text:
Kazuho Fujii

## 1. Introduction

Storage tiering is a technique to create a high performance and cheap storage by combining several kinds of storage devices. Tiered storage system allocates frequently accessed data to high performance but expensive devices, and it allocates rarely accessed data to cheaper but slower device. The system collects access statistics and sometimes optimizes data allocation. For example, SSD, SAS HDD and SATA HDD are usually used for each tiers.

Caching is a similar technique with tiering. Caching duplicates data from poor performance devices to high performance device when the data is accessed. Compared to tiering, caching reallocates data more frequently. Additionally we can use volatile memory for caching tiers such as RAM.

Only kinds of devices do not make storage tiers. Tiers can be made with data formats (e.g., compressed data or raw data) and data positions (e.g., over the network or not). All schemes of saving data can be regarded as tiers. That is, storage tiering is not technique only for storage systems; all computer system is related to it.

Most of tiering technologies concern about data replacement. It is a popular problem which data should be on high performance storage. Many schemes are proposed that predict the next accessed data and enhance probability of "cache hit". LRU (least recently used) method and LFU (least frequently used) method are well known.

In contrast constitution of tiers has not been concerned. This is because configuration of tiers was static; it could not be changed flexibly. However, vitalization is general technology now. We can create virtual tiered storage consisted as we want.

Although there are many kinds of storage devices (including data format), it is unclear what kinds of devices we should combine or how we should decide the ratio of devices. If we can know them, we can design effective storage.

In this report, I theoretically derive the optimum configuration of tiered storage. The rest of this report is organized as follows: Section 2 setups the analyzed system; Section 3 introduces the assumption of the Pareto distribution access; Section 4 calculates the mean performance of storage under the Pareto distribution access; Section 5 derives theoretically optimized configuration of tiered storage; Section 6 discusses about the condition of devices for effective tiering; Section 7 describes the conclusion.

## 2. Setup

Now I think about creating tiered storage with $N$ kinds of recording devices respectively named Device-$1$, Device-$2$, $\cdots$, Device-$N$. Device-$1$ has the lowest performance but it is the most cost-effective. Device-$N$ has the highest performance but it is the most expensive.

I introduce performance values of devices. The performance value of Device-$i$ is expressed with $q_i$. Then, $0 < q_1 < q_2 < \cdots < q_N$. The performance value of a device represents any performance characteristic of the device (e.g., random read, sequential read) with a single value. Of course measuring this value is difficult, but defining it is convenient for a theoretical discussion.

Similarly I express the cost per a unit capacity of Device-$i$ with $c_i$. Then, $0 < c_1 < c_2 < \cdots < c_N$. Caching copy data from a device to another device. In this case $c_i$ includes costs of both of the devices.

I express the capacity ratio of Device-$i$ to the total capacity of the volume with $r_i$. By definition, \begin{align} \sum_{i=1}^N r_i = 1. \label{sum_ratio} \end{align} The mean cost per a unit capacity $\bar{c}$ is \begin{align} \bar{c}=\sum_{i=1}^N r_i c_i. \label{mean_cost} \end{align}

The goal is to get the optimum values of $r_i$. Values $q_i, c_i$ are given as catalog specifications of storage devices. And $\bar{c}$ is given as a budget. Hence, the required values are $r_i$ where the tiered storage get the highest performance.

## 3. Pareto Distribution

The optimum values of $r_i$ are different with access pattern to the storage. We have to assume a probability distribution of access to blocks of the storage.

Here, I assume that the access obeys the Pareto distribution. It is sometimes called the law of 80:20 or the Zipf's law. Under the Pareto distribution just few data are accessed mostly and most of data are rarely accessed. This distribution can express "hot spots" and "long tails". Tiering is effective in such cases.

In the Pareto distribution, the probability $F$, where the number of access to a block is not more than $x$, is \begin{align} F = 1 - \left(\frac{x_0}{x}\right)^a,\ (a > 1,\ x_0 > 0). \end{align} $a$ and $x_0$ are parameters of the probability distribution. $a$ means bias of access. It is called Pareto coefficient. As $a \approx 1$ access is concentrated to a single block; as $a \to \infty$ access is uniformly distributed, that is the number of access at each block is $x_0$.

The probability density function $f$ is \begin{align} f := \frac{\partial F}{\partial x} = \frac{a x_0^a}{x^{a+1}}. \end{align}

## 4. Mean Performance

For the next, I calculate the mean performance of the storage under the Pareto distribution access pattern.

I assume that the tiering is perfect. That is the higher accessed block is always allocated to the higher performance device. Let Device-$i$ to have blocks which access count $x$ is $x_{i-1} \leq x < x_i$. Note that $x_N = \infty$. Then, the relationship between $r_i$ and $x_i$ is \begin{align} r_1+r_2+\cdots+r_i=1-\left(\frac{x_0}{x_i}\right)^a. \end{align}

The number of access to the tier of Device-$i$ equals $\int_{x_{i-1}}^{x_i} x f dx$, and the total number of access to the storage equals $\int_{x_0}^{\infty} x f dx$. Therefore, the probability where an access is to the tier of Device-$i$ is \begin{align} p_i &= \frac{\displaystyle{\int_{x_{i-1}}^{x_i} x f dx}}{\displaystyle{\int_{x_0}^{\infty} x f dx}} \nonumber\\ &= \left(1-r_1-\cdots-r_{i-1}\right)^\frac{a-1}{a} -\left(1-r_1-\cdots-r_i\right)^\frac{a-1}{a}. \end{align} Consequently, I can get the mean performance of tiered storage $\bar{q}$. That is \begin{align} \bar{q}&=\sum_{i=1}^N p_i q_i \nonumber\\ &=\sum_{i=1}^N \left\{ \left(1-r_1-\cdots-r_{i-1}\right)^\frac{a-1}{a} -\left(1-r_1-\cdots-r_i \right)^\frac{a-1}{a} \right\} q_i. \label{mean_quality} \end{align}

## 5. The Optimum Ratio of Devices

In this section I derive the optimum values of $r_i$. Now I want to derive the ratio of each drives $r_1, r_2, \cdots, r_N$ which maximizes the mean performance value $\bar{q}$ under the constraints \eqref{sum_ratio} \eqref{mean_cost}. I can use the method of Lagrange multiplier for this problem. With multiplier $\mu$ and $\lambda$, I define a function $g$ as \begin{align} g := \bar{q} - \lambda \left( \bar{c} - \sum_{i=1}^{N} r_i c_i \right) -\mu \left( 1- \sum_{i=1}^N r_i \right). \end{align} I can obtain $r_1, r_2, \cdots ,r_N$ where $\bar{q}$ get an extreme value by solving the simultaneous equations of \eqref{sum_ratio} \eqref{mean_cost} and following $N$ equations: \begin{align} \frac{\partial g}{\partial r_i} = 0 ,\ (i = 1, 2, \cdots , N). \label{lagrange_theory} \end{align}

By solving the simultaneous equations \eqref{sum_ratio} \eqref{mean_cost} \eqref{mean_quality} \eqref{lagrange_theory}, I can get the optimum ratio of each device $r_1,r_2,\cdots,r_N$ as following: \begin{align} r_1&=(\bar{c}-c_1)\left\{ \sum_{i=1}^{N-1}\Delta c_i \left(\frac{\Delta c_i}{\Delta q_i} \right)^{-a}\right\}^{-1} \left\{ - \left(\frac{\Delta c_1}{\Delta q_1} \right)^{-a} \right\}+1, \label{r1_c}\\ r_i&=(\bar{c}-c_1)\left\{ \sum_{i=1}^{N-1}\Delta c_i \left(\frac{\Delta c_i}{\Delta q_i} \right)^{-a}\right\}^{-1} \left\{ \left(\frac{\Delta c_{i-1}}{\Delta q_{i-1}} \right)^{-a} - \left(\frac{\Delta c_i}{\Delta q_i} \right)^{-a} \right\}, \label{ri_c}\\ r_N&=(\bar{c}-c_1)\left\{ \sum_{i=1}^{N-1}\Delta c_i \left(\frac{\Delta c_i}{\Delta q_i} \right)^{-a}\right\}^{-1} \left\{ \left(\frac{\Delta c_{N-1}}{\Delta q_{N-1}} \right)^{-a} \right\}, \label{rN_c} \end{align} where \begin{align} \Delta c_i &:= c_{i+1} - c_i, \\ \Delta q_i &:= q_{i+1} - q_i. \end{align} Note that $r_i$ in the equation \eqref{ri_c} are in cases of $i = 2, 3, \cdots, N-1$. In this case, $\bar{q}$ gets the maximum value \begin{align} \bar{q} = q_1 + \left(\bar{c}-c_1\right)^\frac{a-1}{a} \left\{\sum_{i=1}^{N-1} \Delta c_i \left(\frac{\Delta c_i}{\Delta q_i}\right)^{-a} \right\}^{-\frac{a-1}{a}} \left\{\sum_{j=1}^{N-1} \Delta q_j \left(\frac{\Delta c_j}{\Delta q_j}\right)^{-a+1} \right\}. \end{align}

## 6. Discussion

Seeing equations \eqref{r1_c} \eqref{ri_c} \eqref{rN_c}, $x_0$ doesn't appear but only the Pareto coefficient $a$ appears. Therefore, we have to decide only $a$ when assuming the probability distribution. It differs in situations where storage used.

Discussing the condition where $r_i > 0$, we can know which devices we should select.

The condition where $r_1 > 0$ in the equation \eqref{r1_c} is \begin{align} \bar{c} < c_1 + \left(\frac{\Delta c_1}{\Delta q_1}\right)^a \sum_{i=1}^{N-1}\Delta c_i \left(\frac{\Delta c_i}{\Delta q_i}\right)^{-a}. \label{over_budget} \end{align} This means that the lowest performance device has no effect when having enough budget. In this case we should remove the lowest performance device and apply equations \eqref{r1_c}\eqref{ri_c}\eqref{rN_c} again.

The condition where $r_i > 0\ (i = 2, 3, \cdots, N-1)$ in the equation \eqref{ri_c} is \begin{align} \frac{\Delta c_{i-1}}{\Delta q_{i-1}} < \frac{\Delta c_i}{\Delta q_i}. \label{satuation} \end{align} This condition means that points $(c_i, q_i)$ must be on a convex curve in a graph where $x$ and $y$ axes are respectively cost and performance. We can not get effective tiered storage if we combine drives that do not meet this condition. The combination of SSD, SAS HDD and SATA HDD might not meet this condition because SSD becomes cheaper.

The equation \eqref{rN_c} always satisfies $r_N > 0$. Hence, the highest performance device always has effect. When a higher performance device newly appears, we should use it.

## 7. Conclusion

In this report the optimum ratio of devices are derived that maximize a tiered storage performance. It can be applied to designing tiered storage or developing a software that automatically adjusts tiering system configurations.

Additionally the fact is cleared that kinds of devices in tiers require some conditions for effective performance. Some tiered storage systems might take a penalty by unsatisfying the conditions.

## Appendix: The Case Provided a Performance Requirement

In the case where a required performance value $\bar{q}\ (q_1 \leq \bar{q} \leq q_N)$ is provided, we can get the optimum ratio with the same method. It is the ratio $r_1,r_2,\cdots,r_N$ which minimize mean costs $\bar{c}$ under the constraints \eqref{sum_ratio} \eqref{mean_quality}.

The result is as follows: \begin{align} r_1 &= \left(\bar{q}-q_1\right)^\frac{a}{a-1} \left\{\sum_{i=1}^{N-1}\Delta q_i\left(\frac{\Delta c_i}{\Delta q_i}\right)^{-a+1}\right\}^{-\frac{a}{a-1}} \left\{-\left(\frac{\Delta c_1}{\Delta q_1}\right)^{-a}\right\} + 1, \label{r1_q} \\ r_i &= \left(\bar{q}-q_1\right)^\frac{a}{a-1} \left\{\sum_{i=1}^{N-1}\Delta q_i\left(\frac{\Delta c_i}{\Delta q_i}\right)^{-a+1}\right\}^{-\frac{a}{a-1}} \left\{\left(\frac{\Delta c_{i-1}}{\Delta q_{i-1}}\right)^{-a} -\left(\frac{\Delta c_i}{\Delta q_i}\right)^{-a}\right\}, \label{ri_q}\\ r_N &= \left(\bar{q}-q_1\right)^\frac{a}{a-1} \left\{\sum_{i=1}^{N-1}\Delta q_i\left(\frac{\Delta c_i}{\Delta q_i}\right)^{-a+1}\right\}^{-\frac{a}{a-1}} \left\{\left(\frac{\Delta c_{N-1}}{\Delta q_{N-1}}\right)^{-a}\right\}. \label{rN_q} \end{align} In this case, the mean cost gets the minimum value \begin{align} \bar{c} = c_1 + \left(\bar{q}-q_1\right)^\frac{a}{a-1} \left\{\sum_{i=1}^{N-1}\Delta q_i\left( \frac{\Delta c_i}{\Delta q_i}\right)^{-a+1}\right\}^{-\frac{a}{a-1}} \left\{\sum_{j=1}^{N-1}\Delta c_j \left( \frac{\Delta c_{j}}{\Delta q_{j}}\right)^{-a}\right\}. \end{align}


text:
# 장기 노출 $SO_2$ 간이 샘플러 개발에 관한 연구

• 이동인 (부산수산대학교 대기과학과)
• Published : 1993.09.01

#### Abstract

The concentrations of $SO_2$ and $SO_3$ were measured to estimate a new developed long time exposure $SO_2$ sampler at Onsan industrial area considering the meteorological factors from June to October, 1992. The mean concentration of $SO_3$ by $PbO_2$ method was 0.924 mg $SO_3 / 10cm^2$ $PbO_2$/day and their high values were shown in the center of the industrial area, which show potential pollution due to the increase of industrial activities and micrometeorological factors in and around the sites. As a result of statistical correlation between $SO_2$ concentration by new sampling method and $SO_3$ concentration by $PbO_2$ method in July and August, 1992, correlation coefficients were high (r=0.87, 0.91) and shown more than 0.83 value in the high concentration data set, which was arbitrarily divided into 7~10${\mu}l$$SO_2$ concentration in an attempt to further investigate these relationships. Therefore, use of new developed long time exposure TEX>$SO_2$ sampler is good for TEX>$SO_2$ measurement and valuable for estimation of air quality in the urban and industrial area. Key Words : a new developed long time exposure TEX>$SO_2$ sampler, correlation coefficients, high, $SO_2$ measurement, estimation of air Quality.

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text:
Ab initio with multiple metal ions

20 posts / 0 new
Ab initio with multiple metal ions
#1

Hi everyone,

I am a beginner in Rosetta and I am wondering if Rosetta could do ab initio folding of a metalloprotein with multiple metal ions.

I read the article by Wang et al.,(Prediction of structures of zinc-binding proteins through explicit modeling of metal coordination geometry, 2010, Protein Science) and was able to set constraints for one metal binding site in residue_pair_jump_cst file and run AbinitioRelax without any issues. However I have no idea how to set things up when two or more metal ions are involved. Do I also have to define every jump between those metal ions?

I would be grateful if someone could help.

Post Situation:
Sun, 2013-12-29 17:58
cheyuk

From best I can tell, things should be able to handle multiple zinc atoms.

For your residue_pair_jump_cst file, simply give a block (each block starts with a BEGIN line and ends with an END line) for each zinc atom you want to model. You have to repeat the geometry lines, but that should be more-or-less a simple copy and paste.

You'll likely also have to update any other references to zinc (e.g. in the -constraints:cst_file and -constraints:cst_fa_file) so that you have a set of references to each zinc atom (each to their own CYS residue).

Tue, 2013-12-31 09:35
rmoretti

I ran the AbinitioRelax application on a metallothionein (PDB code: 2L62) with the input files as attached. Just want to see if Rosetta could reproduce the native fold. The fasta file:

>2L62
GSGC[CYZ]DDKC[CYZ]GC[CYZ]AVPC[CYZ]PGGTGC[CYZ]RC[CYZ]TSARZ[ZN]Z[ZN]

This protein has 26 aa and I appended two ZN ions to its fasta file. Constraints files were modified accordingly. However, I met the following error:

protocols.abinitio.AbrelaxApplication: read fasta sequence: 66 residues
GSGC[CYZ]DDKC[CYZ]GC[CYZ]AVPC[CYZ]PGGTGC[CYZ]RC[CYZ]TSARZ[ZN]Z[ZN]
protocols.evaluation.ChiWellRmsdEvaluatorCreator: Evaluation Creator active ...
core.chemical.ResidueTypeSet: Finished initializing centroid residue type set. Created 1981 residue types
core.io.fragments: reading fragments from file: 2L62_09_05.200_v1_3 ...
core.io.fragments: rosetta++ fileformat detected! Calling legacy reader...
core.fragments.ConstantLengthFragSet: finished reading top 25 9mer fragments from file 2L62_09_05.200_v1_3
core.io.fragments: reading fragments from file: 2L62_03_05.200_v1_3 ...
core.io.fragments: rosetta++ fileformat detected! Calling legacy reader...
core.fragments.ConstantLengthFragSet: finished reading top 200 3mer fragments from file 2L62_03_05.200_v1_3
core.scoring.constraints.util: Constraint choice: 2L62.cen_cst
core.fragment: compute strand/loop fractions for 28 residues...
protocols.jumping: read ResiduePair jump-definitions from 2L62.residue_pair_jump_cst
core.chemical.ResidueTypeSet: Finished initializing fa_standard residue type set. Created 9281 residue types
core.scoring.ScoreFunctionFactory: SCOREFUNCTION: score13_env_hb
core.scoring.etable: Starting energy table calculation
core.scoring.etable: smooth_etable: changing atr/rep split to bottom of energy well
core.scoring.etable: smooth_etable: spline smoothing lj etables (maxdis = 6)
core.scoring.etable: smooth_etable: spline smoothing solvation etables (max_dis = 6)
core.scoring.etable: Finished calculating energy tables.
basic.io.database: Database file opened: scoring/score_functions/PairEPotential/pdb_pair_stats_fine
basic.io.database: Database file opened: scoring/score_functions/hbonds/sp2_elec_params/HBPoly1D.csv
basic.io.database: Database file opened: scoring/score_functions/hbonds/sp2_elec_params/HBEval.csv
basic.io.database: Database file opened: scoring/score_functions/P_AA_pp/P_AA
basic.io.database: Database file opened: scoring/score_functions/P_AA_pp/P_AA_n
basic.io.database: Database file opened: scoring/score_functions/P_AA_pp/P_AA_pp
basic.io.database: Database file opened: scoring/score_functions/rama/Rama_smooth_dyn.dat_ss_6.4
core.pack.dunbrack: Dunbrack 2010 library took 0.31 seconds to load from binary
Segmentation fault (core dumped)

Then I ran Rosetta in debug mode and saw the following error:

protocols.abinitio.AbrelaxApplication: read fasta sequence: 66 residues
GSGC[CYZ]DDKC[CYZ]GC[CYZ]AVPC[CYZ]PGGTGC[CYZ]RC[CYZ]TSARZ[ZN]Z[ZN]
protocols.evaluation.ChiWellRmsdEvaluatorCreator: Evaluation Creator active ...
core.chemical.ResidueTypeSet: Finished initializing centroid residue type set. Created 1981 residue types
core.io.fragments: reading fragments from file: 2L62_09_05.200_v1_3 ...
core.io.fragments: rosetta++ fileformat detected! Calling legacy reader...
core.fragments.ConstantLengthFragSet: finished reading top 25 9mer fragments from file 2L62_09_05.200_v1_3
core.io.fragments: reading fragments from file: 2L62_03_05.200_v1_3 ...
core.io.fragments: rosetta++ fileformat detected! Calling legacy reader...
core.fragments.ConstantLengthFragSet: finished reading top 200 3mer fragments from file 2L62_03_05.200_v1_3
core.scoring.constraints.util: Constraint choice: 2L62.cen_cst
core.fragment: compute strand/loop fractions for 28 residues...
protocols.jumping: read ResiduePair jump-definitions from 2L62.residue_pair_jump_cst
core.chemical.ResidueTypeSet: Finished initializing fa_standard residue type set. Created 9123 residue types
core.scoring.ScoreFunctionFactory: SCOREFUNCTION: score13_env_hb
core.scoring.etable: Starting energy table calculation
core.scoring.etable: smooth_etable: changing atr/rep split to bottom of energy well
core.scoring.etable: smooth_etable: spline smoothing lj etables (maxdis = 6)
core.scoring.etable: smooth_etable: spline smoothing solvation etables (max_dis = 6)
core.scoring.etable: Finished calculating energy tables.
basic.io.database: Database file opened: scoring/score_functions/PairEPotential/pdb_pair_stats_fine
basic.io.database: Database file opened: scoring/score_functions/hbonds/sp2_elec_params/HBPoly1D.csv
basic.io.database: Database file opened: scoring/score_functions/hbonds/sp2_elec_params/HBEval.csv
basic.io.database: Database file opened: scoring/score_functions/P_AA_pp/P_AA
basic.io.database: Database file opened: scoring/score_functions/P_AA_pp/P_AA_n
basic.io.database: Database file opened: scoring/score_functions/P_AA_pp/P_AA_pp
basic.io.database: Database file opened: scoring/score_functions/rama/Rama_smooth_dyn.dat_ss_6.4
core.pack.dunbrack: Dunbrack 2010 library took 0.56 seconds to load from binary
AbinitioRelax.linuxgccdebug: src/utility/pointer/owning_ptr.hh:225: T* utility::pointer::owning_ptr::operator->() const [with T = protocols::jumping::ResiduePairJump]: Assertion p_ != 0' failed.
Aborted (core dumped)

Looks like something wrong in ResiduePairJump? Anyone could help with this? Thanks in advance.

Wed, 2014-01-01 18:07
cheyuk

Unfortunately, there's not much to go on there, and it's a little difficult to debug locally without the fragment files.

Could you run gdb and get a backtrace? To do this, run "gdb AbinitioRelax.linuxgccdebug". Then at the gdb prompt, do a "catch throw", followed by a "run ". The program will run for a while, and then halt at the prompt. If it halts before the place where you saw the error (frequently in the hbond lines), do a "continue". Once it prints the "Dunbrack 2010 library took XXX seconds to load from binary" line and halts, do a "backtrace", and post the resulting lines here. You can then do a "quit" to exit the gdb environment.

Alternatively, if you could upload the fragment files to some (publicly accessible) external site (something like Google Drive or Dropbox), and post or PM me a link, I can download them and test the issue myself. (Include the complete commandline that you used to run things - I'm assuming it might be "AbinitioRelax.linuxgccdebug @flags", though differences might change things.)

Also, which version of Rosetta are you using? Is it one of the weekly releases?

Thu, 2014-01-02 09:33
rmoretti
Thanks rmoretti. Now I attached all the files including fragment files in dropbox: https://www.dropbox.com/s/kcxx0rlw08c8hpp/2L62.zip The command line I used is ~/rosetta/main/source/bin/AbinitioRelax.linuxgccdebug @flags -database ~/rosetta/main/database" The Rosetta version is 2013wk50. Another thing I noticed is that the Robetta server will replace the residue name Z (which refers to ZN in the fasta file) to Q (Gln?). This is observed both in the fasta file (though I don't need it to run Abinitio) and psipred_ss2 file. Will this cause any issue?
Thu, 2014-01-02 17:38
cheyuk

I ran gdb and got the following backtrace:

(gdb) backtrace
#0 0x000000329c2328e5 in raise () from /lib64/libc.so.6
#1 0x000000329c2340c5 in abort () from /lib64/libc.so.6
#2 0x000000329c22ba0e in __assert_fail_base () from /lib64/libc.so.6
#3 0x000000329c22bad0 in __assert_fail () from /lib64/libc.so.6
#4 0x00007ffff465a37b in utility::pointer::owning_ptr::operator-> (this=0x7fffffffc530)
at src/utility/pointer/owning_ptr.hh:225
this=0x2ba2ff0, fname="2L62.residue_pair_jump_cst")
at src/protocols/jumping/ResiduePairJumpSetup.cc:91
#6 0x00007ffff452f1d3 in protocols::abinitio::AbrelaxApplication::setup_jumps
(this=0x7fffffffdf20, extended_pose=...)
at src/protocols/abinitio/AbrelaxApplication.cc:1370
#7 0x00007ffff452fe86 in protocols::abinitio::AbrelaxApplication::setup_fold (
this=0x7fffffffdf20, extended_pose=..., prot_ptr=...)
at src/protocols/abinitio/AbrelaxApplication.cc:1468
#8 0x00007ffff4537b6d in protocols::abinitio::AbrelaxApplication::run (
this=0x7fffffffdf20) at src/protocols/abinitio/AbrelaxApplication.cc:2156
#9 0x000000000040307c in main (argc=2, argv=0x7fffffffe158)
at src/apps/public/AbinitioRelax.cc:57

Thu, 2014-01-02 17:52
cheyuk

It looks like your first issue is the blank line between the BEGIN and END blocks in 2L62.residue_pair_jump_cst - removing that fixes the error that you saw.

You get an additional error later on, though. This is due to the foldtree setup. It's (incorrectly) thinking that there's going to be a polymeric bond between the two zincs. I think this can be solved by changing the cutpoint range of the second residue from "26 26" to "27 27". (Basically, tell Rosetta you want to break the chain between the two zincs.)

But we're not done yet. There's a third error. Now it's trying to insert fragments across the zincs, which involves changing their backbone coordinates. (Yeah, that works about as well as it sounds.). The issue here is that it looks like you generated fragment files with the full 29 residue sequence, including the two zincs. I'd recommend just generating fragments for just the protein portion of the complex (or equivalently, deleting off the last two positions on the current fragment files), which should work as long as the non-protein portion is at the very end of the pose.

Those three fixes result in a successful run with a nice compact structure, with the two zincs near their cysteines. It looks nothing like 2L62, but who can say what will happen over a tens to hundreds of thousands output structure production run.

Also, I'm not sure why things are changing your Z's to Q's, (was this for fragment generation? If so, the reason would be the fragment generator doesn't know about non-protein residues), but it's likely to be an issue if you use that output for input later on. If you see that happen, you'll want to change it back. The single letter code of the residue should match the one given by the IO_STRING line of the residue type params file in the database (which for zinc is Z).

Fri, 2014-01-03 09:01
rmoretti

Thanks for your reply. A follow-up question, is it necessary and feasible to set up any constraints between zinc atoms in those constraints files?

Wed, 2014-01-01 00:56
cheyuk

You can certainly try it without the 2L62.cen_cst and 2L62.fa_cst files, though I'm guessing it won't go as well as with them. In the residue_pair_jump_file there's some specification of geometry, but I'm not sure in how many stages of the protocol that carries through. Adding the constraints to specify the geometry makes sure that the desired geometry is enforced in more locations.

An additional note regarding scoring: In your flags files your explicitly calling for the "score13_env_hb" weights file. Along with the switch from 3.5 to weekly releases, we also did some significant changes to the way the scorefunction works. Using "-score:weights score13_env_hb" with the weekly releases isn't going to work the same as it would with 3.5 and earlier. I'd recommend doing one of two things. The first is just remove the -score:weights line and use the new default scorefunction (talaris2013), or alternatively add the flag "-restore_pre_talaris_2013_behavior" to your flags file to reverse the changes made, going back to (mostly) what was the 3.5 scorefunction behavior.

Fri, 2014-01-03 09:10
rmoretti

Thanks so much for your help. Now it is working all right. I also tried this protocol with four metal ions and it worked just as well.

Only one small question: Does Rosetta strictly comply with the constraints we provide? When violations of constraints occur (Not all constraints can be satisfied), will Rosetta still outputs decoys as best it can?

Sun, 2014-01-05 17:55
cheyuk

"Constraints" in Rosetta are more properly called "restraints" in the Molecular Dynamics world. That is, Rosetta "constraints" aren't strictly enforced by Rosetta, instead there's just an energy penalty when they're violated. So Rosetta will indeed output decoys the best it can when not all constraints can be satisfied at the same time.

You can see this in the scorefile output. Normally there will be columns corresponding to the various constraint scoreterms in the scorefile. If one or more of the "constraints" are "violated", then those scoreterms will have a non-zero energy. Typically this isn't too bad in most cases (less than an REU per constraint), but when you start getting larger values, then you know something is off about the structure. Often you'll get a mix of high and low constraint values, and you can simply ignore/discard those decoys which have bad constraint violations.

If you expect that only a subset of your constraints will be valid at any one time, you may want to look into AmbigiousConstraints and KofNConstraints, which allow you to pick a certain best scoring subset of constraints.

Mon, 2014-01-06 07:21
rmoretti

Thanks a million for your help, rmoretti. Your explanations are so clear and spot-on. I really appreciate it.

Mon, 2014-01-06 19:12
cheyuk

Hello everyone,

how can I generate fragments with the zinc-atom?! I believe Robetta dont know the Z at the end of the sequence. For the 1dsv-metallo-protein(30 amino acids), which is given in Rosetta, I calculated structures with rmsd-values between 6-12 . Is it normal or could sometime wrong with my structures?!

What mean the values from the 1dsvA.cen_cst file?!

AtomPair CB 5 ZN 32 HARMONIC 3.2 0.2 (CYS)
AtomPair CB 8 ZN 32 HARMONIC 3.2 0.2 (CYS)
AtomPair CB 13 ZN 32 HARMONIC 3.5 0.2 (HIS)
AtomPair CB 18 ZN 32 HARMONIC 3.2 0.2 (CYS)

For Cystein it could be the distance between CB and Zn. But for Histidin I think the value is to small. Or are there radians?!

In the 1dsvA.residue_pair_jump_cst file stand something from A and B. What are this for parameters?!

disAB: 2.20
angleA: 68.0
angleB: 70.5
dihedralA: -150.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0
dihedralAB: -150.0 -120.0 -90.0 -60.0 -30.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0
dihedralB: 120.0
END

Thanks

Fri, 2014-11-28 11:37
masterofpuppets

As long as your zinc atom is at the very end of a pose (or more precisely, so long as the protein portion you wish to generate fragments for is all a the beginning of the pose), you can just simply generate the fragments for the protein portion. (That is, give Robetta just the protein sequence without the zinc residue.) Most protocols which use fragments should work if you have "extra" residues at the end of the pose which aren't in the fragments file, which will then be ignored for fragment insertion portions of the run. (Which is what you probably want for your zinc residue.)

The 1dsvA.cen_cst file is a centroid (low resolution) constraints file. Here you're specifying extra terms to add to the scoring function. Particularly, you're saying that the distance between atom pairs involving the ZN atom on residue 32 and the CB atoms on residues 5, 8, 13, and 18 should be restrained. (AtomPair is a distance constraint: https://www.rosettacommons.org/docs/latest/constraint-file.html ) HARMONIC and what follows tells how to do that - you want a harmonic (parabolic) restraint, centered on either 3.2 or 3.5 Ang, with a deviation parameter of 0.2 Ang. (See the constraint file documentation ) for details, but basically the smaller the value the more closely Rosetta will require that distance to be. -- I think you may be right about the histidine distance. One limitation is that in centroid mode you don't have individual sidechain atoms besides CB, so you have to approximate distances.

For 1dsvA.residue_pair_jump_cst, the A and B are the two sides of the jump that are being sampled. A being the first (fixed) side, whereas B is the second (movable) side. The values listed are the ones which specify what the allowable values for the respective parameters are. So in this case you exactly specify the distance, the two angles, and the dihedral on the zinc side of the bond, but allow the dihedral on the protein side and the one around the "bond" connecting the two sides to vary in 30 degree increments.

Sun, 2014-11-30 11:17
rmoretti

Thank you.
I find the 3.5 value for histidin in the 1dsv example from Rosetta. It might be that the value must be 5.3?!
A stands for the first aminoacid, from which the jump is definded and B stands for the other three residues?! I dont understand which angles are concretely that should be.

Sat, 2015-01-03 09:43
masterofpuppets

The 3.5 value probably works in that situation because that's what the value is in the experimental structure for that particular system. A different system may need a more general/loose value to match up with reality.

What A and B are depend on your setup. In this case, I believe that the A residue is your amino acid, and the B residue is the metal ion. So distanceAB is the distance between the two atoms, one in A and one in B, which connect the two residues. (Note that this is the logical connection, rather than any physical/chemical connection.) The angleA is the angle on the "A" side of that connection, so two atoms on the amino acid (the connection atom and the "parent" of the connection atom), and the connection atom on the metal. The angleB is the other way around. (The metal ion residue has "virtual" atoms for orientation, which are the ones used for the second atom on the metal ion residue.) dihedralAB is the dihedral for the "bond" between the two residues, so one where there are two atoms from each residue. dihedralA is the dihedral where there are three atoms on the "A" residue (the connection atom, the parent of the connection atom, and the parent of the parent) and one (the connection atom) from the metal ion. dihedralB is the other way around - which explains why it's not really sampled. It's effectively just rotating the virtual atoms on the metal ion.

If your mental picture of it isn't too good, you might want to dust off your old organic chemistry model building kit and play around with it to see how moving bonds/angles/dihedrals will change how things are oriented. (Just keep in mind that changes propagate toward the "B" side of the interaction.)

Tue, 2015-01-06 10:11
rmoretti

Thanks a lot.
For Example my Aminoacid is cystein:
angle A is: CB-SG-ZN
angle B: SG-V-ZN
dihedral AB: CB-SG-V-ZN
dihedral A: CA-CB-SG-ZN
dihedral B: ZN-SG-CB-CA
in my file i found the fixed value of 68.0 but in my strucure i measure a angle of 110, whats wrong?!
I believe that dihedral B is wrong, because it is equal to dihedral B, but I don´t know which angle it should be.

Fri, 2015-01-16 08:36
masterofpuppets

Hello again,
I also wondering about an additional virtual atom by the cystein residue. I believe that I dont define this atom. Why is this atom there, when I make the zinc coordination. I have the other four virtual atoms to define the zinc coordination.

Tue, 2015-01-20 03:33
masterofpuppets

Is the virtual atom on the cysteine residue, or the zinc residue? If it is actually on the cysteine residue, om is there probably to help define geometries in certain situations. I wouldn't necessarily include it with your constraint files. The virtual atoms on the zinc residue are there to help orient the zinc residue in space. They're supplied automatically by Rosetta even in the case where the input structure doesn't provide them. They are needed to define the geometry of the interaction, but probably shouldn't be constrained, or constrained loosely if at all.

In your situation, I'd recommend looking for a chain of six atoms, three on the amino acid side, and three on the zinc side. From these six atoms, you can build each of your geometries.

So in your case it would be something like (CA-CB-SG)-(ZN-V1-V2)

The distance is from the atoms between them: SG ZN
Angle A is an angle two on the cysteine side, and one on the zinc side: CB SG ZN
Angle B is an angle one on the cysteine side, and two on the zinc side: SG ZN V1
Dihedral A is three on the cysteine side, one on the zinc side: CA CB SG ZN
Dihedral AB is two on the cysteine side, two on the zinc side: CB SG ZN V1
Dihedral B is one on the cysteine side, three on the zinc side: SG ZN V1 V2

Note that order matters, and all of the measures are going to use the same path of six, so there aren't going to be any rearrangements between the measures.

Angle B and dihedrals AB and B are used mainly to orient the virtual atoms on the zinc residue. In something like a spherically symmetric zinc ion, they probably don't really matter all that much, so do not need to be constrained heavily. The first atom on the B side of the chain (the zinc) will be placed with distance, angle A and dihedral A alone.

Mon, 2015-01-26 13:50
rmoretti

Thank you very much. I am first wondering which V are mean, but through your explanation I understand it.

Tue, 2015-01-27 04:26
masterofpuppets


text:
Elementary Algebra 2e

Elementary Algebra 2e1.6 Add and Subtract Fractions

### Learning Objectives

By the end of this section, you will be able to:

• Add or subtract fractions with a common denominator
• Add or subtract fractions with different denominators
• Use the order of operations to simplify complex fractions
• Evaluate variable expressions with fractions

### Be Prepared 1.6

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions.

### Add or Subtract Fractions with a Common Denominator

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

If $a,b,andca,b,andc$ are numbers where $c≠0,c≠0,$ then

$ac+bc=a+bcandac−bc=a−bcac+bc=a+bcandac−bc=a−bc$

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

### Manipulative Mathematics

Doing the Manipulative Mathematics activities “Model Fraction Addition” and “Model Fraction Subtraction” will help you develop a better understanding of adding and subtracting fractions.

### Example 1.77

Find the sum: $x3+23.x3+23.$

### Try It 1.153

Find the sum: $x4+34.x4+34.$

### Try It 1.154

Find the sum: $y8+58.y8+58.$

### Example 1.78

Find the difference: $−2324−1324.−2324−1324.$

### Try It 1.155

Find the difference: $−1928−728.−1928−728.$

### Try It 1.156

Find the difference: $−2732−132.−2732−132.$

### Example 1.79

Simplify: $−10x−4x.−10x−4x.$

### Try It 1.157

Find the difference: $−9x−7x.−9x−7x.$

### Try It 1.158

Find the difference: $−17a−5a.−17a−5a.$

Now we will do an example that has both addition and subtraction.

### Example 1.80

Simplify: $38+(−58)−18.38+(−58)−18.$

### Try It 1.159

Simplify: $−29+(−49)−79.−29+(−49)−79.$

### Try It 1.160

Simplify: $59+(−49)−79.59+(−49)−79.$

### Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

### Least Common Denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

### Manipulative Mathematics

Doing the Manipulative Mathematics activity “Finding the Least Common Denominator” will help you develop a better understanding of the LCD.

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

### Example 1.81

#### How to Add or Subtract Fractions

Add: $712+518.712+518.$

### Try It 1.161

Add: $712+1115.712+1115.$

### Try It 1.162

Add: $1315+1720.1315+1720.$

### How To

1. Step 1. Do they have a common denominator?
• Yes—go to step 2.
• No—rewrite each fraction with the LCD (least common denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
2. Step 2. Add or subtract the fractions.
3. Step 3. Simplify, if possible.

When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.

In Example 1.81, the LCD, 36, has two factors of 2 and two factors of $3.3.$

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3.

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2.

We will apply this method as we subtract the fractions in Example 1.82.

### Example 1.82

Subtract: $715−1924.715−1924.$

### Try It 1.163

Subtract: $1324−1732.1324−1732.$

### Try It 1.164

Subtract: $2132−928.2132−928.$

In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.

### Example 1.83

Add: $35+x8.35+x8.$

### Try It 1.165

Add: $y6+79.y6+79.$

### Try It 1.166

Add: $x6+715.x6+715.$

We now have all four operations for fractions. Table 1.26 summarizes fraction operations.

 Fraction Multiplication Fraction Division $ab·cd=acbdab·cd=acbd$Multiply the numerators and multiply the denominators $ab÷cd=ab·dcab÷cd=ab·dc$Multiply the first fraction by the reciprocal of the second. Fraction Addition Fraction Subtraction $ac+bc=a+bcac+bc=a+bc$Add the numerators and place the sum over the common denominator. $ac−bc=a−bcac−bc=a−bc$Subtract the numerators and place the difference over the common denominator. To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed.
Table 1.26

### Example 1.84

Simplify: $5x6−3105x6−310$ $5x6·310.5x6·310.$

### Try It 1.167

Simplify: $3a4−893a4−89$ $3a4·89.3a4·89.$

### Try It 1.168

Simplify: $4k5−164k5−16$ $4k5·16.4k5·16.$

### Use the Order of Operations to Simplify Complex Fractions

We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division. We simplified the complex fraction $34583458$ by dividing $3434$ by $58.58.$

Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.

### Example 1.85

#### How to Simplify Complex Fractions

Simplify: $(12)24+32.(12)24+32.$

### Try It 1.169

Simplify: $(13)223+2.(13)223+2.$

### Try It 1.170

Simplify: $1+42(14)2.1+42(14)2.$

### How To

#### Simplify Complex Fractions.

1. Step 1. Simplify the numerator.
2. Step 2. Simplify the denominator.
3. Step 3. Divide the numerator by the denominator. Simplify if possible.

### Example 1.86

Simplify: $12+2334−16.12+2334−16.$

### Try It 1.171

Simplify: $13+1234−13.13+1234−13.$

### Try It 1.172

Simplify: $23−1214+13.23−1214+13.$

### Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

### Example 1.87

Evaluate $x+13x+13$ when $x=−13x=−13$ $x=−34.x=−34.$

### Try It 1.173

Evaluate $x+34x+34$ when $x=−74x=−74$ $x=−54.x=−54.$

### Try It 1.174

Evaluate $y+12y+12$ when $y=23y=23$ $y=−34.y=−34.$

### Example 1.88

Evaluate $−56−y−56−y$ when $y=−23.y=−23.$

### Try It 1.175

Evaluate $−12−y−12−y$ when $y=−14.y=−14.$

### Try It 1.176

Evaluate $−38−y−38−y$ when $y=−52.y=−52.$

### Example 1.89

Evaluate $2x2y2x2y$ when $x=14x=14$ and $y=−23.y=−23.$

### Try It 1.177

Evaluate $3ab23ab2$ when $a=−23a=−23$ and $b=−12.b=−12.$

### Try It 1.178

Evaluate $4c3d4c3d$ when $c=−12c=−12$ and $d=−43.d=−43.$

The next example will have only variables, no constants.

### Example 1.90

Evaluate $p+qrp+qr$ when $p=−4,q=−2,andr=8.p=−4,q=−2,andr=8.$

### Try It 1.179

Evaluate $a+bca+bc$ when $a=−8,b=−7,andc=6.a=−8,b=−7,andc=6.$

### Try It 1.180

Evaluate $x+yzx+yz$ when $x=9,y=−18,andz=−6.x=9,y=−18,andz=−6.$

### Section 1.6 Exercises

#### Practice Makes Perfect

Add and Subtract Fractions with a Common Denominator

425.

$6 13 + 5 13 6 13 + 5 13$

426.

$4 15 + 7 15 4 15 + 7 15$

427.

$x 4 + 3 4 x 4 + 3 4$

428.

$8 q + 6 q 8 q + 6 q$

429.

$− 3 16 + ( − 7 16 ) − 3 16 + ( − 7 16 )$

430.

$− 5 16 + ( − 9 16 ) − 5 16 + ( − 9 16 )$

431.

$− 8 17 + 15 17 − 8 17 + 15 17$

432.

$− 9 19 + 17 19 − 9 19 + 17 19$

433.

$6 13 + ( − 10 13 ) + ( − 12 13 ) 6 13 + ( − 10 13 ) + ( − 12 13 )$

434.

$5 12 + ( − 7 12 ) + ( − 11 12 ) 5 12 + ( − 7 12 ) + ( − 11 12 )$

In the following exercises, subtract.

435.

$11 15 − 7 15 11 15 − 7 15$

436.

$9 13 − 4 13 9 13 − 4 13$

437.

$11 12 − 5 12 11 12 − 5 12$

438.

$7 12 − 5 12 7 12 − 5 12$

439.

$19 21 − 4 21 19 21 − 4 21$

440.

$17 21 − 8 21 17 21 − 8 21$

441.

$5 y 8 − 7 8 5 y 8 − 7 8$

442.

$11 z 13 − 8 13 11 z 13 − 8 13$

443.

$− 23 u − 15 u − 23 u − 15 u$

444.

$− 29 v − 26 v − 29 v − 26 v$

445.

$− 3 5 − ( − 4 5 ) − 3 5 − ( − 4 5 )$

446.

$− 3 7 − ( − 5 7 ) − 3 7 − ( − 5 7 )$

447.

$− 7 9 − ( − 5 9 ) − 7 9 − ( − 5 9 )$

448.

$− 8 11 − ( − 5 11 ) − 8 11 − ( − 5 11 )$

Mixed Practice

In the following exercises, simplify.

449.

$− 5 18 · 9 10 − 5 18 · 9 10$

450.

$− 3 14 · 7 12 − 3 14 · 7 12$

451.

$n 5 − 4 5 n 5 − 4 5$

452.

$6 11 − s 11 6 11 − s 11$

453.

$− 7 24 + 2 24 − 7 24 + 2 24$

454.

$− 5 18 + 1 18 − 5 18 + 1 18$

455.

$8 15 ÷ 12 5 8 15 ÷ 12 5$

456.

$7 12 ÷ 9 28 7 12 ÷ 9 28$

Add or Subtract Fractions with Different Denominators

In the following exercises, add or subtract.

457.

$1 2 + 1 7 1 2 + 1 7$

458.

$1 3 + 1 8 1 3 + 1 8$

459.

$1 3 − ( − 1 9 ) 1 3 − ( − 1 9 )$

460.

$1 4 − ( − 1 8 ) 1 4 − ( − 1 8 )$

461.

$7 12 + 5 8 7 12 + 5 8$

462.

$5 12 + 3 8 5 12 + 3 8$

463.

$7 12 − 9 16 7 12 − 9 16$

464.

$7 16 − 5 12 7 16 − 5 12$

465.

$2 3 − 3 8 2 3 − 3 8$

466.

$5 6 − 3 4 5 6 − 3 4$

467.

$− 11 30 + 27 40 − 11 30 + 27 40$

468.

$− 9 20 + 17 30 − 9 20 + 17 30$

469.

$− 13 30 + 25 42 − 13 30 + 25 42$

470.

$− 23 30 + 5 48 − 23 30 + 5 48$

471.

$− 39 56 − 22 35 − 39 56 − 22 35$

472.

$− 33 49 − 18 35 − 33 49 − 18 35$

473.

$− 2 3 − ( − 3 4 ) − 2 3 − ( − 3 4 )$

474.

$− 3 4 − ( − 4 5 ) − 3 4 − ( − 4 5 )$

475.

$1 + 7 8 1 + 7 8$

476.

$1 − 3 10 1 − 3 10$

477.

$x 3 + 1 4 x 3 + 1 4$

478.

$y 2 + 2 3 y 2 + 2 3$

479.

$y 4 − 3 5 y 4 − 3 5$

480.

$x 5 − 1 4 x 5 − 1 4$

Mixed Practice

In the following exercises, simplify.

481.

$23+1623+16$ $23÷1623÷16$

482.

$−25−18−25−18$ $−25·18−25·18$

483.

$5n6÷8155n6÷815$ $5n6−8155n6−815$

484.

$3a8÷7123a8÷712$ $3a8−7123a8−712$

485.

$− 3 8 ÷ ( − 3 10 ) − 3 8 ÷ ( − 3 10 )$

486.

$− 5 12 ÷ ( − 5 9 ) − 5 12 ÷ ( − 5 9 )$

487.

$− 3 8 + 5 12 − 3 8 + 5 12$

488.

$− 1 8 + 7 12 − 1 8 + 7 12$

489.

$5 6 − 1 9 5 6 − 1 9$

490.

$5 9 − 1 6 5 9 − 1 6$

491.

$− 7 15 − y 4 − 7 15 − y 4$

492.

$− 3 8 − x 11 − 3 8 − x 11$

493.

$11 12 a · 9 a 16 11 12 a · 9 a 16$

494.

$10 y 13 · 8 15 y 10 y 13 · 8 15 y$

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

495.

$2 3 + 4 2 ( 2 3 ) 2 2 3 + 4 2 ( 2 3 ) 2$

496.

$3 3 − 3 2 ( 3 4 ) 2 3 3 − 3 2 ( 3 4 ) 2$

497.

$( 3 5 ) 2 ( 3 7 ) 2 ( 3 5 ) 2 ( 3 7 ) 2$

498.

$( 3 4 ) 2 ( 5 8 ) 2 ( 3 4 ) 2 ( 5 8 ) 2$

499.

$2 1 3 + 1 5 2 1 3 + 1 5$

500.

$5 1 4 + 1 3 5 1 4 + 1 3$

501.

$7 8 − 2 3 1 2 + 3 8 7 8 − 2 3 1 2 + 3 8$

502.

$3 4 − 3 5 1 4 + 2 5 3 4 − 3 5 1 4 + 2 5$

503.

$1 2 + 2 3 · 5 12 1 2 + 2 3 · 5 12$

504.

$1 3 + 2 5 · 3 4 1 3 + 2 5 · 3 4$

505.

$1 − 3 5 ÷ 1 10 1 − 3 5 ÷ 1 10$

506.

$1 − 5 6 ÷ 1 12 1 − 5 6 ÷ 1 12$

507.

$2 3 + 1 6 + 3 4 2 3 + 1 6 + 3 4$

508.

$2 3 + 1 4 + 3 5 2 3 + 1 4 + 3 5$

509.

$3 8 − 1 6 + 3 4 3 8 − 1 6 + 3 4$

510.

$2 5 + 5 8 − 3 4 2 5 + 5 8 − 3 4$

511.

$12 ( 9 20 − 4 15 ) 12 ( 9 20 − 4 15 )$

512.

$8 ( 15 16 − 5 6 ) 8 ( 15 16 − 5 6 )$

513.

$5 8 + 1 6 19 24 5 8 + 1 6 19 24$

514.

$1 6 + 3 10 14 30 1 6 + 3 10 14 30$

515.

$( 5 9 + 1 6 ) ÷ ( 2 3 − 1 2 ) ( 5 9 + 1 6 ) ÷ ( 2 3 − 1 2 )$

516.

$( 3 4 + 1 6 ) ÷ ( 5 8 − 1 3 ) ( 3 4 + 1 6 ) ÷ ( 5 8 − 1 3 )$

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

517.

$x+(−56)x+(−56)$ when
$x=13x=13$
$x=−16x=−16$

518.

$x+(−1112)x+(−1112)$ when
$x=1112x=1112$ $x=34x=34$

519.

$x−25x−25$ when $x=35x=35$ $x=−35x=−35$

520.

$x−13x−13$ when $x=23x=23$ $x=−23x=−23$

521.

$710−w710−w$ when $w=12w=12$ $w=−12w=−12$

522.

$512−w512−w$ when $w=14w=14$ $w=−14w=−14$

523.

$2x2y32x2y3$ when $x=−23x=−23$ and $y=−12y=−12$

524.

$8u2v38u2v3$ when $u=−34u=−34$ and $v=−12v=−12$

525.

$a+ba−ba+ba−b$ when $a=−3,b=8a=−3,b=8$

526.

$r−sr+sr−sr+s$ when $r=10,s=−5r=10,s=−5$

#### Everyday Math

527.

Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs $1212$ yard of print fabric and $3838$ yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

528.

Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs $1212$ cup of sugar for the chocolate chip cookies and $1414$ of sugar for the oatmeal cookies. How much sugar does she need altogether?

#### Writing Exercises

529.

Why do you need a common denominator to add or subtract fractions? Explain.

530.

How do you find the LCD of 2 fractions?

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

Do you know how you learn best?
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text:
# How to align multiline equations?

I am working with equations that are should be broken to multiple lines. The code I wrote is as follows:

\begin{aligned} O_{max} = w_1 \Sigma_{a=1}^{m} \Sigma_{b=a+1}^{n} (-\left|CPT_a - CPT_b\right|)\\ & + w_2 \Sigma_{j=1}^{m} (DIF_j) + w_3 \Sigma_{j=1}^{m} (INT_j/\Sigma_{x=1}^{n} x_{ij}) \end{aligned} \label{equ:ho}

and the other one is

 \begin{align}
\begin{split}
nr(G_i,r) =
\begin{cases}
1 &  \text{r is naturally played by one member of $G_i$}\\
-2 &  \text{r is not naturally played in $G_i$} \\
-p & \text{r is naturally played by p members in $G_i$}\\
\end{cases}
\end{split} \nonumber \\
\begin{split}
nb(G_i) = \Sigma_{r=1}^{9} nr(G_i,r)\\
& max_{\forall G\in C} \left( b(G) =
\frac{\Sigma_{i=1}^{g}nb(G_i)}{g}\right)
\end{split}
\label{equ:yannibelli}
\end{align}

Alignment of each equation is not the same (second part goes to the right and out of the margin of the pages). How can I fix them, please??

• Should be i=1 in the last sum? Dec 31 '19 at 21:11
• Welcome to TSE. I suggest that you post a minimal working example, instead of code snippets. Dec 31 '19 at 21:21

EDIT: 2020/01/01 for the comment of the user @Mico. The first alignment with the important suggestion of the user @Sigur is:

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\begin{document}
\begin{aligned} O_{\max}& = w_1 \sum_{a=1}^{m} \sum_{b=a+1}^{n} (-\lvert\text{CPT}_a -\text{CPT}_b\rvert)\\ &\quad + w_2 \sum_{j=1}^{m} (\text{DIF}_j) + w_3 \sum_{j=1}^{m} (\text{INT}_j/\sum_{x=1}^{n} x_{ij}) \end{aligned} \label{equ:ho}
\end{document}

For the second code my proposal to align is:

    \documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\begin{document}
\begin{aligned} \mathrm{nr}(G_i,r) & = \begin{cases} 1 & \text{r is naturally played by one member of G_i}\\ -2 & \text{r is not naturally played in G_i} \\ -p & \text{r is naturally played by p members in G_i}\\ \end{cases}\\[3pt] \mathrm{nb}(G_i) & = \sum_{r=1}^{9} \mathrm{nr}(G_i,r)\\ & = \max_{\forall G\in C} \left( b(G) = \frac{\sum\limits_{r=1}^{9}\mathrm{nb}(G_i)}{g}\right) \label{equ:yannibel} \end{aligned}
\end{document}

• I'd use &{}\quad + in the second line. Dec 31 '19 at 21:02
• @Sebastiano - In the first row of the first formula, please replace \left| and \right| with \lvert and \rvert. Implementing this change makes a big difference for the spacing between the leading - (minus) symbol and the first vertical bar: With -\lvert, but not with -\left|, one gets the spacing appropriate for a unary operator.
– Mico
Jan 1 '20 at 3:02
• Thanks a lot. It works. Jan 1 '20 at 5:53
• @Mico I give you my best wishes of happy 2020!!! Many thanks. Jan 1 '20 at 12:43
• @Mico, well observed. Done. Jan 2 '20 at 12:59

Alignments require the use of alignment symbols - & - to properly indicate the horizontal position within successive lines that will be used for alignment purposes. You don't have any of them in the first construction. Here's how it should look:

\begin{aligned} O_{\max} &= w_1 \sum_{a = 1}^m \sum_{b = a + 1}^n (-\lvert CPT_a - CPT_b \rvert) \\ &\phantom{{}={}} + w_2 \sum_{j = 1}^m (DIF_j) + w_3 \sum_{j = 1}^m (INT_j / \sum_{x = 1}^n x_{ij}) \end{aligned}

Note the use of \max and \sum instead of max and \Sigma. You may also consider defining \CPT and \DIF as math operators.

Then, for the second construction, there seems to be no need for split here. You can use a nested equation-aligned just like in the first:

\begin{aligned} nr(G_i, r) &= \begin{cases} 1 & \text{r is naturally played by one member of G_i}\\ -2 & \text{r is not naturally played in G_i} \\ -p & \text{r is naturally played by p members in G_i}\\ \end{cases} \\ nb(G_i) &= \sum_{r = 1}^9 nr(G_i, r) \\ & \max_{\forall G \in C} \left( b(G) = \frac{\sum_{i = 1}^g nb(G_i)}{g} \right) \end{aligned}

It really depends on how you want the alignment to occur, together with the numbering. My assumption here is based on a single numbering for each construction.

• Please do not get mad at me but the output makes only limited sense. At least use $r$ is naturally played by $p$ members in $G_i$, i.e. wrap p in \$ signs, and do something with the \max_{\forall G \in C} (add an equality before or whatever, and drop the \forall). I personally would also add punctuations and so on.
– user194703
Dec 31 '19 at 21:52
• @Schrödinger'scat: Ahhh, forgot about the p. I'm not sure what the intent of the \max equation is in the OP's request, so I'll leave that as-is.
– Werner
Dec 31 '19 at 21:56

I'd do like this (using \sum instead of \Sigma):

\documentclass{article}
\usepackage{amsmath,amssymb}

\begin{document}
\begin{aligned} O_{\max} &= w_1 \sum_{a=1}^{m} \sum_{b=a+1}^{n} (-|CPT_a - CPT_b|) \\ &\quad + w_2 \sum_{j=1}^{m} DIF_j + w_3 \sum_{j=1}^{m} \Bigl(INT_j/\sum_{i=1}^{n} x_{ij}\Bigr) \end{aligned} \label{equ:ho}

or another version with \verb|\mathrm{}|:

\begin{aligned} O_{\max} &= w_1 \sum_{a=1}^{m} \sum_{b=a+1}^{n} (-|\mathrm{CPT}_a - \mathrm{CPT}_b|) \\ &\quad + w_2 \sum_{j=1}^{m} \mathrm{DIF}_j + w_3 \sum_{j=1}^{m} \Bigl(\mathrm{INT}_j/\sum_{i=1}^{n} x_{ij}\Bigr) \end{aligned} \label{equ:ho}
\end{document}

• Here, I have used \sum before but the output present a dot not the summation symbol. That is what direct me to use \sigma! If you know what is the problem, please let me know.. Thanks again. Jan 1 '20 at 5:55


text:
# Preprints

 2017-07-2711:10 [FZJ-2017-05279] Preprint et al Isoscalar Single-Pion Production in the Region of Roper and $d^*(2380)$ Resonances Exclusive measurements of the quasi-free $pn \to pp\pi^-$ and $pp \to pp\pi^0$ reactions have been performed by means of $pd$ collisions at $T_p$ = 1.2 GeV using the WASA detector setup at COSY. Total and differential cross sections have been obtained covering the energy region $T_p = 0.95 - 1.3$ GeV ($\sqrt s$ = 2.3 - 2.46 GeV), which includes the regions of $\Delta(1232)$, $N^*(1440)$ and $d^*(2380)$ resonance excitations. [...] OpenAccess: PDF PDF (PDFA); 2017-07-2711:05 [FZJ-2017-05278] Preprint et al Determination of N* amplitudes from associated strangeness production in p+p collisions We present the first determination of the energy-dependent production amplitudes of N$^{*}$ resonances with masses between 1650 MeV/c$^{2}$ and 1900 MeV/c$^{2}$ for an excess energy between $0$ and $600$ MeV. A combined Partial Wave Analysis of seven exclusively reconstructed data samples for the reaction p+p $\rightarrow pK\Lambda$ measured by the COSY-TOF, DISTO, FOPI and HADES collaborations in fixed target experiments at kinetic energies between 2.14 and 3.5 GeV is used to determine the amplitude of the resonant and non-resonant contributions.. OpenAccess: PDF PDF (PDFA); 2017-07-2710:59 [FZJ-2017-05275] Preprint et al Strange Hadron Spectroscopy with a Secondary KL Beam at GlueX We propose to create a secondary beam of neutral kaons in Hall D at Jefferson Lab to be used with the GlueX experimental setup for strange hadron spectroscopy. A flux on the order of 3 x 10^4 KL/s will allow a broad range of measurements to be made by improving the statistics of previous data obtained on hydrogen targets by three orders of magnitude. [...] OpenAccess: PDF PDF (PDFA); 2017-07-2516:47 [FZJ-2017-05176] Preprint et al Ab initio calculations of the isotopic dependence of nuclear clustering Nuclear clustering describes the appearance of structures resembling smaller nuclei such as alpha particles (4He nuclei) within the interior of a larger nucleus. While clustering is important for several well-known examples, little is known about the general nature of clustering in nuclei. [...] OpenAccess: PDF; 2017-07-2516:32 [FZJ-2017-05174] Preprint Zhu, S.-l. Possible hadronic molecules composed of the doubly charmed baryon and nucleon We perform a systematical investigation of the possible deuteron-like bound states with configuration $\Xi_{cc}N (\bar{N})$, where $N(\bar{N})$ denotes the nucleon (anti-nucleon), in the framework of the one-boson-exchange-potential model. In the spin-triplet sector we take into account both the ${}^3S_1$ and ${}^3D_1$ channels due to non-vanishing tensor force. [...] OpenAccess: PDF PDF (PDFA); 2017-07-2510:03 [FZJ-2017-05103] Preprint et al A novel FbFP-based biosensor toolbox for sensitive in vivo determination of intracellular pH Journal of biotechnology S0168165617302080 (2017) [10.1016/j.jbiotec.2017.05.006] 2017-04-2619:14 [FZJ-2017-03272] Preprint et al LFP beta amplitude is predictive of mesoscopic spatio-temporal phase patterns 2017-03-2011:20 [FZJ-2017-02231] Preprint et al Neutron-proton scattering at next-to-next-to-leading order in Nuclear Lattice Effective Field Theory We present a systematic study of neutron-proton scattering in Nuclear Lattice Effective Field Theory (NLEFT), in terms of the computationally efficient radial Hamiltonian method. Our leading-order (LO) interaction consists of smeared, local contact terms and static one-pion exchange. [...] OpenAccess: PDF PDF (PDFA); 2017-03-2011:10 [FZJ-2017-02229] Preprint et al Ab initio calculations of the isotopic dependence of nuclear clustering Nuclear clustering describes the appearance of structures resembling smaller nuclei such as alpha particles (4He nuclei) within the interior of a larger nucleus. While clustering is important for several well-known examples, little is known about the general nature of clustering in nuclei. [...] OpenAccess: PDF; 2017-03-1712:08 [FZJ-2017-02203] Preprint Helias, M. Distributions of covariances as a window into the operational regime of neuronal networks


text:
Question

# Find the function f given that the slope of the tangent line at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point. f '(x)=5(2x − 1)^{4}, (1, 9)

Functions
Find the function f given that the slope of the tangent line at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point. $$f '(x)=5(2x − 1)^{4}, (1, 9)$$
Start by finding the slope of f(x). To do this, integrate f'(x). The integral comes out to $$\frac{(2x-1)^{5}}{2}+C$$. Now solve for C by solving $$f(1)=\frac{(2\times1−1)^{5}}{2}+C=9$$, which gives us a C value of $$\frac{17}{2}$$, thus the function $$f(x)=\frac{(2x−1)^{5}}{2}+\frac{17}{2}$$


text:
# constructing a 95% confidence interval - manipulating inequalities

given the asymptotic distribution of $\hat{\theta_1}$ construct a 95% confidence interval for $\theta$ for large samples:

$\hat{\theta_1} = \frac{\hat{\theta_1}-\theta}{\frac{\theta}{\sqrt{n}}}$

I know that the confidence interval will be :

$-1.96 \leq \frac{\hat{\theta_1}-\theta}{\frac{\theta}{\sqrt{n}}} \leq 1.96$

and hence:

$\frac{\hat{\theta_1}}{1+\frac{1.96}{\sqrt{n}}} \leq \theta \leq \frac{\hat{\theta_1}}{1-\frac{1.96}{\sqrt{n}}}$

Question:

How to get from $-1.96 \leq \frac{\hat{\theta_1}-\theta}{\frac{\theta}{\sqrt{n}}} \leq 1.96$ to $\frac{\hat{\theta_1}}{1+\frac{1.96}{\sqrt{n}}} \leq \theta \leq \frac{\hat{\theta_1}}{1-\frac{1.96}{\sqrt{n}}}$ ? I am aware it is just inequality manipulation, however i am not able to solve it. Please provide detailed steps as my mathematical background is rather weak.

• You can get displayed equations by enclosing them in double dollar signs instead of single dollars signs. Especially with nested fractions, that makes them a lot easier to read. – joriki Jul 28 '18 at 9:57

## 2 Answers

For ease, let $\hat{\theta_1}=x, \theta=y$.

Divide by $\sqrt{n}$: $$-\frac{1.96}{\sqrt{n}}\le \frac{x}{y}-1 \le \frac{1.96}{\sqrt{n}}.$$ Add $1$: $$1-\frac{1.96}{\sqrt{n}}\le \frac{x}{y} \le 1+\frac{1.96}{\sqrt{n}}.$$ Raise to power $-1$: $$\frac{1}{1-\frac{1.96}{\sqrt{n}}}\ge \frac{y}{x} \ge \frac{1}{1+\frac{1.96}{\sqrt{n}}}.$$ Note: $2<3 \iff \frac12>\frac13$.

Now multiply by $x$ to get the final result.

Treat each of the inequalities like you'd treat an equation: Multiply both sides by the denominator; bring mulitples of $\theta$ to one side and everything else to the other; divide by the coefficient of $\theta$. The only thing specific to inequalities is to make sure you reverse the inequality when you multiply or divide by a negative quantity. In the present case, assuming that $\theta\gt0$ and $n\ge4$, no reversal occurs. In the end, you can put the two inequalities back together.

• thanks for the answer, but i still cannot solve this. I am getting stuck on the fact that there is 2 inequality signs and i do multiply by negative -1.96, therefor reversal occurs. I must be doing something wring. I would very much appreciate an little more elaborate answer that shows the aforementioned operations – user1607 Jul 28 '18 at 11:19
• @user1607: There shouldn't be two inequality signs if you follow the answer. You should be treating $A\le B\le C$ as two separate inequalities $A\le B$ and $B\le C$ and performing all operations exclusively on single inequalities up to the very end where you put them together again. You shouldn't be multiplying by negative $-1.96$, either -- in which of the steps in the answer does that happen? – joriki Jul 28 '18 at 11:22
• $$-1.96 \leq \frac{\hat{\theta_1}-\theta}{\frac{\theta}{\sqrt{n}}}$$ then $$-1.96 \frac{\theta}{\sqrt{n}} \leq \hat{\theta_1} - \theta$$ and here i devide by -1.96 so $$\frac{\theta}{\sqrt{n}} + \theta >= \frac{\hat{\theta_1}}{-1.96}$$ – user1607 Jul 28 '18 at 11:32
• @user1607: First, irrespective of what I suggested in the answer, that's not a valid transformation -- you'd have to divide the entire inequality by $-1.96$, including $\theta$. But it's also not what I suggested to do -- the next step at this point was to divide by the coefficient of $\theta$. What's the coefficient of $\theta$ in $-1.96\frac\theta{\sqrt n}+\theta$? – joriki Jul 28 '18 at 11:52
• @user1607: Please note the slight correction I made to the answer -- the statement that no reversal occurs was only correct for $n\ge4$, since otherwise $1-1.96/\sqrt n$ will be negative. – joriki Jul 28 '18 at 11:59


text:
# Partial Fraction Decomposition

## Overview

### Description

A rational expression can be written in different forms. Some forms are more useful than others when trying to solve problems about rational expressions. A partial fraction decomposition is a way to write a rational expression as a sum of simpler rational expressions that all have constants or linear expressions for numerators. Each decomposition is formed by factoring the denominator and using the factors of the denominator as the denominators of the partial fractions. One use of partial fractions is finding the area under a curve that is a complicated rational expression. This application of partial fractions is encountered in calculus.


text:
# How many permutations of [9] are there in which exactly 5 numbers are in their original position?

How many permutations of [9] are there in which exactly 5 numbers are in their original position?

My professor does not mean equivalence class, she means the set of positive integers less than or equal to 9: so {${1,2,3,4,5,6,7,8,9}$}.

This looks like a derangement problem. So my first intuition is to count the number of possible permutations of all 9 numbers, which is simply $9!$, then subtract the number of derangements which would be $D_5$. So my answer would be $9!-D_5$.

Am I thinking of this the correct way? What's a better way of going about solving problems like this Any tips?

• $D_5$ is rather small, so you are claiming that almost every permutation qualifies. That is clearly not the case. – Ross Millikan Jul 16 '17 at 17:12

• So, let's suppose all the odd numbers stay in their original position. There's only one way to do that. That means, we need to derange the remaining 4. So there are 9 derangements total for those 4 numbers ($D_4$)? – John Locke Jul 16 '17 at 17:14
• Cool that makes sense. When do we use complementary counting, to subtract $D_4$ from $9!$? Because the answer to this problem is just $D_4$? – John Locke Jul 16 '17 at 17:29


text:
Stock price prediction has always been a challenging task for the researchers in financial domain. While the Efficient Market Hypothesis claims that it is impossible to predict stock prices accurately, there are work in the literature that have demonstrated that stock price movements can be forecasted with a reasonable degree of accuracy, if appropriate variables are chosen and suitable predictive models are built using those variables. In this work, we present a robust and accurate framework of stock price prediction using statistical, machine learning and deep learning methods We evaluate MULBERRY GROUP PLC prediction models with Modular Neural Network (Market Direction Analysis) and Multiple Regression1,2,3,4 and conclude that the LON:MUL stock is predictable in the short/long term. According to price forecasts for (n+4 weeks) period: The dominant strategy among neural network is to Hold LON:MUL stock.

Keywords: LON:MUL, MULBERRY GROUP PLC, stock forecast, machine learning based prediction, risk rating, buy-sell behaviour, stock analysis, target price analysis, options and futures.

## Key Points

1. How do you pick a stock?
2. Trust metric by Neural Network
3. How can neural networks improve predictions?

## LON:MUL Target Price Prediction Modeling Methodology

With the up-gradation of technology and exploration of new machine learning models, the stock market data analysis has gained attention as these models provide a platform for businessman and traders to choose more profitable stocks. As these data are in large volumes and highly complex so a need of more efficient machine learning model for daily predictions is always looked upon. We consider MULBERRY GROUP PLC Stock Decision Process with Multiple Regression where A is the set of discrete actions of LON:MUL stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4

F(Multiple Regression)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Modular Neural Network (Market Direction Analysis)) X S(n):→ (n+4 weeks) $\stackrel{\to }{S}=\left({s}_{1},{s}_{2},{s}_{3}\right)$

n:Time series to forecast

p:Price signals of LON:MUL stock

j:Nash equilibria

k:Dominated move

a:Best response for target price

For further technical information as per how our model work we invite you to visit the article below:

How do AC Investment Research machine learning (predictive) algorithms actually work?

## LON:MUL Stock Forecast (Buy or Sell) for (n+4 weeks)

Sample Set: Neural Network
Stock/Index: LON:MUL MULBERRY GROUP PLC
Time series to forecast n: 14 Sep 2022 for (n+4 weeks)

According to price forecasts for (n+4 weeks) period: The dominant strategy among neural network is to Hold LON:MUL stock.

X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)

Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)

Z axis (Yellow to Green): *Technical Analysis%

## Conclusions

MULBERRY GROUP PLC assigned short-term Ba1 & long-term B2 forecasted stock rating. We evaluate the prediction models Modular Neural Network (Market Direction Analysis) with Multiple Regression1,2,3,4 and conclude that the LON:MUL stock is predictable in the short/long term. According to price forecasts for (n+4 weeks) period: The dominant strategy among neural network is to Hold LON:MUL stock.

### Financial State Forecast for LON:MUL Stock Options & Futures

Rating Short-Term Long-Term Senior
Outlook*Ba1B2
Operational Risk 5370
Market Risk5952
Technical Analysis8741
Fundamental Analysis7438
Risk Unsystematic8654

### Prediction Confidence Score

Trust metric by Neural Network: 74 out of 100 with 549 signals.

## References

1. J. Harb and D. Precup. Investigating recurrence and eligibility traces in deep Q-networks. In Deep Reinforcement Learning Workshop, NIPS 2016, Barcelona, Spain, 2016.
2. V. Borkar and R. Jain. Risk-constrained Markov decision processes. IEEE Transaction on Automatic Control, 2014
3. Imai K, Ratkovic M. 2013. Estimating treatment effect heterogeneity in randomized program evaluation. Ann. Appl. Stat. 7:443–70
4. Vapnik V. 2013. The Nature of Statistical Learning Theory. Berlin: Springer
5. Imai K, Ratkovic M. 2013. Estimating treatment effect heterogeneity in randomized program evaluation. Ann. Appl. Stat. 7:443–70
6. S. Devlin, L. Yliniemi, D. Kudenko, and K. Tumer. Potential-based difference rewards for multiagent reinforcement learning. In Proceedings of the Thirteenth International Joint Conference on Autonomous Agents and Multiagent Systems, May 2014
7. Challen, D. W. A. J. Hagger (1983), Macroeconomic Systems: Construction, Validation and Applications. New York: St. Martin's Press.
Frequently Asked QuestionsQ: What is the prediction methodology for LON:MUL stock?
A: LON:MUL stock prediction methodology: We evaluate the prediction models Modular Neural Network (Market Direction Analysis) and Multiple Regression
Q: Is LON:MUL stock a buy or sell?
A: The dominant strategy among neural network is to Hold LON:MUL Stock.
Q: Is MULBERRY GROUP PLC stock a good investment?
A: The consensus rating for MULBERRY GROUP PLC is Hold and assigned short-term Ba1 & long-term B2 forecasted stock rating.
Q: What is the consensus rating of LON:MUL stock?
A: The consensus rating for LON:MUL is Hold.
Q: What is the prediction period for LON:MUL stock?
A: The prediction period for LON:MUL is (n+4 weeks)

## People also ask

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text:
## Introduction

It is a fundamental problem in wave physics to continuously boost the transmission capacity and speed of the information carried by waves, which are highly desired for a plethora of applications. A representative example is the underwater communication dominated by acoustic wave due to the low absorption and scattering effects as compared to its electromagnetic counterparts1,2,3,4,5. Given the inherently low frequency and speed of acoustic waves and their lack of polarization, however, breaking through the current information transmission efficiency has been a long-standing challenge crucial for acoustic communications. Recently orbital angular momentum6,7,8,9 (OAM)-based acoustic communication is proposed and implemented which uses space as a new dimension independent of phase and amplitude for encoding information. For vortex beam imprinted with specific OAM, its typical spiraling phase can be mathematically expressed as $$\exp ({{{\rm{i}}}}l\theta)$$, where l is the topological charge and θ is the azimuthal angle. Based on the orthogonality of OAM, different modes of OAM beams carrying independent information can be multiplexed into a single path without inter-modal crosstalk. This enables significant boost of the acoustic communication capacity but needs postprocessing of signals received with complicated sensor arrays10 or multi-layered metamaterials with de-multiplexing capability limited by insertion loss and diffraction effect11.

More importantly, the OAM multiplexing12,13,14,15 strategies rely on multiple coaxially-overlapped twisted beams used as orthogonal communication channels, while leaving the potential of non-coaxial beams to further boost acoustic communication efficiency unexplored. In such cases, the channel number has to be increased by including more high-order OAM modes, leading to more severe diffraction and spatial aliasing effect, which limit the maximum number of available communication channels16,17,18. Although the possibility of information multiplexing/demultiplexing based on non-coaxial OAM beams has been proven in optics19,20,21,22, it is not feasible to directly translate the same mechanism to acoustics considering the fact that complicated equalization algorithm23,24,25 based on time-consuming computation is needed for counteracting the diffraction-induced crosstalk effect which is much stronger for acoustic waves. On the other hand, the passive devices previously proposed for decoding optical OAM beams in real-time are too bulky to straightforwardly apply to acoustic waves with macroscopic wavelength26,27,28,29,30,31. These pose fundamental limits on the further boosting of the spatial information density (defined as the number of available channels per unit transmitter/receiver area) one can achieve with the existing mechanisms of acoustic communications.

In this article, we theoretically propose and experimentally demonstrate a mechanism that uses multi-path twisting of acoustic waves for realizing real-time free-space communication in a passive postprocessing-free and sensor-scanning-free paradigm. By combining the advantages of both coaxial and non-coaxial vortex beam multiplexing, we break through the limits of information transfer capacity in the current spatial multiplexing-based mechanisms. Besides, the introduction of non-coaxial acoustic vortex beam as a new encoding degree of freedom also helps to reduce the dependence on high-order OAM beam and significantly increase the effective communication distance and signal-to-noise rate (SNR). A monolayer metamaterial-based demultiplexer is designed to decode the information carried by such synthesized beams in real-time despite its compact size in all three dimensions, resulting in unprecedentedly high information density in space. In particular, such metamaterial-enabled32,33 mechanism is innately robust to the multi-path effect that would otherwise be significant for a small-scaled system and notably lower the communication accuracy by weakening the contrast between different paths in conventional designs. Furthermore, our strategy is still compatible with the traditional equalization mechanism used in conventional multiple-input multiple-output (MIMO) spatial multiplexing techniques, offering the possibility to further enhance the communication efficiency. We demonstrate the performance of a high spatial information density communication system designed based on our proposed mechanism both numerically and experimentally via distinct example of a complicated image transmission, which shows the high accuracy and high efficiency of our proposed mechanism.

## Results

### The theory of multipath twisting of acoustic wave

Our proposed mechanism for establishing a real-time acoustic communication with high spatial information density is schematically illustrated in Fig. 1a. In our multi-path system, N transmission paths are established between N pairs of transmitting and receiving units, referred to as Ti and Ri respectively with $$i\,=\,1,\,2,\,{{{\ldots }}},\,N$$. Each transmitting unit emits a synthesized acoustic beam formed by coaxially overlapping vortices with different orders lm with $$m\,=\,1,\,2,\,{{{\ldots }}},\,M$$, which are used as independent channels for encoding different information. As a consequence of combining coaxial and non-coaxial vortex beams, the total number of communication channels reaches $${{{{NM}}}}$$, which is a remarkable boost in comparison to the existing acoustic communication mechanisms10,11,34 such as those using OAM-multiplexing/de-multiplexing. Although OAM states reside in an infinite dimensional Hilbert space, the production and recognition of vortex beams with high order, which needs to be $${{{{NM}}}}$$ for achieving equivalent information capacity, is challenging due to the strong diffraction effect17. It should be noted that, however, the diffraction effect of vortex beam is also unavoidable in our mechanism, which leads to spreading out of beam during propagation and causes the receiving unit Ri to also pick up the information carried by encoded vortices emitted from transmitting units other than Ti. This is responsible for the channel crosstalk (or interference) effect that degrades the received signals.

Despite the seeming similarity, it is worth stressing that our proposed mechanism differs intrinsically from the existing spatial multiplexing-based mechanisms of information transfer. One of the most important features of our system is the record-breaking high spatial information density which is ensured by more available channels and the compactness of both the transmitting and receiving terminals. Physically, the shrinking of the transverse size of emitting source and the increase in vortex order dramatically increase the diffraction of information-carrying beams and subsequent crosstalk effect, which conventionally needs to be counteracted via complicated postprocessing of the signal measured by large-scaled sensor array and hinders downscaling of the transmitting/receiving units. In contrast to the existing optical communication systems with bulky size, our mechanism uses a rationally designed metamaterial with compact size in all three dimensions to receive and decode the information carried by the transmitting acoustic beam. Such metamaterial-enabled de-multiplexing mechanism can be simply understood as follows. The whole metamaterial layer comprises N subunits, each of which is judiciously designed to redistribute the energy of a synthesized vortex beam normally impinging on it such that the lm-th OAM beam is effectively untwisted and converged into a prescribed focal point (marked by Fm) at an output plane, as illustrated in Fig. 1b. It can therefore be expected that the acoustic signal received by a single sensor placed at Fm has the original encoded information and an amplitude augmented by a constant factor, which simultaneously enhances the SNR and maintains the compatibility with the conventional multiplexing techniques using phase, amplitude, frequency, etc35,36,37.

More significantly, the designed metamaterial has strong spatial selection since any oblique incidence of vortex beam will not notably affect the acoustic intensity at the prescribed focal points. With innate robustness to the growth in undesired crosstalk effect, our mechanism allows design of a simple and low-cost passive system for real-time communication of information with high spatial density in a sensor scanning-free and postprocessing-free paradigm. It is also noteworthy that our proposed mechanism is inherently compatible with the conventional MIMO signal processing to further equalize the crosstalk between different channels and substantially reduce the computational complexity by simplifying the channel matrix as ensured by the orthogonality among the OAM beams with M different modes (see Supplementary Note 1 for more details). Furthermore, thanks to the monolayer metamaterial design with high transmission and parallel interaction between all its unit cells and the incident acoustic beam, the proposed system can be readily extended to have larger number of communication channels while reducing the increased hardware complexity and energy consumption in conventional communication mechanisms.

In our designed acoustic communication system, the information-carrying acoustic beam emitted from the transmitting unit Ti is generated by using an active transducer array with each transducer being individually controlled by a single-chip microcomputer Arduino Mega 2560, which independently modulates the amplitude and phase of the signal (see Supplementary Note 2 for more details), as shown in Fig. 1a. Then the acoustic pressure distribution on an individual transmitting unit is $$p(\theta,t)=\mathop{\sum }\nolimits_{m=1}^{M}{A}_{m}(t){e}^{{{{{{\rm{i}}}}}}({l}_{m}\theta+{\phi }_{m}(t))}$$, which produces a multiplexed acoustic vortex beam that will be subject to diffraction effect when propagating in free space. Here the amplitude Am(t) and phase ϕm(t) are time-dependent and can be modulated by existing modulation techniques such as phase-shift keying (PSK), amplitude-shift keying (ASK) and quadrature amplitude modulation (QAM) for further improving the data capacity and spectral efficiency. In the current study, we choose to use binary amplitude-shift keying (2ASK) modulation technology to encode information into the amplitude of the produced acoustic vortex beams as different sequences of “0” and “1”. At the receiving end, the transfer function of the single metamaterial-based demultiplexer can be expressed as

$${\Psi }=\exp ({{{{{\rm{i}}}}}}\varphi )=\mathop{\sum }\limits_{m=1}^{M}\exp ({{{{{\rm{i}}}}}}k{r}_{m}-{{{{{\rm{i}}}}}}{l}_{m}\theta )$$
(1)

where k is the wavevector in free space and $${r}_{m}=\sqrt{{(x-{x}_{m})}^{2}+{(y-{y}_{m})}^{2}+{(z-{z}_{m})}^{2}}$$ is the distance between the demultiplexer and the focal point (xm, ym, zm) of the lm-th OAM (detailed design of demultiplexer can be seen in Supplementary Note 3). This transfer function simultaneously removes the OAMs of all the vortex beams and converges the energy of different OAM modes into their respective detection points38. Note that our mechanism eliminates the coupling among demultiplexed beams and thus allows freely manipulating the propagation path of every output beam regardless of the topological charge. Based on this, we further extend the output region to the whole three-dimensional (3D) space behind the demultiplexer, which boosts the channel capacity as well as lowers the crosstalk effect. As a practical implementation, each metamaterial unit cell of the demultiplexer is designed as a hybrid structure composed of a side-loaded resonating cavity array and a straight pipe for modulating the propagation phase of transmitted wave and enhancing the transmission efficiency respectively. We choose to modulate the height h of straight pipes to effectively realize the manipulation. The resulting metamaterial unit cell can thus be modified to produce arbitrary phase shift within the full 0-to-$$2\pi$$ range while keeping a near-unity transmission as required by our mechanism (see Fig. 1c). Instead of continuously varying the structural parameters of each unit cell, in our experiments we only use four kinds of hybrid structures as basic building blocks (marked by four dots in Fig. 1c), which substantially simplifies the design and reduces the possible experimental errors (see Supplementary Note 4 for details). Based on this design, the metamaterial can precisely decode the data carried by synthesized vortex beam and the time-domain signals in NM channels can be received by $${{{{NM}}}}$$ microphones located at focal points, as will be verified via numerical simulations and experimental measurements in what follows.

### Experimental realization of real-time communication using multipath multiplexed vortex beams

The experimental setup for demonstrating the effectiveness of our proposed mechanism of free-space communication with multipath twisting of acoustic wave is depicted in Fig. 2a. The measurement is carried out in an anechoic chamber for eliminating the undesired reflections from the boundaries. For simplicity while without losing generality, in the current study we choose to build a communication system with four independent channels, which includes two pairs of transmitting and receiving units with each emitting two co-axially overlapped OAM beams with l1 = +1 and l2 = −1 (viz., N = M = 2). In our simulations and experiments, the working frequency f0 is set to be 3430 Hz (corresponding to a wavelength λ0 of 10 cm in air). The distance between the centers of T1 and T2 is 0.7 m. The multiplexed vortex beams emitted from T1 and T2 with radial sizes of 0.15 m are propagating toward the receiving units R1 and R2 over a transmission distance of 1.2 m (12λ0), and the spreading and overlapping of vortex beams persist during propagation which may result in channel crosstalk such as in conventional MIMO communication (see Supplementary Note 5). However, our mechanism with spatial selectivity offers the robustness to diffraction-effect-induced crosstalk and ensures the accuracy of communication, as will be discussed later. At the receiving end, two demultiplexers are placed in alignment as a single-layered metasurface to detect and decode the data of four channels. For each OAM mode, we choose a specific focal point located in the 3D space behind the metamaterial layer, i.e., point F1 (0, 0.15 m, 0.2 m) for OAM mode l1 = +1 from T1, F2 (0, −0.15 m, 0.2 m) for OAM mode l2 = −1 from T1, F3 (0.7 m, 0.15 m, 0.2 m) for OAM mode l1 = +1 from T2 and F4 (0.7 m, −0.15 m, 0.2 m) for l2 = −1 from T2. After interacting with the compact metamaterial with each subunit of thickness and width of λ0/2 and 5λ0 respectively (i.e., 20 × 20 hybrid structure unit cells), the two-path multiplexed vortex beams are demultiplexed and converged into four focal points and finally received by four microphones. Such a device with transverse size of 5λ0 × 5λ0 can realize effective demultiplexing of the multiplexed vortex beams after long-distance propagation while ensuring high spatial information density. Without complicated and time-consuming post-processing algorithm or sensor scanning, our methodology enables direct information decoding by using a single sensor to pick up the sound signal in the focus. Besides, thanks to the remarkable enhancement of local acoustic intensity, the SNR of receiving signal increases dramatically, ensuring longer effective transmission distance and lower bit error rate (BER) in acoustic communication process.

In experiment, the spatial distribution of sound intensity on the output plane after the metamaterial layer is measured by using a 1/4-in. free-field microphone (Brüel & Kæjr type-4961) which is attached to a 3D stepping motor to scan the target regions point by point. Figure 2b shows the comparison between the simulated and measured spatial distributions of acoustic intensity of four 20 × 20 cm2 square regions centered at four focal points on the output plane when only a single communication channel is “turned on”, e.g., in the upper figure we only transmit acoustic signal encoded into a vortex beam with l1 = +1 from T1. Next, we also measured the sound intensity profile of the output plane when all the four OAM channels are turned on simultaneously and plotted the typical results in Fig. 2c. By separately inspecting the output wave field produced by each vortex beam, we prove both numerically and experimentally that the designed metamaterial focuses the acoustic energy into the target focal point with high efficiency and precision regardless of the number of working channels. Notice that the practical focal positions in the detection plane slightly offset especially for multi-mode OAM demultiplexing, which is owing to the superposition of target mode and additional OAM mode generated by the demultiplexer. However, due to the null center of OAM mode the sound signals at the foci received by the fixed single microphone will not be affected by the additional OAM mode despite the foci offset. The high spatial selectivity of metamaterial on the incidence direction of vortex beam is verified by the fact that the acoustic intensities at the rest focal points remain negligible as expected. This accounts for the strong robustness of our mechanism against the crosstalk effect caused by inter-channel interference, as will be demonstrated in what follows.

To quantitatively evaluate the strength of inter-channel crosstalk in our designed system, we experimentally measured the power leakage between the four communication channels and calculate the crosstalk of each channel as listed in Table 1. The crosstalk for a specific OAM channel lm can be calculated by $${P}_{l\ne {l}_{m}}/{P}_{l={l}_{m}}$$, where $${P}_{l\ne {l}_{m}}$$ is the sum of the received power of channel lm when all channels except for channel lm are opened separately, and $${P}_{l={l}_{m}}$$ is the received power of channel lm when only channel lm is turned on. We observe that the crosstalk of four OAM channels is below −10 dB, which ensures the high accuracy and robustness of our communication system. Notice that the crosstalk effect caused by the diffraction effect is not trivial in the system shown in Fig. 2a, which would otherwise degrade the received signal in the absence of our proposed de-multiplexing mechanism. On the other hand, the crosstalk effect is not perfectly uniform for all the communication channels as in an ideal model, which may result from the experimental error such as imperfect sample fabrication and alignment.

We first demonstrate the effectiveness of our proposed mechanism via a simple example of independent transmission of four 50-bit data streams in parallel over a communication distance of 12λ0, as illustrated in Fig. 3. In our implementation, four independent channels in two paths, marked as Ch1 to Ch4, are utilized to transmit two-path synthesized vortex beams, with which we could encode four data streams into the amplitude of each vortex beam in 2ASK format (see Supplementary Note 6 for details). The data-carrying signal is in pulse modulation with a pulse period of 20T0 (T0 refers to the period of carrier wave) and each pulse cycle contains 1-bit data such that the communication speed of our system is 686 bit/s. The time-domain signals received at four focal points are also plotted in Fig. 3, corresponding to the four data streams respectively. For the purpose of extracting data streams from the time-domain signals, we perform the cross-correlation by multiplying the received signals with an ideal sinusoidal signal of identical frequency (see Supplementary Note 7 for details). The decoded data streams in four channels are plotted in Fig. 3 (marked by dots) in comparison with the target signal. Obviously, the decoded information streams are consistent well with the input ones without any distortion, indicating that our proposed mechanism allows to achieve real-time and high-precision decoding of information transmitted over four independent channels even in the absence of complicated postprocessing indispensable for conventional mechanisms.

Notice also that the number of channels, working frequency, and digital modulation techniques in our scheme, are chosen for facilitating the proof-of-concept experiment rather than attaining an optimal performance, thereby leading to the relatively limited communication speed. Thanks to its compatibility with existing communication techniques, our mechanism enables significantly boosting the communication capacity by NM-fold. This suggests that by adjusting the working frequency and digital modulation techniques, the communication speed of our design can be easily improved to outperform the conventional acoustic spatial multiplexing communication systems (see Supplementary Note 8 for design example of realizing higher communication speed). Therefore, our mechanism opens an avenue to high-capacity communication paradigm, where the high capacity is realized by a larger number of achievable OAM channels and the compatibility of our proposed mechanism with existing modulation techniques.

Next, we experimentally showcase a more practical example of real-time transmission of a complicated binary image including 462 × 368 pixels. As shown in Fig. 4a, the target image of a school badge is divided into four parts and transmitted through four OAM channels simultaneously. Therefore, each channel will transmit 42504-bit 2ASK data stream. Figure 4b shows the enlarged view of a specific square region marked by the red square in Fig. 4a, where the binary information “0” and “1” represent two different colors of each pixel. The decoded information of four lines L1, L2, L3, and L4 extracted from the time-domain signals is depicted in Fig. 4c, where the target data stream is also presented for comparison. By assembling four measured data streams together, we obtain the output image and plot the results in Fig. 4d, which exhibits the high quality of image transmission. Thus, by utilizing two-path twisting of acoustic wave we realize the real-time transfer of school badge image with high spatial information density. The real-time communication is evidenced by the fact that there is no time delay caused by time-consuming computer-aided postprocessing as in the conventional active mechanisms, except for the time of wave propagation between the transmitting and receiving terminals.

In order to give a quantitative estimation of the accuracy of data transfer in our proposed mechanism under longer transmission distances, we increase the transmission distances from 12λ0 to 20λ0 and employ BER to characterize the performance of our communication system. Here, the BER is defined as the ratio of the number of error symbols to the total number of all transmitted symbols. Figure 5a shows the measured BERs for the transmission of the binary school badge image as a function of the transmission distance. It is apparent that the measured BER curve is much lower than the forward error correction (FEC) limit39 of 3.8 × 10−3. Interestingly, when the transmission distance of our communication system reaches 20λ0 the mainlobe of vortex beam has already exceeded the receiving region and non-coaxial vortex beams have overlapped severely in space. However, the demultiplexer can still work and keep superior performance, which suggests our metasurface-enabled demultiplexing mechanism has inherent robustness to channel crosstalk. To further characterize the performance of our communication system, we vary the SNR of the received signal by increasing the ambient noises and experimentally measure the BERs of recovered binary image versus SNR under a transmission distance of 12λ0. From the typical results in Fig. 5b, we can clearly observe that the BERs of four channels increase with the decrease of SNR, as expected. Nevertheless, the BER of the whole communication system is still much lower than the FEC limit even at extremely low SNR level, which demonstrates the important antijamming capability of our proposed system. Although here we only demonstrate a communication system with a relatively short transmission distance of 20-wavelength in experiment for facilitating the experimental fabrication and measurements, our mechanism can apply to realize accuracy and remote communication over a km-scale distance with optimal transmitting/receiving array dimensions (see Supplementary Note 9). Due to the fact that the OAM beams with small topological chargers are less affected by the ocean turbulence, our proposed mechanism eliminating the dependence on high-order OAM beam is also to a certain extent robust against the complex factors in underwater environment, which is crucial for its practical implementation in long-distance underwater communication. In our scheme, we have implemented four channels in a receiving area of 12λ0 × 5λ0, and the number of channels achieved in per unit area (normalized by $${\lambda }_{0}^{2}$$) is about one order of magnitude higher than most OAM-based communication systems, which suggests a high spatial information density of our proposed mechanism. The spatial information density can be further improved by decreasing the distance between subunits (see Supplementary Note 10) and increasing the number of OAM modes in each transmission path. It is also noteworthy that the effective transmission distance of our proposed system can be further improved with no need of increasing the transmitter aperture by imposing low-diffraction vortex beam such as Bessel beam40 (see Supplementary Note 11).

## Discussion

In summary, we have proposed and experimentally demonstrated a mechanism using multipath twisting of acoustic wave to realize real-time free-space communication with high spatial information density in a passive postprocessing-free and sensor-scanning-free paradigm. The metamaterial, employed as the passive and efficient demultiplexing component for data decoding, not only has good performance in resisting inter-channel crosstalk but also can be compatible with conventional MIMO signal processing to further improve the accuracy of data transmission. The effectiveness of our proposed mechanism is demonstrated via real-time transmission of a complicated binary image. Despite the low number of transmission paths and short communication distance we choose to demonstrate here for simplifying the experimental fabrication and measurement, it is obvious that our scheme is general and can be straightforwardly applied to the design and implementation of large-scaled communication systems with a larger number of transmission paths and OAM modes. Moreover, our mechanism can apply to demultiplex the vortex beam of oblique incidence by compensating an additional deflection phase on the demultiplexer according to the generalized Snell’s law, which is crucial for underwater acoustic communication with multipath effect. Note also that the communication capacity and spectral efficiency can be further improved by using high-order shift keying to encode data. In addition to the extendability to break the limited bandwidth of underwater communication and ocean exploration, we also anticipate our proposed strategy to have far-reaching implications in diverse fields and to give important inspiration for relevant researches for other classical waves, such as integrated light communication devices with significant application potential to all-optical on-chip applications.

## Methods

### Numerical simulation

Throughout the paper, the simulations are conducted with a finite element method based on COMSOL MULTIPHYSICS software. The background medium is air, for which the mass density and sound speed are 1.21 kg/m3 and 343 m/s, respectively. The solid material for building the hybrid structure is chosen as UV resin whose mass density and sound speed are 1400 kg/m3 and 2000 m/s, respectively.

### Experimental configuration

The experiment is performed in an anechoic chamber without interference from other obstacles. The samples of two metamaterial-based demultiplexers are fabricated via 3D printing technology. Ten loudspeakers (HiVi, model B2S) are circularly-arranged on an acrylic glass plate and a single-chip microcomputer Arduino Mega 2560 is used to control the amplitude and phase of the drive signal of each loudspeaker. Four microphones (Brüel & Kæjr type-4961) are placed at the focal points to detect the signals. The real-time signals are recorded with Brüel & Kæjr PULSE 3160-A-042 4-channel analyzer.