text:
0
Q:

# Sound travels at 330 metres per second. The distance (in kilometres) of a thunder cloud when its sound follows the flash after 10 seconds

 A) 0.33 km B) 3.3 km C) 33 km D) 33.3 km

Explanation:
Q:

A car covers 25 km at a uniform speed. If the speed had been 8 knv/h more, it would have taken 10 hours less for the same journey. What is the speed of the car (in km/h)?

 A) 4 B) 3 C) 2.5 D) 2

Explanation:

0 48
Q:

The ratio between the speeds of two trains is 5 : 7. If the first train covers 300 km in 3 hours, then the speed (in km/h) of the second train is:

 A) 100 B) 150 C) 140 D) 120

Explanation:

0 140
Q:

The ratio between the speeds of two trains is 2 : 5. If the first train runs 350 km in 5 h, then the difference between the speeds (in km/h) of both the trains is:

 A) 350 B) 105 C) 165 D) 180

Explanation:

2 399
Q:

A man divided his journey into three parts of distances of 18 km, 20 km and 27 km. He travelled the distances at the speeds of 6 km/h, 5 km/h and 9 kn/h,respectively. What was his average speed during the entire journey?

 A) 6.5 km/h B) 4.5 km/h C) 7.5 km/h D) 5.5 km/h

Explanation:

2 650
Q:

The distance between two railway stations is 1176 km. To cover this distance, an express train takes 5 hours less than a passenger train while the average speed of the passenger train is 70 km/h less than that of the express train. The time taken by the passenger train to complete the travel is:

 A) 12 hours B) 23 hours C) 17 hours D) 18 hours

Explanation:

2 746
Q:

The distance between the places H and O is D units. The average speed that gets a person from H to O in a stipulated time is S units. He takes 20 minutes more time than usual if he travels at 60 km/h, and reaches 44 minutes early if he travels at 75 km/h. The sum of the numerical values of D and S is:

 A) 376 B) 358 C) 344 D) 384

Explanation:

0 481
Q:

A train that is running at the speed of 72 km/h crossesan electric pole in 36 seconds. The length of the train (in metres) is:

 A) 720 m B) 460 m C) 360 m D) 620 m


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# Thread: find n such that T(n) = 16

1. ## find n such that T(n) = 16

Find the smallest positive integer that has exactly 16 positive divisors.

I kinda know how to approach this problem but I'm not sure how to answer it in a formal way.

2. Let $n = \prod_{i=1}^k p_i^{e_i}$ be the prime factorization of $n$.

If you recall the properties of $\tau (n)$, we want: $\tau (n) = \prod_{i=1}^k \left(e_i+1\right) = 16$

Now, $16 = 2^4$ can be written as a product of integers in only 5 ways:
• $2^4 = 16$
• $2^3 \cdot 2 = 8 \cdot 2$
• $2^2 \cdot 2^2 = 4 \cdot 4$
• $2^2 \cdot 2 \cdot 2 = 4 \cdot 2 \cdot 2$
• $2 \cdot 2 \cdot 2 \cdot 2$

So our possibilities:
• $\tau (n) = e_1 + 1 = 16 \ \Rightarrow \ e_1 = 15$
• $\tau (n) = (e_1 + 1)(e_2 + 1) = (8)(2) \ \Rightarrow \ e_1 = 7, \ e_2 = 2$
• $\tau (n) = (e_1 + 1)(e_2 + 1) = (4)(4) \ \Rightarrow \ e_1 = 3, \ e_2 = 3$
• $\tau (n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) = (4)(2)(2) \ \Rightarrow \ e_1 = 3, \ e_2 = 1, \ e_3 = 1$
• $\tau (n) = (e_1 + 1)(e_2 + 1)(e_3 + 1)(e_4 + 1) = (2)(2)(2)(2) \ \Rightarrow \ e_1 = 1, \ e_2 = 1, \ e_3 = 1, \ e_4 = 1$

Since we want the smallest integer, we consider the smallest primes in the factorization of $n$. So this narrows our list down to:
$n = 2^{15} \quad 2^7 \cdot 3^1\quad 2^3 \cdot 3^3\quad 2^3 \cdot 3^1 \cdot 5^1\quad 2 \cdot 3 \cdot 5 \cdot 7$

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# Prob. 10, Sec. 3.5, in Kreyszig's Functional Analysis: How to show that this set is at most countable?

This is Prob. 10, Sec. 3.5, in the book Introductory Functional Analysis With Applications by Erwine Kreyszig:

Let $X$ be an inner product space, let $M$ be an uncountable orthonormal subset of $X$, let $x \in X$ such that $x$ is not the zero vector $\mathbf{0}_X$ in $X$, and let $S(x)$ be the subset of $M$ defined as follows: $$S(x) \colon= \left\{ \ v \in M \ \colon \ \langle x, v \rangle \neq 0 \ \right\}.$$ Then how to show that this set $S(x)$ is at most countable?

And, this is Prob. 8, Sec. 3.4, in the same book:

Let $\left( e_k \right)_{k \in \mathbb{N} }$ be an orthonormal sequence in an inner product space $X$, let $x \in X$, let $m \in \mathbb{N}$, let $A_m(x)$ be the subset of $\mathbb{N}$ defined as follows: $$A_m(x) \colon= \left\{ \ k \in \mathbb{N} \ \colon \ \left\lvert \left\langle x, e_k \right\rangle \right\rvert > \frac{1}{m} \ \right\},$$ and let $n_m$ be the cardinality of $A_m(x)$. Then $$n_m \leq m^2 \lVert x \rVert^2.$$

The proof of this result involves the Bessel's inequality and goes as follows:

As $\left( e_k \right)_{k \in \mathbb{N}}$ is an orthonormal sequence in the inner product space $X$, so by Theorem 3.4-6 (Bessel Inequality) in Kreyszig, for any $x \in X$, the series $\sum \left\lvert \left\langle x, e_k \right\rangle \right\rvert^2$ converges in $\mathbb{R}$, and $$\sum_{k=1}^\infty \left\lvert \left\langle x, e_k \right\rangle \right\rvert^2 \leq \lVert x \rVert^2.$$

Let $x\in X$, let $m$ be a given natural number, and let $A_m(x)$ be ths subset of $\mathbb{N}$ given by $$A_m(x) \colon= \left\{ \ k \in \mathbb{N} \ \colon \ \left\lvert \left\langle x, e_k \right\rangle \right\rvert > \frac{1}{m} \ \right\}.$$

Let $n_m$ denote the cardinality of the set $A_m(x)$ (which is the same as the number of elements in the set $A_m(x)$ if $A_m(x)$ is finite ), where $$n_m \in \{ \ 0 \ \} \cup \mathbb{N} \cup \{ \ \aleph_0 \ \},$$ where $\aleph_0$ (pronounced aleph null'') denotes the cardinality of the set $\mathbb{N}$ of natural numbers, because the set $A_m(x)$ can be empty, non-empty but finite, or countably infinite. Furthermore, as $A_m(x) \subset \mathbb{N}$ and as $\mathbb{N}$ is countable, so $A_m(x)$ cannot be uncountable.

If $x = \mathbf{0}_X$, the zero vector in $X$, then $$\left\langle x, e_k \right\rangle = 0$$ for all $k \in \mathbb{N}$, and so the set $A_m(x)$ is empty, and therefore $$n_m = 0 = m^2 \cdot 0 = m^2 \lVert x \rVert^2.$$

So let's suppose that $x$ is not the zero vector in $X$, and suppose also that $n_m \geq m^2 \lVert x \rVert^2$. Then as $$\left\lvert \left\langle x, e_k \right\rangle \right\rvert^2 \geq 0$$ for all $k \in \mathbb{N}$ and as $$\left\lvert \left\langle x, e_k \right\rangle \right\rvert^2 > \frac{1}{m^2}$$ for all $k \in A_m(x)$, so we note that \begin{align*} \sum_{k=1}^\infty \left\lvert \left\langle x, e_k \right\rangle \right\rvert^2 &= \sum_{k \in \mathbb{N} } \left\lvert \left\langle x, e_k \right\rangle \right\rvert^2 \\ &= \sum_{k \in A_m(x) } \left\lvert \left\langle x, e_k \right\rangle \right\rvert^2 + \sum_{k \in \mathbb{N} - A_m(x) } \left\lvert \left\langle x, e_k \right\rangle \right\rvert^2 \\ &\geq \sum_{k \in A_m(x) } \left\lvert \left\langle x, e_k \right\rangle \right\rvert^2 \\\ &> \frac{n_m}{m^2} \\ &\geq \frac{m^2 \lVert x \rVert^2 }{m^2} \\ &= \lVert x \rVert^2, \end{align*} which contradicts the Bessel's inequality. Hence we must have $$n_m < m^2 \lVert x \rVert^2,$$ as required.

In the above calculation, we have used the equality $$\sum_{k=1}^\infty \left\lvert \left\langle x, e_k \right\rangle \right\rvert^2 = \sum_{k \in \mathbb{N} } \left\lvert \left\langle x, e_k \right\rangle \right\rvert^2.$$ This is because of Theorem 3.55 in the book \emph{Principles of Mathematical Analysis} by Walter Rudin, 3rd edition, which says that if a series of complex numbers converges absolutely, then, by altering the order of the terms of that series in any way whatsoever, we obtain a series that also converges absolutely and has the same sum as the sum of the original series.

Is this proof correct? If so, then can we use the latter result to establish the former?

If not, then where have I erred in this proof? And, how to give an independent proof of the former result?

• Can't the countability of $S(x)$ be shown more directly using Bessel's inequality? For the left hand side of $\sum_{v \in M} \left\lvert \left\langle x, v \right\rangle \right\rvert^2$ to be finite, the number of non-zero terms needs to be countable. – md2perpe Jun 29 '17 at 12:46
• @md2perpe can you please present your argument in a detailed proof, using the notation similar to mine? What you say is intuitively clear, but I'm unable to come up with a rigorous way of presenting it. – Saaqib Mahmood Jun 29 '17 at 14:11

For a arbitrary set $A$ of non-negative numbers we define $$\sum_{a \in A} a = \sup \{ \sum_{a \in F} a \mid F \subset A \text{ finite} \}$$
For every finite (even countable) $F \subset M$ we know that $$\sum_{v \in F} | \langle x, v \rangle |^2 \leq \|v\|^2$$ and therefore also $$\sum_{v \in M} | \langle x, v \rangle |^2 = \sup_{F \text{ finite}} \sum_{v \in F} | \langle x, v \rangle |^2 \leq \|v\|^2$$
Now it's well-known that a sum of non-negative values can only be finite if the number of non-zero terms is at most countable. Otherwise there would be an infinite number of terms greater than $\epsilon$ for some $\epsilon>0$. But that would make the sum greater than $n \epsilon$ for every $n \in \mathbb N$, and so the sum would be infinite.
Let $\{ e_{\alpha} \}_{\alpha \in \Lambda}$ be an orthonormal subset of an inner product space $X$. Let $x\in X$ be given, and define $$E_n = \{ \alpha \in \Lambda : |(x,e_{\alpha})| \ge 1/n \},\;\; n=1,2,3,\cdots.$$ Then it is easy to argue that $$\bigcup_{n=1}^{\infty} E_n = \{ \alpha : (x,e_{\alpha}) \ne 0 \}.$$ Indeed, if $(x,e_{\alpha})\ne 0$ for some $\alpha$, then $|(x,e_{\alpha})| \ge 1/n$ for some $n$, which puts $\alpha \in E_n$ and, hence, in the union on the left; conversely, $E_n \subseteq \{ \alpha : (x,e_{\alpha}) \ne 0 \}$ clearly holds for all $n$. By Bessel's inequality, each set $E_n$ must empty or finite because $\{ \alpha_1, \alpha_2, \cdots, \alpha_N \}\subseteq E_n$ implies $\|x\|^2 \ge \sum_{k=1}^{N}|(x,e_{\alpha_k})|^2 \ge N/n$, which bounds $N$ by $n\|x\|^2$. So each $E_n$ is finite or empty, which makes their union empty, finite, or countably infinite.


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# Crossing number (knot theory)

Trefoil knot without 3-fold symmetry with crossings labeled.
A table of all prime knots with seven crossing numbers or fewer (not including mirror images).

In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot. It is a knot invariant.

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## Examples

By way of example, the unknot has crossing number zero, the trefoil knot three and the figure-eight knot four. There are no other knots with a crossing number this low, and just two knots have crossing number five, but the number of knots with a particular crossing number increases rapidly as the crossing number increases.

## Tabulation

Tables of prime knots are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant (this sub-ordering is not based on anything in particular, except that torus knots then twist knots are listed first). The listing goes 31 (the trefoil knot), 41 (the figure-eight knot), 51, 52, 61, etc. This order has not changed significantly since P. G. Tait published a tabulation of knots in 1877.[1]

Square knot (cr(6)) = trefoil (cr(3)) + trefoil reflection (cr(3)).

There has been very little progress on understanding the behavior of crossing number under rudimentary operations on knots. A big open question asks if the crossing number is additive when taking knot sums. It is also expected that a satellite of a knot K should have larger crossing number than K, but this has not been proven.

Additivity of crossing number under knot sum has been proven for special cases, for example if the summands are alternating knots[2] (or more generally, adequate knot), or if the summands are torus knots.[3][4] Marc Lackenby has also given a proof that there is a constant N > 1 such that ${\displaystyle {\frac {1}{N}}(\mathrm {cr} (K_{1})+\mathrm {cr} (K_{2}))\leq \mathrm {cr} (K_{1}+K_{2})}$, but his method, which utilizes normal surfaces, cannot improve N to 1.[5]

## Applications in bioinformatics

There are connections between the crossing number of a knot and the physical behavior of DNA knots. For prime DNA knots, crossing number is a good predictor of the relative velocity of the DNA knot in agarose gel electrophoresis. Basically, the higher the crossing number, the faster the relative velocity. For composite knots, this does not appear to be the case, although experimental conditions can drastically change the results.[6]

## Related invariants

There are related concepts of average crossing number and asymptotic crossing number. Both of these quantities bound the standard crossing number. Asymptotic crossing number is conjectured to be equal to crossing number.

Other numerical knot invariants include the bridge number, linking number, stick number, and unknotting number.

## References

1. ^ Tait, P. G. (1898), "On Knots I,II,III'", Scientific papers, 1, Cambridge University Press, pp. 273–347.
2. ^ Adams, Colin C. (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, Providence, RI: American Mathematical Society, p. 69, ISBN 9780821836781, MR 2079925.
3. ^ Gruber, H. (2003), Estimates for the minimal crossing number, arXiv:math/0303273, Bibcode:2003math......3273G.
4. ^ Diao, Yuanan (2004), "The additivity of crossing numbers", Journal of Knot Theory and its Ramifications, 13 (7): 857–866, doi:10.1142/S0218216504003524, MR 2101230.
5. ^ Lackenby, Marc (2009), "The crossing number of composite knots", Journal of Topology, 2 (4): 747–768, arXiv:0805.4706, doi:10.1112/jtopol/jtp028, MR 2574742.
6. ^ Simon, Jonathan (1996), "Energy functions for knots: Beginning to predict physical behavior", in Mesirov, Jill P.; Schulten, Klaus; Sumners, De Witt, Mathematical Approaches to Biomolecular Structure and Dynamics, The IMA Volumes in Mathematics and its Applications, 82, pp. 39–58, doi:10.1007/978-1-4612-4066-2_4.


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A boat crosses a river from port 𝐴 to port 𝐵, which are just on the opposite side. The speed of the water is 𝑉𝑊 and that of boat is 𝑉𝐵 relative to still water. Assume 𝑉𝐵 = 2𝑉𝑊. What is the time taken by the boat, if It has to cross the river directly on the 𝐴𝐵 line

Question : A boat crosses a river from port 𝐴 to port 𝐵, which are just on the opposite side. The speed of the water is 𝑉𝑊 and that of boat is 𝑉𝐵 relative to still water. Assume 𝑉𝐵 = 2𝑉𝑊. What is the time taken by the boat, if It has to cross the river directly on the 𝐴𝐵 line

(A) $\dfrac{2D}{V_{B}\sqrt{3}}$

(B) $\dfrac{\sqrt{3}D}{2V_{B}}$

(C) $\dfrac{D}{V_{B}\sqrt{2}}$

(D) $\dfrac{D\sqrt{2}}{V_{B}}$


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# Question #fec06

Apr 9, 2015

Rate enhancement would be about ${10}^{5}$ fold

The rate enhancement can be calculated if you know the temperature and the activation energy for both the catalyzed and uncatalyzed reactions.

$\text{rate enhancement} = {e}^{\frac{\left(\Delta {G}_{1} - \Delta {G}_{2}\right)}{R T}}$

$\text{rate enhancement} = {e}^{\frac{\left(30000 - 15000\right)}{R \left(273 + 25\right)}}$

$\text{rate enhancement} = {e}^{\frac{\left(285000\right)}{8.3145 \left(298\right)}}$

$\text{rate enhancement"=98963 " or about } {10}^{5}$

Notes
- The universal gas constant R is written with Joules - to obtain the correct answer, the activation energies given need to be converted from kJ

• I could have used 273.15 for my Celsius to Kelvin conversion (I simplified) but you must convert temperatures to Kelvin


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User Manual | Methods | Time Course Calculation | Stochastic Simulation

# Stochastic Simulation

### The Next-Reaction-Method

This stochastic simulation method utilizes the algorithm developed by Gibson and Bruck [Gibson00]. For each reaction a putative stochastic reaction time is calculated and the reaction with the shortest reaction time will be realized. The set of reactions is organized in a priority queue to allow for the efficient search for the fastest reaction. In addition, by using a so-called dependency graph only those reaction times are recalculated in each step, that are dependent on the reaction, which has been realized. This simulation method requires all the reactions to be irreversible. However, COPASI provides a tool, that converts all reversible reactions into irreversible ones. Because the algorithm internally works on discrete particle numbers rather than concentrations, the particle numbers in the system must not exceed a value of $2^{64}-1$ which is approximately $1.84 \cdot 10^{19}$.

The current implementation of the Next Reaction Method is very inefficient when the model contains assignment rules, which leads to extended calculation times.

There is a restriction regarding global quantities: If a differential equation is provided for a global quantity (the quantity is of type "ode") the model cannot be simulated stochastically with the current version of COPASI. If the global quantity is of type "assignment" stochastic simulation is possible but not as efficient as for models without assignments. No restrictions apply for "fixed" global quantities.

#### Options for the Next-Reaction-Method

Max Internal Steps
This parameter is a positive integer value specifying the maximal number of internal steps the integrator is allowed to take before the next desired reporting time. The default value is '1000000'.
Subtype
This parameter is ignored in the current version of COPASI.
Use Random Seed
This flag can be '0' or '1' and determines if the user-defined random seed should be used for the calculation. The default is '0' meaning that the random seed is set to a random value before each run and consecutively calculated trajectories will be different. If the value of this flag is set to '1', the user-defined random seed will be used and each calculated trajectory will be the same for the same value of the given random seed.
Random Seed
This unsigned integer is used as random seed in the calculations, if the flag Use Random Seed is set to '1'. The default value is '1'.

### The Direct-Method

This stochastic simulation method implements the Gillespie's direct method as described in [Gillespie76].

#### Options for the Direct-Method

Max Internal Steps
This parameter is a positive integer value specifying the maximal number of internal steps the integrator is allowed to take before the next desired reporting time. The default value is '1000000'.
Use Random Seed
This flag can be '0' or '1' and determines if the user-defined random seed should be used for the calculation. The default is '0' meaning that the random seed is set to a random value before each run and consecutively calculated trajectories will be different. If the value of this flag is set to '1', the user-defined random seed will be used and each calculated trajectory will be the same for the same value of the given random seed.
Random Seed
This unsigned integer is used as random seed in the calculations, if the flag Use Random Seed is set to '1'. The default value is '1'.

### The τ-leap Method

This stochastic simulation method implements the τ-leap method as described in [Gillespie01]

#### Options for the τ-leap Method

Max Internal Steps
This parameter is a positive integer value specifying the maximal number of internal steps the integrator is allowed to take before the next desired reporting time. The default value is '10000'.
Use Random Seed
This flag can be '0' or '1' and determines if the user-defined random seed should be used for the calculation. The default is '0' meaning that the random seed is set to a random value before each run and consecutively calculated trajectories will be different. If the value of this flag is set to '1', the user-defined random seed will be used and each calculated trajectory will be the same for the same value of the given random seed.
Random Seed
This unsigned integer is used as random seed in the calculations, if the flag Use Random Seed is set to '1'. The default value is '1'.


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# Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames

Mischa P. Woods1 and Álvaro M. Alhambra2

1Institute for Theoretical Physics, ETH Zurich, Switzerland
2Perimeter Institute for Theoretical Physics, Waterloo, Canada

### Abstract

Following the introduction of the task of $\textit{reference frame error}$ $\textit{correction}$ [1], we show how, by using reference frame alignment with clocks, one can add a continuous Abelian group of transversal logical gates to $any$ error-correcting code. With this we further explore a way of circumventing the no-go theorem of Eastin and Knill, which states that if local errors are correctable, the group of transversal gates must be of finite order. We are able to do this by introducing a small error on the decoding procedure that decreases with the dimension of the frames used. Furthermore, we show that there is a direct relationship between how small this error can be and how accurate quantum clocks can be: the more accurate the clock, the smaller the error; and the no-go theorem would be violated if time could be measured perfectly in quantum mechanics. The asymptotic scaling of the error is studied under a number of scenarios of reference frames and error models. The scheme is also extended to errors at unknown locations, and we show how to achieve this by simple majority voting related error correction schemes on the reference frames. In the Outlook, we discuss our results in relation to the AdS/CFT correspondence and the Page-Wooters mechanism.

Quantum error correction is the field that aims to devise schemes that protect quantum information as it is being processed in computations. This is a monumental task, which needs to overcome fundamental physical constraints unique to quantum physics such as the no-cloning theorem. The codes must then present different features that makes them both scalable and robust. A crucial one is the existence of "transversal gates", which are logical operations that can be implemented by acting separately on the different subsystems within the code.

Previously, a number of no-go theorems heavily restricted the possibility of such transversal gates, establishing that the set of them must in general be finite. Here, we show that one can in fact enhance these sets to continuous or infinite groups by appending a so-called "quantum clock" to the codes. This is only possible if one allows for a small error in the decoding procedure — an idea that had been overlooked in much of the previous literature. We show that the error here in fact quickly vanishes as the size and precision of the clocks increases, so that in principle it can be made as small as required in the different implementations. Quantum error correcting codes have recently found numerous applications in physics beyond their initial purpose, and we explain what our results could imply for these connections.

### ► References

[1] Patrick Hayden, Sepehr Nezami, Sandu Popescu, and Grant Salton. Error correction of quantum reference frame information. ArXiv:1709.04471, 2017. URL https://arxiv.org/abs/1709.04471.
arXiv:1709.04471

[2] Bei Zeng, Andrew Cross, and Isaac L Chuang. Transversality versus universality for additive quantum codes. IEEE Trans. Inf. Theory, 57 (9): 6272–6284, 2011. 10.1109/TIT.2011.2161917.
https://doi.org/10.1109/TIT.2011.2161917

[3] Sergey Bravyi and Robert König. Classification of topologically protected gates for local stabilizer codes. Phys. Rev. Lett., 110 (17): 170503, 2013. 10.1103/PhysRevLett.110.170503.
https://doi.org/10.1103/PhysRevLett.110.170503

[4] Fernando Pastawski and Beni Yoshida. Fault-tolerant logical gates in quantum error-correcting codes. Phys. Rev. A, 91 (1): 012305, 2015. 10.1103/PhysRevA.91.012305.
https://doi.org/10.1103/PhysRevA.91.012305

[5] Tomas Jochym-O'Connor, Aleksander Kubica, and Theodore J Yoder. Disjointness of stabilizer codes and limitations on fault-tolerant logical gates. Phys. Rev. X, 8 (2): 021047, 2018. 10.1103/PhysRevX.8.021047.
https://doi.org/10.1103/PhysRevX.8.021047

[6] Benjamin J Brown, Daniel Loss, Jiannis K Pachos, Chris N Self, and James R Wootton. Quantum memories at finite temperature. Rev. Mod. Phys., 88 (4): 045005, 2016. 10.1103/RevModPhys.88.045005.
https://doi.org/10.1103/RevModPhys.88.045005

[7] Bryan Eastin and Emanuel Knill. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett., 102 (11): 110502, 2009. 10.1103/PhysRevLett.102.110502.
https://doi.org/10.1103/PhysRevLett.102.110502

[8] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A, 71 (2): 022316, 2005. 10.1103/PhysRevA.71.022316.
https://doi.org/10.1103/PhysRevA.71.022316

[9] Emanuel Knill, Raymond Laflamme, and W Zurek. Threshold accuracy for quantum computation. ArXiv:quant-ph/9610011, 1996. URL https://arxiv.org/abs/quant-ph/9610011.
arXiv:quant-ph/9610011

[10] Héctor Bombín and Miguel Ángel Martín-Delgado. Topological computation without braiding. Physical review letters, 98 (16): 160502, 2007. 10.1103/PhysRevLett.98.160502.
https://doi.org/10.1103/PhysRevLett.98.160502

[11] Adam Paetznick and Ben W Reichardt. Universal fault-tolerant quantum computation with only transversal gates and error correction. Phys. Rev. Lett., 111 (9): 090505, 2013. 10.1103/PhysRevLett.111.090505.
https://doi.org/10.1103/PhysRevLett.111.090505

[12] Tomas Jochym-O'Connor and Raymond Laflamme. Using concatenated quantum codes for universal fault-tolerant quantum gates. Phys. Rev. Lett., 112 (1): 010505, 2014. 10.1103/PhysRevLett.112.010505.
https://doi.org/10.1103/PhysRevLett.112.010505

[13] Héctor Bombín. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes. New J. Phys., 17 (8): 083002, 2015. 10.1088/1367-2630/17/8/083002.
https://doi.org/10.1088/1367-2630/17/8/083002

[14] Theodore J Yoder, Ryuji Takagi, and Isaac L Chuang. Universal fault-tolerant gates on concatenated stabilizer codes. Phys. Rev. X, 6 (3): 031039, 2016. 10.1103/PhysRevX.6.031039.
https://doi.org/10.1103/PhysRevX.6.031039

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### Cited by

[1] Shishir Khandelwal, Maximilian P.E. Lock, and Mischa P. Woods, "Universal quantum modifications to general relativistic time dilation in delocalised clocks", Quantum 4, 309 (2020).

[2] Dong-Sheng Wang, Yun-Jiang Wang, Ningping Cao, Bei Zeng, and Raymond Laflamme, "Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes", New Journal of Physics 24 2, 023019 (2022).

[3] Dong‐Sheng Wang, "A comparative study of universal quantum computing models: Toward a physical unification", Quantum Engineering 3 4(2021).

[4] John Bostanci and Aleksander Kubica, "Finding the disjointness of stabilizer codes is NP-complete", Physical Review Research 3 4, 043192 (2021).

[5] Dong-Sheng Wang, "A prototype of quantum von Neumann architecture", Communications in Theoretical Physics 74 9, 095103 (2022).

[6] Sam Cree, Kfir Dolev, Vladimir Calvera, and Dominic J. Williamson, "Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted", PRX Quantum 2 3, 030337 (2021).

[7] Rhea Alexander, Si Gvirtz-Chen, and David Jennings, "Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system", New Journal of Physics 24 5, 053023 (2022).

[8] Sami Boulebnane, Mischa P. Woods, and Joseph M. Renes, "Waveform Estimation from Approximate Quantum Nondemolition Measurements", Physical Review Letters 127 1, 010502 (2021).

[9] Kun Fang and Zi-Wen Liu, "No-Go Theorems for Quantum Resource Purification: New Approach and Channel Theory", PRX Quantum 3 1, 010337 (2022).

[10] Yuxiang Yang, Yin Mo, Joseph M. Renes, Giulio Chiribella, and Mischa P. Woods, "Optimal universal quantum error correction via bounded reference frames", Physical Review Research 4 2, 023107 (2022).

[11] Sisi Zhou, Zi-Wen Liu, and Liang Jiang, "New perspectives on covariant quantum error correction", Quantum 5, 521 (2021).

[12] ChunJun Cao and Brad Lackey, "Approximate Bacon-Shor code and holography", Journal of High Energy Physics 2021 5, 127 (2021).

[13] Kyungjoo Noh, S. M. Girvin, and Liang Jiang, "Encoding an Oscillator into Many Oscillators", Physical Review Letters 125 8, 080503 (2020).

[14] Philippe Faist, Sepehr Nezami, Victor V. Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, and John Preskill, "Continuous Symmetries and Approximate Quantum Error Correction", Physical Review X 10 4, 041018 (2020).

[15] Bartosz Regula and Ryuji Takagi, "Fundamental limitations on distillation of quantum channel resources", Nature Communications 12 1, 4411 (2021).

[16] Linghang Kong and Zi-Wen Liu, "Near-Optimal Covariant Quantum Error-Correcting Codes from Random Unitaries with Symmetries", PRX Quantum 3 2, 020314 (2022).

[17] Aleksander Kubica and Rafał Demkowicz-Dobrzański, "Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem", Physical Review Letters 126 15, 150503 (2021).

[18] Dong-Sheng Wang, Guanyu Zhu, Cihan Okay, and Raymond Laflamme, "Quasi-exact quantum computation", Physical Review Research 2 3, 033116 (2020).

[19] Hyukjoon Kwon and M. S. Kim, "Fluctuation Theorems for a Quantum Channel", Physical Review X 9 3, 031029 (2019).

[20] Tamara Kohler and Toby Cubitt, "Toy models of holographic duality between local Hamiltonians", Journal of High Energy Physics 2019 8, 17 (2019).

[21] Philippe Faist, Mischa P. Woods, Victor V. Albert, Joseph M. Renes, Jens Eisert, and John Preskill, "Time-energy uncertainty relation for noisy quantum metrology", arXiv:2207.13707.

[22] Alexey Milekhin, "Quantum error correction and large N", SciPost Physics 11 5, 094 (2021).

[23] Hayata Yamasaki, Kosuke Fukui, Yuki Takeuchi, Seiichiro Tani, and Masato Koashi, "Polylog-overhead highly fault-tolerant measurement-based quantum computation: all-Gaussian implementation with Gottesman-Kitaev-Preskill code", arXiv:2006.05416.

[24] Ning Bao and Newton Cheng, "Eigenstate thermalization hypothesis and approximate quantum error correction", Journal of High Energy Physics 2019 8, 152 (2019).

[25] Mischa P. Woods and Michał Horodecki, "The Resource Theoretic Paradigm of Quantum Thermodynamics with Control", arXiv:1912.05562.

[26] Sami Boulebnane, Mischa P. Woods, and Joseph M. Renes, "Approximate quantum non-demolition measurements", arXiv:1909.05265.

The above citations are from Crossref's cited-by service (last updated successfully 2022-09-24 05:37:22) and SAO/NASA ADS (last updated successfully 2022-09-24 05:37:23). The list may be incomplete as not all publishers provide suitable and complete citation data.


text:
#### Vol. 13, No. 10, 2019

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
The construction problem for Hodge numbers modulo an integer

### Matthias Paulsen and Stefan Schreieder

Vol. 13 (2019), No. 10, 2427–2434
##### Abstract

For any integer $m\ge 2$ and any dimension $n\ge 1$, we show that any $n$-dimensional Hodge diamond with values in $ℤ∕mℤ$ is attained by the Hodge numbers of an $n$-dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of $n$-dimensional smooth complex projective varieties besides the ones induced by the Hodge symmetries, which answers a question raised by Kollár in 2012.

##### Keywords
Hodge numbers, Kähler manifolds, construction problem
##### Mathematical Subject Classification 2010
Primary: 32Q15
Secondary: 14C30, 14E99, 51M15


text:
# NerdTool over GeekTool

GeekTool went up on the Mac App Store on August 21, which explains why my various GeekTool posts have seen a jump in traffic.

I’ve been using NerdTool since I saw this Brett Terpstra post several months ago. I had to abandon it for a while on my MacBook Air when I upgraded to OS X 10.7, but I’m running it again now that Mutable Code has made it Lion-compatible.

Since GeekTool is free, and Brett seems to have gone back to it, I figured I’d run the two programs head-to-head to see how they compare. Based on a short test, I’ll keep using NerdTool, mainly because of this:

I suppose I shouldn’t be concerned about a hundred megabytes or so, but I’ve had trouble with GeekTool eating up memory in the past and just don’t want to worry about that again. For reference, both Tools were running my old Date and Time scripts and this set of Spotify scripts I wrote a couple of months ago.

Another point in NerdTool’s favor is its support for shadowed text. The Finder uses shadowed text for files on the Desktop and for good reason—it definitely makes the text easier to read.

GeekTool doesn’t have a checkbox to shadow the text, although I’ve found that you can edit an exported Geeklet file (it’s just a plist) and turn on shadowing. The code to look for is this:

xml:

Just change that false to a true and reimport the file to get shadowing. Unfortunately, GeekTool’s shadowing looks like shit with smaller fonts:


text:
If $g$ and $g\circ f$ are graded homomorphisms, must $f$ be graded?

Question

Let $A$ be a graded ring (always commutative with identity) and $M,N$ and $P$ be graded $A$-modules. Let $f:M \longrightarrow N$ and $g:N \longrightarrow P$ be $A$-module homomorphisms with $g$ being graded such that $g\circ f$ is graded. Is $f$ graded?

My thinking:

$\forall$ $m_k \in M_k$ with degree $k$. Therefore, $g\circ f(m_k) \in P_k$ since $g\circ f$ is graded. Suppose $f(m_k) = n_1 + \dots + n_r$ with $deg(n_i) = i$.

Therefore, $g\circ f(m_k) = g(n_1) + \dots + g(n_r)$. Because $g$ is graded, $deg(g(n_i)) = i$ for each $i$. We have $g(n_i) = 0$ if $i \neq k$.

That is, if $n_i \neq 0$ for some $i \neq k$ ,then $g(n_i) \neq 0$ (I think this statement has trouble...)

-
Is the main problem that $g$ may not be injective? – Peter Hu May 29 '12 at 5:35
I think that statement is not true definitely, but I can't find a concrete counterexample. – Peter Hu May 29 '12 at 5:40

The answer is negative: take $f$ arbitrary and $g=0$.