text:
# A grid of images to fill A4 -paper for economic printing with some space for scissors

I want to print a A4 paper full of the same QR image. I want more elegant way to this answer here, yes I can get it working like that but I want to leave just enough space for scissors and no space near the edges and equal spacing (and is it possible to do that with for -loop or otherwise simpler?) -- possible?

-

Look, ma! No loops! :)

Borrowed from the TeXbook, exercise 21.8 (p. 225).

\documentclass{minimal}
\usepackage[a4paper,margin=2mm]{geometry}
\usepackage{graphicx}

\sbox0{%
\includegraphics{image}%
\kern2mm
\vrule width 0pt depth 2mm}

\setlength{\parindent}{0pt}

\begin{document}

\null

\end{document}

-
I need to learn to save votes for moments like this one. :) A tear of joy fell from my eye. A true capolavoro! I wish I could vote it 10 times. :) –  Paulo Cereda Jan 16 '12 at 23:44
@PauloCereda Just DEK's code adapted to the situation. –  egreg Jan 16 '12 at 23:49
Your image has white space around it. Try putting it into an \fbox to see how much. With the macros I proposed, there are exactly 2mm between images. –  egreg Jan 17 '12 at 0:34
@hhh GIF files are not supported by TeX programs. Convert it to PNG. –  egreg Jan 17 '12 at 21:36
@hhh Probably the image is too big. Check its dimensions with the \fbox method I mentioned. –  egreg Jan 17 '12 at 21:49

Here is one way using the forloop package which implements an elementary nested/recursive for loop using counters. Given some counter <cnt>, the interface

\forloop[<step>]{<cnt>}{<start>}{<condition>}{<loop content>}

initializes <cnt> to <start> and executes <loop content> until <condition> is met. An optional first parameter <step> (default is 1) can be set to modify the counter step size:

\documentclass{article}
\usepackage[margin=2cm,a4paper]{geometry}% http://ctan.org/pkg/geometry
\usepackage{forloop}% http://ctan.org/pkg/forloop
\usepackage{pst-barcode}% http://ctan.org/pkg/pst-barcode
\begin{document}
\setlength{\baselineskip}{2cm}%
\newcounter{qrcodesA}\newcounter{qrcodesB}%
\forloop{qrcodesA}{0}{\value{qrcodesA}<12}{% Rows
\noindent%
\forloop{qrcodesB}{0}{\value{qrcodesB}<9}{% Columns
\psbarcode{tex.stackexchange.com}{}{qrcode}\hspace{2cm}}\hspace*{\fill}\\%
}%
}
\end{document}

Using \psbarcode directly creates a dimensionless object (width and height of 0pt). The spacing of 2cm is arbitrary, as is the setting for \baselineskip. You can modify this to suit your scissor-cutting expertise.

This requires a latex->dvips->ps2pdf or xelatex compilation sequence.

-
...this covers the corners/sides apparently +1. –  hhh Jan 17 '12 at 0:33
@hhh: You can modify the row and column counters, as well as the margins set by geometry. I chose margin=1cm, but you can reduce that to your liking as long as your printer can handle it. Is that what you're referring to by "covering the corners/sides"? –  Werner Jan 17 '12 at 1:09
there is no "includegraphics"? This is nice! It generates the images directly from the text! (needless to have middle image) Is it possible to output the ps to image formats such as png? –  hhh Jan 17 '12 at 21:08
@hhh: The picture is generated by LaTeX via the package pst-barcode. It has a function (or command) \psbarcode{...}{}{qrcode} which generates the output since the structure for producing QR codes is known. The benefit of producing it on-the-fly means that you can literally create your own QR code for every image that you produce, rather than have them all the same. My code produces the exact same QR code, pointing to TeX.SX. Yes, you can port the output to an image form. –  Werner Jan 17 '12 at 21:14
@hhh: You won't be able to output directly to PNG, but you can output to PS or PDF and convert to PNG. Also, this holds for either the entire page (as in my example), or a single QR code. –  Werner Jan 17 '12 at 21:16

You can loop using Knuth's \loop and \repeat. Here some minimal code. You will need to experiment with the gutter between the images, as well as the numbers and scaling of the image.

\documentclass[a4paper]{article}
\usepackage[top=.1cm, left=0.1cm,bottom=.1cm,right=0.1cm]{geometry}
\usepackage{graphicx}
\def\qcode{\includegraphics{./graphics/qrcode}}
\parindent0pt
\pagestyle{empty}
\begin{document}
\newcounter{cnt}
\setcounter{cnt}{0}
\loop
\qcode
\stepcounter{cnt}
\ifnum\thecnt<185
\repeat
\end{document}

Here is the result. You can also print the qrcode with LaTeX using pstricks.

-

run with xelatex or latex->dvips->ps2pdf

\documentclass{article}
\usepackage{multido,pst-barcode}
\def\xstrut{\rule{0pt}{2cm}}
\parindent=0pt
\pagestyle{empty}
\begin{document}
\multido{\iA=0+1}{54}{%
\makebox[2cm]{\xstrut\psbarcode{tex.stackexchange.com}{}{qrcode}} }
\end{document}

-

One way would be to compute the number that can fit on one line based on the \WidthOfImage, and the number of rows based on the \HeightOfImage (adjusting for the spacing required by the scissors):

\documentclass{article}

\usepackage[demo]{graphicx}
\usepackage[showframe]{geometry}
\usepackage{pgf}
\usepackage{pgffor}

\newlength{\Separation}%
\setlength{\Separation}{0.1cm}% Space for the "scissors"

\newlength{\WidthOfImage}%
\setlength{\WidthOfImage}{2.0cm}%

\newlength{\HeightOfImage}%
\setlength{\HeightOfImage}{2.0cm}%

\pgfmathtruncatemacro{\NumberOfRows}{\textheight/(\HeightOfImage+\Separation)}%
\pgfmathtruncatemacro{\NumberOfColumns}{\textwidth/(\WidthOfImage+\Separation)}%

\begin{document}
%NumberOfRows = \NumberOfRows\par
%NumberOfColumns = \NumberOfColumns\par
%
\foreach \x in {1,...,\NumberOfRows}{\noindent%
\foreach \y in {1,...,\NumberOfColumns}{%
\includegraphics[width=\WidthOfImage,height=\HeightOfImage]{image}%
\hspace{\Separation}%
}%
\newline%
}%
\end{document}

-


text:
Solvable group

# Solvable group

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, more specifically in the field of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, a solvable group (or soluble group) is a group that can be constructed from abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s using extensions
Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...

. That is, a solvable group is a group whose derived series terminates in the trivial subgroup.

Historically, the word "solvable" arose from Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

and the proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

of the general unsolvability of quintic equation. Specifically, a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

if and only if the corresponding Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

is solvable.

## Definition

A group is called solvable if it has a subnormal series whose factor groups are all abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

, that is, if there are subgroups such that is normal in , and is an abelian group, for .

Or equivalently, if its derived series, the descending normal series
where every subgroup is the commutator subgroup
Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group....

of the previous one, eventually reaches the trivial subgroup {1} of G. These two definitions are equivalent, since for every group H and every normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

N of H, the quotient H/N is abelian if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

N includes H(1). The least n such that is called the derived length of the solvable group G.

For finite groups, an equivalent definition is that a solvable group is a group with a composition series
Composition series
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence...

all of whose factors are cyclic groups of prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

. This is equivalent because a finite abelian group has finite composition length, and every finite simple
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

abelian group is cyclic of prime order. The Jordan–Hölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to nth roots (radicals) over some field. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic
Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...

to Z itself, it has no composition series, but the normal series {0,Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.

## Examples

All abelian groups are solvable - the quotient
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

A/B will always be abelian if A is abelian. But non-abelian groups may or may not be solvable.

More generally, all nilpotent group
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...

s are solvable. In particular, finite p-groups
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...

are solvable, as all finite p-groups
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...

are nilpotent.

A small example of a solvable, non-nilpotent group is the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

S3.
In fact, as the smallest simple non-abelian group is A5, (the alternating group of degree 5) it follows that every group with order less than 60 is solvable.

The group S5 is not solvable — it has a composition series {E, A5, S5} (and the Jordan–Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to A5 and C2; and A5 is not abelian. Generalizing this argument, coupled with the fact that An is a normal, maximal, non-abelian simple subgroup of Sn for n > 4, we see that Sn is not solvable for n > 4. This is a key step in the proof that for every n > 4 there are polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s of degree n which are not solvable by radicals (Abel–Ruffini theorem
Abel–Ruffini theorem
In algebra, the Abel–Ruffini theorem states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.- Interpretation :...

). This property is also used in complexity theory in proof of Barrington's theorem
NC (complexity)
In complexity theory, the class NC is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem is in NC if there exist constants c and k such that it can be solved in time O using O parallel processors...

.

The celebrated Feit–Thompson theorem
Feit–Thompson theorem
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...

states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.

Any finite group whose every p-Sylow subgroups is cyclic is a semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...

of two cyclic groups, in particular solvable. Such groups are called Z-group
Z-group
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups:* in the study of finite groups, a Z-group is a finite groups whose Sylow subgroups are all cyclic....

s.

## Properties

Solvability is closed under a number of operations.
• If G is solvable, and there is a homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

from G onto H, then H is solvable; equivalently (by the first isomorphism theorem), if G is solvable, and N is a normal subgroup of G, then G/N is solvable.
• The previous property can be expanded into the following property: G is solvable if and only if both N and G/N are solvable.
• If G is solvable, and H is a subgroup of G, then H is solvable.
• If G and H are solvable, the direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

G × H is solvable.

Solvability is closed under group extension
Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...

:
• If H and G/H are solvable, then so is G; in particular, if N and H are solvable, their semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...

is also solvable.

It is also closed under wreath product:
• If G and H are solvable, and X is a G-set, then the wreath product
Wreath product
In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. Wreath products are an important tool in the classification of permutation groups and also provide a way of constructing interesting examples of groups.Given two groups A and H...

of G and H with respect to X is also solvable.

For any positive integer N, the solvable groups of derived length at most N form a subvariety
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...

of the variety of groups, as they are closed under the taking of homomorphic
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

images, subalgebra
Subalgebra
In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...

s, and (direct) products
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.

## Burnside's theorem

Burnside's theorem states that if G is a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

of order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

where p and q are prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s, and a and b are non-negative integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, then G is solvable.

### Supersolvable groups

As a strengthening of solvability, a group G is called supersolvable (or supersoluble) if it has an invariant normal series whose factors are all cyclic. Since a normal series has finite length by definition, uncountable groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group A4 is an example of a finite solvable group that is not supersolvable.

If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:
cyclic
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

< abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

< nilpotent
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...

< supersolvable
Supersolvable group
In mathematics, a group is supersolvable if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability.-Definition:Let G be a group...

< polycyclic
Polycyclic group
In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups...

< solvable < finitely generated group.

### Virtually solvable groups

A group G is called virtually solvable if it has a solvable subgroup of finite index. This is similar to virtually abelian. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.

### Hypoabelian

A solvable group is one whose derived series reaches the trivial subgroup at a finite stage. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. A group whose transfinite derived series reaches the trivial group is called a hypoabelian group
Hypoabelian group
*"Hypoabelian group" is an archaic name for an orthogonal group over a field of characteristic 2.*A hypoabelian group can mean a group whose perfect core is trivial...

, and every solvable group is a hypoabelian group. The first ordinal α such that G(α) = G(α+1) is called the (transfinite) derived length of the group G, and it has been shown that every ordinal is the derived length of some group .

format: plain


text:
main-content

## Über dieses Buch

This volume contains selected papers which had been presented during CISCON 2018. The papers cover the latest trends in the fields of instrumentation, sensors and systems, industrial automation & control, image and signal processing, robotics, renewable energy, power systems and power drives, with focus on solving the current challenges faced in the field of instrumentation and control engineering. This volume will be of use to academic and industry researchers and students working in this field.

## Inhaltsverzeichnis

### Dynamic Analysis of an Integrated Reformer-Membrane-Fuel Cell System with a Battery Backup and Switching Controller for Automotive Applications

Abstract
Fuel cells have been considered as an ideal source of energy in the future power generation applications due to its pollution-free nature, noise-free operation and better efficiency. Direct storage of hydrogen in specially designed tanks for automobiles running on fuel cells is not a viable option due to several drawbacks associated with safety and space limitations. To overcome the challenges of direct onboard storage of hydrogen, storing hydrocarbons rich in hydrogen and suitably reforming it to produce hydrogen using several reforming techniques seems to be an acceptable option. Using available gas purification techniques such as palladium membrane-based gas separation, pure hydrogen gas can be extracted from a mixture of other gases and can be fed to the fuel cell for generating power. In this work, a mathematical model of the battery system is analyzed along with a switching controller operating based on an energy management policy. The switching controller switches between battery and fuel cell to ensure a delay-free delivery of the power to the external load. A case study on the dynamic behavior of the integrated system under set point changes in the power demand is analyzed in the presence of a battery backup and a switching controller.
P. S. Pravin, Ravindra D. Gudi, Sharad Bhartiya

### Design and Implementation of Fuzzy Logic Controller on MPSoC FPGA for Shell and Tube Heat Exchanger

Abstract
In this paper design of fuzzy logic controller (FLC) on system on chip field programmable gate array (SoC FPGA) for Shell and Tube Heat Exchanger (STHE) is described. STHE is interfaced with Compact Reconfigurable Input Output (CRIO) based FPGA through data acquisition (DAQ) card. For designing controller on chip, National Instruments (NI) CRIO-9101 is reconfigured with FLC code. Interfacing is carried out with NI analog input-output device embedded with NI CRIO-9012 microcontroller and NI CRIO-9101 FPGA module. A graphical program is developed for real-time control of the process plant using CRIO based FPGA. FPGA acts as a standalone processor instead of PC for controlling the STHE. The fuzzy logic controller designed on FPGA is compared with benchmark controller in real time temperature control of STHE. It is demonstrated that the controller proposed outperformed conventional controller.
Rajarshi Paul, C. Shreesha

### Simultaneous Exploration and Coverage by a Mobile Robot

Abstract
In this paper, we propose a problem of simultaneous exploration and coverage for a mobile robot, combining the problems of area coverage with exploration and mapping. The primary task here is to completely cover an initially unknown region. Here we combine the advantages of online and off-line coverage path planning algorithms by using the exploration as an aid. The robots perform intermittent exploration during coverage in order to update the map of the environment, which in turn is used to generate the coverage path. We illustrate and demonstrate the problem using the off-line version of Spanning Tree Coverage algorithm with a frontier-based exploration strategy. The simulation results demonstrate that the robot successfully achieves complete and non-repetitive coverage.

### Tracking Control and Deflection Suppression of an AMM Modelled TLFM Using Backstepping Based Adaptive SMC Technique

Abstract
Manipulators are widely used in all areas of science and technology. Effective trajectory tracking and quick deflection suppression are the two main aspect of research for a flexible manipulator. The paper reports aperiodic signal like trajectory tracking control for a planar assumed modes modelled two-link flexible manipulator (TLFM). The aperiodic chaotic signal is used as a desired trajectory for the TLFM. Thus, designing of a robust controller for the aperiodic signal tracking control is a challenging task. A backstepping based adaptive SMC technique is designed for the considered problem. In adaptive SMC, the gain of the switching control law is estimated online. The effectiveness of the considered controller is compared to an available backstepping controller. It is found that the designed backstepping based adaptive SMC perform better in terms of smaller tracking time, quick tip deflection suppression and lesser, smoother control efforts. Proposed trajectory strategy is validated on a two-link flexible manipulator in MATLAB simulation environment.
Kshetrimayum Lochan, Jay Prakash Singh, Binoy Krishna Roy

### Multi-robot Coverage Using Voronoi Partitioning Based on Geodesic Distance

Abstract
In this paper we propose Geodesic-VPC, a “partition” and “cover” strategy for a multi-robot system using Voronoi partitioning based on geodesic distance metric in the place of the usual Euclidean distance. Each robot is responsible for covering the corresponding geodesic-Voronoi cell using a single-robot coverage strategy. The proposed partitioning scheme ensures that Voronoi cells are contiguous even in the presence of obstacles. We demonstrate that if the single-robot coverage strategy is capable of providing a complete and non-repetitive coverage, then the proposed Geodesic-VPC strategy provides a complete and non-repetitive coverage. We use spanning tree-based coverage algorithm as the underlying single-robot coverage strategy for the purpose of demonstration, though any existing single-robot coverage algorithm can be used.
Vishnu G. Nair, K. R. Guruprasad

### Secure Communication Using a New Hyperchaotic System with Hidden Attractors

Abstract
Objectives of the paper are to (i) develop a new hyperchaotic system having hidden attractors and (ii) to show the applications using the new system in the form of secure communication. New system proposed in the paper has a stable equilibrium, hence considered under the class of the hidden attractors dynamical system. Dynamical characteristics of the novel system is confirmed using some numerical means like phase portrait, Poincaré map and Lyapunov spectrum plot. The applications of the new system are shown by encrypting and decrypting a sinusoidal signal and sound wave. Secure communication is achieved by designing a proportional integral (PI) based sliding mode control (SMC). MATLAB simulation results validate and ensure that the objectives are satisfied.
Jay Prakash Singh, Kshetrimayum Lochan, Binoy Krishna Roy

### Manhattan Distance Based Voronoi Partitioning for Efficient Multi-robot Coverage

Abstract
In this paper we address the problem of area coverage using multiple cooperating robots. One of the main concerns of using multiple robots is of avoiding repetitive coverage apart from complete coverage of the given area. Partitioning the area to be covered into cells and allotting one each cell to each of the robots for coverage is a simple and elegant solution for this problem. However, the spacial partitioning may lead to additional problems leading to either incomplete coverage or coverage overlap near the partition boundary. We propose a manhattan distance based Voronoi partitioning scheme of $$2D\times 2D$$ gridded region, where D is the size of the robot footprint. We show that the proposed partitioning scheme completely eliminates coverage gaps and coverage overlap using illustrative results.
Vishnu G. Nair, K. R. Guruprasad

### Deposition of ZnO Thin Film at Different Substrate Temperature Using RF Sputtering for Growth of ZnO Nanorods Using Hydrothermal Method for UV Detection

Abstract
The growth properties of ZnO Nanorods was studied on different seed layers by the deposition of Zinc oxide (ZnO) thin film on SiO2/Si substrate by RF sputtering at two different conditions, i.e., one at room temperature and another at 400 °C substrate temperature. Surface morphology of the seed layer was studied by X-Ray Diffraction and Atomic Force Microscopy (AFM). Low cost hydrothermal method was employed for the growth of ZnO Nanorods on both the seed layers. The structural properties of ZnO nanorods were characterized by Field Emission Scanning Electron Microscope (FESEM). The FESEM images showed the proper alignment and orientation of ZnO nanorods grown on both the seed layers. The I-V measurements were carried out at room temperature under dark light and Ultraviolet (UV) light source. In order to examine the UV detection, MSM (Metal–Semiconductor–Metal) photodetector was fabricated and responsivity was measured for the nanorods grown on both seed layers. The better responsivity and contrast ratio of ZnO nanorods based UV detector was observed in case of 150 nm seed layer deposited at 400 °C.
Basavaraj S. Sannakashappanavar, C. R. Byrareddy, Sanjit Varma, Nandini A. Pattanshetti, Aniruddh Bahadur Yadav

### 3D Printable Modules for Manually Reconfigurable Manipulator with Desired D-H Parameters

Abstract
Modular robots are designed to increase the utilization of robots by modularizing their architecture. We discuss manually reconfigurable manipulators, where a manipulator of desired kinematic configuration is built by assembling the available modules. In the case of a serial-link manipulator with revolute joints, the joint angle is a variable. Out of the remaining three D-H parameters, namely, link-length, link-offset and link-twist, the twist angle influences the workspace the most. This work proposes a conceptual design and fabrication of individual modules which can be assembled to obtain a modular manipulator with desired kinematic configuration in terms of twist angles between any two consecutive joints. We also discuss possible provisions for length adjustment of a link. Designed modules are fabricated using 3D printer. As we focus on manually reconfigurable manipulators, simplicity of individual modules, in terms design, fabrication, and assembly, has been given higher priority, in contrast to similar designs available in the literature.

### FIR Filter Design Technique to Mitigate Gibb’s Phenomenon

Abstract
A unique methodology employing a linear phase finite impulse response (FIR) low pass filter (LPF) was proposed with an attempt to mitigate passband and stopband ripples due to Gibb’s phenomenon. The three regions of the filter response in the frequency domain are approximated using trigonometric functions. The proposed filter model achieved a sharp transition of 2π, fairly flat passband and a stopband attenuation of 40 dB. Our algorithm suppressed the oscillations near the edge of the transition region as well as in the passband region, reducing the Gibb’s phenomenon from the conventional passband ripples from 18% to as low as 2%. Thus a threefold satisfactory performance was achieved in all the three bands namely passband, transition and stopband. Our proposed linear phase FIR LPF was effectively used to filter out power line interference and higher unwanted frequencies from the real time electroencephalogram signals.
Niyan Marchon, Gourish Naik

### PLS-Based Multivariate Statistical Approach for Soft Sensor Development in WWTP

Abstract
Multivariate Statistical Process Control is a projection method of projecting a high- dimensional model space with a number of measured variables to a low-dimensional space. The different methods include Principal Component Analysis, Principal Component Regression, Partial Least Squares Regression, Finite Impulse response, Autoregressive exogenous input and autoregressive moving average, etc. The advantage of using Multivariate Statistical Process Control is that it identifies low-dimensional quality process data while reducing the variability in the process and increasing the product quality. The paper aims to find the low- dimensional information-rich space for soft sensor design using Partial Least Squares-based Multivariate Statistical Process Control technique for the controlling variables in a Wastewater Treatment Plant. The input considered here is the stored data of the actual process variables obtained from the plant. This is carried out using the 14 days data for the three weather conditions, dry rain and storm available from the benchmark model. The performance of the applied method is verified using scatter plot and R-squared.


text:
# Quod Erat Demonstrandum

## 2008/11/24

### what an abundant Saturday

Filed under: Family,Report — johnmayhk @ 4:13 下午
Tags:

What an abundant Saturday (2008-11-22).

In the morning, I’d attended the seminar organized by the Hong Kong Association for Science and Mathematics Education (香港數理教育學會). Both speakers were vice principals, mr. 葉志豪 and mr. 潘偉強 shared something about the students’ performance in recent CE mathematics paper one and two. Although mathematics teachers may found the subject matter discussed was quite familiarized, it was still very useful for us, teachers, to adjust the way of teaching.

Some topics were not taught deeply in the syllabus but students may come across, as for example, polar coordinates and centres of triangles.

The topic of polar coordinate system was discussed in a short period of time in F.1, and there will be no further discussion throughout the following years. For centres of triangles, like centroid, circum-centre, in-centre, students had seen them in F.3, but the way of teaching may not fully equip them to cope with problems like: is circumcentre always lie inside the triangle, say. They suggested that teachers should not spend much time going into details in junior forms, just give extra exercises during the revision sections in F.5 classes.

In the Question and Answer part, one teacher (not a young one) queried something about the marking scheme of a mathematics CE paper.

1. Why 3 marks should be granted to the following easy question?

Simplify $\frac{a^3b^{-2}}{a^5}$ (sorry, I forget about the exact indices)

2. Why not giving 0.5 mark or setting the total score of mathematics paper to be 200 for detailed marking?

The speakers expressed their opinions, many teachers just felt relaxing and laughed, but I looked at the questioner’s face, he was taking it so serious, no smile.

OK, I don’t say too much, just feel a bit annoying about the OSM (on-screen marking).

What a ‘kai’ guy I was, I wore a hat and brought my heavy guitar to attend that seminar. Then I ran to join my wife to attend another 3-hour seminar. The topic was “好丈夫好太太是訓練出來的”. The speaker was 理盧幼慈教授. More than 300 people took part in the talk.

I suggested this seminar to my wife because I think it is worthwhile. Joining the talk was not representing that we have some problems in marriage, it is just a kind of reflection.

My wife and I also felt that we learnt something useful in the talk, as for example “Same day request is always no". Norma Leben (理盧幼慈) was still presenting in a humorous way, though it was the 40th (something) talk on the same topic, she claimed. “Power of the last speaker" is an old ‘trick’ I’d come across some times ago. Anyway, it was joyful to take seminar together with my wife.

After the seminar, my wife and I went to a friend’s home to have some practice about the singing performance in a wedding ceremony. I had not played my guitar for ages, I felt a bit pain in my fingers. We finished the practice in more than one and a half hour, then my wife and I had a rich dinner and she needed to buy accessaries for the wedding. Urm, quite a lot of time spent. But it was comprehensible, it was a great chance to be ‘free from our son’ and did something hurry!

Getting home, we were really exhausted.

## 2 則迴響 »

1. 容我指出一個常見文法錯誤.
I felt a bit “painful" in my fingers.

迴響 由 lanven — 2008/11/25 @ 1:44 上午 | 回覆

2. Thank you lanven, 受教了。

迴響 由 johnmayhk — 2008/11/25 @ 10:38 上午 | 回覆


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				   1. Find the set of Points where the function of f(x)=tan[∏2]x-tan[-∏2]x
2. Let a real function f is given by f(x) =n(x1/n-1), x>0. If f(x) satisfies 97f(1/x)+Kf(x)=0, then find the value of K
Solution to the above two problems may pl be give?n? ?i?f? ?p?o?s?s?i?b?l?e?.
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6 years ago

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										If 5f(x)+3f(1/x)=x+2 and y=xf(x), then dy/dx at x=1
5f(x)+3f(1/x)=x+2
25f(x)+15(1/x)=5x+10…………………..(i)
Replacing x with 1/x,
5f(1/x)+3(x)=1/x+2
15f(1/x)+9f(x)=3/x+6………………………(ii)
(i)-(ii)
16f(x)=5x-3/x+4
f(x)=5x/16-3/16x+1/4
y=xf(x)=5x2/16-3/16+x/4
dy/dx=10x/16+1/4
dy/dx at x=1 is 10/16+1/4=7/8

6 years ago

# Other Related Questions on Algebra

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The answer is -1+ as shown below......................................................................................

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if points wid position vectors 60i+3j,40i-8j,ai-52j are collinear find a

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More Questions On Algebra


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# Thevenin equivalent with dependent source

Well, I'm trying to find the Thevenin equivalent of this circuit:

As you can see, the only current source is a dependent source. However, the control variable is in a wire where, if nothing is connected between A and B, is $i_\Delta = 0$ A, and that makes the source 0 A. However, if I connect an impedance $Z$ to the circuit, I will have current, and a Thevenin equivalent. I don't know how to solve this problem. I've bee searching here for a while, and I found this: Thévenin equivalent with a dependent source . I can see how is solved that case. However, I still don't know how to apply that method to my circuit.

Any help about finding the Thevenin equivalent is appreciated n.n

It's actually quite straightforward to find the Thevenin impedance of this circuit.

The equivalent impedance looking into the port ab is defined by:

$$Z_{ab} \equiv \frac{V_{ab}}{I_{ab}} = \frac{V_{ab}}{i_{\Delta}} = Z_{th}$$

But you can write by inspection a simple KVL equation for $V_{ab}$ in terms of $i_{\Delta}$:

$$V_{ab} = i_{\Delta}(-jX_{1uF} + 10k\Omega) + 200 i_{\Delta} \cdot 100 \Omega) = i_{\Delta}(-jX_{1uF} + 30k)\Omega$$

Generally speaking, to find the Thevenin equivalent of a circuit with only dependent source(s), you must be sure to "activate" the dependent source(s) with a test source.

This is what was done above. We solved for the voltage across the port due to a test current source, $I_{ab}$ which, in this case, equals the controlling variable thus making this problem particularly easy to solve.

• Great! Thank you! Jan 5, 2014 at 19:53


text:
# QED Lagrangian lead to self-interaction?

1. Jul 16, 2008

### per.sundqvist

Hi, I'm not very good at QFT, but starting from Dirac's QED Lagrangian (look for example at: http://en.wikipedia.org/wiki/Quantum_electrodynamics" [Broken] )

$$L=\bar{\psi}\left(\gamma^{\mu}\left(i\hbar\frac{\partial}{\partial\mu}-q(A^{ext}_{\mu}+A_{\mu})\right)+ mc\gamma^0 \right)\psi-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}$$

From here we derive Dirac's equation and Maxwell's equations. Now omitting all derivation steps, every thing could be summarized into the non-relativistic limit as:

$$\left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}- \frac{q^2}{4\pi\epsilon}\int\frac{\mid\psi(\vec{r})\mid^2}{\mid \vec{r}-\vec{r}^{\prime} \mid}d^3\vec{r}^{\prime}\right)\psi=E\psi$$

This is in fact what the QED Lagrangian result in (ignoring the very small contribution from the magnetic vector potential A for simplicity), but the effective Schrödinger equ looks more like Kohn-Sham equation for a single particle. But is this correct?

The Coulomb integral suggest that the electron is self-interacting with itself! (compare coulomb blocking) I thought that the correct limit has to be Schrödinger equation, but with some funny coupling term (but not so strong as coulomb self-interaction). Is it the field operators that are needed in some way to vanish this term, when it is not "appropriate"?

Last edited by a moderator: May 3, 2017
2. Jul 17, 2008

### akhmeteli

I hoped somebody else would answer this question:-) I am afraid I have neither a clear answer nor time to sort this out, but I suspect this question is considered in Barut's works on self-field electrodynamics and in nightlight's posts in this forum.

Last edited by a moderator: May 3, 2017
3. Jul 18, 2008

### Maaneli

YES!!! you guys are correct! That is the correct form of the equation. Indeed this is Barut's SFQED.

Now you just have to put in the self-interaction from the vector potential, and you can do all of QED with a first-quantized formalism!

Last edited: Jul 18, 2008
4. Jul 18, 2008

### Maaneli

Also, you'll notice that on this view, the physically correct Schroedinger equation for a charged particle always has those self-interaction terms, and therefore the physically correct Schroedinger (or Dirac) equation is not actually a linear equation, but a nonlinear integro-differential equation. This has profound implications for the interpretations of QM. Most notably, the Everett MWI interpretation is not consistent with a Schroedinger equation like this with self-interaction terms.

Last edited by a moderator: May 3, 2017
5. Jul 18, 2008

### akhmeteli

If you mean the standard second-quantized QED in its entirety, I suspect this statement requires a few caveats. First, Barut introduced the Feynman propagator for the electromagnetic field (which corresponds to a complex Lagrangian) - that was an additional assumption. Second, to implement Pauli's exclusion principle in his theory, Barut had to modify the Lagrangian. Furthermore, Barut did not claim his theory exactly reproduced all the results of the standard QED, rather he hoped experiments would favor his theory if and where it differed from QED.

6. Jul 18, 2008

### Crosson

I have always suspected that this is the case, but have never seen the equations interpreted that way in a mainstream context.

If such a result was more widely accepted, it could have profound implications for quantum computing. Mark Oskin, from the University of Washington CS department, explains the time evolution of the quantum state in terms of a unitary operator:

ψ' = Uψ

"The fact that U cannot depend ψ and only on t1 and t2 is a subtle and disappointing fact. We will see later that if U could depend on ψ then quantum computers could easily solve NP complete problems!"

Amazing!

7. Jul 18, 2008

### Maaneli

I mean you can calculate all radiative effects that are calculate with 2nd quantized QED, but with Barut's 1st quantized formalism.

The Feynman propagator he uses, yes, was an additional assumption, but can't really be called "2nd quantization", and is not necessarily problematic.

The exclusion principle part I'm not sure about.

Even though Barut hoped his theory could be differentiated from standard QED, that was only a hope and he had no real physics-based reason to believe this would be the case (in fact there is reason to think the theories are empirically equivalent). And to the extent that SFQED was applied to all radiative processes in QED, it made the same predictions. That's why I said you could do all of QED with his theory.

8. Jul 18, 2008

### Maaneli

That would be interesting if this was an implication for quantum computing, but I don't know enough about this.

9. Jul 18, 2008

### Crosson

I found the link I was looking for to the paper containing the statement:

"www.cs.washington.edu/homes/oskin/quantum-notes.pdf"[/URL]

I don't think that the quantum computing people will take notice until someone comes up with a method that could in theory exploit the nonlinear self-interaction term to create a general logic gate that transforms qubits with regard for their current state.

Last edited by a moderator: Apr 23, 2017
10. Jul 18, 2008

### Maaneli

Last edited by a moderator: Apr 23, 2017
11. Jul 18, 2008

### per.sundqvist

Thanks. I have to check this guy Barut I think.

I solved the self-consistent problem today for a free electron in vacuum. You then got a bound state and at the same time, the electric field that wave function produces also contain that energy, so energy is conserved. The electron is then a blob on its own and not a wave. But isn't this unphysical?

12. Jul 18, 2008

### Maaneli

What do you mean it is a blob on its own? The wavefunction solution to the nonlinear S.E. is still a Fourier expansion of waves.

By the way, Barut et al. have treated the case of the nonrelativisic free particle of their theory:

Quantum electrodynamics based on self-fields, without second quantization: A nonrelativistic calculation of g-2
A. O. Barut, Jonathan P. Dowling, J. F. van Huele
http://prola.aps.org/abstract/PRA/v38/i9/p4405_1

13. Jul 18, 2008

### akhmeteli

"not necessarily problematic" - maybe, but some knowledgeable people (such as Bialynicki-Birula) contend the use of the Feynman propagator is equivalent to 2nd quantization of the electromagnetic field in the absence of external photon lines.

I'll try to find a reference if and when I have time. Same for the reference to Bialynicki-Birula.

Actually, it was the word "just" in your phrase "Now you just have to put in the self-interaction from the vector potential, and you can do all of QED with a first-quantized formalism!" that triggered my previous post.

Maybe I missed something, but I'm not sure he "hoped his theory could be differentiated from standard QED", my impression was he would have preferred if the results had been the same. Maybe I'm wrong though.

14. Jul 18, 2008

### Maaneli

I have already read the Birula paper. I meant two things: that the matter field in Barut is not 2nd quantized (that's obvious), and that although Feynman and Birula say the use of the Feynman propagator is equivalent to 2nd quantization of the EM field in the absence of external photon lines, this is not the same as saying that the self-field is 2nd quantized, in my opinion. Maybe it's just semantics, but it's not so clear to me what "2nd quantized" means with respect to the self-field. If they just mean use of the complex-valued Feynman propagator instead of the real-valued classical Green's propagator, well, OK, but then both parts of the term "2nd quantized" seem to me a misnomer. I mean, the self-field is not an operator-valued field, nor is it decomposable into quantized harmonic oscillators. One could certainly however say that it is a 1st quantized self-field because after all the electron charge is coupled to the 1st quantized matter current density, but that's about it as far as I can see.

OK, for the Pauli exclusion principle, thanks.

I have asked Dowling about this and read some papers where they pretty much say they did hope it would be a different theory than standard perturbative QED. The only reasons to think this is that their method of solution is different, being a nonperturbative iteration procedure with Mellin-Barnes transforms, as opposed to asymptotic expansions with renormalization. But all the QED phenomena they did treat in their theory gave the same results to lowest orders in Z*alpha. Moreover, the equivalence of eliminating the 2nd quantized free field with the self-field as Feynman and Birula mention would also suggest to me an empirical equivalence for QED phenomena, even if the methods of solution are different.

There is however one place where a difference of predictions does seem to exist between the two theories (and I think it suggests that perhaps perturbative QED is after all an approximation to the Barut theory), namely, the old cosmological constant problem. Perturbative QED predicts an infinite vacuum energy density (even in the absence of matter) whose absolute value induces infinite spacetime curvature according to the Einstein field equation. But the Barut theory does not predict any such infinite vacuum energy density, with or without the presence of matter. So it easily solves the old cosmological constant problem. That to me seems like a significant difference, but one based on an intertheoretic consideration. Actually, this also ties into the fact that SFQED gives finite answers whereas perturbative QED gives infinite bare values. So maybe you could indeed have good reason to say that perturbative QED is an approximation to SFQED.

Last edited: Jul 18, 2008
15. Jul 18, 2008

### Maaneli

I should also add that Barut and Dowling did in fact extend their approach to 2nd quantized matter fields and got the same answers.

16. Jul 18, 2008

### reilly

The idea of self energy makes great sense, and is, in fact, forced on us by the basic structure of the QED interaction, for example. It is evident in Poynting's thrm, and in the old adiabatic assembling of a charge. That self energy shows up is no surprise, so the issue is what do you do with it? And, the jury is still out.

Fortunately, our inability to deal with this concept has not precluded great advances in QED, the Standard Model and on.... What we've learned from QED is that the corrections due to self energy and polarization of the vacuum and charge screening (Corrections due to vertex diagrams) are very small, and have virtually no effect on physics at an atomic or molecular or nuclear scale. Non-corrected theory works just fine in those regions of physics. So, typically we throw out the self energy terms, with a nod to empirical justification. In the relativistic case, we throw away the infinities that plague us, but in a way that is astonishingly accurate.

It ain't pretty, but it's the best we have. Great opportunity indeed.

Regards,
Reilly Atkinson

17. Jul 18, 2008

### Maaneli

I strongly disagree with this characterization of corrections due to self energy as having "virtually no effect on physics at an atomic or molecular or nuclear scale". The Lamb shift, spontaneous emission, corrections to g-2, and cavity QED effects, are all examples of highly nontrivial physical phenomena in various parts of AMO and nuclear physics. Moreover, Barut's self-field approach is the most explicit example of how self-energy is indispensable to said QED phenomena.

18. Jul 19, 2008

### akhmeteli

So here's the reference.

A.O. Barut, "Foundations of Self-Field Quantumelectrodynamics", in: "New Frontiers in Quantum Electrodynamics and Quantum Optics", Ed. by A.O. Barut, NATO ASI Series V.232, 1991, p. 358:

"For two identical particles we use the postulate of the first quantized quantum theory that the field is symmetric or antisymmetric under the interchange of all dynamical variables of identical particles. In our formulation we go back to the original action principle and assume that the current $j_\mu$ is antisymmetric in the two fields
$j_\mu=\frac{1}{2}e(\bar{\psi_1}\gamma_\mu\psi_2-\bar{\psi_2}\gamma_\mu\psi_1)$,
$e_1=e_2=e$ (52)

This implies in the interaction action

$W_{\textrm{int}}=\frac{1}{4}e^2\left[\int dx dy \bar{\psi_1}(x)\gamma_\mu\psi_2(x)D(x-y) \bar{\psi_1}(y)\gamma_\mu\psi_2(y)-\int dx dy \bar{\psi_1}\gamma_\mu\psi_2D(x-y) \bar{\psi_2}\gamma_\mu\psi_1+(1\leftrightarrow2)\right]$ (53)"

19. Jul 19, 2008

### Maaneli

Thanks for this reference! I have seen this paper before, and only vaguely recall this part. I don't really consider this an "additional assumption". It is simply a consequence of staying in the first quantized matter formalism. In any case, thanks.

20. Jul 19, 2008

### akhmeteli

In my book, if one changes the Lagrangian, he does introduce an additional assumption. So I guess we disagree on this point.

Another thing. You see, to "stay in the first quantized matter formalism", Barut "smuggles" second quantization for the electromagnetic field by using the Feynman propagator, and "smuggles" something similar for the electron field by changing the Lagrangian. I am not trying to criticize Barut, I am just trying to say that his self-field electrodynamics is an unfinished business. It is not easy to determine its exact status. It took me quite some time to sort it out, and I am not sure I have a clear picture now. So I try to tread carefully. It is too easy to say something that is not quite accurate. Sometimes I had to admit in this forum that I had made a mistake. Things are just too often not quite what they look.


text:
# 2020

Dec 14
Preprint released at arXiv: Semi-Supervised Few-Shot Learning
Dec 01
Konstantinos Tertikas begins his PhD thesis on 3D Shape Retrieval
Dec 01
Shashanka Venkataramanan begins his PhD thesis on Metric Learning
Nov 16
I begin teaching the 4th iteration of Deep Learning for Vision at UR1
Nov 01
Bill Psomas begins his PhD thesis on Incremental Learning
Oct 22
Bill Psomas defends his MSc thesis on Deep Metric Learning
Oct 11
Two papers accepted at ICPR 2020: Few-Shot Few-Shot Learning and Local Propagation
Oct 01
Felipe Torres Figueroa begins his PhD thesis on Interpretable Visual Recognition
Sep 10
Oriane Siméoni defends her PhD thesis on Visual Search, Discovery and Recognition
Sep 01
Deniz Engin begins her PhD thesis on Multimodal Question Answering
Sep 01
Research grant MQA begins
Sep 01
Timothée Neitthoffer defends his MSc thesis on Neural Architecture Search
Aug 09
Paper accepted at TIFS: Boundary Projection Attack
Jul 03
HDR Thesis thesis defended at University of Rennes 1: Exploring and Learning from Visual Data
Jul 02
Paper accepted at ECCV 2020: Few Clean and Many Noisy Labels
Jun 29
Preprint released at arXiv: Asymmetric Metric Learning
Jun 22
Paper accepted at ICPR 2020: Deep Active Learning
Apr 01
Research grant MEERQAT begins
Feb 18
Paper accepted at JIS: Smooth Adversarial Examples
Feb 18
Preprint released at arXiv: Few-Shot Few-Shot Learning
Feb 17
I begin teaching the 2nd iteration of Computer Vision at NKUA
Jan 01
Research grant UNLIR begins

# 2019

Dec 04
Preprint released at arXiv: Boundary Projection Attack
Dec 01
Preprint released at arXiv: Nano-Supervised Object Detection
Dec 01
Research grant GRAPES begins
Nov 19
Preprint released at arXiv: Rethinking Deep Active Learning
Nov 18
I begin teaching the 3th iteration of Deep Learning for Vision at UR1
Oct 01
Preprint released at arXiv: Few Clean and Many Noisy Labels
Sep 06
I become a father for a second time
Aug 05
I am an Outstanding Reviewer of ICCV 2019
Jul 13
Two research grants accepted: MEERQAT and UNLIR
May 15
Preprint released at arXiv: Deep Spatial Matching
May 15
Research grant GRAPES accepted
Apr 22
I am a Senior Member of IEEE
Apr 10
Preprint released at arXiv: Deep Label Propagation
Mar 29
Preprint released at arXiv: Smooth Adversarial Examples
Mar 13
Preprint released at arXiv: Dense Classification + Implanting
Feb 26
New website online at avrithis.net, replacing IVA
Feb 25
Feb 19
I begin teaching the 1st iteration of Computer Vision at NKUA
Feb 08
Paper accepted at MVA: Particular Object Discovery

# 2018

Nov 20
Paper accepted at CVIU: Planar Shape Decomposition
Nov 19
I begin teaching the 2nd iteration of Deep Learning for Vision at UR1
Oct 10
Research grant BnF begins
Sep 21
Paper accepted at ACCV 2018: Hybrid Diffusion
Sep 01
Research grant MobilAI begins
Jul 23
Preprint released at arXiv: Hybrid Diffusion
May 10
Mar 29
Preprint released at arXiv: Revisiting Oxford and Paris
Mar 29
Preprint released at arXiv: Mining on Manifolds
Mar 22
I give a talk on Deep Image Retrieval at Safran Tech
Feb 28
Three papers accepted at CVPR 2018: Fast Spectral Ranking, Revisiting Oxford and Paris and Mining on Manifolds
Feb 19
Yann-Raphaël Lifchitz begins his PhD thesis on Few-Shot Learning
Feb 19
Research grant Few-Shot begins
Jan 19
Paper accepted at WACV 2018: Unsupervised Object Discovery

# 2017

Nov 20
I begin teaching the 1st iteration of Deep Learning for Vision at UR1
Oct 22
I am an Outstanding Reviewer of ICCV 2017
Sep 14
Preprint released at arXiv: Unsupervised Object Discovery
Sep 01
Hanwei Zhang begins her PhD thesis on Adversarial Learning
Aug 22
Paper accepted at CEFRL 2017: Discriminative Parts
Jul 21
I am an Outstanding Reviewer of CVPR 2017
May 24
Apr 21
Preprint released at arXiv: Panorama to Panorama
Apr 12
Preprint released at arXiv: Unsupervised Part Learning
Mar 29
Paper accepted at ICMR 2017: Panorama to Panorama
Mar 20
Preprint released at arXiv: Fast Spectral Ranking
Mar 17
Two papers accepted at CVPR 2017: Unsupervised Part Learning and Diffusion on Region Manifolds
Mar 07
Yahoo! Research chooses LOPQ to index and provide a similar image search functionality on its entire Flickr collection - covered by The Verge, Engadget, Petapixel, Digital Trends and Venture Beat

# 2016

Nov 27
I serve as Area Chair of EUSIPCO 2017
Nov 15
Preprint released at arXiv: Diffusion on Region Manifolds
Nov 14
Preprint released at arXiv: Discriminative Parts
Oct 20
Sep 02
Hoel Kervadec defends his MSc thesis on Deep Hough Networks
Sep 01
Oriane Siméoni begins her PhD thesis on Visual Search, Discovery and Recognition
Aug 23
I serve as Area Chair of ACM-MM 2017
Mar 31
Mar 31
Preprint released at arXiv: k-d Randomized Forests
Mar 25
Paper accepted at CGI 2016: k-d Randomized Forests
Mar
Christos Varytimidis completes his PhD thesis on Local Feature Detection
Feb 01
I am a Research Scientist at Inria Rennes-Bretagne Atlantique

# 2015

Dec 16
Paper accepted at CVA: Dithering-Based Sampling
Nov 25
Oct 26
I become a father
Oct
Vasileios Chatzipanos completes his Diploma thesis on Compact Representations for Retrieval
Sep 03
Paper accepted at ICCV 2015: Web-Scale Image Clustering
Aug 28
Paper accepted at PR: $\alpha$-Shapes for Local Features
Jul 03
Paper accepted at BMVC 2015: Planar Shape Decomposition
Jun
Giorgos Mitsis completes his Diploma thesis on Object Proposals
Mar 26
Mar 05
Paper accepted at IJCV: Selective Match Kernels
Mar 02
Paper accepted at CVPR 2015: Early Burst Detection
Mar 01
I am a Research Scientist at National and Kapodistrian University of Athens

# 2014

Nov
Yannis Kalantidis completes his PhD thesis on Web-Scale Search and Clustering
Oct 31
Research grant IS-Helleana ends
Oct 02
Oct
Georgios Perakis completes his Diploma thesis on License Plate Recognition
Sep 05
Paper accepted at ACCV 2014: Dithering-Based Sampling
May 26
I am a Scientific Advisor at Odd Concepts Inc.
May
Athanasios Papadopoulos completes his Diploma thesis on Geometric Descriptor Aggregation
Mar 01
Paper accepted at CVPR 2014: Locally Optimized Product Quantization
Mar
I begin teaching the 9th iteration of Signals and Systems at NTUA

# 2013

Dec 25
Paper accepted at CVIU: Geometry Indexing
Oct 12
Paper accepted at IJCV: Hough Pyramid Matching
Oct
Giorgos Tolias completes his PhD thesis on Particular Object Retrieval
Sep 02
Two papers accepted at ICCV 2013: Quantize and Conquer and To Aggregate or not
Oct
I begin teaching the 7th iteration of Image and Video Analysis at NTUA
Sep 26
I am married to Maria
Jun 07
Paper accepted at TMM: Movie Summarization
Mar
I begin teaching the 8th iteration of Signals and Systems at NTUA

# 2012

Oct
Agni Delvinioti completes her Diploma thesis on Classification by Spatial Matching
Oct
I begin teaching the 6th iteration of Image and Video Analysis at NTUA
Jul
Christos Arvanitis completes his Diploma thesis on Augmented Reality
Jun 25
Jun 25
Paper accepted at ACM-MM 2012: SymCity
Apr 27
Mar
I begin teaching the 7th iteration of Signals and Systems at NTUA
Mar
Charalampos Moustafelos completes his Diploma thesis on Visual Synonyms
Mar
Spyros Leonardos completes his Diploma thesis on Image Segmentation

# 2011

Nov 01
Research grant IS-Helleana begins
Oct
I begin teaching the 5th iteration of Image and Video Analysis at NTUA
Jun 30
Two papers accepted at ICCV 2011: Relaxed Spatial Matching and Medial Feature Detector
Mar 31
Research grant WeKnowIt ends
Mar 11
Paper accepted at CC: Spatiotemporal Features
Mar
I begin teaching the 6th iteration of Signals and Systems at NTUA
Feb 23
Paper accepted at ICMR 2011: Logo Recognition

# 2010

Dec 31
Nov 16
Paper accepted at MTAP: VIRaL
Oct
I begin teaching the 4th iteration of Image and Video Analysis at NTUA
Jul 31
Research grant JUMAS ends
Jul 05
Two papers accepted at ACM-MM 2010: Landmark and Non-Landmark Images and Feature Map Hashing
Sep 07
Paper accepted at SGA 2010: Regions from Edges
Mar
I begin teaching the 5th iteration of Signals and Systems at NTUA
Feb 28
Research grant X-Media ends

# 2009

Nov
Evaggelos Spyrou completes his PhD thesis on Semantic Visual Search
Nov
Thanos Athanasiadis completes his PhD thesis on Knowledge-Based Image Analysis
Oct
I begin teaching the 3th iteration of Image and Video Analysis at NTUA
Apr
Research grant Imagination ends
Mar 20
Paper accepted at SP:IC: Spatiotemporal Saliency
Mar
I begin teaching the 4th iteration of Signals and Systems at NTUA
Mar
George Koumoulos completes his Diploma thesis on Local Feature Detection
Feb 28
Research grant MESH ends
Feb 28
Research grant BOEMIE ends
Feb 27
Two papers accepted at CBMI 2009: Compound Document Analysis and VIRaL
Feb 23
Paper accepted at CVPR 2009: Spatiotemporal Feature Points
Jan 22
I serve as Associate Editor of JIVP
Jan 20
Paper accepted at TMM: High-Level Concept Detection
Jan 07
Jan
Yannis Kalantidis completes his Diploma thesis on Visual Features and Geometry

# 2008

Dec 31
Research grant K-Space ends
Dec 31
Research grant OntoMedia ends
Dec 15
Paper accepted at ICASSP 2009: Video Event Detection
Nov 08
Nov
I serve as General Chair of CBMI 2009
Sep 15
Oct
I begin teaching the 2nd iteration of Image and Video Analysis at NTUA
Jul
I serve as Program Chair of CIVR 2009
Jun 18
Paper accepted at MMSP 2008: Semantic Video Segmentation
Jun
Christos Varytimidis completes his Diploma thesis on Object Detection
Apr 25
Paper accepted at ICIP 2008: Movie Summarization
Apr 01
Research grant WeKnowIt begins
Apr
Konstantinos Rapantzikos completes his PhD thesis on Visual Saliency
Apr
Phivos Mylonas completes his PhD thesis on Context-Based Analysis
Mar 20
Mar
I begin teaching the 3th iteration of Signals and Systems at NTUA
Feb 01
Research grant JUMAS begins
Jan 25
Paper accepted at WIAMIS 2008: Semantic Video Analysis
Jan
I serve as Program Chair of MMM 2009


text:
Rotational Spring

Ideal spring in mechanical rotational systems

Library

Mechanical Rotational Elements

Description

The Rotational Spring block represents an ideal mechanical rotational linear spring, described with the following equations:

`$T=K·\phi$`
`$\phi ={\phi }_{init}+{\phi }_{R}-{\phi }_{C}$`
`$\omega =\frac{d\phi }{dt}$`

where

 `T` Torque transmitted through the spring `K` Spring rate φ Relative displacement angle (spring deformation) φinit Spring preliminary winding (spring offset) φR, φC Absolute angular displacements of terminals R and C, respectively ω Relative angular velocity `t` Time

The block positive direction is from port R to port C. This means that the torque is positive if it acts in the direction from R to C.

Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Parameters

Spring rate

Spring rate. The default value is `10` N*m/rad.

Ports

The block has the following ports:

`R`

Mechanical rotational conserving port.

`C`

Mechanical rotational conserving port.

Version History

Introduced in R2007a


text:
# Tag Info

11

No. The wikipedia article is in my honest opinion misrepresenting this article on a reduced round attack on the SHA-2 family of hashes. Although these attacks improve upon the existing reduced round SHA-2 attacks, they do not threaten the security of the full SHA-2 family. In other words, no collisions have been found in any of the SHA-2 hashes. The ...

5

Password strength is typically measured in bits of entropy, or in layman's terms, the amount of "true randomness" in the system. This is measured by the process of how the password is generated rather than by the number of bits in the output. It's a simple extension of Kerckhoff's principle: assume your attacker knows your process, and the only information ...

1

One problem not mentioned here is that CRC collisions are a certainty. If you were using a cryptographically secure hash, you would never encounter a false positive where both solutions were possible. In this scheme, every 256 messages would yield identical CRC values, and your different versions would be indistinguishable. You might be able to "stutter" ...

4

This started as a comment to @Poncho's fine answer, and grew over the 600-char limit. Point is: a careful choice of the definition of V2 messages can keep some the existing capabilities of the original CRC to always detect some kinds of errors. Foremost, we are interested in short error bursts (where all bits in error are within a small number of ...

3

If there was a better known way than brute force, I suppose someone would already have made a paper about this (or lets this be secret to use it for attacks – but then this person wouldn't enter in such a competition). So it looks like just brute force. Why are these people better than you? Some guesses (and most likely it is a combination of these): They ...

4

Since the output of the configuration of Skein is 1024 bits, we would expect to get 512 bits correct just by random hashing. So the question is, what is the probability of getting 640 bits correct? From that we can figure out how long it would take for that to occur. There are ${1024 \choose 640}\approx 4.16\cdot 10^{292}\approx 2^{972}$ ways this can ...

1

CRCs are not cryptographically secure. If you need cryptographic security, replace the CRC with a message authentication code (MAC). If you don't need cryptographic security, then your question is off-topic for Crypto.SE and you should probably flag it to ask the moderators to migrate it to Computer Science.SE.

8

First of all, if your goal is to keep the garbled messages to "once every hundred years", well, you already don't meet that goal, even before the change. With an 8 bit CRC, a random change has a probability 1/256 of being accepted; hence if your wireless network has a transmission error at least once every three months (which, to me, sounds like an ...

2

I wouldn't even go so far as to say that a good hash function must have "as few collisions as possible" as you state in your question. The identity function has no collisions if we restrict the domain to be the inputs having the same size as the digest. But it's clearly a lousy hash function. To be "good" as a hash function is hard to define (which is ...

4

If a hash function is a random oracle, which is about as perfect as can be expected for a hash function, then we actually expect a lot of collisions. If the hash function output is in a space of size $N$ (e.g. $N = 2^n$ for a "$n$-bit output") then collisions are expected, on average, when about $\sqrt{N}$ distinct inputs are hashed (see the birthday problem ...

Top 50 recent answers are included


text:
# Is this powerset right?

1. Mar 17, 2012

### charmedbeauty

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

if A ={a}, B={b,c}, C={d,e,f,g,h,i,j}

Find P(A*B)
and |P(B*C)|

2. Relevant equations

3. The attempt at a solution

so for the first one I had

[{(a,b)}, {(a,c)}, {(a,b),(a,c)}, ∅]

and secondly 2^14.

is this right I dont know what is meant by the use of this |..| around a power set?

2. Mar 18, 2012

### LCKurtz

What does A*B mean?

3. Mar 18, 2012

### charmedbeauty

Sorry its meant to mean A times B

dont know what the proper sign is ? or do I just use ()

4. Mar 18, 2012

### LCKurtz

Sets have union and intersection operations. What is the definition of A "times" B?

5. Mar 18, 2012

### charmedbeauty

Its a Cartesian product

6. Mar 18, 2012

### LCKurtz

Ok, now that I understand your notation your answers look fine. The absolute value signs around a set just means how many elements it contains. So your $2^{14}$ is correct.


text:
Euclidean distance

Calculate the Euclidean distance between shops A, B and C, where:
A 45 0.05
B 60 0.05
C 52 0.09

Wherein the first figure is the weight in grams of bread and second figure is price in USD.

Correct result:

|AB| =  15
|BC| =  8.0001
|AC| =  7.00011

Solution:

$\mathrm{\mid }AB\mathrm{\mid }=\sqrt{\left(45-60{\right)}^{2}+\left(0.05-0.05{\right)}^{2}}\doteq 15$
$\mathrm{\mid }BC\mathrm{\mid }=\sqrt{\left(60-52{\right)}^{2}+\left(0.05-0.09{\right)}^{2}}\doteq 8.0001$
$\mathrm{\mid }AC\mathrm{\mid }=\sqrt{\left(45-52{\right)}^{2}+\left(0.05-0.09{\right)}^{2}}\doteq 7.00011$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Please write to us with your comment on the math problem or ask something. Thank you for helping each other - students, teachers, parents, and problem authors.

Tips to related online calculators
Pythagorean theorem is the base for the right triangle calculator.

You need to know the following knowledge to solve this word math problem:

We encourage you to watch this tutorial video on this math problem:

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text:
# How many ways to choose $k$ out of $n$ numbers with exactly/at least $m$ consecutive numbers?

How many ways to choose $k$ out of $n$ numbers is a standard problem in undergraduate probability theory that has the binomial coefficient as its solution. An example would be lottery games were you have $13983816$ ways to choose $6$ numbers out of $49$.

My question is: How many ways are there to choose $k$ out of $n$ numbers with exactly/at least $m$ consecutive numbers?

An example would be how many ways are there to choose $6$ out of $49$ numbers with exactly/at least $5$ consecutive numbers, e.g. $\{2,3,4,5,6,26\}$? I read that the answer here is $1936$ ways for the "at least"-case.

I would like to have a general formula and if possible a derivation of it. Good references are also welcome. Thank you.

-
The "exactly/at least" wording is confusing, perhaps it could be worded as "a minimum of $m$ consecutive...". I think the derivation starts by noting that the consecutive constraint really reduces the number of "choices" by $m-1$... –  abiessu Sep 8 '13 at 10:11
@abiessu: It’s an abbreviation for two separate questions, one asking for the number of ways to choose them so that exactly $m$ are consecutive, the asking for the number of ways to choose them with at least $m$ consecutive. –  Brian M. Scott Sep 8 '13 at 19:57

First we are going to make some obvious restriction:

$$m \le k \le n$$

Note that if we choose the first number then the following $m-1$ numbers can't be chosen. For example in your example if we choose $2$ to be our first number, then it implies that $3,4,5,6$ are the second, third, fourth and fifth number respectivly.

There are $n-m+1$ to choose the first number, because if we choose a number greater that $n-m$ then we can't choose $m$ consecutive number starting with that one.

Now it's we need to chose $n-m$ random numbers for $k-m$ places.

Because you've mentioned lottery tickets, where the order isn't important we'll assume that in the calculation the order isn't important.

First I want you to note something. I'll again use your example. We take numbers from 1-5 as consecutive and we'll place them in the first 5 places, and then some random number but for the purpose of presentation we'll choose 6. Then the set will be:

$${1,2,3,4,5,6}$$

Now let's think like these, we'll choose the numbers from 2-6 and will place them in the last 5 places and the random number will be 1. The set then will be;

$${1,2,3,4,5,6}$$

Aren't this sets the same? But we choose the using 2 different ways, so that means we'll make some restrictions and we'll make 2 cases.

Case 1: $k=m$

Obviously we have space only to place the consecutive number so let $P(n,k,m)$ represent the number of ways to choose $k$ numbers out of $n$ numbers with $m$ consecutive numbers, so we have:

$$P(n,k,m) = n-m+1$$

Case 2: $k \ge m+1$

Note that if $1$ is the first number of the series we can't choose the predecessor, because there isn't one. So we have:

$$P(n,k,m) = (n-m) \times \binom{n-m-1}{k-m} + \binom{n-m}{k-m}$$

And if we calculate for your example we have:

$$P(49,6,5) = (49-5) \times \binom{49-5-1}{6-5} + \binom{49-5}{6-5}$$ $$P(49,6,5) = 44 \times \binom{43}{1} + \binom{44}{1}$$ $$P(49,6,5) = 44 \times 43 + 44$$ $$P(49,6,5) = 1936$$

-

Given a subset $S$ of $[n]$, let us denote by $x_i$ (resp. $y_i$) the lengths of the successive runs of elements inside (resp. outside) $S$. For example, if $n=7$ and $S=\{1,2,4,5,6\}$, then you have $x_1=2$ (corresponding to $1,2\in K$), $y_1=1$ (corresponding to $3\notin K$), $x_2=3$ and $y_2=1$.

Using the correspondence above, we have that any $k$-subset $S$ of $[n]$ with at least $m$ consecutive elements is uniquely determined by a unique pair $\bigl((x_i),(y_i)\bigr)$ of (finite) sequences of positive integers such that $\sum_ix_i=k, \sum_iy_i=n-k, x_i,y_i\geq1$ for each $i$ and $x_i\geq m$ for some $i$. If the sequence of $x$s has length $r$ (that is, if our subset $S$ has $r$ runs of consecutive elements), then the mutually excluding possibilities for the "braiding" of the sequence $(y_i)$ with the sequence $(x_1,\dots,x_r)$ are the following:

• If $1,n\in S$ then the braiding is of the form $x_1,y_1,x_2,y_2,\dots,x_{r-1},y_{r-1},x_r$.

• If $1,n\notin S$ then the braiding is of the form $y_1,x_1,y_2,x_2,\dots,y_r,x_r,y_{r+1}$.

• If $1\in S$ and $n\notin S$ then the braiding is of the form $x_1,y_1,x_2,y_2,\dots,x_r,y_r$.

• If $1\notin S$ but $n\in S$ then the braiding is of the form $y_1,x_1,y_2,x_2,\dots,y_r,x_r$.

Recall that the number of solutions in $\mathbb N$ of the equation $z_1+\cdots+z_p=q$ is precisely $\binom{p+q-1}{p-1}$, and the number of solutions in $\mathbb Z^+$ of the same equation is $\binom{q-1}{p-1}$ (because changing $z_i$ by $z_i-1$, then you are looking for solutions in $\mathbb N$ of $z_1+\cdots+z_p=q-p$). Since each $x_i$ must be $\geq1$, then changing $x_i$ by $x_i-1$ we see that we want solutions in $\mathbb N$ of $x_1+\cdots+x_r=k-r$ with $x_i\geq m-1$ for some $i$, so we must apply inclusion-exclusion. For each $j$ let $A_j=\bigl\{(x_1,\dots,x_r): x_1+\cdots+x_r=k-r, x_j\geq m-1\bigr\}$. If $1\leq j_1<\cdots<j_\ell\leq r$, then by changing $x_i$ by $x_i-(m-1)$ for $i=j_1,\dots,j_\ell$ we see that the elements in $A_{j_1}\cap\cdots\cap A_{j_\ell}$ correspond to solutions in $\mathbb N$ of $x_1+\cdots+x_r=k-r-\ell(m-1)$, and so $\bigl|A_{j_1}\cap\cdots\cap A_{j_\ell}\bigr|=\binom{k-\ell(m-1)-1}{r-1}$. Therefore the number of solutions $(x_1,\dots,x_r)$, for fixed $r$, is equal to

$$\sum_{\ell=1}^r(-1)^{\ell-1}\sum_{1\leq j_1<\cdots<j_\ell\leq r}\bigl|A_{j_1}\cap\cdots\cap A_{j_\ell}\bigr|=\sum_{\ell=1}^r(-1)^{\ell-1}\binom r\ell\binom{k-\ell(m-1)-1}{r-1}\,.$$

On the other hand, the number of solutions $(y_1,\dots,y_s)$ in $\mathbb Z^+$ of $y_1+\cdots+y_s=n-k$ is equal to $\binom{n-k-1}{s-1}$. Note that $s=r$ in two instances of the problem, $s=r+1$ in one instance and $s=r-1$ in the remaining instance. Thus, for fixed $r$, the total number of tuples $(y_i)$ that can be braided with a fixed sequence $(x_1,\dots,x_r)$ is equal to

\begin{align*} &\,\Biggl[\,\binom{n-k-1}{r-1}+\binom{n-k-1}{r}\Biggr]+\Biggl[\,\binom{n-k-1}{r}+\binom{n-k-1}{r+1}\Biggr]\\ =&\,\binom{n-k}{r}+\binom{n-k}{r+1}\\ =&\,\binom{n-k+1}{r+1}\,. \end{align*}

Consequently, the number of $k$-subsets of $[n]$ with at least $m$ consecutive members is equal to

$$N(n,k,m)=\sum_{r\geq1}\binom{n-k+1}r\sum_{\ell=1}^r(-1)^{\ell-1}\binom r\ell\binom{k-\ell(m-1)-1}{r-1}$$

and the number of $k$-subsets of $[n]$ with $m$ consecutive members but without $m+1$ consecutive members will be equal to $N(n,k,m)-N(n,k,m+1)$. I am assuming that when you ask for "exactly" $m$ consecutive members, you are allowing to have two or more such sequences of consecutive numbers (i.e., according to the notation above, $x_i=m$ can occur for more than one value of $i$); otherwise the problem is a lot more challenging.

Actually the restricted problem is not more difficult than the version above. Suppose that you want to construct a $k$-subset of $[n]$ containing exactly one run of $m$ consecutive numbers, and the other runs being of length less than $m$. Let $r\geq1$ be the number of runs of consecutive elements in your subset. As before, the possibilities for the runs of consecutive elements outside your subset contributes (multiplicatively) $\binom{n-k+1}{r+1}$ to the number of subsets. On the other hand, with $x_1,\dots,x_r$ as before, then exactly one of the numbers $x_i$ is equal to $m$. This can be made into $r$ ways. And now you are looking for solutions in $\mathbb Z^+$ of $z_1+\cdots+z_{r-1}=k-m$ (the $z_k$ are simply the variables $x_j$ with $j\ne i$) with $z_k<m$ for each $k$. This number can also be calculated through inclusion-exclusion: in fact, we want to calculate $\bigl|(A_1\cup\cdots\cup A_{r-1})^{\,c}\bigr|$, where $A_k=\bigl\{(w_1,\dots,w_{r-1})\in\mathbb N^{r-1}: w_1+\cdots+w_{r-1}=k-m-(r-1), w_k\geq m-1\bigr\}$ ($w_j=z_j-1$). Reasoning as in the first part of my answer we conclude that the number of desired sequences $(z_1,\dots,z_{r-1})$ is equal to

\begin{align*} &\,\binom{k-m-1}{r-2}-\sum_{\ell=1}^{r-1}(-1)^{\ell-1}\binom{r-1}\ell\binom{k-m-\ell(m-1)-1}{r-2}\\ =&\,\sum_{\ell=0}^{r-1}(-1)^\ell\binom{r-1}\ell\binom{k-m-\ell(m-1)-1}{r-2}\,. \end{align*}

Consequently the number of $k$-subsets of $[n]$ with exactly one run of $m$ consecutive members and all the other runs of consecutive members with length strictly less than $m$, is equal to

$$\sum_{r\geq1}r\,\binom{n-k+1}{r+1}\sum_{\ell=0}^{r-1}(-1)^\ell\binom{r-1}\ell\binom{k-m-\ell(m-1)-1}{r-2}\,.$$

-

After some further googling I found the following reference which gives a general formula and a derivation:

Lottery combinatorics by McPherson & Hodson

-


text:
# 18.1. Introduction: Test Cases¶

A test case expresses requirements for a program, in a way that can be checked automatically. Specifically, a test asserts something about the state of the program at a particular point in its execution.

We have previously suggested that it’s a good idea to first write down comments about what your code is supposed to do, before actually writing the code. It is an even better idea to write down some test cases before writing a program.

There are several reasons why it’s a good habit to write test cases.

• Before we write code, we have in mind what it should do, but those thoughts may be a little vague. Writing down test cases forces us to be more concrete about what should happen.

• As we write the code, the test cases can provide automated feedback. You’ve actually been the beneficiary of such automated feedback via test cases throughout this book in some of the activecode windows and almost all of the exercises. We wrote the code for those test cases but kept it hidden, so as not to confuse you and also to avoid giving away the answers. You can get some of the same benefit from writing your own test cases.

• In larger software projects, the set of test cases can be run every time a change is made to the code base. Unit tests check that small bits of code are correctly implemented. Functional tests check that larger chunks of code work correctly. Running the tests can help to identify situations where a change in code in one place breaks the correct operation of some other code. We won’t see that advantage of test cases in this textbook, but keep in mind that this introduction to test cases is setting the stage for an essential software engineering practice if you are participating in a larger software development project.

Now it’s time to learn how to write code for test cases.

Python provides a statement called assert.

• Following the word assert there will be a python expression.

• If that expression evaluates to the Boolean False, then the interpreter will raise a runtime error.

• If the expression evaluates to True, then nothing happens and the execution goes on to the next line of code.

Why would you ever want to write a line of code that can never compute anything useful for you, but sometimes causes a runtime error? For all the reasons we described above about the value of automated tests. You want a test that will alert that you that some condition you assumed was true is not in fact true. It’s much better to be alerted to that fact right away than to have some unexpected result much later in your program execution, which you will have trouble tracing to the place where you had an error in your code.

Why doesn’t assert print out something saying that the test passed? The reason is that you don’t want to clutter up your output window with the results of automated tests that pass. You just want to know when one of your tests fails. In larger projects, other testing harnesses are used instead of assert, such as the python unittest module. Those provide some output summarizing tests that have passed as well as those that failed. In this textbook, we will just use simple assert statements for automated tests.

To write a test, we must know what we expect some value to be at a particular point in the program’s execution. In the rest of the chapter, we’ll see some examples of assert statements and ideas for what kinds of assertions one might want to add in one’s programs.

Note

A note to instructors: this chapter is deliberately structured so that you can introduce testing early in the course if you want to. You will need to cover chapter 8, on Conditionals, before starting this chapter, because that chapter covers Booleans. The subchapters on testing types and testing conditionals can be covered right after that. The subchapter on testing functions can be delayed until after you have covered function definition.

When assert x==y is executed and x and y have different values, what will happen?

• A runtime error will occur
• The expression x==y evaluates to False, so a runtime error will occur
• A message is printed out saying that the test failed.
• If the assertion fails, a runtime error will occur
• x will get the value that y currently has
• x==y is a Boolean expression, not an assignment statement
• Nothing will happen
• The expression x==y evaluates to False
• A message is printed out saying that the test passed.
• When an assertion test passes, no message is printed. In this case, the assertion test fails.

When assert x==y is executed and x and y have the same values, what will happen?

• A runtime error will occur
• The expression x==y evaluates to True
• A message is printed out saying that the test failed.
• The expression x==y evaluates to True
• x will get the value that y currently has
• x==y is a Boolean expression, not an assignment statement
• Nothing will happen
• The expression x==y evaluates to True
• A message is printed out saying that the test passed.
• When an assertion test passes, no message is printed.

Test cases are a waste of time, because the python interpreter will give an error message when the program runs incorrectly, and that’s all you need for debugging.

• True
• You might not notice the error, if the code just produces a wrong output rather generating an error. And it may be difficult to figure out the original cause of an error when you do get one.
• False
• Test cases let you test some pieces of code as you write them, rather than waiting for problems to show themselves later.


text:
# Sides Extended !!!

Geometry Level 3

$ABCD$ is a square. Sides $AD$ and $DC$ are extended to $E$ and $F$ respectively such that $DE=AD$ and $CF=DC$. $BE$ and $AF$ intersect at point $O$. If $\frac{OE}{OB} = \frac{m}{n}$; where $m$ and $n$ are relatively prime, then find $m+n$.

×


text:
# Residues and Mittag-Leffler sequence

Let $X$ be a compact Riemann surface, $\omega$ a meromorphic differential on $X$ and $f$ a meromorphic function on $X$ with poles only over the points $P_1,\dots,P_d$. The product $\;f\cdot\omega\;$ is still a meromorphic differential, so by the residue theorem we know that $$\sum_{P\in X} res_P(\omega) = \sum_{P\in X} res_P(\;f\cdot\omega) = 0.$$

## My question is:

what can we say about $\sum_{i=1}^{d} res_{P_i}(\;f\cdot\omega)$ ?

Is this sum also zero, or is it there any condition on $\omega$ that forces it to be zero?

## $$\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ\circ$$

$$\heartsuit \quad \heartsuit \quad \heartsuit$$

## Remark (to make it more interesting)

If we think of $X$ as an algebraic curve over $\mathbb{C}$ and we set $D=P_1+\dots+ P_d$, we can consider the exact sequence $$H^0(\mathcal{O}_X(D)) \to H^0(\mathcal{O}_X(D)\otimes \mathcal{O}_D)\to H^1(\mathcal{O}_X)$$ where the first map is the restriction map $\alpha$ and the second one is the coboundary $\delta$ coming from the Mittag-Leffler sequence of invertible sheaves $$\mathcal{O}_X \to \mathcal{O}_X(D)\to \mathcal{O}_X(D)\otimes \mathcal{O}_D\;.$$

In this language the above meromorphic $f$ is an element of $\mathcal{O}_X(D)$ and, if $v\in H^0(\mathcal{O}_X(D)\otimes \mathcal{O}_D)$ is such that $\alpha(f)=v$, then we have $\delta v = 0$.

One can define a perfect pairing $$\langle \bullet, \bullet \rangle : H^1(\mathcal{O}_X) \otimes H^0(K) \to \mathbb{C}$$ by setting $$\langle \delta v, \omega \rangle := \sum_{P\in D} res_P (\omega\cdot v_P).$$

My question arises from the fact that I would expect $\delta v = 0$ to imply that $\langle \delta v, \omega \rangle = 0$ for every differential $\omega$.

The answer to your first question is, "no." Let $X$ be the Riemann sphere $\mathbb{C}P^1$ with the meromorphic differential $\omega = dz/z = -dw/w$, where $w=1/z$. Let $f$ be the meromorphic function $f(z) = z/(z-z_0)$ for some $z_0 \neq 0$. The only pole of $f$ is at $z_0$, and the residue of $f\omega$ at that pole, i.e., the residue of $dz/(z-z_0)$ at $z_0$, is nonzero.


text:
# tslearn.metrics.lcss_path¶

tslearn.metrics.lcss_path(s1, s2, eps=1, global_constraint=None, sakoe_chiba_radius=None, itakura_max_slope=None)[source]

Compute the Longest Common Subsequence (LCSS) similarity measure between (possibly multidimensional) time series and return both the path and the similarity.

LCSS is computed by matching indexes that are met up until the eps threshold, so it leaves some points unmatched and focuses on the similar parts of two sequences. The matching can occur even if the time indexes are different, which can be regulated through the sakoe chiba radius parameter that defines how far it can go.

To retrieve a meaningful similarity value from the length of the longest common subsequence, the percentage of that value regarding the length of the shortest time series is returned.

According to this definition, the values returned by LCSS range from 0 to 1, the highest value taken when two time series fully match, and vice-versa. It is not required that both time series share the same size, but they must be the same dimension. LCSS was originally presented in [1] and is discussed in more details in our dedicated user-guide page.

Parameters: s1 A time series. s2 Another time series. eps : float (default: 1.) Maximum matching distance threshold. global_constraint : {“itakura”, “sakoe_chiba”} or None (default: None) Global constraint to restrict admissible paths for LCSS. sakoe_chiba_radius : int or None (default: None) Radius to be used for Sakoe-Chiba band global constraint. If None and global_constraint is set to “sakoe_chiba”, a radius of 1 is used. If both sakoe_chiba_radius and itakura_max_slope are set, global_constraint is used to infer which constraint to use among the two. In this case, if global_constraint corresponds to no global constraint, a RuntimeWarning is raised and no global constraint is used. itakura_max_slope : float or None (default: None) Maximum slope for the Itakura parallelogram constraint. If None and global_constraint is set to “itakura”, a maximum slope of 2. is used. If both sakoe_chiba_radius and itakura_max_slope are set, global_constraint is used to infer which constraint to use among the two. In this case, if global_constraint corresponds to no global constraint, a RuntimeWarning is raised and no global constraint is used. list of integer pairs Matching path represented as a list of index pairs. In each pair, the first index corresponds to s1 and the second one corresponds to s2 float Similarity score

lcss
Get only the similarity score for LCSS
lcss_path_from_metric
Compute LCSS using a user-defined distance metric

References

 [1] M. Vlachos, D. Gunopoulos, and G. Kollios. 2002. “Discovering Similar Multidimensional Trajectories”, In Proceedings of the 18th International Conference on Data Engineering (ICDE ‘02). IEEE Computer Society, USA, 673.

Examples

>>> path, sim = lcss_path([1., 2., 3.], [1., 2., 2., 3.])
>>> path
[(0, 1), (1, 2), (2, 3)]
>>> sim
1.0
>>> lcss_path([1., 2., 3.], [1., 2., 2., 4.])[1]
1.0