In my math book, everywhere the author has used "non-ascending" instead of descending and "non-descending" instead of ascending. I was wondering if there is some special meaning or use associated with it?  Because the way I see it is that it just increases the number of words and wastes ink.
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In english based math language it seems that non-increasing $\Longleftrightarrow$ less or equal  (non-strict decreasing) decreasing     $\Longleftrightarrow$ strict less    (    strict decreasing) Is that correct ? If so, how does it make sense ? precision I should note that even very good math teacher are making mistakes about this. Actually I asked this question after watching Boyd's video on convex optimization where even him is confused about this.... So I imagine many many people are, and there must be classes and tests about this absurd and buggy concept, which yields absolutely nothing interesting. So I just wonder if I really am missing something, or if, yes, some people decided to create an abstraction that is leaky (not not increasing $\neq$ increasing ?) verbose (4 words, with special negation logic, instead of using the word 'strict'  and keeping the usual well defined predicate logic rules) absurdity of the concept This notation is absurd for the following reason :  when dealing with element instead of functions we dont apply the same logic : we dont phrase $x &lt; y$ as  &quot;$x$ is less than $y$&quot; nor  &quot;$x\leq y$&quot; as &quot;$x$ is not-more than $y$&quot;. (If we did though, at least it would not be so harmful as not not-more would mean more)  you have to define functions using a not notation, $f$ is non-increasing function $\Longrightarrow$ if $x$ is not-less than $y$, say 0.3 feet and 2.5 inches, then $f(x)$ is not-more than $f(y)$   This also violates a very basic tenet in programming style 101, which is here for a reason : never define or use something with a negation, it is confusing.  To apply composition rules between functions, you better be buckled up with all the not. must be a fluff of cases  More profoundly, this violates a fundamental principle of logic which is that given some ambiguity, you should assume the most general case apply. It is way worse than measuring things with non integral units. This is violating logical rules, and leaving a very basic concept obfuscated.
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When somebody uses the phrase "not uncommon", do they mean "common" because of the double negative?  I was recently informed that "not uncommon" could imply that something was anything but uncommon (e.g. rare), whereas I thought that rare would be a subset of uncommon and that saying it was not uncommon must mean it is common.
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I am studying General Relativity and some basic QFTs. It bothers me a lot that different books use different metric signatures, i.e. $(-+++)$ and $(+---).$ Can anyone tell me the advantages of using both of them, or why some people prefer the former while some prefer the latter? I would like to know which one should I while tackling different problems.
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While googling about whether "atheist" should be capitalized, and skimming past posts by people blogging about atheism rather than about English grammar, I came across    It says     [Capitalize] Religions and religious terms       Examples:      Catholicism, Hinduism, Buddha, the Bible, Christian, the Ten   Commandments, Baptist church, Mt. Zion Methodist Church      NOTE: Do not capitalize the following:       Examples:      church, communion, atheist, agnostic, spirituality   but I don't recall hearing of such a prescription before. Wouldn't most of the examples listed be capitalized because they're proper nouns? Are there good examples of words being capitalized solely because they're religious terms?  I'm mainly interested in modern English, rather than days of past where a lot more words were capitalized than they are nowadays.  Somewhat related:  - use "God" when referring to the name of the god of Judaism and Christianity (a proper noun), but "god" when referring to the general concept of a deity (common noun).
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I am not good at mathematics and have been learning ML from Udacity. In its tutorial video, the tutor says(I concluded it in a short way): using Gradient to get the steepest slope to go upwards, so in order get minimized loss we go the opposite. Here is the video starting from time 3:00:  my question is : it is not necessary the exactly opposite direction of the gradient(steepest ascent) that the steepest descent goes. As it's 3D, if the gradients goes 0 degree, it doesn't necessary mean the 180 degree points you the steepest descent. So I assume his statement is wrong? Could someone correct me(using plain language even a laymen can understand, thanks) if I got wrong somewhere? Thanks update After reading John's answer, my understanding is: At the particular point P we can draw a tangent line A. Let's assume that the 0 degree of A points to the fastest direction for ascending, but the actual fastest direction for descending is 150 degree at point P. Since the tangent line can ONLY have 2 directions which are 0 and 180, so we can only take 180 degree of the line A as our next direction for descending. Is my understanding correct?
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I just read somewhere that there are as many even numbers as natural numbers! Is there an intuitive way of coming to terms with this fact? Because honestly, it's quite baffling. Also, does the concept of odd/even apply to negative integers as well? By definition, it should. But I have never seen a negative multiple of two being used as an example of an even number.
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I was reading this today:  We look forward to following your progress.  Am I correct in thinking that it is missing a be or that -ing should be removed?  So  We look forward to be following your progress.  or  We look forward to follow your progress.  Interestingly I get the same jarring feeling when reading the examples here
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a. His two previous books  b. His previous two books   .............   His 2 latest previous books His 2 previous latest books His previous latest 2 books His latest previous 2 books His previous 2 latest books His latest 2 previous books  I am wondering what is the difference could be between these in meaning?  Which one do you use? which one you don't use?  I think just especially native speakers or learned ones in English can provide the most proper answer.  Thanks
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Are there rules of usage when using the ampersand "&amp;" instead of "and"?  Are they completely interchangeable?   The ampersand seems more casual, but I'm not sure.
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After reading the question , I got the impression that (at least in the writers' field) the "corresponding author" flag could carry an implied meaning, such as "did more work than the other authors", or "is a more senior author". Is it the case?  I am in mathematics, a field where we typically sort the authors alphabetically, and I've never given any importance to that role (apart from "this author is willing to answer a couple of extra e-mails, if needed").
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I hope this qualifies as a programming question, as in any programming tutorial, you eventually come across 'foo' in the code examples. (yeah, right?)  what does 'foo' really mean?  If it is meant to mean nothing, when did it begin to be used so?
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Why the disparity? And why use 'No.'? Is it from the French?  And the hash or pound sign seems a weird choice too, is there a history or any reason involved?
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I've just wondered by it is "description", but "to describe". It looks as if "to descripe" would be a more consistant choice. Is there any other explanation besides "this is how it is written for years"?  edit: Or alternatively "to describe" and "describtion".
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I have a question regarding the use of hyphens in words with certain prefixes. For example,     Neo-Nazism   Neo-Confucianism   Neo-liberal     are words with a hyphen, but     Neoplatonism   Neocolonialism   are words without hyphens. I'm especially confused with neo-Confucianism and neoplatonism: one contains  a hyphen and a capital letter and the other doesn't.  Is there a specific rule for hyphens in words with prefixes or is it purely arbitrary?
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What's the meaning of 90-plus? More than 90? If so, please tell me more ways to say "more than 90". Thank you!
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I still can't wrap my head around this one and have to consult references every time I want to be safe:   Is it "0.4 point" or "0.4 points"?   Is it "1.0 point" or "1.0 points"?   I know that I can work around that by using four tenths of a point or one point instead, but sometimes writing 0.4 point/points or 1.0 point/points is called for, especially in technical documents.  I gathered that this mostly goes by the style guide I choose to (or have to) use, and it's quite possible that there is no universally accepted convention for this. Then again, in light of potential benefits for learners here, I think it's worth asking, and I hope that we can lay out some rules of thumb, something that can be used as a general guideline, especially on writing tests and exams.  Addendum: another case that's worth mentioning in answers, for completeness, is "0.0 point" or "0.0 points"? In my humble opinion, it's safe to assume that most learners on our site know what to use when the quantity falls out of the range between -1.0 and 1.0.
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What does the number 2 mean in this picture?    Thanks.   Sorry I'm not a math expert.  I need this to make a program in C to generate a trinomial triangle for a guy who asked it on StackOverflow (it is his homework). Gotta earn that reputation. You know how it is!