text:
TEXT SIZE

CrossRef (0)
Repair policies of failure detection equipments and system availability

Seongryong Na1,a, Sung-Hwan Banga

aDivision of Data Science, Yonsei University, Korea
Correspondence to: 1 Division of Data Science, Yonsei University, 1 Yonseidai-Gil, Wonju, Gangwon-Do 26493,
Korea. E-mail : nasr@yonsei.ac.kr
Received June 11, 2021; Revised August 20, 2021; Accepted August 24, 2021.
Abstract
The total system is composed of the main system (MS) and the failure detection equipment (FDE) which detects failures of MS. The analysis of system reliability is performed when the failure of FDE is possible. Several repair policies are considered to determine the order of repair of failed systems, which are sequential repair (SQ), priority repair (PR), independent repair (ID), and simultaneous repair (SM). The states of MS-FDE systems are represented by Markov models according to repair policies and the main purpose of this paper is to derive the system availabilities of the Markov models. Analytical solutions of the stationary equations are derived for the Markov models and the system availabilities are immediately determined using the stationary solutions. A simple illustrative example is discussed for the comparison of availability values of the repair policies considered in this paper.
Keywords : system availability, failure detection equipment, repair policy, reliability, Markov process
1. Introduction

A repairable system can conceptually consist of a main system (MS) and a failure detection equipment (FDE). The MS performs system functions and the FDE detects the failures of MS. This conceptual formation can be found in telecommunication systems and automated plant facilities under remote monitoring and control. Failures of FDE delay the detection of MS failures and affect system availability. The reliability analysis considering the detection of failures of safety systems has been done in Zhang et al. (2003) and Guo and Yang (2008). Recently, Na (2011) and Na and Bang (2013) has studied the effect of FDE on system availability using Markov modeling.

We assume various situations about the occurrence of failures and the repair policies. For example, when the MS is under repair, it can be assumed that the FDE does not break down (Na, 2011). However, if it happens that the FDE fails after MS’s failure, several repair policies can be considered. In Na and Bang (2013), the system availability has been derived with the scheme of sequential repair. In this paper, we consider several repair policies such as priority repair, independent repair and simultaneous repair and compare the results of system availability with previous one.

This paper is constructed as follows. In Section 2, the assumptions for the MS-FDE system are stated and the results of previous studies are provided. The system availability results of various repair policies are given in Section 3. Concluding remarks are given in Section 4.

2. Markov modeling of main system (MS)-failure detection equipment (FDE) system

The total system consists of the main system (MS) and the failure detection equipment (FDE). The Markov modeling of the system is possible using the following assumptions.

• (A1) MS stays in normal state during an exponential time of mean $λM-1$ before failure.

• (A2) The repair time of MS in failure is exponentially distributed with mean $μM-1$.

• (A3) The sojourn times of MS in normal state and the repair times are independent.

• (A4) The repair of MS starts immediately when the failure of MS is detected.

• (A5) The failure of MS is immediately detected when FDE operates.

If the FDE is assumed to operate with no failure, the system availability Av (RN) is given as following by the elementary property of alternating renewal processes (Ross, 1996).

$Av (RN)=λM-1λM-1+μM-1=μMλM+μM.$

The FDE can be out of order in real world and the following Markovian assumptions are about the failure.

• (B1) FDE operates normally during an exponential time of mean $λD-1$ before failure.

• (B2) The repair time for FDE is exponentially distributed with mean $μD-1$.

• (B3) The operation times and repair times of FDE are independent.

• (B4) The failure of FDE is immediately detected.

The assumptions of repair policy and the failure mechanism of FDE are required to represent the MS-FDE system exactly. The FDE is shut down and does not fail when the MS is detected to fail and is being repaired. In this situation, the following assumption is possible.

• (NF) The failure of FDE does not occur when the failure of MS is detected and MS is in repair.

If we represent the normal state as 1 and the state in failure as 0, the state of MS and FDE can be represented as (1, 1), (0, 1), (1, 0), or (0, 0). The MS-FDE system assuming the conditions (A1)–(A5), (B1)–(B4) and (NF) can be defined as a continuous-time Markov process with the state space {(1, 1), (0, 1), (1, 0), (0, 0)} (Na, 2011).

The transition diagram is Figure 1 and its system availability Av (NF) is given by the following equation.

$Av (NF)=μMλM+μM·μD(λM+μM)(λM+λD+μD)μD(λM+μM)(λM+λD+μD)+λMλDμM.$

Note that since the failure of FDE is assumed, the system availability (2.2) is less than (2.1).

It would be practical to assume that FDE can fail during the repair of MS if it costs to shut down FDE. The state (0, 0) can be divided into (0, 0)A where MS fails after FDE and (0, 0)B where FDE fails after MS. The sequential repair is the policy that MS or FDE is repaired in the order of failure and detection. The following is assumed for the sequential repair.

• (SQ) FDE fails with rate $λD′$ when MS is in repair and the sequential repair is performed.

The MS-FDE system with the assumptions (A1)–(A5), (B1)–(B4) and (SQ) can be represented as a continuous-time Markov process with the state space {(1, 1), (0, 1), (1, 0), (0, 0)A, (0, 0)B} and the transition diagram is given by

The system of Figure 2 has been studied in Na and Bang (2013) and the system availability is given by,

$Av (SQ)=μMμD(λM+λD+μD)(λD′+μM)·C-1,$

where the normalizing constant is equal to the following equation.

$C=μMμD (λD′(λM+λD)+λDμM)+λMμMμD (λM+λD+μD)+μMμD (λMμM+μMμD+λD′μD) +λMλD′μD (λM+λD+μD)+λMμM (λD′(λM+λD)+λDμM).$

It can be seen that the system availability (2.3) of sequential repair is less than (2.2). Meanwhile, the mean time between failures (MTBF) is given by the following equation regardless of the repair policy (NF) or (SQ).

$MTBF=λM-1.$

Another important reliability measure mean time to repair (MTTR) is equal to,

$MTTR=μM-1+μD-1·λDλM+λD+μD,$

under the condition (NF) and

$MTTR=μM-1+μD-1·λD(μD+λD′)+λMλD′(μM+λD′)(λM+λD+μD),$

under the condition (SQ). It is obvious that the MTTR increases as the failure of FDE is assumed to occur more frequently. Refer to Na (2011) and Na and Bang (2013) for details.

3. Repair policies for system reliability analysis

We can consider various policies for repair order when both MS and FDE get into breakdown. The repair policies in real situations are flexibly determined in consideration of system characteristics, operation mechanisms, the number of repairmen, and so on. In this section, the reliability results according to various repair policies such as priority repair, independent repair and simultaneous repair are examined in addition to sequential repair.

First, we consider the method of priority repair that the failed MS is repaired preferentially. It is assumed about the policy of priority repair that the failure of MS is immediately detected and the failed MS is repaired in advance of FDE. That is, if MS fails while FDE is in repair, the repairman repairs MS instead of FDE. This can be a practical policy in case that a special monitoring of the failed MS is performed and repairmen are not sufficient. The following assumption is about the priority repair.

• (PR) FDE fails with rate $λD′$ when MS is in repair and the priority repair is performed.

The MS-FDE system with the conditions (A1)–(A5), (B1)–(B4) and (PR) can be represented as a continuous-time Markov process with the state space {(1, 1), (1, 0), (0, 1), (0, 0)A, (0, 0)B} and the transition diagram is given by Figure 3. It can be also seen that the Markov process with Figure 3 is reduced to the process with Figure 4 where two states (0, 0)A and (0, 0)B are combined into (0, 0). The system availability is provided in the following theorem by analyzing the Markov process of

### Theorem 1

If we assume the MS-FDE system with the conditions (A1)–(A5), (B1)–(B4) and (PR), then the system availability is equal to

$Av(PR)=μMλM+μM.$
Proof

We obtain the following stationary equations for the Markov process of

• (1,1) : p(11)(λM + λD) = p(10)μD + p(01)μM

• (1,0) : p(10)(λM + μD) = p(11)λD + p(00A)μM + p(00B)μM

• $(0,1):p(01)(μ+λD′)=p(11)λM$

• (0,0)A : p(00A)μM = p(10)λM

• $(0,0)B:p(00B)μM=p(01)λD′$

Combining two states (0, 0)A and (0, 0)B into (0, 0), we obtain the following stationary equations which correspond to

• (1,1) : p(11)(λM + λD) = p(10)μD + p(01)μM

• (1,0) : p(10)(λM + μD) = p(11)λD + p(00)μM

• $(0,1):p(01)(μ+λD′)=p(11)λM$

• $(0,0):p(00)μM=p(01)λD′+p(10)λM$

The following equation is obtained by summing the stationary equations for states (1,1) and (1,0).

$p(11)λM+p(10)λM=p(01)μM+p(00)μM.$

Finally, the system availability is equal to

$Av (PR)=p(11)+p(10)=1-(p(01)+p(00))=μM(λM+μM).$

Note that the availability (3.1) is equal to (2.1). The state of MS behaves as in the alternating renewal process under the assumption (PR) that MS is repaired preferentially.

Under the independent repair policy, MS and FDE are repaired independently whenever the failures are detected. It is assumed that the repair of FDE starts additionally when FDE fails in the course of the repair of MS. This system can be possible when sufficient repairmen work.

• (ID) FDE fails with rate $λD′$ when MS is in repair and MS and FDE are repaired independently.

The MS-FDE system with conditions (A1)–(A5), (B1)–(B4) and (ID) can be represented as a continuoustime Markov chain with the state space {(1, 1), (1, 0), (0, 1), (0, 0)A, (0, 0)B} and the transition diagram is given by Figure 5. The system availability under the independent repair policy is given in the following theorem.

### Theorem 2

If we assume the MS-FDE system with the conditions (A1)–(A5), (B1)–(B4) and (ID), then the system availability is equal to,

$Av (ID)=C-1μMμD(λM+λD+μD)(λD′+μM+μD),$

where the normalizing constant is given by

$C=μMμD((λM+μD)(μM+μD)+λD′μD)+μMμD(λD(μM+μD)+λD′(λM+λD)) +λMμD(λM+λD+μD)(μM+μD)+λMμM(λD(μM+μD)+λD′(λM+λD))+λMλD′μD(λM+λD+μD).$
Proof

The following stationary equations can be obtained for the Markov process of

• (1,1) : p(11)(λM + λD) = p(10)μD + p(01)μM

• (1,0) : p(10)(λM + μD) = p(11)λD + p(00B)μM

• $(0,1):p(01)(μM+λD′)=p(11)λM+p(00A)μD+p(00B)μD$

• (0,0)A : p(00A)μD = p(10)λM

• $(0,0)B:p(00B)(μM+μD)=p(01)λD′$

We obtain the following equations from the above stationary equations.

• $p(10)=p(11)·λD(μM+μD)+λD′(λM+λD)(λM+μD)(μM+μD)+λD′μD$

• $p(01)=p(11)·λM(λM+λD+μD)(μM+μD)μM((λM+μD)(μM+μD)+λD′μD)$

• $p(00A)=p(11)·λM(λD(μM+μD)+λD′(λM+λD))μD((λM+μD)(μM+μD)+λD′μD)$

• $p(00B)=p(11)·λMλD′(λM+λD+μD)μM((λM+μD)(μM+μD)+λD′μD)$

Using that the probability sum is equal to 1, we have the following results.

• $p(11)=C-1μMμD((λM+μD)(μM+μD)+λD′μD)$

• $p(10)=C-1μMμD(λD(μM+μD)+λD′(λM+λD))$

• p(01) = C1λMμD(λM + λD + μD)(μM + μD)

• $p(00A)=C-1λMμM(λD(μM+μD)+λD′(λM+λD))$

• $p(00B)=C-1λMλD′μD(λM+λD+μD)$

Finally, the system availability for the policy of independent repair becomes

$Av (ID)=p(11)+p(10)=C-1μMμD(λM+λD+μD)(λD′+μM+μD).$

We observe that the system availability (3.2) of independent repair policy is similar to (2.3) of sequential repair. Meanwhile, it is easily expected that (3.2) is larger than (2.3). Note that the Markov process of Figure 5 of the policy of independent repair is not reversible, which is apparent from the stationary equations of (0, 0)A and (0, 0)B.

We can consider the policy of simultaneous repair under which failed MS and FDE are repaired simultaneously. In particular, if FDE fails in the course of the repair of MS, both MS and FDE are repaired together. This policy seems to be appropriate when both systems are initiated at the same time after repair. The transition diagram of simultaneous repair policy is given by

• (SM) FDE fails with rate $λD′$ when MS is in repair and MS and FDE are repaired simultaneously with rate μT.

The system availability is provided in the following theorem.

### Theorem 3

The system availability of the MS-FDE system under the policy of simultaneous repair satisfying the conditions (A1)–(A5), (B1)–(B4) and (SM) is given by

$Av (SM)=C-1μDμT(λM+λD+μD)(λD′+μM),$

where the normalizing constant is equal to

$C=μDμT(λM+μD)(λD′+μM)+λDμDμT(λD′+μM)+λMμDμT(λM+λD+μD) +λMλDμT(λD′+μM)+λMλD′μD(λM+λD+μD).$
Proof

The following stationary equations are valid for the Markov process of

• (1,1) : p(11)(λM + λD) = p(10)μD + p(01)μM + p(00B)μT

• (1,0) : p(10)(λM + μD) = p(11)λD

• $(0,1):p(01)(μM+λD′)=p(11)λM+p(00A)μD$

• (0,0)A : p(00A)μD = p(10)λM

• $(0,0)B:p(00B)μT=p(01)λD′$

The following equations are obtained from the above stationary equations.

• $p(10)=p(11)·λDλM+μD$

• $p(01)=p(11)·λM(λM+λD+μD)(λM+μD)(λD′+μM)$

• $p(00A)=p(11)·λMλDμD((λM+μD)$

• $p(00B)=p(11)·λMλD′(λM+λD+μD)μT(λM+μD)(λD′+μM)$

Here, we derive the following stationary distribution.

• $p(11)=C-1μDμT(λM+μD)(λD′+μM)$

• $p(10)=C-1λDμDμT(λD′+μM)$

• p(01) = C1λMμDμT (λM + λD + μD)

• $p(00A)=C-1λMλDμT(λD′+μM)$

• $p(00B)=C-1λMλD′μD(λM+λD+μD)$

Finally, the system availability is obtained as follows.

$Av (SM)=p(11)+p(10)=C-1μDμT(λM+λD+μD)(λD′+μM).$

The results of the system availability corresponding to various repair policies can be compared. It has been verified that Av (RN) > Av (NF) > Av (SQ). Meanwhile, it can be expected that Av (PR) > Av (SM) > Av (ID) but its direct verification seems difficult. The actual values of system availability under a simple condition that all failure rates and repair rates are equal are given as follows,

$Av (RN)=Av (PR)=12, Av (NF)=Av (SM)=37, Av (ID)=922, Av (SQ)=25.$

It can be seen that the above values are consistent with the intuitive expectation. A detailed comparison of usual situations can be done in a further study.

4. Conclusion

Various repair policies can be considered for the analysis of system reliability when FDE fails. System characteristics, operation method and the number of repairmen affect the determination of repair policy. In this paper, the system availability has been analyzed by assuming priority repair, independent repair and simultaneous repair. The state of MS-FDE system of each repair policy can be represented by a Markov process. We solved the stationary equations of each Markov process and derived the system availability for MS-FDE of each repair policy using the stationary probabilities. It can be observed that the system availability has different value for each repair policy.

The analysis of system reliability can be performed more efficiently by applying the system availability of exact MS-FDE reflecting real situations. In this view we have considered various repair policies in this paper. A new attempt can be made for the analysis of reliability systems with non-Markovian property, which may result from the non-exponential assumption of the system working time and repair time, and a numerical approach may be helpful. For complex systems, new reliability modeling which contains additional subsystems rather than MS and FDE can be possible.

Figures
Fig. 1. Transition diagram of MS-FDE system where no failure (NF) of FDE is assumed when MS is in repair. The state (i, j) denotes that MS is in state i and FDE is in state j for i, j =1(normal) or 0(failed).
Fig. 2. Transition diagram of MS-FDE system where sequential repair (SQ) is assumed. The state (i, j) denotes that MS is in state i and FDE is in state j for i, j =1(normal) or 0(failed).
Fig. 3. Transition diagram of MS-FDE system where priority repair (PR) is assumed. The state (i, j) denotes that MS is in state i and FDE is in state j for i, j =1(normal) or 0(failed).
Fig. 4. Transition diagram of MS-FDE system where priority repair (PR) is assumed.(reduced states) The state (i, j) denotes that MS is in state i and FDE is in state j for i, j =1(normal) or 0(failed).
Fig. 5. Transition diagram of MS-FDE system where independent repair (ID) is assumed. The state (i, j) denotes that MS is in state i and FDE is in state j for i, j =1(normal) or 0(failed).
Fig. 6. Transition diagram of MS-FDE system where simultaneous repair (SM) is assumed. The state (i, j) denotes that MS is in state i and FDE is in state j for i, j =1(normal) or 0(failed).
References
1. Guo H and Yang X (2008). Automatic creation of Markov models for reliability assessment of safety instrumented systems. Reliability Engineering and System Safety, 93, 807-815.
2. Na S (2011). Reliability analysis of repairable systems considering failure detection equipments. The Korean Journal of Applied Statistics, 24, 515-521.
3. Na S and Bang S (2013). The effect of failure detection equipment on system reliability. The Korean Journal of Applied Statistics, 26, 111-118.
4. Ross SM (1996). Stochastic Processes (2nd Ed), New York, Wiley.
5. Zhang T, Long W, and Sato Y (2003). Availability of systems with self-diagnostic components ? applying Markov model to IEC 61508-6. Reliability Engineering and System Safety, 80, 133-141.


text:
# $f(z) = z^k + kz$ is injective in the unit disc

How would I go around proving that $f(z) = z^k + kz$ in injective in the open unit disc in $\mathbb{C}$, for each natural $k$?

I don't really know where to start. I observed that its derivative is nonzero, but that was all that I came up with.

Any help is appreciated. Thanks.

-
More generally, if $\mathrm{Re}\,f'>0$ on a convex domain, then $f$ is injective. And you have $f'(z)=k(z^{k-1}+1)$. – 40 votes Jul 20 '13 at 21:53

If $z_1^k+kz_1=z_2^k+kz_2$ then
$$(z_1-z_2)\left( z_1^{k-1}+z_1^{k-2}z_2+...+z_2^{k-1}+k\right)=0$$
Hint if $z_1,z_2$ are in the open unit disk, then by the triangle inequality
$$\left| z_1^{k-1}+z_1^{k-2}z_2+...+z_2^{k-1} \right| < k \,.$$

format: plain


text:
# 30.3 Bohr’s theory of the hydrogen atom  (Page 6/14)

 Page 6 / 14

But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is what we call semiclassical . The orbits are quantized (nonclassical) but are assumed to be simple circular paths (classical). As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not explain that some spectral lines are doublets (split into two) when examined closely. We shall examine many of these aspects of quantum mechanics in more detail, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation for all of atomic physics that has since evolved.

## Phet explorations: models of the hydrogen atom

How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.

## Section summary

• The planetary model of the atom pictures electrons orbiting the nucleus in the way that planets orbit the sun. Bohr used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula
$\frac{1}{\lambda }=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right),$
where $\lambda$ is the wavelength of the emitted EM radiation and $R$ is the Rydberg constant, which has the value
$R=\text{1.097}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{-1}\text{.}$
• The constants ${n}_{i}$ and ${n}_{f}$ are positive integers, and ${n}_{i}$ must be greater than ${n}_{f}$ .
• Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by
$\Delta E=\text{hf}={E}_{\text{i}}-{E}_{\text{f}},$
where $\Delta E$ is the change in energy between the initial and final orbits and $\text{hf}$ is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
• Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum given by
$L={m}_{e}{\text{vr}}_{n}=n\frac{h}{2\pi }\left(n=1, 2, 3 \dots \right),$
where $L$ is the angular momentum, ${r}_{n}$ is the radius of the $n\text{th}$ orbit, and $h$ is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by
${r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}\text{(allowed orbits}\phantom{\rule{0.25em}{0ex}}n=1, 2, 3, ...\right),$
$Z$ is the atomic number of an element (the number of electrons is has when neutral) and ${a}_{\text{B}}$ is defined to be the Bohr radius, which is
${a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}\text{.}$
• Furthermore, the energies of hydrogen-like atoms are given by
${E}_{n}=-\frac{{Z}^{2}}{{n}^{2}}{E}_{0}\left(n=1, 2, 3 ...\right)\text{,}$
where ${E}_{0}$ is the ground-state energy and is given by
${E}_{0}=\frac{{2\pi }^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}=\text{13.6 eV.}$
Thus, for hydrogen,
${E}_{n}=-\frac{\text{13.6 eV}}{{n}^{2}}\left(n,=,1, 2, 3 ...\right)\text{.}$
• The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.

## Conceptual questions

How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.

In Inelastic collision cunculate the vilocity
explain how a body becomes electrically charged based on the presence of charged particles
induction
babar
induction
DEMGUE
definitely by induction
Raymond
induction
Raymond
induction
Shah
induction
Korodhso
please why does a needle sinks in water
DEMGUE
induction
Korodhso
induction
Auwal
what are the calculations of Newton's third law of motiow
what is dark matter
(in some cosmological theories) non-luminous material which is postulated to exist in space and which could take either of two forms: weakly interacting particles ( cold dark matter ) or high-energy randomly moving particles created soon after the Big Bang ( hot dark matter ).
Usman
if the mass of a trolley is 0.1kg. calculate the weight of plasticine that is needed to compensate friction. (take g=10m/s and u=0.2)
what is a galaxy
what isflow rate of volume
flow rate is the volume of fluid which passes per unit time;
Rev
flow rate or discharge represnts the flow passing in unit volume per unit time
bhat
When two charges q1 and q2 are 6 and 5 coulomb what is ratio of force
When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?
nehemiah
why is it proportional
i don't know
y
nehemiah
what are the relationship between distance and displacement
They are interchangeable.
Shii
Distance is scalar, displacement is vector because it must involve a direction as well as a magnitude. distance is the measurement of where you are and where you were displacement is a measurement of the change in position
Shii
Thanks a lot
Usman
I'm beginner in physics so I can't reason why v=u+at change to v2=u2+2as and vice versa
Usman
what is kinematics
praveen
kinematics is study of motion without considering the causes of the motion
Theo
The study of motion without considering the cause 0f it
Usman
why electrons close to the nucleus have less energy and why do electrons far from the nucleus have more energy
Theo
thank you frds
praveen
plz what is the third law of thermodynamics
third law of thermodynamics states that at 0k the particles will collalse its also known as death of universe it was framed at that time when it waa nt posible to reach 0k but it was proved wrong
bhat
I have not try that experiment but I think it will magnet....
Hey Rev. it will
Jeff
I do think so, it will
Chidera
yes it will
lasisi
If a magnet is in a pool of water, would it be able to have a magnetic field?.
yes Stella it would
Jeff
formula for electric current
Fokoua
what are you given?
Kudzy
what is current
Fokoua
I=q/t
saifullahi
Current is the flow of electric charge per unit time.
saifullahi
What are semi conductors
saifullahi
materials that allows charge to flow at varying conditions, temperature for instance.
Mokua
these are materials which have electrical conductivity greater than the insulators but less than metal, in these materials energy band Gap is very narrow as compared to insulators
Sunil
materials that allows charge to flow at varying conditions, temperature for instance.
Obasi
wao so awesome
Fokoua
At what point in the oscillation of beam will a body leave it?
Atambiri
what is gravitational force


text:
# Need verification to solution for trig function limit question

1. Jun 24, 2008

### kylera

As I do not have the solution book and classes are over for the summer, I don't have any other way of verifying whether this is right or not.

1. The problem statement, all variables and given/known data
lim(x->0) (sin 4x) / (sin 6x)

2. Relevant equations

3. The attempt at a solution
If I may temporarily remove the lim sign...
$$\frac{sin 4x}{sin 6x} = \frac{sin4x}{4x} \times \frac{4x}{sin6x}$$

$$= \frac{sin4x}{4x} \times \frac{6x}{sin6x} \times \frac{4x}{6x}$$

By Limit Laws, take the limit of each fraction to get
$$1 \times 1 \times \frac{4}{6}$$


text:
# Find Aut(Z n ) for cases n = 2,3,4, and 5

1. Feb 18, 2009

### zcdfhn

Find Aut(Zn) for cases n = 2,3,4, and 5

My question is to find Aut(Zn) for cases n = 2,3,4, and 5

First, I am a little confused about the case Aut(Z), the integers. I know that the only functions that satisfy Aut(Z) are f(x) = x, and f(x) = -x, but what is wrong with f(x) = x+1?

It would be great if you explain to me how to find automorphic functions, and why the previous example works and doesn't work.

My attempts at Aut(Z2): I think that f(x) = kx and f(x) = x+k, k $$\in$$ Z are the only functions that apply in this case.

The rest, I believe, are similar, but I just need help before I commit to an answer.

$$f \in Aut (Z)$$ implies $$f:Z \rightarrow Z$$ and $$f^{-1}:Z \rightarrow Z$$ should be injective. Thus, both the kernels of $$f, f^{-1}$$ should be identity. That implies the identity is mapped to an identity in Z.
Generally speaking, the set of automorphisms forms a group and the order of $$Aut(Z_n)$$ is $$\phi(n)$$ (Euler totient function).
Once you find an automorphism, you need to make sure that the order of each element is same with $$x \in Z_n$$ and $$f(x) \in Z_n$$, where f is an automorphism of $$Z_n$$.


text:
# ten ants lie on the real line

Question:

Ten ants are on the real number line. At time $t=0$, the $k-th$ ant starts at the point $k^2$ and travelling at uniform speed, reaches the point $(11-k)^2$ at time $t=1$. The number of distinct times at which at least two ants are at the same location is:

Options:
1. $45$
2. $11$
3. $17$
4. $9$

My method:- i found out the velovity of the k th particle and it turned out to be $11(11-2k)$

so the first ant move 99 m second moves 11 m and so on so the first ant crosses 9 other ants second crosses 8 and so on therefore the answer is $9+8+7...+1=45$

but the given answer is $17$

Note:-this question was asked in the exam and i want to challenge its key.So can anybody tell what the correct answer is?

• The question is not asking how many crosses happen between the ants, why are you assuming your $45$ crossings all happen at different times? Nov 10, 2016 at 14:36
• no i have taken it to be crossing between 0 to 1 sec Nov 10, 2016 at 14:38
• as the 10th ant moves to the left the first ant moves to the right and it has to overtake or cross evrybody Nov 10, 2016 at 14:38

The timetable of ant $A_k$ $(1\leq k\leq 10)$ is given by $$x_k(t)=(1-t)k^2+t(11-k)^2\ .$$ Two ants $A_k$ and $A_l$ with $l>k$ could in principle meet at a certain time $t$ which is the solution of the equation $$x_k(t)=x_l(t)\ \tag{1}$$ A priori this solution could lay in the exterior of the $t$-interval $[0,1]$. In this case the ants would not meet during the experiment.
Now analyze how many different admissible values for $t$ you can get by solving equations of the type $(1)$.
• @MathMan, you are correct that all $45$ pairs of ants will meet one another at some time or other. The point is, some of these pairwise encounters happen simultaneously. The problem wants you to count the number of distinct times the encounters occur. Nov 10, 2016 at 15:13


text:
# Young’s Double Slit Experiment

Show/Hide Sub-topics (Superposition | A Level)

The source emits monochromatic light.(comprising waves of one frequency only)

The single slit ensures that the light waves incident on the double slits are coherent.

• Coherent sources must have same frequency and constant phase difference. Constant phase difference means that the two sources always differ in phase by the same amount.

$x \approx \frac{\lambda D}{a}$, X is only valid for values of D >> a and when $\theta$ is very small. This equation cannot be used for diffraction grating!

$a \, sin \, \theta = n \lambda$

Factors affecting appearance of fringes due to double slit interference of light:

• If the single slit together with the light source is moved closer to the double slits, the bright fringes produced are brighter as the intensity of the light passing through the double slit is greater. The fringe separation remains the same.
• If one of the slit is closed, a different pattern is obtained. The central maximum is now much brighter and broader compared to the other maxima at the sides. This is the diffraction pattern for single slit.
• If white light is used, the central fringe is white and the fringes on either side are coloured. Recall that $\lambda_{red} \, > \, \lambda_{blue}$, so $x_{red} \, > \, x_{blue}$. Blue is the colour nearer to the central fringe and red is farther away.
• If the space between the double slit and screen is filled with a transparent medium of refractive index higher than that of air or vacuum, the fringes produced on screen will be narrower. This is because the wavelength is smaller in a medium where the refractive index is higher than that of air or vacuum.

The following video can help you understand: (I recommend that you stop at 1 min 50 seconds of the video unless you understand wave-particle duality)


text:
# How do you find the derivative of (e^(2*x)-e^(-2*x))/(e^(2*x)+e^(-2*x))?

Apr 3, 2018

$2 \left(1 - {\left({e}^{\left(2. x\right) - {e}^{- 2. x}} / \left({e}^{2. x} + {e}^{- 2. x}\right)\right)}^{2}\right)$

#### Explanation:

isn't that just $\tanh 2 x$?

and the derivative of $\tanh x$ = $1 - {\left(\tanh x\right)}^{2}$
http://math2.org/math/derivatives/more/hyperbolics.htm

thus the derivative of
$\tanh 2 x$

is

$2 \left(1 - {\left(\tanh x\right)}^{2}\right)$


text:
A sum of eigenvalues

Let $X$ be an $n\times n$ symmetric matrix. Suppose $\lambda_1(X)\geq \lambda_2(X) \geq \cdots \geq \lambda_n(X)$ are eigenvalues of $X$. Let $r$ be any integer with $1\leq r\leq n$. It is well-known that $\sum_{i=1}^r \lambda_i(X)$ is convex. Now, my question is: Is the following function convex? $$\sum_{i=1}^r \max(0,\lambda_i(X))$$

Thanks!

-

The answer is yes, because your function (let me call it $f$) is the maximum of convex functions. As such, it is convex. The formula : $$f(X)=\max\left(0,\max_{1\le r\le n}\sum_{j=1}^r\lambda_j(X)\right).$$

-
Yes, your idea is correct. But the reason should be revised to be $$f(X) = \max\{0,\sum_{i=1}^1 \lambda_i(X), \cdots, \sum_{i=1}^r \lambda_i(X)\}.$$ –  user11870 May 30 '12 at 12:42
@wmmiao. I agree. –  Denis Serre May 30 '12 at 12:54

The matrix is assumed to be real, correct?

The answer seems to be trivially yes. Indeed, assume that $\lambda_{k} \geq 0$ and $\lambda_{k+1}<0$ and $r \geq k+1$. Then $\sum_{i=1}^{r}{\max(0,\lambda_{i})}=\sum_{i=1}^{k}{\lambda_{i}}$.

-
@Felix. Your argument does not work because your $k$ depends upon $X$. –  Denis Serre May 30 '12 at 12:21
Yes, assume to be real. But what we want to prove is for any 0≤α≤1, $$\alpha \sum_{i=1}^r \max(0, \lambda_i(X))+(1-\alpha)\sum_{i=1}^r \max(0,\lambda_i(Y)) \leq \sum_{i=1}^r \max(0,\lambda_i(\alpha X+(1-\alpha)Y)).$$It is not trivial. –  user11870 May 30 '12 at 12:24
Oh dear. Well, I had a feeling something was amiss. :) –  Felix Goldberg May 30 '12 at 13:41


text:
## anonymous 3 years ago find the partial derivatives with respect to x and y, if f(x,y) = $\sum_{n=0}^{}$ (xy)^n

1. anonymous

|xy| <1

2. anonymous

So, wrt to one variable, you treat the other as a constant. So how would you differentiate (constant*x)^n?

3. anonymous

(n)x^(n-1)? but what about the sum?

4. anonymous

@First question - Yep. @Second question - What's the upper limit? Also, what's the |xy| < 1 for? Sorry, I'm not quite understanding the question fully.

5. anonymous

the upper limit is infinity. |xy|<1 is the condition that goes along with the question

6. anonymous

Ah! Makes sense. So, basically, for x, it would be: $\displaystyle\sum_{n=0}^{\infty} nx^{n-1}.$At this point, I'll just wait for another response. I really have no idea myself. Sorry. :( All the best!

7. anonymous

patial derivative wrt to x $\sum_{n=0}^{n=\infty}nx ^{n-1}y ^{n}$

8. anonymous

and wrt to y $\sum_{n=0}^{n=\infty}x ^{n}(ny ^{n-1})$

9. anonymous

any query?

10. anonymous

I dont think that is quite right. I dont get how you take a partial of a sum and how you would account for |xy| < 1

11. anonymous

ok first tell me the partial derivative of xy^3

12. anonymous

wrt to x??????

13. anonymous

just find out and tell me

14. anonymous

find 2 different partials, one with relation to x and one with relation to y

15. anonymous

ya u find


text:
fastq file error
1
1
Entering edit mode
5.0 years ago
Pals ★ 1.3k

Hi,

After removing rRNAs using sortmerna, 2/42 files failed fastqc with error:

I checked the fastq file using fastq_stats which suggested the fastq file is malformed

Malformed fastq record (length mismatch) at line 69042283 Segmentation fault (core dumped)

can anybody please tell me what's the problem in this file?

zcat 4590_END_sortmerna_R2.fastq.gz | sed -n '69042275,69042295p'

+
BBBFFFFFFFFFFIIIIIIIIIIIIIIIIFIIFIIIIIIFIIIIIFIIIIIIIIIIIIIFFFFFFFFFFFFFFFFFFFFFFFFFFBFFFFFFFFBFFFFFF
@HWI-ST1072:210:C5LGGACXX:4:1309:19821:97176 2:N:0:AGTTCC
CTGGAATCCCTCGATCACTGAACAGGAAGGAAACCTGATGCAGAGACTCAGGACGCAGGCTCCAGAAGTCCCAGACCATGTCCGGATCCTTCAGGTGTGTT
+
BBBFFFFFFFFFFIIIIIIIIIIIIIIIIIIIIIIFFIIIIIIIBFIIIIFFFIIIIIFFFFFFFFFBBBBBFFBFBFFFFFFBFFFFFFFFBBBBBBBBB
@HWI-ST1072:210:C5LGGACXX:4:1309:19994:97188 2:N:0:AGTTCC
CTCATGTACCGGGGGTTACTGGATGCTCTGCTGCAGACGGCTCGGACAGAGGGCATTTTTGGCATGTACAAGGGAATAGGTGCCTCCTACTTCCGCCTTGG
+
BBBFFFFFFFFFFIIFFIIIIIIIIIIIIIIIIIIIIIFFFFFFFBBBFB<BB7<BFFFFFFBBBBBFFFFFFFBBFBFBBBBFFFFFFFFFFFFFFFFBB
@HWI-ST1072:210:C5LGGACXX:4:1309:19837:97199 2:N:0:AGTTCC
CCTACTTCTCTTTTAATTAAAATGTAGCCACTATATAACAGATATGGTTTACGATCTGTTTCAAAAACATTTACCTAAGTGACTCTGTAAAAACCTCAGCA
+
BBBFFFFFFFFFFIIIIIIIIIIIIIIIIIIIIIIIIIIIIIFIIIIFIIBFFIIIIIIIIIIIIIIIIIIIIIIIIIIFFFFFFFFFFFFFFFFFFFFFB
@HWI-ST1072:210:C5LGGACXX:4:1309:19815:97203 2:N:0:AGTTCC
GGGCCATTCGGGGTTGGCTGGGGGTGAGTGGCAGAGCAAGCTTTCAGCACAGGGCGGCATGTGCCAAAAGCACGGTGGCCTGAGGCTGCCCCTCGTGAGCA
+
BBBFFFFFFFFFFFIIIIIIIIFIFFFIFIFIIIIIIFFFFFBFFFFFFF<BBBBBFFFFBBBBFFFFFFBFFFFFFFBFFFFFFF0BBFFFBFFBF<BB0
@HWI-ST1072:210:C5LGGACXX:4:1309:19975:97229 2:N:0:AGTTCC
AGAGACTGCATGGCTTTCATGACGTGAAGATTGGGCACATTCTTGTCTGCCAGCTCCGGGTGCTTGGGCATGTGGACATCCTTCTTGGCCACCATCACCCC
+

Thanks a lot!

fastq fastqc sortmerna RNA-Seq • 5.1k views
0
Entering edit mode

I would just like to also report the same problem. I also had corrupted fastq file output after SortMeRNA run and it is also missing lines with quality scores. If someone has sorted out this issue would be grateful if you could share your solution. Thanks

1
Entering edit mode
5.0 years ago
GenoMax 110k

Purely looking at the line in question nothing seems to be wrong but obviously the error is there with FastQC. This appears to be a paired end dataset so you could use repair.sh from BBMap suite to see if you can identify the malformed read pairs and remove them.

To verify proper pairing: reformat.sh in1=r1.fq.gz in2=r2.fq.gz vpair
To repair: repair.sh in1=r1.fq.gz in2=r2.fq.gz out1=fixed1.fq.gz out2=fixed2.fq.gz outsingle=singletons.fq.gz

0
Entering edit mode

After the following error, the output file size is 3.1 compared to 3.3 (original). It seems sortmerna steps created the problem in reads.

Started output stream.
java.lang.AssertionError:
Error in 4563_END_sortmerna_R2.fastq.gz, line 194876135, with these 4 lines:
BBBFFFFFFFFFFIIIIIIIIIFIIIIIIIIIIIIIFIIFIIIIIIIIIIIIIIIIIIIIIFFFFFFFFF7<BFF<FFFFFFFFFFFFFFFFFFFFFFFFF
@HWI-ST1072:210:C5LGGACXX:2:2305:2793:35345 2:N:0:CAGATC
CGGGGATTCTCCATTCCGGATGTGTTTCGTGGAGTGCATCGATACCTGCGCAATGCCTATGCTAGGGAAGAGTTTGCTTCCACCTGTCCAGATGATGAGGA
+

at stream.ConcurrentGenericReadInputStream$ReadThread.readLists(ConcurrentGenericReadInputStream.java:656) at stream.ConcurrentGenericReadInputStream$ReadThread.run(ConcurrentGenericReadInputStream.java:635)

Set cris2Active=false
java.lang.AssertionError:
Error in 4563_END_sortmerna_R1.fastq.gz, line 194876135, with these 4 lines:

@HWI-ST1072:210:C5LGGACXX:2:2305:2793:35345 1:N:0:CAGATC
ATTTTGGAGTGTGTCCGTTGGGTAGTATGTGGAAACCACCCAGGGCCTTTGTGGAGAAAATGGAGGGGGGTGCCGGGGGGCCCTAGGAAGGGCCTTATTTG
+

at stream.ConcurrentGenericReadInputStream$ReadThread.readLists(ConcurrentGenericReadInputStream.java:656) at stream.ConcurrentGenericReadInputStream$ReadThread.run(ConcurrentGenericReadInputStream.java:635)

Set cris1Active=false
at jgi.SplitPairsAndSingles.process3_repair(SplitPairsAndSingles.java:562)
at jgi.SplitPairsAndSingles.process2(SplitPairsAndSingles.java:310)
at jgi.SplitPairsAndSingles.process(SplitPairsAndSingles.java:236)
at jgi.SplitPairsAndSingles.main(SplitPairsAndSingles.java:45)

0
Entering edit mode

What step is this error from? Before running repair.sh or while running repair.sh?

I thought you were able to repair the files before.

0
Entering edit mode

While running the repair.sh

0
Entering edit mode

Did you check the files with vpair first? Is the error from that step or actual repair?

I am going to tag Brian Bushnell . He should be along in 2-3 hours. It seems to me that some fastq records themselves may be broken (in addition to pairing).

0
Entering edit mode

Here is the error from vpair step:

/homeappl/home/kipokh/appl_taito/mytools/bbmap/reformat.sh in1=4563_END_sortmerna_R1.fastq.gz in2=4563_END_sortmerna_R2.fastq.gz vpair

java -ea -Xmx200m -cp /homeappl/home/kipokh/appl_taito/mytools/bbmap/current/ jgi.ReformatReads in1=4563_END_sortmerna_R1.fastq.gz in2=4563_END_sortmerna_R2.fastq.gz vpair

No output stream specified.  To write to stdout, please specify 'out=stdout.fq' or similar.
Set INTERLEAVED to false
Input is being processed as paired
java.lang.AssertionError:
Error in 4563_END_sortmerna_R1.fastq.gz, line 194876135, with these 4 lines:

@HWI-ST1072:210:C5LGGACXX:2:2305:2793:35345 1:N:0:CAGATC
ATTTTGGAGTGTGTCCGTTGGGTAGTATGTGGAAACCACCCAGGGCCTTTGTGGAGAAAATGGAGGGGGGTGCCGGGGGGCCCTAGGAAGGGCCTTATTTG
+

at stream.ConcurrentGenericReadInputStream$ReadThread.readLists(ConcurrentGenericReadInputStream.java:656) at stream.ConcurrentGenericReadInputStream$ReadThread.run(ConcurrentGenericReadInputStream.java:635)
java.lang.AssertionError:
Error in 4563_END_sortmerna_R2.fastq.gz, line 194876135, with these 4 lines:
BBBFFFFFFFFFFIIIIIIIIIFIIIIIIIIIIIIIFIIFIIIIIIIIIIIIIIIIIIIIIFFFFFFFFF7<BFF<FFFFFFFFFFFFFFFFFFFFFFFFF
@HWI-ST1072:210:C5LGGACXX:2:2305:2793:35345 2:N:0:CAGATC
CGGGGATTCTCCATTCCGGATGTGTTTCGTGGAGTGCATCGATACCTGCGCAATGCCTATGCTAGGGAAGAGTTTGCTTCCACCTGTCCAGATGATGAGGA
+

at stream.ConcurrentGenericReadInputStream$ReadThread.readLists(ConcurrentGenericReadInputStream.java:656) at stream.ConcurrentGenericReadInputStream$ReadThread.run(ConcurrentGenericReadInputStream.java:635)
There appear to be different numbers of reads in the paired input files.
The pairing may have been corrupted by an upstream process.  It may be fixable by running repair.sh.

1
Entering edit mode

It appears that you have records with bases but no quality scores. Those are not valid fastq records. It looks like maybe, some of the records have 2 newlines after the "+" instead of 1.

Probably, sortmerna is doing something inappropriate that results in corrupted output.

0
Entering edit mode

@Brian: Is there something in BBMap to check and fix broken fastq records in files?

0
Entering edit mode

Well... I tried to add some capability to do so in some limited ways, but it doesn't work very well. The problem is that there are an infinite number of ways a file could be broken, so the software will always end up guessing. So it's impossible to write a program that can correctly fix any broken fastq file, or even that can ensure that the corrections it did were correct. You can only write a program that will fix very specific problems, like a correctly-formatted fastq file with interleaved reads out of order.

0
Entering edit mode

Thanks a lot Brian and genomax2. I had also asked this problem to Jenya Kopylov (author of sortmerna) and I decided to skip sortmerna step for now. By the way, I found BBTools really useful :)

0
Entering edit mode

Are you able to actually fix the broken pairing by running reformat.sh (second command)?

One additional thing to test would be to validate the individual files to see if the fastq records themselves are broken: http://genome.sph.umich.edu/wiki/FastQValidator


text:
## bio mechanics

Assuming that the capillaries are 8 ?m in diameter, estimate the total length of capillaries in the body (excluding the lungs).


text:
# Problem: Calculate the formula weight for each of the following compounds:AgHSO3C3H5N3O3(NH4)3PO4

###### Problem Details

Calculate the formula weight for each of the following compounds:

AgHSO3

C3H5N3O3

(NH4)3PO4


text:
# Introducing Elliptic Curves

With all the recent revelations of government spying and backdoors into cryptographic standards, I am starting to disagree with the argument that you should never roll your own cryptography. Of course there are massive pitfalls and very few people actually need home-brewed cryptography, but history has made it clear that blindly accepting the word of the experts is not an acceptable course of action. What we really need is more understanding of cryptography, and implementing the algorithms yourself is the best way to do that. [1]

For example, the crypto community is quickly moving away from the RSA standard (which we covered in this blog post). Why? It turns out that people are getting just good enough at factoring integers that secure key sizes are getting too big to be efficient. Many experts have been calling for the security industry to switch to Elliptic Curve Cryptography (ECC), because, as we’ll see, the problem appears to be more complex and hence achieves higher security with smaller keys. Considering the known backdoors placed by the NSA into certain ECC standards, elliptic curve cryptography is a hot contemporary issue. If nothing else, understanding elliptic curves allows one to understand the existing backdoor.

I’ve seen some elliptic curve primers floating around with all the recent talk of cryptography, but very few of them seem to give an adequate technical description [2], and legible implementations designed to explain ECC algorithms aren’t easy to find (I haven’t found any).

So in this series of posts we’re going to get knee deep in a mess of elliptic curves and write a full implementation. If you want motivation for elliptic curves, or if you want to understand how to implement your own ECC, or you want to understand the nuts and bolts of an existing implementation, or you want to know some of the major open problems in the theory of elliptic curves, this series is for you.

The series will have the following parts:

1. Elliptic curves as elementary equations
2. The algebraic structure of elliptic curves
3. Points on elliptic curves as Python objects
4. Elliptic curves over finite fields
5. Elliptic curve cryptography and random number generation

Along the way we’ll survey a host of mathematical topics as needed, including group theory, projective geometry, and the theory of cryptographic security. We won’t assume any familiarity with these topics ahead of time, but we do intend to develop some maturity through the post without giving full courses on the side-topics. When appropriate, we’ll refer to the relevant parts of the many primers this blog offers.

A list of the posts in the series (as they are published) can be found on the Main Content page. And as usual all programs produced in the making of this series will be available on this blog’s Github page.

The first post will be published on Monday 2014-02-10. Hope you enjoy it!

[1] Okay, what people usually mean is that you shouldn’t use your own cryptography for things that actually matter, but I think a lot of the warnings are interpreted or extended to, “Don’t bother implementing cryptographic algorithms, just understand them at a fuzzy high level.” I imagine this results in fewer resources for people looking to learn cryptography and the mathematics behind it, and at least it prohibits them from appreciating how much really goes into an industry-strength solution. And this mindset is what made the NSA backdoor so easy: the devil was in the details.
[2] From my heavily biased standpoint as a mathematician.

# Thinking about Graduate School? Consider Mathematical Computer Science at UI Chicago!

It’s that time of year where senior undergraduates are considering whether to go to graduate school. And I wouldn’t be surprised if many students were afraid of the prospect, perhaps having read that popular genre of articles these days that tell you graduate school will turn you into an emotional wreck and that only a psychopathic masochist would put themselves through it.

The problem with these articles is they’re usually written by both outliers and those who put themselves in situations with no other options. I’ve felt my time at UI Chicago, however, has provided me nothing but options and excitement! So if you’re thinking about graduate school in mathematics or theoretical computer science, here’s my pitch for

## Why you should come to UI Chicago and study theoretical computer science

We’re social.

In fact, UI Chiago’s mathematics department is the most social of any math department I’ve ever heard of. I think this is the biggest benefit for me. On my first day here, I was surprised that everyone was totally normal and not the typical weird antisocial stereotype one associates with people who like math. Our department has a huge list of seminars going on every day of the week, and a small party every Friday called “Tea” that has a large attendance. We often go out to bars and restaurants, and have other outings. We even have a Facebook group (for grad students only) and a ping pong league that the professors sometimes join. We currently have over 150 graduate students in our department, and I know around 70 by name.

We have world-class faculty.

Some of my colleagues came to UIC specifically to work with David Marker on model theory, or Lou Kauffman on knot theory. At least one researcher here has over two hundred publications! We have big names in algebraic geometry, hypergraph combinatorics, dynamical systems, low-dimensional topology, and a very active logic group. Our theoretical computer science group (mixed with our combinatorics group) is small but vibrant and growing fast. We just got three new mathematical computer science students this year, and I’m doing everything I can to convert some of the other students over to our side.

We’re in the middle of a thriving intellectual community.

Chicago is the center of the Midwest US, and there are a ton of universities not only in the city but within a few hours drive. There are regular seminars and colloquia at the University of Chicago, Northwestern, and other smaller institutions like the Toyota Technical Institute (which has very strong researchers). Then there are the universities of Wisconsin, Indiana, and Michigan which all have strong theoretical computer science groups (and of course other mathematics groups) and we get together for conferences like Midwest Theory Day.

Our department is not cutthroat competitive.

I hear rumors about top mathematics and computer science programs that (unintentionally) pit students against each other for the attention of a few glorified professors. That simply doesn’t happen here. Everyone is friendly and people regularly collaborate. You can approach any professor and ask to do a reading course with them or ask them what kinds of open problems they’re thinking about, and most of them will gladly sit down with you and explain all the neat ideas in their heads. Even the hardest, most sarcastic professors genuinely care about their students. I think, along with being social, this makes our department one of the friendliest and most stress-free places to get a PhD.

We’re in a great city.

Our department staff is very supportive.

Our director and assistant director of graduate studies are extremely helpful at getting new students situated and ensuring they have funding. It’s not uncommon for students who start in the PhD program to decide after one or two years that a PhD is not right for them. Usually they will stop with the requirements for a master’s degree, and there are no hard feelings. Students who do this are even encouraged to return if they decide they want to finish their PhD later. In the mean time, our department guarantees tuition waivers and stipends to all of its teaching assistants (and there are alternatives to teaching as well), so you can focus on your studies and not have to think too much about money.

And even more, if you decide to study theoretical computer science at UI Chicago you get a whole bunch of other benefits:

You get to hang out and do research with me!

(Okay maybe that’s not a serious benefit to consider)

Jobs are hard to come by for the purest of pure mathematics researchers. Research positions are in short supply, and unless you want to go into industry with an applied math degree the remaining option is to teach at a 4-year institution. But if you study theoretical computer science, now you are qualified to do all kinds of things. Work at industry research labs like Microsoft Research, Google Research, or Yahoo! Research. Work at government labs like Lincoln Labs and Lawrence Livermore National Labs, both of which I interned at. You can shoot for a professorship or do a postdoc like a regular mathematics PhD would. If you’re hand with Python you could go into the software industry and get a high-demand job at any major company in cryptography or operations research (both of which depend on ideas from TCS). And you always keep the option of teaching at a 4-year.

You have many options for internships during summers.

I, my colleagues, and even my advisor did research internships during the Summers at various research labs and industry companies. This is a particularly nice benefit of doing mathematical computer science in grad school, because it augments your normal graduate student stipend by enough to live much more comfortably than otherwise (that being said, for extra money a lot of my pure math colleagues will tutor on the side, and tutoring comes at a high price these days). It’s not uncommon to receive additional funding through these opportunities as well.

You get to travel a lot.

The main publication venue in computer science is the conference, and that means there are conferences happening all over the world all the time. In fact, I just got back from a conference in Aachen, Germany, earlier this year I was at Berkeley and Stanford, I am helping to run a conference in Florida early next year, and I am looking at conferences in Beijing and Barcelona next Summer. All of the trips you take to present your published research is paid for, so it’s just pure awesome.

You enjoy the breadth of problems in computer science.

Computer science is unique in that it connects to almost every field of mathematics.

1. Like statistics? There’s statistical machine learning and randomized algorithm design.
2. Like real analysis and dynamical systems? There’s convex optimization, support vector machines, and tons of computational aspects of PDE’s.
3. Like algebra or number theory? There’s cryptography.
4. Like combinatorics? There’s combinatorial optimization.
5. Like game theory? I just got back from a conference on algorithmic game theory.
6. Like geometry and representation theory? There’s a Geometric Complexity Theory program working toward P vs NP.
7. Like logic? You might be surprised to know that the cleanest proofs of the incompleteness theorems are via Turing machines.
8. Like topology? There are researchers (not at UIC) working on computational topology, like persistent homology which we’ve been slowly covering on this blog.

The list just goes on and on, and this isn’t even mentioning the purely pure theoretical computer science topics which have a flavor of their own.

Programming options exist, but you aren’t forced to write programs.

Some of the greatest computer science researchers cannot write simple computer programs, and if you’re just interested in theory there is plenty of theory to go around. On the other hand, we have researchers in our department studying aspects of supercomputing, and options for collaboration with researchers in the (engineering) computer science department. Over there they’re studying things like biological networks, machine learning and robotics, and all kinds of hands-on applied stuff that you might be interested in if you read this blog.

So if you’re interested in joining us for next year and have any questions, feel free to drop me or the professors in the MCS group or the director of graduate studies an email.

# Status Update, and Boston Python Project Night

Hey everyone, this is just a quick note to say what’s going on.

Recently I’ve been slowing down a bit on blog posts. I’ve been travelling and settling in at my Summer job in Boston where we’re looking at a lot of really fascinating questions w.r.t. clustering and learning theory. For various reasons (my proper hardware isn’t with me, time constraints, etc.), I haven’t been able to get down to the meaty posts I’ve been hoping to work on, such as the follow-up to Bezier & Picasso and more on category theory. I will do my best to continue posting, but until about September most of the posts will be on the mathy side with no hand-made images and no Youtube videos.

On a brighter note, I will be attending this upcoming Monday’s meeting of the Boston Python User’s Group! It’s going to be pretty fun, and as of this writing there are 43 spots left. So if you’re in the area, if you enjoy free pizza, and if you want to come and meet people and code Python for a few hours, I’ll be there working on some stuff related to stable marriages. You’ll know me as the guy who keeps talking about math.

As a side note, I noticed there is no comparable meetup in Chicago. Despite my already busy life (i.e., maybe in a year or two), I was considering starting a Python meetup in Chicago. Then I thought about how small the software scene is in Chicago, and wondered if I would be able to find enough sponsors and people to give talks. It’s more of a pipe dream right now, but maybe someday it can become a reality. At least I can start by signing up to give a talk (cf. evangelize mathematics) at the Boston meetup later next month.

In other news, I’ve got a small surprise tomorrow in the form of a fun guest post on a friend’s blog. Keep an eye out for that.

Until next time!

# Why Theoretical Computer Scientists Aren’t Worried About Privacy

There has been a lot of news recently on government surveillance of its citizens. The biggest two that have pervaded my news feeds are the protests in Turkey, which in particular have resulted in particular oppression of social media users, and the recent light on the US National Security Agency’s widespread “backdoor” in industry databases at Google, Verizon, Facebook, and others. It appears that the facts are in flux, as some companies have denied their involvement in this program, but regardless of the truth the eye of the public has landed firmly on questions of privacy.

Barack Obama weighed in on the controversy as well, being quoted as saying,

You can’t have 100% security and 100% privacy, and also zero inconvenience.

I don’t know what balance the US government hopes to strike, but what I do know is that privacy and convenience are technologically possible, and we need not relinquish security to attain it.

Before I elaborate, let me get my personal beliefs out of the way. I consider the threat of terrorism low compared to the hundreds of other ways I can die. I should know, as I personally have been within an $\varepsilon$ fraction of my life for all $\varepsilon > 0$ (when I was seven I was hit by a bus, proclaimed dead, and revived). So I take traffic security much more seriously than terrorism, and the usual statistics will back me up in claiming one would be irrational to do otherwise. On the other hand, I also believe that I only need so much privacy. So I don’t mind making much of my personal information public, and I opt in to every one of Google’s tracking services in the hopes that my user experience can be improved. Indeed it has, as services like Google Now will, e.g., track my favorite bands for me based on my Google Play listening and purchasing habits, and alert me when there are concerts in my area. If only it could go one step further and alert me of trending topics in theoretical computer science! I have much more utility for timely knowledge of these sorts of things than I do for the privacy of my Facebook posts. Of course, ideologically I’m against violating privacy as a matter of policy, but this is a different matter. One can personally loathe a specific genre of music and still recognize its value and one’s right to enjoy it.

But putting my personal beliefs aside, I want to make it clear that there is no technological barrier to maintaining privacy and utility. This may sound shocking, but it rings true to the theoretical computer scientist. Researchers in cryptography have experienced this feeling many times, that their wildest cryptographic dreams are not only possible but feasible! Public-key encryption and digital signatures, secret sharing on a public channel, zero-knowledge verification, and many other protocols have been realized quite soon after being imagined. There are still some engineering barriers to implementing these technologies efficiently in large-scale systems, but with demand and a few years of focused work there is nothing stopping them from being used by the public. I want to use this short post to describe two of the more recent ideas that have pervaded the crypto community and provide references for further reading.

## Differential Privacy and Fully Homomorphic Encryption

There are two facts which are well known in theoretical computer science that the general public is not aware of. The first is about the privacy of databases:

There is a way to mine information from a database without the ability to inspect individual entries in the database.

This is known as differential privacy. The second is no less magical:

There are secure encryption schemes which allow one to run programs on encrypted data and produce encrypted results, without the ability to decrypt the data.

This is known as fully homomorphic encryption.

The implications of these two facts should be obvious: search engines need not know our queries but can still fetch us search results and mine our information to serve ads, Facebook need not have access to our personal data but may still accurately predict new friends, grocery stores can even know what products to place side by side without knowing what any individual customer has purchased. Banks could process our transactions without knowing the amounts involved, or even the parties involved. Perhaps most importantly, governments can have access to databases (in the form of differentially private queries) and mine for the existence of threats without violating any individual user’s privacy. If they get an indication of a terrorist threat, then they can use the usual channels (court orders) to get access to specific individual data.

It’s easy to argue that these techniques will never become mainstream enough for individuals to benefit from it. Indeed, we’ve had cryptography for many years but few average users actively encrypt their communication for a lack of convenience. And then there are questions of policy: why would any company relinquish the ability to work directly with user data? And the cost of rearchitecturing existing services to utilize these technologies would be enough to dissuade most business leaders.

But the point of all this is that these are problems of policy that could in principle be solved without waiting for governments and corporations to get their act together. With enough demand for such services and with enough technologically-minded entrepreneurs (I’m looking at you, Silicon Valley), it would be just a matter of time before the world was differentially private. Mathematics cannot be revoked or legislated away.

## Fully Homomorphic Encryption

A fully homomorphic encryption scheme is a normal encryption scheme (two functions “enc” and “dec” to encrypt and decrypt) with one additional function, which we’ll call “eval.” Very roughly, eval accepts as input the text of a program and a ciphertext, and produces as output a ciphertext such that the following diagram commutes:

That is, $m$ is our message, and $\textup{eval}$ runs $f$ on the encrypted version of our message. In practice this happens by lifting two operations, multiplication and addition, from plaintexts (which are usually number-representations of letters) to ciphertexts (again usually numbers). Once this is done one can simulate the functionality of an arbitrary circuit on the encrypted data without decrypting it. Those readers who have been following our category theory series will recognize these sorts of diagrams as being functorial. [Actually, at the time of this writing we have yet to look at functors, but we will soon!] So perhaps a better term would be “functorial encryption.”

I should emphasize: a truly homomorphic encryption scheme has the ability to run any computable function on the encrypted data. There is no loss of functionality in preserving the privacy from the program runner. The main use of this is to maintain privacy while deferring large computations to the cloud. We do this all the time, e.g. a search query, but it also applies to big websites like Reddit, which operate entirely on Amazon Web Services.

Fully homomorphic encryption was first envisaged by Rivest, Adleman (two of the inventors of RSA), and Dertouzos in the late seventies, mainly because the RSA encryption scheme is close to being homomorphic (one can multiply ciphertexts, but not add them). In 2009, Craig Gentry released the first working fully-homomorphic scheme based on the mathematical theory of ideal lattices, and later that year he (with a group of other researchers) came up with a second system that is arguably as simple as RSA; it operates on integers with modular arithmetic.

Gentry has produced a lot of research since then in homomorphic encryption, but the interested reader should probably start with his tutorial paper describing his arithmetic-based system. From there, there are existing implementations in Python (using Sage) and C++, both of which are freely available on github.

## Differential Privacy

The main idea of differential privacy is that one can add noise to statistical data to protect the identities of individual records. Slightly more rigorously, a randomized algorithm $f$ is said to be $\varepsilon$-differentially private if for all possible datasets (inputs) $D_1, D_2$ which differ on a single record, and all possible collections of outputs $y$ of $f$, the probability of correctly guessing $D_1$ from $y$ is not significantly different from that of $D_2$. In particular, their quotient is at most $e^{\varepsilon}$ (this choice of using $e$ is arbitrary, but makes the analysis nicer).

The motivation for differential privacy came from two notable events in which companies released “anonymized” data which was partially de-anonymized because it was too specific. The first was the million-dollar Netflix Prize contest to develop a better recommendation algorithm, and the second was the release of the Massachusetts Group Insurance Commission medical database. As such, many companies are very strict with how they handle their user data, and information sharing the medical community is practically nonexistent.

There are many known differentially private algorithms, and they’re much stronger than one would imagine at first. One can run random forests of decision trees, network trace analysis, query-click analysis, certain forms of clustering, and a whole host of combinatorial optimization problems. For a gentle introduction to differential privacy, see Christine Task’s lecture video, a Practical Beginner’s Guide to Differential Privacy. There is also an influential survey from Microsoft Research of Dwork. These go into much more detail about the abilities and inabilities of differential privacy than I could do here.

If there’s one thing to take away from this discussion, it’s that efficient protocols for ensuring privacy are out there waiting to be implemented in software. So while we complain and listen to others complain about governments violating our liberties (and indeed, this discussion is extremely important to have), let’s do a little mathematics, do a little computer science, and figure out how to make privacy the standard of measure in software.

Until next time!


text:
### MeshHierarchy coarsening over mutliple levels?

341
views
1
12 months ago by
Hello,

by applying the coarsening functionality of the MeshHierarchy class, the finest level gets coarsened by one level.
However, I was wondering if it is somehow possible to coarsen the resulting mesh once more.

I tried to set the parent for the finest level manually to say the coarsest level but failed.

mh = MeshHierarchy(mesh)
cell_markers_coarsen = CellFunction("bool", mh.finest())
...
mh = mh.coarsen(cell_markers_coarsen)

mh.finest().set_parent(mh[0])

cell_markers_coarsen = CellFunction("bool", mh.finest())
...
mh = mh.coarsen(cell_markers_coarsen)

Any ideas?
Community: FEniCS Project

4
12 months ago by
The MeshHierarchy coarsening is very simplistic. It requires that you start from a coarse mesh, and refine down to the finest level that you need. Then you can go back up one level. In practice, you might as well just keep copies of all the Meshes, and if you need to "coarsen", just go back to an earlier one.
I'm a bit confused with "coarsen" and "unrefine".
If I understand it correctly, unrefine goes back completely one level, whereas coarsen goes back one level and then re-refines the level, but without refining the parents of the cells marked for coarsening. (At least I got the impression from the source code that this is what happens).

The problem then is, if cells stay the same over two levels, there is no chance to coarsen them!

If this is the case then in order to be able to coarsen cells back to the 0-th level, I probably need to take the initial mesh and refine a few times manually? This seems very unefficient....
written 12 months ago by Lukas O.
0
12 months ago by
I'm a bit confused with "coarsen" and "unrefine".
If I understand it correctly, unrefine goes back completely one level, whereas coarsen goes back one level and then re-refines the level, but without refining the parents of the cells marked for coarsening. (At least I got the impression from the source code that this is what happens).

The problem then is, if cells stay the same over two levels, there is no chance to coarsen them!

If this is the case then in order to be able to coarsen cells back to the 0-th level, I probably need to take the initial mesh and refine a few times manually? This seems very unefficient....


text:
# Wayland KDE X11

## KDE Impressions

These days, I often hear a lot about Wayland. And how much of effort is being put into it; not just by the Embedded world but also the usual Desktop systems, namely KDE and GNOME.

In recent past, I switched back to KDE and have been (very) happy about the switch. Even though the KDE 4 (and initial KDE 5) debacle had burnt many, coming back to a usable KDE desktop is always a delight. It makes me feel home with the elegance, while at the same time the flexibility, it provides. It feels so nice to draft this blog article from Kwrite + VI Input Mode

Thanks to the great work of the Debian KDE Team, but Norbert Preining in particular, who has helped bring very up-to-date KDE packages into Debian. Right now, I’m on a Plamsa 5.21.1 desktop, which is recent by all standards.

## Wayland

Almost all the places in the Linux world these days are busy with integrating Wayland as the primary display service. Not sure what the current status on the GNOME side is but I definitely keep trying KDE + Wayland with every release.

I keep trying with every release because it still is not prime for daily use. And it makes me get back to X11, no matter how dated some may call. Fact is, X11 still shines to me as an end-user.

Glitches with Wayland still are (Based on this week’s test on Plasma 5.21.1):

• Horrible performance compared to X11
• Very crashy, especially when hotplugging secondary display. Plasma would just crash. X11 is very resilient to such things, part of the reason I can think is the age of the codebase.
• Many many applications still need to be fixed for Wayland. Or Wayland needs to accomodate them in some way. XWayland does not really feel like the answer.

And while KDE keeps insisting users to switch to Wayland, as that’s where all the new enhancements and fixes are put in, someone like me still needs to stick to X11 for the time being. So to get my shiny new LG 27" 4K Monitor (3840x2160 60.00*+) to work without too much glitch, I had to live with an alias:

$alias | grep xrandr alias rrs_xrandr_lg='xrandr --output DP-1 --mode 3840x2160 --scale .75x.75' 18:31 ♒ ॐ ♅ ♄ ⛢ ☺ 😄  ## Plasma 5.21 On the brighter side, the Plasma 5.21.1 release brings some nice enhancements in other areas. • I’m now able to make use of tighter integration with systemd/cgroups, with better organization and management of processes overall. • The new Plasma theme, Breeze Twilight, is a good blend of Light + Dark. I also appreciate the work put in by Michail Vourlakos. The KDE project is lucky to have a developer/designer like him. His vision and work into the KDE desktop is well beyond a writing by me. $ usystemctl status plasma-plasmashell.service
● plasma-plasmashell.service - KDE Plasma Workspace
Active: active (running) since Fri 2021-02-26 18:34:23 IST; 13s ago
Main PID: 501806 (plasmashell)
Memory: 759.8M
CPU: 13.706s
CGroup: /user.slice/user-1000.slice/user@1000.service/session.slice/plasma-plasmashell.service
└─501806 /usr/bin/plasmashell --no-respawn

Feb 26 18:35:00 priyasi plasmashell[501806]: qml: recreating buttons
Feb 26 18:35:21 priyasi plasmashell[501806]: qml: recreating buttons
Feb 26 18:35:49 priyasi plasmashell[501806]: qml: recreating buttons
Feb 26 18:35:57 priyasi plasmashell[501806]: qml: recreating buttons
18:36 ♒ ॐ ♅ ♄ ⛢     ☺ 😄

## OBS - Open Build Service

I should also thank the OpenSUSE folks for the OBS work. It has enabled the close equivalent (or better, in my experience) of PPAs for Debian. And that is what has enabled developers like Norbert to easily and quickly be able to deliver the entire KDE suite.

### OBS - Some detail

Christian asked for some more details on the OBS side of things, of my view. I’m updating this article with it because the comment system may not always be reliable and I hate losing content.

Having been using OBS myself, and also others in the Debian community who are making use of it, I surely think we as project should consider making use of OBS.

Given that OBS is Free Software, it is a perfect fit for Debian. Gitlab is another example of what we’ve made available in Debian.

OBS is divided into multiple parts

• OBS Server
• OBS DoD service
• OBS Publisher
• OBS Workers
• OBS Warden
• OBS Rep Server

For every Debian release I care about, I add an OBS project per release. So I have OBS projects for: Sid, Bullseye, Buster, Jessie.

Now, say you have a package, foo . You prep your package and enable all the releases that you want to build the package for. So the same package gets built, in separate clean environments, for every release I mentioned as an example above. You don’t have to manually trigger the build for every release/every architcture. You should add the release (as projects) in OBS, set their supported architectures, and then add those enabled release projets as bits to your package.

Every build involves:

• Creating a new chroot for each build
• Building the package

Builds can be scattered across multiple hosts, known as workers in OBS terminology. Your workers are independent machine entities, supporting different architectures. The machines can be Bare-Metal ones, VMs, even containers. So this allows for very nice scale-in and scale-out. There may be auto-scaling too but that is something worth investigating.

Think of things like cross architecture builds. Let’s assume the cloud vendors decide to donate resources to the Debian project. We could enable OBS worker instances on the respective clouds (different architectures) and plug them into the master OBS instance that Debian hosts. Fully distributed. Similarly, big hardware vendors willing to donate compute resources could house them in their premises and Debian could just easily establish a connection to them. All of this just a TCP connection away.

So when I look at the features of OBS, from the point of view of Debian, I like it more. Extensibility won’t be an issue. Supporting a new Debian release would just be a matter of bootstrapping the Debian release as a project in OBS, and then all done. The single effort of setting of the target release project is a one time job, and then all can leverage it.

The PPA was a long craved feature missing in Debian, in my opinion. OBS allows to not just fulfil that gap but also extend it in a very easy way.

Andrew Lee had put in a nice video presentation about the same @ Debconf 20


text:
The Magic of Romanian Folktales Starts with the First Words

While a grew up I thought that the magic and musicality of Romanian folktales was so much better than that of the Grimm’s fairy tales. Because I was sure of their truthfulness.

Although both sources shared the same well-known prologue, “Once upon a time,” the Romanian ones went on with “for if it didn’t happen it couldn’t be told,” thus proving that some truth was at the bottom of the folktale about to unravel, since nothing can be told that didn’t happen

Continue reading “The Magic of Romanian Folktales Starts with the First Words”


text:
# How many significant figures are in the measurement 811.40 grams?

Aug 6, 2016

$5$

#### Explanation:

In order to be able to say how many sig figs are used for the measurement $\text{811.40 g}$, you need to know that

• non-zero digits are always significant
• a trailing zero that follows a decimal point and a non-zero digit is always significant

In your case, the measurement

$\textcolor{b l u e}{811} . \textcolor{b l u e}{4} \textcolor{red}{0}$

has four non-zero digits and one significant trailing zero.

$\textcolor{b l u e}{8 , 1 , 1 , 4} \to$ non-zero digits

$\textcolor{w h i t e}{8 , 1 , 1 ,} \textcolor{red}{0} \to$ significant trailing zero

Therefore, you can say that this measurement has five significant figures

$811.40 \to$ five significant figures


text:
# findroot, derivative in defined functions

I apologize for my unclear question, I will write it in a more detailed way.

I first define:

g[τ_, ϵ_] := c /. FindRoot[2. c^4 - 1.6931471805599454 c^4 τ -
0.8611473146305157 τ^2 +
0.25 τ^2 Log[ϵ^(1/6)/c^(
2/3)] + (1. c^4 +
0.6732867951399863 τ) τ Log[τ] -
0.125 τ^2 Log[τ]^2 -
0.25 τ^2 Log[-0.6931471805599453 + Log[τ]], {c, 1000}]

The function I am trying to solve is as follows:

]1

I can plot $\zeta_2 (r)$ in the region $1<r<\tau/2$ for different values of $a, \tau, \epsilon$, where $a>0$, $\tau>2$, $10^{-7}<\epsilon<10^{-2}$. what I need to do is to find the smallest value of r satisfying $\zeta_2(r)=0$. I use Findroot to do it. but it does not work well all the way through the parameter region.

The second question is that supposing we find the smallest value of $r$, which is $h[a,\tau,\epsilon]$, I need to do a integration, which is defined as:

w[a_, τ_, ϵ_] := Integrate[Evaluate[{Subscript[ζ, 2][r]^2} /. f[a, τ, ϵ]], {r, 1, h[a,τ, ϵ]}]

As a test, I choose $a=1, \tau=5, \epsilon=10^{-6}, h(a,\tau,\epsilon)=1.2$, but $w(1,5,10^{-6})$ does not return me a number. So how should I define $w$ to get a number?

The last question is if I have got the right definition of $w$, set the derivative $D[w,{a,1}]=0$, and find $a$ in terms of $\tau, \epsilon$. I am getting stocked by doing the derivative of functions. Thanks very much. By the way, I hope this time that Mr.Wizard won't wasting time on revising my writing. I am really sorry

-

I have never understood why WRI would not incorporate Ted Ersek's RootSearch routine into Mathematica. It is true that RootSearch is restricted to 1-dimensional roots along a path and FindRoot will handle multi-dimensional roots. Also many roots can be obtained by Solve or Reduce and probably various algebraic methods. But these are often not obvious to use. RootSearch is easy to use to find numerical roots, intuitive and it applies to many common problems. So here is an example. Since you haven't supplied us with an example I've made up one that should be good enough.

<< ErsekRootSearch

f[x_] := Sin[100 x] + Cos[50 x] + x - 1.830
Plot[f[x], {x, 0, 1}]

roots = RootSearch[f[x] == 0, {x, 0, 1}, PrecisionGoal :> 25]
{{x -> 0.1342401536562234144657613},
{x -> 0.1425470784657371848184752},
{x -> 0.2576641634599237715747900},
{x -> 0.2703039062616023447786570},
{x -> 0.3816011520392444127374391},
{x -> 0.3975359955533713429773951},
{x -> 0.5057681048545612636327423},
{x -> 0.5245246522191860737400116},
{x -> 0.6300648404511711207196194},
{x -> 0.6513679694342461184637879},
{x -> 0.7544406174842106100577210},
{x -> 0.7781141014717457963000968},
{x -> 0.8788641445373034149609143},
{x -> 0.9047910841068439265967640}}

Here we take the first root and check it.

f[x] /. First[roots]
0.*10^-24

-


text:
# “MeasureDump” appears when using NIntegrate with decimal parameters

I have had a strange problem with NIntegrate. I have this function NInt2 depending on two complex parameters:

In[303]:=
Nint2[z1_, z3_] :=
Quiet[NIntegrate[
ComplexExpand[-2 I*z3/(z1 - z2)*1/(1 - Conjugate[z2]*z3), {z1, z2,
z3}] /. (z2 -> x + I*y), {x, y} \[Element] Disk[]]]
z1 = 1/2 - 0.1 I
z3 = I/2
Nint2[z1, z3]
2 Pi*Arg[1 - z1*Conjugate[z3]]
2 Pi*Arg[1 - z1*Conjugate[z3]] // N

Out[304]= 0.5 - 0.1 I

Out[305]= I/2

Out[306]= 3.14159 ((
0.5 + 0.1 I)/((0.26 + 1. Im[MeasureDumpX$178265[2.]] + Im[MeasureDumpX$178265[2.]]^2) (0. +
0.25 Re[MeasureDumpX$178265[1.]]^2)) - ((0.25 - 0.95 I) Im[ MeasureDumpX$178265[1.]])/((0.26 +
1. Im[MeasureDumpX$178265[2.]] + Im[MeasureDumpX$178265[2.]]^2) (0. +
0.25 Re[MeasureDumpX$178265[1.]]^2)) - ((0. + 0.5 I) Im[ MeasureDumpX$178265[1.]]^2)/((0.26 +
1. Im[MeasureDump`X$178265[2.]] + (and so on...) Out[307]= 1.46865 Out[308]= 1.46865 However if I define both numbers to have rational values then the problem disappears: In[292]:= Nint2[z1_, z3_] := Quiet[NIntegrate[ ComplexExpand[-2 I*z3/(z1 - z2)*1/(1 - Conjugate[z2]*z3), {z1, z2, z3}] /. (z2 -> x + I*y), {x, y} \[Element] Disk[]]] z1 = 1/2 z3 = I/2 Nint2[z1, z3] 2 Pi*Arg[1 - z1*Conjugate[z3]] // N Out[293]= 1/2 Out[294]= I/2 Out[295]= 1.53925 + 0.190458 I Out[296]= 1.53925 Do you have an idea of what this problem could be, and how to work around it? I would like to use random numbers as input eventually. • I think this depends on where you decide to apply the$(x,y)\$ replacement in your code. Try to do so within ComplexExpand, rather than after it. Also, why do you have Quiet? What are you suppressing? – MarcoB Feb 17 '17 at 15:52

To understand what is happening, consider

ComplexExpand[z2, z2]
(* I Im[z2] + Re[z2] *)

% /. (z2 -> x + I*y)
(* -Im[y] + Re[x] + I (Im[x] + Re[y]) *)

The code of Nint2, although more complicated, is doing the same thing. NIntegrate, not knowing what to do with the Re and Im operating on the variables of integration, returns the strange result shown in the question. (A more graceful response would be preferable.)

This problem can be eliminated by correcting the code to read

Nint2[z1_, z3_] := NIntegrate[ComplexExpand[-2 I*z3/(z1 - z2)*1/(1 - Conjugate[z2]*z3)
/. (z2 -> x + I*y), {z1, z3}], {x, y} ∈ Disk[]]

or more simply

Nint2[z1_, z3_] := NIntegrate[-2 I*z3/(z1 - z2)/(1 - Conjugate[z2]*z3)
/. (z2 -> x + I*y), {x, y} ∈ Disk[]]

since ComplexExpand is unnecessary here. Either way, the function with z1 and z3 as defined in the question yields

(* 1.46866 + 0.479793 I *)

along with warnings of poor convergence, presumably because the integrand is singular at a point in the domain of integration. However, the singularity appears to integrable, and the answer is insensitive to WorkingPrecision, so it probably is correct.

• Thanks a lot! You are right I should not use ComplexExpand here... It's only there because I copied the code from a another integral that suffered from instabilities without it, never mind why. – Wernli Feb 24 '17 at 14:27


text:
# Linear Diff. Equ. of Second Order

• December 9th 2009, 08:18 AM
kjchauhan
Linear Diff. Equ. of Second Order

(1) $\frac{d^2y}{dx^2}+\frac{2}{x}\frac{dy}{dx}-n^2y=0$

(2) $\frac{d^2y}{dx^2}+\frac{2}{x}\frac{dy}{dx}+n^2y=0$

(3) $(x-3)\frac{d^2y}{dx^2}-(4x-9)\frac{dy}{dx}+3(x-2)y=0$

(4) $(1-x^2)\frac{d^2y}{dx^2}+3x\frac{dy}{dx}+y=0$

(5) $\frac{d^2y}{dx^2}+4x\frac{dy}{dx}+4x^2y=0$

(6) $x\frac{d^2y}{dx^2}-(2x-1)\frac{dy}{dx}+(x-1)y=0$

(7) $\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y=f(x)$

(8) $(x\sin x+\cos x)\frac{d^2y}{dx^2}-x\cos x\frac{dy}{dx}+ y\cos x$

Method of transforming the independent variables, I took $z=\int e^{-\int Pdx} \; dx$
But I didn't gt it properly.. and $Q_1 \ne constant$
I also supposed $Q_1=constant$, but in this case I also didint get it.