (like helvetica) for the text of a LATEX document. What's harder is using the same font for maths - the font will lack many of the special characters required for maths. An easy, reasonable option is to use the mathptmx package to put the maths into the postscript Times and symbol fonts where possible.
Alternatively, use
• the mathpazo package (loads Palatino as the text font family and a mixture of the Pazo and CM fonts for math).
• the mathpple package (loads Palatino as the text font family and a mixture of artificially obliqued Euler fonts and CM fonts for math).
Commercial and free alternatives are under development. See Stephen G. Hartke's for details.
### 1.13 Matlab and LaTeX
has some support for LaTeX production. For example
latex('(sin(x)+2*x+3*x^2)/(5*x+6*x^2)','math.tex')
puts the LaTeX representation of the expression into a file called math.tex. Type "help latex" inside matlab for details.
### 1.14 Examples
• \begin{eqnarray}
{\cal M}^2(\hat{\theta},\theta) &=& E[(\hat{\theta} - \theta)^2]
\nonumber \\
{\cal M}^2(\hat{\theta},\theta) &=& {\rm var}^2(\hat{\theta}) +
{\cal B}^2(\hat{\theta}).
\end{eqnarray}
gives
M2( ^ θ ,θ)
=
E[( ^ θ − θ)2]
M2( ^ θ ,θ)
=
var2( ^ θ ) + B2( ^ θ ).
(7)
• $$\hat{W}_{s}(t,\omega;\phi) \bydefn \int\limits_{-\infty}^{\infty} {\hat{\cal R}_s(t,\tau;\psi) e^{-j\omega \tau} \, d \tau }$$
gives
^ W s (t,ω;ϕ) def= ∞⌠⌡−∞ ^ ℜ s (t,τ;ψ)e−jωτ d τ
(8)
• \begin{eqnarray}
{\cal B}(t,\omega) & \approx &
{1 \over 4\pi}
{\cal D}_t^2 W_{\bf S}(t, \omega)
{{{\scriptstyle \infty} \atop
{\displaystyle \int \! \int \!
}}\atop {\scriptstyle -\infty}}
t_1^2
\phi(t_1,\omega_1) \, dt_1 d\omega_1
\nonumber \\
&& +
{1 \over 4\pi}
{\cal D}_\omega^2 W_{\bf S}(t, \omega)
{{{\scriptstyle \infty} \atop
{\displaystyle \int \! \int \!
}}\atop {\scriptstyle -\infty}}
\omega_1^2
\phi(t_1,\omega_1) \, dt_1 \, d\omega_1.
\label{F4}
\end{eqnarray}
gives
B(t,ω)
≈
1
Dt2 WS(t, ω)
⌠⌡ ⌠⌡
−∞
t12ϕ(t11) dt11
+ 1
Dω2 WS(t, ω)
⌠⌡ ⌠⌡
−∞
ω12ϕ(t11) dt1 dω1.
(9)
• \newsavebox{\DERIVBOXZLM}
\savebox{\DERIVBOXZLM}[2.5em]{$\Longrightarrow\hspace{-1.5em} \raisebox{.2ex}{*} \hspace{-.7em}\raisebox{-.8ex}{\scriptsize lm}\hspace{.7em}$}
\newcommand{\Deriveszlm}{\usebox{\DERIVBOXZLM}}
\Deriveszlm
gives
⇒*lm
## 2 Graphics
Graphics can be produced
• from within LaTeX - pictures can be drawn in a picture environment, but you'll find graph paper handy (xfig can create code for the picture environment). The pstricks packages is far more powerful.
• by any program that can produce Postscript files (for LaTeX) or JPEG, PNG andPDF (for pdflatex).
Whatever graphics you want to add, you should use the figure environment so that LATEX can cope sensibly with situations where, for example, you attempt to insert near the bottom of a page a graphic that's half a page high. The figure environment will float the graphic to the top or bottom of the page, or on the next page, or here (where you asked for it).
h here t top of page b bottom of page p on a page with no text
The order you put these letters in doesn't matter, but whether you include them does. Putting ! as the first argument in the square brackets will encourage LATEX to do what you say, even if the result's sub-optimal. Below is a simple (and usually sufficient) usage. See the online hints for further details.
\begin{figure}[htbp]
\vspace{0.5in}
\caption{0.5 inch of space}
\end{figure}
Figure 1: 0.5 inch of space
It's possible to have more than one graphic in a figure. See the example later on.
pdflatex supports JPEG, PNG andPDF images - but not postscript. latex supports Postscript files as long as they have a proper bounding box comment; i.e. LATEX requires full
#### 2.0.1 psfrag: adding maths to postscript files
Many packages that produce postscript output don't provide good maths facilities. It's often easier to add the maths in later using the psfrag package. This lets you replace text in a postscript file (produced with xfig, matlab, etc) by a fragment of LATEX. For example, doing
\usepackage{psfrag}
...
\begin{figure}
\psfrag{MATHS}{$x^2$}
\includegraphics{foo.eps}
\end{figure}
would display the file with MATHS replaced by x2. See the psfrag documentation for details.
### 2.1 Scaling, rotation, clipping, wrap-around and shadows
The graphicx package includes routines that are useful even without graphics. \reflectbox{Reflect} produces and \resizebox{3cm}{0.2cm}{Stretched} produces . To scale imported graphics, use some optional arguments
\includegraphics[width=5cm,height=10cm]{yourfile.jpg}
would rescale the postscript so that it was 5cm wide and 10cm high. To make the picture 5cm wide and scale the height in proportion use
\includegraphics[width=5cm]{yourfile.png}
To rotate anticlockwise by the specified number of degrees, use
\includegraphics[angle=150]{yourfile.pdf}
These options can be combined - note that order matters. The following examples demonstrate how to combine these features and how to use the subfig package to have more than one graphic in a figure.
Figure 2: Tigers
\centering
\begin{figure}[hbtp]
\includegraphics[height=40mm]{tiger.jpg}
\includegraphics[angle=120, height=20mm]{tiger.jpg}
\caption{Tigers}
\end{figure}
% remember to do \usepackage{subfig} at the top of the document!
\begin{figure}[hbtp]
\centering
\subfloat[Medium]
\subfloat[Large]
{\includegraphics[height=50mm]{tiger.jpg}}
\caption{3 crests}
\end{figure}
Figure
Figure 3: 3 crests
To clip images use the viewpoint argument. The following fragment would display only part of the image. The viewport coordinates are in the same units as the bounding box.
\begin{figure}[htbp]
\includegraphics[viewport=200 400 400 600,width=5cm,clip]%
{tiger.jpg}
\end{figure}
% Use the floatflt package
\begin{floatingfigure}[l]{4cm}
\includegraphics[width=2cm]{tiger.jpg}
\caption{Using floatingfigure}
\end{floatingfigure}
Figure 4: Using floatingfigure
The package lets you insert a graphic and have the text wrap around it. You can provide 2 arguments to the floatingfigure command: the first (l or r) selects whether you want the graphic to be on the left or right of the page. The 2nd argument gives the width of the graphic. Not all text will flow perfectly around (for example, verbatim text fails, as illustrated below) so check the final output carefully.
Using the fancybox package gives you access to \shadowbox, \ovalbox, \Ovalbox and \doublebox commands, which can be used with text or with graphics. For example, \shadowbox{shadow package} produces and
\ovalbox{\includegraphics[height=10mm]{crest.jpg}}
`
produces . Unfortunately, the fancybox package as supplied suppresses the table of contents. The locally produced contentsfancybox solves this, but may introduce graphics problems.
File translated from TEX by TTH, version 4.03.
On 11 May 2012, 11:47.
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2014-12-20 00:05:22
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https://docs.microej.com/en/latest/KernelDeveloperGuide/featuresCommunication.html
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# Communication between Features¶
Features can communicate together through the use of shared interfaces. The mechanism is described in Chapter Shared Interfaces of the Application Developer’s Guide.
## Kernel Type Converters¶
The shared interface mechanism allows to transfer an object instance of a Kernel type from one Feature to an other. To do that, the Kernel must register a new converter (See Kernel.addConverter() method).
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2021-08-05 10:48:10
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https://projecteuclid.org/euclid.aos/1458245724
|
## The Annals of Statistics
### Estimating sparse precision matrix: Optimal rates of convergence and adaptive estimation
#### Abstract
Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained $\ell_{1}$ minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces. The estimator, called ACLIME, is easy to implement and performs well numerically.
A major step in establishing the minimax rate of convergence is the derivation of a rate-sharp lower bound. A “two-directional” lower bound technique is applied to obtain the minimax lower bound. The upper and lower bounds together yield the optimal rates of convergence for sparse precision matrix estimation and show that the ACLIME estimator is adaptively minimax rate optimal for a collection of parameter spaces and a range of matrix norm losses simultaneously.
#### Article information
Source
Ann. Statist. Volume 44, Number 2 (2016), 455-488.
Dates
Revised: June 2013
First available in Project Euclid: 17 March 2016
https://projecteuclid.org/euclid.aos/1458245724
Digital Object Identifier
doi:10.1214/13-AOS1171
Mathematical Reviews number (MathSciNet)
MR3476606
Zentralblatt MATH identifier
1341.62115
Subjects
Primary: 62H12: Estimation
Secondary: 62F12: Asymptotic properties of estimators 62G09: Resampling methods
#### Citation
Cai, T. Tony; Liu, Weidong; Zhou, Harrison H. Estimating sparse precision matrix: Optimal rates of convergence and adaptive estimation. Ann. Statist. 44 (2016), no. 2, 455--488. doi:10.1214/13-AOS1171. https://projecteuclid.org/euclid.aos/1458245724
#### References
• Bickel, P. J. and Levina, E. (2008a). Regularized estimation of large covariance matrices. Ann. Statist. 36 199–227.
• Bickel, P. J. and Levina, E. (2008b). Covariance regularization by thresholding. Ann. Statist. 36 2577–2604.
• Cai, T. T. and Jiang, T. (2011). Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. Ann. Statist. 39 1496–1525.
• Cai, T. and Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation. J. Amer. Statist. Assoc. 106 672–684.
• Cai, T., Liu, W. and Luo, X. (2011). A constrained $\ell_{1}$ minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106 594–607.
• Cai, T. T. and Yuan, M. (2012). Adaptive covariance matrix estimation through block thresholding. Ann. Statist. 40 2014–2042.
• Cai, T. T., Zhang, C.-H. and Zhou, H. H. (2010). Optimal rates of convergence for covariance matrix estimation. Ann. Statist. 38 2118–2144.
• Cai, T. T. and Zhou, H. H. (2012). Optimal rates of convergence for sparse covariance matrix estimation. Ann. Statist. 40 2389–2420.
• d’Aspremont, A., Banerjee, O. and El Ghaoui, L. (2008). First-order methods for sparse covariance selection. SIAM J. Matrix Anal. Appl. 30 56–66.
• El Karoui, N. (2008). Operator norm consistent estimation of large-dimensional sparse covariance matrices. Ann. Statist. 36 2717–2756.
• Friedman, J., Hastie, T. and Tibshirani, T. (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9 432–441.
• Lam, C. and Fan, J. (2009). Sparsistency and rates of convergence in large covariance matrix estimation. Ann. Statist. 37 4254–4278.
• Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Oxford Univ. Press, New York.
• Liu, H., Lafferty, J. and Wasserman, L. (2009). The nonparanormal: Semiparametric estimation of high dimensional undirected graphs. J. Mach. Learn. Res. 10 2295–2328.
• Liu, W.-D., Lin, Z. and Shao, Q.-M. (2008). The asymptotic distribution and Berry-Esseen bound of a new test for independence in high dimension with an application to stochastic optimization. Ann. Appl. Probab. 18 2337–2366.
• Liu, H., Han, F., Yuan, M., Lafferty, J. and Wasserman, L. (2012). High-dimensional semiparametric Gaussian copula graphical models. Ann. Statist. 40 2293–2326.
• Meinshausen, N. and Bühlmann, P. (2006). High-dimensional graphs and variable selection with the lasso. Ann. Statist. 34 1436–1462.
• Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York. Translated from the Russian by A. A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82.
• Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. Oxford Univ. Press, New York.
• Ravikumar, P., Wainwright, M. J., Raskutti, G. and Yu, B. (2011). High-dimensional covariance estimation by minimizing $\ell_{1}$-penalized log-determinant divergence. Electron. J. Stat. 5 935–980.
• Rothman, A. J., Bickel, P. J., Levina, E. and Zhu, J. (2008). Sparse permutation invariant covariance estimation. Electron. J. Stat. 2 494–515.
• Saon, G. and Chien, J. T. (2011). Bayesian sensing hidden Markov models for speech recognition. In Acoustics, Speech and Signal Processing (ICASSP), 2011 IEEE International Conference 5056–5059. Prague.
• Thorin, G. O. (1948). Convexity theorems generalizing those of M. Riesz and Hadamard with some applications. Comm. Sem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 9 1–58.
• Xue, L. and Zou, H. (2012). Regularized rank-based estimation of high-dimensional nonparanormal graphical models. Ann. Statist. 40 2541–2571.
• Yuan, M. (2010). High dimensional inverse covariance matrix estimation via linear programming. J. Mach. Learn. Res. 11 2261–2286.
• Yuan, M. and Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model. Biometrika 94 19–35.
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2018-01-21 04:55:14
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http://math.stackexchange.com/questions/95323/smooth-curves-of-genus-3
|
# Smooth Curves of genus 3
Let $X$ be a smooth projective curve of genus 3 over an algebraically closed field of characteristic 0. How do I show that any curve like this is hyperelliptic or a plane curve of degree 4? Why is $K(X)$ isomorphic to $k(t)[y]$ for $y^{2}=f$ with $f \in k[t]$?
-
Under your hypotheses, the Riemann–Roch theorem and the embedding theorem are applicable. If I recall correctly this theorem is a case bash involving the canonical divisor and the aforementioned theorems. – Zhen Lin Dec 31 '11 at 1:14
The canonical divisor has degree $4 = 2g-2$ and $3$ (the genus) linearly independent global sections. If the canonical divisor is very ample, then the imbedding it defines must land on a plane quartic. If not, then the curve is hyperelliptic. (Recall that a curve is hyperelliptic if and only if the canonical bundle is not very ample.)
The second question is equivalent to asking whether the curve is a double cover of $\mathbb{P}^1$ (or equivalently, that the corresponding field is a degree two extension of $k(t)$), and that's the definition of hyperellipticity.
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2015-04-27 17:37:00
|
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|
https://www.maa.org/press/books/old-and-new-unsolved-problems-in-plane-geometry-and-number-theory
|
# Old and New Unsolved Problems in Plane Geometry and Number Theory
### Victor Klee and Stan Wagon
Print ISBN: 978-0-88385-315-3
340 pp., Paperbound, 1991
List Price: $32.00 Member Price:$24.00
Series: Dolciani Mathematical Expositions
Part of the broad appeal of mathematics is that there are simply stated questions that have not yet been answered. These questions are plentiful in the areas of plane geometry and number theory, and the purpose of this book is to discuss some unsolved problems in these fields.
The presentation is organized around 24 central problems, many of which are accompanied by other, related problems. Each problem is placed in its historical and mathematical context and is presented in two parts. The first gives an elementary overview, discussing the history and both solved and unsolved variants of the problem. Part Two contains more details, including a few proofs of related results, a wider and deeper survey of what is known about the problem and its relatives, and a large collection of references.
The book will appeal to a range of mathematicians―teachers at all levels, students (both undergraduate and graduate) as well as researchers. It could be used as a text in a course about unsolved problems, and also in courses in geometry or number theory. High school teachers interested in learning about developments in modern mathematics and the status of famous problems such as those dealing the Riemann hypothesis, perfect and prime numbers, tilings of the plane, or illumination of polygons will find the book very useful.
Preface
Chapter 1: Two Dimensional Geometry
Chapter 2: Number Theory
Chapter 3: Interesting Real Numbers
Hints and Solutions: Two-Dimensional Geometry
Hints and Solutions: Number Theory
Hints and Solutions: Interesting Real Numbers
Glossary
Index of Names
Subject Index
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2020-04-03 12:11:39
|
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https://it.mathworks.com/help/comm/ref/comm.phasenoise-system-object.html
|
# comm.PhaseNoise
Apply phase noise to baseband signal
## Description
The `comm.PhaseNoise` System object™ adds phase noise to a complex signal. This object emulates impairments introduced by the local oscillator of a wireless communication transmitter or receiver. The object generates filtered phase noise according to the specified spectral mask and adds it to the input signal. For a description of the phase noise modeling, see Algorithms.
To add phase noise using a `comm.PhaseNoise` object:
1. Create the `comm.PhaseNoise` object and set its properties.
2. Call the object with arguments, as if it were a function.
## Creation
### Syntax
``phznoise = comm.PhaseNoise``
``phznoise = comm.PhaseNoise(Name,Value)``
``phznoise = comm.PhaseNoise(level,offset,samplerate)``
### Description
````phznoise = comm.PhaseNoise` creates a phase noise System object with default property values.```
example
````phznoise = comm.PhaseNoise(Name,Value)` creates a phase noise object with the specified property `Name` set to the specified `Value`. You can specify additional name-value pair arguments in any order as (`Name1`,`Value1`,...,`NameN`,`ValueN`).```
````phznoise = comm.PhaseNoise(level,offset,samplerate)` creates a phase noise object with the phase noise level, frequency offset, and sample rate properties specified as value-only arguments. When specifying a value-only argument, you must specify all preceding value-only arguments.```
## Properties
expand all
Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the `release` function unlocks them.
If a property is tunable, you can change its value at any time.
Phase noise level in decibels relative to carrier per hertz (dBc/Hz), specified as a vector of negative scalars. The `Level` and `FrequencyOffset` properties must have the same length.
Data Types: `double`
Frequency offset in Hz, specified as a vector of positive increasing values. The `Level` and `FrequencyOffset` properties must have the same length.
Data Types: `double`
Sample rate in samples per second, specified as a positive scalar. To avoid aliasing, the sample rate must be greater than twice the largest value specified by `FrequencyOffset`.
Data Types: `double`
Source of the random stream, specified as `'Global stream'` or `'mt19937ar with seed'`.If `RandomStream` is set to `'mt19937ar with seed'`, the mt19937ar algorithm is used for normally distributed random number generation, in which case the reset method reinitializes the random number stream to the value of the `Seed` property.
Data Types: `char` | `string`
Initial seed for `RandomStream`, specified as a positive scalar less than 232.
#### Dependencies
To enable this property, set `RandomStream` to `'mt19937ar with seed'`.
Data Types: `double`
## Usage
### Syntax
``out = phznoise(in)``
### Description
example
````out = phznoise(in)` adds phase noise, specified by the `phznoise` System object, to the input signal. The result is returned in `out`.```
### Input Arguments
expand all
Input signal, specified as an NS-by-1 vector of complex values. NS is the number of samples.
Data Types: `double`
Complex Number Support: Yes
### Output Arguments
expand all
Output signal, returned as an NS-by-1 vector of complex values. NS equals the number of samples in the input signal.
Data Types: `double`
Complex Number Support: Yes
## Object Functions
To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named `obj`, use this syntax:
`release(obj)`
expand all
`visualize` Visualize spectrum mask of phase noise
`step` Run System object algorithm `release` Release resources and allow changes to System object property values and input characteristics `reset` Reset internal states of System object
## Examples
collapse all
Add a phase noise vector and frequency offset vector to a 16-QAM signal. Then plot the signal.
Create a phase noise System object.
`pnoise = comm.PhaseNoise('Level',-50,'FrequencyOffset',20);`
Generate modulated symbols.
```M = 16; % From 16-QAM data = randi([0 M-1],1000,1); modData = qammod(data,M);```
Use `pnoise` to apply phase noise. Plot the impaired data.
```y = pnoise(modData); scatterplot(y)```
View the effects of phase noise on a 10 MHz sine wave by using a spectrum analyzer. Adjust the resolution bandwidth of the spectrum analyzer to see its impact on the visualized spectral noise.
Initialize variables for the simulation.
```fc = 1e6; % Carrier frequency in Hz fs = 4e6; % Sample rate in Hz. phNzLevel = [-85 -118 -125 -145]; % Phase noise level in dBc/Hz phNzFreqOff = [1e3 9.5e3 19.5e3 195e3]; % Phase noise frequency offset in Hz Nspf = 6e6; % Number of Samples per frame freqSpan = 400e3; % Frequency span in Hz for spectrum computation```
Create sine wave, phase noise, and spectrum analyzer objects.
```sinewave = dsp.SineWave('Amplitude',1,'Frequency',fc,'SampleRate',fs, ... 'SamplesPerFrame',Nspf,'ComplexOutput',true); pnoise = comm.PhaseNoise('Level',phNzLevel, ... 'FrequencyOffset',phNzFreqOff,'SampleRate',fs); spectrumscopeRBW1 = dsp.SpectrumAnalyzer('NumInputPorts',2, ... 'SampleRate',fs,'FrequencySpan','Span and center frequency', ... 'CenterFrequency',fc,'Span',freqSpan,'RBWSource','Property', ... 'RBW',1,'SpectrumType','Power density','SpectralAverages',10, ... 'SpectrumUnits','dBW','YLimits',[-150 10], ... 'Title','Resolution Bandwidth 1 Hz','Position',[79 147 605 374]); spectrumscopeRBW10 = dsp.SpectrumAnalyzer('NumInputPorts',2, ... 'SampleRate',fs,'FrequencySpan','Span and center frequency', ... 'CenterFrequency',fc,'Span',freqSpan,'RBWSource','Property', ... 'RBW',10,'SpectrumType','Power density','SpectralAverages',10, ... 'SpectrumUnits','dBW','YLimits',[-150 10], ... 'Title','Resolution Bandwidth 10 Hz','Position',[685 146 605 376]);```
To analyze the spectrum and phase noise, the example includes two spectrum analyzer objects, with 1 Hz and 10 Hz resolution bandwidths, respectively. The spectrum analyzer objects use the default `Hann` windowing setting, the units are set to `dBW/Hz`, and the number of spectral averages is set to `10`.
```x = sinewave(); y = pnoise(x);```
When the resolution bandwidth is 1 Hz, the `dBW/Hz` view for the spectrum analyzer shows the tone at 0 dBW/Hz. The spectrum analyzer object corrects for the power spreading effect of the Hann windowing. Results show the visual average of the phase noise match the specified phase noise spectrum.
`spectrumscopeRBW1(x,y)`
When the resolution bandwidth is 10 Hz, the `dBW/Hz` view for the spectrum analyzer shows the tone at -10 dBW/Hz. The tone energy of the sine wave is now spread across 10 Hz instead of 1 Hz, so the sine wave PSD level reduces by 10 dB. With the resolution bandwidth at 10 Hz, the visual average of the phase noise still achieves the phase noise defined by the phase noise object.
With the resolution bandwidth increased from 1 Hz to 10 Hz, the spectrum analyzer object still corrects for the power spreading effect of the Hann window, and it achieves better spectral averaging with the wider resolution bandwidth. For more information, see Why Use Windows?.
`spectrumscopeRBW10(x,y)`
Calculate the RMS phase noise in degrees between the pure and noisy sine waves. In the general case, the pure signal must be time aligned with the noisy signal to accurately determine the phase error. However, in this case, the periodicity of the sine wave makes this step unnecessary.
```ph_err = unwrap(angle(y) - angle(x)); rms_ph_nz_deg = rms(ph_err)*180/pi(); sprintf('The computed RMS phase noise is %3.2f degrees.\n',rms_ph_nz_deg)```
```ans = 'The computed RMS phase noise is 0.18 degrees. ' ```
## Algorithms
The output signal, yk, is related to input sequence xk by yk=xkejφk, where φk is the phase noise. The phase noise is filtered Gaussian noise such that φk=f(nk), where nk is the noise sequence and f represents a filtering operation.
To model the phase noise, define the power spectrum density (PSD) mask characteristic by specifying scalar or vector values for the frequency offset and phase noise level.
• For a scalar frequency offset and phase noise level specification, an IIR digital filter computes the spectrum mask. The spectrum mask has a 1/f characteristic that passes through the specified point.
• For a vector frequency offset and phase noise level specification, an FIR filter computes the spectrum mask. The spectrum mask is interpolated across log10(f). It is flat from DC to the lowest frequency offset, and from the highest frequency offset to half the sample rate.
IIR Digital Filter
For the IIR digital filter, the numerator coefficient is
`$\lambda =\sqrt{2\pi {f}_{offset}{10}^{L/10}}\text{\hspace{0.17em}},$`
where foffset is the frequency offset in Hz and L is the phase noise level in dBc/Hz. The denominator coefficients, γi, are recursively determined as
`${\gamma }_{i}=\left(i-2.5\right)\frac{{\gamma }_{i-1}}{i-1}\text{\hspace{0.17em}},$`
where γ1 = 1, i = {1, 2,..., Nt}, and Nt is the number of filter coefficients. Nt is a power of 2, from `2`7 to `2`19. The value of Nt grows as the phase noise offset decreases towards 0 Hz.
FIR Filter
For the FIR filter, the phase noise level is determined through log10(f) interpolation for frequency offsets over the range [df, fs / 2], where df is the frequency resolution and fs is the sample rate. The phase noise is flat from 0 Hz to the smallest frequency offset, and from the largest frequency offset to fs / 2. The frequency resolution is equal to $\frac{{f}_{s}}{2}\left(\frac{1}{{N}_{t}}\right)$, where Nt is the number of coefficients, and is a power of 2 less than or equal to `2`16. If Nt < `2`8, a time domain FIR filter is used. Otherwise, a frequency domain FIR filter is used.
The algorithm increases Nt until these conditions are met:
• The frequency resolution is less than the minimum value of the frequency offset vector.
• The frequency resolution is less than the minimum difference between two consecutive frequencies in the frequency offset vector.
• The maximum number of FIR filter taps is `2`16.
## References
[1] Kasdin, N. J., "Discrete Simulation of Colored Noise and Stochastic Processes and 1/(f^alpha); Power Law Noise Generation." The Proceedings of the IEEE. Vol. 83, No. 5, May, 1995, pp 802–827.
## Extended Capabilities
### Blocks
Introduced in R2012a
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2021-06-22 13:50:55
|
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http://physics.stackexchange.com/questions/63893/how-to-find-speed-when-accelerating-down-a-slanted-wire/63894
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# How to find speed when accelerating down a slanted wire
I saw this picture on one of my social media sites with the caption, "I'd do this in a heart beat! Who's with me!"
I was about to go balls to the walls and say, "I'm in! When and where??" But then I got to thinking, how fast would I be going when I hit the water? If I were going too fast, would it hurt me?
SO I was trying to figure this out, and I'm not very good at physics so I was wondering if you guys could help me out.
I estimate the guy is 90 kg in mass, the wire is angled pi/6 from the horizontal and he's about 50 meters above the water when he starts (all estimates...).
What is the formulas I need to figure out the speed the guy will be going once he hits the water? I know there's some calculus in there, and I'm pretty good at calculus.
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It just looks like he's 150 feet above the water, so I was just estimating. – OghmaOsiris May 9 '13 at 3:36
Sorry wrong estimate :), it would be about 100Km/hr – ABC May 9 '13 at 3:39
The only force which works is gravity$^1$. So, change in gravitational potential energy equals final Kinetic energy(assume initial is zero). $$mgh=mv^2/2$$ $$v=\sqrt{2gh}$$
here $h$ is vertical height traversed.See the velocity does not depend on angle of string, mass of body too..
Let's see the kinematics of body.
The length of string is $h cosec\theta$ ($\theta$ being angle with horizontal assumed $\pi/6$)
acceleration of body along the string=$g\sin\theta$
Now $\text{using} : v^2=u^2+2as$
$$v^2=0+2\times h cosec\theta\times g \sin\theta$$ $$v=\sqrt{2gh}$$
Working in differentials
for $v$ along the rope. $$dv/dt=v\dfrac{dv}{dx}=a$$ $$\int_0^{v_f} v.dv=\int_0^{hcosec\theta} a.dx=ax\Bigg|_0^{hsosec\theta}$$ $$\dfrac{v_f^2}2=gsin\theta.hcosec\theta \ \ ; \ \ a=gsin\theta$$
$1)$Assuming the pulley being used to slide to be friction less.Though not possible.Also the rope is assumed to be in-extensible and straight.
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Would the acceleration due to gravity be the same when going at an angle, though? I thought it would be a lot less than 9.81 m/s – OghmaOsiris May 9 '13 at 3:41
@OghmaOsiris See I worked out energies , the gravitational potential energy just change in vertical displacement.So, the work done is just $mgh_{\text{vertical}}$ – ABC May 9 '13 at 3:43
I was hoping there woul be some integration involved :(... I really like calculus lol. – OghmaOsiris May 9 '13 at 4:05
@007 It looks like you integrated the velocity anyways. And I understand things better when put into the reference frame of math. The way you explained it isn't intuitive at all to me. It seemed more intuitive to integrate the velocity along a path. – OghmaOsiris May 9 '13 at 4:37
As an aside to physiology, be careful about hitting the water at that speed in that position. You could wind up forcing water to go where it really shouldn't... and can't... – DJohnM May 9 '13 at 6:32
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2016-05-01 17:49:28
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https://stats.stackexchange.com/questions/48378/difference-of-gamma-random-variables
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# Difference of Gamma random variables
Given two independent random variables $X\sim \mathrm{Gamma}(\alpha_X,\beta_X)$ and $Y\sim \mathrm{Gamma}(\alpha_Y,\beta_Y)$, what is the distribution of the difference, i.e. $D=X-Y$?
If the result is not well-known, how would I go about deriving the result?
• I think may be relevant: stats.stackexchange.com/q/2035/7071 – Dimitriy V. Masterov Jan 23 '13 at 21:14
• Unfortunately not relevant, that post considers the weighted sum of Gamma random variables where the weights are strictly positive. In my case the weights would be +1 and -1 respectively. – FBC Jan 23 '13 at 21:17
• The Moschopoulos paper claims that the method can be extended to linear combinations, but you are right that the rescaling seems to be restricted to weights greater than 0. I stand corrected. – Dimitriy V. Masterov Jan 23 '13 at 21:41
• There's little hope of deriving anything simple or in closed form unless the two scale factors are the same. – whuber Jan 23 '13 at 21:41
• Just a small remark: for the special case of exponentially distributed rvs with the same parameter the result is Laplace (en.wikipedia.org/wiki/Laplace_distribution). – Ric Mar 28 '13 at 9:57
I will outline how the problem can be approached and state what I think the end result will be for the special case when the shape parameters are integers, but not fill in the details.
• First, note that $X-Y$ takes on values in $(-\infty,\infty)$ and so $f_{X-Y}(z)$ has support $(-\infty,\infty)$.
• Second, from the standard results that the density of the sum of two independent continuous random variables is the convolution of their densities, that is, $$f_{X+Y}(z) = \int_{-\infty}^\infty f_X(x)f_Y(z-x)\,\mathrm dx$$ and that the density of the random variable $-Y$ is $f_{-Y}(\alpha) = f_Y(-\alpha)$, deduce that $$f_{X-Y}(z) = f_{X+(-Y)}(z) = \int_{-\infty}^\infty f_X(x)f_{-Y}(z-x)\,\mathrm dx = \int_{-\infty}^\infty f_X(x)f_Y(x-z)\,\mathrm dx.$$
• Third, for non-negative random variables $X$ and $Y$, note that the above expression simplifies to $$f_{X-Y}(z) = \begin{cases} \int_0^\infty f_X(x)f_Y(x-z)\,\mathrm dx, & z < 0,\\ \int_{0}^\infty f_X(y+z)f_Y(y)\,\mathrm dy, & z > 0. \end{cases}$$
• Finally, using parametrization $\Gamma(s,\lambda)$ to mean a random variable with density $\lambda\frac{(\lambda x)^{s-1}}{\Gamma(s)}\exp(-\lambda x)\mathbf 1_{x>0}(x)$, and with $X \sim \Gamma(s,\lambda)$ and $Y \sim \Gamma(t,\mu)$ random variables, we have for $z > 0$ that \begin{align*}f_{X-Y}(z) &= \int_{0}^\infty \lambda\frac{(\lambda (y+z))^{s-1}}{\Gamma(s)}\exp(-\lambda (y+z)) \mu\frac{(\mu y)^{t-1}}{\Gamma(t)}\exp(-\mu y)\,\mathrm dy\\ &= \exp(-\lambda z) \int_0^\infty p(y,z)\exp(-(\lambda+\mu)y)\,\mathrm dy.\tag{1} \end{align*} Similarly, for $z < 0$, \begin{align*}f_{X-Y}(z) &= \int_{0}^\infty \lambda\frac{(\lambda x)^{s-1}}{\Gamma(s)}\exp(-\lambda x) \mu\frac{(\mu (x-z))^{t-1}}{\Gamma(t)}\exp(-\mu (x-z))\,\mathrm dx\\ &= \exp(\mu z) \int_0^\infty q(x,z)\exp(-(\lambda+\mu)x)\,\mathrm dx.\tag{2} \end{align*}
These integrals are not easy to evaluate but for the special case $s = t$, Gradshteyn and Ryzhik, Tables of Integrals, Series, and Products, Section 3.383, lists the value of $$\int_0^\infty x^{s-1}(x+\beta)^{s-1}\exp(-\nu x)\,\mathrm dx$$ in terms of polynomial, exponential and Bessel functions of $\beta$ and this can be used to write down explicit expressions for $f_{X-Y}(z)$.
From here on, we assume that $s$ and $t$ are integers so that $p(y,z)$ is a polynomial in $y$ and $z$ of degree $(s+t-2, s-1)$ and $q(x,z)$ is a polynomial in $x$ and $z$ of degree $(s+t-2,t-1)$.
• For $z > 0$, the integral $(1)$ is the sum of $s$ Gamma integrals with respect to $y$ with coefficients $1, z, z^2, \ldots z^{s-1}$. It follows that the density of $X-Y$ is proportional to a mixture density of $\Gamma(1,\lambda), \Gamma(2,\lambda), \cdots, \Gamma(s,\lambda)$ random variables for $z > 0$. Note that this result will hold even if $t$ is not an integer.
• Similarly, for $z < 0$, the density of $X-Y$ is proportional to a mixture density of $\Gamma(1,\mu), \Gamma(2,\mu), \cdots, \Gamma(t,\mu)$ random variables flipped over, that is, it will have terms such as $(\mu|z|)^{k-1}\exp(\mu z)$ instead of the usual $(\mu z)^{k-1}\exp(-\mu z)$. Also, this result will hold even if $s$ is not an integer.
• +1: Having looked at this problem before, I find this answer fascinating. – Neil G Mar 28 '13 at 5:57
• I'm going to accept this answer even though there appears to be no closed form solution. It's as close as it gets, thanks! – FBC Jan 9 '14 at 16:26
• I love the reasoning here, but I'm wondering if there is any measure where the second step breaks, I.e., $f_{-Y}(\alpha) ≠ f_{Y}(-\alpha)$? – mpacer Sep 29 '15 at 6:53
• @mpacer No, $f_{-Y}(\alpha) = f_{Y}(-\alpha)$ always holds. It is a general result that does not require any assumptions (normality, Gamma-eity, positive RV etc). For the special case of a positive random variable (that is, $P\{Y > 0\} = 1$), $-Y$ is a negative random variable that takes on values less than $0$ with probability $1$. – Dilip Sarwate Sep 29 '15 at 14:16
• @mpacer If $Y$ is a positive random variable with density $f_Y(\alpha)$, then it is not true that $f_Y(\alpha)$ is undefined for $\alpha<0$. In fact, $f_Y(\alpha)$ is defined as having value $0$ for $\alpha<0$. Thus, $f_{-Y}(\alpha)=f_Y(\alpha)=0$ for all positive numbers $\alpha$, and the density of $Y$ is the density of $Y$ "flipped over" with respect to the origin (or vertical axis if you prefer.) I am not "interpreting" the $-$ operator differently, it is you who is demanding an "appropriate" notion of $-$ that will support your idea that the domain of $f_Y$ is $\mathbb R^+$ only – Dilip Sarwate Sep 30 '15 at 20:48
To my knowledge the distribution of the difference of two independent gamma r.v.’s was first studied by Mathai in 1993. He derived a closed form solution. I will not reproduce his work here. Instead I will point you to the original source. The closed form solution can be found on page 241 as theorem 2.1 in his paper On non-central generalized Laplacianness of quadratic forms in normal variables.
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2020-08-15 09:27:49
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http://lt-jds.jinr.ru/record/58383?ln=en
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/ Experiment-HEP arXiv:1205.0594
Measurement of the Lambda(b) cross section and the anti-Lambda(b) to Lambda(b) ratio with Lambda(b) to J/Psi Lambda decays in pp collisions at sqrt(s) = 7 TeV
Abstract: The Lambda(b) differential production cross section and the cross-section ratio for anti-Lambda(b)/Lambda(b) production are measured as functions of transverse momentum pt(Lambda(b)) and rapidity y(Lambda(b)) in pp collisions at sqrt(s) = 7 TeV using data collected by the CMS experiment at the LHC. The measurements are based on Lambda(b) decays reconstructed in the exclusive final state J/Psi Lambda, with the subsequent decays J/Psi to an opposite sign muon pair and Lambda to proton pion, using a data sample corresponding to an integrated luminosity of 1.9 inverse femtobarns. The product of the cross section times the branching ratio for Lambda(b) to J/Psi Lambda versus pt(Lambda(b)) falls faster than that of b mesons. The measured value of the cross section times the branching ratio for pt(Lambda(b)) > 10 GeV and abs(y(Lambda(b))) < 2.0 is 1.02 +/- 0.06 +/- 0.12 nb, and the integrated cross section ratio for anti-Lambda(b)/Lambda(b) is 1.02 +/- 0.07 +/- 0.09, where the uncertainties are statistical and systematic, respectively.
Note: * Temporary entry *; Submitted to Physics Letters B
Total numbers of views: 1911
Numbers of unique views: 760
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2020-04-02 04:36:42
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http://blog.vmchale.com/3
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# blog
Vanessa McHale's blog (3)
• ## Edit Distance in ATS
As ATS remains a somewhat obscure language, I figured I'd write up a recent success and give some commentary on some advanced features of ATS.
• ## Benchmarking Code Produced by Different Versions of GHC
Though it does not often get mentioned in the Haskell community, simply bumping to a new version of GHC can drastically improve the performance of your code. Here, I have several examples from my fast-arithmetic package, which will hopefully give an idea of just how much work has gone into optimizing code produced by GHC.
Initially, I had written hackage-fetch to see if there was any use of coelgot anywhere on Hackage. At the time, there was not, but this has changed due to my gmpint package. As of writing, it is not surprisingly the only use of co-(Elgot algebra)s on the entirety of Hackage.
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2019-05-23 00:42:09
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http://archive.numdam.org/item/BSMF_2001__129_1_91_0/?source=PMIHES_1990__72__63_0
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Perturbation singulière en dimension trois : canards en un point pseudo-singulier nœud [ Singular perturbation, tridimensional case: canards on a pseudo-singular node point ]
Bulletin de la Société Mathématique de France, Volume 129 (2001) no. 1, p. 91-113
We study singularly perturbed system of differential equations like $\left\{\begin{array}{ccc}\hfill \stackrel{˙}{x}& =& f\left(x,y,z,\epsilon \right),\hfill \\ \hfill \stackrel{˙}{y}& =& g\left(x,y,z,\epsilon \right),\hfill \\ \hfill \epsilon \stackrel{˙}{z}& =& h\left(x,y,z,\epsilon \right),\hfill \end{array}$ where $f$, $g$ and $h$ are analytic functions. In known papers, regular points of the slow surface $h=0$ are studied. At this point, the fast flow (vertical) is tranverse to the slow surface. The Tikhonov’s theorem can be applied here. In other papers, fold points and cusps of the slow surface were studied. The list of generic singularities contains also the pseudo-singular points which are connected to the turning points in lower dimension. They are (generically) saddle, focus or node. In the neighborhood of focus points, nothing happens, the saddle were studied in their papers, but the node points were never studied in the litterature. In this paper, whe prove that generally, there exist two overstable (i.e. regular with respect $\epsilon$) solutions. When the ratio between two eigenvalues is an integer, a resonance appears, and one of the two overstable solutions disappears. Technically, we transform first the system into a more canonical equation. After that, we prove the existence of formal solutions, and, using the implicit function theorme on Banach spaces of Gevrey series, we can prove that the formal solution is Gevrey. The theory of summation of Gevrey series gives the over-stable solutions.
On étudie les systèmes différentiels singulièrement perturbés de dimension 3 du type $\left\{\begin{array}{ccc}\stackrel{˙}{x}\hfill & =& f\left(x,y,z,\epsilon \right),\hfill \\ \stackrel{˙}{y}\hfill & =& g\left(x,y,z,\epsilon \right),\hfill \\ \epsilon \stackrel{˙}{z}\hfill & =& h\left(x,y,z,\epsilon \right),\hfill \end{array}$$f$, $g$, $h$ sont analytiques quelconques. Les travaux antérieurs étudiaient les points réguliers où la surface lente $h=0$ est transverse au champ rapide vertical. C’est le domaine d’application du théorème de Tikhonov. Dans d’autres travaux antérieurs, on étudiait les singularités de certains types : plis et fronces de la surface lente, ainsi que certaines singularités plus compliquées, analogues aux points tournants en dimension inférieure : les points pseudo-singuliers cols. Génériquement, les seules singularités génériques non encore étudiées dans la littérature sont les points pseudo-singuliers nœuds. Dans cet article, on étudie les points pseudo-singuliers nœuds où on montre l’existence d’une ou deux solutions surstables (c’est-à-dire assez régulières en $\epsilon$). Quand le rapport de deux valeurs propres est entier, un phénomène nouveau et intéressant apparaît : la résonance. Techniquement, on se ramène d’abord à une forme plus canonique, puis on montre l’existence de solutions formelles, en utilisant le théorème des fonctions implicites sur un opérateur entre espaces de Banach de séries Gevrey. Les séries obtenues sont alors Gevrey, et les théories de sommation de ces séries donnent les solutions surstables recherchées.
DOI : https://doi.org/10.24033/bsmf.2387
Classification: 34E15, 34M30, 34M60, 34E05, 34C20
Keywords: ordinay differential equation, singular perturbation, turning point, Gevrey series, overstable solution, canard
@article{BSMF_2001__129_1_91_0,
author = {Beno\^\i t, \'Eric},
title = {Perturbation singuli\ere en dimension trois : canards en un point pseudo-singulier n\oe ud},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {129},
number = {1},
year = {2001},
pages = {91-113},
doi = {10.24033/bsmf.2387},
zbl = {0992.34072},
mrnumber = {1871979},
language = {fr},
url = {http://www.numdam.org/item/BSMF_2001__129_1_91_0}
}
Benoît, Éric. Perturbation singulière en dimension trois : canards en un point pseudo-singulier nœud. Bulletin de la Société Mathématique de France, Volume 129 (2001) no. 1, pp. 91-113. doi : 10.24033/bsmf.2387. http://www.numdam.org/item/BSMF_2001__129_1_91_0/`
[1] V. ArnolʼD - Chapitres supplémentaires de la théorie des équations différentielles ordinaires, “Mir”, Moscow, 1980, Translated from the Russian by Djilali Embarek. | MR 626685 | Zbl 0956.34502
[2] É. Benoît - « Canards et enlacements », Inst. Hautes Études Sci. Publ. Math. (1990), no. 72, p. 63-91 (1991). | Numdam | MR 1087393 | Zbl 0737.34018
[3] É. Benoît - « Systèmes lents-rapides dans ${ℝ}^{3}$ et leurs canards », Third Schnepfenried geometry conference, Vol. 2 (Schnepfenried, 1982), Astérisque, vol. 109, Soc. Math. France, Paris, 1983, p. 159-191. | MR 753147 | Zbl 0529.34037
[4] -, « Canards de ${ℝ}^{3}$ », Thèse, Université de Nice, 1984.
[5] É. Benoît & J.-L. Callot - « Chasse au canard. », Collect. Math. 32 (1981), no. 2, p. 37-119. | MR 653889 | Zbl 0529.34046
[6] É. Benoît, A. Fruchard, R. Schäfke & G. Wallet - « Solutions surstables des équations différentielles complexes lentes-rapides à point tournant », Ann. Fac. Sci. Toulouse Math. (6) 7 (1998), no. 4, p. 627-658. | Numdam | MR 1693589 | Zbl 0981.34084
[7] M. Canalis-Durand, J. P. Ramis, R. Schäfke & Y. Sibuya - « Gevrey solutions of singularly perturbed differential equations », J. Reine Angew. Math. 518 (2000), p. 95-129. | MR 1739408 | Zbl 0937.34075
[8] C. K. R. T. Jones - « Geometric singular perturbation theory », Dynamical systems (Montecatini Terme, 1994), Lecture Notes in Math., vol. 1609, Springer, Berlin, 1995, p. 44-118. | MR 1374108 | Zbl 0840.58040
[9] C. H. Lin - « The sufficiency of the Matkowsky condition in the problem of resonance », Trans. Amer. Math. Soc. 278 (1983), no. 2, p. 647-670. | MR 701516 | Zbl 0513.34055
[10] C. Lobry, T. Sari & S. Touhami - « On Tykhonov's theorem for convergence of solutions of slow and fast systems », Electron. J. Differential Equations (1998), p. No. 19, 22 pp. (electronic). | MR 1631397 | Zbl 0897.34052
[11] B. Malgrange & J.-P. Ramis - « Fonctions multisommables », Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, p. 353-368. | Numdam | MR 1162566 | Zbl 0759.34007
[12] B. Malgrange - « Sur le théorème de Maillet », Asymptotic Anal. 2 (1989), no. 1, p. 1-4. | MR 991413 | Zbl 0693.34004
[13] F. Takens - « Constrained equations ; a study of implicit differential equations and their discontinuous solutions », Structural Stability, the theory of catastrophes and applications in the sciences, Lecture Notes in Math., vol. 525, Springer Verlag, 1976. | MR 478236 | Zbl 0386.34003
[14] W. Wasow - Linear turning point theory, Applied Mathematical Sciences, vol. 54, Springer-Verlag, New York, 1985. | MR 771669 | Zbl 0558.34049
[15] M. Wechselberger - « Singularly perturbed folds and canards in ${ℝ}^{3}$ », Thèse, Universität Wien, 1998.
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2020-06-06 21:55:23
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https://cs.gmu.edu/~gjstein/2021/7/action-items-should-have-well-defined-en/
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# Action items should have well-defined end conditions
Fri 16 Jul 2021
I have a lot of meetings, many of which are with students pursuing research objectives as part of their PhD. Part of my job is making sure students are on a path to success, ensuring that they have a good idea of what needs to happen next if they are to make progress towards their ultimate goal. It is common for our meetings to produce action items that we add to something that functions like a persistent task list: a collection of intermediate objectives for the students to explore in the pursuit of their research.
Note that I don't usually police these task lists, but my students and I have so far found it helpful to keep a shared workspace (on Notion) where we can record what we've agreed sound like good next steps.
Research requires making progress in multiple areas at once—reading papers, brainstorming new ideas, writing papers of one's own, writing/running code, etc.—and so the tasks we generate after a meeting will be similarly varied. Yet if we aren't careful, some tasks stay on the list indefinitely, leading to clutter and reducing how effective the list is for motivating progress and keeping track of work that has yet to be completed.
One common trend I have discovered: many tasks that are never crossed off our list lack well-defined end states. For many of these tasks, it is clear how to make progress towards completing those tasks, but nothing that indicates when the task should be marked as done. For instance, a task to conduct a literature review on XYZ is easy to write down, but how do I know when I've completed my literature review? Instead, I recommend including some way of quantifiable progress in the task itself, adding either an amount of time (e.g., look for related papers for 2.5 hours) or a target number (e.g., add 4 papers to my annotated bibliography). Once the task is complete, we can decide whether or not it was sufficient and augment it if necessary.
Finally, it should be easy to get started making progress towards a task. If making progress towards Task A requires that some other pre-work be done first, that pre-work should be its own task that exists as a prerequisite of Task A.
I generally follow a Getting Things Done methodology, in which tasks are grouped into "projects" and assigned either a NEXT (unblocked) TODO (blocked) keywords.
I have found that tasks that implicitly require pre-work feel harder to sit down and start working towards, which can make the difference between whether or not I choose that task to work on.
I try to make sure that my tasks are bite-sized and are both easy to start making progress towards and have clear completion criteria. Working towards complex, long-horizon objectives—as is commonplace for a researcher—can be daunting, but this procedure of making small, well-defined tasks (and occasionally splitting up larger ones) helps me and my students make continual incremental progress and staves off procrastination.
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2022-05-19 02:55:38
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