text:
# Why interviewing in tech is broken (and the direction it is going)

I think interviewing is a largely broken system that is very poorly optimized from both a technical (engineer) and non-technical (recruiter) standpoint. After applying/interviewing for 4 years straight I have some opinions on tech interviews. The fallacy is that the most qualified and most intelligent person is hired but that is rarely the case. I would like to think that I am going up against other candidates in a person against person battle of skills and who can handle pressure, but that simply isn’t true. It is more subjective due to the human nature of assessing interviews, and it is flawed on asking simple algorithm questions, it is based on faith that a person who can present themselves in an interview can work as a software engineer in that team. All those things are not respective of the job, but unfortunately there is no great alternative at this point. The Googles and Facebooks have technical bars so the industry copied it and followed suit.

Equating it to regular jobs, it hinges more on WHO you know rather than what you know. And the referrals can go a long way or straight up be duds. Sometimes job hunting and interviewing just feels like blindly throwing darts at a dartboard.

### Problems of interviewing (summarized)

• technical recruitment and resume screening is a broken process
• the most technical/best/smartest person doesn’t always get hired
• memorizing coding questions is a bad practice that still works
• the level of perfection expected usually causes false negatives/positives
• interviewing is hardly a fair process

## The Problems

### Recruiters screening

Where do I start. Non-technical recruiters often have no idea what projects or experience mean on a resume. How do they know who to pick? (This is AFTER your resume has been passed through an filtering software)

Technical recruiters supposedly have a little more insight, but even then how many times do you think they check if you full-stack web app posted on your github was not just a copy and paste from some other repo? (which unfortunately has been a real story from /r/uwaterloo) Or that you didn’t just put a bunch of buzzwords to pass through the resume screening software. (Which is actually a valid way of trying to get picked) Those may be overly cynical takes but it’d be easy to create false-positives on a resume for the sake of impressing recruiters. The line between embellishment and exaggeration can often be blurred. In an industry like tech with a lot of different technical skills beyond just “coding”, it isn’t a surprise if you feel screened out in cases where you feel extremely-qualified and vice versa.

The next “easy” cutaway is looking at companies you interned at and basing a person’s worth on that. There are a multitude of issues here. How do you equate impact or actual technical competency from looking at their job/job description. An example of this is a thought process along the lines of “xyz interned at Google, they surely must be a reputable engineer. They passed their technical bar and did an internship there.” We are forced to extrapolate data based on this experience but it doesn’t tell the whole story. We have very little context into how they were hired (were they asked some easy question or something they saw before?) and we have little context in their internship other than what they wrote about it. (did they perform poorly? Were they unable to work in a team setting?)

This is more of a gripe with recruiters not doing their research but I recently had someone on LinkedIn messaged me this:

Hey John,
We at <COMPANY> are ex-Googlers and ex-Amazon engineers, working on AI powered search to own local shopping.We’re funded and growing.
Nike CF Sephora Ulta are our customers.
I’m looking for super smart engineers like yourself to join our team.

I’m not sure how much more general you can get. He only mentions my LinkedIn internship and then talks about his company. He mentions ex-Googlers and ex-Amazon employees and a fairly vague company description. It probably is a copy-and-pasted headhunt message but this shows the lack of depth into candidate research recruiters sometimes do. If I asked what they liked about my internship at LinkedIn, or why they reached out to me, I’m not sure I’d get a well-thought out answer. They cannot tell my actual impact from just glancing briefly at my LinkedIn profile.

I could go on and on about recruiter and recruitment practices but it can be boiled down to a few statements. A lot of times they have no idea who to look for, who is right for the role, or what makes a good engineer. (You could make the case only software engineer managers can really do those things, but their time is usually spent managing software projects not searching for prospective candidates)

### Memorizing questions, question banks, etc.

Sites like leetcode got popular by posting specific lists of questions companies would ask and people then memorizing those specific questions. This always frustrated me because questions like “reversing a linked list” is really dumb but I got asked it 3-4 times. I’ve been asked “coin change” 4-5 times, “kth largest number” 5-6 times, “2d-islands” or some variant 3-4 times, etc.

I did not go out and attempt to memorize these questions. There is a lack of creativity and actual gauging of algorithmic ability in interviews (especially at companies that aren’t big N size). Making questions that test problem solving skills, data-structures and algorithms, and making it the right difficulty is hard. Imagine you make a great question, only for it to be leaked on Glassdoor, Leetcode, or any other derivative. The once novel algorithmic test has now been turned into public domain knowledge. (for those willing to read the questions)

A lot of companies ask textbook questions in interviews hoping to litmus test candidates on this, and feel pressured due to big tech companies following same procedures. (A few are moving away from this, but more on that later). In reality a good startup or smaller-sized company should tailor questions to their business, therefore making questions weed out people who are not interested and also get a much more focused group of people rather than people who know the “Top 100 most common interview questions”.

If everyone is going out and memorizing questions on how to find cycles in linked lists, two sum, and largest contiguous subarray why should companies ask this? They are just playing a game of “what do you know from leetcode” and “what have your studied previously”. It in turn has become to the stages of doing research in the “currently asked questions” and “most updated questions from the question bank” for the highest level of solving success.

I have always said that people should learn the techniques of how to use various data structures and algorithms (arrays, linked lists, trees, graph traversals, sliding window, dynamic programming) and apply them to solve the problems. If you learn the fundamentals you should be able to apply them to different problems

I used to dislike the notion of asking dynamic programming in an interview but lately I’ve had a change of opinion. Dynamic programming questions are hinged on implementation. You cannot implement a dynamic programming question WITHOUT understanding the recurrence relation (which often is not TOO complicated unless the interviewer picks a fiendish problem on purpose). If you memorized a recurrence relation for a question and you don’t actually understand what the question is asking it is evident when you need to explain what is going on. (For the simple DP questions, you may get away with this, but in this day and age people should not even be asking coin change.)

People memorizing questions to pass the bar creates a situation where you potentially have a “leetcoder” joining your team. Great at memorizing niche problems like “how to create a bst from preorder traversal” or “inverting a binary tree” but can they code your web app? Might be a yikes situation depending on their actual aptitude to learn, willingness to work in a team, and ability to grow as a software engineer. In the best case, they practiced and studied knowing that they need to pass the algorithms section to pass the interview.

I always felt it was weird how KMP string matching was a technical report paper in 1970, but it is now expected in some interviews to write it out. If it took extremely smart computer scientists time to create it, why would I regurgitate something I memorized purely for the sake of memorizing it. This is in the same vein of “memorized purely for sake of recall in an interview”.

Some companies may ask (technical) trivia questions like “what is a decorator in python” or “what are some issues with multithreading in python”. For frequent users of the language this may be fine, but if you mainly work in a different language, tech stack, or even frontend/backend/ios/etc. side of the product these are largely unknown.

This is a personal gripe because I don’t have the greatest memory for things I don’t use frequently so I dislike trivia. I don’t know the exact number for doing a chmod on a file to make it read only. If I did a 4 second google search I could find it out. If I did it often enough I would know off the top of my head. I barely even chmod files, and I don’t do it regularly enough to know the exact bits that are set.

There is a lot of trivia that I do know by virtue of general use. I got asked about “difference in public, private, and protected in objects”, I got asked about “eventual consistency and strict consistency”, etc. Some stuff was from university classes, some was just from using it often enough. If I don’t know a trivia question, like 99% of people, I would go google it and find the answer. Memorizing the trivia would save that time, but in all honesty I am not a huge hash-table lookup that memorizes every bit of information I receive.

### Expecting perfection in interviews

Something that I don’t like is that perfection is expected in answers for solving technical interviews. Personally, I have never passed an interview where I did not code a problem (or in the case of multiple rounds, every round) to 100% perfection. Even places where I coded to 100% perfection they would still say no. It transcends perfection in cases, failure or really mediocrity in most cases aren’t tolerated.

One place I got asked 2sum (trivial), 3sum (harder but still trivial), and 4sum (n^3 is trivial. Looking back ok it isn’t even that hard. But interviewer didn’t nudge me in that direction). Since I didn’t solve all 3 questions optimally in 45 minutes I got rejected. That’s tight constraints. I’ve found you need perfection a lot of times in all facets.

It is disappointing because in my opinion someone who gets stuck but prevails and solves it in the end, or is solving organically (as opposed to memorizing a question and having it be perfect on the first try) is more valuable in my opinion, but it is extremely devalued in the interview. If I had a candidate who solved it with struggles on their own in 25 minutes versus someone who coded it perfectly in 10 minutes, at face value you want the person who did it perfectly. My argument is that struggling is a natural step in solving questions because everyday coding is met with unlimited struggles with various constraints (speed of delivery versus depth of implementation, dependency issues, navigating tech debt, etc.) and NO ONE is always perfect.

For an easy question sometimes it should be solved in a short time frame, but I am generalizing it to the point that a real solving experience is more valuable for an interviewer than some quick perfect solve of something (they may have seen beforehand).

### Explaining code to interviewer (and making sure they understand)

If an interviewer asks me a question in an interview, I hope they understand the question. (This has unfortunately not always been the case. I have had to argue about heapify runtimes previous)

I try my best to outline my algorithm to make sure the interviewer is on the same page as me. What I deem as important is the understanding that the interviewer knows what I am doing. For one, they are writing my interview notes and feedback so for them to understand is paramount.

In the off chance they don’t understand what is going on, or don’t understand my thought process I try to explain it to them the best I can without losing lots of time.

Due to unavoidable social barriers sometimes people just don’t connect socially. Sometimes, you can think of concepts and code in ways that they don’t think of (or isn’t as natural for their coding style). Some interviewers demand you solve a problem THEIR way. Any other way is suboptimal (when it may not be) or is not acceptable (which is also dumb). When this occurs you are stuck. You are forced to do it THEIR way because they are the interviewer.

### Some interviewers want you to succeed and help you…some really want you to suffer

The majority of interviewers I have interacted with were very polite, understanding, and willing to help when I got stuck.

On the off-chance, sometimes interviewers were flat-out terrible. I’ve had ones berate my code, ask me why I was putting comments to explain “production code” (in the interview I perceive it as “interview code” so I will put comments for extra clarity), eating loudly during the interview, make fun of my hobbies etc. If these are the people interviewing me and writing feedback on me, how am I even supposed to show them my skills and what I coded?

This stems from the fact that a lot of interviewers are engineers first and not really “interviewers”. Most of the time your evaluations are on the projects you do and the code you write. Interviewing and hiring can often be an afterthought for some due to the lack of real recognition in their role. It is a little disappointing this occurs since hiring for the company is pretty important (in some cases you may work with this person).

### Not really that fair of a system

I am not going to touch on this in depth but the system is not fair and probably rigged against you. You often say “____ got an interview/offer and I didn’t. I’m much more qualified”. Whether or not if that is actually true recruiters make mistakes in who they choose. I’ve seen it happen both ways.

What really needs to be thought about is who are the people a company is trying to hire? Do they want the most qualified candidates or do they want a person with a diverse life story? Are they in 3rd year and above? (Looking at age), or even if they contain visa sponsorship to work in that country (Looking to save money from visa sponsorship). Some places may ask information like this and choose based on criteria like this. They often ask it in online applications.

### Parity

There really is no parity in job hunting so people will do what they deem as necessary. Here are some scenarios that you may come across:

• Is lying about your accomplishments on your resume (putting different companies or roles) okay since everyone else embellishes their resumes?
• Is it cheating on a code challenge if the code was posted on leetcode?
• Is memorizing all of a company’s questions using leetcode premium before your interview fair? All I had to do was pay for leetcode premium to see the questions
• Can I google the coding question they asked me DURING the interview? They can’t see me if it is virtual. (BIG yikes)

These are huge grey areas people face and there is little to no boundaries other than people’s morals. When software engineer positions and internships are so coveted some people are willing to do whatever it takes to get those jobs. The competition is present in any profession but here so much money is at stake for many people.

## The Future of Interviewing

Where I think interviewing is going. Hopefully in a positive direction, but there is still a lot to fix in the very broken process.

### Previous landscape

A few years ago the questions being asked at Big N companies are now widely known. Questions have risen in difficulty. No longer does Google ask brainteasers. Topics like dynamic programming are more commonstay. Lesser known topics are being more common like sliding window, prefix sum, disjoint sets, monoqueues, segment trees, etc. (some of the latter ones not so much but we shall see in the future) It is hard to predict what is to come, but I think places like India where the competition is much higher, a lot more people are competing for less positions (compared to US), and talent pool is much higher is going to cause question difficulty to skyrocket.

### More custom questions focusing on concepts

I’ve seen more follow-up questions focusing on understanding now. Specifically under hypothetical space constraints, dealing with distributed systems, etc.

I’ve had places say “ok now we can’t load it all in memory, what can we do if…” or “How does this function run with a stream of data…”. These are usually more nuanced statements meant to probe deeper in the fundamentals of the problem solving. Which also means if you memorized a problem’s solution and nothing beyond it, that is a steep cliff that stands before you. I think these are fine in terms of learning more of a person’s thoughts on a solution. They are usually more open-ended as they initiate discussion.

### System Design

I think the future of tech interviews lies in system design questions. They are less memoizable with leetcode and other dedicated problem sites. I think leetcode questions may be obsolete in a few years due to the bar being that “people just memorize questions”. It has come to the point where you just read CTCI (cracking the coding interview) and hope a big name overlaps a question with one you saw in the book.

System design (which even I have areas to improve on it) encompasses object-orientated design, real-world distributed system constraints, and a lot more fluidity for thought from backend and frontend. I think the negative aspect is that it pushes more towards generalists that need to be familiar with all facets, but also means you need to understand how your API webhook fits into the giant landscape of the product. There are negatives and positives to basing cs knowledge on system design more and more. (which becomes more prevalent as your move upwards).

## Final Thoughts

These are just my thoughts after doing interviews for a few years. I still fail lots of interviews. These are some key things to remember:

• The criteria that companies ask is not standardized. If you fail somewhere just remember there are lots of other places.
• Interviewing like life is sometimes daunting, depressing, and tiring. You just need to keep trying to find a place that is a good match for you.


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Can't one load media with spaces in their file names?

I appear to have successfully installed media9. My first attempt to use it (listed below) has led to an error, of course: in this case a complaint that 'car' is not a known option. Is it impossible to use spaces in the file name of the media? This would be a serious problem for me, since all my hundreds of file names have spaces in them - usually lots of spaces! (I added quotes to the file name, but that didn't help.)

\documentclass{article}
\usepackage{media9}
\begin{document}
\includemedia[
width=0.4\linewidth,
height=0.3\linewidth,
addresource="D:Audio/67/67A1_Snippets/67_321 fast, car, go here fast.mov",
activate=onclick,
deactivate=onclick,
flashvars={source="D:Audio/67/67A1_Snippets/67_321 fast, car, go here fast.mov"},
]{}{VPlayer.swf}
/end{document}

P.S. All my media files are MOVs, but I believe they are MPEG-4 inside.

• The simple fact is that filenames with spaces, commas and many other non-alphanumeric characters will often cause you problems. See What are the allowed characters in filenames? and \includegraphics: Dots in filename. You would likely be better normalizing your filenames to exclude these characters. – Alan Munn Jun 16 '15 at 15:07
• Try with addresource={"D:Audio/67/67A1_Snippets/67_321 fast, car, go here fast.mov"}, – egreg Jun 16 '15 at 15:08
• Did you try @egreg's suggestion? If it works it would constitute an answer, but I'm unsure. – Joseph Wright Jul 28 '15 at 6:06
• I just tried the simplified version shown below. There were no errors, but the video didn't play. What I got instead was a blue screen with a message at the bottom saying "LOADING 0%" – Peyton Todd Jul 29 '15 at 15:06
• The rules on this site don't enable me to press Enter without closing the comment, so I've not been able to insert code (and it will no longer enable me to edit the abortive version just above). Suffice it to say that a simplified version of the above code with @egreg's suggestion included built the PDF, but trying to play the video led either to the blue screen just described or to the message Unable to play '{"40 631.MP4"}', where 40 631.MP4 is the name of the video, depending on whether I went with VPlayer.swf or VPlayer9.swf. Just the bare video WITH THE SPACES (!) did play. Go figure... – Peyton Todd Jul 29 '15 at 15:19

addresource={"D:Audio/67/67A1_Snippets/67_321 fast, car, go here fast.mov"}

format: plain


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# Sugarcane Genomics Review

Posted on Mon 29 January 2018 in genomics

I have recently started working on a sugarcane genome assembly project, so I figured I would share some interesting points from my literature review. Sugarcane is obviously a very economically important crop, having applications in both food and energy production. Despite this economic importance, genomic resources are lacking for sugarcane due to the complexity of the genomes of modern cultivars. Let's look into the specifics of why that is.

## Taxonomy¶

Sugarcane is in the Poaceae family, the same mega-grass family as major crops such as rice, barley, maize, and wheat. When broadly referring to sugarcane, people are often talking about the Saccharum genus, in which there are a few prominent species. From my reading, it seems that the two key species that have influenced the modern production of sugarcane are S. officinarum and S. spontaneum, with many cultivars being interspecific hybrids between them. Specifically, around the late 19th century, important traits from S. spontaneum such as disease resistance and high yield were introgressed into S. officinarum varieties which already had high sugar content. Given that sugar cane has relatively long breeding cycles (10-15 years), there are relatively few generations separating modern varieties from these interspecific hybrid events1,2,3.

In trying to anticipate potential challenges for genome assembly, this taxonomy raises some red flags. Interspecific hybrids can often be challenging to assembly given the relatively high degree of structural variation (and heterozygosity in general). Sugarcane takes this notion to the extreme.

## Ploidy and Genome Size¶

Here are some numbers off-the-bat

Table 1
Species Chromosome Number Ploidy
S. officinarum 2n=8x=80 Decaploid
S. spontaneum 2n =5x-16x=40-128 Octoploid
Modern Hybrids 2n=100-130 Aneuploid

Modern interspecific hybrids between S. officinarum and S. spontaneum exhibit allopoloploidy as well as aneuploidy. They are allopolyploids in that distinct portions of the genome are contributed to by these two ancestors. Specifically, about 70-80% (I've also read 75-85%) of the modern cultivars genome are of S. officinarum ancestry, while the rest comes from S. spontaneum 1. Perhaps most interesting is the aneuploidy. For lack of a better way of putting it, this means that each chromosome has a distinct number of homologous/homeologous chromosomes (i.e. chromosome 1 may have 10 homologous/homeologous chromosomes, while chromosome 2 has 13 homologous/homeologous chromosomes). This is best visualized in the diagram below provided by Grivet et al, where each row represents a different chromosome group.

"Schematic organization of the genome of a current sugarcane cultivar. Each bar represents a chromosome; yellow coloring represents regions originating from S. officinarum and green coloring from S. spontaneum. Chromosomes aligned within the same row are homologous (or homoeologous). The key characteristics of this genome are the high level of ploidy, the aneuploidy, the bispecific origin of the chromosomes (with at least three-quarters inherited from S. officinarum), the existence of structural differences between chromosomes of the two origins, and the occurrence of interspecific chromosome recombinants." 1

The above diagram represents one particular cultivar, though the exact structure may change from individual to individual.

Naturally, along with this ploidy comes a large genome size. Rough genome size estimates I have read for modern cultivars is around 10 Gbp (for context, the human genome size is roughly 3.3 Gbp). This puts it in the same class as other monster genomes that have recently been assembled by some of my colleagues here at Johns Hopkins. These include loblolly pine and hexploid wheat.

## Genome Assembly Strategy¶

Given the immense genome size of modern sugarcane cultivars as well as the aneuploidy/polyploidy, some refer to sugar cane as the hardest economically important genome to assemble. A quick NCBI search yeilds one assembly for the Saccharum genus with a contig N50 of 8.4 kbp and a total assembly size of ~ 1 Gbp. Though this is almost a comically low N50, I think it speaks more to the difficulty of assembling this genome more than anything else.

Obtaining long-read data alone (such as pacbio) is quite expensive and time-consuming. To achieve 40X coverage, a decent starting place for noisy read assemblers such as FALCON and canu, one must obtain 400Gbp of sequencing data! Then, one must deal with the unexpected behavior of genome assemblers given this genome size. I have already had some issues with k-mer counting using canu (perhaps more on this later). The aforementioned pine and wheat genomes are great starting places for me to research how to work with exceptionally large genomes.

The most interesting and challenging aspect of this genome assembly project is the ploidy. For those not familiar with genome assembly, I won't get into it here, but polyploid genome assembly presents many challenges compared to diploid genome assembly. To mitigate this, a common strategy when working with polyploids is to sequence diploid progenitor species of the species in question. For example, genomic resources for such a progenitor species in wheat were used for the hexaploid wheat assembly. However, such a resource is not currently available for modern sugarcane cultivar progenitor species. This is presumably because both progenitor species are polyploids (see Table 1) and would be difficult to assemble in their own right.

Sugarcane, however, is not completely devoid of genetic resources. There is an abundance of linkage maps available for both progenitor Saccharum species as well as a few modern cultivars4,5. Here is a map of Saccharum spontaneum ‘SES 208 from Al-Janabi, Salah M., et al. published in 1993.

And here are just a few linkage groups of the Louisiana cultivar from a more recent (2016) study from Liu, Pingwu, et al.

I will certainly be investigating how we might be able to leverage the abundance of linkage maps available in the Saccharum to aid with the assembly process, specifically anchoring whatever contigs we might produce. Of course, this will still be pretty challenging since any assembly we produce will probably be fairly fragmented, which means fewer contigs will be anchored using linkage maps.

# References¶

1. Grivet, Laurent, and Paulo Arruda. "Sugarcane genomics: depicting the complex genome of an important tropical crop." Current Opinion in Plant Biology 5.2 (2002): 122-127.

2. Butterfield, M. K., A. D’hont, and N. Berding. "The sugarcane genome: a synthesis of current understanding, and lessons for breeding and biotechnology." Proc S Afr Sug Technol Ass. Vol. 75. 2001.

3. Raboin, Louis-Marie, et al. "Analysis of genome-wide linkage disequilibrium in the highly polyploid sugarcane." Theoretical and Applied Genetics 116.5 (2008): 701-714.

4. Al-Janabi, Salah M., et al. "A genetic linkage map of Saccharum spontaneum L. 9SES 2089." Genetics 134.4 (1993): 1249-1260.

5. Liu, Pingwu, et al. "Identification of quantitative trait loci controlling sucrose content based on an enriched genetic linkage map of sugarcane (Saccharum spp. hybrids) cultivar ‘LCP 85-384’." Euphytica 207.3 (2016): 527-549.

In [ ]:


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# Using diagonalization to find A^k

1. Nov 17, 2012

### 1up20x6

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

$$A = \begin{pmatrix} 1 & 4\\ 2 & -1 \end{pmatrix}$$

Find $A^n$ and $A^{-n}$ where n is a positive integer.

2. Relevant equations

3. The attempt at a solution

$$(xI - A) = \begin{pmatrix} x-1 & -4\\ -2 & x+1 \end{pmatrix}$$
$$det(xI - A) = (x-3)(x+3)$$
$$λ_1 = 3\quad λ_2 = -3$$

$$\begin{pmatrix} 2 & -4\\ -2 & 4\end{pmatrix} \begin{pmatrix} a\\ b \end{pmatrix} = \begin{pmatrix} 0\\ 0 \end{pmatrix}\quad \begin{pmatrix} a\\ b \end{pmatrix} = \begin{pmatrix} 2\\ 1 \end{pmatrix}$$
$$\begin{pmatrix} -4 & -4\\ -2 & -2\end{pmatrix} \begin{pmatrix} c\\ d \end{pmatrix} = \begin{pmatrix} 0\\ 0 \end{pmatrix}\quad \begin{pmatrix} c\\ d \end{pmatrix} = \begin{pmatrix} -1\\ 1 \end{pmatrix}$$
$$P = \begin{pmatrix} λ_1 & λ_2 \end{pmatrix} = \begin{pmatrix} 2 & -1\\ 1 & 1 \end{pmatrix}$$
$$P^{-1} = \begin{pmatrix} \frac{1}{3} & \frac{1}{3}\\ \frac{-1}{3} & \frac{2}{3} \end{pmatrix}$$
$$D = \begin{pmatrix} λ_1 & 0\\ 0 & λ_2 \end{pmatrix} = \begin{pmatrix} 3 & 0\\ 0 & -3 \end{pmatrix}$$
$$PD^nP^{-1} = A^n = \begin{pmatrix} 2 & -1\\ 1 & 1 \end{pmatrix} \begin{pmatrix} 3^n & 0\\ 0 & -3^n \end{pmatrix} \begin{pmatrix} \frac{1}{3} & \frac{1}{3}\\ \frac{-1}{3} & \frac{2}{3} \end{pmatrix}$$
$$= \begin{pmatrix} 2(3^n) & -(-3)^n\\ 3^n & (-3)^n \end{pmatrix} \begin{pmatrix} \frac{1}{3} & \frac{1}{3}\\ \frac{-1}{3} & \frac{2}{3} \end{pmatrix}$$
$$= \begin{pmatrix} \frac{2}{3}(3^n) + \frac{1}{3}((-3)^n) & \frac{2}{3}(3^n) - \frac{2}{3}((-3)^n)\\ \frac{1}{3}(3^n) - \frac{1}{3}((-3)^n) & \frac{1}{3}(3^n) + \frac{2}{3}((-3)^n) \end{pmatrix}$$

I think that everything I've done so far is correctly, but I can't find any way to simplify this equation any further, and I don't think that I could find $A^{-n}$ with an equation this complicated, so I must be missing something.

2. Nov 17, 2012

### lurflurf

There are different ways of writing that out, you might not have pick the most simple. Another way is something like

A^n=(3^n)(1/2)((1+(-1)^n)I+(1/3)(1-(-1)^n)A)

which you can see is lot like yours, note all that (-1)^n stuff is just to unify the even and odd terms

even A^n=(3^n)I

odd A^n=(3^n)(1/3)A

I=A^0 the 2x2 identity matrix

3. Nov 18, 2012

### HallsofIvy

Staff Emeritus
Frankly, the first thing I would have done would be to note that
$$A^2= \begin{pmatrix}1 & 4 \\ 2 & -1\end{pmatrix}^2= \begin{pmatrix}9 & 0 \\ 0 & 9\end{pmatrix}= 9\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$$

From that it follows immediately that if n is even, $A^n= 3^nI$ and if n is odd, $A^n= 3^{n-1}A$

4. Nov 18, 2012

### 1up20x6

Thanks for your responses. Are those equations self-evident or is there a proof that they apply for all values of $n$?


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## Precalculus: Mathematics for Calculus, 7th Edition

For a quadratic function in standard form: f(x) = a(x-h)^2 + k (a) the vertex is always (h,k) (b) If a$\gt$0 this means that the parabola opens UPWARD with the vertex being on the bottom, thus if a$\gt$0 the vertex is the MINIMUM point of the parabola (c) If a$\lt$0 this means the parabola opens DOWNWARD with the vertex on the top, therefore if a$\lt$0 the vertex is the MAXIMUM point of the parabola.


text:
# Troposphere

Last updated

The troposphere is the lowest layer of Earth's atmosphere, and is also where nearly all weather conditions take place. It contains approximately 75% of the atmosphere's mass and 99% of the total mass of water vapor and aerosols. [2] The average height of the troposphere is 18 km (11 mi; 59,000 ft) in the tropics, 17 km (11 mi; 56,000 ft) in the middle latitudes, and 6 km (3.7 mi; 20,000 ft) in the polar regions in winter. The total average height of the troposphere is 13 km.

Weather is the state of the atmosphere, describing for example the degree to which it is hot or cold, wet or dry, calm or stormy, clear or cloudy. Most weather phenomena occur in the lowest level of the atmosphere, the troposphere, just below the stratosphere. Weather refers to day-to-day temperature and precipitation activity, whereas climate is the term for the averaging of atmospheric conditions over longer periods of time. When used without qualification, "weather" is generally understood to mean the weather of Earth.

An atmosphere is a layer or a set of layers of gases surrounding a planet or other material body, that is held in place by the gravity of that body. An atmosphere is more likely to be retained if the gravity it is subject to is high and the temperature of the atmosphere is low.

Mass is both a property of a physical body and a measure of its resistance to acceleration when a net force is applied. The object's mass also determines the strength of its gravitational attraction to other bodies.

## Contents

The lowest part of the troposphere, where friction with the Earth's surface influences air flow, is the planetary boundary layer. This layer is typically a few hundred meters to 2 km (1.2 mi; 6,600 ft) deep depending on the landform and time of day. Atop the troposphere is the tropopause, which is the border between the troposphere and stratosphere. The tropopause is an inversion layer, where the air temperature ceases to decrease with height and remains constant through its thickness. [3]

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:

In meteorology the planetary boundary layer (PBL), also known as the atmospheric boundary layer (ABL) or peplosphere, is the lowest part of the atmosphere and its behaviour is directly influenced by its contact with a planetary surface. On Earth it usually responds to changes in surface radiative forcing in an hour or less. In this layer physical quantities such as flow velocity, temperature, moisture, etc., display rapid fluctuations (turbulence) and vertical mixing is strong. Above the PBL is the "free atmosphere", where the wind is approximately geostrophic, while within the PBL the wind is affected by surface drag and turns across the isobars.

A landform is a natural feature of the solid surface of the Earth or other planetary body. Landforms together make up a given terrain, and their arrangement in the landscape is known as topography. Typical landforms include hills, mountains, plateaus, canyons, and valleys, as well as shoreline features such as bays, peninsulas, and seas, including submerged features such as mid-ocean ridges, volcanoes, and the great ocean basins.

The word troposphere is derived from the Greek tropos (meaning "turn, turn toward, change") and sphere (as in the Earth), reflecting the fact that rotational turbulent mixing plays an important role in the troposphere's structure and behaviour. Most of the phenomena associated with day-to-day weather occur in the troposphere. [3]

Greek is an independent branch of the Indo-European family of languages, native to Greece, Cyprus and other parts of the Eastern Mediterranean and the Black Sea. It has the longest documented history of any living Indo-European language, spanning more than 3000 years of written records. Its writing system has been the Greek alphabet for the major part of its history; other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic, and many other writing systems.

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.

## Pressure and temperature structure

### Composition

By volume, dry air contains 78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide, and small amounts of other gases. Air also contains a variable amount of water vapor. Except for the water vapor content, the composition of the troposphere is essentially uniform. The source of water vapor is at the Earth's surface through the process of evaporation. The temperature of the troposphere decreases with altitude. And, saturation vapor pressure decreases strongly as temperature drops. Hence, the amount of water vapor that can exist in the atmosphere decreases strongly with altitude and the proportion of water vapor is normally greatest near the surface of the Earth.

Nitrogen is a chemical element with symbol N and atomic number 7. It was first discovered and isolated by Scottish physician Daniel Rutherford in 1772. Although Carl Wilhelm Scheele and Henry Cavendish had independently done so at about the same time, Rutherford is generally accorded the credit because his work was published first. The name nitrogène was suggested by French chemist Jean-Antoine-Claude Chaptal in 1790, when it was found that nitrogen was present in nitric acid and nitrates. Antoine Lavoisier suggested instead the name azote, from the Greek ἀζωτικός "no life", as it is an asphyxiant gas; this name is instead used in many languages, such as French, Russian, Romanian and Turkish, and appears in the English names of some nitrogen compounds such as hydrazine, azides and azo compounds.

Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group on the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as well as with other compounds. By mass, oxygen is the third-most abundant element in the universe, after hydrogen and helium. At standard temperature and pressure, two atoms of the element bind to form dioxygen, a colorless and odorless diatomic gas with the formula O
2
. Diatomic oxygen gas constitutes 20.8% of the Earth's atmosphere. As compounds including oxides, the element makes up almost half of the Earth's crust.

Argon is a chemical element with symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in the Earth's atmosphere, at 0.934%. It is more than twice as abundant as water vapor, 23 times as abundant as carbon dioxide, and more than 500 times as abundant as neon. Argon is the most abundant noble gas in Earth's crust, comprising 0.00015% of the crust.

### Pressure

The pressure of the atmosphere is maximum at sea level and decreases with altitude. This is because the atmosphere is very nearly in hydrostatic equilibrium so that the pressure is equal to the weight of air above a given point. The change in pressure with altitude can be equated to the density with the hydrostatic equation [4]

Mean sea level (MSL) is an average level of the surface of one or more of Earth's oceans from which heights such as elevation may be measured. MSL is a type of vertical datum – a standardised geodetic datum – that is used, for example, as a chart datum in cartography and marine navigation, or, in aviation, as the standard sea level at which atmospheric pressure is measured to calibrate altitude and, consequently, aircraft flight levels. A common and relatively straightforward mean sea-level standard is the midpoint between a mean low and mean high tide at a particular location.

In fluid mechanics, a fluid is said to be in hydrostatic equilibrium or hydrostatic balance when it is at rest, or when the flow velocity at each point is constant over time. This occurs when external forces such as gravity are balanced by a pressure-gradient force. For instance, the pressure-gradient force prevents gravity from collapsing Earth's atmosphere into a thin, dense shell, whereas gravity prevents the pressure gradient force from diffusing the atmosphere into space.

${\displaystyle {\frac {dP}{dz}}=-\rho g_{n}=-{\frac {mPg_{n}}{RT}}}$

where:

Altitude or height is defined based on the context in which it is used. As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context. Although the term altitude is commonly used to mean the height above sea level of a location, in geography the term elevation is often preferred for this usage.

Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the pressure relative to the ambient pressure.

The gas constant is also known as the molar, universal, or ideal gas constant, denoted by the symbol R or R and is equivalent to the Boltzmann constant, but expressed in units of energy per temperature increment per mole, i.e. the pressure–volume product, rather than energy per temperature increment per particle. The constant is also a combination of the constants from Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. It is a physical constant that is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation.

Since temperature in principle also depends on altitude, one needs a second equation to determine the pressure as a function of altitude as discussed in the next section.

### Temperature

The temperature of the troposphere generally decreases as altitude increases. The rate at which the temperature decreases, ${\displaystyle -dT/dz}$, is called the environmental lapse rate (ELR). The ELR is nothing more than the difference in temperature between the surface and the tropopause divided by the height. The ELR assumes that the air is perfectly still, i.e. that there is no mixing of the layers of air from vertical convection, nor winds that would create turbulence and hence mixing of the layers of air. The reason for this temperature difference is that the ground absorbs most of the sun's energy, which then heats the lower levels of the atmosphere with which it is in contact. Meanwhile, the radiation of heat at the top of the atmosphere results in the cooling of that part of the atmosphere.

Environmental Lapse Rate (ELR)
Altitude RegionLapse rateLapse Rate
(m)(Kelvin/km)(°F/1000 feet)
0 – 11,0006.53.57
11,000 – 20,0000.00.0
20,000 – 32,000-1.0-0.55
32,000 – 47,000-2.8-1.54
47,000 – 51,0000.00.0
51,000 – 71,0002.81.54
71,000 – 85,0002.01.09

The ELR assumes the atmosphere is still, but as air is heated it becomes buoyant and rises. The dry adiabatic lapse rate accounts for the effect of the expansion of dry air as it rises in the atmosphere and wet adiabatic lapse rates includes the effect of the condensation of water vapor on the lapse rate.

When a parcel of air rises, it expands, because the pressure is lower at higher altitudes. As the air parcel expands, it pushes the surrounding air outward, transferring energy in the form of work from that parcel to the atmosphere. As energy transfer to a parcel of air by way of heat is very slow, it is assumed to not exchange energy by way of heat with the environment. Such a process is called an adiabatic process (no energy transfer by way of heat). Since the rising parcel of air is losing energy as it does work on the surrounding atmosphere and no energy is transferred into it as heat from the atmosphere to make up for the loss, the parcel of air is losing energy, which manifests itself as a decrease in the temperature of the air parcel. The reverse, of course, will be true for a parcel of air that is sinking and is being compressed. [3]

Since the process of compression and expansion of an air parcel can be considered reversible and no energy is transferred into or out of the parcel, such a process is considered isentropic, meaning that there is no change in entropy as the air parcel rises and falls, ${\displaystyle dS=0}$. Since the heat exchanged ${\displaystyle dQ=0}$ is related to the entropy change ${\displaystyle dS}$ by ${\displaystyle dQ=TdS}$, the equation governing the temperature as a function of height for a thoroughly mixed atmosphere is

${\displaystyle {\frac {dS}{dz}}=0}$

where S is the entropy. The above equation states that the entropy of the atmosphere does not change with height. The rate at which temperature decreases with height under such conditions is called the adiabatic lapse rate.

For dry air, which is approximately an ideal gas, we can proceed further. The adiabatic equation for an ideal gas is [5]

${\displaystyle p(z)T(z)^{-{\frac {\gamma }{\gamma -1}}}={\text{constant}}}$

where ${\displaystyle \gamma }$ is the heat capacity ratio (${\displaystyle \gamma }$=7/5, for air). Combining with the equation for the pressure, one arrives at the dry adiabatic lapse rate, [6]

${\displaystyle {\frac {dT}{dz}}=-{\frac {mg}{R}}{\frac {\gamma -1}{\gamma }}=-9.8^{\circ }\mathrm {C} /\mathrm {km} }$

If the air contains water vapor, then cooling of the air can cause the water to condense, and the behavior is no longer that of an ideal gas. If the air is at the saturated vapor pressure, then the rate at which temperature drops with height is called the saturated adiabatic lapse rate. More generally, the actual rate at which the temperature drops with altitude is called the environmental lapse rate. In the troposphere, the average environmental lapse rate is a drop of about 6.5 °C for every 1 km (1,000 meters) in increased height. [3]

The environmental lapse rate (the actual rate at which temperature drops with height, ${\displaystyle dT/dz}$) is not usually equal to the adiabatic lapse rate (or correspondingly, ${\displaystyle dS/dz\neq 0}$). If the upper air is warmer than predicted by the adiabatic lapse rate (${\displaystyle dS/dz>0}$), then when a parcel of air rises and expands, it will arrive at the new height at a lower temperature than its surroundings. In this case, the air parcel is denser than its surroundings, so it sinks back to its original height, and the air is stable against being lifted. If, on the contrary, the upper air is cooler than predicted by the adiabatic lapse rate, then when the air parcel rises to its new height it will have a higher temperature and a lower density than its surroundings, and will continue to accelerate upward. [3] [4]

The troposphere is heated from below by latent heat, longwave radiation, and sensible heat. Surplus heating and vertical expansion of the troposphere occurs in the tropics. At middle latitudes, tropospheric temperatures decrease from an average of 15 °C at sea level to about −55 °C at the tropopause. At the poles, tropospheric temperature only decreases from an average of 0 °C at sea level to about −45 °C at the tropopause. At the equator, tropospheric temperatures decrease from an average of 20 °C at sea level to about −70 to −75 °C at the tropopause. The troposphere is thinner at the poles and thicker at the equator. The average thickness of the tropical troposphere is roughly 7 kilometers greater than the average tropospheric thickness at the poles. [7]

### Tropopause

The tropopause is the boundary region between the troposphere and the stratosphere.

Measuring the temperature change with height through the troposphere and the stratosphere identifies the location of the tropopause. In the troposphere, temperature decreases with altitude. In the stratosphere, however, the temperature remains constant for a while and then increases with altitude. This coldest layer of the atmosphere, where the lapse rate changes from positive (in the troposphere) to negative (in the stratosphere), is defined as the tropopause. [3] Thus, the tropopause is an inversion layer, and there is little mixing between the two layers of the atmosphere.

## Atmospheric flow

The flow of the atmosphere generally moves in a west to east direction. This, however, can often become interrupted, creating a more north to south or south to north flow. These scenarios are often described in meteorology as zonal or meridional. These terms, however, tend to be used in reference to localised areas of atmosphere (at a synoptic scale). A fuller explanation of the flow of atmosphere around the Earth as a whole can be found in the three-cell model.

### Zonal flow

A zonal flow regime is the meteorological term meaning that the general flow pattern is west to east along the Earth's latitude lines, with weak shortwaves embedded in the flow. [8] The use of the word "zone" refers to the flow being along the Earth's latitudinal "zones". This pattern can buckle and thus become a meridional flow.

### Meridional flow

When the zonal flow buckles, the atmosphere can flow in a more longitudinal (or meridional) direction, and thus the term "meridional flow" arises. Meridional flow patterns feature strong, amplified troughs of low pressure and ridges of high pressure, with more north-south flow in the general pattern than west-to-east flow. [9]

### Three-cell model

The three cells model of the atmosphere attempts to describe the actual flow of the Earth's atmosphere as a whole. It divides the Earth into the tropical (Hadley cell), mid latitude (Ferrel cell), and polar (polar cell) regions, to describe energy flow and global atmospheric circulation (mass flow). Its fundamental principle is that of balance – the energy that the Earth absorbs from the sun each year is equal to that which it loses to space by radiation. This overall Earth energy balance, however, does not apply in each latitude due to the varying strength of the sun in each "cell" as a result of the tilt of the Earth's axis in relation to its orbit. The result is a circulation of the atmosphere that transports warm air poleward from the tropics and cold air equatorward from the poles. The effect of the three cells is the tendency to even out the heat and moisture in the Earth's atmosphere around the planet. [10]

## Synoptic scale observations and concepts

### Forcing

Forcing is a term used by meteorologists to describe the situation where a change or an event in one part of the atmosphere causes a strengthening change in another part of the atmosphere. It is usually used to describe connections between upper, middle or lower levels (such as upper-level divergence causing lower level convergence in cyclone formation), but also be to describe such connections over lateral distance rather than height alone. In some respects, teleconnections could be considered a type of forcing..

### Divergence and convergence

An area of convergence is one in which the total mass of air is increasing with time, resulting in an increase in pressure at locations below the convergence level (recall that atmospheric pressure is just the total weight of air above a given point). Divergence is the opposite of convergence – an area where the total mass of air is decreasing with time, resulting in falling pressure in regions below the area of divergence. Where divergence is occurring in the upper atmosphere, there will be air coming in to try to balance the net loss of mass (this is called the principle of mass conservation), and there is a resulting upward motion (positive vertical velocity). Another way to state this is to say that regions of upper air divergence are conducive to lower level convergence, cyclone formation, and positive vertical velocity. Therefore, identifying regions of upper air divergence is an important step in forecasting the formation of a surface low pressure area.

## Related Research Articles

The tropopause is the boundary in the Earth's atmosphere between the troposphere and the stratosphere. It is a thermodynamic gradient stratification layer, marking the end of troposphere. It lies, on average, at 17 kilometres (11 mi) above equatorial regions, and above 9 kilometres (5.6 mi) over the polar regions.

The atmosphere of Earth is the layer of gases, commonly known as air, that surrounds the planet Earth and is retained by Earth's gravity. The atmosphere of Earth protects life on Earth by creating pressure allowing for liquid water to exist on the Earth's surface, absorbing ultraviolet solar radiation, warming the surface through heat retention, and reducing temperature extremes between day and night.

The lapse rate is the rate at which an atmospheric variable, normally temperature in Earth's atmosphere, changes with altitude. Lapse rate arises from the word lapse, in the sense of a gradual change. It corresponds to the vertical component of the spatial gradient of temperature. Although this concept is most often applied to the Earth's troposphere, it can be extended to any gravitationally supported parcel of gas.

Equivalent potential temperature, commonly referred to as theta-e , is a quantity that is conserved during changes to an air parcel's pressure, even if water vapor condenses during that pressure change. It is therefore more conserved than the ordinary potential temperature, which remains constant only for unsaturated vertical motions.

Alpine climate is the average weather (climate) for the regions above the tree line. This climate is also referred to as a mountain climate or highland climate.

The density of air or atmospheric density, denoted ρ, is the mass per unit volume of Earth's atmosphere. Air density, like air pressure, decreases with increasing altitude. It also changes with variation in atmospheric pressure, temperature and humidity. At 1013,25 hPa (abs) and 15°C, air has a density of approximately 1.225 kg/m³ according to ISA.

The barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure of the air changes with altitude. The pressure drops approximately by 11.3 Pa per meter in first 1000 meters above sea level.

The pressure-gradient force is the force which results when there is a difference in pressure across a surface. In general, a pressure is a force per unit area, across a surface. A difference in pressure across a surface then implies a difference in force, which can result in an acceleration according to Newton's second law of motion, if there is no additional force to balance it. The resulting force is always directed from the region of higher-pressure to the region of lower-pressure. When a fluid is in an equilibrium state, the system is referred to as being in hydrostatic equilibrium. In the case of atmospheres, the pressure gradient force is balanced by the gravitational force, maintaining hydrostatic equilibrium. In Earth's atmosphere, for example, air pressure decreases at altitudes above Earth's surface, thus providing a pressure gradient force which counteracts the force of gravity on the atmosphere.

In meteorology, convective available potential energy, is the amount of energy a clump of air (parcel) would have if lifted a certain distance vertically through the atmosphere. CAPE is effectively the positive buoyancy of an air parcel and is an indicator of atmospheric instability, which makes it very valuable in predicting severe weather. It is a form of fluid instability found in thermally stratified atmospheres in which a colder fluid overlies a warmer one. An air mass will rise if it is less dense than the surrounding air. This can create vertically developed clouds due to the rising motion, which could lead to thunderstorms. It could also be created by other phenomena, such as a cold front. Even if the air is cooler on the surface, there is still warmer air in the mid-levels, that can rise into the upper-levels. However, if there is not enough water vapor present, there is no ability for condensation, thus storms, clouds, and rain will not form.

In various scientific contexts, a scale height is a distance over which a quantity decreases by a factor of e. It is usually denoted by the capital letter H.

The potential temperature of a parcel of fluid at pressure is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure , usually 1000 millibars. The potential temperature is denoted and, for a gas well-approximated as ideal, is given by

A hot tower is a tropical cumulonimbus cloud that reaches out of the lowest layer of the atmosphere, the troposphere, and into the stratosphere. In the tropics, the border between the troposphere and stratosphere, the tropopause, typically lies at least 15 kilometres (9.3 mi) above sea level. These formations are called "hot" because of the large amount of latent heat released as water vapor condenses into liquid and freezes into ice. The presence of hot towers within the eyewall of a tropical cyclone can indicate possible future strengthening.

In meteorology, convective instability or stability of an air mass refers to its ability to resist vertical motion. A stable atmosphere makes vertical movement difficult, and small vertical disturbances dampen out and disappear. In an unstable atmosphere, vertical air movements tend to become larger, resulting in turbulent airflow and convective activity. Instability can lead to significant turbulence, extensive vertical clouds, and severe weather such as thunderstorms.

The lifted condensation level or lifting condensation level (LCL) is formally defined as the height at which the relative humidity (RH) of an air parcel will reach 100% with respect to liquid water when it is cooled by dry adiabatic lifting. The RH of air increases when it is cooled, since the amount of water vapor in the air remains constant, while the saturation vapor pressure decreases almost exponentially with decreasing temperature. If the air parcel is lifting further beyond the LCL, water vapor in the air parcel will begin condensing, forming cloud droplets. The LCL is a good approximation of the height of the cloud base which will be observed on days when air is lifted mechanically from the surface to the cloud base.

A reference atmospheric model describe how the ideal gas properties of an atmosphere change, primarily as a function of altitude, and sometimes also as a function of latitude, day of year, etc. A static atmospheric model has a more limited domain, excluding time. A standard atmosphere is defined by the World Meteorological Organization as "a hypothetical vertical distribution of atmospheric temperature, pressure and density which, by international agreement, is roughly representative of year-round, midlatitude conditions."

Atmospheric thermodynamics is the study of heat-to-work transformations that take place in the earth's atmosphere and manifest as weather or climate. Atmospheric thermodynamics use the laws of classical thermodynamics, to describe and explain such phenomena as the properties of moist air, the formation of clouds, atmospheric convection, boundary layer meteorology, and vertical instabilities in the atmosphere. Atmospheric thermodynamic diagrams are used as tools in the forecasting of storm development. Atmospheric thermodynamics forms a basis for cloud microphysics and convection parameterizations used in numerical weather models and is used in many climate considerations, including convective-equilibrium climate models.

Atmospheric instability is a condition where the Earth's atmosphere is generally considered to be unstable and as a result the weather is subjected to a high degree of variability through distance and time. Atmospheric stability is a measure of the atmosphere's tendency to discourage or deter vertical motion, and vertical motion is directly correlated to different types of weather systems and their severity. In unstable conditions, a lifted thing, such as a parcel of air will be warmer than the surrounding air at altitude. Because it is warmer, it is less dense and is prone to further ascent.

The moist static energy is a thermodynamic variable that describes the state of an air parcel, and is similar to the equivalent potential temperature. The moist static energy is a combination of a parcel's enthalpy due to an air parcel's internal energy and energy required to make room for it, its potential energy due to its height above the surface, and the latent energy due to water vapor present in the air parcel. It is a useful variable for researching the atmosphere because, like several other similar variables, it is approximately conserved during adiabatic ascent and descent.

Vertical pressure variation is the variation in pressure as a function of elevation. Depending on the fluid in question and the context being referred to, it may also vary significantly in dimensions perpendicular to elevation as well, and these variations have relevance in the context of pressure gradient force and its effects. However, the vertical variation is especially significant, as it results from the pull of gravity on the fluid; namely, for the same given fluid, a decrease in elevation within it corresponds to a taller column of fluid weighing down on that point.

## References

1. "STS-130 Shuttle Mission Imagery". NASA . Retrieved 21 September 2012.
2. "Troposphere". Concise Encyclopedia of Science & Technology. McGraw-Hill. 1984. It contains about four-fifths of the mass of the whole atmosphere.
3. Danielson, Levin, and Abrams (2003). Meteorology. McGraw Hill.CS1 maint: Uses authors parameter (link)
4. Landau and Lifshitz, Fluid Mechanics, Pergamon, 1979
5. Landau and Lifshitz, Statistical Physics Part 1, Pergamon, 1980
6. Kittel and Kroemer, Thermal Physics, Freeman, 1980; chapter 6, problem 11
7. Paul E. Lydolph (1985). "The Climate of the Earth". Rowman and Littlefield Publishers Inc. p. 12.Missing or empty |url= (help)
8. "American Meteorological Society Glossary – Zonal Flow". Allen Press Inc. June 2000. Archived from the original on 2007-03-13. Retrieved 2006-10-03.
9. "American Meteorological Society Glossary – Meridional Flow". Allen Press Inc. June 2000. Archived from the original on 2006-10-26. Retrieved 2006-10-03.
10. "Meteorology – MSN Encarta, "Energy Flow and Global Circulation"". Encarta.Msn.com. Archived from the original on 2009-10-31. Retrieved 2006-10-13.


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## 957 Reputation

14 years, 312 days
University of Kent
Dr

## A tiny latex (from Maple) superscripts p...

Maple 18

v1:=[seq(tau[lambda,i]^2,i=1..5)];

latex(v1);

gives:

[{\tau_{{\lambda,1}}}^{2},{\tau_{{\lambda,2}}}^{2},{\tau_{{\lambda,3}}
}^{2},{\tau_{{\lambda,4}}}^{2},{\tau_{{\lambda,5}}}^{2}]

which produces a non natural output in latex, see below.

I wonder it could be improved in the future release?

'latex' command is really useful when i have large algebric output. But having to change this tiny thing at many places is just very time consuming. I'd really appreciate if this can be fixed in updates or future versions of Maple.

Casper

## simplify and function with arguments...

Maple 18

Hi all,

I will use the following dummy example.

with function,f

f:=(xid,yid)->sum(x[i],i=1..xid)*sum(y[i],i=1..yid);

and a complicated term, myterm

myterm:=(f(3,4)+f(2,2))*f(1,1):
myterm:=expand(myterm);

'if' i have some previous knowledge, or know a bit of the term, i can find the structure by doing this

repar:=[f(1,1),f(2,2),f(3,4)];  # Or with more f(xid,yid) terms

tmp:=seq(repar[i]=ff[i],i=1..3);
simplify(myterm,{tmp});          # This is fine, gives me what i want

But, can we go further, and more 'obvious'

Given the fucntion f, same as before, and the same 'myterm'

can I have this
restart:
iwant:=(f(3,4)+f(2,2))*f(1,1);  # as a result, straightforward

so I dont have to go back to 'repar' and find that the terms exactly are.

Thanks,

## return the indicis of a value from list...

Maple 18

Hi, with a list

l:=[1,1,1,2,3,3,4];

What's the best way to get the index(s) for the values equal to '1'?

Say for x=1, we want

[1,2,3]

for x=2, we want

[4]

ect.

## How to take away each element in a Vecto...

Maple 18

Hi say I have the vector V1.

V1:=Vector([a,b,c,d,e,f,g]):

and function myfun.

how do i use it as the input to the function my fun, by taking away each element in turn?
myfun(V1[2..]);              # 1st element removed
myfun(V1[[1,3..]]);         # 2nd element removed
myfun(V1[[1,2,4..]]);      # 3rd element removed

and so go

is there a more efficient way?

Many thanks,

## pick up by the sum of the list elements...

Maple 18

Say I have this list,

tmp:=[[0, 1, 2], [1, 0, 2], [1, 1, 2], [1, 2, 0]];

and the sums of each element (list),

map(x->convert(x,+),tmp);

How do I quickly pickup the elements, where they sum to 3? Like this:

wanttohave:=[[0, 1, 2], [1, 0, 2],  [1, 2, 0]];

Thanks,

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text:
# Find the critical numbers of the function on the interval 0 ≤ θ < 2π. g(θ) = 4 θ - tan(θ)

By definition, critical numbers of a function `g` are the values of argument at which `g '` is zero or does not exist.

The given function `g ( theta ) = 4 theta - tan ( theta )` itself is defined everywhere except `theta = pi / 2 + n...

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

By definition, critical numbers of a function `g` are the values of argument at which `g '` is zero or does not exist.

The given function `g ( theta ) = 4 theta - tan ( theta )` itself is defined everywhere except `theta = pi / 2 + n pi , n in ZZ.`

Its first derivative is `g ' ( theta ) = 4 - 1 / cos ^ 2 ( theta ),` which also exists everywhere except `theta = pi / 2 + n pi , n in ZZ.` Where does it equal zero?

`4 - 1 / cos ^ 2 ( theta ) = 0` is the same as `cos ^ 2 ( theta ) = 1 / 4,` which is the same as `cos ( theta ) = +- 1 / 2 .` Such points are `+- pi / 3 + 2 n pi, n in ZZ` and `+- ( 2 pi ) / 3 + 2 n pi , n in ZZ.`

The last task is to determine which of these points belong to the given interval (its inner part `( 0 , 2 pi )`). They are `pi / 3 , ( 2 pi ) / 3 , ( 4 pi ) / 3 , ( 5 pi ) / 3.`

Approved by eNotes Editorial Team


text:
# How to find the algebraic multiplicity given the eigenvalues and eigenspaces?

Let A be a 4x4 matrix with eigenvalues $\lambda$ = 2,3 and eigenspaces

$E_{\lambda=2} = \operatorname{span} \left\{ {\begin{bmatrix} 1\\0\\0\\1\end{bmatrix}, \begin{bmatrix} 1\\0\\1\\0\end{bmatrix}}\right\}$ and $E_{\lambda=3} = \operatorname{span} \left\{ {\begin{bmatrix} 0\\1\\0\\1\end{bmatrix}, \begin{bmatrix} 0\\0\\1\\1\end{bmatrix}}\right\}$

(A) Can you calculate the algebraic multiplicities of $\lambda$ = 2 and $\lambda$ = 3? If yes, do it.
(B) Can you calculate A? If yes, how?

I know how to calculate the algebraic multiplicities given the matrix A, but I do not know where to start when I am only given the eigenvalues and eigenspaces. My guess as to how to find A is to use the theorem A $\vec x$ = $\lambda$$\vec x$
For example, later in the problem I am given $\vec x = \begin{bmatrix} 3\\1\\4\\6\end{bmatrix}$
Solving A $\vec x$ = $\lambda$ $\vec x$, I get $A\begin{bmatrix} 3\\1\\4\\6\end{bmatrix} = \begin{bmatrix} 6\\3\\11\\16\end{bmatrix}$

How do I find A from here?

Additionally, I know that I cannot recover A from the eigenpairs if A is deficient, so I know I need to solve (a) first in order to solve (b). However, I have no idea how to find the algebraic multiplicities of A without first knowing A and finding the characteristic polynomial.

• Get us started, Sarah... – JohnD Dec 9 '14 at 22:23
• What are your thoughts on the problem? Do you think that either can be calculated, any thoughts as to how? Did you have any other ideas? – Omnomnomnom Dec 9 '14 at 22:23
• Essential: Down with Determinants! (axler.net/DwD.html) or why the algebraic multiplicity is as geometric as the geometric multiplicity. – Martín-Blas Pérez Pinilla Dec 11 '14 at 8:20

Recall that the geometric multiplicities are always at most the algebraic multiplicities; in other words $A(\lambda)\geq G(\lambda)$. By the information we already have, we know that $A(2)\geq 2$ and $A(3)\geq 2$.
Moreover, some general theory: we know that the characteristic polynomial has degree $n$, and by the Fundamental Theorem of Algebra every degree-$n$ real polynomial has $n$ complex roots. Therefore, $\sum_\lambda A(\lambda)=n$.
Applying the general theory to our situation, we have $A(2)+A(3)=4$.


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# math

posted by Emma

A sector in a circle of radius 8.1 m has an area of 53.5 m^2. The central angle in a sector must be _____o. Round to one decimal place.

1. MathMate

Hint:
a sector is a pie-piece, similar to a slice of pizza.

The area of a sector of a given radius is proportional to the central angle.

Area of Sector
--------------
Area of circle

central angle
=-------------
360°

2. Emma

okay thanks!

3. MathMate

You're welcome!

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Rapid developments in streaming data technologies are continuing to generate increased interest in monitoring human activity. Wearable devices, such as wrist-worn sensors that monitor gross motor activity (actigraphy), have become prevalent. An actigraph unit continually records the activity level of an individual, producing a very large amount of data at a high-resolution that can be immediately downloaded and analyzed. While this kind of \textit{big data} includes both spatial and temporal information, the variation in such data seems to be more appropriately modeled by considering stochastic evolution through time while accounting for spatial information separately. We propose a comprehensive Bayesian hierarchical modeling and inferential framework for actigraphy data reckoning with the massive sizes of such databases while attempting to offer full inference. Building upon recent developments in this field, we construct Nearest Neighbour Gaussian Processes (NNGPs) for actigraphy data to compute at large temporal scales. More specifically, we construct a temporal NNGP and we focus on the optimized implementation of the collapsed algorithm in this specific context. This approach permits improved model scaling while also offering full inference. We test and validate our methods on simulated data and subsequently apply and verify their predictive ability on an original dataset concerning a health study conducted by the Fielding School of Public Health of the University of California, Los Angeles.

Alaimo Di Loro, P., Mingione, M., Lipsitt, J., Batteate, C.M., Jerrett, M., Banerjee, S. (2021). Bayesian hierarchical modeling and analysis for physical activity trajectories using actigraph data.

### Bayesian hierarchical modeling and analysis for physical activity trajectories using actigraph data

#### Abstract

Rapid developments in streaming data technologies are continuing to generate increased interest in monitoring human activity. Wearable devices, such as wrist-worn sensors that monitor gross motor activity (actigraphy), have become prevalent. An actigraph unit continually records the activity level of an individual, producing a very large amount of data at a high-resolution that can be immediately downloaded and analyzed. While this kind of \textit{big data} includes both spatial and temporal information, the variation in such data seems to be more appropriately modeled by considering stochastic evolution through time while accounting for spatial information separately. We propose a comprehensive Bayesian hierarchical modeling and inferential framework for actigraphy data reckoning with the massive sizes of such databases while attempting to offer full inference. Building upon recent developments in this field, we construct Nearest Neighbour Gaussian Processes (NNGPs) for actigraphy data to compute at large temporal scales. More specifically, we construct a temporal NNGP and we focus on the optimized implementation of the collapsed algorithm in this specific context. This approach permits improved model scaling while also offering full inference. We test and validate our methods on simulated data and subsequently apply and verify their predictive ability on an original dataset concerning a health study conducted by the Fielding School of Public Health of the University of California, Los Angeles.
##### Scheda breve Scheda completa Scheda completa (DC)
2021
Alaimo Di Loro, P., Mingione, M., Lipsitt, J., Batteate, C.M., Jerrett, M., Banerjee, S. (2021). Bayesian hierarchical modeling and analysis for physical activity trajectories using actigraph data.
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Presentation Open Access

# Evolution of magnetic activity on the main sequence as a function of spectral type using Kepler data

Mathur, Savita; et al.

### DataCite XML Export

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<identifier identifierType="DOI">10.5281/zenodo.5571808</identifier>
<creators>
<creator>
<creatorName>Mathur, Savita</creatorName>
<givenName>Savita</givenName>
<familyName>Mathur</familyName>
<nameIdentifier nameIdentifierScheme="ORCID" schemeURI="http://orcid.org/">0000-0002-0129-0316</nameIdentifier>
<affiliation>Instituto de Astrofisica de Canarias</affiliation>
</creator>
<creator>
<creatorName>et al.</creatorName>
</creator>
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<titles>
<title>Evolution of magnetic activity on the main sequence as a function of spectral type using Kepler data</title>
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<publisher>Zenodo</publisher>
<publicationYear>2021</publicationYear>
<dates>
<date dateType="Issued">2021-10-15</date>
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<relatedIdentifier relatedIdentifierType="URL" relationType="IsPartOf">https://zenodo.org/communities/plato2021</relatedIdentifier>
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<rights rightsURI="info:eu-repo/semantics/openAccess">Open Access</rights>
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<descriptions>
<description descriptionType="Abstract">
Stellar magnetic activity studies are very important for different fields of astrophysics. Several spectroscopic surveys have been aimed at characterizing the&nbsp;magnetic activity of solar-like stars, especially to look for cycles. These surveys were mostly led to put the Sun into context and in time compared to other stars. By investigating the magnetic activity of other stars with different conditions (rotation periods, metallicity&hellip;), we can provide additional constraints to dynamo models. Stellar magnetic activity has a direct impact on the habitability of exoplanets hosted by those stars. Consequently,&nbsp;it is important to understand how magnetic activity evolves&nbsp;in time and as a function of spectral type.

The recent catalog of rotation periods and photometric magnetic activity proxies for more than 55,000 stars observed by the Kepler mission opens the possibility to study the surface magnetic activity of a large number of stars. In this talk, we will present a subsample of main-sequence stars in order to compare the Sun to Sun-like stars and show the effect of metallicity using high-resolution spectroscopic data. While we see an interesting behavior as a function of metallicity, we also find that the magnetic activity of the Sun is comparable to the one of stars selected to be very similar to the Sun based on effective temperature, metallicity, and Rossby number, which is the ratio of rotation period and the convective turnover time and is a key parameter in dynamo theory. For all the stars of our sample, we also compute ages based on models taking into account the most recent theory of angular momentum transport that reproduce rotation rates for the Kepler asteroseismic sample. This allows us to study the evolution of magnetic activity as a function of Rossby number and age, providing a more complete picture to understand the changes in the dynamo behaviors during the life of the star until the terminal age main sequence.
</description>
</descriptions>
</resource>

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### #Actualharshman_chris

Posted 17 January 2013 - 10:25 AM

I like warning free code, especially for a dll, these warnings are the only left right now I have to deal with. I am not sure how to deal with them however. If someone could point me in the right direction that would be great.

std::basic_string<_Elem,_Traits,_Ax>
std::map<_Kty,_Ty>
std::vector<_Ty>
std::list<_Ty>'
std::_List_iterator<_Mylist>

Update:

So to compound this, I get this error when setting up my .exe project:

Error 49 error C2491: 'ParticlesEngine::Singleton<T>::instance' : definition of dllimport static data member not allowed

The Code:

#ifndef _SINGLETON_H
#define _SINGLETON_H

#define NULL 0

#ifdef PARTICLESENGINE_DLL
#define PARTICLESENGINE_API __declspec( dllexport )
#else
#define PARTICLESENGINE_API __declspec( dllimport )
#endif

namespace ParticlesEngine
{

template <typename T>
class PARTICLESENGINE_API Singleton
{
public:
Singleton<T>(void)
{
}

static inline T* getInstance(void)
{
if(instance == NULL)
{
instance = new T();
}
return instance;
}

static inline bool exists( void ) { return instance != 0; }

virtual ~Singleton( void ) { instance = 0; }

protected:
static T* instance;
};

template <typename T>
T* Singleton<T>::instance = 0;

}

#endif

### #1harshman_chris

Posted 17 January 2013 - 09:55 AM

I like warning free code, especially for a dll, these warnings are the only left right now I have to deal with. I am not sure how to deal with them however. If someone could point me in the right direction that would be great.

std::basic_string<_Elem,_Traits,_Ax>
std::map<_Kty,_Ty>
std::vector<_Ty>
std::list<_Ty>'
std::_List_iterator<_Mylist>

PARTNERS


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Protect Relay issues - Sansui 881

MisterBill2

Joined Jan 23, 2018
14,232
If probing some point with the meter causes the relay to operate then I would suspect an open connection someplace. A few years back I was given an amplifier that "just quit working one day", with no smoke or noise or anything. Although it had a Kenwood logo and a Kenwood part number, Kenwood denied ever maaking it, so no circuit information was available at all. Then I examined it very closely and saw some solder joints did not look good. So I resoldered every joint on the PCB, which took almost 20 minutes, because there wer a lot of them. After that the stereo amplifier worked perfectly, and still works perfectly. On some occasions, resoldering all of the connections is a cheap and easy fix, if one is good at soldering. Certainly there are some who can do mare damage with a soldering tip than other could do with a big hammer. THEY should avoid this advice.

My whole point being that a failed solder connection can cause all kinds of problems.

Thread Starter

rfjohnsto

Joined Dec 31, 2020
15
If probing some point with the meter causes the relay to operate then I would suspect an open connection someplace. A few years back I was given an amplifier that "just quit working one day", with no smoke or noise or anything. Although it had a Kenwood logo and a Kenwood part number, Kenwood denied ever maaking it, so no circuit information was available at all. Then I examined it very closely and saw some solder joints did not look good. So I resoldered every joint on the PCB, which took almost 20 minutes, because there wer a lot of them. After that the stereo amplifier worked perfectly, and still works perfectly. On some occasions, resoldering all of the connections is a cheap and easy fix, if one is good at soldering. Certainly there are some who can do mare damage with a soldering tip than other could do with a big hammer. THEY should avoid this advice.

My whole point being that a failed solder connection can cause all kinds of problems.

Yes, good point!

KeepItSimpleStupid

Joined Mar 4, 2014
5,088
These were something like 34.5Mv R and 13.5Mv L or similar on all three.
Those numbers are fine. You start to worry at 750mV. My amp is below 5mv which is very unusual.

VR1 adjusts the offset and Vr3 adjusts the idling current. Most amps don;t have a way to adjust the offset voltage.

There probably is a low value resistor between O and P and Q and P. This is where you measure the idle current. With no signal, the idle current should not run away.

The voltage across Q7e and Q7c matters. It will change based on VR3 and heatsink temperature. You can force it to be zero, but it should not be zero or you'll get crossover distortion. When troubleshooting, I sometimes force this to be zero (short).
All of the low valued resistors matter. You have to get all of the bad parts replaced.

This is where a variac helps. Monitoring the difference between Q7e and Q7c with a signal is the best turn on possibilty. I;ll take that back. With Q7e - Q7c = zero and no load is the best initial turn on possibility

MisterBill2

Joined Jan 23, 2018
14,232
If that amplifier is direct coupled then the excess offset voltage might be coming from an earlier stage. That was common on the direct coupled amps: if the $1 transistor of the input stage goes leaky than the$45 output module gets fried. And that was a 1985 price for the module.

KeepItSimpleStupid

Joined Mar 4, 2014
5,088
C11's, etc (4 caps) working voltage need to be upgraded to 2x the supply. 100V not 50.

Thread Starter

rfjohnsto

Joined Dec 31, 2020
15
If that amplifier is direct coupled then the excess offset voltage might be coming from an earlier stage. That was common on the direct coupled amps: if the $1 transistor of the input stage goes leaky than the$45 output module gets fried. And that was a 1985 price for the module.
I think this one resembles that design... Ouch ..... finger crossed !!

Thread Starter

rfjohnsto

Joined Dec 31, 2020
15
C11's, etc (4 caps) working voltage need to be upgraded to 2x the supply. 100V not 50.
I was using the manual spec when ordering these..

Thread Starter

rfjohnsto

Joined Dec 31, 2020
15
Something's come up today, so can't do any work on it just now. Will be back in day or so.. thx again..

sagor

Joined Mar 10, 2019
797
Looking at TR04 and 05, should the collector of TR05 also be close to 29,6V? With all 3 pins at 0V on TR05, it seems suspicious. It suggests TR05 is conducting, but there should be a small voltage difference between Collector and Emitter.

KeepItSimpleStupid

Joined Mar 4, 2014
5,088
I built the leach amp http://leachlegacy.ece.gatech.edu/lowtim/ and they specified 100 uF/50v in about the same location as yours. The rails were +-50V. One explosively popped.

I had really big filter caps too and a slow turn-on circuit. My slow turn-on circuit died because of large spikes across the power supply. I had to put ZNR's across the main power supply caps. 50V is to close to the power rail in my opinion.

I worked on a surround sound receiver that had come back to a shop 2x. I was moonlighting there.
The Amp IC's were operating too close to the power supply rails and they would eventually die.

sghioto

Joined Dec 31, 2017
3,940
Looking at TR04 and 05, should the collector of TR05 also be close to 29,6V?
No. The collector of TR05 is connected to the base of TR04. On a working unit the voltage there is probably only about 1 volt depending on the current drawn by the relay coil.
The problem was a shorted capacitor C12 which explained all the zero readings.

sagor

Joined Mar 10, 2019
797
No. The collector of TR05 is connected to the base of TR04. On a working unit the voltage there is probably only about 1 volt depending on the current drawn by the relay coil.
The problem was a shorted capacitor C12 which explained all the zero readings.
My error on that 29V, I did not notice the resistor divider network between the 100k and the 22k resistors that the TR05 collector is connected to. When conducting, it should be a very low voltage. When not conducting, it should be around 5V. TR05 is just a driver for TR04. However, my point was that there should not be 0V at the collector (maybe 1V, maybe 5V - does not matter directly), and a shorted C12 does explain it.

sghioto

Joined Dec 31, 2017
3,940
When not conducting, it should be around 5V. TR05 is just a driver for TR04.
TR04 is normally always conducting so the voltage is never above appx 1.5 volts. TR05 is not a driver. TR05 shuts off TR04
when the offset dc is too high by shorting the emitter to base junction. This is the only time TR04 is not conducting. When that happens the voltage on the base of TR04 is practically zero. So you will never see 5 volts on the base of TR04.

rjonst

Joined Nov 4, 2020
1
Hello again,
Thanks for all the replies. The set is working again, thanks to your input, I made a set of test point sheets and constantly measured voltages after replacing a few transistors. The one rail was still low on voltage. Led me to the Driver board, where there were 8 tiny diodes were bad. Four on each side (-)(+). After that, system took off like it was new.


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# Chapter 2 - Chemical Compounds - Questions for Review and Thought - Topical Questions - Page 90b: 46c

Ammonium Phosphate

#### Work Step by Step

Here, we must recognize Phosphate ($PO_4^{3-}$) as well as ammonium $(NH_4^+)$. Based on these charges (which can be obtained from the table), it makes sense that there are three ammonium molecules in the compound for every one phosphate molecule. We see that the name of the compound is: Ammonium Phosphate

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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# what is the frequency selective channel and doubly-selective channel

Could anyone please help me to explain the frequency-selective channel and doubly-selective channel and the difference between them in brief and easy way. I have tried to check on internet, I couldn't get a clear explanation about them.

thanks

Example. For a channel that can be modeled by LTI system, we send an impulse-like signal and receive channel impulse reponse $h(\tau)$ where $\tau$ is delay.

Frequency-flat channel means $h(\tau) \sim \delta(0)$ or for discrete-time version, $h[n] \sim \delta_0$.

Frequency-selective channel means $h(\tau) \neq \delta(0)$ or for $\tau > 0,h(\tau) \neq 0$.

Now if we do the measure at time $t=t_1$ and $t=t_2$, $h(\tau,t_1) = h(\tau,t_2)$, the channel is called non time-selective (anyone remember the name?).

If we do the measure at time $t=t_1$ and $t=t_2$, $h(\tau,t_1) \neq h(\tau,t_2)$, the channel is time-selective.

Doubly selective is simply both time-selective and frequency-selective.

• @Eng.Badr why is that? – AlexTP May 21 '18 at 11:36
• the best channel is AWGN where it is easier to estimate channel. In general, selective characteristics makes channels unpredictable that in turn makes everything worse. Non time selective = no Doppler = no ICI (assuming frame/sample/frequency offsets are correctly estimated and compensated). Maybe I misunderstood what you mean. Again, why do you say that? – AlexTP May 21 '18 at 12:35
• It's ok Alex, .. do you have any contact, If I have a question I will chat with you directly. .. if possible. – New_student May 21 '18 at 12:56
• @Eng.Badr we don't do direct consultations here. If you want to ask something, please do it publicly, so that we can all help you, and everyone can profit from the exchange of knowledge (you're of course free to ask, it's just that asking for private consultation is simply not how this site works). – Marcus Müller May 28 '18 at 16:30