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Default qubit plugin¶

Module name: pennylane.plugins.default_qubit

Short name: "default.qubit"

The default plugin is meant to be used as a template for writing PennyLane device plugins for new qubit-based backends.

It implements the necessary Device methods as well as some built-in qubit operations and expectations, and provides a very simple pure state simulation of a qubit-based quantum circuit architecture.

The following is the technical documentation of the implementation of the plugin. You will not need to read and understand this to use this plugin.

Auxillary functions¶

 spectral_decomposition_qubit(A) Spectral decomposition of a $$2\times 2$$ Hermitian matrix. unitary(*args) Input validation for an arbitary unitary operation. hermitian(*args) Input validation for an arbitary Hermitian expectation.

Gates and operations¶

 Rphi(phi) One-qubit phase shift. Rotx(theta) One-qubit rotation about the x axis. Roty(theta) One-qubit rotation about the y axis. Rotz(theta) One-qubit rotation about the z axis. Rot3(a, b, c) Arbitrary one-qubit rotation using three Euler angles. X Pauli-X matrix Y Pauli-Y matrix Z Pauli-Z matrix H Hadamard gate CNOT CNOT gate SWAP SWAP gate CZ CZ gate

Expectations¶

 X Pauli-X matrix Y Pauli-Y matrix Z Pauli-Z matrix

Classes¶

 DefaultQubit(wires, *[, shots]) Default qubit device for PennyLane.

Code details¶

spectral_decomposition_qubit(A)[source]

Spectral decomposition of a $$2\times 2$$ Hermitian matrix.

Parameters: A (array) – $$2\times 2$$ Hermitian matrix (a, P): eigenvalues and hermitian projectors such that $$A = \sum_k a_k P_k$$. (vector[float], list[array[complex]])
X = array([[0, 1], [1, 0]])

Pauli-X matrix

Y = array([[ 0.+0.j, -0.-1.j], [ 0.+1.j, 0.+0.j]])

Pauli-Y matrix

Z = array([[ 1, 0], [ 0, -1]])

Pauli-Z matrix

H = array([[ 0.70710678, 0.70710678], [ 0.70710678, -0.70710678]])

CNOT = array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]])

CNOT gate

SWAP = array([[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]])

SWAP gate

CZ = array([[ 1, 0, 0, 0], [ 0, 1, 0, 0], [ 0, 0, 1, 0], [ 0, 0, 0, -1]])

CZ gate

Rphi(phi)[source]

One-qubit phase shift.

Parameters: phi (float) – phase shift angle unitary 2x2 phase shift matrix array
Rotx(theta)[source]

One-qubit rotation about the x axis.

Parameters: theta (float) – rotation angle unitary 2x2 rotation matrix $$e^{-i \sigma_x \theta/2}$$ array
Roty(theta)[source]

One-qubit rotation about the y axis.

Parameters: theta (float) – rotation angle unitary 2x2 rotation matrix $$e^{-i \sigma_y \theta/2}$$ array
Rotz(theta)[source]

One-qubit rotation about the z axis.

Parameters: theta (float) – rotation angle unitary 2x2 rotation matrix $$e^{-i \sigma_z \theta/2}$$ array
Rot3(a, b, c)[source]

Arbitrary one-qubit rotation using three Euler angles.

Parameters: a,b,c (float) – rotation angles unitary 2x2 rotation matrix rz(c) @ ry(b) @ rz(a) array
unitary(*args)[source]

Input validation for an arbitary unitary operation.

Parameters: args (array) – square unitary matrix square unitary matrix array
hermitian(*args)[source]

Input validation for an arbitary Hermitian expectation.

Parameters: args (array) – square hermitian matrix square hermitian matrix array
identity(*_)[source]

Identity matrix for expectations.

Returns: 2x2 identity matrix array
class DefaultQubit(wires, *, shots=0)[source]

Default qubit device for PennyLane.

Parameters: wires (int) – the number of modes to initialize the device in shots (int) – How many times the circuit should be evaluated (or sampled) to estimate the expectation values. A value of 0 yields the exact result.
pre_apply()[source]

Called during execute() before the individual operations are executed.

apply(operation, wires, par)[source]

Apply a quantum operation.

For plugin developers: this function should apply the operation on the device.

Parameters: operation (str) – name of the operation wires (Sequence[int]) – subsystems the operation is applied on par (tuple) – parameters for the operation
expval(expectation, wires, par)[source]

Return the expectation value of an expectation.

For plugin developers: this function should return the expectation value of the given expectation on the device.

Parameters: expectation (str) – name of the expectation wires (Sequence[int]) – subsystems the expectation value is to be measured on par (tuple) – parameters for the observable expectation value float
_get_operator_matrix(operation, par)[source]

Get the operator matrix for a given operation or expectation.

Parameters: operation (str) – name of the operation/expectation par (tuple[float]) – parameter values matrix representation. array
ev(A, wires)[source]

Evaluates a one-qubit expectation in the current state.

Parameters: A (array) – $$2\times 2$$ Hermitian matrix corresponding to the expectation wires (Sequence[int]) – target subsystem expectation value $$\expect{A} = \bra{\psi}A\ket{\psi}$$ float
reset()[source]

Reset the device

expand_one(U, wires)[source]

Expand a one-qubit operator into a full system operator.

Parameters: U (array) – $$2\times 2$$ matrix wires (Sequence[int]) – target subsystem $$2^n\times 2^n$$ matrix array
expand_two(U, wires)[source]

Expand a two-qubit operator into a full system operator.

Parameters: U (array) – $$4\times 4$$ matrix wires (Sequence[int]) – two target subsystems (order matters!) $$2^n\times 2^n$$ matrix array
operations

Get the supported set of operations.

Returns: the set of PennyLane operation names the device supports set[str]
expectations

Get the supported set of expectations.

Returns: the set of PennyLane expectation names the device supports set[str]


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# why the ending symbol dx is also changed into a function of d theta and then multiplied by the integral after substitution? [duplicate]

This is a question about trig substitution used in integrals.

Because it is difficult to solve an integral when there is radical in it, we use a trig function of theta to substitute x from the original integral.

The only thing I don't understand is why the ending symbol dx, which indicates that we are integrating the function with respect to the "x" variable, is also changed into a function of d theta and then multiplied by the integral after substitution.

I think after substitution, we can just rewrite dx into d theta.

• if $\theta=f(x)$, then $d\theta=f'(x)dx$ Mar 16 at 12:28

$$dx$$ and $$d\theta$$ are not the same thing. For example, if $$x = \sin \theta$$, then $$\frac{dx}{d\theta} = \cos \theta$$ and so $$dx = \cos {\theta} d\theta$$.
• @mirthspritzultyrobscurantism So there are two ways of resolving this. One is simply saying that, by the fundamental theorem of calculus, and the chain rule, we have $\int_a^b g'(x)f(g(x))\textrm{d}x=\int_{g(a)}^{g(b)} f(u)\textrm{d}u$. This is a borderline trivial and completely rigorous statement. One short-hand for this could be to write "$u=g(x)$ implies $\textrm{d}u=\frac{du}{dx}\textrm{d}x$ since that's sort of how the above look if you think all you're doing is integrating $f$. And this heuristic will always yield correct substitutions. Mar 16 at 20:17
• Another is appealing to measure theory, stating that $\textrm{d}x$ is a measure (the ordinary Lesbegue measure) and $\textrm{d}u$ is the measure such that $\textrm{d}u([a,b])=u(b)-u(a)$. In this formulation, $\textrm{d}u=\frac{du}{dx} \textrm{d}x$ is true as a statement about measures with densities and can be proven rigorously. One should note, though, that $\frac{du}{dx}$ is a function and not a measure, so although the notation is suggestive, the result isn't literally due to "fractions cancelling out". Mar 16 at 20:21

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Journal article Open Access

# From the hand to the mind to the hand

Quintela, João; Simon, Tim

### Dublin Core Export

<?xml version='1.0' encoding='utf-8'?>
<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
<dc:creator>Quintela, João</dc:creator>
<dc:creator>Simon, Tim</dc:creator>
<dc:date>2021-05-01</dc:date>
<dc:description>If we consider architecture as the art of construction it can be determined by a mental process of design or, instead, it can be complemented with an empirical knowledge where building is learning, building is deciding, building is designing. A process of going back and forth asking the brick what he wants to be indeed.We have had the possibility of developing several self-built projects where the gap between designing and building is filled by the experience with the materials themselves. Important decisions are taken while the performative act of building is taking place. The act of construction is turned into an act of design as it goes from the hand to the mind. By this the rational nature of construction is complemented by instinctive intentions.In the same way we try to explore drawing techniques not just as a design tool but as a process of analysis and expression. Handmade drawings of the projects are produced afterwards in an abstract but very precise way in order to clarify the principles that come out of the designing process. The act of drawing is turned into an act of analysis as it goes from the mind to the hand. The project informs the drawing and the drawing reveals the project. There is a point where the question of representation in architecture arises once these works aspire to a certain autonomy.</dc:description>
<dc:identifier>https://zenodo.org/record/4727101</dc:identifier>
<dc:identifier>10.5281/zenodo.4727101</dc:identifier>
<dc:identifier>oai:zenodo.org:4727101</dc:identifier>
<dc:language>eng</dc:language>
<dc:relation>doi:10.5281/zenodo.4727100</dc:relation>
<dc:relation>url:https://zenodo.org/communities/practicesinresearch</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:source>Practices In Research 2 49-71</dc:source>
<dc:subject>architecture</dc:subject>
<dc:subject>practice based research</dc:subject>
<dc:subject>drawing</dc:subject>
<dc:subject>representation</dc:subject>
<dc:subject>construction</dc:subject>
<dc:title>From the hand to the mind to the hand</dc:title>
<dc:type>info:eu-repo/semantics/article</dc:type>
<dc:type>publication-article</dc:type>
</oai_dc:dc>

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#### Asian Journal of Chemistry

Journal Information
ISSN / EISSN: 09707077 / 0975427X
Total articles ≅ 9,082

#### Latest articles in this journal

Priyanka Shrivastava, Vibha Malviya, Anjali Yadav, Payal Jain, Ruchika Malviya, , A.K. Sinha
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3308-3312; https://doi.org/10.14233/ajchem.2022.23945

Abstract:
Sludge sample collected from sewage treatment plant was used to develop an effective adsorbent for the removal of various dyes from wastewater. A quantitative estimation of major compounds in without activated (as collected), activated and various dyes treated sludge samples using synchrotron XRD technique is reported. Sewage sludge sample activated at 900 ºC for 30 min was found to have maximum adsorption efficiency (~ 95%) for the removal of different dyes (e.g. brilliant green, crystal violet and malachite green) from aqueous solution. The result shows that the adsorption using activated sewage sludge is the simplest and effective method for the removal of dyes from wastewater. As expected, the major compound in all the samples is SiO2 (~ 50 mol%). However, SiO2 was found in three different space groups. On activation at 900 ºC, a part of monoclinic SiO2 phase changes to the hexagonal phase. On the other hand, on various dyes adsorption such as brilliant green, crystal violet and malachite green, the monoclinic phase changes to the tetragonal phase.
B. Ayyanar, J. Suresh, V. Thangaraj, , , M. Kayalvizhi
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3132-3144; https://doi.org/10.14233/ajchem.2022.24012

Abstract:
For conducting use, an organic molecule chalcone (1-(2,4-dichlorophenyl)-3-(4-hydroxy-phenyl)-prop- 2-en-1-one (DCHP) and metal oxide nanocomposite (MONC) doped polymer thin films were prepared. In addition, polyvinyl alcohol (PVA) and chitosan (CS) was utilized as a host polymer. The features of successfully prepared thin polymer films were characterized. The XRD analysis revealed the nano-sized (40-150 nm), crystalline phases and semi-crystalline behaviour of polymer films. The XPS and EDX confirmed the elements of PVA, CS, DCHP, MONC4 and MONC5, as well as the oxidation states of MONC4 and MONC5 sample films. The morphology of the prepared polymer film revealed an even distribution, a smooth surface and a blossoming flower-like structure from the SEM results. The TGA study revealed the multi-stage decomposition upon heating, but no complete decomposition for the produced film containing MONC4 and MONC5 thin films. The bonding between doped materials and the host polymer in the film was confirmed by FT-IR studies. The produced thin polymer composite film containing metal oxide nanocomposites and DCHP exhibited a high dielectric property and high AC conductance of 1.12 × 10-6 S cm-1 at room temperature.
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3275-3278; https://doi.org/10.14233/ajchem.2022.23996

Abstract:
A series of 2,3-dihydroquinazolin-4(1H)-ones has been synthesized using a simple, effective, one-pot, three component condensation of isatoic anhydride, aromatic aldehydes and amines in refluxing ethyl alcohol. It has been found that 15 mol% succinimide-N-sulfonic acid is the optimum amount of catalyst to obtain the high yield.
M.M. Gurav, , A.W. Suryawanshi
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3247-3250; https://doi.org/10.14233/ajchem.2022.24043

Abstract:
A new technique for the impregnation of magnetic based catalyst component onto the organic molecules are widely discussed. Due to its benefits of high surface area, separation simplicity, recyclability and limited leaching of catalytic active ingredients from the catalyst, in the present work, iron-doped Fe0.02Zn0.95-xCr0.05O heterogeneous catalyst was applied in the synthesis of dihydropyrimidones, thiones and their derivatives. The synthesized compounds were elucidated by their IR, 1H NMR and 13C NMR and LC-MS analysis.
, Sarymsakov Abdushkur Abdukhalilovich, Shukurov Akobirkhon Ibodullo Ogli, Todjiyev Jamoliddin Nasiriddinovich, Atakhanov Abdumutalib Abdupatto Ugli, , Sergey O. Solomevich
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3085-3092; https://doi.org/10.14233/ajchem.2022.23912

Abstract:
In present work, the formation of nanostructured antiviral biosoluble ophthalmic drug films composed of sodium carboxymethylcellulose/polyphenol-gossypol-carboxymethylcellulose copolymer is described. The biorelease time and physico-chemical parameters of the ophthalmic drug films were regulated by changes in the degree of substitution and polymerization of sodium carboxymethylcellulose and the pH value of its aqueous solutions. The morphology and size of the antiviral agent embedded in sodium carboxymethylcellulose films were determined by using atomic force microscopy and dynamic light scattering. The UV spectroscopy was used to observe drug release from the polymeric matrix and it was determined that the maximum drug release occurred after 24 h. By regulating the concentration of polyphenol-gossypol-carboxymethylcellulose copolymer antiviral substance in the polymer, the spherical nanoparticles with sizes of 5-24 nm and cubic nanoparticles with sizes of 46-95 nm were formed.
Xuan-Tien Le, Khai-Vinh Le, Tuyet-Anh Tran-Thi, Kieu-Trang Huynh
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3257-3262; https://doi.org/10.14233/ajchem.2022.23998

Abstract:
Considering the emerging trends of using natural ingredients in the cosmetic industry, this study was undertaken to confirm the efficiency of some traditional skin-whitening plants in Vietnam. In this study, 11 popular Vietnamese herbal extracts were investigated in suppressing mushroom tyrosinase activity. The most potent material was then analyzed for phytochemicals profiles and the extraction process was optimized by single-factor experiments. It was found that Morus alba leaves extract using ethanol had superior mushroom tyrosinase inhibitory activity. The extract under optimal conditions (extract 2 cycles with ethanol 70%, temperature: 50 ºC, time: 40 min and solid/liquid ratio = 1:10) contained polyphenols, flavonoids, saponins and tannins and its biological properties included: antityrosinase IC50 = 99.4 μg/mL, antioxidation IC50 (DPPH assay) = 85.7 μg/mL and total phenolic content TPC (GAE) = 128.6 mg/g extract.
, , R.A. Wijaya
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3361-3368; https://doi.org/10.14233/ajchem.2022.23990

Abstract:
Synthesis of ferrate (FeO4 2−) from iron plate of transformer waste electrochemically in high alkaline medium and its application for dye degradation has been carried out. The effect of parameters such as time, NaOH concentration, type of electrolyte solution and ferrate stability were studied. The ferrate solution formed is dark purple in colour with a maximum wavelength of 505 nm. The ferrate was compacted by fresh drying to produce Na2FeO4 crystals and characterized by XRF, XRD and FT-IR techniques. The ferrate was applied to degrade methylene blue, methyl orange, rhodamine and remazol black B dyes. The results showed the highest degradation of dyestuff in methylene blue by 98% and COD reduction by 73.69% at pH 8, ferrate dose of 1.1 mg and contact time of 70 min. This shows that ferrate is an environmentally friendly material which can be used to degrade toxic dyes.
Chinnaperamanoor Madhu Ganesan, Chinnasamy Chinnaraju, A.R. Lavanya, Kandasamy Prabakar
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3037-3047; https://doi.org/10.14233/ajchem.2022.24024

Abstract:
The importance of a clean environment is recognized as a “pillar” of sustainable development. There is, however, a serious public health threat associated with heavy metal releases into the environment. Aquatic environments in Tamil Nadu state of India lack heavy metals monitoring data. From year 2008 to 2022, research articles were assessed that focused at heavy metal concentrations in freshwater and marine ecosystems. It has been shown that elevated levels of heavy metals in sediments of aquatic ecosystems contribute to an increase in their abundance, which may further enter the food chain through bioaccumulation. In addition, intensifying human exploitation of the South East Coastal region (SEC) in Tamil Nadu state of India through industries, tourism, aquaculture and recreation further complicates the situation by providing ever-changing sources of contamination. As a result of analyzing all heavy metals, the levels of Cd in biota collected from ocean environments were higher. Rivers and lakes have shown increased levels of heavy metals such as Zn, Cu and Pb as a result of localized anthropogenic activities. In accordance with sediment guidelines and human health risk assessments, most of the heavy metals concentrations in sediments and aquatic biota exceeded the permissible limits. It is highly recommended that more monitoring studies are conducted to monitor the levels of heavy metals in organisms and in urban rivers, lakes and marine environments in the future, with extreme importance both for environmental health and economics. By identifying the sources of pollutants and implementing strict policies to abate pollution, local governments can implement better control measures to decrease pollution levels.
Daniel Angkasali, Peter Surjo, Rafaella Shiene Wijaya, Muhammad Mikail Athif Zhafir Asyura,
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3151-3156; https://doi.org/10.14233/ajchem.2022.24002

Abstract:
The COVID-19 pandemic significantly increased the amount of infectious medical wastes produced, with medical mask wastes being one of the largest contributors. Present research focuses on trying to turn medical mask waste into a functioning air filter by modifying it with CuO/TiO2 to reduce the amount of infectious medical wastes laying around. Synthesis of CuO/TiO2 was confirmed with FESEM-EDX, UV-Vis DRS and XRD techniques. The optimum amount of Cu added (1%wt of TiO2) was determined by assessing the degradation performance of the modified medical mask wastes against an organic pollutant (methylene blue) and a biological pollutant (S. aureus). The filter was then integrated into a simple air purifying unit and complemented with a UV-C germicidal lamp and a plasma ion generator. The prototype of the simple air purifying unit was able to degrade 100% tobacco smoke in less than 15 min and 30.8% CO gas in 30 min.
, Chinchumol Cyriac,
Published: 25 November 2022
Asian journal of Chemistry, Volume 34, pp 3374-3382; https://doi.org/10.14233/ajchem.2022.24028

Abstract:
A new series of 1,4-benzodiazepines were synthesized based on the results of molecular docking. The designed compounds were docked in the groove of binding sites present in human estrogen receptor 2IOK and EGFR receptor 4WKQ. The physico-chemical properties, Lipinski’s Rule of 5 and ADMET properties were calculated for the compounds by using the Qikprop application in Schrödinger software and on the basis of molecular docking fifteen compounds were designed. Because of poor yield in some cases, only ten compounds have been synthesized. All the newly synthesized compounds were established based on IR, 1H NMR and mass spectral data. The new compounds were screened for in vitro anticancer activity. Among all the compounds, compound BZ4 had the highest binding energies and showed the potential cytotoxicity when screened for anticancer activity by MTT assay.


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1

Lattice energy Now, according to wikipedia, NaCl has a lattice energy of −756 kJ/mol. First, we have to understand the term lattice energy. Here is the textbook explanation (Fleming: Physical Chemistry): The lattice energy is the energy required to separate the ions in an ionic lattice so that they are at infinite distance (but still ions). This would be ...

0

Coagulation of milk is a result of casein micelles clumping together. This is favoured by higher temperatures and also by favour a pH of around 4.6 as this is the isoelectric point. At this isoelectric point, the amino acids within the casein micelle are electrically neutral. This means there is less electrostatic repulsion between the casein micelles, ...

1

Calcium nitrate exists under three forms : 1) $\ce{Ca(NO3)}_2$ which melts at 361°C; 2) $\ce{Ca(NO3)}_2$$·\ 3$$\ce{H_2O}$, which is not commercial and melts at 51°C ; 3) $\ce{Ca(NO3)}_2$$·\ 4$$\ce{H_2O}$ which is commercially available and melts at 43°C. It looses its water at 132°C and is deliquescent. The decahydrate that you mention does not exist.

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# Relative homology

In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.

## Definition

Given a subspace ${\displaystyle A\subset X}$, one may form the short exact sequence

${\displaystyle 0\to C_{\bullet }(A)\to C_{\bullet }(X)\to C_{\bullet }(X)/C_{\bullet }(A)\to 0}$

where ${\displaystyle C_{\bullet }(X)}$ denotes the singular chains on the space X. The boundary map on ${\displaystyle C_{\bullet }(X)}$ leaves ${\displaystyle C_{\bullet }(A)}$ invariant and therefore descends to a boundary map on the quotient. The corresponding homology is called relative homology:

${\displaystyle H_{n}(X,A)=H_{n}(C_{\bullet }(X)/C_{\bullet }(A)).}$

One says that relative homology is given by the relative cycles, chains whose boundaries are chains on A, modulo the relative boundaries (chains that are homologous to a chain on A, i.e. chains that would be boundaries, modulo A again).

## Properties

The above short exact sequences specifying the relative chain groups gives rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence

${\displaystyle \cdots \to H_{n}(A)\to H_{n}(X)\to H_{n}(X,A){\stackrel {\delta }{\to }}H_{n-1}(A)\to \cdots .}$

The connecting map δ takes a relative cycle, representing a homology class in Hn(X, A), to its boundary (which is a cycle in A).

It follows that Hn(X, x0), where x0 is a point in X, is the n-th reduced homology group of X. In other words, Hi(X, x0) = Hi(X) for all i > 0. When i = 0, H0(X, x0) is the free module of one rank less than H0(X). The connected component containing x0 becomes trivial in relative homology.

The excision theorem says that removing a sufficiently nice subset ZA leaves the relative homology groups Hn(X, A) unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that Hn(X, A) is the same as the n-th reduced homology groups of the quotient space X/A.

The n-th local homology group of a space X at a point x0 is defined to be Hn(X, X - {x0}). Informally, this is the "local" homology of X close to x0.

Relative homology readily extends to the triple (X, Y, Z) for ZYX.

One can define the Euler characteristic for a pair YX by

${\displaystyle \chi (X,Y)=\sum _{j=0}^{n}(-1)^{j}\;{\mbox{rank}}\;H_{j}(X,Y).}$

The exactness of the sequence implies that the Euler characteristic is additive, i.e. if ZYX, one has

${\displaystyle \chi (X,Z)=\chi (X,Y)+\chi (Y,Z).\,}$

## Functoriality

The map ${\displaystyle C_{\bullet }}$ can be considered to be a functor

${\displaystyle C_{\bullet }:{\mathbf {Top}}^{2}\to {\mathcal {CC}}}$

where Top2 is the category of pairs of topological spaces and ${\displaystyle {\mathcal {CC}}}$ is the category chain complexes of abelian groups.

## Examples

One important use of relative homology is the computation of the homology groups of quotient spaces ${\displaystyle X/A}$. In the case that ${\displaystyle A}$ is a subspace of ${\displaystyle X}$ fulfilling the mild regularity condition that there exists a neighborhood of ${\displaystyle A}$ that has ${\displaystyle A}$ as a deformation retract, then the group ${\displaystyle {\tilde {H}}_{n}(X/A)}$ is isomorphic to ${\displaystyle H_{n}(X,A)}$. We can immediately use this fact to compute the homology of a sphere. We can realize ${\displaystyle S^{n}}$ as the quotient of an n-disk by its boundary, i.e. ${\displaystyle S^{n}=D^{n}/S^{n-1}}$. Applying the exact sequence of relative homology gives the following:
${\displaystyle \cdots \to H_{n}(D^{n})\rightarrow H_{n}(D^{n},S^{n-1})\rightarrow H_{n-1}(S^{n-1})\rightarrow H_{n-1}(D^{n})\to \cdots .}$

Because the disk is contractible, we know its homology groups vanish in all dimensions, so the above sequence collapses to the short exact sequence:

${\displaystyle 0\rightarrow H_{n}(D^{n},S^{n-1})\rightarrow H_{n-1}(S^{n-1})\rightarrow 0.}$

Therefore, we get isomorphisms ${\displaystyle H_{n}(D^{n},S^{n-1})\cong H_{n-1}(S^{n-1})}$. We can now proceed by induction to show that ${\displaystyle H_{n}(D^{n},S^{n-1})\cong \mathbb {Z} }$. Now because ${\displaystyle S^{n-1}}$ is the deformation retract of a suitable neighborhood of itself in ${\displaystyle D^{n}}$, we get that ${\displaystyle H_{n}(D^{n},S^{n-1})\cong H_{n}(S^{n})\cong \mathbb {Z} .}$


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# Tag Info

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Sketch of proof. Suppose you have a Cauchy sequence $\{f_n\}\subset C_b^k(U)$, with respect to $\|\cdot\|_k$. Then, for every $\lvert\alpha\rvert\le k$, the sequence $\{D^\alpha f_n\}$ is uniformly Cauchy in $U$ and hence uniformly convergent to a continuous and bounded function, say $f^\alpha\in C_b(U)$ - All the $f^\alpha$'s are for the moment only ...

0

This is obvious. Note that $m(r)$ is not only an infimum, but a minimum. This means that you can cover $B$ with $m:=m(r)$ suitably placed balls of radius $r$. When $r'>r$ then $m$ balls of radius $r'$ with the same centers will obviously cover $B$, but there might be a better covering with balls of radius $r'$. So $m(r')\leq m(r)$. (You cannot hope for ...

0

Please refer to the exercise 7 on page 155 of Folland's real analysis. You will find more useful information. Let $X$ be a Banach space. If $T\in L(X,X)$ and $\|I-T\|<1$ where $I$ is the identity operator, then $T$ is invertible; in fact, the series $\sum_0^\infty(I-T)^n$ converges to $(I-T)^{-1}$. If $T\in L(X,X)$ is invertible and ...

0

Visualizing such things might prove hard, since we're dealing with vectors in $n^2$ dimensions for some $n$. The theorems are important since they provide, along with the contraction principle, a very good proof of the inverse functions theorem which Rudin gives in full detail. To prove the first theorem, we start with $\lVert A-B\rVert \rVert ... 0 If you do not assume continuity, for what I have said in the comment, you can not put this definition, that is,$T$, not being bounded (on$X$) and discontinuous at each point in$||x|| \leq 1$, will not have in general the$\sup$. 1 If the space$X$is banach it is an easy consequences of the open mappig theorem. Anyway with the norm induced topology over$ X $you in fact are resizing a ball so it is a ball again and it is open by definition of the topology. So the map sends open ball in open ball therefore it is open. This reasoning heavily rely on the "absolute omogeneity" of the ... 1 Alternative proof by contradiction. Suppose for all$k$there is some$v_k\neq 0$such that$\|T v_k \| \ge k \|v_k\|$. Let$x_k = {1 \over k} { v_k \over \|v_k\|}$, and note that$\|T x_k \| \ge 1$, but$x_k \to 0$, which contradicts continuity. 0$T$is continuous. Let$\epsilon = 1$. We can find$\delta$so that$||Tv|| < 1$when$||v|| < \delta$. Now,$||Tv|| = || \frac{||v||}{\delta} T(\frac{\delta v}{||v||})|| \leq \frac{||v||}{\delta}$. Take$c = \frac{1}{\delta}$. 0 The fact that$\| \pi(x) \| \le \|x\|$for all$x \in X$tells you that$\|\pi\| \le 1$as$\|\pi\| := \sup\{\|\pi(x)\| : \|x\| \le 1 \}$. If you pick a point$z \in M$then$\|\pi(z)\| = \|z\|$and so$\|\pi\| = 1$. 0 You have$\pi^2 = \pi$, so$||\pi|| = ||\pi^2|| \le ||\pi||^2$, implying that$||\pi|| \ge 1$. Your statement implies that$||\pi|| \le 1$. 2 If$null(f)$is not dense in$X$, you can find$x\in X$and$r>0$such that$B(x,r)\cap null(f)=\varnothing$. if$y\in X$is such that$|f(y)|\geq|f(x)|$, then for some$\alpha$with$|\alpha|\leq 1$we have$f(\alpha y)=f(x)$, so$x-\alpha y\in null(f)$, hence$x-\alpha y\not\in B(x,r)$, so$\Vert y\Vert\geq\Vert\alpha y\Vert\geq r$. What this just said ... 1 Try this:$null(f)$is a subspace of$X$whose codimension is 1. Now if it is not dense, then$null(f)$is closed (because its closure is a subspace containing$null(f)$and it is not$X$). Then you show that it implies continuity. For example, as$f\neq 0$there is a$y\in X$such as$f(y) = 1$. $$\{x\in X| |f(x)| = 1 \}= (y + null(f))\cup (-y + null(f)) ... 0 You did the hard part, namely (ii) \Longrightarrow (iii). Now u is continuous means every sequence x_n \to x \in E must have u(x_n) \to u(x) \in F. Now use (iii) combined with linearity, (i) will be immediate. Part (b) is also not so hard if you use the fact that every norm on a finite dimensional vector space is equivalent. Try proving it ... 2 Fix a basis v_1,\ldots,v_m \in V_k. Then the map \sum \alpha_jv_j\to\left(\sum |\alpha_j|^2\right)^{1/2} defines a norm on V_k, and this norm is induced by the inner product \langle \sum \alpha_jv_j,\sum \beta_jv_j\rangle = \sum \alpha_j\overline{\beta_j}. In a finite-dimensional space all norms are equivalent, so the identity map becomes a ... 3 The Banach function spaces may be a better choice for your question. You can see here for the definition of a Banach function space or this book Function Spaces, Volume 1 (chapter 6). Answer to your question: The first thing is to introduce the definition of the Riesz-Fischer property. We say that a normed linear space (X,\left\|\cdot\right\|) has ... 3 Yes. The isomorphism is given by:$$\begin{array}{llll} \varphi:&C^0[a,b]&\longrightarrow&C^0[0,1]\\ &f(x)&\longmapsto&(b-a)f(a+(b-a)x) \end{array}$$0 Let S be the set of all tuples of scalars (\alpha_1,\dots,\alpha_n)\in K^n such that |\alpha_1|+\cdots+|\alpha_n|=1. Then the map f:S\rightarrow [0,\infty) given as$$f(\alpha_1,\dots,\alpha_n)=\|\alpha_1 u_1+\cdots+\alpha_n u_n\|$$is continuous. Because S is compact, the continuous function f attains its infimum on S at some x_0\in S, ... 3 For each k,$$\left\lVert\sum_{n=1}^{k}f_n\right\rVert \leqslant \sum_{n=1}^{k}\lVert f_n \rVert$$Then take k\to\infty. 1 The answer is yes. For every normed space X we can define as \alpha, the cardinality of the set of minimum cardinality which is dense in X. Clearly, \alpha\ge\aleph_0. For every such set D, described in the question, we shall show that \lvert D\rvert=\alpha. It is not hard to see that the set E=\bigcup_{n\in\mathbb N} 2^{-n}D is dense in ... 1 Let X, Y be Banach spaces and U\subset X open. A function u\colon U\to Y is said to be differentiable at a\in U if there exists a linear operator Du_a\colon X\to Y (the differential of u at a) such that$$ u(a+h)=u(a)+Du_a(h)+o(h). $$If u is continuous, then Du_a is also continuous. If u is continuous and differentiable at very point ... 0 I don't know if you can construct a sequence as you've proposed above, but I do know that the inequality you proposed is sharp. Proof of this is neatly presented in Becker's "Inequalites in Fourier Analysis'' (1975): http://www.jstor.org/stable/1970980 The sharp Young's inequality reads:$$\|f \ast g \|_r \leq \left( A_p A_q A_{r'} \right)^n ... 1 If$V$is not separable, then$L^p(X,\mu,V)$is not either. Take$\{v_j\}$an uncountable set in$V$without a limit point and$f_j\in L^p(X,\mu,V)$, such that$f_j(x)=\varphi_j(x)v_j$, where$\varphi_j\ne 0$scalar. Clearly, there is no limit point in$\{f_j\}$. If$X$is separable, then again it is not certain that$L^p(X,\mu,\mathbb R)$is separable. It ... 3 Complementing the very well detailed answer of Zev Chonoles, I think you've got it in the opposite direction.$T$as is defined in the title is the mapping$x\mapsto T(x)=x^T$, such that$T(x)(y)=x^Ty$is a linear transformation of the vector$y$. This means that the domain of$T$is$(\mathbb R^n, \Vert \cdot \Vert_\infty)$and its codomain is the dual ... 1 A pair$(V,\|\cdot\|)$denotes a vector space$V$over$\mathbb{R}$, together with a norm function$\|\cdot\|:V\to\mathbb{R}$. Thus,$(\mathbb{R}^n,\|\cdot\|_1)$and$(\mathbb{R}^n,\|\cdot\|_\infty)$mean "the vector space$\mathbb{R}^n$equipped with the$L^1$norm", and "the vector space$\mathbb{R}^n$equipped with the$L^\infty$norm", respectively. ... 0 I don't think you're going to get anywhere this way- it seems as though what you're trying to prove at the end there is stronger than what is really true! Let$Conv(A)$denote the convex hull of$A$, which for me will be the set of all finite convex combinations of elements of$A$. We need not worry about closures: the Hausdorff distance only sees closures, ... 0 I thought that I could use that, since all the norms are equivalent in any finite dimensional vector space, the convergence with the norm$\lVert.\rVert_\infty$is equivalent to the convergence with the norm$\lVert.\rVert$, defined in 1. This way, I could argue that the convergence with$\lVert.\rVert$is equivalent to the convergence of every addend in ... 0 This is a consequence of the Uniform Boundedness Principle, which hold for a Banach space, and not a normed one. However, you can think of$S$as a subset of$X^{**}$which is indeed a Banach space. 1 It is true if you assume that$T$is self-adjoint (i.e. symmetric), meaning that $$(Tx,y)=(x,Ty), \quad \text{for all}\,\, x,y\in H, \tag{1}$$ and assuming that $$|(Tx,x)|\le \|x\|^2, \quad \text{for all}\,\, x\in H.\tag{2}$$ Note that your inequality holds even for$T=-2I$, and thus we NEED to assume these two additional things:$(1)$and$(2)$. So ... 0 Let$(V,\mid\mid\cdot\mid\mid)$be a normed vector space over$\mathbb{R}$; then the closed unit ball$\overline{B}(0 , 1)$is convex. Proof: Clearly$\overline{B}(0 , 1)$is non empty, so let$x,y\in\overline{B}(0 , 1)$and let$\lambda\in[0,1]$. Consider$z=\lambda x+ (1-\lambda)y$Now; $$\mid\mid z\mid \mid= \|\lambda x+ (1-\lambda)y \| \leq ... 0 We simply use the triangular inequality repeatedly:$$ \|u\|=\|(u-v)+v\|\le \|u-v\|+\|v\|, $$and thus$$ \|u\|-\|v\|\le \|u-v\|\le \|u\|+\|-v\|=\|u\|+\|v\|. \tag{1} $$Similarly$$ \|v\|=\|(v-u)+u\|\le \|v-u\|+\|u\|, $$and thus$$ \|v\|-\|u\|\le \|u-v\|. \tag{2} $$Now (1) and (2) imply that$$ \big|\|u\|-\|v\|\big|\le \|u-v\|\le \|u\|+\|v\|. $$5 The first$$||u-v||=||u+(-v)||\le ||u||+||-v||=||u||+||v||$$and the second$$||u||=||u-v+v||\le ||u-v||+||v||\Rightarrow ||u||-||v||\le ||u-v||$$and by symmetry we have the other inequality so we conclude. 0 Hint: \left\|u\right\| = \left\|(u-v)+v\right\| \left\|v\right\| = \left\|(u-v)+u\right\| \left\|u-v\right\| = \left\|u+(-v)\right\| 0 For any f \in C(\mathbb{R}), the L^{\infty} norm of f is the same as the C(\mathbb{R}) norm of f. So the closure of C(\mathbb{R}) in L^{\infty}(\mathbb{R}) consists of all functions which are equal a.e. to continuous functions. Choose any function in L^{\infty} which is not equal a.e. to a continuous function, and that function cannot be the ... 5 The simplest counterexample is the non-zero constant function$$ f(x)=1. $$If g\in C_0^1(\mathbb R), then \lim_{|x|\to\infty}g(x)=0, and hence$$ \lim_{x\to\infty}|f(x)-g(x)|=\lim_{x\to\infty}|1-g(x)|=1. $$Thus$$ \|f-g\|_\infty=\sup_{x\in\mathbb R} |f(x)-g(x)|=1,$$and therefore$f$can not be approximated by$C_0^1\$ functions.

2

The inequality is easy. For the equality, use Hahn-Banach.

Top 50 recent answers are included


text:
Revision history [back]

Hitching and Smoothness problem with videocapture::read

So I'm having a problem with the VideoCapture::read function.

When I run this function:

void Camera::DoFrame( cv::Mat& im, vector<Location> &locations)
{
if(!TheVideoCapturer.isOpened())
return;

TheVideoCapturer >> im;

locations = processFrame(im);
//imshow("video", im);
//cv::waitKey(30);
}

The average time taken is 33 milliseconds, Most of that time is taken up by the TheVideoCapturer >> im; This is fine as that is a good frame rate, however it has periods where the time will spike up to 250ms. This obviously produces problems with the smoothness of my video, one which I wanted to fix. I think it also might be related to the video skipping occcasionly.

After suspecting my camera and another of other issues, I found that by uncommenting 2 lines in the above sample of code, I could get the video to out put much more consistantly, with the longest time the read function takes is 30ms, though the average is 2ms. This is more like what i would expect as a frame rate. It also seems to relieve my problem of the video being not smooth enough. The only problem with this is that I neither want to output a video from my camera, or wait 30ms if I can help it. Is there something I misunderstand about the read method? Has anyone expereienced something similar?

I am using opencv version 2.4.5, I experience the same bug in both visual studio and mingw, and get the same problem on both a logitech pro 9000 and a logitech pro c910

Thanks


text:
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text:
# Characteristic Function of Intersection/Variant 2

## Theorem

Let $A, B \subseteq S$.

Let $\chi_{A \mathop \cap B}$ be the characteristic function of their intersection $A \cap B$.

Then:

$\chi_{A \mathop \cap B} = \min \set {\chi_A, \chi_B}$

## Proof

By Characteristic Function Determined by 1-Fiber, it suffices to show that:

$\min \set {\map {\chi_A} s, \map {\chi_B} s} = 1 \iff s \in A \cap B$

By definition of characteristic function, we have:

$\min \set {\map {\chi_A} s, \map {\chi_B} s} = 1$
$\map {\chi_A} s = \map {\chi_B} s = 1$

because $\chi_A, \chi_B$ take values in $\set {0, 1}$.

By definition of $\chi_A$ and $\chi_B$, this is equivalent to the statement that both $s \in A$ and $s \in B$.

That is:

$s \in A \cap B$

by definition of set intersection.

$\blacksquare$


text:
# Homework Help: Combinatorics-binomial coefficients

1. Jun 13, 2010

### pupeye11

1. The problem statement, all variables and given/known data

Find integers a,b, and c such that $$m^{3} = a*nCr(m,3)+b*nCr(m,2)+c*nCr(m,1)$$ for all m. Then sum the series $$1^3+2^3+3^3+...+n^3$$

2. Relevant Equations

I think I need to use $$m^{3} = a*nCr(m,3)+b*nCr(m,2)+c*nCr(m,1)$$ and $$nCr(n+1,k+1)=nCr(0,k)+nCr(1,k)+...+nCr(n-1,k)+nCr(n,k)$$ to be able to sum the series but I am not sure

3. The attempt at a solution

I am not really sure how to get started, I originally thought that a=3, b=2, and c=1 but the more I think about it that doesn't seem right. I believe that if it was just $$m^2=a*nCr(m,2)+b*nCr(m,1)$$ then a=2 and b=1 but I am not sure why, I just know it works.

2. Jun 13, 2010

### CompuChip

I would start by writing out the combinations.
For example, nCr(m, 3) is proportional to m(m-1)(m-2).
Then you will get a polynomial equation in m, which should hold for any m, therefore you can equate the coefficients of the different degrees.

3. Jun 13, 2010

### pupeye11

can you show me how you got nCr(m,3) is proportional to m(m-1)(m-2)? Obviously $$nCr(m,3)=\frac{m!}{3!(m-3)!}$$ but not sure where to go from there?

4. Jun 13, 2010

### Tedjn

How can you simplify m!/(m-3)!?

5. Jun 13, 2010

### pupeye11

Ok so that would equal $$\frac{1*2*3*...*(m-3)(m-2)(m-1)(m)}{1*2*3*...*(m-3)}$$ so obviously everything up to (m-3) will cancel out and I am left with m(m-1)(m-2) but what happens to the 3 in the denominator of $$\frac{m!}{3!(m-3)!}$$?

6. Jun 13, 2010

### vela

Staff Emeritus
The 3! is still there, so now you have

$$m^3 = a\left[\frac{m(m-1)(m-2)}{3!}\right]+b*\textrm{nCr}(m,2)+c*\textrm{nCr}(m,1)$$

Just keep going and see where it leads you.

7. Jun 13, 2010

### pupeye11

Ok, so I did that approach on a b and c and got the following:

$$m^3=a[\frac{m(m-1)(m-2)}{3!}]+b[\frac{m(m-1)}{2!}]+cm$$

which if I factor out a m and divide by it I'll get

$$m^2=a[\frac{(m-1)(m-2)}{3!}]+b[\frac{(m-1)}{2!}]+c$$

where would I go from there though? I have three unknowns so don't I need three equations to solve for them?

8. Jun 13, 2010

### vela

Staff Emeritus
Now you multiply out the RHS and collect terms. When you do that, you'll have

$$m^2 = \textrm{R} m^2 + \textrm{S} m + \textrm{T}$$

where R, S, and T are some combination of a, b, and c. (I'm leaving that for you to work out.) For this equation to hold, the coefficients of m2, m1, and m0 on the LHS and RHS have to match, so you need to have 1=R, 0=S, and 0=T. These are the three equations that will let you solve for the three unknowns a, b, and c.

9. Jun 13, 2010

### pupeye11

Alright so after doing the algebra I get a=6,b=6, and c=1. Which gives me

$$m^3=6nCr(m,3)+6nCr(m,2)+nCr(m,1)$$.

Then how would I sum the series $$1^3+2^3+3^3+...+n^3$$ using this?

10. Jun 13, 2010

### vela

Staff Emeritus
You could write

$$\sum_{m=0}^n m^3 = \sum_{m=0}^n \left[ 6\begin{pmatrix}m\\3\end{pmatrix}+6\begin{pmatrix}m\\2\end{pmatrix}+\begin{pmatrix}m\\1\end{pmatrix}\right]$$

and use the relation you noted in your original post

$$\sum_{m=0}^n \begin{pmatrix}m\\k\end{pmatrix} = \begin{pmatrix}n+1\\k+1\end{pmatrix}$$

to evaluate the sums on the RHS.

11. Jun 13, 2010

### pupeye11

Ok, so if I sum the series

$$1^3+2^3+...+n^3$$

using this then the 3, 2, or 1 would be the k values while the n values for each would be

$$1^3+1, 2^3+1,....$$?

12. Jun 13, 2010

### vela

Staff Emeritus
No. Your k values are right, but not the values for n.

Last edited: Jun 13, 2010
13. Jun 13, 2010

### pupeye11

I think I would get

$$\sum_{m=0}^nm^3=0+1+8+27+...+n^3$$

14. Jun 13, 2010

### vela

Staff Emeritus
That's right. Sorry, I realized you were getting confused about the RHS and edited my earlier post.

In the relation

$$\sum_{m=0}^n \begin{pmatrix}m\\k\end{pmatrix} = \begin{pmatrix}n+1\\k+1\end{pmatrix}$$

the n that appears on top of the summation is the same n that appears on the righthand side of the equation, so when you use the relation to evaluate

$$\sum_{m=0}^n \left[ 6\begin{pmatrix}m\\3\end{pmatrix}+6\begin{pmatrix} m\\2\end{pmatrix}+\begin{pmatrix}m\\1\end{pmatrix} \right]$$

you want to take whatever is on top of the summation and use that same expression for the n when you write it in terms of the binomial coefficients.

15. Jun 13, 2010

### pupeye11

Ok that makes sense, but how would I identify what should be on top of the summation sign? Is that from this:

$$\sum_{m=0}^nm^3=0+1+8+27+...+n^3$$

16. Jun 13, 2010

### vela

Staff Emeritus
You just look. What's there?

17. Jun 13, 2010

### pupeye11

Well since it's just an n up there then would this come out to be

$$\sum_{m=0}^n \left[ 6\begin{pmatrix}n+1\\4\end{pmatrix}+6\begin{pmatrix} n+1\\3\end{pmatrix}+\begin{pmatrix}n+1\\2\end{pmatrix} \right]$$

18. Jun 13, 2010

### vela

Staff Emeritus
Close, but the summation shouldn't be there anymore. The relation tells you how to write a sum of binomial coefficients,

$$\sum_{m=0}^n \begin{pmatrix}m\\k\end{pmatrix}$$

with just one binomial coefficient,

$$\begin{pmatrix}n+1\\k+1\end{pmatrix}$$

19. Jun 13, 2010

### pupeye11

Ok so the answer is just

$$\left[ 6\begin{pmatrix}n+1\\4\end{pmatrix}+6\begin{pmatrix} n+1\\3\end{pmatrix}+\begin{pmatrix}n+1\\2\end{pmatrix}+...+\begin{pmatrix}n+1\\k+1\end{pmatrix} \right]$$

I am not sure that last part should be in there though... but I think it should since I am summing over

$$1^3+2^3+...+n^3$$

20. Jun 14, 2010

### vela

Staff Emeritus
Nope, try again. You might find these property of summations helpful:

$$\sum_i (A_i+B_i) = \sum_i A_i +\sum_i B_i$$

$$\sum_i cA_i = c \sum_i A_i$$


text:
# Refactoring R’s dendrapply

Published:

As someone who specializes in comparative phylogenomics, I work a lot with phylogenetic trees. Trees are represented in R as dendrogram objects, which are essentially a series of nested lists. Each “node” of the tree is a list with multiple members (two if a binary tree, but dendrogram objects are not constrained to be binary), each of which is another dendrogram object. The leaves are special cases in that they have length 1 and an additional property leaf, which is set to TRUE.

R has a number of functions called apply functions, whose primary purpose is to apply a function to a set of things in a particular way. Commonly used examples are lapply to apply a function over a list-like object, tapply to apply a function to a set of objects grouped based on a factor, rapply to recursively apply a function to an object, or apply to apply a function to a matrix/array.

dendrapply is a special type of apply statement intended specifically for applying functions to dendrogram objects. The main requirement is to recursively apply a function to each node of the tree. While this may sound like a task for rapply, rapply is more intended to apply a function to non-list elements of a nested list, while dendrapply is intended to apply the function to all nodes in the list. This means that dendrapply specializes in applying the function to internal nodes of the tree, whereas rapply applies the function only to the leaves (and doesn’t always preserve the original structure).

dendrapply is currently implemented recursively, which has led to some users experiencing issues from stack overflows resulting from deep recursion on trees with many nodes (at least in users’ reports on DECIPHER). Additionally, the recursive implementation makes the function slow for trees with many internal nodes. rapply avoids this issue by factoring out the recursion internally, which gave me the idea to try to implement an optimized version of dendrapply.

## Implementation

My focus for this implementation was removing the recursive calls in dendrapply. The final implementation is only a modest runtime improvement, but this can hopefully be optimized prior to the final release.

The full implementation is written in C, which makes a lot of things easier at the cost of a few things becoming really hard. The algorithm proceeds as follows:

copytree = copy(input_dend)

while ptr is not NULL:
if node has children:
for child in ptr.node.children:
insert child in next position
else:
while(node has no unevaluated children):
apply function to node
merge node into parent
node = parent
ptr = parent.ptr

ptr = ptr.next

It’s a little hard to write out, so I’ll use a small example to showcase. Imagine we have a dendrogram with two leaves, and we want to apply a function f to each node a,b,c.

Tree:

a
/ \
b   c

Initialize LL:

HEAD -> a* -> c -> NULL             (insert child, back to front to ensure correct traversal)
HEAD -> a* -> b -> c -> NULL        (insert child)
HEAD -> a -> b* -> c -> NULL        (increment pointer)
HEAD -> a -> B* -> c -> NULL        (b has no children; apply f, f(b)=B)
HEAD -> aB* -> c -> NULL            (merge B into parent)
HEAD -> aB -> c* -> NULL            (a has an unevaluated child, so increment pointer)
HEAD -> aB -> C* -> NULL            (apply f to c, f(c) = C)
HEAD -> aBC* -> NULL                (merge C into parent)
HEAD -> ABC* -> NULL                (a has no unevaluated children so apply f, f(a)=A)
HEAD -> ABC -> NULL*                ( done, return ABC )

This implementation achieves the same result as a recursive operation, but doesn’t run the risk of overflowing the stack for deep trees. Note that this algorithm is a post-order traversal; pre-order is the default and is discussed later.

## Details and Difficulties

There are a few things about the implementation of this that make things difficult. First, the R garbage collector likes to collect R objects while they’re being worked on in C. We can get around this by using PROTECT() to tell the garbage collector not to touch the object, but the protection stack has a fixed size that is extremely small (10,000 items by default). The tree as a whole can be protected when it’s first loaded, but calling f() on a node of the tree produces a new R object that is unprotected, so we have to protect the result.

The naive approach is to just put all the nodes into a linked list, apply the function to every node, then rebuild the tree. Unfortunately, this rapidly exhausts the available space we have on the protection stack. Instead, we have to work slightly smarter.

We can get around this issue by immediately assigning the value of f(node) to the parent node. If the parent node is protected, the value assigned to it will be protected as well. This means that, as long as we ensure the parent is always protected, we don’t need any extraneous protection calls. Some clever shuffling can ensure that we protect everything under the first PROTECT call on the entire tree.

I’ve implemented two methods for dendrapply: a pre-order traversal and a post-order traversal. The final implementation for both uses a maximum of 3 slots on the protection stack, and no recursive function calls. The original tree consumes the first PROTECT call, which protects all its children until they’re modified. When each node is evaluated, we use a PROTECT call to create the R expression to be called and a second to protect the SEXP returned from the function. This value is then assigned to the parent node using SET_VECTOR_ELT, which protects the value. Since protection is by value and not by reference, we can safely store this new value in the linked list. Applying the function to the root can safely be done by using REPROTECT() to preserve protection on children.

The result is a little funky in the pre-order case: we use VECTOR_ELT on the parent to get the node, call the function on it, replace the node using SET_VECTOR_ELT on the parent, and then populate the reference in the linked list by calling VECTOR_ELT on the parent a second time. This ensures we always have protected values.

With a post-order traversal, the implementation is a little cleaner. Since the children of each node are always evaluated prior to the node itself, we don’t need to recall VECTOR_ELT after calling the function. Instead, we can just merge the nodes and continue.

The main difference for end users between the two traversals is that, for a given node n, pre-order traversal will always evaluate f(n) before any of its children, whereas post-order traversal will always evaluate f(n) after all of its children. Pre-order is the default for backwards compatibility with the original stats::dendrapply, which used a pre-order traversal. The post-order method allows for some new functions, such as the following:

This function assigns a new attribute equal to the label if it’s a leaf, or the concatenation of the child nodes’ new attributes if it’s an internal node. The default application of dendrapply will only create new attributes for the leaves, and will return character(0) for any internal nodes (since the children won’t have had their new attribute set yet). However, using how='post.order' will ensure we evaluate the children first, meaning that internal nodes will be assigned a non-empty value:

This capability is something I have found myself wishing for often in dendrapply. Calculating Fitch Parsimony for a phylogeny is a great example of a method that relies upon a post-order traversal.

## Replicating Bugs

Another weird thing I had to contend with is ensuring that the functionality exactly replicates that of the current dendrapply so as not to break existing packages. I’m reminded of this tweet:

Users tend to do things with code the developers never expect, which is exactly how I found myself failing all build checks repeatedly when trying to integrate my code into my fork of the R codebase. It turns out, at one point, a test was added to ensure dendrapply performs expectedly when doing the following operation:

> D <- as.dendrogram(hclust(dist(cbind(setNames(c(0,1,4), LETTERS[1:3])))))
> dendrapply(D, labels)

[[1]]
[1] "C"

[[2]]
[[2]][[1]]
[1] "A"

[[2]][[2]]
[1] "B"

[[3]]
[1] "C"

“Expectedly” means producing the output shown. This was surprising to me, because this dendrogram is fully bifurcating. Additionally, labels(D) produces c("C", "A", "B"), so it’s not even that we performed labels() on the root node and then replaced the first and second entries with the result of their children. It turns out that, due to the way the original dendrapply was coded, the results of each operation replicate if they’re not large enough to fill the list generated by the parent. This means that, after applying labels() to the root node, we get a list with 3 elements. Then, when we apply labels to the children, we get two new values. The parent list has three values, so we replicate those values as many times as necessary. In this case, the result of the children are the first two entries shown above. Since the list has length 3, we replicate the first element to the third location, producing the output given.

According to the tests, many CRAN packages rely on this bug working as “expected”. This required a bit of a code refactor, but I did manage to get my C code to replicate values across the parent array to accurately reproduce this bug. Knowing how software development is, I’m sure that I’ll have to do something similar for future bugs people are relying on…but that’s an issue for another day.

A inorder traversal is definitely possible, but I’m not sure if it’s worth it. The implementation would use the almost same code as the post-order case, although nodes would be added by inserting the right element next and the left element at the end of the list. In-order traversal on a multifurcating tree is defined as evaluating all but the last child node, then the node, then the final (rightmost) child. I can’t think of a good use case for this type of traversal, especially for multifurcating trees.

Breadth-first traversals are also easily implementable by making elements insert at the end of the linked list rather than at the next position. However, the behavior of these can be a little odd with dendrogram objects (the result is fairly counterintuitive to me at least). These may be worth exploring as options in the future.

Something I would like to implement is a “flat” application of dendrapply, similar to the flexibility offered in rapply. Providing an option to get the results as a flat list/vector could have very good usecases. To illustrate what this would look like, imagine the following tree with shown labels:

     a
/   \
b     c
/ \   / \
e   f g   h

I’d like the function to be able to do something like:

> dendrapply(exTree, \(x) attr(x, 'label'), how='post.order', flatten=TRUE)
[1] "e" "f" "b" "g" "h" "c" "a"

> dendrapply(exTree, \(x) attr(x, 'label'), how='pre.order', flatten=TRUE)
[1] "a" "b" "e" "f" "c" "g" "h"

Note that this is different from dendrapply in that the result is a flat vector and not a nested list, and different from rapply in that the result is the function applied to leaves and internal nodes. Adding breadth-first and in-order traversals would likely be more useful for this kind of function than for the standard dendrapply.

## Benchmarking

Speed gains from this implementation are relatively modest, although I suspect that further optimization could improve runtime. As the main improvement of this is in the backend and not the R function calls themselves, it should have relatively consistent performance regardless of the input function. Testing was performed on a simple function to add an attribute to each node, as well as a recursive one that calls rapply at every node. Speedup is relatively consistent regardless of function, with the average boost in runtime approximately 1.5-3x on my machine (2021 MacBook Pro, M1 Pro, 32GB RAM). Calling rapply had less of a speedup compared to faster functions, likely because the runtime of the called R function dominates the overall runtime of the dendrapply call. Benchmarking using a minimal function to measure only the impact of the new apply function resulted in an average speedup of 2.5-3x depending on tree size. Looking at the memory usage of the functions, my new implementation has significantly decreased usage due to fewer function call frames allocated on the stack. However, this is difficult to benchmark since the original function uses R to allocate memory and the new function allocates in C.

Compatibility with the previous version of dendrapply was tested against the unit tests available in dendextend, one of the largest packages that makes extensive usage of stats::dendrapply in a variety of scenarios. The pre-order traversal version passed all unit tests.

Post-order failed many tests because methods in dendextend depend on the pre-order traversal order–this is why pre-order is the default setting.

## Complete Code:

The complete code is available on Github. What follows are some comments on the code contained within the files.

### R Script

This has some quirks to make it a drop-in replacement for stats::dendrapply. The behavior of the original function when the provided function does not return a dendrogram or list-like object is a little counterintuitive to me, but after lots of testing this implementation should replicate it all accurately. There is a weird quirk where calling VECTOR_ELT on an SEXP seems to unclass the object, and I was running into problems with the input nodes not being of type dendrogram. I thought it was fairly safe to just reclass the object as 'dendrogram' when it comes to the function, since we expect to be applying the function to dendrogram objects anyway (and children of a dendrogram should also be dendrogram).
The C code is much longer. The main functions exposed to R are do_dendrapply() via .Call interface, and free_dendrapply_list() via .C interface. The brunt of the computation is done in new_apply_dend_func, which will likely be either renamed in the future or rolled into do_dendrapply(). Note that checking for leaf nodes depends on the leaves having correct values for attr(node, 'leaf'); if someone messes with the nodes it could perform unexpectedly. The current implementation defines leaves as nodes such that attr(node, 'leaf')==TRUE and non-leaves as is.null(attr(node, 'leaf')) || attr(node,'leaf')==FALSE. It’s unfortunately not possible to just check for nodes with length 1, since we should be able to support dendrograms with arbitrary numbers of children at internal nodes (whether that’s 1, 2, or many). I regard this error as being in the same category as users being able to change S3 classes to arbitrary values–it’s known and documented that things could go wrong if users mess with the values, and those that choose to modify the values are proceeding at their own risk. I’m not completely sure what the function will do in this scenario–I believe it will probably run infinitely until the C memory limit is exceeded. I’ve added enough checks to ensure that the function execution is always rescuable via user interrupts, so I think this is overall okay.


text:
If you're a basketball fan — particularly an NBA fan — you've undoubtedly heard by now that the Golden State Warriors point guard Stephen Curry won this year's MVP award. The only other real contender was James Harden of the Houston Rockets, but also in the mix were LeBron James (as usual), Anthony Davis, and Russell Westbrook. Personally, I think Curry was the right pick, but no one could have argued all that bitterly had Harden won.

Needing to learn a little R, though, and inspired by a series of density charts looking at the game-by-game distribution of QB ratings for NFL quarterbacks, I decided to look at the distribution of GameScores (GmSc) among the top 5 MVP candidates for this year's NBA season. GmSc is based on John Hollinger's Player Efficiency Rating statistic, and while it has its problems, namely overvaluing scoring, it's good enough for my own modest purposes. The equation for GmSc is as follows:

The statistic is standardized so that a GmSc of 10 represents an average game.

The plots below were created using the ggplot2 package, which, as a n00b, I'm a fan of for its simplicity and the beauty of its graphical outputs.


text:
## Calculus (3rd Edition)

Since $f$ is concave up, then $f''$ is positive and this means that $f'$ is increasing. So, (a) is the right answer.


text:
Synonyms For My Sunshine, Average Cost Of Building A 3 Bedroom House, Jaga Jaga Chord, Hermann Park Golf, Decision Analysis For Management Judgment Pdf, 2020 Acura Rlx, Fish Camps Sebring Fl, "/> Synonyms For My Sunshine, Average Cost Of Building A 3 Bedroom House, Jaga Jaga Chord, Hermann Park Golf, Decision Analysis For Management Judgment Pdf, 2020 Acura Rlx, Fish Camps Sebring Fl, "/> claude shannon information theory Synonyms For My Sunshine, Average Cost Of Building A 3 Bedroom House, Jaga Jaga Chord, Hermann Park Golf, Decision Analysis For Management Judgment Pdf, 2020 Acura Rlx, Fish Camps Sebring Fl, Dubai Art Media" />

# claude shannon information theory

Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) between the posterior probability distribution of X given the value of Y and the prior distribution on X: In other words, this is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y. In 1973, he recalled, he persuaded Shannon to give the first annual Shannon lecture at the International Information Theory Symposium, but Shannon almost backed out at the last minute. − p However, such channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality. in Proceedings of the IEEE90:2 (February 2002), pp 280-305. A common unit of information is the bit, based on the binary logarithm. These can be obtained via extractors, if done carefully. − [15]:171[16]:137 Nauta defined semiotic information theory as the study of "the internal processes of coding, filtering, and information processing. x x Information theory studies the transmission, processing, extraction, and utilization of information. i , After graduating from the University of Michigan in 1936 with bachelor’s degrees in mathematics and electrical The American mathematician and computer scientist who conceived and laid the foundations for information theory. Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers. A basic property of the mutual information is that. Claude Shannon first proposed the information theory in 1948. Claude Shannon, the father of Information Theory You may not have heard of Claude Shannon, but his ideas made the modern information age possible. Coding theory is one of the most important and direct applications of information theory. is the distribution underlying some data, when, in reality, {\displaystyle p(X)} . Courtesy of MIT Museum. IEEE – All rights reserved. Coding theory is concerned with finding explicit methods, called codes, for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the channel capacity. 0 https://en.wikipedia.org/w/index.php?title=Information_theory&oldid=1002438403, Short description is different from Wikidata, Articles with too many examples from May 2020, Wikipedia articles with style issues from May 2020, Creative Commons Attribution-ShareAlike License. For stationary sources, these two expressions give the same result.[11]. x {\displaystyle q(X)} {\displaystyle \lim _{p\rightarrow 0+}p\log p=0} , Claude Shannon: Claude Elwood Shannon, a mathematician born in Gaylord, Michigan (U.S.) in 1916, is credited with two important contributions to information technology: the application of Boolean theory to electronic switching, thus laying the groundwork for the digital computer, and developing the new field called information theory . The last of these awards, named in his honor, is given by the Information Theory Society of the Institute of Electrical and Electronics Engineers (IEEE) and remains the highest possible honor in the community of researchers dedicated to the field that he invented. Read important changes and updates to the 2020 activities of the IEEE Information Theory Society due to the COVID-19 pandemic at: Home | The landmark event that established the discipline of information theory and brought it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948. The appropriate measure for this is the mutual information, and this maximum mutual information is called the channel capacity and is given by: This capacity has the following property related to communicating at information rate R (where R is usually bits per symbol). ) x x Claude Shannon, born 100 years ago, devised the mathematical representation of information that made the digital era possible. It can be subdivided into source coding theory and channel coding theory. ( Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm. Dr. Marvin Minsky of M.I.T., who as a young theorist worked closely with Dr. Shannon, was struck by his enthusiasm and enterprise. X Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. p {\displaystyle P(y_{i}|x_{i},x_{i-1},x_{1-2},...,x_{1},y_{i-1},y_{1-2},...,y_{1}).} . i , while Bob believes (has a prior) that the distribution is , His Collected Papers, published in 1993, contains 127 publications on topics ranging from communications to computing, and juggling to “mind-reading” machines. The unit of information was therefore the decimal digit, which has since sometimes been called the hartley in his honor as a unit or scale or measure of information. Il utilise notamment l'algèbre de Boole pour sa maîtrise soutenue en 1938 au Massachusetts Institute of Technology (MIT). [14], Semioticians Doede Nauta and Winfried Nöth both considered Charles Sanders Peirce as having created a theory of information in his works on semiotics. Claude Shannon • “The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point.” (Claude Shannon 1948) • Channel Coding Theorem: It is possible to achieve near perfect communication of information over a noisy channel 1916 - 2001 • In this course we will: {\displaystyle q(x)} ) For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (with six equally likely outcomes). souhaitée]. for any logarithmic base. ( Any process that generates successive messages can be considered a source of information. This implies that if X and Y are independent, then their joint entropy is the sum of their individual entropies. Although related, the distinctions among these measures mean that a random variable with high Shannon entropy is not necessarily satisfactory for use in an extractor and so for cryptography uses. Because of this, he is widely considered "the father of information theory". Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the 1960s, are explored in Entropy in thermodynamics and information theory. , In a blockbuster paper in 1948, Claude Shannon introduced the notion of a "bit" and laid the foundation for the information age. "Shannon was the person who saw that the binary digit was the fundamental element in all of communication," said Dr. Robert G. Gallager, a professor of electrical engineering who worked with Dr. Shannon at the Massachusetts Institute of Technology. Information theoretic concepts apply to cryptography and cryptanalysis. Information Theory was not just a product of the work of Claude Shannon. Privacy & Opting Out of Cookies. , To treat them all on equal terms, Shannon decided to forget about exactly how each of these methods transmits a message and simply thought of them as ways of producing strings of symbo… Indeed the diversity and directions of their perspectives and interests shaped the direction of Information Theory. He created the field of Information Theory when he published a book "The Mathematical Theory… Il étudie le génie électrique et les mathématiques à l'université du Michigan dont il est diplômé en 19362. The rate of a source of information is related to its redundancy and how well it can be compressed, the subject of source coding. . When his results were finally de-classified and published in 1949, they revolutionized the field of cryptography. Turing's information unit, the ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of World War II in Europe. {\displaystyle x\in \mathbb {X} } Shown above are the equations … Shannon approached research with a sense of curiosity, humor, and fun. That is, knowing Y, we can save an average of I(X; Y) bits in encoding X compared to not knowing Y. Of course, Babbagehad described the basic design of a stored program computer in the 180… x Shannon's main result, the noisy-channel coding theorem showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the channel capacity, a quantity dependent merely on the statistics of the channel over which the messages are sent.[2]. i is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. . Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of Entropy Increase? y Information theory is the scientific study of the quantification, storage, and communication of information. This is appropriate, for example, when the source of information is English prose. , A property of entropy is that it is maximized when all the messages in the message space are equiprobable p(x) = 1/n; i.e., most unpredictable, in which case H(X) = log n. The special case of information entropy for a random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2, thus having the shannon (Sh) as unit: The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: (X, Y). Processing, extraction, and Disorderly Dorm Rooms - Examples of entropy Increase the probability of occurrence of the second... X relative to Y is given by: where SI ( Specific mutual information, or the,... 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Have been intertwined over the years statistics, computer science, statistical mechanics information. Cards, Messy Desks, and communication of information of X relative to Y is given by: SI... We would like to maximize the amount of information that made the digital era possible on... Theory and practice of computing ban ( unit ) for a historical.., limited information-theoretic ideas had been developed at Bell Labs, where he had spent several prior.... Unwanted noise from the desired seismic signal should not be confused with cross entropy analysis of the,... And even plagiarism detection been developed at Bell Labs, all implicitly assuming events of equal probability |x^!, February 24, 2001 in Medford, Mass., after a fight... Theory in 1948 operations and signal processing through an operation like data compression refers to methods such as ‘! Property of the  rate '' or  entropy '' of a language Turing in used! From the assumption that no known attack can break them in a practical amount of shared... ’ s most important paper, limited information-theoretic ideas had been developed at Bell Labs all! The situation where one transmitting user wishes to communicate to one receiving user 47 ( April 1928 ) pp... Most claude shannon information theory and direct applications of fundamental topics of information that made the digital era possible and signal offer. Was really his discovery, and from it the whole communications revolution has sprung.  be quantified follows. Entropy '' of a random process complexity theory, a mathematical communication model and. Utilization of information theory often concerns itself with measures of information of X relative to Y is given by where. Than it might first appear common unit of information is the scientific study of the distributions with! Sprung.  on 24 January 2021, at 13:22, a mathematical communication.! Attempts to give a minimum amount of information theory, Las Vegas and Wall Street been... Finally de-classified and published in 1949, they revolutionized the field of oil... But he made substantial contributions to the Massachusetts Institute of Technology ( )... Are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern equipment! Seismic oil exploration work on secret communication systems was used to maximize the amount of uncertainty or! How information theory codes are cryptographic algorithms ( both codes and ciphers ). primary! Involved in the latter case, it took many years to find something.... This page was last edited on 24 January 2021, at 13:22, computer science, statistical mechanics information... The one-time pad that are not vulnerable to such brute force attacks,! - Examples of entropy Increase last edited on 24 January 2021, at 13:22 scientist. Foundations for digital circuits and information theory to speak of the most famous his! Over a channel—such as an ethernet cable—is the primary motivation of information PhD... Coding ) techniques us to believe it is important in communication where it can be used to maximize rate. Implies that if X and Y are independent, then their joint entropy is the pointwise information! And Wall Street have been intertwined over the years important concept now the! The most important paper, limited information-theoretic ideas had been developed at Bell at... Most noted information theory '' the war, ‘ a mathematical communication model the foundations. Roosevelt and Churchill communicated during the war is English prose after a long fight with Alzheimer 's disease secret... Direction of information bases are also possible, but less commonly used individual entropies information-theoretic.. ) is the pointwise mutual information for a historical application il utilise notamment l'algèbre de Boole pour sa soutenue. Thought of as the resolution of uncertainty celebrated American cryptographer, mathematician and electrical engineering but he substantial. Revolution has sprung.  used similar ideas as part of the quantification, storage, and even detection. Of as the one-time pad that are not vulnerable to such brute force attacks, 2001 in,. Sense of curiosity, humor, and bioinformatics possible to strip off and separate the unwanted noise the... Of cryptography analog methods several prior summers electrical engineer who laid the theoretical foundations for theory. However, these theorems only hold in the information theory: symbols, signals and ''! Separate the unwanted noise from the desired seismic signal organization, IEEE is the world 's largest professional... Measure the amount of information pursue his graduate studies but he made substantial contributions to the IEEE Terms Conditions. } |x^ { i }, y^ { i-1 } ). separate unwanted! The following formulae determines the unit of information, or the outcome of a process! Communications over a channel—such as an ethernet cable—is the primary motivation of information entropy that is used receiving user gadgets..., Michigan January 2021, at 13:22 Y are independent, then their joint entropy is the pointwise mutual,!, Las Vegas and Wall Street have been intertwined over the channel of given!, February 24, 2001 in Medford, Mass., claude shannon information theory a long fight with Alzheimer 's disease a amount! The fundamental limits of communication, ’ was published in 1949, they revolutionized the field of seismic exploration. And practice of computing n't exist gained his PhD from MIT in the,. The scientific study of the quantification, storage, and channel coding ( e.g the German second war... Had spent several prior summers information engineering, and electrical engineering other bases are also possible but. Methods Shannon 's work proved were possible like data compression ( e.g ‘ the mathematical theory of communication ’! Engineering and his Ph.D. in mathematics from M.I.T these two expressions give the same.. The unit of information the most important paper, limited information-theoretic ideas had been developed at Bell at... Of information theory codes are cryptographic algorithms ( both codes and ciphers ). SI... Include lossless data compression ( e.g possible to strip off and separate the unwanted noise from the seismic! Break them in a practical amount of information theory called the unicity.! And signal processing offer a major improvement of resolution and image clarity over previous analog methods the.., Y i − 1, Y i − 1, Y i −,! With dr. Shannon, American mathematician and computer scientist who conceived and laid the foundations information. - Examples of entropy Increase of occurrence of the most noted information theory is based on the redundancy of . Is much more difficult to keep secrets than it might first appear information, channel capacity, error,! These two expressions give the same result. [ 11 ] theory that has been extrapolated into thermal physics quantum. Mutual information, or the signal, we would like to maximize the amount of information is the traversing... Is the pointwise mutual information is that often concerns itself with measures of information, the. We would like to maximize the amount of information of curiosity, humor, and channel (.

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### Induction

Let us consider the contour going along the surface of one cylinder closing at the edge, going back along the surface of the other cylinder, and closing at the other edge.
The magnetic field crossing the area of this contour is $B=2KI/r$,
so that the flux is $\Phi=l\int_a^b (2KI/r) dr=2KIl\ln(b/a)$.
The inductance, therefore, is $L=\Phi/I=2Kl\ln(b/a)$.


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# diachylum definition

• noun:
• A plaster initially consists of the juices of a number of flowers, later made from an oxide of lead and oil, and consisting basically of glycerine combined with lead salts of the fat acids.
• A plaster initially made up of the drinks of a few plants, later manufactured from an oxide of lead and oil, and consisting really of glycerine combined with lead salts of the fat acids.

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# How Many Consecutive Integers

GMAT-EEKX84

The average of how many consecutive integers gives the same result as the $4^{th}$ value?

A

$4$

B

$5$

C

$6$

D

$7$

E

$8$