Suppose that $a$, $b$ and $c$ are positive real numbers and that $a \leq b + c$. By cross multiplying or otherwise, show that  $$\frac{a}{1 + a} \leq \frac{b}{1 + b} + \frac{c}{1 + c}$$  Anyone able to get this to work out? I cross multiplied them but am unable to manipulate it into an expression that verifies the statement.
With $x+y\ge z$ $(x,y,z\ge0)$, prove that: $$\frac{x}{1+x}+\frac{y}{1+y}\ge\frac{z}{1+z}$$  I'm aware that using analytic view this is easy since $f(x)=\frac{x}{1+x}$ is concave in $[0,\infty)$. However I want to prove it using merely algebraic techniques. Is that possible?
Beamer how to change color of infolines and frame title How could I change the color of color of infolines and frame title in beamer. I am using this now   \documentclass[xcolor=dvipsnames]{beamer} \usetheme{CambridgeUS} \useinnertheme{rectangles} \useoutertheme{infolines} \usecolortheme[named=Brown]{structure}    I need to change the color from red to brown. [named=Brown] doesn't work for \useoutertheme{infolines}. This is my current output. I also need to change the background of frame title (mouse pointer) NSString instance reports its class as NSCFString My objective here is really simple -- I'm trying to set an NSString to some test data, then return the class, which should be NSString. Here's my code:  NSString* stringer = [NSString stringWithFormat: @"Test"]; NSLog(@"%@", [stringer class]);   The log says that the class is NSCFString, not NSString. What's going on here? One adjective for two or more nouns Could we use one adjective for two nouns? Is the following sentence correct?     It has low speed and voltage. I was watching the film A Game of Shadows starring Robert Downey Junior and Jude Law when this line came up, "...neither you nor I is in control..." (I can’t remember the exact words that ended the phrase; if anyone can supply them, it would be much appreciated.)   It immediately struck me as being odd. When I mentioned this to my boyfriend; an English native speaker, with a PhD in something sciency, he told me it was correct. When I asked why, he couldn't explain but put forward the following sentence as an example. "Neither of us is in control..." However, I disagreed with the wording and said the sentence should have been: "Neither one of us is in control..." The genderless third person, one, is singular, and hence, so too the verb that follows. And we say, "one is" NOT "one are".      "Neither you nor I are in control over ..."    I believe the above sounds better, but it is more logical to say:     "Neither you nor I am in control..."    because the verb agrees with the subject closest to it; however, I'm not sure if I have ever read or heard this solution. So, I tried inverting the subject and auxiliary verb order and the resulting phrase was:    "Neither you nor am I in control..."    which sounds very formal, pompous, and ... odd. Am I imagining things?  The two subjects you and I are two separate individuals who are not joined together, I agree the verb should be in the singular; but why should we use "is"? We don't say, "you is" or "I is". Before anyone rushes off to claim my question is a duplicate of a previous Stackexchange one:  I want to explain why the answer to that question fails to explain why my boyfriend and the film makers consider the  sentence below correct:     Neither you nor I is in control...   As I see it, Michael and Albert are two separate individuals, it therefore stands to reason that the verb should be singular, and "is" agrees with the subject. If I rephrase the neither and nor construction, I would get:          Michael is not correct but neither is Albert   OR          Michael is not correct and Albert isn't either.      Which have the same meaning, (either being the negative equivalent of neither) the sentences are grammatical which leads me to conclude that the verb must be singular and hence         Neither Michael nor Albert is correct      However, when I rephrase the "neither you nor I" sentence following the same procedure I have to say this:        You are not in control but neither am I (‘I am’ is ungrammatical)   OR              You are not in control and I’m not either.      which leads me to surmise that the sentence below must be grammatically correct        Neither you nor I am in control"      So, why does my boyfriend insist that the sentence ought to be:     Neither you nor I is in control   Which one of us is right and why? Should "each" be followed by a singular or plural possessive? If a possessive noun, which is plural, is preceded by "each", then should it use the singular or plural possessive form?  For example, which of the following is correct?    spend time in each other's presence spend time in each others' presence Is there any adjustment required when computing the correlation coefficient between:   continuous and discrete variable  two discrete variables ?    Thanks ! I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?   The standard rule you always hear:     “If a person knows which item you are talking about then use "the"   . . . doesn’t clear things up for me, as I have no idea whether or not they know. 2 separate lights on two separate switches on one line - issue Will some parents be offended when being asked, "Is it male or female?" If I ask the parent about a baby's gender, will it be impolite or not appropriate to say, "Is it male or female?"  Is there any subtle difference, in terms of politeness, among   "Is it a boy or girl?" "Is it he or she?" "is it male or female?" Correlations between continuous and categorical (nominal) variables I would like to find the correlation between a continuous (dependent variable) and a categorical (nominal: gender, independent variable) variable. Continuous data is not normally distributed. Before, I had computed it using the Spearman's $\rho$. However, I have been told that it is not right.  While searching on the internet, I found that the boxplot can provide an idea about how much they are associated; however, I was looking for a quantified value such as Pearson's product moment coefficient or Spearman's $\rho$. Can you please help me on how to do this? Or, inform on which method would be appropriate?  Would Point Biserial Coefficient be the right option? Do gravitational lenses work on gravitational waves? Do gravitational lenses work on gravitational waves? Could we get an Einstein cross of gravity? If you copy the contents of   httpd.conf   and then paste it into a cat command.. such as this one..  #!/bin/bash cat &gt; /test &lt;&lt; EOF pasted here.. EOF   you run into this error:  -bash: command substitution: line 1: unexpected EOF while looking for matching `'' -bash: command substitution: line 4: syntax error: unexpected end of file   perhaps the solution is to escape the dollar signs and maybe even quotes and so on..  but given that this is such a large file.. can dollar signs be automatically be escaped ?  is my only option to escape them via another program and then give it to the cat command ? In the chapter on Spellcasting, there is a section called . However, despite the seeming generality of the title, the text of this section only talks about magical effects from spells:     The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect--such as the highest bonus--from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.   It makes sense that this section would only talk about spells, since it is in the chapter about spellcasting. However, this seems to leave ambiguous the question of how magical effects from sources other than spells combine. Such an issue could arise, for example, if two different items (or two of the same item) grant a bonus to attack rolls, or a bonus to the same saving throw.  Is there a published rule or official ruling that governs how non-spell magical effects combine?  (This question was inspired by  about the interaction between +X magical ammunition and Oil of Sharpness, and could be considered a generalization of that question.) How to create Motion Trail in Cycles I want to create a motion trail effect for a sword like known in games look at that example if you have an solution that looks not as fancy as in the picture don't mind to post it. I'll accept anything useful. I have a contact in a generic sObject in apex and while I'm able to easily get detail field values using   sObject.get('MyContactField__c')  I'm not able to get fields off the related parent object, and I'm sure the related field is in memory, based on the debug logs. This returns a runtime error:  sObject.get('Account.MyAccountField__c')  with the error     Invalid field Account.MyAccountField__c' for Contact   How do I get a field value off of a parent on a generic sObject? Why is a ConcurrentModificationException thrown and how to debug it I am using a Collection (a HashMap used indirectly by the JPA, it so happens), but apparently randomly the code throws a ConcurrentModificationException. What is causing it and how do I fix this problem? By using some synchronization, perhaps?  Here is the full stack-trace:  Exception in thread "pool-1-thread-1" java.util.ConcurrentModificationException         at java.util.HashMap$HashIterator.nextEntry(Unknown Source)         at java.util.HashMap$ValueIterator.next(Unknown Source)         at org.hibernate.collection.AbstractPersistentCollection$IteratorProxy.next(AbstractPersistentCollection.java:555)         at org.hibernate.engine.Cascade.cascadeCollectionElements(Cascade.java:296)         at org.hibernate.engine.Cascade.cascadeCollection(Cascade.java:242)         at org.hibernate.engine.Cascade.cascadeAssociation(Cascade.java:219)         at org.hibernate.engine.Cascade.cascadeProperty(Cascade.java:169)         at org.hibernate.engine.Cascade.cascade(Cascade.java:130) Consider the equation $$|2^x-3^y|=1$$ in the unknowns $x \in \mathbb{N}$ and $y \in \mathbb{N}$. Is it possible to prove that the only solutions are $(1,1)$, $(2,1)$ and $(3,2)$? Am I required to pay UK/ USA Visa fees if I have a German residence permit? Here is the situation. I am non-european but staying in Germany with a Residence permit. I can visit all Schengen listed countries with no issue. Assuming that I would want to travel to a non-Schengen country e.g. UK or USA, can I use my residence permit to travel there or am I required to apply as a citizen of my country? Assuming Germans are not required to pay Visa fees to that country, will I have to pay since I only have a residence permit? Third case, Europeans can fly directly to the airport of some countries and apply for a visa from there. In some other countries, you are required to book a Visa appointment at the embassy, which may be granted or not. Does that apply for one with a residence permit? Have we managed to make a perfect vacuum? Have we managed to make a device without any atom inside it on earth? I was reading about vacuum , and I found in the examples part  that even on the best man made vacuum devices, there are still millions of "molecules per $cm^3$". So my other question here is, what are these molecules and what is the mass of them? MonoTouch.Dialog: How to set limit of number of characters for EntryElement