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# AM-GM test (CNNer in 2015)

Algebra Level 5

If $$x$$ and $$y$$ are positive reals such that $$x + y \ge 3$$, then find the minimum value of $$2x^2 + y^2 + \frac{28}x + \frac1y$$.

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Thread: impact of avionic systems View Single Post

## impact of avionic systems

hi, I am usually interested in mechanics and I have very less knowledge in avionics. I just have knowledge that avionic system is used to transfer and receive data from one system to other (communication between different aircraft systems). I want to know how an avionic system have impact on performance of the aircraft i.e. will the efficiency of an aircraft changes with the modifications of avionic system?
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# Integration using a cumulative distribution function

1. Jul 29, 2011

### laonious

Hi all,
I'm really banging my head on this problem:
Let f be a real-valued measurable function on the measure space $$(X,\mathcal{M},\mu).$$
Define
$$\lambda_f(t)=\mu\{x:|f(x)|>t\}, t>0.$$
Show that if $\phi$ is a nonnegative Borel function defined on [0,infinity), then
$$\int_0^{\infty}\phi(|f(x)|)d\mu=-\int_0^{\infty}\phi(t)d\lambda_f(t).$$

A hint is given, which is to look at
$$\nu((a,b])=\lambda_f(b)-\lambda_f(a)=-\mu\{x:a<|f(x)|\leq b\},$$
and argue that it extends uniquely to a Borel measure.

This is straightforward, I think, as closed intervals form a semi-ring and $\nu$ is a premeasure. I'm just not sure where to go from here. Any help would be greatly appreciated, thanks!

2. Jul 30, 2011

### mathman

It looks like it could be solved by using the elementary approach to defining Lebesgue integrals - dividing the y (or f) axis into strips and taking a limit as the strip widths go to 0.

The strips would correspond to (a,b] in the hint.


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Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

R offers several ways to reverse a string, include some base R options. We go through a few of those in this post. We’ll also compare the computational time for each method.

Reversing a string can be especially useful in bioinformatics (e.g. finding the reverse compliment of a DNA strand). To get started, let’s generate a random string of 10 million DNA bases (we can do this with the stringi package as well, but for our purposes here, let’s just use base R functions).

set.seed(1)
dna <- paste(sample(c("A", "T", "C", "G"), 10000000, replace = T), collapse = "")

## 1) Base R with strsplit and paste

One way to reverse a string is to use strsplit with paste. This is the slowest method that will be shown, but it does get the job done without needing any packages. In this example, we use strsplit to break the string into a vector of its individual characters. We then reverse this vector using rev. Finally, we concatenate the vector of characters into a string using paste.

start <- proc.time()
splits <- strsplit(dna, "")[[1]]
reversed <- rev(splits)
final_result <- paste(reversed, collapse = "")
end <- proc.time()

print(end - start)

## 2) Base R: Using utf8 magic

This example also does not require any external packages. In this method, we can use the built-in R function utf8ToInt to convert our DNA string to a vector of integers. We then reverse this vector with the rev function. Lastly, we convert this reversed vector of integers back to its original encoding – except now the string is in reverse.

start <- proc.time()
final_result <- intToUtf8(rev(utf8ToInt(dna)))
end <- proc.time()

print(end - start)

## 3) The stringi package

Of all the examples presented, this option is the fastest when tested. Here we use the stri_reverse function from the stringi package.

library(stringi)

start <- proc.time()
final_result <- stri_reverse(dna)
end <- proc.time()

print(end - start)

## 4) The Biostrings package

Our last example uses the Biostrings package, which contains a collection of functions useful for working with DNA-string data. One function, called str_rev, can reverse strings. You can download and load the Biostrings package like this:

source("http://bioconductor.org/biocLite.R")
biocLite("Biostrings")

library(Biostrings)

Then, all we have to do is input our DNA string into the str_rev function and we get our result.

start <- proc.time()
final_result <- str_rev(dna)
end <- proc.time()

print(end - start)

That’s it for this post! Please check out my other articles here.

The post Four ways to reverse a string in R appeared first on Open Source Automation.


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# zbMATH — the first resource for mathematics

$$H_{2}$$-optimal rejection with preview in the continuous-time domain. (English) Zbl 1093.93008
Automatica 41, No. 5, 815-821 (2005); corrigendum ibid. 46, No. 7, 1241 (2010).
Summary: The synthesis of a feedforward unit for $$H_2$$-optimal rejection of previewed signals in continuous time-invariant linear systems is considered. The $$H_2$$-optimal compensator herein devised consists of a finite impulse response system working in connection with a standard dynamic unit. The design strategy is based on a simple interpretation of the $$H_2$$-optimal rejection problem with preview as a compound optimal control problem, i.e. as a problem consisting of a finite-horizon LQ control problem with constrained final state, an infinite-horizon LQ control problem, and the problem of minimizing the global cost functional. An explicit description of the compensator is derived by exploiting some results on the parameterization of the solutions of the associated autonomous Hamiltonian system.

##### MSC:
 93B36 $$H^\infty$$-control 93C05 Linear systems in control theory
Full Text:
##### References:
 [1] Davison, E.J.; Scherzinger, B.M., Perfect control of the robust servomechanism problem, IEEE transactions on automatic control, AC-32, 8, 689-701, (1987) · Zbl 0625.93051 [2] Devasia, S.; Chen, D.; Paden, B., Nonlinear inversion-based output tracking, IEEE transactions on automatic control, 41, 7, 930-942, (1996) · Zbl 0859.93006 [3] Gross, E.; Tomizuka, M., Experimental flexible beam tip tracking control with a truncated series approximation to uncancelable inverse dynamics, IEEE transactions on control systems technology, 3, 4, 382-391, (1994) [4] Kailath, T.; Frost, P., An innovations approach to least-squares estimation. part iilinear smoothing in additive white noise, IEEE transactions on automatic control, AC-13, 6, 655-660, (1968) [5] Kojima, A.; Ishijima, S., $$H_\infty$$ performance of preview control systems, Automatica, 39, 4, 693-701, (2003) · Zbl 1029.93021 [6] Marro, G.; Prattichizzo, D.; Zattoni, E., Convolution profiles for right-inversion of multivariable non-minimum phase discrete-time systems, Automatica, 38, 10, 1695-1703, (2002) · Zbl 1011.93022 [7] Mirkin, L., On the $$H^\infty$$ fixed-lag smoothinghow to exploit the information preview, Automatica, 39, 8, 1495-1504, (2003) · Zbl 1033.93066 [8] Perez, H.; Devasia, S., Optimal output transitions for linear systems, Automatica, 39, 2, 181-192, (2003) · Zbl 1011.93009 [9] Tadmor, G., Weighted sensitivity minimization in systems with a single input delaya state space solution, SIAM journal on control and optimization, 35, 5, 1445-1469, (1997) · Zbl 0968.93021 [10] Tomizuka, M. (1975). Optimal continuous finite preview problem. IEEE Transactions on Automatic Control, pp. 362-365. · Zbl 0301.93076 [11] Zattoni, E. (2004). $$H_2$$-optimal decoupling with preview in the continuous-time domain. Technical report DEIS-GCG-02-2004. University of Bologna, Bologna, Italy. [12] Zou, Q.; Devasia, S., Preview-based stable-inversion for output tracking of linear systems, Journal of dynamic systems, measurement and control, 121, 625-630, (1999) [13] Zou, Q., & Devasia, S. (2002). Preview-based optimal inversion for output tracking: application to scanning tunneling microscopy. In Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, USA.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Almost sure convergence and equivalent definition

First of all this question has related answers here and here, but I am still struggling to understand the proof and the nuances that go along with it.

Show that a.s convergence of a sequence of random variables $X_k$ to a constant $\mu$ is equivalent to the condition that for all $\epsilon > 0$: $$\lim_{n\to\infty}\mathbb{P}[\omega:\sup_{k>n}|X_k(\omega) - \mu|>\epsilon] = 0$$

I have interpreted the proof in the second link as follows. I fixed $\epsilon>0$ as given and defined,

$${A_k}: = \left\{ {\left| {{X_k} - \mu } \right| \geq \epsilon } \right\}\qquad {\text{and }}\qquad {B_n}: = \bigcup\limits_{k \geqslant n} {{A_k}}$$

Then if $B = \bigcap\limits_{n \geq 1} {{B_n}}$ we have $P\{B\} = 0$ since, $$P\left\{ {\mathop {\lim }\limits_{n \to \infty } {X_n} = \mu } \right\} = 1\quad \Rightarrow \quad P\left\{ {\mathop {\lim }\limits_{n \to \infty } {X_n} \ne \mu} \right\} = 0{\text{ }}\quad {\text{or}}\quad {\text{ }}\mathop {\lim }\limits_{n \to \infty } {X_n}=\; \left({\text{D.N.E}}\right)$$ Then, \begin{aligned} P\left\{ B \right\} & = 0 \\ & = P\left\{ {\bigcap\limits_{n \geqslant 1} {\bigcup\limits_{k > n} {{A_k}} } } \right\} = P\left\{ {\mathop {\lim \sup }\limits_{k \to \infty } {A_k}} \right\} = P\left\{ {\mathop {\lim }\limits_{n \to \infty } \mathop {\sup }\limits_{k > n} {A_k}} \right\} \\ &= \mathop {\lim }\limits_{n \to \infty } P\left\{ {\mathop {\sup }\limits_{k > n} \left| {{X_k} - \mu } \right| \geqslant \varepsilon } \right\} \hfill \\ \end{aligned} But I am not sure how the last step - pulling the limit out of the probability - is justified.

I understand that $B_n$ are decreasing sets, but the continuity of probability result tell us that the limit outside becomes a union or intersection when taken inside. But this seems to be a limit remaining a limit when taken outside. Is it true that we can always pull a limit outside?

As for the converse I do not understand it at all, and my attempts to define similar sets $A_k,B_n$ etc. have failed. I know that the condition gives us, $$\mathop {\lim }\limits_{n \to \infty } P\left\{ {\mathop {\sup }\limits_{k > n} \left| {{X_k} - \mu } \right| < \varepsilon } \right\} = 1 \quad \forall \varepsilon > 0$$ But I don't know how to proceed from here. My attempts have been to take the limit inside and show somehow that $\mathop {\lim \sup }\limits_{k \to \infty } \left| {{X_k} - \mu } \right| \leq 0$. Since $\mathop {\lim \inf }\limits_{k \to \infty } \left| {{X_k} - \mu } \right| \geq 0$ if we show the previous, then the limit exists and the desired equality : ${\mathop {\lim }\limits_{k \to \infty } \left| {{X_k}} \right| = \mu }$ is achieved.

Any help would be much appreciated. I have been mulling over this for three days, and I loathe to post questions that are similar to ones already I asked, but I really have not been able to crack it or find a source that enumerates and explains each step and why it is justified.

• It's probably easier to understand the last line from the first equality. The inside is an intersection of a decreasing sequence of sets. By monotonicity properties of finite measures you can write the probability of the intersection as the limit of the probabilities. Now it's just a matter of showing that $\bigcup_{k\ge n}A_k = \{\sup_{k\ge n} |X_k-\mu| \ge \epsilon\}$ – spaceisdarkgreen Feb 17 '17 at 4:58

As mentioned in the comment, the key is to show $$\lim \sup_{k \to \infty} A_k = \{\lim \sup_{k \to \infty} |X_k - \mu| \geq \epsilon \}$$ You can do that in a couple steps, showing $\cup_{k \geq n} A_k = \{\omega \, : \,\, \sup_{k \geq n} |X_k(\omega) - \mu| \geq \epsilon \}$ holds for all $n$ using a double-inclusion argument.

Then the step of pulling the limit outside the probability is OK, as you said, from monotone limit properties of probability measures.

For the other direction, take $U_N = \{\lim \sup_{k \to \infty} |X_k - \mu| \geq 1/N \}$. Then you get $$P(U_N) = \lim_n P(\sup_{k \geq n} |X_k - \mu| \geq 1/N) = 0$$ for any $N$ by assumption.

$U_N \uparrow \{\lim \sup_{k \to \infty} |X_k - \mu| > 0 \} = U$, so $P(U) = \lim_N P(U_N) = 0$. That means

$$0 \leq \lim \inf_{k \to \infty} |X_k - \mu| \leq \lim \sup_{k \to \infty} |X_k - \mu| = 0$$ with probability one, so the limit exists and is zero.

About your question in italics, it is always true that for a real-valued random variable sequence $\{f_k\}$ you have for any $a \geq 0$ $$\lim \sup_{k \to \infty} \{|f_k| \geq a \} = \{\lim \sup_{k \to \infty} |f_k| \geq a \}$$

which is what you would prove in the step mentioned in the comment. But if you switch the inequality direction it doesn't need to hold.


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# 33. Creating a StationXML file (without instrument responses) from Scratch¶

Creating a custom StationXML file is a task that sometimes comes up in seismology. This section demonstrates how to it with ObsPy. Please note that this is not necessarily easier or more obvious then directly editing an XML file but it does provider tighter integration with the rest of ObsPy and can guarantee a valid result at the end.

Note that this assumes a certain familiarity with the FDSN StationXML standard. We’ll create a fairly simplistic StationXML file and many arguments are optional. ObsPy will validate the resulting StationXML file against its schema upon writing so the final file is assured to be valid against the StationXML schema.

Please note that this tutorial does not cover the authoring of instrument responses which would be a larger endeavour.

The following illustration shows the basic structure of ObsPy’s internal representation.

Each big box will be an object and all objects will have to be hierarchically linked to form a single Inventory object. An inventory can contain any number of Network objects, which in turn can contain any number of Station objects, which once again in turn can contain any number of Channel objects. Instrument Response objects are part of the channels and not discussed here.

from obspy.core.inventory import Inventory, Network, Station, Channel, Site
import obspy

# We'll first create all the various objects. These strongly follow the
# hierarchy of StationXML files.
inv = Inventory(
networks=[],
# The source should be the id whoever create the file.
source="ObsPy-Tutorial")

net = Network(
# This is the network code according to the SEED standard.
code="XX",
# A list of stations. We'll add one later.
stations=[],
description="A test stations.",
# Start-and end dates are optional.
start_date=obspy.UTCDateTime(2016, 1, 2))

sta = Station(
# This is the station code according to the SEED standard.
code="ABC",
latitude=1.0,
longitude=2.0,
elevation=345.0,
creation_date=obspy.UTCDateTime(2016, 1, 2),
site=Site(name="First station"))

cha = Channel(
# This is the channel code according to the SEED standard.
code="EHZ",
# This is the location code according to the SEED standard.
location_code="",
# Note that these coordinates can differ from the station coordinates.
latitude=1.0,
longitude=2.0,
elevation=345.0,
depth=10.0,
azimuth=0.0,
dip=-90.0)

# Now tie it all together.
inv.networks.append(net)
net.stations.append(sta)
sta.channels.append(cha)

# And finally write it to a StationXML file. We also force a validation against
# the StationXML schema to ensure it produces a valid StationXML file.
#
# Note that it is also possible to serialize to any of the other inventory
# output formats ObsPy supports.
inv.write("station.xml", format="stationxml", validate=True)


text:
# Tag Info

77

1 Easy Proposition Let $f:X\to Y$ be a continuous map of topological spaces, $\mathscr F$ a sheaf of abelian groups on $X$ such that $R^jf_*\mathscr F=0$ for $j>0$. Then for all $i\geq 0$ there exists a natural isomorphism $$H^i(Y, f_*\mathscr F)\simeq H^i(X,\mathscr F)$$ Proof Apply the composition rule for the derived functors of $G=\Gamma(Y, \_ )$ ...

43

This is not really an answer, but rather a meta-answer as to why there exist many conventions in the first place. The symmetric monoidal category $\mathit{sVect}$ of super-vector spaces has a non-trivial involution $J$. The symmetric monoidal functor $J:\mathit{sVect}\to \mathit{sVect}$ is the identity at the level of objects and at the level of morphisms. ...

40

The category of sets admits precisely nine model category structures, no more no less. I learned this fact from Tom Goodwillie's comments on a different MO question. It always shocks people when I mention it to them, so I guess it is surprising. I am not sure which is more surprising, that you can actually compute all the model structures or that there are ...

38

This is a result in algebraic topology, where we study the structure of topological spaces $X$. One early way to do this is to calculate a thing called $H_*(X)$, the ordinary homology of $X$. Later people discovered various "extraordinary homologies", which give more precise information. There are very many different extraordinary homologies, including ...

37

It seems that the archives have been taken down because of "legal issues related to the ownership of the content." See http://publishing.mathforge.org/discussion/169/clockss-and-portico/#Item_18 for further discussion. Unfortunately, nobody involved seems to be publicly discussing the situation, and I have not yet been able to learn exactly what's going on....

37

Perhaps one of the biggest ideas that VV was pursuing is the Initiality Conjecture for Martin-Löf type theory (with universes). A rough idea is that from the rules for a type theory, one can define a category (the syntactic category) that has certain structure depending on which rules are admitted to the type theory. For instance, dependent products ...

36

I think Abdelmalek Abdesselam and William Stagner are completely correct in their interpretation of the words "Behind" and "one immutable source" as describing one theory, the theory of symmetric functions, being the central core of another. The issue that led to this question instead comes from misunderstanding this sentence: Today it is K-theory ...

31

If you are given a homotopy functor $L$ from spaces to spectra, the assembly map is a natural map of spectra $$H_\bullet(X;L) \to L(X) ,$$ where the domain is a homology theory. This homology theory is the "homology of $X$ with coefficients in $L$," that is, $X_+ \wedge L(\text{pt})$. This map is a universal approximation to $L$ on the left by a ...

30

Aside from his work on the foundations of mathematics, which others have already elaborated on, earlier in his career Voevodsky also proved the Milnor conjecture in algebraic geometry. The Milnor conjecture relates Milnor K-theory to Galois cohomology (or, equivalently, etale cohomology, since we are dealing with the case of a field). It is related to ...

27

Of course, in any spectral sequence $E_{r+1}$ is a subquotient of $E_r$ (the kernel of $d_r$ divided by the image of $d_r$). And in general new torsion can appear in the sense of torsion elements in $ker/im$ that are not represented by torsion elements in $ker$. But this cannot happen when the spectral sequence is rationally trivial, that is, when the ...

26

There is a very nice article by Peter May et al. which is all concerned with this question: H. Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html) The article cleanly identifies three different flavors of ...

26

There is a basic way to see whether things like this should be true. Any bounded double complex of vector spaces over a field $k$ is (noncanonically) the direct sum of complexes of the following two sorts: Squares: $$\begin{matrix} k & \rightarrow & k \\ \uparrow & & \uparrow \\ k & \rightarrow & k \end{matrix}$$ Staircases:$$\... 26 I agree with Ryan that Serre's proof can be viewed as perfectly conceptual, but here is a modern version. Accept from Serre that the homotopy groups of spheres are finitely generated. Let k\colon S^n \longrightarrow K(\mathbf{Z},n) be the canonical map. We know how to rationalize spaces and maps. The rationalization of k is a map k_{0}\colon S^n_{... 26 Here are some thoughts, gathered from reading many texts about algebraic K-theory. Let me start with some historical remarks, then try to give a more revisionist motivation of the plus construction. First of all, it's true as you say that the already-divined definitions of the lower K-groups made it seem like the higher K-groups, whatever they might be, ... 25 I was the only editor of the old "K-theory" who did not join the editorial board of the "Journal of K-theory". A decision I have never regretted. On the other hand, I do regret that it is no longer possible to access the old K-theory papers online. I do not believe the fault is with Springer. I urge all authors to make their papers available online. If you ... 24 I suspect the name just arose naturally (for obvious reasons) but that it would be tough to trace back to any single person. After Cartan-Eilenberg proved it in 1956 (Homological Algebra, p.40) the first mention I see in English is by Tate in 1966/67 (p-divisible groups, p.178) followed by Hartshorne in 1968 (Cohomological Dimension of Algebraic Varieties, p.... 23 If Y is a connected CW-complex of finite type which is both an H-space and a co-H-space, then Y has the homotopy type of S^1, S^3, S^7 or a point. This is a result of Robert West: Robert W. West, H-spaces which are co-H-spaces, Proc. Amer. Math. Soc. 31 (1972), 580--582. It follows that if X is a finite type CW-complex such that \Sigma X ... 21 You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least the following properties K-theory of an affine scheme \mathrm{Spec}\,R is given by the algebraic K-theory of projective R-modules in the sense of Quillen ... 20 Since for any two spectra E,F we have$$ E_n(F)=[\mathbb{S}^n,E\wedge F]\cong [\mathbb{S}^n,F\wedge E] = F_n(E), $$you may as well ask about H_n(KU;\mathbb{Z}), the integral homology of the complex K-theory spectrum. This calculation is carried out e.g. in Chapter 16 of the book Switzer, Robert M., Algebraic topology – homology and homotopy., ... 20 We have KU_*(H\mathbb Z) = \pi_*(KU\wedge H\mathbb Z); this ring is concentrated in even dimensions and carries an isomorphism between the additive and multiplicative formal group law, hence is rational. Thus the map$$ KU\wedge H\mathbb Z\to (KU\wedge H\mathbb Z)\wedge H\mathbb Q\cong KU\wedge H\mathbb Q $$is an isomorphism. The Chern character KU\to \... 19 I don't really know if this helps, but you can in effect give the plus-construction definition of K-groups without explicitly mentioning homotopy groups, and without ever doing the plus construction: H_1BGL(R) is K_1(R). Map BGL(R) to an Eilenberg-MacLane space BK_1(R) and consider the homotopy fiber. This space has trivial H_1. Its H_2 is ... 18 Concerning the definition of symplectic K-theory: there are various possible definitions, the homotopy groups of the plus-construction of the classifying space of the infinite symplectic group is one such definition. Other definitions can be given using categories of forms (similar to Q-construction or S^{-1}S-construction for algebraic K-theory) or a ... 18 I think you are misinterpreting the quote. In the last sentence, the word "source" does not mean "source of these theories (K-theory, categories, group representations", but "source of the theory of symmetric functions". Rota is not claiming that K-theory, etc. have "at their core the ordinary, crude definitions of symmetric functions", but that "the theory ... 17 I think I have been able to reproduce the "argument by Wagoner" (perhaps it was removed from the published version?). It certainly holds in more generality that what I have written below, using the notion of "direct sum group" in Wagoner's paper (which unfortunately seems to be a little mangled). Let M be a homotopy commutative topological monoid with \... 17 You can refine this. Let's take k=2 to give the idea. To a based set X you can associate (X\wedge X)/X, a based set with free action of \Sigma_2. This leads to an operation going from stable homotopy of S^0 to stable homotopy of B\Sigma_2, such that when followed by transfer it gives the difference between the identity and squaring. This leads ... 17 The operation which sends a finite set S to its set of k-element subsets, \binom{S}{k}, gives rise to the k-th stable Hopf invariant. There is additional structure in this: the set \binom{S}{k} has a canonical k-fold covering so the operation is better viewed as a map$$ QS^0 \to Q(B\Sigma_k)_+ $$rather than as a map$$ Q S^0 \to QS^0 ,  ...

17

Thanks to a librarian's listserv here an update: Since recently K-Theory is now available through Portico (only available to members) - see http://www.portico.org/digital-preservation/news-events/news/general-news/portico-now-provides-access-to-k-theory-content But no news if there are any plans to change the situation at the Springer website...

17

Here is one conceptual description of the relationship. (I wouldn't call it an explanation; I don't really know why it's true, except that it's because Bott periodicity is true.) KO-theory is the first Weiss-derivative of the $K$-theory of Clifford algebras. More precisely: given a real inner product space $V$, we get a category $M(V)=\mathrm{Mod}(Cl(V))$ ...

17

This answer probably won't be coming from the perspective that you want, since it'll use even more stable homotopy theory than Adams did. But I think it's pretty clear in its own way. First let me make a small correction to Neil's comment. Actually, from the J-homomorphism construction you only get a map of spectra from the connective cover of \$\Sigma^{-1}...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Permutation matrix

Matrices describing the permutations of 3 elements
The product of two permutation matrices is a permutation matrix as well.

These are the positions of the six matrices:

(They are also permutation matrices.)

In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. Each such matrix represents a specific permutation of m elements and, when used to multiply another matrix, can produce that permutation in the rows or columns of the other matrix.

## Definition

Given a permutation π of m elements,

$\pi : \lbrace 1, \ldots, m \rbrace \to \lbrace 1, \ldots, m \rbrace$

given in two-line form by

$\begin{pmatrix} 1 & 2 & \cdots & m \\ \pi(1) & \pi(2) & \cdots & \pi(m) \end{pmatrix},$

its permutation matrix is the m × m matrix Pπ whose entries are all 0 except that in row i, the entry π(i) equals 1. We may write

$P_\pi = \begin{bmatrix} \mathbf e_{\pi(1)} \\ \mathbf e_{\pi(2)} \\ \vdots \\ \mathbf e_{\pi(m)} \end{bmatrix},$

where $\mathbf e_j$ denotes a row vector of length m with 1 in the jth position and 0 in every other position.[1]

## Properties

Given two permutations π and σ of m elements and the corresponding permutation matrices Pπ and Pσ

$P_{\sigma} P_{\pi} = P_{\pi\,\circ\,\sigma}$

This somewhat unfortunate rule is a consequence of the definitions of multiplication of permutations (composition of bijections) and of matrices, and of the choice of using the vectors $\mathbf{e}_{\pi(i)}$ as rows of the permutation matrix; if one had used columns instead then the product above would have been equal to $P_{\sigma\,\circ\,\pi}$ with the permutations in their original order.

As permutation matrices are orthogonal matrices (i.e., $P_{\pi}P_{\pi}^{T} = I$), the inverse matrix exists and can be written as

$P_{\pi}^{-1} = P_{\pi^{-1}} = P_{\pi}^{T}.$

Multiplying $P_{\pi}$ times a column vector g will permute the rows of the vector:

$P_\pi \mathbf{g} = \begin{bmatrix} \mathbf{e}_{\pi(1)} \\ \mathbf{e}_{\pi(2)} \\ \vdots \\ \mathbf{e}_{\pi(n)} \end{bmatrix} \begin{bmatrix} g_1 \\ g_2 \\ \vdots \\ g_n \end{bmatrix} = \begin{bmatrix} g_{\pi(1)} \\ g_{\pi(2)} \\ \vdots \\ g_{\pi(n)} \end{bmatrix}.$

Now applying $P_\sigma$ after applying $P_\pi$ gives the same result as applying $P_{\pi\circ\sigma}$ directly, in accordance with the above multiplication rule: call $P_\pi\mathbf{g} = \mathbf{g}'$, in other words

$g'_i=g_{\pi(i)}\,$

for all i, then

$P_\sigma(P_\pi(\mathbf{g})) = P_\sigma(\mathbf{g}') = \begin{bmatrix} g'_{\sigma(1)} \\ g'_{\sigma(2)} \\ \vdots \\ g'_{\sigma(n)} \end{bmatrix} = \begin{bmatrix} g_{\pi(\sigma(1))} \\ g_{\pi(\sigma(2))} \\ \vdots \\ g_{\pi(\sigma(n))} \end{bmatrix}.$

Multiplying a row vector h times $P_{\pi}$ will permute the columns of the vector by the inverse of $P_{\pi}$:

$\mathbf{h}P_\pi = \begin{bmatrix} h_1 \; h_2 \; \dots \; h_n \end{bmatrix} \begin{bmatrix} \mathbf{e}_{\pi(1)} \\ \mathbf{e}_{\pi(2)} \\ \vdots \\ \mathbf{e}_{\pi(n)} \end{bmatrix} = \begin{bmatrix} h_{\pi^{-1}(1)} \; h_{\pi^{-1}(2)} \; \dots \; h_{\pi^{-1}(n)} \end{bmatrix}$

Again it can be checked that $(\mathbf{h}P_\sigma)P_\pi = \mathbf{h}P_{\pi\circ\sigma}$.

## Notes

Let Sn denote the symmetric group, or group of permutations, on {1,2,...,n}. Since there are n! permutations, there are n! permutation matrices. By the formulas above, the n × n permutation matrices form a group under matrix multiplication with the identity matrix as the identity element.

If (1) denotes the identity permutation, then P(1) is the identity matrix.

One can view the permutation matrix of a permutation σ as the permutation σ of the columns of the identity matrix I, or as the permutation σ−1 of the rows of I.

A permutation matrix is a doubly stochastic matrix. The Birkhoff–von Neumann theorem says that every doubly stochastic real matrix is a convex combination of permutation matrices of the same order and the permutation matrices are precisely the extreme points of the set of doubly stochastic matrices. That is, the Birkhoff polytope, the set of doubly stochastic matrices, is the convex hull of the set of permutation matrices.[2]

The product PM, premultiplying a matrix M by a permutation matrix P, permutes the rows of M; row i moves to row π(i). Likewise, MP permutes the columns of M.

The map Sn → A ⊂ GL(n, Z2) is a faithful representation. Thus, |A| = n!.

The trace of a permutation matrix is the number of fixed points of the permutation. If the permutation has fixed points, so it can be written in cycle form as π = (a1)(a2)...(ak)σ where σ has no fixed points, then ea1,ea2,...,eak are eigenvectors of the permutation matrix.

From group theory we know that any permutation may be written as a product of transpositions. Therefore, any permutation matrix P factors as a product of row-interchanging elementary matrices, each having determinant −1. Thus the determinant of a permutation matrix P is just the signature of the corresponding permutation.

## Examples

### Permutation of rows and columns

When a permutation matrix P is multiplied with a matrix M from the left PM it will permute the rows of M (here the elements of a column vector),
when P is multiplied with M from the right MP it will permute the columns of M (here the elements of a row vector):

 P * (1,2,3,4)T = (4,1,3,2)T (1,2,3,4) * P = (2,4,3,1)

Permutations of rows and columns are for example reflections (see below) and cyclic permutations (see cyclic permutation matrix).

### Permutation of rows

The permutation matrix Pπ corresponding to the permutation :$\pi=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 2 & 5 & 3 \end{pmatrix},$ is

$P_\pi = \begin{bmatrix} \mathbf{e}_{\pi(1)} \\ \mathbf{e}_{\pi(2)} \\ \mathbf{e}_{\pi(3)} \\ \mathbf{e}_{\pi(4)} \\ \mathbf{e}_{\pi(5)} \end{bmatrix} = \begin{bmatrix} \mathbf{e}_{1} \\ \mathbf{e}_{4} \\ \mathbf{e}_{2} \\ \mathbf{e}_{5} \\ \mathbf{e}_{3} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \end{bmatrix}.$

Given a vector g,

$P_\pi \mathbf{g} = \begin{bmatrix} \mathbf{e}_{\pi(1)} \\ \mathbf{e}_{\pi(2)} \\ \mathbf{e}_{\pi(3)} \\ \mathbf{e}_{\pi(4)} \\ \mathbf{e}_{\pi(5)} \end{bmatrix} \begin{bmatrix} g_1 \\ g_2 \\ g_3 \\ g_4 \\ g_5 \end{bmatrix} = \begin{bmatrix} g_1 \\ g_4 \\ g_2 \\ g_5 \\ g_3 \end{bmatrix}.$

## Explanation

A permutation matrix will always be in the form

$\begin{bmatrix} \mathbf{e}_{a_1} \\ \mathbf{e}_{a_2} \\ \vdots \\ \mathbf{e}_{a_j} \\ \end{bmatrix}$

where eai represents the ith basis vector (as a row) for Rj, and where

$\begin{bmatrix} 1 & 2 & \ldots & j \\ a_1 & a_2 & \ldots & a_j\end{bmatrix}$

is the permutation form of the permutation matrix.

Now, in performing matrix multiplication, one essentially forms the dot product of each row of the first matrix with each column of the second. In this instance, we will be forming the dot product of each row of this matrix with the vector of elements we want to permute. That is, for example, v= (g0,...,g5)T,

eai·v=gai

So, the product of the permutation matrix with the vector v above, will be a vector in the form (ga1, ga2, ..., gaj), and that this then is a permutation of v since we have said that the permutation form is

$\begin{pmatrix} 1 & 2 & \ldots & j \\ a_1 & a_2 & \ldots & a_j\end{pmatrix}.$

So, permutation matrices do indeed permute the order of elements in vectors multiplied with them.

## References

1. ^ Brualdi (2006) p.2
2. ^ Brualdi (2006) p.19


text:
# Commutators in finite groups

I get stuck with this question :

Let $g$ be an element in a finite group $G$, and let $k$ be an integer coprime to $|g|$. Prove that $g$ is a commutator in $G$ if and only if $g^k$ is a commutator.

It is obvious that if $g$ is a commutator, i.e., $g\in G'$ then $g^k\in G'$ since $G'$ is a subgroup of $G$. I dont know how to proceed the converse. Do you have any hint?

• "$g$ is a commutator" is not the same thing as "$g$ is in $G'$". The subgroup $G'$ is generated by the commutators, but it may contain elements that are not, themselves, commutators. – Gerry Myerson Oct 21 '12 at 23:12
• $g=[a,b]=a^{-1}b^{-1}ab$ for some $a,b\in G$. Then $g^k=k^{-1}a^{-1}b^{-1}abk=k^{-1}a^{-1}kk^{-1}b^{-1}kk^{-1}akk^{-1}bk=[a^k,b^k]$. Just kidding. Anyway, I just answered what is evidently problem 2.12 over here math.stackexchange.com/questions/218302/… if that would help you. – Alexander Gruber Oct 22 '12 at 22:35


text:
# getri_batch#

Computes the inverses of a batch of LU-factored matrices determined by getrf_batch.

Description

getri_batch supports the following precisions.

T

float

double

std::complex<float>

std::complex<double>

## getri_batch (Buffer Version)#

Description

The buffer version of getri_batch supports only the strided API.

Strided API

The routine computes the inverses $$A_i^{-1}$$ of general matrices $$A_i$$. Before calling this routine, call the Strided API of the getrf_batch (Buffer Version) function to factorize $$A_i$$.

Syntax

namespace oneapi::mkl::lapack {
void getri_batch(cl::sycl::queue &queue, std::int64_t n, cl::sycl::buffer<T> &a, std::int64_t lda, std::int64_t stride_a, cl::sycl::buffer<std::int64_t> &ipiv, std::int64_t stride_ipiv, std::int64_t batch_size, cl::sycl::buffer<T> &scratchpad, std::int64_t scratchpad_size)
}

Input Parameters

queue

Device queue where calculations will be performed.

n

Order of the matrices $$A_i$$ ($$0 \le n$$).

a

Result of the Strided API of the getrf_batch (Buffer Version) function.

lda

Leading dimension of $$A_i$$ ($$n\le \text{lda}$$).

stride_a

Stride between the beginnings of matrices $$A_i$$ inside the batch array a.

ipiv

Arrays returned by the Strided API of the getrf_batch (Buffer Version) function.

stride_ipiv

Stride between the beginnings of arrays $$\text{ipiv}_i$$ inside the array ipiv.

batch_size

Number of problems in a batch.

Scratchpad memory to be used by routine for storing intermediate results.

Size of scratchpad memory as a number of floating point elements of type T. Size should not be less than the value returned by the Strided API of the getri_batch_scratchpad_size function.

Output Parameters

a

Inverse $$n \times n$$ matrices $$A_i^{-1}$$.

This routine shall throw the following exceptions if the associated condition is detected. An implementation may throw additional implementation-specific exception(s) in case of error conditions not covered here.

oneapi::mkl::lapack::batch_error

oneapi::mkl::unimplemented

oneapi::mkl::unsupported_device

oneapi::mkl::lapack::invalid_argument

The info code of the problem can be obtained by info() method of exception object:

If info = -n, the $$n$$-th parameter had an illegal value.

If info equals to value passed as scratchpad size, and detail() returns non zero, then passed scratchpad is of insufficient size, and required size should be not less then value returned by detail() method of exception object.

If info is not zero and detail() returns zero, then there were some errors for some of the problems in the supplied batch and info code contains the number of failed calculations in a batch.

## getri_batch (USM Version)#

Description

The USM version of getri_batch supports the group API and strided API.

Group API

The routine computes the inverses $$A_i^{-1}$$ of general matrices $$A_i$$, $$i \in \{1...batch\_size\}$$. Before calling this routine, call the Group API of the getrf_batch (USM Version) function to factorize $$A_i$$. Total number of problems to solve, batch_size, is a sum of sizes of all of the groups of parameters as provided by group_sizes array.

Syntax

namespace oneapi::mkl::lapack {
cl::sycl::event getri_batch(cl::sycl::queue &queue, std::int64_t *n, T **a, std::int64_t *lda, std::int64_t **ipiv, std::int64_t group_count, std::int64_t *group_sizes, T *scratchpad, std::int64_t scratchpad_size, const std::vector<cl::sycl::event> &events = {})
}

Input Parameters

queue

Device queue where calculations will be performed.

n

Array of group_count $$n_g$$ parameters specifying the order of the matrices $$A_i$$ ($$0 \le n_g$$) belonging to group $$g$$.

a

Result of the Group API of the getrf_batch (USM Version) function.

lda

Array of group_count $$\text{lda}_g$$ parameters specifying the leading dimensions of the matrices $$A_i$$ ($$n_g \le \text{lda}_g$$) belonging to group $$g$$.

ipiv

Arrays returned by the Group API of the getrf_batch (USM Version) function.

group_count

Number of groups of parameters. Must be at least 0.

group_sizes

Array of group_count integers. Array element with index $$g$$ specifies the number of problems to solve for each of the groups of parameters $$g$$. So the total number of problems to solve, batch_size, is a sum of all parameter group sizes.

Scratchpad memory to be used by routine for storing intermediate results.

Size of scratchpad memory as a number of floating point elements of type T. Size should not be less than the value returned by the Group API of the getri_batch_scratchpad_size function.

events

List of events to wait for before starting computation. Defaults to empty list.

Output Parameters

a

Inverse $$n_g \times n_g$$ matrices $$A_i^{-1}$$.

Return Values

Output event to wait on to ensure computation is complete.

Throws

This routine shall throw the following exceptions if the associated condition is detected. An implementation may throw additional implementation-specific exception(s) in case of error conditions not covered here.

oneapi::mkl::lapack::batch_error

oneapi::mkl::unimplemented

oneapi::mkl::unsupported_device

oneapi::mkl::lapack::invalid_argument

The info code of the problem can be obtained by info() method of exception object:

If info = -n, the $$n$$-th parameter had an illegal value.

If info equals to value passed as scratchpad size, and detail() returns non zero, then passed scratchpad is of insufficient size, and required size should be not less then value returned by detail() method of exception object.

If info is not zero and detail() returns zero, then there were some errors for some of the problems in the supplied batch and info code contains the number of failed calculations in a batch.

Strided API

The routine computes the inverses $$A_i^{-1}$$ of general matrices $$A_i$$. Before calling this routine, call the Strided API of the getrf_batch (USM Version) function to factorize $$A_i$$.

Syntax

namespace oneapi::mkl::lapack {
cl::sycl::event getri_batch(cl::sycl::queue &queue, std::int64_t n, T *a, std::int64_t lda, std::int64_t stride_a, std::int64_t *ipiv, std::int64_t stride_ipiv, std::int64_t batch_size, T *scratchpad, std::int64_t scratchpad_size, const std::vector<cl::sycl::event> &events = {})
};

Input Parameters

queue

Device queue where calculations will be performed.

n

Order of the matrices $$A_i$$ ($$0 \le n$$).

a

Result of the Strided API of the getrf_batch (USM Version) function.

lda

Leading dimension of $$A_i$$ ($$n \le \text{lda}$$).

stride_a

Stride between the beginnings of matrices $$A_i$$ inside the batch array a.

ipiv

Arrays returned by the Strided API of the getrf_batch (USM Version) function.

stride_ipiv

Stride between the beginnings of arrays $$\text{ipiv}_i$$ inside the array ipiv.

batch_size

Number of problems in a batch.

Scratchpad memory to be used by routine for storing intermediate results.

Size of scratchpad memory as a number of floating point elements of type T. Size should not be less than the value returned by the Strided API of the getri_batch_scratchpad_size function.

events

List of events to wait for before starting computation. Defaults to empty list.

Output Parameters

a

Inverse $$n \times n$$ matrices $$A_i^{-1}$$.

Return Values

Output event to wait on to ensure computation is complete.

Throws

This routine shall throw the following exceptions if the associated condition is detected. An implementation may throw additional implementation-specific exception(s) in case of error conditions not covered here.

oneapi::mkl::lapack::batch_error

oneapi::mkl::unimplemented

oneapi::mkl::unsupported_device

oneapi::mkl::lapack::invalid_argument

The info code of the problem can be obtained by info() method of exception object:

If info = -n, the $$n$$-th parameter had an illegal value.

If info equals to value passed as scratchpad size, and detail() returns non zero, then passed scratchpad is of insufficient size, and required size should be not less then value returned by detail() method of exception object.

If info is not zero and detail() returns zero, then there were some errors for some of the problems in the supplied batch and info code contains the number of failed calculations in a batch.

Parent topic: LAPACK-like Extensions Routines


text:
# Question #ae517

Oct 24, 2016

$x = \pm 2 \sqrt{7}$

#### Explanation:

$f \left(x\right) = g \left(x\right)$

$\implies x + 5 = \frac{3}{x - 5}$

$\implies \left(x + 5\right) \left(x - 5\right) = 3$

$\implies {x}^{2} - 25 = 3$

At this point it would be simplest to just add $25$ to both sides and take a square root, but as the question refers to the quadratic formula, we will instead put it in standard form and proceed in that manner.

$\implies {x}^{2} - 28 = 0$

$\implies 1 {x}^{2} + 0 x - 28 = 0$

$\implies x = \frac{- 0 \pm \sqrt{{0}^{2} - 4 \left(1\right) \left(- 28\right)}}{2 \left(1\right)}$

$= \frac{\pm 4 \sqrt{7}}{2}$

$= \pm 2 \sqrt{7}$


text:
# RD Sharma Class 12 Ex 11.2 Solutions Chapter 11 Differentiation

Here we provide RD Sharma Class 12 Ex 11.2 Solutions Chapter 11 Differentiation for English medium students, Which will very helpful for every student in their exams. Students can download the RD Sharma Class 12 Ex 11.2 Solutions Chapter 11 Differentiation book pdf download. Now you will get step-by-step solutions to each question.

## RD Sharma Class 12 Ex 11.2 Solutions Chapter 11 Differentiation

### Question 1. Differentiate y = sin (3x + 5) with respect to x.

Solution:

We have,

y = sin (3x + 5)

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 2. Differentiate y = tan2 x with respect to x.

Solution:

We have,

y = tan2 x

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 3. Differentiate y = tan (x + 45°) with respect to x.

Solution:

We have,

y = tan (x + 45°)

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 4. Differentiate y = sin (log x) with respect to x.

Solution:

We have,

y = sin (log x)

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 5. Differentiate y = esin √xwith respect to x.

Solution:

We have,

y = esin √x

On differentiating y with respect to x we get,

On using chain rule, we have

On using chain rule again, we have

### Question 6. Differentiate y = etan x with respect to x.

Solution:

We have,

y = etan x

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 7. Differentiate y = sin2 (2x + 1) with respect to x.

Solution:

We have,

y = sin2 (2x + 1)

On differentiating y with respect to x we get,

On using chain rule, we have

On using chain rule again, we have

As sin 2A = 2 sin A cos A, we get

### Question 8. Differentiate y = log7 (2x − 3) with respect to x.

Solution:

We have,

y = log7 (2x − 3)

As , we have

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 9. Differentiate y = tan 5x° with respect to x.

Solution:

We have,

y = tan 5x°

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 10. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 11. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 12. Differentiate y = logx 3 with respect to x.

Solution:

We have,

y = logx 3

As , we get

y =

On differentiating y with respect to x we get,

On using chain rule, we have

As , we get

### Question 13. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 14. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 15. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 16. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 17. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 18. Differentiate y = (log sin x)2 with respect to x.

Solution:

We have,

y = (log sin x)2

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 19. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

On using quotient rule, we have

### Question 20. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

On using quotient rule, we have

### Question 21. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using product rule, we have

On using chain rule, we have

### Question 22. Differentiate y = sin(log sin x) with respect to x.

Solution:

We have,

y = sin(log sin x)

On differentiating y with respect to x we get,

On using chain rule, we have

On using chain rule again, we have

### Question 23. Differentiate y = etan 3x with respect to x.

Solution:

We have,

y = etan 3x

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 24. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 25. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

On using quotient rule, we have

### Question 26. Differentiate y =  with respect to x.

Solution:

We have,

y =

y =

As , we get

y =

On differentiating y with respect to x we get,

On using chain rule, we have

On using quotient rule, we have

### Question 27. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 28. Differentiate y =  with respect to x.

Solution:

We have,

I =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 29. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using quotient rule, we have

On using product rule, we have

### Question 30. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 31. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using quotient rule and chain rule, we get

### Question 32. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using quotient rule and chain rule, we get

### Question 33. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 34. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 35. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 36. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 37. Differentiate y =  with respect to x.

Solution:

We have,

y =

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 38. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 39. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using quotient rule, we have

On using product rule and chain rule, we have

### Question 40. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using product rule and chain rule, we have

### Question 41. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using product rule and chain rule, we have

### Question 42. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using quotient rule, chain rule and product rule we get,

### Question 43. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we get

As 2 sin A cos A = sin 2A, we get

### Question 44. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using product rule and chain rule, we have

### Question 45. Differentiate y =  with respect to x.

Solution:

We have,

y =

On rationalizing we get,

y =

y =

y =

y =

y =

y =

On differentiating y with respect to x we get,

### Question 46. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

### Question 47. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule, we have

On using chain rule again, we have

### Question 48. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using chain rule and quotient rule, we get

### Question 49. Differentiate y =  with respect to x.

Solution:

We have,

y =

On differentiating y with respect to x we get,

On using quotient rule, we get

On using product rule, we get

I think you got complete solutions for this chapter. If You have any queries regarding this chapter, please comment in the below section our subject teacher will answer you. We tried our best to give complete solutions so you got good marks in your exam.

If these solutions have helped you, you can also share rdsharmasolutions.in to your friends.


text:
# Institut de Physique Nucléaire de Lyon

Domaine scientifique de la Doua
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• ### [in2p3-01063466] Superheavy magic nuclei in Relativistic Hartree-Fock-Bogoliubov theory

19 avril, par ano.nymous@ccsd.cnrs.fr.invalid (J. Li), J. Li
The occurrence of the spherical shell closures in superheavy nuclear systems is explored within the relativistic Hartree-Fock-Bogoliubov (RHFB) theory with density- dependent meson-nucleon couplings. We use the two-nucleon gaps δ2n(p) and pairing gaps Δv(π) to characterize the shell effects. The (...)
• ### [in2p3-00838514] Superheavy magic structures and relativistic pseudo-spin symmetry

19 avril, par ano.nymous@ccsd.cnrs.fr.invalid (J. J. Li), J. J. Li
We have explored the occurrence of the spherical shell closures for superheavy nuclei in the framework of the relativistic Hartree-Fock-Bogoliubov (RHFB) theory and the shell effects in terms of two-nucleon gaps $\delta_2n(p)$. Although the results depend slightly on the forces used, the (...)

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 P: 106 Hey folks I'm looking for a way to find the characteristic equation for a second order coupled system of differential equations such as... $$\ddot{x} + A\dot{y} + Bx = 0$$ $$\ddot{y} + C\dot{x} + Dy = 0$$ Where x and y are functions of time. I know I can solve it by setting x and y to standard results (trig, exponential) but I'd like to know a method to solving this rather than plug and solve for coefficients. Specifically I'd like to know how to find the characteristic equation for this. I've tried setting it to a first order system but I can't see it leading anywhere (or perhaps I just did it wrong...). I don't want a full answer, just the name of a method or something like that.


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# Revision history [back]

### OpenCV Window Resize with Trackbar

Hi Guys,

I have set my windows to CV_WINDOW_NORMAL but whenever I add a track-bar to a window, it expends that window to the full width and I cannot re-size properly. It can be re sized vertically and slightly horizontally till the end of the slider. Is there anyway around it? Perhaps limiting the max length of the slider or having the slider automatically re-size to the window width?

An example:

    namedWindow("Threshold", CV_WINDOW_NORMAL);
createTrackbar("Min", "Threshold", &threshMin, 255, on_trackbar);
imshow("Threshold", imageThresh);

And this is what I get: See image:

If it helps, I'm using Visual Studio 2013 running on Windows 7. When the trackbar is removed from the window, it can be re-sized normally.


text:
# Difference between revisions of "Manuals/calci/COTH"

COTH(x)

• where x is any real number.
• COTH() returns the inverse hyperbolic tangent of a number.

## Description

• This function gives the hyperbolic Cotangent of 'x'.
• It's also called as Circular function.
• Let z is any real number.
• COTH is the reciprocal of TANH function.i.e.COTH(z)=
• i.e or iCOT(iz).where 'i' is the imaginary unit and .
• Also relation between Hyperbolic & Trignometric function is &

## Examples

COTH(x)

• x is any real number.
 COTH(x) Value COTH(1) 1.3130352854993312 COTH(30) 1 COTH(-45) -1

Hyperbolic COT


text:
# HowTo Adding a new transport equation

Suppose you want to solve and additional scalar transport equation for the scalar ψ by adding it to an existing solver. The equation has the form

$\frac{\partial}{\partial t} \left(\rho \psi \right) + \nabla \cdot \phi \psi - \nabla \cdot \Gamma \nabla \psi = S_{\psi}$

where ρ and Γ are defined as dimensionedScalar and are retrieved from a dictionary using the lookup member function. Sφ is the source term.

To implement the equation, just add the line:

dimensionedScalarField psi;

to the createFields.H file of the solver,

Then, solve the equation by adding the following lines in the point of the solver where you want the equation to be solved.

solve
(
fvm::ddt(rho, psi)
+ fvm::div(phi, psi)
- fvm::laplacian(gamma, psi)
==
S_psi
);

In this example, the source term is treated explicitly. To manage it implicitly, OpenFOAM provides the Sp and the SuSp functions. Suppose we can write it as Sk = Κψ. The code lines to solve the same equation with an implicit treatment of the source term are:

solve
(
fvm::ddt(rho, psi)
+ fvm::div(phi, psi)
- fvm::laplacian(gamma, psi)
==
fvm::Sp(kappa, psi)
);

SuSp(kappa, psi) can be used to discretise the source term implicitly or explicitly according to the sign of kappa.


text:
# A line segment has endpoints at (7 ,8 ) and (3 ,5 ). If the line segment is rotated about the origin by (3 pi)/2 , translated vertically by -2 , and reflected about the y-axis, what will the line segment's new endpoints be?

Nov 6, 2016

to $A \left(7 , 8\right)$ corresponds $A ' ' ' \left(- 8 , - 9\right)$
to $B \left(3 , 5\right)$ corresponds B'''(-5;-5)

#### Explanation:

A rotation of $3 \frac{\pi}{2}$ anticlockwise is equivalent to a rotation of $\frac{\pi}{2}$ clockwise. In practice we we have to swap the abscissa with the ordinata payng attention to rigth sign to choose for each coordinate according to quadrant of destination after the rotation!

In our case, if $A \left(7 , 8\right)$ its corrispondent after rotation is $A ' \left(8 , - 7\right)$
and if $B \left(3 , 5\right)$ its corrispondent after rotation is $B ' \left(5 , - 3\right)$

If we translate vertically by -2, the rotated segment $A ' B '$, we have to lower the segment by 2 units. In practice we have to subtract 2 to the ordinates and we get
$A ' ' \left(8 , - 9\right)$ and #B''(5,-5)

The reflection about the y-axis implies to change the signs only to the abscissas
If we reflect about the y-axis, we get
$A ' ' ' \left(- 8 , - 9\right)$ and $B ' ' ' \left(- 5 , - 5\right)$


text:
Note that there are several notations in common use for the complex conjugate. It observes that there are some significant differences between the real/complex matrix equations and the complex conjugate matrix equations. quat = [quaternion (randn (2,4)),quaternion (randn (2,4))] quat= 2×2 quaternion array 0.53767 - 2.2588i + 0.31877j - 0.43359k 3.5784 - 1.3499i + 0.7254j + 0.71474k 1.8339 + 0.86217i - 1.3077j + 0.34262k 2.7694 + 3.0349i - 0.063055j - 0.20497k. Create a matrix of quaternions and compute its complex conjugate transpose. “The book is well organized and presents the most important notions of iterative solutions, explicit solutions and applications in systems and control. \(\hspace{60px} A\hspace{130px}\bar{A}\\. Create a matrix of quaternions and compute its complex conjugate transpose. / Matrix Transform. ...you'll find more products in the shopping cart. Complex conjugate for a complex number is defined as the number obtained by changing the sign of the complex part and keeping the real part the same. The notation A^* is sometimes also used, which can lead to confusion since this symbol is also used to denote the conjugate transpose. Calculates the conjugate matrix. If a complex number is represented as a 2×2 matrix, the notations are identical. Springer is part of, Please be advised Covid-19 shipping restrictions apply. Applied physics and engineering texts tend to prefer z^*, while most … numpy.matrix.conjugate¶ matrix.conjugate ¶ Return the complex conjugate, element-wise. The complex conjugate of a complex number $${\displaystyle z}$$ is written as $${\displaystyle {\overline {z}}}$$ or $${\displaystyle z^{*}\!}$$. The book is the first book on complex matrix equations including the conjugate of unknown matrices. quatTransposed = quat'. The book is the first book on complex matrix equations including the conjugate of unknown matrices. The study of these conjugate matrix equations is motivated by the investigations on stabilization and model reference tracking control for discrete-time antilinear systems, which are a particular kind of complex system with structure constraints. When a scalar real-valued function depends on a complex-valued matrix parameter, the necessary conditions for optimality can be found by either setting the derivative of the function with respect to the complex-valued matrix parameter or its complex conjugate to zero. We need to consider what the cofactor matrix corresponds to. 218.100.84.67. (CCE), Over 10 million scientific documents at your fingertips. For example, the solvability of a real Sylvester matrix equation can be characterized by matrix similarity; however, the solvability of the con-Sylvester matrix equation in complex conjugate form is related to the concept of con-similarity. Please review prior to ordering, Offers iterative solutions and explicit solutions for some complex conjugate matrix equations, Observes the differences between the real/complex matrix equations and complex conjugate matrix equations, Provides a full picture of this field for researchers and introduces some new concepts, such as conjugate products and con-Sylvester-sum as well, ebooks can be used on all reading devices, Institutional customers should get in touch with their account manager, Usually ready to be dispatched within 3 to 5 business days, if in stock, The final prices may differ from the prices shown due to specifics of VAT rules. For example, the solvability of a real Sylvester matrix equation can be characterized by matrix similarity; however, the solvability of the con-Sylvester matrix equation in complex conjugate form is related to the concept of con-similarity. For a matrix, the complex conjugate … The second is preferred in physics, where dagger (†) is used for the conjugate transpose, while the bar-notation is more common in pure mathematics. It observes that there are some significant differences between the real/complex matrix equations and the complex conjugate matrix equations. https://doi.org/10.1007/978-981-10-0637-1, Springer Science+Business Media Singapore 2017, COVID-19 restrictions may apply, check to see if you are impacted, Hierarchical-Update-Based Iterative Approaches. The book is the first book on complex matrix equations including the conjugate of unknown matrices. }}}$$means$${\displaystyle e^{i\varphi }+e^{-i\varphi }}$$. For example, writing$${\displaystyle e^{i\varphi }+{\text{c.c. In addition, the new concept of conjugate product for complex polynomial matrices is also proposed in order to establish a unified approach for solving a type of complex matrix equation. The result of the sequential application of these two operations is called conjugate transpose (or Hermitian transpose). The study of these conjugate matrix equations is motivated by the investigations on stabilization and model reference tracking control for discrete-time antilinear systems, which are a particular kind of complex system with structure constraints. Matrix Complex Conjugate Transpose. The book is suitable for senior undergraduate and graduate students as well as practical engineers, scientist and researchers interested in iterative, explicit solutions and of applications in systems and control.” (Seenith Sivasundaram, zbMATH 1371.15003, 2017), Hierarchical-Update-Based Iterative Approaches. The operation also negates the imaginary part of any complex numbers. The operation also negates the imaginary part of any complex numbers. In some texts, the complex conjugate of a previous known number is abbreviated as "c.c.". These facts, however, are not sufficient to rewrite the expression as a complex number identity. It proposes useful approaches to obtain iterative solutions or explicit solutions for several types of complex conjugate matrix equation. book series It proposes useful approaches to obtain iterative solutions or explicit solutions for several types of complex conjugate matrix equation. Is abbreviated as  c.c.  well organized and presents the most important notions of iterative solutions or solutions... In the Wolfram Language as conjugate [ z ] previous known number is abbreviated as  c.c..... ( \hspace { 60px } A\hspace { 130px } \bar { a } \\ what cofactor. Matrix and conjugate of matrix, transpose and take the complex conjugate transpose of a number... Price for Spain ( gross ), © 2020 Springer Nature Switzerland AG ( or transpose... And take the complex conjugate transpose of matrix and conjugate of matrix -.. As  c.c.  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Nature Switzerland AG the matrix taking the complex conjugate matrix equations in common use for the complex conjugate unknown... Numpy.Matrix.Conjugate¶ matrix.conjugate ¶ Return the complex conjugate is implemented in the Wolfram Language as conjugate [ z.. Wu, Ai-Guo, Zhang, Ying pure imaginary matrix equation returns on! Its complex conjugate matrix  a '' compute its complex conjugate matrix equation, the notations are identical only. In matrix algebra that we need complex conjugate matrix both transpose and conjugate of unknown matrices transpose a. The main diagonal \bar { a } \\... you 'll find more products in the shopping cart ( Hermitian. ¶ Return the complex conjugate transpose is only defined elementwise in general. several notations in use. Despite the existence of conj complex matrix equations for Systems and Control et des millions livres.: complex_conjugate function calculates conjugate of a matrix is obtained by replacing each with... 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Of each element, reflecting the elements across the main diagonal several types of complex conjugate of unknown matrices expression! Is part of any complex numbers, Ai-Guo, complex conjugate matrix, Ying pure imaginary free! B=0, z is real, when a=0, we say that z pure! A conjugate matrix of a complex number online, transpose of a matrix interchanges row. Obtained by replacing each term with its complex conjugate, element-wise a complex number reflecting the elements across main! Known number is represented as a complex number corresponds to the reciprocal the! Of any complex numbers of, Please be advised Covid-19 shipping restrictions apply in Wolfram... The sequential application of these two operations is called conjugate transpose be advised Covid-19 shipping apply... Stock sur Amazon.fr important notions of iterative solutions, explicit solutions for several types complex! If a complex number online of each element of  a '' is the matrix representation of matrix. Titles in Popular Science \ ( \hspace { 60px } A\hspace { 130px } {! Reflecting the elements across the main diagonal ) continues to be defined, despite existence! Is real, when a=0, we say that z is real, a=0... Javascript is currently disabled, this site works much better if you enable javascript in your browser z!

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text:
# Access from included files to environments in AUCtex [duplicate]

When I am writing slides, I usually use one master file and several dependent files; as a macro package I use beamer. I would like to access the environments defined by the beamer package also in the dependent files (which I include through \input) through C-c C-e. I can access them in the master file. By the same token, I would like to access the macros in my macro packages, which I also include through \input. My .emacs file contains the lines

(setq-default TeX-master nil)
(setq TeX-parse-self t)
(setq TeX-auto-save t)

Any suggestions?

-
Welcome to TeX.sx! You don't have to sign with your name since it automatically appears in the lower right corner of your post. –  lockstep Dec 11 '12 at 19:49
Does setting the variable LaTeX-default-style to beamer help, e.g. putting (setq LaTeX-default-style beamer)` in your emacs or by putting -*- mode: LaTeX; LaTeX-default-style: beamer; -*- in the first line of the dependent files? –  N.N. Dec 12 '12 at 9:51


text:
0
Research Papers

Solar Water Disinfection Studies With Supported $TiO2$ and Polymer-Supported Ru(II) Sensitizer in a Compound Parabolic Collector

[+] Author and Article Information
Juan Rodríguez

Facultad de Ciencias, Universidad Nacional de Ingeniería, P.O. Box 31-139, Lima 31, Perú; Universidad de Tarapacá, Avenue General Velásquez No. 1775, Arica, Chile

Clido Jorge, Patricia Zúñiga, Silvia Ponce, Walter Estrada

Facultad de Ciencias, Universidad Nacional de Ingeniería, P.O. Box 31-139, Lima 31, Perú

Javier Palomino, Pedro Zanabria

Centro de Capacitación para el Desarrollo (CECADE), Urbanización COVIDUC H-16 San Sebastián, Cusco, Perú, Cusco 84, Perú

José L. Solís

Facultad de Ciencias, Universidad Nacional de Ingeniería, P.O. Box 31-139, Avenue Tupac Amaru 210, Lima 31 Peru; Instituto Peruano de Energía Nuclear, Avenue Canadá 1470, Lima, Perú

J. Sol. Energy Eng 132(1), 011001 (Nov 09, 2009) (5 pages) doi:10.1115/1.4000328 History: Received October 11, 2007; Revised March 19, 2009; Published November 09, 2009

Abstract

Solar water disinfection was performed using $TiO2$ and a Ru(II) complex as fixed catalysts located in a compound parabolic collector photoreactor. Studies were performed in the laboratory as well as at a greenfield site. Under laboratory conditions, natural water contaminated with cultured bacteria was photocatalytically treated and the influence of the photolysis as well as of both catalysts was studied. Experiments were performed with contaminated water flowing at 12 l/min; under these conditions, photocatalytic experiments performed with a supported heterogeneous photocatalyst (Ahlstrom paper impregnated with $TiO2$) showed it to be effective in degrading bacteria in water. The Ru complex catalyst, however, showed no clear evidence for disinfecting water, and its efficiency was comparable to the simple photolysis. Under on-site experiments, bacteria contaminated water from the Yaurisque river at Cusco, Peru was treated. As a general trend, after photocatalytic treatment a reduction in the E-coli population present in water was observed. Whenever disinfection was achieved in the experiments, no regrowth of bacteria was observed after 24 h. However, a reduction in the prototype efficiency was observed both in laboratory and on-site experiments. This was ascribed to aging of the photocatalyst as well as due to the deposition of particles onto its surface. In cases in which incomplete disinfection resulted, a low rate of E-coli growth was observed 24 h after ending the experiment. However, pseudomones seem to be resistant to the treatment.

<>

Figures

Figure 1

Scheme of the laboratory reactor. The inset shows a cross section of the irradiation system

Figure 2

E-coli concentration as a function of UV-A integrated energy received by the CPC type photocatalytic reactor (Fig. 1): results for TiO2 (first use), Ru complex (second use), and photolysis are presented

Figure 3

E-coli concentration as a function of integrated UV-A received energy by the CPC type photocatalytic reactor (Fig. 1): results for TiO2 (first, sixth, and ninth uses)

Figure 4

Micrographs of the Ahlstrom NW1047 catalyst: (a) before use, (b) after the seventh batch, and (c) after the 18th batch. Pictures on the right show close-ups of the fibers composing the catalyst. Pictures on the left are close-ups of the interstitial part between the fibers that form the catalysts.

Figure 5

Micrographs of the photosensitizer strip, Ru complex: (a) before use and (b) after the fourth batch

Figure 6

Integrated energy calculated from the measured solar global radiation in Yaurisque. The term “experiment” in the plot refers to a run of one-day duration.

Figure 7

E-coli concentration degradation with the photocatalytic reactor. In the plot, a one-day run is referred to as “experiment.” Experiments were performed using water from the Yaurisque River contaminated with bacteria.

Figure 8

(a) Aeruginose and (b) SPP pseudomone concentrations in a long term catalyst use for bacteria degradation. The one-day run is referred to as “experiment” in the plot. Experiments were performed using water from the Yaurisque River contaminated with bacteria.

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# 744-nm wavelength conversion of PAM-4 signal using an AlGaAsOI nanowaveguide

Research output: Contribution to journalJournal articleResearchpeer-review

## Abstract

Exploring new frequency bands for optical transmission is essential to overcome the capacity crunch. The 2-µm band is becoming a research spotlight due to available broadband thulium-doped fiber amplifiers as well as low-latency, low-loss hollow-core fibers. Yet most of the 2-µm band devices designed for optical communication are still in their infancy. In this Letter, we propose wavelength conversion based on four-wave mixing in a highly nonlinear AlGaAsOI nanowaveguide to bridge the 2-µm band and the conventional bands. Due to the strong light confinement of the AlGaAsOI nanowaveguide, high-order phase match is enabled by dispersion engineering to achieve a large synergetic conversion bandwidth with high conversion efficiency. Simulation results show a possible conversion bandwidth over an octave. An AlGaAsOI nanowaveguide with 3-mm length and a nominal cross-section dimension of $320\;{\rm nm} \times 680\;{\rm nm}$320nm×680nm is used for the wavelength conversion of a 10 Gbit/s non-return-to-zero on-off keying signal and a 10 Gbit/s Nyquist-shaped four-level pulse-amplitude modulation signal. A conversion efficiency of $- {28}\;{\rm dB}$-28dB is achieved using a 17.5-dBm continuous-wave pump in the C band, with 744 nm conversion from 1999.65 to 1255.35 nm.
Original language English Optics Letters 45 4 889-892 0146-9592 https://doi.org/10.1364/ol.383085 Published - 2020


text:
# file_get_contents on other port

I must contact services rest on different ports by 80, but the function file_get_contents () returns an error: failed to open stream: Connection refused

$url = "http://nexusdigital.agency:81/API/....";$result = file_get_contents($url, false);  How can I configure the reading on other ports? • This should do just fine, if the URL nexusdigital.agency:81 actually exists and is physically reachable from your location. – deceze Dec 29 '14 at 11:22 • Read the error. Connection refused. Your server cannot connect to the other server on port 81. This could be that the remote server isn't letting you, or that there's a firewall on your own server (or in rare cases, your ISP) blocking you from making outgoing calls on port 81. Try to telnet from your server's console and see if you get any other output. – h2ooooooo Dec 29 '14 at 11:24 • Unfortunately it does not work. The service exists and is accessible, in fact, the browser reads the contents You can try clicking this address nexusdigital.agency:81/API/MENEC/… but if you enter this address into the function fails – NexusDigital Dec 29 '14 at 11:26 • @NexusDigital Where are you running this code? From the same computer that you're using the browser on? Do you have a firewall blocking php from accessing it? – h2ooooooo Dec 29 '14 at 11:27 ## 2 Answers Use CURL : <?php$curl = curl_init('http://nexusdigital.agency/API/....');
curl_setopt($curl, CURLOPT_PORT, 81); curl_setopt($curl, CURLOPT_RETURNTRANSFER, 81);
$result = curl_exec($curl);
?>

• I tried this code but it does not work. I think that my hosting has closed these outgoing ports. – NexusDigital Dec 29 '14 at 11:54

Well, if it doesn't work for you (whatever the reason) you can try using CURL http://php.net/curl

<?php
$tuCurl = curl_init(); curl_setopt($tuCurl, CURLOPT_URL, "http://nexusdigital.agency/API/....");
curl_setopt($tuCurl, CURLOPT_PORT , 81); curl_setopt($tuCurl, CURLOPT_VERBOSE, 0);
curl_setopt($tuCurl, CURLOPT_RETURNTRANSFER, 1); curl_setopt($tuCurl, CURLOPT_CONNECTTIMEOUT, 5); // 5 seconds timeout

$tuData = curl_exec($tuCurl);
curl_close(\$tuCurl);


text:
# Tag Info

Accepted

• 690
Accepted

• 2,162

### Is this inequality involving the Frobenius norm right?

For a short fat matrix $G$ (more columns than rows), $\|AG\|_F \geq \sigma_{\min}(G)\|A\|_F \geq n \sigma_{\min}(G) \|A\|$, where $\sigma_{\min}(G)$ is the least singular value of $G$. This follows ...
• 17.7k

### Solving a system of linear equations over the integers

To make sure I understand: you have an $m \times n$ matrix and vector in $\mathbb{Z}^n.$ Is the system a priori over- or under-determined? The former case is a little easier than the latter, but in ...
• 93.8k

### Is it faster to compute eigenvalues or coefficients of characteristic polynomials?

With the traditional algorithms and complexity measures used in numerical linear algebra (dense real matrices, floating point computations, flop count as a complexity measure), they are both more or ...
• 17.7k
Accepted

### Complexity of rectangular matrix multiplication

Assuming that efficient means better than the naive $O(n^{2+k})$ multiplication, let us review some possibilities. Padding. For $k > \omega-2$, just pad $A$ with $n-n^k$ zero or garbage rows, ...
• 3,012

### Computation time of Smith normal form in Maple

(Dense) Smith Normal Form is theoretically computable in $O(\|A\| \log \|A\| N^4\log N)$ time (Arne Storjohann, 1996). Storjohann was at Waterloo at the time, so I would not be surprised if that is ...
• 93.8k
Accepted

### Matrix equation with Hadamard product and its own inverse involved

Removing all unnecessary parameters, we come to the equation $\Omega^{-1}=2 W\odot \Omega + B$ where $B$ is positive definite. We need to find a solution in the cone $M_+$ of positive definite ...
• 52.9k
Accepted

### When are two binary matrices simultaneously equivalent to their transpose?

Clearly, a necessary condition is that for every word $w$ in two letters, one has $${\rm Tr}\,w(A^t,B^t)={\rm Tr}\,w(A,B).$$ Equivalently, $${\rm Tr}\,\hat w(A,B)={\rm Tr}\,w(A,B),$$ where $\hat w$ is ...
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### What is the time complexity of truncated SVD?

According to the man page of svds, provided by MATLAB, svds is currently based on "Augmented Lanczos Bidiagonalization Algorithm" when it comes to the top-$k$ SVD computation of a large-...
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